Magneto-optical absorption spectroscopy
In the previous section, ARPES as surface sensitive probe is shown to be a powerful tool allowing us to study the electronic band structure of Dirac materials. In this section, infrared magneto-optical absorption spectroscopy, the technique we used in this thesis to probe and characterize Dirac matter, will be described. When the crystal surface is subjected perpendicular to an applied magnetic√field , 2the+ electron states will be quantized into relativistic Landau levels dispersing as or , where and are band parameters. This is a typical characteristic feature of Dirac fermions. The optical transitions occurring between these Landau levels give important information about the physical parameters of the electronic band structure of bulk states as well as surface states. Infrared spectral range is chosen thanks to the energy compatibility for probing semiconductors of which the energy gap is less than 1 eV. This is the primary reason why magneto-optical absorption spectroscopy in the infrared domain is primarily used in this thesis to investigate Dirac matter.
Figure 1.3. Photo of magneto-optical spectroscopy experimental setup.
Fig. 1.3 shows the whole experimental setup used to probe Dirac fermions. The principal element is the Oxford Instruments 1.5K/17T cryostat, situated at the center of the photo, equipped with a superconducting coil. It allows us to do experiments in the temperature range 1.5 K < < 220 K and under magnetic fields = 0-17 T. The Fourier transform infrared (FTIR) interferometer (Bruker VERTEX 80V), located in the upper left hand corner of the photo, is employed as the infrared light source and the spectral analysis apparatus at the same time. These two essential elements are connected by a coupler containing a parabolic mirror. Detailed information about the experimental setup and the data acquisition will be described in the following subsections.
Sample preparation for measurement
Samples are first prepared and attached at the bottom of the sample probe for magneto-optical absorption measurement. In this subsection, three important parts will be described: the sample probe, the sample holder and the bolometer used as a detector of transmitted signals.
Figure 1.4. Different parts of the sample probe. (a) The sample is placed at the bottom of the sample probe for measurement. The sample probe envelope is used to avoid any contact between the sample and the exterior environment. (b) Zoom of the top of the sample probe.
A 1.5 m long sample probe was designed to mount samples for magneto-optical absorption measurement and to maintain three electrical channels carried by two nonmagnetic coaxial cables (Fig. 1.4(b)). The inner walls of the sample probe guide the incident infrared light reaching towards the sample. The three channels are used for the ±18 V bolometer power supply, the signal acquisition of the bolometer and the ground. The sample probe envelope (Fig. 1.4(a)) is necessary for protecting the cables, the bolometer and the sample from the exterior environment when the sample probe is immersed in the cryostat filled with liquid helium~.×After the sample is mounted and sealed, the sample probe is primarily evacuated down to 1 10-2 mbar. It is then filled with helium exchange gas, up to a pressure of 80-800 mbar at room temperature to ensure sample thermalisation, before being put into the variable temperature insert (VTI) of the cryostat for measurement. The pressure in the sample probe is maintained owing to a diamond window located at the connection between the sample probe and the coupler. The diamond window enables also an optimal passage of the transmitted signals throughout the infrared range. For the investigation of Dirac fermions in graphene and topological insulators, the pressure of the helium exchange gas is about 100-120 mbar at room temperature.
Fig. 1.5(a) shows two kinds of sample holders: sample holder with one hole and rotating sample holder with two holes. The sample holder used for the transmission experiment has several diameters. An appropriate diameter for a given sample is chosen for maximizing the transmitted signals. Fig. 1.5(b) shows a sample bonded on a sample holder. The rotating sample holder can mount two pieces of samples as seen in Fig. 1.5(c). The rotating system (Fig. 1.4(b)) joining the rotating sample holder allows switching from one sample to another sample in situ. This is very practical for a measurement requiring a normalization between two consecutive transmission spectra at the same applied field. To glue a sample on a sample holder, we use silver paste or PMMA (Poly(methyl methacrylate)).
Figure 1.5. Examples of sample holders. (a) Examples of a sample holder (with one hole) and a rotating sample holder (with two holes). There are several diameters adapting to the dimension of the sample. (b) A sample glued with silver paste to the sample holder. (c) A sample and a substrate glued with silver paste to the rotating sample holder.
Figure 1.6. Infrared Si-composite bolometer with a diamond window. The Si bolometer and the sample are attached to the bottom of the sample probe.
