Al'tshuler-Aronov correction to the conductivity of a large metallic square network

TL;DR

This paper analyzes the electron-electron interaction correction to conductivity in a large metallic network, revealing a behavior that interpolates between 1D and 2D regimes and closely resembles 2D logarithmic behavior.

Contribution

It provides a novel analysis of the Al'tshuler-Aronov correction in large metallic networks, connecting spectral determinants to conductivity corrections across dimensions.

Findings

Correction varies with thermal length similarly to weak localization.

Results interpolate between 1D and 2D behaviors.

Interaction parameter exhibits 1D characteristics.

Abstract

We consider the correction $\Delta\sigma_\mathrm{ee}$ 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 $\Delta\sigma_\mathrm{ee}(L_T)$ as a function of the thermal length $L_T$ is found very similar to the variation of the weak localization $\Delta\sigma_\mathrm{WL}(L_\phi)$ as a function of the phase coherence length. Our result for $\Delta\sigma_\mathrm{ee}$ interpolates between the known 1d and 2d results, but the interaction parameter entering the expression of $\Delta\sigma_\mathrm{ee}$ keeps a 1d behaviour. Quite surprisingly, the result is very close to the 2d logarithmic behaviour already for $L_T\sim{a}/2$, where $a$ is the lattice parameter.

Figures

Equations

$$\Delta {\sigma }_{\mathrm{WL}}$$

$$=$$

$${\mathcal{M}}_{\alpha \beta }={\delta }_{\alpha \beta }\left({\lambda }_{\alpha }+\sqrt{\gamma }\sum _{\mu }{a}_{\alpha \mu }\coth \sqrt{\gamma }{l}_{\alpha \mu }\right)$$

$$-a_{\alpha \beta }\sqrt{\gamma }\frac{{\text{mbox}e}^{-\mathrm{i}{\theta }_{\alpha \beta }}}{\sinh \sqrt{\gamma }{l}_{\alpha \beta }}$$

$$S(\gamma )=\prod _{(\alpha \beta )}\frac{\sinh \sqrt{\gamma }{l}_{\alpha \beta }}{\sqrt{\gamma }}\det \mathcal{M}$$

$$S(\gamma )={\left(2\frac{\sinh \sqrt{\gamma }a}{\sqrt{\gamma }}\right)}^{N_x{N}_y}$$

$$\times \prod _{n=1}^{N_x}\prod _{m=1}^{N_y}\left(2\cosh \sqrt{\gamma }a-\cos \frac{2\pi n}{N_x}-\cos \frac{2\pi m}{N_y}\right).$$

$${\int }_0^{2\pi }\frac{\mathrm{d}x\mathrm{d}y}{(2\pi {)}^2}\frac{1}{2A+\cos x+\cos y}=\frac{1}{\pi A}\mathrm{K}(1/A),$$

$$\frac{1}{\mathrm{Vol}}\frac{\partial }{\partial \gamma }\ln S(\gamma )$$

$$\Delta {\sigma }_{\mathrm{WL}}=-\frac{2e^2}{h}\left(L_{\phi }-\frac{L_{\phi }^2}{2a}+O\left({\text{mbox}e}^{-2a/L_{\phi }}\right)\right)$$

$$\displaystyle\frac{1}{{\rm Vol}}\frac{\partial}{\partial\gamma}\ln S(\gamma)=\frac{a}{2\pi}\bigg{[}\ln(4L_{\varphi}/a)+\frac{\pi}{6}$$

$$\displaystyle+O\left(\frac{a^{2}}{L_{\varphi}^{2}}\ln\frac{L_{\varphi}}{a}\right)\bigg{]}$$

$$\Delta {\sigma }_{\mathrm{WL}}\simeq -\frac{2e^2}{h}a\left[\frac{1}{\pi }\ln (L_{\phi }/a)+C_{\mathrm{WL}}\right]$$

