Phase coherence and diffusion of electrons and photons :
a simple picture
E. Akkermans and G. Montambaux
Introduction:
Conductance
mesoscopic physics, phase coherence
weak localization of electrons,
coherent backscattering of light
universal conductance fluctuations, speckle fluctuations
g = transmission
coefficient
T
Classical transport :
add intensities
conductance = ratio of two volumes
Quantum corrections :
add amplitudes
1/g corrections
Weak localisation and coherent backscattering
Universality of conductance or speckle fluctuations
Conclusion
Other examples : P(g), noise, interactions
1
Mesoscopic diffusive regime
le
Bergmann, 84
Régime diffusif mésoscopique
Lφ
λF
le
L
Lφ
R
Lφ ∼ 1 − 10µ m
Corrections to classical transport
Violation of Ohm’s law
B
Weak localization : quantum correction to the classical resistance : increase of the resistance
suppressed by a magnetic field
negative magnetoresistance
Universal conductance fluctuations
2
Universal conductance fluctuations
Reproducible « fluctuations » as a function of an external parameter
Lee,Stone, Fukuyama 87
« Magnetofingerprint »
Au
Si
numérique
G
The amplitude of fluctuations is universal
h
= 25800 Ω
2
e
e2
δG = δG = G − G ∼
h
2
2
2
if L
Lφ
In a good metal, δG << G
3
Universal conductance fluctuations (2)
Mailly, Sanquer 92
G
« 46 magnetofingerprints »
G
Average
Weak localization correction
δ G2
G = Gcl + ∆G
∆G is suppressed by the magnetic field
Variance
δ G2
The variance is reduced by a factor 2, in a magnetic field
B
4
Speckle fluctuations
G. Maret
a
Tab
b
Transmission from a channel a to a channel b
Rayleigh law: relative speckle fluctuations are large, of order 1
δ T = Tab
2
ab
2
What is the equivalent of universal conductance fluctuations
δG << G
5
Conductance = transmission
a
Landauer formula
b
e2
e2
G = T = ∑ Tab
h
h a ,b
a, b : incoming and outgoing channels
bb
aa
Optics : measures Tab , Ta , or T
(a)
Electronics : measures
T = ∑ Tab
aa
bb
a'
a’
b'
b’
(b)
aa
Ta = ∑ Tab
(c)
b
T = ∑ Tab
(d)
a ,b
a ,b
6
Diffusion : length and time scales
L
le
le
D
elastic mean free path
Diffusive motion :
diffusion coefficient
2
r ∼ Dt
1
D = vle
3
L2 ∼ D τ D
τD
t
τD
Traversal or Thouless time
t
τD
7
Conductance = ratio of two volumes
Conductivity (Drude)
ne 2τ e
σ=
= e2 Dρ
m
Conductance (Ohm’s law)
Dimensionless conductance
S
G =σ
L
D=
v F le
3
S = W d −1
e2
G=g
h
W d −1
D V
V
=
∼ d −1
g∼
d −1
d −1 2
vF λ
L
vF λ L λ v Fτ D
D
8
Conductance = ratio of two volumes
V
g ∼ d −1
λF v F τ D
Volume of the sample
Volume of a tube of section λ
diffusing from one edge of the sample to the other
9
Conductance and transmission
The conductance is related to the probability of transferring electrons from
one side of the sample to the other
Introduce
in r
P(r,r’,t), probability to find a particle in r’ , if is has been injected
Quantum amplitude
G ( r, r ') = ∑ Aj ( r, r ')
j
j
r
Aj ( r, r ') = | Aj ( r, r ') | e
iϕ j ( r , r ')
r’
The probability is the squared modulus of the amplitude
2
P( r, r ') =
∑ A (r, r ')
j
j
10
Conductance and transmission (2)
Disorder average
Two types of contribution to the probability
P ( r, r ') = ∑ Aj ( r, r ') + ∑ Aj ( r, r ') A*j ' ( r, r ')
2
j≠ j '
j
Classical term
