Vijay B. Shenoy - Department of Physicsshenoy/LectureNotes/ti_icmp.pdfTowards Understanding...
Transcript of Vijay B. Shenoy - Department of Physicsshenoy/LectureNotes/ti_icmp.pdfTowards Understanding...
Towards Understanding Topological Insulators
Vijay B. Shenoy
Centre for Condensed Matter Theory, IISc [email protected]
ICTS ICMP 2010, Bangalore
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Acknowledgement
S.-Q. Shen (Hong Kong), Lectures at ICTS Topology School
Amal Medhi, IISc (for discussions)
Generous funding by DAE and DST
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Goal and Outline
Goal – Answer the question: What the !@@! is a topological insulator(TI)?
Where do TIs fit into condensed matter physics?
Symmetries (revise time reversal)
Band theory – metals and insulators
Integer quantum Hall effect
2D TIs
3D TIs
Why the fuss? What can you do with TIs?
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Prerequisites
Graduate quantum mechanics (including second quantization)
Basic solid state physics (@Kittel)
...
A keen desire to understand things and put in the necessary effort
Follow the maxim (source: T. V. Ramakrishnan):There are no stupid questions, only stupid answers!
DISCLAIMER: The speaker is NOT an expert! Focus is on elementaryconcepts...not on calculations
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References
RMP by Hassan and Kane
Physics Today article by Qi and Zhang
Annual Reviews of Condensed Matter by Hassan and Moore
PRB Papers by Rahul Roy
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Condensed MatterOperative definition of condensed matter: A collection/aggregate ofatoms/ions in the non-relativistic regime“Raw materials” for condensed matter
Solids
ArtificiallyPrepared
LiquidsGases
58CeCerium
140.116
5.5387
°
AtomicNumber
Symbol
Name
Ground-stateConfiguration
Ground-stateLevel
IonizationEnergy (eV)
†Based upon 12C. () indicates the mass number of the most stable isotope.
AtomicWeight†
P E R I O D I C T A B L E
Atomic Properties of the Elements
29CuCopper63.546
7.7264
11NaSodium
22.989770
5.1391
12Mg
Magnesium24.3050
7.6462
13Al
Aluminum26.981538
5.9858
14SiSilicon
28.0855
8.1517
15P
Phosphorus30.973761
10.4867
16S
Sulfur32.065
10.3600
17Cl
Chlorine35.453
12.9676
18ArArgon39.948
15.7596
1 2S1/2
HHydrogen1.00794
13.5984
4BeBeryllium9.012182
9.3227
37RbRubidium85.4678
4.1771
55CsCesium
132.90545
3.8939
42Mo
Molybdenum95.94
7.0924
41NbNiobium
92.90638
6.7589
86RnRadon(222)
10.7485
74W
Tungsten183.84
7.8640
43Tc
Technetium(98)
7.28
75ReRhenium186.207
7.8335
44Ru
Ruthenium101.07
7.3605
76OsOsmium190.23
8.4382
45RhRhodium
102.90550
7.4589
77Ir
Iridium192.217
8.9670
46Pd
Palladium106.42
8.3369
78Pt
Platinum195.078
8.9588
47Ag
Silver107.8682
7.5762
79Au
Gold196.96655
9.2255
48Cd
Cadmium112.411
8.9938
80HgMercury200.59
10.4375
60Nd
Neodymium144.24
5.5250
62SmSamarium
150.36
5.6437
63Eu
Europium151.964
5.6704
64Gd
Gadolinium157.25
6.1498
65TbTerbium
158.92534
5.8638
61Pm
Promethium(145)
5.582
66Dy
Dysprosium162.500
5.9389
67HoHolmium
164.93032
6.0215
68ErErbium167.259
6.1077
69TmThulium
168.93421
6.1843
49InIndium
114.818
5.7864
50Sn
Tin118.710
7.3439
51Sb
Antimony121.760
8.6084
52Te
Tellurium127.60
9.0096
53I
Iodine126.90447
10.4513
81Tl
Thallium204.3833
6.1082
82Pb
Lead207.2
7.4167
83Bi
Bismuth208.98038
7.2855
84Po
Polonium(209)
8.414
85At
Astatine(210)
58CeCerium140.116
5.5387
59Pr
Praseodymium140.90765
5.473
70Yb
Ytterbium173.04
6.2542
90ThThorium
232.0381
6.3067
92U
Uranium238.02891
6.1941
93Np
Neptunium(237)
6.2657
94Pu
Plutonium(244)
6.0260
95AmAmericium
(243)
5.9738
96Cm
Curium(247)
5.9914
91Pa
Protactinium231.03588
5.89
97Bk
Berkelium(247)
6.1979
98Cf
Californium(251)
6.2817
99Es
Einsteinium(252)
6.42
100FmFermium
(257)
6.50
101Md
Mendelevium(258)
6.58
102NoNobelium
(259)
6.65
° ° °
° °
° °
°
°
° ° ° ° ° ° °
?°
°
°
° °
° °
° ° ° °°
105 107 106 108 109 111 110 112DbDubnium
(262)
SgSeaborgium
(266)
HsHassium
(277)
BhBohrium
(264)
MtMeitnerium
(268)
UunUnunnilium
(281)
UuuUnununium
(272)
°
1s
114 116
3
1s22s
LiLithium6.941
5.3917
10Ne
Neon20.1797
21.5645
2HeHelium
4.002602
24.5874
9O
Oxygen15.9994
13.6181
8F
Fluorine18.9984032
17.4228
7N
Nitrogen14.0067
14.5341
6C
Carbon12.0107
11.2603
5B
Boron10.811
8.2980
57La
Lanthanum138.9055
5.5769
89AcActinium
(227)
5.17
71LuLutetium174.967
5.4259
103Lr
Lawrencium(262)
4.9 ?
