En)

T he topological matter is nowadays one of the most discussed topics of condensedmatter-physics. The conical bands present in various form in these structures allow us to study relativistic-like effects which were before almost exclusive for high-energy physics. In recent times, the topological family expanded rapidly with the new materials being discovered on an almost yearly basis.

In this thesis, we apply THz and infrared magneto-spectroscopy to study selected topological materials. Such a spectroscopy technique, working at low photon energies, is a relevant tool to probe of low-energy electronic excitations which are directly connected with specific topological order. In particular, we focus on the study of intra-and interband Landau level (LL) transitions. Such experiments allow us, among other, to determine basic material parameters of the explored topological matter, including the energy band gap and velocity parameters which describe the slope and anisotropy of conical bands. The magneto-optics is combined with absolute reflectivity and magneto-transport, which serve as useful complementary techniques. This thesis is focused on experimental investigations of Dirac nodal-line and Weyl semimetals, representative members of the topological matter family. Our primary goal was deepen our current knowledge about the complex electronic band structures of these appealing materials. The first part of the thesis is dedicated three particular compounds: BaNiS 2 , NbAs 2 and NbSb 2 which belong to the class of Dirac nodal-line semimetals. BaNiS 2 is a non-symmorphic material (space group P4/nmm) and can be viewed as one of the simplest Dirac nodal-line semimetals. The results obtained in magneto-optics experiments comprised a well-defined set of intra-and interband inter-LL excitations that were explained analyzed using the model of massive Dirac electrons. The extracted band gap and the velocity parameter were in line with the predictions of DFT calculations. The optical studies without any magnetic field applied revealed a broad interval of photon energies in which the optical conductivity was practically temperature independent. This so-called isosbestic line is typical of Dirac nodal-line materials and was reproduced using both analytical and numerical approaches.

NbAs 2 and NbSb 2 are other two Dirac nodal-line semimetals investigated. The nodal-line structure in these materials is more complex as compared to BaNiS 2 . The nodal lines are open-ended and propagate through multiple Brillouin zones. The same experimental approach was applied to both materials. Multiple crystallographic facets were measured using magneto-reflectivity in order to trace the relatively strong anisotropy of

the magneto-optical response. We found that the optical band gap of NbAs 2 extracted using the zero-field extrapolation of inter-LL excitations is highly facet-dependent. This surprising observation was interpreted as a direct consequence of the Lorentz-boost-driven renormalization of two-dimensional system of massive Dirac electrons. The additional polarization-resolved measurement without magnetic field corroborated the determination of the local direction of the nodal-line structure in both materials.

The second part of the thesis is dedicated to three-dimensional Weyl semimetals. Three materials: TaAs, TaP and NbAs, from the well-established group of noncentrosymmetric transition-metal pnictides were investigated. An unusual set of inter-Landau level excitations which decrease their energy with the increasing magnetic field were found in TaP and NbAs. The observed effect was associated with the inversion of the electronic bands which is present at the Σ point in the Brillouin zone and which is responsible for the formation of the W1 nodes. The corresponding effective-Hamiltonian model was developed and first applied to TaP. Later on, it was generalized for other monopnictide Weyl semimetals. Optical and magneto-optical studies performed on different crystallographic facets of TaAs. The surprisingly rich magneto-reflectivity data were analyzed using a semiclassical approach, by applying the Lifshitz-Onsanger formula. This procedure allowed us to identify the response due to excitation at the W2 cones which are characterized by a relatively high isotropy and which extend over a few hundred of meV. In contrast, the optical response associated with the vicinity of the W1 cones was found highly anisotropic and optical excitations directly in the W1 cones were not observed, most likely due to the occupation effect, implying the Fermi energy well above (or below) the W1 cones. Intriguingly, even though TaAs is a well-known three-dimensional Weyl semimetal, the magneto-optical response observed on some of the explored facets resembled a conventional narrow gap semiconductor.

Resumé (FR) Resumé (FR) L a matière topologique est aujourd'hui l'un des sujets les plus discutés de la physique de la matière condensée. Les bandes coniques présentes sous diverses formes dans ces structures nous permettent d'étudier des effets de type relativiste qui étaient auparavant presque exclusivement réservés à la physique des hautes énergies. Ces derniers temps, la famille topologique s'est rapidement développée avec la découverte de nouveaux matériaux presque chaque année.

Dans cette thèse, nous appliquons la magnéto-spectroscopie THz et infrarouge pour étudier certains matériaux topologiques. Cette technique de spectroscopie, qui fonctionne à de faibles énergies photoniques, est un outil pertinent pour sonder les excitations électroniques de faible énergie qui sont directement liées à un ordre topologique particulier. En pratique, nous nous concentrons sur l'étude des transitions de niveaux de Landau (LL) intra et inter-bandes. Ces expériences nous permettent, entre autres, de déterminer les paramètres matériels de base des matériaux topologiques explorés, y compris la bande d'énergie interdite et les paramètres de vitesse qui décrivent la pente et l'anisotropie des bandes coniques. La magnéto-optique est combinée à la réflectivité absolue et au magnéto-transport, qui sont des techniques complémentaires utiles.

Cette thèse se concentre sur les investigations expérimentales des semi-métaux de Weyl ou à ligne nodale de Dirac, membres représentatifs de la famille des composés topologiques. Notre objectif principal est d'approfondir nos connaissances actuelles sur les structures de bandes électroniques complexes de ces matériaux captivants. La première partie de la thèse est consacrée à trois composés particuliers: BaNiS 2 , NbAs 2 et NbSb 2 qui appartiennent à la classe des semi-métaux à ligne nodale de Dirac. BaNiS 2 est un matériau non-symorphique (groupe spatial P4/nmm) et peut être considéré comme l'un des semi-métaux à ligne nodale de Dirac les plus simples. Les résultats obtenus dans les expériences de magnéto-optique comprenaient un ensemble bien défini d'excitations intra et inter-bandes de niveaux de Landau qui ont été expliquées et analysées en utilisant le modèle des électrons à masse non nulle de Dirac. La bande interdite et le paramètre de vitesse extraits sont en accord avec les prédictions des calculs DFT. Les études optiques en l'absence de champ magnétique ont révélé un large intervalle d'énergies dans lequel la conductivité optique est pratiquement indépendante de la température. Cette ligne dite isosbétique est typique des matériaux à ligne nodale de Dirac et a été reproduite en utilisant des approches analytiques et numériques.

NbAs 2 et NbSb 2 sont deux autres semi-métaux à ligne nodale de Dirac étudiés.

CONTENTS

La structure de la ligne nodale dans ces matériaux est plus complexe que celle de BaNiS 2 . Les lignes nodales sont ouvertes et se propagent, dans la zone de Brillouin, suivant plusieurs directions. La même approche expérimentale a été appliquée aux deux matériaux. De multiples facettes cristallographiques de ces échantillons ont été mesurées en utilisant la magnéto-réflectivité afin de charactériser l'anisotropie relativement forte de la réponse magnéto-optique. Nous avons montré que la bande interdite optique du NbAs 2 extraite en utilisant l'extrapolation à champ nul des excitations inter-niveaux de Landau est fortement dépendante de la facette. Cette observation très surprenante a été interprétée comme une conséquence directe de la renormalisation du système bidimensionnel d'électrons de Dirac massifs, induite par la poussée de Lorentz. Une mesure supplémentaire résolue en polarisation, à champ magnétique nul, a confirmé la détermination de la direction locale de la structure de la ligne nodale dans les deux matériaux.

La deuxième partie de la thèse est consacrée aux semi-métaux de Weyl tridimensionnels. Trois matériaux, TaAs, TaP et NbAs, appartenant au groupe bien établi des pnictides de métaux de transition non centrosymétriques, ont été étudiés. Nous avons trouvé, dans TaP et NbAs, un ensemble inhabituel d'excitations entre niveaux de Landau qui diminuent leur énergie avec l'augmentation du champ magnétique. L'effet observé a été associé à l'inversion des bandes électroniques qui est présente au point de la zone de Brillouin et qui est responsable de la formation des noeuds W1. Un modèle d' Hamiltonien effectif correspondant a été développé et appliqué d'abord à TaP. Par la suite, il a été généralisé à d'autres semi-métaux de Weyl monopnictides.

Des études optiques et magnéto-optiques ont été réalisées sur différentes facettes cristallographiques du TaAs. Les données de magnéto-réflectivité, étonnamment riches, ont été analysées par une approche semi-classique, en appliquant la formule de Lifshitz-Onsanger. Cette procédure nous a permis d'identifier la réponse due à l'excitation au niveau des cônes W2 qui sont caractérisés par une isotropie relativement élevée et qui s'étendent sur quelques centaines de meV. En revanche, la réponse optique associée au voisinage des cônes W1 s'est révélée fortement anisotrope et les excitations optiques directement dans les cônes W1 n'ont pas été observées, très probablement en raison de l'effet d'occupation, impliquant l'énergie de Fermi bien au-dessus (ou en dessous) des cônes W1. De manière intriguante, même si le TaAs est un semi-métal de Weyl tridimensionnel bien connu, la réponse magnéto-optique observée sur certaines des facettes explorées ressemblait à un semi-conducteur conventionnel à bande interdite étroite.

1

Introduction to topological matter

The topological matter is nowadays a well-established class containing a vast number of materials. The modern history of relativistic-like effects in the condensed-matter physics starts with the discovery of graphene, which will be shortly discussed at the beginning of the chapter. The basic concepts of topological insulators will be presented together with the prior experiments confirming their existence. In the second section, we move to the brief description of the interaction of light and matter with and without an external magnetic field.

Graphene

After the discovery of graphene and its first experimental isolation in 2004 [1], the science community was confronted with a new concept to be introduced to solid-state physics. Graphene, the first truly stable two-dimensional (2D) crystal, is composed of the carbon atoms bonded together into a 2D hexagonal honeycomb-like lattice with sp 2 hybridization. From the mechanical point of view, graphene is one of the most robust materials thanks to the three strong σ bonds between neighbor atoms formed by the p x and p y orbitals. The measured Young's modulus reaches almost 1 TPa (second strongest material after diamond) with an intrinsic tensile strength as large as 130 GPa [2,3]. On the other hand, the p z orbital is delocalized due to the 2D nature of the system hybridizing in π * and π bands, whose symmetry-protected at the K and K ′ points of the Brillouin zone ensuring a gapless conical dispersion (Fig. 1.1) [4].

The conical bands at the corners of the Brillouin zone are responsible for the relativistic-like properties of graphene. It was demonstrated that the low-energy excitations in graphene can be conveniently described using relativistic-like Hamiltonian for massless electrons [1,6]. Graphene was the first compound in which a relativistic-like narrative was widely accepted by the scientific community and it served as a good platform to test various concepts of high-energy physics. This groundbreaking discovery brought a lot of motivation to search for other materials, which could host conical dispersion in their electronic band structure.

Topological insulators

The ability to conduct electric current is one of the main characteristics of a solidstate system. The class of topological insulators presents the unique phase, which may combine both, conducting and insulating behavior. They are characterized as materials, Fig. 1.1. Energy dispersion of graphene. Linear band dispersion of graphene. The valence (π) and conduction (π * ) bands intersect at K and K ′ points of Brillouin zone. Adapted from [5].

which behave as ordinary insulators in bulk, but conducts via edges (2D compounds) or surfaces (3D compounds) [7,[START_REF] Moore | The birth of topological insulators[END_REF]. Even though the earliest models of 3D topological insulators date back to the mid-eighties [START_REF] Volkov | Two-dimensional massless electrons in an inverted contact[END_REF][START_REF] Pankratov | Supersymmetry in heterojunctions: band-inverting contact on the basis of Pb 1-x Sn x Te and Hg 1-x Cd x Te[END_REF], the experimental confirmations and more extensive theoretical descriptions highlighting the significance of the time-reversal symmetry came during the 2000s [START_REF] Kane | Quantum Spin Hall Effect in Graphene[END_REF][START_REF] Kane | Z 2 Topological Order and the Quantum Spin Hall Effect[END_REF][START_REF] Bernevig | Quantum Spin Hall Effect[END_REF][START_REF] Bernevig | Quantum spin Hall effect and topological phase transition in HgTe quantum wells[END_REF][START_REF] König | Quantum spin Hall insulator state in HgTe quantum wells[END_REF].

To better understand the concept of a topological insulator, let us first discuss the simplest topological phenomenon of the quantum Hall effect, which arises in systems with a 2D electron gas subjected to the magnetic field strong enough to induce the quantization into Landau levels. The electrons in such a structure undergo a cyclotron motion perpendicular to the direction of the magnetic field. In the quantum mechanical approach, these orbits are quantized into so-called Landau levels (LL), the energy spectrum of which can be determined by solving an appropriate Hamiltonian. The energy spacing of LLs is proportional to the magnitude of the applied magnetic field. This implies two scenarios depending on the position of the Fermi energy (E F ): (i) the material behaves as a conductor if E F is set within some LL, (ii) the material is insulating if E F is situated in the energy gap between LLs. In 1980, Klaus von Klitzing found out that even in the insulating case, the Hall conductivity remains to be non-zero, and moreover, quantized into [START_REF] Klitzing | New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance[END_REF]:

G = e 2 ν h , (1.1)
where e is the elementary charge, h is the Plank's constant, and ν is either integer representing the filling factor of LLs. The quantized Hall conductance originates from the boundary electrons with opened cyclotron orbits. This results in effectively gapless electronic states propagating in one direction without any dissipation of energy (Fig. 1.2b).

Even though, the observation of quantum Hall effect usually requires relatively low temperatures, the room temperature quantum Hall effect was reported in graphene [START_REF] Novoselov | Room-temperature quantum hall effect in graphene[END_REF] (Fig. 1.2e).

Topological insulators host similar gapless states even in the absence of the external magnetic field. The strong magnetic field needed for a realization of the quantum Hall effect is here replaced by the spin-orbit interactions (SOC), which couples the spin of electrons with their orbital motion [7,[START_REF] Moore | The birth of topological insulators[END_REF]. In the quantum spin Hall phase of the 2D topological insulator, two conducting gapless spin edge states (spin-up and spin-down in Fig. 1.2c) are created and they propagate in the opposite directions. Such counterpropagating states were theoretically predicted to exist in HgTe/Hg x Cd 1-x Te quantum wells by tuning the thickness of the layers [START_REF] Bernevig | Quantum spin Hall effect and topological phase transition in HgTe quantum wells[END_REF]. Tuning the width of a HgTe quantum well to a critical value d c ∼ 64 Å triggers a quantum phase transition of valence and conduction band inversion [START_REF] Bernevig | Quantum spin Hall effect and topological phase transition in HgTe quantum wells[END_REF]. The conductance of 2e 2 /h of the quantum spin Hall states was experimentally confirmed by the group of Prof. Molenkamp at the University of Wurzburg in 2007 [START_REF] König | Quantum spin Hall insulator state in HgTe quantum wells[END_REF] (Fig. 1.2f).

The discovery of a 3D topological insulator came soon after. In the 3D topological insulator, the gapless states are realized at the surface of the material in the form of a 2D conical band similar to graphene. Due to strong spin-orbit interaction, the direction of the real-space motion is coupled (locked) with the spin orientation. The gapless surface states (e) Room-temperature quantum Hall effect realized in graphene. The Hall conductivity (σ xy -blue) and resistivity (ρ xy -red) plotted as a function of gate voltage V g . The Hall conductivity plateaus are quantized by 2e 2 /h for both electrons (positive V g ) and holes (negative V g ). The inset shows LL quantization of Dirac electrons with E F set in the gap ∆E. (f) Quantum spin Hall effect detected in the HgTe/Hg x Cd 1-x Te quantum well heterostructure. Devices III and IV with higher thickness than d c ≈ 64 Å exhibit a quantized conductance of 2e 2 /h. (g) ARPES measurement of Bi 2 Se 2 surface state resembling the 2D Dirac cone of graphene. Adapted from [7,[START_REF] Moore | The birth of topological insulators[END_REF][START_REF] Novoselov | Room-temperature quantum hall effect in graphene[END_REF][START_REF] Xia | Observation of a large-gap topological-insulator class with a single Dirac cone on the surface[END_REF].

were predicted in the semiconducting alloys of bismuth with strong SOC in 2007 prior to the experiment [START_REF] Fu | Topological insulators with inversion symmetry[END_REF][START_REF] Fu | Topological Insulators in Three Dimensions[END_REF]. In contrast to its 2D counterparts, the gapless state contribution of the 3D topological insulator is something difficult to detect in transport experiments since the bulk conductivity does not vanish in many topological insulators. On the other hand, angle-resolved photoemission spectroscopy (ARPES) appeared as a perfectly suited technique to study electronic states on the surface of topological materials. The ARPES is a surface sensitive technique with a penetration depth of only few nm. The principle of ARPES is based on an outer photoelectric effect. The high-intensity photon beam (UV to X-ray range) ejects the electrons from the sample surface. The analysis of their energy, momentum, and spin leads to the reconstruction of the band dispersion of the material. The first 3D topological insulator Bi 1-x Sb x was confirmed by this technique in 2008 [START_REF] Hsieh | A topological Dirac insulator in a quantum spin Hall phase[END_REF], followed by Bi 2 Se 3 and Bi 2 Te 3 in 2009 [START_REF] Xia | Observation of a large-gap topological-insulator class with a single Dirac cone on the surface[END_REF][START_REF] Chen | Experimental realization of a three-dimensional topological insulator, Bi 2 Te 3[END_REF] (Fig. 1.2g).

Both edge and surface states are topologically protected by the time-reversal symmetry in topological insulators. The gapless states are robust to any small perturbation (e.g., crystal impurities, strain) until the time-reversal symmetry is preserved [7,[START_REF] Fu | Topological insulators with inversion symmetry[END_REF][START_REF] Fu | Topological Insulators in Three Dimensions[END_REF]. The relativistic-like effects similar to the ones mentioned above were also found in the class of topological crystalline insulators [START_REF] Fu | Topological Crystalline Insulators[END_REF][START_REF] Dziawa | Topological crystalline insulator states in Pb (1-x) Sn (x) Se[END_REF][START_REF] Hsieh | Topological crystalline insulators in the SnTe material class[END_REF]. In this case, the surface states are protected by a certain crystal symmetry rather than the time-reversal symmetry.

Classification of topological materials

The family of topological materials was expanding rapidly over the last years. Numerous algorithmic procedures were developed by both physicists and chemists that allows the determination of topological properties of solids [START_REF] Bradlyn | Topological quantum chemistry[END_REF][START_REF] Kruthoff | Topological Classification of Crystalline Insulators through Band Structure Combinatorics[END_REF][START_REF] Po | Symmetry-based indicators of band topology in the 230 space groups[END_REF]. It was found out that approximately 27% of all solids are topological and roughly 12% of them are topological insulators [START_REF] Vergniory | A complete catalogue of high-quality topological materials[END_REF]. These attempts resulted in multiple catalogs and interactive web-based topological databases (e.g., [START_REF] Bradlyn | Topological quantum chemistry[END_REF] or [START_REF] Talirz | Materials Cloud, a platform for open computational science[END_REF]). The topological band theory provides us with useful tools that allow for the classification such as Chern numbers, Berry curvatures, or Z 2 invariants. However, these non-trivial theoretical constructs are not always practical when dealing with experimental phenomena. The less rigorous classification can be done focusing mostly on the description of the conical structure, which is responsible for the majority of the relativistic-like effects. Conical bands can be categorized by their dimensionality (1-3D) and their degeneracy (spin and/or valley). The simplified cone classification in various classes of topological matter together with the key publications and particular examples is presented in Fig. 1

Interaction of light with condensed-matter systems

Let us briefly summarize the basic description of light and matter interaction. In the 1860s, James Clerk Maxwell published a series of articles linking the previous studies of electricity and magnetism into one comprehensive theory describing light as electromagnetic wave. The set of four equations describes the connections between electromagnetic quantities such as electric and magnetic field strengths (E, H), electric displacement field (D), magnetic flux density (B), electrical current (j), and the charge density (ρ). In order to derive the quantities, which reflect dissipative properties of materials such as the dielectric function, optical conductivity or refraction index, we utilize the procedure described in [START_REF] Dressel | Electrodynamics of Solids: Optical Properties of Electrons in Matter[END_REF]. The Maxwell equations in the presence of matter read as:

∇ × E + ∂ B ∂t = 0, (1.2a) ∇ × H - ∂ D ∂t = j, (1.2b)
∇ • D = ρ, (1.2c)
∇ • B = 0. (1.2d)
The equations above are accompanied by the material relations and Ohms law in the continuum form:

D = ε 1 E, B = µ 1 H, j = σ 1 E, (1.3)
where ε 1 is the dielectric constant, µ 1 is permeability and σ 1 is conductivity. The material parameters 1.3 allows us to rewrite Eq.1.2b into:

∇ × H = -iωε 1 E + σ 1 E = -iωεE. (1.4)
When the complex representation is used, the complex dielectric function ε(ω) can be defined with the assumption of harmonic time dependence of electric displacement field ∂ D/∂t = -iω as:

ε(ω) = ε 1 (ω) + i σ 1 (ω) ε 0 ω = ε 1 (ω) + iε 2 (ω), (1.5)
where ε 0 is permittivity of vacuum. The complex optical conductivity σ (ω) can be established in a similar way as ε(ω):

σ (ω) = σ 1 (ω) + iσ 2 (ω). (1.6)
The relation between ε(ω) and σ (ω) then reads as:

ε(ω) = 1 + iσ (ω) ε 0 ω . (1.7)
The dissipation of light interaction with matter can be also described by using the complex refractive index:

N(ω) = n(ω) + ik(ω), (1.8)
where n(ω) and ik(ω) represent the (real) index of refraction and extinction coefficient, respectively. The complex refractive index is related to the dielectric function as:

ε(ω) = N(ω) 2 . (1.9)
Optical experiments often consist of frequency-dependent measurement of reflectivity. The complex refractive index outers reflectivity in the following way:

R(ω) = 1 -N(ω) 1 + N(ω) 2 = (1 -n(ω)) 2 + k(ω) 2 (1 + n(ω)) 2 + k(ω) 2 .
(1.10)

Optical transitions in crystalline solids

The main contribution to the optical response of a solid usually stems from intraband (Drude-like) and interband excitations of electrons (Fig. 1.4). The intraband contribution can be modeled classically by using well-established Drude model. The Drude model treats electrons of the density N as a dilute gas of particles moving freely in any direction. The system relaxes to equilibrium by collisions with impurities, crystal defects but also with other particles. Such relaxation is characterized by the scattering time τ (or scattering rate Γ = 1/τ). The system is brought out of equilibrium by applying the electric field E(t) = E 0 e -iωt . The equation of motion describing interaction of a single electron with the electrical component of radiation reads:

m d 2 r dt 2 + m τ dr dt = -eE(t). (1.11)
The solution comes in the form of complex optical conductivity:

σ (ω) = σ 0 1 -iωτ , (1.12)
where σ 0 is the optical conductivity in dc limit (at ω = 0):

σ 0 = Ne 2 τ ε 0 m = ω 2 p τ, (1.13)
where plasma frequency ω p defines a limit until which the material is fully reflecting.

In contrast to the free-carrier response, the interband excitations require a fully quantum-mechanical approach. The absorption of light, due to electrons promoted between electronic states |α⟩ and |β ⟩ is described by the real part of optical conductivity σ 1 , which may be approximately described using Kubo-Greenwood formula:

σ (ω) = πe 2 ω h2 ∑ α̸ =β d 3 k (2π) 3 |⟨α| vi |β ⟩| 2 × [ f (E α ) -f (E β )] × δ [hω -(E β -E α )]. (1.14)
Let us now highlight the three relevant terms in the above formula:

• Selection rules. |⟨α| vi |β ⟩| 2 are matrix elements of the velocity operator vi which describe the probability (absorption rate) of an optical transition between states |α⟩ and |β ⟩.

• Joint density of states. The number of optical transitions at the given frequency ω obtained by a sum (integral) over the relevant part of the reciprocal space. The joint density of states is defined as: • Occupation factor (Pauli blocking). The term f (E α )f (E β ) represents a joint occupation factor, which ensures that only transitions from occupied to non-occupied states are allowed. In a gapless system with a full electron-hole symmetry, it is the Fermi level which defines the onset of interband absorption at 2E F . In a gapped system (2∆), again with the electron-hole symmetry preserved, the onset of interband absorption is at the photon energy of 2(∆ + E F ), for the Fermi energy counted from the band edge.

∑ k δ [hω -(E β -E α )]. (1.15)
In general, the real part of the optical conductivity follows the jDOS as σ 1 (ω) ≈ jDOS(ω)/ω, thus is drastically affected by the system dimensionality and band dispersion (e.g., parabolic vs. linear). As an example, the optical conductivity of 3D Weyl cone reads as [START_REF] Tabert | Optical and transport properties in three-dimensional Dirac and Weyl semimetals[END_REF][START_REF] Pronin | Nodal semimetals: a survey on optical conductivity[END_REF](Fig. 1.4c):

σ 1 (ω) = e 2 N W 12h ω v , (1.16)
where v is velocity parameter representing the slope of the cone and N W is the total number of Weyl cones. On the other hand, the optical conductivity due to interband excitations is frequency-independent in 2D conical bands which are present, for instance, in graphene or Dirac nodal-line semimetal. The optical response of 2D cones will be discussed in a greater detail in Sec. 3.

Crystalline solids in magnetic fields

The effect of the magnetic field applied to a solid-state system is directly reflected in its optical properties. The sufficiently strong magnetic field forces charged carries to exhibit a cyclotron motion following a path defined by the band structure of the material. The energy of the cyclotron orbits is quantized into so-called Landau levels (LLs). The Landau level spectroscopy, which is this main experimental technique of this thesis, is a powerful technique providing useful insights into the band structure of various materials in the framework of intra-and interband excitations of Landau-quantized electrons.

The LL spectrum is closely linked to the particular profiles of electronic bands 1.2. Interaction of light with condensed-matter systems and can be obtained by diagonalization of corresponding Hamiltonians. Let us begin with the quantum description of a parabolic band using the time-independent Schrödinger equation. The Hamiltonian describing a single free electron subjected to the magnetic field B = (0, 0, B) reads:

2Δ E(k) k y k x 𝐸 𝐹 𝐸 𝐹 𝐿 2 𝐿 3 𝐿 1 𝐿 0 𝐿 -1 𝐿 -2 𝐿 -3 E(k) k y k x 2Δ 𝐸 𝐹 𝐿 2 𝐿 3 𝐿 1 𝐿 0 𝐿 -1 𝐿 -2 𝐿 -3 𝐿 0 E(k)
ĤS = 1 2m 0 (p -eA) 2 , (1.17)
where m 0 is the electron mass, p is the momentum operator, and A is the vector potential A = (0, Bx, 0) assuming the asymmetric Landau gauge. Rewriting Eq. 1.17 with the respect to the orientation of the magnetic field and introducing a classical cyclotron frequency: ω c = eB/m 0 , we obtain a Hamiltonian in the form similar to a quantum harmonic oscillator:

ĤS = - h2 2m 0 d 2 dx 2 + 1 2 m 0 ω 2 c (x -x 0 ) 2 , (1.18)
where x 0 = hk y /m 0 ω c is the center of oscillator. The energy spectrum of this Hamiltonian is given by the formula [START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF]:

E S,n = hω c n + 1 2 , (1.19)
where n is the Landau level index. The Landau levels of parabolic bands are thus spaced equidistantly in energy and develop strictly linearly with the increasing magnetic field (Fig. 1.5a).

A very different result is obtained when the Landau quantization of electrons in conical bands, present in many topological materials, is considered. Let us describe such a conical band, with an energy band gap of 2∆ possibly opened, using the following Hamiltonian, which is referred to as the Hamiltonian of massive Dirac electrons:

ĤD =   ∆ hv(q x + iq y ) hv(q x -iq y ) -∆   , (1.20)
in which the parameter v plays the role of an asymptotic velocity. To obtain the LL spectrum of this Hamiltonian, we add the magnetic field (0,0,B) using the standard Peirls substitution and obtain:

ĤD = hv   ∆ hv(p x -iq y + ieBx) hphv(q x + iq y -ieBx) -∆   . (1.21)
The energy spectrum for the valence (negative values of n) and conduction (positive values of n) band is derived as:

E D,n = sgn(n)v 2ehB |n| + ∆ 2 , n = 0, ±1, ±2, ±3, . . . . (1.22)
The optical contribution consists of both intraband and interband transitions. Assuming conventional selection rules for the isotropic system, the electric-dipole-active transitions occur with the change of the Landau level index ∆n = ±1:

E Inter D,n = 2hev 2 B|n + 1| + ∆ 2 + 2hev 2 B|n| + ∆ 2 , (1.23) E Intra D,n = 2hev 2 B|n + 1| + ∆ 2 -2hev 2 B|n| + ∆ 2 . (1.24)
Let us note that selection rules stem from the matrix elements present in the Kubo-Greenwood formula in the previous chapter. The non-conventional selection rules with a higher LL index difference can be observed in an anisotropic system.

