hartmut buhmann - uspmacbeth.if.usp.br/~gusev/spinhallquantico.pdf · h. buhmann band structure...

H. BuhmannH. Buhmann

Hartmut Buhmann

Physikalisches Institut, EP3

Universität Würzburg

Germany


Outline

• Insulators and Topological Insulators

• HgTe quantum well structures

• Two-Dimensional TI

• Quantum Spin Hall Effect

• experimental observations:

• quantized conductivity

• non-locality

• spin polarization

• Three-Dimensional TI

• Quantum Hall Effect

H. Buhmann

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

C.L.Kane and E.J.Mele, PRL 95, 146802 (2005)


A.Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)

The Insulating States

Topologically Generalized

H. Buhmann

Covalent Insulator

Characterized by energy gap: absence of low energy electronic excitations

The vacuumAtomic Insulator

e.g. solid Ar

Dirac

VacuumEgap ~ 10 eV

e.g. intrinsic semiconductor

Egap ~ 1 eV

3p

4s

Silicon

Egap = 2 mec2

~ 106 eV

electron

positron ~ hole

The Usual Boring Insulating State

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The Insulating State – Topologically

Generalized

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2D Cyclotron Motion, Landau Levels

gap cE E

Energy gap, but NOT an insulator

The Integer Quantum Hall State

H. Buhmann

bulk

insulating

Quantized Hall conductivity :

y xy xJ E

2

xyh

ne

Integer accurate to 10-9

Edge States

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Perfect Conductor

The QHE state spatially separates

the right and left moving states

(spinless 1D liquid)

suppressed backscattering

1d dispersionmagnetic field

k

kF-kF

E

-

-

-

-

-

-

first example of a new

Topological Insulator

H. Buhmann

Topological Insulator

H. Buhmann

21( ) ( )

2 BZn d u u

i 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

Insulator : n = 0

IQHE state : xy = n e2/h

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

( ) :H kBloch Hamiltonians

Brillouiw

n zone (torus) ith energy gap

u(k) = Bloch wavefunction

Topological Band Theory

H. Buhmann

Geometry and Topology

Gauss and Bonnet

S

gKdA )1(22

1

S: surface

K-1: product of the two radii of curvatures

dA: area element

g: number of handles

geometric

topological

Example: Mug to Torus

H. Buhmann

Vacuum

Dn = # Chiral Edge Modes

This approach can actually be generalized to a spinfull QHE at zero magnetic field:

the Quantum Spin Hall Effect

Edge States

The TKNN invariant can only change at a phase

transition where the energy gap goes to zero

Vacuum

n = 0 n = 3 n = 0

H. Buhmann





The Insulating State – Topologically

Generalized


A two-dimensional topological insulator

B

quantum Hall effect topological invariant

broken time reversal symmetry (TRS)



B



B

helical edge states

invariant under time reversal

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Protected Edge States

suppressed backscattering:

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Protected Edge States

spin rotation:

Dj = 2

Interference is destructive

suppressed backscattering:

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x

x

Quantum Spin Hall Effect

Stability of Helical Edge States:

backscattering is only possible by time reversal symmetry breaking processes

(for example external magnetic fields)

sksk ,, 21,

x

x

x

x

or if more states are present

h

eGQSHE

22

H. Buhmann

Topological Insulator :

A New B=0 Phase

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There are 2 classes of 2D time reversal invariant band structures

Z2 topological invariant: n = 0,1

n=0 : Conventional Insulator n=1 : Topological Insulator

Kramers degenerate at

time reversal

invariant momenta

k* = k* + G

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

Edge States for 0<k</a

even number of bands

crossing Fermi energyodd number of bands

crossing Fermi energy

Topological Insulator :

A New B=0 Phase

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Graphene edge states

QSHE in Graphene

• Graphene – spin-orbit coupling strength is too weak gap only about 30 meV.

