Title Page
Abstract
Contents
Chapter 1. Introduction 22
Chapter 2. Theoretical Background 25
2.1. Building the Commensurate Moiré Cell 25
2.2. Tight-Binding model (TB model) 28
2.2.1. Effective TB model 28
2.2.2. Full TB model 30
2.3. Molecular Dynamics simulations for the atomic relaxations 41
2.4. Lanczos recursion 41
Chapter 3. Graphene on h-BN, BG and BG/h-BN benchmark calculations 43
3.1. Graphene on h-BN 43
3.2. Bilayer graphene and BG/h-BN 46
Chapter 4. Angle-Dependent Primary and Secondary Gaps for G/h-BN 48
Chapter 5. Layer Polarization and Charge Transfer in Encapsulated BG 58
5.1. Rigid and relaxed h-BN-encapsulated BG graphene 58
Chapter 6. Conclusion 67
Chapter 7. Appendices 70
A. TB model and electronic structure considerations 70
A.1. Bilayer vs single layer F2G2 70
A.2. Interlayer-distance dependent intralayer moiré terms 71
A.3. Inter-layer Tunneling Maps 71
A.4. Electronic band structures 71
A.5. Probability distribution of bandgap edge states 73
B. Commensurate simulation cells 77
C. Total energy calculations 78
D. Local rotation angle θR[이미지참조] 79
E. Random phase convergence 81
F. Charge transfer sliding maps 82
References 84
국문초록 94
Table 2.1. Average hopping strengths tijn corresponding to the intrasublattice terms with structure factors gn and intersublatice interactions with structure fac-...[이미지참조] 33
Table 2.2. Honsite from Eq. (2.24) at different stacking configurations AA, AB and BA represented here by A, B and C labels. The average, equal to C0ii in...[이미지참조] 33
Table 2.3. Fitting parameters entering Eq. (2.27) to take into account the inter-layer distance-dependence on the parameters from Table 2.2. The suitable range... 35
Table 2.4. Fitting parameters p and q entering Eq. (2.33) for the four different possible interactions. The same parameters fit both the real and imaginary part... 40
Table B.1. Commensurate angles for G/h-BN with their corresponding four indices in Eq. (2.4), including the actual αBN value as well as the number of atoms...[이미지참조] 77
Table B.2. Commensurate angles for h-BN/G/h-BN with their corresponding six indices in Eq. (2.5), including the actual αBNL₁ and αBNL₄ values. These are used...[이미지참조] 78
Table C.1. Commensurate angles for G/h-BN and their corresponding four indices in Eq. (2.4), with more accurate lattice constant αBN than Table B.1. These...[이미지참조] 79
Figure 2.1. Visual illustration of the different systems considered in this system, namely BG, G/h-BN, BG/h-BN and h-BN/BG/h-BN. Due to the inequivalent B... 26
Figure 2.2. (color online) Matrix elements Hii(dij : K) (top panels) and Hij(dij : K) (bottom panels) from Eq. (2.24) and Eq. (2.28) respectively. C₁ and C₂ are...[이미지참조] 36
Figure 2.3. (top panels) d-dependent tunneling map for an interlayer distance of 3.35 Å, focusing here on the real part of the C₁ - B interaction. The left panel... 38
Figure 3.1. Electronic band structure for the rigid configuration (dashed lines) and the suspended configuration (solid lines). The increase in primary gap and... 44
Figure 3.2. Electronic band structures and corresponding layer-resolved DOS for BG and BG/h-BN systems. BG is represented both using the 4-atom unit cell as... 45
Figure 4.1. (a) Primary (red) and secondary (blue) gap estimates and average mass term m (green) for G/BN with respect to the twist angle θ. In the upper...[이미지참조] 49
Figure 4.2. (a) z-coordinate of the atoms in the top and bottom layers as well as their subtraction illustrating the interlayer distance indicating that the most stable... 52
Figure 4.3. (top row) Maximum value from the twist angle θ-dependent local rotation angle θR at the center of the AA (counter-clockwise) and AB (clock-...[이미지참조] 56
Figure 4.4. Energies renormalized per atom for for three different relaxation types. The values are globally shifted by -7.07020 eV/atom to bring the sus-... 57
Figure 5.1. Electronic band structures and corresponding layer-resolved DOS for the 4 flavors of h-BN/BG/h-BN in their most stable stacking configuration as... 59
Figure 5.2. (top) Charge transfer amplitudes that indicate that Type II and type III show the largest charge transfer regardless of their sliding configuration, while... 62
Figure 5.3. (left) θ₁₂/θ₄₃-dependent charge transfer for the rigid systems using the effective model from Eq. (2.9) and (right) corresponding maps for the re-... 63
Figure 5.4. Charge transfer amplitudes in presence of an applied electric field using onsite energy shifts in Eq. (2.9) for the different allotropes for their different... 65
Figure A.1. Electronic bandstructure comparisons between full DFT calculations and TB model where the intralyer terms use either monolayer G and h-BN... 72
Figure A.2. Comparison between the band structures of aligned h-BN when the interlayer-distance dependent parametrization of the DFT calculation is considered... 73
Figure A.3. Additional comparisons between DFT and TB tunneling maps as first shown in Fig. 2.33 for C₁ - B repeated in the first row here. The other rows... 74
Figure A.4. Band structure of rigid G/h-BN and suspended G/h-BN for different twisted angles. The horizontal lines define the edges of the bands that define the... 75
Figure A.5. Probability distribution maps for aligned suspended G/h-BN for states at both the valence and conduction band edges of the bandgap as well... 76
Figure D.1. (color online) (a) Continuum and (b) real space calculation of the local rotation angles for θ = 0˚ and θ = 0.558˚ where for the real-space we... 80
Figure E.1. (main) Convergence study on the number of random phases NRP using θ₃₂ = 0 for Type I for different twist angles. (inset) Same as main, where... 82
Figure F.1. Same data as in Fig. 5.2 but with a different colormap range representation. 83