H. BuhmannH. Buhmann

Hartmut Buhmann

Physikalisches Institut EP3Physikalisches Institut, EP3Universität Würzburg

Germany

H. BuhmannQuantum Hall Effect

1 h

Nobel Prize K. von Klitzing 1985

Hall resistance 2

1eh

ienB

sxy ==ρ

ρxy

semiconductor 2DEG

ρxx

zero longitudinal resistance

H. BuhmannEdge Channels

magnetic field

-

g

-

-

- -

-

hi l d t tchiral edge states

H. BuhmannQuantum Spin Hall Effect

• The QSH state can be thought of as two copies of QH states one forB two copies of QH states, one for each spin component, each seeing the opposite magnetic field. (Bernevig and Zhang PRL 2006)

effB

(Bernevig and Zhang, PRL, 2006)

B effB

H. BuhmannQuantum Spin Hall Effect

• The QSH state can be thought of as two copies of QH states one fortwo copies of QH states, one for each spin component, each seeing the opposite magnetic field. (Bernevig and Zhang PRL 2006)(Bernevig and Zhang, PRL, 2006)

• The QSH state does not break the time reversal symmetry, and can exist without any external magnetic field.

−−+ sksk ,, ,ψψ

backscattering between Kramers’ doublets is forbidden

insulating bulk

H. BuhmannQuantum Spin Hall Effect

no magnetic fieldmagnetic field

- -- spin up

g

-

-

-

--

-

- - --spin down

chiral edge states helical edge states

H. BuhmannQSHE in Graphene

Graphene edge states

C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)

• Graphene – spin-orbit coupling strength is too weak gap only about 10-3 meV. p p p g g g p y

• not accessible in experiments

H. BuhmannQSHE in HgTe

Helical edge statesfor inverted HgTe QW

E

H1

E1

kπ 0 π

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

H. BuhmannH. Buhmann

H. BuhmannHgTe Quantum Well Structures

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

gate Au

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

gate

insulator

Au

100 nm Si N /SiO3 4 2

cap layer

barrierdoping layer

25 nm HgCdTe x = 0.7

10 nm HgCdTe x = 0 7 9 nm HgCdTe with I10 nm HgCdTe x = 0.7

doping layer

barrier

barrierquantum well

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

symmetric or asymmetricd i

buffer

substrate CdZnTe(001)

25 nm CdTe10 nm HgCdTe x = 0.7 doping

( )

H. Buhmannn to p Transitions

nmax = 1.35 x 1012 cm-2

100

150

H1

pmax = 3.2 x 1011 cm-20

50

E /

meV H2

150

-100

-50

E130000

T = 1.5 K

0,0 0,1 0,2 0,3 0,4 0,5-150

k / nm-1

10000

20000VGate

+5,0 V +4,5 V+4,0 V

Q21638 nm QW 0V

10000

0

+3,5 V +3,0 V +2,5 V +2,0 V+1,5 V

Rxy

/ Ω +5V

gate

insulator

cap layer

doping layer

-20000

-10000 +1,0 V +0,5 V 0 V -2,0 V-2V

doping layer

barrier

barrierquantum well

doping layer

buffer

0 2 4 6 8-30000

B / T

substrate

H. BuhmannHgTe

band structuresemi-metal or semiconductorsemi metal or semiconductor

1000

0

500

eV) 8

E

-500E(m

e

6

Eg

-1 0 -0 5 0 0 0 5 1 0-1500

-1000

7

fundamental energy gap

1.0 0.5 0.0 0.5 1.0k (0.01 )

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

fundamental energy gap

meV 30086 −≈− ΓΓ EE

H. BuhmannHgTe-Quantum Wells

BarrierQW

1000HgTe Hg0.32Cd0.68Te

1000

500

8

5006

-500

0

E(m

eV) 8

6 -500

0

E(m

eV)

8VBO

-1000

7

-1000

7

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

7

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

7

VBO = 570 meVVBO = 570 meV

H. BuhmannHgTe-Quantum Wells

Typ-III QW

d

HgTe

Γ6

HgCdTeHgCdTe

g

HH1E1

Γ8

band structure

invertednormal

H. BuhmannBand Gap Engineering

ΓH CdTHgTe

Γ6

Zero Gap for 6.3 nm QW structures

HgCdTeHgTe

Γ6

H1E1

HgCdTe

Γ8

E1H1

Γ8

Γ8

normal

inverted

direct band gaps between80 ... 0 and 0 ... - 30 meV

H. BuhmannQSHE in HgTe

Helical edge statesfor inverted HgTe QW

E

H1

E1

kπ 0 π

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

H. BuhmannMass domain wall

invertedband structure

EH1

states localized on the domain wall which disperse along the x-direction

E1

y

m>0m<0

k3π 0 π

m

m>0y

x

m 0m 0

m0

m<0

x

m>0

x

helical edge states

H. BuhmannSimplified Picture

normal

m > 0 m < 0

insulator QSHE

bulkbulk

bulkinsulating

entire sampleinsulating

H. BuhmannH. Buhmann

H. Buhmann

Experiment

ε ε

k k

H. BuhmannSmaller Samples

LL

W

(L x W) μm

2.0 x 1.0 μm1.0 x 1.0 μm1.0 x 0.5 μm

H. BuhmannQSHE Size Dependence

106 (1 x 1) μm2

non-inverted10

Ω

non inverted

105

2

Rxx

/ Ω

104G = 2 e2/h

(1 x 1) μm2(2 x 1) μm2

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0103 (1 x 0.5) μm2

(VGate- Vthr) / V

H. BuhmannQSHE in inverted HgTe-QWs

20

in 4-terminal geometry!?

