Al'tshuler-Aronov correction to the conductivity of a large metallic
  square network
Al’tshuler-Aronov correction to the conductivity of
a large metallic square network
Christophe Texier1, 2 and Gilles Montambaux2
1Laboratoire de Physique Théorique et Modèles Statistiques,
UMR 8626 du CNRS, Université Paris-Sud, F-91405 Orsay Cedex, France.
2Laboratoire de Physique des Solides, UMR 8502 du CNRS,
Université Paris-Sud, F-91405 Orsay Cedex, France.
(Dated: April 5, 2007)
We consider the correction ∆σee due to electron-electron interaction to the conductivity of a
weakly disordered metal (Al’tshuler-Aronov correction). The correction is related to the spectral
determinant of the Laplace operator. The case of a large square metallic network is considered. The
variation of ∆σee(LT ) as a function of the thermal length LT is found very similar to the variation
of the weak localization ∆σWL(Lϕ) as a function of the phase coherence length. Our result for
∆σee interpolates between the known 1d and 2d results, but the interaction parameter entering
the expression of ∆σee keeps a 1d behaviour. Quite surprisingly, the result is very close to the 2d
logarithmic behaviour already for LT ∼ a/2, where a is the lattice parameter.
PACS numbers: 73.23.-b ; 73.20.Fz ; 72.15.Rn
I. INTRODUCTION
At low temperature, the classical (Drude) con-
ductivity of a weakly disordered metal is affected
by two kinds of quantum corrections : the first one
is the weak localization (WL) correction, a phase
coherent contribution that originates from quan-
tum interferences between reversed electronic tra-
jectories. This contribution to the averaged con-
ductivity depends on the phase coherence length
Lϕ and the magnetic field : ∆σWL(B, Lϕ). The
temperature manifests itself through Lϕ, since
phase breaking may depend on temperature, e.g.
if it originates from electron-electron1 or electron-
phonon2 interaction.
In a metal, an electron is not only elastically
scattered on the disordered potential, but, due to
the electron-electron interaction, is also scattered
by the electrostatic potential created by the other
electrons. At low temperatures, when the elas-
tic scattering rate (1/τe) dominates the electron-
electron scattering rate (1/τee(T )), the motion of
the electron is diffusive between scattering events
with other electrons. In this regime, electron-
electron interaction is responsible for a small deple-
tion of the density of states at Fermi energy (called
the DoS anomaly or the Coulomb dip) and a cor-
rection to the averaged conductivity as well, the so-
called Al’tshuler-Aronov (AA) correction3,4,5,6,7,8,9
(see Refs.10,11,12 for a recent discussion). AA and
WL corrections are of the same order (but this
latter vanishes in a magnetic field). However, con-
trary to the WL, the AA correction is not sensitive
to phase coherence and involves another important
length scale : the thermal length LT =
(~ = kB = 1). The AA correction, denoted be-
low ∆σee(LT ), has been measured in metallic wires
in several experiments14,15,16,17. From the exper-
imental point of view, AA correction allows to
study interaction effects in weakly disordered met-
als, but also furnishes a local probe of temperature
in order to control Joule heating effects15,17, which
is crucial in a phase coherent experiment.
All the works aforementioned refer to the quasi-
one-dimensional (wire) or two-dimensional (plane)
situations. Quantum transport has also been stud-
ied in more complex geometries like networks of
quasi-1d wires. For example several studies of WL
have been provided on large regular networks in
honeycomb and square metallic networks18,19, in
square networks etched in a 2DEG20, and in square
and dice silver networks21. Theoretical studies of
WL on networks have been initiated by the works
of Douçot & Rammal22,23 and improved by Pas-
caud & Montambaux24 who introduced a powerful
tool25 : the spectral determinant of the Laplace
operator, that will be used in the following (see
also Ref.26).
The aim of this paper is to study how the AA
correction can be computed in networks. In a first
part we briefly recall how the spectral determinant
can be used to compute the WL. Then in a second
part we will consider the AA correction.