87Fr
Francium(223)
4.0727
88RaRadium(226)
5.2784
104 ?
RfRutherfordium
(261)
6.0 ?
72Hf
Hafnium178.49
6.8251
40Zr
Zirconium91.224
6.6339
39Y
Yttrium88.90585
6.2173
38Sr
Strontium87.62
5.6949
56BaBarium
137.327
5.2117
73Ta
Tantalum180.9479
7.5496
54XeXenon
131.293
12.1298
19K
Potassium39.0983
4.3407
20CaCalcium40.078
6.1132
21Sc
Scandium44.955910
6.5615
22Ti
Titanium47.867
6.8281
30Zn
Zinc65.409
9.3942
31GaGallium69.723
5.9993
32Ge
Germanium72.64
7.8994
33AsArsenic
74.92160
9.7886
34Se
Selenium78.96
9.7524
35Br
Bromine79.904
11.8138
36KrKrypton83.798
13.9996
23V
Vanadium50.9415
6.7462
24Cr
Chromium51.9961
6.7665
25Mn
Manganese54.938049
7.4340
26Fe
Iron55.845
7.9024
27CoCobalt
58.933200
7.8810
28NiNickel
58.6934
7.6398
UubUnunbium
(285)
UuqUnunquadium
(289)
UuhUnunhexium
(292)
NIST SP 966 (September 2003)
Perio
d
1
6
5
4
3
2
7
For a description of the data, visit physics.nist.gov/data
2S1/2
1s22s2
2S1/2
2S1/2
[Ne]3s2
1S0
[Ne]3s
1S0
1S0
2S1/21S0
2S1/21S0
2S1/21S0
[Ar]4s2[Ar]4s
[Kr]5s2[Kr]5s
[Xe]6s2[Xe]6s
[Rn]7s2[Rn]7s
1G4
[Xe]4f5d6s2
2D3/23F2
2D3/23F2
3F2
3F2
[Ar]3d4s2 [Ar]3d24s2
[Kr]4d5s2 [Kr]4d25s2
[Xe]4f145d26s2
[Rn]5f146d27s2?
