Diverse quantization phenomena in layered materials

TL;DR

This paper reviews diverse quantization phenomena in layered 2D materials, highlighting their unique electronic and optical properties, theoretical models, and experimental confirmations across various systems like graphene, silicene, phosphorene, and topological insulators.

Contribution

It provides a comprehensive overview of quantization effects in layered materials, integrating theoretical models with experimental results, and exploring a wide range of 2D systems and their unique behaviors.

Findings

Distinct quantization phenomena observed in various layered 2D materials.

Theoretical models align well with high-resolution experimental measurements.

Unique electronic and optical properties depend on stacking and structural configurations.

Abstract

The diverse quantization phenomena in 2D condensed-matter systems, being due to a uniform perpendicular magnetic field and the geometry-created lattice symmetries, are the focuses of this book. They cover the diversified magneto-electronic properties, the various magneto-optical selection rules, the unusual quantum Hall conductivities, and the single- and many-particle magneto-Coulomb excitations. The rich and unique behaviors are clearly revealed in few-layer graphene systems with the distinct stacking configurations, the stacking-modulated structures, and the silicon-doped lattices, bilayer silicene/germanene systems with the bottom-top and bottom-bottom buckling structures, monolayer and bilayer phosphorene systems, and quantum topological insulators. The generalized tight-binding model, the static and dynamic Kubo formulas, and the random-phase approximation, are developed/modified to thoroughly explore the fundamental properties and propose the concise physical pictures. The different high-resolution experimental measurements are discussed in detail, and they are consistent with the theoretical predictions.

Equations

$$H=$$

$$+\frac{i}{3\sqrt{3}}\sum _{⟨⟨I,J⟩⟩,\alpha ,\beta ,l}{\lambda }_l^{SOC}{\gamma }_l{v}_{IJ}{c}_{I\alpha }^{†l}{\sigma }_{\alpha \beta }^z{c}_{J\beta }^l$$

$$-\frac{2i}{3}\sum _{⟨⟨I,J⟩⟩,\alpha ,\beta ,l}{\lambda }_l^R{\gamma }_l{u}_{IJ}{c}_{I\alpha }^{†l}(\vec{\sigma }\times {\hat{d}}_{IJ}{)}_{\alpha \beta }^z{c}_{J\beta }^l.$$

$$⟨A_m^{1\uparrow }|H|A_m^{1\uparrow }⟩=-i{\lambda }_1^{SOC}(Q_1{P}_4-Q_1^{\ast }{P}_4^{\ast })+E_z{l}_1,$$

$$⟨A_m^{1\uparrow }|H|A_m^{1\downarrow }⟩=i{\lambda }_1^R(Q_1{P}_4-Q_1^{\ast }{P}_4^{\ast }),$$

$$⟨A_m^{1\uparrow }|H|B_m^{1\uparrow }⟩=⟨A_m^{1\downarrow }|H|B_m^{1\downarrow }⟩={\gamma }_0(Q_3{P}_2+Q_3^{\ast }{P}_3),$$

$$⟨A_m^{1\downarrow }|H|A_m^{1\downarrow }⟩=i{\lambda }_1^{SOC}(Q_1{P}_4-Q_1^{\ast }{P}_4^{\ast })+E_z{l}_1,$$

$$⟨B_m^{1\uparrow }|H|B_m^{1\uparrow }⟩=i{\lambda }_2^{SOC}(Q_2{P}_4+Q_2^{\ast }{P}_4^{\ast })+E_z{l}_2+{\gamma }_4,$$

$$⟨B_m^{1\uparrow }|H|B_m^{1\downarrow }⟩=i{\lambda }_2^R(Q_2{P}_4-Q_2^{\ast }{P}_4^{\ast }),$$

$$⟨B_m^{1\downarrow }|H|B_m^{1\downarrow }⟩=-i{\lambda }_2^{SOC}(Q_2{P}_4-Q_2^{\ast }{P}_4^{\ast })+E_z{l}_2+{\gamma }_4,$$

$$⟨A_m^{2\uparrow }|H|A_m^{2\uparrow }⟩=i{\lambda }_1^{SOC}(Q_1{P}_4-Q_1^{\ast }{P}_4^{\ast })-E_z{l}_1,$$

$$⟨A_m^{2\uparrow }|H|A_m^{2\downarrow }⟩=-i{\lambda }_1^R(Q_1{P}_4-Q_1^{\ast }{P}_4^{\ast }),$$

$$⟨A_m^{2\uparrow }|H|B_m^{2\uparrow }⟩=⟨A_m^{2\downarrow }|H|B_m^{2\downarrow }⟩={\gamma }_0{P}_1,$$

$$⟨A_m^{2\downarrow }|H|A_m^{2\downarrow }⟩=-i{\lambda }_1^{SOC}(Q_1{P}_4-Q_1^{\ast }{P}_4^{\ast })-E_z{l}_1,$$

$$⟨B_m^{2\uparrow }|H|B_m^{2\uparrow }⟩=i{\lambda }_2^{SOC}(Q_5{P}_4-Q_5^{\ast }{P}_4^{\ast })-E_z{l}_2+{\gamma }_4,$$

$$⟨B_m^{2\uparrow }|H|B_m^{2\downarrow }⟩=i{\lambda }_2^R(Q_5{P}_4+Q_5^{\ast }{P}_4^{\ast }),$$

$$⟨B_m^{2\downarrow }|H|B_m^{2\downarrow }⟩=i{\lambda }_2^{SOC}(Q_5{P}_4-Q_5^{\ast }{P}_4^{\ast })-E_z{l}_2+{\gamma }_4,$$

$$⟨A_m^{1\uparrow }|H|A_m^{2\uparrow }⟩={\gamma }_1,$$

$$⟨A_m^{1\uparrow }|H|B_m^{2\uparrow }⟩={\gamma }_2{P}_1,$$

$$⟨A_m^{1\downarrow }|H|A_m^{2\downarrow }⟩={\gamma }_1,$$

$$⟨A_m^{1\downarrow }|H|B_m^{2\downarrow }⟩={\gamma }_2{P}_1,$$

$$⟨B_m^{1\uparrow }|H|A_m^{2\uparrow }⟩={\gamma }_2(Q_3{P}_2^{\ast }-Q_3^{\ast }{P}_3^{\ast }),$$

$$⟨B_m^{1\uparrow }|H|B_m^{2\uparrow }⟩={\gamma }_3(Q_6{P}_2+Q_6^{\ast }{P}_3),$$

$$⟨B_m^{1\downarrow }|H|A_m^{2\downarrow }⟩={\gamma }_2(Q_3{P}_2^{\ast }-Q_3^{\ast }{P}_3^{\ast }),$$

$$⟨B_m^{1\downarrow }|H|B_m^{2\downarrow }⟩={\gamma }_3(Q_6{P}_2+Q_6^{\ast }{P}_3),$$

$$⟨A_m^{1\uparrow }|H|A_{m+1}^{1\uparrow }⟩=i{\lambda }_1^{SOC}(P_6{Q}_4^{\ast }+P_5{Q}_4),$$

$$⟨A_m^{1\uparrow }|H|A_{m+1}^{1\downarrow }⟩=i{\lambda }_1^R(P_5{Q}_4-P_6{Q}_4^{\ast }),$$

$$⟨A_m^{1\downarrow }|H|A_{m+1}^{1\uparrow }⟩=i{\lambda }_1^R(P_5{Q}_4{e}^{i\pi /6}+P_6{Q}_4^{\ast }{e}^{-i\pi /6}),$$

$$⟨A_m^{1\downarrow }|H|A_{m+1}^{1\downarrow }⟩=-i{\lambda }_1^{SOC}(P_6{Q}_4^{\ast }+P_5{Q}_4),$$

$$⟨B_m^{1\uparrow }|H|A_{m+1}^{1\uparrow }⟩={\gamma }_2{P}_1,$$

Full text

1. 1 Introduction

2. 2 Experimental measurements on magnetic quantization

3. 2.1 Scanning tunneling spectroscopy

4. 2.2 Magneto-optical spectroscopies

5. 2.3 Quantum transport apparatus, and differential thermal calorimeter/laser flash analysis

6. 2.4 Electron energy loss spectroscopy and light inelastic scatterings

7. 3 Theoretical models

8. 3.1 The generalized tight-binding models for typical layered systems

9. 3.2 Magneto-optical excitation theory

10. 3.3 Transport theories

11. 3.3.1 Quantum Hall effects

12. 3.3.2 Magneto-heat capacity

13. 3.4 Modified Random-phase approximation for magneto-electronic Coulomb excitations

14. 4 Concluding remarks

15. 5 Future perspectives and open issues

1 Introduction

Shih-Yang Lin,e,§ Thi-Nga Do,b,c,∗ Chiun-Yan Lin,a Jhao-Ying Wu,d,†

Po-Hsin Shih,a,‡ Ching-Hong Ho,d,¶ Ming-Fa Lina,♯

How to diversify the quantization phenomena is one of the main-stream issues in physics science. Up to now, they can be achieved by the intrinsic lattice symmetries[1], the distinct dimensions[2, 3], the diminished scales[4], the various stacking configurations[1, 2], the mechanical strains[5], the uniform and/or modulated magnetic fields[6, 1, 2], and the electric fields[2, 3, 7]. The focuses of this book are the magnetically quantized behaviors of the emergent layered materials, in which they will be clearly revealed in the electronic properties, optical absorption spectra, quantum Hall transports, and magneto-Coulomb excitations. Such 2D condensed-matter systems are very suitable for exploring the rich and unique essential properties, such as, the twisted bilayer graphene systems[8, 9, 10], the stacking-modulated bilayer graphene ones[11, 12], the sliding bilayer graphenes[3, 13], the few-layer graphenes with the AAA[3, 2], ABA[3, 2], ABC[3, 2] and AAB stackings[3], the silicon-doped graphene systems[1], the AA- and AB-stacked bilayer silicene/germanene systems with the bottom-top and bottom-bottom bucking structures[15, 117], the monolayer and bilayer phosphorene systems [24, 25], and the quantum topological insulators[16]. To thoroughly present and comprehend the diverse magnetic quantization, the previous theoretical modes need to be further developed and modified. The significant magnetic Hamiltonians, which cover the single- or multi-orbital chemical bondings, the intralayer $\&$ interlayer hopping integrals, the important spin-orbital couplings, and the external field, will be efficiently solved by the generalized tight-binding model[2, 1]. Its direct combinations with the dynamic Kubo formula is reliable in calculating the magneto-optical excitation spectra and analyzing the selection rules under the gradient approximation[3]. Another linking with the static one can investigate the unusual quantum conductivities[17]. Moreover, the random-phase approximation (RPA), which is available for the electronic excitations in all the condensed-matter systems, requires the exact modifications being closely related to the layered structures[15, 18, 19]. The detailed comparisons between the theoretical predictions and the high-resolution experimental measurements are also made in this work.