II. SPECTRAL DETERMINANT AND
WEAK LOCALIZATION
Interferences of reversed electronic trajec-
tories are encoded in the Cooperon, solu-
tion of a diffusion-like equation (∂t − D[∇ −
2ieA(x)]2)Pc(x, x′; t) = δ(x − x′)δ(t), where A(x)
is the vector potential. On large regular networks,
when it is justified to integrate uniformly the
Cooperon over the network (see Ref.27 for a discus-
sion of this point) it is meaningful to introduce the
space-averaged Cooperon Pc(t) =
Pc(x, x; t)
http://arxiv.org/abs/0704.0741v1
∆σWL = −
dt e−t/τϕ Pc(t) (1)
= −2e
lnS(γ) (2)
where τϕ = L
ϕ/D is the phase coherence time.
The factor 2 stands for spin degeneracy. We
have omitted in (1,2) a factor 1/s where s is the
cross-section of the wires. The parameter γ is re-
lated to the phase coherence length γ = 1/L2ϕ
(note that description of the decoherence due to
electron-electron interaction in networks requires
a more refined discussion28,29). The spectral de-
terminant of the Laplace operator is formally de-
fined as S(γ) = det(γ −∆) =
n(γ + En) where
{En} is the spectrum of −∆ [in the presence of
a magnetic field, ∆ → (∇ − 2ieA)2]. The inter-
est in introducing S(γ) is that it can be related
to the determinant of a V × V -matrix, where V is
the number of vertices, that encodes all informa-
tion on the network (topology, length of the wires,
magnetic field, connection to reservoirs). We la-
bel vertices by greek letters. lαβ designates the
length of the wire (αβ) and θαβ the circulation of
the vector potential along the wire. The topology
is encoded in the adjacency matrix : aαβ = 1 if
α and β are linked by a wire, aαβ = 0 otherwise.
λα = ∞ if α is connected to a reservoir and λα = 0
if not. We introduce the matrix
Mαβ = δαβ
aαµ coth
−iθαβ
where the aαµ constraints the sum to run over
neighbouring vertices. Then24
S(γ) =
γlαβ√
detM (4)
where the product runs over all wires. We now con-
sider a large square network of size Nx ×Ny made
of wires of length lαβ = a ∀(αβ). For simplicity we
impose periodic boundary conditions (topology of
a torus), which is inessential as soon as the to-
tal size of the network remains small compared
to Lϕ. At zero magnetic field the spectrum of
the adjacency matrix is ǫn,m = 2 cos(2nπ/Nx) +
2 cos(2mπ/Ny), with n = 1, · · · , Nx and m =
1, · · · , Ny. Therefore
S(γ) =
)NxNy
2 cosh
γa− cos 2πn
− cos 2πm
The calculation of lnS(γ) involves a sum that can
be replaced by an integral when Nx, Ny ≫ Lϕ/a.
Using
(2π)2
2A+ cosx+ cos y
K(1/A), (6)
where K(x) is the complete elliptic integral of first
kind30, yields20
lnS(γ) =
γa− 1√
where the volume of the network is Vol = 2NxNya.
We recover the expression of the WL first derived
by Douçot & Rammal23. Figure 1 displays the
dependence of the WL correction as a function of
the phase coherence length Lϕ. We now discuss
two limiting cases.
0 1 2 3 4 5Lϕ/a
0.1 1.0 10.0
FIG. 1: ∆σWL in unit of 2e
2/h as a function of Lϕ/a
(at zero magnetic field). The dashed line is the 1d re-
sult. The dotted line is the 2d limit eq. (10).
1d limit.– In the limit Lϕ ≪ a (i.e.
γa ≫ 1) :
∆σWL = −
−2a/Lϕ
We compare with the result for a wire of length a
connected at its extremities : ∆σwireWL ≃ − 2e
). As we can see the dominant terms coincide.
Deviations appear when Lϕ/a increases since tra-
jectories begin to feel the topology of the network.
This is already visible by comparing the second
terms of the expansions.
2d limit.– In the limit Lϕ ≫ a (i.e.
