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Appendix A. Decimation techniques with G (1) 33 = (E+iη−H(1) 33 )−1 , and so on. Notice that with each decimation step, only the cells that were in direct contact with the decimated cell are affected. To simplify the notation, we introduce: H (n) LL := H (n) 11 H (n) LR := H(n) 1,n+2 H (n) RL := H (n) n+2,1 H (n) RR := H(n) n+2,n+2 Now, with H (0) i,j = Hi,j, we can run an iteration over n = 0, 1, . . . , N − 2 following the steps: G (n) XX := E + iη − H (n) RR −1 H (n+1) LL = H (n) LL + H(n) LRG(n) XXH(n) RL H (n+1) LR = H (n) LRG(n) XXH(0) n+1,n+2 H (n+1) RL = H (0) n+2,n+1G(n) XXH(n) RL H (n+1) RR = H (0) n+2,n+2 + H(0) n+2,n+1G(n) XXH(0) n+1,n+2 and arrive at the desired quantities: H eff 11 H eff 1,N H eff N,1 Heff N,N = H (N−2) LL H (N−2) RL H (N−2) LR H (N−2) RR A simple implementation of this algorithm is given in App. C.6. A.3.1. Thoughts about efficiency The straightforward way to obtain the quantities G11 (E), G1,N (E), GN,1 (E) and GN,N (E) would be to invert the matrix E + iη − H completely and read out the data from the corners. The general matrix inversion itself, however, is a order N 3 operation in time and N 2 in memory. The algorithm above, on the other hand is order N in both time and memory. For any system that can be split into more than a few cells, this will give a gain in speed and memory consumption that easily outweighs the slightly increased complexity. It has been tried before to take the decimation method one step further and split the system into individual atoms, decimating one atom after 164
A.4. Periodic systems the other. This, however, will only bring additional gain if the H (0) i,j block matrices themselves are sparse, i.e., if there is low connectivity inside each cell. For dense block matrices, it is generally more efficient to construct the block matrices once and let some optimized library do the inversion. A.4. Periodic systems A periodic quasi-1D system with a localized basis has an infinite Hamiltonian of the form: H = ⎛ ⎞ . .. . .. . .. . .. . .. · · · 0 ⎜ ⎟ ⎜ ⎟ ⎜ 0 H10 H00 H01 0 . ⎟ ⎜ ⎟ ⎜ ⎝ . 0 H10 H00 H01 0 ⎟ ⎠ . 0 · · · .. . .. . .. . .. . .. cutting this system in halves, we get two semi-infinite surface systems: HL = HR = ⎛ . .. ⎜ ⎝ . . .. 0 . .. H10 0 . .. H00 H10 . .. H01 H00 ⎞ 0 ⎟ 0 ⎟ ⎠ H01 ⎛ 0 · · · 0 H10 H00 ⎞ ⎜ ⎝ H00 H01 0 · · · 0 H10 H00 H01 0 0 H10 H00 H01 0 0 . .. As always, the Green functions have the same dimensionalities as the corresponding Hamiltonian, but now, the entries in the bulk and at the . .. . .. . .. . . .. ⎟ ⎠ 165
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Quantum Transport in Carbon-based N
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Contents Introduction 9 1. Carbon-b
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Contents A. Decimation techniques 1
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Introduction Carbon is the most ver
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Chap. 3: Quantum transport The theo
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Chapter 1. Carbon-based nanostructu
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1.1. Hybridization of carbon orbita
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1.1. Hybridization of carbon orbita
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Geometry 1.2. Graphite Figure 1.4.:
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1.3.1. Isolation by exfoliation 1.3
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1.3. Graphene Figure 1.10.: The hon
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1.4. Graphene nanoribbons Figure 1.
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1.5. Carbon nanotubes Figure 1.17.:
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1.6. Fullerenes Figure 1.23.: Struc
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Chapter 2. Electronic structure The
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2.1. The tight-binding approximatio
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2.1. The tight-binding approximatio
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2.2. Band structure of graphene nan
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2.3. Band structure of single-wall
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2.3. Band structure of single-wall
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2.5. Beyond tight-binding: Density
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2.5. Beyond tight-binding: Density
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Chapter 3. Theory of quantum transp
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3.1. Mesoscopic length scales In cr
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3.1. Mesoscopic length scales Figur
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3.2. The transport regimes 3.2. The
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3.4. The quantum mechanical transmi
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3.5. Beyond coherent transport: int
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Chapter 4. Electrical contacts to n
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4.1. Conventional contact models in
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4.2. Extended contacts Figure 4.2.:
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4.2. Extended contacts Figure 4.10.