2

Experimental techniques

The Fourier-transform infrared spectroscopy (FTIR) was the main experimental technique used in this thesis. The history background and the basic principles of this technique will be discussed. The main part of the data acquisition was carried out using the magneto-reflectivity setup at the LNCMI-CNRS. Complementary measurements of absolute reflectivity at zero magnetic field were conducted at the University of Fribourg in the group of Prof. Ana Akrap. Magneto-transport experiments were performed at the LNCMI-CNRS in collaboration with Dr. B. A. Piot.

Infrared spectroscopy 2.1.1 Fourier-transform infrared spectroscopy

Fourier-transform infrared spectroscopy (FTIR) was used as the main experimental technique in this thesis. The FTIR is a powerful method suitable for a wide range of material characterization, ranging from the exploration of the vibration and rotation spectra of molecules up to the study of the electronic structure of condensed-matter systems. The history of infrared spectroscopy dates back to the year 1893 when the first measurement of the infrared transmission was performed on various substances [START_REF] Nichols | A study of the transmission spectra of certain substances in the infra-red[END_REF]. Over the years, infrared spectroscopy evolved from the use of the dispersive spectrometer towards interferometry utilizing the Fourier-transformation (FT). The fast Fourier-transform (FFT) algorithm and the high computation power of modern computers can render the resulting frequency spectrum almost instantly. The fundamental advantages of the FTIR over the conventional dispersive spectroscopy can be summarized as follows [START_REF] Griffiths | Fourier Transform Infrared Spectrometry[END_REF]:

• Multiplex (Fellgett's) advantage. The FTIR can collect simultaneously a broad range of photon frequencies. The dispersive spectrometer has to measure the same spectrum m times longer than the FTIR to achieve the same signal-to-noise ratio.

• Throughput (Jacquinot's) advantage. The monochromator used in dispersive spectroscopy utilizes entrance and exit slits, which restrict the intensity of light coming through. The FTIR does not use any kind of slits and the amount of light is limited only by the size of the aperture and mirrors.

• Wavelength accuracy (Connes') advantage. The dispersive setup consists of numerous adjustable mechanical parts such as diffraction grating, slits, and other optical elements. A wrong alignment could resolve in an inaccurate readout of the photon frequency. In contrast, the only movable part of most FTIRs is the interferometer mirror, which position is precisely calibrated by a laser passing through the interferometer.

The Michelson interferometer can serve as one of the simplest examples among a large variety of interferometers. It consists of one movable mirror, one fixed mirror, and a beam splitter (Fig. 2.1). The infrared radiation generated by the mercury lamp, Globar TM or tungsten filament, passes through the interferometer. The beam path is divided at the beam splitter into two branches. The length of the first branch is modified by the movable mirror, while the length of the second branch remains constant. Hence, the constructive and destructive interference of individual components of radiation (wavelengths) defines the spectral profile of the outgoing beam. The intensity of light at each mirror positions required by the wanted sampling (in fact, defining the spectral resolution) is measured by a detector. The resulting interferogram of the mirror displacement is then transformed by the FT into the domain of frequencies.

Let us now discuss how the FTIR energy spectrum is obtained in a greater detail. The wavenumber ∼ ν 0 (cm -1 ) is defined as an inverse value of the wavelength of the

I(δ)

Frequency spectrum S(ν 0 ) ~δ Fig. 2.1. Michelson interferometer. The beam from the IR source is divided into two branches at the beam splitter. The beam paths merge again in the constructive or destructive interference after the reflection from the fixed and movable mirror. The light passes or is reflected from the sample and it is detected by the active element of the bolometer. The resulting signal is the intensity interferogram as a function of δ . The interferogram is then treated by the FT algorithm into the frequency spectrum of photon energy. radiation source λ 0 (cm):

∼ ν 0 = 1 λ 0 . (2.1)
Let us denote the displacement of the mirror by δ , which represents the optical path difference (retardation) from the zero path difference (ZPD -the fixed and movable mirror have the same distance from the beam splitter). The beam is modified in the following way depending on δ :

• δ = 0. The beams are in phase and interfere constructively for the ZPD.

• δ = λ 0 /2. The beams are out of phase and interfere destructively.

• δ = λ 0 . The beams are in phase with constructive interference.

The beam is then transmitted or reflected from the sample and collected by the detector. In our case, the bolometer is the most frequently used detector of light in far-and mid-infrared detection. The active element of a bolometer changes its resistivity with irradiation and the response is compared with the load resistor of known resistivity. The measured voltage is recorded at the same time as the mirror displacement δ , resulting in the intensity interferogram as a function of I(δ ). In general, the resulting frequency spectrum S( ∼ ν 0 ) is obtained after applying the Fourier transform algorithm.

The energy resolution ∼ ν 0 of the spectrometer is proportional to the maximal displacement of the movable mirror ∼ ν 0 ∝ 1/δ . The nowadays available commercial research spectrometers can provide the resolution ranging from 0.2 cm -1 (e.g., base Bruker Vertex 80v) up to the high resolution spectrometers with resolution better than 0.0009 cm -1 (e.g., Bruker IF125) with the movable mirror displacement exceeding 11 meters.

Reflectivity setup for Landau level spectroscopy

To realize Landau level spectroscopy experiments presented in this thesis, only the reflectivity configuration has been used. These experiments were carried out at the LNCMI-CNRS in Grenoble. The infrared spectra were obtained by the commercial spectrometer Bruker Vertex 80v [START_REF]VERTEX 80/80v FT-IR Spectrometer[END_REF] with the combination of two polychromatic thermal sources (Hg-lamp, Globar TM ) and three different beam splitters (T230, T222, KBr). The whole optical path is evacuated to suppress the undesired IR-active components coming from air molecules. The output of the spectrometer can be separated from the rest of the beam path using the valve gate. This prevents the breaking of the vacuum if the external part of the setup has to be modified.

The output beam of the interferometer is focused by a gold-plated parabolic mirror attached to the removable stage on the top of the probe (Fig. 2.2a). The silicon beam splitter (Edmund optics 1-mm thick silicon wafer polished from both sides) is located right below the mirror stage. The silicon beam splitter also serves as a seal of the probe. The light then propagates through the gold-plated waveguide (1/2 inch diameter) and it is focused by an optical cone on the surface of the sample. The reflected light is collected by the same waveguide and reflected by the top silicon beam splitter towards the bolometer detector, which is placed inside the separate cryostat. The whole probe area is kept under low helium pressure ensuring sufficient heat transfer between the probes inside and the helium bath of the magnet cryostat. The superconducting solenoid (Oxford Instruments) is capable of reaching 16 T at 4.2 K and 18 T at 2.2 K.

The bolometer cryostat contains two detection units, each with a separate optical window and condensing cone. The first one is suitable for the far-infrared range (TPX window), the second one is for the mid-infrared range (KRS5 window). With the combination of two different sources, three beam splitters, and two bolometer detectors, one can cover a range from ∼ 25 to 7000 cm -1 . The final (in our case) reflectivity spectrum is composed of the combination of all measured ranges (see Table 2.1). To achieve the best possible signal-to-noise ratio, the bolometer is cooled down to 1.6 K by pumping the evaporating helium gas. In the used configuration, light propagates parallel to the applied magnetic field. This configuration is usually referred to as the Faraday geometry (Fig. 2.2b). The magnetic field is always applied perpendicular to the sample surface.

The high magnetic field installation at the LNCMI-CNRS in Grenoble can generate the magnetic field up to 37 T (magnet M9) using a 24 MW power supply. The resistive coil consists of two parts, the outer Bitter coil and the inner co-centric helical insert with a bore diameter of 34 mm. The current flowing through the coil reaches over 30 kA and generates a large amount of Joule heating. Two hydraulic pumps (each 150 l/s flow) are running water through the magnet installation to evacuate the dissipated heat. The high magnetic field reflectivity setup is operating on the same principle as the superconductive setup. To keep the magnetic field at the spectrometer location as low as possible, the 4meter-long oversized waveguide is used.

Reflectivity setup without magnetic field

The absolute reflectivity is a suitable complementary method to relative magnetoreflectivity -the main experimental method of this thesis. The majority of such complementary experiments, presented in this thesis, were carried out in close collaboration with the group of Prof. Akrap at the University of Fribourg. The reflectivity in the spectral range from far-to near-infrared was measured using the Bruker 70v spectrometer. The sample is mounted on a cold finger in the helium flow cryostat that is capable of reaching the temperature ranging from 5 to 300 K. The polarization-resolved measurements of NbAs 2 and NbSb 2 were realized at a near-normal angle of incidence with the light linearly polarized along the a or b crystallographic axes. The absolute reflectivity was determined using the custom-build in-situ gold evaporation technique. At high photon energies, the phase is fixed using ellipsometry (Woolam VASE ellipsometer). Optical conductivity was deduced from the reflectivity data using the Kramers-Kronig analysis. The Hagen-Rubens technique was used for the low-energy extrapolation [START_REF] Homes | Technique for measuring the reflectance of irregular, submillimeter-sized samples[END_REF]. At high photon energies, reflectivity is estimated using the method developed by Tanner [START_REF] Tanner | Use of x-ray scattering functions in Kramers-Kronig analysis of reflectance[END_REF] which calculates the reflectivity from atomic X-ray scattering cross-sections from 10 to 60 eV, followed by a 1/ω 4 free-electron termination.

Magneto-transport

The magneto-transport technique was another complementary method employed in this thesis. Resistivity measurements at low temperatures and high magnetic fields enable us to observe quantum oscillations, from which band structure parameters and Fermi surfaces can be extracted. Other quantities such as carrier concentrations and mobilities can also be determined from electronic transport. The magneto-transport measurements were realized at the LNCMI-CNRS in Grenoble in close collaboration with Dr. Benjamin A. Piot and Dr. Yuriy Krupko. With their help, we realized magneto-transport experiments on the Dirac nodal-line semimetal NbAs 2 and followed the angular dependence of Shubnikov-de Haas oscillations in this material. These results are discussed in Sec 3.4.5. To this end, four electric contacts were deposited using the silver paste at the edges of a square-like NbAs 2 single crystal, thus working in the Van der Pauw geometry. The experiment in magnetic fields up to 11 T and at various temperatures was carried out using a standard low-frequency lock-in technique (Fig. 1.3). The resistor (1 kΩ) is used to fix the current (I = 2 mA) flowing through the semimetallic sample. Care was taken to avoid heating of samples via current. The mechanical in-situ rotation stage was used for the angular dependence.

Lock-in Amplifier

3

Dirac nodal-line semimetals

In this section, the class of Dirac nodal-line semimetals will be introduced. The main concepts and properties of this group of topological materials will be described together with their optical signatures. The experimental work on three materials will be presented. The first material BaNiS 2 represents the group of nonsymmorphic materials (space group P4/nmm) and serves as one of the simplest examples of Dirac nodal-line semimetals and will be followed by two particular cases of transition metal di-pnictides NbAs 2 and NbSb 2 . The concept of the Lorentz boost will be discussed as a potential explanation of the unusual response of these two materials.

State-of-the-art

Band structure of nodal-line semimetals

The class of Dirac nodal-line semimetal (DNLSM) is a rather special category of topological matter, where the exotic electronic states are mostly related to the bulk band structure [START_REF] Pronin | Nodal semimetals: a survey on optical conductivity[END_REF]. The DNLSM class was theoretically predicted in 2011 by Burkov et al. [START_REF] Burkov | Topological nodal semimetals[END_REF] and then observed in 2016 in the case of PtSn 4 [START_REF] Wu | Dirac node arcs in PtSn 4[END_REF]. Unlike other topological classes such as Dirac or Weyl semimetals, where the linear band dispersion touches in a single point, in DNLSMs the touching point is extended to a 1D line propagating throughout the Brillouin zone of the material. The main characteristic feature of the electronic structure of such nodal lines is that if a 2D cut is made perpendicular to the local direction of the nodal line, the resulting electronic band structure resembles a 2D Dirac cone similar to the one found in graphene. Thus, the Dirac nodal-line semimetals exhibit 2D electronic states in their 3D bulk band structure.

A large variety of different nodal-line geometries can be found in the several crystallographic groups of solid state compounds, described both experimentally and theoretically. Similarly to other classes of topological matter, new materials are usually proposed based on symmetry arguments [START_REF] Bradlyn | Topological quantum chemistry[END_REF][START_REF] Vergniory | A complete catalogue of high-quality topological materials[END_REF][START_REF] Vergniory | All Topological Bands of All Stoichiometric Materials[END_REF][47][START_REF]Topological Materials Database[END_REF]. The best known representative of this topological class are the materials crystallizing in the nonsymmorphic space group P4/nmm. The presence of dispersive nodal lines in this compound group was predicted before the topological matter became a widely spread topic in condensed matter physics [START_REF] Tremel | Square Nets of Main Group Elements in Solid-State Materials[END_REF]. Even though the word "nodal" is often used, majority of the materials of this class exhibit a small spin-orbit gap instead of a pure band crossing.

ZrSiS represents one of the best-known examples in the class. The nodal lines and nodal surfaces were studied by different experimental techniques including angle resolved photo-electron spectroscopy (ARPES), magneto-transport, optics and magnetooptics [START_REF] Pezzini | Unconventional mass enhancement around the Dirac nodal loop in ZrSiS[END_REF][START_REF] Schilling | Flat Optical Conductivity in ZrSiS due to Two-Dimensional Dirac Bands[END_REF][START_REF] Uykur | Magneto-optical probe of the fully gapped Dirac band in ZrSiS[END_REF][START_REF] Fu | Dirac nodal surfaces and nodal lines in ZrSiS[END_REF][START_REF] Schoop | Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS[END_REF]. The electronic properties are protected by the time-reversal and spaceinversion symmetry of the crystal. ZrSiS is composed of alternating Zr, Si, and S layers. The material can be cleaved before experiment along c axis to ensure that the fresh sur- Transition metal di-pnictides (TMdP) is another group of compounds which hosts the nodal lines in their band structure. The group consists of six materials with the formula XPn 2 (X = Ta, Nb, Pn = P, As, Sb) crystallizing in the monoclinic system with the centrosymmetric space group C12/m1 [START_REF] Bensch | Nbas2[END_REF] (Fig. 3.2a). The presence of a nodal line region was revealed by first-principle calculations in all TMdP materials, with a small spin-orbit gap (Fig. 3.2b) [START_REF] Xu | Electronic structures of transition metal dipnictides XPn2 (X=Ta, Nb; Pn= P, As, Sb)[END_REF][START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF]. Two spin degenerated non-equivalent nodal lines are located with the respect to the mirror plane of the crystal. All TMdP nodal lines are open and propagate through multiple Brillouin zones. In contrast to the previous group of nonsymmorphic P4/nmm materials, there are not many works dealing with nodal lines in TMdP. The most detailed studies were reporting the anisotropic optical response of the NbAs 2 [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF] and magneto-transport measurements [START_REF] Yuan | Large magnetoresistance in compensated semimetals TaAs 2 and NbAs 2[END_REF][START_REF] Shen | Fermi surface topology and negative longitudinal magnetoresistance observed in the semimetal NbAs 2[END_REF][START_REF] Wang | Resistivity plateau and extremely large magnetoresistance in NbAs 2 and TaAs 2[END_REF][START_REF] Peramaiyan | Anisotropic magnetotransport and extremely large magnetoresistance in NbAs 2 single crystals[END_REF]. NbAs 2 was also found to host a pressure-induced superconductivity without any structural phase transition up to the pressure of 29.8 GPa [START_REF] Li | Pressure-induced superconductivity in topological semimetal NbAs2[END_REF].

This work is mainly focused on magneto-optics of three different materials. The first compound is BaNiS 2 representing the group of nonsymmorphic P4/nmm materials and it is one of the simplest examples of a dispersive nodal line. Then the attention is paid to two particular representatives of TMdP: NbAs 2 and NbSb 2 with a more complex nodal-line structure.

Optical response of Dirac nodal-line semimetals

The optical spectroscopy techniques in the far-and mid-infrared range have been widely used as a sensitive probe for bulk electronic states in the crystalline solids. With its penetration depth and high energy resolution one can determine precisely the electronic band structure of topological materials. The linear band dispersion is usually spread over the range of tens to hundreds of meV, thus the high energy resolution is one of the biggest advantages over other widely applied condensed-matter techniques such as the ARPES. Optical excitation with or without the presence of the magnetic field allows us to determine the crucial material parameters and the overall bulk properties of the system such as band gap, velocity parameters and to qualify the departure from the linearity of the cone, or the direction of the propagation of the nodal line in the Brillouin zone.

The optical studies of DNLSMs are mostly realized by the measurement of the reflectivity response with or without magnetic field. The reflectivity measurement done in the wide range of frequencies (often combined with ellipsometry) allows one to deduce the optical conductivity, σ (ω) = σ 1 (ω) + iσ 2 (ω), by Kramers-Kronig relations. The real part of the optical conductivity σ 1 (ω) reflects the dissipate properties of a given solid and can be often interpreted straightforwardly. Let us now review the basic aspects of the real part of the interband optical conductivity of the nodal line materials. The simplest case of electron-hole symmetrical Dirac or Weyl cone with a dimensionality of d follows a power-law frequency dependence [START_REF] Schilling | Flat Optical Conductivity in ZrSiS due to Two-Dimensional Dirac Bands[END_REF][START_REF] Carbotte | Dirac cone tilt on interband optical background of type-I and type-II Weyl semimetals[END_REF][START_REF] Pronin | Nodal Semimetals: A Survey on Optical Conductivity[END_REF]:

σ 1 (ω) ∝ ω (d-2) .
(3.1)

In the 3D case (d = 3), the full expression reads:

σ 1 (ω) = e 2 N 12h ω v Θ (hω -2E F ) , (3.2)
where v is the velocity parameter given by the slope of the linear band dispersion, E F represents the Fermi level. Below 2E F , the optical transitions are being blocked by Pauli exclusion principle. This is described by the Heaviside step function Θ(hω -2E F ) (Fig 3 .3a,d). If the electron scattering rate Γ is considered, the Heaviside function can be substituted by:

1 2 + 1 2 arctan ω -2E F /h Γ . (3.3)
In the special case of flat gapless nodal-line with the dimensionality of the bands d = 2, the observed optical conductivity becomes independent of the frequency, similarly to the one observed in graphene where the universal optical conductivity is given by πe 2 /(2h) per a graphene sheet [START_REF] Mak | Measurement of the optical conductivity of graphene[END_REF][START_REF] Nair | Fine structure constant defines visual transparency of graphene[END_REF] (Fig 3 .4a). The 3D nature of the nodal line is reflected by the length k 0 in the reciprocal space. Thus the frequency dependence can be described as (Fig 3 .3b, e):

σ f lat 1 (ω) = e 2 16h k 0 Θ (hω -2E F ) . (3.4)
In the majority of cases, the nodal lines are actually gapped due to the spin-orbital coupling which is inherently present in all topological materials. In such case, the above formula is valid only approximately when the photon energies approach the value of the band gap 2∆.

The line can disperse in energy along its propagation in reciprocal space. Such lines are referred to as dispersive nodal-lines. The linearly dispersing part is characterized by the velocity parameter v ∥ = 1/h dE(k)/dk line . Then the constant parameter k 0 in Eq. 3.4 has to be now replaced by the function depending on the energy profile of the line k 0 (ω). The resulting optical conductivity increases linearly with the frequency until the point k max , where dispersive part turns into a flat one with v ∥ = 0 [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF].

The polarization-resolved infrared spectroscopy can be used for the determination of the direction of the nodal-line propagation within the plane measured. It was theoretically predicted [START_REF] Ahn | Electrodynamics on Fermi Cyclides in Nodal Line Semimetals[END_REF][START_REF] Mukherjee | Transport and optics at the node in a nodal loop semimetal[END_REF][START_REF] Carbotte | Optical response of a line node semimetal[END_REF] and experimentally confirmed [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF] that in the case of an open nodal line propagating throughout the Brillouin zone in a certain direction k line , the optical conductivity vanishes if the polarization of the incident light is aligned with the line direction (E ∥ k line ). On the other hand, the response is strongest in the case of the polarization perpendicular to the k line (E ⊥ k line ).

Almost frequency independent optical conductivity was observed in the recent study of ZrSiS performed in the group of Prof. Dressel in Stuttgart. It serves as an almost perfect example of the quasi-2D system [START_REF] Schilling | Flat Optical Conductivity in ZrSiS due to Two-Dimensional Dirac Bands[END_REF] (Fig 3 .4b). The optical conductivity reached the constant value of σ 1 (ω) ≈ 6600 Ω -1 cm -1 over the total range of nearly 300 meV. The approximate length of the line was extracted as k 0 = 4.3 Å -1 , which is in agreement with the band structure calculation and the total size of the Brillouin zone. Since the Fermi energy is not exactly known, the onset of interband absorption in optical conductivity give us only upper bound of the band gap 2∆ (below 30 meV).

Magneto-optical spectroscopy is a suitable technique to extract the band gap and velocity parameter of DNLSMs with great precision. For magnetic fields strong enough, the electronic states becomes quantized in Landau levels, which can be well described using 2D massive Dirac model (example of ZrSiS case in Figs 3 .4c,d). When the nodalline direction is aligned with the applied magnetic field, the Landau level spectrum reads:

E ±n = ± 2eh|n|Bv 2 + 2∆ 2 , (3.5)
where the v is the averaged velocity parameter.

For a non-zero angle θ between nodal line and magnetic field, the spectrum scales Chapter 3. Dirac nodal-line semimetals with a cosine function:

E ±n = ± 2eh|n|Bv 2 cos θ + 2∆ 2 . (3.6)
The combination of the optical methods serves as one of the most precise procedures to study the properties of Dirac electrons of the nodal line and to describe the nodal-line propagation through the Brillouin zone. To give an example, the band gap of ZrSiS was theoretically predicted as 10 meV [START_REF] Schoop | Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS[END_REF], the ARPES measurement were reporting values between 15-60 meV [START_REF] Chen | Dirac line nodes and effect of spin-orbit coupling in the nonsymmorphic critical semimetals MSiS (M = Hf, Zr)[END_REF] and optical conductivity measurements below 30 meV [START_REF] Schilling | Flat Optical Conductivity in ZrSiS due to Two-Dimensional Dirac Bands[END_REF]. The values extracted from magneto-optics are ∆ = (26 ± 2) meV and the averaged velocity parameter v = (3.0 ± 0.2) × 10 5 m/s [START_REF] Uykur | Magneto-optical probe of the fully gapped Dirac band in ZrSiS[END_REF].

Relativistic-like effects in dispersive nodal lines

After the rise of graphene and topological matter, the relativistic-like description of solids had became a relevant and an integral part of condensed matter physics [START_REF] Bansil | Colloquium: topological band theory[END_REF][START_REF] Chiu | Classification of topological quantum matter with symmetries[END_REF][START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF]. Such a type of relativity persist beyond the dispersion of the low-energy bands in many solids, with the examples such as Klein tunneling [START_REF] Klein | Die reflexion von elektronen an einem potentialsprung nach der relativistischen dynamik von Dirac[END_REF][START_REF] Katsnelson | Chiral tunnelling and the Klein paradox in graphene[END_REF][START_REF] Young | Quantum interference and Klein tunnelling in graphene heterojunctions[END_REF] or the chiral anomaly [START_REF] Nielsen | The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal[END_REF][START_REF] Son | Chiral anomaly and classical negative magnetoresistance of Weyl metals[END_REF][START_REF] Xiong | Evidence for the chiral anomaly in the Dirac semimetal Na 3 Bi[END_REF][START_REF] Huang | Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs[END_REF][START_REF] Burkov | Chiral anomaly without relativity[END_REF][START_REF] Reis | On the search for the chiral anomaly in Weyl semimetals: the negative longitudinal magnetoresistance[END_REF][START_REF] Jia | Weyl semimetals, Fermi arcs and chiral anomalies[END_REF][START_REF] Ong | Experimental signatures of the chiral anomaly in Dirac-Weyl semimetals[END_REF].

The particular dependence of energy on the frame of reference described by the Lorentz transformations is one of the most profound aspects of relativity. Originally in the special theory of relativity, the Lorentz transformations describes the relations between two mutually in motion reference frames. The main consequences of Lorentz boost are the dilation of the time and the length contraction through the Lorentz factor γ. In the frame of condensed matter physics, we regularly use Dirac Hamiltonians and their variants to describe electronic states of solids. Such Hamiltonians are thus naturally relativistic-like, having a Lorentz covariant forms. One may wonder whether this fact has some directly measurable impact on electrical or optical properties of particular solids.

One of the salient aspects of relativity is the particular dependence of energy on the frame of reference: for a particle of mass m moving at a speed u lower than the speed of light c, a Lorentz boost to the co-moving frame of reference changes the particle's energy from E to E/γ = mc 2 , in terms of the Lorentz factor γ = 1/ 1β 2 and the rapidity β = u/c. A natural question that one may ask is the following: can one observe this relativistic renormalization equally in topological materials governed by the Dirac Hamiltonian or a variant of it, upon the replacement of c by a characteristic velocity v?

While the effects of Lorentz covariance have been theoretically studied, namely in the framework of systems with tilted conical bands, both in two [START_REF] Katayama | Pressure-induced zero-gap semiconducting state in organic conductor α-(BEDT-TTF) 2 I 3 salt[END_REF][START_REF] Goerbig | Electric-field-induced lifting of the valley degeneracy in α-(BEDT-TTF) 2 I 3 Dirac-like Landau levels[END_REF][START_REF] Sari | Magneto-optics of quasirelativistic electrons in graphene with an inplane electric field and in tilted dirac cones in α -(BEDT TTF) 2 I 3[END_REF] and three [START_REF] Soluyanov | Type-II Weyl semimetals[END_REF][START_REF] Tchoumakov | Magnetic-field-induced relativistic properties in type-I and type-II Weyl semimetals[END_REF][START_REF] Zhang | Landau quantization in tilted Weyl semimetals with broken symmetry[END_REF] dimensions, an experimental verification is yet lacking. The key finding of this section is that Lorentz boosts have important experimental consequences in Dirac materials. Most notably and unexpectedly, we find that the optical band gap extracted from magnetooptical measurements in the dispersive nodal-line Dirac semimetal niobium diarsenide (NbAs 2 ) depends on the orientation of the explored crystal (presented in Sec.3.4). As we show below, this orientation defines a particular Lorentz boost, by fixing the angle between the applied magnetic field and the direction of the spectroscopically relevant part of the nodal line.

To appreciate the link between magnetic-field and Lorentz boosts in tilted cones, let us first consider a 2D conical band, characterized by a gap 2∆ and an asymptotic velocity v, which is tilted by an additional velocity parameter u (in Fig. 3.5, u ∥ x). Such a system is described by the following variant of a 2D massive Dirac Hamiltonian:

Ĥ2D = hu • k1 +   ∆ hv(k x -ik y ) hv(k x + ik y ) -∆   . (3.7)
In an out-of-plane magnetic field, B ⊥ u, the tilt can be formally viewed as a drift velocity of electrons in the crossed magnetic and effective electric fields, E = uB. The

E(k) = -huk x ± ∆ 2 + h2 v 2 k 2 .
The dashed lines correspond to asymptotic dispersions. (a) 2D Dirac cone, which can be also viewed as a reference frame. (b) Tilted type-I cone with tilt parameter u representing the relativistic particle in motion with the respect to the material characteristic slope velocity v. The system is referred as in magnetic regime. (c) Overtilted type-II cone -the Landau level quantization collapse and the system transfers from magnetic regime to electric one. drift is in the direction perpendicular to both the tilt and the magnetic field.

In this specific case, the problem of the electron motion in a tilted cone becomes mathematically equivalent to the dynamics of a relativistic charge carrier in the crossed electric and magnetic fields [START_REF] Goerbig | Electric-field-induced lifting of the valley degeneracy in α-(BEDT-TTF) 2 I 3 Dirac-like Landau levels[END_REF][START_REF] Tchoumakov | Magnetic-field-induced relativistic properties in type-I and type-II Weyl semimetals[END_REF][START_REF] Lukose | Novel electric field effects on Landau levels in graphene[END_REF]. This motion is therefore governed by fully Lorentz-covariant Dirac and Maxwell equations. This covariant formulation, and thus the use of Lorentz transformations, allows us to calculate the energy spectrum in a reference frame where the (effective) electric field vanishes, meaning u = 0 [START_REF] Goerbig | Electric-field-induced lifting of the valley degeneracy in α-(BEDT-TTF) 2 I 3 Dirac-like Landau levels[END_REF][START_REF] Sari | Magneto-optics of quasirelativistic electrons in graphene with an inplane electric field and in tilted dirac cones in α -(BEDT TTF) 2 I 3[END_REF][START_REF] Lukose | Novel electric field effects on Landau levels in graphene[END_REF]. A similar relativistic-like approach has been invoked in the past, in order to understand behaviour of narrow-gap semiconductors [START_REF] Aronov | Light absorption in semiconductors in crossed electric and magnetic fields[END_REF][START_REF] Zawadzki | Semirelativistic behavior of electrons in InSb in crossed magnetic and electric fields[END_REF][START_REF] Zawadzki | Inversion electrons on narrow-band-gap semiconductors in crossed electric and magnetic fields[END_REF] in real crossed electric and magnetic fields.