The bolometer is a photo-detector used for spectrum measurement. Its operating principle is to convert the energy of the incident electromagnetic radiation on the surface of a metallic or semiconductor absorber into heat. A Si-composite bolometer (Infrared Laboratories) equipped with a diamond window was used in this thesis for infrared magneto-optical absorption measurement. It is used to collect the transmitted light directly below the sample (Fig. 1.6). The signal from the bolometer passes through an external preamplifier before being transmitted to the FTIR interferometer for analysis. It is possible to adjust the amplification factor (200, 2,000 and 5,000) of the preamplifier to obtain satisfying signal intensity.
Fourier transform infrared (FTIR) interferometer
Infrared magneto-optical spectroscopy is a technique employed to obtain transmission spectra of a sample (intensity as a function of energy) in the infrared domain (30-7500 cm-1 or 4-930 meV). The laboratory is equipped with a Bruker VERTEX 80V Fourier transform infrared (FTIR) interferometer monitored by the OPUS operating software. This spectrometer plays two essential roles as infrared light source and spectral analysis tool.
Operating principle of the FTIR interferometer
As represented in Fig. 1.7, The FTIR interferometer possesses two infrared light sources: far-infrared (FIR) source for 30-700 cm-1 and mid-infrared (MIR) source for 700-7500 cm-1. The light beam is collimated and directed towards a beam splitter and a system of associated mirrors. The half portion of the signal is transmitted to a mobile mirror which can move on nitrogen cushion thanks to a motor. When the mirror moves, each wavelength is periodically blocked or transmitted by the interferometer by interference phenomenon. Finally, the light emerging from the spectrometer is sent towards the cryostat using a vacuum coupler with a parabolic mirror. The incident light is then focused on the sample placed above the bolometer. The detector measures the light intensity remaining after passing through the sample and sends the transmitted signal, after amplification, to the FTIR interferometer for spectral analysis.
Figure 1.8. OPUS control window showing an interferogram. The central peak corresponds to the zero path difference (ZPD) position of the mobile mirror at which the maximum of light passes through the interferometer towards the detector.
In order to get a good signal/noise ratio, each final spectrum is obtained after acquisition and average of several spectra. The number of averaged spectra is proportional to a parameter which is the number of scans. It can typically be selected among the values of 64, 128 or 256 scans. Furthermore, the maximal spectral resolution can be adjusted up to 0.2 cm-1. The spectral resolution chosen for our magneto-optical absorption experiment is 5 cm-1. Note that the vacuum is essential during the measurement in the FTIR interferometer, the entire optical path and inside the coupler in order to avoid the absorption of the infrared light beam by the atmospheric gases (H2O, O2, CO2 , etc.).
Cryostat and superconducting coil
As illustrated in Fig. 1.9(a), the cryogenic storage dewar of total volume of 85 L contains a superconducting coil at the bottom of the cryostat and a variable temperature insert (VTI), resulting finally in a capacity of 46 L of liquid helium. The VTI is separated from the exterior container by the inner vacuum shield, consequently, the temperature of the sample can be varied to be different from the temperature of liquid helium (4.2 K). To decrease the temperature below 4.2 K, we introduce liquid helium from the exterior container into the VTI via the needle valve and then pump out the pressure in the VTI. To increase the temperature above 4.2 K, we use the Oxford Instruments ITC503 automated control/heater apparatus that allows us to fix the desired temperature. The sample at the bottom of the sample probe is placed at the heart of the superconducting coil as seen in Fig. 1.9(b). The sample holder is surrounded by the sample probe envelope to avoid any direct contact between the sample and liquid helium. The control and power supply of the superconducting coil are provided by the Oxford Instruments IPS120-10 apparatus, enabling to work at fixed magnetic fields and to sweep the field with a maximum speed of 1 T/minute.
Figure 1.9. Schematics of the cryostat equipped with a superconducting coil. (a) The dewar consists of two containers: an interior one or the variable temperature insert (VTI) and an exterior one containing the superconducting coil immersed in liquid helium. The maximum and minimum filling levels of liquid helium are indicated. The opening of the needle valve lets flow liquid helium from the exterior container into the VTI. (b) Zoom of the superconducting coil and the bottom of the sample probe. The heat exchange between the sample and the VTI is via a helium exchange gas of a pressure of 80-800 mbar at room temperature.
