Topological insulators: Dirac equation in condensed matters
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Heidelberg [u.a.]
Springer
2012
|
| Series: | Springer series in solid-state sciences
174 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XIII, 225 S. Ill., graph. Darst. |
| ISBN: | 9783642328572 |
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MARC
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| 245 | 1 | 0 | |a Topological insulators |b Dirac equation in condensed matters |c Shun-Qing Shen |
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Record in the Search Index
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|---|---|
| adam_text | Contents
1
Introduction
................................................................. 1
і
. 1
From the Hall Effect to Quantum Spin Hall Effect
................. 1
1.2
Topologica!
Insulators as Generalization of Quantum
Spin Hall Effect
...................................................... 5
.3
Topologica!
Phases in Superconductors and Superfluids
........... 7
.4
Dirac Equation and Topological Insulators
......................... 8
.5
Summary: The Confirmed Family Members
........................ 9
.6
Further Reading
...................................................... 9
References
.................................................................... 10
Starting from the Dirac Equation
........................................ 13
2.1
Dirac Equation
........................................................ 13
2.2
Solutions of Bound States
........................................... 15
2.2.1
Jackiw-Rebbi Solution in One Dimension
................ 15
2.2.2
Two Dimensions
............................................ 18
2.2.3
Three and Higher Dimensions
............................. 19
2.3
Why Not the Dirac Equation
......................................... 19
2.4
Quadratic Correction to the Dirac Equation
........................ 19
2.5
Bound State Solutions of the Modified Dirac Equation
............ 20
2.5.1
One Dimension: End States
................................ 20
2.5.2
Two Dimensions: Helical Edge States
..................... 22
2.5.3
Three Dimensions: Surface States
......................... 24
2.5.4
Generalization to Higher-Dimensional
Topological Insulators
...................................... 26
2.6
Summary
.............................................................. 26
2.7
Further Reading
...................................................... 26
References
.................................................................... 26
Minimal Lattice Model for Topological Insulator
....................... 29
3.1
Tight Binding Approximation
....................................... 29
3.2
From Continuous to Lattice Model
.................................. 31
3.3
One-Dimensional Lattice Model
.................................... 33
vii
viii Contents
3.4 Two-Dimensional Lattice Model.................................... 36
3.4.1 Integer Quantum Hall
Effect
............................... 36
3.4.2 Quantum
Spin Hall Effect
.................................. 38
3.5 Three-Dimensional Lattice Model................................... 38
3.6
Parity at the Time Reversal Invariant Momenta
.................... 40
3.6.1
One-Dimensional Lattice Model
........................... 40
3.6.2
Two-Dimensional Lattice Model
.......................... 42
3.6.3
Three-Dimensional Lattice Model
......................... 43
3.7
Summary
.............................................................. 45
References
.................................................................... 45
4
Topologica!
Invariants
...................................................... 47
4.1
Bloch Theorem and Band Theory
................................... 47
4.2
Berry Phase
........................................................... 48
4.3
Quantum Hall Conductance and Chern Number
.................... 51
4.4
Electric Polarization in a Cyclic Adiabatic Evolution
.............. 55
4.5
Thouless Charge Pump
............................................... 56
4.6
Fu
-Капе
Spin Pump
.................................................. 58
4.7
Integer Quantum Hall Effect: Laughlin Argument
................. 61
4.8
Time Reversal Symmetry and the Z2 Index
......................... 63
4.9
Generalization to Two and Three Dimensions
...................... 67
4.10
Phase Diagram of Modified Dirac Equation
........................ 70
4.11
Further Reading
...................................................... 73
References
.................................................................... 73
5
Topological Phases in One Dimension
.................................... 75
5.1
Su-Schrieffer-Heeger Model for Poly acetylene
..................... 75
5.2
Ferromagnet with Spin-Orbit Coupling
............................. 80
5.3
/7-Wave Pairing Superconductor
..................................... 80
5.4
Ising Model in a Transverse Field
................................... 82
5.5
One-Dimensional Maxwell s Equations in Media
.................. 83
5.6
Summary
.............................................................. 84
References
.................................................................... 84
6
Quantum Spin Hall Effect
................................................. 85
6.1
Two-Dimensional Dirac Model and the Chern Number
............ 85
6.2
From Haldane Model to
Капе
-Mele
Model
......................... 86
6.2.1
Haldane Model
.............................................. 86
6.2.2
Капе
-Mele
Model
.......................................... 88
6.3
Transport of Edge States
............................................. 92
6.3.1 Landauer-Büttiker
Formalism
.............................. 92
6.3.2
Transport of Edge States
................................... 95
6.4
Stability of Edge States
.............................................. 98
6.5
Realization of Quantum Spin Hall Effect in HgTe/CdTe
Quantum Well
........................................................ 99
6.5.1
Band Structure of HgTe/CdTe Quantum Well
............. 99
Contents ix
6.5.2
Exact Solution of Edge States
.............................. 102
6.5.3
Experimental Measurement
................................ 105
6.6
Quantum Hall Effect and Quantum Spin Hall Effect:
A Case Study
......................................................... 107
6.7
Coherent Oscillation Due to the Edge States
....................... 109
6.8
Further Reading
......................................................