γa ≪ 1), we
obtain
lnS(γ) =
ln(4Lϕ/a) +
The conductivity reads
∆σWL ≃ −
ln(Lϕ/a) + CWL
with CWL =
2 ln 2
≃ 0.608. As noticed in the
beginning of the section, eqs. (8,10) should be di-
vided by the cross-section s of the wires. In the 2d
limit, diffusive trajectories expand over distances
larger than a and feel the two dimensional nature
of the system, being the reason why (10) is reminis-
cent of the 2d result. It is interesting to point that
the network provides a natural cutoff (the length
of the wires, a) while the computation of the WL
for a plane in the diffusion approximation requires
to introduce a cutoff by hand for lower times in
eq. (1), which is the elastic scattering time τe. In
this latter case the constant added to the logarith-
mic behaviour is not well controlled since it de-
pends on the cutoff procedure (the computation of
the constant for a plane requires to go beyond the
diffusion approximation and leads to31 ∆σ
plane
ln(2L2ϕ/ℓ
e + 1) ≃ − 2e
ln(Lϕ/ℓe) +
ln 2]
since ℓe ≪ Lϕ).
III. AL’TSHULER-ARONOV
CORRECTION
At first order in the electron-electron interac-
tion, the exchange term is the dominant contribu-
tion to the correction to the conductivity8,10,11,32
∆σee = −
dπVol
ω coth
D~q 2
U(~q, ω)
(−iω +D~q 2)3 (11)
where U(~q, ω) is the dynamically screened inter-
action. Within the RPA approximation and in
the small ~q and ω limit, the interaction takes the
form33 U(~q, ω) ≃ 1
−iω+D~q 2
D~q 2
where ρ0 is the den-
sity of states per spin channel. Replacing the
Drude conductivity by its expression σ0 = 2e
and performing an integration by parts, we get
∆σee = −
πdVol
ω coth
−iω +D~q 2 (12)
After Fourier transform, the result can be cast in
the form11 :
∆σee = −λσ
sinhπT t
Pd(t) ,
For the exchange term considered here, one finds
λσ = 4/d. Further calculation yields
8 λσ ≃
F , where F is the average of the interaction
on the Fermi surface (see definition in Refs.8,9).
This expression of λσ is valid in the perturba-
tive regime, F ≪ 1 ; nonperturbative expression
is given in Refs.6,7,8,9. Pd(t) is the space inte-
grated return probability Pd(t) =
Pd(x, x; t),
where Pd(x, x′; t) is solution of a classical diffusion
equation similar to the equation for Pc(x, x′; t),
apart that it does not feel the magnetic field :
[∂t−D∆]Pd(x, x′; t) = δ(x−x′)δ(t). Therefore the
Laplace transform of Pd(t) is given by ∂γ lnS(γ)
with θαβ = 0. It is interesting to point out that
(13) has a similar structure to (1) with a different
cutoff procedure for large time. It also involves
a different scale : the temperature dependence of
∆σee is driven by the length scale LT instead of
Lϕ for the weak-localization correction ∆σWL.
Up to eq. (13) the discussion is rather general
and nothing has been specified on the system.
We have seen in section II that the WL for the
square network presents a dimensional crossover
from 1d to 2d by tuning Lϕ/a. A similar dimen-
sional crossover occurs for the AA correction by
tuning LT /a as we will see. This remark raises the
question of the dimension d in eq. (11). To answer
this question we should return to the origin of the
factor 1/d : the current lines in the conductivity
σij produce a factor qiqj replaced by δij
~q 2 after
angular integration. Since in a network the diffu-
sion in the wires has a 1d structure (provided that
W ≪ LT ∼
D/ω, where W is the width of the
wires), the dimension in λσ is d = 1. Therefore we
have for the network λnetworkσ ≃ 4− 32F .
If one now expands the thermal function in (13)
sinh y
= 4y2
n e−2ny , (14)
we can also relate ∆σee to the spectral determi-
nant. We obtain :
∆σee = −λσ
lnS(γ)
γ= 2nπ
which is the central result of this paper. It is the
starting point of the discussion below.
Application to the case of the square network.– We
have to compute γ2 ∂
lnS(γ). We start from (7)
and compute its second derivative. We obtain after
some algebra :
∆σee = −λσ
where the function ϕ(x) is given by :
ϕ(x) = − 8
2x coshx
sinh3 x
sinh2 x
3 cothx
3 tanhx
coshx
3− 2x
sinh 2x
coshx
, (17)
E(x) being the complete elliptic integral of second
kind30. The function ϕ(x) is plotted in figure 2
and its limiting behaviours are easily obtained30 :
ϕ(x) =
+O(x2) for x → 0 (18)
+O(xe−2x) for x → ∞ (19)
The LT dependence of AA correction on a square
network is displayed on figure 3, where we have
plotted ∆σee(LT ) given by eq. (16). The di-
mensional crossover now occurs by tuning the ra-
tio LT/a. We consider the two limits.
