M Zinkiewicz

A O Slobodeniuk

T Kazimierczuk

P Kapuściński

K Oreszczuk

M Grzeszczyk

M Bartos

K Nogajewski

K Watanabe

T Taniguchi

C Faugeras

P Kossacki

M Potemski

A Babiński

M R Molas

Neutral and charged dark excitons in monolayer WS 2

published or not. The documents may come

Neutral and charged dark excitons in monolayer WS 2

I. INTRODUCTION

Monolayers (MLs) of semiconducting transition metal dichalcogenides (S-TMDs) MX 2 where M=Mo or W and X=S, Se or Te, are direct band gap semiconductors with the minima (maxima) of conduction (valence) band located at the inequivalent K + and K -points of their hexagonal Brillouin zone (BZ) 1,2 . The strong spin-orbit interaction in the crystal lifts the spin-degeneracy of the bands, which leads, in particular, to the splitting of the valence (∆ v ) and the conduction (∆ c ) bands. While the former splitting is of the order of few hundreds of meV, the latter equals few tens of meV only. Moreover, the ∆ c splitting can be positive or negative. As a result two subgroups of MLs can be distinguished: bright (the excitonic ground state is optically active or bright) and darkish (the excitonic ground state is optically inactive or dark). It is well established that WS 2 and WSe 2 MLs belong to the group of darkish materials 3,4 , while MLs of MoSe 2 and MoTe 2 belong to the family of bright materials 1,3,5 . The assignment of the MoS 2 ML is still under debate as theoretical predictions and experimental results are contradictory 3,5,6 .

The complex electronic structure of S-TMD MLs results in a wide variety of excitonic complexes, which can be formed from carriers at the vicinity of the conduction (CB) and valence band (VB) extrema. Their studies have been largely facilitated by the efforts to improve their optical quality, in particular by encapsultaing MLs of S-TMDs in thin layers of hexagonal BN (hBN). In particular, this improvement allowed to investigate several dark excitonic complexes 5,[7][8][9][10][11][12][13][14][15][16][17][18] . Among them there are spin-or momentum-forbidden dark excitons, which can not recombine optically due to spin or momentum conservation laws. Moreover, these complexes can be neutral and charged. Fig. 1 illustrates schematically the neutral and negatively charged spin-forbidden dark excitons in darkish monolayer (i.e. WS 2 or WSe 2 ). The neutral dark exciton at K + valley is composed of an electron from the lowest-lying level of CB and a hole from the highest-lying level of VB. The negative trion at the K -point is formed similarly by the energetically lowest electron-hole (eh) pair and an extra electron located in the K + valley. It was proposed theoretically and demonstrated experimentally that the spin-forbidden neutral dark excitons exhibit a double (fine) structure comprising so-called grey (X G ) and dark (X D ) complexes. These complexes are characterized by the out-of-plane and zero excitonic dipole momenta 7,12,19 , respectively. Although substantial efforts have been made to study dark excitons in WSe 2 MLs 7,[11][12][13][14][15][16][17][18]20,21 , their properties in WS 2 are still far from complete understanding 11,17,20 . For example, the reported energy difference between the bright and dark excitonic emission is not well determined as it varies significantly (55 meV 20 vs. 46 meV 11 ). This motivates our addressing properties of dark states in WS 2 ML.

In this work, we use polarization resolved photoluminescence spectroscopy with an applied magnetic field to investigate dark excitonic complexes through the magnetic brightening effect in a high quality and naturally n-doped WS 2 ML encapsulated in thin hBN layers. The magnetic field has been applied in different geometries: perpendicular, parallel or at 45 • with respect to the monolayer plane. The emissions from both the neutral and the charged exciton complexes are activated by the in-plane magnetic field component. The double structure of the neutral dark exciton (grey and dark excitons) is observed while the dark trions do not present any fine structure. Our study also shows that the magnetic brightening of the dark excitons and dark trions depends on the carrier concentration.

II. RESULTS

Fig. 2 illustrates the brightening of neutral and charged dark excitons in a monolayer of WS 2 encapsulated in hBN by an in-plane magnetic field. The zero-field PL spectrum is composed of several emission lines. Based on the previous reports, 11,[22][23][24][25] three peaks can be ascribed unquestionably to a bright exciton (X B ) and singlet (T S ) and triplet (T T ) states of negative trions. The application of an in-plane magnetic field B || results in the appearance of three additional lines, labelled X G , X D and T D . This can be appreciated in Fig. 2, which displays the spectrum measured at B || =10 T. The X G and X D peaks correspond to the dark and grey states of the neutral exciton, while the T D peak is related to a dark state of the negative trion. Moreover, the detailed analysis of the X D/G line, shown in the inset to Fig. 2(a), exhibits its fine structure, in line with previous studies on WSe 2 MLs 12,26 . These results can be summarized as follows: (i) the X D/G line is red-shifted by 40 meV from the X B peak; (ii) the energy separation between the X G and X D emissions δ=530 µeV; (iii) the T D resonance is red-shifted by 57 meV from the X B one. Surprisingly, the energy separation between the negatively charged/neutral dark and bright neutral exciton in the studied WS 2 ML, which amounts to 57 meV/40 meV, are very similar to the corresponding values observed in a WSe 2 ML encapsulated in the same dielectric environment (∼57 meV/∼40 meV) 7,[11][12][13][14][15][16]18,20,21 . The obtained value for the fine structure splitting of the neutral darkgrey exciton of about 530 µeV in the studied WS 2 ML is also very similar to that observed in WSe 2 (660 µeV) 7,12 .