Ill
References
....................................................................
Ill
7
Three-Dimensional Topological Insulators
.............................. 113
7.1
Family Members of Three-Dimensional Topological Insulators
... 113
7.1.1
Weak Topological Insulators: Pb v Sn
,
_.v
Te
................
J
13
7.1.2
Strong Topological Insulators: Bii_vSbv
.................. 114
7.1.3
Topological Insulators with a Single Dirac
Cone: Bi2Se3 and
ВІ2ТЄ3
................................... 115
7.1.4
Strained HgTe
............................................... 115
7.2
Electronic Model for Bi2Se3
......................................... 117
7.3
Effective Model for Surface States
.................................. 119
7.4
Physical Properties of Topological Insulators
....................... 122
7.4.1
Absence of Backscattering
................................. 122
7.4.2
Weak Antilocalization
...................................... 123
7.4.3
Shubnikov-de Haas Oscillation
............................ 124
7.5
Surface Quantum Hall Effect
........................................ 125
7.6
Surface States in a Strong Magnetic Field
.......................... 128
7.7
Topological Insulator Thin Film
..................................... 130
7.7.1
Effective Model for Thin Film
............................. 130
7.7.2
Structural Inversion Asymmetry
........................... 133
7.7.3
Experimental Data of
ARPES
.............................. 134
7.8
HgTeThinFilm
...................................................... 136
7.9
Further Reading
...................................................... 137
References
.................................................................... 138
8
Impurities and Defects in Topological Insulators
....................... 141
8.1
One Dimension
....................................................... 141
8.2
Integral Equation for Bound State Energies
......................... 143
8.2.1
¿—potential
.................................................. 144
8.3
Bound States in Two Dimensions
................................... 145
8.4
Topological Defects
.................................................. 149
8.4.
1 Magnetic Flux and Zero-Energy Mode
.................... 149
8.4.2
Wormhole
Effect
............................................ 151
8.4.3
Witten
Effect
................................................ 152
8.5
Disorder Effect to Transport
......................................... 155
8.6
Further Reading
...................................................... 157
References
.................................................................... 157
x
Contents
9 Topological Siiperconductors and
Superflui
ds..........................
159
9.1
Complex
{ρ
+
//»)-Wave Superconductor of
Spinless
or Spin-Polarized
Fermions
.......................................... 160
9.2
Spin-Triplet Pairing Superfluidity:
3He-A
and 3He-B Phases
...... 164
9.2.1
3He: Normal Liquid Phase
................................. 164
9.2.2
^He-B Phase
................................................ 165
9.2.3
3He-A Phase: Equal Spin Pairing
.......................... 167
9.3
Spin-Triplet Superconductor: Sr2RuO4
............................. 169
9.4
Superconductivity in Doped Topological Insulators
................ 170
9.5
Further Reading
...................................................... 171
References
.................................................................... 171
10
Majorana
Fermions
in Topological Insulators
.......................... 173
10.1
What Is the
Majorana Fermion?