0 5 10 15
FIG. 2: The function ϕ(x) of eq. (17).
1d limit.– For LT ≪ a we can replace the expan-
sion (19) in the series (16). Therefore
∆σee ≃ −λσ
3ζ(3/2)
3ζ(3/2)
≃ 0.782. The dominant term again
coincides with the one for a connected wire8,10,11
while the second differs by a factor 2, as for the WL
[see discussion after eq. (8)].
2d limit.– In the limit LT ≫ a we introduce
N = (LT /a)2 and cut the sum (16) in two pieces :
N . It is clear from the limit be-
0 1 2 3 4 5LT/a
ee 0.1 1.0 10.0
FIG. 3: The continuous line is ∆σee in unit of λσ
as a function of LT /a (the series (16) is computed nu-
merically). The dashed line is the 1d limit, eq. (20),
and the dotted curve is the 2d limit, eq. (21).
haviours of ϕ(x) that the first sum diverges loga-
rithmically with N while the second brings a neg-
ligible contribution of order N 0. Therefore :
∆σee ≃ −λσ
ln(LT /a) + Cee
The constant is estimated numerically. We find
Cee ≃ 0.56.
The two eqs. (20,21) should be divided by the
cross-section s of the wires.
The two functions ∆σWL(B = 0, Lϕ) (figure 1)
and ∆σee(LT ) (figure 3) are very similar. Apart
from the prefactors 2e2/h and λσe
2/h which ac-
count respectively for the spin degeneracy and the
interaction strength, the linear behaviours at the
origin have a different slope (1 and 0.782) and
the logarithmic behaviours are slightly shifted :
CWL ≃ 0.61 and Cee ≃ 0.56.
IV. COMPARISON WITH EXPERIMENTS
The AA correction has been recently measured
by Mallet et al34 in networks of silver wires with
3 104 and 105 cells, lattice spacing a = 0.64 µm
and diffusion constant D ≃ 100 cm2/s. The dif-
fusion constant D has been measured separately
(through measurement of the Drude conductance),
therefore we can compare our result (16) with ex-
periment using one fitting parameter only : the
interaction parameter λσ. The 2d logarithmic
behaviour (21) has been observed in the range
100mK< T < 1K from which the value λexpσ ≃ 3.1
was extracted, in agreement with similar mea-
surements performed on a long silver wire for
which34,35 λexp, wireσ ≃ 3.2. We now compare with
the theoretical value : for silver Fermi wavelength
is k−1F = 0.083 nm and Thomas-Fermi screening
length κ−1 = 1/
8πρ0e2 = 0.055 nm. In the
Thomas-Fermi approximation, the parameter F is
given by11 F = ( κ
)2 ln[1 + (2kF
)2], therefore
F ≃ 0.58. Using the 1d nonperturbative expres-
sion8 λσ = 4 +
1 + F/2 − 1 − F/4), we get
λthσ ≃ 3.24, close to the experimental value.
V. CONCLUSION
Equations (15,16) are our main results. The first
one shows that AA and WL can be formally re-
lated :
∆σee(LT ) =
∆σWL(Lϕ)
= 2nπ
The validity of this relation is the same as for
eqs. (1,2) : the system should be such that it
is meaningful to average uniformly the nonlocal
conductivity σ(r, r′) to get the local conductivity
drdr′
σ(r, r′). A similar discussion has been
proposed to relate WL and conductivity fluctua-
tions (see appendix E of Ref.29).
Our starting point (11) is a formulation in the
Fourier space, what implicitly assumes translation
invariance. Whereas this assumption seems rea-
sonable for a large regular network such as the
square network studied in this article, its valid-
ity is not clear for networks of arbitrary topology,
what would need further developments.
We have computed the AA correction in a large
square network and shown that the result inter-
polates between the 1d, eq. (20), and a 2d result,
eq. (21). Interestingly, the 2d limit in a network
involves a 1d constant λnetworkσ ≃ 4 − 32F , what
is confirmed by experiments, as discussed in sec-
tion IV.
