Download Wave in disordered media and localisation phenomena

Master iCFP
Dominique Delande
Laboratoire Kastler-Brossel,
UPMC-Paris 6, 4 Place Jussieu, F-75005 Paris
Christophe Texier
Laboratoire de Physique Th´eorique et Mod`eles Statistiques,
Universit´e Paris-Sud, Bˆat. 100, F-91405 Orsay
Wave in disordered media and localisation phenomena
16
G. Bergmann, Weak localization in thin films
54.0
0.1
Au
~
16%Au
8%Au
Mg
14.0
°°
-1.0
4%Au
7.1
2%Au
~
0
0
3.8
1%Au
R~91Q
T’4.6K
0.5
-0.1
1.0
0.5
-0.8
-0.6
-0.4
0%Au
-0.2
0
H(T)
0.2
0.4
06
0.8
Fig. 2.10. The magneto-resistance of a thin Mg-film at 4.5 K for different coverages with Au. The Au thickness is given in % of an atomic layer on
the right side of the curves. The superposition with Au increases the spin-orbit scattering. The points are measured. The full curves are obtained
with the theory by Hikami et al. The ratio T
1/T5, on the left side gives the strength of the adjusted spin-orbit scattering. It is essentially proportional
to the Au-thickness.
because the quenched condensation yields homogeneous films with high resistances. The points are
measured. The spin-orbit scattering of the pure Mg is determined as discussed above. The different
experimental curves for different temperatures are theoretically distinguished by their different H, (i.e.
the inelastic lifetime). This is the only adjustable parameter for a comparison with theory (after H,,~,is
determined). The ordinate is completely fixed by the theory in the universal units of L00 (right scale).
The full curves give the theoretical results with the best fit of H, which is essentially a measurement of
H1. The agreement between the experimental points and the theory is very good. The experimental
result proves the destructive influence of a magnetic field on QUIAD. It measures the area in which the
coherent electronic state exists as a function of temperature and allows the quantitative
determination
2 law for Mg
as is shown
1
of
scattering time T1. The temperature dependence is given by a T
in the
fig. coherent
2.12.
For other metal films where the nuclear charge is higher than in Mg one finds even in the pure case
the substructure caused by spin-orbit scattering. In fig. 2.13 the magneto-resistance curves for a thin
May 3, 2015
Figures :
Left : Weak localisation correction to resistance of thin metallic films (Mg dopped by Au). From
Ref. [25].
Right : Speckle pattern for light scattered by a turbid granular medium. Time reversal symmetry
is destroyed by using the Faraday effect. From Ref. [51].
2
Contents
I
Introduction : Disorder is everywhere (Lecture 1, DD)
I.A Disorder in condensed matter . . . . . . . . . . . . . . . . . . . . . . .
I.B Importance of disorder : how would be the world without disorder ? .
1)
Free electrons in a metal . . . . . . . . . . . . . . . . . . . . . .
2)
Fermions in a periodic potential : Bloch oscillations . . . . . .
I.C The physics of the diffusion . . . . . . . . . . . . . . . . . . . . . . . .
I.D How to model disorder ? What is measured ? What should be studied
1)
Electronic transport . . . . . . . . . . . . . . . . . . . . . . . .
2)
Scattering of light by disorder . . . . . . . . . . . . . . . . . . .
3)
Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I.E Outline of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem : Bloch oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7
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II Anderson localisation in one dimension (Lecture 2, CT)
II.A How to model disorder ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1)
Discrete models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2)
Continuum models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II.B Transfer matrix approach – The Furstenberg theorem . . . . . . . . . . . . . . .
1)
Preliminary : Furstenberg theorem . . . . . . . . . . . . . . . . . . . . . .
2)
A discrete model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3)
A continuum model : δ-impurities . . . . . . . . . . . . . . . . . . . . . .
4)
Preliminary conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II.C Detailed study of a solvable continuum model . . . . . . . . . . . . . . . . . . . .
1)
Definition of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2)
Riccati analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3)
Localisation – Weak disorder expansion . . . . . . . . . . . . . . . . . . .
4)
Lifshits tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5)
Conclusion : universal versus non universal regimes . . . . . . . . . . . .
6)
Relation between spectral density & localisation : Thouless formula . . .
II.D Self-averaging and non self-averaging quantities – Conductance distribution . . .
II.E Experiment with cold atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II.F Transfer matrix and scattering matrix : Anderson localisation versus Ohm’s law
II.G The quasi-1D situation (multichannel case) . . . . . . . . . . . . . . . . . . . . .
Appendix : Langevin equation and Fokker-Planck equation . . . . . . . . . . . . . . .
Problem : Weak disorder expansion in 1D lattice models and band center anomaly . .
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10
11
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15
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25
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29
30
30
32
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TD 2 : Thouless relation
37
TD 3 : Localisation for the random Kronig-Penney model – Concentration expansion
and Lifshits tail
39
III Scaling theory : qualitative picture
III.A Several types of insulators . . . . .
III.B Models of disorder . . . . . . . . .
III.C What is localisation ? . . . . . . .
III.D Length scales . . . . . . . . . . . .
III.E Scaling theory of localisation . . .
(Lecture
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41
41
41
42
43
44
IV Weak disorder : perturbative (diagrammatic) approach (Lectures 4-6, CT)
IV.A Introduction : importance of the weak disorder regime . . . . . . . . . . . . . . .
1)
Multiple scattering – Weak disorder regime . . . . . . . . . . . . . . . . .
2)
Interference effects in multiple scattering . . . . . . . . . . . . . . . . . . .
3)
Phase coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4)
Motivation : why coherent experiments are interesting ? . . . . . . . . . .
IV.B Kubo-Greenwood formula for the electric conductivity . . . . . . . . . . . . . . .
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46
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49
51
51
3
3, DD)
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1)
Linear response theory . . . . . . . . . . . . . . . . . . .
2)
Conductivity . . . . . . . . . . . . . . . . . . . . . . . .
3)
Conductivity in terms of Green’s function . . . . . . . .
IV.C M´ethode des perturbations et choix d’un mod`ele de d´esordre .
1)
D´eveloppements perturbatifs . . . . . . . . . . . . . . .
2)
Vertex et r`egles de Feynman . . . . . . . . . . . . . . .
3)
Self ´energie . . . . . . . . . . . . . . . . . . . . . . . . .
IV.D La conductivit´e `
a l’approximation de Drude . . . . . . . . . . .
IV.E Corr´elations entre fonctions de Green . . . . . . . . . . . . . . .
IV.F Diagrammes en ´echelle – Diffuson (contribution non coh´erente)
IV.G Quantum (coherent) correction : weak localisation . . . . . . .
1)
Small scale cutoff . . . . . . . . . . . . . . . . . . . . . .
2)
Large scale cutoff (1) : the system size . . . . . . . . . .
3)
Large scale cutoff (2) : the phase coherence length . . .
4)
Magnetic field dependence – Path integral formulation .
IV.H Scaling approach and localisation (from the metallic phase) . .
1)
The β-function . . . . . . . . . . . . . . . . . . . . . . .
2)
Localisation length in 1D and 2D . . . . . . . . . . . . .
IV.I Conductance fluctuations . . . . . . . . . . . . . . . . . . . . .
1)
Disorder averaging in large samples . . . . . . . . . . . .
2)
Few experiments . . . . . . . . . . . . . . . . . . . . . .
3)
Conductivity correlations/fluctuations . . . . . . . . . .
4)
Averaging : AB versus AAS oscillations . . . . . . . . .
IV.J Conclusion : a probe for quantum coherence . . . . . . . . . .
Exercices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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51
51
54
54
55
56
57
59
62
63
66
67
68
69
72
75
75
76
77
77
78
79
80
82
83
TD 4 : Classical and anomalous magneto-conductance
Problem 4.1 : Anomalous (positive) magneto-conductance . . . . . . . . . . . . .
Problem 4.2 : Green’s function and self energy . . . . . . . . . . . . . . . . . . .
1)
Propagator and Green’s functions . . . . . . . . . . . . . . . . . . .
2)
Green’s functions in momentum space and average Green’s function
3)
Self energy : stacking . . . . . . . . . . . . . . . . . . . . . . . . . .
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84
84
86
86
86
86
TD 5 : Magneto-conductance of thin metallic films and wires – Spin-orbit
and weak anti-localisation
Problem 5.1 : Magneto-conductivity in thin metallic films . . . . . . . . . . . . . . .
Problem 5.2 : Magneto-conductance in narrow wires . . . . . . . . . . . . . . . . . .
Problem 5.3 : Spin effects and weak antilocalisation in thin metallic films . . . . . .
scattering
. . . . . .
. . . . . .
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87
87
89
91
TD 6 : Conductance fluctuations and correlations in narrow wires
VI.K Preliminary : weak localisation correction and role of boundaries . . . . . . . . . . . . . .
VI.L Fluctuations and correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
93
93
VIIUniversal conductance fluctuations (Lecture 7, CT)
100
VIII
Coherent back-scattering (Lecture 8, DD)
101
IX Toward strong disorder – Self consistent theory
IX.A Self-consistent theory of localization . . . . . . .
IX.B Importance of symmetry properties . . . . . . . .
IX.C Transport by thermal hopping . . . . . . . . . . .
X Dephasing and decoherence (Lecture 10, DD)
(Lecture
. . . . . .
. . . . . .
. . . . . .
9, DD)
. . . . . . . . . . . . . . . . .
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102
102
102
102
103
XI Interaction effects (2014, CT)
104
XI.A Quantum (Altshuler-Aronov) correction to transport . . . . . . . . . . . . . . . . . . . . . 104
XI.B Decoherence by electronic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
XI.C More advanced topics (???) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4
TD 8 : Decoherence by electronic interactions – Influence functional approach
Bibliography109
106
Bibliography
Books
On probability and random processes (useful tools) : Gardiner [59], Risken [96] and van
Kampen [115].
Localisation in 1D : J.-M. Luck [80].
More complete but less pedagogical : Lifshits, Gredeskul & Pastur [79].
Presentation of mathematical aspects : Bougerol & Lacroix [30, 32].
Random matrix approach of quantum transport : the very well writtten review of Beenakker
[20] and the book of Mello & Kumar [86].
Introductory text for mesoscopic physics : Datta [41]
Advanced textbook (emphasize on weak disorder), both for photons and electrons : Akkermans
& Montambaux [2, 3].
A pedagogical presentation of the field theoretical approach : the book of Altland & Simons [4]
Quantum Hall effect : two books [94, 67]
Review articles
• Introductory texts :
Altshuler and Lee [11] and Washburn and Webb [122].
Sanchez-Palencia and Lewenstein [97] (emphasize on cold atom physics).
• Experiment oriented :
The review on WL measurements in thin films of Bergmann [25].
A review article focused on conductance fluctuations and AB oscillations in normal metals : Washburn and Webb [121].
A review on quantum oscillations (AAS oscillations in metals and oscillations in superconductors) : Aronov and Sharvin [16].
• Theory oriented :
A pedagogical presentation on weak localisation using path integral : Chakravarty and
Schmid [33].
A nice overview with emphasize on scaling theory : Kramer and MacKinnon [74].
Interaction effects in weakly disordered metals : Lee and Ramakrishnan [77] ; more
difficult (but more detailed) is the review article by Altshuler and Aronov [6] (part of a
book edited by Efreos and Shklovskii on interaction with disorder [49]).
A review (quite technical) on transmission correlations in optics : van Rossum and
Nieuwenhuizen [116].
5
I
Introduction : Disorder is everywhere
Aim : Illustrate the importance of disorder in practical situations.
Introduce the notion of random potential (modelizing randomness).
Discuss : averaged quantities vs fluctuations.
Localisation : wave + disorder
I.A
Disorder in condensed matter
Real crystals (give illustrations)
• structural disorder
• substitutional disorder
I.B
1)
Importance of disorder : how would be the world without disorder ?
Free electrons in a metal
Imagine that disorder is absent, hence motion of electrons is ballistic, i.e. momentum is conserved
p~(t) → p~. Linear response gives the conductivity (response of current density to electric field)
Z ∞
ine2 1 ne2
σ(ω) =
dt eiωt
h vˆx (t) , x
(I.1)
ˆ i=
| {z }
~
−iω m
0
=ˆ
px /m
diverges as ω → 0!
Classical theory – Drude conductivity.—
rates at low frequency
σ(ω) =
Experiments show that the conductivity satu-
ne2
1
ne2 τ
−→ σDrude =
m 1/τ − iω ω→0
m
(I.2)
The transport scattering time encodes the effect of collisions which limits the ballistic propagation of electrons in the metal. These collisions can be of different nature : collisions with
phonons (lattice vibrations), collision with impurities (lattice defects), spin scattering, etc. 1 The
different rates characterizing these scattering processes are simply added (Matthiesen law) :
1
1
1
=
+
+ ···
τ (T )
τe τe−ph (T )
(I.3)
Going to low temperature, degrees of freedom, like phonons, are frozen and the total rate (as
the resistivity) saturates :
lim 1/τ (T ) = 1/τe .
(I.4)
T →0
In general, electron-phonon scattering provides the dominant scattering mechanism [17].
This leads to the following resistivity (Bloch-Gr¨
uneisen formula)
1
ρ(T ) =
+A
σ0
T
TD
5 Z
TD /2T
dx
0
1
x5
sinh2 x
(I.5)
Note that collisions among electrons themselves do not affect the conductivity because a collision between
two electrons conserves the total current of electrons.
6
2
where TD is the Debye temperature and σ0 = nemτe is the residual conductivity). The power 5
comes from the temperature dependence of the electron-phonon scattering rate [17] τe−ph (T ) ∝
T −5 . Note that electron-phonon scattering is strongly anisotropic, being the reason why transport time 2 deviates from the total scattering rate τe−ph, tot (T ) ∝ T −3 (involved for example in
phase coherence properties).
Few order of magnitudes for Gold are given in the table.
Gold
m∗ /me = 1.1
k3
n = 3πF2 = 55 nm−3
kF−1 = 0.085 nm
6
F
vF = ~k
m∗ = 1.25 × 10 m/s
(∗)
εF = 5.5 eV
Work function W = 5.1 eV
∗ kF
= 1.07 × 1047 J−1 m−3 = 17 eV−1 nm−3
DoS ν0 = 2s m
2π 2 ~2
Resistivity ρ(T → 0) = 1/σ0 = 0.022 × 10−8 Ω.m [85]
`e = 4 µm (in bulk) (∗∗)
τe = 3.2 ps
D = 13 vF `e = 1.7 m2 /s (from σ0 = e2 ν0 D)
Debye temperature TD = 170 K [17]
melting point 1338 K and boiling point 3135 K [85]
~2 k2
Table 1: Few orders of magnitude for Gold. (∗) Note that 2mF∗ = 4.9 eV. (∗∗) In thin films the
mean free path is strongly reduced : e.g. for a gold wire of thickness, 50 nm it was observerd that
`e ' 22 nm [31].
Electronic transport is dominated by disorder at low temperature (T . 1 K)
and by inelastic scattering processes at high temperature.
2)
Fermions in a periodic potential : Bloch oscillations
• semiconducting superlattices
• Experiments with cold atoms falling on a periodic potential ⇒ oscillations !
I.C
The physics of the diffusion
Disorder leads to multiple scattering and diffusion.
Introduce few scales : τe , `e , D = vF `e /d,...
Einstein relation
σ = e2 ν0 D
(I.6)
where ν0 is the density of states. Using that the electronic density is n ∝ kFd , we have ν0 =
nd/(2εF ).
Diffusion equation (free)
(∂t − D∆) Pt (r|r0 ) = δ(t)δ(r − r0 )
2
Conductivity involves transport times.
7
(I.7)
Propagator of the diffusion
Pt (r|r0 ) =
(r−r 0 )2
θH (t)
− 4Dt
e
(4πDt)d/2
(I.8)
or its (spatial) Fourier transform
2
Pb(q; t) = θH (t) e−Dt q
or
Pe(q; ω) =
I.D
1)
1
−iω + Dq 2
(I.9)
(I.10)
How to model disorder ? What is measured ? What should be studied ?
Electronic transport
Hamiltonian for one electron
H=
p~ 2
+ Vcrystal (~r) + Vdisorder (~r)
| {z }
2me
(I.11)
fluctuates from sample to sample
Any observable depends on the disorder configuration
E.g. : conductance of a metallic wire G[Vdisorder ]
• In practice, we observe that in a long wire the conductance is not fluctuating from sample
to sample! It is given by the Ohm’s law G = σ Ls (usual feature of statistical physics :
fluctuations are washed out at macroscopic scale).
• Fluctuations may be observed by going to mesoscopic scale, L . few µm and T . few K
Observe reproducible fluctuations (not experimental noise) as a function of an external
parameter (here the magnetic field). These fluctuations are a signature of the presence of
the disorder. Such a curve is called the magnetofingerprint of the mesoscopic sample.
Manifestation of (quantum) interferences
due to scattering of wave by disordered potential
2)
Scattering of light by disorder
Speckle pattern, etc
3)
Calculations
From the theoretical point of view : what can be computed ? We can only compute averaged
quantities. Or study numerically some statistical properties of some observables.
I.E
Outline of the course
Main purpose of the course : analyse the interplay between wave character and disorder.
Where are we going ? Give an overview.
Problem : Bloch oscillations
8
16.5
G
G (( ee22/h
/h ))
16.4
(a)
16.3
16.2
16.1
16.0
15.9
200
600
1000
1400
1800
-4
B ( 10-4
Teslas )
16.5
G
G (( ee22/h
/h ))
16.4
(b)
16.3
16.2
16.1
16.0
15.9
200
600
1000
1400
1800
B ( 10 Teslas )
-4
-4
var
var (G)
(G) (10
(10-3-3(( ee22/h
/h ))22 ))
4.00
(c)
3.00
2.00
1.00
0.00
200
600
1000
1400
1800
-4
B ( 10-4
Teslas )
Figure 1: Left : UCF : 46 MC curves obtained with a short (coherent) wire etched in a 2DEG
(T = 45 mK). The different curves are obtained by modifying the disorder configuration. Right :
Averaging the MC leads to the weak localisation. Figures of Ref. [83].
II
Anderson localisation in one dimension
The main object of this course is the study of the wave dynamics (optical wave, electronic wave,
etc) in a disordered medium. For concreteness we can think at an electronic wave in a cristalline
structure. In practice, crystals are not perfect but subject to randomness, i.e. some atoms
might be replaced by atoms of other nature (random alloy), or the structure of the crystal can
present some defects (structural disorder). We will first have to discuss how we can model the
9
presence of the disorder, i.e. a time independent potential describing sample to sample
fluctuations. This will be achieved by assuming that the (static) potential is random in space.
As a consequence of the presence of disorder, the translational symmetry is broken in a
given sample (but restored after disorder averaging). Whereas eigenstates of a perfect crystal are
extended Bloch waves forming energy bands (continuous spectrum), the presence of the disorder
changes drastically the nature of the eigenstates, as we will see.
Aim : Although the 1D situation is somehow particular (dimension is crucial in localisation
problems), it allows to solve many problems, sometimes exactly, thanks to some powerful nonperturbative methods. The aim of this chapter is to demonstrate, in the frame of several particular
models, that the presence of disorder leads to the localisation of all eigenstates, i.e. their
exponential decay in space.
Reminder (some useful notions of probability) :
• Generating function, moments, cumulants
• Central limit theorem versus large deviations
• Stochastic calculus (SDE and FPE) (cf. appendix)
I refer to the books of Gardiner [59] and van Kampen [115]. More advanced topics of probability
theory can be found in the classical monograph of Feller [53].
II.A
How to model disorder ?
In this section we start by discussing in details how to model disorder. We introduce several
models that will be used in the following.
1)
Discrete models
As localisation theory has arised in the context of condensed matter physics [14], a popular
and widely studied model is the tight binding model (Anderson model) describing an electron
moving on a lattice of atoms, each characterised by a single orbital. In 1D, after projection on
the orbitals | n i at position x = na, where a is the lattice spacing, the Schr¨odinger equation
H| ψ i = ε| ψ i takes the form :
−t∗n ψn+1 + Vn ψn − tn−1 ψn−1 = ε ψn ,
(II.1)
def
where ψn = h n | ψ i is the component of the wave function on the site n, tn = −h n + 1 |H| n i
def
describes the coupling between nearest neighbour atoms and Vn = h n |H| n i a potential.
At this level, two options are
• Consider the case of random potentials Vn (“diagonal disorder”) describing a random alloy.
• Consider the case of random couplings tn (“off-diagonal disorder”) corresponding to the
case of structural disorder.
In the following, we will focus on the first case for simplicity, and set the couplings to unity, t = 1.
10
Distribution of disordered potential Vn .– As mentioned, translation invariance is broken
by the presence of disorder, i.e. for one configuration of the random potential. However, assuming homogeneity, translation invariance is restored after averaging over the disorder. As a
consequence the distribution of the potential Vn on site n is independent on n :
Pn (V ) = Proba[Vn = V ]
transl. inv.
−→
P (V )
(II.2)
We can give few examples of distributions :
1. The binary alloy : consider a lattice of atoms of type A. With probability p, one atom A
is replaced by an atom B with an energy Vn = W , i.e. P (V ) = (1 − p) δ(V ) + p δ(V − W )
2. A widely used distribution is the box distribution P (V ) =
1
W
θH (W/2 − |V |)
3. The choice of a more regular distribution will lead to more regular results (for the spectral
2
2
density, etc). Two convenient choices are : a Gaussian distribution P (V ) = √ 1 2 e−V /2W
or P (V ) =
2πW
1 −|V |/W
.
2W e
4. etc.
Correlations.– The statistical properties of the set of potentials {Vn } is not only characterised
by the marginal law P (V ), but in general depends on the correlations between potentials, partly
encoded in the correlation function hVn Vm i = C(|n − m|). A natural assumption is that the
correlations decay fast with the distance (exponentially fast). Studying the large scale properties,
a simplification will be to assume that potentials are uncorrelated hVn Vm i ∝ δn,m .
2)
Continuum models
Helmholtz equation with random dielectric constant.– In some cases, it can be more
natural to consider a continuum model for the propagation of a wave. For example, the analysis
of the propagation of an optical wave in a medium with fluctuating dielectric constant (~r) =
+ δ(~r) leads to consider the Helmholtz equation
−∆E(~r) − k 2
δ(~r)
E(~r) = k 2 E(~r)
(II.3)
describing the eigenmodes of the system (in the weak disorder regime, it is justified to decouple
the polarisation effect and thus treat in a first step the electric field as a scalar field [2]).
Schr¨
odinger equation with a random potential.– In a condensed matter physics context,
lattice models are more natural, however their large scale properties can be more conveniently
described by continuum models. For example, if one considers the 1D Anderson model
−t ψn+1 + Vn ψn − t ψn−1 = ε ψn
(II.4)
we encounter two interesting limits. In the absence of the random potential, the eigenstates are
simple plane waves ψn = √12π eikn of energy
εk = −2t cos k
for k ∈ [−π, π]
(II.5)
In the band edge, |k| 1, the dispersion relation may be simplified as εk ' −2t + t k 2 . This
quadratic behaviour corresponds to a non relativistic particle of mass m with ~2 /(2m) = ta2 ,
11
where a is the lattice spacing. In other terms, we expect that the low energy properties of the
model are conveniently described by the Schr¨odinger equation with a random potential
H=−
~2 d2
+ V (x)
2m dx2
(II.6)
in the following we will set ~2 /2m = 1 (hence [E] = L−2 ). This discussion holds if the disorder
is weak Vn t, so that it does only couple low energy states among themselves.
Distribution of V (x) – Generating functionals.– The notion of a “random function”
leads to a small technical complication : the distribution of the potential
must be encoded in
R
a functional DV (x) P [V ] and averaging involves functional integrals DV (x) P [V ] (· · · ). This
little difficulty can be circumvent by making use of the notion of generating functional. We
define
D R
E
def
(II.7)
G[b] = e dx V (x) b(x)
V (x)
which is usually more easy to handle. The correlation functions can be deduced by functional
derivation :
δ n G[b]
hV (x1 ) · · · V (xn )i =
(II.8)
δb(x1 ) · · · δb(xn ) b=0
It is usually more convenient to consider the connected correlation functions (cumulants)
δ n W [b]
hV (x1 ) · · · V (xn )ic =
where W [b] = ln G[b]
(II.9)
δb(x1 ) · · · δb(xn ) b=0
Two examples :
• General Gaussian disorder
1
P [V ] = N exp −
2
Z
dxdx V (x)A(x, x )V (x )
0
0
0
(II.10)
where
If we introduce the inverse of the integral kernel, defined by
R 0 N is 00a normalisation.
dx A(x, x )C(x00 , x0 ) = δ(x − x0 ), we find the generating functional
Z
1
0
0
0
G[b] = exp
dxdx b(x)C(x, x )b(x ) .
(II.11)
2
δ 2 G[b] δb(x)δb(x0 ) b=0
= C(x, x0 ) = hV (x)V (x0 )i is the two point correlation function.3
d2
Example : the Gaussian measure with A(x, x0 ) = σ1 δ(x − x0 ) 1 − `2c dx
leads to correla2
σ −|x−x0 |/`c
0
tions exponentially suppressed with the distance C(x, x ) = 2`c e
.
• Random uncorrelated impurities. A natural model of disorder is the case of localised
impurities at random positions xn :
X
V (x) =
vn δ(x − xn ) .
(II.12)
n
3
If this looks too formal, think that it is just a generalisation of usual matrix manipulations (with discrete indices)
indices. A general Gaussian measure would take the form P (· · · , Vx , · · · ) =
Pto continuous N exp − 21 x,x0 Vx Ax,x0 Vx0 . The Generating function is then given by a simple Gaussian integration in
P
P
RN : G(· · · , bx , · · · ) = he x Vx bx i = exp 12 x,x0 bx Cx,x0 bx0 . Functional derivation corresponds with partial
2
derivative ∂ G = Cx,x0
∂bx ∂bx0 b=0
12
A natural assumption is that the positions xn ’s are independent random variables distributed with a uniform mean density ρ. Fixing the averaged density, the number N of
N
independent impurities in a volume V is given by the Poisson distribution PN = (ρVN )! e−ρV .
The calculation of the generating functional follows straightforwardly :
*N
+
Z
N
∞
Y
X
dxn D vn b(xn ) E
vn b(xn )
G[b] =
e
=
PN
e
(II.13)
vn
V V
n=1
N =0
N, {xn }, {vn }
Z
vn b(x)
(II.14)
= exp ρ dx he
ivn − 1
or more conveniently
Z
W [b] = ρ
dx hevn b(x) ivn − 1 .
(II.15)
Functional derivations lead straightforwardly to
hV (x)i = ρ hvi i
hV (x1 ) · · · V (xn )ic = ρ
hvin i
(II.16)
δ(x1 − x2 ) · · · δ(x1 − xn )
(II.17)
(note that the cumulants of the potential are controlled by the moments of weights vi ’s).
Statistics of uncorrelated events - Poisson process.– In the calculation of the generating
functional (II.13), we have taken the point of view that xn ’s are independent random variables
distributed with a uniform mean density ρ. I.e., when dropped on the interval [0, L], their joint
distribution is P (x1 , · · · , xN ) = L1N .
Another point of view corresponds to order the random independent variable as 0 < x1 <
x2 < · · · < xn < · · · . We now ask the question : what is the number N (x) of such impurities on
the interval [0, x] ? The random non decreasing process N (x) ∈ N is called a “Poisson process”.
Its distribution function, PN (x) = Proba{N (x) = N }, can be easily studied (be careful : N
refers to the random variable whereas x is a parameter). We obtain
PN (x) =
(ρx)N −ρx
e
,
N!
(II.18)
which is proved in the exercice below.
-
Exercice II.1 Distribution of the Poisson process : The occurence of “events” (impurity positions) are independent. On the infinitesimal interval of width dx, the probability to
find one impurity is ρdx.
a) Writing P0 (x + dx) in terms of P0 (x), deduce a differential equation for P0 (x) and solve it.
b) Proceeding in a similar way for PN (x), show that these probabilities solve the infinite set of
coupled differential equations
d
PN (x) = ρ PN −1 (x) − ρ PN (x) .
dx
(II.19)
c) These equations can be solved by the generating function technics. Introduce
∞
N (x) X
=
sN PN (x) ,
G(s; x) = s
def
(II.20)
N =0
where s is a complex number. Deduce a differential equation for G(s; x). What is G(s; 0) ?
Deduce PN (x).
d) Compute the mean hN (x)i and the variance Var(N (x)).
13
Band center – Disordered Dirac equation.– The Schr¨odinger equation with a random
potential is not the only continuum limit of the lattice model. If, instead of studying the
properties of the Anderson model at the band edge, one consider the low energy properties in
the band center, since the spectrum εk = −2t cos k is linear for k ∼ ±π/2, the emerging theory
is a relativistic wave equation. Wave vectors close to +π/2 are described by one field ϕ and
wave vectors close to −π/2 by another field χ. If the random potential Vn presents a modulation
on the scale of the lattice spacing, we write Vn = A0 (na) + (−1)n m(na), where A0 (x) and m(x)
are two smooth functions. The function A0 (x) is related to small transfer of wave vector, i.e.
does not couple the two fields ϕ and χ. On the other hand, the modulated part corresponds
to large transfers of wavector δk ∼ ±π and describes the coupling between the two fields. The
natural wave equation modelizing this situation is therefore the Dirac equation with a random
scalar potential and a random mass for the bi-spinor Ψ = (ϕ, χ)
for HD = −iv0 σ3 ∂x + σ1 m(x) + A0 (x) ,
HD Ψ(x) = ε Ψ(x)
(II.21)
where σi ’s are Pauli matrices. The mapping is discussed in details in the problem at the end of
the chapter.
II.B
Transfer matrix approach – The Furstenberg theorem
The concept of transfer matrix is a very important one in statistical physics [19]. We show in
this section that the study of the models introduced previously within this formulation leads
straightforwardly to the conclusion that the wave functions are exponentially localised, provided
the knowledge of an important theorem of the theory of random matrix product.
1)
Preliminary : Furstenberg theorem
Let us consider a sequence of independent and identically distributed (i.i.d.) random matrices
Mn ’s, according to a suitable measure µ(dM ) defined over some group. 4 Then we form the
product of such matrices
Π N = MN · · · M 2 M1
(II.23)
A natural question is : given µ(dM ) what is the distribution of ΠN ? This corresponds to
look for some generalisation of the central limit theorem for non commuting objects. 5 The
Furstenberg theorem states that the matrix elements of the product ΠN grow exponentially
with N as N → ∞.
Norm.– In order to measure the growth of the matrix elements, it is convenient to define the
norm of the matrix. Several definitions are possible and the precise choice is not essential for
the following. For example (for n × n matrices) :
def
||M || = Sup{|M x| ; x ∈ Rn ; |x| = 1}
(II.24)
4
If one considers a group elements can be labelled by some parameters depending on the choice of the group
representation. Consider for example SL(2, R), a three-parameter-group ; it is well known that matrices of SL(2, R)
can be decomposed thanks to the Gauss decomposition
b
1 a
e
0
1 0
M (a, b, c) =
(II.22)
0 1
c 1
0 e−b
The measure µ(dM ) over SL(2, R) corresponds to some joint distribution P (a, b, c).
5
Consider some i.i.d. random variables xn . The statistical properties of the
P product of these random variables,
pN = xN · · · x2 x1 are clearly given by the central limit theorem for ln pN = n ln xn .
14
where |x| is the usual norm in Rn . Another choice could be
Z
def
dx |M x| .
||M || =
(II.25)
|x|=1
def
Furstenberg theorem (1963).– Assuming that ln+ ||Mn || < ∞, where ln+ x = θH (x −
1) ln x, the Furstenberg theorem [57] states that
def
ln ||ΠN ||
>0
N →∞
N
γ = lim
(II.26)
(note that averaging is not needed). γ is called the (maximum) “Lyapunov exponent” of the
random matrix product. It depends on the group and on the probability measure. Given this
information, it is however difficult to compute it in general.
