users.lps.u-psud.fr/montambauxGilles Montambaux, Université Paris-Sud, Orsay, France

Quantum transport in 2D

GRAPHENE & CO, Cargèse April 2-13, 2018

Landauer-Büttiker formalismof quantum transport

Quantum transport : what is conductance?

Landauer-Büttiker : conductance = transmission

metallic ring atomic contact nanotube

2D gas graphene wire network

Landauer formula

2

2 eG Th

Landauer-Büttiker multiterminal formalism

R. Landauer (1927-1999)

M. Büttiker (1950-2013)

Conductance = transmission

1V

2V

1V 2V

4V

3V

M. Büttiker, Four terminal Phase-coherent conductor, PRL 1986

reservoircontactterminal

1V 2V

The electronic transport between the two reservoirs isa wave transmission through a potential barrier

lead

scatterer

Example : carbon nanotube

1 1eV 2 2eV

The 1D wire

4

reservoircontactterminal

1V 2V

Hypotheses :

• A terminal absorbs electrons and inject them at a given potential and a giventemperature.• No phase relation between incoming and outgoing electrons in eachterminal.• the scatterer is elastic.• The resistance of the reservoirs is negliglible.

lead

scatterer

A problem of 1D quantum mechanics…

1 1eV 2 2eV

The 1D wire

5

1V 2Vscatterer

Current carried by an electron in a state k : kevj TL

1 1eV 2 2eV

1 20

2 ( )[ ( ) ( )]eI T f eV f eV dh

Summation over all filled states :

(Remarkable result : the velocity has disappeared ! )

T () : transmission coefficient

The 1D wire

6

1V 2V

Linear regime :

1 1eV 2 2eV

2

0

2 ( )e fG T dh

Low temperature :

22 ( )FeG Th

2

1/(25812,807 )eh

Conductance quantum :

The 1D wirescatterer

(Remarkable result : the velocity has disappeared ! )7

1V 2V

22 ( )FeG Th

Landauer formula

No scatterer (infinite conductivity ?)

22eGh

The conductance is finite and quantized !!!

The 1D wirescatterer

8

1V 2V

Where is the potential drop ?

Where is power dissipated ?

How to measure the conductance of the scatterer itself ?

22eGh

The perfect conductor has a finite and quantized conductance !!!

9

No resistance in the sample

« contact » resistance

Power dissipated in the contacts

124

mV V hI e

cR

22

1 22 ( )eP V Vh

Potential profile

1V

2V

T = 1AV

x

BV

Ballistic

potential drop AT the contacts

22

2

1

2 I eG

V V h

4 A B

IGV V

Potential profile

1V

2V

T = 1AV

x

BV

Ballistic

potential drop AT the contacts

2

2 2 eGh

4G

H. Pothier et al., Energy distribution of electrons in an out-of-equilibrium metallic wire, Phys. B 103, 313-318 (1997)

« 2 terminal » resistance

vs « 4 terminal » resistance

sample reservoirsleads

I1V 2V

2

1 22 ( )eI V Vh

Perfect sample : VA=VB

AV BV

2

2 2 eGh

4G

4 vs 2 terminals

13

( )A BI V V

I1V 2V

22 1( )I G V V 4 ( )A BI G V V

With a scatterer VA=VB

scatterer

AV BV

2

2 2 eG Th

4 ???G

4 vs 2 terminals

14

1V

2V

T = 1AV

x

BV

T < 12V

AVBV

1V

Ballistic

One scatterer

potential drop AT the contacts

2

2 2 eGh

4G

2

4 21

e TGh T

2

2 2 eG Th

No dissipation in the wire

Potential profile

15

1 + R

T

1

1 I1V 2V

2

1 22 ( )eI T V Vh

2

2 ( )A Be TI V Vh R

2

2 2 eG Th

2

4 2 e TGh R

« 2 terminal » conductance

AV BV

4 vs 2 terminals

« 4 terminal » conductance

1 2( )A BV V R V V

1 2 1 2 A A B BV V V V V V V V

I I I I2R

2 c 4 cR R R R

The two-terminal resistance is the addition in seriesof the four-terminal resistance and the two contact resistances.