• not accessible in experiments

gap

H. Buhmann

QSHE in HgTe

Helical edge states

for inverted HgTe QW

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

k

E

H1

E1

0

H. Buhmann

band structure

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

semiconductor

fundamental energy gap

meV 30086 EE

semi-metal

HgTe

-1.0 -0.5 0.0 0.5 1.0

k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV) 8

6

7

Eg

D

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CdTe and HgTe

H = H1 + HD + Hmv + Hso

CdTe HgTe

H. Buhmann

-1.0 -0.5 0.0 0.5 1.0

k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV)

HgTe

8

6

7

-1.0 -0.5 0.0 0.5 1.0

k (0.01 )

Hg0.32Cd0.68Te

-1500

-1000

-500

0

500

1000

E(m

eV)

6

8

7

VBO

BarrierQW

VBO = 570 meV

HgTe-Quantum Wells

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Typ-III QW

band structure

HgCdTeHgTe

6

8

inverted

HgTe-Quantum Wells

HgCdTe

HH1E1

d

normal

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Bandstructure

HgCdTe

HgTe

6

8

HgCdTe

HgTe

6

8

Zero Gap for 6.3 nm QW structures

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4 6 8 10 12 14

-80

-40

0

40

80 E2

H4

H3

H2

H1

E [

me

V]

d [nm]

E1

dc

0.0 0.1 0.2 0.3 0.4 0.5

-100

-50

0

50

100

in-plane direction

(kx,k

y)=(1,0)

(kx,k

y)=(1,1)

H2

H1

E1

E [

me

V]

k [nm-1]

Dirac Band Structure

H. Buhmann

Zero Gap

-1 0 1 2 3-2

-1

0

1

2

3

4

5

6

7

xy [

e²/

h]

VG - V

Dirac [V]

B = 5 T

0

1k

2k

3k

xx [

-0,5 0,0 0,5 1,0

0

2

4

6

8

10

12

14

B = 1 T

xx [

VG - V

Dirac [V]

xy [

e²/

h]

0

10k

20k

30k

40k

50k

60k

70k

80k

B. Büttner et al., Nature Physics 7, 418 (2011)

H. Buhmann

Kane model

effective model

Parameters:

M ~ 0 (-0.035 meV)

D = -682.2 meV nm²

G = -857.3 meV nm²

A = 372.9 meV nm

LLchart

E(VG) conversion

taken from

appropriate

Kane model

LL-Dispersion


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Finite Mobility

4 8 12 16 20 240.2

0.4

0.6

0.8

1.0

1.2

1.4 measurement

theory

xx in

e2/h

T [K]

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60

20k

40k

60k

80k

xx

VG-V

Dirac [V]

xx,min

= 0.36 e²/h

DVG


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Bandstructure

HgCdTe

HgTe

6

8

HgCdTe

HgTe

6

8

B.A Bernevig et al., Science 314, 1757 (2006)

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Effective Tight Binding Model

square lattice with 4 orbitals per site

, , , , ( ), , ( ),x y x ys s p ip p ip

nearest neighbor hopping integral

basis: 1 1 1 1, , ,E H E H

*

( ) 0( , )

0 ( )eff x y

h kH k k

h k

H. Buhmann

Effective Tight Binding Model

square lattice with 4 orbitals per site

, , , , ( ), , ( ),x y x ys s p ip p ip

nearest neighbor hopping integral

basis: 1 1 1 1, , ,E H E H

*

( ) 0( , )

0 ( )eff x y

h kH k k

h k

relativistic Dirac equation with tunable parameter M (band-gap)

at the -pointHgTe

2

2

)(sinsin

sinsin)()(

kkMkikA

kikAkkMkh

xx

xx

)(

)(0

kMikkA

ikkAkMk

xx

xx

QW inverted

QW normalfor

0

0

M

M

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Mass domain wall

states localized on the domain wall

which disperse along the x-direction

M

y

0

M>0

M<0

M>0

y

x

x

M>0M<0

inverted

band structure

helical edge states

k

E

H1

E1

0

H. Buhmann

Quantum Spin Hall Effect

normal

insulator

bulk

bulk

insulating

entire sample

insulating

M > 0 M < 0

QSHE


x

x

h

eG

22

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Experiment

k

e

k

e

H. Buhmann

Small Samples

L

W

(L x W) mm

2.0 x 1.0 mm

1.0 x 1.0 mm

1.0 x 0.5 mm

H. Buhmann

-1.0 -0.5 0.0 0.5 1.0 1.5 2.010

3

104

105

106

G = 2 e2/h

Rxx /

(VGate

- Vthr

) / V

QSHE Size Dependence

(1 x 0.5) mm2

(1 x 1) mm2(2 x 1) mm2

(1 x 1) mm2

non-inverted

König et al., Science 318, 766 (2007)