The impact of the tilt u -or in the sense of reasoning above, the impact of the Lorentz boost -on the Landau quatization is profound [START_REF] Goerbig | Electric-field-induced lifting of the valley degeneracy in α-(BEDT-TTF) 2 I 3 Dirac-like Landau levels[END_REF][START_REF] Tchoumakov | Magnetic-field-induced relativistic properties in type-I and type-II Weyl semimetals[END_REF]. We obtain the Landau level (LL) spectrum that is typical of 2D massive Dirac electrons, but whose energy band gap and velocity parameter are renormalized by the Lorentz factor, γ = 1/ 1u 2 /v 2 :

E n = ± (∆/γ) 2 + 2eBnhv 2 /γ 3 , n = 0, 1, 2 . . . (3.8)
For large tilts, u ≥ v, the spectrum collapses and marks a transition between regimes referred to as magnetic and electric [START_REF] Tchoumakov | Magnetic-field-induced relativistic properties in type-I and type-II Weyl semimetals[END_REF][START_REF] Takeuchi | Relativistic E × B acceleration[END_REF]. In the semi-classical picture, this crossover corresponds to a transition from closed towards open cyclotron orbits in type-I and type-II conical bands (Figure 3.5b, c).

Landau levels

To describe electronic states at the nodal line and to see its connection with the above discussed tilted cones, let us consider the following minimal Hamiltonian for a dispersive nodal line:

Ĥ = (ε 0 + hwq line )1 +   ∆ hv(q x -iq y ) hv(q x + iq y ) -∆   , (3.9)
where the wave vector q = (q x , q y , q line ) = kk line is defined with respect to any point k line on the nodal line. The velocity parameter w describes the slope of the dispersing nodal line. In the plane (q x , q y ) perpendicular to the local direction τ τ τ, the Hamiltonian (3.9) becomes that of a 2D massive Dirac electron, with the gap 2∆ and the asymptotic velocity v. In contrast, when the bands are cut in a plane that is not perpendicular to τ τ τ, the corresponding dispersion shows the tilt described by (3.7) and schematically shown in Figure . 3.5.

Let us now explore the impact of the magnetic field in the detail. For a sake of simplicity we will consider only one spin sector (spin up) of one of the two nodal lines (ξ = +1). Due to the isotropy of the Hamiltonian (3.9) in the x-y plane, the magnetic field is defined as B = B(cos θ e line + sin θ e x ) = Be B for 0 < θ < π/2 and a new orthogonal basis with the third component aligned with the magnetic field can be introduced without any loss of generality: {e x ′ , e y ′ , e z ′ ≡ e B } = {cos θ e xsin θ e line , -e y , cos θ e line + sin θ e x }.

(3.10)

When this new basis is used, the Hamiltonian reads:

H = hw (q B cos θ -q x ′ sin θ ) + hv (q x ′ cos θ + q B sin θ ) σ x -hvq y ′ σ y + ∆σ z . (3.11)
The energy spectrum in the presence of a magnetic field will be derived in the next few steps with the emphasis on the relativistic properties, which are embedded in the above Hamiltonian. When the dispersion along the direction of the magnetic field is neglected (q B = 0), the Hamiltonian becomes that of a tilted 2D massive Dirac electron:

H = hwq x ′ sin θ + hvq x ′ cos θ σ x -hvq y ′ σ y + ∆σ z , (3.12)
where the tilt velocity along the x ′ direction is w sin θ and the Fermi velocity along the x ′ and y ′ directions are v cos θ and -v, respectively. However, this Hamiltonian is not that for the Lorentz-invariant Dirac equation but a variant of it due to the tilt term. To unveil the relativistic properties of the system, one has to introduce the magnetic field through the standard Landau gauge A = -By ′ e x ′ as:

H -hwq x ′ sin θ = -hw sin θ eBy ′ + hv cos θ (q x ′ -eBy ′ )σ x -hvq y ′ σ y + ∆σ z , (3.13)
where an effective electric field Bw sin θ is aligned with the y ′ direction. Most importantly, the right hand side is now that of the standard Lorentz invariant Dirac equation except the anisotropy of the speed of light in the x ′ and y ′ directions. Therefore, one encounters now the problem of a relativistic 2D massive Dirac electron in the presence of an electric field of Bw sin θ in the y ′ direction and a magnetic field of B in the z ′ direction, thus moving with a drift velocity of w sin θ in the x ′ direction assuming a small drift velocity. Remark that the speed of light in the x ′ direction is replaced by v cos θ , so the rapidity is defined as β = w tan θ /v. The electric field could be always canceled in the co-moving frame of an electron under influence of a crossed electric and magnetic field in the case when the drift velocity is smaller than the speed of light, implying the rapidity β < 1 [START_REF] Sari | Magneto-optics of quasirelativistic electrons in graphene with an inplane electric field and in tilted dirac cones in α -(BEDT TTF) 2 I 3[END_REF][START_REF] Tchoumakov | Magnetic-field-induced relativistic properties in type-I and type-II Weyl semimetals[END_REF].

Deriving the co-moving frame, with the drift velocity w, and with the use of a Lorentz boost, the problem is thus simplified to that of an electron subjected only to magnetic field, which is the case of Landau quantization. Once the Landau levels (LL) are found in the co-moving frame, the spectrum in the lab frame (the original frame of reference), could be immediately concluded by doing the inverse Lorentz boost from the co-moving frame back to the lab frame. Technically, this can be conveniently accomplished using hyperbolic transformation, which is the 2D representation of Lorentz boost in the Lorentz group. In our case, the hyperbolic transformation of the aforementioned Lorentz boost reads M = exp(φ σ x /2), with the rapidity

β = tanh φ = ω tan θ /v.
The Hamiltonian now can be solved with the magnetic field already incorporated by the above Landau gauge:

H = hw q B cos θ -(q x ′ -eBy ′ ) sin θ + hv (q x ′ -eBy ′ ) cos θ + q B sin θ σ x -hvq y ′ σ y + ∆σ z . (3.14)
Using the same hyperbolic transformation, M = exp(φ σ x /2), with the rapidity β = tanh φ = w tan θ /v, the final Hamiltonian is obtained:

H T = MHM = hwv v * q B + hw 2 + v 2 v * q B sin θ cos θ σ x + hv * (q x ′ -eBy ′ )σ x -hvq y ′ σ y + ∆σ z (3.15)
where the velocity v * is defined as:

v * 2 = v 2 cos 2 θ -w 2 sin 2 θ with γ = 1 1 -β 2 = cosh φ = v cos θ v * (3.16)
with γ > 1 is the relativistic Lorentz factor.

It should be noted that H T does not have the same spectrum as H for the reason that we are now working in the co-moving frame. In the basis of |ψ T ⟩ = γ -1/2 M -1 |ψ⟩ given the eigenstate |ψ⟩ of H, one can construct from H T another Hamiltonian H E = (H T -E sinh φ σ x )/ cosh φ with the same spectrum as H:

H E = hwq B cos θ + 1 γ   ∆ h√ 2vv * ℓ B a † E h√ 2vv * ℓ B a E -∆   , (3.17)
where ℓ B = h/(eB) is the magnetic length and a pair of energy-dependent ladder operators is defined as:

a E = - 1 √ 2vv * v * ℓ B (y ′ -⟨y ′ ⟩ E ) + iℓ B vq y ′ , a † E = - 1 √ 2vv * v * ℓ B (y ′ -⟨y ′ ⟩ E ) -iℓ B vq y ′ , ⟨y ′ ⟩ E = ℓ 2 B v * v * q x ′ + q B w 2 + v 2 v * sin θ cos θ -E w sin θ hv * , (3.18) with [a E , a † E ] = 1.
The subscript E indicates the dependence on energy E. In particular, the center of cyclotron ⟨y ′ ⟩ E shifts with energy.

Since the energy-dependent term in H E is absorbed in the definition of the ladder operators, the energy spectrum is self-consistently found in the eigenstates of energy E λ n in the form:

|ψ T,n,λ ⟩ =   cos ζ n,λ |n, E λ n ⟩ sin ζ n,λ |n -1, E λ n ⟩   , (3.19)
where ζ n,λ is an angle depending on n and the sign of energy λ = ±, |n ′ , E λ n ⟩ is the wave function of the one-dimensional quantum harmonic oscillator defined by the previous ladder operators. Given the Landau level index n, n ′ = n or n -1. The final Landau level spectrum is obtained:

E ± n = hw q B / cos θ ± (∆/γ) 2 + 2nheBv 2 cos θ /γ 3 for n > 0, E 0 = hw q B / cos θ + ∆ γ for n = 0, (3.20)
which has the form typical of massive Dirac electrons. Nevertheless, with the gap and velocity renormalized by the pseudo-relativistic Lorentz factor:

2∆ → 2∆/γ and v → v √ cosθ γ 3/2 . (3.21)
Importantly, for the other line index ξ = -1 or the spin down we obtain an analogous LL spectrum replacing ∆ by -∆.

It is worth noting that the discussed Lorentz-boost-driven renormalization of the apparent band gap may, to certain extent, resemble the well-known Franz-Keldysh effect [START_REF] Franz | Einfluß eines elektrischen Feldes auf eine optische Absorptionskante[END_REF][START_REF] Keldysh | Behavior of non-metallic crystals in strong electric fields[END_REF] which finds its practical use in electro-optical modulation [START_REF] Liu | Review and perspective on ultrafast wavelength-size electro-optic modulators[END_REF]. Even a closer analogy appears when the Franz-Keldysh effect is explored in crossed electric and magnetic fields [START_REF] Aronov | Light absorption in semiconductors in crossed electric and magnetic fields[END_REF]. Nevertheless, in our case, there is no real electric field applied to the explored system. It only appears in an effective way, in the pseudo-relativistic Dirac-type Hamiltonian, being proportional to the magnetic-field component perpendicular to local direction of the dispersive nodal line τ τ τ D .

Velocity operators

To study the magneto-optical properties of dispersive Dirac nodal-line semimetals theoretically, one has to evaluate the matrix element of the corresponding velocity operators: ⟨ψ n | vk |ψ m ⟩. In this case, it is more practical to work with the basis |ψ T,n ⟩ already in hands by remarking that:

⟨ψ n |∇ k H|ψ n ′ ⟩ = γ⟨ψ T,n |∇ k H T |ψ T,n ′ ⟩ (3.22)
thanks to the fact that M is k-independent. Finally, in the basis |ψ T ⟩, the velocity operators for H T are:

vT,x = v * σ x , (3.23) vT,y = -vσ y , (3.24)
vT,z = wv v * +

w 2 + v 2 v * sin θ cos θ , (3.25)
where one notices an emergent anisotropy of the velocity parameter induced by the applied magnetic field.

Selection rules

With the velocity operators, one can a priori derive the selection rules for electricdipole inter-LL transitions which are active in the Faraday configuration (the configuration with the wave vector of light parallel to B). The process could be analogous to Landau-quantized tilted 3D cones [START_REF] Sari | Magneto-optics of quasirelativistic electrons in graphene with an inplane electric field and in tilted dirac cones in α -(BEDT TTF) 2 I 3[END_REF]. When calculating the matrix elements such as ⟨ψ T,n | vT,x |ψ T,m ⟩, one has to deal with terms such as ⟨n ′ , E n |m ′ , E m ⟩ which are no longer 0 or 1. This is due to the mismatch of their energy-dependent orbital center (see Eq. 3.18). The typical selection rules n → n ± 1 are not obtained in general [START_REF] Landwehr | Landau level spectroscopy[END_REF]. Instead, all direct transitions become in principle possible, as long as the occupation of states (Pauli principle) allows. The transitions stemming from the selection rule other than n → n ± 1 proliferates in the optical conductivity. By the sum rule, this would dilute the prominent Landau fan which reflects the n → n ± 1 rule. Therefore, the usual clear-cut Landau fan is blurred.

To illustrate the evolution of electric-dipole selection rules with the angle θ D between the magnetic field and the local nodal-line direction of the dispersive part τ τ τ, we proceed in the way analogous to [START_REF] Sari | Magneto-optics of quasirelativistic electrons in graphene with an inplane electric field and in tilted dirac cones in α -(BEDT TTF) 2 I 3[END_REF][START_REF] Tchoumakov | Magnetic-field-induced relativistic properties in type-I and type-II Weyl semimetals[END_REF][START_REF] Scully | Quantum Optics[END_REF]. In Fig. 3.6, we plot normalized squares of matrix elements, such as |⟨ψ T,n | vT,x /v * |ψ T,m ⟩| 2 , for different interband inter-LL excitations (for indices n = 0 . . . 6) and parameters deduced for the dispersive part of the nodal line in NbAs 2 . As expected, one obtains the standard selection rules in isotropic systems for θ D = 0 • : n → n ± 1. This is because the energy-dependence in the cyclotron center is canceled by sin θ (see in Eq. 3.18). With increasing θ D , additional inter-LL excitations emerge and their relative strength gradually increases. At relatively large angles θ D , the selection rules differ significantly and transitions, n → n ± 1, originally dominating the response, effectively disappear, except the transition 0 → 1 that stays well-defined and relatively strong (as long as β < 1).

Magneto-optics of BaNiS 2

As mentioned earlier, the Dirac nodal-line semimetals may host the nodal lines of various forms. In this class of materials, BaNiS 2 is one of the simplest examples. It belongs to the group of materials crystallizing in the non-symmorphic tetragonal P4/nmm space group. The crystal is composed of puckered NiS 5 pyramids (Fig. 3.7a). The nodal lines in this material were found to be slightly dispersive and propagating parallel to the k z direction (tetragonal axis). The Dirac nodal line crosses the Fermi level at k z = 0 and rising above Fermi level with increasing k z (Fig. The ab-intio calculations were conducted using the DFT approach on the QUAN-TUM ESPRESSO package with plane-wave implementation [START_REF] Giannozzi | Quantum espresso: a modular and open-source software project for quantum simulations of materials[END_REF][START_REF] Giannozzi | Advanced capabilities for materials modelling with quantum espresso[END_REF]. The precision of the calculation was enhanced by the modified hybrid Heyd-Scuseria-Ernzerhof (HSE) functional [START_REF] Heyd | Hybrid functionals based on a screened Coulomb potential[END_REF][START_REF] Heyd | Erratum: "Hybrid functionals based on a screened Coulomb potential[END_REF][START_REF] Krukau | Influence of the exchange screening parameter on the performance of screened hybrid functionals[END_REF]. The resulting band structure and density of states (DOS) are plotted in Fig 3 .7c. From the calculated Fermi surface, band structure and the density of states (DOS), the two pockets at the Fermi energy can be identified in agreement with the previous studies [START_REF] Santos-Cottin | Rashba coupling amplification by a staggered crystal field[END_REF][START_REF] Klein | Importance of nonlocal electron correlation in the BaNiS 2 semimetal from quantum oscillations studies[END_REF]. The electron pocket lies at the Γ -Z connection in the Brillouin zone (purple part in Fig 3 .7b). The hole pockets are located halfway along the Γ -M line (brown part in Fig 3.7b). This type of the band structure is typical of the square-lattice compounds [START_REF] Tremel | Square Nets of Main Group Elements in Solid-State Materials[END_REF][START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF]. As in other topological materials, the inclusion of spin-orbit coupling opens a small gap in the Dirac cone. With only one type of nodal line dispersing in the k z direction, BaNiS 2 is one of the simplest DNLSM, compared to the other widely explored materials such as ZrSiS (nodal loop + dispersive nodal line in k z ) or transitionmetal di-pnictades materials (curved dispersive line in the a-b plane of the crystal with multiple crossing with E F ). This makes BaNiS 2 a perfect case study for demonstration of a nodal-line fingerprint in both optics and magneto-optics.

Magneto-reflectivity

The magneto-optical spectroscopy was used for precise determination of the band gap of the material and for extraction of the velocity parameter. The initial measurements were performed using superconducting solenoid (up to 16 T). The data set was then extended up to 34 T using resistive coil in order to explore the further development of interand intra-LL transitions. To make sure that both of the experiments are consistent, the overlap between both setups was made in the region between 12 T and 16 T. The sample was kept under the low-pressure helium exchange gas. The measurements were conducted at the temperature of 4.2 K.

The magnetic field was applied along the nodal-line direction i.e., perpendicular to the (001) facet. The experiment was conducted in the Faraday configuration, when the wave vectors of the incident and reflected light are parallel to the applied magnetic field. The obtained magneto-reflectivity spectra were normalized by the response at B = 0 to visualize the B induced changes (Fig. 3.8). A series of electric-dipole active excitations is observed and development of the transitions with the increasing magnetic field is assigned to the inter-and intra-band transitions between LLs of 2D massive Dirac electrons:

ε ±n (B) = ± 2hev 1 v 2 Bn + ∆ 2 , n = 0, 1, 2 . . . , (3.26)
Assuming the standard n → n ± 1 selection rules, valid for isotropic bands, the energies of the excitations read:

E Inter n = 2heBv 1 v 2 (n + 1) + ∆ 2 + 2hv 1 v 2 (n) + ∆ 2 , (3.27)
E Intra n = 2heBv 1 v 2 (n + 1) + ∆ 2 -2hv 1 v 2 (n) + ∆ 2 , (3.28)
where ∆ is the half band gap and v 1,2 are the asymptotic velocity parameters for two mutually perpendicular directions. The lowest interband excitation corresponding to the Landau level indexes E Inter -1→0 and E Inter 0→1 is the most profound one. In the data, one may find traces of an intraband (cyclotron resonance-like) transition, which originates from the hole doped part and can be found from the region below ∼20 meV.

To simplify the analysis, the resonance energies were directly associated with the maxima of the relative magneto-reflectivity R B /R 0 . This approximation is valid due to the fact that ε 2 is larger in this energy range than ε 1 , thus dominates over the reflectivity response. From the zero-field limit extrapolation, the band gaps 2∆ = (16 ± 2) meV is extracted. The slope of the transitions dE/dB corresponds to the asymptotic velocity pa-

rameter v = √ v 1 v 2 = (1.55 ± 0.05) × 10 5 m • s -1 .
With the band gaps only (16±2) meV, BaNiS 2 has one of the lowest band gap among all known Dirac nodal-line semimetals up to now. The band gap is three times smaller as compared to ZrSiSe [START_REF] Shao | Electronic correlations in nodal-line semimetals[END_REF], twice smaller than ZrSiS [START_REF] Schoop | Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS[END_REF], and seven times smaller than NbAs 2 [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF].

Optical conductivity

The conclusions of magneto-optical studies realized at the LNCMI-CNRS in Grenoble can be confronted with the zero-field optical response measured by Santos-Cottin et al. [START_REF] Santos-Cottin | Optical conductivity signatures of open Dirac nodal lines[END_REF]. The optical response of the (001) oriented facet, without any external magnetic field applied, was measured in the configuration of near-normal-incidence reflectivity in the energy range from 4 meV to 1.85 eV at the temperature ranging between 5-300 K. Two spectrometers were used for the data acquisition (Bruker IFS113/v for the far infrared range and Bruker IFS 66/v for the mid/near infrared region). The dataset was extended up to 5.45 eV at room temperature using AvaSpec 14x2048 optical fiber spectrometer (Fig. 3.9). A high quality single crystal of the size about 2×2×0.3 mm 3 was grown by a self-flux method [START_REF] Santos-Cottin | Rashba coupling amplification by a staggered crystal field[END_REF][START_REF] Shamoto | Single crystal growth of BaNiS2[END_REF]. Before the measurement, the samples a-b plane was cleaved in order to probe the fresh surface. The sample was mounted on a cold finger of an ARS Helitran cryostat using an overfilling technique [START_REF] Homes | Technique for measuring the reflectance of irregular, submillimeter-sized samples[END_REF] providing the absolute reflectivity accuracy better than 0.5 % and the relative accuracy between different temperatures about 0.1 %. In order to deduce the complex optical conductivity, the low and high frequency limit was extrapolated using the Hagen-Rubens formula and by the Tanner method, respectively [START_REF] Tanner | Use of x-ray scattering functions in Kramers-Kronig analysis of reflectance[END_REF].

The real part of optical conductivity is the temperature independent in the range of 0.1 -0.3 eV. The so-called isosbestic point, which is common for gapped semiconductors [START_REF] Braslavsky | Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations 2006)[END_REF], is in a direct contrast to the isosbestic line of BaNiS 2 . The main feature of such line is that the spectral weight (SW, area under the σ 1 (ω)) is transferred between the low-energy region (below 0.1 eV) and the higher energy region (above 0.25 eV). The SW is depleted from the high energy region to the low energy part with rising temperature, leaving the isosbestic line intact. No SW transfer is observed above 0.6 eV. The amount of the SW transferred at different energy cuts is shown in the inset of Fig. 3.10b. An ideal nodal line should give optical conductivity independent of the frequency, just scaling with the length of the line (k 0 ), similar to an ideal 2D Dirac system. In the case of BaNiS 2 , the slope of optical conductivity (region 0.1 to 0.3 eV) is non-zero which suggests a slight dispersion of the nodal line along k z (Fig. 3.10c). Taking into account the DFT calculated band structure, the band dispersion model can be approximated as:

ε ± = ± ∆ 2 + v 2 1 k 2 1 + v 2 2 k 2 2 + v z k z , (3.29)
where k 1 and k 2 are momentum vectors perpendicular to the nodal line direction, ∆ is spin-orbit half-gap, and v 1,2 are asymptotic velocities defining the slope of an anisotropic conical band at large momentum. The DFT band structure calculation predicts the velocity parameters v 1 = 2.61 eVÅ and v 2 = 0.94 eVÅ, thus implying a profound anisotropy of the Dirac cone. The averaged velocity parameter

v i = √ v 1 v 2 = 1.
57 eVÅ is in good agreement with the experimental value extracted from magneto-optics v = 1.02 eVÅ.

The optical conductivity due to a the dispersive nodal line was modeled using Eq.3.29 as:

σ i NL (ω) = N 8 e 2 h k 0 (ω) v 2 i v 1 v 2 1 + 4∆ 2 ω 2 Θ(ω -2∆) , (3.30)
where N is the degeneracy of the nodal line (the number of nodal lines and the spin multiplicity) in the first Brillouin zone, while 2∆ stands for the local optical gap. The Θ stands for the Heaviside step function, which allows one to fit the experimental onset of σ 1 due to interband transitions. In the used model, the electric field of incident light is polarized in the i direction. Since no polarization was used in experiments the velocity parameter v i will be considered with the electric field averaged in the plane of incidence, for the purpose of the data evaluation. The segment k(ω) is an effective length of the nodal-line contributing at the photon energy:

k 0 (ω) =      0, ω < 2∆, (k max -k 0 ) 2πE max ω, 2∆ ≤ ω ≤ E max , (k max -k 0 ) 2π , ω > E max .
(3.31)

The revealed nodal line features can be directly linked to the proposed model eq. 3.30. The E max parameter describes the saturation of the linear band dispersion and can be roughly estimated from the optical and magneto-optical data. The 2D Dirac model follows experimental data well until ≈ 80 meV (Fig. 3.10). The same deviation can be seen in σ 1 thus the best estimate of E max is about 80 meV. The Dirac cone network is fourfold degenerated in the first Brillouin zone and twice degenerate with the spin which implies N = 8. The k max parameter represents the whole length of the nodal line in the first Brillouin zone and it is defined as:

k max = π I c , (3.32)
where I c is the total length in k z direction leading to I c = 8.897 Å. The gap and velocity parameter are extracted from our magneto-optical measurement as 2∆ = 16.5 meV and v = 1.02 eVÅ. Velocity v 1 parameter could be extracted from the ARPES as v 1 = 1.85 eVÅ [START_REF] Santos-Cottin | Rashba coupling amplification by a staggered crystal field[END_REF]. Finally remaining v 2 = v 2 /v 1 = 0.56 eVÅ.

Magneto-optics of NbAs 2

NbAs 2 is so far the most studied transition metal di-pnictide. Magneto-transport studies revealed the presence of several Fermi surfaces, as well as extremely large and strongly anisotropic magneto-resistance [START_REF] Yuan | Large magnetoresistance in compensated semimetals TaAs 2 and NbAs 2[END_REF][START_REF] Shen | Fermi surface topology and negative longitudinal magnetoresistance observed in the semimetal NbAs 2[END_REF][START_REF] Wang | Resistivity plateau and extremely large magnetoresistance in NbAs 2 and TaAs 2[END_REF][START_REF] Peramaiyan | Anisotropic magnetotransport and extremely large magnetoresistance in NbAs 2 single crystals[END_REF]. Optical measurements performed at zero magnetic field identified the high anisotropic response in the mid-infrared region depending on the polarization of incident light with respect to crystallographic orientation. Nearly vanishing optical conductivity deduced for light polarized parallel with the b axis of NbAs 2 was interpreted as a signature of a nodal line extending indefinitely through multiple Brillouin zones [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF]. Up to this day, the magneto-optical response of NbAs 2 has been barely explored [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF], let alone understood, which calls for its detailed investigations.

NbAs 2 is a nearly compensated semimetal, which makes this material a great candidate for optical and magneto-optical studies. It was proven both theoretically and experimentally that this material hosts two nodal lines [START_REF] Xu | Electronic structures of transition metal dipnictides XPn2 (X=Ta, Nb; Pn= P, As, Sb)[END_REF][START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF]. These nodal lines are opened and propagate through the whole Brillouin zone and are mirror symmetrical with the respect to the Γ -Y -Z plane. The low-energy electronic excitations can be described, in the plane perpendicular to the local nodal-line direction τ τ τ, using a model for 2D massive Dirac electrons (Fig. 3.5a). The corresponding velocity parameter v and the spin-orbit gap 2∆, typical of all currently studied nodal-line or nodal-loop semimetals [START_REF] Shao | Electronic correlations in nodal-line semimetals[END_REF][START_REF] Schoop | Dirac cone protected by nonsymmorphic symmetry and three-dimensional Dirac line node in ZrSiS[END_REF][START_REF] Takane | Dirac-node arc in the topological line-node semimetal Hf-SiS[END_REF], may vary along the line (Fig.

Samples overview and characterization

For the purpose of this study the four NbAs 2 samples were used for magnetooptical and optical measurements. All samples were synthesized by Dr. Raman Sankar from Taipei, Academica Sinica. The similar outcomes were obtained from the crystal synthesized by Prof. Mario Novak (University of Zagreb) but for the sake for consistence all presented results were obtained on samples from Dr. Raman Sankar. The same synthesis procedure was used for both NbAs 2 and NbSb 2 (results obtained from NbSb 2 are presented in the section 3.5). Two step chemical vapor transport processes were used to synthesize and grow single crystals of NbX 2 (X= Sb, As). A quartz ampoule with a length of 30-40 cm was used for the synthesis and growth. At first, stoichiometric amounts of 5N purity precursors of Nb, Sb or As in a molar ratio of 1:2 were sealed in an evacuated quartz ampule. The vacuum-sealed quartz ampule containing the binary mixture was treated at 950 °C for two days and then cooled to room temperature, yielding polycrystalline NbX 2 (X= Sb, As). Secondly, the polycrystalline powder of NbX 2 (X= Sb, As) was mixed with I 2 in a weight ratio of 100:5 and vacuum-sealed in a two-zone tube furnace having a thermal gradient of about 950-850 °C within 40 cm. The resulting NbX 2 (X= Sb, As) single Fig. 3.12. X-ray characterization of NbAs 2 samples. Normalized diffraction patterns of all facets studied in magneto-optical experiments. The (001)-oriented facet of Sample 1 was also used for reflectance measurements at B = 0. crystals possessed multiple shiny few millimeters facets suitable for optical experiments.

The identification of each facet was carried out by the Bragg-Brentano diffractometer equipped with Cu X-ray tube, channel-cut germanium monochromator and scintillation detector. The orientation was identified from the resulted X-ray diffraction patterns for ten facets in total Fig. 3.12. The Sample 1 was used for acquisition of polarizationresolved measurement without magnetic field. The orientation of its crystallographic axes a and b was deduced from the pole diagram figure. The magneto-reflectivity data with B oriented perpendicular to the ( 101) and (607) crystallographic planes were collected using the nearby (403) facet on Sample 1 rotated respectively by 5 • and 7 • in the a-c plane.

Optical response of NbAs 2 at B=0

To characterize the optical response of NbAs 2 at B = 0, infrared reflectivity was measured in a close collaboration with the group of Prof. Ana Akrap (Fribourg University) on Sample 1, on the (001)-oriented facet using radiation polarized linearly along the a and b crystallographic axes. The broad range of photon energies were measured by the Vertex 70v FTIR spectrometer equipped with custom-built in-situ gold evaporation technique. The high energy limit was extended using ellipsometry measurements. The complex optical conductivity σ (ω) was obtained by the standard Kramers-Kronig analysis. The measured reflectivity and the deduced optical conductivity are presented in Fig. 3.13, 3.14.