$$\Delta {\sigma }_{\mathrm{ee}}$$

$$\Delta {\sigma }_{\mathrm{ee}}=-{\lambda }_{\sigma }\frac{e^{2}D}{\pi }{\int }_0^{\infty }\mathrm{d}t{\left(\frac{\pi Tt}{\sinh \pi Tt}\right)}^2{\mathcal{P}}_d(t),$$

$${\left(\frac{y}{\sinh y}\right)}^2=4y^2\sum _{n=1}^{\infty }n{\text{mbox}e}^{-2ny},$$

$$\Delta {\sigma }_{\mathrm{ee}}=-{\lambda }_{\sigma }\frac{e^2}{\pi \mathrm{Vol}}\sum _{n=1}^{\infty }\frac{1}{n}{\left[{\gamma }^2\frac{{\partial }^3}{\partial {\gamma }^3}\ln S(\gamma )\right]}_{\gamma =\frac{2n\pi }{L_T^2}}$$

$$\Delta {\sigma }_{\mathrm{ee}}=-{\lambda }_{\sigma }\frac{e^2}{h}\frac{a}{8}\sum _{n=1}^{\infty }\frac{1}{n}\phi \left(\sqrt{2n\pi }\frac{a}{L_T}\right)$$

$$\phi (x)=-\frac{8}{x^2}+\frac{2x\cosh x}{{\sinh }^{3}x}+\frac{3}{{\sinh }^{2}x}+\frac{3\coth x}{x}$$

$$+\frac{2}{\pi }\left\{\left[\frac{3\tanh x}{x}-3\right]\mathrm{K}\left(\frac{1}{\cosh x}\right)+\left[3-\frac{2x}{\sinh 2x}\right]\mathrm{E}\left(\frac{1}{\cosh x}\right)\right\},$$

$$\phi (x)$$

$$=$$

$$\Delta {\sigma }_{\mathrm{ee}}\simeq -{\lambda }_{\sigma }\frac{e^2}{h}\left(\frac{3\zeta (3/2)}{4\sqrt{2\pi }}{L}_T-\frac{\pi }{12}\frac{L_T^2}{a}\right)$$

$$\Delta {\sigma }_{\mathrm{ee}}\simeq -{\lambda }_{\sigma }\frac{e^2}{h}a\left[\frac{1}{\pi }\ln (L_T/a)+C_{\mathrm{ee}}\right]$$

$$\Delta {\sigma }_{\mathrm{ee}}(L_T)$$

Full text

I Introduction

At low temperature, the classical (Drude) conductivity 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 quantum interferences between reversed electronic trajectories. This contribution to the averaged conductivity depends on the phase coherence length $L_{\varphi}$ and the magnetic field : $\Delta\sigma_{\mathrm{WL}}(\mathcal{B},L_{\varphi})$. The temperature manifests itself through $L_{\varphi}$, since phase breaking may depend on temperature, e.g. if it originates from electron-electronAltAroKhm82 or electron-phononChaSch86 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 elastic scattering rate ($1/\tau_{e}$) dominates the electron-electron scattering rate ($1/\tau_{\mathrm{ee}}(T)$), the motion of the electron is diffusive between scattering events with other electrons. In this regime, electron-electron interaction is responsible for a small depletion of the density of states at Fermi energy (called the DoS anomaly or the Coulomb dip) and a correction to the averaged conductivity as well, the so-called Al’tshuler-Aronov (AA) correction AltAro79 ; AltKhmLarLee80 ; AltAro83 ; Fin83 ; CasCasLeeMa84 ; AltAro85 ; LeeRam85 (see Refs. AleAltGer99 ; AkkMon04 ; footnote1 for a recent discussion). AA and WL corrections are of the same order (but this latter vanishes in a magnetic field). However, contrary to the WL, the AA correction is not sensitive to phase coherence and involves another important length scale : the thermal length $L_{T}=\sqrt{D/T}$ ($\hbar=k_{B}=1$). The AA correction, denoted below $\Delta\sigma_{\mathrm{ee}}(L_{T})$, has been measured in metallic wires in several experiments WhiTinSkoFla82 ; EchGerBozBogNil94 ; PieGouAntPotEstBir03 ; BauMalSchMaiEskSam05 . From the experimental point of view, AA correction allows to study interaction effects in weakly disordered metals, but also furnishes a local probe of temperature in order to control Joule heating effects EchGerBozBogNil94 ; BauMalSchMaiEskSam05 , 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 studied 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 networks PanChaRamGan85 ; DolLicBis86 , in square networks etched in a 2DEG FerAngRowGueBouTexMonMai04 , and in square and dice silver networks SchMalMaiTexMonSamBau07 . Theoretical studies of WL on networks have been initiated by the works of Douçot & Rammal DouRam85 ; DouRam86 and improved by Pascaud & Montambaux PasMon99 who introduced a powerful toolfootnote2  : the spectral determinant of the Laplace operator, that will be used in the following (see also Ref. AkkComDesMonTex00 ).