..................................... 173
10.2
Majorana
Fermions
in /»-Wave Superconductors
................... 174
10.2.1
Zero-Energy Mode Around a Quantum Vortex
........... 174
10.2.2
Majorana
Fermions
in Kitaev s Toy Model
............... 177
10.2.3
Quasi-One-Dimensional Superconductor
................. 179
10.3
Majorana
Fermions
in Topological Insulators
...................... 182
10.4
Detection of
Majorana
Fermions
.................................... 183
10.5
Sau-Lutchyn-Tewari-Das
Sarma
Model for
Topological Superconductor
......................................... 185
10.6
Non-Abelian Statistics and Topological Quantum Computing
.... 188
10.7
Further Reading
...................................................... 189
References
.................................................................... 190
11
Topological Anderson Insulator
........................................... 191
11.1
Band Structure and Edge States
..................................... 191
11.2
Quantized Anomalous Hall Effect
................................... 193
11.3
Topological Anderson Insulator
..................................... 195
11.4
Effective Medium Theory for Topological Anderson Insulator
__ 197
11.5
Band Gap or Mobility Gap
........................................... 198
11.6
Summary
.............................................................. 199
11.7
Further Reading
...................................................... 201
References
.................................................................... 201
12
Summary: Symmetry and Topological Classification
.................. 203
12.1
Ten Symmetry Classes for Noninteracting Fermion Systems
...... 203
12.2
Physical Systems and the Symmetry Classes
....................... 205
12.2.1
Standard (Wigner-Dyson) Classes
......................... 205
12.2.2
Chiral Classes
............................................... 206
12.2.3
Bogoliubov-de Gennes (BdG) Classes
.................... 206
12.3
Characterization in the Bulk
......................................... 207
12.4
Five Types in Each Dimension
...................................... 208
12.5
Conclusion
............................................................ 208
12.6
Further Reading
...................................................... 209
References
.................................................................... 209
Contents xi
A Derivation of Two Formulae
............................................... 211
A.
1
Quantization of the Hall Conductance
.............................. 211
A.
2
A Simple Formula for the Hall Conductance
....................... 213
В
Time Reversal Symmetry
.................................................. 217
B.
1
Classical Cases
....................................................... 217
B.2 Time Reversal Operator
θ
........................................... 218
B.3 Time Reversal for a Spin-
j
System
................................. 219
Index
............................................................................... 221
Springer Series
in Solid-State
Sciences
174
Shun-Qing Shen
Topologica!
Insulators
Dirac
Equation in Condensed Matters
Topological insulators are insulating in the bulk, but process metallic states around
its boundary owing to the topological origin of the band structure. The metallic edge
or surface states are immune to weak disorder or impurities, and robust against the
deformation of the system geometry. This book,
Topologica!
insulators, presents a uni¬
fied description of topological insulators from one to three dimensions based on the
modified Dirac equation. A series of solutions of the bound states near the boundary-
are derived, and the existing conditions of these solutions are described. Topological
invariants and their applications to a variety of systems from one-dimensional
poh
-
acetalene, to two-dimensional quantum spin Hall effect and p-wave superconductors,
and three-dimensional topological insulators and superconductors or superfluids are
introduced, helping readers to better understand this fascinating new field.
This book is intended for researchers and graduate students working in the field of
topological insulators and related areas.
Shun-Qing Shen is a Professor at the Department of Physics, the University of Hong
Kong, China.
|
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| id | DE-604.BV040501666 |
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| indexdate | 2024-07-10T00:25:07Z |
| institution | BVB |
| isbn | 9783642328572 |
| language | English |
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| physical | XIII, 225 S. Ill., graph. Darst. |
| publishDate | 2012 |
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| series | Springer series in solid-state sciences |
| series2 | Springer series in solid-state sciences |
| spelling | Shen, Shun-Qing Verfasser (DE-588)103018741X aut Topological insulators Dirac equation in condensed matters Shun-Qing Shen Heidelberg [u.a.] Springer 2012 XIII, 225 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in solid-state sciences 174 Topologischer Isolator (DE-588)1035519526 gnd rswk-swf Topologischer Isolator (DE-588)1035519526 s DE-604 Erscheint auch als Online-Ausgabe 978-3-642-32858-9 Springer series in solid-state sciences 174 (DE-604)BV000016582 174 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025348433&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025348433&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Shen, Shun-Qing Topological insulators Dirac equation in condensed matters Springer series in solid-state sciences Topologischer Isolator (DE-588)1035519526 gnd |
| subject_GND | (DE-588)1035519526 |
| title | Topological insulators Dirac equation in condensed matters |
| title_auth | Topological insulators Dirac equation in condensed matters |
| title_exact_search | Topological insulators Dirac equation in condensed matters |
| title_full | Topological insulators Dirac equation in condensed matters Shun-Qing Shen |
| title_fullStr | Topological insulators Dirac equation in condensed matters Shun-Qing Shen |
| title_full_unstemmed | Topological insulators Dirac equation in condensed matters Shun-Qing Shen |
| title_short | Topological insulators |
| title_sort | topological insulators dirac equation in condensed matters |
| title_sub | Dirac equation in condensed matters |
| topic | Topologischer Isolator (DE-588)1035519526 gnd |
| topic_facet | Topologischer Isolator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025348433&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025348433&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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