E

k=a k=b

Quantum Spin Hall Effectand Topological Insulators

Laurens W. MolenkampPhysikalisches Institut (EP3), Universität Würzburg

Quantum Spin Hall Effect and Topological Insulators

I. Introduction- Topological Classification of Insulators - Edge States with and w/o Time Reversal Symmetry

II. Two Dimensional TI : Quantum Spin Hall Effect

- Transport in HgTe quantum wellsIII. Three Dimensional TI :

- Topological Insulator & Surface States- Photoemission on BixSb1-x and Bi2Se3- Transport in strained bulk HgTe

C.L.Kane and E.J. Mele, Science 314, 1692 (2006)

C.L.Kane and E.J.Mele, PRL 95, 146802 (2005)C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)A.Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)

The Insulating State – Topologically Generalized

The Usual Boring Insulating State

Covalent Insulator

Characterized by energy gap: absence of low energy electronic excitations

The vacuumAtomic Insulatore.g. solid Ar

Dirac Vacuum

Egap ~ 10 eV

e.g. intrinsic semiconductor

Egap ~ 1 eV3p

4s

Silicon

Egap = 2 mec2

~ 106 eV

electron

positron ~ hole

C.L.Kane and E.J. Mele, Science 314, 1692 (2006)

C.L.Kane and E.J.Mele, PRL 95, 146802 (2005)C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)A.Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)

The Insulating State – Topologically Generalized

The Integer Quantum Hall State

2D Cyclotron Motion, Landau Levels

gap cE E

Energy gap, but NOT an insulator

bulkinsulating

Edge States (as experimentalists see them)

Quantized Hall conductivity :

y xy xJ E2

xy hn e

Integer accurate to 10-9

Topological Band Theory

g=0 g=1

21 ( ) ( )2 BZ

n d u ui

k kk k k

The distinction between a conventional insulator and the quantum Hall state is a topological property of the manifold of occupied states

Analogy: Genus of a surface : g = # holes

Insulator : n = 0IQHE state : xy = n e2/h

The TKNN invariant can only change at a phase transition where the energy gap goes to zero

Classified by Chern (or TKNN) integer topological invariant (Thouless et al, 1982)

( ) :H k Bloch Hamiltonians Brilloui wn zone (torus) ith energy gap

u(k) = Bloch wavefunction

Edge StatesGapless states must exist at the interface between different topological phases

IQHE staten=1

Vacuumn=0

Edge states ~ skipping orbits

Bulk – Boundary Correspondence : n = # Chiral Edge Modes

This approach can actually be generalized to a spinfull QHE at zero magnetic field:the Quantum Spin Hall Effect

C.L.Kane and E.J. Mele, Science 314, 1692 (2006)

C.L.Kane and E.J.Mele, PRL 95, 146802 (2005)C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)A.Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)

The Insulating State – Topologically Generalized

Topological Insulator : A New B=0 Phase There are 2 classes of 2D time reversal invariant band structures

Z2 topological invariant: = 0,1 is a property of bulk bandstructure, but can be understood by

from the bulk - boundary correspondence

=0 : Conventional Insulator =1 : Topological Insulator

Kramers degenerate attime reversal

invariant momenta k* = k* + G

k*=0 k*=/a k*=0 k*=/a

Edge States for 0<k</a

Even number of bandscrossing Fermi energy

Odd number of bandscrossing Fermi energy

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.76.6

1.0

1.5

0.5

0.0

-0.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

6.0

5.5

Bandgap vs. lattice constant(at room temperature in zinc blende structure)

Ban

dgap

ene

rgy

(eV

)

lattice constant a [Å]0 © CT-CREW 1999

(Hg,Cd)Te compound semiconductors

band structure

D.J. Chadi et al. PRB, 3058 (1972)

fundamental energy gap

meV 30086 EE meV 30086 EE

semi-metal or semiconductor

HgTe bulk band structure

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV) 8

6

7

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV) 8

6

7

Eg

Layer Structure

gate

insulator

cap layer

doping layer

barrier

barrierquantum well

doping layer

buffer

substrate

Au

100 nm Si N /SiO

3 4 2

25 nm CdTe

CdZnTe(001)

25 nm CdTe10 nm HgCdTe x = 0.79 nm HgCdTe with I10 nm HgCdTe x = 0.74 - 12 nm HgTe10 nm HgCdTe x = 0.7 9 nm HgCdTe with I10 nm HgCdTe x = 0.7

symmetric or asymmetricdoping

Carrier densities: ns = 1x1011 ... 2x1012 cm-2

Carrier mobilities: = 1x105 ... 1.5x106 cm2/Vs

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 80

100

200

300

400

500

µ=1.06*106cm2(Vs)-1

nHall=4.01*1011cm-2

Q2134a_Gate

B[T]

Rxx

[]

-15000

-10000

-5000

0

5000

10000

15000

Graph2

Rxy

[]