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 trajectories are encoded in the Cooperon, solution of a diffusion-like equation $(\partial_{t}-D[\nabla-2{\rm i}{}eA(x)]^{2})\mathcal{P}_{c}(x,x^{\prime};t)=\delta(x-x^{\prime})\delta(t)$, 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. TexMon04 for a discussion of this point) it is meaningful to introduce the space-averaged Cooperon $\mathcal{P}_{c}(t)=\int\frac{{\rm d}{x}}{\mathrm{Vol}}\mathcal{P}_{c}(x,x;t)$ then

[TABLE]

where $\tau_{\varphi}=L_{\varphi}^{2}/D$ 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 $\gamma$ is related to the phase coherence length $\gamma=1/L_{\varphi}^{2}$ (note that description of the decoherence due to electron-electron interaction in networks requires a more refined discussion LudMir04 ; TexMon05b ). The spectral determinant of the Laplace operator is formally defined as $S(\gamma)=\det(\gamma-\Delta)=\prod_{n}(\gamma+E_{n})$ where $\{E_{n}\}$ is the spectrum of $-\Delta$ [in the presence of a magnetic field, $\Delta\to(\nabla-2{\rm i}eA)^{2}$]. The interest in introducing $S(\gamma)$ is that it can be related to the determinant of a $V\times{V}$-matrix, where $V$ is the number of vertices, that encodes all information on the network (topology, length of the wires, magnetic field, connection to reservoirs). We label vertices by greek letters. $l_{\alpha\beta}$ designates the length of the wire $({\alpha\beta})$ and $\theta_{\alpha\beta}$ the circulation of the vector potential along the wire. The topology is encoded in the adjacency matrix : $a_{\alpha\beta}=1$ if $\alpha$ and $\beta$ are linked by a wire, $a_{\alpha\beta}=0$ otherwise. $\lambda_{\alpha}=\infty$ if $\alpha$ is connected to a reservoir and $\lambda_{\alpha}=0$ if not. We introduce the matrix

[TABLE]

where the $a_{\alpha\mu}$ constraints the sum to run over neighbouring vertices. Then PasMon99

[TABLE]

where the product runs over all wires. We now consider a large square network of size $N_{x}\times{N_{y}}$ made of wires of length $l_{\alpha\beta}=a$ $\forall({\alpha\beta})$. For simplicity we impose periodic boundary conditions (topology of a torus), which is inessential as soon as the total size of the network remains small compared to $L_{\varphi}$. At zero magnetic field the spectrum of the adjacency matrix is $\epsilon_{n,m}=2\cos(2n\pi/N_{x})+2\cos(2m\pi/N_{y})$, with $n=1,\cdots,N_{x}$ and $m=1,\cdots,N_{y}$. Therefore

[TABLE]

The calculation of $\ln{S(\gamma)}$ involves a sum that can be replaced by an integral when $N_{x},\,N_{y}\gg{L_{\varphi}/a}$. Using

[TABLE]

where $\mathrm{K}(x)$ is the complete elliptic integral of first kind gragra , yields FerAngRowGueBouTexMonMai04

[TABLE]

where the volume of the network is ${\rm Vol}=2N_{x}N_{y}a$. We recover the expression of the WL first derived by Douçot & Rammal DouRam86 . Figure 1 displays the dependence of the WL correction as a function of the phase coherence length $L_{\varphi}$. We now discuss two limiting cases.

