Magneto-optics in multilayer epitaxial graphene 

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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 1.7. 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.).

Infrared light sources

The typical characteristics of FIR and MIR light sources of the FTIR interferometer are summarized in Table 1.1.

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.
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 of ( ) . 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.

Magneto-optics in multilayer epitaxial graphene

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.

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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.

Ideal graphene

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)).
The electronic band structure of single -layer of graphene was first proposed by P. R. Wallace in 1947 via tight-binding approach for band description in bulk graphite 8. He considered the perpendicular p orbital, forming the bond (Fig. 2.2(b)), that is responsible for the electronic band structure of graphene.

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
2.1.3. 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
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
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 0
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
Conclusion and outlook 


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