Having identified the dark excitons, we now investigate these complexes at higher magnetic fields (up to 30 T) but with unpolarized optical detection. In fact, as can be seen in Fig. 2(b), the application of the in-plane magnetic field B || up to 30 T leads to a strong brightening effect, which is significantly different for the neutral and charged dark excitons. With increasing B || , the intensity of the dark trion increases substantially, while the intensity of the bright exciton and trions stays practically unchanged. As can be seen in Fig. 2(b), the emission of the neutral grey-dark exciton doublet is much smaller as compared to dark trion even at highest magnetic fields (B=30 T). The B || evolution of the intensity of dark excitons is expected to be quadratic I = αB 2 || 3,4,12 . Note that we neglect the zero-field intensity of the grey components of the neutral and charged complexes, as we were not able to resolve it at zero magnetic field. Fig. 3 strates the B 2 || dependence of the relative intensities of the charged and neutral dark excitons in magnetic fields up to 10 T and 30 T, which are accompanied with the quadratic fits. The integrated relative intensities of the T D , X G and X D lines were obtained by fitting the relative spectra defined as (PL B =0 T -PL B=0 T )/I X B (B) using Lorentzian functions. The PL B =0 T and PL B=0 T are correspondingly photoluminescence spectra measured at non-zero and zero magnetic fields, while I X B (B) represents the integrated intensity of the bright neutral exciton as a function of magnetic fields. Note that the division by the I X B (B) parameter allows us to eliminate variation of the signal intensity during measurements, e.g. the measured signal in 30 T setup is affected by the Faraday effect. The fitted α parameters for three analysed excitons observed in both experimental setups are coherent within experimental error. However, as it is shown in Figs 3(a) and 3(b), there is a well pronounced difference between the fitted α parameters for three analyzed excitons, beyond experimental error. The difference between the two neutral dark components: α X D =0.23/T 2 versus α X G =0.15/T 2 , can be explained by the thermal occupation of these two states at T =10 K corresponding to our experimental conditions. The calculated population ratio is e -δ/k B T =0.54, which is in reasonable agreement with the measured ratio α X G /α X D =0.65.

demon-

The brightening rate of the dark trion with respect to dark excitons is much faster with α T D ∼ 24α X G . This large difference may be understood considering the finite free electron concentration of 10 11 cm -2 in the WS 2 ML 27 . The formation of trions and of dark trions is then favored compared to neutral excitons and this is reflected in the much more pronounced brightening effect for T D than for X D/G . Note that in our previous studies devoted to the dark excitonic complexes in the WSe 2 ML 12 , we observed the opposite situation, i.e. the brightening of the X D/G line was significantly larger as compared to the one of the charged complex. This may be understood if the doping level of the WSe 2 ML was close to the neutrality point, which favours the creation of the neutral dark excitons. The obtained results highlight a role of doping in MLs on the brightening of the neutral and charged dark exciton emission in the in-plane magnetic field. The polarization properties of magnetically brightened neutral and charged dark excitons were also analysed (see Fig. 4(a)). The integrated PL intensity for each peak measured at B || =10 T as a function of the linear polarization is shown in Fig. 4(b). The detection angle is measured between the polarizer axis and the direction of the B || field. The X G and X D intensities are expected to follow the I(φ d ) ∼ I 0 sin 2 (φ d + θ) dependence, where φ d represents the detection angle, while I 0 and θ are fitting parameters. Our measurements show that the neutral dark (grey) exciton at B || =10 T is linearly polarized along (perpendicularly to) the direction of the in-plane magnetic field, in line with our previous study of magnetic brightening in a monolayer of WSe 2 12 . In contrast to dark excitons, the emission from dark trions do not show sizable linear polarization.

In order to describe the T D polarization hallmarks, we analyse the observed sine-square intensity profiles of neutral dark excitons presented in Fig. 4(b). The evolution results from two effects: i) the coupling between dark and bright excitons of the same valley by an in-plane magnetic field; and ii) the coupling of dark excitons of opposite valleys by short-range exchange interaction. The first effect allows to observe dark excitons by transferring part of the optical activity from bright to dark excitons through the mixing of the the two spin states of the conduction band. Then, dark excitons from K ± valleys get the ability to decay radiatively by emitting σ ± polarized photons in the direction, perpendicular to the monolayer's plane. The exchange interaction mixes the "valley" dark exciton states into new "grey" and "dark" states 19 . The new states are no more degenerated in energy, are split by the exchange interaction 20 , and are orthogonal to each other. As a result, dark excitons recombine by emitting linearly polarized light with orthogonal polarization. The linear polarization results in the sine-square profile of the PL intensity versus the detection angle, while the orthogonality is responsible for the π/2 phase shift of corresponding curves. The lowest energy dark exciton state is more populated than the grey exciton state and hence emits more photons 12 . The difference between the corresponding intensities becomes significant at low temperatures due to Boltzmann factor exp(-δ/k B T ), as it is observed in our study, see Figs 3 and4. Note that in the absence of the exchange interaction both sine-square curves would be characterised by the same intensity profile and energy. In such case the valley dark excitonic doublet would remain doubly degenerate and the PL intensity of dark exciton line would become angle-independent, since sin 2 (φ d ) + sin 2 (φ d + π/2) = 1. In fact, such a situation is realized in the case of negative dark trions. They also form the valley doublet, which is brightened by in-plane magnetic field. However, since the dark trions from opposite valleys are not influenced by exchange interaction 26 , their emission intensity is not polarized, as it was observed in the experiment (see green points in Fig. 4(b)).

In addition, it is expected that dark trions are characterized by an out-of-plane exciton dipole momentum, which leads to the reported in-plane emission at 55 meV below the X B in WS 2 monolayer 20 .

To determine the dark trion g-factor, it is necessary to apply both an in-plane magnetic field allowing for its direct observation in a photoluminescence experiment and, at the same time, to apply a magnetic field perpendicular to the layer plane to couple to its magnetic moment. This is achieved in our experiment by tilting the layer by 45 • with respect to the magnetic field direction. Fig. 5(a) shows the evolution of the photoluminescence response as a function of the out-of-plane component of the magnetic field (B ⊥ ), in the form of colour-coded map. The T D emission grows due to the in-plane field and additionally splits into components due to the valley Zeeman effect. The extracted emission energies of the negative dark trion are presented in the Fig. 5(b). The energy evolutions (E(B)) in external out-of-plane magnetic fields (B ⊥ ) can be described as E(B) = E 0 ± 1 2 gµ B B ⊥ , where E 0 is the energy of the transition at zero field, g denotes the g-factor of the considered excitonic complex and µ B is the Bohr magneton. The black solid lines in Fig. 5(b) are fits to our experimental data using this equation. We find that the T D g-factor in the studied WS 2 monolayer is on the order of -8.9, which is very similar to the reported g-factors for dark trions in WSe 2 MLs 13,14,18,21 , about two times bigger than the g-factors of the X B , T S and T T lines (for details see Supplementary Material (SM)).

