Introduction to Topological Insulators - JAEA...Introduction to Topological Insulators •Quantum...

Introduction to Topological Insulators

Lecture at JAEA 1/23/2017 RIKEN

Kentaro Nomura (IMR, Tohoku)

• Quantum Hall effect• Z2 topological insulators• Electromagnetic responses

outline

K. von Klitzing 1980

heNRxyxy

Z = {..,‐1, 0, 1, 2, …}N

Quantum Hall effects

Hall conductivity

p eA(r) 2

[ix e(By)]2 y2

Landau gauge

A (By,0,0)

translational symmetry in x-direction

p eA(r) 2

[ix e(By)]2 y2

ix kx 2Lx

2( y yn)

A (By,0,0)

n=1,2, …, N

Landau gauge

translational symmetry in x-direction

p eA(r) 2

[ix e(By)]2 y2

ix kx 2Lx

2( y yn)

n=1,2, …, N

Wave function

p eA(r) 2

[ix e(By)]2 y2

ix kx 2Lx

2( y yn)

n=1,2, …, N

Wave function

B 0 B 0

insulator (gapped)

2( y yn)

metal (gapless)

n=1,2, …, N

Landau準位

Wave function

insulator (gapped)

Insulators (gapped)

Current does not flow

j 0 (?)

Landau準位

Quantized Hall current is carried in the ground state

p e[A A ] 2

[ix e(By Lx

)]2 y2

yn nyy Ly / Ny

n=1,2, …, N

Laughlin (1982)

Ex Lx 0

Tt 0 T 00

jy (e)LxT

e(h / eEx )

xy yx e2

Faraday’s law

n=1,2, …, N

Hall conductivity

Laughlin (1982)

insulator (gapped)

Quantum Hall insulators

The ground state carries the current

No Joule heating

Disorder effects

extended

Density of states

localized

xy [-e2/h]

jy (e)LxT

e(h / eEx )

[ix e(By)]2 y2 U (y)

2( y yn)

2 U ( yn)

Edge states

potential term

yn ℓ B2 kx

Edge states

E(k) EF vF (k kFR)

‐right moving modes

yn ℓ B2 kx

vF dE(k)

dk kkFR

Edge states

E(k) EF vF (k kFR)

E(k) EF vF (k kFL)

vF dE(k)

dk kkFR

0‐right moving modes

‐left moving modes

yn ℓ B2 kx

vF dE(k)

dk kkFL

Edge states

E(k) EF vF (k kFL)

R(x,t) eikFRxR(x,t)

L(x,t) eikFRxL(x,t)

E(k) EF vF (k kFR)

Edge states

NR dx RR

NL dx LL

2nd quantizationformalism

hdx Ex

anomaly equation

Berry’s phase

| n,R ein [C ] | n,R

Berry’s phase

Berry connection

Berry curvature

Berry’s phase

Berry connection

Berry curvature

Gauge transformation

Berry’s phase

RiRRiRRR

H σRR][

Berry’s phase

|,R ei /2 ei /2 cos 2

ei /2 sin 2

Berry’s phase

N (r )

S (r )

Berry’s phase

N (r )

S (r )

Geometry and Quantum Mechanics

differential

connection

cuarvature

| n,Rei (x)

jik (x) ek (x) jei (x) A(R) in,R |R | n,R

R R iA DRiVj iV

j ikjV k DiV

DxDy DyDx iBz(DaDb DbDa )V j Riabj V i

Curved space Quantum system

j = 0 No current flows

Trivial band insulator

Non-trivial band insulator

j = 1(e2/h)E

topologically trivial topologically nontrivial

Topology?

j = 1(e2/h)E

Hall conductivity

topologically trivial topologically nontrivial

j = 1(e2/h)E

Hall conductivity

Perturbation theory： HE H0 eEy y

Hall conductivity

Perturbation theory： HE H0 eEy y

Heisenberg equation

Hall conductivity

Heisenberg equation

Hall conductivity

Heisenberg equation

Hall conductivity

H unk Enk unk

Hall conductivity

H unk Enk unk

Hall conductivity

H unk Enk unk

Hall conductivity

Hall conductivitya(k) i

n unk k unk

hd 2k2

hd 2k2 k a(k) z

Hall conductivitya(k) i

n unk k unk

hd 2k2

hd 2k2 k a(k) z

Thouless, Kohmoto, Nightingale, Nijs, PRL 49, 405 (1982).Kohmoto, Ann. Phys. 160 355 (1985).