In this thesis, all experimental results were obtained from infrared magneto-optical absorption measurement. Fig. 1.10 displays the whole experimental setup used to probe Dirac fermions in graphene and topological insulators. The process of spectra acquisition is as follows. The infrared light beam generated from FIR or MIR sources passes by the beam splitter and the system of associated mirrors in the vacuum FTIR interferometer and is then transmitted to the entrance of the sample probe using the vacuum coupler. The parabolic mirror inside the coupler bends the light beam to propagate directly to the sample placed at the center of the superconducting coil. The magnetic field is oriented perpendicular to the sample surface in Faraday geometry and can be varied up to = 17 T. Each measurement is performed at a constant magnetic field. The temperature is fixed at 4.5 K. The Si bolometer detects the transmitted light directly below the sample. The transmission signals are acquired, then amplified and sent to the FTIR interferometer for spectral analysis. The corresponding interferogram is obtained after the analysis and will then be converted by Fourier transform calculation to the transmission spectrum.
In this work, the study of Dirac matter was first devoted to graphene: the first truly two-dimensional crystal, composed of carbon atoms, ever found in nature. The fundamental study of the theoretical aspects and experimental realization of graphene has always retained this research area active in condensed matter physics after the 2010 Nobel Prize in Physics was awarded jointly to A. K. Geim and K. S. Novoselov for « groundbreaking experiments regarding the two-dimensional material graphene ». In particular, the most intriguing typical characteristic of graphene, at low energies, is that its unusual linear energy-momentum dispersion is similar to the physics of quantum electrodynamics for massless fermions but the Dirac velocity of these particles is 300 times smaller than the speed of light. This completely differs from ordinary electrons when subjected to magnetic fields. Graphene is thus a model system of Dirac matter allowing us to study the relativistic behavior of Dirac fermions in analogy with high-energy physics.
In this chapter, the electronic properties of an ideal graphene and graphene stacks will be addressed by magneto-optical spectroscopy. Different methods of graphene fabrication will be briefly described. We will essentially focus on the behavior of Dirac fermions in multilayer epitaxial graphene, fabricated by thermal decomposition of SiC substrates, which were investigated using infrared magneto-optical absorption measurements. Experimental results of multilayer epitaxial graphene on the C-terminated and Si-terminated faces of SiC substrates will be shown.
Electronic properties of graphene
From a purely theoretical point of view, graphene is a two-dimensional (2D) one-atom-thick allotrope of carbon. As represented in Fig 2.1(a), graphene is the mother for other carbon materials in different dimensionalities owing to the flexibility of the carbon-carbon bonding present in its honeycomb lattice structure. One can obtain a fullerene molecule (0D) from wrapped-up graphene with the introduction of pentagons (Fig. 2.1(b)), a carbon nanotube (1D) by rolling up graphene along a chosen direction (Fig. 2.1(c)), and a graphite (3D) by stacking many graphene layers connected by van der Waals force (Fig. 2.1(d)).
Figure 2.1. Allotropes of carbon. (a) Graphene is a 2D honeycomb lattice structure of carbon atoms. It is a mother building material for carbon materials in other dimensionalities. (b) Fullerene (C60) is a 0D buckyball molecule constructed by wrapping graphene with the introduction of pentagons on the hexagonal lattice. (c) Carbon nanotube is a 1D material that can be obtained by rolling up a graphene layer. (d) Graphite is a 3D structure consisting of several graphene layers electronically connected by van der Waals force. Adapted from 2.
Graphene was isolated for the first time, in the experiment carried out by K. S. Novoselov and A. K. Geim in 2004, by repeated peeling or mechanical exfoliation of pyrolytic graphite allowing to obtain few-layer graphene to measure its optical effects on top of the Si/SiO2 substrate 1. They found that the electronic properties of their graphene with few layers on the Hall bar devices are different from those of 3D graphite. After this discovery, graphene has attracted great interest in both its fundamental physics study and enormous range of promising applications 2–7. Graphene was shown to possess remarkable physical properties which are fundamentally different from those of metals and conventional semiconductors such as transparency, elasticity, impermeability to any gases, outstanding intrinsic strength, high electronic and thermal conductivities, and high carrier mobility. As a consequence, graphene has become a candidate material for a wide range of applications, for example, a new generation of nanoscale ultra-fast transistors or flexible displays.