4F3/27S3
6D1/27S3
4F3/25D0
[Xe]4f145d36s2 [Xe]4f145d46s2
[Kr]4d45s [Kr]4d55s
[Ar]3d34s2 [Ar]3d54s
6S5/25D4
[Ar]3d54s2 [Ar]3d64s2
6S5/2
6S5/2
[Xe]4f145d56s2
[Kr]4d55s2
4F9/2
[Ar]3d74s2
4F9/2
[Kr]4d85s
3F42S1/2
5F5
[Kr]4d75s
5D4
[Xe]4f145d66s2
4F9/2
[Xe]4f145d76s2
2S1/2
[Kr]4d105s
1S0
[Kr]4d10
3D3
[Xe]4f145d96s
2S1/2
[Xe]4f145d106s
1S02P1/2
1S0
[Kr]4d105s2 [Kr]4d105s25p
[Xe]4f145d106s2
1S0
[Hg]6p
2P1/2
1s22s22p
1S0
1s2
3P0
1s22s22p2
4S3/2
1s22s22p3
3P2
1s22s22p4
2P3/2
1s22s22p5
1S0
1s22s22p6
2P1/23P0
4S3/23P2
2P3/21S0
3P04S3/2
3P22P3/2
1S0
2P1/23P0
4S3/23P2
2P3/21S0
2P1/23P0
4S3/23P2
2P3/21S0
[Ar]3d104s24p[Ar]3d104s2[Ar]3d84s2 [Ar]3d104s [Ar]3d104s24p2
[Kr]4d105s25p2
[Ar]3d104s24p3
[Kr]4d105s25p3
[Ar]3d104s24p4
[Kr]4d105s25p4
[Ar]3d104s24p5
[Kr]4d105s25p5
[Ar]3d104s24p6
[Kr]4d105s25p6
[Hg]6p2 [Hg]6p3 [Hg]6p4 [Hg]6p5 [Hg]6p6
2D3/2
[Xe]4f145d6s2
1S0
[Xe]4f146s2
[Ne]3s23p [Ne]3s23p2 [Ne]3s23p3 [Ne]3s23p4 [Ne]3s23p5 [Ne]3s23p6
[Xe]4f136s2[Xe]4f126s2[Xe]4f116s2[Xe]4f106s2[Xe]4f96s2[Xe]4f75d6s2[Xe]4f76s2[Xe]4f66s2[Xe]4f56s2[Xe]4f46s2[Xe]4f36s2[Xe]4f5d6s2[Xe]5d6s2
[Rn]5f147s27p?[Rn]5f147s2[Rn]5f137s2[Rn]5f127s2[Rn]5f117s2[Rn]5f107s2[Rn]5f97s2[Rn]5f76d7s2[Rn]5f77s2[Rn]5f67s2[Rn]5f46d7s2[Rn]5f36d7s2[Rn]5f26d7s2[Rn]6d27s2[Rn]6d7s2
2D3/21G4
2D3/23F2
4I9/25I4
5I84I15/2
°4I15/25I8
6H5/27F0
8S7/29D2
6H15/22F7/2
3H6
2P1/21S0
2F7/23H6
9D26H15/2
7F08S7/2
6L11/25L6
4K11/2
Lant
hani
des
Act
inid
es
Group1IA
2IIA
3IIIB
4IVB
5VB
6VIB
7VIIB
9VIII
8 10 11IB
12IIB
13IIIA
14IVA
Standard ReferenceData Groupwww.nist.gov/srd
PhysicsLaboratoryphysics.nist.gov
15VA
16VIA
17VIIA
18VIIIA
Frequently used fundamental physical constants
1 second = 9 192 631 770 periods of radiation corresponding to the transition
speed of light in vacuum 299 792 458 m s−1
Planck constant 6.6261 × 10−34 J s elementary chargeelectron mass
proton massfine-structure constant 1/137.036Rydberg constant 10 973 732 m−1
Boltzmann constant 1.3807 × 10−23 J K −1
cheme
k
For the most accurate values of these and other constants, visit physics.nist.gov/constants
between the two hyperfine levels of the ground state of 133Cs (exact)
0.5110 MeV
13.6057 eV
RR cR hc
( /2 )
mec2
mp
1.6022 × 10−19 C 9.1094 × 10−31 kg
1.6726 × 10−27 kg
3.289 842 × 1015 Hz
6 / 36
Condensed Matter
What happens when we aggregate atoms? Many things...
In fact, the same atoms will give you very different things ifaggregated differently! Eg., carbon atoms give
Different arrangement of atoms leads to very different emergentproperties!
Again: More is different... Different mores are more so!
Natural question: What are all the different “emergentstates/things/properties” can we obtain by aggregating atoms?
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Quantum Condensed Matter PhysicsWe will dwell mainly on electrons in materialsElectrons in materials experience various things:
...this defines the “space of Hamiltonians” for the electronsThe ground state (and concomitant excitations) of the electronsdepend on where in the Hamiltonian space you are! There are many“electronic phases”One encounters (quantum) phase transitions as one moves about inthe Hamiltonian space! 8 / 36
Electronic Phases
Electrons in materials can organize themselves in many different ways– “phases/states”...we have
I MetalsI Semi-metalsI Insulators/SemiconductorsI Topological insulatorsI SuperconductorsI MagnetsI Charge density wave systemsI Spin liquidsI ...
Each of these have a common set of characteristics...(similar in sprintto: all liquids flow). In this sense each of the above is an “electronicphase”!
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Electronic Phases
The goal of condensed matter research is to discover and understand“all” the phases of electrons!
Think of a “character table”
...idea is to tabulate phases and their properties...and add to thistable!
This is useful ...and most interesting!
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Characterization of Phases
Question: How do we characterize phases?
Key idea: Symmetry...many electronic phases break the symmetries ofthe Hamiltonian...e.g., a ferromagnet breaks time reversal and spinrotation symmetry
Not all phases can be distinguished by symmetry...