The signature features of a nodal-line semimetal are apparent at the lower photon energies in the absence of the magnetic field [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF][START_REF] Mele | Dowsing for nodal lines in a topological semimetal[END_REF]. The local optical band gaps of the NbAs 2 were identified in the optical conductivity measurement from the clearly distinguishable onset of interband absorption (cf. Fig. 3.11 and 3.13b). The energies of the band gaps were deduced and interpreted in line with Shao et al. [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF] as 2∆ D = (88±2) meV and 2∆ F = (113 ± 2) meV for the dispersive and flat parts, respectively. Above this onset, the optical conductivity increases linearly in ω. While such behavior resembles systems with 3D conical bands [START_REF] Timusk | Threedimensional Dirac fermions in quasicrystals as seen via optical conductivity[END_REF][START_REF] Orlita | Observation of three-dimensional massless Kane fermions in a zinc-blende crystal[END_REF][START_REF] Neubauer | Interband optical conductivity of the [001]-oriented Dirac semimetal Cd 3 As 2[END_REF], in NbAs 2 , the linear dependence is caused by the occupation effect of Pauli blocking along the dispersive part of the nodal line [START_REF] Pronin | Nodal semimetals: a survey on optical conductivity[END_REF][START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF][START_REF] Santos-Cottin | Optical conductivity signatures of open Dirac nodal lines[END_REF][START_REF] Mele | Dowsing for nodal lines in a topological semimetal[END_REF]. The polarization-resolved experiment allows to determine the approximate orientation of the local direction of each part of the nodal line in the (001) crystallographic plane. The detailed modeling of the local direction and further analysis of the optical conductivity is presented in the next section with the focus on the zero-field response and modeling of the nodal line direction (section 3.4.4).

The anisotropy of the optical response (Fig. 3.13b) reflects the orientation of the nodal lines, implying significantly higher Drude-type and interband absorption strength for the radiation polarized along the b axis (approximately perpendicular to τ τ τ) [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF]. The strong in-plane anisotropy was also suggested by the magneto-transport measurement, see [START_REF] Shen | Fermi surface topology and negative longitudinal magnetoresistance observed in the semimetal NbAs 2[END_REF][START_REF] Wang | Resistivity plateau and extremely large magnetoresistance in NbAs 2 and TaAs 2[END_REF][START_REF] Peramaiyan | Anisotropic magnetotransport and extremely large magnetoresistance in NbAs 2 single crystals[END_REF] and Sec. 3.4.5. The smearing of the plasma edge ∼ 80 meV is apparent from the temperature dependent optical measurements (Fig. 3.14). The optical conductivity of the nodal line region is nearly temperature independent and increasing linearly with the frequency. As was demonstrated in Sec. 3.3.2, such behavior is one of the typical fingerprints of the Dirac nodal-line response. So-called isosbestic line is the direct consequence of the 2D nature of the Dirac cone [START_REF] Santos-Cottin | Optical conductivity signatures of open Dirac nodal lines[END_REF]. The nearly flat optical conductivity spreads over the region 200-400 meV and it is reminiscent of the universal optical conductivity of graphene [START_REF] Nair | Fine structure constant defines visual transparency of graphene[END_REF].

Magneto-optical response

In total, ten different crystallographic facets of NbAs 2 were measured using the tools of magneto-optics. The reflectivity measurements were conducted using the nonpolarized infrared radiation. The magnetic field was applied perpendicular to the selected facets. All experiments were carried out in the Faraday geometry with the incident and reflected beam parallel to the orientation of the magnetic field. The investigation of the multiple facets allowed us to probe the angular response of the material under the influence of the magnetic field with the electrons undergo the cyclotron motion always perpendicular to the orientation of the magnetic field. Each facet provides a different fixed angle between the field and the local direction of the flat (τ τ τ F ) or dispersive (τ τ τ D ) part of nodal line. To keep the geometry as simple as possible, we selected facets with a zero middle index (n 0 m). Thus the vector B always lied in the mirror plane of the NbAs 2 crystals and forms identical angles θ F and θ D with the local directions, τ τ τ F and τ τ τ D , of the two mirror-symmetric nodal lines at the flat and dispersive parts, respectively, where the Fermi level is crossed. Applying the magnetic field perpendicular to the facets with a non-zero middle index would not respect the mirror-symmetry, thus leading to two local direction angles for each part of the nodal line.

The observed response comprises a series of sharp resonances with a weakly sublinear dependence on B that can be directly associated with interband Landau level (inter-LL) excitations (Fig. 3.15). The observed response -in position, spacing and relative/absolute intensity of inter-LL transitions -strongly varies with the explored facet. Across the board, this response includes two characteristic sets of inter-LL excitations:

1. The lower set of transitions extrapolate, depending on the particular facet, to the energy equal to or lower than 2∆ D = (88 ± 2) meV (see B = 0 extrapolations using yellow dashed lines in Fig. 3.15). The band gap change is approximately 15 meV across the measured facets, traceable throughout the lowest inter-LL transition. It is worth noting that the lower set is not observable on all facets.

2. The upper set contains transitions that always extrapolate to the energy of 2∆ F = (113 ± 2) meV in the zero-field limit and thus are clearly associated with the flat crossing of the nodal line (grey transparent line in Fig. 3.15). The band gap is not changed and in all cases the response of 2D massive Dirac electrons is observed. The velocity parameter, extracted from the slope of the transition v = d(hω)/dB, varies for each measured facet.

This observation -an orientation-dependent gap -is the main finding of our work on NbAs 2 and it ventures beyond the common knowledge in LL spectroscopy of solids [START_REF] Landwehr | Landau level spectroscopy[END_REF]. In conventional materials, the slope d(hω)/dB, and the spacing of inter-LL excitations may depend on the orientation of the crystal with respect to the magnetic field, as well as on the experimental configuration (e.g., Faraday versus Voigt). Nevertheless, the apparent band gap obtained from the zero-field extrapolation of interband inter-LL excitations is widely used as an unambiguous estimate of the separation between electronic bands. Quite unexpectedly, such an apparently evident approach fails in the case of NbAs 2 .

The apparent change of the band gap may be explained in the view of relativistic properties of the material described by the Hamiltonian 3.9 introduced in section 3.2. Let us assume that the externally applied magnetic field makes an angle θ with the local direction of the nodal line τ τ τ. The electrons are driven by the conventional Lorentz force in the specific plane cut of the band dispersion perpendicular to B. Thus, for non-zero angles θ , we study the magneto-optical response of electrons in tilted anisotropic conical bands (described in depth in section 3.2). The corresponding Landau level spectrum then gets the Lorentz-boost-renormalized form:

E n = ± (∆/γ) 2 + 2eBnhv 2 /γ 3 , n = 0, 1, 2, ..., (3.33)
modified by the dispersive term hwq B / cos θ , where q B is the wave vector along the applied magnetic field and w is the velocity parameter dispersing along local direction τ τ τ in the vicinity of the k D line representing the tilt of the cone. The ratio between the tilt (w sin θ ) and asymptotic velocities (v cos θ ) defines the rapidity β , which is angular dependent and reads:

β = w v tan θ . (3.34)
The effective angle-dependent Lorentz factor reads:

γ = 1 1 -w 2 v 2 tan 2 θ . (3.35)
Therefore, a pseudo-relativistic decrease of the band gap and the velocity parameter is anticipated when the magnetic field orientation deviates from the local direction of the nodal line:

2∆ → 2∆ eff = 2∆ γ and v → v eff = v γ 3/2 .
(3.36)

Taking various facets of the crystal (different θ ), one can effectively tune the strength of the renormalization effects up to the point when the angle exceeds the critical value of tan -1 (v/w). In this case the β > 1 marking the transition into type II overtilted cone and the Landau level quantization collapses [START_REF] Tchoumakov | Magnetic-field-induced relativistic properties in type-I and type-II Weyl semimetals[END_REF].

The lowest interband Landau level transition LL Inter 0↔1 was used to extract the experimental parameters such as the gap and velocity. As suggested in section 3.2, the selection rules are alternated by the pseudo-relativistic renormalization as well [START_REF] Sari | Magneto-optics of quasirelativistic electrons in graphene with an inplane electric field and in tilted dirac cones in α -(BEDT TTF) 2 I 3[END_REF]. Nevertheless, for any β < 1, the LL Inter 0↔1 transition is always expected to be relatively strong. The energy of the lowest inter-LL transitions is then defined as:

LL Inter 0↔1 = ∆ eff + (∆ eff ) 2 + 2ehB(v eff ) 2 . (3.37)
Any optically active part of the nodal line could be modelled by the proposed equations 3.37, 3.36 and 3.35. The tilt parameter w = 0 for the flat part at k F line gives the γ = 1, thus no renormalization is expected with the change of the magnetic field orientation. On the other hand, the dispersive part around k D line with the tilt w ̸ = 0 implies a facet-dependent pseudo-relativistic decrease of the optical gap. Let us now discuss the selection rules elaborated theoretically in Sec. 3.2.3 in the context of experimental data. In addition to the Lorentz-boost renormalization of the spectrum, our model for the dispersive nodal line implies a departure from the conventional electric-dipole selection rules, n → n ± 1, which are generally valid for all isotropic systems [START_REF] Landwehr | Landau level spectroscopy[END_REF]. Some additional transition are expected with the higher LL index change. The matrix elements have been numerically calculated for the parameters of the dispersive part of NbAs 2 . The relative intensities of transitions are graphically visualized in panels above the detailed analysis of each facet in Figs 3.17 The facets can be classified into three groups based on their characteristic magnetooptical response. The first set of measurements -facets (40-1), (20-1), ( 100) and (10-1) -is presented in Fig. 3.17. It was found out by an analysis of the inter-LL excitations corresponding to the dispersive part and by the minimization fit searching for the local direction of both τ τ τ D and τ τ τ F , normals of these facets are relatively close to the τ τ τ D with the angles θ D = 26 • , 27.5 • , 34 • and 49 • , respectively. The massive Dirac model fit is plotted over the false-color map of R B /R 0 respectively for each facet. Clear agreement was found between the flat part fit (grey dashed lines) and the experimental data using the parameters shown in Fig 3 .16a,b. At least four inter-LL transitions are resolved following the standard dipole active selection rules n → n ± 1. Analyzing the transitions corresponding to the dispersive part, for small angles θ D , the magneto-optical response is dominated by n → n ± 1 transitions (red dotted lines), although, other excitations emerge as well (e.g., n → n ± 2, orange dotted lines). As for the higher θ D , new sets of inter-LL transitions with a greater change of the LL index n start to gain on intensity. Nevertheless, for these four particular facets, the relative intensities of the transitions stays mostly within n → n ± 1 and n → n ± 2 (matrix elements plotted above each panel). Thus the renormalization effect of the Lorentz-boost stays relatively weak for low angles θ D .

To explore and validate the Lorentz-boost-driven gap renormalization and see the further development of the inter-LL transitions, a high magnetic field experiment up to 34 T was carried out on the second group, which consists of facets ( 201) and (403). With the θ D = 51 • and 56 • , respectively (Fig. 3.18a, b), the effect of gap renormalization is getting stronger and one can observe the zero field limit extrapolation of inter-LL to the lower band gap values than the original non-renormalized 2∆ D . In contrast to the previous set of measurements (Fig. 3.17), for larger angles θ D one finds a plethora of optical transitions apart from the lowest 1 ↔ 0. Theoretically the dominant ones follow the ruleof-thumb selection rules n → αn and n → n/α, where α is an integer (α = 4 -6, panels of Fig. 3.18c, d), in agreement with preceding works on tilted 3D cones [START_REF] Sari | Magneto-optics of quasirelativistic electrons in graphene with an inplane electric field and in tilted dirac cones in α -(BEDT TTF) 2 I 3[END_REF][START_REF] Tchoumakov | Magnetic-field-induced relativistic properties in type-I and type-II Weyl semimetals[END_REF]. As seen in Fig. 3.18c, d, transitions lowest in energy (1 ↔ 0) stay strong as long as β < 1 and were used to deduce the effective parameters 2∆ eff D and v eff D . These parameters can be used to identify excitations between Landau levels with higher indices and thus give experimental insights into the selection rules. The experimental data collected with B perpendicular to the (201) and (403) crystallographic planes were compared in Fig. 3.18a, b. The expected positions of selected interband inter-LL excitations (dotted lines) calculated using the corresponding effective gap and velocity parameters. The same color framing is used for the selection rules intensities plotted next to panels for a clear comparison to the experimental data. When the magnetic field is applied perpendicularly to these crystallographic planes (Fig. 3 of ( 201) and (403). The flat part transitions are not visible at all in these two cases due to the high θ F = 87 • and 90 • , respectively. On the other hand the two remaining facets (20-3) and (001) do not show any trace of the lower set of transitions implying the collapse of the Landau level quantization for these two particular cases. The transitions from the flat part can be traced easily from both of them.

Detailed analysis of optical response at B=0

The detailed analysis of the polarization-resolved optical response without magnetic field underlines our estimates of the local directions τ τ τ F and τ τ τ D based on the magnetooptical data. To describe the optical response of NbAs 2 at B = 0, the simple model for electronic states in a nodal-line semimetal proposed in the section 3.2 was used. The proposed Hamiltonian 3.9, now with both line index ξ = ±1 and spin included, reads: Ĥ = (ε 0 + hwq line )1 + hv(ξ q x σ x + q y σ y ) + ξ ∆σ z τ z , (3.38) where σ i (i = x, y, z) and τ z are standard Pauli matrices for orbital and spin degrees of freedom, respectively. This Hamiltonian describes electronic states in the vicinity of any point at the nodal line, k = k line + q. The used orthogonal coordinate system, q = (q x , q y , q line ≡ q z ), has the third component always aligned with the local direction of the nodal line τ τ τ (see Fig. 3.20) which is roughly parallel with the a crystallographic axis [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF]. The other nodal line is located symmetrically in momentum space with respect to the Γ -Y -Z mirror plane. Similar to the local direction τ τ τ, the Hamiltonian parameters ε 0 , v, ∆ and w also vary, smoothly and weakly along the nodal line, and therefore, can be viewed as a function of q line .

To simplify the problem at hand, the nodal line zone could be divided into four segments I, II, III and IV, which approximately describe its propagation through a half of the Brillouin zone. The locations of these segments, as well as their conduction/valence band energies (at k line ), are shown in Figs. 3.21b,d,f. The suggested profile of the nodal lines approach the results of DFT simulations and overall conclusions presented by Shao et al. [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF]. In each segment, the nodal line is reduced down to a straight line in momentum space with the length of k i . The corresponding segment is characterized by a set of parameters: ε i 0 , v i , ∆ i and w i , where i = I, II, III and IV, and two angles θ i a and θ i b which encodes the local direction of the given segment τ τ τ i with respect to a and b crystallographic axes, cf. Figs. 3.21d, f. Within each segment, these parameters are supposed to be constant except ∆ III which is considered to vary linearly with q line to ensure the continuity of the nodal line in energy.

With such simplifications, the diagonal component of optical conductivity is ob-Fig. 3.20. Nodal lines in momentum and real space -schematics. The position of two nodal lines in NbAs 2 with respect to the real space crystal axes a, b, c and the corresponding reciprocal space axes k a , k b , k c . The local axes along the nodal line: q line is parallel to the nodal line and q x,y is always orthogonal to it. tained by integration along the chosen segment i, using the local basis of the Hamiltonian:

ℜ σ i xx (ω) = Ne 2 8h dq line 1 + 4(∆ i ) 2 (hω) 2 [ f (ε i 0 + hw i q line -hω/2) -f (ε i 0 + hw i q line + hω/2)]Θ(hω -2∆ i ) (3.39)
and, by the isotropy of the model, σ i xx (ω) = σ i yy (ω). The other diagonal component reads:

ℜ σ i zz (ω) = Ne 2 4h(v i ) 2 dq line ∂ ∆ i ∂ q line 2 1 - ∆ i hω 2 [ f (ε i 0 + hw i q line -hω/2) -f (ε i 0 + hw i q line + hω/2)]Θ(hω -2∆ i ). (3.40)
In the equations above, N = 2 is the number of the nodal lines in the Brillouin zone (dou-ble degeneracy due to already included spin), f is the Fermi-Dirac distribution and Θ is the Heaviside function. The main contribution to the optical conductivity comes from Eq. 3.39. Eq. 3.40 leads to a non-zero contribution to the optical conductivity only when the gap parameter changes with q line . Such a possibility is not explicitly included in the Hamiltonian 3.38. Nevertheless, as mentioned above, it is relevant in our approximation for the segment III (see Fig. 3.21b). Anyway, in practice the contribution from Eq. 3.40 is negligible due to the small ratio between h-1 ∂ q line ∆ i and v i .

In our reflectivity measurements at B = 0 (Fig. 3.21a, c, e), the incident radiation was polarized linearly along the a or b axes. To calculate the experimentally probed optical conductivities ℜ [σ aa (ω)] and ℜ [σ bb (ω)], one has to make a corresponding projection for each segment (i = I, II, III and IV) and sum their contributions:

ℜ [σ aa (ω)] = ∑ i cos 2 θ i a ℜ σ i zz (ω) + sin 2 θ i a ℜ σ i xx (ω) , (3.41
)

ℜ [σ bb (ω)] = ∑ i cos 2 θ i b ℜ σ i zz (ω) + sin 2 θ i b ℜ σ i xx (ω) . (3.42)
Even though a relatively crude approximation for the profile of the nodal lines is used, there still remains a number of free parameters to be tuned. For each segment with its length k i in momentum space, there are four Hamiltonian parameters v i , ∆ i , w i and ε i 0 as well as two angles θ a and θ b . The Fermi energy is an additional parameter common to all segments. To reduce this number, the following points can be considered:

1. When the anisotropy of the dispersion perpendicular to the nodal line is neglected (i.e., v i x = v i y ), the main contribution to optical conductivity (Eq. 3.39) becomes independent of the velocity parameter v i .

2. For the parameters ∆ i and w i as well as the corresponding angles, θ i a and θ i b , for i = II and IV, there are solid estimates coming from the analysis of the magneto-optical response, due to the dispersive and flat parts of the nodal line, respectively: e.g.,

∆ II = ∆ D , ∆ IV = ∆ F , w II = w, w IV = 0.
3. Additional constraints on w i , ∆ i , ε i 0 and k i , appear because the nodal lines are continuous in momentum as well as in energy. For instance, the size of the Brillouin zone approximately fixes the sum of segments lengths and the gap parameter ∆ III has to evolve smoothly from ∆ II = ∆ F to ∆ IV = ∆ D . In the latter case, the simplest approximation (linear in q line ) was chosen.

4. Due to the Pauli occupation effect, the expected profile of the nodal line [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF] allows us to neglect the contribution of interband excitations coming from the segment I. This is because our interest lies only in the optical response at relatively low photon energies (below hω ≈ 400 meV, Fig. 3.21a, c, e).

Using the constraints (1) to (4), the number of freely tunable parameters is strongly reduced down to the segments' lengths. The theoretically calculated conductivity approaches fairly well the experimentally measured curves, see Fig. 3.21a. The contributions of individual segments are shown using dotted lines and the corresponding color coding in Figs. 3.21c, e. Notably, the agreement could be further improved by considering velocity anisotropy (v i

x ̸ = v i y ) which alters the relative strength of individual segments.

Such approach has been successfully used by Shao et al. [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF]. Nevertheless, these additional three tuning parameters cannot be directly determined from the magneto-optical experiments and they will be not considered in the fitting procedure. The obtained values of the used tuning parameters -the lengths of the segments k i and the angles θ

Transport measurement

Magneto-transport experiments provided another, complementary characterization of the explored NbAs 2 crystals. To this end, a monocrystal from the same batch as Sample 1 was chosen. The electrical contacts were deposited using silver paint in the corners of the (001)-oriented facet of a rectangular shape. The current was always applied along b axis of the sample and the longitudinal magneto-resistance R xx measured at selected (low) temperatures. Two sets of experiments were performed, with the magnetic field applied perpendicular to the a-b plane and along the a axis, see Fig. 3.22. In both cases, a profound magneto-resistance R xx (B) was observed [START_REF] Yuan | Large magnetoresistance in compensated semimetals TaAs 2 and NbAs 2[END_REF]. While the frequency F a ≈ 300 T seems to be rather independent of the magnetic field direction, other frequencies, F b and F c , indicate a large degree of anisotropy and correspondingly non-spherical Fermi surfaces. The damping of oscillations with temperature has been used to get estimates of the effective (cyclotron) masses m * . The damping of the most prominent frequencies, i.e., F a = 276 T and F c = 120 T for the magnetic field applied perpendicular to the (001) plane and F a = 280 T with B parallel to the a axis, was analyzed using the Lifshitz-Kosevich formula:

∆ 0 /∆(T 0 ) = αT m * / B sinh(αT m * / B) , (3.43)
where B stands for the mean magnetic field, T 0 is the lowest measured temperature and α = 2π 2 k B m 0 /(eh) = 14.69 T/K. The fitted effective masses are similar for all three frequencies: m * = (0.25 ± 0.05)m 0 where m 0 is the bare electron mass. This result is in agreement with values reported in the literature [START_REF] Yuan | Large magnetoresistance in compensated semimetals TaAs 2 and NbAs 2[END_REF][START_REF] Shen | Fermi surface topology and negative longitudinal magnetoresistance observed in the semimetal NbAs 2[END_REF][START_REF] Wang | Resistivity plateau and extremely large magnetoresistance in NbAs 2 and TaAs 2[END_REF][START_REF] Peramaiyan | Anisotropic magnetotransport and extremely large magnetoresistance in NbAs 2 single crystals[END_REF].

To explore this anisotropy in a greater detail, we followed the longitudinal magnetoresistance of NbAs 2 as a function of the magnetic-field direction with respect to the crystal, while keeping the current flowing along the b axis. In Figs. 3.24, the magnetic field was applied perpendicular to the a and b axes and the sample rotated around the a and b crystallographic axes, respectively. In the latter case, the frequencies F b and F c exhibit a fairly pronounced angle dependence, with the maximum appearing roughly at θ B,a ∼ 90 • (i.e., B ⊥ a) and with the maximum-to-minimum ratio reaching nearly 10. This indicates strongly elongated, cigar-like, Fermi surfaces oriented approximately along the a axis. In contrast, the frequencies F b and F c remain nearly constant when rotating the sample around the a axis. Such behavior agrees with conclusions of the preceding magnetotransport studies [START_REF] Yuan | Large magnetoresistance in compensated semimetals TaAs 2 and NbAs 2[END_REF][START_REF] Shen | Fermi surface topology and negative longitudinal magnetoresistance observed in the semimetal NbAs 2[END_REF][START_REF] Wang | Resistivity plateau and extremely large magnetoresistance in NbAs 2 and TaAs 2[END_REF][START_REF] Peramaiyan | Anisotropic magnetotransport and extremely large magnetoresistance in NbAs 2 single crystals[END_REF], and importantly, also with the presence of the nodal lines propagating roughly parallel to the a crystallographic axis. The sketch of geometry for each measurement with the direction of magnetic field and applied current.

Magneto-optics of NbSb 2

Niobium di-antimonide is another nodal-line Dirac semimetal explored in the scope of this thesis. Based on a DFT study, one may expect the band structure of NbSb 2 resembles the one of NbAs 2 , with two nodal lines. Two calculated cuts of the beginning (Z -I 1 ) and the end of the nodal line (Y -X 1 ) are close to the Fermi level and thus their low energy excitation can be probed by optical and magneto-optical methods Fig. 3.25 [START_REF] Xu | Electronic structures of transition metal dipnictides XPn2 (X=Ta, Nb; Pn= P, As, Sb)[END_REF]. NbSb 2 is almost electron-hole compensated, similarly to NbAs 2 and TaAs 2 . The material has a mirror symmetry with the respect to the Γ -Y -Z.

In the contrast to NbAs 2 , there are no extensive studies found in the literature apart from one DFT and one magneto-transport publications, which are focused on the comparison of multiple TMdP [START_REF] Xu | Electronic structures of transition metal dipnictides XPn2 (X=Ta, Nb; Pn= P, As, Sb)[END_REF][START_REF] Wang | Anisotropic giant magnetoresistance in NbSb 2[END_REF][START_REF] Guo | Extreme magnetoresistance and SdH oscillation in compensated semimetals of NbSb 2 single crystals[END_REF]. This leaves the details of the NbSb 2 band structure mostly uncovered. In this section, we will show how the magneto-optical response of different facets and the detailed analysis of the optical response at B = 0 may allow us to determine the local direction of the nodal line at each crossing with the Fermi energy. This knowledge along with the comparison to the DFT results and transport experiment will then be used to determine the nodal line band structure with a great precision.

Overview of studied samples

Two single crystals of NbSb 2 were investigated. They were grown by Raman Sankar from Academica Sinica in Taipei. The charge vapor transport procedure similar to the synthesis of NbAs 2 was employed (see details in Sec. 3.4.1). The synthesized crystals are of a cylindrical shape and possess a few millimeter-size shiny facets appropriate for magneto-optical and optical studies. At the end, only one facet of Sample 1 was used for the study, the rest of the measurement was done on the Sample 2. Both samples came from the same batch. The X-ray diffraction was used to identify the individual facets of both samples. The pole diagram analysis and the geometrical analysis with the respect to the facets of the whole crystal (visual inspection) was used to determine the a or b crystallographic axes, respectively.

Optical response

The optical response of the NbSb 2 was investigated in the analogous way as for NbAs 2 . The same configuration was used for polarization-resolved experiment performed in the group of Prof. Akrap at the University of Fribourg, as described already in Sec. 3.4.2. The radiation on the (001) oriented surface of Sample 2 was polarized along the a or b crystallographic axes, respectively. From the extracted real part of optical conductivity σ 1 (ω), it is seen that the system exhibits a high anisotropy, similarly to the NbAs 2 . From the temperature dependent measurements done in the range from 10 to 300 K (Fig. 3.27), one can see the characteristic optical response of an open Dirac nodal-line system with an isosbestic line spreading over the frequency range from ∼100 -400 meV. A narrow Drude component and the area above the nodal line region (>∼600 meV) are the only temperature affected parts of the spectrum.

Polarization-resolved measurements revealed several relevant features at low photon energies (below 400 meV). In total, three clearly visible steps in the optical conductivity can be distinguished and assigned to the onset of interband absorptions at the different locations of the nodal line (marked by the black arrows in Fig. 3.28b). All three onsets are visible in the configuration with the polarization of the light aligned with the b crystallographic axis (E ∥ b). On the other hand, only one interband onset is clearly seen in the configuration with the polarization along the a crystallographic axis (E ∥ a), suggesting that the major part of the line propagates mostly parallel to the a crystallographic axis, with only a small deviation from it. Apart from that, a sharp peak arises in the spectrum for E ∥ a around 95 meV. The specific spectral shape of the feature may remind a phononlike response, however, no phonon is expected in this particular energy range [START_REF] Jin | Raman scattering study of large magnetoresistance semimetals TaAs 2 and NbAs 2[END_REF]. A sharp artifact can appear in the real part of the optical conductivity as a consequence of the complex Kramers-Kronig analysis. Nevertheless, a sharp dip is seen in the plasma edge at the same photon energy in the reflectivity measurement. It may resemble the nonuniform carrier density similarly to the one reported for the case of Cd 2 As 3 [START_REF] Crassee | Nonuniform carrier density in Cd 3 As 2 evidenced by optical spectroscopy[END_REF].

Considering the previous DFT studies and the high similarity to NbAs 2 , one may assume that the nodal line profile will not be drastically different. Assuming a similar profile of the nodal-lines as in NbAs 2 , the nodal-line can be divided into segments as i = I, II, III, IV, with i = I for the start of the nodal line at the plane cut Z -I 1 and i = IV for the end cut Y -X 1 (see section 3.4.4). Each onset corresponds to the local optical band gap which also includes Moss-Burstein effect in most cases [START_REF] Moss | The Interpretation of the Properties of Indium Antimonide[END_REF]. The general propagation of the nodal line throughout the whole Brillouin zone is described by k line . The local direction vector τ τ τ i can be assigned to the each point (in our simplified case to each line segment) separately describing the propagation of the nodal line with the respect to the deviation from a-b or a-c crystallographic planes.

In total, six different crystallographic facets were measured using magneto-optics in order to investigate the angular dependence of the system and find the orientation of the local direction of the relevant parts of nodal line (Fig 3.29). The beam of non-polarized light was aligned perpendicularly to exposed facets, propagating parallel with the magnetic field (Faraday geometry). This configuration allows us to investigate electrons undergoing the cyclotron motion in planes (parallel to facets), which make different angles θ i with the respect to the local nodal-line directions τ τ τ i .