The magnetic quantization can be greatly diversified by modulating/changing the various geometric structures of condensed-matter systems and the external field forms. The diverse phenomena have been predicted/observed in the main-stream or emergent materials. For example, the hexagonal symmetry, stacking configuration play critical roles in graphene-related honeycomb lattices. Under a uniform perpendicular magnetic field [$B_{z}\hat{z}$], the low-lying Landau-level energies of monolayer graphene are proportional to the square root of $nB_{z}$ ($n$ quantum number)[20], the sliding bilayer graphene systems exhibit three kinds of Landau Levels [the well-behaved, perturbed and undefined Landau levels][13], the twisted ones present the fractal energy spectra during the variation of the magnetic-field strength[21], the AAA-stacked graphene systems only display the quasi-monolayer Landau levels with the different initial energies for the distinct groups (the vertical Dirac=cone structure), the trilayer ABA stacking shows the unusual superposition of monolayer- and bilayer-like Landau levels[20], the irregular Landau-level energy spectra and the frequent anti-crossing $\&$ crossings come to exist in the ABC-stacked graphene systems[22], the strongly, significantly and weakly $k_{z}$-dependent Landau subbands, respectively, appear in the AA-, AB- and ABC-stacked 3D graphites[2]. The low-energy quasi-Landau-levels survive in the 1D graphene nanotubes[4], and the dispersionless Landau levels are absent in carbon nanotubes/carbon tori[23]. Such diverse Landau levels are only created by the significant single-2pz orbital hybridizations of carbon atoms. Furthermore, the multi-orbital chemical bondings and/or the non-negligible spin-orbital interactions, which can create new categories of Landau levels, are frequently revealed in the emergent layered materials, such as, few-layer silicene [26, 31], germanene [28, 29, 30] stanene [31, 32], bismuthene [33], antimonene [34], GaAs [35], and MoS2-related systems [36, 37]. The above-mentioned Landau levels are drastically changed by a uniform perpendicular electric field [the layer-dependent Coulomb potentials], e.g., the induced Landau-level splittings, crossings, and anti-crossings. When a opposite external field, a uniform perpendicular magnetic field accompanied with a spatially modulated magnetic/electric one, is present in 2D systems, the non-uniform magnetic quantization will lead to the abnormal energy dispersions without the very high state degeneracy and the irregular wave functions[6]. The previous theoretical studies are focused on monolayer graphene, and the similar methods could be generalized to 2D emergent materials. The up-to-date calculations on the main features of Landau levels and Landau subbands are covered in the following chapters. Some magnetic issues, being associated with the novel geometries and the adatom chemisorptions, are worthy of the systematic studies, e.g., the Landau-level characteristics in the 1D folded[80], curved[4, 39] and scrolled[40] graphene nanoribbons[4], amorphous and defect-enriched graphene systems[41, 42], adatom-adsorbed graphene systems[43].

Various types of experimental equipment can accurately examine and then identify the diverse magnetic quantization phenomena in layered materials. Their measurements cover the magneto-electronic energy spectra $\&$ Landau level/Landau subband wave functions[2, 3, 1], magneto-optical absorption spectra[3], Hall conductivities[17], magnetic-field-dependent specific heat[44, 45], and magneoplasmon modes[46]. First, the energy-related and energy-fixed measurements of scanning tunneling spectroscopy [STS; details in Chap. 2.1] are, respectively, available in exploring the main features of Landau-level energy spectra[49, 48, 49] and probability distributions[50]. Most of the experimental verifications are conducted on the Former, while the opposite is true for the latter, e.g., the various $B_{z}$ dependencies for few-layer graphene systems with different stacking configurations[2, 3, 1]. The direction identifications of the number of zero points and distribution symmetries are worthy of the systematic studies under the great enhancement in STS measurement techniques[49, 48, 49]. Second, the transmission[51], absorption[52], reflection[53] and Rayleigh[54] optical spectroscopies, accompanied by an external magnetic field, are very useful in examining the special structures and selection rules of electronic excitation spectra [details in Chap. 3.2]. Up to now, the high-resolution measurements have confirmed and the symmetric absorption peaks and the specific magneto-selection rule of $\Delta n=\pm 1$ [$n$ quantum number] for the well-behaved Landau levels in monolayer-like graphene systems[51]. However, the different excitation categories and the extra selection rules, which are, respectively, due to the multi-group Landau levels and the abnormal spatial distributions, require the further experimental test. Third, the Hall transport measurements are frequently utilized to investigate the normal and unusual quantum conductivities arising from the static scatterings of Landau-level states. Specifically, the experimental and theoretical studies are consistent with each other for monolayer[55], AB[56] and ABC[57] stacking graphene systems. For example, magnetic transport measurements. have verified the unconventional half-integer Hall conductivity $\sigma_{xy}=(m+1/2)4e^{2}/h$, in which $m$ is an integer and the factor of 4 stands for the spin- and sublattice-induced Landau-level state degeneracy. This novel quantization is attributed to the quantum anomaly of $n=0$ Landau levels corresponding to the Dirac point. Fourth, a delicate calorimeter, which is sensitive to the temperature and magnetic-field strength, can accurately identify the very high Landau-level degeneracy, a weak but significant Zeeman splitting, and few Landau levels nearest to the Fermi level. Such measurements on graphene-related materials are absent up to date. However, they have been done a 3D electron gas in GaAs-GaALAs[144, 59], showing the non-monotonous $T$- and $B_{z}$-dependences. Finally, in general, the electronic Coulomb excitations could be measured by the electron energy loss spectroscopy (EELS) and the hard-X ray inelastic scatterings. The EELS measurements are very difficult to the collective excitations in the presence of a magnetic field, since the incident electron beam is easily perturbed by such field. The light scattering experiments are very suitable for the full exploration of magnetoplasmon modes, e.g., the clear verifications on the magnetoplasmon modes due to electron gases in the doped semiconductor compounds[60].

Up to now, there exist the effective-mass approximation[64, 63, 62, 61] and the generalized tight-binding model[2, 3, 1], being frequently utilized in exploring the rich magnetic quantization of layered materials. It should be noticed that the first-principles method can solve the electronic properties[65, 66], but not the magneto-electronic ones, as a result of the enlarged unit cell by the vector potential [details discussed later in Chap. 3.1]. Such calculations could provide the reliable energy bands at the high symmetry points [e.g., the K and $\Gamma$ point] and thus the reliable hopping integrals related to the single- or multi-orbital hybridizations in chemical bonds by using the Wanner functions[67]. As to the low-energy approximation, this method is suitable in dealing with the magneto-electronic properties, if the layered systems have the simple and monotonous valence and conduction bands near the Fermi level, e.g., monolayer graphene[55], silicene[68] and phosphorene[70, 69], and the bilayer AA- and AB-stacked graphene systems[62, 61]. The zero-field and magnetic Hamiltonian matrices have the same dimension, so the main features of Landau levels could be solved very quickly. However, it might create the high barriers in calculating the other essential physical properties [detailed comments in Chap. 3.1], e.g., magneto-optical absorption spectra[3], unusual Hall conductivities[70, 69], and magnetoplasmons[71, 72]. On the other side, the generalized tight-binding model covers all the intrinsic interactions, the intralayer and interlayer hopping integrals due to the single- or multi-orbital hybridizations and the spin-orbital couplings; that is, such model does not ignore the significant atomic and spin interactions, e.g., the complicated non-vertical interlayer hopping integrals in the trilayer ABC-stacked graphene[22]. Such interactions and the magnetic field (the composite field) are included in the numerical calculations simultaneously. The concise physical pictures could be proposed to clarify how many groups of the different sublattice-dominated Landau levels can be classified[2, 3, 1]. Furthermore, the magneto-electronic energy spectra and the Landau-level wave functions are very useful in the further understanding of the other physical properties.

