Anatolie Mitioglu

Atsuhiko Miyata

Alessandro Surrente

Michal Baranowski

Zhuo Yang

Keywords: hybrid perovskites, photovoltaics, phase transition, exciton binding energy, effective mass iii perovskites hybrides, photovoltaïque, transition de phase, én- transmission en fonction de la température . . . . B.5.3 Mesure de magnéto-transmission à

Investigation of the excitonic properties of hybrid and fully inorganic perovskite using magneto-spectroscopy

Perovskites form a wide class of materials having the same crystal structure as calcium titanate (CaTiO 3 ), named after the Russian mineralogist Lev Perovski in 1839 [Wenk 2016]. Perovskites are interesting for their flexibility of chemical substitution and have a wide range of industrial applications. For example, BaTiO 3 is a well recognized material for applications in ferroelectrics [Megaw 1945], YBa 2 Cu 3 O 7 opened a new approach to high-temperature superconductivity [Wu 1987], and Ba(Zn 1/3 Ta 2/3 )O 3 , with a high dielectric constant, is widely used as frequency filters in wireless communications [Tamura 1986].

The general chemical formula of a perovskite structure is ABX 3 . The unit cell consists of a large cation A, a small cation B and three anions X. A schematic view of the unit cell of a perovskite structure in the cubic phase is presented in Fig. 1.1(a). The A cation sits at the corner positions, the B cation is located at the body center, and the X anions are located at the center of the cubic face. In a perovskite structure, one B cation and six X anions form an octahedra (BX 6 octahedra), where the B cation is located at the center of the octahedra and X anions are located at the corners around the B cation (6-fold coordination). In fact, the perovskite structure can be described as follows, the BX 6 octahedra forms a three dimensional network by connecting all the corners, while the A cations fill the holes among the octahedra (12-fold coordination), as shown in Fig. 1.1(b).

Tolerance factor

The ideal cubic perovskite structure is quite rare and can be found only in a few compounds, such as CaRbF 3 or SrTiO 3 [START_REF] Johnsson | Crystallography and chemistry of perovskites. Handbook of magnetism and advanced magnetic materials[END_REF]]. In an ideal cubic perovskite structure, the length of the unit cell (a) is related to the ionic radii by:

a = √ 2(r A + r X ) = 2(r B + r X ), (1.1)
where r A , r B and r X are the ionic radii of the A, B cations and the X anion.

For the majority of perovskite compounds, the ionic radii of the A and B cations and the X anion do not fulfill Eq. 1.1. In this case, the BX 6 octahedron are distorted (tilted or rotated), and the symmetry of the crystal is lower than the ideal cubic structure. In order to describe the degree of the distortion, Goldschmidt's tolerance factor t is introduced:

t = r A + r X √ 2(r B + r X ) , (1.2)
Due to the different degrees of distortion, perovskite structures can be loosely classified in the following way: (i) Orthorhombic, with the highest distortion and lowest symmetry, characterized by a tolerance factor t < 0.87. (e.g. CaTiO 3 , GdFeO 3 ) [START_REF] Johnsson | Crystallography and chemistry of perovskites. Handbook of magnetism and advanced magnetic materials[END_REF], Sachdev 2006]. (ii) Tetragonal, with less distortion and a higher tolerance factor being in the range of 0.81 < t < 0.87. (iii) Cubic, which is expected to be stable in the range of 0.89 < t ≤ 1 (e.g. SrTiO 3 , BaTiO 3 ) [Buttner 1992[START_REF] Johnsson | Crystallography and chemistry of perovskites. Handbook of magnetism and advanced magnetic materials[END_REF], and (iv) Hexagonal, which occurs for t > 1 (e.g BaNiO 3 ) [START_REF] Johnsson | Crystallography and chemistry of perovskites. Handbook of magnetism and advanced magnetic materials[END_REF]]. Since the tolerance factor mainly considers the ionic bonds, such a classification of the perovskite structure using this parameter alone is only a rough approximation.

Hybrid perovskite

In recent years, a large family of the hybrid organic-inorganic perovskites have gained considerable attention due to their potential applications, for example lightemitting diodes (LED) [Cho 2015], lasers [Xing 2014] and perhaps most importantly, high efficiency photovoltaic devices [Kojima 2009, Liu 2013].

Hybrid organic-inorganic perovskites are formed when the inorganic A cation is replaced by an organic one. Due to the limited space between the inorganic octahedra, only small organic cations, consisting of three or less C-C or C-N bonds, can fit inside the perovskite structure. The most common organic A cations are Methylammonium (MA: CH 3 NH + 3 ) or Formamidinium (FA: CH(NH 2 ) + 2 ). The B cations are usually Pb 2+ or Sn 2+ , while X anions come from the group VII halides ( Br -, Cl -, I -, or a mixed composition of those).

It has been suggested, that hybrid perovskites have a strong self-assembling character during the crystal formation; (i) the inorganic ions are favorable to form the octahedral network, and (ii) the organic molecules are favorable to occupy the vacancies between the octahedra [Cheng 2010]. This character make this kind of materials easy to synthesize through a variety of fabrication methods, such as sequential deposition [Burschka 2013], solution-processed [Yang 2015b], vapor-assisted deposition [START_REF][END_REF]], and vacuum evaporation techniques [Liu 2013].

Another property of the hybrid perovskites is that it is relatively easy to substitute A, B cations or halide ions by other elements. Using chemical substitution, one can tune the band gap energy over a wide range from 1.1 eV to 3.1 eV [Suarez 2014[START_REF] Hoke | Reversible photo-induced trap formation in mixed-halide hybrid perovskites for photovoltaics[END_REF][START_REF] Zhao | [END_REF], which covers the entire visible, and part of the near infrared range of the spectrum.

In addition to the properties described above, hybrid perovskites are also characterized by their strong broad band absorption, long diffusion length [Dong 2015] and long carrier life time [Stranks 2013]. All these properties make hybrid perovskite very promising as light harvesters for low-cost and high-efficiency solar cells. MAPbI 3 , as a prototype of this family, was first used as an active layer in the solar cell devices in 2009 [Kojima 2009].

Tuning the band gap

The band gap of the perovskites can be easily tuned by chemical substitution. In hybrid perovskite material, the conduction band minimum (CBM) and valence band maximum (VBM) are mainly made up from the p or s orbital of the B cation and X anions. The organic cations mainly contribute to the flat bands in the band structure, which are far from the CBM and VBM [Borriello 2008, Jin 2012]. Thus, the inorganic octahedra cage play an essential role in determining the band gap energy. Although the organic cation does not directly contribute to the band gap, their steric effect (size and symmetry) influences the inorganic octahedra, and therefore influences the band gap [START_REF] Katan | [END_REF], Amat 2014].

Influence of the halide ions

By mixing the composition of the halogen atoms in MAPbX 3 , it has been shown that the bandgap can be tuned to cover the entire visible spectral range [Noh 2013[START_REF] Hoke | Reversible photo-induced trap formation in mixed-halide hybrid perovskites for photovoltaics[END_REF], Huang 2014]. For example, at room temperature, MAPbI 3 has a bandgap E g ≈ 1.6 eV, while MAPbBr 3 has a bandgap E g ≈ 2.2 eV [START_REF] Hoke | Reversible photo-induced trap formation in mixed-halide hybrid perovskites for photovoltaics[END_REF]]. Using a mixture of the I and Br atoms gives a tunable bandgap from 1.6 eV to 2.2 eV, which is shown in Fig. 1.2 [Noh 2013]. Panel (a) shows the UV-visible absorption spectra of MAPb(I 1-x Br x ) 3 for the pure tri-iodide, via mixed iodide and bromide, to pure tri-bromide. With the increasing bromide compositions, the absorption energies of MAPb(I 1-x Br x ) 3 shift toward higher energy. Panel (b) shows photographs of the MAPb(I 1-x Br x ) 3 with 11 different halide compositions. With increasing bromine composition, the color of the MAPb(I 1-x Br x ) 3 changes from a dark brown (MAPbI 3 ) to a light yellow (MAPbBr 3 ), which simply reflects the change of the band gap. By incorporating lighter chlorine ions, it is even possible to tune the band gap to the UV range (E g ≈ 3.1 eV for MAPbCl 3 ) [Suarez 2014]. Similar phenomenon of band gap tuning by changing the halide composition are observed for FA based perovskite compounds [Eperon 2014].

Influence of the B cation

The band gap of MAPbX 3 only varies from 1.6 eV (MAPbI 3 ) to 3.1 eV (MAPbCl 3 ), corresponding to the spectra range from 540 nm to 780 nm. To extend the band gap toward the mid-infrared range, we have to modify the chemical composition of the B cations. MASnI 3 compound is reported to exhibit a band gap around 1.2 eV [Noel 2014]. Several reports suggest that substituting partially Pb 2+ by Sn 2+ in MAPbI 3 can substantially change the band gap [Hao 2014a, Ke 2017[START_REF] Zhao | [END_REF]]. Zhao et al. [START_REF] Zhao | [END_REF]] have reported a band gap tuning for MA(Sn x Pb 1-x )I 3 in the Sn composition range 0 < x < 1. The band gap energy of MA(Sn x Pb 1-x )I 3 as a function of Sn content is shown in Fig. 1.3. With higher Sn content, the band gap shifts toward lower energy. In contrast to the band gap tuning by the halide composition, which exhibits a monotonic change in the band gap, the alloy of B cation in MA(Sn x Pb 1-x )I 3 shows a band gap bowing with a local minimum. The lowest band gap (1.19 eV) is obtained when Sn content reaches 80% [START_REF] Zhao | [END_REF]]. Similar band gap bowing behavior is observed by Hao et al [Hao 2014b]. However, they reported a minimal band gap of MA(Sn x Pb 1-x )I 3 occurs at 50% Sn composition. It is worth mentioning here that the Sn/Pb mixed perovskites are not always stable and sometimes undergo a phase separation. The difference in Sn composition for the minimum bandgap may arise from the way of the sample is prepared and the chemical composition. Various sample preparation procedures have been used. Hao et al. [Hao 2014b] first mixed MAI and MX 2 (M = Sn, Pb) in an aqueous HI/H 3 PO 2 solvent at 373 K, and then slowly cool down the solution to room temperature to form the polycrystalline thin film. Zhao et al. [START_REF] Zhao | [END_REF]] first prepared the pure lead and tin precursor solution. The MA(Pb x Sn 1-x )I 3 perovskite was prepared by mixing the corresponding ratio of pure lead and tin perovskite precursor solutions. In addition to the differences in the thin film preparation procedure, the chemical composition is also slightly different. Zhao et al. dissolved PbI 2 and MACl to form a pure lead precursor solution. However, this inevitably induces Cl -in the MA(Pb x Sn 1-x )I 3 perovskites. On the other hand, Hao et al. used pure I -as the halide anions.

Influence of the organic cation

With different A-cations, hybrid perovskite exhibit a different band gap. For example, at room temperature, the band gap of FAPbI 3 is 1.5 eV [Eperon 2014], while the band gap of MAPbI 3 is 1.64 eV. [START_REF] Pellet | [END_REF]] demonstrated a linear dependence of the band gap with FA compositions in mixed cation MA x FA 1-x PbI 3 . Moreover, the organic cations do not directly influence the band gap. Borriello et al. [Borriello 2008] suggest that the band gap is influenced by the size of the embedded organic cation. They claim that the band gap is very sensitive to any modification of the octahedra. The effective ionic radii of FA is higher than that of MA. The size of the organic cation influences the volume, tilting angle and deformation of the inorganic cage, and therefore, influences the band gap energy [Borriello 2008]. In a more complete study, Amat et al. [Amat 2014] take the exact form of the structural phase and spin-orbital coupling into consideration to investigate the influence of the A-cations on the band gap. The organic cation usually has a non-spherical geometry and rotates constantly in the lattice. Fig. 1.4 shows the schematic representations of MA and FA [Amat 2014]. Compared to the MA molecule, the FA molecule has a higher number of hydrogen bonds. As a consequence, FAPbI 3 exhibits a trigonal/pseudocubic crystal structure at room temperature, while MAPbI 3 exhibits a tetragonal crystal structure [Amat 2014]. The trigonal structure amplifies the ionic character of Pb-I bonds, which enhances the Pb character of the conduction band, and thus increases the effect of spin-orbit coupling, which lower the band gap energy of FAPbI 3 [Amat 2014]

Structural phase transition

As described in Section 1.1.1, perovskite structures are classified into different phases due to the degree of distortion, namely orthorhombic, tetragonal, cubic and hexagonal structures. The classification according to the tolerance factor described in Section 1.1.1 is highly inaccurate for hybrid perovskites [Kieslich 2014, Kieslich 2014]. The tolerance factor was essentially developed for inorganic oxide perovskites, and mainly consider the ionic bonds [Li 2016]. When organic cations are incorporated into the perovskite structure, their polar character leads to an interaction between the organic cations and the inorganic cage, resulting in additional disorder [Amat 2014]. The structural phase of the hybrid perovskite material depends mainly on the temperature induced distortion. At different temperatures, perovskite material exhibit a different lattice constant, degree of distortion, and thus, different structural phases. For example, MAPbX 3 can be in three different structural phases. When the temperature increases, they undergo structural phase transitions (second order) from orthorhombic to the tetragonal phase, and then to the cubic phase. The phase transition temperature varies for different chemical compositions. Table 1.1 summarizes the temperature dependence of the crystal structure of MAPbX 3 .

To understand the behavior of organic cation and inorganic octahedra in different phases, we show in Fig. 1.5 the crystal structure of (a) orthorhombic, (b) tetragonal, and (c) cubic structure of MAPbI 3 [Brivio 2015]. MAPbI 3 perovskite crystals exhibit a orthorhombic phase at low temperature up to 161 K. In the orthorhombic phase, the PbI 6 octahedra are strongly deformed (as shown in Fig. 1.5(a)), which restricts the rotational degrees of freedom of MA [Motta 2015]. In this case, the organic cation is fully ordered. The ordering may be sensitive to the material preparation or cooling rate into this phase [Brivio 2015]. Around 160 K, the orthorhombic structure is transformed to the tetragonal phase. In the tetragonal structure, the MA cations are partially ordered [START_REF] Whitfield | [END_REF]] and start to have the freedom to rotate. The PbI 6 octahedra are less distorted and tilted, as shown in Fig. 1.5(b). Above 330 K, MAPbI 3 exhibits a cubic symmetry. In the cubic phase, the MA cations are fully disordered and free to rotate between the octahedra. NMR measurements have shown that the exact location of the MA cations cannot be determined [Knop 1990]. The structural phase transitions influence the electronic properties of hybrid perovskites. The most straightforward influence is the change of the band gap. Depending on the compound, the phase transition from the orthorhombic to the tetragonal phase can be accompanied by tens of meV decrease of the band gap energy [Yamada 2015, Kunugita 2015]. Another influence of the structural phase transition is the dielectric screening. In the orthorhombic phase, the organic cations are frozen and cannot rotate, while the organic cations start to move and rotate in the tetragonal phase. The movement of the organic cations increases the dielectric screening, which should result in a lower exciton binding energy in the tetragonal phase. This phenomenon was proposed by Even et al [Even 2014b[START_REF] Poglitsch | Dynamic disorder in methylammoniumtrihalogenoplumbates (II) observed by millimeter-wave spectroscopy[END_REF][START_REF] Onoda-Yamamuro | [END_REF], and confirmed by the magneto-optical measurements of Miyata et al. [Miyata 2015].

In polycrystalline perovskite films, it is worth to mention that the orthorhombic/tetragonal phase transition does not happen uniformly over the whole film. It has been reported that part of the polycrystalline film remains in the tetragonal phase at temperatures even below the phase transition. Indeed, rem-nants of the tetragonal phase can be found at temperatures as low as T = 4 K [Wehrenfennig 2014, Galkowski 2017]. In addition, it is also reported that the phase transition temperature strongly depends on the size of the crystalline grains, when the crystalline grain size is below 500 nm [Li 2016].

Morphology

Nowadays, a variety of fabrication methods are employed to prepare the perovskite materials, such as single-step [Kim 2016], sequential deposition [Burschka 2013] or gas assisted techniques [Huang 2014]. This leads to a variety of morphologies for perovskite materials. For the polycrystalline thin films, the crystal grain size can vary from tens of nanometers to microns. With a low temperature supersaturated solution approach, it is even possible to obtain millimeter size single crystals [Dong 2015, Shi 2015]. The morphology plays a very important role in perovskite materials, since it can influence the diffusion length or carrier lifetime and in consequence the efficiency of the solar cells.

Several reports have shown that the crystal grain size play a non-negligible role for the carrier diffusion length. For example, Xiao et al. [Xiao 2014] have shown that the the carrier diffusion length in MAPbI 3 thin film can increase to 1 µm, when the annealing procedure enlarges the crystal grain size. Indeed, Dong et al. show that the carrier diffusion length > 175 µm are obtained in a MAPbI 3 single crystal [Dong 2015].

The morphology also has an influence on the carrier life time. Due to the inhomogeneous character of the thin films, the emission does not occur uniformly over the film. Vorphl et al. [Vorpahl 2015] have demonstrated that the MAPbI 3 (Cl) thin films are composed of dark and bright regions. A confocal PL image of a 3 µm by 3 µm region of MAPbI 3 (Cl) is shown in Fig. 1.6(A). The image clearly demonstrate that bright and dark regions co-exist in the same film. The PL spectra from dark and bright region are shown in Fig. 1.6(B). The PL intensity from a dark region is significantly lower than that of a bright region. Fig. 1.6(C-E) shows the PL decay of the indicated dark and bright region at low, medium and high excitation fluences, respectively. The PL decay curve from the bright region clearly show a single, long decay, while the dark region show a double exponential PL decay curve, composed of a short and a long decays. This is mainly due to the fact that, compared to the bright regions, dark regions have a higher density of deep trap states, resulting in a shorter carrier life time and low PL intensity [Vorpahl 2015]. These results point out that perovskite material have significant nonradiative losses due to the inhomogeneity, which are detrimental for the overall performance of the perovskite devices. Therefore, it is essential to remove these nonradiative pathways to obtain uniform perovskite film with only bright grains.

In addition to the parameters discussed above, it was also suggested, by Grancini et al. [Grancini 2015], that the film morphology has an influence on the Coulomb interaction between the electron and holes. Analyzing the results of temperature dependent time resolved transmission spectra, they claim that the exciton binding energy can vary depending upon the microstructure of the perovskite film. As we will see, the results presented in this thesis clearly demonstrate that this is not the case.

Electronic properties of hybrid perovskite

To further improve the performance of the hybrid perovskite material in solar cells, a deep understanding of the fundamental properties is important. Here, we summarize the known electronic properties of the hybrid perovskite materials.

Band structure

The calculation of the band structure of hybrid perovskites is mainly performed by using density functional theory (DFT) [Even 2012, Even 2013, Even 2014b].

Here, the DFT simulation of MAPbI 3 in the cubic phase, performed by Even et al. [Even 2015], is taken as an example to illustrate the band structure in hybrid perovskites. The cubic phase of MAPbI 3 has a P m3m space group symmetry, and the space group I4/mcm is used to describe the tetragonal phase [START_REF] Whitfield | [END_REF]].