TKNN formula

a(k) in unk k unk

hd 2k2

hd 2k2 k a(k) z

Hall conductivity

a(k) in unk k unk

hd 2k2

hd 2k2 k a(k) z

Hall conductivity

= 0 !?

a(k) in unk k unk

hd 2k2

hd 2k2 k a(k) z

Hall conductivity

= 0 !?

a(k1 )

a(k) in unk k unk

hd 2k2

hd 2k2 k a(k) z

Hall conductivity

a I (k) a II (k)k (k)

Hall conductivity

a I (k) a II (k)k (k)2N

unkI ei (k ) unk

Hall conductivity

“hole”

Hall conductivity

“hole”

Hall conductivity

“hole”

Hall conductivity

m > 0 m < 0

“hole”

Hall conductivity

m > 0 m < 0

“hole”

Hall conductivity

m > 0 m < 0

AN (R) 1 cos2Rsin

“hole”

Hall conductivity

“hole”

Hall conductivity

The gap closesat the transition point

v=0 v=1

Transition between different topological phasesfor example = 0 and 1

How topology changes?

The gap closesat the boundary

= gapless edge modes

Two topologically distinct insulators attached with each other

gapped state gapped state

Edge states

Two topologically distinct insulators attached with each other

gapped state gapped state

Edge states

outline

Trivial insulator topological insulator

What is topology insulators?

QHE up spin

QHE down spin

Quantum Hall Effect (QHE) is realized when time‐reversal symmetry is broken

Basic idea

QHE up spin

QHE down spin

TheoryKane-Mele (2005)Bernevig-Zhang (2006)Bernevig-Hughes-Zhang (2006)

ExperimentMolemkamp goup (2007)

Quantum spin Hall effect (QSHE)

HgTe QW

Basic idea

Spin-Orbit Coupling (SOC)

SOC in an atomMoving electrons feel

an effective magnetic fieldBeff

Hso = L S

Beff v E

Analogy with a magnetic field HBorbital em

H (peA)2

Spin-Orbit Coupling (SOC)

SOC in an atomMoving electrons feel

an effective magnetic field

Insulator (Semiconductor)

3D bulk

p-band

s-band

L =1Lz=+1, 0, -1

3-fold degeneracy(without spin)

3D bulk

p-band

s-band

L =1Lz=+1, 0, -1

3-fold degeneracy(without spin)

3D bulk 2D quantum well

| s,, | s,

| p ,, | p ,

Hso = L S

J2 L2 S2

p-band

s-band

J = L + S

j =3/2jz=+3/2, -3/2 (heavy hole band)jz=+1/2, -1/2 (light hole band)

j =1/2jz=+1/2, -1/2 (split off band)

3D bulk 2D quantum well

| s,, | s,

| p ,, | p ,

J = L + S

j =3/2jz=+3/2, -3/2 (heavy hole band)jz=+1/2, -1/2 (light hole band)

j =1/2jz=+1/2, -1/2 (split off band)

Hso = L S

J2 L2 S2

2D (quantum well)

p-band

s-band

2dTI in HgTe/CdTe quantum well

2D quantum well

| s,, | s,

| p ,, | p ,

2D quantum well

| s,, | s,

| p ,, | p ,

Bernevig, Hughes, Zhang (2006)