More information can be found in the monograph by Bougerol and Lacroix [30].
2)
A discrete model
The study of the tight binding equation (II.4) for t = 1 can be reformulated in terms of transfer
matrices as follows
ψn
ψn+1
Vn − ε −1
=
.
(II.27)
ψn−1
ψn
1
0
{z
}
|
=Mn ∈SL(2,R)
The behaviour of the wave function is controlled by those of the product of transfer matrices
ψn ∼ ||Πn ||
where Πn = Mn · · · M1
(II.28)
The Furstenberg theorem immediatly tells us that, given some initial values (ψ1 , ψ0 ), the wave
function 6 grows exponentially
ln |ψn | ∼ γ n
(II.29)
n→∞
√
up to fluctuations of order n, where γ is the Lyapunov exponent of the transfer matrices. This
furnishes a possible definition of the localisation length
def
ξ = 1/γ
3)
(II.30)
A continuum model : δ-impurities
Another interesting model is the delta impurity model (random Kronig-Penney model)
H=−
X
d2
+
vn δ(x − xn )
dx2
n
(II.31)
We set ~2 /(2m) = 1 for convenience. Let us gather the wave function and its derivative in a
vector
0 ψ (x)
(II.32)
k ψ(x)
We may encode the evolution of the vector through the action of transfer matrices of two types
(figure 2) :
6
We already stress that we consider here the solution of the initial value problem, and not the real normalised
eigenstates solution of the spectral problem.
15
• Between impurity n and n + 1, the evolution is free, hence ψ(x) = An sin(kx + ϕn ), and
−
ψ 0 (x)/k = An cos(kx + ϕn ). This makes clear that the vectors at x+
n and xn+1 are related
by a rotation of positive angle θn = k`n where `n = xn+1 − xn .
• Evolution through the impurity n corresponds to ψ 0 (xn +) − ψ 0 (xn −) = vn ψ(xn ), with
ψ(x) continuous.
Therefore the evolution of the vector (II.32) involves a sequence of random matrices of the form
cos θn − sin θn
1 un
Mn =
∈ SL(2, R)
(II.33)
sin θn cos θn
0 1
with random positive angles θn = k`n > 0 and random coefficients un = vn /k. We have
ψ 0 (x1 −)
ψ 0 (xn+1 −)
where Πn = Mn · · · M1 ,
(II.34)
= Πn
k ψ(x1 −)
k ψ(xn+1 −)
ψ(x)
impurity
free evolution
ψ ’(x)
Figure 2: Transformation of the vector (ψ 0 , kψ).
10
lnÈÈPnÈÈ
20
5
15
10
0
5
0
0
-5
-5
0
5
10
2000
4000
6000
8000
10 000
n
15
Figure 3: Evolution of the random matrix product. Left : Motion of the vector under the
action of the 1000 random matrices of the form (II.33). the angle are exponentially distributed
1 −|u|/a
with θ = 0.02 and the coefficients un under the symmetric PDF p(u) = 2a
e
with a = 0.1.
Right : Evolution of the modulus of the vector with n for two realisations of the disorder : slope
indicates the Lyapunov exponent.
Note that the fact that the matrix Mn ∈ SL(2, R) follows from the conservation of probability
current : denoting Ψ(x) the vector (II.32), the current density is expressed as Jψ = Im[ψ ∗ ψ 0 ] =
1
Ψ† σ2 Ψ. Hence current conservation between two points related by a transfer matrix M is
− 2k
ensured by the condition M † σ2 M = σ2 . This implies that M ∈ SL(2, R), up to a global phase.
16
Impurities may form a lattice (fixed distances `n = 1/ρ) or be randomly dropped; e.g.
dropped independently and uniformly for a mean density ρ, i.e. p(`) = ρe−ρ` . Weights vn may
be random or not. The Furstenberg theorem again immediaty shows that the wave function
envelope increases exponentially
ln |ψ(x)|
lim
(II.35)
=γ
x→∞
x
where γ coincides with the Lyapunov exponent of matrices (II.33) (up to a factor ρ coming from
def
n ∼ ρx). This gives again a definition of the localisation length ξ = 1/γ in this continuum
model.
-
Exercice II.2 : The study of the perfect lattice of impurities is instructive. We consider
transfer matrices (II.33) for `n = a and vn = v, ∀ n.
a) Show that Bloch theorem implies that the transfer matrix M has eigenvalue eiK .
b) Deduce the quantization equation and show that spectrum of eigenstates forms energy bands
(hint : use a graphical resolution).
c) Wheck that the eigenvalues of M can be written as e+Ω and e−Ω . Show that for the perfect
crystal we have either Ω ∈ R or Ω ∈ iR. Deduce the expression of the Lyapunov expression and
of the integrated density of states.
Solution : exercice 6.10 of [107].
The exercice illustrates that for a non disordered problem, the Lyapunov vanishes on the spectrum, i.e. states are extended.
Is the Lyapunov exponent always a good measure of localisation ? Consider the initial
value (Cauchy) problem :
−ψ 00 (x; E) + V (x)ψ(x; E) = E ψ(x; E)
0
ψ(0; E) = 0 & ψ (0; E) = 1
(II.36)
(II.37)
that exists ∀ E. The transfer matrix approach permits to describe the evolution of ψ(x; E),
hence the Lyapunov exponent γ of the random matrices characterizes the Cauchy problem.
On the other hand, the real eigenstates, solution of a Sturm-Liouville problem, −ϕ00 (x) +
V (x)ϕ(x) = E ϕ(x) with ϕ(0) = ϕ(L) = 0, exist for discrete values of the energy E ∈
{E0 , E1 , · · · } ; they are built from the solutions of the Cauchy problem, ϕn (E) = ψ(x; En ).
Defining the localisation length as 1/ξ = γ, we have therefore assumed that the second
boundary condition, which constraints the wavefunction to return to ψ(L) = 0, does not affect
its statistical properties. This is not obvious. Although this is a reasonnable assumption for
high energy wave functions rapidly oscillating, when oscillations and growth of the envelope
decouple, it is more questionable for low energy states (see the discussion in the review [39] and
references therein).
4)
Preliminary conclusion
The use of the Furstenberg theorem has immediatly provided the nature of the eigenstates,
shown to be exponentially localised. The argument is very general because any (linear) wave
equation can be formulated with the help of transfer matrices (even in the multi-channel case ;
strictly 1D is not a restriction). However it does not give the quantitative information about the
localisation length. In general, given the measure µ(dM ) characterizing the random matrices,
finding the Lyapunov exponent is a very hard task. From the point of view of the disordered
model (II.4), the question would be : given the distribution P (V ) of the random potentials Vn ,
17
ψ(x)
ψ(x)
x
x
ξ
Figure 4: Extended and localised states. Left : Typical shape of the extended states
in a perfect 1D crystal. Right : In the presence of disorder, the eigenstates typically vanish
exponentially over a typical distance ξ, called the localisation length.
what is the Lyapunov exponent γ ? A review on how to answer to this precise question can be
found in the first part of the book by Jean-Marc Luck [80].
We analyse below a continuum model where such an analysis will be facilitated.
II.C
1)
Detailed study of a solvable continuum model
Definition of the model
Let us consider the Schr¨
odinger Hamiltonian
d2
+ V (x)
dx2
for the simplest model of random potential, namely the Gaussian white noise
Z
1
P [V ] ∝ exp −
dx V (x)2
⇒
V (x)V (x0 ) = σ δ(x − x0 ) .
2σ
H=−
(II.38)
(II.39)
-
Exercice II.3 Relation to other models :
a) Relate σ to the strength of the disordered potentials of the lattice model hVn Vm i = W 2 δn,m .
b) Show that the model with the Gaussian white noise potential may be obtained by considering
the high density limit (ρ → ∞) with weak impurities (vn → 0) of the random impurity model
introduced previously. What is the precise relation between the two models ? I.e. how can one
relates σ to the parameters of the impurity model ?
Hint : Find the proper non trivial limit of the characteristic functional (II.15).
-
Exercice II.4 Dimensional analysis :
a) Check that the dimension of disorder strength is [σ] = L−3 .
b) We consider the Lyapunov exponent γ(E) (inverse localisation length) and the integrated
density of states per unit length N (E). Use dimensional analysis to express γ and N in terms
of two scaling functions.
c) Deduce how the zero energy localisation length behaves with the disorder strength ? What
is the amount of states that have migrated from R+ to R− ?
2)
Riccati analysis
We now show that the localisation properties and the spectral properties of the Schr¨odinger
equation with a disordered potential can be obtained by studying a random process related to
the solution of the Cauchy problem (the wavefunction), denoted ψ(x) henceforth. The ideas will
be applied to the case of the Gaussian white noise potential below. We introduce the Riccati
variable
0
def ψ (x)
z(x) =
(II.40)
ψ(x)
18
We will see that the study of this random process can be easily handled by standard technics
(Langevin and Fokker-Planck equations). From −ψ 00 + V ψ = Eψ, we obtain that the Riccati
variable obeys a first order non linear differential equation
d
z(x) = −E − z(x)2 + V (x) = F(z(x)) + V (x)
dx
(II.41)
where we introduce the “force” F(z) = −E − z 2 and the related “potential” :
Z
1
U(z) = − dz F(z) = Ez + z 3 .
3
(II.42)
Equation (II.41) has the structure of the Langevin equation describing a fictitious Brownian
particle of “position” z(x) at “time” x, in the overdamped regime where the speed is proportional
to the force.
+
+
U(z)=Ez+ z
3
U(z)=Ez+ z
3
3
z
z
E>0
−
3
z
E=0
E<0
Figure 5: Dynamics of the Riccati variable under the action of the deterministic force.
The introduction of the Riccati variable will be very useful in order to analyze the dynamics
of the process. We can already make the following observations :
• When E > 0, the “deterministic force” alone, i.e. the force F(z) = −E − z 2 induces a flow
of the process from +∞ to −∞. In the absence of the potential, the Riccati variable takes
a finite “time” to cross R. Let us start from z(x0 ) = +∞ (that corresponds to a node of
the wave function, ψ(x0 ) = 0). Then the next divergence (i.e. the next node of ψ) occurs
after a “time” ` given by
Z
x0 +`
x0
dx = −
Z
−∞
+∞
dz
E + z2
⇒
π
`= √
E
(II.43)
The distance between the nodes of the wave function is the inverse of the integrated density
of states (IDoS) per unit length N0 (E) = 1/`. This is the result of a general theorem : the
oscillation theorem (or Sturm-Liouville theorem, cf. 6).
√
For E < 0, the potential traps the Riccati variable at z+ = −E.
• The random potential plays the role of a random (Langevin) force.
For E > 0, it induces fluctuations of the time needed to go from +∞ to −∞,
what is
√
1
responsible for a deviation of the IDoS N (E) from the free IDoS N0 (E) = π E.
For E < 0, a negative fluctuation of the potential may allow the process to escape the
potential well and induces a finite flow of the Riccati.
19
Localisation length.— Our definition of the localisation length is the growth rate of the
solution of the Cauchy problem
Z x
ln |ψ(x)|
dt
γ = lim
= lim
z(t) .
(II.44)
x→∞
x→∞
x
0 x
Assuming some ergodic properties, we can express the Lyapunov exponent as an integral of the
Riccati’s stationary distribution f (z) as
Z
γ = r dz z f (z)
def
⇒
ξ = 1/γ .
(II.45)
We have introduced a principal part
Z
Z +∞
def
r
dx h(x) = lim
−∞
+R
R→+∞ −R
Z
+∞
dx h(x) =
dx
−∞
h(x) + h(−x)
2
in order to account for the possible slow power law decay of the Riccati distribution.
Spectal properties.— In the absence of the random potential V (x), we have shown that
the “time” needed by the process z(x) to go from +∞ to −∞ coincides with the IDoS per
unit length. This property remains true in the presence of the disordered potential, provided
we consider the average time, i.e. N (E) = 1/`. This relation may be proven as follows : The
density 1/` is the average density of nodes of the wave function per unit length. From the
oscillation (Sturm Liouville) theorem this coincides precisely with the IDoS per unit length. It
will also be convenient for the following to notice that the density of the Riccati’s infinitudes
can be interpreted as the stationary probability current −J :
N (E) = J = 1/`
(II.46)
...
J = number of infinitudes of the Riccati process per unit length (“time”)= number of nodes of
ψ per unit length= IDoS per unit length (oscillation theorem).
11
00
00
11
00
11
000
11
ϕ2( x)
11
00
00
11
00
11
000
11
ϕ1( x)
ϕ0( x)
11
00
00
11
00
11
000
11
1
0
0
1
0
1
L1
0
1
0
0
1
0
1
0
L1
1
0
0
1
0
1
L1
0
Figure 6: Oscillation (Sturm-Liouville) theorem. In 1D, the number of nodes of the wavefunction coincides with the excitation degree n, i.e. the IDoS.
Application to the model with Gaussian white noise disorder.— If the disordered
potential is a Gaussian white noise, Eq. (II.39), then (II.41) is the usual Langevin equation with
20
a force uncorrelated in “time” . Then we can directly relate this Langevin equation to a Fokkerdef
Planck equation for the probability density of the Riccati variable f (z; x) = hδ(z − z(x))i (cf.
appendix) :
∂
∂
σ ∂2
−
f (z; x) =
F(z) f (z; x) .
(II.47)
∂x
2 ∂z 2 ∂z
It can be conveniently recast under the form of a conservation equation
∂
σ ∂
∂
f (z; x)
f (z; x) = − J (z; x) where J (z; x) = F(z) −
∂x
∂z
2 ∂z
(II.48)
is the probability current, given by the sum of a drift term and a diffusion current.
The Fokker-Planck involves the (forward) generator
G† =
σ d2
d
−
F(z) ,
2 dz 2 dz
(II.49)
2
d
d
adjoint of the generator of the diffusion G = σ2 dz
2 + F(z) dz (i.e. the backward generator). We
prove in the exercice below that it has a discrete spectrum of eigenvalues. From this observation,
we deduce that the distribution of the Riccati variable converges to a limit distribution on a
finite length scale `c (the correlation length of the Riccati process)
f (z; x) −→ f (z) ,
x→∞
(II.50)
where f (z) obeys
σ d
2
+ E + z f (z) = N (E)
2 dz
(II.51)
where we have used that the steady current
J (z; x) −→ −N
(II.52)
x→∞
is the IDoS per unit length.
- Exercice II.5 Spectrum of the generator : The aim of the exercice is to show that
f (z; x) reachs a limit distribution f (z) over a characteristic scale `c (the correlation length).
Assuming initial condition z(0) = z0 , we can write formally the propagator as 7
X
†
L
−x En
f (z; x) = h z |ex G | z0 i =
ΦR
(II.53)
n (z) Φn (z0 ) e
n
R
L
L
in terms of left/right eigenvectors, G † ΦR
n (z) = −En Φn (z) and G Φn (z) = −En Φn (z).
a) Consider the nonunitary transformation
1
1
H+ = e σ U (z) (−G † )e− σ U (z)
This shows that G † and H+ are isospectral. Chek that
σ d2
[F(z)]2 F 0 (z)
σ d
F(z)
d
F(z)
H+ = −
+
+
=
+
−
+
2 dz 2
2σ
2
2 dz
σ
dz
σ
(II.54)
(II.55)
b) Analyse the shape of the effective potential for the Schr¨odinger operator H+ and deduce the
nature of the spectrum of eigenvalues {En } (discrete/continuous).
7
Propagator : Consider a Schr¨
odinger operator H, the propagator is the evolution operator in imaginary
time
Kt (x|x0 ) = θH (t) h x |e−tH | x0 i, Green’s function of the time dependent Schr¨
odinger equation ∂t + H Kt (x|x0 ) =
δ(t) δ(x − x0 ).
21
The Hamiltonian (II.55) is called a supersymmetric operator in reference with the particular
structure (generalisation of the harmonic oscillator a† a) [69, 37].
-
Exercice II.6 Stationary solution : a) Construct explicitly the solution of (II.51).
b) Rice formula.– Show that the asymptotic behaviour of the distribution is f (z)
N (E)/z 2 .
'
z→±∞
c) Show that the normalisation provides an integral representation of the IDoS (one of the two
integrals can be done).
3)
Localisation – Weak disorder expansion
It is possible to compute exactly the Lyapunov exponent (see exercice II.8 below), however we
now only present a more simple perturbative analysis (in the disorder strength σ), valid in the
high energy limit, E σ 2/3 . Let us assume that the stationary distribution and the IDoS have
a perturbative expansion :
f (z) = f (0) (z) + f (1) (z) + · · ·
N =N
(0)
+N
(1)
+ ···
(II.56)
(II.57)
Inject this form in (II.51) leads to
N (0)
(II.58)
z 2 + k2
N (n)
σ
d (n−1)
f (n) (z) = 2
−
f
(z)
(II.59)
z + k 2 2(z 2 + k 2 ) dz
√
all the normalisation, hence we recover the
where we have introduced k = E. f (0) carries
√
R
E
(0)
well-known IDoS for the free case N
= π . At first order we must have dz f (1) (z) = 0,
d (0)
hence N (1) = 0 and f (1) (z) = − 2(z 2σ+k2 ) dz
f (z), etc. Finally we get
f (0) (z) =
f (z) =
k/π
σk
z
+
+ O(σ 2 )
2
2
+k
π (z + k 2 )3
z2
that can be introduced into (II.45) :
Z
σk
z2
σ
γ =0+
dz 2
+ O(σ 2 ) =
+ O(σ 2 )
π
(z + k 2 )3
8E
(II.60)
(II.61)
The important conclusion of this calculation is the fact the the Lyapunov exponent is different
from zero for any (positive) energy. Therefore all eigenstates get localised by the disorder,
how weak it is. The localisation length however increases with energy as
ξE
'
Eσ 2/3
8E
σ
(II.62)
We may interpret this important result as follows :
• in the absence of disorder (σ = 0), the spectrum of eigenvalues
√ is R+ . Negative energies
correspond to evanescent (non normalisable solutions)
exp(±
−E x), hence the Lyapunov
√
exponent is non zero outside of the spectrum, γ = −E for E < 0, and vanishes on the
spectrum, γ = 0 for E > 0.
• In the presence of disorder (σ 6= 0), the Lyapunov exponent becomes non zero on the
spectrum. Correlatively, the disorder drags some states in the negative part of the spectrum
(figure 7).
22
2.0
1.5
1.5
ΠNHEL & ΓHEL
ΠNHEL & ΓHEL
2.0
1.0
0.5
1.0
E e-8 E
0.5
32
3 Σ
ӐH8EL
0.0
0.0
-4
-2
0
2
4
-4
E
-2
0
2
4
E
Figure 7: Halperin model.– IDoS (black continuous line) and Lyapunov exponent (dashed red
line) in the absence of disorder (left) and with Gaussian white noise potential (right). From
Eq. (II.80)
Generalisation : The perturbative formula for the Lyapunov exponent may be generalised
to arbitrary random potentials with finite correlation function. It leads to [15, 79]
Z
1
γ ' 2 dx hV (x)V (0)i cos 2kx
as E = k 2 → ∞ .
(II.63)
8k
-
Exercice II.7 : In the experiment with cold atoms, the disordered potential is realised
with a speckle pattern (Fig. 10). The potential felt by the atoms is proportional to the light
intensity V (x) ∝ |E(x)|2 , where the electric field presents correlations hE(x)E ∗ (x0 )i(speckle) =
0 |/` )
c
I0 sin(|x−x
|x−x0 |/`c . What are the potential correlations ? Deduce the Lyapunov exponent from the
perturbative formula (II.63). What can you expect when this expression vanishes ?
See Ref. [81]
Remark : V (x) with infinite moments.– The formula (II.63) emphasizes that the high
energy behaviour of the localisation length ξE ∝ E is pretty universal for random potentials
uncorrelated in space. This behaviour however obviously requires that the second moment of
the potential is finite, i.e. hV (x)V (0)i < ∞. If this is not the case, large fluctuations of the
random potential can induce unusual localisation properties, like a non linear dependence of the
localisation length with E or the phenomenon of superlocalisation [27] (a clear discussion on
sublocalisation and superlocalisation and references can be found in Ref. [29]).
4)
Lifshits tail
Another feature common to disordered systems is the existence of the low energy non analytic
density of states (non analytic in the disorder strength σ). The fraction of states with energy
E < 0 may be estimated by a simple dimensional analysis as
N (0) ∼ σ 1/3
(II.64)
hence the DoS per unit length get finite at zero energy ρ(0) ∼ σ −1/3 (it diverges as disorder
disappears, as it should). We may obtain the low energy behaviour of the IDoS by a very simple
argument, remembering the interpretation of the IDoS as a current of the Riccati variable : for
large E = −k 2 , the potential U(z) developes a local minimum at z = +k, with a potential barrier
∆U = U(−k) − U(k) = 4k 3 /3, that traps the process a very long time ∝ exp 2∆U
σ (Arrhenius
law). The current is expected to be diminished in inverse proportion, hence
8
3/2
N (E) ∼ exp − (−E)
.
(II.65)
E→−∞
3σ
23
This interpretation was first suggested by Jona-Lasinio [68].
Ordered spectral statistics and energy level correlations.– It allows to obtain not only
the average density of states per unit length, but although the order statistics of eigenvalue
(distribution of ground state energy, first excited state, etc) [106]. I have shown in this paper
that this ordered statistics problem for a priori correlated variables (the eigenvalues of a r andom
operator) coincides with Gumbel laws describing uncorrelated random variables. This shows that
in the strongly localised phase, eigenvalues behave as independent random variables [89].
5)
Conclusion : universal versus non universal regimes
We have encountered two regimes :
• Universal results (high energy/weak disorder) do not depend on details of P (V ), but only
on few properties, like hV 2 i. Anderson localisation takes place in this regime : although
the energy of the particle is much higher than the potential, E σ 2/3 , eigenstates are
localised (particle would be free classically).
• Non universal results (low energy/strong disorder), such as Lifshits tails, depend on the
full distribution P (V ), as usual for large deviations. 8 In the low energy limit, localisation
of eigenstates is somehow obvious and corresponds to trapping of the particle by deep
potential wells (what would also occur classically).
2
1
0
-1
-10
-5
0
5
10
x
Figure 8: The ground state and some high energy states of the Anderson model with Gaussian
random potentials for W/t = 0.79.
Final remark : Although we have developed the Riccati analysis in the frame of a continuous
model, these ideas can be formulated in the context of the discrete Anderson model (II.67). In
def
this case a natural definition for the Riccati variable is Rn = ψn+1 /ψn which obeys the recursion
Rn = Vn − ε −
1
.
Rn−1
A pedagogical description can be found in the book of J.-M. Luck [80].
8
Cf. Refs. [38, 61] for reviews on low energy spectral singularities in 1D disordered systems.
24
(II.66)
6)
Relation between spectral density & localisation : Thouless formula
We show that the IDoS and the Lyapunov exponent are closely related.
An elementary derivation for the discrete Anderson model.— Let us consider the
tight-binding Hamiltonian (for hopping t = 1)
−ψn+1 + Vn ψn − ψn−1 = ε ψn
(II.67)
on a finite interval n ∈ {1, · · · , N }. We consider the initial boundary conditions ψ0 = 0
(Dirichlet-like) and ψ1 = 1. Wave function on the next sites can be computed from ψn+1 =
(Vn − ε) ψn − ψn−1 . It will be useful to remark for the following that
ψn+1 ' (−ε) ψn
for ε → ∞
(II.68)
The spectrum of the finite system is obtained by imposing a second boundary condition, say
ψN +1 = 0. This equation is nothing but the quantification equation, whose solutions are the
eigenvalues {εα }α=1,··· ,N of the Schr¨
odinger Hamiltonian. We conclude that the solution ψn of
the initial value problem ψ0 = 0 & ψ1 = 1 can be written as a function of the energy
ψN +1 =
N
Y
(εα − ε) ,
(II.69)
α=1
where the prefactor has been fixed thanks to (II.68). Taking the logarithm of this equation, we
obtain
Z
1
ln |ψN +1 | = dε0 ρ(ε0 ) ln |ε0 − ε|
(II.70)
N
P
def
where ρ(ε) = N1 N
α=1 δ(ε − εα ) is the density of states per site. Note that this equation is
general and has not required any assumption on the potential. Now considering a disordered
potential, the left hand side can be identitified with the Lyapunov exponent in the limit N → ∞
when the spectrum becomes dense, thus
Z
γ(ε) =
dε0 ρ(ε0 ) ln |ε0 − ε|
(for Anderson model).
(II.71)
This formula is known as the Thouless formula [111].
Exactly the same idea can be performed for a continuum model by considering the wave
function ψ(x; E) solution of the initial value problem ψ(0; E) = 0 and ψ 0 (0; E) = 1. The
quantification equation now reads ψ(L; E) = 0 providing the spectrum on a finite interval of
length L with Dirichlet boundary conditions. A little technical difficulty however arises due
to the fact that the spectrum is not bounded from above and involves an infinite number
of eigenvalues. One can however show that the wavefunction is proportional to the functional
determinant 2ψ(L; E) = det(E +∂x2 −V (x)) (see [73] for a pedagogical introduction to functional
determinants, or [64]).
Analytical properties of the complex Lyapunov exponent (continuum model).—
The IDoS and the Lyapunov exponent are the real and imaginary part of a single analytic
function defined in the upper complex plane. It follows that it is possible to write a KramersKronig relation relating one to the other (the Thouless formula), as was well illustrated by the
exercice below.
25
This might be understood in a simple way by considering the Fourier transform of the
stationary distribution
Z
ˆ
fE (q) = dz e−iqz fE (z)
(II.72)
For example, in the case of the Schr¨
odinger equation with Gaussian white noise potential V (x),
ˆ
it follows from (II.51) that fE (q) obeys the differential equation
d2
iσ
− 2 + E + q fˆE (q) = 2π N (E) δ(q)
(II.73)
dq
2
For other models of random potential, we obtain a different differential equation. We can see
that
fˆE0 (0+) = −π N (E) − i γ(E)
(II.74)
thus the integrated density of states and the Lyapunov exponent are real and imaginary part
of an analytic function of the energy in the upper complex plane, denoted as the characteristic
function, or the complex Lyapunov exponent
Ω(E) = γ(E) − iπ N (E) .
(II.75)
Kramers-Kronig relation can be obtained by performing some soustraction in order to deal with
an asymptotically decaying function :
p
def
e
Ω(E)
= Ω(E) − Ω0 (E)
where Ω0 (E) = −E − i0+
(II.76)
is the complex Lyapunov exponent in the free case. Considering the appropriate closed contour
in the complex half plane gives :
Z
e
dω Ω(ω)
e
Ω(E)
= −i r
R π ω−E
(II.77)
i.e., taking the real part
Z
N (E 0 ) − N0 (E 0 )
γ(E) − γ0 (E) = − r dE 0
,
(II.78)
E0 − E
√
√
where γ0 (E) = θH (−E) −E and πN0 (E) = θH (E) E. We can rewrite the relation in terms
of the density of states ρ = N 0 as
γ(E) − γ0 (E) =
Z
dE 0 ρ(E 0 ) − ρ0 (E 0 ) ln |E 0 − E|
(for Schr¨odinger equation). (II.79)
Up to the soustraction, required due to the fact that the spectrum is unbounded from above,
this is the same relation as (II.71). Analyticity shows that real and imaginary parts of Ω(E) are
related through a Hilbert transform.
-
Exercice II.8 Halperin’s result (1965) : In exercice II.6, we have obtained an integral
representation of the IDoS, due to Frisch and Lloyd [55]. An integral representation of the
Lyapunov exponent could be obtained similarly from the knowledge of the stationary distribution
f (z). In this exercice we show that γ and N can be expressed in terms of Airy functions. This
result is due to Halperin [63].
R
a) Fourier transform the differential equation (II.51) : fˆ(q) = dz e−iqz f (z).
26
b) Solve the differential equation for fˆ(q) on R+ (find the solution decaying for q → +∞).
Deduce that the complex Lyapunov exponent is
σ 1/3 Ai0 (ξ) − i Bi0 (ξ)
Ω(E) = γ(E) − iπ N (E) =
2
Ai(ξ) − i Bi(ξ)
2/3
2
where ξ = −
E.
σ
(II.80)
c) Recover the asymptotic behaviour for the Lyapunov exponent and the low energy density of
states (use that the Wronskian of the two Airy functions is W [Ai, Bi] = 1/π).
Appendix :
Airy equation f 00 (z) = z f (z) admits
independent
real solutions Ai and Bi
asymptotic
2 two 3/2
2 with3/2
1
π
−1
behaviours Ai(z) ' √π (−z)1/4 cos 3 (−z) − 4 and Bi(z) ' √π (−z)1/4 sin 3 (−z) − π4 for
z → −∞, and Ai(z) ' 2√π1z 1/4 exp − 23 z 3/2 and Bi(z) ' 2√π1z 1/4 exp 32 z 3/2 for z → +∞.
II.D
Self-averaging and non self-averaging quantities – Conductance distribution
Importance of fluctuations.– Note that the Furstenberg theorem does not characterize the
fluctuations of the wave function, but only the average growth rate. Generalised central limit
theorems have been developed for products of random matrices [58, 70, 76] (see [30, chapter
√
V]). Generalised central limit theorems state that fluctuations grow like x. Precisely, if one
introduces a Wiener process 9 W (x), we can write
“ ln |ψ(x)| = γ x +
√
γ2 W (x) ” .
(II.81)
The presence of the “· · · ” is here to remind that this representation only holds over large scales
x `c , where `c is the correlation length, and neglect the rapid oscillations of the wave function
on short scale. 10 Relative fluctuations of ln |ψ(x)| are negligible, however, when considering
the wave function ψ(x) itself, fluctuations appear in the exponential and can therefore have a
very strong effect on physical quantities, like the local DoS [12]. For example they dominate the
moments of the wave function
1 2
q
h|ψ(x)| i = exp
q γ2 + qγ x
(II.82)
2
Single parameter scaling (SPS) hypothesis.– The analysis of the fluctuations have plaid
a very important role in the theory of disordered systems since the early steps of the scaling
theory. The most simple scaling hypothesis was assumed, i.e. a single parameter scaling theory
[1, 13]. This was to assume that the full distribution of random quantities (like the conductance)
is controlled by a single length scale, i.e. γ = γ2 (note that this property is only asymptotically
for E → ∞).
It was later realised that such an assumption only holds in the weak disorder regime/high
energy, whereas the study of the strong disorder/low energy regime requires a two parameter
scaling theory [34] (i.e. average and fluctuations are controlled by two different scales γ > γ2 ).
9
A Wiener process W (x) is a normalised Brownian motion, i.e. a Gaussian process characterised by hW (x)i = 0
and hW (x)W (x0 )i = min(x, x0 ).
10
More rigorously we can follow the phase formalism : we perform the change
of variable (ψ, ψ 0 ) → (ξ, θ)
p
according to ψ = eξ sin θ and ψ 0 = keξ cos θ. Considering the envelope eξ = ψ 2 + (ψ 0 /k)2 is a natural way to
√
eliminate the oscillations. We write more corretly ξ(x) = γ x + γ2 W (x) (although this relation is only true over
scale larger than the correlation length).
27
2
1.2
1
1.5
and
1
2
0.8
2
/
1
0.6
1
0.4
0.5
0
0.2
0
-8
-6
-4
-2
0
2
4
-6
-4
-2
0
2
4
6
E
E
Figure 9: The two cumulants of γ1 = limx→∞ x1 ln |ψ(x)| (red cross) and γ2 =
d2
limx→∞ x1 Var ln |ψ(x)| (blue squares) for the model − dx
2 + V (x) as a function of the energy.
2/3
Whereas γ1 ' γ2 for E σ , they strongly deviates at low energy γ1 γ2 for −E σ 2/3 .
From [95].