The interest of the network compared to the
plane is to control the constant Cee of eq. (21) :
for a plane, a cutoff must be introduced in eq. (13)
at short time t ∼ τe and the constant Cee is
replaced by a number that depends on the pre-
cise cutoff procedure. Experimentally, it would
be interesting to observe the crossover from (20)
to (21) by varying LT/a. This was not pos-
sible in experiments of Mallet et al34 described
in section IV because measurements are compli-
cated by the fact that electron-phonon interaction
also brings a temperature-dependent contribution,
∆σe−ph, at high temperature (above few Kelvins).
The conductivity is given by σ = σ0 + ∆σWL +
∆σee + ∆σe−ph. The WL can be suppressed by a
magnetic field however the electron-phonon contri-
bution is difficult to separate from ∆σee. There-
fore the network should be patterned in a way such
that the crossover 1d-2d remains below T ∼ 1 K
where ∆σe−ph is negligible. As an example we con-
sider the silver networks studied in Ref.21 for which
LT = 0.27 × T−1/2 (LT in µm and T in K). In
order to see clearly the 1d and the 2d regimes it
would be convenient to study two networks with
different lattice spacings. If temperature is con-
strained by 10 mK< T < 1 K, for a = 0.5 µm we
have 0.5 . LT /a . 5, which probes the 2d regime
over one decade. A second lattice with a ∼ 5 µm
would allow to probe the 1d regime since in this
case 0.05 . LT/a . 0.5.
Acknowledgements
We have benefitted from stimulating discussions
with Christopher Bäuerle, Hélène Bouchiat, Meydi
Ferrier, François Mallet, Laurent Saminadayar and
Félicien Schopfer.
1 B. L. Altshuler, A. G. Aronov, and D. E. Khmel-
nitsky, Effects of electron-electron collisions with
small energy transfers on quantum localisation,
J. Phys. C: Solid St. Phys. 15, 7367 (1982).
2 S. Chakravarty and A. Schmid, Weak localization:
the quasiclassical theory of electrons in a random
potential, Phys. Rep. 140(4), 193 (1986).
3 B. L. Al’tshuler and A. G. Aronov, Contribu-
tion to the theory of disordered metals in strongly
doped semiconductors, Sov. Phys. JETP 50(5), 968
(1979).
4 B. L. Al’tshuler, D. E. Khmel’nitzkĭı, A. I. Larkin,
and P. A. Lee, Magnetoresistance and Hall effect
in a disordered two-dimensional electron gas, Phys.
Rev. B 22(11), 5142 (1980).
5 B. L. Altshuler and A. G. Aronov, Fermi-liquid the-
ory of the electron-electron interaction effects in
disordered metals, Solid State Commun. 46, 429
(1983).
6 A. M. Finkel’shtĕın, Influence of Coulomb interac-
tion on the properties of disordered metals, Sov.
Phys. JETP 57(1), 97 (1983).
7 C. Castellani, C. Di Castro, P. A. Lee, and
M. Ma, Interaction-driven metal-insulator transi-
tions in disordered fermion systems, Phys. Rev. B
30(2), 527 (1984).
8 B. L. Altshuler and A. G. Aronov, Electron-electron
interaction in disordered conductors, in Electron-
electron interactions in disordered systems, edited
by A. L. Efros and M. Pollak, page 1, North-
Holland, 1985.
9 P. A. Lee and T. V. Ramakrishnan, Disordered elec-
tronic systems, Rev. Mod. Phys. 57, 287 (1985).
10 I. L. Aleiner, B. L. Altshuler, and M. E. Gershenson,
Interaction effects and phase relaxation in disor-
dered systems, Waves RandomMedia 9, 201 (1999).
11 É. Akkermans and G. Montambaux, Mesoscopic
physics of electrons and photons, Cambridge Uni-
versity Press, 2007.
12 It is worth mentioning that a similar effect exists at
higher temperature, in the ballistic regime (τee ≪
τe) ; Ref.
13 provides a nice review on this point and
describes the crossover between the two regimes.
13 G. Zala, B. N. Narozhny, and I. L. Aleiner, Interac-
tion corrections at intermediate temperatures: Lon-
gitudinal conductivity and kinetic equation, Phys.