To get more information on the dark trion, we have performed time-resolved measurements at in-plane magnetic field of B || =10 T. Fig. 6 displays the normalized time-resolved traces of the bright singlet (T S ) and triplet (T T ), together with the dark (T D ) charged excitons. Monoexponential fits of these traces show that the decay time of the two bright trions is of 30 ps, in line with previous studies for monolayers of other S-TMD 13,21 . Dark trions are characterized by a much slower decay time, reaching values close to 0.5 ns. The obtained decay times of the bright and dark trions are in good agreement with previously reported values for other S-TMD monolayer 13,21 .

III. METHODS

The studied sample is composed of WS 2 ML encapsulated in hBN flakes and supported by a bare Si substrate. The structure was obtained by two-stage polydimethylsiloxane (PDMS)-based 28 mechanical exfoliation of WS 2 and hBN bulk crystals. A bottom layer of hBN in the hBN/WS 2 /hBN heterostructure was created in the course of a non-deterministic exfoliation. The assembly of the hBN/WS 2 /hBN heterostructure was realized via succesive dry transfers of WS 2 ML and capping hBN flake from PDMS stamps onto the bottom hBN layer.

Low-temperature micro-magneto-PL experiments are performed in the Voigt, Faraday and tilted geometries, i.e. magnetic field oriented parallel, perpendicular, and 45 • with respect to ML's plane. Measurements (spatial resolution ∼2 µm) were carried out with the aid of two systems: a split-coil superconducting magnet and a resistive solenoid producing fields up to 10 T and 30 T using a free-beam-optics arrangement and an optical-fiber-based insert, respectively. The sample was placed on top of a x-y-z piezo-stage kept at T =10 K or T =4.2 K and was excited using a laser diode with 532 nm or 515 nm wavelength (2.33 eV or 2.41 eV photon energy). The emitted light was dispersed with a 0.5 m long monochromator and detected with a charge coupled device (CCD) camera. The linear polarizations of the emissions in the Voigt geometry were analyzed using a set of polarizers and a half wave plate placed directly in front of the spectrometer. In the case of the Faraday and tilted-field configurations, the combination of a quarter wave plate and a linear polarizer placed in the insert were used to analyse the circular polarization of signals (the measurements were performed with a fixed circular polarization, whereas reversing the direction of magnetic field yields the information corresponding to the other polarization component due to time-reversal symmetry). For time-resolved measurements, a femtosecond pulsed laser with excitation at 570 nm (2.18 eV photon energy) from frequency-doubled Ti:Sapphire Coherent Mira-OPO laser system operated at a repetition rate of 76 MHz and a synchroscan Hamamatsu streak camera were used correspondingly for excitation and detection. Note that the excitation power for experiments performed in magnetic fields up to 30 T and with time resolution was adjusted based on the comparison of the measured PL spectrum and the one obtained under excitation of laser with 532 nm in measurement in fields up to 10 T.

IV. SUMMARY

We have presented a photoluminescence investigation of dark exciton complexes in a monolayer of WS 2 encapsulated in hBN. Based on polarization resolved measurements, we have identified both the dark and the grey excitons which are brightened by the in-plane component of the magnetic field. The dark trion is also observed in this magneto-brightening experiment. In contrast to dark excitons, dark trions do not show any fine structure nor linear polarization due to the absence of coupling between dark trions in the two valleys. Time resolved measurements indicate that the decay time of dark trions is close to 0.5 ns, which is more than two orders of magnitude larger than that of bright trions in the same monolayer. This study also indicates that monolayer of WS 2 encapsulated in hBN is naturally n-doped as revealed by the much higher brightening rate of the dark trion with respect to dark excitons, in contrast to previous results obtained in WSe 2 . excitons, i.e. complexes for which recombining an electron and a hole posses the same sign of the spin, are very close to the theoretically predicted value of 41 .

FIG. 1 .

 1 FIG. 1. Schematic illustration of possible configurations for the neutral (blue shade) and the negatively charged dark (green shade) excitons located at the K + and K -valleys, respectively. The grey (orange) curves indicate the spin-up (spin-down) subbands. The electrons (holes) in the conduction (valence) band are represented by blue (white) circles.

FIG. 2 .

 2 FIG. 2. (a) Low-temperature PL spectra measured on a WS2 ML encapsulated in hBN flakes at zero magnetic field and at B || =10 T. Grey curve represents unpolarized detection, while the blue and orange curves correspond to the detection of two linear polarizations aligned parallel and perpendicular to the direction of magnetic field, respectively. The inset shows the relative intensities of the grey and dark exciton emissions defined as (PLB=10 T -PLB=0 T)/PLB=0 T. (b) False-color map of the PL response as a function of B || from zero field to 30 T. The intensity scale is normalized to the T T intensity to remove the Faraday effect affecting the experimental data. White curves superimposed on the map represent the PL spectra normalized to the most intense peaks recorded at selected values of B || .

FIG. 3 .

 3 FIG. 3. The effect of the in-plane magnetic field B || on the integrated relative intensities of the charged and neutral dark excitons measured in magnetic fields up to (a) 10 T and (b) 30 T. The spectra were detected in (a) two linear polarizations for X G and X D emissions (see Fig. 2(a) for details), and otherwise detection was unpolarized. Note that the X D/G intensities in both panels were multiplied by factor 5 for clarity. The solid black curves represent quadratic fits.

FIG. 4 .

 4 FIG. 4. (a) False-color map of the charged (left panel) and neutral (right panel) dark exciton emissions as a function of the detection linear polarization angle measured at B || =10 T. (b) The integrated intensities of the luminescence peaks measured at B || =10 T as a function of the polarization. Solid black curves represent sine fits.