Distribution of the conductance/transmission probability.– If we consider the solution
of the initial value problem ψ(x), say ψ(0) = 1 and ψ 0 (0) = 0 for example, we can express the
dimensionless conductance (i.e. the transmission probability) of a long wire as
g ≈ |ψ(L)|−2
(II.83)
This relation is only true for long L ξE . In principle the conductance involves a property of
the eigenstates (the scattering state) that is constructed by satisfying some appropriate matching
at the boundaries [15, 105], however over large scales, the wave function behaves exponentially
and loose the memory of its initial condition. We immediatly deduce that ln g is distributed
according to a normal distribution. Moreover, assuming SPS, i.e. considering the weak disorder
√
regime, energydisorder, we use ln |ψ(x)| = γx + γ W (x) (SPS) [15, 105] leading to
PL (ln g) ' √
(ln g + 2γL)2
1
exp −
.
8γL
8πγL
(II.84)
The distribution of the conductance itself is log-normal. The moments of the conductance are
immediatly deduced from (II.82), which shows that they increase exponentially fast :
hg n i ' e(2n−1)nγL .
(II.85)
In particular the relation hg n i1/n ' e(2n−1)γL shows that the moment of order n characterises
fluctuations exponentially larger than the moment of order n − 1.
Conclusion :
• g is not self averaging.
•
II.E
1
L
ln g is self averaging.
Experiment with cold atoms
Anderson localisation is an interference effect, and thus requires coherence on a large scale.
In metals, Anderson localisation competes with interaction whose effect is twofold : first it
breaks phase coherence in the weak disorder regime, second it can induce localisation (Mott
transition) in the strong localisation regime.
28
In 1958, P.W. Anderson predicted the exponential wave functions in space has been reported in condensed
localization1 of electronic wave functions in disordered matter.
crystals and the resulting absence of diffusion. It has been
realized later that Anderson localization (AL) is
ubiquitous in wave physics2 as it originates from the
interference between multiple scattering paths, and this
has prompted an intense activity. Experimentally,
In optics,
Anderson localisation is difficult (but not impossible) to distinguish from absorplocalization has been reported in light waves3,4,5,6,7,
tion. microwaves8,9, sound waves10, and electron11 gases but to1
our knowledge there is no direct observation of
Recently,
a spatial
beautiful
observation
of (electrons
localisation of matter waves has been achieved in a cold
exponential
localization
of matter-waves
n of Anderson atom
localization
ofHere,
matter-waves
in aiscontrolled
or others).
we report
the observation
of exponential
experiment.
An
atomic
gas
trapped
by a harmonic well (figure 10.a). A disordered polocalization of a Bose-Einstein condensate (BEC) released
tential into
is superimposed,
resulting
a speckle
pattern (blue) together with a one-dimensional
a one-dimensional waveguide
in from
the presence
of a
12
controlled
disorderAt
created
by laser
speckle
. We the
operate
1
1
1
1
confinment
(pink).
some
initial
time
harmonic
well is released (figure 10.b) : a fraction
se1, Zhanchun Zuo1, Alain
Bernard
,
Ben
Hambrecht
,
Pierre
Lugan
,
David
Clément
,
in 1a regime allowing AL: i) weak disorder such that
1
, Philippe Bouyer1 & Alain
Aspect
of the
gas
remains
the disordered
localization
resultslocalised
from many by
quantum
reflections of potential (green profile on figure b), exhibiting
small amplitude; ii) atomic density small enough that
exponential
tails.
ThisCampus
experiment
is therefore
ry de l'Institut d'Optique,
CNRS and
Univ.
Paris-Sud,
RD atomic
128, F- a rather direct observation of exponential localinteractions
are negligible.
We Polytechnique,
image directly the
rance
profiles
vs time, and Fig.
find that
weak (although
disorder can what is observed is not a wave function but a
isation density
of wave
functions,
10.d,
lead to the stopping of the expansion and to the formation
densityofwave
profile).
remarkable
control
on amicroscopic parameters in cold atom experiment
n predicted the exponential
functionsThe
in space
has localized
been reported
condensed
a stationary
exponentially
wave infunction,
wave functions in disordered
matter.
direct
of AL. Fitting
the exponential wings, we
allows for
asignature
quantitative
analysis.
bsence of diffusion. It has been extract the localization length, and compare it to
derson localization (AL) is theoretical calculations. Moreover we show that, in our
cs2 as it originates from the one-dimensional speckle potentials whose noise spectrum
iple scattering paths, and this has a high spatial frequency cut-off, exponential
nse activity. Experimentally, localization occurs only when the de Broglie wavelengths
ported in light waves3,4,5,6,7, of the atoms in the expanding BEC are larger than an
s10, and electron11 gases but to effective mobility edge corresponding to that cut-off. In
of exponential localization. a) A
no direct observation of the opposite case, we find that the density profiles decay Figure 1. Observation
4
small BEC (1.7 x 10 atoms) is formed in a hybrid trap, which
tion of matter-waves (electrons algebraically, as predicted in ref 13. The method
is the combination of a horizontal optical waveguide ensuring
the observation of exponential presented here can be extended to localization of atomic a strong transverse confinement, and a loose magnetic
ein condensate (BEC) released quantum gases in higher dimensions, and with controlled longitudinal trap. A weak disordered optical potential,
aveguide in the presence of a interactions.
transversely invariant over the atomic cloud, is superimposed
by laser speckle12. We operate
(disorder amplitude VR small compared to the chemical
i) weak disorder such that
The transport of quantum particles in non ideal potential !in of the atoms in the initial BEC). b) When the
many quantum reflections of material media (e.g. the conduction of electrons in an longitudinal trap is switched off, the BEC starts expanding and
ic density small enough that imperfect crystal) is strongly affected by scattering from the then localises, as observed by direct imaging of the
We image directly the atomic impurities of the medium. Even for weak disorder, semi- fluorescence of the atoms irradiated by a resonant probe. On
a and b, false colour images and sketched profiles are for
d find that weak disorder can classical theories, such as those based on the Boltzmann
illustration purpose, they are not exactly on scale. c-d)
expansion and to the formation equation for matter-waves scattering from the impurities, Density profile of the localised BEC, 1s after release, in linear
lly localized wave function, a often fail to describe transport properties2, and fully quantum or semi-logarithmic coordinates. The inset of Fig d (rms width
ting the exponential wings, we approaches are necessary. For instance, the celebrated ot the profile vs time t, with or without disordered potential)
length, and compare it to Anderson localization1, which predicts metal-insulator shows that the stationary regime is reached after 0.5 s. The
oreover we show that, in our transitions, is based on interference between multiple diamond at t=1s corresponds to the data shown in Fig c-d.
otentials whose noise spectrum scattering paths, leading to localized wave functions with Blue solid lines in Fig c are exponential fits to the wings,
equency cut-off, exponential exponentially decaying profiles. While Anderson's theory corresponding to the straight lines of Fig d. The narrow profile
hen the de Broglie wavelengths applies to non-interacting particles in static (quenched) at the centre represents the trapped condensate before
release (t=0).
ding BEC are larger than an disordered potentials1, both thermal phonons and repulsive
14,15
rresponding to that cut-off. In inter-particle interactions significantly affect AL . To our
Figure 1. Observation of exponential localization. a) A
that the density profiles decay knowledge,
4
no direct
of of
exponentially
localized
Figure 10:
Strong
atoms
localised by the random potential resulting from a
small BEC
(1.7
x localisation
10 observation
atoms) is formed
incold
a hybrid
trap,
which
ed in ref 13. The method is the combination of a horizontal optical waveguide ensuring
2008-04-14,09:04:00
ended to localization ofSpeckle
atomic preprint.doc,
pattern.
From
Ref.
[28].
a strong transverse confinement, and a loose magnetic
imensions, and with controlled longitudinal trap. A weak disordered optical potential,
transversely invariant over the atomic cloud, is superimposed
(disorder amplitude VR small compared to the chemical
potential !in of the atoms in the initial BEC). b) When the
longitudinal trap is switched off, the BEC starts expanding and
then localises, as observed by direct imaging of the
fluorescence of the atoms irradiated by a resonant probe. On
a and b, false colour images and sketched profiles are for
illustration purpose, they are not exactly on scale. c-d)
Density profile of the localised BEC, 1s after release, in linear
or semi-logarithmic coordinates. The inset of Fig d (rms width
ot the profile vs time t, with or without disordered potential)
shows that the stationary regime is reached after 0.5 s. The
diamond at t=1s corresponds to the data shown in Fig c-d.
Blue solid lines in Fig c are exponential fits to the wings,
corresponding to the straight lines of Fig d. The narrow profile
at the centre represents the trapped condensate before
release (t=0).
uantum particles in non ideal
conduction of electrons in an
y affected by scattering from the
Even for weak disorder, semithose based on the Boltzmann
Anderson localisation is an interference effect.
scattering from the impurities,
rt properties2, and fully quantum
For instance, the celebrated
Discuss paper by Berry & Klein [26]
which predicts metal-insulator
interference between multiple
localized wave functions with
files. While Anderson's theory
particles in static (quenched)
thermal phonons and repulsive
In practice, the strictly one-dimensional description may not be relevant, in particular in the
gnificantly affect AL14,15. To our
context of electronic transport. Most of the experiments on quantum wires deal with wires
vation of exponentially localized
II.F
Transfer matrix and scattering matrix : Anderson localisation versus
Ohm’s law
II.G
The quasi-1D situation (multichannel case)
with a large number of conducting channels Nc 1 (metallic wires are equivalent, for
electronic waves, to optical waveguides for electromagnetic waves). Only few experimental
setup correspond to the 1D situation : conducting carbon nanotubes are effectively 1D with
4 degenerate channels, the cleaved edge overgrowth technics also allows to realise 1D devices
(spin degenerate) [42]. However, in these cases, the existence of interaction drastically affect the
electronic properties and the 1D electron liquid cannot be described like a Fermi liquid.
29
Some of the ideas introduced for the strictly 1D case can be extended to the multichannel
case : it is possible to develop a transfer matrix formalism, known as the DMPK formalism
[44, 87, 45]. An excellent review article is the one written by Beenakker [20] (also see the book
[86]). Assuming some maximal entropy principle (i.e. conducting channels are all mixed with
equal probability after an elementary slice of disorder medium) it is possible to obtain a spectrum
of Lyapunov exponents characterising the transfer matrix. One obtains a linear spectrum [20]
γn =
1 1 + β (n − 1)
αd `e 2 + β (Nc − 1)
for n ∈ {1, · · · , Nc }
(II.86)
where β ∈ {1, 2, 4} is the Dyson index 11 and `e the elastic mean free path. αd = Vd /Vd−1 is
a number of order 1 (Vd = π d/2 /Γ(d/2 + 1) being the volume of the unit sphere). The smallest
Lyapunov exponent is interpreted as the inverse localisation length, which thus scales as
ξ ∼ Nc `e
(II.87)
An interesting outcome of this analysis is the identification of a new regime that was not
present in the strictly 1D situation. Considering a wire of finite length L with a large number
of channels Nc 1, there exists a room between the localised regime (L ξ ∼ Nc `e ) and
the ballistic regime (`e L) for a new regime : the diffusive regime (`e L ξ). This
regime will be the main subject of investigation of the chapter IV.
Further reading
Localisation and spectral properties of 1D disordered discrete models are described in the
pedagogical book of J.-M. Luck [80].
More diverse but less pedagogical reference is the book by Lifshits, Gredeskul & Pastur [79].
The reader interested in mathematical aspects can have a look on Bougerol, Carmona &
Lacroix’ monographs [30, 32].
A recent review on Lyapunov exponent for 1D localisation is [39].
For the study of the multichannel disordered wires : the pedagogical review of Beenakker [20]
or the more complete book of Mello & Kumar [86].
A nice little review article on multichannel disordered wires in the presence of interaction is
[88].
11
In random matrix theory, Dyson index corresponds to the symmetry of the problem : β = 1 corresponds to
a problem with TRS (time reversal symmetry). β = 2 in the presence of a strong magnetic field breaking TRS.
β = 4 when the disorder breaks the spin rotational symmetry (with TRS).
30
Appendix : Langevin equation and Fokker-Planck equation
a) Preliminary : Gaussian white noise and Wiener process
We introduce a normalised Gaussian white noise η(t), distributed according to the measure
Z
1
(II.88)
dt η(t)2 ,
P [η] = exp −
2
i.e. such that hη(t)i = 0 and hη(t)η(t0 )i = δ(t − t0 ). The solution of the differential equation
dW (t)
= η(t)
dt
(II.89)
with initial condition W (0) = 0 is a Wiener process. It is obvious that W (t) =
Gaussian process, characterised by the correlation function
W (t) W (t0 ) = min t, t0 .
Rt
0
dt0 η(t0 ) is a
(II.90)
Rescaling the time : let us consider the mapping t = f (u), where f is a function increasing
monotenously. We have
δ(u − u0 )
η(f (u)) η(f (u0 )) =
(II.91)
f 0 (u)
what we rewrite under the form of an “equality in law ”
η(u)
(law)
η(f (u)) = p
f 0 (u)
(II.92)
meaning that the two sides of the equation have the same statistical properties (they are both
Gaussian and have the same two point correlation function).
As a trivial example we choose f (t) = α t, where α is a positive number
1
(law)
η(α t) = √ η(t)
α
⇒
(law)
W (α t) =
√
α W (t) ,
(II.93)
which is the usual scaling property of the Brownian motion.
b) Stochastic differential equation and Fokker-Planck equation
We recall some basic tools for homogeneous random process (process with no jumps). Consider
the stochastic differential equation (SDE) 12
d
x(t) = F(x(t)) + B(x(t)) η(t) ,
dt
(II.94)
p
interpreted within the Stratonovich convention. F(x) is a force (drift) and B(x) = 2D(x)
describes an inhomogeneous diffusion constant. The SDE is related to the Fokker Planck equation
∂
1
P (x; t) = G † P (x; t) where G = B(x)∂x B(x)∂x + F(x)∂x
(II.95)
∂t
2
is the generator of the diffusion, i.e. the backward generator, adjoint of the forward generator
G † = 12 ∂x B(x)∂x B(x) − ∂x F(x). The Fokker-Planck equation may be written under the form
of a conservation equation
∂
∂
1
P (x; t) = − J(x; t) where J(x; t) = F(x) − B(x)∂x B(x) P (x; t)
(II.96)
∂t
∂x
2
12
which Mathematicians prefer to write dx(t) = F(x(t)) dt + B(x(t)) dW (t).
31
is the current density (in 1D, current density is a current).
Generalization to higher dimensions is rather natural ; it may be found in standard monographs like [59]. Consider the set of SDEs
d
xi (t) = Fi (~x(t)) + Bij (~x(t)) ηj (t)
dt
(II.97)
with implicit summation over repeated indices. ηi ’s are identical and independent noises hηi (t)ηj (t0 )i =
δij δ(t − t0 ). The generator of the diffusion is now
1
G = Bik ∂i Bjk ∂j + Fi ∂i ,
2
(II.98)
where ∂i ≡ ∂/∂xi .
c) The case of a constant diffusion constant
Let us focus on the simpler case where the diffusion constant is uniform in space : B(x)2 = 2D.
In 1D, the drift (the force) can always be written as the derivative of a potential : F(x) = −U 0 (x).
In this case it is useful to write
G† = D
1
d 0
d
d2
d 1
+
U (x) = D e− D U (x) e D U (x)
2
dx
dx
dx
dx
(II.99)
Stationary state.– This form of the generator immediatly allows to find the solution of
G † P (x) = 0, i.e. the stationary state. Three possibilities :
1. The solution
1
Peq (x) = C e− D U (x)
(II.100)
is normalisatble. This describes an equilibrium state.
2. Peq (x) is non normalisable, however the solution with finite current
Z
J − 1 U (x) x 0 1 U (x0 )
D
PJ (x) = − e
dx e D
D
(II.101)
is normalisable, now describing an out-of-equilibrium stationary state.
3. If none of the solutions is normalisable, the problem considered simply does not have a
stationary state.
Examples :
k
1. Ornstein-Uhlenbeck process.– For F(x) = −2k x, the equilibrium solution Peq (x) = C e− D x
is normalisable.
2
2. For F(x) = F0 − x2 , the potential is unbounded from below U(x) = −F0 x + 31 x3 , however
the stationary distribution PJ (x) is normalisable.
3. Brownian motion.– For F(x) = 0, none of the two solutions Peq (x) and PJ (x) is normalisable and there exist no stationary state.
32
-
Exercice II.9 Mapping between the Wiener and Ornstein-Uhlenbeck processes :
d
Consider a Wiener process, described by the SDE dt
x(t) = η(t), where η(t) is a Gaussian white
noise. Check that the transformation
τ = ln t
x(t)
y(τ ) = √
t
(II.102)
(II.103)
maps the Wiener process onto the Ornstein-Ulhenbeck process
d
1
y(τ ) = − y(τ ) + η(τ ) .
dτ
2
33
(II.104)
Problem : Weak disorder expansion in 1D lattice models and band center
anomaly
Let us consider the 1D Anderson (tight binding) model
−t ψn+1 + Vn ψn − t ψn−1 = ε ψn
(II.105)
where t is the coupling between nearest neighbouring sites and Vn a random potential.
In the absence of potential, eigenstates are plane waves ψn = eikn of energy εk = −2t cos k,
where the (dimensionless) wavevector belongs to the Brillouin zone k ∈ [−π, π]. Plane waves
are characterised by the following density of states and Lyapunov exponent :
1
ρ0 (ε) = p
π (2t)2 − ε2
⇒ N0 (ε) =
1 1
− arccos(ε/2t)
2 π
γ0 (ε) = argcosh|ε/2t|
(II.106)
(II.107)
1/ Recover rapidly these properties. Check that γ0 (ε) and N0 (ε) are real and imaginary part of
an analytic function of ε.
A natural question is the search for weak disorder expansions of the density of states and
the Lyapunov exponent under the form of a weak disorder expansion (perturbative
like). For
2
uncorrelated site potentials, hVn Vm i ∝ δn,m , with finite moments, hVn i = 0, Vn < ∞, etc, one
finds [43, 80]
3
Vn
+ O(Vn5 )
(II.108)
N (ε) = N0 (ε) −
3
3
3
2 24π t sin k
V
(II.109)
γ(ε) = 2 n 2 + O(Vn4 )
8t sin k
where k parametrizes the energy ε = −2t cos k.
2/ Plot the qualitative ρ(ε) and γ(ε) expected from these formulae (with the one corresponding
to Vn = 0).
It was however observed, numerically by Czycholl, Kramer & MacKinnon [40], and by Kappus
and Wegner [71], that these formulae fail to describe accuratly the band center properties (ε → 0,
i.e. k → π/2). This is at first sight surprising : although it is clear that the expansion has some
problem at the band egdes, k → 0 ou π, where disorder is non perturbative, this seems not to
be the case in the band center. Inspection of the full series however shows that it breaks down
at the band center. It was shown by Derrida and Gardner [43] that the correct weak disorder
expansion is
Γ(3/4) 2 Vn2
γ(ε) =
+ O(ε)
(II.110)
Γ(1/4)
t2
showing a small difference with the previous expansion, 0.125 vs 0.114... This phenomenon,
denoted the “band center anomaly” (or “Kappus Wegner anomaly”), is a lattice effect due
to commensurability. In order to clarify this, we now develop a continuum model. This is
conveniently done by Fourier transforming the difference equation (II.105).
X
˜
ψ(k)
=
ψn e−ikn
(II.111)
n
Z
+π
ψn =
−π
dk ˜
ψ(k) eikn
2π
34
(II.112)
(we recall the Poisson formula
ikn
ne
P
= 2π
P
n δ(k
− 2πn)).
3/ Rewrite the Schr¨
odinger equation in Fourier space.
4/ We write Vn = A0 (na) + (−1)n m(na) where a is the lattice spacing ; A0 (x) and m(x) are
two smooth functions. Discuss the qualitative structure of the Fourier transform V˜ (k). The
low energy properties of the model involves the Fourier components of the wave function at
k ∼ ±π/2. Argue that only the potential’s Fourier components |q| 1 and |π ± q| 1 are of
importance in this case.
5/ For |κ| 1, we introduce
˜
ψ(π/2
+ κ) = ϕ(κ)
˜
(II.113)
˜
ψ(−π/2
+ κ) = χ(κ)
˜
(II.114)
Show that the ϕ(κ)
˜
and χ(κ)
˜
obeys two coupled equations. Going back to real space, deduce
that the low energy properties of the Anderson model are described by the Dirac equation
for HD = −iσ3 v0 ∂x + σ1 m(x) + A0 (x)
HD Ψ(x) = ε Ψ(x)
(II.115)
where Ψ = (ϕ, χ). At the light of this result, can you explain the breakdown of the weak disorder
expansion at the band center ?
6/ The large scale (low energy) properties of the model are conveniently described by considering
white noises :
A0 (x) A0 (x0 ) = g0 δ(x − x0 )
(II.116)
0
0
m(x) m(x ) = gπ δ(x − x )
(II.117)
(II.118)
The zero energy Lyapunov exponent has been computed exactly in this case [36, 95] and is given
by elliptic integrals (now v0 = 1)
1 E(ξ)
1
γ(0) = gπ
− 1 + 1 with ξ = p
(II.119)
2
ξ
K(ξ)
1 + g0 /gπ
What is the value of g0 /gπ describing the case originally studied of uncorrelated site potentials
hVn Vm i ∝ δn,m ? How can one makes the small anomaly large ?
Appendix :
we recall few properties of the Elliptic integrals
Z π/2
dt
p
K(k) =
0
1 − k 2 sin2 t
Z π/2 p
E(k) =
dt 1 − k 2 sin2 t
For k → 1− we have (setting k 0 =
√
(II.121)
0
1 − k2 )
2
K(k) = ln(4/k 0 ) + O(k 0 ln k 0 )
1 2
4
E(k) = 1 + k 0 ln(4/k 0 ) − 1/2 + O(k 0 ln(1/k 0 ))
2
We give
(II.120)
√
E(1/ 2)
2Γ(3/4) 2
√ −1=
2
Γ(1/4)
K(1/ 2)
35
(II.122)
(II.123)
(II.124)
Master iCFP - Wave in random media
May 3, 2015
TD 2 : Thouless relation
We demonstrate a relation between the Lyapunov exponent (i.e. the localisation) and the density
of states. This relation relies on the existence of an analytic function encoding both informations.
A. Discrete tight binding model.– We study the Sch¨odinger equation on a regular 1D
lattice :
−ψn+1 + Vn ψn − ψn−1 = ε ψn
(II.125)
We first consider the problem on N sites n ∈ {1, · · · , N } and impose the boundary conditions
ψ0 = ψN +1 = 0 (the quantification equation) resulting in the spectrum of N eigenvalues {εα }.
1/ We denote by Ψn (ε) the solution of the initial value problem Eq. (II.125) with Ψ0 (ε) = 0
and Ψ1 (ε) = 1. Argue that ΨN +1 (ε) is the polynomial of degree N
ΨN +1 (ε) =
N
Y
(εα − ε) .
(II.126)
α=1
2/ Thouless relation.– We define the density of states per site, which is a continuous function
in the limit N → ∞, and the Lyapunov exponent
def
N
1 X
δ(ε − εα )
N →∞ N
ρ(ε) = lim
and
def
ln |ΨN +1 (ε)|
.
N →∞
N
γ(ε) = lim
(II.127)
dε0 ρ(ε0 ) ln |ε0 − ε|
(II.128)
α=1
Deduce the relation
Z
γ(ε) =
3/ Complex Lyapunov exponent (characteristic function).– We consider the analytic
function
ln ΨN +1 (ε)
def
Ω(ε) = lim
for ε ∈ C .
(II.129)
N →∞
N
Discuss its analytic properties. Show that the Lyapunov exponent and the density of states are
related to the real and imaginary parts of Ω(ε) on the real axis. What is the relation of this
observation with the Thouless relation ?
Hint : we recall limη→0± ln(x + iη) = ln |x| ± iπ θH (−x).
4/ As a first (trivial) application of the concept of complex Lyapunov exponent, we consider the
free problem (Vn = 0).
a) What is the spectrum of the Schr¨
odinger equation (II.125) in this case ?
b) We consider an energy ε = −2 cosh q < −2. Find Ψn (ε) and deduce Ω(ε).
c) By an analytic continuation, deduce the spectral density ρ(ε). Plot on the same graph the
Lyapunov exponent γ(ε) and the spectral density ρ(ε).
Hint :
√
√
argcosh(x) = ln(x + x2 − 1) for x ∈ R+ and arcsin(x) = −i ln(ix + 1 − x2 ) for x ∈ [−1, +1].
d ) In the presence of a weak disordered potential, what behaviours do you expect for ρ(ε) and
γ(ε) ?
36
B. Continuous model.– In this part we consider the continuous model
d2
− 2 + V (x) ϕ(x) = E ϕ(x)
dx
(II.130)
where V (x) is a Gaussian white noise of zero mean with hV (x)V (x0 )i = σ δ(x − x0 ). We denote
by ψ(x; E) the solution of the Cauchy problem satisfying ψ(0; E) and ψ 0 (0; E) = 1, where
0 ≡ ∂ . We will make use of the concept of complex Lyapunov exponent in order to get analytic
x
expressions for the Lyapunov exponent and the spectral density.
def
Riccati analysis.– The Riccati variable z(x; E) = ψ 0 (x; E)/ψ(x; E) obeys the Langevin equation z 0 = −E − z 2 + V (x). The probability current of the Riccati variable through R coincides
with the integrated density of states (IDoS) per unit length N (E). It solves
σ d
2
(II.131)
+ E + z f (z; ) = N (E)
2 dz
where f (z; E) is the (normalised) probability density for the stationary process z(x; E).
1/ Using only the definition of z(x; E), show that its average hzi coincides with the Lyapunov
exponent
ln |ψ(x; E)|
γ(E) = lim
.
(II.132)
x→∞
x
2/ We consider the Fourier transform of the distribution
Z
fˆ(q; E) = dz e−iqz f (z; E)
(II.133)
Argue that Im[fˆ0 (0+ ; E)] = −γ(E). Fourier transforming Eq. (II.131), show that Re[fˆ0 (0+ ; E)] =
− Re[fˆ0 (0− ; E)] = π N (E). Deduce the relation between fˆ0 (0+ ; E) and the complex Lyapunov
exponent
Ω(E) = γ(E) − iπ N (E) .
(II.134)
ˆ
3/ Justify that fˆ(0; E) = 1 and that
that the solution on R+∗
f (q; E) decays at infinity. Show
vanishing at infinity is fˆ(q; E) = C Ai(−E − iq) − i Bi(−E − iq) (for σ/2 = 1).
4/ Recover the asymptotic behaviours for the Lyapunov exponent and the low energy density
of states.
Appendix :
Airy equation f 00 (z) = z f (z) admits
independent
real solutions Ai and Bi
asymptotic
2 two 3/2
2 with3/2
1
π
−1
√
√
behaviours Ai(z) ' π (−z)1/4 cos 3 (−z) − 4 and Bi(z) ' π (−z)1/4 sin 3 (−z) − π4 for
z → −∞, and Ai(z) ' 2√π1z 1/4 exp − 23 z 3/2 and Bi(z) ' 2√π1z 1/4 exp 32 z 3/2 for z → +∞.
The Wronskian of the two Airy functions is W [Ai, Bi] = Ai Bi0 − Ai0 Bi = 1/π.
Further reading : • J.-M. Luck, Syst`emes d´esordonn´es unidimensionnels, coll. Al´ea Saclay, (1992).
• The Thouless relation was introduced in : D. J. Thouless, A relation between the density of states and
range of localization for one-dimensional random systems, J. Phys. C: Solid St. Phys. 5, 77 (1972).
• Exact calculation of N (E) for the continuous model has been performed in : B. I. Halperin, Green’s
Functions for a Particle in a One-Dimensional Random Potential, Phys. Rev. 139(1A), A104–A117
(1965).
• The interest of the method we have exposed here is to relate the determination of the complex Lyapunov exponent to the resolution of a differential equation. This has been recently exploited for a more
general class of models in : A. Grabsch, C. Texier and Y. Tourigny, One-dimensional disordered quantum
mechanics and Sinai diffusion with random absorbers, J. Stat. Phys. 155(2), 237–276 (2014).
37
Master iCFP - Wave in random media
May 3, 2015
TD 3 : Localisation for the random Kronig-Penney model – Concentration expansion and Lifshits tail
The aim of the problem is to study a 1D disordered model introduced by Frisch and Lloyd in
Ref. [55]. This model is a random version of the Kronig-Penney model
H=−
X
d2
+
vn δ(x − xn ) .
dx2
n
(II.135)
Impurity positions xn ’s are distributed identically and independently for a uniform mean density
ρ. We consider the case where the weights vn ’s are also independent random variables. 13 We
prove the Anderson localisation of eigenstates and analyse the low energy density of states.
1/ The positions are ordered as x0 = 0 < x1 < x2 < · · · < xn < · · · Denoting by `n =
xn+1 − xn > 0 the distance between consecutive impurities, recall the distribution of these
lengths.
2/ Riccati.– We denote by ψ(x; E) the solution of the initial value problem Hψ(x; E) =
E ψ(x; E) with ψ(0; E) = 0 and ψ 0 (0; E) = 1. Derive the stochastic differential equation condef
trolling the evolution of the Riccati variable z(x) = ψ 0 (x; E)/ψ(x; E).
−
3/ Describe the effect of the random potential on its dynamics (i.e. relate z(x+
n ) to z(xn )).
4/ We denote by f (z; x) = hδ(z − z(x))i the probability distribution of the random process.
Show that the distribution of the Riccati variable obeys the integro-differential equation
∂x f (z; x) = ∂z (E + z 2 )f (z; x) + ρ hf (z − v; x) − f (z; x)iv
(II.136)
where h· · ·iv denotes averaging over the random weights vn ’s.
Hint : analyse the effects of the two terms of the SDE in order to relate f (z; x + dx) to f (z 0 ; x).
5/ Probability current and stationary distribution.– Rewrite (II.136) under the form of
a conservation equation ∂x f (z; x) = −∂z J(z; x), where J(z; x) is the probability current density.
We have seen that the disribution reaches a stationary distribution f (z; x) −→ f (z) for a
x→∞
steady current J(z; x) −→ −N , related to the Integrated density of states per unit length of
x→∞
the disordered Hamiltonian. Show that the stationary distribution obeys the integral equation
Z z
2
0
0
N (E) = (E + z )f (z) − ρ
dz f (z )
.
(II.137)
z−v
v
What is the condition on the weights vn for having a non vanishing density of states for E < 0 ?
6/ High density
limit.– We consider the case hvn i = 0. Discuss the limit ρ → ∞ and vn → 0
with σ = ρ vn2 fixed (no calculation).
7/ Small concentration expansion.– We now discuss the opposite limit when ρ vn . We search
for the solution of the integro-differential equation under the form of an expansion f (z) =
13
Note that Frisch and Lloyd considered the case of random positions and fixed weights. The case of non
random positions (on a lattice) and fixed weights was considered earlier by Schmidt in [99]. The case of random
positions and random weights was also considered in several papers, e.g. by Nieuwenhuizen [90].
38
f (0) (z) + f (1) (z) + f (2) (z) + · · · where f (n) = O(ρn ). Accordingly the density of states presents
a similar expansion N = N (0) + N (1) + · · · We recall that the Lyapunov exponent is given by
Z
γ = r dz z f (z)
R
where
Z
Z
def
r dz h(z) = lim
R
+R
R→+∞ −R
Z
dz h(z) =
dz
R
h(z) + h(−z)
(II.138)
2
a) Compute f (1) and deduce that the Lyapunov exponent at lowest order in ρ is
v 2 ρ
n
γ=
ln 1 +
+ O(ρ2 )
2
2k
vn
(II.139)
Hint : We give the integral
Z
π t
(arctan(t) − arctan(t − x)) = ln 1 + (x/2)2 ,
(II.140)
dt 2
t +1
2
R
i−t
which could be computed by writing arctan(t) = 2i1 ln i+t
and using the Residue’s theorem.
b) Study the limiting cases, setting E = k 2 :
(i) High energy limit k vn , ρ.