The space group I4/mcm is a subgroup of P m3m, in other words, tetragonal and cubic structure obey a group-subgroup relation [Even 2015]. The ground state optical transition is predicted to be at the R point between the E 1/2g valence band and E 1/2u conduction band states. Several other transitions at the R point are also optical allowed. Among them, F 3/2g → E 1/2u and E 1/2g → F 3/2u are the transition with the lowest energy. One may expect that the photocreated carriers at R point are trapped in F 3/2u and F 3/2g states. However, due to the optical phonon assisted relaxation, the carriers further relax through the path F 3/2u → E 1/2u (holes) and F 3/2g → E 1/2g (electrons) [Even 2014b]. Thus, the hybrid perovskites are characterized by broad band absorption, due to the numerous optical allowed transitions discussed above. From high temperature to low temperature, the MAPbI 3 transforms from cubic structure to tetragonal structure, then to orthorhombic structure. The electronic band structures for the orthorhombic, cubic and tetragonal phase of MAPbI 3 are shown in Fig. 1.9(a)-(c) [Even 2015]. In the orthorhombic and tetragonal phase, the tilts of inorganic octahedra and the reduced symmetry result in an increased band gap. Moreover, the ground state transitions are of a direct type at the Γ-point instead of R-point and M-point. The electronic states from R and M-points in the Brillouin zone of cubic structure are folded back to Γ-point in the Brillouin zone of tetragonal and orthorhombic structure. It is worth mentioning that the spin-orbital coupling effect is not taken into account in Fig. 1.9.

Spin-orbital coupling

In hybrid perovskites, the bandgap is dominated by a strong spin-orbit coupling (SOC) effect, particularly in the conduction band. In the band structure of hybrid perovskites, reported by Papavassiliou et al. [Papavassiliou 1994], the conduction band is constructed mainly from the p-orbitals of the B cations, whereas the valence band arises from the p-orbitals of the halide atom and the s-orbital of the B cations. The hybridization of the p-orbital of the B cations and the halide induces spin-orbit split conduction and valence bands in perovskite materials [START_REF] Katan | [END_REF]]. Due to the SOC effect, the band gap of perovskite materials exhibits a blue shift with the increasing temperature. This is quite counterintuitive, since majority of semiconductor materials shows a red shift with the increasing temperature [START_REF] Varshni | Temperature dependence of the energy gap in semiconductors[END_REF]]. The underlying reason is that, when the temperature increases, the unit cell expand, resulting in a larger B-X distance. As a consequence, the strength of the SOC is reduced, resulting in a smaller splitting and therefore larger band gap [START_REF] Katan | [END_REF]].

The optical absorption in semiconductors is mainly related to the formation of the exciton and transitions involving free carriers. The simulation based on DFT only deal with the band structure, and therefore describes only the transitions of free carriers. In semiconductors, when the spin-orbit coupling effect is combined with the loss of centrosymmetry, it results in a k-dependent spin splitting of the band structure, namely, the Rashba effect. Several theoretical reports have predicted a Rashba effect in the hybrid perovskites [Even 2014b, Brivio 2014[START_REF] Katan | [END_REF]. By combining ab initio molecular dynamics, DFT and relativistic quasi-particle GW theory, McKechnie et al. [McKechnie 2017] investigated the electronic band structure of MAPbI 3 and CsPbI 3 . They claim that the Rashba effect may arise from the local asymmetry induced by the lead halide octahedra distortion.

Rashba effect

The hybrid perovskites are characterized by a large density of defects at the surface or grain boundaries [Shkrob 2014]. In principle, these defects are supposed to act as a carrier trap center, which will significantly reduce the carrier life time. However, hybrid perovskites exhibit an extremely long carrier life time [Stranks 2013], which is quite surprising. The Rashba effect is proposed to be one of the explanations for the long carrier life time in the hybrid perovskites [Kepenekian 2017]. To understand the influence of Rashba effect on the carrier life time, we show in Fig. 1.11 the schematic view of band dispersion with (a) and without (b) the Rashba effect. Originally, the band is doubly degenerate, as shown in Fig. 1.11(a) [Kepenekian 2017]. When the Rashba effect is induced, the degeneracy in momentum is lifted, the original parabolic band split into two branches, as shown in Fig. 1.11(b). When the Rashba effect is observed in the valence band or the conduction band, it result in a direct to indirect band gap transition. The indirect band gap suppresses the momentum conserving carrier recombination, thus enhancing the carrier life time.

To date, the Rashba effect has only been observed in MAPbBr 3 single crystal by Niesner et al. using angle-resolved photo emission spectroscopy (ARPES) [Niesner 2016]. The ARPES technique only probes the surface of the sample and is mainly used to investigate the states in valence band [Dil 2009]. However, the hybrid perovskites exhibit a strong SOC in the conduction band [Even 2014b]. Moreover, to date, no other group has managed to repeat the ARPES observation of the Rashba effect. Therefore, the Rashba effect in perovskite materials is still under debate, and more detailed experimental investigations are needed.

Fully inorganic perovskite

MAPbI 3 , although it has many advantages, has limited practical use due to its unstable character under ambient conditions. For example, it can easily dissociate into PbI 2 and MAI under environmental stress (moisture or temperature) [Wang 2016, Philippe 2015]. FAPbI 3 has a lower band gap than MAPbI 3 , resulting in a higher efficiency of the solar cell devices. However, due to the large ionic radii of FA + , FAPbI 3 , it is favorable to form a photo-inactive phase at room temperature [Eperon 2014]. Although the photo-inactive and photo-active phase transition of FAPbI 3 can be reversed by annealing, the long term stability still limit the application of FAPbI 3 in photovoltaic devices.

To overcome the instability problems of hybrid perovskites, cesium can be incorporated to form mixed cation perovskites, which show better stability and higher efficiency. In this mixed cation perovskites, Cs + plays an essential role due to its smaller size. Before discussing the mixed cation perovskites, we first summarize the electronic properties and stability of CsPbX 3 .

CsPbI 3

Of the fully inorganic perovskites, CsPbI 3 has received most attention due to the fact that cubic CsPbI 3 exhibits a band gap around 1.7 eV, which is very close to the optimal band gap energy for the top cell of tandem solar cell structure [Shah 1999]. Nevertheless, the CsPbI 3 compound also has stability problems, which limit its Figure 1.12: Two morphologies of CsPbI 3 : non-perovskite "yellow" phase (left) and "black" perovskite cubic phase (right). After [Eperon 2015]. practical applications. The ionic radius of the Cs + cation (1.73 Å) is comparable to that of the I -anion, which results in a rather low values of Goldschmidt's tolerance factor. The cubic structure are formed at the temperature much higher than the temperature of operational conditions. That is, in CsPbI 3 it is favorable to form a cubic perovskite structure above 583 K as shown in Fig. 1.12(b). The cubic phase has a black color with band gap around 1.7 eV, thus, it is also called "black" phase. At room temperature, CsPbI 3 compounds reveal a high level of structural distortion, with a non-perovskite, orthorhombic "yellow" phase with band gap around 2.8 eV, as shown in Fig. 1.12 [Eperon 2015]. Under ambient conditions, the black phase of CsPbI 3 transforms to the yellow phase almost immediately [Beal 2016].

Band structure of CsPbI 3

The density of states of CsPbI 3 shows that the origin of its band gap is largely the same as MAPbI 3 . That is, the p-orbitals of I -forms the valence band maximum, and the p-orbitals of Pb 2+ forms the conduction band minimum [Brgoch 2014]. The band structure of the yellow and black phases of CsPbI 3 are presented in Fig. 1.13(a) and (b) [Brgoch 2014]. The red solid and the black dashed lines represent the band structure with and without the SOC effect. It is clear that SOC significantly reduces the band gap of CsPbI 3 , as for MAPbI 3 . In the yellow phase of CsPbI 3 , the PbI 6 octahedra network shows a low degree of connectivity (see Fig. 1.12(a)), leading to an extremely flat band, as shown in Fig. 1.13(a) [Brgoch 2014]. On the other hand, the band structure of the black phase CsPbI 3 (see Fig. 1.13) is similar to that of cubic MAPbI 3 . The direct band gap is related to the R point. This similarity is due to the fact that the band gap of APbI 3 is comprised only of the states from the PbI 6 octahedra. Moreover, the black phase CsPbI 3 shows a higher band dispersion compared to the yellow phase CsPbI 3 , resulting in a lower band gap [Brgoch 2014]. As discussed above, only the lower band gap of CsPbI 3 (black phase) is suitable for photovoltaic devices. The main challenge of CsPbI 3 for applications is to maintain the black cubic perovskite phase at room temperature. Recent reports suggests that, with a large surface-to-volume ratio, the CsPbI 3 nanocrystals can remain in the black cubic phase under ambient conditions [Swarnkar 2016, Lai 2017, Waleed 2017]. Swarnkar et al. [Swarnkar 2016] have fabricated the CsPbI 3 quantum dots with size varies from 3.4 nm to 12.5 nm. Due to the quantum confinement effect, the band gap shifts toward high energy with decreasing crystal size. Notably, the CsPbI 3 quantum dots with ≈9 nm size remain in the cubic phase for more than 60 days [Swarnkar 2016]. Moreover, Waleed et al. andLai et al. [Lai 2017, Waleed 2017] demonstrated that CsPbI 3 nanowires are also able to maintain the black cubic phase under ambient condition. The underlying reason for maintaining the black cubic phase of CsPbI 3 nanocrystals is still under debate. One possibility is that nanocrystals have a larger surface energy due to the high surface-volume ratio. This surface energy com-pensates the required energy difference when going from the yellow to the black phase [START_REF][END_REF]]. Using CsPbI 3 quantum dots films as the photoative material, Swarnkar et al. fabricated a stable photovoltaic device with a photon conversion efficiency of 10.7% [Swarnkar 2016].

CsPbBr 3

CsPbBr 3 remains in a cubic phase at temperatures above 403 K. When the temperature is lower than 403 K, it transforms to a tetragonal phase before transforming to an orthorhombic phase at 361 K. However, all three morphologies of CsPbBr 3 have a perovskite structure with a similar band gap in the range of 2.3-2.4 eV [Kulbak 2015]. MAPbBr 3 have a similar band gap as CsPbBr 3 (E g of MAPbBr 3 : 2.32 eV). By using a variety of characterization methods, Kulbak et al. [Kulbak 2015] have shown that CsPbBr 3 shows a high thermal stability of the perovskite structure and enhanced device life time. Despite these advantages, CsPbBr 3 is limited for the photovoltaic applications due to its large band gap.

Although several approaches have been investigated to enhance the phase stability for CsPbX 3 compounds, the long term stability still remains a problem for the photo-active phase of CsPbX 3 compounds. Moreover, the PCE of CsPbX 3 is relatively low, compare to the hybrid perovskites (to date, 11.8% with CsPbI 2 Br films [Chen 2017]). Thus, it is desirable to investigate a more stable perovskite material. In next section, we summarize the recent development on the mixed cation perovskite material with a higher efficiency, which exhibit higher efficiency and stability than their pure A cation counterpart.

Evolution of the mixed cation perovskite

The Goldschmidt tolerance factor is an empirical index for predicting stable crystal structures of the perovskite materials [Li 2015, Correa-Baena 2017]. We recall that a value of the tolerance factor between 0.8 and 1.0 is favorable for the cubic perovskite. Fig. 1.14 shows the correlation between the tolerance factor and the A cation of APbI 3 perovskite materials. The inset images show the cation structures. Only APbI 3 with Cs, MA, FA cations have a tolerance factor between 0.8 and 1.0, and can form a cubic structure. The size of Na, K, Rb cation is too small, and the imidazolium (IA), ethylamine (EA), guanidinium (GA) are too large for APbI 3 to form a cubic structure. Although CsPbI 3 and FAPbI 3 can form a cubic structure (cubic FAPbI 3 above 333 K [Eperon 2014], cubic CsPbI 3 above 583 K [Eperon 2015]), they favor the formation of a photoinactive polymorph at room temperature, which limit the their application for the photovoltaic devices. MAPbI 3 degrades easily in the presence of moisture or at high temperature. In general, even when these materials are incorporated into a full solar cell architecture, they degrade within days to weeks [Leijtens 2015].

As discussed above, the pure perovskite compounds suitable for PV applications exhibit a variety of instability problems. It has been suggested that sub-Figure 1.14: Tolerance factor of APbI 3 perovskites as a function of A cations. The perovskites with a tolerance factor between 0.8 and 1.0 show a photo-active cubic phase (solid circles). The cation structures are depicted in the inset images. After [Correa- Baena 2017] stituting a small amount of I -by Br -is highly beneficial for the stability of mixed halide perovskites [Jacobsson 2016]. However, when large amount of Br - (about 50%) are incorporated in the mixed halide perovskites, it results in a partial phase segregation, leading to the formation of a small amount of an iodine rich phase [Correa-Baena 2017]. This iodine rich phase has a lower band gap and acts as an efficient recombination center, which is detrimental for the device performance [Jacobsson 2016].

Since solid-state alloying is a common tool for tailoring the optical absorption and chemical stability of perovskite material, one way to solve these stability problems is to mix FA based perovskite compounds with some smaller A-cations (MA, Cs or Rb) to reduce the tolerance factor.

Mixed cation compounds in single junction solar cells

Since different solar cell structure have different requirements for the band gaps, we divide the discussion of the evolution of mixed cation perovskite material into two parts, according to the solar cell structure. In this section, we discuss the recent achievement (stability, efficient et al.) using mixed cation perovskites in the single junction solar cells.

(FAPbI 3 ) 1-x (MAPbBr 3 ) x . It has been reported that mixing FAPbI 3 with MAPbBr 3 can enhance the stability of the photo-active perovskite phase of FAPbI 3 . The PCE and structural stability of (FAPbI 3 ) 1-x (MAPbBr 3 ) x has been investigated with the ratio x ranging from 0 to 0.3 [Jeon 2015]. In this work, (FAPbI 3 ) 0.87 (MAPbBr 3 ) 0.13 , with a PCE of 18%, was reported to be the optimal composition for the stabilization of the photo-active perovskite phase [Jeon 2015, Baena 2015]. Moreover, (FAPbI 3 ) 1-x (MAPbBr 3 ) x compounds exhibit improved crystal morphology comparing to FAPbI 3 , revealed in smoother surface with reduced voids between the grain boundaries [Jeon 2015] .

Cs 0.05 (FA 0.83 MA 0.17 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 . Although the perovskite materials with MA/FA mixed cation exhibit a higher stability than FAPbX 3 in the photoative phase, the presence of MA does not prevent this MA/FA mixed cation material from turning into the yellow phase occasionally [START_REF] Pellet | [END_REF]]. Saliba et al. have started to investigate triple Cs/MA/FA cation mixture, where Cs is used to improve the stability of MA/FA perovskite compounds [Saliba 2016b]. They found that a small amount of Cs is sufficient to effectively suppress the yellow phase and induces highly uniform perovskite grains. Cs x (FA 0.83 MA 0.17 ) 1-x Pb(I 0.83 Br 0.17 ) 3 with cesium content at x = 0.05 is proposed to be the optimal composition. The perovskite solar cells using this compounds have a PCE as high as 21.1% and an output at 18% under operational condition after 250 hours.

RbCsMAFA compounds. The ionic radii of Rb + is slightly smaller than Cs + , resulting in a lower tolerance factor for RbPbI 3 than CsPbI 3 . The RbPbI 3 cannot form a black perovskite phase at any temperature [Saliba 2016a]. However, Saliba et al. has demonstrated that the incorporation of Rb cations in the triple cation perovskite can significantly improve the solar cell performance [Saliba 2016a]. The RbCsMAFA compound based devices have a PCE up to 21.6% on small areas (average value: 20.2%) [Saliba 2016a]. By adjusting the halide composition, the RbCsMAFA compound based devices can reach a PCE of 22.1% (highest PCE to date) [Yang 2017a]. In addition, compounds incorporating Rb also exhibit better stability. After 500 hours at 358 K under the continuous illumination, the PCE of the devices only decreases by 5%.

Mixed cation compounds in Si/perovskite tandem solar cells

Due to the Shockley-Queisser limit, traditional single junction solar cells have a maximum theoretical efficiency of 30% [Shockley 1961]. To overcome this limitation, multi junction solar cells are used. Theoretically, an infinite number of junctions with different band gaps would have a limiting efficiency of 86.8% under highly concentrated sunlight [Green 2001]. Recently, the concept of perovskite based tandem solar cells was proposed to fully exploit the excellent absorption properties of perovskite materials [Bailie 2015, Sivaram 2015]. The tandem architecture is a two junction solar cell, which is composed of a rear subcell, and a front subcell. A schematic view of tandem solar cell architecture is represented in Fig. 1.15. The rear subcell has a narrow band gap, to absorb the low energy range of the solar spectrum, while front subcell has a wide band gap, to absorb the high energy range of the solar spectrum.

The perovskite based tandem solar cells are mainly divided into two types, namely, Si/perovskite and perovskite/perovskite. In this section, we first discuss the Si/perovskite tandem solar cell. In this kind of tandem solar cell, the front subcell is made of perovskite material, while the rear subcell is made by silicon with a band gap of 1.1 eV [Sivaram 2015]. The optimal band gap energy of front subcell is proposed to be 1.75 eV [Shah 1999].

It is possible to tune the band gap by changing the halide composition of FAPb(I 1-x Br x ) 3 or MAPb(I 1-x Br x ) 3 . FAPb(I 1-x Br x ) 3 compounds have a problem with a photo-inactive phase, while, for MAPb(I 1-x Br x ) 3 , soaking it with light induces a halide segregation within the perovskite [START_REF] Hoke | Reversible photo-induced trap formation in mixed-halide hybrid perovskites for photovoltaics[END_REF]]. To fulfill the requirements of band gap tuning and stability, Cs/FA based perovskite compounds have been developed [START_REF] Mcmeekin | [END_REF], Beal 2016, Lee 2015]. [START_REF] Mcmeekin | [END_REF]] have reported a mixed cation perovskite with composition of CsFAPb(I x Br 1-x ) 3 . They have shown that partially substituting the FA cation with Cs leads to the result that the phase instability was entirely eliminated in the iodide to bromide compositional range. Moreover, by tuning the composition of alloyed Cs and Br, it is possible to obtain a perovskite material with an optical band gap around 1.74 eV. The optimal composition is reported to be Cs 0.17 FA 0.83 Pb(I 0.6 Br 0.4 ) 3 [START_REF] Mcmeekin | [END_REF]]. Based on this building block, Bush et al. [Bush 2017] reported Si/perovskite tandems with a PCE up to 23.6% by optimizing the interfaces and the band gap of the top cell.

Mixed cation compounds in perovskite/perovskite tandem solar cells

As mentioned above, another strategy for the perovskite tandem solar cells is to use perovskite/perovskite structure. As the name suggests, for the perovskite/perovskite tandem solar cells, both the front subcell and rear subcell are made of perovskite material. Eperon et al. [Eperon 2016] has fabricated monolithic tandem solar cells with two mixed cation perovskites. The SEM of the monolithic perovskite/perovskite tandem solar cell is shown in Fig. 1.16. In this structure, the top cell is made of FA 0.75 Cs 0.25 Sn 0.5 Pb 0.5 I 3 with a band gap around 1.2 eV,

Absorption in semiconductors

In this thesis, we mainly show and discuss the results of magneto-absorption measurement on different perovskite materials. The optical absorption in semiconductors are mainly related to the formation of the excitons and the transition of free carriers. In this section, we introduce a short summary of the main features of excitons. Then, we discuss the influence of a magnetic field on the free carriers and the free excitons in the perovskite materials.