FIG. 5

 5 FIG. 5. a) False-color map of the PL response as a function of out-of-plane component (B ⊥ ) of the applied tilted magnetic field. Note that the positive and negative values of magnetic fields correspond to σ ± polarizations of detection. The intensity scale is normalized to subtract the Faraday effect affecting the experimental data. White dashed superimposed on the investigated transitions are guides to the eyes. (b) Transition energies of the σ +/-(red/blue points) components of the T D line as a function of the out-of-plane magnetic field. The solid lines represent fits according to the equation described in the text.

FIG. 6 .

 6 FIG. 6. Low-temperature normalized time-resolved PL of the T T , T S and T D lines measured on a WS2 ML encapsulated in hBN flakes at in-plane magnetic field B || =10 T.

  FIG. S2.Transition energies of the σ +/-(blue/red points) components of the X B , T S and T T lines as a function of the outof-plane magnetic field. The solid lines represent fits according to Eq. 1.

M. Koperski, M. R. Molas, A. Arora, K. Nogajewski, M. Bartos, J. Wyzula, D. Vaclavkova, P. Kossacki, and M. Potemski,

2D Materials 6, 015001 (2019). 2 A. V. Stier, K. M. McCreary, B. T. Jonker, J. Kono, and S. A. Crooker, Nature Communications 7, 10643 (2016).3 G. Plechinger, P. Nagler, A. Arora, A. Granados del Águila, M. V. Ballottin, T. Frank, P. Steinleitner, M. Gmitra, J. Fabian, P. C. M. Christianen, R. Bratschitsch, C. Schüller, and T. Korn, Nano Letters 16, 7899 (2016). 4 R. Schmidt, A. Arora, G. Plechinger, P. Nagler, A. Granados del Águila, M. V. Ballottin, P. C. M. Christianen, S. Michaelis de Vasconcellos, C. Schüller, T. Korn, and R. Bratschitsch, Phys. Rev. Lett. 117, 077402 (2016). 5 P. Kapuściński, D. Vaclavkova, M. Grzeszczyk, A. O. Slobodeniuk, K. Nogajewski, M. Bartos, K. Watanabe, T. Taniguchi, C. Faugeras, A. Babiński, M. Potemski, and M. R. Molas, arXiv:2005.03522 [cond-mat.mes-hall]. (2020).

ACKNOWLEDGEMENTS

The work has been supported by the the National Science Centre, Poland (grants no. 2017/27/B/ST3/00205, 2017/27/N/ST3/01612 and 2018/31/B/ST3/02111), EU Graphene Flagship project (no. 785219), the ATOM-OPTO project (TEAM programme of the Foundation for Polish Science, co-financed by the EU within the ERD-Fund), the Nano fab facility of the Institut Néel, CNRS UGA, and the LNCMI-CNRS, a member of the European Magnetic Field Laboratory (EMFL). P. K. kindly acknowledges the National Science Centre, Poland (grant no. 2016/23/G/ST3/04114) for financial support for his PhD. M. B. acknowledges the financial support of the Ministry of Education, Youth and Sports of the Czech Republic under the project CEITEC 2020 (Grant No. LQ1601). K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, (grant no. JPMXP0112101001), JSPS KAK-ENHI (grant no. JP20H00354), and the CREST (JP-MJCR15F3), JST.

Supplemental Material: Neutral and charged dark excitons in monolayer WS 2 M. Zinkiewicz, 1 A. O. Slobodeniuk, 2 T. Kazimierczuk, 1 P. Kapuściński, 3,4 K. Oreszczuk, 1 M. Grzeszczyk, 1 M. Bartos, 3,5 K. Nogajewski, 1 K. Watanabe, 6 T. Taniguchi, 7 C. Faugeras, 3 P. Kossacki, 1 M. Potemski, 1,3 A. Babiński, 1 and M. R. Molas 1

S1. G-FACTORS OF BRIGHT EXCITONIC COMPLEXES

To get more information on the properties of the bright excitons emissions in the studied WS 2 monolayer, we performed the magneto-photoluminescence experiment in magnetic fields up to 10 T oriented perpendicular to ML's plane. Fig. S1 illustrates the measured PL spectra as a function of magnetic fields in the form of colour-coded map. Upon application of an out-of-plane magnetic field, the excitonic emissions split into two circularly polarized components due to the excitonic Zeeman effect 1 . Their energies evolutions (E(B)) in external out-of-plane magnetic fields (B ⊥ ) can be described as:

where E 0 is the energy of the transition at zero field, g denotes the g-factor of the considered excitonic complex and µ B is the Bohr magneton. The results of fitting to the experimental results denoted by red and blue points, are presented in Fig. S2 as solid black curves. We found that the g-factors for the bright exciton (X B ) and singlet (T S ) and triplet (T T ) states of negative trions are of about -3.5, -4.0 and -3.9, respectively. The obtained values are in in very close agreement to previous measurements on WS 2 monolayer 1-5 . Summarizing, the g-factors of the bright


Dirk Wulferding

Youngsu Choi

Seung-Hwan Do

Chan Hyeon Lee

Peter Lemmens

Clément Faugeras

Yann Gallais

Kwang-Yong Choi

Magnon bound states vs. anyonic Majorana excitations in the Kitaev honeycomb magnet α-RuCl 3

The pure Kitaev honeycomb model harbors a quantum spin liquid in zero magnetic fields, while applying finite magnetic fields induces a topological spin liquid with non-Abelian anyonic excitations. This topological phase has been much sought after in Kitaev candidate materials, such as α-RuCl 3 . Currently, two competing scenarios are discussed for the intermediate field phase of this compound (B = 7 -9 T) based on experimental as well as theoretical results, involving (i) conventional multiparticle magnetic excitations of integer quantum number and (ii) non-trivial Majorana fermionic excitations of possibly non-Abelian nature with a fractional quantum number. To discriminate between these scenarios a detailed spectroscopic investigation of magnetic excitations over a wide field-temperature phase diagram is essential. Here we present a Raman spectroscopic study revealing magnon-and multimagnon excitations on top of a fractionalized continuum in an antiferromagnetic (B < 7 T) and a fieldpolarized phase (B > 9 T). These conventional excitations are contrasted by lowenergy quasiparticles emerging at intermediate fields, which are closely linked to a continuum of fractionalized excitations. By tracing their temperature evolution, we find fingerprints of the formation of bound states out of Majorana fermionic excitations.

