Magneto-optics in multilayer epitaxial graphene 

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Angle-resolved photoemission spectroscopy (ARPES)

Angle -resolved photoemission spectroscopy (ARPES) is one of the photoelectron spectroscopy techniques based on the photoelectric effect first observed by H. Hertz 1 and later explained by A. Einstein for the quantum description of light 2. In other words, this technique relies fundamentally on the detection of photoemitted electrons allowing to probe directly the momentum-dependent electronic band structure and provide detailed information about the band dispersion and Fermi surface of solids. To obtain ARPES spectra, the kinetic energy and angular distribution of the electrons photoemitted from a material under sufficiently high-energy illumination are measured and analyzed. Up to now, ARPES reaches 2 meV energy resolution and 0.2° angular resolution 3 which lead to better reveal the behavior of the electrons propagating inside a material through a penetration length of a few nanometers. This improvement has played a key role in enhancing the potential of ARPES to become a more sophisticated precision tool for the investigation of complex phenomena.
As mentioned earlier, the fundamental objective of an ARPES experiment is to detect the photoemission from the photoelectric effect occurring in a material. Within the non-interacting electron scheme and the energy conservation law, one can thus relate the kinetic energy ( ) of the photoelectron in vacuum to the binding energy ( ) of the electronic state inside the material by the following expression: = ℎ − − ǀ ǀ (1.1) where ℎ is the Planck constant, is the photon frequency, and is the material work function. The latter represents the potential barrier at the surface that prohibits the valence electrons from escaping.
Fig. 1.1 schematically illustrates the photoemission occurring in a sample using ARPES probe under ultra-high vacuum (UHV) condition ( < 4 10-11 mbar) in order to minimize surface eV are, for example,ℎ path of the emitted electrons 3. A monochromatized contamination and maximize the mean free × light beam of energy is incident on the sample. Light sources of energy between 10 and 200 plasma helium discharge lamp, synchrotron radiation, or lasers. As a consequence, the electrons will be emitted by photoelectric effect and escape in all directions in vacuum.
The electronic band structure of a(�material⃗) can be established by studying the momentum- dependent binding energy ( ). Only occupied electronic states can be observed by ARPES. Fig. 1.2 represents ARPES spectra obtained from different Dirac systems accommodating 2D and 3D topological Dirac fermions. These ARPES data clearly show the surface electronic structure dispersion map for the 2D topological surface Dirac cone in 3D topological insulator Bi2Se3 5, 3D tunable topological insulators TlBi(S1-xSex)2 with x = 0.5 6 and (Bi1-xInx )2Se3 with x = 0.04 7, and the 3D bulk Dirac cone in 3D Dirac semimetal Cd3As2 8. Moreover, ARPES measurement allows us to estimate the surface state band velocity from the experimental slope of the Dirac cone structure.
In summary, ARPES is an ideal surface sensitive probe used to investigate the electronic band structure of quantum materials due to the improvements of energy and angle resolutions and data acquisition efficiency. There are also many exciting developments trying to add new dimensions into this technique leading to the spin-resolved ARPES and time-resolved ARPES. With the efforts put into its development, this powerful tool will continue playing an irreplaceable role in the search for novel phenomena of complex materials. The ARPES results that will be presented later in this work were obtained by our collaborators.

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.
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.

Sample probe

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.

Sample holder

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.

Bolometer

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.

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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 1.7. Schematic representation of the FTIR interferometer (Bruker VERTEX 80V).
The FTIR interferometer obtains the signal from the bolometer as an interferogram (the transmitted light as a function of mobile mirror position) (Fig. 1.8) and then changes it into a spectrum (the transmitted light as a function of energy) using the calculation of the Fourier spectrum as a function of frequency . ( ) transform (Eq. 1.6). Here, is the intensity of the interferogram as a function of phase 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.

Data acquisition

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.
The transmission spectra measured at different magnetic fields will be manipulated in order to obtain and analyze the relative transmission and the transmittance. As a result, we are able to extract valuable quantitative information about the physical properties, for instance, the Dirac velocity, the Dirac mass or the energy gap of a Dirac material. The relative transmission is defined to be the normalization of the sample transmission at a given magnetic field by a zero-field sample transmission . This indicates the absorption due to the transitions. This allows us to( )
carriers between different Landau levels. The transmittance at a fixed magnetic field is defined (0) ( ) normalized by the corresponding substrate transmission as the sample transmission gain the information about the absorption of the free carriers and to determine the absorption threshold of the sample.
Infrared magneto-optical absorption spectroscopy represents the powerful ability to investigate the volume of a quantum solid. It is shown to be a bulk efficient sensitive probe, yet not blind to the surface, used to reveal the electronic band structure of solids via physical parameters obtained from the measurement.

Table of contents :

Introduction 
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
2.1.3. Bolometer
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
References
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
5. Conclusion
References
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 insulators: 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
7.1. Pb1-xSnxSe
7.2. Pb1-xSnxTe
8. Absence of the band gap closure across the topological phase transition in Pb1-xSnxTe
9. Conclusion and perspectives
References 
Conclusion and outlook 
Appendix

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