Towards Understanding Topological Insulators

Vijay B. Shenoy

Centre for Condensed Matter Theory, IISc Bangaloreshenoy@physics.iisc.ernet.in

ICTS ICMP 2010, Bangalore

1 / 36

Acknowledgement

S.-Q. Shen (Hong Kong), Lectures at ICTS Topology School

Amal Medhi, IISc (for discussions)

Generous funding by DAE and DST

2 / 36

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?

3 / 36

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

4 / 36

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

5 / 36

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?

7 / 36

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”!

9 / 36

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!

10 / 36

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!

11 / 36

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

12 / 36

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

13 / 36

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

14 / 36

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

15 / 36

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

16 / 36

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

18 / 36

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

32 / 36

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