I. INTRODUCTION

The search for Majorana fermions in solid-state systems has led to the discovery of several promising candidate materials for exchange-frustrated Kitaev magnets 1-6 . One of the closest realizations of a Kitaev honeycomb lattice is α-RuCl 3 [START_REF] Plumb | α-RuCl 3 : A spin-orbit assisted Mott insulator on a honeycomb lattice[END_REF][START_REF] Sears | Magnetic order in α-RuCl 3 : A honeycomb-lattice quantum magnet with strong spin-orbit coupling[END_REF] , where the spin Hamiltonian is dominated by Kitaev interaction K. Nevertheless, non-Kitaev interactions, such as Heisenberg (J) and off-diagonal symmetric exchange terms ("Γ-term"), as well as stacking faults in α-RuCl 3 lead to an antiferromagnetically ordered zigzag ground state below T N ≈ 7 K [START_REF] Sears | Magnetic order in α-RuCl 3 : A honeycomb-lattice quantum magnet with strong spin-orbit coupling[END_REF] . The exact values of the Hamiltonian terms have not been pinpointed, yet a general consensus on the minimal model has emerged about a ferromagnetic K ∼ -6 --16 meV, as well as Γ ∼ 1 -7 meV, and J ∼ -1 --2 meV [START_REF] Do | Majorana fermions in the Kitaev quantum spin system α-RuCl 3[END_REF][START_REF] Suzuki | Effective model with strong Kitaev interactions for α-RuCl 3[END_REF] . This set of the magnetic parameters partly obscures the presence of Majorana fermions, but the fermionic quasiparticles are well preserved at high energies and finite temperatures [START_REF] Rousochatzakis | Quantum spin liquid at finite temperature: Proximate dynamics and persistent typicality[END_REF] . Indeed, many independent and complementary ex-perimental techniques have been used to confirm and better understand the emergence of itinerant Majorana fermions and localized gauge fluxes from the fractionalization of spin degrees of freedom [START_REF] Do | Majorana fermions in the Kitaev quantum spin system α-RuCl 3[END_REF][START_REF] Glamazda | Relation between Kitaev magnetism and structure in α-RuCl 3[END_REF][START_REF] Kasahara | Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid[END_REF] .

A promising route in understanding Kitaev physics might be the suppression of long-range magnetic order in a magnetic field, with the possibility of generating an Ising topological quantum spin liquid at a critical field of B c ∼ 6 -7 T [START_REF] Sears | Phase diagram of α-RuCl 3 in an in-plane magnetic field[END_REF][START_REF] Wolter | Field-induced quantum criticality in the Kitaev system α-RuCl 3[END_REF][START_REF] Baek | Evidence for a Field-Induced Quantum Spin Liquid in α-RuCl 3[END_REF] . Higher magnetic fields lead to a trivial spin polarized state. In the intermediate field range of 7-9 T, the magnetic order melts into a quantum disordered phase, in which the half-integer quantized thermal Hall conductance is observed. This remarkable finding may be taken as evidence for a field-induced topological spin liquid with chiral Majorana edge states and the central charge q = ν/2 (Chern number ν = 1) [START_REF] Kasahara | Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid[END_REF] . However, it is less clear whether such a chiral spin liquid state can be stabilized against a relatively large field and non-Kitaev terms in α-RuCl 3 . In the original Kitaev honeycomb model, non-Abelian Majorana excitations are created upon breaking time-reversal symmetry, e.g., through applying a magnetic field [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF] . These composite quasiparticles correspond to bound states of localized fluxes and itinerant Majorana fermions (see the sketch of Fig. 2a) [START_REF] Théveniaut | Bound states of fractionalized excitations in a modulated Kitaev spin liquid[END_REF] . The composite bound states are of neither fermionic nor bosonic character, but instead follow anyonic statistics that acquire an additional phase in the wavefunction upon interchanging particles [START_REF] Stern | Non-Abelian states of matter[END_REF] . For the perturbed Kitaev system bound itinerant Majorana fermions are possible, yet the stability of the topological spin liquid is poorly understood. In particular, a remaining open issue is whether the observed thermal Hall effect is a fingerprint of a non-Abelian phase featuring anyonic Majorana excitations.

There exists another scenario in which the intermediate phase is smoothly connected to the fully polarized phase and the transition through B c involves conventional multiparticle excitations due to anisotropic interactions 20 . To resolve these opposing views, one needs to clarify the nature of quasiparticle excitations emergent in the intermediate-to-high-field phase. While neutron scattering 21 , THz spectroscopy [START_REF] Wang | Magnetic Excitations and Continuum of a Possibly Field-Induced Quantum Spin Liquid in α-RuCl 3[END_REF] , and electron spin resonance [START_REF] Ponomaryov | Unconventional spin dynamics in the honeycomb-lattice material α-RuCl 3 : High field electron spin resonance studies[END_REF] have revealed a significant reconfiguration of the magnetic response through B c , these methods do not suffice for obtaining a complete picture of individual quasiparticles due to limited accessible ranges of experimental parameters (energy, temperature, and field). In this work, we employ Raman spectroscopy capable of sensing magnetic excitations over the sufficiently wide ranges of temperatures T = 2-300 K, fields B = 0-29 T, and energies ω = 1-25 meV (8 -200 cm -1 ). At low fields (B < B c ) and low temperatures (T < T N ) we observe a number of magnetic modes superimposed onto a continuum of fractionalized excitations.

Towards higher fields above 10 T, the magnetic continuum opens progressively a gap and its spectral weight is transferred to well-defined sharp excitations that correspond to magnon bound states, marking the crossover to a field-polarized phase. In the intermediate phase, a weakly bound state emerges. The bound state is formed via a spectral transfer from the fractionalized continuum through an isosbestic point around 8.75 meV, and does not smoothly connect to the magnon bound states in the high-field phase. Our results suggest that this weakly bound state carries Majorana flavor [START_REF] Rousochatzakis | Quantum spin liquid at finite temperature: Proximate dynamics and persistent typicality[END_REF][START_REF] Takikawa | Impact of off-diagonal exchange interactions on the Kitaev spin liquid state of α-RuCl 3[END_REF] and that the intermediate-field phase of α-RuCl 3 is a promising platform for testing anyonic statistics.

