Hartmut Buhmann - meso12.itp.ac.rumeso12.itp.ac.ru/Meso12_Talk_Slides/Buhmann_talk.pdf · H. Buhmann C.L.Kane and E.J. Mele, Science 314, 1692 (2006) ... gap 2= 2 m e c 6~ 10 eV electron

H. Buhmann H. Buhmann

Hartmut Buhmann

Physikalisches Institut, EP3

Universität Würzburg

Germany


TV-Series: „The Big Bang Theory“, S04, E14 (2011)


Outline

• Insulators and Topological Insulators

• HgTe

• crystal structure

• quantum wells

Two-Dimensional TI

• Quantum Spin Hall Effect

• quantized conductivity

• non-locality

• spin polarization

• strained layers

Three-Dimensional TI

• quantum Hall effect

• super currents

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 vacuum Atomic Insulator

e.g. solid Ar

Dirac

Vacuum Egap ~ 10 eV

e.g. intrinsic semiconductor

Egap ~ 1 eV

3p

4s

Silicon

Egap = 2 mec2

~ 106 eV

electron

positron ~ hole

The Usual Insulating State

H. Buhmann





The Insulating State – Topologically

Generalized

H. Buhmann

2D Cyclotron Motion, Landau Levels

gap cE E

Energy gap bulk insulator

The Integer Quantum Hall State

H. Buhmann

Edge States

skipping orbits chiral edge states

H. Buhmann

Quantum Hall Effect

Hall resistance

2

1

e

h

ien

B

s

xy xy

xx

zero longitudinal resistance

semiconductor 2DEG

Nobel Prize K. von Klitzing 1985

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 : n > 0 (integer) sxy = 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

(Thouless, Kohmoto, Nightingale & Den Nijs)

H. Buhmann

Vacuum

Dn = # Chiral Edge Modes

Edge States

The TKNN invariant can only change with a phase

transition where the energy gap goes to zero

Vacuum

n = 0 n = 3 n = 0

H. Buhmann

bulk

insulating

Edge States

1d dispersion

k

kF-kF

E

The QHE state spatially separates

the right and left moving states

(spinless 1D liquid)

suppressed backscattering

first example of a new

Topological Insulator

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

the Quantum Spin Hall Effect

H. Buhmann





The Insulating State – Topologically

Generalized

H. Buhmann

Quantum Spin Hall Effect

effB

effB

Quantum Spin Hall State

consisting of two counter propagating spin

polarized edge channels

and an insulating bulk

broken time reversal symmetry (TRS)

invariant under TR

TR copy

H. Buhmann

Topological Insulator :

A New B=0 Phase

H. Buhmann

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 degeneracy

at points with

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 energy odd number of bands

crossing Fermi energy

Topological Insulator :

A New B=0 Phase

H. Buhmann

There are 4 surface Dirac Points due to Kramers degeneracy

Surface Brillouin Zone

2D Dirac Point

E

k=La k=Lb

E

k=La k=Lb

n0 = 1 : Strong Topological Insulator

Fermi circle encloses odd number of Dirac points

Topological Metal : 1/4 graphene

Berry’s phase

Robust to disorder: impossible to localize

n0 = 0 : Weak Topological Insulator

Related to layered 2D QSHI ; (n1n2n3) ~ Miller indices

Fermi surface encloses even number of Dirac points

OR

L4

L1 L2

L3

EF

How do the Dirac points connect? Determined

by 4 bulk Z2 topological invariants n0 ; (n1n2n3)

kx

ky

kx

ky

kx

ky

3D Topological Insulator


H. Buhmann

Bi1-xSbx

Theory: 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 n0;(n1,n2,n3) = 1;(111)

• 5 surface state bands cross EF

between G and M

ARPES Experiment : Y. Xia et al., Nature Phys. (2009).

