users.lps.u-psud.fr/montambaux

Disorder and mesoscopic physics

Gilles Montambaux, Université Paris-Sud, Orsay, France

Lecture 3

Weak localizationCoherent backscattering in optics

2

( , ')P r r

conductance ~ transmission ~ probability

1

( ) 1dF D

DF

gvpV

classical diffuson quantum corrections

quantum crossing 1/g correction

2

2

classical transport

quantum effects

egh

eh

Summary of previous lecture

int ( )P t 3

Quantum correction

Classical conductance

Time reversed trajectories

clG

One crossing One loop

Crossing

= distribution of number of loops with time t = return probabililty

Weak localization

int

22 ( )e P th

G

(0, )clP L

(0, )P L

4

Weak-Localization

Nb loops and return probabilityMagnetic field, phase coherenceWeak-localisation in dimension d

A few solutions of the diffusion equation and WLMagnetic field and negative magnetoresistanceMagnetic field in quasi-1D wiresAAS oscillations

int

22 ( )e P th

G

int int( ) ( , , ) dP t P r r t d r

int ( )P t

Classical return probability

Diffuson

Interference term

Cooperon =

If time reversal invariance

int ( , , ) ( , , )clP r r t P r r t

Weak localization : how to calculate ?

int

22 ( )e P th

G

int ( )P t ( )clP t

6

Important difference :

( , ', )clP r r t

int ( , ', )P r r t

*jA *T

jA

jA TjA .p dl

If time reversal invarianceint ( , , ) ( , , )clP r r t P r r t

paired trajectories follow the same direction

paired trajectories follow opposite directions

have the same phase

jA jADiffuson Cooperon

If phase coherence between the reversed trajectories is preserved

7

The return probability P(t) increases for small d

Coherent effets are more important in low dimension

/int 2( ) ( )(4 )cl d

dLP t P tDt

Weak localization

int int

2 2

( ) ( )2 2e D

e eh

P t dtPh

G t

phase coherence time

elastic collision timeDt

efor

time spent in the sample2

DLD

2LD

volume explored after time t

8

2

0i

/ /nt4 ( ) et t

D

e dtG P t e eh

2 2

int int( ) (2 )2e D

e e dtGh h

P t P t

Qualitative result

correct result

Long trajectories are cut because of loss of phase coherence beyond

Measurement of this quantum correction gives access to the coherence length

Weak localization correction : exact result

9

2

/ /

0int4 ( ) et

D

te dtG P t e eh

Macroscopic limit L L D /2( )(4 )d

VP tDt

/ 2e

d

dtt

1 1

e

lne

e 1 ( 1 )d quasi D

2d

3d

Weak localization : dependence on dimensionality

10

Mesoscopic system L L D

1 ( 1 )

2

3

d quasi D

d

d

2

2

2

( )

( )ln

2

e

e

L TeG sh L

L TeG sh le LG s

h l

2

2

2

ln

2

e

e

eG she LG sh le LG s

h l

1 ( 1 )

2

3

d quasi D

d

d

Correction more important for small dbecause return probability is enhanced

Weak localization : dependence on dimensionality

11

1

2

3

d

d

d

( )

( )1 ln

12

e

e

L Tg

LL T

gl

Lgl

Correction more important for small dbecause return probability is enhanced

Weak localization : dependence on dimensionality

11( )

(2 )d

dd

Feg k WA lL

22

3

e

F e

F e

lg ML

k lg

k l Lg

12

1

2

3

d

d

d

( )

( )1 ln

12

e

e

L Tg

LL T

gl

Lgl

Weak localization : dependence on dimensionality

11( )

(2 )d

dd

Feg k WA lL

22

3

e

F e

F e

lg ML

k lg

k l Lg

2 2

ln( / )2

1

e

e

F e

F e

Lgg M l

L lgg k lgg k l

1gg

1

22

F e

D e

k l

D e

M l

l e

defines a new length scale at which perturbation breaks down

Localization length :

13

M.E. Gershenson et al.