The measured magneto-optical spectra are presented in a form of false-color maps in Fig. 3.29. All spectra were normalized with the respect to the response at B = 0 (R B /R 0 ), to visualize the magnetic field induced changes. Two sets of inter-Landau level transitions emerge. Similarly to NbAs 2 , they exhibit a certain dependence on the explored facet and indicate a slightly varying band gap. We classify these two sets following their properties:

1. Lower set. The lower energy set is detectable on the facets (20-1), ( 100) and . The extrapolated band gap in the zero field limit deduced 2∆ D = 115 meV for facets (20-1) and ( 100). The slight change of the band gap by 3 meV can be found in the case of (10-1). Some traces of the lower energy set can be also seen in the cases of (20-3) and ( 201) with a very low intensity. Without the knowledge of the band gap renormalization due to the Lorentz-boost presented in the section 3.4, this change of the band gap would be interpreted as an experimental error or not noticed at all. Let us move forward in this speculative direction. In the view of our work on NbAs 2 , it may be seen as a direct consequence of the dispersive nodal line section of the system. The response thus originates from the sector i = II as in the case of NbAs 2 and represents the dispersive part of the nodal line.

2. Upper set. The higher set was observed at four measured facets. On the facets (20-1), (10-1) and (20-3), the extracted band gap does not change considerably. We assign this set to the flat part at the end of the nodal line with the index i = IV. This part is not dispersing with the energy along the general direction of the nodal line with the tilt parameter w = dE/dk line = 0. The velocity parameter of the slope of the massive Dirac dispersion scales with the cos θ F .

Local direction fit

A more detailed analysis of the magneto-optics and optical conductivity may allow us to determine the electronic properties (direction, gap, velocity parameters) of the nodal line. Each experimental observation adds a different information about the studied band structure in the following way:

• Band gap. The effective optical band gap can be precisely extracted from the magneto-optical measurement using the zero-field limit extrapolation. The band gap renormalization with the respect to the different facets then allows us to find the local direction of the dispersive part. The strength of the renormalization effect, on the other hand, is determined by the steepness of the velocity parameter w parallel to the propagation of the dispersive part as the renormalization is driven by the Lorenz factor γ. The effective renormalized band gap reads:

2∆ eff = 2∆ γ , (3.44)
with the Lorentz factor defined as:

γ = 1 1 -w 2 v 2 tan 2 θ , (3.45)
where w is a tilt (slope) parameter of the dispersive part, v is the velocity parameter of the 2D linear dispersion perpendicular to the nodal line propagation and θ is the angle between local direction τ τ τ and the applied magnetic field.

• Velocity parameter. The velocity parameter can be extracted from the slope of B dependence of inter-LL transition as v i = d(hω)/dB. The highest effective velocity can be found in the case when magnetic field is perfectly aligned with the local direction of a particular section of the nodal line. From the scaling of the velocity parameters, one can guess the local direction orientation for each relevant segment of the nodal line. The velocity parameters of the non-tilted 2D Dirac cone scales with the cos θ i . The dispersive part scales with the Lorentz factor implemented as

v eff D = v i /γ 3/2 .
• Real part of the optical conductivity. The characteristic response of the flat nodal line is a constant dependence of the real part of the optical conductivity on the frequency of probing light. The dispersive part, on the other hand, gives rise to optical conductivity, which increases linearly with with ω as shown in the case of NbAs 2 or BaNiS 2 . Optical conductivity onsets corresponds to the optical gaps. The comparison of the optical band gap and the band gap extracted from the magneto-optics may help us in the determination of the Fermi level E F . • Polarization dependence. The polarization-resolved measurement was used to find the direction of the nodal line in the a-b crystallographic plane. The spectral weight is highest when the polarization of the light is perpendicular to the propagation of the nodal line (E ⊥ k line ). On the other hand, if the polarization of the light is aligned with the direction of the line (E ∥ k line ), the optical conductivity vanishes.

According to these assumptions, the ratio between the spectral weights (area below the curve) of E ∥ a and E ∥ b serves as deduction for the direction of the nodal line, or its particular part. Such selection rules are valid for a non-tilted 2D Dirac cone. In tilted cones, where complex selection rules apply and activate transitions with a higher change of the LL index. Nevertheless, the lowest transition 0 → 1 or 1 → 0 should always remain visible.

• Anisotropy of the magneto-optical response. Dependence on the facet orientation and the magnetic field is an effective way to determine the electronic band structure and the local direction of a certain part of the nodal line. If the θ i angle is high enough -tan -1 (v/w) -, the Landau level spectrum collapses in the case of dispersive part.

The upper set is observed at four different facets (20-1), (10-1), (20-3) and (001), presented in Figs. 3.30a, b, c and Fig. 3.31a, respectively. On the other hand, no traces of transitions belonging to upper set are observed on the facets ( 100) and (201), shown in Fig. 3.30d and Fig. 3.31b. This suggests that the local direction of this part should lay in the approximate direction between (10-1) and (20-1), for which the excitations are the most intense and the velocity parameter is the highest. The presence of only electricdipole active transitions with the change of Landau level index by n → n ± 1, suggests that this part is not dispersive, or disperses only weakly, along the direction of the nodal The analysis of the extracted band gap and velocity parameter from magneto-optics of dispersive part. The decreasing band gap with the higher angle between the magnetic field and the local direction of the dispersive nodal line is explained on a speculative level by the renormalization through the Lorentz factor γ. The velocity parameter scales with v D cos(θ )/γ (3/2) . The band gaps of (20-3) and (201) were extracted from the weak transition of unknown Landau level index (marked purple). Parts (d), (e): Band gap and velocity parameter of the higher energy set associated with the transitions originated from the flat part at the end of the nodal line. The band gap values lies within the range of experimental error 2∆ F = 144 ± 2 meV. The velocity parameter scales with √ θ of the deviation of the magnetic field and local direction, which is characteristic for 2D Dirac model. line. The extracted band gap from magneto-optics is not changed drastically and can be assigned to the onset of the zero-field optical conductivity at ∼ 144 meV. The response of the flat part is present in both polarization-resolved experiments suggesting the deviation from the general direction of the a crystallographic axis. The velocity parameter scales as cos(θ F ) with the angle θ F between the applied magnetic field to the local direction. This is typical of the 2D Dirac electrons. The fitting of the velocity parameter and the polarization-resolved optical conductivity gives the local direction τ τ τ F . We found that the best agreement between the model and experimental data is achieved with this set of parameters: v F = (5.2 ± 0.1) × 10 5 m/s, 2∆ F = (144 ± 2) meV, and for local direction deviating from the a-c plane by τ τ τ ac F = (30±15) • and from the a-b plane by τ τ τ ab F = (21±10) The result of the fitting procedure is shown in the Fig. 3.32 and the parameters summary in Table 3.1.

The lower set is clearly observed in the magneto-reflectivity data collected on the facet which are (20-1), (10-1) and ( 100) presented in Fig 3 .30a,b,d. The traces of some transitions can be also found for facets and ( 201), plotted in Fig 3 .30d and Fig 3 .31b, respectively. From the relative intensity of the transitions and their steepness (with B), one may speculate that the local direction of the dispersive part lies between normals of the (20-1) and ( 100) facets. Rather narrow spacing of inter-LL transitions suggests that some additional selection rules to the standard dipole ones (n → n ± 1) exists. It was found that additional transitions with higher Landau level index change (n → n ± 2) agree well with the experimental data. The transitions with a higher change of LL index are seen mostly for the dispersive parts of the nodal line and they are allowed with the higher tilt parameter w of the nodal line or the high deviation angle θ D (as described in section 3.2). The dispersive part also implies the band gap renormalization due to the Lorentz transformation. The slight change of the band gap can be observed in the case of the (10-1) facet simultaneously with the lower intensity of all transitions. Some traces of transitions which extrapolate under the original band gap 2∆ D = (116 ± 2) meV can be found on facets (201) and . The minimization fit of the local direction angles, band gap and the slopes of the transition, was used in order to find the best suitable parameters (results are presented in Table 3.1). The gap and velocity parameter dependence with the respect to the theoretical model is shown in Fig. 3.32.

More detailed modeling of the optical conductivity was done by the procedure introduced in section 3.4.4 for NbAs 2 . Taking the parameters ∆ F , ∆ D , w, v D , and v F extracted from the magneto-optics and the local direction of the flat and dispersive parts τ τ τ F and τ τ τ D , one can reduce the fitting parameters to the bare minimum needed to explain the deduced optical conductivity. However, we still deal with a relatively large number of free parameters and the found set of parameters does not have to be necessarily a unique one. The high anisotropy of the polarization-resolved measurement suggests that the nodal line is considerably bending in the a-b crystallographic plane. The rich response in the case of E ∥ b can be understood in terms of four different contributions. The lowest onset of optical conductivity at ∼ 116 meV is assigned to the dispersive part of the nodal line (i = II). This part completely vanishes in the E ∥ a configuration suggesting the propagation along the a crystallographic axis. The slight mismatch was found between the tilt parameter w extracted from the magneto-optical and optical measurements (see Table 3.1 for detail). The second onset ∼ 142 meV represents the flat part at the end cut Y -X 1 with i = IV. This part can be seen with a relatively high spectral weight in both polarization configurations. This fact is in line with the deviation in the a-b plane fitted from the magneto-optics by (21±10) • .

The interband excitations in the parts i = I, IV do not contribute to the magnetooptical response below ∼ 150 meV due to the Pauli blocking. The third onset at ∼ 190 meV originates in the beginning of the nodal line at the cut Z -Y 1 (i = I). The length of this part is relatively small in comparison to the rest of the nodal line and its located below the Fermi level. The local direction τ τ τ I is aligned with the a crystallographic axis and deviates by (26 ± 15) • from the a-b plane. The last part i = III represents the second dispersive part which is above the Fermi level. At this point, the nodal line starts to deviate from the direction of the previous two parts (i = I, II) in both a-c and a-b cuts. They are responsible for the linear in ω contribution in optical conductivity observed in the range above 150 meV.

4

Weyl semimetals

The beginning of this chapter will be dedicated to the brief overview and classification of the topological class of Weyl semimetals. The effect of crystal symmetries and its consequences will be discussed together with the examples of topological effects found in this group of materials. The main focus will be put on the comparison of three particular representatives of non-centrosymmetric compounds of transition-metal mono-pnictides TaAs, TaP, NbAs. The exotic down-dispersing interband Landau level transitions observed in magneto-optics will be interpreted as possible effect of a band-inverted electronic structure.

State-of-the-art

Weyl semimetals are the next addition to the broad family of topological materials. In 1929, Hermann Weyl introduced a simplified version of Dirac equation describing the massless fermions distinguishable by their chirality [START_REF] Weyl | GRAVITATION AND THE ELECTRON[END_REF]. This approach was initially intended to explain the behavior of neutrinos, which were considered massless at that time period. Nevertheless, it was found out later that neutrinos mass never vanishes and this discovery was awarded by the Nobel prize in 2015 [START_REF] Kajita | Nobel Lecture: Discovery of atmospheric neutrino oscillations[END_REF][START_REF] Mcdonald | Nobel Lecture: The Sudbury Neutrino Observatory: Observation of flavor change for solar neutrinos[END_REF]. With the neutrinos out of the game, the scientific community continued the pursuit for a 3D material class which could provide the massless particles with a defined chirality. It was predicted that the certain crystals with the combination of spin-orbital coupling could host so-called Weyl cones in the band structure with low energy excitations behaving as massless electrons [START_REF] Hasan | Topological insulators, topological dirac semimetals, topological crystalline insulators, and topological kondo insulators[END_REF][START_REF] Herring | Accidental Degeneracy in the Energy Bands of Crystals[END_REF]. Nowadays, the Weyl semimetals are well established theoretically [START_REF] Wan | Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates[END_REF][START_REF] Weng | Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides[END_REF][START_REF] Bansil | Colloquium: Topological band theory[END_REF][START_REF] Chiu | Classification of topological quantum matter with symmetries[END_REF][START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF][START_REF] Herring | Accidental Degeneracy in the Energy Bands of Crystals[END_REF][START_REF] Burkov | Weyl Semimetal in a Topological Insulator Multilayer[END_REF] and have been confirmed by a large variety of experimental techniques including angle resolved photo-electron spectroscopy, optics, magneto-optics, magneto-transport and scanning tunneling microscopy [START_REF] Xu | Discovery of a Weyl fermion semimetal and topological Fermi arcs[END_REF][144][START_REF] Lv | Observation of Weyl nodes in TaAs[END_REF][START_REF] Yang | Weyl semimetal phase in the non-centrosymmetric compound TaAs[END_REF][START_REF] Xu | Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide[END_REF][START_REF] Huang | A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class[END_REF][START_REF] Liu | Evolution of the Fermi surface of Weyl semimetals in the transition metal pnictide family[END_REF].

The key difference between Dirac and Weyl semimetals is the spin degeneracy of the conical bands. In general, the bands are expected to be double spin degenerated, if the spin-orbital coupling is missing. If it is present, the double spin degeneracy only appears when the crystal possess both time-reversal and space-inversion symmetry. The spin degeneracy of a Dirac cone can be lifted by breaking one of the specific symmetries with the combination of spin-orbital coupling. This may lead to the creation of pairs of Weyl cones in the band structure assigned with the opposite chirality. The material can be considered as a Weyl semimetal, if the Fermi level is sufficiently close to the cones' apices. The topological phase transition from a 3D Dirac cone to spin non-degenerated Weyl cones is driven by the breaking of time-reversal or space-inversion symmetry (Fig. 4.2). The time-reversal symmetry of the crystal implies:

E n,↑ (k) = E n,↓ (-k), (4.1)
where two electronic states of the same energy at k and -k in reciprocal space possess an opposite spin. On the other hand, in systems with the space-inversion symmetry, the certain property σ remains same at k and -k as:

E n,σ (k) = E n,σ (-k). (4.2)
New Weyl semimetal candidates are firstly predicted by searching through the symmetries of real crystals -those which satisfy the conditions above. Similarly to other topological materials, Weyl nodes cannot be removed by any small external perturbation, and thus are topologically protected. Weyl nodes can vanish only if two cones with opposite chiralities meet in reciprocal space and annihilate. The Weyl semimetals are classified with the respect to its symmetry as follows:

• Broken time-reversal symmetry. With the time-reversal symmetry broken and space-inversion symmetry preserved, the minimal number of Weyl points is two. Due to the opposite chirality, the total number of Weyl cones has to be always even. The inversion symmetry assures that they are located at ±k at the same energy, ensuring the same vicinity to Fermi level for both cones. This group comprises a large variety of materials, including magnetic Weyl semimetals and Weyl semimetals in Kagome lattices or Heusler compounds [START_REF] Okamura | Giant magneto-optical responses in magnetic Weyl semimetal Co 3 Sn 2 S 2[END_REF][START_REF] Liu | Topological phase transition in a magnetic Weyl semimetal[END_REF][START_REF] Shekhar | Anomalous Hall effect in Weyl semimetal half-Heusler compounds RPtBi (R = Gd and Nd)[END_REF][START_REF] Manna | Heusler[END_REF].

• Broken space-inversion symmetry. The main difference to the previous group is that the Weyl cones emerge in quartets, which is a direct consequence of the timereversal symmetry preserved. The time-reversal symmetry guarantees that each Weyl cone at k is transformed into a Weyl cone with the same chirality at -k [START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF]. The total chirality of the Weyl cones network has to disappear, hence the whole system must be at least fourfold degenerated.

• Both symmetries broken. A special case of both symmetries broken was discussed, for example, in the case of layered super-lattice hetero-structure made from alternating layers of topological and ordinary insulators. Weyl cones are then expected to appear at different energies, while preserving the topological properties of Weyl semimetals [START_REF] Zyuzin | Weyl semimetal with broken time reversal and inversion symmetries[END_REF].

The systems with a broken space-inversion symmetry are further classified with the respect to the tilt of the Weyl cones into two subgroups (Fig 4 .2):

• Weyl type-I. As discussed in the previous section about Dirac nodal-line semimetals, the conical dispersion can be tilted in a certain direction. If this tilt, described by an additional velocity parameter w, is smaller than the asymptotic velocity parameter v (w < v), the Weyl semimetal is referred to as type I, with the Lorentzsymmetry conserved. The Fermi surfaces of such Weyl cones are closed. One of the main topic of the thesis is the group of non-centrosymmetric materials, which is the best known representative of type-I Weyl semimetals.

• Weyl type-II. Weyl cones become over-tilted, if the tilt parameter w is higher than

Consequences of Weyl semimetal topology

The generation of Berry flux between the cones of opposite chirality is one of the main consequences of the Weyl semimetals topology. In 3D solids, the Berry flux behaves as a dual magnetic field. In contrast to the real magnetic field, Berry flux allows for the existence of monopoles. Weyl points have to be always present in pairs, as required by the symmetry. Each Weyl point acts as a source or a sink of the Berry flux in the reciprocal space [START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF][START_REF] Yan | Topological Materials: Weyl Semimetals[END_REF]. The Chern number C = ±1 can be assigned to each Weyl point referring to the different chirality. Nevertheless, the total Chern number (total chirality of the system) of all Weyl points in the Brillouin zone has to be zero [START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF][START_REF] Yan | Topological Materials: Weyl Semimetals[END_REF].

The presence of the Berry flux monopoles implies the appearance of exotic surface states called Fermi arcs. These may be viewed as a direct consequence of the Berry flux penetrating from the bulk state into surface states. In contrast to ordinary topological insulators, where the bulk is gapped and the surface gapless, Weyl semimetals are by definition gapless in their bulk band structure. If one makes a 2D surface Brillouin zone cut within the Weyl points, which are in the vicinity of Fermi level, the resulting surface structure will consist of the unclosed line representing Fermi arc connecting two Weyl points with the opposite chirality (Fig 4 .3a,b). The exotic Fermi arc line is in a direct contrast to other material classes such as topological insulators, ordinary metals or insulators, where the Fermi surface is always represented by a closed loop. If the Fermi level is set higher than the intersection of the cone, the Fermi arc position is then displaced in k-space with respect to the cone slope defined by the velocity parameter (Fig 4 .3c). The Fermi arc is preserved until the energy at which the cones merge [START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF].

Chiral anomaly (also referred as Adler-Bell-Jackiw anomaly [START_REF] Bell | A PCAC puzzle: π0→γγ in the σ -model[END_REF][START_REF] Adler | Axial-Vector Vertex in Spinor Electrodynamics[END_REF]) is another effect related to the topology of Weyl semimetals. The models from high-energy physics predict that the total number of chiral Weyl fermions is not conserved in the presence of external magnetic field and electric field, applied in parallel or at least not orthogonal to each other. The chiral current is induced between the Weyl points with the opposite chirality as a direct consequence of two Berry curvature monopoles [START_REF] Nielsen | The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal[END_REF]. With the magnetic field applied, the system becomes quantized into Landau levels. The zeroth Landau level is then expected to be chiral. The zeroth LL can be shown to be chiral, i.e., it propagates only along or opposite to the field direction, with reversed velocities at the two opposite ), respectively. In the presence of magnetic and electric field it is expected that the electrons will be pumped from one cone to the another with two distinguishable Fermi levels E ±1 F .

chirality Weyl nodes [START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF]. The consequence of chiral pumping leads to the creation of two different Fermi levels in Weyl cones of an opposite chirality (Fig. 4.4). The effect is expected to be strongest for the B ∥ E and vanishing for B ⊥ E. In magneto-transport experiments, the chiral anomaly is expected to manifest itself in a pronounced negative magneto-resistance [START_REF] Parameswaran | Probing the Chiral Anomaly with Nonlocal Transport in Three-Dimensional Topological Semimetals[END_REF]. Indeed, such an effect was reported in a number of experimental works. However, current jetting, which led to similar behavior, was also discussed in literature [START_REF] Ong | Experimental signatures of the chiral anomaly in Dirac-Weyl semimetals[END_REF][START_REF] Liang | Experimental Tests of the Chiral Anomaly Magnetoresistance in the Dirac-Weyl Semimetals Na 3 Bi and GdPtBi[END_REF]. The presence of chiral anomaly effect was also reported by several magneto-optical studies [START_REF] Yuan | Chiral Landau levels in Weyl semimetal NbAs with multiple topological carriers[END_REF][START_REF] Yuan | The discovery of dynamic chiral anomaly in a Weyl semimetal NbAs[END_REF][START_REF] Levy | Optical evidence of the chiral magnetic anomaly in the Weyl semimetal TaAs[END_REF].

Non-centrosymmetric Weyl type I semimetals

In the search for Weyl semimetals, one has to look for materials which are close to the band inversion, which lack the space-inversion or time-reversal symmetry, and which exhibit strong spin-orbital coupling. It was theoretically predicted that the group of non-centrosymmetric compounds crystallizing into a body-centered tetragonal lattice with space group I4 1 md (109) satisfy all these requirements [START_REF] Weng | Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides[END_REF][START_REF] Huang | A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class[END_REF]. One has to search for the regions of the conduction and valence band inversion in order to find Weyl cones in the band structure. If the spin-orbital coupling is not considered, the band inversion forms a gapless nodal line ring. The nodal lines can be found in the mirror planes M x and M y close to the high symmetry points Σ, N and S (green lines in Fig. 4.5). The nodal line becomes gapped when the spin-orbital coupling is (virtually) switched on. The resulting gapless Weyl cones are then formed further from the mirror planes but in the vicinity of the original high symmetry points [START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF][START_REF] Yan | Topological Materials: Weyl Semimetals[END_REF].

The group of transition-metal pnictides (four materials in total: TaAs, TaP, NbAs and NbP) is a remarkable example of Weyl type-I semimetals. The Weyl cones of this group are almost not tilted, thus respecting the Lorentz symmetry. The first experimental observation of a Weyl semimetal was reported on TaAs using the ARPES technique, showing both the bulk cone structure and the surface Fermi arcs [START_REF] Yang | Weyl semimetal phase in the non-centrosymmetric compound TaAs[END_REF]. There are 24 Weyl cones in these materials which can be classified into two groups as:

• Weyl 1. Four pairs of Weyl 1 (W1) cones are located close to the high-symmetry point Σ, within k z = 0 plane, with the respect to the C 4 rotation symmetry (Fig. 4.5).

In general, the distance between pairs of W1 nodes is smaller as compared to the W2 ones. The W1 cones are almost isotropic in the k x -k y plane, while the dispersion along the k z direction is flatter. It is worth noting that the nomenclature of W1 and W2 cones is not consistent in the literature (cf. [START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF] and [START_REF] Xu | Observation of Weyl nodes and Fermi arcs in tantalum phosphide[END_REF]). The different lattice parameters of all materials of this Fig. 4.7. Weyl cone separation and chirality. (a) Weyl cones profile in TaAs. The Fermi level is set close to intersection of both cones below the saddle point, so the chirality of both W1 and W2 can be distinguished. (b) Weyl cones of TaP. Due to the high energy separation of the cones, Fermi level is set higher that the W1 saddle point and the Chern number of Fermi surface is 0. The cones chirality is not differentiable anymore. Adapted from [START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF][START_REF] Xu | Observation of Weyl nodes and Fermi arcs in tantalum phosphide[END_REF].

class have a direct impact on the Weyl cones shape and location within reciprocal space. The energy separation between W1 and W2 cones is smaller in the case of arsenic-based materials. On the other hand, higher separation in energy can be found in the phosphorusbased compounds. Position of the Fermi level together with the energy separation of cones plays an important role in the applicability of experimental techniques. For example, in ARPES measurements of TaAs, both W1 and W2 cones can be distinguished on the Fermi surface. However, in the case of TaP, W1 cones are believed to be buried under the Fermi level and the cones cannot be separated due to the fact that both cones are placed within the same Fermi surface (Fig. 4.7) [START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF][START_REF] Xu | Observation of Weyl nodes and Fermi arcs in tantalum phosphide[END_REF]. The distance of cones increases with the strength of spin-orbital coupling (Fig. 4.6) [START_REF] Liu | Evolution of the Fermi surface of Weyl semimetals in the transition metal pnictide family[END_REF]. Although ARPES is one of the most used technique to study the topological matter, there are huge limitation in the kspace and energy resolution. Another drawback comes from the fact that ARPES is only able to study the band structure below the Fermi level and the measurements cannot be performed in the presence of the magnetic field due to the detector limitations.

Band inversion in Weyl semimetals

Band inversion model of Weyl semimetals

This section is dedicated to a specific approach how to describe electronic and optical properties of time-reversal-invariant Weyl semimetals beyond the common approximation of conical bands. This procedure was developed in the close collaboration with M. O. Goerbig (Laboratoire de Physique des Solides) and it was used to explain the unusual magneto-optical features found in TaP [START_REF] Polatkan | Magneto-Optics of a Weyl Semimetal beyond the Conical Band Approximation: Case Study of TaP[END_REF]. The proposed model deals with a quartet of Weyl nodes and will be later on applied to describe four Weyl cones that appear in the close vicinity of the high symmetry Σ point. The model was proposed to describe the impact of the band inversion on the magneto-optical response of 3D Weyl semimetals which includes, among others, an unusual set of inter-LL excitations which decrease their energy with the applied magnetic field. Later on, the implications of this model are confronted with the results of magneto-optical experiments conducted on two representatives of the non-centrosymmetric family of 3D Weyl semimetals: TaP and NbAs.

Let us start with the following 2×2 Hamiltonian which describes a quartet of Weyl nodes in a time-reversal-invariant Weyl semimetal:

Ĥ =   ∆ - h2 q 2 2M γ(q) γ * (q) -∆ + h2 q 2 2M   . (4.3)
The energy spectrum of this Hamiltonian preserves the electron-hole symmetry and reads:

E(q) = ± [∆ -h2 q 2 /(2M)] 2 + |γ(q)| 2 . ( 4.4)
The form and strength of the off-diagonal coupling parameter γ(q) is decisive for the general nature of the described system:

• γ = 0 → the system described is a nodal-line semimetal.

• γ = const. → the system is a gapped nodal-line semimetal [START_REF] Pal | Chemical potential asymmetry and quantum oscillations in insulators[END_REF].

• γ(q) = hvq x + iq y → the system is topological insulator [START_REF] Bernevig | Quantum spin Hall effect and topological phase transition in HgTe quantum wells[END_REF][START_REF] Liu | Model Hamiltonian for topological insulators[END_REF].

To obtain Weyl cones nearby the time-reversal-invariant momentum point q = 0, the following non-isotropic shape of the off-diagonal coupling has to be taken γ(q):

γ(q) = γ 0 - h2 q 2 x 2m x - h2 q 2 y 2m y + ihv z q z . (4.5)
In the following, γ 0 is considered to be a real positive parameter here to simplify the discussions. An imaginary part of γ 0 would only shift the plane, which contains the nodal line, in the z direction by q 0 z = -i(γ 0 )/hv z . Four Weyl cones appear, within the q xq y plane at E(q w ) = 0, for a particular set of tunable parameters in the Hamiltonian 4.3. The locations of the Weyl cones q = (±q w

x , ±q w y , 0) corresponds to the intersection points of the diagonal term of Hamiltonian 4.3, having the form of a circle in the reciprocal space (blue line in Fig. 4.8a):

∆ = h2 2M + (q 2 x + q 2 y ) (4.6) (c)

W1 quartet

Fig. 4.8. Band inversion scheme. (a) q xq y momentum plane. The Weyl cones are created on the intersection of circle and ellipse, which represents diagonal and off-diagonal terms of Hamiltonian 4.3. (b) Band-inverted electronic structure with four Weyl nodes located at q = (±q w

x , ±q w y , 0), with full electron-hole symmetry. Saddle points are located at the local extrema at q = 0. (c) Close-packed W1 group of cones with q = 0 = Σ. Part (c) adapted from [START_REF] Weng | Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides[END_REF]. and off-diagonal term hat represent an ellipse (red line in Fig. 4.8a):

γ 0 = h2 q 2 x 2m x + h2 q 2 y 2m y . (4.7)
Taking all tuning parameters M, m x , m y , ∆ positive, two additional conditions must 4.2. Band inversion in Weyl semimetals be fulfilled to create intersection points of the ellipse and circle:

m x > M ∆ γ 0 > m y or m x < M ∆ γ 0 < m y . (4.8)
The positions of Weyl nodes are then expressed as:

hq w x = 2m x m x -m y (M∆ -γ 0 m y ), (4.9)
and hq w y =

2m y m y -m x (M∆ -γ 0 m x ). (4.10)
For appropriately chosen parameters, the conduction and valence bands meet at four particular points, thus creating four anisotropic 3D Weyl nodes. With increasing (or decreasing) energy, these four Weyl cones merge via two types of saddle points. At q = 0, two local extrema, a minimum for the valence and a maximum for the conduction band are present in the band structure. The time-reversal symmetry implies that the cones located at q w and -q w have the same chirality. The sum of chiralities of this Weyl-cone quarter is zero.