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

• n0;(n1,n2,n3) = 1;(000) : Band inversion at G • Energy gap: D ~ .3 eV : A room temperature

topological insulator

• Simple surface state structure :

Similar to graphene, except

only a single Dirac point EF

3D Topological Insulator

H. Buhmann


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

DEg = 30…60 meV

(typically)

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

-

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 G-point

--

-

2

2

)(sinsin

sinsin)()(

kkMkikA

kikAkkMkh

xx

xx

-

-

)(

)(0

kMikkA

ikkAkMk

xx

xx

QW inverted

QW normalfor

0

0

M

M

H. Buhmann

Quantum Spin Hall Effect

normal

insulator

bulk

bulk

insulating

entire sample

insulating

M > 0 M < 0

QSHE

H. Buhmann

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>0 M<0

inverted

band structure

helical edge states

k

E

H1

E1

0



H. Buhmann

band structure

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

band inversion

meV 30086 -- GG EE

semi-metal

or

“topological 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

H. Buhmann

CdTe and HgTe

H = H1 + HD + Hmv + Hso

CdTe HgTe

H. Buhmann

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

Semi-metal with bulk band inversion

How to open the bulk band gap?

1. quantum confinement

2D TI

2. strain

3D TI


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

Barrier QW

HgTe-Quantum Wells

H. Buhmann

Typ-III QW

band structure

HgCdTe HgTe

G6

G8

inverted

HgTe-Quantum Wells

HgCdTe

HH1 E1

d

-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

LH1

H. Buhmann

Typ-III QW

band structure

HgCdTe HgTe

G6

G8

inverted

HgTe-Quantum Wells

HgCdTe

HH1 E1

d

normal

H. Buhmann

Bandstructure

HgCdTe

HgTe

G6

G8

HgCdTe

HgTe

G6

G8

Zero Gap for 6.3 nm QW structures B. Büttner et al., Nature Physics 7, 418 (2011)

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



x

x

h

eG

22

H. Buhmann

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)


1 mm 2 mm

1 2 3

6 5 4

H. Buhmann

-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

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

-

-

generally 22

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)

I 1

2 3

4

5 6

V

(1 x 0.5) mm2

(2 x 1) mm2

I 1

2 3

4

5 6

V

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

I 1

2 3

4

5 6

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

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

-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

Spin Polarizer

2D topological insulator attached to four metallic contacts/leads

electron drain

voltage contacts

electron source

H. Buhmann

Spin Polarizer

50%

50% 1 : 4


electron drain

electron source

H. Buhmann

Spin Polarizer

50%

100 %

100%

100%


electron drain

electron source

electron drain

I = 0

H. Buhmann

Spin Injector

100 % 100%

spin

injection

even with backscattering!


H. Buhmann

Spin Detection

100 % 100%

spin

detection

Dm

H. Buhmann

Spin Manipulation

Dm

spin injection and

spin manipulation by SO induced

spin rotation (Rashba effect)


QSHE

SHE


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

n-type injector

p-type injector

SHE

QSHE

QSHI-type detector

-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

QSHI-type detector

injector sweep

detector sweep

H. Buhmann

QSHE Spin-Injector

5x

50x

n-type detector

p-type detector

-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

-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

QSHI-type injector

detector sweep

injector sweep


C. Brüne et al., Nature Physics, 8, 485 (2012)

H. Buhmann

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

Spin Diffusion Length



2d Dirac system

• hybridization of the top and bottom surface lead to a gapped

states on these two surfaces

• free 1d Dirac electrons at the edges of the layer


2d to 3d transition

• hybridization of the top and

bottom surface is lifted

• gapless surface state are

recovered


3d Dirac system

free 2d Dirac electrons on all six surfaces


H. Buhmann

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

-400

-300

-200

-100

0

100

EF

E, m

eV

k, nm-1

Bulk HgTe

x-ray diffraction (115 reflex)

fully strained

70 nm HgTe

on CdTe substrate

Band Structure

- gap openning (10…20 meV)

- two Dirac cones on

different surfaces

L1

H1

E1

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

dga

p e

ner

gy

(eV

)

lattice constant a [Å]0© CT-CREW 1999

H. Buhmann

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

QHE in a 3D TI system

H. Buhmann

2 4 6 8 10 12 14 160

2

4

6

8

10

12

14

s

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)


h

enxy

2

2

1

s

single Dirac

surface

H. Buhmann

2 4 6 8 10 12 14 160

2000

4000

6000

8000

10000

Rxx in

Oh

m

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


B

I

H. Buhmann

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


H. Buhmann


0 2 4 6 8 10 12 14 16

-1

0

1

2

3

4

5 0

5

10

15

20

25

Vgate [V

]

B [T]

Rx

y [

k

]

Rxy from -1 V to 5 V n = 1

n = 3

n = 5

flat band condition at -1 V top and bottom surface exhibit approx. the same

carrier density.

h

enxy

2

12 s for ntop = nbottom

H. Buhmann

0 2 4 6 8 10 12 14 16

-1

0

1

2

3

4

5 0

5

10

15

20

25

Vgate [V

]

B [T]

Rx

y [

k

]

QHE in a 3D TI system Rxy and Rxx from -1 V to 5 V

0 2 4 6 8 10 12 14 16

-1

0

1

2

3

4

5 0

10

20

30

40

Vgate [T

]

B [T]

Rx

x [

k

]

0 5 10 15

-20

-10

0

10

20

30

n=3.2*1011

cm-2

E, m

eV

B, T

0.00 0.05 0.10 0.15 0.20 0.25

-150

-100

-50

0

50

n=3.2*1011

cm-2

E, m

eV

k, nm-1

Band structure and Landau level at the symmetry point

H. Buhmann


0 5 10 15

-20

-10

0

10

20

30

n=3.2*1011

cm-2

E, m

eV

B, T

0.00 0.05 0.10 0.15 0.20 0.25

-150

-100

-50

0

50

n=3.2*1011

cm-2

E, m

eV

k, nm-1

Band structure and Landau level at the symmetry point

20 40 60 80 100

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

6.3

5.45

3.6

2

0

|(k

F)|

2

n, 10

11 c

m-2

z, nm

development of the top and bottom

surface state


Parameters devices A and B (similar design) HgTe: Thickness mesa: D 70 nm (structure etched 108 nm deep) Dimensions junction: L x W L x 2000 nm with LA 1000 nm LB,C 350 nm LD,E,F 150 nm

HgTe

E F

devices

L W

D C B A

Nb: Thickness: d 68 nm Width: w 500 nm Length: l 2000 nm (crossing the complete width of the HgTe mesa) Capping layer: 10 nm Al and 10 nm Ru

Nb Nb Nb Nb Nb Nb Nb Nb

G

H. Buhmann

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

0

20

40

60

80

100

120dV/dI(B)

B

dV

/dI

T 25 mK Iex 100 nA

Super Current in a 3D TI

-0.01 -0.005 0 0.005 0.01 0.015-10

0

10

20

30

40

50

60

70

80

90dV/dI(B) T = 25 mK

B

dV

/dI

T 25 mK Iex 21 nA

H. Buhmann

Summary

observation in 2D systems

– quantum spin Hall effect

• spin polarized quantized edge channel transport

observation in 3D systems

– quantum Hall effect of two Dirac surfaces

– proximity induced super conductivity

H. Buhmann

Acknowledgements

Quantum Transport Group (Würzburg, H. Buhmann)

C. Brüne

C. Ames

P. Leubner

L. Maier

M. Mühlbauer

C. Thienel

J. Mutterer

D. Knott

A. Astakhova

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

R. Schaller

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:

Penn. State University

CX. Liu


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)

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Documents

Hartmut Buhmann - meso12.itp.ac.rumeso12.itp.ac.ru/Meso12_Talk_Slides/Buhmann_talk.pdf · H. Buhmann C.L.Kane and E.J. Mele, Science 314, 1692 (2006) ... gap 2= 2 m e c 6~ 10 eV electron