Excitons in semiconductors

In a semiconductor material, the absorption of a photon creates an electron in the conduction band and a hole in the valence band. With opposite charge, the electron and hole attract each other via the coulomb interaction, and form a quasi particle, known as an exciton. Excitons were predicted in 1931 by Yakov Frenkel [Frenkel 1931] and observed for the first time in 1950 by Hayashi and Katsuki in the absorption spectra of CuO 2 [Hayashi 1950].

Excitons are well described by the hydrogen atom-like model. A hydrogen atom is composed of a proton with positive charge and an electron with negative charge, which are used as an analogy for the hole and the electron in the exciton. The energy levels in hydrogen atom are described by,

E n = - -me 4 2 2 n 2 = - R y n 2 , (1.3)
where R y = me 4 /2 2 = 13.6 eV is the Rydberg constant. Here, n is the principle quantum number. R y indicates the ionization energy of the ground state (n = 1) of the hydrogen atom, which is also called the binding energy. The principle quantum number is related to the radial quantum number k and azimuthal quantum number l by,

n = k + l + 1, (1.4)
For each principle quantum number n, the allowed values of l are l = 0, 1, ..., n -2, n-1,. The states with l = 0, 1, 2, 3, 4, ... are commonly referred as s, p, d, f, g, h, ... states. For instance, 1s denotes the state (n = 1, l = 0); 2s and 2p denote the l = 0 and l = 1 states at n = 2; 3s, 3p and 3d denote the l = 0, 1, 2 states at n = 3, and so on. One of the differences between a hydrogen atom and an exciton is the dielectric screening. For the hydrogen atom, the relative dielectric constant is r = 1. However, for the excitons in a semiconductor, the Coulomb interaction between the electron and hole pair is screened by the presence of the mobile charges in the crystal, thus the relative dielectric constant r is larger than 1. Another differences is the reduced mass. The mass of the proton in hydrogen atoms is much higher than the mass of electron. In this case, the reduced mass of hydrogen atom is similar to the mass of a free electron. However, as the hole and electron effective masses are both smaller than the free carrier mass, the reduced mass of exciton µ = (1/m * e + 1/m * h ) -1 is much smaller than the mass of an electron. These two differences result in a lower binding energy and larger Bohr radius for the exciton in comparison to the hydrogen atom. In this case, R * y = µe 4 /2 2 2 r is the energy difference of exciton between ground states (n = 1) and the continuum states (n = ∞, free carriers), which is the effective binding energy of the exciton. Thus, the energy levels in an exciton are given by,

E X n = - R * y n 2 , (1.5)
In perovskites, the excitons are Wannier-Mott excitons with the electron and hole separated over many unit cells, which can move freely through the crystal. In Fig. 1.17(a), we show a schematic view of the free exciton in a two-dimensional crystal lattice. The distance between the electron and hole is several times larger than the crystal lattice constant, and is given by

r = n 2 a * , (1.6)
where a * = 2 r /µe 2 is the effective Bohr radius, which denotes the distance between electron and hole in the ground state (n = 1). In Fig. 1.17(b), we show schematically the energy states of the exciton.

In addition to the Wannier-Mott exciton, there is another kind of excitons called Frenkel excitons. For a Frenkel exciton, the Bohr radius is approximately restricted to a single unit cell. These excitons mainly occur in molecular crystals and are localized at a specific atom or molecule [Pelant 2012].

Free carriers in a magnetic field

Let us consider a free carrier moving in an external magnetic field. Due to the magnetic field, a free carrier experiences a Lorentz force. The Peierl's substitution p -→ p -q  can be used to express the Lorentz force acting on the carrier. The Hamiltonian of the free carrier moving in a magnetic field can then be written as:

Ĥ = 1 2m (p -q Â) 2 , (1.7)
where q and m are the charge and mass of the free carrier, p = -i ∇ is the momentum operator, while  is the magnetic vector potential related to the magnetic field B = ∇ × Â. Assuming the magnetic field is applied along the z-axis, B = (0, 0, B), we choose the Landau gauge:

 =   0 Bx 0   , (1.8)
Then, the resulting Hamiltonian is expressed as:

Ĥ = p2 x 2m + 1 2m (p y -qBx) 2 + p2 z 2m , (1.9)
Considering that Ĥ is independent of the y and z coordinates component of the position operator, it commutes with the py and pz operators. In other words, py and pz can be replaced by their eigenvalues, resulting in:

Ĥ = p2 x 2m + ( k y -qBx) 2 2m + 2 k 2 z 2m , (1.10)
Subsequently, the Hamiltonian can be written in the form describing a harmonic oscillator along the x-axis and a free motion along the z-axis:

Ĥ = p2 x 2m + mω c 2 (x - k y mω c ) 2 + 2 k 2 z 2m , (1.11)
Here, the cyclotron frequency of the free carrier, ω c = qB/m, is introduced to simplify the expression. By solving the eigenvalue problem, we obtain:

E N = ω c N + 1 2 + 2 k 2 z 2m , (1.12)
where N = 0, 1, 2, 3... is the orbital quantum number. ω c (N + 1 2 ) presents the discrete energy levels in two dimensional plane (x -y plane), while 2 k 2 z /2m results from the kinetic energy dispersion along z-axis. The combination of these two leads to a density of states with van Hove singularities (the density of states diverges) at energies:

E N = ω c N + 1 2 , (1.13)
where the states E N are called Landau levels.

The movement of the carrier in semiconductors, in the presence of the magnetic field, can be described by the same formula as Eq. 1.13. The Landau levels for the holes in the valence band and the electrons in the conduction band in a semiconductor can be calculated based on Eq. 1.13, replacing m by the effective mass of the hole m * h and electron m * e , respectively. A schematic view of the Landau levels for N 4 in the conduction and valence bands is represented in Fig. 1.18(a). In this work, we measure the interband transitions of the electron from the Landau levels in the valence band to the conduction band, as marked by the orange arrows in Fig. 1.18(a). For perovskite material, the optical selection rule for dipole allowed transitions is ∆N = 0 [Even 2012]. Using equation 1.13, the calculated energy of the interband transition is given by,

∆E N = E g + E e N + E h N = E g + ( qB m * e + qB m * h )(N + 1 2 ) = E g + ω r c (N + 1 2 ) (1.14)
where ω r c = qB/µ with 1/µ = 1/m * h + 1/m * e , here µ is nothing other than the usual reduced exciton mass. Fig. 1.18(b) shows the energy of the Landau level interband transition as a function magnetic field. Compared to the Hamiltonian of the free carriers in a magnetic field, the Hamiltonian of the free excitons in magnetic field has an additional term corresponding to the Coulomb potential:

Free excitons in a magnetic field

Ĥ = 1 2µ (p 2 -e Â) 2 - e 2 r r , (1.15)
where r is the distance between the electron and hole. Here, by choosing the cylindrical polar coordinates (ρ, φ, z) and the cylindrical gauge  = 1 2 ( B × r), the magnetic field is along the z direction and r = (ρ 2 + z 2 ) 1/2 . Then the Hamiltonian can be written as: .16) where γ = ω c /2R * y is a dimensionless parameter, indicating the ratio between cyclotron frequency ω c and the effective binding energy R * y . In 1986, Makado and McGill calculated the energy eigenvalues E n (γ) using a numerical solution for values of γ in the range 0 to 10 and n up to 4 [Makado 1986]. E n (γ) are the energies of the excitonic states, given as the energy shifts with respect to the band gap energy E g , in the effective Rydberg units.

Ĥ = -∇ 2 - 2 r -iγ ∂ ∂φ + 1 4 γ 2 ρ 2 , ( 1
In Fig. 1.19, we present the calculated excitonic transition energies E n (B) as a function of magnetic field using the solutions of Makado and McGill. The 1s, 2s, 2p, 3d and 4f excitonic state energies are calculated using R * = 16 meV and m * = 0.1 m 0 with the band gap E g = 0 meV. The inset of Fig. 1.19 shows an expanded view of the excitonic transition energies at low magnetic field. It is worth to mention that, at zero magnetic field, the energy between the 1s transition and the 2s transition is three quarters of the binding energy. This follows from Eq. 1.5, This chapter presents the experimental techniques used in this thesis. A brief introduction of the pulsed magnetic field generation in LNCMI-T is followed by a detailed description of the magneto transmission and reflectivity techniques.

E X 2s -E X 1s = - R * y 4 + R * y = 3 4 R * y , (1.17

Magnetic field generation

Currently, the high magnetic field generation techniques for research can be divided into two categories, steady state (typical duration of several hours) and pulsed magnetic field techniques (typical duration of 1 s). Steady state magnetic field can be generated by immersing a superconducting coil in liquid helium, which maintains the temperature of the superconducting coil blow the critical temperature T c . In commercially available systems, the value of the magnetic field in the superconducting coil is limited to 22 T by the critical current of the superconducting wire. Further increase of the magnetic field can be achieved using resistive coils. In this case, magnetic fields of 38 T can be reached today, limited by the thermal (cooling) and mechanical constraints. A hybrid magnet, a combination of an outer superconducting and an inner resistive coil, generates a magnetic field beyond the limitation of a purely resistive coil, up to 45 T [Miller 1994].

To reach magnetic fields above 45 T, we have to turn to pulsed magnetic field techniques. The principle of the pulse magnetic field is relatively simple. A capacitor bank is charged to a certain voltage and then discharged into the resistive coil. The electrical energy is transfered from the capacitors to the coils, which creates a current pulse in the resistive coil. The maximum magnetic field of the pulse proportional to the voltage to which the capacitor bank was charged.

The Laboratoire National des Champs Magnétiques Intenses-Toulouse (LNCMI-T) is one of the world leading laboratories, which provides up to 90 T non-destructive long pulse ( 1 s) and up to 150 T semi-destructive, ultrashort ( 10 µs), pulsed magnetic field. LNCMI-T serves as a user facility providing open access to researchers from all over the world. During this thesis, all the spectroscopy measurements in high magnetic field have been performed at LNCMI-T.

Generator

The energy required to energize the resistive coils is stored in three different capacitor banks (14 MJ, 6 MJ, 1.15 MJ). All capacitor banks can be charged up to 24 kV. Fig. 2.1(a) shows a panoramic view of the 14 MJ capacitor bank. It consists of 14 capacitor units. Each unit stores energy up to 1 MJ, with total capacitance of 48 mF. The 6 MJ capacitor bank consists of two 3 MJ units, with total capacitance of 21 mF. In addition, there is a 1.15 MJ mobile capacitor bank with a capacitance of 4 mF. These three capacitor banks can be combined with different resistive coils, resulting in a different duration and value of the magnetic field pulse. The resistive coils are designed, and manufactured in house in the LNCMI-T technical workshop. In order to sustain the high magnetic field pressure originating from the Lorentz force and to minimize coil heating, the material of the coil has to fulfill three important condition, (i) high intrinsic strength, (ii) high ductility and (iii) high electrical conductivity. However, high strength conductors usually have a low ductility and a reduced electrical conductivity. To overcome this problem, special reinforcement of the copper stainless steel-(CuSS) wires with Zylon is used. The CuSS wires are used to wind the coil with approximately 1000 turns, as shown in Fig. 2.1(b). Afterwards, additional protection and contacts are made on the winding wires to form a complete resistive coil. A complete 70 T coil is shown in Fig. 2

Resistive coils

.1(c).

To avoid overheating during the pulse of the magnetic field, the resistive coils are immersed in liquid nitrogen in a specially designed nitrogen cryostat as presented in Fig. 2.1(d). Cooling the coil also reduces the resistance of the metallic copper wires. During each pulse of the magnetic field, the temperature of the resistive coil increases due to the Joule effect. Immersing the resistive coils in liquid nitrogen prevents destructive explosion of the coil due to overheating. The time required to cool the coils between magnetic field pulses varies from several minutes to several hours depending on the strength of magnetic field and the length of the pulse. LNCMI-T has developed a fast cooling channel method, which reduces the cooling time by a factor of 3 [Frings 2008].

Different combination of the resistive coil with the different generators result in a duration of the magnetic field pulses varying from a few tens to several hun- dreds of milliseconds, accompanied with the maximum magnetic field from 60 T to 90 T. Fig. 2.2 shows the magnetic field as a function of time obtained by different combination of the resistive coil and the generator. The inner bore diameter of the resistive coil and the energies required for each magnetic field pulse are marked next to each curve. For example, the inner bore diameter of the 70 T magnet is 13 mm (See orange solid line in Fig. 2.2). The typical duration of a 70 T pulse is 150 ms if the 6 MJ capacitor bank is used for charging the coil. Magnets with a maximum magnetic field below 80 T are single coil systems, while the 90 T magnet is a dual-coil system. The outer magnet, energized by 1.15 MJ mobile capacitor bank, generates a 34 T long background pulse, while the inner magnet combined with 14 MJ capacitor bank generates a short 62 T pulse [Béard 2013]. Synchronizing the maximum field of the two pulses creates a magnetic field of 90.2 T with a duration of 9.2 ms (see the blue solid line in Fig. 2.2). In this thesis, all the magneto-optical measurement have been performed using a standard 70 T magnet with fast cooling channels.

Cryogenics

For the pulsed magnetic field measurement, the liquid helium cryostat is placed in the magnet bore, as schematically shown in Fig. 2.3(a). Due to the high magnetic field environment and the limited space in the magnet bore, special helium cryostats are required. As presented in Fig. 2.3(a), the liquid helium cryostat consists of top and bottom baths connected by a capillary and a needle valve. The top bath is used to store liquid helium transfered from a helium dewar. The bottom bath has a long extension (tail) which is inserted directly into the magnet bore. The sample holder is placed inside the tail. A resistive heater made from manganin wires is placed around the tail. It allows to control the temperature from 4.2 K (liquid helium) to 300 K by adjusting the input power of the heater [START_REF] Lncmp-Team | The LNCMP-team. The LNCMP: a pulsed-field user-facility in Toulouse[END_REF]. According to the diameter of the magnet bore, three type of helium cryostats are used. The main difference between them is the inner diameter of the tail. Table 2.1 shows the diameter of the magnet bore and the corresponding inner diameter of the tail for three different magnets. Photo of a typical experimental setup is presented in Fig. 2.3(b) One of the experimental techniques used in this thesis is magneto-transmission, which is suitable to investigate thin samples on a transparent substrate. We have measured the spectra of the transmitted light through the sample, which is placed in the center of the field. The general idea of the magneto transmission measurements is presented in Fig. 2.4.

Magneto transmission measurements

Due to the limited space in the helium cryostat, a special transmission probe has been designed. The photo of the transmission probe used in our experiment is shown in Fig. 2.5(a). The material of the probe inserted into the magnet has to be insulating to avoid eddy current heating and magnetic forces during the pulse of the magnetic field. The top part of the probe consists a hexagonal head and a 19 mm rod made of the stainless steel. The bottom part of the probe consist a 7 mm rod and a sample holder. The rod is made of low thermal expansion coefficient epoxy fiberglass, while the sample holder is made of ceramic. The entry of the probe to the helium cryostat is sealed by a movable o-ring on the stainless steel rod, which allows the probe to be moved vertically in the cryostat to adjust the position of the sample to be exactly in the center of the magnetic field.

A schematic representation of the optical path in the probe is shown in Fig. 2.5(c). Two multi-mode fibers are used to guide light, one is 200 µm diameter fiber used to guide the incident light beam from a broad-band tungsten-halogen lamp. At the end of the fiber, lens ( 1) is used to collimate the beam. Two gold plated mirrors are used to reflect the beam to the sample space. Then, the beam, transmitted through the sample, is focused by the lens (2) on the second fiber with 400 µm diameter, which guides the signal to the spectrometer. This probe is equipped with a thermometer and pick-up coil, as shown in Fig. 2.5(b). To precisely measure the temperature of the sample, we use a Cernox thermometer. Three pairs of 100 µm diameter coper wires are used to connect the thermometer and the pick-up coil to the socket of the probe head. We use twisted wire pairs to reduce the voltage induced by the pulsed magnetic field.

Pick-up coil

In order to measure precisely the value of the magnetic field on the sample during the magnetic field pulse, a small coil is wound around the sample holder, referred to as a pick-up coil (see Fig. 2.5(b)). It is made of 50-100 turns of 100µm diameter insulated copper wire. The pulse of the magnetic field induces a voltage (V) in the pick-up coil,

V = - dφ dt = -N × S dB dt , (2.1)
where φ is the magnetic flux, N is the number of turns and S is the surface of one spire. N × S is the effective area of the whole pick-up coil, which is determined as follow. The pick-up coil is placed in the center of 1 m long calibration coil with known B (I) dependence. The oscillator output of a lock-in amplifier is used to generate a sinusoidal magnetic field in the calibration coil. The pick-up coil voltage is recorded by the lock-in amplifier at the same time. If the magnetic field is known, then the area of the pick-up coil can be calculated using equation 2.1.

Resistance thermometer

A Cernox thin film resistor is used as a thermometer on the probe. Cernox resistors are convenient due to their small package size, which makes them easily fit to the limited space in the probe (see Fig. 2.5(b)). In order to obtain a precise resistance value, we use four-terminal sensing to measure the resistance. The schematic diagram of four-terminal sensing is shown in the inset of Fig. 2.6. Separated measurements of the current and voltage eliminates the contact and lead resistance, leading to a precise measurement of the resistance values.

The resistance/temperature relationship varies between different Cernox. Thus, the resistance/temperature relation of the Cernox has to be calibrated for each thermometer. It is typically done in a following way. The Cexnox to be calibrated is placed on a calibration probe next to a calibrated Cernox. The calibration probe is immersed in a helium cryostat. As the liquid helium slowly evaporates and the cryostat warms up, the temperature increases from 4.2 K to 293 K. The resistance of both Cernox are recorded by a Lakeshore temperature controller. Typically, these data are record every 5 or 10 seconds. An example of calibrated resistance/temperature relationship is represented in Fig. 2.6. The calibrated curve obtained is uploaded to the Lakeshore temperature controller connected to our probe. The data has to be interpolated due to the limited data points that can be stored by the controller (typically 200 data points).

Magneto reflectivity measurements

For thick samples, we have used a magneto-reflectivity technique. The schematic illustration of the reflectivity set-up is presented in Fig. 2.7. In this case, the incident white light is directly reflected by the sample, and the reflected light is guided to the spectrometer. We have designed and built a probe adapted for such an experiment. Fig. 2.8(a) shows the photo of the reflectivity probe. As before, the probe is mainly made of stainless steel and fiber glass. We use a fiber bundle composed by 7 fibers with 100 µm diameter. The cross-section of the fiber bundle is shown in the inset of Fig. 2.8(b). The fiber in the center is used to guide the excitation light to the sample, then the reflected signal is gathered by the 6 collection fibers surrounding the central fiber. Fig. 2.8(b) shows the bottom of the probe, where the positions of pick-up coil and thermometer are indicated. The thermometer is located in a slightly higher position than pick-up coil and is covered by the teflon tape for protection.

The sample holder consists of a 2.5 mm diameter ferrule and a sleeve made of ceramic (see the inset of Fig. 2.8(c)). The size of the sample should be less than 2.5 mm, and the sample is placed on the top of the ferrule, then the ferrule and the probe head are then connected by the sleeve. The assembled probe and the sample holder are shown in Fig. 2.8(c). It is worth to mention that the diameter of the white fiber glass rod is only 4 mm, which allows the probe to be used in both the 100 T helium cryostat (4 mm diameter) and 70 T helium cryostat (7 mm diameter). Since the 70 T magnetic field pulse only lasts 150 ms and the typical time to acquire a spectrum using the CCD camera is 1-2 ms, it is crucial to synchronize the magnetic field pulse with the CCD acquisition. A system was designed for this temporal synchronization with a precision of 0.1 ms. The schematic diagram is shown in Fig. 2.9. A typical sequence of the events is listed as follow: (1) First, the repeater sends an optical continuous pulse signal as a pre-trigger to the delay generator (marked as trigger box) via a multi-mode fiber.