Type-III QW

VBO = 570 meV

HgCdTeHgCdTeHgTe

HgCdTe

HH1E1

QW < 63 Å

HgTe

inverted normal

band structure

conduction band

valence band

HgTe-Quantum Wells

123456

k (0.01 -1)

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

Ener

gyE(

k)(e

V)

k || (1,1)k || (1,0)k = (kx,ky)

k || (1,1)k || (1,0)k = (kx,ky)

4 nm QW 15 nm QW

normal

semiconductor

inverted

semiconductor

1 2 3 4 5 6

k (0.01 -1)

-0.20

-0.15

-0.10

-0.05

0.00

0.50

0.10

0.15

0.20

E2

H1H2

E1L1

0.6 0.8 1.0 1.2 1.4

dHgTe (100 )

E2E2

E1E1H1H1

H2H2H3H3

H4H4 H5H5

H6H6L1L1

QW Band Structure from k.p Model

Dirac Bandstructure near dc

E

k

E1

H1

invertedgap

4.0nm 6.2 nm 7.0 nm

normalgap

H1

E1

B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)

Edge StatesGapless states must exist at the interface between different topological phases

Egap

Domain wall bound state 0

n=1 n=0

Band inversion – Dirac Equation

x

y

M<0

M>0

Smooth transition : gap must pass through zero

Jackiw, Rebbi (1976)Su, Schrieffer, Heeger (1980)

This is the zero B-field generalization of the Quantum Hall effect:the Quantum Spin Hall Effect

QSHE, Simplified Picture

normalinsulator

bulk

bulkinsulating

entire sampleinsulating

m > 0 m < 0

QSHE

Experimental Signature

normal insulator state

QSHI

Need small Samples

L

W

(L x W) m

2.0 x 1.0 m1.0 x 1.0 m1.0 x 0.5 m

First Observation of QSHI state

M. König et al., Science 318, 766 (2007).

Multi-Terminal Probe

210001121000012100001210000121100012

T

heIG

heIG

t

t

2

23

144

2

14

142

232

generally

22 2)1(

ehnR t

3exp4

2 t

t

RR

heG t

2

exp,4 2

Landauer-Büttiker Formalism normal conducting contacts no QSHE

0.0 0.5 1.0 1.5 2.00

5

10

15

20

25

R (k

)

V* (V)

I: 1-4V: 2-3

1

3

2

4

R14,23=1/4 h/e2

R14,14=3/4 h/e2

Non-Local data on H-bar

A. Roth et al., Science 325, 294 (2009).

Configurations would be equivalent in quantum adiabatic regime

-1 0 1 2 30

5

10

15

20

25

30

35

40

R (k

)

V* (V)

I: 1-4V: 2-3

R14,23=1/2 h/e2

R14,14=3/2 h/e2

I: 1-3V: 5-6

R13,13=4/3 h/e2

R13,54=1/3 h/e2

-1 0 1 2 3 4

V* (V)

Multi-Terminal Measurements

A. Roth et al., Science 325, 294 (2009).

Metallic Spin-Hall Effect

Intrinsic SHE

Rashba effect

J.Sinova et al.,Phys. Rev. Lett. 92, 126603 (2004)

H-bar for detection of Spin-Hall-Effect

(electrical detection through inverse SHE)

E.M. Hankiewicz et al ., PRB 70, R241301 (2004)

200 nm 200 nm

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00

100

200

300

400

500

0

2

4

6

8

10

12

R12

,36

()

Vg (V)

I (n

A)

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

T = 2 K

Rno

nloc

al /

k

VGate / V

I / n

A

n-conductingp-conducting

insu

latin

g

– Suppress non-local QSHE using long leads or narrow wires

– Intrinsic metallic SHE only shows up for holes: larger spin-orbit

– Amplitude in agreement with modeling (E. Hankiewicz, J. Sinova)

H-bar experiments

C. Brüne et al., Nature Physics 6, 448 (2010).

QSHE and iSHE as spin injector and detector

Split-gated H-bar

Detect iSHE through QSHI edge channels

I

U

Gate in 3-8 leg is scanned, 2-9 leg is n-type metallic,

current passed between contacts 2 and 9.

C. Brüne et al., Nature Physics 8, 486–491 (2012)

Detect QSHI through inverse iSHE

I

U

Gate in 3-8 leg is scanned, 2-9 leg is n-type metallic,

current passed between contacts 3 and 8 C. Brüne et al., Nature Physics 8, 486–491 (2012)

From traffic jam to info-superhighwayon chip

Traffic jam inside chips today Info highways for the chips in the future

3D Topological InsulatorsThere are 4 surface Dirac Points due to Kramers degeneracy

Surface Brillouin Zone

2D Dirac Point

E

k=a k=b

E

k=a k=b

0 = 1 : Strong Topological Insulator

Fermi circle encloses odd number of Dirac pointsTopological Metal :

1/4 grapheneBerry’s phase Robust to disorder: impossible to localize

0 = 0 : Weak Topological Insulator

Related to layered 2D QSHI ; (123) ~ Miller indicesFermi surface encloses even number of Dirac points