It is well known that the geometric symmetries are one of the critical factors in determining the fundamental physical, chemical and material properties. The condensed-matter systems, which are purely made up of carbon atoms, include the significant ${sp^{3}}$[73, 74, 75, 76], ${sp^{2}}$ [73, 75, 76], and ${sp}$ chemical bondings[73, 77], being, respectively, revealed as the 3D bulk, 2D surface, and 1D line structures, such as, diamond, graphene and carbon chain. Apparently, the diversified orbital hybridizations in C-C bonds originate from the four half occupied orbitals of ${[2s,2p_{x},2p_{y},2p_{z}]}$. Among carbon allotropes, the open/closed surface geometries might appear in the planar/nonplanar forms, covering graphites[78], layered graphenes[79], graphene nanoribbons[80] and quantum dots[81] [3D-0D]/carbon nanotubes[82], C60-related fullerenes[83] and carbon onions [21] [1D-0D]. These unusual materials possess the unique honeycomb lattices with the hexagonal symmetries; therefore, the arrangements of normal hexagons are expected to play important roles in diversifying the various essential properties. A perfect 2D graphene crystal, with three nearest neighbors, is responsible for the isotropic Dirac-cone band structures[79]. Very interesting, the achiral or chiral arrangement on a cylindrical nanotube surface/two open boundaries of nanoribbon has been clearly identified to dominate the metallic or semiconducting behaviors across the Fermi level [Refs[85, 86]/Refs[87]]. The similar diverse phenomena are verified/predicted to appear in bilayer graphene systems through the twisting[9, 8] or sliding effects[13]. The former behave as a Moire superlattice, with a lot of carbon atoms in an enlarged unit cell of bilayer graphene, while the latter only have four ones being identical to those in AA or AB stackings [details in Chap. 9.1]. The distinct geometric structures of twisted bilayer graphenes are accurately confirmed through the high-resolution STM measurements [details in Chap. 2.1], e.g., the twisted angles between two honeycomb lattices of ${\theta\,=1.4^{\circ}}$, 3.5*∘, 6.4∘, 9.6∘* and 1.79∘[88]. In addition, the spatially resolved Raman spectroscopy also provides the geometric information, a bilayer system with a mixture of AB stacking and twisted structure[89, 90, 91]. Simultaneously, the STS examinations clearly identify three kinds of van Hove singularities in density of states, covering the V-shape across the Fermi level, shoulders, and logarithmically divergent peaks at the negative and positive energies. The measured results are consistent with the theoretical calculations of the first-principles method[92, 93] and tight-binding model[94, 95]. Such special structures indicate the linear and parabolic energy dispersions, as verified from the angle-resolved photo-emission spectroscopy[96]. The main features of geometric and electronic properties are directly reflected in optical absorption spectra[97] and quantum Hall transports[98]. Very interesting, the specific bilayer system, with a magic angle of ${\theta\,\sim\,1.1^{\circ}}$, is thoroughly examined and verified to exhibit the superconducting characteristics [almost vanishing resistance and diamagnetism] at the critical temperature of ${T_{c}=1.7}$ K[99]. However, the systematic theoretical investigations on the essential properties are absent up to date. For example, the magneto-electronic states, being closely related to the unusual optical selection rules and Hall conductivities, need to be well characterized through the magnetic wave functions with the normal or irregular spatial oscillations. This will be completely explored by the generalized tight-binding model in Chap. 4.

In addition to the variations among the well-behaved stacking configurations[2, 1, 3], a creation of the non-uniform sublattices will lead to the dramatic changes in the fundamental physical properties, especially for the thorough transformation from the 2D to quasi-1D behaviors. The modulated geometries in few-layer graphene systems have been successfully synthesized using the STM tips[100], mechanical exfoliation[101], and chemical vapor deposition[102]. From the theoretical point of view, the stacking symmetries of bilayer graphene systems can be drastically altered by the significant sliding[13], twisting[95], and generation of domain walls[12]. When only the low-lying electronic states due to the C-2pz orbitals are taken into account, the first and second systems are predicted to exhibit two pairs and many pairs of valence and conduction bands along the high symmetry points in the hexagonal Brillouin zone, respectively. Energy bands have the regular dispersion relations with the 2D wave vectors. Furthermore, their band-edges states possess the 2D-specified structures in van Hove singularities of density of states [12] and the similar ones in absorption spectral structures[12]. The magnetic quantization in the first [second] system creates the highly degenerate Landau states with the well-behaved, perturbed, or undefined oscillation modes [the normal ones]. Apparently, the mandatory modulation of stacking configuration, which are due to the changes of bond lengths, have generate a large unit cell and the non-uniform intralayer and interlayer hopping integrals. Even in the zero-field case, the dependence of essential properties on the wave vector along the modulated direction is expected to be negligible. That is, the quasi-1D phenomena come to exist in certain special cases, e.g., plenty of 1D energy subbands with the unusual dispersion relations, and many significant band-edge states and absorption structures[12]. A uniform magnetic field strongly affects the commensurate periods and the vector-potential-induced Peierls phases on the neighboring atomic interactions. As a result, the non-uniform magnetic quantization can create a plenty of the low-degenerate Landau subbands with the sufficiently wide oscillating energy widths and the strong couplings[6], being in sharp contrast with the high-degenerate Landau levels. It is worthy of making a systematic comparison for the main features of Landau subbands in stacking-modulated bilayer graphene systems[12], graphene nanoribbons[4], and Bernal and rhombohedral graphites[2].

Obviously, the 3D silicon crystals, with the ${sp^{3}}$ chemical bondings, have shown the critical roles in the basic sciences[103], applied engineering[105, 104], and high-technique semiconductor applications[106]. From the experimental and theoretical viewpoints, the reduction in dimension should be a very effective way in diversifying the various physical phenomena. The researches on layered silicene systems are getting into one of the main-stream topics in 2D emergent materials. In general, the epitaxial growth is frequently utilized to synthesize monolayer and bilayer silicene systems on the different substrates, e.g., the former grown on Ag(111)[107, 108], Ir(111)[109] and ZrB2(0001)[110], with the ${4\times\,4}$, ${\sqrt{3}\times\,\sqrt{3}}$ and ${2\times\,2}$ unit cells, respectively. The STM [details in Chap. 2.1] and low-energy electron diffraction can identify a buckled single-layer honeycomb lattice, as well as an enlarged unit cell arising from the significant atomic interactions between the host and guest atoms. Such geometry clearly indicates the dominating ${sp^{2}}$ bonding and its competition with the weak, but important ${sp^{3}}$ one. The phenomenon of orbital hybridizations are further reflected in the low-lying valence and conduction bands. The former, which possesses a slightly separated Dirac-cone structure [approximately a zero-gap semiconductor; the monolayer-graphene-like behavior] is confirmed through the high-resolution angle-resolved photoemission spectroscopy [ARPES; [108]]. However, it is very difficult to successfully generate the pure bilayer silicene systems through the various experimental methods. The previous study shows that there exist three types of bilayer silicenes after treating the calcium-intercalated monolayer silicene [CaSi2] around a BF${{}_{4}^{-}}$-based ionic liquid. That is to say, the bilayer silicenes might consist of the four-, five- and six-membered silicon rings with three nearest neighbors[107], where the third ones correspond to the AB-bottom-top stacking configuration [AB-bt]. In addition, the experimental evidences of AA stackings are absent up to now. Such images are accurately achieved by a high-angle annular dark field scanning transmission electron microscopy [HAADF-STEM]. Moreover, the measured absorption spectrum suggests an indirect optical gap of ${1.08}$ eV [or a threshold excitation frequency]. Energy gaps are quite different from in monolayer and bilayer material; therefore, the stacking configurations, the interlayer hopping integrals, and the spin-orbital couplings play a critical role in the fundamental physical properties. Most important, the multi-metastable geometric structures come to exist simultaneously, clearly illustrating the very active chemical environments due to the four half-occupied orbitals of silicon atoms [${(3s,3p_{x},3p_{y},3p_{z})}$]. Few-layer silicene systems are expected to become the 2D materials in the next generation of electronic devices[111], while their structural instabilities need to be overcome for the highly potential applications.

Also, a lot of theoretical predictions on 2D layered silicene materials present very interesting results, especially for the diversified physical phenomena. Generally speaking, their essential properties are thoroughly investigated by the first-principles calculations, ${\it ab}$ ${\it initio}$ molecular dynamics, the tight-binding model and the effective-mass approximation. The optimal geometric structures are accurately presented through the first and second methods. For example, monolayer silicene has a buckled honeycomb lattice, with a significant height difference between A and B sublattice[107], and bilayer ones possesses the distinct members in silicon rings and the various stacking configurations[107], such as, the metastable configurations of AA-bt, AA-bb, AB-bt and AB-bb in bilayer materials. Specifically, the Vienna ab initio simulation package is quite powerful in predicting electronic properties, covering band structures, spatial charge distributions, atom-, orbital- and spin-projected density of states, magnetic moments, and spin distribution configuration. For example, the chemisorption-diversified fundamental properties are clearly revealed in the hydrogenated and oxidized silicene systems[115, 114]. For pristine bilayer silicenes, their band properties across the Fermi level are predicted to be very sensitive to the stacking configurations, e.g., a semimetal with some free carrier densities[116], and a finite indirect-gap/direct-gap semiconductor [${\sim,0.5-1.0}$ eV[107, 117], being in sharp contrast with a very narrow gap of monolayer system [${\sim,5}$ meV; [108]]. On the other hand, the third and fourth methods are quite efficient in studying the diverse physical phenomena after getting a set of reliable parameters for all the intrinsic interactions. It should be noticed that an outstanding fitting in non-monolayer silicene systems, which is done by a detailed comparison with the first-principles calculations, will become a giant barrier. The more complex interlayer hopping integrals and enhanced spin-orbital couplings, due to the rich bucklings and stackings, are very difficult to obtain even for bilayer silicenes. Up to date, the effective-mass can only deal with the fundamental properties for monolayer silicene under the external electric and magnetic fields, covering the in-orbital-induced slight separation of Dirac points, gate-voltage-created state splitting[117], and the magnetic-field-generated degenerate Landau levels and specific selection rule[117]. There are no excellent parameters for two pairs of low-lying valence and conduction bands in AA V and AB-stacked systems. However, the first pair nearest to the Fermi level is well fitted for the non-monotonous energy dispersions and the irregular/normal valleys. Apparently, the diversified Coulomb excitations and magnetic quantizations are effectively solved by the generalized tight-binding model, e.g., the diverse [momentum, frequency]-phase diagrams in monolayer silicene under the composite effects of spin-orbital interactions, electric potentials and magnetic fields, the inter-Landau-level and electron-gas-like magnetoplasmon modes with the rather strong electron-hole dampings for the same system[118], and the spin- and valley-split Landau levels of bilayer silicenes in the frequent non-crossings, crossings and anti-crossings[117]. The systematic investigations will be made on the fundamental properties of AA- and AB-stacked bilayer systems.