Synchronization

(2) Trigger box converts the optical continual signal to electrical signal pulse signal with 500 ms duration. Voltage of this electrical signal is TTL (5 V).

(3) The input signal for triggering the CCD cameras accepts only 3 V TTL signal, therefore, a voltage converter between the trigger box and the spectrometer is used.

(4) After the spectrometer is triggered, it sends a sequence of multiple acquisitions to the delay box and a digital oscilloscope (Hioki). Between the spectrometer and Hioki, there is a LED and a photo diode. The LED, outside of the experimental room, converts the electrical signal to optical signal. The photo diode, inside of the experimental room, converts the optical signal to electrical signal. The principle here is that all the connection from outside to inside of the experimental room has to avoid using metallic cable, optical fibers are responsible for all the connections.

(5) Delay box is used to adjust the delay to have a maximum of the magnetic field in the middle of one of the CCD acquisition frames (see inset of Fig. 2.10). The delay box converts the electrical multiple acquisition signal to 500 ms single optical signal and sends it to the generator. Next, the generator discharges into the coil giving a high magnetic field pulse.

(6) The voltage of the pick-up coil on the probe, during the pulse of the magnetic field, is recorded by a Hioki. The magnetic field as a function of time is calculated by integrating the voltage. Typical results are shown in Fig. 2.10, where we show the voltage of pick-up coil (purple line) and the sequence of CCD acquisition (green line) as a function of time during a 68 T pulse. Based on equation 2.1 which shows that voltage of the pick-up is just the derivative of the magnetic field, the magnetic field is calculated by integrating the voltage of pick-up. The magnetic field as a function of time is shown in Fig. 2.10 as black solid line. Each rectangle of the green line represents a CCD acquisition with a typical duration of 2 ms, which ensures that the spectra are acquired at an essentially constant magnetic field. In the case of Fig. 2.10, the delay time is 24 ms, which ensures the 7th acquisition corresponds to the maximum field (see the inset of Fig. 2.10).

Chapter 3

Magneto-transmission on MAPbI 3 polycrystalline films In this chapter, we describe magneto-transmission measurements, which lead to the precise determination of the exciton binding energy and reduced mass in thin film perovskite materials. By investigating thin films with a wide range of crystal grain sizes, varying from tens of nanometers to microns, we elucidate the impact of the microstructure on the Coulomb interaction between photo created electrons and holes in MAPbI 3 and Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 materials. The results presented in this chapter have been published in Energy Environ. Sci., 10, 1358-1366 (2017).

Introduction

Due to the different fabrication methods, thin film perovskites exhibit a variety of morphologies, resulting in crystal grain sizes which vary from tens of nanometers to tens of microns [Huang 2014, D'Innocenzo 2014b]. In this chapter, we investigate the correlation between the morphology of the thin film perovskites and the excitonic properties. To do so, we perform magneto-transmission measurements to extract the binding energy and reduce mass of excitons in MAPbI 3 and Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 thin films, with a wide range of the crystal grain sizes.

Magneto-transmission spectroscopy on MAPbI 3

First, we will discuss the results for MAPbI 3 . We have performed low temperature magneto-transmission spectroscopy to investigate MAPbI 3 thin films with different morphologies, using long magnetic field pulse techniques as described in detail in chapter 2. The investigated materials are MAPbI 3 thin film perovskites with a wide range of grain sizes from tens of nanometers to hundreds of nanometers. The experimental approach and data analysis used for all the samples will be explained in detail using as an example results for the LPC sample. Typical magnetotransmission spectra of the LPC sample measured at the indicated values of magnetic field at low temperature (2 K) are presented in Fig. 3.2(a). For each spectrum, we observe a minimum, which corresponds to the 1s excitonic transition. The 1s states blue shifts with increasing magnetic field. For an unambiguous identification of the weaker optical transition, we show in Fig. 3.2(b), (c) the ratio of the transmission spectra at high magnetic field divided by the transmission spectrum obtained at zero magnetic field field. In Fig. 3.2(b), we present the ratio spectra measured at intermediate field (19-30 T), which shows clearly the 2s excitonic transitions. A sequence of the transitions marked with arrows start to be visible from 30 T in a higher energy range, as presented in Fig. 3.2(c). These transitions become more pronounced with increasing magnetic field. Their energies have a larger field-induced blue shift than the low-energy transitions, which suggests that they are mainly free carrier transitions between Landau levels in the valence and conduction bands. To follow the free carrier transitions over an extended magnetic field range (70 T-150 T), we collaborated with Dr. Atsuhiko Miyata to perform Mega Gauss measurement at LNCMI-T. In this technique, the transmission of monochromatic (laser) light through the sample is measured as a function of the magnetic field. In Fig. 3.3, we show the typical results for the LPC sample. All the spectra show clear minima as a function of magnetic field, which correspond to the resonant transition between the free carrier Landau levels. The energy of the excitation laser is marked below each curve.

Description of the samples

Magneto-transmission spectra on LPC sample

Data analysis

For the long pulse measurement, we analyzed each transmission spectrum to find the energy of the absorption at the indicated magnetic field. To do so, we take the energy at the center of each absorption minimum. After finding absorption energies for all the spectra, we plot them as a function of the magnetic field, as shown as colored spheres in Fig. 3.4(a).

For the Mega Gauss measurement, we analyzed each transmission spectrum to find the magnetic field for resonant absorption at the energy of excitation. To do so, we take the magnetic field at the center of each absorption minimum. We plot the excitation energies as a function of the magnetic fields, as shown as black stars in Fig. 3.4(a).

Assuming that free carriers transitions dominate in the high magnetic field range, we first fit the interband transitions between Landau level in the conduction and valence bands using Eq. 1.14. The reduced mass µ is the only fitting parameter for all the Landau levels. Modeling the high energy range and the high magnetic field range of the data allows a precise determination of the reduced mass. The results of a typical fit of free carriers transition up to N = 4 for the LPC sample are shown in Fig. 3.4(b). For this sample, the reduced mass µ is estimated to be 0.102 ± 0.002m 0 .

Then, as the second step, we use Makado and McGill's numerical solutions for the hydrogen-like Hamiltonian to fit the data. The fitting parameters of this solution are the exciton reduced mass µ and the exciton binding energy R * . We use the same reduced mass µ extracted from the interband Landau level transitions. The fitting results are shown as dashed lines in Fig. 3.4(c). In this sample, we observed three excitonic transitions -1s, 2s and 2p. The observation of both 1s and 2s states gives a strong constraint on the value of exciton binding energy. The binding energy of exciton in LPC sample is estimated to be 16 ± 1 meV.

Magneto-transmission spectra on SPC, SC and MP samples

Next, we performed the same measurement and analysis on other MAPbI 3 samples (SPC, SC, MP) with different grain size. The typical magneto-transmission spectra of three different samples measured at 2 K are shown in Fig. 3.5. Panel (a)-(c) shows the transmission spectra measured at indicated magnetic field for SPC, SC and MP samples, respectively. As before, for each sample, 1s excitonic transitions dominate the spectra. Unfortunately, no 2s excitonic transition are visible for any of these samples. Panel (d)-(f) shows the ratios of the transmission spectra in magnetic field T (B) to that measured at zero magnetic field T (0) for SPC, SC and MP samples, respectively. With the increase of the crystal size, the line-width of the 1s transition become smaller and more Landau levels are observed. In the absence of a clear observation of the 2s excitonic transition for the SPC, SC and MP samples, a global fit to the excitonic and free-carrier transitions is employed. To visualize the negligible change in the optical response under high magnetic fields of the MAPbI 3 with different morphology, the overlaid fan diagrams of the complete set of excitonic and free carrier optical transitions, after shifting by the corresponding bandgap offsets, are presented in Fig. 3.6. The fitting parameters for the global fit are R * y = 16±1meV and µ = 0.106±0.003m 0 . Here we clearly show that these MAPbI 3 samples, although composed of dramatically different grain sizes and morphologies, exhibit very similar magneto-optical behavior.

Results and discussion

In addition to the global fit shown in Fig. 3.6, we also fitted the data for the four morphologies of MAPbI 3 separately, as shown in Fig. 3.7. The spheres present the data points from long pulse measurement. The stars present the data points from Mega Gauss measurement performed by A. Miyata. The solid and dashed lines are the fits to the set of Landau levels and the excitonic transitions, respectively. The fitting parameter for the four samples are summarized in Table 3.1. In Fig. 3.8, we show a graphical representation of the summary in Table 3.1. The reduced mass and binding energy are presented with red circles and blue squares, respectively. The SEM image for each sample are inserted at the bottom of the figure. The reduced constraint on the exciton binding energy, resulting in a more precise determination of the exciton binding energy.

Recent reports have suggested that the permanent dipole moment of organic cation in small and defects in the crystals generates an electrostatic potential gradient, which causes an enhanced dielectric screening. This results in a very small value of the exciton binding energy of the order of 2-5 meV [Lin 2015, Grancini 2015]. On the other hand, large and defect free crystals exhibit an order of magnitude higher value of exciton binding energy [D'Innocenzo 2014a, Grancini 2015]. However, in our results, the value of exciton binding energy from the MAPbI 3 polycrystalline thin films with different crystal grain sizes are all identical. In addition, our results are in good agreement with the previous work of our group [Miyata 2015], which shows that the exciton binding energy and reduced mass of thin film MAPbI 3 are (16±2) meV and 0.104 m 0 , respectively. This indicates that, contrary to the reports in the literature [D'Innocenzo 2014a, Grancini 2015], the exciton binding energy is independent of the crystal grain size for the MAPbI 3 perovskite material. In fact, this is expected from the relative size of the exciton Bohr radius and the crystal grains. The Bohr radius of the exciton is 5 nm for all the MAPbI 3 morphologies in the low temperature phase [Soufiani 2017], which is substantially less even than the smallest crystal size (tens of nanometer) of thin film perovskite. This rules out any possibility of quantum confinement effect, which has an influence on the exciton binding energy.

3.3 Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 thin films To generalize our finding for MAPbI 3 to other perovskite compositions, we have performed magneto-transmission measurement on a state-of-the-art triple-cation lead mixed-halide perovskite, Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 [Saliba 2016b]. The samples were fabricated by Mojtaba Abdi Jalebi, at the Cavendish Laboratory, University of Cambridge. For simplicity, only two considerably different morphologies have been investigated, namely (i) a planar polycrystalline thin film (Planar) with the crystal grain size around 100-300 nm and (ii) a film with the perovskite infiltrated into a thick mp-Al 3 O 2 scaffold (Mesoporous), with a resulting crystal grain size < 50 nm. The top-view SEM images of two morphologies taken by Mojtaba Abdi Jalebi are shown in Fig. 3.9.

Magneto-transmission spectra

Typical results of the magneto-transmission spectra measured at 2 K for Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 samples are presented in Fig. 3.10. Panel (a) and (d) show the as measured transmission spectra for planar and mesoporous sample, respectively. Both spectra are dominated by a pronounced minimum, which corresponds to the 1s excitonic transition. The transition blue shifts with the increasing magnetic field. Panel (b) and (e) show the ratio transmission spectra for planar sample at 15 T and mesoporous sample at 33 T. At this intermediate magnetic field, we observe the 2s excitonic transition for both samples, which is critical for the accuracy of the fitting procedure. Panel (c) and (f) show the ratio transmission spectra for the planar and mesoporous sample measured at the indicated magnetic field. We observed the 2p excitonic transition and 3 Landau levels for the planar sample, while only the 2p excitonic transition is observed for Mesoporous sample in panel (f). This is probably due to the larger line width of the mesoporous sample, indicating a lower crystal quality.

Data analysis

As for the analysis of the MAPbI 3 samples, we take the transition energy at the minimum of the absorption line and construct the energy fan chart. We show the overlaid fan chart for both planar and mesoporous sample in Fig. 3.11. The green and blue spheres are the transition energies for mesoporous and planar samples, respectively. The solid and dashed lines are the fits to the set of Landau levels and the excitonic transitions, respectively. R * y = 13 ± 2 meV and µ = 0.096 ± 0.002 m 0 are employed as the fitting parameters. The excitonic transitions of the two morphologies show very good overlap. These results allow us to extend our conclusions to the wider alloyed perovskite family, giving strong proof that the micro structure has a negligible impact on the excitonic properties at low temperature. The solid lines and the dashed lines are the fits to the set of Landau levels and the excitonic transitions, respectively.

We also fit the energy fan chart for both morphologies separately, as shown in Fig. 3.12 . A summary of the fitting parameter for both morphologies is shown in Table 3.2. The reduced mass µ = 0.096 ± 0.008 m 0 and binding energy Ry * = 13 ± 2 meV are identical within experimental error for both morphologies. These values are slightly less than the values for MAPbI 3 . It has been suggested that the band structure of perovskite material can be described by a simple semiempirical two band k.p Hamiltonian approach [Even 2015]. Assuming the effective mass for valence and conduction band are the same which predicts that the binding energy and reduced mass scale linearly with the bandgap. Consider the bandgap of Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 is less than MAPbI 3 , our finding on Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 is in a good agreement with our previous work [Galkowski 2016].

Conclusions

In this chapter, we have investigated the relationship between excitonic properties and microstructure of the MAPbI 3 and Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 perovskite using magneto-transmission spectroscopy. At low temperature, the excitonic binding energy and reduced mass are the same for each sample with a variety of crystal grain size. Thus, we conclude that microstructure truly play a negligible role in the exciton properties of mixed organic-inorganic lead halide-based perovskites, regardless of the thin film deposition technique and final morphology. This confirms the hypothesis, that the wide range of exciton binding energies found in the literature [D'Innocenzo 2014a] cannot be attributed to different morphologies or grain sizes. This chapter presents the results of a magneto-reflectivity investigation of a MAPbI 3 single crystal. We have determined the binding energy of the exciton in the low (orthorhombic) and high temperature (tetragonal) phases to be 16 ± 2 meV and 12 ± 4 meV, respectively. The binding energy of the exciton in the single crystal at low temperature is identical to that of the polycrystalline thin film, confirming the conclusion in Chapter 3 that the microstructure plays a negligible role in the excitonic properties of perovskite materials. Moreover, at room temperature, the binding energy of the exciton in MAPbI 3 single crystal is lower than the thermal energy, indicating a non-excitonic behavior. The results presented in this chapter have been published in J. Phys. Chem. Lett. 8, 1851Lett. 8, -1855Lett. 8, (2017) )

Introduction

In this work, we have performed magneto-reflectivity studies on a millimeter size MAPbI 3 high quality single crystal. First, we describe the data and the analysis, which was performed to obtain the energies of the absorption resonance from the reflectivity spectra. We then deduce the exciton binding energy and reduced mass from the reflectivity measurements in the high magnetic field for both the low temperature orthorhombic, and the high temperature tetragonal phases.

MAPbI 3 single crystal

A single crystal of MAPbI 3 was grown by Dr. Amir-Abbas Haghighirad at the Clarendon Laboratory of the University of Oxford. The growth was performed from a small seed crystal in a supersaturated solution [Nayak 2016]. A photo of the single crystal used in our measurement is shown in Fig. 4.1. The sample has a lateral size of ≈4 mm with a ≈3 mm thickness. The single crystal character was confirmed by X-ray diffraction measurement performed by Dr. Amir-Abbas Haghighirad at the University of Oxford.

Data and analysis

We have performed magneto-reflectivity measurement on a MAPbI 3 single crystal using the reflectivity technique described in Section 2.3. In this configuration, the reflected signal from the single crystal is mixed with photoluminance (PL) emission. Since the reflectivity spectra have a dispersive shape, we first discuss the procedure, used to extract the energies of the absorption resonances from the reflectivity spectra.

A typical [0, 1] normalized reflectivity spectrum measured at 60 T and at 2 K is presented as a black line in Fig. 4.2. The MAPbI 3 single crystal exhibits strong PL emission even when illuminated with a broad band white light source, thus, the reflectivity spectrum is composed of both PL emission and the reflected light. To distinguish the PL and absorption in the reflectivity spectrum, we have performed PL measurement on the single crystal independently, with the same configuration as the reflectivity measurement. To do so, we simply change the excitation source from the broad band white light source to a laser emitting at 685 nm. The PL spectrum of the MAPbI 3 single crystal measured at the same magnetic field is presented as the red line in Fig. 4.2. Both reflectivity and PL spectra show a dominant peak around 1646 meV (marked as PL1), which originates from bound exciton recombination. In a lower energy range, two weaker peaks with energies around 1626 meV and 1612 meV are observed (marked as PL2, PL3), which probably originate from donoracceptor bound exciton, as proposed by Diab et al. and Phuong et al. [Diab 2016, Figure 4.2: Typical normalized reflectivity (black line) and PL (red line) spectra of a single crystal measured in 60 T. Phuong 2016]. In addition to these three peaks, a very weak peak (PL4) with a peak energy around 1658 meV is observed on the high energy side of PL1. To better see this feature, we show an expanded view of the PL spectrum in the inset of Fig. 4.2. The peak energy of PL4 (1658±1 meV) corresponds very well to the absorption energy of the 1s + state (which will be discussed below). Thus, we deduce that PL4 originates from the high energy component of the Zeeman split exciton emission, while the lower energy component is immersed within the PL1 peak.

First we discuss the difference between PL and reflectivity spectra. The reflectivity spectrum clearly shows two minima on the high energy side of the PL1 peak, which are absent in the PL spectrum. This suggests that they come from the absorption resonance. In fact, these two minima correspond to the Zeeman split 1s exciton absorptions, as marked in Fig. 4.2. It is worth to mention that this is the first time we observe the Zeeman splitting of the 1s transition in MAPbI 3 perovskite material without using polarization optics. This is due to the extremely narrow linewidth of the 1s absorption in the single crystal (3.8 meV at 0 T, which is shown in the inset of Fig. 4.4(a)). This should be compared to the much larger value of 20-40 meV in MAPbI 3 polycrystalline films [Miyata 2015].

In transmission spectra, the minimum of the absorption line corresponds to the absorption energy, while the reflectivity spectra have a typical dispersive line shape, so the energy of the absorption corresponds to the position of the maximum slope (gradient) of the reflectivity dispersive curve [Cardona 2005]. To determine the absorption energy from the reflectivity spectrum, Korona et al. [Korona 1996] have proposed the following formula to fit the reflectivity data:

r(E) = r 0 + j A j E j -E + jΓ j Γ 2 j + (E -E j ) 2 e iθ j , (4.1)
where r 0 is a constant indicating the total amplitude of the reflectivity signal (background), θ j is a phase, A j , E j and Γ j are the amplitude, energy and broadening parameter for each of the resonances.