...we go from the half filled state to the fully filled state by tuning thechemical potential
There is no symmetry difference, but the states are very different!
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Characterization of PhasesPhases with the same symmetry can have very different responsesThis is the idea behind classification as “metals” (σDC > 0) and“insulators” (σDC = 0)Abstractly: The phase is characterized by how “test particles behavein the phase (system)”...the “most natural” test particle –light(photon)!In somewhat more technical language: ψ – “matter” fields, Aµ gaugefields associated with light
S[ψ,Aµ] =
∫ (L(ψ?, ψ,Aµ)− 1
4FµνFµν
)∨
Integrate out ψ∨Seff [Aµ] =
∫d1 d2 Aµ(1)Kµν(1, 2)Aν(2)− 1
4FµνFµν
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Characterization of Phases
Kµν(1, 2) is the electromagnetic response function which can becalculated by the Kubo formula
Kµν(1, 2) ∼ −〈T jµ(1)jν(2)〉where jµ is the current...the mean is taken over the ensemble
describing an (equilibrium) state of the fermionic system Find out more:
Kubo formula is actually just second order perturbation theory.
Kµν(q, ω) (written in Fourier space) characterizes the system...Ametal has a very different characteristic functional form of Kµν thanan insulator or superconductor!
The DC conductivity of the system is related to Kµν
σDC ∼ limω→0, q→0
1
iωKµν(q, ω)
Key point: The nature of the state of electrons modifies how anelectromagnetic wave propagates in the system...this can be used tocharacterize the phase that the electrons organize themselves in
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Insulators
An insulating state is characterized by σDC = 0
This can arise in many ways
Non-interacting systemsI Band insulatorsI Anderson insulators (due to disorder)
Interacting systemsI Mott insulators
These states have σDC = 0, but very different σ(ω)...electromagneticresponse can distinguish these phases!
We are interested in band insulators...non-interacting electrons!These come in two varieties:
Ordinary (trivial) insulators (OI)
Topological insulators (TI)
Immediate goal: Look at one-electron physics in crystals
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One-Electron Physics in Crystals
Preliminaries
Symmetries – mainly time reversal
Tight binding models
Metals and insulators
Example: Graphene
Example: Insulator (related to graphene)
Edge states
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One-Electron Physics in Crystals: PreliminariesUnit cell with many atoms (unit cell vectors a1, a2, a3)
Hamiltonian
H = T + V + HSO
I T = P2
2 – kinetic energyI V – potential due to ionsI HSO spin-orbit interaction
(keep aside for now)Subspace of focus in the Hamiltonian space
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SymmetriesSymmetry – Quick Revision
System ≡ A Hilbert space + Hamiltonian HA symmetry operation U on a system is a function on the Hilbertspace, such that if |φ〉 = U(|φ〉) and |ψ〉 = U(|ψ〉), then
|〈φ|ψ〉| = |〈φ|ψ〉| for all |φ〉 and |ψ〉 Question: What are some symm. ops. on R3?
Wigner’s theorem: A symmetry operation U is either a linear/unitary
or an anti-linear/unitary operator Find out more: Look at Gottfried and Yan
A symmetry operation U is a symmetry of the system if U−1HU = H,i. e., the Hamiltonian is unchanged by the symmetry operation
Hilbert space: space spanned by |rσ〉 = |r〉 ⊗ |σ〉 where r runs overpoints in a box of volume Ω = NV, N number of unit cells,|σ〉, σ =↑, ↓ span the spin sector
Hamiltonian: H = T + V ≡ −∇2
2 + V (r)Symmetries
I Lattice translationI Time reversalI Parity (Inversion) – Not always, but in most crystals 17 / 36
Lattice Translations
Translation operator T (a) = e−ia·P; T (a)|r〉 = |r + a〉Every lattice translation T (l), l =
∑α nαaα is a symmetry of our
Hamiltonian Exercise: Show that T (l) form a group. Is it a non-Abelian group?
