Mesoscopic Physics for Beginners Gilles Montambaux Laboratoire de Physique des Solides Université Paris-Sud, Université Paris-Saclay Orsay

µεσος

GDR Physique Quantique Mésoscopique, Aussois, déc. 2015

Mesoscopic physics = Phase coherence

Breakdown of classical laws of electronic transport

1 2R R R= +

LRS

ρ=

1 2G G G= +

SGL

σ=

1R 2R

2G1G

cf. Two path interferometer…

1GR

=

Phase coherence

dimensionality disorder

interactions

The mesoscopic triangle

H =p2

2m+ V (~r) H = ¹hc~¾:~p + V (~r)

Ã(~r) fÃA(~r); ÃB(~r)g

4

Domain of mesoscopic physics, deviations to classical transport Length scales, different regimes Conduction = transmission Landauer-Buttiker Quantization of conductance Universal conductance fluctuations Weak-localization What limits phase coherence ?

5

e F el v τ=

Lφ

Mean free path : distance between elastic collisions

Phase coherence length

el

( )L Tφ

Interaction with an external degree of freedom (phonons, electrons, spin impurities… breaks phase coherence

interference

L Dφ ϕτ=

Elastic collisions do not break phase coherence

F elλ

6

Ohm’s law I GV= SGL

σ=

G conductance, σ conductivity

2ene

mτσ = Drude-Sommerfeld formula

Validity ? Diffusive regime No quantum effects

L Lφ>

elL

7

int10

22cosI I I I πφφ

= ++

R. Webb (IBM, 1985) THE founding experiment of mesoscopic physics

1µm

1I

2II

Classical physics Ohm’s law :

2G1G

φ

8

int10

22cosI I I I φπφ

= ++

R. Webb (IBM, 1985) THE founding experiment of mesoscopic physics

Interferences between electronic waves (cf. Young’s slits)

1µm

1I

2II

Aharonov-Bohm effect (1959) el L Lφ<

0he

φ =

2eG Gh

δ

φ

9

int10

22cosI I I I φπφ

= ++

R. Webb (IBM, 1985) THE founding experiment of mesoscopic physics

Interferences between electronic waves (cf. Young’s slits)

1µm

1I

2II

Aharonov-Bohm effect (1959) el L Lφ<

0he

φ =φ

2eG Gh

δ

10

Reproducibles conductance fluctuations el L Lφ<

IGV

= ,ij

ij klkl

IG

V=

Exp. measures a conductance and not a conductivity Drude-Einstein conductivity provides an average description, valid if How to go beyond this average and describe interferences, fluctuations ?

L Lφ <

B(T )

2eGh

δ

Conductance depends not only on the system to be studied, but also on its connection to the outside world

11

Is the conductance of this Au atomic contact in any way related to the conductivity of gold ? NO new concepts, new tools

What is conductance?

SGL

σ=

12

Quantization of the conductance (1988)

W

W

G W∝

Classically

ballistic « Quantum Point Contact » QPC

13

2 2 22 2 int[ ]F

e e WG Mh h λ

= =

2eh

Quantum of conductance

W

W

Quantization of the conductance (1988)

« Quantum Point Contact » QPC ballistic

At which scale do we need new concepts ?

14

macroscopic 1nm 10 1000nm− 1 mµ

nanoscopic

Lφel

Mean free path : distance between elastic collisions

Phase coherence length

mesoscopic

ballistic diffusive

el

( )L Tφ

15

What is conductance?

Landauer-Büttiker : conductance = transmission

metallic ring atomic contact nanotube

2D gas graphene wire network

16

a b

Analogies electronics - optics

Aharonov-Bohm oscillations Young’s slits UCF Speckle Weak-localization Coherent backscattering

Conductance – transmission coefficient

… electron quantum optics…

17

1D wire

Reservoir Contact Terminal

1V 2V

Hypothesis : coherent transport in the wire, dissipation in the reservoirs

Lead

scatterer

Problem of 1D quantum mechanics

T

1 2( )I G V V= −

I

•A reservoir absorbs electrons and emits them at its own chemical potential and temperature. • No phase relation between ingoing and outcoming electrons in a reservoir. •The scatterer is elastic. •The resistance of the reservoirs is negligible.

