1

Cours -TD 2

Localisation faibleBoucles et croisements quantiques Nombre de boucles et probabilité de retour à l’origineChamp magnétique, cohérence de phaseLocalisation faible en dimension d

Quelques solutions de l’équation de diffusion et localisation faibleChamp magnétique et magnétorésistance négativeChamp magnétique dans des fils quasi-1Doscillations AAS

Transport quantique dans les systèmes désordonnés

Gilles Montambaux

2

( , ')clP r r

Summary of lecture 1

conductance ~ transmission ~ probability

1

( ) 1dF D

DF

gvpV

ττ λ −

× ∼ ∼

classical diffuson quantum corrections

quantum crossing 1/g correction

2

2

classical transport

quantum effects

egh

eh

∝

∝

3

Weak localization

int ( )P t

Quantum correction

Classical conductance

Time reversed trajectories

clG

One crossing One loop

Crossing

int

1

( )e

F

cl

dF v P tV

GG

dtτ

λ −Δ − ∫∼

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

4

(-) sign

k 'k

k 'kel

Fλ'k k−∼

5

Weak localization

int ( )1ecl D

P tG dtgG τ τ

− Δ

∫∼

int

1

( )e

F

cl

dF v P tV

GG

dtτ

λ −Δ − ∫∼

int ( )P t ?

6

How to calculate ? 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=

7

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

8

In a magnetic field, dephasing between time reversed trajectoriesThe cooperon vanishes at large field

Phase coherence magnetic flux

Diffuson Cooperon

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

φ φ0

2 φπφ

0

2 φπφ

0

2 φπφ

−

0

4 φπφ

phase difference

9

04 )

i t

(

n ( ) ( )c

i

l

t

P t P t eφπφ=

2(( ) )R tt B BDtφ ∼ ∼ / Bte τ−∼

Trajectories which enclose more than one flux quantum do not contribute to int ( )P t

0( )tφ φ<

0( )tφ φ> 0BBDτ φ=

Uniform magnetic field (qualitative)

10

magnetic impuritieselectron-phonon, electron-electron interactions

Phase coherence

Diffuson Cooperon

If some process breaks Phase Coherence, only trajectories with contribute

t φτ<

** * *

/int ( ) ( )cl

tP t P t e φτ−= 0

(4 )i t

eπ φ

φ

11

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

int

2

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

δ δτ

⎡ ⎤⎛ ⎞∂⎢ ⎥+ − ∇+ = −⎜ ⎟⎜ ⎟∂⎢ ⎥⎝ ⎠⎣ ⎦

Phase coherence time Vector potential

Effective charge

'r r=

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

δ δ∂⎛ ⎞− Δ = −⎜ ⎟∂⎝ ⎠

12

( )2

/t

/

0in2 ( ) et

D

te dtG s P t e eh

φτ τ

τ

∞− −= −Δ −∫

Weak localization correction

int ( )1ecl D

P tG dtgG τ τ

− Δ

∫∼

Exact result :

13

( )2

/t

/

0in2 ( ) et

D

te dtG s P t e eh

φτ τ

τ

∞− −= −Δ −∫

Macroscopic system L Lφ D φτ τ/ 2

( )4

dDP tt

τπ

⎛ ⎞= ⎜ ⎟⎝ ⎠

/ 2e

d

dtt

φτ

τ∫

1 1

eφτ τ−

lne

φττ

eφτ τ− 1 ( 1 )d quasi D= −

2d =

3d =

Dependence on dimensionality

14

Mesoscopic system L Lφ D φτ τ

1 ( 1 )

2

3

d quasi D

d

d

= −

=

=

Dependence on dimensionality2

2

2

( )

( )ln

2

e

e

L TeG sh L

L TeG sh l

e LG sh l

φ

φ

π

π

Δ = −

Δ = −

Δ = −

2

2

2

ln

2

e

e

eG she LG sh l

e LG sh l

π

π

Δ −

Δ −

Δ −

∼

∼

∼

1 ( 1 )

2

3

d quasi D

d

d

= −

=

=

Correction more important for small dbecause return probability is enhanced

15

Solving diffusion equation

( ) nt

n

EP t e−= ∑

where are the eigenvalues of the diffusion equationnE n n nD Eψ ψ− Δ =

Example : uniform magnetic field in 2D

1 82n

eBDnh

E π⎛ ⎞= +⎜ ⎟⎝ ⎠

0

0

/( )sinh 4 /

BP tBDtφ

π φ=

Quasi-1D wire : boundary conditions (TD)