1d limit.– In the limit $L_{\varphi}\ll{a}$ (i.e. $\sqrt{\gamma}a\gg 1$) :

[TABLE]

We compare with the result for a wire of length $a$ connected at its extremities : $\Delta\sigma_{\mathrm{WL}}^{\mathrm{wire}}\simeq-\frac{2e^{2}}{h}\,(L_{\varphi}-\frac{L_{\varphi}^{2}}{a})$. As we can see the dominant terms coincide. Deviations appear when $L_{\varphi}/a$ increases since trajectories 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 $L_{\varphi}\gg{a}$ (i.e. $\sqrt{\gamma}a\ll 1$), we obtain

[TABLE]

The conductivity reads

[TABLE]

with $C_{\mathrm{WL}}=\frac{2\ln 2}{\pi}+\frac{1}{6}\simeq 0.608$. As noticed in the beginning of the section, eqs. (8,10) should be divided 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 reminiscent 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 $\tau_{e}$. In this latter case the constant added to the logarithmic behaviour is not well controlled since it depends on the cutoff procedure (the computation of the constant for a plane requires to go beyond the diffusion approximation and leads to CasSha94 $\Delta\sigma^{\mathrm{plane}}_{\mathrm{WL}}=-\frac{e^{2}}{\pi h}\ln(2L_{\varphi}^{2}/\ell_{e}^{2}+1)\simeq-\frac{2e^{2}}{h}[\frac{1}{\pi}\ln(L_{\varphi}/\ell_{e})+\frac{1}{2\pi}\ln 2]$ since $\ell_{e}\ll{L}_{\varphi}$).

III Al’tshuler-Aronov correction

At first order in the electron-electron interaction, the exchange term is the dominant contribution to the correction to the conductivity AltAro85 ; footnote3 ; AleAltGer99 ; AkkMon04

[TABLE]

where $U(\vec{q},\omega)$ is the dynamically screened interaction. Within the RPA approximation and in the small $\vec{q}$ and $\omega$ limit, the interaction takes the formfootnote4 $U(\vec{q},\omega)\simeq\frac{1}{2\rho_{0}}\frac{-{\rm i}\omega+D\vec{q}\,^{2}}{D\vec{q}\,^{2}}$ where $\rho_{0}$ is the density of states per spin channel. Replacing the Drude conductivity by its expression $\sigma_{0}=2e^{2}\rho_{0}D$ and performing an integration by parts, we get

[TABLE]

After Fourier transform, the result can be cast in the form AkkMon04  :

[TABLE]

For the exchange term considered here, one finds $\lambda_{\sigma}=4/d$. Further calculation yields AltAro85 $\lambda_{\sigma}\simeq\frac{4}{d}-\frac{3}{2}F$, where $F$ is the average of the interaction on the Fermi surface (see definition in Refs. AltAro85 ; LeeRam85 ). This expression of $\lambda_{\sigma}$ is valid in the perturbative regime, $F\ll 1$ ; nonperturbative expression is given in Refs. Fin83 ; CasCasLeeMa84 ; AltAro85 ; LeeRam85 . $\mathcal{P}_{d}(t)$ is the space integrated return probability $\mathcal{P}_{d}(t)=\int\frac{{\rm d}{x}}{\mathrm{Vol}}\mathcal{P}_{d}(x,x;t)$, where $\mathcal{P}_{d}(x,x^{\prime};t)$ is solution of a classical diffusion equation similar to the equation for $\mathcal{P}_{c}(x,x^{\prime};t)$, apart that it does not feel the magnetic field : $[\partial_{t}-D\Delta]\mathcal{P}_{d}(x,x^{\prime};t)=\delta(x-x^{\prime})\delta(t)$. Therefore the Laplace transform of $\mathcal{P}_{d}(t)$ is given by $\partial_{\gamma}\ln{S}(\gamma)$ with $\theta_{\alpha\beta}=0$. 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 $\Delta\sigma_{\mathrm{ee}}$ is driven by the length scale $L_{T}$ instead of $L_{\varphi}$ for the weak-localization correction $\Delta\sigma_{\mathrm{WL}}$.