A commonly employed method to analyze the reflectivity spectra is to calculate the derivative. This is mainly based on two reasons, (i) the reflectivity spectra have a dispersive line shape, so that the absorption energy is located at the gradient extreme of the reflectivity spectra. Calculating the derivative make the spectra easier to analyze. (ii) Some weak features are hardly observed in the as measured spectra. Calculating the derivative make the weak features more pronounced in the spectra. Thus, instead of fitting the reflectivity spectrum directly, we first calculate the negative derivative of the reflectivity spectrum, as shown with black line in Fig. 4.3. There are two dominant peaks corresponding to the two absorptions in Fig. 4.2. Then, we fit this spectrum with the negative derivative of Eq. 4.1, which is given by,

- dr dE = r 0 + j A j (Γ 2 j -(E -E j ) 2 )cos(θ j ) + 2Γ j (E -E j )sin(θ j ) (Γ 2 j + (E -E j ) 2 ) 2 , (4.2)
where r 0 and A j indicate the background and amplitude for the negative differential spectra. The fitting curve is shown as violet dashed line in Fig. 4.3. We limit the fitting to the range from 1646 meV to 1680 meV, which covers the two resonances. The absorption energies determined from the fitting are 1650.0±0.5 meV and 1658.1±0.5 meV, which are identical to the energy in the maximum of the peaks. Thus, we conclude that, within experimental error, the peak position in negative derivative of the reflectivity spectra (-dr/dE) gives the same energies of the resonant absorption as those obtained from fitting formula 4.2 to the derivative of the data. In the following data analysis, we simply take the energy of the maximum in the negative derivative, rather than fitting the differential reflectivity spectrum.

Magneto-reflectivity measurement on single crystal MAPbI 3

In this section, we show the results of the magneto-reflectivity measurement on MAPbI 3 single crystal at both low temperature (2 K) and high temperature (168 K), and the results of temperature dependence reflectivity measurements. The measurements were performed at LNCMI-T using a long magnetic pulse technique. The detailed experimental setup is introduced in the section 2.3. A full set of the differential magneto-reflectivity spectra measured at 2 K are shown in Fig. 4.4(a) for the magnetic field up to 66 T. At low magnetic field, the dominant peak at 1640 meV corresponds to the 1s excitonic transition. The Zeeman splitting of the 1s state becomes visible around 32 T. On the higher energy side of the 1s peak, an additional weak absorption peak is attributed to the transition of the 2s excitonic state. To observe additional absorption peaks in the higher energy range, we first divided the measured spectra at high magnetic field by the zero-field spectrum. In such way, we obtained the ratio spectra. Then, in order to obtain the correct transition energies at the maxima of the peaks, we calculated the negative derivative of the ratio spectra, as presented in Fig. 4.4(b). One excitonic transition (2p) and three interband transitions between Landau levels in the conduction and valence bands (N = 2, 3, 4) are observed. Note that the 2p excitonic transition is forbidden at zero magnetic field, but becomes dipole allowed at high magnetic field once the cyclotron energy exceeds the exciton binding energy [Makado 1986].

Low temperature magneto-reflectivity

In the inset of Fig. 4.4(a), we show the reflectivity spectrum measured at zero magnetic field. We clearly observe 1s and 2s excitonic transitions, as marked in the insert. However, in the MAPbI 3 polycrystalline thin films, the 2s excitonic transition is absent at zero magnetic field, and only starts to be visible in high magnetic field [Miyata 2015]. This is due to the fact that MAPbI 3 polycrystalline thin films exhibit a lower crystal quality, resulting in a much broader linewidth of the 1s excitonic transition, which prevents the observation of the 2s excitonic transition. This phenomenon is consistent with the results obtained for other materials, such as MAPbBr 3 . Galkowski at al. [Galkowski 2016] did not observe 2s excitonic transition in MAPbBr 3 polycrystalline thin films, while the 2s excitonic transitions are clearly observed by Tilchin et al. [Tilchin 2016] in the reflectivity spectra of a MAPbBr 3 single crystal. To analyze the data, we plot the energy of the absorption as a function of the magnetic field, as presented in Fig. 4.5. We fit the data using the same models discussed as Chapter 3, namely, the dashed lines present the fit of the hydrogen-like transitions in magnetic field based on the work of [START_REF] Pc Makado | Energy levels of a neutral hydrogen-like system in a constant magnetic field of arbitrary strength[END_REF]]. The results of linear fits of the interband transition between Landau levels are presented as blue solid lines. The three excitonic transitions and three inter-Landau level transitions allow us to precisely determine the exciton binding energy and the reduced mass. The splitting of the 1s excitonic transitions is included in this model by adding the Zeeman term (±g ef f µ B B/2) to the excitonic transition energies, where g ef f and µ B are the effective g-factor and Bohr magneton. The energy splitting increases linearly with the magnetic field, which gives an effective g factor g ef f = 2.6 ± 0.1. This value is slightly larger than previously found (g ef f = 1.72 ± 0.05) by Tanaka et al. [Tanaka 2003b], possibly due to the improved resolution of our measurement. It is, however, quite similar to the previous results of our group (g ef f = 2.3 ± 0.1) obtained on MAPbI 3-x Cl x thin film sample [Galkowski 2016].

The fitting of the exciton and free carriers transitions allows us to conclude that at 2 K the reduced mass and the exciton binding energy are (0.104 ± 0.005) m 0 and (16 ± 2) meV, respectively. It is important to note that the exciton binding energy obtained from the single crystal is identical to that of polycrystalline thin films discussed in Chapter 3. This result confirms the conclusion in Chapter 3 that the crystal grain sizes play a negligible role on the exciton binding energy for perovskite materials.

The exciton binding energy from our result is significantly smaller than previously reported (37-50 meV) from the magneto-optical studies on MAPbI 3 polycrystalline thin films, where only the 1s excitonic state was observed [Hirasawa 1994, Tanaka 2003a]. We attribute this difference to the much poorer resolution of the excitonic transitions and the underestimation of the value of the dielectric constant in the previous study [Tanaka 2003a]. In this work, the authors assumed that the exciton binding energy is much larger than the energies of the optical phonon modes, so that the dielectric screening is not influenced by the lattice vibration. Thus, the authors used the high frequency dielectric constant ( ef f = 6.5) to calculate the exciton binding energy [Tanaka 2003a]. However, recent Raman scattering experiment shows that optical phonons have energies from 8 meV (vibration of the inorganic cage) to 16 meV (libration of the organic cation) [Quarti 2013]. This suggests that the dielectric screening is enhanced by the lattice vibration, and consequently a value intermediate between the static 0 = 25.7 [Wehrenfennig 2014] and the high frequency ∞ = 5.6 [Brivio 2014, Umari 2014] dielectric constant should be used [Even 2014b]. Our values of the exciton binding energy and reduced mass suggest a dielectric constant ef f ≈ 9.4, which is indeed intermediate between the low and high frequency values, as expected. Our results are also in a good agreement with the exciton binding energy of 15 meV deduced by Even et al. by fitting a zero field exciton absorption spectrum [Even 2014b]. The practical application of perovskite based solar cell devices requires a knowledge of the electronic properties of the high temperature tetragonal phase of these materials. We first measured the temperature dependence of the reflectivity spectra of the single crystal. Fig. 4.6(a) shows the typical differential reflectivity spectra at the indicated temperature. The 1s transition appears as a pronounced peak, while the 2s transition appears as a small shoulder on the high energy side of the 1s transition. Due to the excellent crystalline quality, both 1s and 2s transition are observed at temperatures as high as 35 K. The 1s and 2s transition energies as a function of temperature are represented in Fig. 4.6(b). The 1s and 2s transitions both exhibit a similar trend, blue shifting with increasing temperature. This blue shift is common in perovskite materials [Wu 2014], in contrast to the majority of semiconductors materials, which show a red shift of the band gap energy with the increasing temperature [START_REF] Varshni | Temperature dependence of the energy gap in semiconductors[END_REF]]. The blue shift observed for MAPbI 3 originates from the increased separation between the band extrema (band gap) with increasing temperature [Huang 2013, Singh 2016]. As discussed in Section 1.3.1, the band gap energy of MAPbI 3 is significantly reduced by the spin-orbit interaction. Since MAPbI 3 has a large volume coefficient for thermal expansion (50 times more than crystalline silicon), when the temperature increases, the spin-orbit interactions within the inorganic lattice are significantly weakened, resulting in an increase of the separation between the band extrema.

Temperature dependence reflectivity

It is possible to deduce the binding energy R * from the 1s and 2s transitions, since the energy separation of 1s and 2s gives the 0.75R * , as expected from Eq. 1.17. The deduced R * as a function of temperature is shown in the inset of Fig. 4.6(b). The deduced R * does not show any temperature dependence up to 35 K. This is expected, since there is no structural phase transition for MAPbI 3 in the temperature range from 4 K to 35 K, so that the exciton binding energy should remain the same.

The phase transition in the thin film MAPbI 3 from orthorhombic to tetragonal phase occurs about 150 K [Baikie 2013] and is accompanied by a dramatic red shift of the bandgap (∼100 meV) [D'Innocenzo 2014a, Even 2014b, Yamada 2015]. In our case, around 154 K, there is a abrupt transition to a lower energy, red-shifted by 103 meV. This suggests that the sample has completed the transition from the low temperature orthorhombic to the high temperature tetragonal phase.

We have also studied the broadening parameter Γ of the free exciton absorption line as a function of temperature, by fitting the temperature dependence differential spectra with Eq. 4.2. Examples of the fitted spectra are shown in Fig. 4.7(a). Solid lines show the measured differential reflectivity spectra, while the dashed lines present the fit. Fig. 4.7(b) shows the broadening parameter Γ as a function of the temperature. Schreiber et al. [Schreiber 1982] have investigate the influence of lattice vibration on the absorption line shape of the excitons. They suggest that, for different dimension of the crystal lattice, the absorption line shape of the excitons exhibit different temperature dependence. In other words, the broadening parameter of the exciton absorption depends on the temperature as T 2/3 , T and T 3/2 for one-, two-and three-dimensional crystal lattices, respectively [Schreiber 1982]. In our case, apart from the temperature region of the phase transition (orthorhombic/tetragonal), Γ shows a T 3/2 dependence, with a residual broadening parameter of 3.6 meV (44 K) at low temperature. According to Schreiber's work, this temperature dependence is in a good agreement with the three dimensional crystal lattice of MAPbI 3 perovskite. In the temperature region of the phase transition, an additional broadening is observed. This may due to the fact that the phase transition induces an additional distortion in the crystal structure, which leads to an additional inhomogeneous broadening. To study the electronic property of the MAPbI 3 single crystal in the tetragonal phase, we repeat the magneto-reflectivity measurement at 168 K. The differential reflectivity spectra measured up to 66 T are shown in Fig. 4.8(a). The 1s excitonic states appears as a pronounced peak in the reflectivity spectrum with a considerable broadening (FWHM of 22 meV). The 2s states appear as a shallow peak at magnetic fields higher than 39 T. However, at such a high temperature, no higher excitonic or Landau levels transition are observed. Fig. 4.8(b) represents the energy fan chart at 168 K. We have fitted the data for the 1s and 2s transition. From our previous work [Miyata 2015], the reduced masses of thin films perovskite remain almost the same for both orthorhombic phase and tetragonal phase. Thus, we assume that reduced mass of single crystal at tetragonal phase is the same as at low temperature, µ 0.104 m 0 . With the same fitting procedure described for Fig. 4.5, we find an exciton binding energy of 12±4 meV comparable to that estimated by similar high temperature magneto-transmission experiments on a thin film [Miyata 2015]. The value of exciton binding energy decreases by few meV when going from the orthorhombic phase to the tetragonal phase. This is consistent with the fact that the phase transition increases the dynamic disorder related to the rotational motion of the organic cation, resulting in an increase of the dielectric constant [START_REF] Poglitsch | Dynamic disorder in methylammoniumtrihalogenoplumbates (II) observed by millimeter-wave spectroscopy[END_REF], Even 2014b].

High temperature magneto-reflectivity

Whilever the MAPbI 3 remains in the tetragonal phase, the dielectric response does not change, so we assume that the exciton binding energy remains unchanged in tetragonal phase independent of the temperature. The thermal energy at room temperature is ≈ 26 meV, which is larger than the exciton binding energy in the single crystal. In this case, the electrons and holes in excitons are dissociated by the thermal energy at room temperature, which suggests that the free carriers (rather than excitons) are dominant in the single crystal at room temperature.

Conclusions

We performed magneto-reflectivity measurement on a high quality MAPbI 3 single crystal. The value of the exciton binding energy and effective reduced mass of the single crystal sample at low temperature are identical to that of polycrystalline thin film sample. The estimated value of exciton binding energy of single crystal sample in tetragonal phase is smaller than the thermal energy at room temperature, indicating that MAPbI 3 shows non-excitonic behavior at room temperature, which is important for the photovoltaic applications. In this chapter, we describe the results of the spectroscopic measurements of the fully inorganic CsPbX 3 perovskites. The temperature dependence of the absorption indicates the absence of a structural phase transition for all the investigated materials, up to room temperature. By performing the magneto-transmission measurements, we determine the binding energy and the reduced mass of the excitons in CsPbX 3 . The results presented in this chapter have been published in ACS Energy Lett., 2 (7), 1621-1627 (2017).

Introduction

It has been recently demonstrated that introducing small amount of inorganiccesium cations into the organic lattice of the hybrid perovskite (e.g. Cs 0.05 (FA 0.83 MA 0.17 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 ) greatly improves their properties resulting in a superior morphology, high solar cell efficiency and improved stability under standard operation conditions [Saliba 2016b, Saliba 2016a[START_REF] Mcmeekin | [END_REF]. Despite the impressive performance of the mixed cation perovskite, very little is known about the influence of the inorganic cesium cation on the electronic properties of the mixed compounds. In order to answer this question, we have investigated the electronic properties of fully inorganic CsPbX 3 perovskites.

We present the results of optical studies of the electronic properties of fully inorganic CsPbX 3 thin films, where X = Br, I or an alloy of the two. The temperature dependence of the absorption measurements suggest that CsPbX 3 does not undergo a structural phase transition in the range from 4 K to 270 K, which is in striking contrast with organic-inorganic perovskites. We also performed magneto-transmission measurements to determine the binding energy and reduced mass of the exciton in CsPbX 3 . By comparing the dielectric constant of both hybrid and fully inorganic perovskites, we show that, at low temperature, the dielectric screening is essentially the same for both hybrid and inorganic materials and is mainly dominated by the relative motion of the atoms within the lead halide cage. Before we introduce the pre-experimental treatment, we first discuss the basic properties of the three materials. CsPbI 3 is known to be unstable under ambient conditions. This is mainly due to the fact that, the small size of the Cs + cation results in a tolerance factor too low to sustain a cubic perovskite structure. In fact, it has a "black" cubic perovskite phase with a band gap of around 1.73 eV (stable above ≈ 593 K) and a "yellow" non-perovskite phase with a band gap of around 2.8 eV at room temperatures [Vorpahl 2015, Eperon 2015]. At room temperature and in air, a full conversion for CsPbI 3 from the black phase to the yellow phase completes within a few minutes, or almost immediately in a humid environment. It has been shown that CsPbI 2 Br is significantly more stable in the cubic phase than their fully iodide counterparts [START_REF] Ma | [END_REF]]. This is mainly because Br -has a smaller ionic radius than I -, resulting in a higher tolerance factor for CsPbI 2 Br than CsPbI 3 . CsPbI 2 Br starts to exhibit a cubic phase with a band gap of 1.9 eV above 338 K [Niezgoda 2017]. Below this temperature, CsPbI 2 Br covert to a non perovskite phase within 10 min. CsPbBr 3 is even more stable. Indeed, it does not have a non-perovskite phase. This compound can crystallize in three different structure, namely, orthorhombic, tetragonal and cubic. All of these three polymorphs have a perovskite structure [START_REF] Sakata | [END_REF]], and a similar band gap in the range of 2.3-2.4 eV, corresponding to their orange color [Kulbak 2015]. This compound has a phase transition from the orthorhombic phase to a tetragonal phase at 361 K, and a further phase transition to a cubic phase at 403 K. [Kulbak 2015, Stoumpos 2013b].

Description of the samples and pre-experimental treatment

Due to their limited stability in air at room temperature [Eperon 2015], CsPbX 3 samples were systematically annealed before the experiment. The aim of this annealing procedure is to recover the cubic perovskite phase for all the compounds, because CsPbX 3 compounds with cubic phase is of interest for their potential in photovoltaic devices, as described in Section 1.4. Initially, CsPbI 3 and CsPbI 2 Br are transparent, due to the fact that they are in a non-perovskite phase and have a bandgap near the UV range. CsPbBr 3 has a yellow color, due to the fact that it is in an orthorhombic phase with a band gap in 2.3-2.4 eV. The temperatures of the phase transition for these three compounds are different (as discussed above), thus, they are annealed at different temperatures. CsPbI 3 was annealed at 623 K for 10 min. CsPbI 2 Br and CsPbBr 3 were annealed at 523 K for 5-7 min. Fig. 5. 1(a-c) presents the photos of CsPbBr 3 , CsPbI 2 Br and CsPbI 3 after annealing procedure. After annealing, CsPbBr 3 shows a yellow color suggesting a band gap in the range of 2.3-2.4 eV, however the crystal structure of annealed CsPbBr 3 is hard to specify from the color. CsPbI 2 Br and CsPbI 3 exhibit a brown color suggesting the band gap in the range of 1.7-1.9 eV. After annealing, the samples were immersed in liquid helium in a cryostat within 4 minutes. In Fig. 5.2(a-c), we show the typical transmission spectra measured at 4 K of annealed CsPbBr 3 , CsPbI 2 Br and CsPbI 3 samples, respectively. All the spectra show a clear minimum corresponding to the 1s excitonic absorption. The absorption energies of CsPbI 3 and CsPbI 2 Br are around 1.72 and 1.87 eV, suggesting that they are in a black perovskite phase (cubic phase) [Beal 2016]. In the case of CsPbBr 3 , the absorption energy is around 2.32 eV, suggesting that the CsPbBr 3 is also in a perovskite phase. As described before, after annealing, the samples were directly put into liquid helium (4 K), and this procedure freezes the crystal structure. Therefore, we conclude that the sample pre-treatment procedure preserves the perovskite structure obtained from annealing.

Temperature dependent transmission measurement

After the pre-treatment of the samples, we measured the absorption of all the CsPbX 3 compounds as a function of the temperature, using the transmission technique described in Section 2.2.

In Fig. 5.3(a-c), we show typical transmission spectra of CsPbBr 3 , CsPbI 2 Br, CsPbI 3 measured over a wide range of temperatures varying from 4 K to 270 K. Each transmission spectrum shows a clear, single minimum, which corresponds to the 1s excitonic absorption. The minimum blue shifts and broadens with increasing temperature for all the compounds. We plot the energy of the minimum of the absorption resonances as a function of the temperature in Fig. 5. 3(d). The absorption energies shift continuously with increasing temperature. In organic-inorganic perovskite materials, the structural phase transition is always accompanied by a dramatic band gap energy change [Stoumpos 2013a, Baikie 2013, Galkowski 2017]. Thus, the continuous shift of the band gap as a function of the temperature suggests that there is no structural phase transition in CsPbX 3 . This phenomenon is in stark contrast with the hybrid organic-inorganic perovskites. In the temperature range of 4 K-300 K, the hybrid organic-inorganic perovskites exhibit a structural phase transition around 150 K-170 K from low temperature orthorhombic to high temperature tetragonal phase [Stoumpos 2013a, Baikie 2013].

All three polymorphs (orthorhombic, tetragonal and cubic phases) of CsPbBr 3 exhibit similar band gaps and a yellow color. Simply from the color and the band gap energy, it is hard to specify the exact polymorph for the annealed CsPbBr 3 . After annealing and putting the sample into the helium cryostat, the measured CsPbBr 3 can be in any of the three polymorohs. In order to check which polymorph we have measured, we compared the results for annealed and non-annealed CsPbBr 3 . The transmission spectra and absorption energy of the annealed CsPbBr 3 sample are presented as black dashed lines and spheres in Fig. 5.3(a), (d). The transmission data and the evolution of the absorption energy with the temperature are identical for annealed and non-annealed CsPbBr 3 . Non-annealed CsPbBr 3 is in an orthorhombic phase [Beal 2016], suggesting that annealed CsPbBr 3 is in an orthorhombic phase as well. Thus, we conclude that, the annealing procedure does not influence the crystal structure of CsPbBr 3 under the measurement condition used.