(ii) Intermediate energy range, vn k ρ.
(iii) The concentration expansion breaks down at k ∼ ρ. What is the estimate for the saturation
value at E = 0 ?
8/ Lifshits tail.– For positive weights vn , the sectrum is in R+ . An approximation for the low
energy IDoS can be obtained as follows. In the limit vnP→ ∞ the intervals between impurities
∞
2
are disconnected. We introduce the IDoS N (E; `) =
n=1 θH (E − (nπ/`) ) for the interval
of length `. Shows that the IDoS per unit length of the disordered Hamiltonian is given by
N (E) ' ρ hN (E; `)i` for ρ → 0. Deduce an explicit form for N (E) and analyse the low energy
behaviour E ρ2 .
Further reading : This analysis was performed in our article with Tom Bienaim´e, Ref. [27].
More information about the concentration expansion and be found in the book of Lishits, Gredeskul and Pastur [79].
39
III
Scaling theory : qualitative picture
Aim : The analysis of the 1D situation (chapter II) is interesting because it can be performed
with the help of powerful (non perturbative) methods. The drawback is that we completely miss
the rich phenomenology of Anderson localisation because dimensionality plays a crucial role.
We give a first presentation of the famous scaling theory developed by the “gang of four”
[1] inspired by Thouless ideas [112, 13]. This will provide the useful concepts in order to have a
general view on localisation effects.
III.A
Several types of insulators
Several types of insulators in condensed matter :
• Band insulator (Fermi energy in a gap)
• Topological insulator (insulating phase in the presence of a topological constraint).
An interesting aspect : existence of edge states between regions characterised by different
topological properties.
Example : quantum Hall state.
• Mott insulator : transport is forbidden due to a strong Coulomb interaction (example : a
lattice with one electron per site and interaction U → ∞)
• Anderson insulator : eigenstates becomes localised due to disorder.
III.B
Models of disorder
• Distribution of on site potential
• Correlation functions
Anderson model
Lattice model : ~r ∈ Zd . Tight-binding Hamiltonian with nearest neighbour couplings
pure crystal
W < Wc
localised
E
ρ(E) Es
delocalised
delocalised
W =0
(III.1)
µ=1
localised
ρ(E)
δ~r,~r 0 +~eµ + δ~r,~r 0 −~eµ + δ~r,~r 0 V~r
E
weakly disordered crystal
ρ(E)
W > Wc
localised
H~r,~r 0 = −t
d
X
E
strongly disordered crystal
Figure 11: Sketch of the DoS for the 3D Anderson model. Left : In the absence of
disorder. Center : With “weak” disorder, i.e. below the threshold for complete localisation
W < Wc . Existence of a mobility edge at ES . Right : Above the threshold W > Wc (for the
cubic lattice Wc = 16.3 ± 0.5 [74]).
40
Edwards model : imurities
H = −∆ +
III.C
X
n
vn δ(~r − ~rn )
(III.2)
What is localisation ?
Transport properties
Vanishing of the diffusion constant D = 0
~r(t)2
=0
lim
t→∞
t
(III.3)
Inverse partition ratio
In a box of volume Ld , i.e. ~r ∈ Zd where xµ ∈ {1, · · · , L}. Study the behaviour IPR
X
I(L) =
|ψ~r |4
(III.4)
~
r∈ box
with L :
• Delocalised : I(L) ∼ L−d
• Localised : I(L) ∼ ξ −d = L0
Mathematicians
The definition of the mathematicians : absolute continuous spectrum (delocalised) versus pure
point spectrum (localised).
Consider the problem in infinite volume, characterised by the local DoS
X
ρ(~r; ε) =
|ϕn (~r)2 | δ(ε − εn ) .
(III.5)
n
The total density of states is a continuous function, weakly sensitive to the nature of the eigenstates (localised/delocalised), apart on the spectral boundaries.
We now integrate the LDoS in a finite volume v
Z
def
ρv (ε) =
d~r ρ(~r; ε)
(III.6)
v
Two situations are encountered
1. The eigenstates are localised : ρv (ε) only receives the contributions of the eigenstates
localised in the volume v and therefore is a sum of few delta functions. The spectrum is
said to be “pure point”.
2. The eigenstates are delocalised : all eigenstates contribute to ρv (ε), which is a continuous
function of energy. The spectrum is said to be “absolute continuous”.
The point of view of a mathematician on localisation : the course by Werner Kirsch [72].
Let’s forget physics for the Anderson’s paper 50th birthday !
41
III.D
Length scales
• λ : wavelength
• `corr : correlation length
• ` : mean free path
• `∗ : transport mean free path
• ξloc : localisation length
• Lϕ : phase coherence length
• L : size of the system
L`
` L ξloc
ξloc L Lϕ
Lϕ L
ballistic
diffusive
localised (by disorder)
classical
Table 2: Regimes
Conductance
Conductance of a metallic box of section W d−1 and length L
GDrude = σ
W d−1
L
(III.7)
Using Einstein’s relation
σ = 2s e2 ρ0 D
(III.8)
where 2s is the spin degeneracy and ρ0 the density of states per unit volume at Fermi energy
per spin channel.
ρ0 ∼ n/εF ∼
d−2
m∗ kF
,
2
~
we obtain the dimensionless conductance :
e2
G
g=
∼
2s e2 /h
def
d−2
m∗ kF
vF `e
d
~2
e2 /~
W d−1
(kF W )d−1 `e
∼
L
L
(III.9)
Nc ∼ (kF W )d−1 is the number of conducting channels.
More precisely, electon density is
n = 2s
Thus 2π~ρ0 D =
g = cd
(4π)1−d/2 d−1
k `e
2Γ( d2 +1) F
(kF W )d−1 `e
L
Vd kFd
kFd
=
2
.
s
(2π)d
(4π)d/2 Γ( d2 + 1)
(III.10)
and


2
where cd =
= 1/2
2Γ( d2 + 1) 

1/(3π)
(4π)1−d/2
42
in d = 1
in d = 2
in d = 3
(III.11)
Thouless energy
The dimensionless conductance can be expressed in terms of the Thouless energy :
DoS=1/δ
z }| {
W d−1
hD
d−1
g = hρ0 D
= 2 ρ0 W
| {z L}
L
L
(III.12)
Vol
The dimensionless conductance is given by the ratio of two energies : the Thouless energy,
related to the Thouless time τD = L2 /D required to explore the system, and the mean level
spacing δ (inverse of the DoS) :
g = 2π
EThouless
δ
def
where EThouless =
~D
L2
(III.13)
Ioffe-Regel criterion
localisation threshold (for d > 3) :
k F `e ∼ 1
III.E
(III.14)
Scaling theory of localisation
Generalities on scaling theories.
Explicit calculation for 1D and quasi-1D systems.
Scaling theory in arbitrary dimension and possible existence of a metal-insulator transition
(Thouless criterion).
Experimental results in 3D on light, microwave, acoustic waves, electrons, cold atoms. Large
fluctuations in the vicinity of the transition, multifractality of the wavefunctions.
Specific case of dimension 2. Importance of the symmetry class.
Scaling properties of the plateaux of the Integer Quantum Hall Effect (Chalker-Coddington
model).
Definition of the metallic and insulating phases within the scaling theory
In a gedanken experiment, study how the conductance of a cube of size Ld behaves with system
size.
• L % ⇒ g % : metallic phase
• L % ⇒ g & : insulating phase
This can be conveniently described by considering the function
def
β(g) =
d ln g
d ln L
(III.15)
The non trivial assumption is that it depends only on one parameter (here chosen to be the
conductance).
Conclusion of the chapter : Determination of the β(g) function is extremely difficult task.
A natural strategy is to consider (quantum) corrections to the classical (Drude) transport. This
leads to the developement of the perturbative approach (chapter IV)...
43
Figure 12: Gauche : Allure sch´ematique de la fonction β(g) d’un conducteur d´esordonn´e en
dimension d, propos´ee dans la R´ef. [1] (pour une potentiel d´esordonn´e scalaire). La transition
d’Anderson ne se produit qu’en dimension strictement sup´erieure `
a 2. Droite : R´esultat obtenu
num´eriquement dans la R´ef. [82] pour les dimensions d = 2 & 3 et β(g) = −(1 + g) ln(1 + 1/g)
pour d = 1 [13].
W : disorder
Wc /t=16.3
Insulator
Metal
E/t
−5
−1 0 1 2 3 4 5 6 7
` gauche : Diagramme de phases pour le mod`ele d’Anderson 3d sur r´eseau cuFigure 13: A
1
bique avec ´energies sur sites distribu´ees selon p() = W
θ( 12 W − ||) [en l’absence de d´esordre
P
` droite :
la bande d’´etats a pour largeur 12t, o`
u t est le terme de saut (~k = −2t µ cos kµ )]. A
1
Diagrammes de phase obtenus num´eriquement pour les distributions p() = W
θ( 12 W − ||),
gaussienne et lorentzienne ; figure tir´ee de [74].
44
IV
Electronic transport in weakly disordered metals : perturbative (diagrammatic) approach
Aim : In most of real situations, the disorder is weak compare to the energy of the diffusing
particle : this is true for the electron dynamics in a metal or the diffusion of the light in a turbid
medium. The aim of the chapter is to give the tools for a quantitative analysis of a weakly
disordered medium. Although the presentation will mostly concern electronic transport, the
basic concepts can be easily adapted to consider any other types of waves.
IV.A
1)
Introduction : importance of the weak disorder regime
Multiple scattering – Weak disorder regime
The aim of the course is to discuss the interaction of a wave with a static disordered potential,
i.e. the regime of multiple scattering.
The study of simple scattering (by a single scattering center) can be found in many textdσ
books : scattering of a wave of wavelength λ is described by a differential cross-section dΩ
(θ, ϕ)
giving the probabilty
the incident particle to be scattered in the direction (θ, ϕ). The total
R for
dσ
cross-section σ = dΩ dΩ
measures the probability for the incident particle to interact with the
scatterer.
In practice, a target is usually made of many equivalent scattering centers with whom the
incident particle may interact. If the cross-section and/or the thickness of the sample is large
enough, multiple scattering must be taken into account. The typical distance between two
collisions by defects is the elastic mean free path `e (i.e. the incident particle interact with
a scattering center with probability 1 on a distance & `e ). Let us consider a diffusing medium
with a density ni of scattering centers. We expect simple behaviours with σ and ni , precisely
`e ∝ 1/σ and `e ∝ 1/ni , thus
`e = 1/(ni σ) .
(IV.1)
Multiple scattering is therefore characterised by two important length scales : the wave length
λ (wave) and the m.f.p. `e (disorder). This allows to define the weak disorder regime as
λ `e
(IV.2)
In this case, it is possible to treat each collision perturbatively (in the strength of the disordered
potential), however we will have to account for multiple scattering events which will make most
2
results non perturbative. For example, the residual Drude conductivity (at T → 0) σ = nemτe , is
inversely proportional to the rate 1/τe which is the perturbative quantity.
Example : disordered metals.– Let us consider Gold, a good metal : Fermi wavelength is
(bulk)
kF−1 = 0.085 nm, which is much smaller than the elastic mean free path, `e
' 4 µm in bulk,
(film)
or `e
' 20 nm in thin metallic films (thickness . 50 nm).
2)
Interference effects in multiple scattering
If a wave interacts with many scattering centers, we expect that an extremely complex interference pattern is produced. Let us examine the typical situations where this occurs.
First of all, interferences are not necessarily dominant in the situation where a wave diffuses
in a turbid medium. For example, in a cloudy day, the light of the sun diffuses inside the clouds,
what is the reason for the uniformly white sky. This is not an interference effect.
45
When discussing interference effects, it is always useful to come back to the simple Young
double slit experiment. In this case, the two possible paths are each associated with an amplitude
and the total intensity involves the sum of two amplitudes
I = |A1 + A2 |2 .
(IV.3)
x
B
’
S
D
I(x)
S
D
I(B)
Figure 14: Interference : Young experiment (two path interference) vs multiple path
interference.
In a disordered medium, the wave can be scattered by many centers, what defines an extremely large number of scattering paths
X 2 X
X
I = AC =
(IV.4)
AC A∗C 0
|AC |2 +
0
C
| C {z } C6|=C {z }
incoherent
interferences
We now discuss few situations where the interference term manifests itself.
Example 1 (light) : Speckle pattern (fluctuations).– If coherent light (from a laser)
is sent on a disordered medium, the resulting interference pattern is extremely complicate,
exhibiting intensity fluctuations of the same order than the mean intensity (right part of Fig. 15).
The typical result of such an experiment is shown on Fig. 15 and is called a speckle pattern.
In the Young experiment, the specific intensity profile is a signature of the double slit device.
Similarly, the speckle pattern is the fingerprint of the disordered potential.
Example 2 (light) : Albedo (average).– The speckle pattern is a manifestation of (sample
to sample) fluctuations. The complex structure of the interference pattern is due to the rapidly
varying phase of the amplitudes AC ' |AC |eik`C in Eq. (IV.4), where `C is the length of the path
C. A naive guess could be that such interferences does not survive averaging. This is however
not the case. If one considers the configuration represented on Fig. 16, the measurement of
the scattered intensity in the direction θ shows some intensity profile almost flat, apart for an
increase at back-scattering : I(θ) ' I0 + ICBS (θ) where ICBS (θ) is a very narrow function (we
will show that the back-scattering cone has a typical width δθ ∼ 1/k`e ). The experimental data
shows that the intensity is almost exactly doubled I(θ = 0) ' 2I0 (Fig. 16). This phenomenon
is due to constructive interference of reversed trajectories, leading to an increase of the
back-scattering (Fig. 17). Thus, this is an effect of localisation.
This can be understood with the help of the qualitative representation (IV.4). Among all
interference terms, one is related to the interference of the two paths corresponding to the
e Exactly at back-scattering (θ = 0), time
reversed sequences of scattering events (path C and C).
46
VOLUME
PHYSICAL REVIEW LETTERS
61, NUMBER 20
14 NOVEMBER 1988
I.O—
/
p
0.5
0
X
(b)
O
~
0
0
4
~
~
~
O. I
0'
x
O
I
(c)
005
I
I
0
LLJ
20
IO
S
P
I
50
cident, and the correlation function shown is that of the
initial (reference) speckle pattern with itself. This function is computed in terms of the rightward shift in pixels
of a replica of the pattern against the original. For zero
shift, the pattern and its replica are, of course, totally
correlated, but as the shift increases, correlation is rapidly lost because of the apparently random arrangement of
the speckle spots. The 0. 12-mrad full width at half maximum of this autocorrelation
function measures the
mean angular diameter of a speckle spot, and is in closee
agreement with the expected width, which is that of the
diffraction pattern of the 6-mm circular aperture used to
define the incident laser beam. Other interesting correlation effects
can be studied by extending these
methods.
In Figs. 1(b) and l(c) the laser beam is successively
rotated about the sample by a small amount b8. The
transmitted speckle pattern may be observed to follow
the incident beam direction while slowly changing. The
cross correlation function (between the initial reference
pattern and the new pattern) accurately tracks this
motion, while its peak height decreases as correlation is
lost. In Fig. 1(d), the cross-correlation function of two
unrelated patterns is shown, and illustrates the expected
small statistical fluctuations about zero.
The ensemble-averaged correlation function, C(68), is
shown for transmission in Fig. 2, and for reflection in
Fig. 3. These data represent the recording, digitizing,
and analysis of some 250 separate patterns, each containing several hundred speckle spots. As may be seen,
in every instance the data are well approximated by a
25
PIXELS
FIG. 1. Memory effect. The right-hand side shows images
of a small portion of speckle patterns in transmission as the incident laser beam direction is varied.
I
40
FIG. 2. Correlation function C(be) in transmission. 88 is
the angle of rotation of either the laser beam direction or the
sample. The solid lines are least-squares fits with an exponential function. Curve a, ground glass; curve b, 370-pm-thick
opal glass; curve b', theory; curve c, 810-pm-thick opal glass;
curve c', theory.
C3
'0
l
50
Se( d. g)
lK
lK
The arrow at the bottom
each image calls attention to the semicircular arc
Figure 15:ofa bright
Speckle
patterns.as aLeft : Interference pattern resulting from the scattering of a laser
spot (the "bulls eye")
reference for tracking the
of the patterns. The
beam (633 nm)
by
a
thin (370
µm)
opal glass ; Estimated m.f.p. ` ∼ 100 µm. From Ref. [54].
reference pattern produced
a
incident laser
beam
rotated
10
(b) the laser
(a),
20
The
correlation
function
(c)
mdeg.
Right : Scattered
light intensity (532 nm) by a powdered Ti:saphire sample. From Ref. [125].
to the left of each image corresponds to the cross-correlation
visual
initial
is shown in
mdeg, and in
by
enclosing
convenient
which serves
motion
normally
by
while in
is
by
shown
coefficients of the reference pattern with the corresponding image. This correlation is plotted as a function of pattern shift in
pixels, and the peak represents the maximum degree of overlap
of the two patterns. Note that the correlation function tracks
the speckle patterns, which, in turn, "remember,
and therefore
dettrack, the incident laser beam direction. The pattern in
ctorwhich is unrelated to the reference pattern in (a),
(d) iseone
and the correlation function shows the expected small statistical fluctuations about zero.
"
directly on the sample, together with careful system
alignment, ensured that the laser beam did not walk
across the sample face during rotation. The free-space
speckle patterns were recorded with a sensitive chargedcoupled-device video camera, and digitized by an on-line
The sample was a thin sheet (370-pm
microcomputer.
thickness) of opal glass supported on a clear glass substrate. In Fig. 1(a) the laser beam was normally in-
θ
2329
source
disordered medium
Figure 16: Coherent back-scattering of light. Left : Principle of measurement. Right :
Coherent back-scattering of light (wavelength 514 nm) by ZnO powder sample. Mean free path
is `e = 1.9 µm. From Ref. [125]. The smooth background of the intensity profile originates from
the diffusion of light in the disordered medium. The narrow peak, δθ ∼ 0.05 rad, is the CBS.
reversed symmetry implies that AC = ACe, what explains the doubling of the intensity :
I(θ = 0) =
X
C
|AC |2 +
X
C
|
CBS
AC A∗Ce +
{z
}
X
C, C 0 6=C
|
e
(C 0 =C)
and Ce
{z
'0
AC A∗C 0 ' 2
X
C
|AC |2 .
(IV.5)
}
Although the effect is not small exactly at back-scattering,
R it is globally a small effect : the
contribution of interferences to the total scattered intensity dθ ICBS (θ) ∼ I0 /k`e is small.
Example 3 (electrons) : Weak localisation correction (average).– The increase of
back-scattering due to interferences of reversed diffusive trajectories can also be demonstrated
in electronic transport. Quantum transport in metallic films and wires was investigated systematically in several experiments (cf. [25] for films). In this case, the interference pattern must be
revealed by tuning some external parameter, like the chemical potential or the magnetic field.
47
’=
θ
’=
’=
θ
~
~
θ =0
θ
Incoherent
A coherent contribution
Coherent back−scattering
Figure 17: Coherent back-scattering. Left : The classical term can be understood as the
pairing of two equal trajectories C 0 = C. Center and Right : The contribution of reversed
trajectories C 0 = Ce to the interference term.
The typical low temperature resistance of a long and narrow wire 14 as a function of the magnetic
field presents the profile shown on Fig. 18, with a narrow peak around zero field. The experiment
demonstrates a decrease of the resistance with the magnetic field, i.e. an opposite effect to the
classical magneto-resistance (cf. problem page 84). For this reason, this phenomenon is called
“anomalous magneto-resistance”. It is equivalent (but not exactly similar) to the CBS cone in
optical experiments. Being an increase of the electric resistance, this is also a manifestation of
localisation.
Figure 18: Magneto-resistance ∆R(B) of thin metallic Lithium wires. Sample are made
of 27 to 101 Lithium wires of length L = 1mm in parallel. Depending on sample, width W varies
between 1 µm to 0.03 µm, and thickness is ∼ 25 nm. From Ref. [78].
3)
Phase coherence
A crucial point of the above discussion is that all effects aforementioned are interference effects,
requiring that the wave remains coherent in the diffusion process on the static potential. In
practice the incoming wave may also interact with other degrees of freedom, what limits coherent
14
The effect is re-inforced in a narrow wire of sub-micron cross-section. The reason is related to the fact that
localisation effects are stronger in low dimension. The precise criterion for the wire to be “narrow” will be made
clear later.
48
properties. Two examples are
• In optical experiments : the motion of scatterers.
• In electronic transport : the electron-phonon interaction (lattice vibrations), or the electronelectron interaction.
x
x
δφ
source
δφ
δφ
I(x)
δφ
I(x)
δφ
Figure 19: Decoherence. Decoherence may be viewed as contributions of random phases arising
from the interaction with other degrees of freedom.
In a metal, the increase of the temperature leads to the activation of all degrees of freedoms
(thermal fluctuations above the Fermi sea, phonons, etc). Hence interferences of electronic waves
progressively disappear, as illustrated by the experimental results shown on Fig. 20. The phase
coherent electronic transport only manifests itself below few Kelvins.
Figure 20: Magneto-resistance ∆R(B) of thin metallic Lithium wires. As temperature is
increased, phase coherence is limited and magneto-resistance peak is suppressed. From Ref. [78].
The efficiency of decoherence is characterised by the phase coherence time τϕ . I.e. phase
coherence is lost after a time t & τϕ . This allows to define several regimes for coherent physics.
In general the problem is controlled by three time scales : the elastic mean free time τe = `e /vF ,
the phase coherent time τϕ and the Thouless time τTh , which is the time needed to explore the
full system.
• Coherent ballistic regime.– Coherent properties are maintained over distances such
that the motion is ballistic
min(τϕ , τTh ) < τe
In this case τTh = L/vF , where L is the system size, and τϕ = Lϕ /vF , where Lϕ is the
phase coherence length.
• Coherent diffusive regime.– The particle is scattered many times by the disorder before
loosing its coherence :
τe < min(τϕ , τTh )
49
The relation between length and time is characteristic of diffusive motion : τTh = L2 /D
and τϕ = L2ϕ /D.
4)
Motivation : why coherent experiments are interesting ?
Looking at figures 16 and 18, one could legitimately ask the question : what is the interest for
studying such tiny effects ? The CBS cone is extremely narrow. The weak localisation correction
is less than ∆R/R . 1 %.
The motivation for studying these small effects is both practical and fundamental. Being a
manifestation of interference effects, the measurement of these quantities furnishes an experimental probe for coherent properties in diffusive systems. Moreover, the precise knowledge of
how weak localisation depends on the phase coherence length Lϕ provides its possible definition.
Weak localisation measurement is a common tool in condensed matter experiments for probing
the fundamental question of phase coherent (quantum) properties of electrons.
The precise question could be : how to extract the phase coherence length from the magnetoresistance curve ? The answer to this question obviously requires the knowledge of the Bdependence and Lϕ -dependence of the magneto-resistance :
Q : ∆R(B, Lϕ ) =?
IV.B
Kubo-Greenwood formula for the electric conductivity
The remaining of the chapter will focus on electronic transport properties. We first recall some
basic formulae for the electrical conductivity of a metal.
1)
Linear response theory
The starting point is the linear response theory (see for instance [108]). We recall the main
result. If one considers a time dependent perturbation involving an observable A of the system :
ˆ
ˆ 0 − f (t) Aˆ
H(t)
=H
The “response” of any other observable B
Z
ˆ
ˆ
hB(t)if = hBi + dt0 χBA (t − t0 ) f (t0 ) + O(f 2 )
(IV.6)
(IV.7)
is controlled at lowest order in perturbation theory by an equilibrium correlation function
χBA (t) =
i
ˆ
ˆ
θH (t)h[B(t),
A]i
~
(IV.8)
where h· · ·i is the quantum-statistical average related to H0 and h· · ·if the one related to H(t).
2)
Conductivity
The conductivity characterises the response of the current density to the electric field. In principle it probes local properties. However, usual transport experiments probe global properties
of the system, and the measured quantity is usually the conductance. On the other hand, measurements of small contributions are facilitated by the introduction of a small modulation of
the excitation, i.e. measuring a signal at small but finite frequency, what allows to distinguish
more easily the signal from the noise (AC lock-in technic). For these reasons we have to consider the conductivity with the proper limits 15 limω→0 limq→0 σ(q, ω). We will thus consider a
15
Recall that these limits do not commute [108].
50
~
uniform perturbation, i.e. a time-dependent uniform electric field E(t).
Due to gauge invariance, we still have some freedom on the choice of the perturbation R: we can consider a gauge
~ · ~r and A
~ ext = 0, or φext = 0 et A
~ ext (t) = − t dt0 E(t
~ 0 ). We prefer the
φext (~r, t) = −E(t)
second choice, that will lead more directly to a representation of the conductivity in terms of
a current-current
correlator, close in spirit to the classical definition of the diffusion constant
R∞
D = 0 dt hvx (t)vx (0)i (recall that the diffusion constant and the conductivity are related by
Einstein relation σ0 = e2 ν0 D). We consider the single electron Hamiltonian
2
ˆ tot (t) = 1 p~ˆ − eA(
~ ext (t) · ~vˆ + · · ·
~ ~rˆ) − eA
~ ext (t) + U (~rˆ) = 1 ~vˆ 2 + U (~rˆ) −eA
H
|
{z
}
2m
2m
|
{z
}
ˆ
H
def
~
where ~v = (~
p − eA)/m
is the speed.
The spatially averaged current for one electron
16
(IV.9)
ˆ pert (t)
H
is simply
~j = e ~v .
Vol
(IV.10)
In the presence of the external vector potential, the
speed also receives the contribution of the
1
~ − eA
~ ext . Accordingly, the current of the electron gas
external vector potential ~vtot = m
p~ − eA
is splitted in two terms
hjitot (t)iA~ ext = hji (t)iA~ ext −
N e2 ext
A (t) ,
Vol i
(IV.11)
P
where N = α fα is the number of electrons, expressed in terms of the Fermi function fα ≡
f (α ). The first term is given by the linear response theory
Z
e2
0
hji (t)iA~ ext =
dt0 Kij (t − t0 ) Aexp
(IV.12)
i (t ) + · · ·
Vol
and involves a velocity-velocity (i.e. current-current) correlator
Kij (t) = iθ(t)h[vi (t), vj ]i
(IV.13)
(we now set ~ = 1). After Fourier transform, we express the current in terms of the electric field
and obtain the conductivity
Z ∞
1 e2
N
i e2 N
(iω−0+ )t
e
− δij + i
dt e
h[vi (t), vj ]i =
δij − Kij (ω)
(IV.14)
σ
eij (ω) =
iω Vol
m
ω Vol m
0
where we have introduced an infinitesimally small regulator 0+ . Using the spectral representation
of the correlator
X
(vi )αβ (vj )βα
e ij (ω) = −
K
(fα − fβ )
,
(IV.15)
ω + α − β + i0+
α,β
def
where (vi )αβ = h ϕα |ˆ
vi | ϕβ i is a one-particle matrix element, we deduce the real (dissipative) part
of the conductivity. For simplicity we restrict ourselves to the longitudinal resistivity (i = j) :
Re σ
exx (ω) =
πe2 X f (α ) − f (α + ω)
|(vx )αβ |2 δ(ω + α − β )
Vol
ω
(IV.16)
α,β
ˆ
In a one-particle picture, the current density operator is ~j(~r) = e n
ˆ (~r) ~vˆ + ~vˆ n
ˆ (~r) where n
ˆ (~r) = δ(~r − ~rˆ) =
ˆ
ˆ
|~r ih~r | is the density operator, ~r the position operator and ~v the speed operator.
16
51
R
Introducing d δ( − α ) = 1 in the expression, we can rewrite this expression as a trace over
the one-particle Hilbert space :
πe2
Re σ
exx (ω) =
Vol
Z
d
n
o
f () − f ( + ω)
ˆ vˆx δ( − H)
ˆ
Tr vˆx δ( + ω − H)
ω
(IV.17)
def
We can deduce the zero frequency conductivity σ = σ
exx (ω = 0) :
n
Z
o
n
o
πe2
πe2
∂f
ˆ vˆx δ( − H)
ˆ
ˆ vˆx δ(F − H)
ˆ
Tr vˆx δ( − H)
−→
σ=
d −
Tr vˆx δ(F − H)
T →0 Vol
Vol
∂
(IV.18)
Because the function − ∂f
is
a
narrow
function
of
width
k
T
much
small
that
the
Fermi
energy,
B
∂
this expression emphasises that only eigenstates in the close neighbourhood of the Fermi level,
|α −F | . kB T , are involved in longitudinal transport properties. Longitudinal conductivity
is a Fermi surface property (contrary to the transverse conductivity [103, 104]). Finally, let
us remark that the Kubo-Greenwood formula, Eq. (IV.16) or Eq. (IV.17), presents the desired
structure of a speed-speed correlator reminiscent of the diffusion constant.
Remark : More on the correlator K.– The relation between σij and the correlator Kij
can be improved by making use of the f -sum rule.
P |(v )αβ |2
def
1
- Exercice IV.1 : f -sum rule.– Show that 2m
+ β αx−
= 0, where (vx )αβ = h α |vx | β i.
β
Deduce
X
|(vx )αβ |2
1 X
fα +
(fα − fβ )
=0
(IV.19)
m α
α − β
α,β
-
e ij (ω), analyse K
e xx (ω = 0)
Exercice IV.2 : Using the spectral representation for K
Hint : note that when ω = 0, the 0+ becomes useless because the numerator vanishes when
α = β . 17
~k
For free electrons, the matrix elements are : h ~k |~v | ~k 0 i = m
δ~k,~k 0 , where | ~k i denotes a plane
e xx (ω 6= 0) = 0. Deduce the conductivity for free electrons.
wave. Show that K
The f -sum rule expresses conservation of particle number. We have shown that
σ
eij (ω) =
e xx (0) − K
e ij (ω)
ie2 δij K
Vol
ω
(IV.20)
This structure may also be understood as a consequence of gauge invariance, which is not a
surprise since Gauge invariance and charge conservation are closely related by Nœther theorem.
e xx (ω)
The expression (IV.20) must be manipulated with caution. We have indeed seen that K
is discontinuous at ω = 0 for free electrons.
To close the remark, let us point out that, being the correlation function of the electron
e xx (ω) can be related by the fluctuation-dissipation theorem to the power spectrum of
velocity, K
R∞
the electron velocity [108], i.e. the diffusion constant D = 0 dt hvx (t)vx (0)i. Therefore, we can
extract from Eq. (IV.20) the Einstein relation between conductivity and diffusion constant.
17
if f (0) = 0 and |f 0 (0)| < ∞ we have
f (x)
x+i0+
=
f (x)
x
since f (x)δ(x) = 0 and P f (x)
=
x
52
f (x)
.
x
3)
Conductivity in terms of Green’s function
A question of methodology is : how to analyse the Kubo-Greenwood formula (IV.16), in particular with the aim of building a pertubative expansion in the disordered potential ? It is
obviously excluded, because far too complicated, to use the well-known perturbative formulae
for the spectrum of the Hamiltonian {α , | ϕα i} ; moreover the calculation requires the matrix
elements (vx )αβ = h ϕα |vx | ϕβ i. It is therefore essential to introduce a compact object encoding
all spectral information and allowing for a simple perturbative expansion. Such an object is the
Green’s function, i.e. matrix elements of the resolvent operator,
G(~r, ~r 0 ; z) = h~r |
def
X ϕα (~r)ϕ∗ (~r 0 )
1
α
,
|~r 0 i =
z−H
z
−
α
α
(IV.21)
whose poles are located on the eigenvalues of the Hamiltonian with residues related to the
eigenfunctions. The calculations will involve the retarded and advanced Green’s functions :
GR,A (~r, ~r 0 ; E) = G(~r, ~r 0 ; E ± i0+ ) ,
def
(IV.22)
which can be related to the matrix elements appearing the Eq. (IV.17) by noticing that
∆G(~r, ~r 0 ; E) = GR (~r, ~r 0 ; E) − GA (~r, ~r 0 ; E) = −2iπh~r |δ(E − H)|~r 0 i .
def
(IV.23)
Let us point out that the diagonal matrix
elements of this latter function is the local density of
P
1
states ρ(~r; E) = −2iπ
∆G(~r, ~r; E) = α |ϕα (~r)|2 δ(E − α ).