18

1D wire

1V 2V

22eG Th

=Landauer formula

Without scatterer 22eGh

=

2

1/(25812,807 )eh

= Ω

Conductance quantum

T

1 2( )I G V V= −

I

19

1D wire

1V 2V

Very qualitatively…

T

I =charge

time= e

energy

h= e

e¢V

h

1 2( )I G V V= −

I

I =e2

h¢V

22eGh

=

time / 1

T

22eG Th

=

)

)

20

The «multichannel » case

Total current

2

1 22 ( )ab ab

eI T V Vh

= −

Courant related to the transmission from a channel ‘b’ to a channel ‘a’

2

1 2,

2 ( )aba b

eI T V Vh

= −∑

1V 2V

abT

a b

2

,

2ab

a b

eG Th

= ∑

multichannel Landauer formula

21

The «multichannel » case

2

,

2ab

a b

eG Th

= ∑

1V 2V

abT

a b

22

2 2

2 2 int[ ]/ 2F

e e WG Mh h λ

= =

The conductance is proportional to the number of modes transmitted through the wave guide

The «multichannel » case

2

,

2ab

a b

eG Th

= ∑

1V 2V

abT

ab abT δ=

23

Quantization of the conductance (1988)

« Quantum Point Contact »

2 2

2 2 int[ ]/ 2F

e e WG Mh h λ

= =

24

4 vs. 2 terminals

1V 2V

2

1 22 ( )eI V Vh

= − ( ) ( )A BI V V −∞=

AV BV

for perfect sample, VA=VB

2

2 2 eGh

= 4G = ∞

no potential drop in the wire :

I

25

4 vs. 2 terminals

1V 2V

22 1( )I G V V= −4 ( )A BI G V V= −

AV BV

with a scatterer VA=VB

scatterer

2

2 2 eG Th

=

I

2

4 21

e TGh T

=−

Potential profile ε

1V

2V

T = 1 AV

x

BV

T < 1 2V

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

AVBV

1V

2VAV

BV

1V

2VAV

BV

Diffusive regime

Four terminal disorder + interferences

2 1

A

B

ENS,Paris

Landauer formula

Conductance = Transmission

2

2 eG Th

=

Landauer-Büttiker formalism

R. Landauer (1927-1999)

M. Büttiker (1950-2013)

29

conductance = transmission

ak

bk

2

,

2ab

a bG Te

h= ∑

b a

analogy with optics

abT

optics - microwaves electronics

b a

ab abT Tδ =G Gδ

b a

fluctuations ~ average

2eGh

δ

fluctuations << average ??? Mailly,Sanquer

Maret

30

conductance = transmission

ak

bk

2

,

2ab

a bG Te

h= ∑

b a

analogy with optics

abT

optics - microwaves electronics b a

In optics, you can mesure Tab , Ta , or T

a abb

T T= ∑,

aba b

T T= ∑abT

In electronics, you can only mesure T

,ab

a bT T= ∑

The fluctuations of are much smaller than the fluctuations of ,

aba b

T T= ∑ abT

31

Weak-localization = first quantum correction to classical transport

Phase coherence effect The negative correction is cancelled by a magnetic field B Negative magnetoresistance The characteristic field depends on temperature measures the phase coherence

magnetoresistance of a Mg film Bergmann

G = Gcl + ±G(B)

2 *'

, '( , ') ( , ') ( , ')j j j

j j jG A r r A r r A r r∝ + ∑ ∑

32

Conductance = Transmission

( , ')jj

A r r∑2

( , ')jj

G A r r∝ ∑r r’

j

j’

21 2I A A= +

2 2 * *1 2 1 2 2 1I A A A A A A= + + +

1

2 S

2 *'

, '( , ') ( , ') ( , ')j j j

j j jG A r r A r r A r r∝ + ∑ ∑

33

Classical transport: only paired identical trajectoires Aj Aj contribute If paired trajectories are different, the amplitudes Aj et Aj’ are different

Classical term Interference term

Quantum effects

Disorder average

Conductance = Transmission

???