The absence of the phase transition in the CsPbX 3 compunds suggests that the electronic properties, such as exciton binding energy, will not change in the investigated temperature range, as we will demonstrate in Section 5.5.

Low temperature magneto-transmission measurement

We have performed low temperature magneto-transmission spectroscopy to investigate the electronic properties for CsPbX 3 compounds, using long pulsed magnetic field techniques as described in detail in chapter 2. The results are presented and discussed below.

Typical results and data analysis

Typical magneto-transmission spectra measured at 2 K for CsPbX 3 are presented in Fig. 5.4(a)-(c). As for the organic-inorganic compounds, we observe a dominant minimum, which corresponds to the 1s excitonic transition. With the increase of the magnetic field, a blue shift of the 1s transition is observed for all the compounds. To observe the weaker absorption in the higher energy range, we divided the spectra at B 0 T by the spectrum acquired at zero magnetic field, as shown in Fig. 5.4(d)-(f). We observed additional absorption minima at higher energies. To follow the higher energy transitions in an extended magnetic field range (70 T-150 T), A. Miyata has performed monochromatic absorption measurement via Mega Gauss technique for all the three compounds. Typical results are shown in Fig. 5.4(g)-(i). All the spectra show clearly a series of minima as a function of magnetic field, which correspond to the absorption at a given wavelength of the laser transmitted through the thin films.

We analyze the magneto-transmission data for all the Cs compounds in the same way as for MAPbI 3 , as described in Chapter 3. Namely, we take the energy at the minimum of the absorption line and construct the energy fan chart for CsPbI 3 , CsPbI 2 Br and CsPbBr 3 . Typical results are shown in Fig. 5.5(a)-(c). The blue circles represent the data from long pulse measurements, while the black stars represent the data from Mega Gauss measurements performed by A. Miyata. We fit the excitonic transition data in the low energy range with the hydrogen model, as described in detail in Section 1.6.3. For the higher energy range data, we used the formula corresponding to interband transition between the Landau level in the conduction and valence band, as described in details in Section 1.6.2. The orange lines present the fit for the excitonic transitions. The gray lines present the fit for interband transition between Landau levels. We highlight the fit of 1s-2s excitonic transitions by plotting an expanded view of the low field and low energy portion of the fan chart in the panels below the full energy fan chart. The values of the effective mass µ and binding energy R * y obtained by the fitting procedure for all three compounds are marked in each figure and summarized in Table 5.2. Note that the exciton binding energy in CsPbI 2 Br (22 ± 3 meV) is less accurate than that of CsPbBr 3 (15 ± 1 meV) and CsPbI 3 (33 ± 1 meV). This is because we observed the 2s state for CsPbBr 3 and CsPbI 3 . The energy difference between 1s and 2s state gives a strong constraint to the value of exciton binding energy. However, no trace of the 2s state for CsPbI 2 Br is observed, resulting in a less accurate value for the exciton binding energy in CsPbI 2 Br.

In Fig. 5.6, we show the binding energy and reduced mass of the exciton in cesium compounds and organic-inorganic compounds as a function of the bandgap. Brown, red and yellow stars present the results for CsPbI 3 , CsPbI 2 Br and CsPbBr 3 . For comparison, we have added the results of the hybrid organic-inorganic perovskites from our group's previous work marked as circles [Miyata 2015, Galkowski 2016]. The pink and violet circles represent the results for FA and MA based hybrid perovskites. The evolution of exciton binding energy with the band gap is fitted with the equation R * = 0.010 × E g , as shown as black line in Fig. 5.6(a). In Fig. 5.6(b), we observe an increase of the effective mass with the increasing of the band gap. This can be understood in the frame of a simple two band k•p model [Heo 2013]. Assuming the effective mass for valence and conduction band are the same, the reduced mass of the exciton can be written as,

1 µ = 4|P | 2 m 0 E g , (5.1)
where P = Ψ VB |p x |Ψ CB is the momentum matrix element that couples states in the conduction and valence bands. 2|p| 2 /m 0 is the Kane energy [Makado 1986].

The theoretical predictions of the Kane energy are in the range 5.3-6.3 eV for these materials [Even 2014b[START_REF] Fang | [END_REF]]. The evolution of the reduced mass with the band gap is fitted with the Kane energy of 8.3 eV, only slightly larger than theo-

Dielectric screening

With the value of the effective mass and binding energy of the exciton, the effective dielectric constant of the materials can be calculated using,

ef f = e 2 µ 2R * y , (5.2)
Using the values of the exciton effective mass and binding energy obtained from the fitting, we calculated the effective dielectric constants ef f for all the compounds, which are summarized in Table 5.2. In Fig. 5.7, we compared the effective dielectric constant for cesium-based perovskites (stars) with the hybrid organic-inorganic perovskites (circles) measured at 2 K. To understand the results, we first group the data in the way that data points from the same group have the same halide, and different A cation (MA, FA, Cs). The blue dashed lines indicate the average value of dielectric constant in each group. It is clear from the results that the dielectric constant does not change within the experimental error for a given lead-halide cage, regardless of the cation. It has been suggested that the main contributions to the dielectric screening are due to the Pb-X stretching modes and Pb-X-Pb rocking modes [Pérez-Osorio 2015, Sendner 2016]. This is consistent with what is observed in Fig. 5.7. At low temperature, when the organic cation motion is frozen, the dielectric mechanism is essentially the same for both hybrid and inorganic perovksites and is mainly dominated by the relative motion of atoms within the lead halide cage.

High temperature magneto-transmission measurement

We also performed magneto-transmission measurement on CsPbBr 3 at 180 K. Typical transmission spectra are shown in Fig. 5.8. The spectra show a dominant minimum corresponding to the 1s excitonic transition. However, no higher excitonic and free carrier transitions are observed.

In order to compare the exciton binding energy in cesium compounds at low and high temperature, we plot the 1s transition energies as a function of magnetic field at low temperature (4 K) and high temperature (180 K) in the same frame, as shown in Fig. 5.9. The data are vertically offset to compensate the change of the band gap induced by the increase of the temperature. The excellent overlap of the two sets of data indicates that the exciton binding energy does not change within experimental accuracy. The temperature dependent transmission measurement suggests that the CsPbX 3 materials do not undergo a structural phase transition up to room temperature. Hence, the dielectric screening does not change, thus, the exciton binding energy does not vary over the investigated temperature interval. This result is in contrast with the organic-inorganic perovskites. The phase transition of organic-inorganic perovskites allows the rotational motion of the organic cations, which enhances the dielectric screening and reduces the exciton binding energy [START_REF] Poglitsch | Dynamic disorder in methylammoniumtrihalogenoplumbates (II) observed by millimeter-wave spectroscopy[END_REF], Even 2014b, Yang 2017b, Galkowski 2016].

Conclusion

In this chapter, we present the results of magneto-optical investigation of fully inorganic CsPbX 3 compounds (X = I, Br and a mixture of these). The continuous increasing of the bandgap with the temperature indicates the absence of any structural phase transition for fully inorganic perovskite in the temperature range from 4 K to 270 K. Thus, the electronic properties of the CsPbX 3 do not change over this temperature range. By performing the magneto-transmission measurement on CsPbX 3 , we have determined the exciton binding energies and reduced mass with high accuracy. Comparison of the values of dielectric constant for inorganic and hybrid perovskites indicates that, at low temperature when the organic cations are frozen, the dominant contribution to the dielectric screening is related to the relative motion within the lead-halide cage.

Chapter 6

Conclusions

Recently, solar cells based on the hybrid perovskites, with the general chemical formula of ABX 3 , have reached an astonishing power conversion efficiency as high as 23.6%. The hybrid perovskites exhibit many advantages for the application of photovoltaic devices, such as strong broad band absorption, low cost fabrication processes, long carrier diffusion lengths and lifetimes. Although remarkable achievement have been made in this research field, there are still many fundamental issues which are under debate. For example, (i) the impact of the microstructure on the Coulomb interaction between the photo-created electron-hole pairs (excitons) in the perovskite materials. (ii) the impact of fully inorganic cation on the electronic properties of mixed compounds. To address these issues, we have used magneto-optical methods to investigate the exciton binding energy and reduced mass in MAPbI 3 compound with a variety of crystal grain sizes and in CsPbX 3 fully inorganic compounds.

A variety of fabrication methods are employed to prepare the perovskite materials, which leads to a variety of morphologies and different degrees of structural disorder in the perovskite materials. To investigate the correlation between the morphology of the thin film perovskites and the excitonic properties, we have performed magneto-transmission measurement on four different MAPbI 3 polycrystalline thin films with different crystal grain sizes (from 50 nm to 770 nm). By analyzing the magnetic field evolution of energies of the interband transitions between the excitonic and free-carrier states, we extract the value of exciton binding energy and reduced mass of MAPbI 3 . We find that these four MAPbI 3 polycrystalline thin films have the same value of exciton binding energy (R * = 16 ± 4 meV ), and the reduced mass of excitons varies slightly from 0.102 m 0 to 0.109 m 0 . This indicates that the microstructure plays a negligible role in the excitonic properties of the MAPbI 3 compounds. To generalize our finding to other perovskite compounds, we have performed the same measurement on triple-cation lead mixed-halide thin film perovskite Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 with two considerably different morphologies, namely (i) mesoporous with crystal gain size <50 nm, (ii) planar with crystal grain size around 100-300 nm. The extracted value of reduced mass and binding energy of excitons in these two films are identical (R * = 13 ± 2 meV, µ = 0.096 ± 0.008 m 0 ). Thus, we conclude that microstructure truly play a negligible role in the exciton properties of mixed organic-inorganic lead halide-based perovskites, regardless of the thin film deposition technique and final morphology.

In addition, we have performed magneto-reflectivity studies on a millimeter size MAPbI 3 high quality single crystal. We extract the exciton binding energy and reduced mass for MAPbI 3 for both low temperature (T = 2 K) orthorhombic and high temperature (T = 168 K) tetragonal phases. The exciton binding energy in MAPbI 3 single crystal at low temperature is identical to that of MAPbI 3 polycrystalline thin films, which confirms that the excitonic properties do not depend upon the crystal size or morphology. The estimated value of exciton binding energy in MAPbI 3 single crystal in the tetragonal phase (R * = 12 meV) is less than the thermal energy (26 meV) at room temperature, indicating a non-excitonic behavior for MAPbI 3 single crystal at the operational condition.

Finally, we have investigated the electronic properties of fully inorganic CsPbX 3 perovskite, where X = Br, I or an alloy of the two. Before the measurement, the CsPbX 3 polycrystalline thin films are annealed to recover the perovskite structure, and then the thin films are put in the liquid helium immediately. This preexperimental treatment process preserves the perovskite structure of CsPbX 3 compounds at temperatures as low as 4 K. After the pre-treatment of the samples, we have measured the absorption of all the CsPbX 3 compounds as a function of the temperature. The energy of the 1s transition blue shifts continuously with increasing temperature, suggesting that the CsPbX 3 compounds do not undergo a structural phase transition in the temperature range from 4 K to 270 K. We also perform low temperature magneto-transmission spectroscopy on all three CsPbX 3 compounds. By analyzing the data, we estimate the exciton binding energy and reduced mass in CsPbBr 3 , CsPbI 2 Br and CsPbI 3 compounds. Knowing the value of exciton binding energy and reduced mass, we calculate the dielectric constant for all the measured compounds. By comparing the dielectric constant of the fully inorganic and hybrid perovskites at low temperature, we find that the dielectric constant does not change significantly for a given lead-halide cage, regardless of the cation. Thus, we conclude that at low temperature, when the organic cation motion is frozen, the dielectric mechanism is essentially the same for both hybrid and inorganic perovksites and is mainly dominated by the relative motion of atoms within the lead halide cage.

B.1 Introduction

Les pérovskites forment une large classe de matériaux, ayant la même structure cristalline que le titanate de calcium (CaTiO 3 ), nommé d'après le minéralogiste russe Lev Perovski (1839) [Wenk 2016]. Les pérovskites sont intéressants pour leur flexibilité de substitution chimique et ont un large éventail d'applications industrielles. Par exemple, BaTiO 3 est un matériau très utilisé comme ferroélectriques [Megaw 1945]. YBa 2 Cu 3 O 7 ouvre une nouvelle voie vers la supraconductivité à haute température critique [Wu 1987]. Enfin, Ba(Zn 1/3 Ta 2/3 )O 3 est largement utilisé comme filtres de fréquence dans les communications sans fil [Tamura 1986] grâce à une constante diélectrique élevée.

B.1.1 Structure pérovskite

B.1.2 Perovskites hybrides

Au cours des dernières années, une grande famille des pérovskites hybrides organiques-inorganiques a attiré beaucoup d'attention en raison de ses nombreuses applications potentielles, comme les diodes électroluminescentes (LED) [Cho 2015], les lasers [Xing 2014] et surtout les dispositifs photovoltaïques à haut rendement [Kojima 2009, Liu 2013]. Les perovskites hybrides organiques-inorganiques sont formés lorsque le cation inorganique A est remplacé par un organique. En raison de l'espace limité entre les octaèdres inorganique, seulement de petits cations organiques, constitués de trois liaisons C-C ou C-N ou moins, peuvent s'adapter à l'intérieur de la structure pérovskite. Les cations A organiques les plus courants sont Méthylammonium (MA: CH 3 NH + 3 ) ou Formamidinium (FA: CH(NH 2 ) + 2 ). Les cations B sont habituellement Pb 2+ ou Sn 2+ , tandis que les anions X proviennent des halogénures du groupe VII (Br -, Cl -, I -, ou une composition mixte de ceux-ci). Il a été suggéré, que les perovskites hybrides ont un fort caractère d'autoassemblage pendant la formation du cristal; (i) les ions inorganiques sont favorables à la formation le réseau octaédrique, et (ii) les molécules organiques sont favorables à l'occupation des zones vacantes entre les octaèdres [Cheng 2010]. Ce caract Ãĺ re fait que ce genre de matériaux est faciles à synthétiser grâce à une variété de méthodes de fabrication, telles que le dépôt séquentiel [Burschka 2013], le traitement en solution [Yang 2015a] assisté par vapeur dépôt [START_REF][END_REF]], et aux techniques d'évaporation sous vide [Liu 2013].

Une autre propriété des perovskites hybrides est qu'il est relativement facile de substituer les cations A, B ou les ions halogénures par d'autres éléments. En utilisant la substitution chimique, on peut régler l'énergie de la bande interdite sur une large gamme de 1,1 eV à 3,1 eV [Suarez 2014[START_REF] Hoke | Reversible photo-induced trap formation in mixed-halide hybrid perovskites for photovoltaics[END_REF][START_REF] Zhao | [END_REF], qui couvre l'ensemble du visible et une partie du spectre de la gamme proche infrarouge. En plus des propriétés décrites ci-dessus, les pérovskites hybrides se caractérisent également par une forte absorption à large bande, une longue longueur de diffusion [Dong 2015] et une longue durée de vie des porteurs de charches [Stranks 2013]. Toutes ces propriétés rendent les perovskite hybride très prometteurs en tant que capteur de lumière pour les cellules solaires à faible coût et à haut rendement. MAPbI 3 , en tant que prototype de cette famille, a d'abord été utilisé comme couche active dans l'industrie solaire dispositifs cellulaires en 2009 [Kojima 2009].

B.1.3 Motivation

Une variétée de méthodes de fabrication sont utilisées pour préparer les matériaux de type pérovskite, tels que les techniques à une seule étape [Kim 2016], à séquençage séquentiel [Burschka 2013] ou à gaz [Huang 2014]. Cela conduit à une variété de morphologies pour les matériaux pérovskite. La morphologie joue un rôle très important, car elle peut influencer la longueur de diffusion ou la durée de vie des porteurs et, par conséquent, l'efficacité des cellules solaires. En plus des paramètres discutés ci-dessus, il a également été suggéré, par Grancini et al. [Grancini 2015], que la morphologie du film a une influence sur l'interaction de Coulomb entre les électron et les trous. En analysant les résultats des spectres de transmission résolus en fonction de la température, ils affirment que l'énergie de liaison de l'exciton peut varier en fonction de la microstructure du film de pérovskite. Afin de vérifier cela, nous avons effectué une spectroscopie de magnéto-transmission sur des films minces MAPbI 3 avec une variété de tailles de grains cristallins. Comme nous le verrons, les résultats présentés dans cette thèse démontrent clairement que ce n'est pas le cas.

Il a été démontré récemment que l'introduction de petites quantités de cations inorganiques dans le réseau organique de la perovskite hybride améliore grandement leurs propriétés, résultant en une morphologie supérieure, une efficacité élevée des cellules solaires et une stabilité améliorée dans des conditions de fonctionnement standard [Saliba 2016b, Saliba 2016a[START_REF] Mcmeekin | [END_REF]. Malgré les impressionnantes performances des perovskites à cations mixtes, on connaît très peu l'influence du cation de césium inorganique sur les propriétés électroniques des composés mixtes. Afin de répondre à cette question, nous avons étudié les propriétés électroniques de perovskites CsPbX 3 entièrement inorganiques.

B.2 Techniques expérimentales

Le deuxième chapitre décrit la mise en place expérimentale, qui est utilisée pour étudier les propriétés excitoniques des pérovskites. Dans cette thèse, deux types de techniques sont utilisés : à savoir les techniques de magnéto-transmission et de magnéto-réflectivité. La technique de magnéto-transmission est utilisée pour mesurer des échantillons transparents (par exemple des films minces polycristallins MAPbI 3 ), tandis que la technique de magnéto-réflectivité est utilisée pour mesurer un échantillon non transparent (par exemple un monocristal de MAPbI 3 ). Plus de détails peuvent être trouvés dans le chapitre 2.

B.3 Magnéto-transmission sur films polycristallins MAPbI 3

Nous étudions la corrélation entre la morphologie des pérovskites à couches minces et les propriétés excitoniques. Pour ce faire, nous effectuons des mesures de magnétotransmission pour extraire l'énergie de liaison et réduire la masse des excitons dans En supposant que les transitions des porteurs libres dominent dans la gamme des champs magnétiques élevés, nous ajustons d'abord les transitions interbandes entre les niveaux de Landau dans les bandes de conduction et de valence. La masse réduite µ est le seul paramètre approprié pour tous les niveaux de Landau. La modélisation de la plage de haute énergie et de la plage de champs magnétiques élevée des données permet une détermination précise de la masse réduite. Les résultats d'un ajustement typique de porteurs libres jusqu'à N = 4 pour l'échantillon LPC sont montrés sur . La masse réduite varie de 0,102 m 0 à 0,109 m 0 , tandis que l'énergie de liaison reste la même pour tous les échantillons dans l'erreur expérimentale. Ainsi, nous concluons que la microstructure joue un rôle négligeable Pour analyser les données, nous tracons l'énergie de l'absorption en fonction du champ magnétique, comme présenté dans la figure B.9. Nous ajustons les données en utilisant les mêmes modèles discutés que le section B.3, à savoir, les lignes pointillées présentent l'ajustement des transitions de type hydrogène dans le champ magnétique basé sur le travail de Makado et McGill [Makado 1986]. Les résultats des ajustements linéaires de la transition interbandes entre les niveaux de Lan-dau sont présentés sous la forme de lignes continues bleues. Les trois transitions excitoniques et trois transitions de niveau inter-Landau permettent de déterminer précisément l'énergie de liaison de l'exciton et la masse réduite. La division des transitions excitoniques 1s est incluse dans ce modèle en ajoutant le terme Zeeman (±g ef f µ B B/2) aux énergies de transition excitoniques, où g ef f et µ B sont le facteur g efficace et Bohr magneton. La division de l'énergie augmente linéairement avec le champ magnétique, ce qui donne un facteur g efficace g ef f = 2.6 ± 0.1. Cette valeur est légèrement plus grande que celle trouvée précédemment (g ef f = 1.72 ± 0.05) par Tanaka et al. [Tanaka 2003b], probablement en raison de la résolution améliorée de notre mesure. Il est, cependant, assez similaire aux résultats précédents de notre groupe (g ef f = 2.3 ± 0.1) obtenu sur MAPbI 3-x Cl x échantillon de film mince [Galkowski 2016].