Rev. B 64, 214204 (2001).
14 A. E. White, M. Tinkham, W. J. Skocpol, and D. C.
Flanders, Evidence for interaction effects in the
low-temperature resistance rise in ultrathin metallic
wires, Phys. Rev. Lett. 48(25), 1752 (1982).
15 P. M. Echternach, M. E. Gershenson, H. M. Bozler,
A. L. Bogdanov, and B. Nilsson, Temperature de-
pendence of the resistance of one-dimensional metal
films with dominant Nyquist phase breaking, Phys.
Rev. B 50(8), 5748 (1994).
16 F. Pierre, A. B. Gougam, A. Anthore, H. Pothier,
D. Esteve, and N. O. Birge, Dephasing of electrons
in mesoscopic metal wires, Phys. Rev. B 68, 085413
(2003).
17 C. Bäuerle, F. Mallet, F. Schopfer, D. Mailly,
G. Eska, and L. Saminadayar, Experimental Test
of the Numerical Renormalization Group Theory
for Inelastic Scattering from Magnetic Impurities,
Phys. Rev. Lett. 95, 266805 (2005).
18 B. Pannetier, J. Chaussy, R. Rammal, and P. Gan-
dit, First Observation of Altshuler-Aronov-Spivak
effect in gold and copper, Phys. Rev. B 31(5), 3209
(1985).
19 G. J. Dolan, J. C. Licini, and D. J. Bishop, Quan-
tum Interference Effects in Lithium Ring Arrays,
Phys. Rev. Lett. 56(14), 1493 (1986).
20 M. Ferrier, L. Angers, A. C. H. Rowe, S. Guéron,
H. Bouchiat, C. Texier, G. Montambaux, and
D. Mailly, Direct measurement of the phase co-
herence length in a GaAs/GaAlAs square network,
Phys. Rev. Lett. 93, 246804 (2004).
21 F. Schopfer, F. Mallet, D. Mailly, C. Texier,
G. Montambaux, L. Saminadayar, and C. Bäuerle,
Dimensional crossover in quantum networks: from
mesoscopic to macroscopic physics, Phys. Rev. Lett.
98, 026807 (2007).
22 B. Douçot and R. Rammal, Quantum oscillations
in normal-metal networks, Phys. Rev. Lett. 55(10),
1148 (1985).
23 B. Douçot and R. Rammal, Interference effects and
magnetoresistance oscillations in normal-metal net-
works: 1. weak localization approach, J. Physique
47, 973–999 (1986).
24 M. Pascaud and G. Montambaux, Persistent cur-
rents on networks, Phys. Rev. Lett. 82, 4512 (1999).
25 Pascaud & Montambaux have rather considered
thermodynamic properties. The nonlocal effects in
networks have been further investigated in Ref.27.
26 E. Akkermans, A. Comtet, J. Desbois, G. Montam-
baux, and C. Texier, On the spectral determinant
of quantum graphs, Ann. Phys. (N.Y.) 284, 10–51
(2000).
27 C. Texier and G. Montambaux, Weak localization
in multiterminal networks of diffusive wires, Phys.
Rev. Lett. 92, 186801 (2004).
28 T. Ludwig and A. D. Mirlin, Interaction-induced
dephasing of Aharonov-Bohm oscillations, Phys.
Rev. B 69, 193306 (2004).
29 C. Texier and G. Montambaux, Dephasing due
to electron-electron interaction in a diffusive ring,
Phys. Rev. B 72, 115327 (2005).
30 I. S. Gradshteyn and I. M. Ryzhik, Table of in-
tegrals, series and products, Academic Press, fifth
edition, 1994.
31 A. Cassam-Chenai and B. Shapiro, Two dimen-
sional weak localization beyond the diffusion ap-
proximation, J. Phys. I France 4, 1527 (1994).
32 The formula (5.1) of Ref.8 has the wrong sign.
33 This interaction assumes that the screening length
is smaller than the transverse size of the wire.
34 F. Mallet et al, to be published (2007).
35 L. Saminadayar, P. Mohanty, R. A. Webb, P. De-
giovanni and C. Bäuerle, Phase coherence in the
presence of magnetic impurities, Physica E, to be
published (2007).