The chemical modifications, as revealed in the theoretical and experimental researches[119, 120, 121], are one of the most efficient methods in dramatically changing the essential properties of condensed-matter systems. They are very effective for the layered materials, since such systems can provide the outstanding chemical environments. For example, there are a lot of dangling C-2pz orbitals on the planar structures of few-layer graphene systems. Recently, the adatom chemisorptions[119, 120] and the guest-adatom substitutions[121] in the emergent 2D materials are easily achieved during the experimental growths. Apparently, how to generate the uniform adatom/guest-atom distributions in experimental laboratories is the studying focus/issue. Up to now, many graphene-related binary and ternary compounds have been successfully synthesized using various methods. Among them, the silicon carbide compounds, with the sp2 or sp3 multi-orbital chemical bondings, frequently appear in the distinct dimensions, e.g., the 3D bulk compounds[123, 124, 122], the 2D systems[125], and the 1D nanotube ones[127, 126]. Similarly, there also exist the dimension-dependent $B_{x}C_{y}N_{z}$ compounds. The 3D silicon carbide systems possess the sp3 multi-orbital hybridizations, mainly owing to the four half-occupied orbitals of $(3s,3p_{x},3p_{y},3p_{z})$ and $\&$ $(2s,2p_{x},2p_{y},2p_{z})$ orbitals, Such chemical bondings are similar to those in diamond[128], so the SiC materials have the outstanding mechanical properties, as observed in the consistence between the calculated results [129, 130] and experimental measurements[131]. Such SiC-related materials have presented the high potentials in applications, such as, the energy storage[132], and the semi-conducting optical devices[133]. Recently, the planar silicon-carbon compounds have been synthesized in the experimental laboratories[134]. The theoretical predictions under the first-principles calculations clearly show that these 2D compounds display the sp2 and $\pi$ chemical bondings [135]. The latter will be responsible for the low-energy essential physical properties. Specifically, the $\pi$ bondings, which mainly originate from the 2pz-2pz hybridization in C-C bond and the 2pz-3pz mixing in C-Si one, is expected to be significantly distorted by the Si-substitutions. These are very sensitive to the concentrations and distributions of the Si-guest atoms, and so do the other physical, material and chemical phenomena [135]. It is very interesting to fully comprehend the close relations between the substitution-enriched chemical bondings and the magnetic quantization behaviors. The non-uniform ionization potentials, C-C $\&$ Si-C bond lengths and hooping integrals are deduced to be the critical factors in determining how many kinds of Landau levels and magneto-optical selection rules can be created during the variation of the Si-substituted geometric structures[2, 1, 3].

For the quantum transport properties, the 2D graphene-related systems are very suitable for fully exploring the diverse magnetic quantization phenomena. Apparently, such layered materials sharply contrast with the doped semiconductors/2D electron gases revealing the integral and fractional Hall effects[136]. The previous investigations clearly show that honeycomb lattices[55], few layers[56, 57], stacking configurations[57], planar/rippled/curved/folded structures[94], negligible spin-orbital interactions [137], dimensions[138], nanodefects[139], chemical absorptions[140], and guest-atom substitutions[121] are responsible for the rich band structures and thus the unusual magneto-electronic and transport properties. The close and complex relations between the critical factors and the diversified Hall conductivities are the studying focus of Chap. 9. Up to date, the experimental measurements and theoretical predictions [the tight-binding model and the effective-mass approximation have delicately identified the rich characteristics of quantum Hall conductivities in monolayer[55], bilayer[56] $\&$ trilayer AB stackings[57], and trilayer ABC configuration[57] under a low Fermi level and magnetic-field strength. For example, there are one, two $\&$ six, and three Landau levels at/across the Fermi energy, leading to the anomalous structures there in terms of height, width and sequence of quantum plateaus[55, 56, 57]. This clearly illustrates the significant roles due to the well-defined modes, state degeneracies, and normal energy spectra. Obviously, the more complicated transport phenomena, which might be created by the non-well-behaved magneto-electronic states and the frequently crossing $\&$ anti-crossing behaviors, deserve a systematic investigation. The sliding bilayer graphenes and AAB trilayer stacking will be chosen for a model study, in which the former have the structural transformation between two high-symmetry configurations[13, 141], and the low-symmetry latter presents the very complex interlayer hopping integrals[13, 141]. These systems are predicted to exhibit the diversified energy bands and magneto-electronic properties, such as, three kinds of Landau levels [the well-behaved, perturbed and undefined ones] with the complex $B_{z}$-dependent energy spectra. To propose the completely physical pictures, the quantum Hall effects are conducted on the trilayer AAA, ABA, ABC and AAB stacking. On the theoretical side, the linear static Kubo formula is directly combined with the generalized tight-binding model [discussed later in Chap. 3.3.1], based on the model consistence between the gradient approximation in evaluating the electric dipole moment and the sublattice-defined subenvelope functions. Such direct association is expected to be very efficient in solving the open transport issues due to the abnormal Landau-level wave functions and energy spectra[13, 141]. That is to say, the developed theoretical framework is suitable and reliable for any kinds of Landau levels, e.g., the quantum transports arising from the well-behaved, perturbed, and undefined oscillation modes. In addition, the effective-mass perturbation method might be quite difficult in exploring the irregular magneto-electronic states and even the higher-energy/strong-field transport properties of the well-behaved Landau levels, mainly owing to the complicated effects coupled by the intrinsic interactions and the magnetic field.

The magneto-electronic specific heats of conventional quantum Hall systems are very suitable for fully understanding the diverse magnetic quantization phenomena. They have been extensively investigated both theoretically[142] and experimentally[145, 143, 144]. For example, a lot of researches on GaAs VGaAlAs[144, 145] show that the thermal properties are very sensitive to the temperature and magnetic-field strength. The temperature-dependent specific heat reveals an obvious peak structure at a low critical temperature[144, 145]. Furthermore, the magnetic-field dependence exhibits an oscillation behavior[144, 145]. These two kinds of thermal behaviors directly reflect the main features of Landau levels, such as, the filling factor and energy spacing. Also, the high-resolution calorimeter measurements are conducted on the graphene-related materials, such as 1D carbon nanotubes[146] and 3D graphite systems[147]. The former clearly display a linear temperature-dependence in ${C(T)}$, and so do for the latter. These results are deduced to be closely related to the 1D and 3D metals. This book will illustrate the rich thermal excitation phenomena on monolayer graphene, e.g., their unique dependences on the Zeeman splitting energy, doping level of conduction electrons/valence holes, temperature, and magnetic field.

The 2D emergent materials are ideal condensed-matter systems in thoroughly exploring the many-particle Coulomb excitations under the magnetic quantization. The strong competition in magneto-excitation spectra, that due to the longitudinal electron-electron interactions and the transverse cyclotron forces, is a very interesting research focus. Recently, the various experimental methods are delicately developed to successfully synthesize a lot of layered materials, e.g., few-layer germanene, silicene and tinene being, respectively, grown on [Pt(111), Au(111) $\&$ Al(111)][150, 149, 148], [Ag(111), Ir(111) $\&$ ZrB2][107, 109, 108, 110] and Bi2Te3 surfaces. Apparently, the obvious buckled structures, the important orbital hybridizations, the significant spin-orbital interactions, and the external electric and magnetic fields would play critical role in the essential physical properties, especially for their strong effects in creating the diverse magneto-electronic excitation. The previous theoretical studies, which are focused on monolayer graphene and bilayer AA and AB stackings[1, 2, 3] clearly show the rich and unique phenomena: a lot of inter-Landau-level single-particle and collective excitations, the unusual magnetoplasmon modes with the non-monotonic momentum dispersion relations, the very strong dependences on the temperature, doping level and interlater atomic interactions, the stacking-enriched [momentum, frequency]-phase diagrams, and the 2D electron-gas behavior only under a sufficiently high free carrier density. On the experimental side [details in Chap. 2.4], the high-resolution electron energy loss spectroscopy is very powerful in the full examinations of the carbon-orbital-[151], dimension-[152], layer-[153], stacking- [153], doping-[154], and temperature-diversified plasmon modes[155] . Up to now, most of the measurements are consistent with the calculated results[18]. A thorough investigation on monolayer germanene is expected to present the distinct magneto-electronic excitation phenomena, since its band structure might be quite differ from those of graphene and silicene systems[15]. The composite effects arising from the significant spin-orbital coupling, free carrier doping, static Coulomb on-site energies, and magnetic field would play critical roles in diversifying magnetoplasmons.

As the final topic, a topological concern is addressed in view of the extensively and intensively studied topological phases of condensed matter[156, 157, 158]. In this realm, the phases with topological order cannot be characterized by symmetry alone. Moreover, the relevant phase transitions do occur without spontaneous symmetry breaking, beyond the scope of Landau’s theory[159]. The first realization of such phases is the discovery of the integer quantum Hall effect (QHE)[160], which was followed soon by a topological interpretation[161, 162, 163]. Later on, a distinct, half-integer QHE was also found from graphene, which has almost spin degeneracy described by $SU(2)$ symmetry[164, 165]. The previous theoretical predictions[166, 167] were realized in this finding. It has been well understood that the anomaly of this half-integer QHE originates from the presence of $2$D massless Dirac fermions around the zero energy with respect to the original Dirac points (DPs). As described in Chap. 3.3, the Hall conductivity and the associated Landau levels (LLs) are of the relativistic type in graphene. The very characteristic lies in that there exists a topologically robust zero-mode LL that is a constant function of the perpendicular magnetic field. More deeply, this zero-mode LL is protected by the local chiral symmetry (CS), against CS preserving perturbations provided that intervalley scattering between the double DPs is inhibited[168, 169, 170], where the CS arises from the global sublattice symmetry in spinless graphene. Since massless Dirac particles are broadly present in condensed matter with various symmetries, not to mention Dirac bosonic systems, it is of interest to see how about the situations in other systems with $2$D massless Dirac fermions[171, 172, 173, 174]. In practice, one can use such a zero-mode LL to signify the existence of $2$D massless Dirac fermions. This topic is central to Chap. 11 for the subject material of this book, that is, $3$D layered systems.