2

2 2 eG Th

2

4 2 e TGh R

T < 12V

AVBV

1V

17

here,d=2

1V 2V

1V 2V2

222

F

ee WGLhl

2 22 F

e WGh

Ohm-Drude

conductanceI G VConductance = transmission

AVBV

1V

2VAV

BV

2

2 2 / 2

eF

e

lL

k Weh l

G

2 c 4 cR R R R

Sharvin

L < le L > le

Sharvin

Drude-Ohm

λFπW

L

leλF2W

R =λF2W

+λFπW

L

le

contact resistance

in units h2e2DiffusiveBallistic

S. Tarucha et al., Sharvin resistance and its breakdown observed in long ballistic channelsPhys. Rev. B 47, 4064 (1993)

2

2 2 eG Th

2

4 2 e TGh R

Two-terminal conductance

Four-terminal conductance

V3 V4

Landauer-Büttiker formulae

1V

1V

2V

2V

non-invasive voltage probes

20

Four terminal resistance of a ballistic quantum wire (2001)

R. De Picciotto et al., Four terminal resistance of a ballistic quantum wire, Nature 411, 51 (2001)

« cleaved-edge overgrowth »

21

1V 2V

Source Drain

1V 2V

Source Drain

4G

2

BVAV

2

2

2 2 eGh

31

The 2-terminal resistance is quantizedThe 4-terminal resistance is 0

R4pt/R2pt

For non invasive contacts

3 4

3 4

R. De Picciotto et al., Four terminal resistance of a ballistic quantum wire, Nature 411, 51 (2001)23

?

=invasivity

ykW

2

ykW

2W W

The current is the sum of the contribution of the different channelsmodes

1V 2VW

abt

ab

Multichannel Landauer formula

24

sin yk y

Total current

2

1 22 ( )ab ab

eI T V Vh

Current resulting from the transmission of a channel b to a channel a

2

1 2,

2 ( )aba b

eI T V Vh

Multichannel Landauer formula

1V 2VW

abt

ab

2ab abT t

25

Multichannel Landauer formula

2ab abT t

Multichannel Landauer formula

1V 2VW

abt

ab

2

,

2ab

a b

eG Th

26

M is the number of transverse channels

22eG Mh

ab abT

Multichannel Landauer formula : clean wave guide

1V 2VW

ab abt

b b

2

,

2ab

a b

eG Th

ab abT

27

W

Gq = 2e2

hM = 2

e2

hInt2W

λF

wave guide

W

‘channels’‘modes’

transverseM

Conductance of a coherent ballistic system

Van Wees et al. PRL 1988; Wharam et al. J. Phys. C 1988

2e2

hper mode …

see tomorrow’s lectureon Landauer formula

wave guide

Quantum point contact QPC

W

Gq = 2e2

hM = 2

e2

hInt2W

λF

wave guide

W

‘channels’‘modes’

transverseM

Conductance of a coherent ballistic system (finite T)

wave guide

Quantum point contact QPC Van Wees et al. PRL 1988; Wharam et al. J. Phys. C 1988

2

,

2 ( )aba b

e fG T dh

22 ( )e fG M d

h

22 ( )nn

eG fh

( ) ( )nM

Characteristic energy :2 2

** 2 1

2K

m W

here :

Quantization of the conductance : temperature effect

250W nm

2 22

22

n n

mW

Landauer formula

2

2 eG Th

Landauer-Büttiker multiterminal formalism

R. Landauer (1927-1999)

M. Büttiker (1950-2013)

1V

2V

1V 2V

4V

3V

0

0

Current probesVoltage probes

I

12,343

2

4

1I

V VG

M. Büttiker, Four terminal Phase-coherent conductor, PRL 1986

2

2 2 eG Th

2

4 2 e TGh R

Two-terminal conductance

Four-terminal conductance

V3 V4

2

4 2 ?eGh

Landauer-Büttiker formulae

1V

1V

2V

2V

1V 2V

3V 4V

32How many coefficients to characterize the « black box » ?