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 $L_{\varphi}/a$. A similar dimensional crossover occurs for the AA correction by tuning $L_{T}/a$ 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 $\sigma_{ij}$ produce a factor $q_{i}q_{j}$ replaced by $\delta_{ij}\frac{1}{d}\vec{q}\,^{2}$ after angular integration. Since in a network the diffusion in the wires has a 1d structure (provided that $W\ll{L_{T}}\sim\sqrt{{D}/{\omega}}$, where $W$ is the width of the wires), the dimension in $\lambda_{\sigma}$ is $d=1$. Therefore we have for the network $\lambda^{\mathrm{network}}_{\sigma}\simeq 4-\frac{3}{2}F$.

If one now expands the thermal function in (13) as :

[TABLE]

we can also relate $\Delta\sigma_{\mathrm{ee}}$ to the spectral determinant. We obtain :

[TABLE]

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 $\gamma^{2}\frac{\partial^{3}}{\partial\gamma^{3}}\ln S(\gamma)$. We start from (7) and compute its second derivative. We obtain after some algebra :

[TABLE]

where the function $\varphi(x)$ is given by :

[TABLE]

$\mathrm{E}(x)$ being the complete elliptic integral of second kind gragra . The function $\varphi(x)$ is plotted in figure 2 and its limiting behaviours are easily obtained gragra  :

[TABLE]

The $L_{T}$ dependence of AA correction on a square network is displayed on figure 3, where we have plotted $\Delta\sigma_{\mathrm{ee}}(L_{T})$ given by eq. (16). The dimensional crossover now occurs by tuning the ratio $L_{T}/a$. We consider the two limits.

1d limit.– For $L_{T}\ll{a}$ we can replace the expansion (19) in the series (16). Therefore

[TABLE]

with $\frac{3\zeta(3/2)}{4\sqrt{2\pi}}\simeq 0.782$. The dominant term again coincides with the one for a connected wire AltAro85 ; AleAltGer99 ; AkkMon04 while the second differs by a factor $2$, as for the WL [see discussion after eq. (8)].

2d limit.– In the limit $L_{T}\gg{a}$ we introduce $\mathcal{N}=(L_{T}/a)^{2}$ and cut the sum (16) in two pieces : $\sum_{1}^{\infty}=\sum_{1}^{\mathcal{N}}+\sum_{\mathcal{N}}^{\infty}$. It is clear from the limit behaviours of $\varphi(x)$ that the first sum diverges logarithmically with $\mathcal{N}$ while the second brings a negligible contribution of order $\mathcal{N}^{0}$. Therefore :

[TABLE]

The constant is estimated numerically. We find $C_{\mathrm{ee}}\simeq 0.56$.

The two eqs. (20,21) should be divided by the cross-section $s$ of the wires.

The two functions $\Delta\sigma_{\mathrm{WL}}(\mathcal{B}=0,L_{\varphi})$ (figure 1) and $\Delta\sigma_{\mathrm{ee}}(L_{T})$ (figure 3) are very similar. Apart from the prefactors $2e^{2}/h$ and $\lambda_{\sigma}{e}^{2}/h$ which account 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 : $C_{\mathrm{WL}}\simeq 0.61$ and $C_{\mathrm{ee}}\simeq 0.56$.