In addition to the interesting topics in this work, the open issuers, being clearly illustrated through a final chapter [details in Chap. 13], are proposed to promote the understanding of magnetic quantizations and explore the other diverse phenomena in 2D emergent materials. Certain main-stream researches might be very difficult to thoroughly investigate their main features according to the current methods/the present calculations, mainly owing to the complicated geometric structures with the non-uniform/quasi-random chemical environments, or a lot of important intrinsic interactions [such as, the significant couplings of multi-orbital hybridizations and spin-orbital interactions]. For example, a single-layer graphene could present the vacancy and Stone-Wales defects[175]; that is, a defective and amorphous system possesses hexagons, pentagons and heptagons. It is deduced that non-homogenous ${sp^{2}}$-bonding network can create the great modifications on the basic magnetic properties, e.g., the strong dependences of energy spectra and wave functions on the concentrations and distribution configurations of various defects, and the close relationships of the defect-created sublattices and the vector-potential-dependent Peierls phases. Whether the highly degenerate Landau levels appear in the seriously deformed/curved surfaces is one of the main-stream topics. The folded graphene nanoribbons[80], curved ones[176], carbon nanoscrolls[40], carbon toroids[23], and carbon onions[21] are very suitable for the model studies on the curvature effects [the misorientation of carbon-${2p_{z}}$ orbitals and hybridization of $\pi$ and $\sigma$ electrons], critical roles of the position-dependent hopping integrals, the strong competition of the quantum-confinement-induced standing wave functions with the magnetically localized ones, and the diamagnetic, paramagnetic, ferromagnetic and anti-ferromagnetic behaviors[80, 176, 40, 23, 21]. The more complex magnetic quantization phenomena will frequently come to exist in the adatom-chemisorption/substitution layered materials [the chemically modified systems]. The previous first-principles calculations show that the halogenated[177], hydrogenated and oxidized graphenes/silicenes [115, 114] exhibit the main characteristics of Moire superlattices in the presence of multi-orbital chemical bondings. The superposition of guest and host sublattices should be the critical mechanism responsible for the localized magneto-electronic states; therefore, plenty of non-equivalent sublattices can take part in the highly irregular magnetic quantizations. Apparently, the above-mentioned geometry- and chemical-modification-dominated novel materials could serve as the typical models in fully exploring the diversified optical properties[3], quantum Hall transports [17], and Coulomb excitations[18] under any external fields. The studying focuses might cover the different magneto-optical selection rules of Landau levels or subbands due to the combined effects of dimensions, edge structure, curvatures and external fields, the integer or non-integer Hall conductivities with the regular/abnormal dependences on the magnetic-field strength and the Fermi energy[55], and the [momentum. frequency]-phase diagrams of the inter-Landau-level magnetoplasmon modes and electron-hole excitations[71]. Also noticed that the theoretical frameworks of the different physical/chemical/material properties, belonging to the single- and many-particle models, might need to make the great modifications for fully understanding their diverse phenomena. Among them, the suitable Hamiltonians with any number of layers, which clearly show the dramatic transformation in multi-orbital hybridizations from few-layer ${sp^{2}}$-bonding silicenes to 3D diamond-like silicon, are one of the challenges.

This book covers Chaps. 1-10 for fully exploring the diverse magnetic quantization phenomena. Chapter 2 discusses the various high-resolution experimental measurements, including scanning tunneling microscopy [STM], scanning tunneling spectroscopy [STS], optical absorption reflection/transmission spectroscopies, transport/thermal instruments, and light inelastic scattering spectroscopy. The theoretical frameworks, the generalized tight-binding model, its combinations with the static and dynamic Kubo formulas, and the modified random-phase approximation, are systematically developed in Chap. 3. They are, respectively, responsible for the magnetically quantized electronic states, the unusual Hall conductivities, the magneto-optical selection rules, and the magneto-electronic excitations. Chapter 4 will thoroughly investigate the diversified magneto-electronic states and optical properties of bilayer graphene systems due to the very strong cooperations/competitions between the twist-induced Moiré superlattice and the enlarged unit cell by the vector potential. The modulation of stacking configuration, which is created by the domain walls in one graphene layer, is clearly illustrated for the AB-stacked bilayer graphene in Chap. 5. Its significant effects on the electronic and optical properties are studied in detail, and furthermore, the detailed comparisons with the sliding and twisted bilayer graphene systems, graphene nanoribbons, and Bernal and rhombohedral graphites are also made. Chap. 6 fully explores the rich and unique phenomena in an AA-bt bilayer silicene with the very strong interlayer hopping integrals and the negligible spin-orbital couplings; furthermore, the significant differences two- and single-layer systems are made in detail. The composite effects, due to the intrinsic atomic interactions and the magnetic $\&$ electric fields, on the essential properties are the studying focuses through the proposed viewpoint: the valley structures in electronic energy spectra will play critical roles in the fundamental physical properties, such as the diverse magnetic quantizations due to the different valleys in monolayer and bilayer silicene systems. The layer-dependent spin-orbital interactions, as reported in Chap. 7, will come to exist in AB-stacked silicene systems. Apparently, the more complicated effects arising from stacking configuration and buckling structure are expected to diversify the magnetic quantization. Chapter 8 clearly illustrate the dramatically chemical modifications of the fundamental properties by using the silicon guest-atom substitution in monolayer graphene. Four kinds of Landau levels and magneto-optical selection rules will be classified according to the main features of magneto-electronic properties and absorption transition channels. Few-layer graphenes in Chap. 9, with the distinct layer numbers and stacking configurations, can present the rich and unique quantum transport properties, the unusual Hall conductivities and magneto-heat capacity. The available scattering events are clarified by delicately analyzing the dipole matrix elements associated with the initial and final Landau-level-state spatial distributions. The magneto-electronic specific heat of monolayer graphene provides another magnetic quantization phenomenon, especially for its rich dependences on the Dirac cone/the unusual Landau-level energy spectrum, Zeeman splitting effect, temperature and magnetic-field strength. The superposition of the transverse magnetic force and the longitudinal Coulomb one leads to the abnormal electronic excitations, where germanene, silicene and graphene will exhibit the diverse magnetoplasmon modes induced by the significant/negligible spin-orbital couplings, as shown in Chap. 10. Chap. 11 covers the topological characterization for the Landau levels in $3$D layered systems of topological matter. The focus is put, in a topological viewpoint, on the existence and stability of $2$D massless Dirac fermions when an external perpendicular magnetic field is applied. If this is the case, the half-integer QHE can be achieved and dictated by the topologically robust zero-mode LL in the relativistic LL spectrum, as having been understood. This is distinct from the nontrivial topological phase of the normal integer quantum Hall effect (QHE) occurring at any $2$D plane of a topological trivial system. Henceforth, Two systems are considered after introducing the relevant topological background. Chap. 11.1 aims at certain $3$D layered spinless nodal-line topological semimetals (TSMs), exemplified by rhombohedral graphite (RG), a $3$D stack of graphene layers in ABC configuration ($SU(2)$ symmetry assumed), where $2$D massless Dirac fermions are hosted along the nodal lines in the bulk. This topological phase is shown here to be protected by the local chiral symmetry, which arises from the global sublattice symmetry. A comparison is made to graphene, including the $3$D half-integer QHE in RG v.s. the $2$D half-integer QHE in graphene. In Chap. 11.2, time reversal invariant spinful strong topological insulators (TIs), such as Bi2Se3 and Bi2Te3, are covered (spinful with $SU(2)$ symmetry broken). There are gapless surface states corresponding to the nontrivial insulating phase of the bulk. The surface state on each surface layer is manifested as a single Dirac cone, where $2$D massless Dirac fermions are hosted owing to the presence of the time reversal symmetry (TRS). One could by intuition think of gapping the surface DCs with some means for violating the TRS. Nevertheless, as known from existing experiments, this thought is true in the case of magnetic doping or a proximate magnetic material while being false in the case of applying an external perpendicular magnetic field. Thus, the zero-mode LLs still signify $2$D massless Dirac fermions in these strong TIs. This odd fact motivates the content of Chap. 11.2. Finally, Chaps. 12 and 13, respectively, present the concluding remarks, and future perspectives $\&$ open issues.

2 Experimental measurements on magnetic quantization

Shih-Yang Lin,e,§ Thi-Nga Do,b,c,∗ Chiun-Yan Lin,a Jhao-Ying Wu,d,†

Po-Hsin Shih,a,‡ Ching-Hong Ho,d,¶ Ming-Fa Lina,♯

High-resolution experimental equipment has now been developed to fully explore the novel physical phenomena in the emergent layered materials. In general, they can measure the fundamental quantities very accurately and efficiently, covering the geometric, electronic, optical, transport, thermal and Coulomb-excitation properties. Specifically, the observed phenomena will become complex and diverse in the presence of a uniform perpendicular magnetic field. The greatly diversified magnetic quantization is clearly revealed in the distinct physical properties, as thoroughly discussed in the following sections associated with the various experimental measurements.