H. Buhmann

QSHE Temperature Dependence

-2.0 -1.6 -1.2 -0.8 -0.4 0.0-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

T = 1.8 K

T = 2.5 K

T = 4.2 K

T = 6.4 K

T = 9.8 K

T = 19 K

T = 28 K

T = 48 K

R2

-8,

9-1

1(k

)

UG(V)

Q2367 Microhallbar 1 x 1 mm2

I(2-8), U(9-11); B=0 T; T= variabel;

H. Buhmann

QSHE Excitation Voltage Dependence

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.20

2

4

6

8

10

12

14

16

18

1600mV

1400mV

1200mV

1000mV

800mV

600mV

400mV

200mV

R (

kO

hm

)

Ug (V)

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-1.0 -0.5 0.0 0.5 1.0 1.5 2.00

2

4

6

8

10

12

14

16

18

20

G = 2 e2/h

Rxx / k

(VGate

- Vthr

) / V

Conductance Quantization

in 4-terminal geometry!?

V

I

H. Buhmann

Small Samples

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Role of Ohmic Contacts

n-type

metallic

contact areas

and leads

areas

where scattering

takes place

spin relaxation

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Multi-Terminal Probe

210001

121000

012100

001210

000121

100012

T

2

144

3 2

2

142

4 1

2

2

3

t

t

I eG

h

I eG

h

m m

m m

generally22

2

)1(

e

hnR t

3

exp4

2 t

t

R

R

h

eG t

2

exp,4 2

Landauer-Büttiker Formalism normal

conducting contacts

no QSHE

ij

jijiiiii TRMh

eI mm

2

H. Buhmann

QSHE in inverted HgTe-QWs

34

2 t

t

R

R-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0

5

10

15

20

25

30

35

40

G = 2/3 e2/h

G = 2 e2/h

Rxx (

k

(VGate- Vthr) (V)

I1

2 3

4

56

V

(1 x 0.5) mm2

(2 x 1) mm2

I1

2 3

4

56

V

H. Buhmann

Non-locality

1 mm2 mm

1 2 3

6 5 4

H. Buhmann

-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00

2

4

6

8

10

12

14

16

18

20

22

0

2

4

6

8

10

12

14

@6 mK

Rn

l (k

)I (n

A)

UGate

[ V ]

Q2308-02

Non-locality

I: 1-4

Vnl:2-3

1 2 3

6 5 4

V

I1

2 3

4

56

V

A. Roth, HB et al.,

Science 325,295 (2009)

H. Buhmann

Non-locality

-1 0 1 2 3 40

2

4

6

8

10

12

R (

k

)

V* (V)

G=3 e2/h

I: 1-3

Vnl:5-4

1 2 3

6 5 4

V

4-terminal

LB: 8.6 k

A. Roth, HB et al.,

Science 325,295 (2009)

H. Buhmann

Non-locality

A. Roth, HB et al.,

Science 325,295 (2009)

G=4 e2/h

I: 1-4

Vnl:2-3

1

3

2

4

V

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00

1

2

3

4

5

6

7

R14,2

3 / k

Vgate / V

LB: 6.4 k

H. Buhmann

-6,0 -5,5 -5,0 -4,5 -4,0 -3,5 -3,00

1

2

3

4

5

6 R

nl (

k

)

V (V)

LB: 4.3 k

Non-locality

1 2 3

6 5 4

V

A. Roth, HB et al.,

Science 325,295 (2009)

H. Buhmann

Back Scattering

potential fluctuations introduce areas of normal metallic (n- or p-) conductance

in which back scattering becomes possible

QSHE

The potential landscape is modified by gate (density) sweeps!