II. RESULTS

To elucidate the field-evolution of the magnetic excitation spectrum of α-RuCl 3 we performed Raman scattering experiments on oriented single crystals (see Supplementary Information I for a detailed outline of the scattering geometries). Fig. 1a showcases the representative raw spectra obtained at increasing fields aligned along the crystallographic a axis [B // (100)]. Besides two sharp, intense phonon modes at 14.5 meV and 20.5 meV

[marked E g (1) and E g (2)], we observe several magnetic excitations experiencing pronounced field dependence: At zero field, the magnetic Raman response consists of a broad continuum (C; green shading) and a sharp peak (M1, blue line). The latter M1 excitation at 2.5 meV is assigned to one-magnon scattering arising from a spin flip process through strong spin-orbit coupling and enables us to trace a gap of low-lying excitations at the Γ-point as a function of field. The green-shaded continuum agrees with several previous Raman scattering studies [START_REF] Glamazda | Relation between Kitaev magnetism and structure in α-RuCl 3[END_REF][START_REF] Sandilands | Scattering Continuum and Possible Fractionalized Excitations in α-RuCl 3[END_REF][START_REF] Glamazda | Raman spectroscopic signature of fractionalized excitations in the harmonic-honeycomb iridates β-and γ-Li 2 IrO 3[END_REF] and dominantly comprises Majorana fermionic excitations stemming from a fractionalization of spin degrees of freedom in the Kitaev honeycomb model [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF] . Although we cannot exclude an incoherent multimagnon contribution to the continuum, the exotic temperature evolution of the continuum supports the notion of Majorana fermions [START_REF] Nasu | Fermionic response from fractionalization in an insulating two-dimensional magnet[END_REF] . As detailed in the Supplementary Information III, we find further evidence of Majorana fermionic excitations from their more prominent fermionic character at B = 6.7 T (when zigzag order is suppressed) compared to B = 0 T. As the magnetic field increases above 10 T, C becomes gapped and its spectral width narrows down. This leads to a build-up of spectral weight towards the edge of the gap (solid line in Fig. 1b-d). Noteworthy is that the continuum has a finite intensity even at high fields B > B c , while several sharp and well-defined excitations emerge additionally with increasing fields.

The residual spectral weight of C at sufficiently high fields means that spin excitations are richer than the multiparticle magnons. We recall that recent numerical calculations on the field effect of the Kitaev model uncover a wide Kitaev paramagnetic region, namely, far beyond the critical field and at finite temperature [START_REF] Yoshitake | Majorana-Magnon Crossover by a Magnetic Field in the Kitaev Model: Continuous-Time Quantum Monte Carlo Study[END_REF] . In this light, we ascribe the gapped continuum excitations to fractional quasiparticles pertinent to the Kitaev paramagnetic state. The M1 peak is ubiquitous in all measured fields. The M2' excitation (orange line) is split off from the M1 peak above 12 T, while the higher-energy M3' excitation (red line) appears inside the gapped continuum above 10-14 T. In consideration of the spectral form and energy, the M2' and M3' peaks are assigned to two and three magnon bound states, respectively. In previous experimental field-dependent studies on α-RuCl 3 ranging from inelastic neutron scattering (INS) [START_REF] Banerjee | Excitations in the field-induced quantum spin liquid state of α-RuCl 3[END_REF] , to THz absorption [START_REF] Wang | Magnetic Excitations and Continuum of a Possibly Field-Induced Quantum Spin Liquid in α-RuCl 3[END_REF] , to ESR 23 such well-defined magnetic excitations were reported and interpreted in terms of one-magnon or magnon bound states. Raman scattering probes a spin-conserving process (∆S z = 0), which could unravel new aspects of low-energy excitations since the above methods are insensitive to a singlet sector. Furthermore, while first-order Raman scattering processes are restricted to k ≈ Γ, higher-order scattering processes probe regions of large density of states across the Brillouin zone, which distinguishes Raman scattering from methods like THz-or ESRspectroscopy. The evolution of all magnetic excitations as a function of fields up to 29 T is depicted in the color contour plots of Figs. 1b,1c, and 1d together with the as-measured Raman spectra in Figs. 1e,1f, and 1g for field directions along (100), (010), and (110), respectively. A slight anisotropy in magnetic excitations as a function of field-direction becomes apparent, which is highlighted in the plot in Fig. 1h. In particular, the energy and field ranges of M3' are sensitive to the in-field directions, indicating the presence of nonnegligible in-plane anisotropic terms. Our high-field Raman data evidence the existence of multi-magnon bound states and a gapped continuum that characterizes the spin dynamics of the partially polarized phase. However, we can extract little information on the intermediate phase because the base temperature is restricted above T = 5 K.

In order to study the intermediate phase in detail we switched to a magneto-optical cryostat setup, enabling us to reach a base temperature of T = 2 K in a field range of B = 0-10 T. In this scattering geometry the sample is slightly tilted away from the in-plane field geometry by an angle of 18 • , resulting in an additional small but finite out-of-plane field component. Note that here both magnetic field direction as well as light scattering geometry are slightly different from the high-field setup, which prohibits a strict one-to-one comparison. Nonetheless, a good correspondence is found between the high-field B//(110) data at T ≈ 5 K and the magneto-optical data at T = 9 K (see Supplementary Information I). The field-induced intermediate regime in a Kitaev magnet is a promising platform for realizing a non-Abelian phase [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF] , which possesses chiral Majorana states at the edge and anyonic excitations in the bulk. Given that the half-integer thermal Hall conductance is evidence for the chiral state, a detection of the Majorana bound states will provide its ultimate confirmation. In the extended Kitaev system, these bound states can occur in different channels [START_REF] Théveniaut | Bound states of fractionalized excitations in a modulated Kitaev spin liquid[END_REF] , namely, from either binding itinerant Majorana fermions to localized fluxes (sketched in Fig. 2a), or by binding two itinerant Majorana fermions (sketched in Fig. 2b). With this in mind, we inspect the field-dependence of Raman spectra measured at T = 2 K, shown in Fig. 2c. Compared to the T = 5 K high-field data shown in Fig. 1a, the T = 2 K data show a new sharp structure at 5 meV (M2) in addition to the one-magnon excitation (M1) and the fractionalized continuum (C). As the field increases, the respective modes evolve in a disparate manner. Initially (at B = 0 -4.3 T), the M1 and M2 modes are slightly suppressed, while the continuum C is partially renormalized towards lower energies.