2.1 Scanning tunneling spectroscopy

Scanning tunneling microscopy has become one of the most important experimental techniques in resolving the diversified nanostructures since the first discovery by Binnig and Rohrer in 1982[1]. STM is capable of revealing the surface topographies in the real space with both lateral and vertical atomic resolutions, e.g., the nanoscaled bond lengths, crystal symmetries, planar/nonplanar geometries, step edges, local vacancies, amorphous dislocations, adsorbed adatoms $\&$ molecules, and nanoclusters $\&$ nanoislands[1]. The STM instrument is composed of a conducting (semi-conducting) solid surface and a sharp metal tip, where the distance of several angstroms is modulated by piezoelectric feedback devices. A very weak current will be generated from the quantum tunneling effect in the presence of a bias voltage between surface and tip. The tunneling current presents a strong dependence on their distance; furthermore, the relation roughly agrees with the exponential decay form. It flows from the occupied electronic states of tip into the unoccupied ones of the surface under the positive bias voltage $V>0$, and vice versa. This current is required to serve as a feedback signal, so that the most commonly used mode, a constant tunneling one (current and voltage), will be operated to resolve the surface structure very delicately by using a piezoelectric device. By combining the tip of a metal probe and a precise scanning device, the spectroscopic information of surface morphology is thoroughly revealed at the preselected positions with the well-defined conditions. The structural response hardly depends on the background effects, being attributed to an ultra-high vacuum environment. The up-to-date spatial resolution of STM measurements is $\sim 0.1$ $\AA$. The rather accurate measurements on the graphene-related systems have clearly illustrated the complicated relations among the honeycomb lattice, the finite-size quantum confinement, the flexible feature, and the active chemical bondings of carbon atoms. For example, the experimental identifications on the rich and unique geometric structures cover the achiral (armchair and zigzag) and chiral graphene nanoribbons with the nanoscale-width planar honeycomb lattices[2, 3], the curved[4], folded [5], scrolled[6] and stacked graphene nanoribbons [6], the achiral and chiral arrangements of the hexagons on cylindrical surfaces of carbon nanotubes[7], the AB[8], ABC[10, 8, 9] and AAB stackings[12, 11] in few-layer graphene systems, the twisted bilayer graphenes[13, 14], the corrugated substrate and buffer graphene layer[15], the rippled structures of graphene islands[16, 17], the adatom distributions on graphene surfaces[18], and the 2D networks of local defects on graphite surface[19, 20].

Scanning tunneling spectroscopy (STS), an extension of scanning tunneling microscopy in Fig. 2.1, creates the tunneling current through the tip-surface junction under a constant height mode [details in Ref [1]]. The electronic properties are characterized by both $I$-$V$ and $dI/dV$ curves sweeping over the bias voltage $V$. In the STS measurements, the normalized differential conductance, being created by a small AC modulation of $dV$, could be measured by adding a lock-in amplifier. Furthermore, the noise involved in the observed conductance is largely reduced. The up-to-now experiments present the highest resolution of $\sim 10$ pA. STS is a very efficient method for examining the electronic energy spectra of condensed-matter systems. The tunneling differential conductance is approximately proportional to the DOS and directly presents the main features, being useful in identifying the semiconducting, semi-metallic or metallic behaviors. For example, a zero-gap semiconducting monolayer graphene presents a V-shape DOS with a vanishing value at the Fermi level (the Dirac point) [21]. The semi-metallic ABC-stacked graphene systems, a pronounced peak at the Fermi level is revealed in tri-layer and penta-layer cases as a result of the surface-localized states[9, 10]. The high-resolution STS could serve as a powerful experimental method for investigating the magnetically quantized energy spectra of layered graphenes. The measured tunneling differential conductance directly reflects the structure, energy, number and degeneracy of the Landau-level delta-function-like peaks. Part of the theoretical predictions on the Landau-level energy spectra have been confirmed by the STS measurements, such as the $\sqrt{B_{z}}$-dependent Landau-level energy in monolayer graphene[23, 22], the linear $B_{z}$-dependence in AB-stacked bilayer graphene[22], the coexistent square-root and linear $B_{z}$-dependences in trilayer ABA stacking[22], and the 2D and 3D characteristics of the Landau subbands in Bernal (AB-stacked) graphite[24, 25]. Specifically, STS is suitable for identifying the spin-split density of states, when the magnetic tips are utilized in the experimental measurements. The spin-polarized STS (SPSTS) is very powerful in fully comprehending the magneto-electronic properties, e.g., the spin-split Landau levels/Landau subbands due to the significant spin-orbital couplings and the gate voltage (perpendicular electric field, e.g., those in silicene and germanene systems [detailed in Chaps. 6 and 9].

The high-resolution STS measurements under the specific energies can directly map the spatial probability distributions of wave functions in the absence and presence of a uniform perpendicular magnetic field. Up to now, they have confirmed the rich and unique electronic states in graphene-related systems. For example, the normal standing waves along the tubular axis, with the finite zero points, are clearly identified to survive in finite-length carbon nanotubes[26]. The topological edge states are created by the AB-BA domain wall of bilayer graphenes[27]. Moreover, the well-behaved Landau-level wave functions, being similar to those of an oscillator, come to exist in few-layer graphene systems wit the distinct stacking configurations, such as, the $n^{c,v}=0$, and $\pm 1$ Landau levels for the monolayer and AB-stacked bilayer graphenes. The higher-$n$[28], undefined[29], and anti-crossing Landau levels[30] are worthy of a series of experimental examinations. The similar STS measurements are available in testing the theoretical predictions on the non-uniform magnetic quantization, establishing the direct relations between the band-edge state energies and the distribution range of the regular or irregular wave functions quantization, e.g., the main features of Landau subbands in Chap. 5].

2.2 Magneto-optical spectroscopies

Optical spectroscopies are very power in characterizing the absorption[31], transmission[51] and reflectance[33] of any condensed-matter systems, especially for the diverse optical properties of emergent layered materials. They can provide the rather sufficient information on the single-particle and many-body vertical optical excitations, directly reflecting the main features of energy bands and strongly coupled excitons [electron-hole pairs attracted by the longitudinal Coulomb interactions; [51, 31]], e.g., the experimental examinations on the predicted energy gap, the threshold excitation frequency, the band-edge state energies, the special absorption structures, the electric-field effects, and the magneto-optical selection rules. Absorption spectroscopy, which is based on the analytical method of measuring the fraction of incident radiation absorbed by a sample, is one of the most versatile and widely used techniques in the sciences of physics, chemistry and materials. This well-developed tool is available within a wide frequency range. The measured spectral functions are frequency-dependent functions for characterizing the fundamental electronic properties of materials. The experimental setup related to the light source, sample arrangement and detection technique is sensitive to the frequency range and the experimental purpose.

The most common optical setup, as clearly shown in Fig. 2.2, is to emit a radiation beam on the specific sample and then delicately measure the transmitted intensity that is responsible for the significant frequency-dependent absorption spectrum. In general, optical experiments are performed by a broadband light source, where the intensity and frequency could be adjusted within a broad range. According to the frequency range of optical operations, the most utilized light sources are classified into three kinds: (${\bf I}$) the xenon-mercury arc lamp under the high pressure in the far-infrared region[34, 35], (${\bf II}$) the black-body source of the heated SiC element in the mid- to near-infrared spectral range[36, 37, 38] and (${\bf III}$) the tungsten halogen lamp in a continuous spectrum from the visible to near-ultraviolet[39, 40]. These optical sources are commonly adopted for the analytical characterizations of optical properties in emergent materials over a broad spectrum, since they are successfully operated even at a high temperature [$>3000$ K] under the inert-halogen mixture atmosphere.

Generally speaking, the optical absorption spectra are delicately measured using the Fourier transform spectrometer[41, 42], linear photodiode array spectrometer[43, 44], or charge coupled device spectrometer [CCD; [45, 33]]. Fourier transform spectrometer is based on the coherent observations of electromagnetic radiations, in either the time- or space-domain ones, e.g., Bruker IFS125 with the high-resolution linewidths narrower 0.001 cm[41, 42]. On the other side, the photodiode array spectrophotometer, which is built from hundreds of linear high-speed detectors integrated on a single chip, simultaneously measures the dispersive light over a wide frequency range. CCD spectrometer is also a multichannel detector; it can simultaneously detect as many frequency channels as the number of the individual resolution pixels. According to the linear dynamic Kubo formula[45, 33], the optical absorption structures are closely related to electronic properties [energy spectra and wave functions], in which they have finite widths and specific forms arising from the initial and final band-edge states [the critical points in energy-wave-vector space].

Up to now, the high-resolution optical observations[47, 48, 49, 59, 51] on layered graphene systems are consistent with part of theoretical calculations[28] in the absence./presence of a uniform perpendicular electric field [$E_{z}\hat{z}$]. For example, the AB-stacked bilayer graphene, the $\sim 0.3$-eV shoulder structure under a zero field[47, 46], the $E_{z}$-created semimetal-semiconductor transition and two low-frequency asymmetric peaks, and two very prominent $\pi$-electronic absorption peaks at middle frequency[47, 46]. The similar examinations performed for the trilayer ABA-stacked graphene clearly show one shoulder at $\sim 0.5eV$[48], the gapless behavior unchanged by the gate voltages, the $E_{z}$-induced low-frequency multi-peak structures[48], and several strong $\pi$-electronic absorption peaks at higher frequencies[48]. The identified features of absorption spectra in the trilayer ABC stacking cover two low-frequency characteristic peaks[49, 48] and gap opening under the electric field[48]. The above-mentioned strong $\pi$ peaks, which are due to all the valence states of carbon-2pz orbitals, have the most high frequency in Bernal graphite because of the infinite graphene layers[59, 51]. On the other hand, certain theoretical predictions deserve further experimental verifications, e.g., the rich and unique optical properties of the trilayer AAB and AAA stackings [28] and sliding bilayer graphene systems[28, 29].