Because the starting point of the perturbative expansion is the free particle problem, it will
be quite natural to work in a momentum representation, i.e. in the basis of plane waves
1 i~k·~r
ψ~k (~r) = h~r | ~k i = √
e
Vol
(IV.24)
where we choose plane waves normalised in a box (i.e. with quantised wave vectors h ~k | ~k 0 i =
δ~k,~k 0 ). Accordingly we introduce the Green’s function in momentum space (keeping the same
notation as in real space)
def
GR,A (~k, ~k 0 ; E) = h ~k |
1
| ~k 0 i .
E − H ± i0+
(IV.25)
Kubo-Greenwood formula (IV.17) now reads
Re σ
e(ω) = −
2s e2
4πm2
Z
d
f () − f ( + ω) 1 X
kx kx0 ∆G(~k, ~k 0 ; + ω) ∆G(~k 0 , ~k; )
ω
Vol
(IV.26)
~k, ~k 0
where we have taken into account the spin degeneracy 2s . It is useful to emphasize that the
Fermi functions constraint the energy to be close to the Fermi energy F .
IV.C
M´
ethode des perturbations et choix d’un mod`
ele de d´
esordre
La m´ethode ? Maintenant que nous avons introduit le “bon” objet, reste `a savoir comment
le manipuler, i.e. comment effectuer la moyenne sur le d´esordre de produits de fonctions de
Green, supposant donn´ee la distribution DV P [V] du d´esordre. La difficult´e vient notamment
1
. Deux
de la pr´esence du potentiel au d´enominateur dans la fonction de Green G(E) = E−H
53
m´ethodes utilisent un “truc” pour exponentier le d´enominateur et faire apparaˆıtre la fonctionnelle caract´eristique du potentiel : la m´ethode des r´epliques [65] 18 et la m´ethode supersym´etrique [124, 48]. La mise en œuvre de ces deux approches n´ecessite des techniques de
th´eorie des champs assez sophistiqu´ees dans lesquelles nous ne souhaitons pas entrer ici. Une
pr´esentation claire de l’utilisation de la m´ethode des r´epliques pour les syst`emes d´esordonn´es
est propos´ee dans l’ouvrage de Altland et Simons [4].
La troisi`eme m´ethode, que nous suivrons, est l’approche perturbative o`
u G(E) est d´evelopp´e
en puissances du potentiel d´esordonn´e, puis le d´eveloppement moyenn´e. Les r´esultats que nous
cherchons sont non perturbatifs dans le d´
esordre. Cette approche n’est int´eressante que
si nous pouvons resommer les corrections perturbatives. 19 Le cas unidimensionnel permet une
resommation compl`ete des corrections perturbatives dans le cas o`
u le potentiel est un bruit
blanc [22, 23]. En v´erit´e ces techniques assez lourdes ne sont pas indispensables et peuvent ˆetre
´evit´ees au profit de m´ethodes non perturbatives probabilistes plus ´el´egantes [79, 80] (au moins
`a mon goˆ
ut, par exemple cf. [35]). Dans les probl`emes `a plusieurs dimensions nous devrons
nous contenter de resommations partielles qui ne seront valables que dans le domaine de faible
d´esordre. Nous pr´eciserons cette remarque par la suite.
1)
D´
eveloppements perturbatifs
L’int´erˆet d’avoir introduit la fonction de Green est que cet objet est le plus appropri´e pour
1
d´evelopper une th´eorie de perturbation. Si l’on pose G = E−H
avec H = H0 + V on a G =
1
G0 + G0 VG0 + G0 VG0 VG0 + · · · o`
u G0 = E−H0 . La fonction de Green libre est diagonale
dans l’espace de Fourier (je laisserai dor´enavant tomber les fl`eches des vecteurs pour all´eger, sauf
ambigu¨ıt´e)
1
1
hk|
| k 0 i = δk,k0 G0 (k) avec G0 (k) =
(IV.27)
E − H0
E − k
def
2
k
. (Tant que cela n’est pas n´ecessaire nous ne sp´ecifions pas s’il s’agit de la fonction de
o`
u k = 2m
Green avanc´ee ou retard´ee). Pour aller plus loin il nous faut savoir comment mod´eliser les d´efauts
(impuret´es, d´efauts structurels) et le choix du mod`ele a-t’il une influence sur la physique ? En
g´en´eral la r´eponse est oui, ´evidemment, toutefois nous nous int´eresserons `a un r´egime de faible
d´esordre et seul le second cumulant de V sera important. Pour cette raison nous pouvons
nous limiter au choix d’une distribution gaussienne, enti`erement caract´eris´ee par la fonction
de corr´elation V(r)V(r0 ). En pratique il est raisonnable de supposer que les corr´elations du
potentiel d´esordonn´e sont `
a courte port´ee, nous les prendrons de port´ee nulle pour simplifier.
En conclusion, pour l’´etude des propri´et´es de transport dans les m´etaux faiblement d´esordonn´es
il est suffisant de choisir le mod`
ele minimal :
Z
1
DV P [V] = DV exp −
dr V(r)2 .
(IV.28)
2w
R
1
~ φ~2 e− 21 Aφ~2 , o`
= limn→0 dφ
La m´
ethode des r´
epliques [47, 50] repose sur le truc suivant. On ´ecrit : A
u
n
1
~ est un “champ” a
φ
` n composantes (les n r´epliques). On transforme le calcul de h A
i (difficile) en he−kA i (facile).
Le prix a
` payer est de faire tendre le nombre de composantes n vers 0 ! Appliquons le truc au calcul de la fonction
R
~
~ 0 ) −S
φ(x
~ φ(x)·
~
de Green moyenne : h x |(H − E)−1 | x0 i = limn→0 Dφ
e
o`
u φ(x)
est un champ a
` n composantes.
La
nR
R
2
1
moyenne sur un d´esordre gaussien, DV P [V] = DV exp − 2w dx V(x) , g´en`ere une action S = dxL pour une
~ 2 − Eφ
~2 − w φ
~ 4 (chapitre 9 du tome 2 de l’ouvrage d’Itzykson &
th´eorie des champs en interaction L = 21 (∂x φ)
2
8
Drouffe [65]). Si l’on consid`ere un probl`eme invariant par translation (au moins apr`es moyenne sur le d´esordre) on
R
∂
~ e−S de laquelle on peut extraire
peut encore simplifier le calcul en ´ecrivant h x |(H − E)−1 | x i = limn→0 n2 ∂E
Dφ
+
la DoS en faisant E → E + i0 .
19
Si nous calculons la conductivit´e perturbativement dans l’intensit´e du d´esordre, contrˆ
ol´e par le taux de
2 P∞
γe n
collision γe = 1/τe , nous obtenons une s´erie σ(ω) = ine
.
Apr`
e
s
resommation,
la
limite
de fr´equence
n=0 iω
mω
18
nulle conduit au r´esultat σ(0) =
ne2
mγe
=
ne2 τe
m
(ce calcul sera essentiellement effectu´e plus bas).
54
Afin de d´egager les r`egles permettant une construction syst´ematique du d´eveloppement perturbatif, analysons en d´etail les premiers termes correctifs du d´eveloppement de la fonction de
Green moyenne G.
2)
Vertex et r`
egles de Feynman
=
w
0 =
δ
0
Vol kg +kg ,kd +kd
kg k d
’
−k g
k’d
kd’ −
kg’
Vertex pour le d´
esordre.– Comme nous travaillons dans l’espace r´eciproque le calcul fera
intervenir la moyenne des ´el´ements de matrice de V (le vertex ´el´ementaire d´ecrivant l’interaction
entre un ´electron et le potentiel d´esordonn´e pour le calcul de quantit´es moyenn´ees) :
Z
Z
dr i(kd −kg )·r
dr0 i(k0 −kg0 )·r0
0
0
w δ(r − r0 )
(IV.29)
h kg |V| kd ih kg |V| kd i =
e
e d
Vol
Vol
(IV.30)
k’g k’d
o`
u kd et kd0 sont des impulsions entrantes et kg et kg0 des impulsions sortantes. Cette expression,
la TF de la fonction de corr´elation V(r)V(r0 ) = w δ(r − r0 ). On peut interpr´eter physiquement
le corr´elateur comme une double interaction avec un d´efaut localis´e qui transfert une impulsion
a un autre, i.e. d’une fonction de Green `a une autre. 20
kg − kd = kd0 − kg0 d’un point `
Nous commen¸cons par calculer le premier terme non nul en moyenne : G0 VG0 VG0 . Le terme
d’ordre 2 de G(k, k 0 ) s’´ecrit :
h k |G0 VG0 VG0 | k 0 i = G0 (k)
X w
w X
δk,k0 G0 (k 00 ) G0 (k 0 ) = δk,k0 G0 (k)2
G0 (q) .
Vol
Vol q
00
(IV.31)
k
La pr´esence du Kronecker assure la conservation de l’impulsion, ce qui d´ecoule de mani`ere
´ (IV.29). Cette propri´et´e reste vraie `a tous les ordres et on a
´evidente de l’analyse du vertex, Eq.
donc G(k, k 0 ) ≡ δk,k0 G(k), qui traduit le fait que l’invariance par translation est restaur´ee apr`es
moyenne d´esordre. Le premier terme du calcul perturbatif de la fonction de Green moyenne est
finalement
w
"
#
k!q
k!q
w X
δ 2 G(k) = G0 (k)
G0 (q) G0 (k) =
(IV.32)
Vol q
q
k
k
La petite repr´esentation graphique nous permet de d´eduire facilement les r`egles de Feynman
permettant une construction syst´ematique des diagrammes.
20
Si nous avions choisi un mod`ele plus g´en´eral de d´esordre, le cumulant d’ordre p serait associ´e `
a p interactions
P
avec un d´efaut. Pour clarifier cette remarque, consid´erons le mod`ele de potentiel d´esordonn´e V(r) = v0 i δ(r−ri )
o`
u les positions sont distribu´ees selon laP
loi de Poisson (positions d´ecorr´el´eesPpour une densit´e moyenne ni ). Le
calcul du second moment V(r)V(r0 ) = v02 i6=j δ(r − ri )δ(r0 − rj )+v02 δ(r−r0 ) i δ(r − ri ) = (ni v0 )2 +ni v02 δ(r−r0 )
montre que le second cumulant s’interpr`ete comme une double interaction avec une mˆeme impuret´e. Le cumulant
C
V(r1 ) · · · V(rn ) = ni v0n δ(r1 − r2 ) · · · δ(r1 − rn ) correspond a
` n interactions avec une impuret´e, ce que nous
repr´esentons comme :
...
r1 r2 r3 ... rn
55
R`
egles de Feynman.– Les diagrammess sont construits en quatre ´etapes :
` l’ordre 2n des perturbations en V, on dessine les (2n − 1)!! diagrammes : une ligne
1. A
continue repr´esente une fonction de Green libre G0 (k), une ligne pointill´ee une double
interaction avec un d´efaut.
2. On associe des impulsions de fa¸con `a respecter la conservation de l’impulsion `a chaque
vertex.
w
3. Chaque double interaction est associ´ee `a un poids
.
Vol
X
4. On somme sur toutes les impulsions libres selon
.
q
Nous dessinons les trois diagrammes obtenus `a l’ordre suivant :
δ 4 G(k) =
3)
+
+
(IV.33)
Self ´
energie
q
k−
Σ2 =
k−
q
Nous pouvons remarquer que le premier des diagrammes de (IV.33) correspond `a une r´ep´etition
du bloc de la premi`ere correction (IV.32), que nous notons :
q
=
w X
G0 (q)
Vol q
(IV.34)
a` chaque ordre, certaines des corrections contiendront ´egalement ce bloc et un des (2n − 1)!!
termes sera n r´ep´etitions de ce bloc. Il y a un moyen tr`es simple de resommer tous ces termes
(c’est une s´erie g´eom´etrique) :
1
G0
(k)−1
− Σ2
=
+
+
+
+ ···
(IV.35)
Nous pouvons donc regrouper dans un objet que nous appelons la self ´energie et notons Σ(E),
tous les blocs insecables par un seul “coup de ciseau” dans une ligne de fonction de Green (les
diagrammes dit irr´eductibles). Outre que cette remarque permet de resommer une infinit´e de
diagrammes sans effort, l’int´erˆet d’introduire cet objet est de conduire `a une fonction de Green
moyenne pr´esentant une structure similaire `a la fonction de Green libre : 21
G(k) =
1
G0
(k)−1
− Σ(E)
=
1
E − k − Σ(E)
(IV.37)
21
Dyson equation : The resummation of diagrams through the introduction of the self energy is particularly
simple here because it is formulated in the basis that diagonalises the free Hamiltonian. In the more general case
one should write a Dyson equation,
G(E) = G0 (E) + G0 (E)Σ(E)G(E)
which takes the form of an integral equation in real space, for example.
56
(IV.36)
pour le mod`ele avec potentiel nous corr´el´e spatialement, V(r)V(r0 ) ∝ δ(r − r0 ), il est clair que la
self ´energie est ind´ependante du vecteur d’onde. 22 Le calcul du terme d’ordre n de Σ(E) met en
jeu bien moins de diagrammes que celui de G(E). Consid´erer le d´eveloppement perturbatif de
Σ(E) est une mani`ere de prendre en compte des corrections d’ordre n de G(E) se d´eduisant de
mani`ere ´evidente des ordres n0 < n. Par exemple, `a l’ordre w2 , Σ4 sera donn´e par le diagramme
de Σ2 plus deux diagrammes d’ordre w2 correspondant aux deux derniers termes de (IV.33) :
+
Σ4 =
+
(IV.39)
-
Exercice IV.3 : Dessiner tous les diagrammes de Σ6 (la self-´energie `a l’ordre w3 ). Comparer avec δ 2 G(k) + δ 4 G(k) + δ 6 G(k).
Les deux ´echelles du probl`eme sont l’´energie de Fermi F (rappelons-nous que la conductivit´e
longitudinale fait intervenir les fonctions de Green `a E ' F ) et le d´esordre w. La self ´energie
nous permet d’introduire une ´echelle plus pertinente que w :
1 def
= − Im ΣR (E)
2τe
(IV.40)
o`
u ΣR (E) est la self ´energie calcul´ee avec des fonctions de Green retard´ees. La fonction de Green
moyenne pr´esente donc la structure
1
R
G (k) =
E − k +
i
2τe
(IV.41)
La partie imaginaire est la plus importante car elle ´eloigne le pˆole de la fonction de Green de
l’axe r´eel, alors que la partie r´eelle de la self ´energie peut en principe ˆetre absorb´ee dans une
red´efinition de l’´energie. 23 La structure de la fonction de Green moyenne met en ´evidence le
sens de τe , qui s’interpr`ete comme la dur´ee de vie de l’onde plane d’´energie E ∼ F , i.e. le temps
pendant lequel la direction de l’impulsion est conserv´ee (le module de l’impulsion est conserv´e
au cours des processus de collision sur le d´esordre statique). Il s’agit donc du temps de parcours
moyen ´elastique. On peut lui associer le libre parcours moyen ´elastique `e = vF τe , o`
u vF est la
vitesse de Fermi.
Dans le r´egime de faible d´esordre
F τe 1
ou
kF `e 1
(IV.42)
22
q
q
Σ2 (k; E) =
k−
k−
q
Let us consider the general case of a disordered potential with a finite correlation length : V(r)V(r0 ) =
e
C(r
− r0 ) where C(r) decays on a microscopic scale `corr . If we introduce the Fourier transform C(Q)
=
R
−iQ·r
dr C(r) e
, we see that the self energy now depends on the momentum
=
w Xe
C(k − q) G0 (q)
Vol q
(IV.38)
where the argument of the correlation function is the transfered momentum Q = k − q.
23
Nous jetons un voile pudique sur la partie r´eelle de la self ´energie. A priori celle-ci est une fonction r´eguli`ere
de l’´energie et elle pourrait ˆetre ´elimin´ee par une reparam´etrisation du spectre des ´energies. En y regardant de
plus pr`es nous constatons que Re Σ diverge en dimension d > 1. Ce probl`eme vient de la nature singuli`ere des
corr´elations du potentiel en δ de Dirac. La mˆeme difficult´e apparaˆıt dans l’analyse de l’Hamiltonien −∆ + v δ(r)
en dimension d > 1. Une mani`ere de donner un sens au probl`eme a ´et´e propos´ee dans l’article [66]. Pour une
pr´esentation d´etaill´ee du probl`eme : probl`eme 10.3 de mon livre [107].
57
l’´energie des ´electrons est bien sup´erieure au taux de probabilit´e pour quitter l’´etat quantique,
i.e. la notion d’´etat libre (d’onde plane) reste pertinente pour des temps t . τe . Dans ce cas
l’approximation de Born Σ(E) = Σ2 (E) + O(w2 ) est justifi´ee. Fixant dor´enavant E = F , nous
´ecrivons :
1
w X
' − Im ΣR
Im GR
2 (F ) = −
0 (q) = πρ0 w ⇒
2τe
Vol q
1
= 2πρ0 w
τe
(IV.43)
o`
u ρ0 = − π1 Im GR
e d’´etats au niveau de Fermi par unit´e de volume par canal
0 (r, r) est la densit´
de spin.
-
Exercice IV.4 : Montrer que, dans la limite de faible d´esordre, la fonction de Green dans
l’espace r´eel pr´esente la structure
R
0
−R/2`e
G (r, r0 ) ' GR
∝
0 (r, r ) e
1
R(d−1)/2
eikF R−R/2`e
(IV.44)
o`
u R = ||r − r0 ||. (Le calcul est simple en d = 1 et d = 3 ; en d = 2 il n’est valable que pour
kF R 1 et correspond au comportement asymptotique d’une fonction de Bessel).
IV.D
La conductivit´
e`
a l’approximation de Drude
Nous commen¸cons par plusieurs remarques permettant de simplifier (IV.26).
(i) Dans la pratique les fr´equences correspondant `a une situation exp´erimentale sont toujours
telles que ω T (rappelons que T = 1 K correspond `a une ´energie 86 µeV et `a une fr´equence
20.8 GHz). Nous proc´edons `
a la substitution f ()−fω(+ω) → − ∂f
∂ .
(ii) D’autre part l’´elargissement thermique ne joue aucun rˆole dans la calcul de la conductivit´e
moyenne (ceci ne serait pas vrai si nous ´etudiions les fluctuations de la conductivit´e). Nous
fixons T = 0.
(iii) Dans le produit ∆G∆G = (GR − GA )(GR − GA ), on peut montrer que seuls les termes
GR GA et GA GR apportent des contributions importantes, apr`es moyenne. Finalement, notre
point de d´epart est 24 :
σ
e(ω) =
2s e2 1 X
kx kx0 GR (k, k 0 ; F + ω) GA (k 0 , k; F )
2πm2 Vol 0
(IV.45)
k, k
La conductivit´e est donn´ee par un produit de fonctions de Green. La premi`ere chose la plus
simple `a faire consiste `
a n´egliger les corr´elations entre les deux fonctions de Green : GR GA '
R A
R
R
G G . En utilisant G (k, k 0 ) = δk,k0 G (k), nous obtenons
σ
e(ω)
Drude
=
2s e2 1 X 2 R
A
kx G (k; F + ω) G (k; F )
2
2πm Vol
(IV.46)
k
Le calcul de ce type de quantit´e est tout `a fait standard et des expressions similaires apparaˆıtront
par la suite. Nous rempla¸cons kx2 par k 2 /d (isotropie de la relation de dispersion) puis k 2 par
kF2 pour le sortir de la somme (ce qu’on peut justifier plus soigneusement en ´etudiant le calcul
ci-dessous).
P
0
R
0
A
0
Notons que ces trois hypoth`eses conduisent a
` Re σ
e(ω) ∝
k,k0 kx kx Re[G (k, k ; F + ω) G (k , k; F )], ce
qui est un petit peu plus faible que (IV.45), sauf dans le cas ω = 0.
Dans la suite du chapitre, on pourra consid´erer que σ ∼ GR GA et oublier les termes GR GR et GA GA . Notons toutefois que ceci n’est plus vrai lorsque l’on consid`ere la correction venant des interactions ´electroniques
(correction Altshuler-Aronov).
24
58
Calcul de
P
R
k
A
G (k) G (k) :
Nous d´etaillons le calcul de la quantit´e
w X R
A
G (k; F + ω) G (k; F ) .
Vol
k
Le principe du calcul et les approximations s’appliqueront `a d’autres quantit´es similaires.
Dans la limite de faible d´esordre F τe 1, la somme est domin´ee par les ´energies voisines
de F (rappelons que ω F ). Cette observation autorise deux approximations :
R dd k
1 P
(i ) On n´eglige la d´ependance (lente) de la densit´e d’´etats en ´energie Vol
=
k =
(2π)d
R∞
R∞
u ρ0 d´esigne la densit´e d’´etats au niveau de Fermi ρ0 = ρ(F ).
0 dk ρ(k ) −→ ρ0 0 dk o`
(ii ) On ´etend l’int´egrale sur l’´energie `a tout R, ce qui permet une utilisation plus directe du
th´eor`eme des r´esidus.
On a donc
Z
w X R
dk
A
G (k; F + ω) G (k; F ) ' wρ0
.
i
i
Vol
R (F + ω − k + 2τe )(F − k − 2τe )
k
On referme le contour d’int´egration soit par le haut soit par le bas pour utiliser le th´eor`eme des
r´esidus. On obtient :
2iπ
wρ0
ω + i/τe
i.e.
w X R
1
A
G (k; F + ω) G (k; F ) =
Vol
1 − iωτe
(IV.47)
k
Conclusion :
Finalement nous aboutissons `a :
Drude
σ
e(ω)
=
2s e2 kF2
1
2s e2 τe 2F ρ0
1
2πρ
τ
=
0
e
2
2πm d
1 − iωτe
m
d 1 − iωτe
(IV.48)
Nous reconnaissons la densit´e ´electronique, n = 2s 2Fd ρ0 , et finalement
σ
e(ω)
Drude
=
σ0
1 − iωτe
o`
u
σ0 =
ne2 τe
m
(IV.49)
est la conductivit´e de Drude r´esiduelle (`a T → 0). Nous avons donc retrouv´e le r´esultat du
mod`ele semi-classique de Drude-Sommerfeld. Alors que la conductivit´e du syst`eme balistique
pr´esente une divergence `
a ω → 0, σ
e(ω) ∝ i/ω, dont l’origine est le mouvement balistique des
25
´electrons,
les collisions sur les d´efauts sont responsables d’une conductivit´e finie `a fr´equence
nulle.
Remarque : le r´
esultat classique est non perturbatif.– Comme nous l’avions d´ej`a not´e
dans l’introduction, la conductivit´e de Drude est non perturbative dans le d´esordre :
σ0 ∝ τe ∝ 1/w .
(IV.50)
Partant d’un r´esultat divergent `
a basse fr´equence en l’absence de d´esordre, σ
e(ω) ∝ i/ω, il ´etait
clair que la coupure de la divergence pour ω → 0 ne pouvait venir d’un calcul perturbatif.
25
elle correspond a
` un comportement de la r´eponse impulsionnelle σ(t) ∝ θ(t), traduisant l’apparition d’un
courant constant (i.e. la conservation de l’impulsion des ´electrons) suite a
` une impulsion de champ ´electrique.
59
-
Exercice IV.5 : Montrer que la conductivit´e de Drude peut ´egalement se re´ecrire sous la
forme σ0 = 2s e2 ρ0 D (relation d’Einstein) o`
u D = `2e /(τe d) est la constante de diffusion. On
−1
donne kF = 0.85 ˚
A(or), quelle est la distance parcourue par un ´electron balistique en 1 s ? On
(bulk)
(film)
donne `e
' 4 µm et `e
' 20 nm. Calculer D. Quelle est la distance parcourue par un
´electron diffusif en 1 s ?
Effet de la temp´erature : `
a T 6= 0, le r´esultat (IV.49) doit ˆetre convolu´e par la d´eriv´ee d’une
fonction de Fermi −∂f /∂. Comme le r´esultat `a T = 0K d´epend faiblement de l’´energie (`a travers
n et τe ), la convolution est sans effet. Toutefois la temp´erature n’est pas sans effet. En effet,
nous venons de montrer que les collisions (´elastiques) sur le d´esordre, induisent une conductivit´e
finie. Si les ´electrons sont soumis `
a d’autres processus de collission 26 (´electron-phonon,...) la self
´energie de la fonction de Green re¸coit les contributions des diff´erentes perturbations : d´esordre,
interaction ´electron-phonon, etc. Cela correspond `a ajouter les taux de relaxation 27 selon la loi
de Matthiesen
1
1
1
+
=
+ ···
(IV.51)
τ (T )
τe τe−ph (T )
Le calcul de la conductivit´e conduit `
a σDrude =
ne2 τ
m .
Drude conductance.– We can relate the Drude conductivity to the conductance G = σ0 s/L
where s is the cross section of the wire and L its length. It will be useful to introduce the
dimensionless conductance g defined by
G=
2s e2
g.
h
(IV.52)
We first consider the 1D case (long narrow wire) :
gDrude =
~kF kFd−1 s
h σ0 s
∼
τe
2s e 2 L
| m{z } L
(IV.53)
=`e
where we have used n ∼ kFd where d is the real dimensionality of the system. We recognize
Nc ∼ kFd−1 s, the number of conducting channels (i.e. number of open transverse modes of the
wire with energy smaller than εF ). Finally the dimensionless conductance can be written as the
ratio of two lengths
Nc `e
gDrude ∼
(quasi 1D)
(IV.54)
L
26
autres que ´electron-´electron qui conserve le courant total,
Temps de vie et temps de transport.– Nous devons ici apporter une petite pr´ecision sur la nature du
temps de relaxation intervenant dans la conductivit´e. Nous avons introduit le temps de vie des ´etats ´electroniques
0
def R
comme : 1/τ = −2 Im ΣR = 2πρ0 hC(θ)iθ , o`
u C(θ) = dr e−i(k −k)r V(r)V(0) o`
u θ est l’angle entre les deux
vecteurs k et k0 sur la surface de Fermi, ||k|| = ||k0 || = kF (`
a l’approximation de Born, C(θ) est proportionnelle a
`
la section efficace de diffusion dans la direction θ). Le temps qui intervient dans la conductivit´e n’est en g´en´eral
2
pas ce temps mais le temps de relaxation de la vitesse [93, 2, 3] : σ = nemτtr (ceci est li´e a
` la pr´esence de kx kx0
dans les relations (IV.26,IV.45)) ; ce temps est appel´e temps de transport 1/τtr = 2πρ0 hC(θ)(1 − cos θ)iθ . Dans le
cas de la diffusion isotrope, C(θ) = w n’a pas de structure et les deux temps co¨ıncident. Dans le cas de diffusion
anisotrope (lorsque la fonction de corr´elation de V pr´esente une structure), les deux temps peuvent toutefois
diff´erer notablement τtr > τ . Cette discussion s’applique au cas de la diffusion ´electron-phonon (chap. 26 de
[17]) : le temps de collision ´electron-phonon est τe−ph ∝ T −3 mais la diffusion est fortement anisotrope a
` basse
temp´erature ( ωD , la coupure de Debye) et le temps de transport correspondant est τe−ph, tr ∝ T −5 ce qui
conduit `
a la r´esistivit´e ρ(T ) = 1/σ(T ) ' 1/σ0 + C T 5 pour T ωD (loi de Bloch-Gr¨
uneisen).
27
60
The length at the numerator ξloc ∼ Nc `e is the localisation length in weakly disordered wire
[20]. So the validity of the treatment is L ξloc i.e.
gDrude 1
(IV.55)
The 2D case (plane in a 2DEG or thin metallic film) can be analysed by setting s = bL in
the previous expressions where b is the thickness (set b = 1 for the plane, in d = 2). We get
gDrude ∼ kF `e kFd−2 b
(2D)
(IV.56)
In d = 2 (in a DEG) we the Drude dimensionless conductance is simply proportional to the
plane
large parameter kF `e : gDrude
∼ k F `e .
IV.E
Corr´
elations entre fonctions de Green
Nous avons retrouv´e le r´esultat semi-classique en n´egligeant les corr´elations entre les fonctions
R A
de Green dans l’´equation (IV.45), i.e. en ´ecrivant GR GA ' G G . Nous souhaitons maintenant
´etudier l’effet des corr´elations entre fonctions de Green sur la conductivit´e moyenne. Il convient
` cette fin nous commen¸cons par donner
d’identifier quel type de corrections est dominant. A
la repr´esentation diagrammatique de (IV.45), que nous dessinons comme une bulle (figure 21)
constitu´ee par les deux fonctions de Green. Les lignes ondul´ees repr´esentent les ´el´ements de
matrice de l’op´erateur courant moyen, ekx /m et ekx0 /m, et nous rappellent que la conductivit´e
est une fonction de corr´elation courant-courant.
k
k
!F + "
k’
k’
!F
Figure 21: Repr´esentation diagrammatique d’une correction perturbative pour la conductivit´e
moyenne donn´ee par l’´equation (IV.45). Les lignes continues (tir´es) repr´esentent des fonctions
R
A
de Green retard´ees G (avanc´ees G ). Les lignes ondul´ees repr´esentent l’op´erateur de courant
(ekx /m et ekx0 /m). Les lignes pointill´ees reliant les lignes retard´ees et avanc´ees correspondent
`
a corr´eler les fonctions de Green GR et GA dans l’´equation (IV.45).
Dans le cas du terme de Drude les lignes de fonctions de Green sont les fonctions de Green
R
A
moyennes G et G . Les corr´elations entre fonctions de Green sont repr´esent´ees par des diagrammes dans lesquels les lignes de fonctions de Green retard´ee et avanc´ee sont coupl´ees par
des lignes d’impuret´e (figure 21).
ii L´
eg`
ere modification des r`
egles de Feynman pour la conductivit´
e ii.– Dans les
diagrammes, les lignes de fonctions de Green seront d´esormais des fonctions de Green moyennes,
R
A
G et G (au lieu des fonctions de Green libres), ce qui est une mani`ere de resommer une partie
des diagrammes ne couplant pas les fonctions retard´ees et avanc´ees.
Transport classique et localisation faible : pr´
esentation heuristique
Nous avons d´ej`
a not´e au cours du calcul de la conductivit´e `a l’approximation de Drude que
les r´esultats int´eressants sont non perturbatifs et correspondent `a resommer certaines classes de
diagrammes d´ecrivant la physique de la diffusion. La question est donc d’identifier les classes
de diagrammes qui apportent les corrections dominantes. Pour cela nous faisons une petite
61
digression et donnons une image plus qualitative de l’´etude du transport dans un m´etal diffusif.
Lorsqu’on ´etudie la conductance d’un syst`eme reli´e `a deux contacts, on peut montrer que la
conductance est reli´ee `
a la probabilit´e de traverser le syst`eme. Cette probabilit´e s’exprime
comme le module carr´e d’une somme d’amplitudes de probabilit´e :
G=
2s e2
2s e2 X 2
g∼
AC h
h
(IV.57)
C
o`
u la somme porte sur tous les chemins C allant d’un contact `a l’autre, pour un ´electron d’´energie
F . La conductance adimensionn´ee est not´ee g. Cette formulation des propri´et´es de transport
peut ˆetre rendue rigoureuse, c’est l’approche de Landauer-B¨
uttiker. La relation avec la formule
de Kubo se comprend en remarquant que l’amplitude AC est associ´ee `a la fonction de Green
retard´ee, et l’amplitude conjugu´ee A∗CP`a la fonction de Green avanc´ee (formule de Fisher &
Lee GR (contact 1 ←contact 2; F ) ∼ C AC ) : cette expression de la conductance poss`ede la
structure g ∼ |GR (contact 1 ←contact 2; F )|2 ∼ GR GA .
