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



2DEG

K. von Klitzing 1980

heNRxyxy

21

Z = {..,‐1, 0, 1, 2, …}N

Quantum Hall effects

Hall conductivity

B

2DEG


H 12m

p eA(r) 2

12m

[ix e(By)]2 y2

x

y

Landau gauge

A (By,0,0)

translational symmetry in x-direction

B


H 12m

p eA(r) 2

12m

[ix e(By)]2 y2

ix kx 2Lx

n

H mc

2

2( y yn)

2 12m

y2

A (By,0,0)

n=1,2, …, N

2DEG

x

y

B

Landau gauge

translational symmetry in x-direction


H 12m

p eA(r) 2

12m

[ix e(By)]2 y2

ix kx 2Lx

n

H mc

2

2( y yn)

2 12m

y2

n=1,2, …, N

2DEG

x

y

B

y

x

yn

Wave function


B 0 B 0

insulator (gapped)

=2

H mc

2

2( y yn)

2 12m

y2

metal (gapless)

y

x

yn

n=1,2, …, N

Landau準位

2DEG

x

y

B

Wave function


insulator (gapped)

=2

Insulators (gapped)

Current does not flow

j 0 (?)

Landau準位

Quantized Hall current is carried in the ground state


H 12m

p e[A A ] 2

12m

[ix e(By Lx

)]2 y2

mc

2

2y yn

0

y

2

12m

y2

yn nyy Ly / Ny

x

yn

n=1,2, …, N

Laughlin (1982)


y

x

yn

Ex Lx 0

Tt 0 T 00

jy (e)LxT

e(h / eEx )

e2

hEx

xy yx e2

h

Faraday’s law

n=1,2, …, N

Hall conductivity

Laughlin (1982)


insulator (gapped)

=2

Quantum Hall insulators

The ground state carries the current

j

E

No Joule heating

xy e2

h

x

y

Disorder effects

x

y

extended

Density of states

localized

xy [-e2/h]

0

1

2

Ener

gy

Ener

gy

jy (e)LxT

e(h / eEx )

e2

hEx

H 12m

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

12m

y2

mc2

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

0

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

0

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

ddt

NR NL

2

e

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

zyx

yxz

RiRRiRRR

H σRR][

R

Berry’s phase

|,R ei /2 ei /2 cos 2

ei /2 sin 2

Berry’s phase

Berry’s phase

N (r )

S (r )

x y

z

Berry’s phase

N (r )

S (r )

x y

z

N 1

Geometry and Quantum Mechanics

basis

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

j

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

Curved space Quantum system

E

k

E

EF

j = 0 No current flows

Trivial band insulator

E

k

heN

Ej

y

xxy

2

j

E

EF

Non-trivial band insulator

QHE

j = 0

E

j = 1(e2/h)E

E

heN

Ej

y

xxy

2

topologically trivial topologically nontrivial

Topology?

j = 0

E

j = 1(e2/h)E

E

heN

Ej

y

xxy

2

xy jx

Ey

Hall conductivity

topologically trivial topologically nontrivial

j = 0

E

j = 1(e2/h)E

E

heN

Ej

y

xxy

2

xy jx

Ey

Hall conductivity

Perturbation theory： HE H0 eEy y

xy jx

Ey

Hall conductivity

Perturbation theory： HE H0 eEy y

xy jx

Ey

Heisenberg equation

Hall conductivity

H unk Enk unk

Hall conductivity

Hall conductivitya(k) i

n unk k unk

xy e2

hd 2k2

ay

kx

ax

ky

e2

hd 2k2 k a(k) z

a(k)

a(k)

Hall conductivitya(k) i

n unk k unk

xy e2

hd 2k2

ay

kx

ax

ky

e2

hd 2k2 k a(k) z

a(k)

a(k)

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

TKNN formula

a(k) in unk k unk

xy e2

hd 2k2

ay

kx

ax

ky

e2

hd 2k2 k a(k) z

a(k)

a(k)

Hall conductivity

ky

kx

a(k) in unk k unk

xy e2

hd 2k2

ay

kx

ax

ky

e2

hd 2k2 k a(k) z

a(k)

a(k)

Hall conductivity

ky

kx

= 0 !?

a(k) in unk k unk

xy e2

hd 2k2

ay

kx

ax

ky

e2

hd 2k2 k a(k) z

a(k)

a(k)

Hall conductivity

ky

kx

= 0 !?

a(k1 )

k1

a(k) in unk k unk

xy e2

hd 2k2

ay

kx

ax

ky

e2

hd 2k2 k a(k) z

a(k)

a(k)

Hall conductivity

ky

kx

RI

k1

RII

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

Hall conductivity

ky

kx

RI

k1

RII

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

Hall conductivity

ky

kx

RI

k1

RII

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

unkI ei (k ) unk

II

Hall conductivity

“hole”

Hall conductivity

m > 0 m < 0

“hole”

Hall conductivity

m > 0 m < 0

AN

AS

“hole”

Hall conductivity

m > 0 m < 0

AN

AS

AN (R) 1 cos2Rsin

e

“hole”

Hall conductivity

“hole”

Hall conductivity

The gap closesat the transition point

E

EF

v=0 v=1

Transition between different topological phasesfor example = 0 and 1

How topology changes?

v=0

The gap closesat the boundary

= gapless edge modes

Two topologically distinct insulators attached with each other

y

x

v=1

gapped state gapped state

Edge states

v=0

Two topologically distinct insulators attached with each other

y

x

v=1

gapped state gapped state

Edge states



outline



EF

+

_ +

_

Trivial insulator topological insulator

What is topology insulators?

