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Conclusions and perspectives ing metals is also of high relevance and could be based upon the existing studies of similar interfaces. The whole issue of spin transport in ferromagnetically contacted carbon nanotubes was, of course, only scratched on the surface by our study. Any in-depth work on this subject will certainly have to do a far more detailed modeling of the contacts. We believe, however, that our concept of extended contacts will prove valuable also when considering ferromagnetic metals. For disorder in nanotubes and ribbons, the obvious next step will be a detailed investigation of the deviations from the perturbative regime. Some of the effects may already be explained by taking into account the modified density of states in a disordered system. Other effects may be captured by higher order perturbation theory. Alternatively, it will certainly be worthwhile to take a close look at the random-matrix theoretical approach, to see in what aspects our straightforward may yet be improved. Point defects leave even more room for future studies. Individual point defects may have very different effects on the transmission, especially in ribbons, where the exact position also has to be taken into account. Even more complexity, however, is to be expected in the study of the interference between multiple defects. In the study of the multilayer coupling, the most immediate next step would be the search for an improved tight binding parameterization that take into account the special demands of incommensurate systems. Fitting ab initio results of continuously displaced commensurate systems would be a possible step to proceed in this direction. At the time being, such attempts are yet unknown. Finally, for studies of the magnetoelectronic structure, it would be of great interest to observe the effect of superlattice structures on the Landau levels of graphene. At an appropriate scale of such structures, the fractality of the Hofstadter butterfly should be directly experimentally observable and could give an interesting interplay with the anomalous Landau levels of the underlying system. 158
Appendix A. Decimation techniques Decimation is a powerful technique for the efficient handling of extended systems. This tutorial explains the basics as well as two of the more advanced applications. Once the reader is familiar with the basic principle behind these algorithms, one should quickly be able to model new use cases of different geometries of symmetries. All of the following was written with a quasi-one-dimensional singleelectron tight-binding model in mind (more specifically a carbon nanotube in π orbital approximation). The technique, however should be just as useful for many other areas that involve a localized basis and a local Hamiltonian in this basis that allows the exploitation of the locality of each decimation step. A.1. The fundamental equation of decimation Be A an invertible matrix of the form: A = A11 A12 A21 A22 where all for Aij are themselves matrices, A11 and A12 invertible. The inverse B = A−1 can be divided up in the same way: B = B11 B12 B21 B22 Expanding the equation AB = into these block matrices, we get for equations: A11B11 + A12B21 = (A.1) A21B11 + A22B21 = 0 (A.2) A11B12 + A12B22 = 0 A21B12 + A22B22 = 159
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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.
Page 107 and 108: 5.2. The elastic mean free path Fig
Page 109 and 110: 5.3. Strong localization 5.3. Stron
Page 111 and 112: 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: periodic and gives little rise to b
Page 161 and 162: From Eq. (A.4) it follows that: G11
Page 163 and 164: A.3. Finite block tridiagonal syste
Page 165 and 166: A.4. Periodic systems the other. Th
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
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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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