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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
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( , ')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
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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
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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 :
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( )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
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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
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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)
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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
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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
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( )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
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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
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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