B

B

QHE up spin

QHE down spin

k

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

Basic idea

B

B

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

k

E

k

‐k

Basic idea

Spin-Orbit Coupling (SOC)

Sv

SOC in an atomMoving electrons feel

an effective magnetic fieldBeff

v E

1 v2

c2

Hso = L S

E

+

_

Sv

Beff v E

1 v2

c2

E

Analogy with a magnetic field HBorbital em

p A

e2m

r p B

A 12

B r

H (peA)2

2m

Spin-Orbit Coupling (SOC)

+

SOC in an atomMoving electrons feel

an effective magnetic field

_

Insulator (Semiconductor)

)(knE

k

)(knE

k

3D bulk

p-band

s-band

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

3-fold degeneracy(without spin)

)(knE

k

)(knE

k

3D bulk


p-band

s-band

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

3-fold degeneracy(without spin)

)(knE

k

)(knE

k

3D bulk 2D quantum well

| s,, | s,

| p ,, | p ,

Hso = L S

2

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)

)(knE

k

)(knE

k

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

2

J2 L2 S2

2D (quantum well)

p-band

s-band

2dTI in HgTe/CdTe quantum well

)(knE

k

)(knE

k

2D quantum well

| s,, | s,

| p ,, | p ,

EF

+

_ +

_



)(knE

k

)(knE

k

2D quantum well

| s,, | s,

| p ,, | p ,

Bernevig, Hughes, Zhang (2006)


p-band

s-band

tsp=0)(knE

k

)(knE

k

Hso = L S

2

J2 L2 S2


tsp=0 tsp=0

p-band

s-band)(knE

k


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

EF

+

_ +

_



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

E

+

_ +

_

Normalinsulator

Topologicalinsulator



Normalinsulator

p-band

s-band

strong SOCweak SOC

E

+

_ +

_

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, …

2d 3d

Topological insulator

E

kx

ky

HgTe QW

k

E

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

(k 0)

Zhang et al. ’09


H(k 0)

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

(k 0)

k.p theory

i ii kcmm 2

0)(k

H(k)

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

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

(k)


1st order 2nd orderk.p theory

i ii kcmm 2

0)(k

H(k)

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

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

(k)

m0 > 0 m0 < 0

normal insulator topological insulator


m0 = 0The gap vanishes at this point

m0 > 0 m0 < 0




E

_

+ _

+

Dirac semimetalNI TI

m0 (Band gap)

strong SOCweak SOC

m0 > 0 m0 < 0




)(knE

k

Surface Dirac modesrealized in a slab geometry

m0 < 0

m0 > 0 (vacuum)


)(knE

k

Surface Dirac modesrealized in a slab geometry

m0 < 0

m0 > 0 (vacuum)

Surface modes described by the Dirac Hamiltonian:

(m 0)

kk

weak topological insulator strong topological insulator(ordinary insulator)

00

Z2 = { 0, 1 }

2 0 1 3

even odd

Z2 topological insulators

even or odd

kx

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

BiTeI

Ishizaka et al. (2011)

Bi2Se3 Bi1‐xSbx

Z2 topological insulators

Z2 = { 0, 1 }even or odd

3D Topological insulator

E


E

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

non‐magnetic impurities magnetic impurities

Impurity effects


E


E



Impurity effects

KN, Koshino, Ryu, PRL (2007)

Topologically protected fromAnderson localization

Topological Conventional

Z2 odd Z2 even


E


E



Impurity effects


E


E



xy

e2 / h d2k

2 bz(k) 12

bz(k) m

2 k2 m23

E

m

Ideal uniform case

Quantum Anomalous Hall Effect

EF

EF

0

xy

xx

Low T

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)

B E

PM

Response to ElectroMagneticfieldsin normal insulators

M = m E

P = e B

(magnetic moment) (electric field)

(electric polarization) (magnetic field)

PM

E B

Response to ElectroMagneticfieldsin topological insulators

E

B = (4/c) j

j

M = Ehce

22

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

3D TI + magnetic impurities

Response to ElectroMagneticfields


B

+ +++++

+ +

E

Surface QH states

0 0


Surface QH states

B E

+ +++++

+ +

++++


Surface QH states

Qi, Hughes, Zhang ’08

E

j

+ +++++

++++

B

The Action Principle

2 2

0

2 2

0

+ 0

+ 0for constant

The Action Principle

Axion term ( term)

2 2

2

2 2

0

Peccei, Quinn 1977Wilczek 1987

2

E

j

+ +++++

++++

0 0

B

2 2

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

E

+

_ +

_

Normalinsulator


It realizes novel magnetoelectric responese.

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