The positions of the Weyl nodes, their chirality, saddle point location and the strength of band inversion can be derived analytically using the proposed model. The saddle point energy is given by:

E sp x(y) = |M∆ -γ 0 m x(y) | M 2 + m 2 x(y) (4.11)
for the x and y components, respectively. The energies of the local extrema of the valence (-) and conduction (+) bands read:

E(q = 0) = ± ∆ 2 + γ 2 0 . (4.12)
Thanks to the simplicity of the proposed model, we may find an analytic expression for the effective velocity v eff which describes the spacing of Landau levels when the conical band approximation is used for the vicinity of the Weyl nodes (B∥z):

v eff = |v + v -| = q w x q w y (m -1 x -m -1 y )/M. (4.13)
To compare the implications of our model with the results of magneto-optical experiments performed on Landau-quantized Weyl semimetals, the magnetic field has to be introduced in the Hamiltonian 4.3. To this end, the standard Peierls substitution is used in the Hamiltonian 4.3 and the corresponding ladder operators read:

a = l b (q x + iq y )/ √ 2. ( 4.14)
The Landau-quantized Hamiltonian then takes the form:

ĤB =   ∆ -hω c (a † a + 1 2 ) γ(a, a † ) γ * (a, a † ) -∆ + hω c (a † a + 1 2 )   , (4.15)
where ω c is a cyclotron resonance in form of eB/M, l B = h/(eB) is a magnetic length and off-diagonal coupling parameter γ(a, a † ) is represented as:

γ(a, a † ) = γ 0 - (a + a † ) 2 m y l 2 B - (a -a † ) 2 m x l 2 B . (4.16)
Let us note that apart from the particular, linearly dispersing Landau level (n = 0) [START_REF] Ashby | Magneto-optical conductivity of Weyl semimetals[END_REF], the dominant contribution to the magneto-optical response comes from the states at q z = 0. Therefore, our Hamiltonian was limited to this particular case.

To find the energy spectrum of the Landau-quantized Hamiltonian 4.15, a diagonalization must be performed. Unfortunately, the form of the Hamiltonian does not allow us to get an analytical solution. Instead, one has to rely on a numerical approach, in which the infinite-size matrix implied by the Hamiltonian 4.15 is truncated for the chosen number of Landau levels and subsequently diagonalized.

Magneto-optics of TaP

To test the validity of our model, let us confront its predictions with the experimental data collected on TaP which is another representative of the monopnictides family of 3D Weyl semimetals with a broken space-inversion symmetry. Even though there is an ongoing debate in the scientific community about the position, shape, extension as well as about the energy separation of the W1 and W2 nodes in basically all type-I Weyl semimetals, one can deduce some basic trends in the monopnictide family. TaP seems to have the largest separation of the energy between W1 and W2 nodes, and also, the W1 nodes appear to be the closest to the Σ point and form nearly a square in the k x -k y plane.

The model parameters were tuned in order to reproduce the coordinates of the W1 cones known from the literature [START_REF] Weng | Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides[END_REF][START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF]. The resulting Landau level spectrum is presented in Fig. 4.9a as a function of B. The √ B dependence at low energies is a clear signature of massless charge carriers. The Landau levels at low energies are fourfould degenerated as a consequence of the time-reversal symmetry. This degeneracy is gradually lifted with the increasing B, when the Landau levels reach the energy of saddles points (marked by red dashed lines in Fig. 4.9a).

In the conduction band, a rich anti-crossing behavior is observed in the region between the saddle point and the local maximum of the conduction band (orange dotted line in Fig. 4.9a). Exotic down-dispersing Landau levels, clearly visible in this range of energy, are a direct consequence of the inverted band structure. In the limit of the vanishing magnetic field, they extrapolate to the energy of ∆ 2 + γ 2 0 , i.e., to the position of the local maxima of the conduction band at q = 0. Due to the full electron-hole symmetry of the system, analogous behavior is seen in the valence band, with an unusual set of Landau level dispersing towards higher energies with the increasing B and extrapolating to -∆ 2 + γ 2 0 in the zero-field limit.

To get a simulation directly comparable with our magneto-optical experiment, the optical response of the system described by our Landau-quantized Hamiltonian has to be calculated. The optical conductivity of a bulk material in the presence of a magnetic field, for a circularly polarized light, can be calculated as:

σ ± (ω, B) = 2 iG 0 l 2 b ω ∑ n,m,q z ( f m -f n )|⟨m| v± |n⟩| 2 E n -E m -hω + iΓ , (4.17)
where G 0e 2 /(2π h) is the quantum conduction, Γ is the broadening parameter, and v± = vx ± vy are the velocity operators defined as:

h vx = ∂ Ĥ ∂ q x and h vx = ∂ Ĥ ∂ q x . (4.18)
In isotropic materials, the evaluation of matrix elements ⟨m| v± |n⟩ results in the well-known selection rules for electric-dipole-allowed transitions n = m ± 1. In our case, we deal with the system which is not isotropic. Therefore, one may expect that additional transitions beyond the standard rules n = m ± 1 become activated, especially when such transitions involves Landau levels in the vicinity of the saddle points. The occupation factor of Landau levels is governed by the Fermi-Dirac distribution f , in which we assumed zero temperature for simplicity. Then the Landau level indices n and m are iterated throughout all possible initial and final states around q z = 0, to be consistent with the proposed Landau-quantized Hamiltonian 4.15. The initial sum over q z in optical conductivity is then replaced by a profile of the joint density of states, (E m -E nhω) -1/2 , averaged over the interval of (E m -E n ; E m -E n + Γ). The result of our calculations is presented in a form of a false-color plot of relative magneto-reflectivity in the Fig. 4.9b. The used parameters were chosen to approach the position and shape expected for W1 cones of TaP, see below (the same parameters were also used for Fig. 4.8a).

Let us now discuss the experimental data. The magneto-reflectivity data were measured on a TaP monocrystal grown by standard charge vapour transport technique [START_REF] S.-I. Kimura | Optical signature of Weyl electronic structures in tantalum pnictides TaPn (Pn= P, As)[END_REF][START_REF] Arnold | Negative magnetoresistance without well-defined chirality in the Weyl semimetal TaP[END_REF][START_REF] Besser | Pressure tuning of the electrical transport properties in the Weyl semimetal TaP[END_REF]. The facet (001) with a lateral size of ∼ 4mm 2 was probed in the Faraday configuration with magnetic field applied perpendicular to the facet. The sample was kept at temperature of 2 K in the low pressure helium exchange gas. The data was collected by the Fourier-transform infrared spectrometer coupled with the super conduction solenoid (up to 13 T) or the resistive coil (13 -34 T). To measure the response at low photon energies, the Hg lamp was used as a source of THz radiation. At higher energies, the globar was used. The reflected beam was collected by the external bolometer kept at the temperature ∼ 1.6 K. The resulting spectra are plotted as a R B /R 0 ratio (Fig. 4.9c).

The features extracted from the magneto-reflectivity data can be categorized using the color coding introduced in Fig. 4.9d. In total, there are five characteristic features:

1. Dotted dark blue lines. The inter-LL excitations as low photon energies with a √ B to sub-linear in B dependence which originate in the close vicinity of the Weyl nodes and represent a hallmark of conical bands in TaP.

2. Red solid lines. The unusual set of inter-LL excitations which disperse towards lower energies with the increasing magnetic field. They are directly reflecting the inverted nature of electronic band at the Σ point. Our theoretical model implies that they extrapolate to the energy of 2 ∆ 2 + γ 2 0 , which in fact is the separation of local extrema of the conduction and valence bands at q = 0.

3. Green dashed lines. The high-energy inter-LL transitions that follow nearly linear in B dependence. These correspond to interband excitations further (in momentum space) from the Σ point.

4. Grey solid lines. The weak lines dispersing upwards linearly with B. These are associated with the additional parabolic bands at the Σ point (q z = 0 in our notation), the presence of which is required by the time-reversal-invariant nature of this point. For simplicity, these bands are not considered by the proposed model and they are not visualized in 5. Blue thick horizontal line. One weakly dispersing line can be found at the energy ≈ 40 meV. This almost horizontal (flat in B) feature can be seen in both simulated (denoted by the grey arrow) and measured magneto-reflectivity. We associate it with the excitation between the saddle points. It is also visible in the zero-field optical data (Fig. 4.10) as a pronounced dissipative feature. However, in the optical response, it nearly coincides with the plasma frequency, which may also give rise to a similar feature in the magneto-reflectivity (splitting due to classical cyclotron motion of free electrons).

Let us note that fairly good agreement of the experiment and the theoretical prediction could be further improved, e.g., by taking into account the electron-hole asymmetry of the real band structure. Nevertheless, such an approach would necessarily lead to other free parameters, by including higher-order terms in the Hamiltonian 4.3. Hence, better than semi-quantitative agreement could be achieved only at the cost of a significantly higher complexity of the model used.

Let us now discuss the experimental data in the view of the whole, more complex, band structure of TaP. As typical for the whole TaAs family, the electronic band structure consists of two types of Weyl cones, W1 and W2. According to theoretical (DFT) studies, but also ARPES and transport experiments [START_REF] Weng | Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides[END_REF][START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF][START_REF] Lee | Fermi surface interconnectivity and topology in Weyl fermion semimetals TaAs, TaP, NbAs, and NbP[END_REF][START_REF] Grassano | Influence of anisotropy, tilt and pairing of Weyl nodes: The Weyl semimetals TaAs, TaP, NbAs and NbP[END_REF][START_REF] Arnold | Nega-tive magnetoresistance without well-defined chirality in the Weyl semimetal TaP[END_REF][START_REF] S.-I. Kimura | Optical signature of Weyl electronic structures in tantalum pnictides TaPn (Pn = P, As)[END_REF][START_REF] Xu | Experimental discovery of a topological Weyl semimetal state in TaP[END_REF][START_REF] Xu | Observation of Weyl nodes and Fermi arcs in tantalum phosphide[END_REF], the energy distance between W1 and W2 nodes is expected to be the largest in the monopnictide family, reaching ∆E W 2-W 1 ≈ 60 -80 meV. Hence, it is plausible to assume that only one type of Weyl nodes might be occupied by electrons, regardless of the exact position of the Fermi energy (see Fig. 4.11 for a schematic illustration). Placing the Fermi energy within the W1 cones (or alternatively W2 cones), it is the Pauli blocking which limits the contribution of the other Weyl cones to the overall optical response to relatively high energies, above the onset of 2∆E W 2-W 1 ≈ 120 -160 meV. In our case, the closely-packed quartet of Weyl nodes at the Σ point seems to be a reasonable realization of our simple theoretical model in the complex band structure of TaP. This leads us to the conjecture that the low energy response of the TaP crystal studies is in fact dominated by the W1 cones.

The cones position and the strength of the band inversion were set, in our calculations, to respect the results preceding DFT studies [START_REF] Weng | Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides[END_REF][START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF][START_REF] Grassano | Influence of anisotropy, tilt and pairing of Weyl nodes: The Weyl semimetals TaAs, TaP, NbAs and NbP[END_REF]. The final coordinates of the W1 cones used for the model are q w x = 0.4 nm -1 and q w y = 0.2 nm -1 . The band inversion parameter was estimated as 2∆ DFT ≈ 60 meV. This knowledge allows one to estimate the diagonal and off-diagonal masses as M = h2 [(q w x ) 2 +(q w y ) 2 ]/(2∆ DFT ) ≈ 0.25m 0 and m y = m x /12 = m 0 /4, respectively. The velocity parameter v z which only determines the relative strength of excitation in our calculations, and was set to 10 5 m/s in agreement with available literature [START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF]. The zero-field extrapolation of the down dis- persing lines allows to determine the band inversion with a relatively high precision as 2 ∆ 2 + γ 2 0 ≈ 2∆ ≈ 73 meV. The Fermi level was set to zero to keep the model as simple as possible. Thus, the only free (without any theoretical constraints) parameter of the model is γ 0 , which has to satisfy the condition γ 0 < ∆ to ensure the existence of the Weyl nodes in our model.

Modeling and comparison of TaAs family

The above developed model is, in principle, applicable to the whole TaAs family of Weyl semimetals. Now, we will discuss how to find appropriate parameters for other materials beyond TaP. When we inspect the DFT and ARPES studies dealing with other Weyl semimetals from the monopnictide family, we may notice certain characteristic trends that relate the chemical composition with the band structure. Naturally, we should keep in mind that the ARPES and DFT studies may only provide us with rough hints due to their limited resolution in energy and momentum space. The position of the W1 nodes and the saddle point gap are mostly affected by the transition-metal component (Ta or Nb). In contrast, the W1-W2 energy separation seems to be given by the pnictide (As or P). Let us sum up our findings:

• Ta-based. The W1 nodes in materials comprising Ta in their composition are relatively close to the borders of the Brillouin zone. W1 cones form a rectangular shape around the high symmetry point Σ (q = 0 in the model). The saddle point gap is higher as compared to niobium-based materials, likely to higher spin-orbit coupling, and it reaches values of ∆ DFT ≈ 50 -60 meV.

• Nb-based. The W1 nodes are located further from the border of the first Brillouin zone (i.e., from the Σ point). The W1 nodes are located on the corners of a profoundly elongated rectangle. One may view them as two rather distant pairs of nodes, rather than a single quartet. The saddle point gap is ∆ DFT ≈ 20 -25 meV.

• -As based. The energy separation of the W1 and W2 nodes is relatively low, 13 and 37 meV for TaAs and NbAs, respectively.

• -P based. Phosphor-based pnictide Weyls have rather large energy separation of the W1 and W2 cones. Placing the Fermi level in one type of the Weyl cones, the other type may contribute to the optical response only at relatively large photon energies, due to Pauli blocking, roughly above the onset of 2∆E W 2-W 1 .

The locations of the W1 nodes in the k z = 0 plane of NbAs and NbP do not allow us to use the previous Hamiltonian 4.3 in its original form. This is because one cannot reproduce the positions of the W1 nodes, and simultaneously, keep a reasonable strength of the band inversion described by the parameter 2 ∆ 2 + γ 2 0 (the distance of the local band extrema at q = 0). To solve this problem, one may add to the model an anisotropy of the diagonal terms, by introducing different diagonal masses M x and M y . The zero-energy points of the diagonal terms thus no longer form a circle, but instead, an ellipse. Naturally, this flexibility in the model is at the cost of introducing one additional free parameter to Fig. 4.12. W1 and W2 band dispersion -DFT. The energy dispersion of the W1 and W2 cones along k xyz directions. The Fermi level is set to 0 eV. The blue dashed line is a guidance for eye to demonstrate the W1 and W2 energy separation ∆E W 2-W 1 (marked respectively in meV for all materials). The blue and green arrows highlight the transition between the Weyl bands and non-topological bands, respectively. The Fermi level of the real material can drastically differ from the DFT one. Adapted from [START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF].

the Hamiltonian which then reads:

Ĥ =   ∆ - h2 q 2 x 2M x - h2 q 2 y 2M y γ 0 - h2 q 2 x 2m x - h2 q 2 y 2m y -γ 0 + h2 q 2 x 2m x + h2 q 2 y 2m y -∆ + h2 q 2 x 2M x + h2 q 2 y 2M y   . (4.19)
Both the diagonal and off-diagonal masses can be, in fact, taken as scaling factors of the effective mass M which may be roughly estimated from the positions of nodes and the saddle-point gap predicted by DFT calculations:

M = h2 [(q w x ) 2 + (q w y ) 2 ] 2∆ DFT . (4.20)
Hence, the prior knowledge of the approximate node positions and saddle-point gap allows one to guess the initial parameters for the model, namely, via setting properly the locations of the intersection points of both ellipses (see Fig. 4.13). The corresponding magneto-optical response was calculated using the procedure introduced in the previous section in the context of the TaP Weyl semimetal. The results are presented in a form of false-color maps of R B /R 0 in Fig. 4.14. Due to a large set of tunable parameters, same intersection points q x , q y can be achieved by multiple approaches: We may set large ∆, The approach has to be changed to two ellipse intersections for NbAs and NbP in order to reach W1 nodes, which are further from q = 0.

γ 0 or tune the mass factors M x , M y , m x , and m y . For the sake of simplicity, M x and M y are fixed as 1 for the Ta-based material. Only x components of the mass parameter are changed, thus leaving y parameters fixed as 1 (particular parameters used in this section are presented in the Tab.4.1). Clearly, the magneto-optical experiment can provide us with the estimate of the band inversion, which is obtained from the zero-field extrapolations of down-dispersing in B inter-LL excitations, in order to reduce the total number of free parameters.

The inter-LL transitions dispersing down in energy with the increasing magnetic field are predicted to be observable for all materials of the class. They are more apparent in the Ta-based compounds extrapolating to the higher energy in the limit of the vanishing magnetic field than their Nb-based counterparts. This is caused by the lower strength of spin-orbital coupling, thus lower energy difference between the local maximum of conduction band and the local minimum of the valence band [START_REF] Liu | Evolution of the Fermi surface of Weyl semimetals in the transition metal pnictide family[END_REF][START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF].

Magneto-optics of TaAs

TaAs was the first confirmed type-I Weyl semimetal. Numerous studies were conducted on this material in order to investigate the Weyl electrons and related exotic properties or phenomena, such as Fermi arcs or chiral anomaly. In total, TaAs hosts 12 pairs of Weyl cones which are divided into two groups denoted as Weyl 1 (W1, 4 pairs) and Weyl 2 (W2, 8 pairs). While the W2 cones are supposed to be rather isotropic in the reciprocal space, the W1 cones remain isotropic only in the (001) crystallographic plane and flattens significantly in the z (c-axis, tetragonal) direction [START_REF] Weng | Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides[END_REF]144,[START_REF] Lv | Observation of Weyl nodes in TaAs[END_REF][START_REF] Arnold | Chiral Weyl Pockets and Fermi Surface Topology of the Weyl Semimetal TaAs[END_REF]. The W2 cones seem to be extended in the energy as compared to the W1 cones. However, the precise determination of the energy range of both cones, their linearity and distribution in momentum space remains unknown due to the lack of experiments allowing to characterize the cones with a sufficient resolution in momentum space as well as in energy. Magneto-optical experiments in the infrared spectral range, in particular when conducted on high-quality Landau-quantized crystals, may provide us with crucial information about the details of the electronic band structure of TaAs (and other Weyl semimetals from the monopnictide family).

Several single crystals of TaAs were grown using the charge vapor transport technique. The XRD characterization was performed in order to identify individual crystallographic facets: (001), ( 101) and [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF]. The initial characterization included also magnetotransport (quantum oscillations) measurements to verify the high electronic quality of the explored samples. Such measurements were done with the magnetic field applied perpendicular to the (001) and ( 101) facets of two different samples. Both samples displayed a large unsaturated magneto-resistance of 35 000% and 11 000% at 1.8 K and 14 T for (001) and ( 101) facet, respectively. Other parameters were extracted from magneto-transport experiments, using a two-carrier Drude model applied to 1.8 K data, are listed in Table 4.2.

TaAs crystallizes in the body-centred-tetragonal lattice (I4 1 md) which lacks the space-inversion symmetry but preserves the time-reversal symmetry [START_REF] Lv | Observation of Weyl nodes in TaAs[END_REF]. Four pairs of W1 nodes lie in the k z = 0 plane and they are located around the time-reversal-invariant momentum point Σ in the Brillouin zone (see Fig. 4.15 left). Eight pairs of the W2 nodes are located above and below the k z = 0 plane, placed symmetrically with respect to the Γ -Z -Σ mirror planes. The W2 cones are theoretically expected to be highly isotropic. In contrast, the W1 cones are elongated in the k z direction (see the sketch in Fig. 4.15).

Sample (Facet)

n (m 3 ) The lattice parameter c (11.641 Å) in TaAs is almost four times longer than a = b = 3.4348 Å [START_REF] Lv | Observation of Weyl nodes in TaAs[END_REF]. With such an elongated shape of the unit cell, the normals of the ( 101) and ( 112) facets made rather large angles with the normal of the (001) plane (∼ 74 • and ∼ 67 • , respectively). Optical experiments on the (001) plane are relatively often reported in the literature, as well as magneto-transport measurements with the magnetic field perpendicular to this plane -other crystallographic directions are much less explored. Here we aim at exploring three considerably different facets, using the optical and magneto-optical techniques, in order to get insights into the shape, extension and anisotropy of the Weyl cones in TaAs.

p (m 3 ) µ n (m 2 V -1 s -1 ) µ p (m 2 V -1 s -1 ) (

Magneto-optical response

The magneto-optical experiments were done using superconductive solenoid up to 16 T and resistive coil up to 34 T. All measurements were conducted at the temperature of 2 K under low helium gas pressure. The orientation of the magnetic field was perpendicular to the investigated facet and the vector of the incident and reflected beam was parallel to the magnetic field (Faraday geometry). The explored TaAs samples are characterized by relatively small Fermi pockets. This, in the combination with a high carrier mobility, allows us to reach the quantum regime, and likely the quantum limit as well, at relatively low magnetic fields, and to investigate the band structure by observing inter-and intra-LL transitions.

In all data presented here, the magneto-reflectivity was normalized by the response at zero magnetic field, R B /R 0 (Fig 4 .16 top and Fig. 4.17a, e). The observed maxima in the R B /R 0 spectra were directly associated with the positions of resonances, a reasonable assumption in the case of relatively narrow resonances observed. The magnetoreflectivity response exhibit an exceptionally high number of transitions. Some interband inter-LL transitions are observable at magnetic fields as low as 1 T above which the quantum regime (Landau quantization) is present. The majority of transitions observed on the (101) facet exhibits nearly a √ B dependence, a hallmark of massless electrons. TaAs Magneto-reflectivity of all facets -low magnetic field. Top panels: Relative magneto-reflectivity R B /R 0 measured in the Faraday geometry of facets (001), ( 101) and [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF]. Bottom panels: Derivative of R B /R 0 used for resonance extraction.

TaAs is known for its complex multi-band electronic band structure at low energies (close to the Fermi level), comprising the W1 and W2 cones but also topologically trivial (hole) pockets. This implies a relatively rich magneto-optical response with multiple series of transitions -a response too complex to be explained using some simple effective Hamiltonian approach [START_REF] Shao | Optical signatures of Dirac nodal lines in NbAs2[END_REF][START_REF] Polatkan | Magneto-Optics of a Weyl Semimetal beyond the Conical Band Approximation: Case Study of TaP[END_REF][START_REF] Akrap | Magneto-Optical Signature of Massless Kane Electrons in Cd 3 As 2[END_REF][START_REF] Hakl | Cyclotron resonance of Kane electrons observed in Cd 3 As 2[END_REF][START_REF] Martino | Two-Dimensional Conical Dispersion in ZrTe 5 Evidenced by Optical Spectroscopy[END_REF]. Instead, we analyze the response semi-classically, using the Lifshitz-Onsanger formula which indicates the allowed electron orbits in momentum space at the onset of Landau quantization:

S n = πk 2 n = eB h (n + γ). (4.21)
The integer n enumerates individual orbits (Landau levels), B is the externally applied magnetic field and γ is the geometrical factor related to the Berry phase. Further, let us assume that we deal with a system displaying a perfect electron-hole symmetry, and in addition, that the orbit index is conserved (as well as its momentum k) when an electron is promoted by light from the valence to the conduction band. Then, each inter-LL excitation represents the energy separation between the bands at the momentum given by the expression:

k n = (n + γ)2eB/h. (4.22)
As a result, assigning a correct index n and plotting energies of inter-LL transitions as a function of the calculated momentum k becomes equivalent to tracing the conductionvalence band distance. Let us also note that the employed Faraday geometry implies the standard selection rules for electric-dipole-active inter-band inter-LL excitations: n → n ± 1. Hence, neither the LL index, nor the momentum k n is really conserved, and therefore, the geometrical factor γ becomes a tunable parameter rather than originally introduced Berry phase. Let us first apply the described procedure to the data acquired on the (101) facet. A smooth linear profile up to 250 meV is obtained after fitting of all excitations with an appropriately assigned LL index n. This implies that we observed widely extending conical band, with no clear signs of deviations from linearity. The corresponding velocity parameter is v = (2.0 ± 0.2) × 10 5 m/s when a full electron-hole symmetry is assumed. Very similar response, with a nearly identical velocity, is also observed on the [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF] facet which deviated by the angle of 43 • from (101) plane (Fig. 4.18). This allows us to assign this widely extending cone as W2, while the W1 cones do not manifest visibly in the data, most likely due to particular position of Fermi level (Pauli blocking). This assignment is based on comparison with available DFT studies which imply rather isotropic W2, but highly anisotropic W1 cones, the latter significantly flattened along the tetragonal direction [START_REF] Weng | Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides[END_REF][START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF].

A detailed look at the data collected on the (001) oriented facet reveals over 50 different spectral lines, almost all with a nearly linear dependence on B. Such a response clearly stems from inter-LL excitations between parabolic bands and it is highly unexpected for a Weyl semimetal. Four separate series of such transitions were identified in Fig. 4.18. Semiclassical analysis of facets ( 101) and [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF]. Semiclassical momentum analysis using Lifshitz-Onsanger analysis for (a) (101) and (b) [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF]. The fitted line represents the velocity parameter of the cone, which is very similar for both cases. total (Fig. 4.17g, h). The linear B transitions are relatively pronounced and dominate the response, thus overshadowing other lines, most notably those with the √ B transitions, observed on the ( 101) and ( 112) facets. On the other hand, the linear in B transitions were not observed on the ( 101) and ( 112) facets which suggests a high anisotropy of the related (parabolic) bands, and that they might originate in the vicinity of the Σ point where the band structure is expected to be highly anisotropic.

The semi-classical approach based on the Lifshitz-Onsanger formula was used to analyze each series of linear in B transitions observed on the (001) facet. Our conclusions are the following. We refer to each set based on the color-coding used in Fig. 4.17g, h: 2. Black set. The line with the lowest energy in the series emerges above 25 T and reaches the energy of 7 meV at 30 T. All black lines extrapolate to zero energy in the limit of a vanishing magnetic field. We interpret these lines as intraband transitions with an associated effective mass of m * H = 0.5m e . This relatively heavy carries can represent the trivial hole pocket reported by multiple magneto-transport studies [START_REF] Arnold | Chiral Weyl Pockets and Fermi Surface Topology of the Weyl Semimetal TaAs[END_REF][START_REF] Ramshaw | Quantum limit transport and destruction of the Weyl nodes in TaAs[END_REF]. The onset of 25 T for the lowest line is in agreement with the quantum limit of this heavy-hole pocket. The other lines in this series are in-terpreted as cyclotron resonance harmonics which may appear in highly distorted (anisotropic) bands. A similar effect was observed in the case of bulk graphite for electrons in the vicinity of a Lifshitz transition [START_REF] Orlita | Cyclotron Motion in the Vicinity of a Lifshitz Transition in Graphite[END_REF].

3. Blue and Green set. These two series of excitations correspond to the response of other higher energy bands, having neither linear nor strictly parabolic profiles. They are extrapolating to two different finite energy onsets in the zero-field limit and they do not seem to be anyhow related to the orange set.

Importantly, we have not identified any features which could be attributed to the W1 conical bands. The response observed on the (001) facet resembles that of a semiconductor with a narrow energy band gap rather than Weyl semimetal. It seems that for the explored crystals, despite relatively low carrier concentration, excitations within W1 conical bands are not accessible using an optical probe.

An obvious question may be raised when the data collected on TaAs are compared to the previously discussed results on TaP. In the latter material, we observed a series of specific, down-dispersing with B interband inter-LL excitations [START_REF] Polatkan | Magneto-Optics of a Weyl Semimetal beyond the Conical Band Approximation: Case Study of TaP[END_REF], which were interpreted as band inversion in the vicinity of the Σ point -the inversion thanks to which the W1 cones appear in monopnictides Weyl semimetals. In TaAs, no such transitions were found despite our considerable effort. The occupation effect is the only plausible explanation. This is consistent with the fact that no lines with a √ B scaling that could be associated with the W1 cones were found. On the other hand, we identified such lines in the response of NbAs, another monopnictide Weyl semimetal, see Sec. 4.4, which confirms that with B down-dispersing inter-LL excitations are not an unique property of TaP and which corroborates our interpretation.

Optical response at B=0

The study of the optical response without any magnetic field applied performed on different TaAs facets brings a complementary view on the electronic band in this material. The wide energy range measurement of the reflectivity (Fig. 4.19a, b) allows one to deduce the complex optical conductivity σ (ω) = σ 1 (ω) + iσ 2 (ω) using the standard Kramers-Kronig analysis. The real part of the optical conductivity σ 1 (ω) reflects the dissipative properties of the material, to great extent driven by the joint density of states, thus affected by both in-plane and out-of-plane (with respect to the explored facet). The sharpness of the plasma edge, found on different facets around photon energies of 15-20 meV in the reflectivity spectra, implies a relatively low scattering rate 1/τ. This low scattering rate, together with a low carrier concentration allows us to observe the Landau quantization at field as low as 1 T. The barrow Drude peak (1/τ) is in line with this, as well as with the large carrier mobility obtained from magneto-transport experiments. The temperature dependence mostly affects the low energy narrow Drude component with 1/τ following quadratically the rising temperature.