As seen previously, graphene presents generally in the form of a stack of several monolayers electronically disconnected from each other. However, stacking in a regular order can change considerably the electronic properties of layered graphene. In this section, the electronic properties of graphene corresponding to the number of graphene sheets and their stacking order will be described.
An ideal graphene is a 2D single crystal layer consisting of carbon atoms arranged in a hexagonal lattice structure shown in Fig. 2.1(a) as a honeycomb. The physical properties of graphene can be explained by the special arrangement of carbon atoms.
Interestingly, four valence electrons of a carbon atom (1s22s 22p2) in graphene have a particular electron configuration. In other words, three of them form an sp2 hybridization between one s orbital and two p orbitals, and the last electron is arranged in the other p orbital as shown in Fig. 2.2(a). The robustness of the honeycomb lattice structure of graphene results from the formation of a bond, owing to the sp2 hybridization, between two carbon atoms separated by a distance ~ 1.42 as shown in Fig. 2.2(b). Three bonds construct a trigonal bond is fully filled of electrons, planar structure with the angle 120 among them. Since the this covalent bonding Å adjacent carbon atoms is between two thus strong. The p orbital perpendicular to the trigonal planar structure will be bound with the p orbitals of neighboring carbon atoms, forming a half-filled bond which is not strong (Fig. 2.2(b)).
Figure 2.2. Origin of the robustness of graphene. (a) Orbital hybridization of a carbon atom in graphene. Four valence electrons (2s22p2) form three sp2 hybridized orbitals and one Åhalf-filled p orbital. (b) and bonds between° two neighboring carbon atoms separated by a distance ~ 1.42 . The angle between two bonds is 120 , yielding a trigonal planar structure.
The electronic properties of a single layer of graphene or monolayer graphene have been described in the previous subsection. In reality, a monolayer graphene is very difficult to be isolated experimentally. Naturally, graphene presents in the form of several monolayers stacked in a regular order. Fig. 2.5 clearly shows three possible orientations of graphene layers: A, B and C. The influence of particular stacking orders on the electronic properties of graphene will be discussed later in the text 10,11.
Figure 2.5. Schematic of three different orientations of graphene layers. ABA (Bernal) stacking is found in bilayer and trilayer graphenes. ABC (rhombohedral) stacking can be found in trilayer graphene.
Bilayer graphene is constituted of two monolayers of graphene with the AB stacking structure (Fig. 2.5). Since the crystalline structure of bilayer graphene can be considered as an elementary brick for constructing the whole lattice structure of graphite (Fig. 2.6(a)), the Slonczewski-Weiss-McClure (SWM) model developed for describing the electronic band structure of graphite 12,13 can thus be applied in bilayer graphene 14 . In contrast to monolayer graphene seen earlier, the electrons in bilayer graphene are massive Dirac fermions satisfying a parabolic energy dispersion 15 and they exhibit interesting quantum phenomena such as the integer quantum Hall effect with anomalies 16. In the SWM model, six electronic hopping energies associated with overlap and transfer integrals calculated for nearest neighboring atoms are , 1, 2, 3, 4 and 5. Only the parameters , 1 , 3 and 4 denoted by red arrows (Fig. 2.6(a)) are considered in the calculation of the band structure of bilayer graphene. In′ bilayer graphene, interesting physical phenomena take place at the high-symmetry and points of the Brillouin zone.
Figure 2.6. Crystalline structure of bilayer graphene. Bilayer graphene lattice structure in the AB stacking order as an elementary brick of graphite lattice structure. The A atoms of each layer are over each other. Only the Slonczewski-Weiss-McClure (SWM) parameters ,1,3,and 4 corresponding to the hopping energies of nearest neighboring atoms are presented.
Multilayer graphene consists of many graphene layers stacked in a particular sequence following three inequivalent orientations shown in Fig. 2.5. The most common observed stacking sequence is ABA, whereas the ABC order can also be found but less frequently due to the higher energy configuration 26. In this thesis, we will mainly focus on multilayer epitaxial graphene (MEG) fabricated by thermal decomposition on the C-terminated or Si-terminated faces of SiC substrates that we will describe later in the subsection 2.4. The electronic properties of MEG were shown by several investigation techniques to be predominantly similar to those of single-layer graphene 27–30. ARPES spectra in Fig. 2.10 show the Dirac cones as the band structure of the graphene layers in a MEG sample. This can be explained by the existence of a rotational stacking occurring in each pair of two adjacent graphene sheets that make all the graphene layers electronically decoupled and satisfy the Dirac linear dispersion 29,30. The MEG sometimes contains a small contribution of bilayer graphene feature and one can thus observe the signature of the electronic band structures stemming from both monolayer and bilayer graphenes 31–33.