Magneto-optical spectroscopies are available for a full exploration of the various magnetic quantization phenomena in the dimension-dependent condensed-matter systems, e.g., the 1D-3D graphene-related sp2-bonding systems. A uniform magnetic field is generated by using the superconducting magnet, in which a 25-T cryogen-free superconducting magnet are successfully developed at the High Field Laboratories[52, 53]. Furthermore, the super high magnetic fields can be built under a semidestructive single-turn coil technique that provides a pulsed magnetic field larger than 100 T for pulse lengths of tens microseconds[54]. Up to now, the high-resolution magneto-optical measurements have confirmed the well-known Aharnov-Bohm effect due to the periodical boundary condition in cylindrical carbon nanotubes[55]. The infrared transmission spectra clearly identify the $\sqrt{B_{z}}$-dependent absorption frequencies of the interband Landau-level transitions in mono- and few-layer graphene systems[56, 57]. Specifically, the magneto-Raman spectroscopy is utilized to observe the low-frequency Landau-level excitation spectra for the AB-stacked graphenes up to five layers[58]. Only the well-behaved selection rule of $\Delta n=1$ is revealed in the above-mentioned inter-Landau-level excitation experiments. Apparently, the extra magneto-optical selection rules in few-layer graphene systems are worthy of a series of detailed examinations, e.g., those in trilayer ABC[28] and AAB [28] stackings, and the electric-field-applied AB-stacked systems[28]. As for the magneto-optical reflection/absorption/transmission/Raman experiments on the Bernal [AB-stacked] graphite[59, 60, 61, 62, 63], the vertical optical excitations, which mainly originate from a lot of 1D Landau subbands with the significant $k_{z}$-dispersions, are verified to present the monolayer- and bilayer-like graphene behaviors. Such phenomena respectively, correspond to the magnetic quantizations initiated from K and H points [$k_{z}=0$ and $\pi$].

2.3 Quantum transport apparatus, and differential thermal calorimeter/laser flash analysis

In general, the Hall-bar method is available in fully exploring quantum transport properties since its first discovery by Edwin Herbert Hall in 1879 [64]. The measurements of electrical resistivities for the rectangular thin films are carried out by a five-probe technique, as clearly displayed in Fig. 2.3. There exist the typical dimensions of 2 mm in length, 1 mm in width, and 100 nm in thickness [65]. This equipment is very efficient and reliable in simultaneously measuring the longitudinal resistivity and the transverse Hall voltage [the positive or negative charges of free carriers and their density]. Five gold-pads are evaporated onto the contact areas to ensure the excellent electrical contacts for the transverse Hall and longitudinal resistivity measurements using the standard direct current methods. The Hall voltages are delicately analyzed through reversing the field direction at a fixed temperature to eliminate the offset voltage associated with the unbalanced Hall terminals. The Hall coefficients are taken in a magnetic-field range of ${\sim\,1-10}$ T, and a typical dc current density of ${\sim\,10^{3}-10^{4}}$ A/cm2 is applied along the longitudinal direction of the sample. In addition, another method, the four-probe measurement of van der Pauw, is frequently utilized for the irregular finite-width materials.

Recently, there are a lot of magneto-transport measurements on quantum Hall conductivities of few-layer graphene systems with the various layer numbers and stacking configurations under the low temperatures. Monolayer graphene has been clearly identified to present the unconventional half-integer transverse conductivity $\sigma_{xy}=(m+1/2)4e^{2}/h$ [66], where $m$ is an integer and the 4 factor represents the spin and the equivalence of A and B sublattices. This unique quantization mainly comes from the quantum anomaly of $n^{c,v}=0$ Landau levels corresponding to the Dirac point, as discussed later in Chap. 3.3.1. As to bilayer AB stacking [67], the quantum Hall conductivity is verified to be $\sigma_{xy}=4m^{\prime}e^{2}/h$ [$m^{\prime}$ a non-zero integer]. Furthermore, an unusual integer quantum Hall conductivity, a double step height of $\sigma_{xy}=8e^{2}/h$, appears at zero energy under a sufficiently low magnetic field. This mainly comes from $n=0$ and $n=1$ Landau levels of the first group. The low-lying quantum Hall plateaus in trilayer ABA graphene are revealed as $2e^{2}/h$, $4e^{/}h$, $6e^{2}/h$, and $8e^{2}/h$ [68], with a step height of $2e^{2}/h$, especially for the energy range of $\sim\pm 20$ meV. Such observation is consistent with the calculated Landau-level energy spectra [69]. The neighboring- and next-neighboring-layer hopping integrals, respectively, create the separated Dirac cone and parabolic bands, and the valley splitting of the latter. At low energy, these further lead to six quantized Landau levels with double spin degeneracy and thus the quantum Hall step of $2e^{2}/h$. Specifically, the higher-energy quantum transport properties in trilayer ABA stacking are roughly regarded as the superposition of monolayer and AB bilayer graphenes [70]. Moreover, the ABC-stacked trilayer graphene, shows the significant differences in the main features of quantum Hall conductivities, compared with the ABA trilayer system. Its quantum Hall conductivity [70] behaves as a sequence of $\sigma_{xy}=4(\pm|m^{\prime}|\pm 1/2)e^{2}/h$ in the absence of the $\sigma_{xy}=\pm 2e^{2}/h$, $\pm 4e^{2}/h$, and $\pm 8e^{2}/h$ plateaus. Specifically, a $\sigma_{xy}=12e^{2}/h$ step comes to exist near zero energy, being created the $n=0$, 1 and 2 Landau levels the first group due to the quantization of surface-localized flat bands [69]. There also exists the four-fold spin and valley degeneracy in ABC trilayer stacking, resulting in the quantum conductivity step height of $4e^{2}/h$. The above-mentioned interesting quantum transports open the door for fully exploring the configuration-modulated Hall conductivities in other layered graphenes, such as the trilayer AAB stacking, sliding bilayer systems, and twisted bilayer ones.

Up to now, two kinds of high-resolution instruments, differential scanning calorimeter [DSC; [73, 71, 72]] and laser flash analysis [LFA; [75, 76]], are frequently utilized to measure the thermal properties, especially for the specific heats of condensed-matter systems. As to the former [Fig. 2.4(a)], the sample material is subjected to a linear temperature algorithm. The heat flow rate into the sample, which corresponds to the instantaneous specific heat, is under the continuous and delicate measurements [71]. Specifically, two sample holders are symmetrically mounted inside an enclosure being normally held at room temperature. A primary temperature-control system dominates the average temperature of two sample holders, in which it covers platinum resistance thermometers and heating elements embedded in the sample holders. Furthermore, a secondary temperature-control system measures the temperature difference between the two sample holders, and generates the vanishing one by controlling a differential component of the total heating power. This differential power is accurately measured and recorded. Very interesting, an extension of DSC, a modulated differential scanning calorimeter [72, 73], is further developed to provide the more reliable results through measuring the heat capacity of the sample and the heat flow at the same time. MDSC utilizes a sinusoidal temperature oscillation instead of the traditional linear ramp. How to examine the magnetic quantization phenomena in thermal properties is a high-technique measurement. The close association of magnetic-field apparatus and DSC is the so-called PPMS calorimeter [a Quantum Design Physical Property Measurement System for magnetization]; furthermore, this combined instrument can measure the $T$- and $B_{z}$-dependent and specific heat and magnetic susceptibility simultaneously. It should be noticed that the stationary and pulsed magnetic fields can achieve the strengths of 10 T and 50 T in standard laboratories, respectively.

LFA [73], as clearly shown in Fig. 2.4(b), represents a recent technical progress in identifying the crucial thermal phenomena. The sample is positioned on a robot being surrounded by a furnace. For the accurate measurements, the furnace remains at a predetermined temperature and a programmable energy pulse irradiates the back side of the specimen. These create a homogeneous temperature rise at the sample surface. The resulting temperature rise of the surface of the sample is measured by a very sensitive high-speed infrared detector. Both properties, the specific heat and thermal diffusivity, are simultaneously determined from the temperature versus time data. The thermal conductivity could also be evaluated from the estimated carrier density. In short, LFA covers the experimental operation processes: uniform heating by a homogenized laser beam through an optical fiber with a mode mixer, measuring the transient temperature of a sample with a calibrated radiation thermometer, analyzing a transient temperature curve with a curve fitting method, and achieving the differential laser flash calorimetry. Recently, LFA is commonly used for the measurements of thermal properties, mainly owing to the excellent performances in the high accuracy, good repeatability, and wide temperature-range [73, 71, 72].

The high-resolution experimental measurements on specific heats of condensed-matter systems are very useful in examining the theoretical predictions about the electronic and phonon energy spectra, e.g., those measured for the various carbon-related materials. For example, The thermal examinations on the AAA- and ABC-stacked graphites [simple hexagonal and rhombohedral configurations] are lacking so far, while they have been conducted for the AB-stacked graphite [Bernal type], including electronic specific heat [77, 80] and lattice specific heat [DeSorbo;1660;Krumhansl;1663]. The previous experiments clearly show that the former presents a linear T-dependence at very low temperature [${T<1.2}$ K; [77]], and the latter follows the $T^{3}$ law at low temperature and approximately $T^{2}$ law at higher temperature [78]. The applications of graphite systems promise an economical way to a new class of efficient thermal management materials, such as, the relatively high thermal conductivities very suitable for thermal interface applications [81, 82]. Specifically, the magneto-electronic specific heats of conventional quantum Hall systems have been successfully verified for many years. In the research on GaAs VGaAlAs [83], the thermal properties are accurately measured under the influences of temperature and magnetic field. For the temperature-dependent specific heat [84], the specific heat reveals a peak structure at a low critical temperature. Furthermore, the magnetic-field dependence heat obviously presents an oscillation behavior [83]. These two kinds of behaviors directly reflect the main characteristics of the Landau levels, such as the filling factors, the energy spacings and the Zeeman effects. The similar examinations could be generalized to the emergent layered material in fully understanding the diverse thermal phenomena under the various magnetic quantizations.