Dans le cas d’un m´etal faiblement d´esordonn´e (kF `e 1), la somme porte sur les chemins de
collisions sur les impuret´es. La phase d’une amplitude associ´ee `a un de ces chemins de collision
est proportionnelle `
a la longueur `C du chemin de diffusion : AC ∝ eikF `C .
La conductance moyenne est donn´ee en moyennant l’expression :
g∼
X
C
|
|AC |2 +
{z
classique
}
X
C6=C 0
|
AC A∗C 0
{z
quantique
(IV.58)
}
Le premier terme, la somme des probabilit´es, doit correspondre au terme classique (Drude).
C’est le terme dominant car donn´e par une somme de termes positifs. En revanche le terme
d’interf´erences (quantique) est une somme de termes portant des phases AC A∗C 0 ∝ eikF (`C −`C0 ) .
On comprend que ce type de contributions a d’autant plus de mal `a survivre `a la moyenne sur le
d´esordre que la diff´erence `C −`C 0 est grande (le terme classique |AC |2 domine car les deux chemins
correspondent `
a la mˆeme s´equence de collisions (figure 22)). Une fa¸con de minimiser la diff´erence
des longueurs est de consid´erer une s´equence de collisions identiques au milieu de laquelle on
introduit un croisement. 28 Cette construction fait apparaˆıtre une boucle `a l’int´erieur de
laquelle une des trajectoires est renvers´ee (figure 22). Ce terme d´ecrit l’interf´erence entre deux
trajectoires diffusives renvers´ees : dans la boucle, les deux trajectoires subissent les collisions
dans l’ordre inverse, ce qui correspond aux diagrammes maximallement crois´es (figure 23) que
` cause du croisement les phases des deux amplitudes diff`erent l´eg`erement et
nous analyserons. A
cette contribution est petite (une bonne introduction est donn´ee au chapitre 1 de [2, 3]).
IV.F
Diagrammes en ´
echelle – Diffuson (contribution non coh´
erente)
Nous venons de montrer par des arguments heuristiques que les s´equences de collisions identiques
jouent un rˆole important. Si nous traduisons les diagrammes de la figure 22 pour des conductivit´e, le diagramme de gauche prend la forme des diagrammes en ´echelle (“ladder diagrams”) :
σ(ω)
28
Diffuson
=
+
+
Rigoureusement, le croisement devrait ˆetre d´ecrit par une “boˆıte de Hikami”.
62
+ ...
(IV.59)
C’ C
C
Contact 1
Contact 2
C
R
A
Contact 1
R
Contact 2
A
` gauche :contribution au terme classique P |AC |2 . A
` droite : contribution au
Figure 22: A
C
P
∗
terme quantique C6=C 0 AC AC 0 qui minimise `C − `C 0 . La boucle d´ecrit l’interf´erence quantique
de trajectoires renvers´ees.
Chaque contribution peut ˆetre analys´ee facilement `a l’aide des r`egles de Feynman ´enonc´ees plus
` ce stade il est instructif d’analyser pr´ecis´ement le second diagramme :
haut. A
k1
k1−k
k
k’
k’−k1
k
∼
k’
X
R
A
kx kx0 G (k) G (k)
k, k0
k1
w X R
w R 0 A 0
A
G (k1 ) G (k1 )
G (k ) G (k )
Vol
Vol
k1
{z
}
|
Λ(0,ω)
o`
u les fonctions retard´ees sont prises `
a ´energie F + ω et les avanc´ees `a F .
La s´erie de diagrammes en ´echelles, i.e. le milieu de la bulle de conductivit´e (IV.59)
k+ q
k 1 +q
k’+q
=
Γd (q, ω) =
k
+
k’
k 1 +q k 2 +q
+ ... ,
+
k1
k1
(IV.60)
k2
est appel´e le Diffuson. Une impulsion suppl´ementaire q a ´et´e introduite dans les fonctions de
Green retard´ees, ce qui sera utile pour la suite. Il est facile de voir que Γd ob´eit `a l’´
equation
de Bethe-Salpether
w
+ Λ(q, ω) Γd (q, ω) ,
(IV.61)
Γd (q, ω) =
Vol
o`
u Λ(q, ω) est le bloc ´el´ementaire
k +q
w X R
A
Λ(q, ω) =
G (k + q; F + ω) G (k; F ) =
Vol
def
k
(IV.62)
k
Autrement dit, Γd correspond `
a une s´erie g´eom´etrique que nous resommons :
Γd (q, ω) =
w
1
Vol 1 − Λ(q, ω)
-
(IV.63)
Exercice IV.6 Calcul de Λ(q, ω) : Nous calculons Λ(q, ω) dans la limite q → 0 et ω → 0.
R
R
R
R
V´erifier que G (k + q; F + ω) = G (k) + (vk · q − ω)[G (k)]2 + (vk · q)2 [G (k)]3 + · · · o`
u toutes
les fonctions de Green du membre de droite sont prises `a ´energie de Fermi ; vk = k/m (on
R
admettra que le terme q [G (k)]2 d’ordre q 2 peut ˆetre n´eglig´e, ainsi que le terme ω 2 ). Comme
63
R
A
w P
n
m se calculent ais´
nousPl’avons Rsignal´e, les quantit´es Vol
ement en faisant
k [G (k)] [G (k)]
1
29 En d´
→
ρ
d
puis
en
utilisant
le
th´
e
or`
e
me
des
r´
e
sidus.
e
duire
que
0
k
k
Vol
R
Λ(q, ω) = 1 + iωτe − q 2 `2e /d + · · ·
(IV.65)
Ce r´esultat sera utilis´e `
a plusieurs reprises.
Validit´
e de l’approximation : ωτe 1 et q`e 1
Approximation de la diffusIon – Pˆ
ole de diffusion.– D’apr`es (IV.65), nous voyons que,
dans la limite ωτe 1 et q`e 1, le Diffuson co¨ıncide avec la fonction de Green de l’´equation
de diffusion :
w
1/τe
1 e
def w
Γd (q, ω) '
=
Pd (q, ω)
(IV.66)
2
Vol −iω + Dq
Vol Dτe
Nous comprenons qu’il n’est pas possible de tronquer la s´erie perturbative (IV.60) `a un ordre
donn´e car la limite q →
0 et ω → 0 nous conduit au bord du domaine de convergence de la s´erie
w P∞
n
g´eom´etrique Γd = Vol
n=0 Λ .
Nous sommes en mesure de calculer la contribution des diagramnes en ´echelle `a la conductivit´e
2s e2 1 X
Diffuson
R
R
σ(ω)
=
kx kx0 |G (k)|2 Γd (0, ω) |G (k 0 )|2
(IV.67)
2
2πm Vol 0
k,k
L’isotropie de la relation de dispersion conduit `a
1
Vol
Diffuson
σ(ω)
P
k
R
kx |G (k)|2 = 0 et donc
=0
(IV.68)
Ce r´esultat s’interpr`ete facilement : rappelons-nous que la conductivit´
Re ∞est reli´ee `a constante
de diffusion, i.e. `
a la fonction de corr´elation de la vitesse σ ∼ D = 0 dt hvx (t) vx i (ce que
nous rappelle la pr´esence de kx kx0 dans (IV.45)). Le Diffuson (IV.60) d´ecrit une s´equence de
collisions arbitrairement grande, apr`es laquelle la m´emoire de la direction initiale de la vitesse
est perdue. 30
Remarque : pˆ
ole de diffusion et conservation du courant.– Le comportement de Γd
est la signature de la nature diffusive des trajectoires ´electroniques dans le m´etal (on pourrait
s’en convaincre en calculant la fonction de r´eponse densit´e-densit´e du m´etal qui est reli´ee au
Dq 2
Diffuson χ0 (q, ω) ' −ρ0 −iω+Dq
ole de diffusion `a la
2 [6] ; cette remarque permet de relier le pˆ
conservation du nombre de particules [117]).
-
Exercice IV.7 Conductivit´
e non locale : Nous discutons l’importance de l’ordre des
limites limq→0 et limω→0 . Si le syst`eme est soumis `a un
inhomog`ene, la r´eponse sera
R champ
R
caract´eris´ee par une conductivit´e non locale ja (r, t) = dt0 dr0 σab (r, t; r0 , t0 ) Eb (r0 , t0 ). Nous
consid´erons sa transform´ee de Fourier σ(q, ω). On admet que la conductivit´e moyenne `a q 6= 0
2s e2 1 P
0
R
0
A 0
est donn´ee par : σxx (q, ω) = 2πm
2 Vol
k,k0 kx kx G (k + q, k + q; F + ω) G (k , k; F ). Montrer
Diffuson
que la contribution des diagrammes en ´echelle (le Diffuson) est σxx (q, 0)
29
2
= −σ0 qqx2 (qui
These useful integrals are given by
def
fn,m =
(n + m − 2)! n+m−2
w X R
A
[G (k)]n [G (k)]m = i−n+m
τe
.
Vol
(n − 1)!(m − 1)!
(IV.64)
k
30
Notons que dans le cas o`
u la diffusion est anisotrope, la contribution du Diffuson est non nulle. Elle fait
apparaˆıtre le temps de transport τtr lorsqu’elle est ajout´ee a
` la contribution de Drude [2, 3].
64
Diffuson
diff`ere donc de σxx (0, ω)
= 0).
Indication : En utilisant les mˆemes approximations que dans l’exercice IV.6, montrer que
R
A
kF `e
w P
k kx G (k + q)G (k) ' −i d qx .
Vol
IV.G
Quantum (coherent) correction : weak localisation
The discussion of the introduction, § IV.A, has underlined the importance of interferences between time reversed diffusive electronic trajectories. Starting from the Landauer picture of
quantum transport, we have reformulated more precisely this idea in § IV.E. This discussion has
emphasized the role of maximally crossed diagrams : 31 interference of reversed diffusive trajectories corresponds to follow the same sequence of scattering events in reversed order (figure 23)
!c
≡
+ ... =
+
Cooperon
Figure 23: Contribution of maximally crossed diagrams (Cooperon) σ(ω)
.
After twisting the diagram of the left part of Fig. 23, the structure of the central bloc is very
similar to the one involved in the Diffuson (right part of Fig. 23), up to a change in the relative
directions of the two Green’s function lines. It is useful to study carefully the structure of the
first diagram :
k1+k+k’
k
k’
k’
k
−k1 k
k’
∼
X
R
A
kx kx0 G (k) G (k)
k, k0
w X R
w R 0 A 0
A
G (k1 + k + k 0 ) G (−k1 )
G (k ) G (k )
Vol
Vol
k1
|
{z
}
Λ(k+k0 ,ω)
where retarded and advanced lines carry energies F + ω and F , respectively. This analysis
makes clear that the series of maximally crossed diagrams is also a geometric series involving
the quantity Λ(k + k 0 , ω) computed above :
k
k’
Γc (Q = k + k 0 , ω) =
k
=
k
k’
k1
k’
=
k
k’
+
+ ...
(IV.69)
k+ k’−k1
(the single impurity line was already included in the Diffuson). As a consequence, the calculation
is similar, and the resummation of the ladder diagrams is obtained by replacing the wave vector
q in the Diffuson (IV.60) by k + k 0 . This new object is named the Cooperon. As for the
Diffuson, the Cooperon presents a diffusion pole when Q = k + k 0 → 0 :
Γc (Q, ω) =
w
Λ(Q, ω)
w
1/τe
1 e
def w
'
=
Pc (Q, ω) .
Vol 1 − Λ(Q, ω)
Vol −iω + DQ2
Vol Dτe
(IV.70)
31
Much before the development of the theory of weakly disordered metals by Altshuler, Aronov, Khmelniskii,
Lee, Stone, Anderson, etc in the early 1980’, the importance of maximally crossed diagrams was recognised by
Langer & Neal [75]
65
Introducing the Cooperon in (IV.45) we obtain the structure
Cooperon
σ
e(ω)
=
2s e2 1 X
R
R
kx kx0 |G (k)|2 Γc (k + k 0 , ω) |G (k 0 )|2 .
2πm2 Vol 0
(IV.71)
k, k
Contrary to the Diffuson contribution, where the sequence of scattering on disorder decouples the
incoming wavevector k 0 to the outgoing one k, now the diffusion pole of the Cooperon constraints
the two wave vectors to be opposite. The diffusion approximation requires a small momentum
||Q|| 1/`e ; because the two momenta are on the Fermi surface ||k|| ' ||k 0 || ' kF 1/`e , the
sum over k and k 0 is dominated by k + k 0 ' 0 :
X
k, k0
R
R
kx kx0 |G (k)|2 Γc (k + k 0 , ω) |G (k 0 )|2 ' −
X
kF2 X R
|G (k)|4
Γc (Q, ω)
d
(IV.72)
Q
k
Cooperon
The fact that σ
e(ω)
is dominated by the contribution of momenta such that k ' −k 0 is
a manifestation of the enhancement of back-scattering of electrons, due to interferences of
reversed diffusive trajectories. This is the origin of the minus sign of the Cooperon’s contribution
to the conductivity, i.e. this is indeed a localisation effect.
R
w P
4
2
- Exercice IV.8 : Check that f2,2 = Vol
k |G (k)| = 2τe .
Using the result of exercice IV.8, and introducing the notation
def
Cooperon
∆σ(ω) = σ
e(ω)
'σ
e(ω) − σ
e(ω)
Drude
(IV.73)
(the Cooperon contribution is the dominant correction to the classical result). We find
∆σ(ω) = −
2s e2 1 X e
Pc (Q, ω)
π~ Vol
where
Q
Pec (Q, ω) =
1
−iω/D + Q2
(IV.74)
If we set the frequency to zero, the correction to the static
conductivity seems at first sight to
2 1 P
−2 . As usual, the occurence of a
se
involve a divergent quantity : ∆σ(ω = 0) = − 2π~
Q
Q
Vol
divergence hides interesting physical phenomenon.
Remark : Kubo vs Landauer and current conservation.– The comparison of the maximally crossed conductivity diagrams of Fig. 23 with the real space diagram of Fig. 22 leads
to the question of the precise correspondence, since the latter diagrams present two additional
Diffuson legs. The answer to this question lies in the connection between the Landauer picture
(conductance) and the Kubo picture (local conductivity), with some interesting relation to the
fundamental question of current conservation. This has been discussed in our paper [110] where
the weak localisation correction to the non local conductivity was constructed (in real space).
1)
Small scale cutoff
The derivation of (IV.74) has involved several approximations : first of all, assuming weak disorder kF `e 1, we have argued that the quantum correction to the conductivity is dominated by
the maximally crossed diagram (Ladder approximation). Then, the Diffuson and the Cooperon
were computed in the diffusion approximation, i.e. for
||Q|| 1/`e .
(IV.75)
This provides a first ultraviolet cutoff in the calculation of (IV.74). There remains the problem
2 R 1/`e dd Q
se
of the infrared divergence ∆σ(ω = 0) = − 2π~
Q−2 , in low dimension d 6 2.
(2π)d
66
2)
Large scale cutoff (1) : the system size
In a real situation, the transport is studied in finite size system. For example, if we consider
a transport in the Ox direction through a rectangular piece of metal of size Lx × Ly × Lz
(Fig. 24), when solving the diffusion equation, we must account for the geometry of the system ;
as a consequence the wavevectors are quantized. In order to account for the electric contacts
(transport experiment) we assume Dirichlet boundary condition in the Ox direction (absorbing
boundary conditions for an electron touching one contact) and Neumann boundary conditions
in the two other directions (reflexion boundary conditions) :
πnx πny πnz
Q=
,
,
with nx ∈ N∗ , ny ∈ N, nz ∈ N
(IV.76)
Lx Ly Lz
Lx
Ly
I
I
Lz
Figure 24: A metallic wire.
In the very asymmetric case Lx Ly , Lz (quasi-1D geometry) we can neglect the contributions of the transverse components of the wave vector :
∆σ(ω = 0) = −
∞
X
2s e2
1
2s e2 Lx 1
1
=
−
.
π~ Lx Ly Lz
(nx π/Lx )2
π~ Ly Lz 6
(IV.77)
nx =1
We prefer to translate the result in terms of the dimensionless conductance
g=
σ Ly Lz
,
2s e2 /h Lx
(IV.78)
from which we get the universal weak localisation correction
∆g = −
1
3
(IV.79)
independent on the material properties. This universal contribution to the mean conductance
must be compared to the large classical Drude conductance gD ∼ Nc `e /L 1 (where Nc is the
number of conducting channels).
In the quasi 2D geometry L = Lx = Ly Lz (thin film), the details of the wave vector
quantization are inessential and we can write
2s e2 1
∆σ(ω = 0) = −
π~ Lz
Z
~
1/L<||Q||<1/`
e
~ 1
d2 Q
2s e 2 1 1
=
−
~2
(2π)2 Q
π~ Lz 2π
Z
1/`e
1/L
dQ
Q
(IV.80)
In the 2D case, the weak localisation is controlled both by the short scale cutoff, `e , and by the
large scale cutoff, L. We get
∆σ(ω = 0) ' −
2s e2 1 1
ln(L/`e )
π~ Lz 2π
67
(IV.81)
therefore
1
∆g ' − ln(L/`e )
π
(IV.82)
In 3D, we consider a cubic box of size L = Lx = Ly = Lz
2s e2
∆σ(ω = 0) = −
π~
Z
~
1/L<||Q||<1/`
e
~ 1
d3 Q
2s e 2 1
=
−
~2
(2π)3 Q
π~ 2π 2
Z
1/`e
1/L
2s e2 1
dQ = −
h π2
1
1
−
.
`e L
(IV.83)
Finally
1
∆g ' − 2
π
L
−1 .
`e
(IV.84)
Note that in this case, contrary to the 2D case, the prefactor depends on the detail of the
regularization of the sum over Q.
3)
Large scale cutoff (2) : the phase coherence length
Weak localisation correction arises from the (quantum) interferences of reversed trajectories,
i.e. is a coherent phenomenon. The results obtained in the previous paragraph hence describe
systems (of size L) fully coherent. In practice, phase coherence is extremely fragile and is
only preserved over lengths smaller than the phase coherence length Lϕ . This length is of
fundamental importance since it set a frontier between the semi-classical transport regime, where
quantum interferences are negligible, 32 and the quantum regime, where transport properties
depend on the wave nature of the electrons.
We can obtain a direct estimation of the phase coherence length with an experimental setup
realizing the principle of the Young experiment : the Aharonov Bohm (AB) ring (Fig. 25). The
geometry of the ring defines two paths and the interference fringes are revealed by changing the
magnetic field. One observes regular oscillations of the conductance with a period in magnetic
field corresponding to one quantum flux φ0 = h/e in the ring. The amplitude of the AB
oscillations can be studied as a function of the temperature (Fig. 26) : the experimental result
(for a ring of perimeter L ' 0.9 µm) shows that oscillations are only visible below few Kelvins.
We can keep in mind the typical value for the phase coherence length
Lϕ (T ∼ 1 K) ∼ 1 µm .
As decoherence arises from interaction between a given electron (the interfering wave) and
other degrees of freedom (phonons, other electrons, etc), it is rather complicate to modelize
(see chapters X and XI). We will limit ourselves, in a first time, to introduce decoherence
within a simple phenomenological description corresponding to the elimination of trajectories
spreading over scale larger than the phase coherence length Lϕ .
This can be most conveniently achieved by reformulating the calculation of the weak localisation correction in a time-space representation. The expression
∆σ = −
2s e2 D X 1
π~ Vol
DQ2
(IV.85)
Q
32
Eventhough the effect of quantum interferences has disappeared above few Kelvins, the electronic transport
is not really classical : up to high temperatures, the dynamics of the electron liquid in a metal is completly
dominated by the Pauli principle (the Fermi-Dirac distribution).
68
e−
I (φ)
φ
φ0
⇒
φ
’
φ
Figure 25: Aharonov-Bohm oscillations in a metallic ring. Aharonov-Bohm oscillations
realises the Young experiment. Oscillating curve is the magnetoconductance. From [123].
Figure 26: Aharonov-Bohm amplitude. The study of AB amplitude as a function of temperature in a Gold ring (diameter 285 nm) with wire width 37 nm ; R ' 76 Ω. AB amplitude
(right) is obtained by averaging over 45 h/e periods (i.e. B ∈ [−1.1 T, +1.1 T]). From [120].
can be understood as a trace of the diffusion propagator integrated over all time scale
Z ∞ Z
2s e2
dr
∆σ = −
D
dt
Pt (r|r) where Pt (r|r0 ) = h r |etD∆ | r0 i
(IV.86)
π~
Vol
0
|
{z
}
def
Pc (t) =
solves (∂t − D∆)Pt (r|r0 ) = δ(t)δ(r − r0 ). The space averaged return probability Pc (t) counts
the number of closed diffusive trajectories for a given time scale t. This representation makes
clear how
p the large scales can be eliminated : we simply cut the integral at the time τϕ , where
Lϕ = Dτϕ . The most natural way is to introduce an exponential in the time integration 33
Z ∞
2s e2
∆σ = −
D
dt Pc (t) e−t/τϕ
(IV.87)
π~
0
33
This is justified within microscopic models of decoherence or dephasing. See chapter 6 and 7 of [3].
69
This is exactly equivalent to the introduction of a “mass” term in the diffusion propagator
∆σ = −
2s e2 1 X
1
.
2
π~ Vol
1/Lϕ + Q2
(IV.88)
Q
Meanwhile, we can also eliminate the shortest scale contributions (< `e ), where the diffusion
approximation ceases to be valid, by introducing another exponential :
∆σ = −
2s e2
D
π~
Z
0
∞
dt Pc (t) e−t/τϕ − e−t/˜τe
def
with Pc (t) =
Z
dr
Pt (r|r) ,
Vol
(IV.89)
where τ˜e = `2e /D = τe d (we have introduced this new scale to have an expression symmetric with
τϕ = L2ϕ /D). This new representation (IV.89) will be very convenient for subsequent analysis.
Remark : Real space derivation : A direct real space derivation of the result (IV.89) is
possible. It can be found in chapters 4 and 7 of the book [2, 3].
Simple applications.– As an application we immediatly obtain the weak localisation in the
1D and 2D case by using that the return probability is simply 34
Pc (t) =
1
.
(4πDt)d/2
(IV.90)
Using the integral
Z
0
∞
i
d
e−at − e−bt
d h d −1
−1
2
2
dt
=
Γ
1
−
a
−
b
2
td/2
(IV.91)
we deduce the weak localisation correction
2s e2 2 Γ 1 − d2 2−d
2−d
∆σ = −
L
−
`
.
ϕ
e
h
(4π)d/2
(IV.92)
We first consider the case of a long (quasi-1D) wire :
∆σ = −
2s e2
2s e2
(Lϕ − `e ) ' −
Lϕ
h
h
in d = 1
(IV.93)
i.e. the WL correction to the dimensionless conductance
∆g = −
Lϕ
L
(IV.94)
For L ∼ Lϕ we recover the result of the coherent wire ∆g ∼ −1, Eq. (IV.79).
The 2D case can be simply studied by dimensional regularization d = 2(1 − ) with → 0.
Remembering that Γ() ' 1/, we find the logarithmic behaviour
∆σ = −
2s e2 1
ln(Lϕ /`e )
h π
in d = 2
(IV.95)
i.e.
1
∆g = − ln(Lϕ /`e )
π
34
We omit the section of the film for d 6 2 ; in principle we should write Pc (t) =
2
the thickness of the film (d = 2) and W the cross-section of the wire (d = 1).
70
(IV.96)
1
(4πDt)d/2 W 3−d
where W is
We may again check that at the crossover L ∼ Lϕ we recover the result of the coherent plane
Eq. (IV.82). Contrary to the 1D case, controlled by the large scale cutoff only, both short scale
and large scale cutoffs appear in the 2D result.
Finally, we mention the result in 3D for the sake of completeness
2s e 2 1
1
1
∆σ = −
−
in d = 3
(IV.97)
h 2π `e Lϕ
However we recall that the typical order of magnitudes for the elastic mean free path in bulk
(bulk)
makes the coherent properties unlikely to be in the diffusive regime (recall that in Gold `e
'
4 µm most surely exceeds Lϕ in usual situations). The related correction to the dimensionless
conductance is (square box)
1 L
L
∆g = −
−
.
(IV.98)
2π `e Lϕ
Note that the prefactor 1/(2π) differs from the one found above for the coherent wire, Eq. (IV.84),
the reason being that we used different regularizations in the two cases (the prefactor depends
on regularisation in d = 3 but not in d 6 2).
How to identify the WL ?– In d 6 2, the fact that (IV.74) exhibits an infrared (large
scales) divergence has required a regularization of the sum, what makes the weak localisation
∆g strongly dependent on Lϕ (we mean that the dominant term of ∆g depends on Lϕ ). The
measurement of the WL provides therefore a direct information on the phase coherence length.
On the other hand, in d = 3, only the subdominant term of the WL depends in Lϕ ; the WL is
therefore not a good tool to probe phase coherence in this case.
The WL is a very small correction to the classical conductance. The question is : how can
we identify the WL on the top of a large conductance G = GDrude + ∆Gquantum ?
Obviously, one should vary some parameter on which the WL depends.
Because decoherence is activated by increasing the temperature, a first natural suggestion
would be to study the temperature-dependence of the conductance. This is however not a good
idea for the reason that the WL is not the only temperature-dependent quantum correction to
the conductivity. We will see (chapter XI) that the conductivity receives another correction
from the electronic interactions, named the Altshuler-Aronov (AA) correction. This has led to
some confusion in the analysis of the first measurements of temperature-dependent quantum
corrections to transport (like in Ref. [60]).
Another possibility is to vary the magnetic field, on which the WL depends, as we explain
below, but not the AA correction ∆Gee . In summary, the low field and low temperature conductance may be written as
G(B, T ) = GDrude + ∆GWL (B, Lϕ (T )) + ∆Gee (T )
(IV.99)
The understanding of the WL’s magnetic-field dependence is thus a crucial issue in order to
identify the WL correction and extract the phase coherence length. This has been used in many
experiments in thin films [24] and narrow wires [98].
4)
Magnetic field dependence – Path integral formulation
We have shown that the WL correction is related to the interference of time reversed trajectories,
what we write schematically as
X
∆σ ∼ −
AC A∗Ce ,
(IV.100)
C
71
where Ce is the reversed trajectory. In the presence of a weak magnetic field (recall that usual
magnetic field are not able to bend significantly the electronic trajectories on the scale `e ), an
amplitude receives an additional magnetic phase :
ie
AC = |AC | eikF `C e ~
R
C
~
d~
r ·A
.
(IV.101)
The magnetic phase remains unchanged under the combination of complex conjugation and time
reversal, therefore the pair of amplitudes carries twice the magnetic phase
I
2ie
∗
∗
~
d~r · A(~r)
(IV.102)
AC ACe = AC ACe exp
~ C
This discussion, in the spirit of § IV.E, can be made precise by reformulating the analysis
of the Cooperon in time representation, Eq. (IV.89), within the frame of path integration. The
diffusion propagator has the path integral representation
0
Pt (~r|~r ) =
~
r(t)=~
r
Z
~
r(0)=~
r0
D~r(τ ) e
1
− 4D
Rt
0
dτ
d~
r (τ )
dτ
2
(IV.103)
where the integral corresponds to a summation over all diffusive paths ~r(τ ) going from ~r 0 to ~r
Rt
d~
r(τ ) 2 1
dτ
with the Wiener measure D~r(τ ) exp − 4D
. The form of the measure ensures
dτ
0
that paths, which are Brownian motions, are continuous (i.e. the discontinuous functions in the
integral have zero weight) but is not differentiable. The path integral (IV.103) corresponds to a
summation over all diffusing electronic trajectories for a given time scale, i.e. for loops with a
given number of collisions 35 t/τe . In Eq. (IV.89), the remaining integral over time realises the
summation over the trajectories length :
X
C
AC A∗Ce
−→
Z
∞
Z
~
r(t)=~
r
dt
τ˜e
~
r(0)=~
r
D~r(τ ) e−
Rt
0
dτ
1 ˙
~
r(τ )2
4D
(IV.104)
(the time integral is cut off at the time τ˜e = `2e /D below which the diffusion approximation
breaks down).
The path integral representation (IV.103) allows us to add in a natural way the contribution
of the magnetic phase. We obtain the path integral representation of the Cooperon
Pt (~r|~r) =
Z
~
r(t)=~
r
~
r(0)=~
r
Dr(τ ) e−
Rt
0
dτ
1 ˙
~ r)·~
~
r(τ )2 + 2ie
A(~
r˙ (τ )
4D
~
.
From this representation, we deduce that Pt (~r|~r 0 ) solves
"
2 #
∂
2ie
~ −
~ r)
−D ∇
A(~
Pt (~r|~r 0 ) = δ(t) δ(~r − ~r 0 )
∂t
~
(IV.105)
(IV.106)
The presence of the charge 2e reminds us that the Cooperon is a two particle propagator in the
particle-particle channel, which is the reason for the terminology “Cooperon”.
35
In a time t, an electron experiences t/τe collisions, i.e. the time t corresponds to trajectories of length
`C = `e (t/τe ) = dDt/`e . The Brownian motion is the continuum limit of the random walk on defects, `e → 0 ; in
this limit the length of the Brownian curve is infinite `C ∝ t/`e → ∞.
72
Summary.– The weak localisation correction will be conveniently computed with the spacetime representation (IV.89) with (IV.106). The time integral can be performed to obtain the
useful representation
Z
2s e2
d~r
∆σ = −
(IV.107)
Pc (~r, ~r)
π~
Vol
where the Cooperon is the Green’s function of a diffusion-like operator
Pc (~r, ~r 0 ) = h~r |
1
~
− ∇−
1/L2ϕ
2 |~r
2ie ~
A(~
r
)
~
0
i
(IV.108)
This last equation furnishes the pratical method to obtain Pc (~r, ~r 0 ).
Magneto-conductance in a narrow wire.– A precise study of the magneto-conductance
of a narrow wire will be done in the problem page 89. We only give here a first qualitative
description of the magnetic-field dependence of the WL correction ∆g(B).
We first notice that we expect a quadratic weak magnetic field dependence (this results from
the fact that the WL involves a propagator at coinciding point, spatially integrated ; hence it
must be a symmetric function of B) :
∆g(B) ' ∆g(0) + c B 2
B→0
(IV.109)
with c > 0 (this is the positive – anomalous – magneto-conductance, cf. TD 4).
The large field behaviour is obtained as follows : the presence of the B field adds the magnetic
phase (IV.102) to the contributions of the reversed electronic trajectories. The presence of this
phase is responsible for an effective B-dependent cutoff since it eliminates all trajectories
encircling a magnetic flux larger that the quantum flux φ0√= h/e. For a given time scale t, a
diffusive trajectory typically spreads over a distance Lt ∼ Dt along the wire. If the wire has
a section W , the trajectory typical encloses a magnetic flux Φ ∼ BW Lt . Contributions of the
trajectories such that Φ ∼ BW Lt & φ0 are eliminated, what gives the cutoff
LB ∼
~
e|B|W
(narrow wire) .
(IV.110)
For LB < Lϕ , the dominant cutoff is the magnetic field, hence we should perform the substitution
Lϕ → LB in Eq. (IV.94) :
LB
∆g(B) ' −
∝ −1/|B|
(IV.111)
L
at large field. The crossover between the low field and the high field behaviour occurs when
LB ∼ Lϕ , thus the WL curve has a typical width in magnetic field
∆B =
φ0
,
W Lϕ
(IV.112)
corresponding to one quantum flux φ0 = h/e in the area W Lϕ .