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4.2. Extended contacts 4.2.5. Three
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4.2.6. Non-epitaxial contacts 4.2.
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4.2. Extended contacts spacing in g
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4.3. Ferromagnetic contacts and spi
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Chapter 5. Disorder and defects The
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5.1. Anderson model for disorder 5.
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5.2. The elastic mean free path Fig
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5.3. Strong localization 5.3. Stron
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5.4. Vacancies and defects Figure 5
Page 113 and 114: to occur in the same unit cell, ℓ
Page 115 and 116: 5.5. Graphene nanoribbons states, b
Page 117 and 118: Chapter 6. Multilayer graphene and
Page 119 and 120: 6.1. Commensurability up of two arm
Page 121 and 122: 6.2. Modeling the interlayer coupli
Page 123 and 124: 6.3. Bilayer graphene Figure 6.3.:
Page 125 and 126: 6.4. Double-wall carbon nanotubes t
Page 127 and 128: 6.4. Double-wall carbon nanotubes s
Page 129 and 130: 6.5. Telescopic carbon nanotubes st
Page 131 and 132: 6.5. Telescopic carbon nanotubes Th
Page 133 and 134: 6.5. Telescopic carbon nanotubes Fi
Page 135 and 136: 6.5. Telescopic carbon nanotubes Fi
Page 137 and 138: Chapter 7. Magnetoelectronic struct
Page 139 and 140: 7.1. Peierls substitution periodic
Page 141 and 142: 7.2. Hofstadter butterfly Figure 7.
Page 143 and 144: 7.3. Butterfly and anomalous Landau
Page 145 and 146: 7.4. Butterfly of single-wall nanot
Page 147 and 148: 7.5. Graphene nanoribbons for arbit
Page 149 and 150: 7.7. Bilayer graphene total flux pe
Page 151 and 152: 7.8. Butterfly of double-wall nanot
Page 153 and 154: 7.8. Butterfly of double-wall nanot
Page 155 and 156: Conclusions and perspectives In the
Page 157 and 158: periodic and gives little rise to b
Page 159 and 160: Appendix A. Decimation techniques D
Page 161 and 162: From Eq. (A.4) it follows that: G11
Page 163: A.3. Finite block tridiagonal syste
Page 167 and 168: A.4. Periodic systems which can be
Page 169 and 170: A.4. Periodic systems Now, it turns
Page 171 and 172: Appendix B. Analytic derivations B.
Page 173 and 174: B.1. Supersymmetric spectrum of gra
Page 175 and 176: B.2. The linear-chain model of exte
Page 177 and 178: B.2. The linear-chain model of exte
Page 179 and 180: B.3. The elastic mean free path in
Page 181 and 182: B.3. The elastic mean free path in
Page 183 and 184: Appendix C. Numerical implementatio
Page 185 and 186: C.3. Constructing chiral carbon nan
Page 187 and 188: C.3. Constructing chiral carbon nan
Page 189 and 190: C.5. Tight-binding interlayer Hamil
Page 191 and 192: C.6. Decimation of a finite block t
Page 193 and 194: C.8. Periodic gauge C.8. Periodic g
Page 195 and 196: Appendix D. Reprints of publication
Page 197 and 198: 180 S. Krompiewski et al.: Spin tra
Page 199 and 200: 182 S. Krompiewski et al.: Spin tra
Page 201 and 202: PRL 96, 076802 (2006) tion. For bot
Page 203 and 204: PRL 96, 076802 (2006) contact regio
Page 205 and 206: NORBERT NEMEC AND GIANAURELIO CUNIB
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NORBERT NEMEC AND GIANAURELIO CUNIB
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Hofstadter butterflies of bilayer g
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HOFSTADTER BUTTERFLIES OF BILAYER G
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Bibliography [1] S. Krompiewski, N.
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Bibliography [31] D. Bethune, C. Kl
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Bibliography [66] S. Datta. Electri
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Bibliography [100] F. Guinea, A. H.
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Bibliography [135] A. E. Karu and M
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Bibliography [171] A. Maiti and A.
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Bibliography [205] D. Papaconstanto
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Bibliography [239] B. Shan and K. C
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Bibliography [274] S. Wang, M. Grif
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Acknowledgments Although a few word
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