16

Four examples

weak localization in 2 D, negative magnetoresistance

weak localization in a quasi-1d wire

weak localization in a ring

weak localization in a cylinder

17

lne

LG

lφΔ − ∼

Weak localization correction is suppressed when

0

0

/( )sinh 4 /

BSP tBDtφ

π φ=

( )min ,ln B

e

L LG

lφΔ − ∼ 2

0BBL φ ∼

( )4

SP tDtπ

=

In a magnetic field :

20B Lφ φ ∼

BL Lφ∼

Example 1: weak localization in 2 D

( ) 1/L T Tφ ∝

/( )e

t

cl

e dG

P ttG φτ

τ

∞ − Δ

∝ − ∫

R

B

Bergmann, 84

18

( )2

/ /

0

0

0

/sinh4 /

2 e

D

t te dtBBD

eh t

G s eφτ τ

φ τφ

π

∞− −−−Δ = ∫

2 1 12 2 4 2 4eB BeG s

h e D e D φτψ

π τψ

⎡ ⎤⎛ ⎞⎛ ⎞Δ = − + − +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

Example 1: weak localization in 2 D

19

1/ 22

2 2B

eG shL L Lφ

−⎛ ⎞1 1

Δ = − +⎜ ⎟⎜ ⎟⎝ ⎠

/0( ) ( ) BtP t P t e τ−= 0BBWL φ ∼

( )0 1/ 2( )4

LP tDtπ

=

In a magnetic field :

Example 2 : W.L. in a quasi-1D wire/( )

e

t

cl

e dG

P ttG φτ

τ

∞ − Δ

∝ − ∫

2B BL Dτ=

1 1 1

Bφ φτ τ τ → +

2 LeG sh L

φΔ = −

Altshuler,Aronov

20

0BW Lφ φ ∼

B

Weak localization correction is suppressed when

L Lφ∼

Example 2 : W.L. in a quasi-1D wire

1/3L Tφ−∝

Licini,Dolan,Bishop,1980

1/ 22

2 2B

eG shL L Lφ

−⎛ ⎞1 1

Δ = − +⎜ ⎟⎜ ⎟⎝ ⎠

0BBWL φ ∼

21

Et à fréquence finie ? (Pieper, Price, Martinis, cité par B. Plaçais

1/ 22

2 2B

eG shL L Lφ

−⎛ ⎞1 1

Δ = − +⎜ ⎟⎜ ⎟⎝ ⎠

( )2

/ /

0int( ) 2 ( ) et t

D

i te dtG s P t e eeh

φ ττωωτ

∞− − −Δ = −∫

1 1 iφ φ

ωτ τ

→ − 2 2

1 1 iL DLφ φ

ω→ −

1/ 22

2 2( )B

eG shL L L

iDφ

ωω−

⎛ ⎞1 1Δ = − +⎜ ⎟⎜ ⎟

⎝ ⎠−

22

Lm

m

LLG e

Lφφ

−

Δ − ∼

Ring in a Aharonov-Bohm flux :

may be extracted either from the amplitude of the oscillations

Example 3 : weak localization in a ring

2 2 / 4

0

( ) cos 44

m L Dt

m

eP t mDt

φπφπ

−

= ∑

Altshuler, Aronov, Spivak, ‘81

/( )e

t

cl

e dG

P ttG φτ

τ

∞ − Δ

∝ − ∫

/mL Le φ− 1/3L Tφ−∝

Lφ

Dolan,Licini,Bishop, ‘86

23

Lm

m

LLG e

Lφφ

−

Δ − ∼

Example 3 : weak localization in a ring

2 2 / 4

0

( ) cos 44

m L Dt

m

eP t mDt

φπφπ

−

= ∑

Altshuler, Aronov, Spivak, ‘81

may be also extracted from the envelope

0BW Lφ φ ∼Dolan,Licini,Bishop, ‘86

/ Bte τ−

penetration of the field in the ring

Lφ

2 2 2BL L Lφ φ

1 1 1 → +

24

2

00

ln 2 co 4sme m

Le LG s Kl

mh L

mφ

φ

φπφπ

⎡ ⎤⎛ ⎞Δ = − +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∑

Cylinder in a Aharonov-Bohm flux :

Example 4 : weak localization in a cylinder

2 2 / 4/

0

( ) cos 44

B

m L Dtt

m

eP t m eDt

τφππ φ

−−= ∑

Altshuler, Aronov, Spivak, ‘81

/( )e

t

cl

e dG

P ttG φτ

τ

∞ − Δ

∝ − ∫

Sharvin,Sharvin, ‘81

2D diffusion winding of trajectories

Altshuler, Aronov, Spivak, ‘81