( )L Tg

L e

Lgg M l

1 ( 1 )d quasi D

elg ML

20el nm10M

Localization length eM l

L

14

In a magnetic field, dephasing between time reversed trajectoriesThe cooperon oscillates with flux

It cancels in a magnetic field

Diffuson Cooperon

Cooperon: in a magnetic flux, paired trajectories get opposite phases

0

2

0

2

0

2

0

4

phase difference

Phase coherence and magnetic field

0

2 2he

Oscillations of period

( )clP t int ( )P t

Sharvin,Sharvin

15

04 )

i t

(

n ( ) ( )c

i

l

t

P t P t e

2(( ) )R tt B BDt / Bte

Trajectories which enclose more than one flux quantum

do not contribute to int ( )P t

0( )t 0( )t 0BBD

Effect of magnetic field (qualitative)

16

Diffuson(classical)

Cooperon(quantum)

²

Weak-localization = phase coherence

τ ττ

t− 2τ( )clP t int ( )P tLoop of time t

17

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

τ 6= t

2

t

τ =t

2« Suppression » « Revival »

Weak-localization = phase coherence

18Magnetic impurities, e-e interaction, magnetic impuritiesAltshuler, Aronov, Khmelnitskii

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

/int ( ) ( )cl

tP t P t e 04i

e

²

Loop of time t

τ ττ

2t− τ( )clP t int ( )P t

Weak-localization = phase coherence

19

Diffusion equation for Pint(r,r,t) ?

int

2

( , ', ) ( ) )2 (1 'D i P r r t r r tAet

Phase coherence time Vector potential

Effective charge

'r r

( , ', ) ( ') ( )clD P r r t r r tt

20

( ) nt

n

EP t e where are the eigenvalues of the diffusion equationnE

Example : uniform magnetic field in 2D

1 42n

eBDE n

0

0

/( )sinh 4 /

BSP tBDt

Solving diffusion equation

2

2 n nnei A ED

0B

B

( )4

SP tDt

0

4

( )BDt

P t e

21

weak localization in 2 D, negative magnetoresistance

weak localization in a quasi-1d wire

weak localization in a ring

weak localization in a cylinder

Four examples

/( )e

t

cl

e dG

P ttG

22

lne

LG

l

Weak localization correction is suppressed when

0

0

/( )sinh 4 /

BSP tBDt

min ,ln B

e

L LG

l 2

0BBL

( )4

SP tDt

In a magnetic field :

2* 0B L

BL L

Example 1: weak localization in 2 D

( ) 1/L T T

/( )e

t

cl

e dG

P ttG

R

B

Bergmann, 84

*

BfB

23

02

0 0

/ //sinh 4 /

4 et

D

te dtGDt

BB

e eh

2 1 122 2 4 2 4eBeG

h e D e DB

Example 1: weak localization in 2 D

R

B

Bergmann, 84

24

2

00

ln 2 co 4sme m

Le LG s Kl

mh L

m

Cylinder in a Aharonov-Bohm flux :

Example 4 : weak localization in a cylinder

2 2 /4/

0

( ) cos 44

m L Dtt

m

eP t m eDt

Altshuler, Aronov, Spivak, ‘81

/( )e

t

cl

e dG

P ttG

Sharvin,Sharvin, ‘81

2D diffusion winding of trajectories

Altshuler, Aronov, Spivak, ‘81

LmLe

“Sample specific” interference

Oscillations of period

After disorder averaging, only remainsThe contribution of paired trajectories

0 /h e

Phase difference between two trajectories0

2

0

4

Anneau unique (Webb)

cylindre ou moyenne sur différents anneaux (Sharvin, Sharvin)

Oscillations « Aharonov-Bohm »

… which disappear in average

Phase difference

Oscillations of period 0 / 2 / 2h e 26

?

27 28

i0

n/ /

t2 ( ) et tB

D

dtg P t e e

min , , )(

0

/2( )4

D B

d

D

D dtgt

Contributions of closed diffusion trajectories whose size is limited bySize of the system, phase coherence, magnetic field, etc.

