users.lps.u-psud.fr/montambaux Disorder and mesoscopic physics Gilles Montambaux, Université Paris-Sud, Orsay, France Lecture 3 Weak localization Coherent backscattering in optics 2 (, ') P rr conductance ~ transmission ~ probability 1 ( ) 1 d F D D F g v p V classical diffuson quantum corrections quantum crossing 1/g correction 2 2 classical transport quantum effects e g h e h Summary of previous lecture int () P t 3 Quantum correction Classical conductance Time reversed trajectories cl G One crossing One loop Crossing = distribution of number of loops with time t = return probabililty Weak localization int 2 2 () e P t h G (0, ) cl P L (0, ) P L 4 Weak-Localization Nb loops and return probability Magnetic field, phase coherence Weak-localisation in dimension d A few solutions of the diffusion equation and WL Magnetic field and negative magnetoresistance Magnetic field in quasi-1D wires AAS oscillations int 2 2 () e P t h G
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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