We will demonstrate
later that the correct definition for the magnetic length of the narrow
√
wire is LB = ~/( 3|B|W ) and will obtain the precise crossover function interpolating between
(IV.109) and (IV.111), perfectly describing the experimental results like the one of Fig. 18 or
Fig. 27.
73
D = 3500 cm 2/s
w = 1500 nm
75
-15
0
B (G)
5
10
15
(b) 36
375 mK
270 mK
200 mK
140 mK
100 mK
76 mK
60 mK
40 mK
33 mK
Lj L
-0.2
34
~Φ0Lj W
2
-0.4
G (e /h)
DgHBL
-5
-0.6
32
L (µm)
0.0
-10
-0.8
2
30
D = 3100 cm /s
w = 1000 nm
-1.0
-6
-4
-2
0
2
4
-30
6
-20
B
-10
0
B (G)
10
20
(c)
30
600 mK
L (µm)
2
G (e /h)
480 mK
Figure 27: Weak localisation correction of a long wire. Left : Theoretical
result,
375 mK
7
270 mK
Eq. (IV.151). Right : Experimental MC of a long wire (L = 150 µm) etched in a 200
2DEG
;
mK
140 mK
100 mK
width W = 0.63 µm (lithographic width wlitho = 1 µm). From [91].
76 mK
60 mK
6
Magneto-conductance in thin metallic film.– We can repeat the same exercice for a
2
D = 2400
/s
metallic film. The low field dependence is again expected5 to be quadratic,
Eq.cm(IV.109).
The
w = 600 nm
high field dependence is obtained as follows
:
for
a
given
time
scale
t,
a
diffusive
trajectory
-80
-40
0
40
80
√
(G)
typically spreads over a distance Lt ∼ Dt in the plane, and typicalB encloses
a magnetic flux
Φ ∼ BL2t . Longest trajectories, such as Φ ∼ BL2t & φ0 , do not contribute to the WL, what gives
FIG. 12. "Color online# WL curves of "a# 1500, "b# 1000, and "c#
the cutoff
s
600 nm wide wires at D = 3500 cm2 / s, 3100 cm2 / s, and
~ 2400 cm2 / s, respectively. The conductance is normalized by e2 / h.
LB ∼
(thin film) .
(IV.113)
e|B| The broken lines are the best fits to Eq. "10#.
2
= 3500
/ s is plotted as a function of T in Fig. 13. As in
Replacing Lϕ by LB in Eq. (IV.96) gives the D
large
fieldcmbehaviour
the diffusive regime, L! follows a T−1/3 law at low tempera1
tures and 1
varies linearly with T above !1 K. Such a tem∆g(B) ' − ln(LB /`e ) ' + ln |B| + cste .
(IV.114)
perature dependence
is indeed expected in the semiballistic
π
2π
regime.54
As a consequence, the crossover between the low field and high field behaviours occurs on a scale
100
∆B =
2
D = 3500 cm /s
φ0
.
L2ϕ
(IV.115)
IV.H
1)
Lφ (µm)
The precise magneto-conductance curve will be derived in the problem page 87.
10
Scaling approach and localisation (from the metallic phase)
w = 1500 nm
w = 1000 nm
w = 600 nm
experiment
where
The β-function
The starting point of the scaling approach is the gedanken
one analyses the
1
conductance of a cubic box of size Ld connected at two opoosite
sides,
of the
0.01
0.1as a function
1
10 size.
T (K)
The Drude conductance was analysed in the introductory chapter
 online# Phase coherence length of 1500, 1000,
FIG. 13. "Color
2
in d = 1
1−d/2
and
600
nm wide
wires as a function of T at D = 3500 cm2 / s. The
(4π)
(kF W )d−1 `e
gD = cd
where cd = solid lines
(IV.116)
1/2fits with in
=2
are =
the best
Eq.d"8#.
L
2 Γ( d2 + 1) 

1/(3π) in d = 3
FIG. 14.
ferent wires
cients. The o
regime and t
best fits to E
V. DISO
A
In Secs.
ture depend
as the semi
the disorde
purpose, we
dence of th
wire widths
Interestingly
temperature
semiballisti
clear that th
regime depe
regime. Thi
as a functio
as shown in
dependencie
D2/3 law, w
245306-8
The WL correction for a coherent piece of metal is (see above)

1

in d = 1

− 3
1
in d = 2
∆g = − π ln(L/`
e )


1
L
−
−1
in d = 3
2π
`e
74
(IV.117)
(note that, in d = 3, the subdominant term +1/2π is important : it will provide the asymptotic
of the β-function). As noticed, being a negative correction to the conductance, the WL is a
precursor of the localisation effect. We can now analyse the asymptotic behaviour of the scaling
approach’s β-function (i.e. from the conducting phase). We compute the β-function
β(g) =
dln g
.
d ln L
(IV.118)
Note that in numerical calculations, scaling analysis is performed on the Lyapunov exponent,
i.e. consider ln g [74]. In the conducting limit, we can write ln g ' ln(gD ) + ∆g
gD . We deduce the
expression of the β-function in the metallic limit g → ∞

1

in d = 1 (multichannel)
3
ad
1
β(g) ' d − 2 −
where ad = π
(IV.119)
in d = 2
g→∞

g
1
in d = 3
2π
which suggests a monotonous behaviour, which is confirmed by numerical calculations (see
Fig. 12) [74].
2)
Localisation length in 1D and 2D
The weak localisation phenomenon is a precursor of the strong localisation, as suggested by
the behaviour of the β-function. Since gD and ∆g have opposite signs, we can expect that the
localisation length corresponds to the length at which the two quantities become comparable
(although this is in principle out of the range of validity of the calculation of the WL) :
gD ∼ |∆g|
for L ∼ ξloc .
(IV.120)
Narrow wire.– The general expression of the Drude conductance in a wire is gD = kF W `e /2L =
αd Nc `e /L where αd = Vd /Vd−1 (where Vd is the volume of the unit sphere) and Nc the number
of conducting channels : Nc = kF W/π in d = 2, i.e. for a wire of width W etched in a two
dimensional electron gas (2DEG). And Nc = (kF W )2 /4π in d = 3, i.e. in a thin and narrow
metallic wire of section W 2 .
This leads to
(1D)
ξloc ∼ Nc `e
(IV.121)
This expression of the localisation length has been confirmed by other approaches, like the
random matrix approach of Dorokhov-Mello-Pereyra-Kumar (see the review by Beenakker [20]).
Planes and thin films.– In d = 2, i.e. in a 2DEG, we have gD = kF `e /2, therefore the
localisation length
π
(2D)
ξloc ∼ `e e 2 kF `e
(IV.122)
is very large.
In d = 3, i.e. in a thin film of thickness W , we have gD = kF2 W `e /3π, therefore we obtain
1 2
(2D)
ξloc ∼ `e e 3 kF W `e .
75
(IV.123)
IV.I
Conductance fluctuations
An important aspect on which we have been elusive up to now is the question of disorder
averaging. As we have pointed out, the theoretical tools can only provide information on
averaged quantities, like the mean conductivity σ. On the other hand, most of the experiments
are performed on a given sample. Two questions immediatly arise : first, how can we understand
that the weak localisation (an averaged quantity) describes well the magnetoconductance of a
given sample, as it was illustrated by showing several experimental data ? Second, can we
characterize the sample to sample fluctuations (or correlations) of the conductance ?
1)
Disorder averaging in large samples
We can answer to the first question by a simple qualitative argument. If a system is much larger
than Lϕ , electronic wave in different parts separated by Lϕ do not interfere. When the system is
splitted in pieces of size Lϕ , they can therefore be considered as statistically independent. As an
illustration, we consider a long wire of size L Lϕ . We recall that the resistance has additive
properties : we introduce ri , the resistance of a coherent piece, of size Lϕ . The different pieces
being incoherent, the classical law of addition of resistances holds
R'
N
X
ri
i=1
where N = L/Lϕ .
(IV.124)
Weak localisation.– The resistance of each coherent part can be written as ri = ricl + ∆ri
where ricl is the classical resistance and ∆ri the quantum correction. The quantum correction
to the total resistance ∆R = R − Rcl can be related to the conductance correction
P
∆ri
∆R
N ∆r
1
∆G = − 2 = − Pi 2 ' −
∆Gϕ
(IV.125)
2 =
|{z}
R
N
cl
N rcl
r
one sample
i i
where we have introduced the weak localisation correction for the coherent piece ∆Gϕ =
2
−∆r/(rcl )2 . We have shown above that ∆Gϕ = 2she ∆g ϕ ∼ −e2 /h. We have thus recovered the
behaviour of the weak localisation correction
∆g
|{z}
one sample
∼−
Lϕ
1
∆g ϕ ∼ −
N
L
(IV.126)
and shown how the measurement on a given sample, ∆g = g −g cl , can coincide with the averaged
quantity, ∆g ' −Lϕ /L.
Conductance fluctuations.– We can develope a similar argument in order to analyse the
conductance fluctuations. We introduce the (sample to sample) fluctuation δri = ri − ri for the
coherent piece of metal :
P
N δr2
1
i,j δri δrj
2
δG ' P 4 '
(IV.127)
4 = 3 δG2ϕ
N
N
r
i ri
where δG2ϕ is the conductance fluctuation for a coherent metal. We deduce that the conductance
fluctuations of the long wire behaves as
3
Lϕ
δg 2 '
δgϕ2 .
(IV.128)
L
76
The conductance fluctuations of the coherent piece of metal δgϕ2 is controlled by two lengths :
the phase coherence length Lϕ and also the thermal length
def
LT =
p
~D/kB T .
(IV.129)
This second length scale has its origin in thermal fluctuations. We recall that the conductivity
at finite temperature is a convolution of the zero temperature conductivity σ
˜ (F ) :
Z
∂f
σ = d −
σ
˜ () .
(IV.130)
∂
The average conductivity has no structure in energy, therefore the thermal smearing has no
effect on averaged transport properties. On the other hand, the conductivity fluctuations δσ 2
involves correlations at different energies : δ˜
σ ()δ˜
σ (0 ). Correlations in energy decay over a
characteristic energy scale which is the Thouless energy
ETh ∼
D
L2
(IV.131)
(we will prove this below by a detailed analysis of the correlator). This has some consequence
on the temperature dependence of the fluctuations δσ 2 as the two Fermi functions constraint
the two energies to be close, | − 0 | . T .
In a coherent system (Lϕ > L) and for T . ETh , the conductance fluctuations are of order
2
δg ∼ 1. On the other hand, when temperature is larger than the correlation energy, T & ETh ,
i.e. LT . L, different energy shells of width ETh can be considered as independent and the
conductance fluctuations are reduced by a factor δg 2 ∼ ETh /T = (LT /L)2 .
We can repeat the argument in the regime Lϕ < L. If LT Lϕ , the thermal smearing
is negligible, hence δgϕ2 ∼ 1, however in the limit LT Lϕ , it leads to a reduction of the
fluctuations, by a factor (LT /Lϕ )2 . In conclusion, the conductance fluctuations of the long wire
are
(
(Lϕ /L)3
for Lϕ LT L
δg 2 ∼
(IV.132)
2
3
LT Lϕ /L
for LT Lϕ L
2)
Few experiments
Figure 28: Magneto-conductance of a short (coherent) wire in a 2DEG for different gate voltage.
From [102].
Magneto-conductance measurements in coherent wires have been realised by Skocpol et
al [102]. The conductance g(B) of a given sample is shown on figure 28 for different gate voltages.
77
G ( e2/h )
It exhibits fluctuations of order g ∼ 1 (i.e. G ∼ e2 /h). The complicate erratic structure, perfectly reproducible (as soon as the disorder configuration is stable), represents the interference
pattern characteristic of the disorder configuration : it is called the “magneto-fingerprint” of
16.5
the sample.
(a)
16.4 Sanquer [83] who were able to produce
A remarkable experience was realised by Mailly and
a set of 46 magneto-conductance (MC) curves associated
with different disorder configurations.
16.3
After having recorded a magneto-conductance curve, they changed the disorder configuration
16.2
by heating the sample and injecting a high current pulse. The 46 MC curves are plotted on
16.1
the figure. Then they perform some statistical analysis
of the data by extracting the mean
conductance and the variance. The mean conductance
g(B)
shows a deep around zero field
16.0
over a scale Bc , which we can identify with the weak localisation correction. Remarkably, the
15.9
200
600
1000
1400
1800
variance drops by a factor 2 over the same correlation field.
B ( 10-4 Teslas )
16.5
G ( e2/h )
16.4
(b)
16.3
16.2
16.1
16.0
15.9
200
600
1000
1400
1800
B ( 10-4 Teslas )
4.00
G ( e2/h )
16.4
var (G) (10-3( e2/h )2 )
16.5
(a)
16.3
16.2
16.1
(c)
3.00
2.00
1.00
16.0
15.9
0.00
200
600
1000
1400
200
1800
B ( 10-4 Teslas )
600
1000
1400
1800
B ( 10 Teslas )
-4
16.5
(b)
G ( e2/h )
Figure 29: Left
16.4 : UCF : 46 MC curves obtained with a short (coherent) wire etched in a 2DEG
(T = 45 mK).
The different curves are obtained by modifying the disorder configuration. Right :
16.3
Averaging the MC leads to the weak localisation. Figures of Ref. [83].
16.2
Not only16.1the variance of the conductance was extracted from experimental data but also
the correlations
C(∆B) = G(B0 )G(B0 + ∆B) (Fig. 30) : the correlation function was obtained
16.0
from a given MC curve by sweeping the magnetic
field B0 . The authors have assumed the
R B2 dB
15.9
0
200
600
1000
1400
1800
equivalence of the two correlation functions B1 B2 −B
G(B0 )G(B0 + ∆B) and C(∆B) (the
1
-4
B
(
10
Teslas
)
“ergodic hypothesis”).
3)
var (G) (10-3( e2/h )2 )
4.00
(c)
Conductivity
correlations/fluctuations
3.00
Identification of corrections.–
2.00
1.00
0.00
200
600
1000
1400
B ( 10 Teslas )
-4
1800
78
Figure 30: Correlations G(B0 )G(B0 + ∆B) from the MC of a short wire in a 2DEG. From [102].
Effect of magnetic field.–
1
δg(B)2 ' δg(0)2
2
for strong B
(IV.133)
as shown on Mailly and Sanquer’s experiment.
Thermal smearing.– Reduction of fluctuations by a factor ETh /T where ETh = D/L is the
Thouless energy (in a coherent sample ; when L Lϕ we should write ETh = D/L2ϕ ).
4)
Averaging : AB versus AAS oscillations
• Around zero field : h/2e oscillations dominate. These are AAS oscillations of the averaged
conductance.
• At large field (LB Lϕ ), h/e oscillations dominate. These are AB oscillations (fluctuations).
Figure 31: Magneto-conductance of a single ring and a series of 30 rings. Fourier spectrum.
From [114].
79
P H Y S I C A L R E V I EFigure
W L E T32:
T E AB
RS
and AAS amplitudes
as a function of the number of rings. From [114].
week ending
12 JANUARY 2007
PHYSICAL REVIEW LETTERS
PRL 98, 026807 (2007)
which is the key relation fro
results, bearing in mind that
L# are fixed parameters.
Let us first consider larg
sions larger than the phase
L# . Since interfering time re
a typical size L# , they do n
system and therefore !! is s
AAS amplitude varies as !
ratio is constant, this amplit
!gAA
In this regime we also see fro
FIG. 3 (color online). AAS amplitude !gAAS and AB ampliFIG. 2 (color online). Magnetoresistance of a square network
h of several samples of
which
leads to
as
a
function
of
the
number
N
of
plaquettes
for
33: 3000
ABplaquettes:
and AAS
a data;
functiontudeof"gthe
number
of
plaquettes
in
large
arrays.
containing
(a) lowamplitudes
field data; (b) highas
field
ble for the small sample.Figure
AB
different networks of various sizes Lx $ Ly / N, for two top(c),(d) Fourier (FFT) amplitudes of (a),(b), respectively. Data are
From
[100].
"gAB /
taken at 400 mK.
ologies, square (squares) and T 3 (diamonds) [17].
performed at 400 mK;
egime with a relatively
zes the signal to noise
At this temperature, the
d from standard weak
0 nm wide wire fabri’ 6 "m, the diffusion
thermal length LT #
the magnetoresistance
aquettes. At low field
riod B # 50 G, correare identified as the
y higher than the field
we observe a different
ns have a periodicity of
these are AB oscillaifferent periodicity of
ions, we display their
field) and 2(d) (high
he main peak clearly
he low field regime, it
ledge, this is the first
ations are observed on
ion of the amplitude of
s versus the number of
cillations we sweep the
00 G, whereas for the
range of $1200 G. To
he AB oscillations, we
eriods after subtraction
low frequency fluctuaund noise by repeating
ame conditions but at
gain the Fourier trans-
form. The amplitude of the AB signal is then obtained from
the Fourier spectrum after subtraction of the background
spectrum integrated over the same frequency range, in a
similar way used for persistent current measurements [11].
This procedure is only necessary for very large networks
(typically larger than 105 plaquettes) since for smaller
networks the noise is negligible. For the determination of
the AAS amplitude such a procedure is not necessary, as
the background noise is always negligible. However, the
second harmonic (!0 =2) of the AB oscillations has the
same frequency as the first harmonic (!0 =2) of the AAS
oscillations. For small networks (typically N % 100) this
contribution cannot be neglected. In order to extract the
AAS signal, we therefore determine first the amplitude of
the second harmonic of the AB oscillations at high field
and then subtract this amplitude from the first harmonic of
the oscillations measured at low field [12].
In Fig. 3 we display the amplitude of magnetoconductance oscillations (AAS and AB) extracted from the
Fourier spectra as a function of the number N of plaquettes.
For large networks (N * 300), the pamplitude
of the AB
!!!!
oscillations clearly decreases as 1= N , whereas the amplitude of the AAS oscillations are independent of the
number of plaquettes as naively expected. More surprising
is the behavior observed for small networks: when they
contain typically less than N ’ 300 plaquettes, pthe
!!!! amplitude of the AB oscillations varies faster than 1= N . At the
same time the AAS amplitude now depends on N (Fig. 3).
In the following, we will show that this new behavior
results from a dimensional crossover when the transverse
size of the network becomes smaller than the phase coherence length: one then enters a new regime where the
transport properties are effectively one dimensional on
the two-dimensional network.
Let us first recall general considerations on the magnetoconductivity oscillations: on one hand, the AAS oscillations are the Fourier harmonics of the weak localization
correction h!!!B"i to the average conductivity [5]. On the
other hand, the amplitude of the AB oscillations can be
obtained from the correlation function of the conductivity
h"!!B""!!B0 "i. The expressions of these two quantities
are indeed related [13–16]. In the limit where the thermal
length LT is smaller than L# , this yields
"!2AB #
80
e2 4$L2T
!!AAS ;
h 3 Vol
(1)
where "!AB and !!AAS are, respectively, the first harmonics of the AB and AAS oscillations and Vol the volume of
the sample. In this formula, the temperature dependence
originates from the conductivity correlation function
which probes a finite energy scale of width kB T [3,16].
The key feature is the proportionality between "!2AB and
!!AAS . Indeed, both quantities can be written in terms of
the coherent part of the return probability to the origin for a
diffusive particle. Consequently, both must probe in the
same way the influence of the geometry [16].
We consider a network of dimensions Lx $ Ly (see
Fig. 1). It is important to keep in mind that experiments
presented here are performed on several networks of different sizes, but of constant aspect ratio Lx =Ly # 10. The
length and width of the networks thus
p!!!! scale with the
number of plaquettes N as Lx / Ly / N .
The dimensionless conductance g # G=!2e2 =h" of the
network is then related to the conductivity by Ohm’s law
g / !Ly =Lx . Combined with Eq. (1) and given that Vol /
Lx Ly , this yields for the amplitudes of the conductance
oscillations !gAAS and "gAB :
2
This is exactly what is obse
the number of plaquettes is
diffuse on what they feel as
For smaller networks, the
tually becomes smaller than
we enter a regime where th
sally coherent whereas it re
ent: Ly ' L# ' Lx . In this
1D scaling !!AAS / L# =Ly
1=Lx and "g2AB / 1=L3x , wh
!gAAS
"gAB /
This is precisely what is ob
Fig. 3.
It remains now to check
crossover observed on Fig.
the phase coherence length. T
N ’ 300 corresponding to Ly
be compared with the coher
sured at T # 400 mK. This
more than qualitative, suppo
To summarize, the dimen
the scaling of the AB oscillat
ent scaling Ly =L3x ! L# =L
ductance fluctuations, with
useful to compare these depe
chain where the number N o
length Lx of the chain, so t
!gAAS / 1=N. Since the con
yields for thep!!!!relative fluc
"gAB =g / 1= N as was obs
An interesting way of che
Eq. (2): we can see that the r
IV.J
Conclusion : a probe for quantum coherence
The main interest in WL analysis is to provide a practical probe of phase coherence properties.
Moreover, the analysis of ∆σ(B, Lϕ ) defines 36 the phase coherence length Lϕ . We can show that
this method is effective in practical situation : in typical WL measurements, the value of Lϕ
can be obtained from the fit of the magneto-conductance curve at different temperatures, what
provides the temperature dependence of the phase coherent length. The typical temperature
dependence (for a narrow wire) is shown on Fig. 34. The analysis of the temperature dependence provides information on the microscopic mechanisms responsible for the decoherence. For
example, the behaviour τϕ ∝ T −3 above T = 1 K is attributed to elctron-phonon interaction.
Below T = 1 K, the dominant decoherence mechanism is provided by electronic interacions,
−2/3 in 1D.
characterised by the behaviour
τϕ ∝OFT ELECTRONS
DEPHASING
IN MESOSCOPIC METAL WIRES
PHYSICAL REVIEW B 68, 085413
$2003%
cess. Nevertheless, if the exponent of T is left as a fit parameter, better fits are obtained with smaller exponents ranging
from 0.59 for samples Ag$6N%d and Au$6N% up to 0.64 for
sample Ag$6N%c. This issue will be discussed in Sec. V B.
The dashed line in Fig. 3 and Fig. 4 is the quantitative prediction of Eq. $3% for electron–electron interactions in
sample Ag$6N%c. The dephasing times are close, though always slightly smaller, to the theoretical prediction of Eq. $3%.
Table III lists the best fit parameters A, B, together with the
prediction A thy of Eq. $3%.
This data set casts doubt on the claim by Mohanty, Jariwala, and Webb7 $MJW% that saturation of " # is a universal
phenomenon in mesoscopic wires. One can always argue that
the saturation temperature for our silver samples is below 40
mK, hence unobservable in our experiments. However, the
resistivity and dimensions of sample Ag$6N%a are similar to
those of sample Au-3 in the MJW paper,7 which exhibits
saturation of " # starting at about 100 mK, and has a maximum value of " #max"2 ns. In contrast, " # reaches 9 ns in
Ag$6N%a.
FIG. 4. Phase coherence time vs temperature in samples
Ag$6N%a (!), Ag$6N%b ("), Ag$6N%c (#), Ag$6N%d ($), and
Au$6N% $* %, all made of 6N sources. Continuous lines are fits of the
data to Eq. $4%. For clarity, the graph has been split in two part,
shifted vertically one with respect to the other. The quantitative
prediction of Eq. $3% for electron–electron interactions in sample
Ag$6N%c is shown as a dashed line.
B. Silver 5N and copper samples
In silver samples made from a 5N purity source, the phase
coherence time is systematically shorter than predicted by
Eq. $3% and displays an unexpectedly flat temperature dependence below 400 mK. The same is true for all the copper
samples we measured, independently of source purity.18
where ! F is the density of states per unit volume at the Fermi
These trends are illustrated for samples Ag$5N%b and
Figure 34: Phase coherence
length
wires as a fonction
of temperature.
energy,
and S is of
the narrow
cross sectionsilver
of the wire.
Cu$6N%b in Fig. 3.
In
order
to
test
the
theoretical
predictions,
the
measured
length ranging from 135 to 400 µm, thickness from 35 to 55 nm and with
from
65responsible
to 105 nm.
What
can be
for this anomalous behavior?
" # (T) curves were fit with the functional form,
There have been several theoretical suggestions regarding
From [92].
sources of extra dephasing. Some of these, such as the pres" #!1 "AT 2/3#BT 3 ,
$4%
ence of a parasitic high frequency electromagnetic
19
can
be ruled outofpurely
In this chapter Lϕ waswhere
introduced
as a cutoff put by hand. It willradiation,
be the
purpose
theon experimental grounds.
the second term describes electron–phonon scattering,
Indeed some samples do show a saturation of " # , while
2
shownmicroscopic
as continudominant
higher
temperatures.
chapter X to discuss the origin
of atthis
length
scale Fits
onare
more
grounds.
Inresistance
chapter
others
of similar
andXI
geometry, measured in the
lines in Fig. 4 $the fit parameters minimize the distance
same cryostat, do not. This indicates that, in our experiments
we will analyse more deeplyous
the
particular
role
of
electronic
interaction.
between the data points and the fit curve in a log–log plot%.
at least, the observed excess dephasing is not an artifact of
Equation $4% describes accurately the temperature depenthe measurement. The main suggestions to explain the
dence of " # (T) for samples Ag$6N%a, b, c and, with a
anomalous behavior of " # are dephasing by very dilute magslightly reduced fidelity, for samples Ag$6N%d and sample
netic impurities,11,20 dephasing by two-level systems associAu$6N%. In all these samples, fabricated using 6N source
ated with lattice defects,21,22 and dephasing by electron–
materials of silver and gold, " # (T) follows very closely, beelectron interactions through high energy electromagnetic
2/3
low about 1 K, the 1/T dependence expected when the
modes.23
electron–electron interaction is the dominant inelastic proThe correlation between source material purity and excess
dephasing amongst silver samples fabricated using the exact
same process but with either our 5N or 6N source material
suggests that impurities are responsible for the anomalous
temperature dependence of " # . The fact that, among all the
6N silver samples, " # (T) deviates the most from the predicSample
A thy
A
B
tion of electron–electron interactions in Ag$6N%d, fabricated
(ns!1 K!2/3)
(ns!1 K!2/3)
(ns!1 K!3 )
36
in MSU
$see
Fig.same
4% would
Each phase coherent physical property, like AB amplitude, persistent current, etc,
does
the
job. mean
It isthat the 6N silver source
Ag$6N%a
0.73
0.045
material used at MSU contains more ‘‘dangerous’’ impurities
not obvious that they all give the
same length 0.55
scale.
than the one at Saclay.
Ag$6N%b
0.51
0.59
0.05
The phase coherence time in the copper samples is always
Ag$6N%c
0.31
0.37
0.047
almost independent of temperature below about 200 mK
Ag$6N%d
0.47
0.044
81 0.56
down to our base temperature of 40 mK $see Refs. 11,24,25%.
Au$6N%
0.40
0.67
0.069
However, as opposed to silver samples, this unexpected beTABLE III. Theoretical predictions of Eq. $3% and fit parameters
for " # (T) in the purest silver and gold samples using the functional
form given by Eq. $4%.
Exercices
-
Exercice IV.9 : Magn´
etoconductance classique et magn´
etoconductance anormale (d’un fil).– On rappelle que la conductivit´e de Drude varie `a “bas” champ comme
e
∆σclass (B)/σ0 ' −(Bµ)2 o`
u µ = eτ
e. On consid`ere de l’argent avec `e = 30 nm
m est la mobilit´
−1
(on rappelle que kF = 0.83 ˚
A). V´erifier que ∆σclass (B)/σ0 ' −10−5 B 2 (B en Tesla).
On consid`ere un fil d’argent de section S = W × a = 60 nm×50 nm soumis `a un champ
magn´etique perpendiculaire. Calculer la longueur LB pour B = 1 T. V´erifier que la localisation
faible d´ecroˆıt comme ∆σ/σ0 ' −10−5 /B (B en Tesla).
B
-t0
I
0
!0
y
x
FIG.I.
I
L
Figure 35: Magnetoresistance of a hollow metallic cylinder in Oersted (1 Oersted= 1 Gauss=
Thus,accordingto (3), rslrro= 5; the secondterm in (l) can thereforebe neglected
and
10−4 Tesla). From [9].
the phaseof the oscillationscoincideswith the observedvalue.
In order for the spin-orbiteffectsto be insigrrificant,lithium, for which e
must be
Exercice IV.10 :
two ordersof magnitudesmallerthan for Mg, waschosenas the materialfor the speci1/ Ballistic ring – Aharonov-Bohm oscillations.–
We consider a metallic ring of micromens.
metric dimension such that the transport can be considered
fully
coherent
low temperature).
Theresultsof
theex
n t (at "A,lilhi-q1r.flm,
similarto that described
in Ref.4.
withR = 2 kO and D= I cm was obtainedby condensation
Discuss how the conductance depends on a magnetic field.'ls.
-
of lithium with an initial purity of 99.95%on a quartz filament. The helium temperature
in the experiment
was l.l K. Same
current,
to 40 ltA,heatedthe speci.
The measuring
2/ Diffusive ring – Altshuler-Aronov-Sharvin
oscillations.–
question
forequal
a ring
mens by an amount of the order of l0- l K.
made of weakly disordered metal, in the diffusive regime [base your analysis on Eq. (IV.102)].
The dashedline in the figure showsthe resultsof a calculationusingEq. (1) with
The first experiment was performed in a famous experiment
by Sharvin and Sharvin [101, 9].
?so= -, F = O,t= 0.72 pm, I, = 2.32prm,and a= 0.12pm. The quantity2r- a= 1.32
What is the diameter of the cylinder ?
trlmis closeto the value of the diameterof the quartz filament 1.3 pm, determined
Further reading : review by Aronov and
Webb [121].
with
the help of an electronmicroscope.Another checkof the theory is the agreementbetween the[16]
monotonic
decrease
andone
the damping
of the oscillations,
Sharvin
and also
the
by Washburn
and determinedby the
samequantity a. Thus, theory and experimentapparentlyarein good quantitativeagreement,
It should be noted that it waspossibleto observein theseexperimentsthe negative
longitudinal magnetoresistanceof thin films.
-
Exercice IV.11 Complex geometry – Altshuler-Aronov-Spivak oscillations : DeConfirming the validity of the basicideasof the theory of weaklocalizationin quasitailed study
twodimensional systems,experimentsof the type describedabovealsomake it possible
to study electron scatteringmechanismsin thin films.
• Weak localisation correction in a metallic ring.
• Hollow cylinder.
Theory by [8] ; experiment [101, 9]
We are grateful to A. I. Larkin and D. E. KhmelhitskiYfor discussing
the resultsto,
A. V. Danilov for help in analyzing the experimental results,and to P. L. Kapitsa for his
interestin this work and for making it possibleto carry out the experimentsat the Insti'
tute of Physical Problemsof the Academy of Sciencesof the USSR.
l)An "rro, in
(6) in Ref.I shouldbecorrected.
expression
Thetotalcoefficient
shouldbe 1/22',
ratherthan1ln2, and2nZ. shouldbereplaced
by I..