As B c is approached and through 6.7 T, the spectral weight of the continuum is massively redistributed. A new low-energy mode (MB) evolves from the low-field M2 mode with a To analyze the field-induced spectral weight redistribution carefully, we replot phonon-subtracted Raman data taken at T = 2 K in Fig. 3a. With increasing field a distinct transfer of spectral weight from the mid-energy (green-shaded A2) to the low-energy regime (purple-shaded A1) is observed, with an isosbestic point located around ω iso = 8.75 meV, at which the magnetic Raman response is independent of the external field. The systematic field-induced redistribution of spectral weight through the isosbestic frequency suggests an intimate connection between the continuum and the newly formed excitation, and therefore supports the concept that a low-energy Majorana bound state (MB) is formed by a confinement of the high-energy broad continuum of Majorana fermionic excitations. Consistent with the high-field data presented in Fig. 1, we observe a remaining finite continuum of Majorana fermions at 9.5 T, i.e., corresponding to the non-trivial quantum spin liquid phase.

A coexistence of massive Majorana fermions that form a broad, gapped continuum together with a sub-gap MB state is not compatible with the trivial polarized phase that is characterized by the multimagnon bound states. Figure 3b shows the thermal evolution obtained at B = 9.5 T (see Supplementary Information IV for full data set). As the temperature increases, the low-energy mode MB gradually loses in intensity and shifts towards higher energies. Meanwhile, the continuum C slightly gains in intensity. The field-and thermal evolution of magnetic modes is visualized in a contour plot in Fig. 3c, based on fits to the as-measured data. It especially highlights the similar evolution of MB with field and temperature: around 5 meV its spectral weight starts to appear at 4.7 T and grows with increasing field, while it becomes thermally stabilized upon cooling below 12 K at 9.5 T. This suggests that quantum and classical fluctuations play the same role in the confinement-deconfinement crossover due to a proximate Kitaev paramagnetic state at finite temperature [START_REF] Rousochatzakis | Quantum spin liquid at finite temperature: Proximate dynamics and persistent typicality[END_REF] . Fig. 3d plots the energy of the MB mode as a function of field together with the spinwave excitations observed in the zigzag ordered phase as well as in the high field spinpolarized phase. We note a smooth transition from M2 to MB through B c . This weak fielddependence suggests that the MB mode corresponds to an excitation in the singlet sector (∆S z = 0), to which Raman spectroscopy is a natural probe. As the magnon corresponds to a condensation of Majorana fermions, the M2-to-MB mode evolution may be interpreted in terms of a condensation-to-confinement crossover where the multimagnon excitations observed at low field gradually evolve into Majorana bound states. Since the continuum of deconfined Majorana excitations above B c is massively gapped with an onset energy of 4-6 meV (see Figs. 1b-1d, Fig. 2c), the low-energy bound state can be naturally created within the gap due to confinement. Unlike the M2-to-MB mode crossover below B c , the MB mode is not smoothly linked to the M2' bound state for fields above B c . Rather, as the field is lowered through 10 T, the M2' mode merges with the M1 mode, while the MB mode appears. This discontinuous evolution of quasiparticle excitations is contrasted by the rather smooth transition observed in ESR experiments through 10 T [START_REF] Ponomaryov | Unconventional spin dynamics in the honeycomb-lattice material α-RuCl 3 : High field electron spin resonance studies[END_REF] . Recalling the different selection rules for quasiparticle excitations in Raman vs. ESR suggests that excitation MB is of different nature than mode 'B' probed by ESR. Although some discontinuous evolution admittedly may be expected in our data due to the slightly different experimental conditions between the high-field and the magneto-cryostat setups, the temperature difference of ≈ 3 K cannot account for a jump of 1 meV. We also recall that the vanishing thermal Hall conductance around 10 T parallels the disappearance of the MB mode [START_REF] Kasahara | Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid[END_REF] .

Further support of the MB state interpretation comes from the temperature dependence at two different magnetic fields, 6.7 T and 9.5 T (Fig. 3e). We see a clear initial increase in the MB mode energy at both fields as the temperature rises. This is contrasted by conventional (magnon) excitations, which continuously soften with increasing temperature.

For bound states, however, the thermal energy competes with the binding energy [START_REF] Choi | Evidence for Dimer Crystal Melting in the Frustrated Spin-Ladder System BiCu 2 PO 6[END_REF] , until eventually an unbinding takes place. Therefore, a sudden drop in mode energy occurs at around 15 K (at B = 9.5 T), setting the binding energy scale to ≈ 1.3 meV (see Supplementary Information V for details). Correspondingly, in a smaller magnetic field of only 6.7 T an unbinding already occurs around T ≈ 10 K. In contrast, the shoulder of built-up Majorana fermionic excitations gives no clear sign of binding. The thermal evolution of area A1, summarized in Fig. 3f, highlights the gradual melting of the bound state at 9.5 T (dark red triangles) with increasing temperature, while the conventional magnetic excitations at 0 T (blue squares) abruptly vanish above T N . All these observations are consistent with the picture of a quasi-bound state of Majorana fermions that is pulled below the gapped fractionalized continuum by residual interactions of non-Kitaev origin 30 .