The interband excitations in Weyl cones are contributing to the low-energy range of the optical response. In general, the (001) facet displays more sharp spectral features as compared to the (101) facet (Fig. 4.19c, d). This can be explained by the flat band dispersion in the vicinity of Σ point. There are three distinguishable absorption bands visible in optical conductivity of the (001) facet at 10 K (marked by the grey arrows in Fig. 4.19d). The lowest-laying onset at 15 meV corresponds to the gap extracted from the magneto-optics (orange set in Fig. 4.17g, h). The other two peaks at 70 and 128 meV can be associated with the green and blue series of interband inter-LL transitions, respectively. It is unlikely that these features would represent some van Hove singularities from other locations in the Brillouin zone, since the band gap in TaP closes only at the locations of the W1 and W2 cones. Thus, the response of the (001) facet is most likely dominated by the surroundings of the W1 nodes, in line with conclusions of Landau level spectroscopy. Optical conductivity of the (101) facet exhibits much smoother profile as compared to the (001) one. The narrow Drude component can be found in the far-infrared range. There is an additional feature at 120 meV, which is not observed on the (001) facet. It is difficult to directly associate any feature in optical conductivity obtained on the (101) facet with the W2 cones. However, it is apparent that the W2 contribution is not a simple, linear in ω increasing component, from the zero energy up to the cones' saddle points and including effect of Pauli blocking. It is worth noting that the results on the (001) oriented facet are not fully in line with preceding reports which lack the 70 and 128 meV features [START_REF] S.-I. Kimura | Optical signature of Weyl electronic structures in tantalum pnictides TaPn (Pn= P, As)[END_REF][START_REF] Xu | Optical spectroscopy of the Weyl semimetal TaAs[END_REF].

Magneto-optics of NbAs

NbAs is another Weyl semimetal investigated using Landau level spectroscopy in the scope of this thesis. As other mono-pnictide Weyl semimetals, it crystallizes in a body-centered tetragonal Bravais lattice lacking the space-inversion symmetry. The Weyl cones and the surface states in the form of Fermi Arcs were experimentally identified using the ARPES technique [START_REF] Xu | Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide[END_REF]. Preceding optical and magneto-optical studies reported the observation of a dynamic chiral anomaly and the chiral Landau levels with a multiple topological carriers [START_REF] Yuan | Chiral Landau levels in Weyl semimetal NbAs with multiple topological carriers[END_REF][START_REF] Yuan | The discovery of dynamic chiral anomaly in a Weyl semimetal NbAs[END_REF]. None down-dispersing inter-LL transitions, similar to those identified in the response of TaP, were seen in this material so far.

According to the previous ARPES and DFT studies, the TaP and NbAs considerably differ by arrangement of the W1 cones quartet and also by the energy separation of the W1 and W2 nodes. These two materials represent the polar opposite in both categories in the whole non-centrosymmetric Weyl semimetal group. In the Ta-based materials the W1 nodes are close to the point Σ, nearly forming a square. In NbAs, the W1 nodes are almost twice further from the Σ point as compared to the TaP ones (for the comparison of the positions of the W1 points see Fig. 4.13 in Sec. 4.2.3).

The W1 and W2 separation energy is the second lowest among the group after TaAs. The combination of the ARPES studies and DFT reports suggests the value of ∆E W2-W1 = 37 meV. That is approximately twice lower as compared to TaP, where the separation reaches 60 -80 meV [START_REF] Weng | Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides[END_REF][START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF][START_REF] Lee | Fermi surface interconnectivity and topology in Weyl fermion semimetals TaAs, TaP, NbAs, and NbP[END_REF][START_REF] Grassano | Influence of anisotropy, tilt and pairing of Weyl nodes: The Weyl semimetals TaAs, TaP, NbAs and NbP[END_REF][START_REF] Arnold | Nega-tive magnetoresistance without well-defined chirality in the Weyl semimetal TaP[END_REF][START_REF] S.-I. Kimura | Optical signature of Weyl electronic structures in tantalum pnictides TaPn (Pn = P, As)[END_REF][START_REF] Xu | Experimental discovery of a topological Weyl semimetal state in TaP[END_REF][START_REF] Xu | Observation of Weyl nodes and Fermi arcs in tantalum phosphide[END_REF]. The relative energy distance of the W1 and W2 nodes does not allow one to set the Fermi level in the vicinity of only one cone type. This implies that we should always expect simultaneous contribution from both types of cones in our experiments. The maximum onset of interband absorption in the case that the Fermi level would be set precisely at the W1 nodes is 2∆E W2-W1 ≈ 70 -80 meV.

Magneto-optical response

The high-quality single crystals were synthesized by Ernest Arushanov (Academy of Sciences of Moldova). The NbAs single crystals with dimensions up to 2×3×3 mm with well-formed optical-quality facets were grown using the chemical vapor transport method. The single crystals growing process was combined with the synthesis process. As the initial elements, niobium foil 99.99% and crystalline arsenic 99.999% were used. Iodine was used as a transport agent. The synthesis was carried out in sealed quartz ampoules with a diameter of 20 mm and a length of 200 mm, in a furnace with three temperature zones 6100 °C, 8500 °C and 8000 °C. The crystals growth zone with a temperature 8500 °C was in the central part of the ampoule, while arsenic was placed in the zone with a temperature of 6100 °C, and foil of niobium was placed in the zone with a temperature of 8000 °C. X-ray diffraction and X-ray fluorescence methods were used for the confirmation of the NbAs phase. The facets size is bigger than the aperture window of the cone mounted at the end of the probe ensuring no back scattered light from the sample holder. In order to investigate the anisotropic response of the system, two largest facets were chosen for the experiment. The crystallographic orientation of the facets were identified by XRD as (001) and [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF].

The magneto-optical response was measured using the superconducting solenoid up to 16 T and resistive coil up to 34 T. The Faraday geometry was utilized in both cases, with the magnetic field oriented perpendicular to the explored facets. The resulting magneto-optical spectra are presented in a form of relative R B /R 0 magneto-transmission to emphasize the B-induced effects (Fig. 4.20). A fairly rich response, with different series of inter-and intraband inter-LL excitations, was observed.

The high anisotropy of the material is apparent, see Fig. 4.20. The response of the (001) oriented facet shows both, lines with a sub-linear dependence on B as well as those following the √ B dependence, most likely from both type W1 and W2 of the Weyl cones. Additionally, two sets of down-dispersing inter-LL transitions can be found in the response of the (001) facet. Again, similar to TaAs, we may expect that the response of the W1 is strongly suppressed on the (112) facet due to profound anisotropy of the W1 cones that are significantly flattened in the k z (tetragonal) direction [START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF][START_REF] Xu | Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide[END_REF]. The down-dispersing inter-LL transitions are not seen at all on the (112) facet. This is another argument that these transitions are originating in the vicinity of the Σ point. The response of the [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF] oriented facet comprises mostly √ B scaling transitions that may be associated with the W2 cones. At low energies (∼< 30 meV), the response seems to be dominated by cyclotron-resonance modes from both cones (W1 and W2). The inter-LL transitions with a linear dependence can be found in both facets at higher photon energies and they extrapolate to ∼ 110 meV and ∼ 90 meV for the (001) and ( 112) facet, respectively. They seem to correspond to the parabolic bands at higher energies as suggested by DFT [START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF][START_REF] Grassano | Influence of anisotropy, tilt and pairing of Weyl nodes: The Weyl semimetals TaAs, TaP, NbAs and NbP[END_REF].

Let us now discuss the magneto-optical response observed on the (001) facet in a greater detail. Two sets of down dispersing inter-LL are clearly observed. They extrapolate to the values of 28 and 68 meV in the limit of a vanishing magnetic field (highlighted in Fig. 4.21c by the green and red solid lines, respectively). These down-dispersing lines are accompanied by lines which extrapolate to the same zero-field energy, but their energies increase with B (marked by the green and red dashed lines, respectively). In addition, a nearly flat in B transition, marked by a white line in Fig. 4.21c, was observed. It may resemble the saddle point excitation, similar to the one observed in the case of TaP [START_REF] Polatkan | Magneto-Optics of a Weyl Semimetal beyond the Conical Band Approximation: Case Study of TaP[END_REF]. To interpret this behavior, the original Hamiltonian 4.3 proposed for TaP will be modified by including the anisotropy of the diagonal terms. As discussed in Sec. 4.2.3, this modification is needed to describe the position of the W1 nodes located further from the edge of the Brillouin zone. The modified "double-ellipse" Hamiltonian 4.19 (Fig. 4.21d) was then used to simulate the magneto-optical response observed on the (001) oriented facet of NbAs.

The parameters of the model were tuned to reproduce the lower green set of transitions (Fig. 4.21a). The saddle-point-like excitation seen in the magneto-reflectivity (white line in Fig. 4.21c) is reproduced by the simulation fairly well (highlighted by white arrow in Fig. 4.21a). Our calculations also correspond well to the DFT predictions of the saddle point energy ∆ DFT = 20 meV [START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF]. Nevertheless, the simulation predicts a slightly higher energy for the band inversion point (2 ∆ 2 + γ 2 0 = 40.5 meV) as compared to the observed zero-field extrapolation of the down-dispersing inter-LLs. Notably, we should not be too much attached to the values extracted from the DFT calculations (as plotted in the corresponding papers). In fact, the energy of the saddle point, extracted from the DFT calculations, is deduced from the two nearest W1 cones. This does not tell us much about the band structure in between the two pairs of Weyl nodes, i.e., around the Σ point where our down-dispersing line in fact originate (see Fig. 4.21d). The agreement between our theory and experiments could be further improved by introducing an electron-hole asymmetry to our model 4.19, but only at the cost of other freely tunable parameters.

In addition, the second set of down-dispersing inter-LL transitions is observed at higher photon energies (red lines in Fig. 4.21c). They extrapolate to the 68 meV in the limit of a vanishing magnetic field. This set of lines is related to bands not included in our model and thus, not caught in our calculations. At the same time, it is plausible to assume the existence of additional electronic bands at the Σ point, at energies slightly higher and lower as compared to those from which the W1 cones are formed. In fact, the Σ point belongs to time-reversal invariant points at which the bands must be twice (spin) degenerated. This degeneracy is lifted when moving away from such point, leading to (in the very first approximation) parabolic bands touching by their apices. Such feature in the band structure might be responsible for the upper set of down-dispersing inter-LL excitations.

Optical response at B=0

The optical response of the (001) oriented facet of NbAs was measured in the group of Prof. Akrap at the University of Fribourg. The zero-field optical spectra were measured in the high range of frequencies in order to deduce the optical conductivity and dielectric function of the material via Kramers-Kronig relations (Fig. 4.22). The real part of the optical conductivity σ 1 (ω) displays the intra-and interband absorption of the material. The narrow Drude component can be found below 10 meV, thus suggesting a relatively high carrier mobility. The Drude peak then broadens significantly with the increasing temperature.

Saliently, two sharp features emerge in the real part of the optical conductivity, with the maxima at 21 and 68 meV (marked by green and red arrow at Fig. 4.22, respectively). These features match almost perfectly with energies extracted in the limit of a vanishing magnetic field from the down-dispersing nodal lines. In addition, multiple inter-band absorption bands are seen in the range of 100 -250 meV. They are most likely associated with electronic bands not directly relevant to the topological properties of NbAs, as suggested by the ab-initio calculation [START_REF] Grassano | Validity of Weyl fermion picture for transition metals monopnictides TaAs, TaP, NbAs, and NbP from ab initio studies[END_REF].

Dirac nodal-line semimetals

In summary, three representatives of Dirac nodal line semimetals were studied using tools of optical spectroscopy in the far-and mid-infrared ranges. In the first section, the basic overview of nodal-line semimetals was given together with the review of two material groups: (i) nonsymmorphic space group P4/nmm, (ii) the family of transition metal di-pnictides. The optical response, with and without a magnetic field applied, of different types of nodal lines was discussed. The review of experimental techniques and the typical information which can be deduced when they are applied, was illustrated on the best known compound ZrSiS.

The first experimental part was dedicated to BaNiS 2 , which is one of the simplest example of nodal-line semimetals. It is the representative of the nonsymmorphic space group P4/nmm. A basic approach how to study the simple dispersive Dirac nodal-line semimetals using the tools of optics and magneto-optics was demonstrated. The observed magneto-optical response consisted of the series of inter-and intraband transitions, which were modeled using the Landau-quantized 2D massive Dirac model. From the obtained fit the band gap and the averaged velocity parameter were extracted.

The optical response of BaNiS 2 appeared to be the characteristic of a Dirac semimetal with a dispersive nodal line. The deduced optical conductivity was modeled analytically, using the proposed simplified model, as well as numerically, based on DFT, leading to the fair agreement with the experiment. The presence of the temperature independent region, so-called isosbestic line, was interpreted as a solid fingerprint of a nodal line. The spectral weight transfer from the region above the isosbestic line to the lower energies was analyzed. Thanks to its simplicity in the optical response, BaNiS 2 may serve as a textbook example of a Dirac nodal-line semimetal.

The next two sections were dedicated to the NbAs 2 and NbSb 2 , which both are electron-hole compensated semimetals from the family of transition metal di-pnictides. The nodal-line structure of both materials is open-ended. Nodal lines propagate through multiple Brillouin zones. Analogous experiments were conducted for both NbAs 2 and NbSb 2 resulting in similar optical and magneto-optical response observed. The series of magneto-optical measurements together with the polarization-resolved optical experiments without external magnetic field were conducted. Highly anisotropic response was found in the magneto-optics of different crystallographic facets. The discovery of the facet-dependent band gap, deduced from the zero-field extrapolation of inter-LL excitations, was unexpected and the most interesting results. This observation was interpreted as a direct consequence of the Lorentz-boost-driven renormalization of the 2D massive Dirac system. The tilt parameter of the cone represents the velocity of a relativistic particle, while the asymptotic velocity of a non-tilted cone acts as an effective speed of light. The extracted band gap and velocity parameters were fitted using the two-dimensional massive Dirac model, with the Lorentz-boost implemented, achieving a good agreement between the theory and the experimental data. The observed Lorentz-boost-driven renormalization of the band gap can be also viewed as an analogue of the well-known Franz-Keldysh effect in the magnetic field [START_REF] Aronov | Light absorption in semiconductors in crossed electric and magnetic fields[END_REF][START_REF] Franz | Einfluß eines elektrischen Feldes auf eine optische Absorptionskante[END_REF]. On the other hand, there is no real electric field applied in our case. The additional polarization-resolved reflectivity and analysis of the real part of the optical conductivity allowed us to determine the local direction of the nodal line in the (001) plane. Together with the magneto-optical angular dependence, the local directions of all Fermi crossings of the nodal line could be found with a relatively good precision.

Weyl Semimetals

In the Chapter 4, we introduced basic concepts of Weyl semimetals. The materials were classified with the respect to crystal symmetries and the tilt of the Weyl cones. The consequences of the Weyl semimetal topology were illustrated with the help of preceding experimental studies. The main focus of the work was put on the group of transition-metal mono-pnictides. We report on experimental investigations, using optical and magnetooptical spectroscopy in the infrared range, of three particular compounds: TaP, TaAs and NbAs.

A simplified model was introduced for non-centrosymmetric Weyl semimetals in order to explain an unusual set of inter-LL excitations which decrease their energy with the increasing magnetic field. Such transitions were found in the magneto-optical response of TaP and NbAs. The model describes massless excitation in the W1 cones located in the vicinity of the time-reversal-invariant-point Σ. The validity of the model is first verified by a direct comparison with the magneto-optical response of TaP. It is demonstrated that the exotic so-called down-dispersing inter-LL transitions are directly linked to the band inversion of the electronic band structure in TaP. We show how the proposed model can be generalized for other time-reversal-invariant Weyl semimetals.

The next section was dedicated to TaAs, which is currently the best-known noncentrosymmetric Weyl semimetal. The magneto-optical response of this materials is analyzed using a semi-classical approach, by applying the Lifshitz-Onsanger formula. A strong anisotropy has been found in both optics and magneto-optics of three different facets -with orientations (001), ( 101) and [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF]. This anisotropy is mainly related to the W1 cones and the nearby electronic bands which are considerably flattened along the k z (tetragonal) direction. Surprisingly, the magneto-optical response of the (001) facet does not display any transitions with a √ B dependence, typical of massless electrons, and the observed behavior would better corresponds to a narrow band gap semiconductor rather than a Weyl semimetal. This suggests that the Fermi level is set well above the W1 saddle points and the Pauli blocking does not allow us to see any excitations within the the W1 cones. On the other hand, inter-LL transitions with a clear √ B dependence dominate the response observed on the (112) and ( 101) facets. They are associated with the response of the W2 cones, which seems to be surprisingly isotropic in three dimensional space and extend over an energy range exceeding 200 meV.

NbAs was the last compound under the investigation. Two different crystallographic orientations were measured in order to address the anisotropy of the material.

While the magneto-optical response of the [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF] facet was dominated by inter-LL transitions with a √ B dependence, originating from the isotropic W2 cones, the response of the (001) showed two sets of down-dispersing inter-LL transitions. This behavior was linked to the band inversion around the Σ point, similarly to the case of TaP. However, the original Hamiltonian introduced for TaP had to be extended to match our experimental observations.

Conclusion (FR)

Semi-métaux à ligne nodale de Dirac

En résumé, trois représentants des semi-métaux à ligne nodale de Dirac ont été étudiés à l'aide d'outils de spectroscopie optique dans les domaines de l'infrarouge lointain et moyen. Dans la première section, nous avons donné une vue d'ensemble des semi-métaux à lignes nodales ainsi qu'un aperçu de deux groupes de matériaux: (i) le groupe spatial non-symorphique P4/nmm, (ii) la famille des di-pnictides de métaux de transition. La réponse optique, avec et sans application d'un champ magnétique, de différents types de lignes nodales a été discutée. L'examen des techniques expérimentales et des informations typiques qui peuvent être déduites lorsqu'elles sont appliquées, a été illustré sur le composé le plus connu ZrSiS.

La première partie expérimentale a été consacrée au BaNiS 2 , qui est l'un des exemples les plus simples de semi-métaux à lignes nodales. Il est le représentant du groupe d'espace non symétrique P4/nmm. Une approche de base pour étudier les semi-métaux de Dirac à ligne nodale dispersifs simples en utilisant les outils de l'optique et de la magnétooptique a été démontrée. La réponse magnéto-optique observée se compose d'une série de transitions inter et intra-bandes, qui ont été modélisées à l'aide du modèle de Dirac massif 2D quantifié par Landau. L'ajustement obtenu a permis d'extraire la bande interdite et le paramètre de vitesse moyenne.

La réponse optique de BaNiS 2 est apparue comme la caractéristique d'un semimétal de Dirac avec une ligne nodale dispersive. La conductivité optique déduite a été modélisée analytiquement, en utilisant le modèle simplifié proposé, ainsi que numériquement, sur la base de la DFT, ce qui a conduit à un bon accord avec l'expérience. La présence de la région indépendante de la température, appelée ligne isosbétique, a été interprétée comme une empreinte solide d'une ligne nodale. Le transfert de poids spectral de la région au-dessus de la ligne isosbétique vers les énergies inférieures a été analysé. Grâce à la simplicité de sa réponse optique, BaNiS 2 peut servir d'exemple type de semimétal à ligne nodale de Dirac.

Les deux sections suivantes sont consacrées au NbAs 2 et au NbSb 2 , qui sont tous deux des semi-métaux à compensation électron-trou de la famille des di-pnictides de métaux de transition. La structure des lignes nodales de ces deux matériaux est ouverte. Les lignes nodales se propagent à travers de multiples zones de Brillouin. Des expériences analogues ont été menées pour les matériaux NbAs 2 et NbSb 2 , ce qui a permis d'observer des réponses optiques et magnéto-optiques similaires. Une série de mesures magnéto-optiques ainsi que des expériences optiques à résolution de polarisation sans champ magnétique externe ont été réalisées. Une réponse hautement anisotrope a été trouvée dans la magnéto-optique de différentes facettes cristallographiques. La découverte de la bande interdite dépendante de la facette, déduite de l'extrapolation à champ nul des excitations inter-LL, était inattendue et constituait le résultat le plus intéressant. Cette observation a été interprétée comme une conséquence directe de la renormalisation du système de Dirac massif en deux dimensions, induite par la poussée de Lorentz. Le paramètre d'inclinaison du cône représente la vitesse d'une particule relativiste, tandis que la vitesse asymptotique d'un cône non incliné agit comme une vitesse effective de la lumière. Les paramètres extraits de la bande interdite et de la vitesse ont été ajustés à l'aide du modèle de Dirac massif bidimensionnel, avec la mise en oeuvre du Lorentz-boost, ce qui a permis d'obtenir un bon accord entre la théorie et les données expérimentales. La renormalisation de la bande interdite observée, induite par le coefficient de Lorentz, peut également être considérée comme un analogue de l'effet Franz-Keldysh bien connu dans le champ magnétique [START_REF] Aronov | Light absorption in semiconductors in crossed electric and magnetic fields[END_REF][START_REF] Franz | Einfluß eines elektrischen Feldes auf eine optische Absorptionskante[END_REF]. D'autre part, il n'y a pas de champ électrique réel appliqué dans notre cas. La réflectivité supplémentaire résolue en polarisation et l'analyse de la partie réelle de la conductivité optique nous ont permis de déterminer la direction locale de la ligne nodale dans le plan (001). Avec la dépendance angulaire magnétooptique, les directions locales de tous les croisements de Fermi de la ligne nodale ont pu être trouvées avec une précision relativement bonne.

Semi-métaux de Weyl

Dans ce chapitre 4, nous avons présenté les concepts de base des semi-métaux de Weyl. Les matériaux ont été classés en fonction des symétries cristallines et de l'inclinaison des cônes de Weyl. Les conséquences de la topologie des semi-métaux de Weyl ont été illustrées à l'aide d'études expérimentales précédentes. L'accent a été mis sur le groupe des mono-pnictides de métaux de transition. Nous rapportons des études expérimentales, utilisant la spectroscopie optique et magnéto-optique dans la gamme THz et infrarouge, de trois composés particuliers : TaP, TaAs et NbAs.

Un modèle simplifié a été introduit pour les semi-métaux de Weyl non-centrosymétriques afin d'expliquer un ensemble inhabituel d'excitations inter-LL qui diminuent leur énergie avec l'augmentation du champ magnétique. De telles transitions ont été trouvées dans la réponse magnéto-optique du TaP et du NbAs. Le modèle décrit une excitation sans masse dans les cônes W1 situés au voisinage du point d'inversion temporelle Σ. La validité du modèle est d'abord vérifiée par une comparaison directe avec la réponse magnéto-optique du TaP. Il est démontré que les transitions inter-LL exotiques dites de dispersion vers le bas sont directement liées à l'inversion de la structure de la bande électronique dans le TaP. Nous montrons comment le modèle proposé peut être généralisé pour d'autres semi-métaux de Weyl invariants dans le temps.

La section suivante est consacrée au TaAs, qui est actuellement le semi-métal de Weyl non centrosymétrique le plus connu. La réponse magnéto-optique de ce matériau est analysée par une approche semi-classique, en appliquant la formule de Lifshitz-Onsanger. Une forte anisotropie a été trouvée à la fois dans l'optique et la magnéto-optique de trois facettes différentes -avec les orientations (001), ( 101) et [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF]. Cette anisotropie est principalement liée aux cônes W1 et aux bandes électroniques proches qui sont considérablement aplaties le long de la direction k z (tétragonale). De manière surprenante, la réponse magnéto-optique de la facette (001) ne montre aucune transition avec une dépendance de √ B, typique des électrons sans masse, et le comportement observé correspondrait mieux à un semi-conducteur à bande interdite étroite plutôt qu'à un semi-métal de Weyl. Cela suggère que le niveau de Fermi est fixé bien au-dessus des points de selle W1 et que le blocage de Pauli ne nous permet pas de voir d'excitations dans les cônes W1. D'autre part, les transitions inter-LL avec une dépendance claire de √ B dominent la réponse observée sur les facettes [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF] et [START_REF] Nilforoushan | Photoinduced renormalization and electronic screening of quasi-two-dimensional dirac states in BaNiS 2[END_REF]. Elles sont associées à la réponse des cônes W2, qui semble être étonnamment isotrope dans l'espace tridimensionnel et s'étend sur une gamme d'énergie dépassant 200 meV.

Le NbAs a été le dernier composé étudié. Deux orientations cristallographiques différentes ont été mesurées afin d'étudier l'anisotropie du matériau. Alors que la réponse magnéto-optique de la facette [START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF] était dominée par des transitions inter-LL avec une dépendance de √ B, provenant des cônes W2 isotropes, la réponse de la facette (001) a montré deux ensembles de transitions inter-LL se dispersant vers le bas. Ce comportement est lié à l'inversion de bande autour du point Σ, comme dans le cas du TaP. Cependant, l'hamiltonien original introduit pour le TaP a dû être étendu pour correspondre à nos observations expérimentales. 001). Stackplots of relative magneto-reflectivity spectra, R B /R 0 , for selected values of the magnetic field collected on the (20-1) and (001) facets, which make angles 7 • , and 90 • with the a-axis, respectively. The yellow and red dots show R B /R 0 maxima belonging to transitions in the lower and upper set, respectively. The horizontal and vertical gray bars show positions of the two steps in the onset of interband absorption at 2∆ D and 2∆ F at B = 0.

A Appendix 1: Waterfall plots

Fig. 1 . 2 .

 12 Fig. 1.2. Electronic states of topological insulators. Parts (a) -(d): Schemes of different electronic states of matter and their corresponding band structure. Parts (e) -(g): Experimental realization of particular topological effects. (a) Insulating state. The valence and conduction band are separated by an energy band gap. (b) Quantum Hall effect. Electrons exhibit cyclotron motion perpendicular to the magnetic field. The cyclotron orbits are not closed at the edge of the material resulting in gapless conducting edge states. (c) 2D topological insulator. There are two gapless spin states (as required by time-reversal symmetry) propagating in the opposite directions at the edges of the samples. (d) 3D topological insulator. The surface states are similar to the 2D Dirac cone, similar to graphene. The direction of electron motion depends on the direction of the spin. (e) Room-temperature quantum Hall effect realized in graphene. The Hall conductivity (σ xy -blue) and resistivity (ρ xy -red) plotted as a function of gate voltage V g . The Hall conductivity plateaus are quantized by 2e 2 /h for both electrons (positive V g ) and holes (negative V g ). The inset shows LL quantization of Dirac electrons with E F set in the gap ∆E. (f) Quantum spin Hall effect detected in the HgTe/Hg x Cd 1-x Te quantum well heterostructure. Devices III and IV with higher thickness than d c ≈ 64 Å exhibit a quantized conductance of 2e 2 /h. (g) ARPES measurement of Bi 2 Se 2 surface state resembling the 2D Dirac cone of graphene. Adapted from [7, 8, 17, 18].

  Liu et al., Science 2014 Liu et al., Nature Mater. 2014 Lv et al., Nature Phys. 2015 Xu et al., Nature Phys. 2015

Fig. 1 . 3 .

 13 Fig. 1.3. Conical dispersion in topological matter. Classification of the conical bands in solids based on their dimensionality and valley and/or spin degeneracy. Examples of particular compounds and the key publications are highlighted for each class.

Fig. 1 . 4 .

 14 Fig. 1.4. Optical transition of conical band. (a) 3D Weyl cone with intraband and interband absorption marked by orange and blue, respectively. Grey area represents occupied states. (b) Drude contribution of free carriers. (c) Real part of optical conductivity with both intra-and interband absorption. Adapted from [32].

Fig. 1 . 5 .

 15 Fig. 1.5. Landau level quantization. Band dispersion and the associated Landau level spectrum of a parabolic band, Dirac system, and massive Dirac system are presented in panels (a), (b), and (c). The Fermi level is always set in conduction band. The intraband and interband absorptions are highlighted by red and orange arrows.

Fig. 2 . 2 .

 22 Fig. 2.2. Magneto-reflectivity setup. (a) Scheme of magneto-reflectivity setup. The waveguide between FTIR spectrometer and parabolic mirror is changed to the 4-meter version in the high magnetic field setup in order to keep the spectrometer out of the reach of the magnetic field. (b) Faraday geometry. The k vectors of the incident and reflected light are parallel to the direction of the magnetic field.

RFig. 2 . 3 .

 23 Fig. 2.3. Magneto-transport setup. The lock-in amplifier acts as a voltage source. The current is fixed to 2 mA by the load resistor (1 kΩ). The current flows through contacts 1 and 2, the voltage is read out between contacts 3 and 4.

Fig. 3 . 1 .

 31 Fig. 3.1. Band structure of ZrSiS. (a) Crystallography of ZrSiS. (b) Electronic bulk dispersion with the inclusion of spin-orbital coupling. The linear band dispersion of the Dirac nodal line are apparent at the cuts Γ -X, Γ -M, Z -R and A -Z. (c) Brillouin zone with Fermi surface deduced from the magneto-transport experiment. The nodal line loop is highlighted by the red line. The electron and holes pockets are shown up by green and purple color, respectively. Adapted from [43].