L'ajustement des transitions des excitons et des porteurs libres permet de conclure qu'à 2 K la masse réduite et l'énergie de liaison des excitons sont (0.104 ± 0.005) m 0 et (16 ± 2) meV, respectivement. Il est important de noter que l'énergie de liaison d'exciton obtenue à partir du monocristal est identique à celle des couches minces polycristallines discutées dans le section B.3. Ce résultat confirme la conclusion du section B.3 que les tailles des grains cristallins jouent un rôle négligeable sur l'énergie de liaison des excitons pour les matériaux pérovskites. montre les spectres de réflectivité différentielle typiques à la température indiquée. La transition 1s apparaît comme un pic prononcé, alors que la transition 2s apparaît comme une petite épaule du côté haute énergie de la transition 1s. En raison de l'excellente qualité cristalline, on observe des transitions 1s et 2s à des températures aussi élevées que 35 K. Les énergies de transition 1s et 2s en fonction de la température sont représentées sur la figure B.10(b). Les transitions 1s et 2s présentent toutes deux une tendance similaire, le décalage vers le bleu avec l'augmentation de la température. Ce décalage vers le bleu est commun dans les matériaux de type perovskite [Wu 2014], contrairement à la majorité des matériaux semi-conducteurs, qui montrent un décalage vers le rouge de l'énergie de bande interdite avec la température croissante [START_REF] Varshni | Temperature dependence of the energy gap in semiconductors[END_REF]]. Le décalage bleu observé pour MAPbI 3 provient de la séparation accrue entre les extrema de la bande (bande interdite) avec une augmentation de la température [Huang 2013, Singh 2016]. L'énergie de bande interdite de MAPbI 3 est significativement réduite par l'interaction spinorbite. Puisque MAPbI 3 a un grand coefficient de dilatation thermique (50 fois plus que le silicium cristallin), lorsque la température augmente, les interactions spinorbite dans le réseau inorganique sont significativement affaiblies, ce qui entraîne une augmentation de la séparation entre la bande extrema.

B.4.2 Dépendence en temperature de la reflectivité

Il est possible de déduire l'énergie de liaison R * des transitions 1s et 2s, puisque la séparation d'énergie de 1s et 2s donne 0.75R * . La déduction de R * en fonction de la température est montrée dans l'encadré de la Fig. B.10(b). La déduction de R * ne montre aucune dépendance en température jusqu'à 35 K. Ceci est attendu, car il n'y a pas de transition de phase structurelle pour MAPbI 3 dans la gamme de température de 4 K à 35 K, de sorte que l'énergie de liaison d'exciton devrait rester la même.

La transition de phase dans le film mince MAPbI 3 de la phase orthorhombique à la phase tétragonale se produit environ 150 K [Baikie 2013] et est accompagnée d'un dramatique décalage vers le rouge de l'énergie de gap. (∼100 meV) [D'Innocenzo 2014a, Even 2014b, Yamada 2015]. Dans notre cas, vers 154 K, il y a une transition brusque vers une énergie plus faible, décalée de 103 meV. Ceci suggère que l'échantillon a achevé la transition de la phase orthorhombique basse température à la phase quadratique haute température.

B.4.3 Magnéto-réflectivité à haute température

Pour étudier les propriétés électroniques des monocristaux de MAPbI 3 dans la phase tétragonale, nous répétons la mesure de magnéto-réflectivité à 168 K. Les spectres de réflectivité différentielle mesurés jusqu'à 66 T sont montrés sur la figure B.11(a). Les états excitoniques 1s apparaissent comme un pic prononcé dans le spectre de réflectivité avec un élargissement considérable (FWHM de 22 meV). Les états 2s apparaissent comme un pic peu profond sous champs magnétiques supérieurs à 39 T. Cependant, à une température aussi élevée, aucune transition excitonique supérieure ou niveau de Landau n'est observée. .9, nous trouvons une énergie de liaison exciton de 12±4 meV comparable à celle estimée par des expériences similaires de magnéto-transmission à haute température sur un film mince [Miyata 2015]. La valeur de l'énergie de liaison de l'exciton diminue de quelques meV lorsque l'on passe de la phase orthorhombique à la phase tétragonale. Ceci est cohérent avec le fait que la transition de phase augmente le désordre dynamique lié au mouvement de rotation du cation organique, entraînant une augmentation de la constante diélectrique [START_REF] Poglitsch | Dynamic disorder in methylammoniumtrihalogenoplumbates (II) observed by millimeter-wave spectroscopy[END_REF], Even 2014b].

Bien que le MAPbI 3 reste dans la phase tétragonale, la réponse diélectrique ne change pas, donc nous supposons que l'énergie de liaison de l'exciton reste inchangée en phase tétragonale indépendante de la température. L'énergie thermique à la température de la pièce est de l'ordre de 26%, ce qui est supérieur à l'énergie de liaison de l'exciton dans le monocristal. Dans ce cas, les électrons et les trous dans les excitons sont dissociés par l'énergie thermique à température ambiante, ce qui suggère que les porteurs libres (plutôt que les excitons) sont dominants dans le monocristal à température ambiante. Dans la figure B.15, nous montrons l'énergie de liaison et la masse réduite de l'exciton dans les composés de césium et les composés organiques-inorganiques en fonction de la bande interdite. Les étoiles brunes, rouges et jaunes présentent les résultats pour CsPbI 3 , CsPbI 2 Br et CsPbBr 3 . à titre de comparaison, nous avons ajouté les résultats des perovskites hybrides organiques-inorganiques provenant des travaux antérieurs de notre groupe marqués comme des cercles [Miyata 2015, Galkowski 2016]. Les cercles roses et violets représentent les résultats pour les pérovskites hybrides à base de FA et MA. L'évolution de l'énergie de liaison d'excitons avec la bande interdite est ajustée avec l'équation R * = 0, 010E g , comme indiqué en trait noir sur la figure B.15. Dans la Fig B.15(b), nous observons une augmentation de la masse effective avec l'augmentation de la bande interdite. Cela peut être compris dans le cadre d'un modèle k.p simple à deux bandes [Heo 2013]. En supposant que la masse efficace pour la valence et la bande de légèrement supérieure aux prédictions théoriques. L'écart peut provenir de la valeur différente de la masse effective utilisée dans le modèle.

Avec la valeur de la masse effective et de l'énergie de liaison de l'exciton, la constante diélectrique effective des matériaux peut être calculée en utilisant,

1 1 1. 1 . 1 2 1. 2

 111122 Perovskite structure . . . . . . . . . . . . . . . . . . . . . . . . Tolerance factor . . . . . . . . . . . . . . . . . . . . . . . . . . Hybrid perovskite . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Tuning the band gap . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Structural phase transition . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Electronic properties of hybrid perovskite . . . . . . . . . . 10 1.3.1 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.2 Spin-orbital coupling . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.3 Rashba effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Fully inorganic perovskite . . . . . . . . . . . . . . . . . . . . 15 1.4.1 CsPbI 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.2 CsPbBr 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 Evolution of the mixed cation perovskite . . . . . . . . . . . 18 1.5.1 Mixed cation compounds in single junction solar cells . . . . . 19 1.5.2 Mixed cation compounds in Si/perovskite tandem solar cells . 20 1.5.3 Mixed cation compounds in perovskite/perovskite tandem solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6 Absorption in semiconductors . . . . . . . . . . . . . . . . . . 22 1.6.1 Excitons in semiconductors . . . . . . . . . . . . . . . . . . . . 22 1.6.2 Free carriers in a magnetic field . . . . . . . . . . . . . . . . . . 24 1.6.3 Free excitons in a magnetic field . . . . . . . . . . . . . . . . . 26 1.1 Perovskite structure

Figure 1 . 1 :

 11 Figure 1.1: (a) Perovskite structure with the chemical formula ABX 3 . (b) the extended network structure connected by BX 6 octahedra. Red, blue, green spheres represent the A, B cations and X anion. Figure reprinted after reference [Cheng 2010].

Figure 1 . 2 :

 12 Figure 1.2: (a) UV-visible absorption spectra of MAPb(I 1-x Br x ) 3 measured with an integral sphere. (b) Photographs of MAPb(I 1-x Br x ) 3 deposited on glass substrate. After [Noh 2013].

Figure 1 . 3 :

 13 Figure 1.3: Band gap energy of MA(Sn x Pb 1-x )I 3 as a function of Sn content After [Zhao 2017].

Figure 1 .

 1 Figure 1.4: Atomic structure of methylammonium (MA) and formamidinium (FA).Figure reprinted from [Amat 2014].

  Figure 1.4: Atomic structure of methylammonium (MA) and formamidinium (FA).Figure reprinted from [Amat 2014].

Figure 1 .

 1 Figure 1.5: The crystal structure of (a) orthorhombic, (b) tetragonal, and (c) cubic phase of MAPbI 3 . The upper and lower panels are oriented though <100> and <001>, respectively.The PbI 6 octahedra are shaded grey. After [Brivio 2015].

Figure 1 .

 1 Figure 1.6: (A) A 3 µm-3 µm fluorescence image of the MAPbI 3 (Cl) perovskite film. (B) PL spectra of bright (red square) and dark (blue circle) regions. (C)-(E) Time -resolved PL decay curves of bright (red square) and dark (blue circle) regions, measured at indicated excitation power. After [Vorpahl 2015].

Figure 1 .

 1 Figure 1.7: (a) Real space 3D view of cubic crystal structure of the hybrid perovksite. (b) Reciprocal-space 3D view showing the first Brillouin zone. (c) Fermi surface (E = -0.5 eV) in the first Brillouin zone. Points R and M are connected along the edges. After [Even 2015].

Figure 1 . 8 :

 18 Figure 1.8: Electronic band structure for the cubic phase of MAPbI 3 . Vertical arrows show various possible optical transitions close to the band gap energy. Optical transitions along the line between the M and R points generate carriers that easily relax toward the R point. After [Even 2015]

Figure 1 . 9 :

 19 Figure 1.9: Electronic band structures for the (a) orthorhombic (b) cubic (c) tetragonal phase of MAPbI 3 . After [Even 2015]

Figure 1 .

 1 Figure 1.10: Electronic band structure of MAPbI 3 in the low-temperature orthorhombic phase without (a) and with (b) spin-orbit coupling interaction. The origin of the energy scale is taken at the top of the valence band. (c) Overview of the orthorhombic phase crystal structure. After [Even 2013].

Figure 1 .

 1 Figure 1.11: Schematic view of band dispersion in the absence of Rashba effect (a), and band dispersion including the Rashba effect. After [Kepenekian 2017]

Figure 1 .

 1 Figure 1.13: Band structure of (a) yellow and (b) black phase of CsPbI 3 with (red solid lines) and without (black dashed lines) SOC. Horizontal lines represent the energy level of the VBM. After [Brgoch 2014].

Figure 1 .

 1 Figure 1.15: Schematics view of two-terminal and four-terminal tandem solar cell structure. After [Eperon 2016].

Figure 1 .

 1 Figure 1.16: SEM of the monolithic perovskite/perovskite tandem solar cell. After [Eperon 2016].

Figure 1 .

 1 Figure 1.17: (a) Schematic of the free exciton in a two-dimensional crystal lattice with lattice constant a. (b) The energy states of a free exciton showing its bound states and the continuum states. E g is the bandgap and Ry * the effective binding energy. After [Pelant 2012].

Figure 1 .

 1 Figure 1.18: (a) Schematic figure of the Landau levels in the conduction and valence band. (b) Energy of Landau levels interband transitions as a function of magnetic filed.

Figure 1 .

 1 Figure 1.19: The calculated transition energies of hydrogen-like excitonic states as a function of magnetic field for R * = 16 meV and m * = 0.1 m 0 . Inset: a close-up view of excitonic transition at low magnetic field.

Figure 2 . 1 :

 21 Figure 2.1: (a) Panoramic view of 14 MJ capacitor bank; (b) copper wires for the coil; (c) photo of a complete 70 T coil; (d) a standard coil mounted in a liquid nitrogen cryostat.

Figure 2 . 2 :

 22 Figure 2.2: Representative magnetic field pulses as a function of time for different coils systems. Diameter of the magnet bore and the energy required for each systems are marked next to each curve.

Figure 2 .

 2 Figure 2.3: (a) A schematic view and (b) a photo of a pulsed magnetic field experiment setup.

Figure 2 .

 2 Figure 2.4: Schematic drawing of the transmission measurement in pulse magnetic field.

Figure 2

 2 Figure 2.5: (a) Photo of the magneto-transmission probe. (b) Zoom of the sample holder. (c) Schematic of the optical path.

Figure 2

 2 Figure 2.6: Typical calibrated resistance/temperature relationship of resistance thermometer on the transmission probe. Inset: schematic diagram of four-terminal sensing.

Figure 2 .

 2 Figure 2.7: Schematic drawing of the reflectivity measurement in pulse magnetic field.

Figure 2 .

 2 Figure 2.8: (a) Photo of the magneto-reflectivity probe. (b) Zoom on the bottom of the probe, where the pick-up coil and thermometer can be seen clearly. Inset: section of fiber bundle at the bottom of the probe. (c) Sample holder with the sample inside. Inset shows disassembled sample holder.

Figure 2 .

 2 Figure 2.9: Schematic diagram of synchronization sequence for magneto-optical measurements

Figure 2 .

 2 Figure 2.10: Magnetic field (black solid line), pick-up voltage (violate solid line) and CCD acquisition as a function of time during 68 T pulse. The synchronization of CCD acquisition and maximum magnetic field are shown in the inset.

Contents 3 . 1

 31 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Magneto-transmission spectroscopy on MAPbI 3 . . . . . . . 42 3.2.1 Description of the samples . . . . . . . . . . . . . . . . . . . . . 42 3.2.2 Magneto-transmission spectra on LPC sample . . . . . . . . . . 43 3.2.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.4 Magneto-transmission spectra on SPC, SC and MP samples . . 45 3.2.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 thin films . . . . . . . 49 3.3.1 Magneto-transmission spectra . . . . . . . . . . . . . . . . . . . 50 3.3.2 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 3 . 1 :

 31 Figure 3.1: Top view SEM images of MAPBI 3 thin film deposited on glass substrates with various morphologies. (A) Polycrystalline thin film with an average crystal grain size 770±230 nm (LPC). (B) Polycrystalline thin film with grain size 210±60 nm (SPC). (C) Small crystals of MAPBI 3 fabricated with the two-step solution-processed dipping technique, resulting crystal size 290±70 nm (SC). (D) MAPbI 3 infiltrated into a thick mesoporous-Al 2 O 3 , resulting crystal grain size < 50 nm (MP). Scale bars are 1 µm for each panel.

Figure 3 . 2 :

 32 Figure 3.2: Typical magneto-transmission spectra of a LPC sample. (a) Sequences of typical transmission spectra of the LPC sample measured at the indicated magnetic fields. (b) A sequences of the ratios of the transmission spectra measured at intermediate magnetic field, which clearly shows the 2s state. (c) A sequences of the ratios of the transmission spectra in the higher energy range, which clearly shows the Landau level interband transitions.

Figure 3 . 3 :

 33 Figure 3.3: Typical results of the low temperature monochromatic transmission as a function of magnetic field obtained through the Mega Gauss technique performed by Dr. Atsuhiko Miyata.

Figure 3 .

 3 Figure 3.4: (a) Typical results of absorption energies as a function of magnetic field. The blue, red and violet spheres comes from the long pulse magnetic field measurement, while the black stars comes from Mega Gauss measurement measured by Dr. Atsuhiko Miyata in LNCMI-T. (b) Fitting of free carriers interband transition between Landau levels conduction band and valance band (blue solid lines). (c) The plot from panel (b) supplemented with the fitting curve of hydrogen like model for 1s, 2s and 2p (blue, red and black dashed lines).

Figure 3 .

 3 Figure 3.5: Panel (a)-(c) Low temperature transmission spectra measured at indicated magnetic fields for SPC, SC and MP samples, respectively. Panel (d)-(f) The ratios of the low temperature transmission spectra in magnetic field T (B) to that measured at zero field T (0) for for SPC, SC and MP samples, respectively.

Figure 3 .

 3 Figure 3.6: Overlaid fan chart of the LPC, SPC, SC and MP samples. The solid and dashed lines are the fits to the set of Landau levels and the excitonic transitions, respectively. The bottom panel shows the zoomed-in plot of the excitonic transition region of the fan chart.

Figure 3 .

 3 Figure 3.7: (a)-(d) Separate energy fan charts of the MP, SC, SPC and MP samples, respectively. The spheres present the data points from long pulse measurement. The stars present the data points from Mega Gauss measurement performed by A. Miyata. The solid and dashed lines are the fits to the set of Landau levels and the excitonic transitions, respectively.

Figure 3 .

 3 Figure 3.8: Exciton binding energy R * and reduced mass µ for the various morphologies.

Figure 3 .

 3 Figure 3.9: Top-view SEM images of Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 thin films with two morphologies deposited on glass substrates. (A) Polycrystalline planar thin film. (B) Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 infiltrated into a thick MP-Al 2 O 3 . Scale bars are 200 nm.

  Figure 3.9: Top-view SEM images of Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 thin films with two morphologies deposited on glass substrates. (A) Polycrystalline planar thin film. (B) Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17 ) 3 infiltrated into a thick MP-Al 2 O 3 . Scale bars are 200 nm.

Figure 3 .

 3 Figure 3.10: Left panels: typical transmission spectra of the polycrystalline (a) and perovskite-infiltrated mesoporous-Al 2 O 3 (c) samples measured at indicated magnetic field at 2 K. Right panels: The ratio of the transmission spectra in the magnetic field T (B) to that measured at zero field T (0) for both morphologies sample.

Figure 3 .

 3 Figure 3.11: Overlaid fan chart of the planar and mesoporous samples. The solid lines and the dashed lines are the fits to the set of Landau levels and the excitonic transitions, respectively.

2 Figure 3 .

 23 Figure 3.12: (a) Separate energy fan chart of the planar and mesoporous samples. The solid lines and the dashed lines are the fits to the set of Landau levels and the excitonic transitions, respectively.

Figure 4 . 1 :

 41 Figure 4.1: A photo of single crystal MAPbI 3 sample. The scale bar is 2 millimeter.

Figure 4

 4 Figure 4.3: A negative differential reflectivity spectrum (-dr/dE) is shown as black line. The orange dashed line shows the fit to the data.