( , ')jj

A r r∑2

( , ')jj

G A r r∝ ∑

phases are uncorrelated all interference terms disappear in average

r r’

j

j’

34

Quantum correction

Classical conductance

clG

One loop and one crossing

Weak-localization

(0, )clP L∝

G∆

Z(¡~k)(¡d~l) =

Z~kd~l

35

int ( )P t

Quantum correction

Classical conductance

Opposite paired trajetories

clG

One loop and one crossing

crossing

= distribution of loops of time t = return probability

Weak-localization

(0, )clP L∝

(0, )P L∝ ∆G∆

i

2

nt ( )2eh

G P t ∆ −

36

The return probability P(t) is larger at small d Phase coherence effects are more important in low dimension

/2int ( )(4 )

d

d

LP tDtπ

=

2 2

in

,

t int2 ( ) ( )2

e

D

D

e eh h

P dtG t P tφτ τ

τ τ∆ − − = ∫

Time spent in the sample Phase coherence time Elastic collision time eτ

2

DLD

τ =

2LD

φφτ =

volume explored after time t

Weak-localization = return probability

37

/2e

d

dtt

Gφτ

τ

∆ ∝ ∫

/2e

d

dtt

φτ

τ∫

lne

φττ

eφτ τ− 1d =

2d =

L Dφ φτ=

The measure of this quantum correction gives access to the phase coherence time (length)

Weak-localization : importance of dimensionality

Lφ

ln Lφ

38

Oscillation of the WL correction with the flux (cf. oscillations Sharvin-Sharvin)

Diffuson Cooperon

Cooperon: in a field, time reversed trajectories acquire opposite phases

φ φ0

2 φπφ

0

2 φπφ

0

2 φπφ

−

0

4 φπφ

Phase difference 0

2 2he

φ= oscillations with period

int ( ) ( )clP t P t= 04i

eπ φ

φ

Weak-localization = phase coherence and magnetic field

( )clP t int ( )P t

39

Negative magnetoresistance Bergmann

Diffuson Cooperon

Cooperon: in a field, time reversed trajectories acquire opposite phases

φ φ0

2 φπφ

0

2 φπφ

0

2 φπφ

−

0

4 φπφ

Phase difference 0

2 2he

φ= oscillations with period

int ( ) ( )clP t P t= 04i

eπ φ

φ

Weak-localization = phase coherence and magnetic field

( )clP t int ( )P t

40

Diffuson (classical)

Cooperon (quantum)

²

Weak-localization = phase coherence

¿ ¿¿

t¡ 2¿( )clP t int ( )P tLoop of time t

41

Suppression and revival of WL through control of time-reversal symmetry Vincent Josse et al. , Institut d’Optique, PRL 2015

¿ 6= t

2

t

¿ =t

2« Suppression » « Revival »

42 Magnetic impurities, e-e interaction, magnetic impurities

Diffuson (classical)

Cooperon (quantum)

Phase coherence broken after a typical time Only trajectories of time contribute to the return probablity and to the WL

t φτ<φτ

/int ( ) ( )cl

tP t P t e φτ−= 04i

eπ φ

φ

²

Weak-localization = phase coherence

Loop of time t

¿ ¿¿

2t¡ ¿( )clP t int ( )P t

43

( )tϕ+

02

( )i

i te eφπφ ϕ

Random dephasing depends on the position of atoms, other electrons, magnetic impurities,…

0

2 ( )tφπ ϕφ

+

0

2 ( )tφπ ϕφ

− +

0

4 ( ) ( )t tφπ ϕ ϕφ

+ −

04 ( )i i t

eφπ ϕφ

+ ∆

21 ( )( ) 2 /i t tte ee φ

ϕϕ τ− ∆ −∆

Dephasing :

Average on the trajectories and on the dynamics of external degrees of freedom

Dephasing

44

0.1

1

10

100

0.001 0.01 0.1 1 10 100 T (K)

L φ ( µ

m)

3Tφτ −∝e-ph interaction

2/3Tφτ −∝

3012

3014

3016

3018

3020

3022

3024

-200 -150 -100 -50 0 50 100 150 200

R +

offs

et (O

hms)

B (G)

30mK

60mK

2000mK

470mK

10

( ) d

BWLG B f φδ

φ

=

Magnetotransport gives access to the phase coherence length

magnetic impurities

( )L Tφ

2

20

( ) d

BLG B f φδ

φ

=

2/3 31( )

AT B TTφτ

= +

e-phonon e-e

e-e interaction

quasi-1D wires

Grenoble

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