III. DISCUSSION

In the unperturbed Kitaev model, Majorana bound states involve flux pairs combined with matter Majorana fermions in a ∆S z = ±1 channel 17 . However, as flux excitations are largely invisible to the Raman scattering process [START_REF] Knolle | Raman Scattering Signatures of Kitaev Spin Liquids in A 2 IrO 3 Iridates with A = Na or Li[END_REF] , the bound states between the flux and matter Majorana fermions barely contribute to the magnetic Raman signal. In the presence of perturbed terms, the bound state between matter Majorana fermions is proliferated (see the cartoon in Fig. 2b) [START_REF] Théveniaut | Bound states of fractionalized excitations in a modulated Kitaev spin liquid[END_REF] . As Raman scattering probes the ∆S z = 0 channel, we conclude that the MB mode largely consists of the matter Majorana bound states. This interpretation is supported by the smooth crossover from the multi-magnon M2 mode to the bound state MB through B c (see Figure 3d). In such a case, α-RuCl 3 as an apt realization of the K -Γ model (see Supplementary Information IV) can host an exotic intermediate-field phase. In relation to this issue, we mention that a numerical study of the J -K -Γ-Γ ′ model shows an extended regime of a chiral spin liquid for the out-of-plane field. As the magnetic field is tilted significantly towards the in-plane direction, the intermediate topological phase vanishes [START_REF] Gordon | Theory of the field-revealed Kitaev spin liquid[END_REF] . This discrepancy raises the challenging question whether the recently reported field-induced phase has a non-Abelian nature and how the in-plane intermediate phase transits to the alleged chiral spin-liquid phase, if the intermediate phase is of topologically trivial nature.

Our finding demonstrates that a non-trivial crossover from the zigzag through the intermediate to the high-field phase involves a confinement and binding of the Majorana fermions, calling for future work on how the observed matter Majorana bound states in an in-plane intermediate field phase of α-RuCl 3 and the non-Abelian phase predicted for out-of-field directions are related.

FIG. 1 .

 1 FIG.1. Field evolution of magnetic excitations in α-RuCl 3 through field-induced phases. a, As-measured Raman spectra at T ≈ 5 K. For H//a α-RuCl 3 passes successively from a zigzag antiferromagnetic-through an intermediate-to a field-polarized-phase with increasing magnetic fields. The color shading denotes the broad continuum (C) on top of well-defined sharp peaks (M1, M2', M3') and phonon modes (E g (1) and E g (2)). b, c, and d, Color contour plots of the Raman scattering intensity evolution with magnetic fields aligned along (100), (010), and (110), respectively. e-g, Their respective sets of raw Raman data. h, Field-dependence of the sharp low-energy magnetic excitations compared for different field directions.

FIG. 2 .

 2 FIG. 2. Magnetic excitations at intermediate magnetic fields. a, The creation of a bound state from itinerant fermions bound to localized fluxes and b, from binding only itinerant Majorana fermions. c, Evolution of Raman data obtained at T = 2 K (open circles) with increasing fields from 0 T -9.5 T. The shaded regions denote the decomposition of the magnetic excitations into well-defined peaks and a continuum of excitations. The solid line is a sum of all excitations. The M1 (blue) and M2 (purple shading) modes at low fields of 0 -4.3 T correspond to spin wave excitations.The excitation MB (dark red) above the critical field of 6.7 T is assigned to a Majorana bound state.

FIG. 3 .

 3 FIG. 3. Spectral weight redistribution and formation of a bound state through B c . a, Raman spectra obtained at T = 2 K with increasing magnetic fields. b, Raman spectra at B = 9.5 T recorded with increasing temperatures. Phonon modes are subtracted in (a) and (b) for better visualization of the magnetic excitations. c, Contour plot of the magnetic Raman scattering intensity as a function of temperature and field. d, Field-dependence of excitations around the intermediate phase (spheres) together with spin-wave excitations at low and high magnetic fields. e, Thermally induced binding-unbinding crossover at B = 6.7 T and 9.5 T. Grey lines are guides to the eyes. f, Thermal melting of the low-temperature magnetic modes at 0 T, 6.7 T, and 9.5 T. Standard deviations in (e)-(f) are indicated by error bars.

K. Kitagawa, et al., A spin-orbital-entangled quantum liquid on a honeycomb lattice. Nature

ACKNOWLEDGMENTS

We acknowledge the support of LNCMI-CNRS, a member of the European Magnetic Field Laboratory (EMFL), as well as important discussions with Natalia Perkins. This work was supported by "Niedersächsisches Vorab" through the "Quantum-and Nano-Metrology (QUANOMET)" initiative within the project NL-4, DFG-Le967-16, and the Excellence Cluster DFG-EXC 2123 Quantum Frontiers. The work at CAU was supported by the National Research Foundation (NRF) of Korea (Grant No. 2017R1A2B3012642).

IV. EXPERIMENTAL DETAILS

High magnetic fields up to 29 T were generated using the resistive magnet M10 at the LNCMI Grenoble. The sample was kept at a temperature T ≈ 5 -10 K and illuminated

with a 515 nm solid state laser (ALS Azur Light Systems) at a laser power P = 0.4 mW and a spot size of 5 µm diameter. Resulting Raman spectra were collected in Voigt geometry for in-plane fields, and in Faraday geometry for out-of-plane geometry, using a Princeton Instruments TriVista spectrometer equipped with a liquid N 2 cooled Pylon CCD.

Temperature-dependent Raman scattering experiments in intermediate fields of B = 0 -9.5 T were carried out in 90 • scattering geometry using a Horiba T64000 triple spectrometer equipped with a Dilor Spectrum One CCD and a Nd:YAG laser emitting at λ = 532 nm (Torus, Laser Quantum). A λ/4-plate was used to generate left-and right-circularly polarized light. The laser power was kept to P = 4 mW with a spot diameter of about 100 µm to minimize heating effects. A base temperature of T base = 2 K was achieved by fully immersing the sample in superfluid He. Measurements at elevated temperatures were carried out in He gas atmosphere. From a comparison between Stokes-and anti-Stokes scattering we estimate the laser heating to be of 3 K outside the superfluid He environment. The sample temperatures are corrected accordingly. In-plane magnetic fields were applied via an Oxford Spectromag split coil system (T = 1.5 K -300 K, B max = 10 T). * Contributed equally to this work.; Corresponding author: dirk.wulferding@tu-bs.de † Contributed equally to this work. ‡ Corresponding author: kchoi@cau.ac.kr