Seen by noting that V (r + l) = V (r) for all l
T (l) has right eigenstates which are labelled by the crystal momentumk which lives in the 1st Brillouin zone ...this is Bloch theorem
T (l)|φk〉 = e−i l·k|φk〉, ∀l
Every |φk〉 =∑
G φk(G)|k + G〉, where |k + G〉 are plane waves, Gs are
reciprocal lattice vectors, φk(G) are c-numbers Exercise: Prove this statement
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Lattice Translations
Since [T (l),H] = 0, ∀l, find simultaneous eigenstates of H and T (l)
T (l)|ψk〉 = e−i l·k|ψk〉, ∀l and H|ψk〉 = ε|ψk〉Note that ψk(r) = 〈r|ψk〉 = e ik·ruk(r) where uk(r + l) = uk(r)...Bloch
states are modulated plane waves Exercise: Show this
uk(r) satisfies a Schrodinger equation[1
2(−i∇ + k)2 + V (r)
]uk(r) = εuk(r)
which is an eigenvalue problem defined only on the unit cell V!Exercise: Show this
For each k, therefore, there is a discrete spectrum εn(k) andassociated states ukn(r); equivalently
H|ψkn〉 = εkn(k)|ψkn〉, 〈r|ψkn〉 = e ik·rukn(r)
The energy functions εn(k) is the nth band
Diagonalized Hamiltonian H =∑
kn εn(k)|knσ〉〈knσ|19 / 36
Time Reversal (TR) SymmetryTR symmetry Θ requires two things:
I Idea of a time reversed stateI Idea of reverse time evolution
..understand this first in classical mechanics
A state s = (r,p), Θs = s = (r,−p)
Reverse time evolution, integrate Hamilton’s equation “backwards”,i. e., negative times
Understand this for a simple classical system
A system is said to be time reversal invariant if the following diagramcommutes
...system is time reversal invariant ifΘ(s(−t0)) = s(t0) for all s and t0
Examples of systems with and without TR symmetry20 / 36
TI Crash Course: Plan of ActionComplete TR invariance, Kramer’s theoremTight binding models, role of TR invarianceGraphene – masslessGraphene – massiveEdge statesInteger quantum hall effect: landau levels, edge statesLaughlin’s argumentQuantum Hall effect on a lattice: Hofstadter butterflyTKNN formula ≡ Chern number, topological quantizationHatsugai’s connection between Chern number and edge statesQHE without magnetic field – Haldane model and edge statesIntroduction of spin-orbit interactionObtaining a TI by gluing two Haldanes – edge statesGeneral arguments using TR in 2D – Z2 index3D - brief statementsPossibilities with TIsTopology and geometry in physics
21 / 36
Graphene
K+K−
Q− Q+
E− E+
8π/3a
4π/3a
2π/a
Question: Which are the TR invariant momenta?
Thanks: J. P. Vyasanakere
22 / 36
Massive Graphene
Thanks: J. P. Vyasanakere23 / 36
Edge States
a
W
A
B
t
x
y
24 / 36
Integer Quantum Hall Effect
xz
y
jx
E y
B
Source: von Klitzing’s homepage
Puzzle: The plateaus, and why integer?
We will not discuss the resolution of the puzzle in detail... Find out
more: Look at any standard book on QHE, e. g., by Jain
25 / 36
Edge States in Integer Quantum Hall
Venturelli, Ph. D. Thesis, SISSA
26 / 36
Hofstadter Spectrum and Butterfly
p/q = 1/20
Avron et al.
27 / 36
Hofstadter Chern Butterfly
Avron et al.28 / 36
Hofstadter Model Edge States
p/q = 1/20 p/q = 1/20,withedges29 / 36
Haldane ModelEdge states
The states corresponding to the twoedge bands are associated withdifferent edges!
Both bands have non-zero Chernnumber – Chern bands!
For one (spinless) fermion per unitcell, bottom band is filled – bulkinsulator!
...and a quantum Hall state!30 / 36
From MassiveG to HaldaneMassiveG Graphene Haldane
Massive graphene and Haldane differ at the edge!We need to close the bulk gap to go from MassiveG to Haldane, i. e.,via graphene – need a quantum phase transition
31 / 36
Kane-Mele Double Haldane Model
Two time reversed copies (↑, ↓) of theHaldane model glued together
The Chern number of the ↓-band isnegative of that of the ↑ band,σxy = 0...has to be since the system isTR invariant...
But there is a SPIN current onapplication of the voltage!! This isquantized and related to the numberof edge modes! In this example it ise2/h...and is dissipation less!!
We get the quantum spin Hall state,with a fully insulating bulk
This is the simplest realization of aTopological Insulator!
Robust against TR invariantperturbations
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Trivial and Topological Insulator (2D)
Edge State
k
π/a0a−π/
Unoccupied
Occupied
Ordinary (Trivial) Insulator
k
π/a0a−π/
Unoccupied
Occupied
Topological Insulator
33 / 36
3D TIs
Qi and Zhang, Physics Today (2010)34 / 36
TI Possibilities
Qi and Zhang, Physics Today (2010)35 / 36
Trivial or Topological?
36 / 36
Trivial or Topological?
36 / 36
Trivial or Topological?
36 / 36