OR4

1 2

3

EF

How do the Dirac points connect? Determined by 4 bulk Z2 topological invariants 0 ; (123)

kx

ky

kx

ky

kx

ky

Bi1-xSbxTheory: Predict Bi1-xSbx is a topological insulator by exploiting

inversion symmetry of pure Bi, Sb (Fu,Kane PRL’07)

Experiment: ARPES (Hsieh et al. Nature ’08)

• Bi1-x Sbx is a Strong Topological Insulator 0;(1,2,3) = 1;(111)

• 5 surface state bands cross EF between and M

ARPES Experiment : Y. Xia et al., Nature Phys. (2009).Band Theory : H. Zhang et. al, Nature Phys. (2009).Bi2 Se3

• 0;(1,2,3) = 1;(000) : Band inversion at

• Energy gap: ~ .3 eV : A room temperaturetopological insulator

• Simple surface state structure :Similar to graphene, except only a single Dirac point

EF

Bulk HgTe as a 3-D Topological ‚Insulator‘

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV) 8

6

7

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV) 8

6

7

Bulk HgTe is semimetal,

topological surface state overlaps w/ valenceband.

k(1/a)

E-E

F(eV

)

ARPES: Yulin Chen, ZX Shen, Stanford

C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).

70 nm layer on CdTe substrate:coherent strain opens gap

0 2 4 6 8 10 12 14 160

2000

4000

6000

8000

10000

12000

14000

16000

0

2000

4000

6000

8000

10000

12000

14000

Rxx (SdH)

R

xx in

Ohm

B in Tesla

Rxy (Hall)

Rxy

in O

hm

Bulk HgTe as a 3-D Topological ‚Insulator‘

@ 20 mK: bulk conductivity almost frozen out - Surface state mobility ca. 35000 cm2/Vs

C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).

2 4 6 8 10 12 14 160

2

4

6

8

10

12

14

xy [e

2 /h]

B [T]

Bulk HgTe as a 3-D Topological ‚Insulator‘

@ 20 mK: same data, plotted as conductivity

3D HgTe-calculations

2 4 6 8 10 12 14 160

2000

4000

6000

8000

10000

2.73.54.45.67.69.711 33.94.96.78.510.112

experiment

Rxx

in O

hm

B in Tesla

n=3.7*1011 cm-2

n=4.85*1011 cm-2

n=(4.85+3.7)*1011 cm-2

DO

S

Red and blue lines : DOS for each of the Dirac-cones with the corresponding fixed 2D-density,Green line: the sum of the blue and red lines

C. Brüne et al., Phys. Rev. Lett. 106, 126803 (2011).

Experiments on a gated Hallbar

0 2 4 6 8 10 12 14 16

-10

12

34

5 0

5

10

15

20

25

Vgate [V]

B [T]

Rxy

[k

]

Rxy from -1.5V to 5V

FM FMchiral interconnect

3D topological insulator

Applications of TI in IT

Topological chiral interconnects

Majorana Fermions

• Quasiparticles in fractional Quantum Hall effect at =5/2Moore, Read ’91

• s-wave superconductor / Topological Insulator structureFu, Kane ‘08

• semiconductor - magnet - superconductor structuresSau, Lutchyn, Tewari, Das Sarma ‘09

• .... among others

• 2 Majorana bound states = 1 fermion- 2 degenerate states (full/empty) = 1 qubit

• 2N separated Majoranas = N qubits• Quantum Information is stored non locally

- Immune from local decoherence• Adiabatic Braiding performs unitary operations

- Non Abelian Statistics

1 2i

Potential Condensed Matter Hosts :

Topological Quantum Computing Kitaev, 2003

Create

Braid

Measure 12 34 12 340 0 1 1 / 2

t

12 340 0

Superconducting contacts on strained HgTe

dV/dI (Vbias, B) at 20 mK

dV/dI (Ibias, B) at 20 mK

Outlook

•A new electronic phase of matter has been predicted and observed- 2D : Quantum spin Hall insulator in (Hg,Cd)Te QW’s- 3D : Strong topological insulator in Bi2Se3, Bi2Te3 and strained HgTe

•Dissipationless transport in spin-polarized 1D channels•Strong Magnetoelectric Effect; Possibilities for domain wall transport?•Superconductor/Topological Insulator structures host Majorana Fermions

- A Platform for Topological Quantum Computation•Some Challenges in the near future:

- Transport Measurements on topological insulators- Superconducting structures :

- Create, Detect Majorana bound states- Magnetic structures :

- Create chiral edge states, chiral Majorana edge states

Würzburg:Bastian Büttner, Christoph Brüne, Markus König, Andreas Roth, VolkmarHock, Alina Novik, Chaoxing Liu, Ewelina Hankiewicz , Grigory Tkachov, Björn TrauzettelStanford:Xiaoliang Qi, Shoucheng Zhang

Funding: DFG (Schwerpunkt Spintronik, DFG-JST), Humboldt Stiftung, GIF, EU, ERC,DARPA

Acknowledgements