The magnet-electronic specific heats could present another magnetic quantization phenomenon. The Landau levels in monolayer graphene are very suitable for a model investigation. It is well known that the hexagonal crystal structure accounts for the unique low-energy electronic properties, the isotropic and linear valence and conduction bands intersecting at the Dirac point and the square-dependent Landau-level energy spectrum of ${E^{c,v}\propto\,\sqrt{n^{c,v}B_{z}}}$. The Zeeman effect further splits the magneto-electronic states into the spin-up and spin-down ones. In general, the level spacings between two neighboring quantum numbers are much wider than the Zeeman splitting energy. This clearly indicates that the former is insensitive to temperature. Furthermore, the latter might be comparable to the thermal energy at low temperature. The Zeeman effect is expected to plays an important role in the thermal properties. On the other hand, monolayer graphene could be rolled up to become a hollow cylinder, [a single-walled carbon nanotube]. Apparently, the periodic boundary condition is responsible for the creation of one-dimensional energy bands with linear and parabolic energy dispersions. However, it is almost impossible to induce the dispersionless Landau levels in nanotube surfaces under various high magnetic fields [85]. Dimensionality and magnetic quantization have a great influence on the van Have singularities of the density of states (DOS), so the magneto-electronic specific heats would be very different between these two systems.

2.4 Electron energy loss spectroscopy and light inelastic scatterings

Electron energy loss spectroscopy [EELS; [96, 97, 86, 87, 88, 89, 94, 90, 91, 92, 93, 95] and inelastic x-ray scatterings [IXS; [107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124]] are the only two available tools in delicately examining excitation spectra of condensed-matter systems. They have successfully identified the reliable Coulomb excitations and phonon dispersion spectra in any dimensional materials, being supported by the theoretical predictions[88, 89, 94, 90, 92, 95]. In general, EELS is achieved by the scattered reflection [98, 99, 100] and transmission[101, 102, 115, 104, 105, 106] of a narrow electron beam, in which the former is very suitable for thoroughly exploring the low-energy excitation properties lower than 1 eV[98]. RELLS is chosen for a clear illustration. This outstanding instrument extracts the bulk or surface energy loss functions from the backward electrons scattered from a specific sample[115]. If the incident electron beam has a kinetic energy of several hundreds eVs, the very efficient technique can provide the screened response functions with an energy resolution of a few meVs, being sufficient in resolving atomic and electronic excitation modes[115, 104]. Up to date, REELS is widely utilized to explore the rich physical, chemistry and material properties of condensed-matter surfaces [102]. It frequently operates under a 25-meV energy resolution within the energy range between 15 and 70 eV[102], while controlling the momentum resolution down to 0.013 $\AA^{-1}$, about one percent of a typical Brillouin zone[102]. Moreover, the energy resolution could be greatly enhanced to $\sim 1$ 1 meV under the specific condition, in which much weaker electron beams are adopted using a high-resolution monochromator at an ultrahigh vacuum base pressure [$\sim 2\times 10^{-10}$ Torr]. The momentum-dependent dispersion relations of collective excitation modes can be accurately detected by an angle-resolved EELS, which is performed at the low kinetic energy and utilizes an analyzer to measure the scattered electrons. The delicate analyzer consists of a magnetic-prism system, as clearly displayed in Fig. 2.5, where the commercially available Gatan spectrometer is installed beneath the camera and the basic interface and ray paths are shown as well. The surface of this prism is curved to largely reduce the spherical and chromatic aberrations. The scattered electrons in the drift tube are deflected by a uniform magnetic field into a variable entrance aperture (typical variation from 1 to 5 mm in diameter). The electrons that lose more energies, deflect further away from the zero-energy-loss electrons according to the Lorenz force law. Furthermore, all the electrons in any directions are focused on the dispersion plane of the spectrometer. The magnetic prism projects the electron energy-loss spectrum onto a CCD camera, which is straightforward to capture the whole energy distribution simultaneously. It is possible to modulate the resolution of the transferred momentum by varying the half-angle of the incident beam in TEM and the scattering one in the spectrometer. For example, based on the angle variation of a few milliradians, the momentum resolution lies in the order of $\sim 0.01$ $\AA^{-1}$[101, 102, 115].

The high-resolution EELS measurements are very powerful in thoroughly exploring electronic Coulomb excitation modes in various condensed-matter systems. They have clearly identified the significant single-particle and collective excitations in the sp2-bonding graphene-related materials, such as, 3D layered graphites[86, 87, 88], graphite intercalation compounds[89], 2D monolayer and few-layer graphene systems[90, 91, 92, 93], 1D single- and multi-walled carbon nanotubes[94], graphene nanoribbons[95], 0D C60-associated fullerenes[96], and carbon onions [97]. In general, the distinct dimensions, geometric symmetries, stacking configurations, interlayer atomic interactions, temperature, and chemical dopings might induce the high/some/few free conduction-electron [valence-hole] density, leading to the low-frequency acoustic or optical plasmon modes with frequencies lower than 1.5 eV. For room temperature, the AB-stacked graphite presents the low-frequency optical plasmons at $\sim 45-50$ meV and 128 meV under the long wavelength limit, corresponding to the electric polarizations parallel and perpendicular to the $(x,y)$- plane, respectively[86]. Furthermore, the detailed analyses on temperature-dependent energy loss spectra verify the strong temperature effects in the former[86]. The high-density free electrons and holes, which are, respectively, doped in the donor- and acceptor type graphite intercalation compounds, lead to the $\sim 1$-eV optical plasmons[89]. This represents their coherent collective oscillations in the periodical graphitic layers, strongly depending on the transferred momenta[89]. Concerning the layered graphene systems, with the metal-adatom chemisorptions[90, 91, 92, 93], the low-frequency plasmon modes become 2D acoustic ones, i.e., they have the $\sqrt{\bf q}$ dependence at small transferred momenta[90, 91, 92, 93]. The above-mentioned important results are consistent with the theoretical illustrations[\cite[cite]{[@@bibref{}{2PRL89;076402, 89, 94, 90, 92, 2Nature468;1088}{}{}]}].

The inelastic X-ray scatterings [IXS] are capable of directly detecting the behaviors of dynamic charge screenings in crystal/non-crystal systems. They are successfully conducted on a wide range of physical phenomena, e.g., phonon energy dispersions in solids[107, 108], carrier dynamics of disordered materials[109, 110, 111, 112] $\&$ biological structures[107], and electronic excitations in condensed-matter systems[113, 114, 115, 116, 117, 118, 119]. The transferred energy and momentum are available parameters and can cover the whole spectra of the dielectric responses. The uniform x-ray beam line is designed to provide a super-high photon flux within the typical wave-vector range of the first Brillouin zone. Furthermore, photon energy is distributed from 4.9 keV to 15 keV, being accompanied by energy and momentum resolutions of $\sim 70$ meV and $0.02-0.03$ $\AA^{-1}$, respectively. Specifically, the IXS instrument in the Swiss Light Source possesses an extremely good energy resolution of $\sim 30$ meV [125]. The super high resolution, $\sim 10$ meV, is expected to be achieved in the further development of new synchrotron sources. IXS is available in fully exploring all kinds of electronic excitations because the electronic charges strongly interact with the high-energy photon beam under the various transferred momenta and energies. Using hard X-ray synchrotron sources, this spectroscopy becomes a very powerful technique to observe the intrinsic properties of condensed-matter systems, and it is suitable for the external electric and magnetic fields [126]. The analyzer, which is built on the basis of Bragg optics, as depicted in Fig. 2.5, efficiently collects and analyzes the energy and momentum distributions of the scattered photons with a small solid angle [$d\Omega$]. This instrument can provide full information on the screened response function. To maximize the light intensity, a spherically bent analyzer ($\sim 10$ cm in diameter) is frequently utilized to capture all the scattered radiation of the momentum-transferred photons in $d\Omega$. The transferred energy is projected onto a CCD detector, and then the full energy loss spectrum is scanned by varying the Bragg angle of the specific crystal. When IXS is operated in the Rowland circle geometry, the specific measurement of the double differential scattering cross-section corresponds to the screened electronic excitations in the characteristic energy-loss regime by using the dissipation-fluctuation theorem.

There exist certain important differences between EELS and IXS in terms of the physical environments. The incident electron and photon beams can be, respectively, focused on the spatial range of $\sim 10$ $\AA$ [127] and 100 $\AA$ [128]. The former possesses more outstanding resolutions for the transferred momenta and energies during the many-particle interactions. Much more inelastic scattering events are conducted by using the EELS within a short time, i.e., the EELS measurements on the full excitation spectra are performed more quickly and accurately. EELS is very reliable for the low-dimensional materials and nano-scale structures, mainly owing to the simultaneous identifications on their positions. On the other hand, IXS, which arises from the continuous synchrotron radiation sources, presents a rather strong intensity with the tunable energies and momenta. Most importantly, the extreme surrounding environments, induced by the applications of magnetic and electric fields as well various temperatures and pressures, can be overcome under the inelastic light scattering. The external fields have strong effects on the incident charges and the sample chamber is too narrow; therefore, EELS cannot work under such environments. As a result, the high-resolution IXS measurements are very useful in thoroughly understanding the magneto-electronic single-particle and collective excitations in any condensed-matter systems, especially for those in graphene-related materials.