Fig. 3 . 2 .

 32 Fig. 3.2. Crystal and band structure of NbAs 2 . (a) Crystal lattice of transition metal di-pnictides. Red atoms represents As/Sb/P, orange atoms Nb/Ta. (b) Ab-initio band structure calculation for NbAs 2 with and without inclusion of spin-orbital coupling. Blue shaded part marks the beginning and the end of the dispersive nodal line from Z -I 1 to Y -X 1 cuts. DFT figure adapted from [56].

Fig. 3 . 3 .

 33 Fig. 3.3. Optical conductivity of Dirac nodal-line semimetals. Parts (a), (d): Scheme and the corresponding optical response of Weyl cones. Parts (b), (e): Flat nodal line. The optical conductivity is independent on the frequency of the incident light. The optical conductivity vanishes if the polarization of the light is aligned with the direction of the k line . The onset is defined by 2E F , respectively 2(E F + ∆) in the case of gapped nodal line. Parts (c), (f): The response of the gapped dispersive nodal line. The frequency dependence increases until the end of the nodal line k max . Adapted from [56].

Fig. 3 . 4 .

 34 Fig. 3.4. Optical response of flat nodal line -ZrSiS case. (a) Absorbance spectrum of graphene measured on three different samples. The sheet conductivity reaches constant value of πe 2 /(2h) per one graphene layer. (b) Flat frequency dependence of optical conductivity observed in ZrSiS showing the 2D Dirac nature of the low energy optical excitations. (c) Brillouin zone of ZrSiS. If the magnetic field B is aligned with the direction of nodal line k line , the resulting plane cut is a 2D Dirac cone. (d) Landau level spectrum of ZrSiS following the massive Dirac model, which could be used for precise determination of material parameters. Adapted from [50, 51, 64].

Fig. 3 . 5 .

 35 Fig. 3.5. Dispersion of non-tilted, tilted and overtilted 2D conical bands. Schematic sketch of a hyperbolic dispersion for (massive) Dirac electrons with an additional tilt:E(k) = -huk x ± ∆ 2 + h2 v 2 k 2 .The dashed lines correspond to asymptotic dispersions. (a) 2D Dirac cone, which can be also viewed as a reference frame. (b) Tilted type-I cone with tilt parameter u representing the relativistic particle in motion with the respect to the material characteristic slope velocity v. The system is referred as in magnetic regime. (c) Overtilted type-II cone -the Landau level quantization collapse and the system transfers from magnetic regime to electric one.

Fig. 3 . 6 .

 36 Fig. 3.6. Relative strength of inter-band transitions between different pairs of Landau levels (for n = 0 . . . 6) plotted for angles θ D = 0, 20, 40, 60 • , and parameters (w, v D and ∆ D ) deduced experimentally for the dispersive part of the nodal line in NbAs 2 (see Sec. 3.4). For θ D = 0 • , standard selection rules are obtained n → n ± 1 for electric-dipole transitions, typical of all isotropic systems. With increasing θ D , gradually, additional transitions become allowed (n → n ± 2, ±3 . . .). For large angles θ D , but still for β < 1, one finds a plethora of optically active transitions. The dominant ones follow the ruleof-thumb selection rules n → αn and n → n/α, where α is an integer (α = 2 -4 for θ D = 60 • ).

  3.10 b) [101-105]. Due to the tetragonal symmetry, the band structure is fourfold degenerate. The nodal lines are not closed in the loops like in some other materials such as ZrSiS but are open, propagating throughout multiple Brillouin zones. It was found out that the properties of the nodal line can be affected by external factors, such as by the chemical substitutions, which change the position within the k space along the Γ -M [101].

Fig. 3 . 7 .

 37 Fig. 3.7. Crystallography and band structure of BaNiS 2 . (a) Crystal structure of BaNiS 2 . (b) First Brillouin zone of BaNiS 2 . The purple area around Γ point represent calculated electron hole. The nodal line region is located halfway through the Γ -M path and it is fourfold degenerated (yellow parts, also highlighted by the red column). (c) Abintio band structure calculation with employed HSE. The Dirac cone is apparent at Γ -M path.

Fig. 3 . 8 .

 38 Fig. 3.8. Magneto-optics of BaNiS 2 . (a) Relative magneto-reflectivity R B /R 0 of BaNiS 2 measured with the magnetic field applied perpendicular to its (001) facet. Extracted maxima of R B /R 0 were used for the 2D massive Dirac of interband transitions. The lowest two transitions have been identified as intraband transitions. The higher interband transitions were extracted up to Landau level index n=6. Transitions were fitted using 2D massive Dirac model. (b) Sketch of the conical band with E F indicated at the top of the valence band and the scheme of the development of Landau levels of 2D massive Dirac system with the BaNiS 2 fitting parameters.

Fig. 3 . 9 .

 39 Fig. 3.9. Optical response of BaNiS 2 -full range. (a) Full range reflectivity of BaNiS 2 . (b) Deduced real part of the optical conductivity σ 1 (ω).

Fig. 3 .

 3 Fig. 3.10. Temperature-dependent reflectivity of BaNiS 2 . (a) Temperature-dependent in-plane reflectivity of BaNiS 2 below 0.8 eV. (b) Energy profile of dispersive nodal line. (c) Deduced real part of optical conductivity σ 1 (ω). The temperature independent part spreads from ∼ 0.1 eV to ∼ 0.3 eV. The filled grey areas indicates the spectral weight transfer from energies above ∼ 0.3 eV to lower energy region between temperatures 300 K to 5 K. (d) Fitted experimental data with the model eq. 3.30 and DFT. R sel is a radius of the area around the nodal line region given in the ratio of reciprocal lattice unit.

  3.11b). The nodal lines in NbAs 2 disperse with momentum and each approaches the Fermi energy four times within the Brillouin zone (Fig. 3.11a, b). The crossings come in pairs of two different types, one of which is associated with a dispersive and the other one with a flat part [56]. They are located at k D line and k F line , respectively, and characterized by the local directions τ τ τ D and τ τ τ F (Fig. 3.11a).

Fig. 3 .

 3 Fig. 3.11. Brillouin zone and band structure of the NbAs 2 nodal line. (a) Brillouin zone of NbAs 2 with the crossings of the nodal lines with the Fermi level and selected high-symmetry points marked. The blue lines show approximate position of two nodal lines, reflecting the DFT calculations and overall conclusions in Ref.[56]. (b) Schematic view of the nodal-line dispersion over a half of the Brillouin zone along the direction of propagation. The band gaps at the crossing points with the Fermi energy, in the dispersive and the (approximately) flat parts of the nodal line, are 2∆ D ,and 2∆ F , respectively.

Fig. 3 .

 3 Fig. 3.13. Polarization-resolved optical response of NbAs 2 without magnetic field. (a) Reflectance measured with the polarization aligned with the respect to the a or b crystallographic axis. Inset shows the full range of measurement. (b) Optical conductivity for both polarizations. Onsets at 2∆ D = (88 ± 2) meV and 2∆ F = (113 ± 2) meV are highlighted by the grey dashed line.

Fig. 3 .

 3 Fig. 3.14. Optical response of NbAs 2 without magnetic field -temperature dependence. Optical measurements without external magnetic field in the range of temperatures from 10-300 K. Parts (a), (b): Reflectance measured with the polarization aligned with the respect to the a or b crystallographic axis. Parts (c), (d): Optical conductivity of both polarization alignments. The region from ∼80-200 meV show the largest dependence on the temperature. The higher energies >200 meV show little or almost no temperature dependence.

Fig. 3 .

 3 Fig. 3.15. Magneto-optics of NbAs 2 . Magneto-optics of NbAs 2 measured up to 16T with the linear gap extrapolation. Two sets of inter-LL transitions can be observed throughout multiple facets. The two optical band gaps deduced from zero-field optics are highlighted by grey horizontal line with the thickness of the line representing the experimental error. Although the higher energy set of inter-LL transitions extrapolate to the same zero-field limit, the band gap of lower set is varying with different crystal orientation from approximately 89 -74 meV.

Fig. 3 .

 3 Fig. 3.16. Analysis of magneto-optical data. Effective band gap and velocity parameter obtained by a fit of the massive-Dirac model to the lowest inter-LL transition in sets belonging to the flat and dispersive crossings with the Fermi energy: (a),(b) and (c),(d), respectively.

Fig. 3 .

 3 Fig. 3.17. Magneto-optics of NbAs 2 -low θ D . Magneto-optics of four closest facets to the local direction τ τ τ D . Grey horizontal lines represent the non-renormalized optical band gaps 2∆ eff D = (88 ± 2) meV and 2∆ eff F = (113 ± 2) meV. Grey dashed lines fit well for the flat part, which is observable for all four cases and follows 2D massive Dirac model. The dispersive part fit is represented by dotted lines. Apart from conventional n → n ± 1 transitions (red dotted), the n → n ± 2 can be observed as well. The development of the selection rules is graphically represented above each facet angle for respective angle. The fitting parameters for both flat and dispersive part are taken from Fig. 3.16.

  , 3.19 and 3.18 with the respective angle θ D .

Fig. 3 .

 3 Fig. 3.18. Magneto-optics of NbAs 2 -high magnetic field. Parts (a), (b): The analysis of the high magnetic field measurements up to 34 T. The facets (201) and (403) exibit the band gap renormalization. The transitions from the flat part are becoming untraceable, due to high θ F = 74 • and 80 • for (201) and (403), respectively. The calculated selection rules for transition intensity are plotted in panels (c), (d) for (201) and (403), respectively.

Fig. 3 .

 3 Fig. 3.19. Magneto-optics of NbAs 2 -high θ D . Parts (a), (b): Facets (101) and (607) with higher angle θ D with the respect to the local direction of the dispersive part of the nodal line. The band gap renormalization effect is apparent with the zero field extrapolation to 78 and 74 meV. The higher flat part set is not present due to the high θ F . Parts (c), (d): Facets with the θ D approaching/exceeding the critical angle when Landau level quantization collapses. The flat part is traceable in these orientation. Selection rules are plotted above each panel respectively.

Fig. 3 .

 3 Fig. 3.21. Modeling the optical response. Parts (a), (c), (e): Real part of optical conductivity (solid lines) deduced via Kramers-Kronig analysis of the NbAs 2 reflectivity response measured on the (001)-oriented facet at T = 10 K using radiation polarized linearly along the a and b crystallographic axes. The dashed and dotted lines show results of theoretical modeling described in the text. Parts (b), (d), (f): Energy (at k line ) and momentum profiles of the nodal lines considered in the simplified (segment-based) model for optical conductivity of NbAs 2 at B = 0.

III a and θ III b -

 b are visualized graphically in Figs. 3.21d, f in which the nodal line location was projected to the a-c and b-c crystallographic planes, respectively. The vectors k b and k b × k c lies in the a-b plane while the vectors k c , k a as well as k b × k c lie in the a-c crystallographic plane, see Fig. 3.20a.

Fig. 3 .

 3 Fig. 3.22. Magneto-resistence of NbAs 2 . Longitudinal magneto-resistance, R xx and ∆R xx , measured at indicated temperatures measured with the magnetic field applied perpendicularly to the (001) facet of the Sample 1 ((a) and (c)) or parallel with the a crystallographic axis ((b) and (d)). The sketch of the geometry of the field and applied current can be found as an inset in parts (a) and (b).

Fig. 3 .

 3 Fig. 3.23. FFT of magneto-resistence of NbAs 2 . Parts (a) and (b): Deduced fast Fourier transformation from longitudinal magneto-resistance plotted in Fig 3.22a,b respectively for each geometry. The main frequencies and their harmonics are highlighted. Parts (c) and (d): Extracted amplitudes of FFT spectra used for the Lifshitz-Kosevich analysis. The extracted masses are in agreement with the ones reported in the literature [57-60].

Fig. 3 .

 3 Fig. 3.24. FFT frequencies angular dependence. Parts (a), (b): Shubnikov-de Haas oscillation frequencies measured at T = 1.5 K as a function of the angle θ between the magnetic field and the b and a axes, respectively. Parts (c), (d): The sketch of geometry for each measurement with the direction of magnetic field and applied current.

Fig. 3 .

 3 Fig. 3.25. Brillouin Zone and DFT of NbSb 2 . (a) Brillouin zone of NbSb 2 with calculated Fermi surfaces. (b) Band structure of NbSb 2 . The Dirac nodal line starts at the cut made from Z -I 1 to the end at Y -X 1 . Adapted from [55].

Fig. 3 .

 3 Fig. 3.26. XRD of NbSb 2 . X-ray analysis of six different facets of NbSb 2 . The facet (001) of the Sample 2 was used for polarization-resolved optical measurement without external magnetic field. The crystallographic axis were determined by the pole diagram.

Fig. 3 .

 3 Fig. 3.27. Optical response at B=0 of NbSb 2 -temperature dependence. Parts (a), (c): Reflectivity measured in the temperature range from 10 to 300 K. Parts (b), (d): Real part of the optical conductivity deduced by Kramers-Kronig relations. The temperature independent region representing the response of the nodal line is spreading from ∼100 -600 meV.

Fig. 3 .

 3 Fig. 3.28. Polarization-resolved optics without magnetic field of NbSb 2 -detailed view. (a) Reflectance measurement performed without magnetic field for both polarization configuration. The plasma edge is apparent at ∼100 meV. (b) Real part of optical conductivity. There are three visible onsets marked by black arrows as ∆ I,II,IV in E ∥ b configuration. The optical conductivity is heavily suppressed for the E ∥ a configuration leaving only the ∆ IV and the increasing slope

Fig. 3 .

 3 Fig. 3.29. Magneto-optical response of all facets of NbSb 2 . Magneto-reflectivity R B /R 0 of six different facets of NbSb 2 . Two sets of inter-LL transitions are detected with the respect to the different orientation of the magnetic field and explored facet.

Fig. 3 . 30 .

 330 Fig. 3.30. Analysis of the magneto-optics of NbSb 2 -Facets (20-1), (10-1), (100) and (20-3). Magneto-reflectivity R B /R 0 of 4 different facets of NbSb 2 . The grey horizontal lines represents the extracted band gap of the dispersive (2∆ D ) and the flat part (2∆ F ) respectively. The grey dashed lines highlight the interband transitions of the flat part of the nodal line with the dipole active selection rules n → n ± 1. The dispersive part is assigned to the lower energy set. The n → n ± 1 transitions are plotted with red dashed lines, the orange ones corresponds to the additional set of transitions n → n ± 2, which is the second most intensive one. The relative intensity of the transitions originating from dispersive part are plotted above panels for each facet respectively with its θ D angle. In the case of facet (20-3), the dispersive part is represented with only a small trace of undefined transition marked by purple arrows extrapolating to ∼ 104 meV.

Fig. 3 .

 3 Fig. 3.31. Analysis of the magneto-optics of NbSb 2 -Facets (001) and (201). Magnetoreflectivity R B /R 0 of two last facets of NbSb 2 . (a) Facet (001) shows only the contribution of the flat part of the nodal line with weak modulation in magnetic field. (b) Facet (201) with the low intensity traces of the dispersive part. From the first sight the extrapolated band gap is lower than the original one highlighted by the grey horizontal line. The low intensity does not allow to extrapolate the band gap precisely. The sign of the lowest possible transition of lower set is highlighted by the purple arrow.

Fig. 3 .

 3 Fig. 3.32. Local direction graphical representation and the summary of the velocity and gap analysis of NbSb 2 . Part (a): Cut along the b/k b vectors of the Brillouin zone of NbSb 2 leaving the a-c plane cut. The local direction vectors of dispersive and flat part are highlighted with the respect to the direction of the magnetic field perpendicular to each measured facet. Parts (b), (c):The analysis of the extracted band gap and velocity parameter from magneto-optics of dispersive part. The decreasing band gap with the higher angle between the magnetic field and the local direction of the dispersive nodal line is explained on a speculative level by the renormalization through the Lorentz factor γ. The velocity parameter scales with v D cos(θ )/γ(3/2) . The band gaps of (20-3) and (201) were extracted from the weak transition of unknown Landau level index (marked purple). Parts (d), (e): Band gap and velocity parameter of the higher energy set associated with the transitions originated from the flat part at the end of the nodal line. The band gap values lies within the range of experimental error 2∆ F = 144 ± 2 meV. The velocity parameter scales with √ θ of the deviation of the magnetic field and local direction, which is characteristic for 2D Dirac model.

Fig. 3 .

 3 Fig. 3.33. Detailed analysis of real part of optical conductivity of NbSb 2 . Parts (a), (c), (e): The real part of the optical conductivity. The experiment is represented with the solid line and theoretical model by dashed line for the whole contribution. The contribution of the each part of nodal line for i = I, II, III, IV is plotted by dotted lines. All observable features are reproduced by the theoretical modeling. Part (b): The energy profile of the nodal line with highlighted Fermi energy. Parts (d), (f): Graphical representation of the nodal line direction on the a-b and a-c plane cut, respectively.

Fig. 4 . 1 .

 41 Fig. 4.1. Dirac and Weyl semimetals comparison. 3D Dirac cone with spin degenerated band at k 0 . Breaking of a space-inversion and/or a time-reversal symmetry leads to the creation of spin non-degenerated Weyl cones at ±k.

Fig. 4 . 2 .

 42 Fig. 4.2. Weyl semimetals with broken space-inversion symmetry. Parts (a), (b): Weyl semimetal cone Type I and its Fermi surface. The cone intersection is projected as a point at the Fermi surface. Parts (c), (d): The over-tilted Type II cone. Fermi surface can be then distinguished into hole and electron part. Adapted from [155].

Fig. 4 . 3 .

 43 Fig. 4.3. Fermi Arcs. (a) 3D bulk and 2D surface Brillouin zone. Weyl points in bulk structure act as sink or source of Berry flux, which penetrates into surface states in the form of Fermi arcs. (b) Weyl cones with Fermi level at the energy of Weyl points. (c) Evolution of the Fermi arcs with the respect to the increasing Fermi level. The displacement from original position of Weyl points is determined by the cone velocity parameter. Adapted from [73].

Fig. 4 . 4 .

 44 Fig. 4.4. Chiral Anomaly. The Landau levels of Weyl cones. The fermions of the opposite chirality on the zeroth Landau level are presented by red and blue color (Chern numbers ±1), respectively. In the presence of magnetic and electric field it is expected that the electrons will be pumped from one cone to the another with two distinguishable Fermi levels E ±1 F .

• Weyl 2 .Fig. 4 . 5 .

 245 Fig. 4.5. Crystal and Brillouin zone of non-centrosymmetric Weyl semimetals. (a) Crystal structure of TaAs family of materials. (b) Brillouin zone with indicated Weyl cones with assigned chirality as red or blue. The nodal line rings are highlighted by green lines laying in the M x and M y mirror symmetries. (c) View on the Brillouin zone parallel to (010) or (100) direction. (d) Brillouin zone from (001) view. Adapted from [137].

Fig. 4 . 6 .

 46 Fig. 4.6. Fermi arcs of different Weyl semimetals -ARPES measurements. (a) (i) Projection of the W1 cone to the surface as a Fermi Arc. (ii-iv) Fermi surface topology of NbP, TaP and TaAs. Measured Fermi arcs (blue) are in fair agreement with the theoretical prediction (red). (b-d) High resolution ARPES measurement of all three materials. ∆K1 and ∆K2 notes the separation between the Weyl points and Fermi arcs, respectively. (e) Extracted ∆K1 and ∆K2 parameters. It is shown that the cone and Fermi arc splitting increases with the spin-orbital coupling of the material. Adapted from [149].

Fig. 4 . 9 .

 49 Fig. 4.9. Analysis of magneto-optical experiment of TaP. (a) Landau levels calculated from Hamiltonian 4.15 with following parameters: ∆ = 35 meV, γ 0 = 10 meV, Γ = 1 meV, and M = m x = m y /12 = m 0 /4, where m 0 is the free electron mass. The total matrix size (number of Landau levels) used for simulation is N = 700. The saddle point region is highlighted by red dotted line. The band inversion local extremes ((∆ 2 + γ 2 0 ) 1/2 ) are shown by orange dotted lines. (b) Simulation of magneto-reflectivity. All spectra are normalized by the reference spectrum calculated at B = 1 with broadening of Γ = 5 meV. The background dielectric constant was set as ε ∞ = 20. Excitation between the higherenergy saddle points are highlighted by grey horizontal arrow. (c) Magneto-reflectivity of TaP (001) facet measured at temperature of 2 K plotted as R B /R 0 . (d) Scheme of multiple types of transitions identified from R B /R 0 . The meaning of each transition is discussed in the text with the respect of the figure color coding.

Fig. 4 .

 4 Fig. 4.10. TaP measurement at B=0. (a) The temperature dependence of the optical response without external magnetic field. (b) The deduced real part of optical conductivity via Kramers-Kronig relation. (c) Dielectric function.

Fig 4 .

 4 8a, b.

Fig. 4 .

 4 Fig. 4.11. Schematic profile of W1 and W2 cones of TaP. Relatively large separation of the cones is given by ∆E W 2-W 1 . The Fermi level is set to the approximate energy of W1 cones intersection. The small cyan arrows represent the intraband absorption of free electrons. Red arrow corresponds to the excitation related to the inverted band structure. Green dashed arrow and black solid arrow stand for interband excitation of W1 and W2 cone, respectively. The intraband absorption may be observable above the onset of 2∆E W 2-W 1 .

Fig. 4 .

 4 Fig.4.13. W1 cone k-space distribution of the transition-metal monopnictides Weyl semimetals. The k-space positions of W1 cones of all TaAs family Weyl semimetals, q = 0 is assigned to the high symmetry point Σ. W1 coordinates of TaAs and TaP could cross by taking into account circle and ellipse representation of diagonal and off-diagonal parameters. The approach has to be changed to two ellipse intersections for NbAs and NbP in order to reach W1 nodes, which are further from q = 0.

Fig. 4 .

 4 Fig. 4.14. Simulation of the magneto-optical response of TaAs, TaP, NbAs, NbP. Relative magneto-reflectivity of all compounds of TaAs family calculated for the same parameters presented in the Fig. 4.13. The down dispersing inter-LL transitions are observable for all presented materials. Precise parameters used for calculation can be found in Table 4.1.

Tab. 4 . 1 .

 41 Compound ∆ (meV) γ 0 (meV) M (m 0 ) M x (M) M y (M) m x (M) m y (M) Fitting parameters for Weyl semimetals. Corresponding fitting parameters used for W1 cones positions in k-space (Fig.4.13) and for magneto-optical response (Fig.4.14).

Fig. 4 .

 4 Fig. 4.15. Brillouin zone of TaAs. Left: Brillouin zone of TaAs with (001) and (101) plane highlighted by red and blue, respectively. Sketch of W1 and W2 energy dispersion in the (001) (a) and (101) (b) projections. The W1 becomes flatter in (101) while W2 remains isotropic.

Fig. 4

 4 Fig. 4.16.TaAs Magneto-reflectivity of all facets -low magnetic field. Top panels: Relative magneto-reflectivity R B /R 0 measured in the Faraday geometry of facets (001), (101) and[START_REF] Yang | Symmetry demanded topological nodal-line materials[END_REF]. Bottom panels: Derivative of R B /R 0 used for resonance extraction.

Fig. 4 . 17 .

 417 Fig. 4.17. TaAs magneto-optical analysis. Magneto-optics of facets (101) (panels a-d) and (001) (panels e-h). Parts (a), (e): Magneto-reflectivity response normalized by the zero-field spectrum as R b /R 0 measured up to 34 T. Parts (b), (f): Derivative of R b /R 0 used for resonance extraction. Parts (c), (g): Deduced position of inter-LL transitions. Parts (d), (h): Lifshitz-Onsanger analysis of the extracted points using γ = 0.

1 .

 1 Orange set. The most intensive series of transitions extrapolate to 15 meV in the zero-field limit. The majority of these transitions collapse to the almost indistinguishable line when plotted as energy versus momentum k n (Fig.4.17h). It represents a joint electronic band (i.e., difference of two bands) with a nearly parabolic profile and the band gap of 2∆ = 15 meV. The lines follow linear in B dependence and their spacing is almost equidistant. The effective carrier mass of m * ≈ 0.4 m e was extracted from the spacing which reaches 15-20 meV at 30 T. The lowest inter-LL transition 0 → 1 emerges at ∼ 6 -8 T and it is highlighted by red color. The onset of ∼ 6 -8 T can be associated with the size of W1 Fermi pocket. A similar FFT frequency of 7 -7.5 T was extracted from quantum (Shubnikov-de Haas) oscillations and it is in agreement with the values reported in literature[START_REF] Arnold | Chiral Weyl Pockets and Fermi Surface Topology of the Weyl Semimetal TaAs[END_REF][START_REF] Ramshaw | Quantum limit transport and destruction of the Weyl nodes in TaAs[END_REF].

Fig. 4 .

 4 Fig. 4.19. TaAs optics B=0. Parts (a), (b): Temperature dependence of reflectivity of (101) and (001) facets, respectively. Parts (c), (d): Deduced real part of the optical conductivity by Kramers-Kronig relations. The grey arrows indicate three absorption onsets. Inset of panel (c) displays the temperature development of 1/τ for both facets.

Fig. 4 .

 4 Fig. 4.20. Magneto-optics of NbAs. Magneto-reflectivity normalized by the zero-field response R B /R 0 for facets (a) (001) and (b) (112), respectively. White vertical line separates the measurements done on superconducting setup and resistive coil setup.

Fig. 4 .

 4 Fig. 4.21. Detailed analysis of NbAs (001) facet. (a) Simulated magneto-optical response plotted as R b /R 0 . The white arrow highlights the saddle point absorption. The orange dashed line shows the energy of band inversion (difference between valence and conduction band maxima 2 ∆ 2 + γ 2 0 = 40.5 meV). (b) Low energy and low field region of NbAs (001) magneto-reflectivity plotted with higher contrast. (c) Extraction of NbAs (001) features. The down dispersing inter-LL are highlighted by the solid green and red lines extrapolating in zero-field limit to 38 and 68 meV, respectively. The white solid almost horizontal line marks the saddle point transition. Black and purple lines represents the intra-and inter-LL transitions. (d) Scheme of W1 cones distribution in the k-space (purple dots). Red and blue ellipse represents diagonal and off-diagonal component of the Hamiltonian 4.19, respectively.

2 ∆ 2 Tab. 4 . 3 .

 243 + γ 2 0 (meV) ∆ (meV) γ 0 (meV) M (m 0 ) M x (M) M y (M) m x (M) m y (Fitting parameters for NbAs. Corresponding fitting parameters used for NbAs simulations of W1 cones in k-space and for magneto-optical response presented in Fig. 4.21.

Fig. 4 .

 4 Fig. 4.22. Optical response of NbAs (001) at B=0. Parts (a), (c), (e): Temperature dependence of high frequency range of reflectivity, real part of the optical conductivity, and real part of the dielectric function. Parts (b), (d), (f): The corresponding detailed look at low photon energy part at temperature of 10K. The absorptions related to the band inversion excitations are marked by green and red arrows in the real part of optical conductivity.

Fig. A. 1 .Fig. A. 2 .

 12 Fig. A.1. Relative magneto-reflectivity of BaNiS 2 . High field magneto reflectance normalized by zero-field reflectance R B /R 0 for B up to 34 T.

Fig. A. 3 .

 3 Fig. A.3. Waterfall plots of magneto-optics of NbAs 2 -Facets (20-1) and (001). Stackplots of relative magneto-reflectivity spectra, R B /R 0 , for selected values of the magnetic field collected on the (20-1) and (001) facets, which make angles 7 • , and 90 • with the a-axis, respectively. The yellow and red dots show R B /R 0 maxima belonging to transitions in the lower and upper set, respectively. The horizontal and vertical gray bars show positions of the two steps in the onset of interband absorption at 2∆ D and 2∆ F at B = 0.

Fig. A. 4 .

 4 Fig. A.4. Waterfall plots of magneto-optics of NbAs 2 . Waterfall plots of relative magneto-reflectivity spectra of NbAs 2 , R B /R 0 , in the magnetic field applied perpendicular to eight different crystallographic planes: (40 -1), (100), (10 -1), (201), (403), (101), (607) and (20 -3), which make angles 12, 29, 39, 51, 58, 63, 65 and 56 • with the a crystallographic axis, respectively. Vertical gray bars correspond to positions of two steps in the onset of interband excitations at 2∆ D and 2∆ F at B = 0, cf. Fig 3.13. The yellow and red points identify maxima of inter-LL resonances belonging to the upper and lower sets at low magnetic fields, respectively.

  Tab. 4.2. TaAs -extracted parameters from magneto-transport. Summary of parameters extracted from magneto-transport measurement. n and p represents the carrier concentrations, µ n,p their mobilities.

	001)	7.04 × 10 24 5.17 × 10 24	13.1	14.8
	(101)	8.11 × 10 24 2.82 × 10 24	4.5	26.2

Publications

Publications of results presented in this thesis: