Quantum conductance I.A. Shelykh St. Petersburg State Polytechnical University, St. Petersburg,...

Quantum conductance

I.A. Shelykh

St. Petersburg State Polytechnical University, St. Petersburg, Russia

International Center for Condensed Matter Physics, Brasilia, Brazil

ICCMP

Outline

• Overwiew of the classical results • Quantum Point Contacts• 1D Ballistic Conductance and Landauer

Buttiker formula• Quantum interference and Aharonov-Bohm

effect• Integer and Fractional Quantum Hall effect• “0.7 anomaly” and fractional quantization of

ballistic conductance

Classical results

2

2 /

s

s rel eff

e n D

e n m

W

GL

Ohm Law

L

W

Parallel

G1 G2

G-1=G1-1+G2

-1

Consequent

G1

G2

G=G1+G2

/I U R GU

Quantum Point Contacts

Let us consider a very small object (QPC or QWire)

L<<Lfree, W~kF-1

The condition L<<Lfree means that there is no inelastic scattering within the region of the QPC

Is G=∞ then?

Contact resistanceThe condition W~kF

-1 means that in the region of QPC the motion in x-direction is quantized

The origin of the resistance: redistribution of the current among the current-carrrrying modes at the interfaces

For parabolic confinement

μ

E0

Left lead

Right leadQPC

Ballistic conductance

I I I

1 1

2 2

1 00, ( ) 0, ( )

2 00, ( ) 0, ( )

k kk E k k E k

k kk E k k E k

e E eI ev n E

L k h

e E eI ev n E

L k h

R. Landauer. IBM J. Res. Dev., 1, 233 (1957)

2

ds

e eI V

h h

dseV

μ1 μ2

Vds=0Vds>0

I

T=0

2eG

h

22eG

h

Ballistic conductance staircase

22

g

eG N V

h

B. J. van Wees, Phys. Rev. Lett. 60, 848-850 (1988) D. A. Wharam et al, J. Phys. C 21 L209-L214 (1988) If there are N open subbands

WG

L

The role of backscattering

2 2

, 1

2 2N

iji j

e eG T N T

h h

Gi Gc

1 1 12 2

2

1

2 2

2

1

c i

i

h h TG G G

e e T

e TG

h T

Several scatterers

12 1 2

1 1 1 11 2c i i

T TT

G G G G

?

1 1 1 112 1 21 2

12 1 2

1 1 1c i i

T T TG G G G

T T T

2 1 212 1 2 1 2 1 2 1 2 1 2

1 2

...1

TTT TT TT R R TT R R

R R

Effects of quantum interference

12 1 2

12 1 2

1 1 1 11 2

1 1 1

c i iG G G G

T T T

T T T

2 1 212 1 2 1 2

12

1 21F

F

ik L

jik L

t tt t t r e

r er

r

1 22

1 2

1

1 21 (1 )(1 ) 2 (1 )(1 ) cos(2 )F

TTT

T T T T k L

Quantum interference term

Fabry-Perot oscillations of quantum conductance

N.T. Bagraev et al, Semiconductors, 34, 817 (2000)

L<<Lφ

Parallel connection1

1iA e

1 21 2

1 1 2 1 22 2 cos

i iA A e A e

G G G G G

1ANo interference:

N=N1+N2, G=G1+G2

2A 22

iA e

To account for the round trips: scattering matrix

With interference

out inSA = AS=

Aharonov-Bohm effect

1,2 2F

ek R

c

How one can easily change phaseshift between

the electrons propagating in the quantum ring?

Possible way: apply a magnetic flux through the ring Φ

Weak backscattering: AB oscillations

Strong backscattering: AAS half-period oscillations

Classical Hall effect

x xx xy x

y yx yy y

E J

E J

dd

scatt

me

vE v B

s denJ v

UH

y

x

scatt field

d d

dt dt

p p

2/

/

xx yy eff s rel

xy yx s

m e n

B en

zB

Experimental configuration1 2

2 3

0

,

x

H

y

xxx

x xx x y y

Hx

x x

y

V V V

V V V

J

E J

V

V

E

W

J

I L

I

I W

LV1 V2

V3

ρxy

ρxx

B

1

/

1

yxs

H

ss xx s x

d Ien e

dB dV dB

ILe

en n V W

Landau quantization, 0

, , 02 2

x y z

x y z

A By A A

By ByA A A

2

1

2

eH

m c

P A

DOS

20

22

22

/ 2, 0

0

1( , ) ( , )

2

1( , )

1( ) ( )

2

,

1( , )

/ , / /

1

2

x

x

x

x y

ik x

yx

c

q

n c

qik xn k n

c x c

eByP P x y E x y

m c

x y e yL

eByk P y E y

m c

eB

mc

x y e e H q qL

q m y q

E n

k c eB m y

Group velocity

10n

gx

Ev

k

Edge states

2 2

22022 2 c

d my y V y f Ef

m dy

x

y

vg≠0

Ballistic conductance and QHE

2 /1 2Backscattering cW l

scattk V k e

I+

I-

Δμ=eVH

μBackscattering is supressed

2

H

eG N

h

Quantum Hall Effect (QHE)

HH

U BR

I ne

K. v. Klitzing, G. Dorda, and M. Pepper Phys. Rev. Lett. 45, 494-497 (1980)

Classical resultIn the experiment

2H

hR

Ne

Fractional QHE

D.C. Tsui et al, PRL 48, 1559 (1982) H.L. Stormer et al, PRL 50, 1953 (1982)

2

2

2

2

3

2

5

2

3

4

5

H

H

H

H

eG

h

eG

h

eG

h

eG

h

Interpretation of FQHELaughlin wavefunction

Composite fermions

Fractional quantization of the ballistic conductance (« 0.7 anomaly »

K.J. Thomas et al, PRL 77, 135 (1996)

Related with spin!

Singlet and Triplet Scattering

3s dir exU V V

tr dir exU V V

0G

V.V. Flambaum, M.Yu. Kuchiev, PRB 62, R7869 (2000)

T. Rejec et al, J. Phys. Cond. Matt. 12, L233 (2000)

int dir ex e SH V V σ σ

Localised and propagating electrons interact in the region of the QPC

Eigenstates: singlet and triplet configurations. The probabilities of realization:

s

1

43s dir ex

P

E V V

F dir exE V V

3

4t

t dir ex

P

E V V


3s dir exU V V

tr dir exU V V

2

0

3 3

4 2

eG G

h






3dir ex F dir exV V E V V

s

1

43s dir ex

P

E V V

3

4t

t dir ex

P

E V V


3s dir exU V V

tr dir exU V V

22eG

h






3F dir exE V V

s

1

43s dir ex

P

E V V

3

4t

t dir ex

P

E V V

0.75 structure: calculation

22 2 2

0

2 2 2 2 2

2k

S e S e S S e S e S e S e S

k kS e S e S S e S e S e S e S S e S e S S e S e S k k

neG P A k P A k A k

m

n nP A k P A k A k P A k P A k n n

kn

k dk

Consider the case ik xe sin

e

1

1 1 0

0 0 1ik x ik x ik x

e S e S e S e Se B e B e

1 1 2 22

1 1 1 1

1 1 1 1ik x ik x ik x ik xC e D e G e F e

3

1 0

0 1ik x ik x

e S e S e S e SA e A e

1 2

2F B dir ex

mk E g B V V

2 2

23F B dir ex

mk E g B V V

0

3

4G G

Is fractional ballistic conductance universal?

0

3

4G G

D.J. Reilly et al, PRB 63, R121311 (2001)

For short constriction

0

1

2G G

For long wire

?

Supposing the contact containing a total spin J :

QPC with Large SpinI.A. Shelykh et al, PRB 74, 085322 (2005)

Fractional quantization: calculation

2 2 2 22 2

0 1 1 1 1 1 1 1 1; 1 ; 1 ; 1 ; ; ; ; ; 10 2 2 2 2 2 2 2 2

( )4 2 1

J

TJ m J m J m J m J m J m J m J mm

eG E A A A A

h J

2 2

2 2

, 0,2

, 0,2

eff

dir exeff

kx x L

mH

kV V x L

m

σ J

The Hamiltonian Using the following basis

1 2 2 1 4 2

1 1 1 1; ; ; ( 1) ; ; , ;

2 2 2 2... ...m m JJ J m J m J

One represents H in a block-diagonal form

(1) (2), 1 1;lk l lk l l k l kH V V

Diagonalised Hamiltonian reads

0

0

, ,( , ) ( )T

f T EG T G E dE

E

With increase of the length of the wire J increases and conductance decreases- as in experiment!

Spontaneous polarization of 1D electron gas

Chuan-Kui Wang, K.-F. Berggren PRB 57, 4552 (1998)N.T. Bagraev et al PRB 70, 155315 (2004)

Why big J can appear in long quantum wires?

Due to exchange interaction!

3 2;kin exn n

2 21

30.28

8 4D

Ce e

Qualitatively in 1D

Dominant for high density

Dominant for low

density

Calculation gives:

2 2 3 2 21 1 1 1

12ln

12 4D D D D

Ds s s

n n n Re

mg g g

2 for unpolarized

1 for polarized

Critical density

2 2 20 0

1

3 2ln

24 4D

n n Re

m

What happens with holes?Light and Heavy Hole Bands in a QPC

Bands splitted in energy depending on the width of the QPC:

Si / GaAs / Ge

Spin Dependent Scattering for Holes

Initial state:

Conductance at T = 0 (44 transmission amplitudes):

Model:

Matrix form (16x16):

where

Physical Origin of the Plateaus

States presenting total spin ST = 3: 7 states; ST = 1: 3 states; ST = 2: 5 states; ST = 0: 1 states.

Potential Barriers

Ferromagnetic Interaction

Steps at:

Antiferromagnetic Interaction

Steps at:

Ferromagnetic

Si

Antiferromagnetic

SiAntiferromagnetic

Applying an Axial Magnetic Field

Si

Ferromagnetic

Experiment for the holes

L.P. Rokhinson et al, 2006N.T. Bagraev et al 2002 Klochan et al, 2006

????

Thank you for your attention

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Quantum conductance I.A. Shelykh St. Petersburg State Polytechnical University, St. Petersburg,...

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