22 ( )i i ii i ij jj i

eI M R V T Vh

1 11 12 13 142

21 2 22 23 24

31 32 3 33 34

41 42 43 4 44

2M R T T T

T M R T TeGT T M R ThT T T M R

Conductance matrix

I GV

Four terminals

M. Büttiker

1V

2V

4V

3V

0

0

339 transmission coefficients… 34

21 2 22 23 12

31 32 3 33 2

41 42 43 3

200

I T M R T Ve T T M R Vh

T T T V

1 11 12 13 14 12

21 2 22 23 24 2

31 32 3 33 34 3

41 42 43 4 44 4

200

M R T T T VIT M R T T VI eT T M R T VhT T T M R V

4 0V

31 42 32 413 22

T T T ThV Ie D

4V3V

2V1V

4 0V

12,343

2

4

1I

V VG

In general depends on 9 transmission coefficients…

2

31 42 32 414 2 e DG

h T T T T

2

2 2 eG Th

2

4 2 e TGh R

Two-terminal conductance

Four-terminal conductance

Landauer-Büttiker formulae

1V 2V

1V 2V

3V 4V

V3 V4

1V 2V

35

can be negative !

1 11 12 13 142

21 2 22 23 24

31 32 3 33 34

41 42 43 4 44

2M R T T T

T M R T TeGT T M R ThT T T M R

M. Büttiker

36

Time Reversal Symmetry

( ) ( )ij jiT B T B

ij jiT T0B

9 transmission coefficients…

1 11 12 13 142

21 2 22 23 24

31 32 3 33 34

41 42 43 4 44

2M R T T T

T M R T TeGT T M R ThT T T M R

In zero field, the 3 x 3 submatrix is symmetric

6 transmission coefficients…

2( 1)N

( 1)2N N

N terminals4 terminals

2

31 42 32 414 2 e DG

h T T T T

2

2 2 eG Th

2

4 2 e TGh R

Two-terminal conductance

Four-terminal conductance

Landauer-Büttiker formulae

1V 2V

1V 2V

3V 4V

V3 V4

1V 2V

38( ) ( )ij jiT B T B

In general depends on 9 transmission coefficients… ( 6 in zero field)

ij jiT T

Low T : the 4 terminal resistance can be negative

31 42 32 4112,34 22

T T T ThRe D

B. Gao et al., Four-point resistance of individual single-wall carbon nanotubes, Phys. Rev. Lett. 95, 196802 (2005)

21

3

4

4 terminal resistance in a carbon nanotube

Low T : the 4 terminal resistance can be negative, but power dissipate is positive

21

3

4

4 terminal resistance in a carbon nanotube

P = e2

2h

Xi,j

(Tij + Tji)(Vi − Vj)2

( ) ( )G B G B

-500 -500B (G)

( ) ( )T B T B

L. Angers et al., Phys. Rev. B 75, 115309 (2007)

Symmetry of the two-terminal conductance

41A. Benoit et al., Asymmetry in the magnetoconductanceof metal wires and loops, Phys. Rev. Lett. 57, 1765 (1986)

Symmetry of the four-terminal conductance ?

2

3434

1212,

31 42 32 41

2

e DGh T T T TV

I

B(T )

B(T )

14,23 23,14( ) ( )G B G B

B -B

Symmetry of the four-terminal conductance ?

A. Benoit et al., Asymmetry in the magnetoconductanceof metal wires and loops, Phys. Rev. Lett. 57, 1765 (1986)

1V 2V

3V 4V

2

31 42 32 414 2 e DG

h T T T T

Landauer-Büttiker formulae

31 42

32 412

342

12

(1 )

(1 )

T TT TTT

2

42( ) 2 eG

h

In de Picciotto experiment, 6 coefficients reduce to one

1V 2V

3V 4V

2

44

=invasivity

Appl. Phys. Lett. 50, 1289 (1987)

Phase coherence

Non- locality

RK=25 812, 807

Quantum Hall Effect

i=1

i=2i=3

V1 V2

V3 V5

V6V4

B

I2

1H

hRe M

0LR

2

1

0

H

L

hRe M

R

47

Quantum Hall effect

Bulk trajectories are pinned by disorderChiral edge trajectories propagate freely

Bulk insulatorPerfect « chiral » conductor at the edges

« Topological insulator »

B

Quantum Hall effect

Left-going and right-going electrons are spatially separated

B

Dissipation in the arrival terminal

Dissipation in the arrival terminal

Quantum Hall effect

Left-going and right-going electrons are spatially separated

B

Dissipation in the arrival terminal

Dissipation in the arrival terminal

Imaging of the dissipation in quantum Hall effect experimentsU. Klass et al., Z. Phys. B 82, 351 (1991)

This experiment shows that electrons stay at the chemical potential of theinjection reservoir and exchange their energy at the arrival reservoir

dissipation

B=0 B=0

Left-going and right-going electrons are spatially separated

Quantum Hall effect

50