, , )min(c D B

2d

( )cL TgL

( )1 ln c

e

L Tgl

1 ( 1 )d quasi D

c cL D

Summary

Coherent backscattering

G. Maret, constance

Multiple scattering in optics 30

Albedo : reflexion coefficient of a scattering medium

Turbid media: colloids, milk, powders, chalk, teflon, etc…

Example : TiO2 powder

Multiple scattering of light by impurities

31

G. Maret, Constance

There is a mechanism which survives disorder average

one disorderconfiguration

Average

« Coherent backscattering »

angular dependence

In average, the intensity is uniform, except near thebackscattering direction

speckle

Angular dependence of the reflected intensity

Coherent backscattering « cone »

The intensity is doubled in the backscattering direction

Triangular singularityZnO powder, D. Wiersma 1995

Teflon

( 100 mrad ~ 6° )

Angular dependence of the reflected intensity

Also teflon, TiO2, chalk, etc…

Et aussi diffusion multiple de la lumière par un gaz d’atomes froids, par les anneaux de Saturne !

Saturn rings,Cassini 2006, NASA

multiple scattering of microwaves, acoustic waves,sismic waves, etc.

An ubiquitous phenomenon

et aussi téflon, TiO2, craie, etc…

Et aussi diffusion multiple de la lumière par un gaz d’atomes froids, par les anneaux de Saturne !

Anneaux de saturne,

Cassini 2006, NASA

diffusion multiple d’autres ondes, e. m. , acoustiques, sismiques

CBS of light by cold atoms

CBS of cold atoms by light

The measured CBS cone probes the internal degrees of freedom of the atoms

Time evolution of the momentum distribution

c.f. seminar Vincent Josse

Classical diffusion Coherent contribution

Reversed trajectories

dephasing

ik

ek

R

Intensité moyenne

The coherent contribution is maximal if ke=-ki

ik

ek

( ).i ei k k Re

0 1 cos( ).i eR

I I k k R

Angular dependence of the reflected intensity

( ). ( )i ei k k R Re P R d distribution of distances

between the first and the last scatterer( )P R

R

Coherent backscattering « Cone »

Douling of the intensity in the backscattering direction

Triangular singularity

r(θ) =1

(1 + k le |θ|)2

∆θ ∝ λ

leMeasure of the mean free pathfrom the width of the cone

37 38

'rr

Reversed trajectories

Reversed trajectories

Localisation faible

Rétrodiffusion cohérente

Where is the quantum crossing ?

Intensité moyenne Where is the quantum crossing ?

WL

CBS

( ).i ei k k R

Re

,

( ).

i ek k

i ei k k Re

39

'rr'R r r

Rik ek

ik

ek

2/sin

eF

F

R lk R ek R

( , )P R t

R

40

Localisation faible

Rétrodiffusion cohérente

Intensité moyenne Where is the quantum crossing ?

WL

CBS

-100 0 100 200 300

0.8

1.2

1.6

2.0

Scal

ed I

nten

sity

Angle (mrad)

-5 0 51.8

1.9

2.0

1/ g

1/ g

41

Correlations ?

clG G G

GG

42

Fluctuations universelles de conductance

Conductance versus an external parameter (magnetic field)

Au Si numerics( )G B

Lee,Stone, Fukuyama, Universal conductance fluctuations in metals, Phys. Rev. 35, 1039 (1987)

Reproducible fluctuations

Universal conductance fluctuations

The amplitude is universal222 2 eG G G G

h

L L2 25800h

e

In a good metal, G G

If quantum coherence :

Lee,Stone, Fukuyama, Universal conductance fluctuations in metals, Phys. Rev. 35, 1039 (1987)

Universal conductance fluctuations

44

clG G G Average

2GVariance

The WL correction G is suppressed by the magnetic field

In a field, the variance is reduced by a factor 2

G

G

2G

B

D. Mailly, M. Sanquer, J. Physique I, 2, 357 (1992)

Magneto-fingerprints = empreintes digitales magnétiques

each trace represents an interference pattern

unités e2/h

Magneto-fingerprints

a b

Speckle pattern : optical intensityemerging from a disordered medium

Conductance - Transmission

Quantum transport in disordered systems

Analogies with optics

Quantum corrections to classical transport ~ interference effects in optics

G G

Conductance fluctuations

ab abT T

http://leeferg.com/the-green-flash/48