590
JETPLett,Vol.35, No. 11, 5 June1982
A t ' t s h u l e re t a / '
=-._lL
82
590
Master iCFP - Wave in random media
May 3, 2015
TD 4 : Classical and anomalous magneto-conductance
4.1 Anomalous (positive) magneto-conductance
1/ Classical magneto-conductivity.– We first analyse transport coefficients in the presence
of a magnetic field within the semi-classical Drude-Sommerfeld theory of electronic transport.
a) Show that the conductivity tensor in the presence of an external magnetic field is
1
1 + (ωc τ )2
ωc τ
= σ0
1 + (ωc τ )2
σxx = σ0
(IV.134)
σxy
(IV.135)
eB
is the cyclotron pulsation and σ0 =
where ωc = m
∗
resistivity tensor ρ = σ −1 .
ne2 τ
m∗
the Drude conductivity. Deduce the
b) Justify physically the decrease of σxx (B) as B increases.
c) At low temperature, the relaxation time saturates at the elastic mean free time τ → τe . What
is the typical scale of magnetic field needed to decrease significantly σxx (B) ? We give the inverse
A and the elastic mean free path `e = 4 µm in gold (bulk).
of the Fermi wavevector kF−1 = 0.85 ˚
d) In thin metallic films with thickness 50 nm, the elastic mean free path is reduced by two order
of magnitudes ! In thin silver wires, one measures `e ' 20 nm. How large must be the magnetic
field to bend significantly the electronic trajectories between collisions on impurities ?
2/ Coherent enhancement of back-scattering.– In a weakly disordered metal, interferences
of time reversed electronic trajectories enhance the back-scattering of electrons, and therefore
diminishes the conductivity. In the absence of a magnetic field, the phase of probability amplitude is an orbital phase proportional to the length of the diffusive trajectory AC = |AC |eikF `C :
∆σ(B = 0) ∼ −
X
C
AC A∗Ce = −
X 2
AC < 0
(IV.136)
C
where the sum runs over all closed diffusive trajectories.
’=
~
Figure 36: Interference of reversed electronic trajectories C and Ce increases back-scattering (weak
localisation).
a) If a weak magnetic field is applied, what is the magnetic field dependence of the probability
amplitudes AC ?
b) How the right hand side of Eq. (IV.135) is modified ?
c) We consider a thin metallic film, i.e. diffusive electronic motion is effectively two-dimensional.
83
Argue that the presence of the perpendicular magnetic flux introduces a cutoff in the summation
over electronic trajectories (IV.135).
d) Anomalous magneto-conductivity.– Deduce the qualitative behaviour of ∆σ(B) and
discuss the experimental result (Fig. 37).
________________________________________________________
Mg
L00
0~
0
0
~O.O5
0
O.5~
0
0
0
o
17
0.11
~
6.4K
9.4K
A
0.11
~0
0.21
19.9 K
0.5T
1.01
H
Fig. 2.11. The magneto-resistance of a thin Mg-film for different temperatures as a function of the applied field. The units of the field are given on
the right of the curves. The points represent the experimental results. The full curves are calculated with the theory. The small spin-orbit scattering
is taken into account.
2
~ (T/K) 2.5
3
-24 Magnesium film (24 Mg). From
Figure 37: Anomalous magneto-resistance of a thin
12
In(t1 is)
Ref. [25].
20
12s]
t1[10
10
-25
Mg
-26
5
R • 84.60
-27
2
1
5
10
T[K]
20
Fig. 2.12. The inelastic lifetime ~ of a Mg-film as a function of temperature. It obeys a r2-law.
84
4.2 Green’s function and self energy
1)
Propagator and Green’s functions
We introduce the propagator
ˆ
K(~r, t|~r 0 , 0) = −i θH (t) h~r |e−iHt |~r 0 i
(IV.137)
where H is the Hamiltonian operator.
a) Check that K(~r, t|~r 0 , 0) is the Green’s function of the time dependent Schr¨odinger equation.
R +∞
b) Compute the Fourier transform GR (~r, ~r 0 ; E) = −∞ dt eiEt K(~r, t|~r 0 , 0). Check that this is
the Green’s function of the stationary Schr¨odinger equation.
2)
Green’s functions in momentum space and average Green’s function
1/ Free Green’s function.– The free Green’s function in momentum space is
G0R (~k, ~k 0 ) = h ~k |
1
~
| ~k 0 i ≡ δ~k,~k 0 GR
0 (k)
EF − H0 + i0+
(IV.138)
1
where | k i is a plane wave, eigenvector of H0 = − 2m
∆ (the dependence in Fermi energy is
R
0
implicit). Compute explicitly G0 (~r, ~r ) in dimension d = 1 and d = 3.
2
k
Hint : in d = 1, compute G0 (x, x0 ) for a negative energy E = − 2m
and perform some analytic
continuation.
(3D)
(1D)
In d = 3, show that G0 (~r, ~r 0 ) can be related to a derivative of G0 (x, x0 ) (after integrations
over angles).
2/ Average Green’s function in the presence of a weak disorder.– Assuming that the
R
self energy is purely imaginary ΣR = −i/2τe , compute explicitly G (~r, ~r 0 ) for d = 1, 3.
p
Hint : express 2m(EF + i/2τe ) in terms of kF and `e .
3)
Self energy : stacking
1/ Recall the expression of the self energy at lowest order in the disorder, in terms of the free
Green’s function. Express its imaginary part.
2/ We now consider a particular class of diagrams :
ΣR
stack =
=
+
+
+ ···
(IV.139)
Deduce an equation for ΣR
stack and solve it. Analyse the weak disorder limit F 1/τe .
85
Master iCFP - Wave in random media
May 3, 2015
TD 5 : Magneto-conductance of thin metallic films and wires –
Spin-orbit scattering and weak anti-localisation
5.1 Magneto-conductivity in thin metallic films
The fit of anomalous magneto-conductivity curves in thin 2D films and wires is a powerful tool
which has been extensively used in order to extract the phase coherence length Lϕ of metallic
devices at low T (. few K). The fit of ∆σ(B, Lϕ ) is performed at several temperatures what
allows to extract the temperature dependence Lϕ (T ) and identify the microscopic mechanisms
responsible for dephasing and/or decoherence.
DIMENSIONAL¼
PHYSICAL REVIEW B 73, 115318 !2006"
FIG. 2. !Color
Weak as
localization
Figure 38: Magnetoresistance
curves online"
for a 2DEG
a functionmagnetoresistance
of the magnetic field in Gauss
measurements
at
different
temperatures.
The
temperature
range is
−4
(1 Gauss= 10 Tesla). From Ref. [52].
from 4.2 K !top" down to 130 mK !bottom". The black solid lines
are the
best fits to electron
Eq. !3". gas (2DEG) submitted to a perpendicular magnetic
We consider a two
dimensional
field B. In this case it will be convenient to write the Cooperon as an integral of the propagator
4.2 K and $130 mK.Z The solid lines are best fits using Eq.
in time
∞
2s e2 Dthere
!3". In our samples
are no magnetic impurities and the
∆σ = −
dt Pt (~r|~r) e−t/τϕ − e−t/˜τe
(IV.140)
π~ Bso0, Bs $ B!, making #! the only fitting
well is symmetric
parameter. The
values
of the extracted
dephasing
where the second exponential
cut off
the contribution
of small
times, time
that from
are not described by
2
2
Eq. !3" are: plotted
Fig.
3 as
of temperature,
andspin degeneracy.
the diffusion approximation
τϕ = Lϕin/D
and
τ˜e a=function
`e /D. The
factor 2s is the
compared
with
the
theoretical
predictions
given
by
Eqs.
!1"
The time propagator of the diffusion
and
we
and !2". F0% is used as a restricted fittingparameter,
2
%
~ 2ie A
~
Dt
∇−
0
0
the
calculations.
This
use the value
~
Pt (~r|~rF0)==−0.4
θH (t)throughout
h~r |e
|~r i
(IV.141)
value is comparable with the estimate given in Ref. 20, of
solves the diffusion-like
equation
for GaAs samples at our electron concentration
F0% = −0.28
"
# is the theoretical value from
rs % 1.05. Thegreen solid2line
2e
~ − i where
~
Eq. !1",
the
∂t −applicable
D ∇
A
Pt (~rsmall-energy
|~r 0 ) = δ(t)δ(~rtransfer
− ~r 0 ) term
~
dominates, and the blue dashed line is the theoretical value
(IV.142)
from Eq. !2", applicable where the large-energy transfer term
1/ Using the mapping
onto theThe
Landau
problem,
Pt (~r|~r) in theof
plane.
dominates.
red dotted
linecompute
is the combination
the
~2 ~
theoretical
value of
from
Eq. !2" and
the2D
linear
term fromHEq.
Hint : We recall that
the spectrum
eigenvalues
of the
Hamiltonian
Landau = − 2m (∇ −
!1", which represents the ballistic limit with some contribui ~process:
2
mperature calibration
!a"
~ eA) for a homogeneous magnetic field is the Landau spectrum εn = ~ωc (n + 1/2) for n ∈ N,
tion from the small-energy transfer linear term. The cyan
he longitudinal resistance is predashed-dotted line presents the prediction for a28 2DEG igic field for several different tem86
he range 3.9– 5 T !1 ↓ −1 ↑ ". The
!top" to around 75 mK !bottom",
where ωc = eB/m and where each Landau level has a degeneracy proportional to the surface of the plane dLL = eBSurf
h . The partition function of the Landau problem ZLandau =
R
− ~t HLandau
|~r i can be easily calculated.
d~r h~r |e
2/ a) Using the integral given in the appendix, deduce that
∆σ(B) =
1 L2B
1 L2B
2s e2 1
ψ
+ 2 −ψ
+ 2
h 2π
2 Lϕ
2
`e
(IV.143)
where LB will be related to the magnetic field.
b) What is the magnetic field corresponding to LB = 1 µm ? And LB = 20 nm ? Looking at
the range of magnetic field on the experimental curve, argue that it is justified to simplify the
result as
2 LB
1 L2B
2s e2 1
(IV.144)
ψ
+ 2 − ln
∆σ(B) =
h 2π
2 Lϕ
`2e
c) Analyse the zero field value ∆σ(0). Discuss the limiting behaviours of ∆σ(B) − ∆σ(0).
3/ Discuss the experimental data of Fig. 38.
Appendix :
We give the integral (formula 3.541 of Gradshteyn & Ryzhik, Ref. [62])
Z ∞
1
1
b
1
a
e−ax − e−bx
=
ψ
+
−ψ
+
,
(IV.145)
dx
sinh λx
λ
2 2λ
2 2λ
0
d
where ψ(z) = dz
ln Γ(z) is the digamma function. We deduce the functional relation ψ(z + 1) =
1
ψ(z) + z . We give two values ψ(1) = −C ' −0.577215 (Euler-Mascheroni constant) and
ψ(1/2) = −C − 2 ln 2, and the limiting behaviour
ψ(x + 1/2) = ln x +
x→∞
87
1
+ O(x−3 )
24x2
(IV.146)
5.2 Magneto-conductance in narrow wires
The aim of the exercice is to analyse the magneto-conductance of a long wire of section W
submitted to a perpendicular homogeneous magnetic field. For simplicity we consider the twodimensional situation of a wire etched in a two-dimensional electron gas (2DEG). We recall that
the weak localisation correction to the conductivity is given by
"
2 #
2s e2
2e
~
~ −i A
∆σ = −
Pc (~r, ~r) with
γ− ∇
Pc (~r, ~r 0 ) = δ(~r − ~r 0 ) ,
(IV.147)
π~
~
where γ = 1/L2ϕ .
We consider the geometry of a infinitly long quasi-1D wire, i.e. x ∈ R and y ∈ [0, W ].
1/ Relate the conductivity σ of the wire to the conductance G = I/V .
We choose the Landau gauge such that Ax is an antisymmetric function of the transverse
coordinate. If y ∈ [0, W ] we choose Ax (W − y) = −Ax (y), i.e.
Ax (y) = (W/2 − y) B
and
Ay = 0 .
We assume that the confinment imposes Neumann boundary conditions
∂y Pc (~r, ~r 0 )y=0 & W = 0 .
(IV.148)
(IV.149)
2/ Zero field.– The aim is to construct the spectrum of the Laplace operator ∆ = ∂x2 + ∂y2 in
the wire.
a) Use the separability of the problem to find the spectrum of eigenvectors and eigenvalues of
the Laplace operator in the infinitly long wire of width W .
b) Green’s function.– Justify the following representation
Pc (~r, ~r 0 ) =
∞
X
1
| x0 i χn (y 0 )
χn (y) h x |
γ + εn − ∂x2
n=0
{z
}
|
(IV.150)
Pc (x,x0 ) for γ→γ+εn
The functions χn (y) satisfy the differential equation −∂y2 χn (y) = εn χn (y) on [0, W ] with appropriate boundary conditions.
Under what condition on W and Lϕ can the Cooperon be approximated by the 1D Cooperon
Pc (x, x0 ) = h x | γ − ∂x2 )−1 | x0 i ?
3/ Weak magnetic field.– In the diffusion approximation, the Cooperon can be interpreted as
the Green’s function of the operator −(∇ − ~i 2eA)2 , Eq. (IV.146). We recall that this treatment
of the magnetic field in the diffusion approximation supposes that `e Rc , where Rc = vF /ωc
is the cyclotron radius of electrons with energy εF (ωc = eB/m∗ is the cyclotron pulsation). Our
aim is to compute the Cooperon in the weak magnetic field limit.
RW
a) Projecting the differential equation (IV.146) (i.e. 0 dy
W × · · · ), show that the effect of
the magnetic field can be absorbed by a transformation of the phase coherence length in the
one-dimensional cooperon
Z
1
1
1
4e2 W dy
1
def 1
−→ eff
= 2 + 2
where 2 = 2
Ax (y)2 .
(IV.151)
L2ϕ
Lϕ LB
~
W
Lϕ (B)2
LB
0
b) Deduce explicitly LB and discuss the range of validity of this approximation, i.e. what is the
88
condition on B, W and Lϕ ?
c) We recall the expression of the 1D Cooperon Pc (x, x) = h x | 1/L21−∂ 2 | x i = Lϕ /2. Deduce the
ϕ
x
expression of the magneto-conductivity ∆σ(B) of the infinitly long wire and show that the WL
correction to the dimensionless conductance can be written as
∆g(B) = p
∆g(0)
1 + (B/Bϕ )2
(IV.152)
Give the expression of the scale Bϕ and interpret physically this expression.
d ) Discuss the experimental data of Fig. 39 at the light of this calculation. In particular, how
ENCE AT LOW
TEMPERATURES IN…
PHYSICAL REVIEW B 81, 245306 "2010#
can one interpret the evolution of the curve when the sample is cooled down ?
600 mK
480 mK
375 mK
270 mK
200 mK
140 mK
100 mK
76 mK
60 mK
42 mK
36 mK
4
3.5
T = 140 mK
4
3
D = 290 cm2/s
w = 1000 nm
2.5
-60
-30
G (e2/h)
D = 290 cm2/s
T = 36 mK
G (e2/h)
(a)
w = 1000 nm
w = 1500 nm
-300
0
B (G)
0
300
B (G)
30
60
4.5
mK
0
1
2
(b) wire etched in a 2DEG as a function 600
480
Figure
39:
Magnetoconductance
curves
for
a
long
of mK
the
B (T)
375
mK
magnetic field in Gauss (1 Gauss= 10−4 Tesla). Length of the wire is L = 150 µm, lithographic
270 mK
200 mK
4
−2 .
width Wlitho = 1 µm and effective width W = 630 nm. Electronic density is ne = 1.5 × 1015140mmK
nline# Magnetoresistance
curves of 1000 and
100 mK
Left : Resistance over a large window in B field, [−2 T, +2 T]. Right : Conductance over76small
mK
at T = 36 mK and D = 290 cm2 / s.
60 mK
window around zero field, [−6 mT, +6 mT]. From3.5
Niimi et al. Phys. Rev. B 81, 245306
39 mK
T = 140 mK
29 mK
(2010) [91].
G (e2/h)
2
G (e /h)
-1
1 "m. The suppression of the WL effect
3
4
2
D = 170 cm /s
2
fields
are
always
much
B # $ / 2ele . These
= 1500
e) In the “high field” regime, LB < W , what expressionwdo
you nm
expect for-200
the MC
?
0
200
B (G)
ally strong fields B! ! m! / "e%e#.
2.5
-30
-20
-10
0
10
B (G)
20
30
B. Experimental results
FIG. 6. "Color online# WL curves of "a# 1000 and "b# 1500 nm
wide wires at D = 290 cm2 / s and 170 cm2 / s, respectively. The
conductance here is divided by e2 / h. The broken lines are the best
mine the phase
coherence
Remarks
: length L!, we
fits of Eq.
"6#. : The
insets in and
"a# Aronov,
and "b# Sov.
showPhys.
the magnetoconduc• This analysis was
performed in a well-known
paper
by Altshuler
JETP
andard magnetoresistance
measurements
tance
at
T
=
140
mK
in
larger
field
ranges.
(1981)
(Ref.
[5]).
mperature. A typical example for such a
1. Quasi-1D wires
• Semi-ballistic
curve is displayed
in Fig. 5.regime.–
Let us Many
first experiments are performed on long wires etched in a two-dimensional
electron
gas
(2DEG)
at
the
interface
semiconductors (GaAs/GaAl
Asx ). In this case the elastic
field range up to a magnetic field
of 2 T. A of two
a T−1/3 law down to the 1−x
lowest
temperatures for both the
(2D)
mean free path `e
of the original 2DEG is usually larger than the section of the wire. The effective
s due to WL is clearly seen at zero field.
(1D)
wires.
Note
the temperature
40 mKby
hasthebeen corelastic mean free path in the wire is also
larger
thanthat
the section
`e
> W . below
The dephasing
magnetic field
the
WL
peak
disappears
bythemeasuring
electronThis
temperature
magnetic field involves different length rected
scale due
phenomenoninofsitu
flux the
cancellation.
has been of the
f negative magnetoresistance
is
observed
quasi-1D
wire
based
on
e-e
interaction
corrections
described by semiclassical methods by Dugaev and Khmelnitskii [46] and Beenakker and van Houten, as demagnetic focusing.
When
going(Ref.
to [21]).
even
Phys. Rev.
B (1988)
tailed in Sec. VI. The absolute values of L! at low temperaT# the well-known Shubnikov de Haas
tures are different between the two wires, which is expected
appear.
in the AAK
89 theory in Eq. "3#. Similar temperature depenWL peak allows to obtain the phase coherdence of L! has also been observed in GaAs/GaAlAs
44
Fig. 6, we show magnetoconductance
5.3 Spin effects and weak antilocalisation in thin metallic films
The magneto-resistance of films made of usual metals is not well described by the simple theory
presented in the previous problem, i.e. by Eq. (IV.142). The reason for this is that electronic
spin degree of freedom cannot be ignored.
First, the presence of spin-orbit coupling
HSO ∼ −
1
~ )·S
~
(~
p × ∇V
m2 c2
(IV.153)
is responsible for an additional phase originating from the rotation of the electronic spin, while
the electron is scattered on the disordered potential V . Spin-orbit coupling is weak in light
metals, like Lithium of Magnesium, but strong in heavy metals like Silver or Gold. Second, the
presence of residual magnetic impurities is another source of electronic spin rotation :
X
~·
Hmag = −S
Ji ~si δ(~r − ~ri )
(IV.154)
i
where ~ri are the positions of the magnetic impurities and ~si their spins (considered frozen).
The starting point of the study of quantum transport is the Kubo-Greenwood formula
σ
e(ω) =
e2 1
2πm2 Vol
X
A
0
0
kx kx0 GR
σ,σ 0 (k, k ; εF + ω) Gσ 0 ,σ (k , k; εF ) ,
(IV.155)
k, k0 , σ, σ 0
where the Green’s function carry spin indices. Using that the average Green’s function is diagonal
in spin indices, we finally obtain the expression of the weak localisation correction
σ’
∆σ =
X r ,i
σ
σ
σ, σ 0
r’, j
=−
σ’
Γc
e2 X (c)
Pσσ0 ,σ0 σ (~r, ~r)
π
0
(IV.156)
σ, σ
where the Cooperon now propagates a pair of spins (Fig. 40).
r ’, γ
α,r
(c)
Γαβ,γδ (~r, ~r 0 )
=
Γc
β,r
r ’, δ
Figure 40: In the presence of spin flip and spin-orbit scattering, the Cooperon depends on four
spin indices.
(c)
(c)
w
1/ Argue that the Cooperon Γαβ,γδ (~r, ~r 0 ) = Dτ
P
(~r, ~r 0 ) can be written as the sum of two
e αβ,γδ
separate contributions associated with singlet and triplet channels.
2/ The efficiency of spin-orbit coupling and scattering by magnetic impurities are controlled by
two lengths Lso and Lm . Introducing the projector in the singlet space, (Π0 )αβ,γδ = 12 (δαγ δβδ −
δαδ δβγ ), we can write the Cooperon (in the space of the two spins) under the form
P (c) (~r, ~r 0 ) = Pc (~r, ~r 0 ; 1/L2S ) Π0 + Pc (~r, ~r 0 ; 1/L2T ) (1 − Π0 ) ,
90
(IV.157)
16
G. Bergmann, Weak localization in thin films
54.0
0.1
Au
~
16%Au
8%Au
Mg
14.0
°°
-1.0
4%Au
7.1
2%Au
~
0
0
3.8
1%Au
R~91Q
T’4.6K
0.5
-0.1
1.0
0.5
-0.8
-0.6
-0.4
0%Au
-0.2
0
H(T)
0.2
0.4
06
0.8
Fig. 2.10. The magneto-resistance of a thin Mg-film at 4.5 K for different coverages with Au. The Au thickness is given in % of an atomic layer on
the right side of the curves. The superposition with Au increases the spin-orbit scattering. The points are measured. The full curves are obtained
with the theory by Hikami et al. Thecurves
ratio
Figure 41: Magneto-resistance
madescattering.
of Itan
alloy
of Magnesium and
on thefor
left sidemetallic
gives the strength of films
the adjusted spin-orbit
is essentially
proportional
to the Au-thickness.
Gold for difference concentration of gold. From Ref. [25].
T
1/T5,
because the quenched condensation yields homogeneous films with high resistances. The points are
1spin-orbit
the pure Mg is determined as discussed above. The different
where Pc (~r, ~r 0 ; γ)measured.
= h~r The
| curves
|~
r 0 i scattering
is theof Cooperon
in the absence of spin scattering. The two
experimental γ−∆
for different temperatures are theoretically distinguished by their different H, (i.e.
the
inelastic
lifetime).
This
is
the
only
adjustable
parameter
for amagnetic
comparison withimpurities
theory (after H,,~,
lengths LS and LT combine the effect of spin-orbit and
: is
determined). The ordinate is completely fixed by the theory in the universal units of L00 (right scale).
The full curves give the theoretical results with the best fit of H, which is essentially a measurement of
1 between
2 the experimental points and1the theory 2is very good.4The experimental
H1. The agreement
and
+ the2area. in which the
(IV.158)
= 2 influence
= It 2measures
result proves L
the2destructive
of a magnetic field L
on2QUIAD.
3Lm
3Lso determination
m as a function of temperature
S stateLexists
T and allows
coherent electronic
the quantitative
2 law for Mg
as is shown
of fig.
the 2.12.
coherent scattering time T1. The temperature dependence is given by a T
in
Explain why LS >For
LTother
. metal films where the nuclear charge is higher than in Mg one finds even in the pure case
the substructure caused by spin-orbit scattering. In fig. 2.13 the magneto-resistance curves for a thin
quench condensed Cu-film are shown. Again the points represent the experimental results whereas the
3/ Discuss the effect
of time reversal in the singlet and triplet channel. Write ∆σ in terms of
full curves show the theory. At low temperatures the inelastic lifetime is long and therefore the effect of
0
2
Pc (~r, ~r ; 1/LS, T ).the spin-orbit scattering dominates in small fields. At high temperatures the inelastic lifetime becomes
smaller than the spin-orbit scattering time and the magneto-resistance becomes negative because of the
minorspin
role ofstructure
the spin-orbitin
scattering.
For Au-filmsin
the Eq.
spin-orbit
scattering is so strong that it
Hint : examine the
the diagram
(IV.155).
completely dominates the magneto-resistance.
The natural question is, why does weak localization change to weak anti-localization in the presence
4/ Using the formula
(IV.142)
for the magneto-conductance of a thin film, deduce the expression
of spin-orbit
scattering?
of the weak localisation in the presence of spin-orbit and spin flip scattering. Explains (at least
qualitatively) the experimental data of Fig. 41.
Further reading : A review article on weak localisation in thin metallic films is the famous
article by Bergmann, Phys. Rep. 107 (1984) (Ref. [25]). A more intuitive description can be
found in the review article of Chakravarty and Schmid, Phys. Rep. 140 (1986) (Ref. [33]).
91
Master iCFP - Wave in random media
May 3, 2015
TD 6 : Conductance fluctuations and correlations in narrow wires
The purpose of the exercice is to analyse precisely the correlation function δg(B)δg(B 0 ) for the
dimensionless conductance of a narrow wire of length L.
L
I
I
W
Figure 42: Transport through a metallic wire of length L.
VI.K
Preliminary : weak localisation correction and role of boundaries
We recall that the weak localisation correction to the dimensionless conductance of a narrow
wire is given by
Z L
2
∆g = − 2
dx Pc (x, x)
(VI.159)
L 0
where the Cooperon solves γ − ∂x2 Pc (x, x0 ) = δ(x − x0 ) where γ = 1/L2ϕ encodes dephasing.
We account for the connections at the two boundaries by imposing some Dirichlet boundary
conditions Pc (x, x0 )x=0, L = 0.
1/ Show that the weak localisation correction can be expressed in terms of the spectrum of
eigenvalues {λn } of the Laplace operator in the wire (i.e. −∂x2 φn (x) = λn φn (x) for φn (0) =
φn (L) = 0).
2/ Deduce that the weak localisation correction can be written as ∆g = −(2/L) G(γ). Analyse
the limiting behaviours L Lϕ and L Lϕ .
VI.L
Fluctuations and correlations
The correlation function for the narrow wire can be expressed as
Z
Z L
0 2 1
dxdx
4
0
0
2
Pd (x, x ; ω) + Re Pd (x, x ; ω) + ( Pd −→ Pc )
δg(B)δg(B 0 ) = 2 dω δT (ω)
L
L2
2
0
(VI.160)
where δT (ω) is a normalised function of width T such that δT (0) = 1/(6T ). The diffuson and
cooperon solve
−iω/D + γd,c − ∂x2 Pd,c (x, x0 ; ω) = δ(x − x0 ) ,
(VI.161)
where the expressions of the dephasing rates differ for diffuson and cooperon in the presence of
a magnetic field :
1
1
1
1
γd = 2 + 2
and γc = 2 + 2
.
(VI.162)
Lϕ L B−B0
Lϕ L B+B0
√
2
2
where LB = ~/( 3|B|W ).
92
1/ Compare the fluctuations at zero field B = B 0 = 0 to the one at large field (no calculation).
2/ Show that the two diffuson contributions can be written in terms of the spectrum {λn } under
the form
Z
4
(diffuson)
0
δg(B)δg(B )
= 4 dω δT (ω) F(γω )
(VI.163)
L
where γω = γd − iω/D. Express the function F(γω ) as a series.
3/ With the help of the representation obtained in the previous question, show that the expression
p of the correlator involves a new cutoff at the length given by the thermal length
def
LT = D/T . Identify a “low temperature” regime and a “high temperature” regime by comparing the three lengths LT , Lϕ and L.
a) “Low” temperature regime (LT = ∞)
In this case we simplify the calculation by performing the substitution δT (ω) → δ(ω) which
allows for a straightforward integration over frequency.
(diffuson)
1/ Show that the correlator δg(B)δg(B 0 )
can be related to the weak localisation (i.e. that
the two functions F(γ) and G(γ) are related).
2/ Analyse the limiting behaviours for L Lϕ and L Lϕ . Recall the physical origin of the
decay of δg 2 with L/Lϕ when L Lϕ .
3/ Adding the cooperon contribution, analyse the structure of the correlator as a function of
the magnetic fields in the limit Lϕ L.
b) “High” temperature regime (small LT )
1/ In the “high temperature” regime, justify the substitution δT (ω) → δT (0) (the value δT (0)
(diffuson)
was given above). Deduce that the correlator δg(B)δg(B 0 )
i.e. expressed in terms of G(γ).
is simply related to the WL,
2/ Analyse the limiting behaviours for L Lϕ and L Lϕ .
3/ Analyse the magnetic field dependence of the full correlator (diffuson and cooperon) when
Lϕ L.
c) Experiment
The weak localisation correction ∆g(B) and the conductance fluctuations δg(B)2 have been
measured in a short wire etched in a 2DEG with a direct averaging procedure (Fig. 43). Discuss
the experimental data at the light of your calculations.
Appendix
∞
X
n=1
1
(nπ)2 + y 2
coth x =
=
1
2y
1
coth y −
.
y
1 1
1
2 5
+ x − x3 +
x + O(x7 ) .
x 3
45
945
93
16.1
16.0
15.9
200
600
1000
1400
1800
B ( 10-4 Teslas )
16.5
G ( e2/h )
16.4
(b)
16.3
16.2
16.1
16.0
15.9
200
600
1000
1400
1800
B ( 10-4 Teslas )
var (G) (10-3( e2/h )2 )
4.00
(c)
3.00
2.00
1.00
0.00
200
600
1000
1400
1800
B ( 10 Teslas )
-4
Figure 43: WL and CF of a short wire (L ' 10 µm) etched in a 2DEG at T = 45 mK. Curves
from Ref. [83].
Further reading :
• The effect of boundary conditions in wires has been studied in : B. L. Al’tshuler, A. G. Aronov
and A. Yu. Zyuzin, Size effects in disordered conductors, Sov. Phys. JETP 59, 415 (1984) ;
[10].
• A measurement of conductance fluctuations with direct averaging over disorder configurations
has been performed in : Dominique Mailly and Marc Sanquer, Sensitivity of quantum conductance fluctuations and 1/f noise to time reversal symmetry, J. Phys. I France 2, 357 (1992) ;
[83]
94
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Experiments
a list of experimental results or fields which should be discussed during the lectures. This list is
necessarily incomplete, and would certainly requite additional thinking and discussions.
One problem is to avoid repetitions with other courses (mainly H. Bouchiat/T. Kontos and
C. Glatli)
Important experimental results or fields
The order in this list is not completely arbitrary. Still, it is at least weakly disordered.
• Quantization of conductance in nano-channels.
• Mesoscopic rings: Aharonov-Bohm (Washburn, Webb, etc), Sharvin-Sharvin, persistent
currents (L´evy, Bouchiat ; Bleszynski, Harris).
• Weak localization, positive and negative magneto-resistance, including spin-orbit effects
(Bergman)
• Coherent back-scattering, both for light (Maynard,...) and for atomic matter waves.
• Time reversal for acoustic waves (Fink).
• 1D Anderson localization with cold atoms (Billy, Aspect).
• Transport of microwaves in quasi-1D systems (Genack, Chabanov).
• Conductance of 1D and quasi-1D wires (Khavin & Gershenson).
• Universal conductance fluctuations (Mailly & Sanquer).
• Fluctuations in 1D and quasi-1D systems (microwave).
• 2D Anderson localization with light in waveguides, electrons (Kravchenko → controversial)
and cold atoms.
• 3D Anderson localization of light (Maret, Lagendijk).
• 3D Anderson localization of electrons (Katsumoto).
• 3D Anderson localization of cold atoms (de Marco, Aspect).
• 3D Anderson localization of elastic waves (Page).
• 3D Anderson localization of acoustic waves (Hu, Strybulevych, Page, Skipetrov, van Tiggelen, PRL, 2008).
• Large fluctuations and multifractality in the vicinity of a metal-insulator transition.
• Scaling properties of the Integer Quantum Hall effect.
• Bloch oscillations for cold atoms and electrons in superlattices.
• Dephasing and decoherence effects on electrons (weak localization, Aharonov-Bohm...).
• Hopping transport a la Mott.
• Decoherence effects for light transport in cold atomic gases (Labeyrie-Miniatura-Kaiser).
• Dephasing effect for light transport (Diffusive Wave Spectroscopy).
• Mott Insulator/Superfluid transition in cold atomic gases (Greiner, Bloch).
• Bose glass in cold atomic gases (Inguscio).
• Percolation vs. quantum effects for transport of cold atoms in nano-channels (Esslinger).
116