Interference term
Quantum effects
Classical transport : only paired trajectories Aj , Aj contribute
If trajectories are different, the amplitudes Aj et Aj’ are different
uncorrelated phases
the interference vanishes
Pcl ( r, r ') =
Aj
Aj*
+
DIFFUSON
11
Effets quantiques
Classical transport =
Q: How do quantum effects appear ?
A: When trajectories cross
Example : 2 particles from r1,r2 to r1’,r2’
Classically : product of probabilities
correction
Quantum crossing
Exchange of amplitudes
12
Quantum crossing
Volume λ
d −1
le
Simple picture : the diffuson P ( r , r ', t ) is an object of length vt and cross-section λ d −1
* Quantum effects are due to crossings
* Importance of quantum effects => evaluate the probability of crossing
13
Schematic picture
14
Evaluation of the probability of quantum crossing
Simple picture: the diffuson P (r , r ', t ) is an object of length vt and cross-section λ d −1
Probability of crossing during a time t, in a volume V=Ld
p× (t ) =
λ vt
d −1
d
L
Transport phenomena : the wave spends in the sample of size L, a time τD=L2/D
The probability of quantum crossings which affect transport properties is thus
p× (τ D ) =
λ d −1vτ D
Ld
∼
1
g
!!!
15
Coherent effects and quantum crossings
The probability of quantum crossing is 1/g
Quantum corrections
Quantum crossings
Fluctuations or correlations
Classical transport
e2
G = g
h
Quantum effects are of order
1 e2
G× ∼
g
h
In a good metal (g >>1), quantum effects are small
16
Weak localization
Classical conductance
Gcl
Quantum correction
One crossing
One loop
Time reversed trajectories
∆G 1
∼
Gcl
g
τD
∫τ
e
P (t )
dt
τD
Crossing
The return probability P(t) increases for small d
Coherent effets are more important in low dimension
Ld
P (t ) =
(4π Dt ) d / 2
17
Phase coherence
Classical return probability
Diffuson
Interférence term
=
If time reversal invariance
Cooperon
ϕj =
1
∫ p.d l
j
In a magnetic field, dephasing between time reversed trajectories
The cooperon vanishes in a magnetic field
If some processes break phase coherence, only trajectories of length < Lφ contribute
magnetic impurities
electron-phonon, electron-electron interactions
18
Speckle and conductance fluctuations
TabTa ' b ' = Tab Ta ' b ' + δ Tabδ Ta ' b '
a
b
Tab Ta ' b '
a’
a
δ Tabδ Ta ' b '
a’
= Tab Ta ' b ' f ( a, a ', b, b ')
b’
b
b’
Memory effect
f ( a, a ', b, b ') = g ( ∆a )δ ( ∆a − ∆b)
19
C1
Speckle fluctuations vs conductance fluctuations
a’ = a
b’=b
δ Tabδ Tab = Tab
a
a’
2
Loi de Rayleigh
a
a’
b
δ g2 =
∑
δ Tabδ Ta 'b '
a , a ',b ,b '
b’
δ g2 =
∑
Tab Ta ' b ' f (a, a ', b, b ') ∼ 0
a ,a ',b ,b '
No conductance fluctuations !
20
Speckle fluctuations vs conductance fluctuations
C2
1/g
a
a
b
b’
b
a’
a’
δ Tabδ Ta ' b ' =
2
Tab Ta ' b ' F (b, b ')
3g
b’
Angular correlations of intermediate range
δ g2 =
2
Tab Ta ' b ' F (b, b ') ∼ 0
∑
3g a ,a ',b,b '
No conductance correlations !
21
C3
Speckle fluctuations vs conductance fluctuations
1/g2
b
a
a a
b
a’
a’
b’
δ Tabδ Ta ' b ' =
2
T Ta ' b '
2 ab
15g
b’
Long-range angular correlations, with very weak amplitude
δ g2 =
2
15g 2
∑
Tab Ta ' b '
a ,a ',b ,b '
=
2
15
Universal conductance fluctuations
22
Speckle fluctuations vs conductance fluctuations (summary)
C1
C2