H. Buhmann

Back Scattering

area of raised or lowered chem. Pot.the probalility for back scatterring increases

strength

length additional ohmic contact

transition from a two to a three terminal device

H. Buhmann

Additional Scattering Potential

A. Roth, HB et al.,

Science 325,295 (2009)

Transition from

2 e2/h

scattering strength

3/2 e2/h

H. Buhmann

Additional Scattering Potential

A. Roth, HB et al.,

Science 325,295 (2009)

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00

1

2

3

4

5

6

7

R14,2

3 / k

Vgate / V

H. Buhmann

-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.00

1

2

3

4

5

6 R

nl (k

W)

LB: 4.3 k

one additional contact:

• between 2 and 3

3.7 k

Potential Fluctuations

1 2 3

6 5 4

V

1 2 3

6 5 4

V

x

A. Roth, HB et al.,

Science 325,295 (2009)

H. Buhmann

1 2 3

6 5 4

V

-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00

5

10

15

20

Rnl /

k

Vg / V

LB: 12.9 k

different gate sweep direction

1 2 3

6 5 4

V x

LB: 22.1 k

Potential Fluctuations

• Hysteresis effects due to charging of trap states at the SC-insulator interface

J. Hinz, HB et al., Semicond. Sci. Technol. 21 (2006) 501–506

H. Buhmann

HgTe Quantum Well Structures

gate

insulator

cap layer

doping layer

barrier

barrierquantum well

doping layer

buffer

substrate

Au

100 nm Si N /SiO

3 4 2

25 nm HgCdTe x = 0.7

CdZnTe(001)

25 nm CdTe


9 nm HgCdTe with I


4 - 12 nm HgTe


9 nm HgCdTe with I


layer structure

meta

l insula

tor

QW

impurity statesdischarging charging

hysteresis effects:

J. Hinz et al., Semicond. Sci. Technol. 21 (2006) 501–506

H. Buhmann

Spin Polarizer

H. Buhmann

Spin Polarizer

50%

50%1 : 4

H. Buhmann

Spin Polarizer

50%

100 %

100%

100%

H. Buhmann

Spin Injector

100 %100%

spin

injection

even with backscattering!

H. Buhmann

Spin Detection

100 %100%

spin

detector

Dm


SHE-1

QSHE

Spin Hall Effect

as creator and analyzer of

spin polarized electrons

SHE

QSHE

H. Buhmann

Split Gate H-Bar

H. Buhmann

QSHE Spin-Detector

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.00

0.02

0.04

0.06

0.08

0.10

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.0

0.2

0.4

0.6

0.8

1.0

Vdetector-gate / V

5x

50x

-4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.50

1

2

3

4

5

R34

,12/ k

Vinjector-gate

/ V

n-type injector

p-type injector

QSHI-type detector

SHE

QSHE

QSHI-type detector

injector sweep

detector sweep


0.0 0.1 0.2 0.3 0.4 0.5 0.6-75

-50

-25

0

25

50

75

0.0 0.1 0.2 0.3 0.4 0.5 0.6-75

-50

-25

0

25

50

75

Q2198

H2

E1

H1

EF

n =4.42·1011

cm-2

ndl=1.0·10

11 cm

-2

E /

me

V

k / nm-1

Dk

H2

H1

E1

EF

p =3.26·1011

cm-2

ndl=1.0·10

11 cm

-2

E /

me

Vk / nm

-1

n-regime p-regime

,16

xs y F F

eEj p p k

m

D

J.Sinova et al.,

Phys. Rev. Lett. 92, 126603 (2004)

Intrinsic Spin Hall Effekt

Nature Physics 6, 448 (2010)

H. Buhmann

QSHE Spin-Injector

-4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.50

1

2

3

4

5

R12

,34/ k

Vdetector-gate / V

5x

50x

n-type detector

p-type detector

QSHI-type injector

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.00

0.02

0.04

0.06

0.08

0.10

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.0

0.2

0.4

0.6

0.8

1.0

SHE-1

QSHE

QSHI-type detector

detector sweep

injector sweep


V

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00

20

40

60

80

100

Rn

on

loc/

Ugate

/ V

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00

200

400

600

800

1000

Rn

on

loc/

Ugate

/ V

detector sweep

p-type

n-type

V

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00

200

400

600

800

1000

Rn

on

loc/

Ugate

/ V

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00

20

40

60

80

100

Rn

on

loc/

Ugate

/ V

H. Buhmann

Spin-Current Switch

I2 I1

H. Buhmann

Spin-Current Switch

I2 I1

QSHE

H. Buhmann

Spin-Current Switch

I2 I1

ballistic

injection


9/10 11/12 1/2

6/58/7 4/3

B E

ACDF

I I

-3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

5,0

I [n

A]