IV Comparison with experiments

The AA correction has been recently measured by Mallet et al MalSamBau07 in networks of silver wires with $3\>10^{4}$ and $10^{5}$ cells, lattice spacing $a=0.64\>\mu$m and diffusion constant $D\simeq 100\>$cm${}^{2}/$s. The diffusion constant $D$ has been measured separately (through measurement of the Drude conductance), therefore we can compare our result (16) with experiment using one fitting parameter only : the interaction parameter $\lambda_{\sigma}$. The 2d logarithmic behaviour (21) has been observed in the range $100\>$mK$<T<1\>$K from which the value $\lambda^{\mathrm{exp}}_{\sigma}\simeq 3.1$ was extracted, in agreement with similar measurements performed on a long silver wire for which SamMohWebDegBau07 ; MalSamBau07 $\lambda^{\mathrm{exp,\>wire}}_{\sigma}\simeq 3.2$. We now compare with the theoretical value : for silver Fermi wavelength is $k_{F}^{-1}=0.083\>$nm and Thomas-Fermi screening length $\kappa^{-1}=1/\sqrt{8\pi\rho_{0}e^{2}}=0.055\>$nm. In the Thomas-Fermi approximation, the parameter $F$ is given byAkkMon04 $F=(\frac{\kappa}{2k_{F}})^{2}\ln[1+(\frac{2k_{F}}{\kappa})^{2}]$, therefore $F\simeq 0.58$. Using the 1d nonperturbative expression AltAro85 $\lambda_{\sigma}=4+\frac{48}{F}(\sqrt{1+F/2}-1-F/4)$, we get $\lambda^{\mathrm{th}}_{\sigma}\simeq 3.24$, 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 related :

[TABLE]

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 $\sigma(r,r^{\prime})$ to get the local conductivity $\sigma=\int\frac{{\rm d}{r}{\rm d}{r^{\prime}}}{\mathrm{Vol}}\sigma(r,r^{\prime})$. A similar discussion has been proposed to relate WL and conductivity fluctuations (see appendix E of Ref. TexMon05b ).

Our starting point (11) is a formulation in the Fourier space, what implicitly assumes translation invariance. Whereas this assumption seems reasonable for a large regular network such as the square network studied in this article, its validity 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 interpolates between the 1d, eq. (20), and a 2d result, eq. (21). Interestingly, the 2d limit in a network involves a 1d constant $\lambda^{\mathrm{network}}_{\sigma}\simeq 4-\frac{3}{2}F$, what is confirmed by experiments, as discussed in section IV.

The interest of the network compared to the plane is to control the constant $C_{\mathrm{ee}}$ of eq. (21) : for a plane, a cutoff must be introduced in eq. (13) at short time $t\sim\tau_{e}$ and the constant $C_{\mathrm{ee}}$ is replaced by a number that depends on the precise cutoff procedure. Experimentally, it would be interesting to observe the crossover from (20) to (21) by varying $L_{T}/a$. This was not possible in experiments of Mallet et al MalSamBau07 described in section IV because measurements are complicated by the fact that electron-phonon interaction also brings a temperature-dependent contribution, $\Delta\sigma_{\mathrm{e-ph}}$, at high temperature (above few Kelvins). The conductivity is given by $\sigma=\sigma_{0}+\Delta\sigma_{\mathrm{WL}}+\Delta\sigma_{\mathrm{ee}}+\Delta\sigma_{\mathrm{e-ph}}$. The WL can be suppressed by a magnetic field however the electron-phonon contribution is difficult to separate from $\Delta\sigma_{\mathrm{ee}}$. Therefore the network should be patterned in a way such that the crossover 1d-2d remains below $T\sim 1\>$K where $\Delta\sigma_{\mathrm{e-ph}}$ is negligible. As an example we consider the silver networks studied in Ref. SchMalMaiTexMonSamBau07 for which $L_{T}=0.27\times\>T^{-1/2}$ ($L_{T}$ in $\mu$m 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 constrained by $10\>$mK$<T<1\>$K, for $a=0.5\>\mu$m we have $0.5\lesssim{L_{T}/a}\lesssim 5$, which probes the 2d regime over one decade. A second lattice with $a\sim 5\>\mu$m would allow to probe the 1d regime since in this case $0.05\lesssim{L_{T}/a}\lesssim 0.5$.

Acknowledgements

We have benefitted from stimulating discussions with Christopher Bäuerle, Hélène Bouchiat, Meydi Ferrier, François Mallet, Laurent Saminadayar and Félicien Schopfer.