Figure 2.10. ARPES spectra showing the band structure of an 11-layer epitaxial graphene grown on the C-face of 6H-SiC substrate. ARPES× measurement shows the top three graphene layers of the sample. Two unperturbed Dirac cones with ~ 1 106 m/s were observed, evidencing that the graphene layers are electronically decoupled and the MEG sample behaves like an isolated graphene sheet. Adapted from 30.
Fabrication methods of graphene
Graphene has long been theoretically and experimentally demonstrated to exhibit outstanding physical properties as mentioned previously. However, the production of graphene to attain the required properties for applications is extremely challenged. To this day, a large number of existing fabrication methods have been employed and continuously developed in order to prepare graphene with specific properties suitable for applications. Such methods are mainly categorized into two classes: bottom-up and top-down methods. The first one is based on the formation of 2D graphene lattice resulting from the covalent bonding between carbon atoms. The last one depends on the direct exfoliation of graphite. As a result, graphene samples obtained by different methods are in various dimensions and their quality is distinguishing. In this section, we will focus only on certain essential techniques producing scalable graphene samples. The four following production methods of graphene will be described as well as their advantages leading to the feasibility of numerous graphene applications.
The first method to fabricate graphene is mechanical exfoliation. There are several mechanical exfoliation techniques 34 but at this stage we will concentrate only on micromechanical cleavage of graphite generating the first graphene flakes in the real world 1. This technique allows to obtain monolayers of graphene from natural graphite thanks to its particular structure in stacked graphene sheets. Fig. 2.11 shows how to create graphene by this process. One can easily make mechanically exfoliated graphene by peeling a great number of times in different orientations some natural graphite grains on an adhesive piece (Fig. 2.11(b)). The normal force from the peeling applied on the graphite surface plays a dominant role of the exfoliation mechanism (Fig. 2.11(a)). The main objective of the peeling is to mechanically overcome the van der Waals attraction force between two adjacent graphene layers. One will then get on the surface of the adhesive a quasi-homogeneous distribution of graphene monolayers, bilayers, multilayers or graphite micro-grains (Fig. 2.11(c)). Finally, the graphene sample will be transferred to the surface of a substrate for measurements.
The mechanical exfoliation technique is one of the most× promising platforms to achieve high-quality graphene× with the electron mobility > 2 105 cm2/(V.s) at ambient temperature or > 1 106 cm2/(V.s) at low temperatures. The dimension of exfoliated graphene is typically > 1 mm2. Importantly, the graphene production using this technique can be effectuated at an extremely low cost compared to other fabrication methods. However, the research on mechanically exfoliated graphene always remains in laboratories because this technique is highly time-consuming, exfoliated graphene samples extracted from graphite have uncontrollable dimensions and defects, and they are impossible to be scaled up for industrial production. Therefore, it is substantial to improve the mechanical exfoliation efficiency.
Figure 2.11. Illustration of graphene production by mechanical exfoliation: micromechanical cleavage. (a) Normal force denoted by blue arrows is applied on the surface of bulk graphite during the peeling. To exfoliate graphite into graphene flakes, the normal force has to overcome the van der Waals interaction force between two adjacent graphene layers. (b) Natural graphite grains are deposited on an adhesive sheet. (c) Exfoliated graphene flakes obtained from the repeated peeling in different oriented axes. (b) and (c) are adapted from the Ph.D. thesis of J. Guignard defended in 2011.
Graphene flakes can also be prepared from graphite via a variety of chemical approaches followed by exfoliation such as liquid-phase exfoliation, graphene from graphite oxide, electrochemical exfoliation, and supercritical fluid exfoliation 35. These various techniques rely in principle on the intercalation procedure of specific molecules between graphene layers stacked in graphite in order to provoke the delamination through chemical reactions. The most straightforward chemical method allowing the reduction of the van der Waals forces is to dip graphite into a liquid medium. Fig. 2.12 schematically elucidates the liquid-phase exfoliation process of graphite 36. The molecules in N-methylpyrrolidone solvent will insert between two adjacent graphene sheets. Ultrasonication is then used to induce exfoliation, leading to the splitting of graphite into individual graphene layers in the suspension. Nevertheless, the interactions between the solvent and the graphene flakes need to compensate the attractive forces among the graphene sheets. Hence, surfactant or intercalator molecules can be sometimes added in a solvent to avoid graphene re-aggregation caused by van der Waals forces after the sonication.