U (Gate ACDF) [V]

QSHE

ballstic


9/10 11/12 1/2

6/58/7 4/3

B E

ACDF

I I

-3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

5,0

I [n

A]

U (Gate ACDF) [V]

-3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,09

10

11

12

13

14

15

I [n

A]

U (Gate ACDF) [V]


¼ Graphene

H. Buhmann

-0,3 -0,2 -0,1 0,0 0,1 0,2 0,3

-500

-400

-300

-200

-100

0

100

200

EF

E, m

eV

k, nm-1

Bulk HgTe

1670 1680 1690 1700 1710 1720 1730Qx*10000(rlu)

5910

5920

5930

5940

5950

5960

5970Qy*10000(rlu) Q2424-map_1.xrdml

1.6

2.9

5.2

9.3

16.6

29.7

53.3

95.4

171.0

306.3

548.7

982.9

1760.7

3154.0

5650.0

10121.3

18131.1

32479.6

58183.3

104228.3

186712.2

x-ray diffraction (115 reflex)

fully strained

70 nm HgTe

on CdTe substrate

band structure

- gap openning

- two Dirac cones on

different surfaces


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)

Rxx in O

hm

B in Tesla

n=4

n=3

Rxy

(Hall)

Rxy in O

hm

n=2


2 4 6 8 10 12 14 160

2

4

6

8

10

12

14

xy [e

2/h

]

B [T]

0 2 4 6 8 10 12 14 160

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

Rxx [

k

]

B [T]

Rxy [

k

]

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


2 4 6 8 10 12 14 160

2000

4000

6000

8000

10000

Rxx in O

hm

B in Tesla

DO

S

Density of states for two independent surfaces with slightly different density

2-11

2

-211

1

cm 1085.4

cm 107.3

n

n


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

H. Buhmann

Summary

• the QSH effect which consists of

– an insulating bulk and

– two counter propagating spin polarized edge channels

(Kramers doublet)

• quantized edge channel transport

• the QSH effect can be used as an effective

– spin injector and

– spin detector

with 100 % spin polarization properties

• Strained HgTe bulk exhibits two-dimensional surface

states

H. Buhmann

Acknowledgements

Quantum Transport Group (Würzburg, H. Buhmann)

C. Brüne

C. Ames

P. Leubner

L. Maier

M. Mühlbauer

A. Friedel

C. Thienel

H. Thierschmann

R. Schaller

J. Mutterer

A. Astakhova

CX Liu

Ex-QT:C.R. Becker

T. Beringer

M. Lebrecht

J. Schneider

S. Wiedmann

N. Eikenberg

R. Rommel

F. Gerhard

B. Krefft

A. Roth

B. Büttner

R. Pfeuffer

Stanford University

S.-C. Zhang

X.L. Qi

J. Maciejko

T. L. Hughes

M. König

Texas A&M University

J. Sinova

Lehrstuhl für Experimentelle Physik 3: L.W. Molenkamp

Univ. Würzburg

Inst. f. Theoretische Physik

E.M. Hankiewicz

B. Trautzettel

Collaborations:


Quantum Spin Hall Effekt

Science 318, 766 (2007)

The Quantum Spin Hall Effect:

Theory and Experiment

J. Phys. Soc. Jap.

Vol. 77, 31007 (2008)

Intrinsic Spin Hall Effekt

Nature Physics 6, 448 (2010)

Nonlocal edge state transport

in the quantum spin Hall state

Science 325, 294 (2009)

Quantum Hall Effect from the

Topological Surface States of

Strained Bulk HgTe

Phys. Rev. Lett. 106, 126803 (2011)

hartmut buhmann - uspmacbeth.if.usp.br/~gusev/spinhallquantico.pdf · h. buhmann band structure...

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