C3
2 
2
δ Tabδ Ta ' b ' = Tab Ta ' b '  f (a, a ', b, b ') + [ F ( a, a ') + F (b, b ')] +
3g
15g 2 

δ g2 =
2
Tab Ta ' b '
2 ∑
15g a ,a ',b,b '

=
2
15
Universal conductance fluctuations
23
Universal conductance fluctuations
Conductance fluctuations = 2 conductances and 2 crossings
2
e  2 1
e 
δG =   g × 2 ⇒  
g
h
h
2
2
2
2
Universal
24
Universal fluctuations and phase coherence
4 combinations
* A magnetic field kills the cooperon contribution
Pint (t )
δG →
2
δ G2
2
* Incoherent processes kill
Pint (t ) and
Diffuson
« classical »
Cooperon
« interference »
L
Lφ
Pcl (t )
⇒ δ G2 → 0
25
Coherent backscattering
ki
Time reversed trajectories
R
dephasing
ei ( k i + k e ). R
ke
Classical diffusion
Coherent contribution
D. Wiersma et al.
Coherent contribution if
ke=-ki
« Backscattering cone »
26
Coherent backscattering
ki
Where is the quantum crossing ?
Time reversed trajectories
Dephasing
R
ei ( k i + k e ). R
ke
Coherent contribution
cf.
27
Conclusion
Quantum transport of electrons and light in diffusive systems
« Lego »
Classical diffusion (diffuson or cooperon)
Quantum crossings
Simple formulation of phase coherent properties
28
Gaussians fluctuations ?
G=
O( g )
+
O (1)
2 conductances and 2 crossings
δG =
2
g2 ×
3 conductances and 4 crossings
δ G3 =
g3 ×
δ Gn =
1
∼ O (1)
g2
1
1
∼
O
g
g4
 
n conductances and (2n- 2) crossings
δG ∼ g ×
Gaussian fluctuations in the limit
g →∞
n
n
1
g 2 n −2
 1 
∼ O  n −2 
g 
29
« Real diagrams »
30
Quantum crossing = Hikami box
H ({ri }) =
H ({ri }) = h ∫ d r ∑ δ ( r − ri ) ( ∇1.∇3 + ∇ 2 .∇ 4 )
h = 2πρ 0 Dτ e4
le5
h=
48π k 2
31
Classical diffusion equation
Pcl (r , r ') is solution of a classical diffusion equation :
∂

 − D∆  Pcl ( r, r ', t ) = δ ( r − r ')δ (t )
 ∂t

Example : return probability :
d
L
P( r, r, t ) =
(4π Dt ) d / 2
32
Why are the fluctuations universal and weak localization is not ?
∆G 1
∼
Gcl
g
τD
∫τ
e
P (t )
τD
 τD 
P (t ) = 

 4π t 
τD
dt
τe
d /2
→∫
t
δ G2
dt
Universal if d < 2
2
cl
G
1
∼ 2
g
τD
∫τ
e
t P (t )
dt
τ D2
d /2
→∫
dt
τD
τe
t
d / 2 −1
Universal if d < 4
33
Terme C2 du speckle
C1
et
bruit de grenaille : quel rapport ???
C2

C3
2
2 
δ Tabδ Ta ' b ' = Tab Ta ' b '  f (a, a ', b, b ') + [ F ( a, a ') + F (b, b ')] +
3g
15g 2 

Bruit de grenaille

1
S = 2eI ×
3
Facteur de Fano
34
Le bruit dans un conducteur diffusif
e2
G = Tr tt †
h
δ g 2 = (Tr tt † ) − Tr tt †
2
∑t
e2
S = 2 eV Tr tt † (1 − tt † )
h
2
=
2
15
Tr tt †tt † ?
∑
*
ab ba
t ta ' b 'tb*' a '
aba ' b '
∗
*
tabtba
t
t
' a 'b' b'a
aba ' b '
g 2 f a ,b,a ',b '
0
2
g Fb,b '
3
0
2
15
ÅÆ C2 du speckle
2
15
0
g2 ×
e2
1
 2
S = 2 eV g  1 −  = 2eI ×
h
3
 3
35
2
3g
Blanter, Büttiker (97)
0
g2 ×
0
36
2
3g