p-band

s-band

tsp=0)(knE

Hso = L S

J2 L2 S2

tsp=0 tsp=0

p-band

s-band)(knE

Weak SOC Strong SOC

tsp=0 tsp=0

Weak SOC

p-band

s-band

Strong SOC + sp hybridization

tsp=0 tsp=0

Weak SOC

p-band

s-band

Strong SOC + sp hybridization

tsp=0 tsp=0

with boundary

Normal insulator Topological insulator

p-band

s-band

tsp=0 tsp=0

p-band

s-band

Normal insulator Topological insulator

with boundary

p-band

s-band

strong SOCweak SOC

Normalinsulator

Topologicalinsulator

Normalinsulator

p-band

s-band

strong SOCweak SOC

Normalinsulator

Ek = E-k

2-fold degeneracy at k=0 is protected by symmetry

Time-reversal symmetry

k0 0k0

Normal Topological

Moore-Balents, Roy, Fu-Kane-Mele, …

Bi‐Sb, Bi2Se3, Bi2Te3, …

Topological insulator

HgTe QW

From 2d to 3d

3d Topological insulator Bi2Se3

Bi : 6s26p3Se : 4s24p2

5 x 3 (px,py,pz) x 2 (spin) = 30 p‐states

Zhang et al. ’09

Bi : 6s26p3Se : 4s24p2

Zhang et al. ’09

Bi : 6s26p3Se : 4s24p2

4‐bandmodel

Zhang et al. ’09

4‐bandmodel

H(k 0)

m0 0 0 00 m0 0 00 0 m0 00 0 0 m0

Zhang et al. ’09

H(k 0)

m0 0 0 00 m0 0 00 0 m0 00 0 0 m0

k.p theory

i ii kcmm 2

m(k) 0 A1kz A2k0 m(k) A2k A1kz

A1kz A2k m(k) 0A2k A1kz 0 m(k)

1st order 2nd orderk.p theory

i ii kcmm 2

m(k) 0 A1kz A2k0 m(k) A2k A1kz

A1kz A2k m(k) 0A2k A1kz 0 m(k)

m0 > 0 m0 < 0

normal insulator topological insulator

m0 = 0The gap vanishes at this point

m0 > 0 m0 < 0

Dirac semimetalNI TI

m0 (Band gap)

strong SOCweak SOC

m0 > 0 m0 < 0

Surface Dirac modesrealized in a slab geometry

m0 < 0

m0 > 0 (vacuum)

Surface Dirac modesrealized in a slab geometry

m0 < 0

m0 > 0 (vacuum)

Surface modes described by the Dirac Hamiltonian:

weak topological insulator strong topological insulator(ordinary insulator)

Z2 = { 0, 1 }

2 0 1 3

even odd

Z2 topological insulators

even or odd

Hsieh et al. (2009) Hsieh et al. (2008)

Ishizaka et al. (2011)

Bi2Se3 Bi1‐xSbx

Z2 topological insulators

Z2 = { 0, 1 }even or odd

3D Topological insulator

Hsurface = vF(ypx - xpy ) + V0(r) + V(r)

non‐magnetic impurities magnetic impurities

Impurity effects

KN, Koshino, Ryu, PRL (2007)

Topologically protected fromAnderson localization

Topological Conventional

Z2 odd Z2 even

Impurity effects

e2 / h d2k

2 bz(k) 12

bz(k) m

2 k2 m23

Ideal uniform case

Quantum Anomalous Hall Effect

Theory Experiment

KN, Nagaosa (2011)

(201３)

KN, Nagaosa (2011) Checkelsky, Yoshimi, Tsukazaki, et al. (2014)

Theory Experiment

outline

M = m B (magnetization)

P = e E(Electric polarization)

Response to ElectroMagneticfieldsin normal insulators

M = m E

P = e B

(magnetic moment) (electric field)

(electric polarization) (magnetic field)

Response to ElectroMagneticfieldsin topological insulators

B = (4/c) j

M = Ehce

Qi, Hughes, Zhang ’08Essin, Moore, Vanderbilt ’09

3D TI + magnetic impurities

Response to ElectroMagneticfields

+ +++++

Surface QH states

+ +++++

Surface QH states

Qi, Hughes, Zhang ’08

+ +++++

The Action Principle

+ 0for constant

The Action Principle

Axion term ( term)

Peccei, Quinn 1977Wilczek 1987

+ +++++

Axion term ( term)

Qi, Hughes, Zhang 2008

SummaryA topological insulator is a material with a finite bulk gap and gapless excitations at the surface.

strong SOCweak SOC

Normalinsulator

It realizes novel magnetoelectric responese.

Introduction to Topological Insulators - JAEA...Introduction to Topological Insulators •Quantum...

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