Figure 4

 4 Figure 4.4: (a) Differential reflectivity spectra measured at indicated magnetic field values at 2 K. The arrows mark the absorption peaks. Inset: As-measured reflectivity spectrum at zero magnetic field. (b) High magnetic field differential reflectivity spectra after normalization by zero-field reflectivity. Arrows of the same colors indicate the transition between the same Landau levels.

Figure 4 . 5 :

 45 Figure 4.5: Energies for excitonic and Landau level transitions as a function of the applied magnetic field at 2 K (symbols). Blue solid lines are the results of a linear fit of the interband transition between Landau levels in the valence and conduction bands. Dashed lines result from the fit of the hydrogen-like transitions. Inset: Expanded view of the low-energy transitions.

Figure 4

 4 Figure 4.6: (a) Differential reflectivity spectra measured at different temperatures. (b) 1s (green circle) and 2s (red circle) transition energy as a function of the temperature. Inset: Exciton binding energy (estimated here from the 1s-2s splitting) as a function of the temperature.

Figure 4

 4 Figure 4.7: (a) The examples of the differentiate reflectivity fitted using Eq. 4.2 at different temperatures. Solid lines are the differentiate reflectivity spectra, dashed lines present the fitted responses. (b) The broadening Γ of free exciton transitions as a function of temperature determined from the fitting of temperature dependence differentiate reflectivity spectra.

Figure 4

 4 Figure 4.8: (a) Magneto-reflectivity spectra measured at T=168 K differentiated with respect to energy. (b) High-temperature energy fan chart. The dashed lines indicate hydrogenic transitions.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Description of the samples and pre-experimental treatment 68 5.3 Temperature dependent transmission measurement . . . . . 70 5.4 Low temperature magneto-transmission measurement . . . 72 5.4.1 Typical results and data analysis . . . . . . . . . . . . . . . . . 73 5.4.2 Dielectric screening . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5 High temperature magneto-transmission measurement . . . 77 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Figure 5 . 1 :

 51 Figure 5.1: (a)-(c) Photos of the annealed CsPbBr 3 , CsPbI 2 Br and CsPbI 3 samples, respectively.

Figure 5 . 2 :

 52 Figure 5.2: (a)-(c) Transmission spectra measured at 4 K for annealed CsPbBr 3 , CsPbI 2 Br and CsPbI 3 sample, respectively.

Figure 5 . 3 :

 53 Figure 5.3: Typical transmission spectra of (a) CsPbBr 3 , (b) CsPbI 2 Br and (c) CsPbI 3 sample measured at different temperatures. Circle symbols represent the evolution of the 1s absorption with temperature. (d) Energies of the 1s transitions as a function of temperature for all three compounds. The black dashed lines and orange solid lines in panel (a) represent the spectra of annealed and as prepared CsPbBr 3 . The black and orange circles in panel (d) represent the energies of 1s transitions for annealed and as prepared CsPbBr 3 .

Figure 5

 5 Figure 5.4: (a)-(c) Transmission spectra measured at the indicated magnetic field values for all the three compounds. (d)-(f) Transmission spectra in magnetic field divided by the zero field spectrum for three compounds. (g)-(i) Monochromatic transmission as a function of magnetic field for three compounds obtained by the short pulse technique performed by A. Miyata. The measurements were performed at 2 K.

Figure 5 . 5 :

 55 Figure 5.5: Transition energies of the excitons and free carriers as a function of the magnetic field at 2 K for (a) CsPbI 3 , (b) CsPbI 2 Br and (a) CsPbBr 3 . Circles represent the data from long pulse field measurement. Stars are the data from the Mega Gauss measurement performed by A. Miyata. Orange and gray lines are the results of the fit of the hydrogen-like transition and the interband transition between Landau levels. The lower panels show the zoom-in view of the low field and low energy portion of the fan chart.

Figure 5

 5 Figure 5.6: (a) Binding energy and (b) effective mass as a function of the band gap. Brown, red and yellow stars present the results for CsPbI 3 , CsPbI 2 Br and CsPbBr 3 . Pink and violet circles represent the results for the FA and MA based hybrid perovskites, from our previous work [Miyata 2015, Galkowski 2016].

Figure 5 . 7 :

 57 Figure 5.7: Dielectric constant as a function of the band gap. Brown, red and yellow stars present the results for CsPbI 3 , CsPbI 2 Br and CsPbBr 3 . Pink circles represent the results for the FA based hybrid perovskites, while the violet circles represent the results for the MA based hybrid perovskites.

Figure 5

 5 Figure 5.8: Transmission spectra measured at indicated magnetic field and at 180 K for CsPbBr 3 .

Figure 5

 5 Figure 5.9: 1s transition as a function of magnetic field in CsPbBr 3 at two different temperature.

Figure B. 1 :

 1 Figure B.1: (a) Structure pérovskite avec la formule chimique ABX 3 . (b) Le structure de réseau étendu connecté par octahedre BX 6 . Les sphères rouges, bleus, vertes représentent réspectivement les cations A, B et l'anion X. D'après [Cheng 2010].

Figure B. 3 :

 3 Figure B.3: Spectres typiques de magnéto-transmission d'un échantillon LPC. (a) Séquences de spectres de transmission typiques d'échantillon LPC mesurées sous les champs magnétiques indiqués. (b) Une séquence des rapports des spectres de transmission mesurés sous champ magnétique intermédiaire, qui montre clairement l'état 2s. (c) Une séquence des rapports des spectres de transmission dans la gamme d'énergie supérieure, qui montre clairement les transitions interbandes des niveaux de Landau.

Figure

  Figure B.4: Résultats typiques de la transmission monochromatique à basse température en fonction du champ magnétique obtenus par la technique de Mega Gauss réalisée par le Dr Atsuhiko Miyata.

Figure

  Figure B.5: (a) Résultats typiques des énergies d'absorption en fonction du champ magnétique. Les sphères bleues, rouges et violettes proviennent de la mesure du champ magnétique à impulsions longues, tandis que les étoiles noires proviennent de la mesure de Mega Gauss mesurée par le Dr. Atsuhiko Miyata au LNCMI-T. (b) Ajustement des transitions entre les porteurs libres et les niveaux de Landau de la bande de conduction et de la bande de valence (lignes continues bleues). (c) La courbe du panneau (b) complétée par la courbe d'ajustement du modèle de type hydrogène pour 1s, 2s et 2p (lignes pointillées bleues, rouges et noires).

  Fig. B.5(b). Pour cet échantillon, la masse réduite µ est estimée à 0.102 ± 0.002m 0 . Ensuite, dans un second temps, nous utilisons les solutions numériques de Makado et McGill pour que l'hamiltonien ressemblant à l'hydrogène corresponde aux données. Les paramètres d'ajustement de cette solution sont la masse réduite d'exciton µ et l'énergie de liaison d'exciton R * . Nous utilisons la même masse réduite µ extraite des transitions inter-bandes de Landau. Les résultats d'ajustement sont représentés par des lignes pointillées dans la Fig. B.5(c). Dans cet échantillon, nous avons observé trois transitions excitoniques-1s, 2s et 2p. L'observation des deux états 1 et 2 donne une forte contrainte sur la valeur de l'énergie de liaison d'exciton. L'énergie de liaison de l'exciton dans l'échantillon LPC est estimée à 16±1 meV. B.3.3 Résultats de la magnéto-transmission sur les échantillons SPC, SC et MP

Figure

  Figure B.6: (a) -(d) Diagrammes en ventilateurs d'énergie séparer des échantillons MP, SC, SPC et MP, respectivement. Les sphères représentent les points de données à partir d'une mesure d'impulsion longue. Les étoiles représentent les points de données issus de la mesure de Mega Gauss effectuée par A. Miyata. Les lignes pleines et pointillées sont respectivement les ajustements à l'ensemble des niveaux de Landau et les transitions excitoniques,.

Figure

  Figure B.8: (a) Spectres de réflectivité différentielle mesurés aux valeurs de champ magnétique indiquées à 2 K. Les flèches indiquent les pics d'absorption. Encart: Spectres de réflectivités tel que mesurés sous un champ magnétique nul. (b) Spectres de réflectivité différentielle sous champ magnétique élevé après normalisation selon la réflectivité à champ nul. Les flèches de mêmes couleurs indiquent la transition entre les mêmes niveaux de Landau.

Figure B. 9 :

 9 Figure B.9: Energies des transitions excitoniques et des niveaux Landau en fonction du champ magnétique appliqué à 2 K (symboles). Les lignes continues bleues sont le résultat d'un ajustement linéaire des transitions interbandes entre les niveaux de Landau dans les bandes de valence et de conduction. Les lignes pointillées résultent de l'ajustement des transitions de type hydrogène. Encart: Vue agrandie des transitions à basse énergie.

Figure

  Figure B.10: a) Spectres de réflectivité différentielle mesurés à différentes températures. (b) 1s (cercle vert) et 2s (cercle rouge) énergie de transition en fonction de la température. Encart: énergie de liaison de l'exciton (estimée ici à partir de la division 1s-2s) en fonction de la température.

  Fig. B.11(b) représente le diagramme d'énergie à 168 K. Nous avons ajusté les données pour la transition 1s et

Figure

  Figure B.11: (a) Dérivée par rapport à l'énergie des spectres de magnéto-réflectivité mesurés à T = 168 K. (b) Diagramme d'énergieà haute température. Les lignes pointillées indiquent des transitions hydrogéniques.

B. 5

 5 Figure B.12: (a) -(c) Photos des échantillons recuits CsPbBr 3 , CsPbI 2 Br et CsPbI 3 , respectivement.

Figure B. 13 :

 13 Figure B.13: Spectres de transmission typique de (a) l'échantillon CsPbBr 3 , (b) CsPbI 2 Br et (c) CsPbI 3 mesurés à différentes températures. Les symboles de cercle représentent l'évolution de l'absorption de 1s avec la température. (d) Energie des transitions 1s en fonction de la température pour les trois composés. Les lignes pointillées noires et les lignes pleines orange dans le panneau (a) représentent les spectres de CsPbBr 3 recuits et préparés. Les cercles noirs et orange du panneau (d) représentent les énergies des transitions 1s pour CsPbBr 3 recuit et préparés.

Figure B. 14 :

 14 Figure B.14: Les énergies de transition des excitons et des porteurs libres en fonction du champ magnétique à 2 K pour (a) CsPbI 3 , (b) CsPbI 2 Br et (a) CsPbBr 3 . Les cercles représentent les données issues de la mesure du champ de l'impulsion longue. Les étoiles sont les données de la mesure Mega Gauss effectuée par A. Miyata. Les lignes orange et grises sont le résultat de l'ajustement de la transition de type hydrogène et de la transition inter-bande entre les niveaux de Landau. Les panneaux inférieurs montrent le zoom avant de la partie basse et basse énergie du diagramme d'éénergie.

  Figure B.16: Constante diélectrique en fonction de la bande interdite. Les étoiles brunes, rouges et jaunes représentent les résultats pour CsPbI 3 , CsPbI 2 Br et CsPbBr 3 . Les cercles roses représentent les résultats pour les pérovskites hybrides à base de FA, tandis que les cercles violets représentent les résultats pour les pérovskites hybrides à base de MA.

Table 1 .

 1 1: Temperature dependence of the crystal structure of MAPbX 3 (X=Cl, Br or I). Data after the reference [Oku 2015].

	Matrial	Cubic	Tetragonal	Orthorhombic
	MAPbI 3	> 330 K 161 K to 330 K	< 161 K
	MAPbBr 3 > 236 K 154 K to 236 K	< 149 K
	MAPbCl 3 > 177 K 172 K to 177 K	< 172 K

  ) Magnetic field generation . . . . . . . . . . . . . . . . . . . . . 29 2.1.1 Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

	Chapter 2
	Experimental Techniques
	Contents
	2.1

2.1.2 Resistive coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.3 Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Magneto transmission measurements . . . . . . . . . . . . . 33 2.2.1 Pick-up coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.2 Resistance thermometer . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Magneto reflectivity measurements . . . . . . . . . . . . . . . 37 2.4 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Table 2 .

 2 1: The inner diameters of the magnet bore and the extension suitable for three different magnet.

	Magnet Magnet bore (mm) Extension (mm)
	100 T	8	4
	70 T	13	7
	60 T	28	21

Table 3 . 1 :

 31 Summary of the fitting parameters for the four morphologies of MAPbI 3 in the low temperature (2 K).

		Crystal size (nm) E g (meV)	µ (m 0 )	Ry * (meV)
	LPC	770 ± 230	1642 ± 2 0.102 ± 0.002	16 ± 1
	SPC	290 ± 70	1643 ± 2 0.105 ± 0.002	16 ± 4
	SC	210 ± 60	1639 ± 2 0.109 ± 0.002	16 ± 4
	MP	<50	1638 ± 2 0.107 ± 0.003	16 ± 4
	mass varies from 0.102 m 0 to 0.109 m 0 , while the binding energy remain the same
	for all the samples within experimental error. The binding energy of the exciton in
	the LPC sample is (16 ± 1) meV. While in SC, SPC, MP samples, it is (16 ± 4) meV.

The difference in the size of the error bar comes from the observation of 2s state in LPC sample. The energy difference between the 1s and 2s state gives a strong

Table 3 . 2 :

 32 Summary of the fitting parameters for the two different morphologies of Cs 0.05 (MA 0.17 FA 0.83 ) 0.95 Pb(I 0.83 Br 0.17

Table 5 .

 5 1: Parameters determined from the fit of the energy fan chart for CsPbX 3 at 2 K.

	Compound	Phase	E g (meV) R * (meV)	µ (m 0 )
	CsPbI 3	cubic	1723	15 ± 1	0.114 ± 0.01
	CsPbI 2 Br	cubic	1906	22 ± 3	0.124 ± 0.02
	CsPbBr 3	orthorhombic	2342	33 ± 1	0.126 ± 0.01

Table 5 .

 5 2: Calculated effective dielectric constant for CsPbX 3 at 2 K.

	Compound	Phase	E g (meV)	ε eff
	CsPbI 3	cubic	1723	10.0 ± 0.4
	CsPbI 2 Br	cubic	1906	8.6 ± 0.8
	CsPbBr 3	orthorhombic	2342	7.3 ± 0.4

Table B .

 B 1: Résumé des paramètres d'ajustement pour les quatre morphologies de MAPbI 3 à basse température (2 K) Ensuite, nous avons effectué la même mesure et analyse sur d'autres échantillons MAPbI 3 (SPC, SC, MP) avec différentes tailles de grains. Nous avons également adapté les données pour les quatre morphologies de MAPbI 3 séparément, comme le montre la Fig. B.6. Les sphères représentent les points de données de la mesure d'impulsion longue. Les étoiles présentent les points de données de la mesure Mega Gauss effectuée par Dr. A. Miyata. Les lignes pleines et pointillées sont respectivement les ajustements de l'ensemble des niveaux de Landau et les transitions excitoniques. Le paramètre d'ajustement pour les quatre échantillons est résumé dans le tableau B.1. Dans la Fig. B.7, nous montre une représentation du tableau B.1. La masse réduite et l'énergie de liaison sont présentées avec respectivement des cercles rouges et des carrés bleus. L'image MEB de chaque échantillon est insérée en bas de la figure

		Crystal size (nm) E g (meV)	µ (m 0 )	Ry * (meV)
	LPC	770 ± 230	1642 ± 2 0.102 ± 0.002	16 ± 1
	SPC	290 ± 70	1643 ± 2 0.105 ± 0.002	16 ± 4
	SC	210 ± 60	1639 ± 2 0.109 ± 0.002	16 ± 4
	MP	<50	1638 ± 2 0.107 ± 0.003	16 ± 4

Acknowledgements

First of all, I would like to express my gratitude to my supervisors Paulina Plochocka and Duncan Maude, for their guiding and patience during my PhD research. Their serious attitude toward science strongly influences me. Their sense of humor makes a comfortable atmosphere for the work in the lab. I would also like to thank the members of the jury for their valuable comments on this research.

I would also like to acknowledge all my amiable collaborators, Robin J. Nicolas and Rebecca J. Sutton (University of Oxford), Sam Stranks and Baodan Zhao (University of Cambridge), Arman M. Soufiani (University of New South Wales), Anna Fontcuberta i Morral and Gozde Tutuncuoglu ( École Polytechnique Fédérale de Lausanne), Lukasz Klopotowski (Polish Academy of Sciences), for their excellent experimental work and enjoyable discussion. A special thank to Sam Stranks for hosting me one month in Cavendish Laboratory, the University of Cambridge.

A great thank to China Scholarship Council for awarding a 42 months scholarship, which allows me to pursue my study in France

La figure B.2 représente des images de microscopie électronique (MEB) de la vue de dessus des échantillons. Les morphologies varient d'un film polycristallin plan avec des tailles de grain de 770±230 nm (A: polycristallin à gros grains annoté LPC) et un film polycristallin avec des tailles de grain de 210±60 nm (B: petit grain polycristallin annoté SPC), à de petits cristaux de 290±70 nm fabriqués par une technique de trempage en deux étapes. (C: petit cristal annoté SC) et également perovskite infiltré dans un échafaudage d'oxyde d'aluminium mésoporeux épaisse de 2 µm (mp-Al 2 O 3 ) avec des tailles de grains <50 nm (D: cristal mésoporeux annoté MP).

B.3.2 Spectres de magnéto-transmission sur un échantillon de LPC

L'approche expérimentale et l'analyse des données utilisées pour tous les échantillons seront expliquées en détail en utilisant comme exemple des résultats de l'échantillon B.5.2 Mesure de transmission en fonction de la température Dans la Fig. B.13, nous montrons les spectres de transmission typiques de CsPbBr 3 , CsPbI 2 Br, CsPbI 3 mesurés sur une large gamme de températures variant de 4 K à 270 K. Chaque spectre de transmission montre un minimum unique clair, qui correspond à la 1s excitonique absorption. Le bleu minimum se déplace et s'élargit avec l'augmentation de la température pour tous les composés. Nous traçons l'énergie du minimum des résonances d'absorption en fonction de la température de la figure B.13(d). Les énergies d'absorption changent continuellement avec l'augmentation de la température. Dans les matériaux perovskites organiques-inorganiques, la transition de phase structurale s'accompagne toujours d'un changement dramatique d'énergie de bande interdite [Stoumpos 2013a, Baikie 2013, Galkowski 2017]. Ainsi, le décalage continu de la bande interdite en fonction de la température suggère qu'il n'y a pas de transition de phase structurelle dans CsPbX 3 . Ce phénomène est en contraste frappant avec les perovskites hybrides organiques-inorganiques. Dans la plage de température de 4 K-300 K, les pérovskites hybrides organiques-inorganiques présentent une transition de phase structurale autour de 150 K-170 K de la phase orthorhombique basse température à la phase tétragonale haute température [Stoumpos 2013a, Baikie 2013].

B.5.3 Mesure de magnéto-transmission à basse température

Nous avons effectué une spectroscopie de magnéto-transmission à basse température pour étudier les propriétés électroniques des composés CsPbX 3 , en utilisant des techniques de champs magnétiques pulsés longs. Les résultats sont présentés et discuté ci-dessous.

B.5.3.1 Résultats typiques et analyse des données

Nous analysons les données de magnéto-transmission pour tous les composés Cs de la même manière que pour MAPbI où P = Ψ VB |p x |Ψ CB est l'élément de matrice de quantité de mouvement qui couple les états des bandes de conduction et de valence. 2|p| 2 /m 0 est l'énergie de Kane [Makado 1986]. Les prédictions théoriques de l'énergie de Kane sont de l'ordre de 5.