V

16

18

0 V

12

14 G = 2 e2/h

kΩ

I

8

10

Rxx

/ k

2

4

6

-1.0 -0.5 0.0 0.5 1.0 1.5 2.00

2

(V V ) / V(VGate- Vthr) / V

H. BuhmannQSHE in inverted HgTe-QWs

40

30

35G = 2/3 e2/h

I1

2 3

4

V

25

30

(kΩ

) 2 3V 56

15

20

G = 2 e2/h

Rxx

( I1 4

56

5

10(2 x 1) μm2

32 ≈t

RR-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0

(V V ) (V)

(1 x 0.5) μm2

4tR(VGate- Vthr) (V)

H. BuhmannMulti-Terminal Probe

Landauer-Büttiker Formalism normal conducting contacts

no QSHEno QSHE

( ) ⎤⎡∑e2

⎟⎞

⎜⎛− 100012

( ) ⎥⎦

⎤⎢⎣

⎡−−= ∑

≠ijjijiiiii TRM

heI μμ2

⎟⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎜⎛

−−

=001210000121100012

T

214

43 2

2t

I eGhμ μ

⎧= =⎪ −⎪

⎨heG t

2

exp,4 2≈

⎟⎟⎟⎟⎟

⎠⎜⎜⎜⎜⎜

⎝ −−

−

210001121000012100

T 3 22

142

4 1

23t

I eGh

μ μ

μ μ

⎪⇒ ⎨⎪ = =⎪ −⎩

3exp4

2 ≈t

t

RR

⎠⎝ 210001

generally22 2)1(

ehnR t

+=

H. BuhmannNon-locality

Q2308: 250 | 90: 0.1 | 400 | 90 | 400 | 90: 0.1 | 1000 | ns= 3.1x1011, μ =143 000

6 8 9

1 μm2 μm

3 2 11

3

H. BuhmannNon-locality

LB 12 9 kΩ

gate sweep

@6 mK Q2308-02

LB: 12.9 kΩ

16

18

20

22

12

143

10

12

14

16

8

10

Rnl (k

Ω)I (

nA)

4

6

8

2

4

6

-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

0

UGate [ V ]

H. BuhmannNon-locality

LB 8 6 kΩ

@6 mK Q2308-02

LB: 8.6 kΩ

50

60

70

12

143

30

40

50

8

10

Rnl (kΩ

)I / n

A

10

20

30

2

4

6

-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00 0

UGate [ V ]

H. BuhmannNon-locality

LB 4 3 kΩ

LB: 4.3 kΩ

5

63

3

4

nl (k

W)

1

2

Rn

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

H. BuhmannBack Scattering

potential fluctuations introduce areas of normal metallic (n- or p-) conductancepotential fluctuations introduce areas of normal metallic (n or p ) conductancein which back scattering becomes possible

QSHE

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

H. BuhmannPotential Fluctuations

LB 4 3 kΩ

LB: 4.3 kΩ

5

63

3

4

nl (k

W)one additional contact:

• between 8 93.7 kΩ

1

2

R

x

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

3

H. BuhmannNon-locality

different gate sweep direction

@6 mK Strom: 6, 11Spannung: 8 9 Q2308-02

LB: 12,9 kΩ

18

20

22

20

22

24

26Spannung: 8, 9

10

12

14

16

12

14

16

18

Rnl (k

Ω)I (

nA)

4

6

8

4

6

8

10

-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

0

2

UGate [ V ]

H. BuhmannPotential Fluctuations

gate-voltage dependenceimpurity statesdischarging charging

ulat

or

QW

met

al insu

hysteresis effects:

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

H. BuhmannNon-locality

two additional contacts:• between 8 9 and 6 8

19.4 kΩ

different gate sweep direction

Q2308-02

LB: 12,9 kΩxx

16

18

20

22

20

22

24

26

10

12

14

16

12

14

16

18

Rnl (k

Ω)I (

nA)

one additional contact:• between 8 9

2

4

6

8

4

6

8

10between 8 9

22.1 kΩ

x

-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

0

2

UGate [ V ]

H. BuhmannH. Buhmann

H. BuhmannH. Buhmann

H. BuhmannSpin Polarizer

H. BuhmannSpin Polarizer

50%1 : 4

50%50%

1 01 : 0

H. BuhmannComing Soon

QSHE

SHEQSHE

SHE-1

H. Buhmann

Split Gate H-Bar

H. BuhmannSummary: QSH Effect

• the QSH effect is a new non-trivial state of matter which consists ofconsists of – an insulating bulk and– two counter propagating spin polarized edge channelstwo counter propagating spin polarized edge channels

(Kramers doublet)

• the QSH effect shows up in a quantized conductance – the quantized value is strongly dependent on the number of

ohmic contactsohmic contacts – or the quality of the sample (potential fluctuations)

• and is destroyed by time reversal symmetry breaking effects, like a magnetic field, g

H. BuhmannAcknowledgements

Quantum Transport Group (Würzburg)

C. BrüneE. Rupp

A. Roth N. EikenbergR. RommelH Thierschmann

A. AstakhovaM. Mühlbauer

H. Thierschmann

Lehrstuhl für Experimentelle Physik 3: L W MolenkampLehrstuhl für Experimentelle Physik 3: L.W. Molenkamp

Univ. WürzburgInst. f. Theoretische Physik

Collaborations:

Ex-QT:Stanford University

S.-C. ZhangX L Qi

yE.M. Hankiewicz

C.R. BeckerT. BeringerM. LebrechtJ. Schneider

X.L. QiT. L. HughesJ. MaciejkoM KönigT. Spitz

S. WiedmannM. König

H. BuhmannH. Buhmann

Quantum Spin Hall EffektScience 318, 766 (2007)

The Quantum Spin Hall Effect:Theory and ExperimentJ. Phys. Soc. Jap.Vol. 77, 31007 (2008)