Table of contents :
Chapter 1 – Investigation techniques of Dirac matter: ARPES and IR magneto-spectroscopy
1. Angle-resolved photoemission spectroscopy (ARPES)
2. Magneto-optical absorption spectroscopy
2.1. Sample preparation for measurement
2.1.1. Sample probe
2.1.2. Sample holder
2.2. Fourier transform infrared (FTIR) interferometer
2.2.1. Operating principle of the FTIR interferometer
2.2.2. Infrared light sources
2.3. Cryostat and superconducting coil
2.4. Data acquisition
Chapter 2 – Magneto-optics in multilayer epitaxial graphene
1. Electronic properties of graphene
1.1. Ideal graphene
1.2. Bilayer graphene
1.3. Trilayer graphene
1.4. Multilayer graphene
2. Fabrication methods of graphene
2.1. Mechanical exfoliation
2.2. Chemical exfoliation
2.3. Chemical vapor deposition
2.4. Epitaxy by thermal decomposition of SiC substrate
3. Magneto-spectroscopy in graphene
3.1. Ideal graphene
3.2. Bilayer graphene
3.3. Trilayer graphene
4. Experimental results
4.1. C-terminated face multilayer epitaxial graphene
4.1.1. Fabrication of C-terminated MEG samples
4.1.2. Dirac Landau level spectroscopy in monolayer and bilayer graphenes
4.1.3. Disorder effect on magneto-optical transitions
4.2. Si-terminated face multilayer epitaxial graphene
4.2.1. Fabrication of Si-terminated MEG samples
4.2.2. Electronic band structure of trilayer graphene from ARPES experiment
4.2.3. Infrared magneto-transmission results of trilayer graphene
Chapter 3 – A brief overview of topological matter
1. Topological insulators
1.1. Historical overview
1.1.1. Quantum Hall effect
1.1.2. Quantum spin Hall effect
1.2. Theoretical notions of topological states of matter
1.2.1. Berry phase
1.2.2. Topological invariants
1.3. Theoretical prediction and experimental realization of Z2 topological insulators
1.3.1. 2D topological insulator: QSHE in CdTe/HgTe/CdTe quantum wells
1.3.2. 3D topological insulator: Bi-based compounds
2. Topological crystalline insulators
2.1. Crystal structure
2.2. Band inversion
2.3. Topological surface Dirac cones in different bulk Brillouin zone orientations
2.4. Electronic band structure of Pb1-xSnxSe and Pb1-xSnxTe
2.4.1. Electronic band structure of nontrivial Pb1-xSnxTe alloy
2.4.2. Electronic band structure of nontrivial Pb1-xSnxSe alloy
2.5. Valley anisotropy
3. Bernevig-Hughes-Zhang Hamiltonian for topological matter
Chapter 4 – Magneto-optical investigation of topological crystalline ninsulators: IV-VI compounds
1. Dirac Landau levels of IV-VI semiconductors
1.1. Landau levels of the longitudinal valley
1.2. Landau levels of the oblique valleys
1.3. Landau levels of the topological surface states
2. Growth and characterization of (111) Pb1-xSnxSe and Pb1-xSnxTe epilayers
2.1. Molecular beam epitaxy growth
2.2. X-ray diffraction
2.3. Electrical transport characterization
3. Magneto-optical Landau level spectroscopy of Dirac fermions in (111) Pb1-xSnxSe
3.1. Bulk states in (111) Pb1-xSnxSe
3.2. Topological surface states in (111) Pb1-xSnxSe
4. Magneto-optical Landau level spectroscopy of Dirac fermions in (111) Pb1-xSnxTe
4.1. Bulk states in (111) Pb1-xSnxTe
4.2. Topological surface states in (111) Pb1-xSnxTe
5. Magneto-optical determination of a topological index
5.1. (111) Pb1-xSnxSe
5.2. (111) Pb1-xSnxTe
6. Validity of the massive Dirac approximation
7. Valley anisotropy in IV-VI compounds
8. Absence of the band gap closure across the topological phase transition in Pb1-xSnxTe
9. Conclusion and perspectives