Nonlinear conductance in weakly disordered mesoscopic wires: Interaction and magnetic field asymmetry

We study the nonlinear conductance $\mathcal{G}\ensuremath{\sim}{\ensuremath{\partial}}²I/\ensuremath{\partial}{V}²{|}_V=0$ in coherent quasi-one-dimensional weakly disordered metallic wires. Our analysis is based on the scattering approach and includes the effect of Coulomb interaction. The nonlinear conductance correlations can be related to integrals of two fundamental correlation functions: the correlator of functional derivatives of the conductance and the correlator of injectivities (the injectivity is the contribution to the local density of states of eigenstates incoming from one contact). These correlators are obtained explicitly by using diagrammatic techniques for weakly disordered metals. In a coherent wire of length $L$, we obtain $\mathrm{rms}\left(\mathcal{G}\right)\ensuremath{\simeq}0.006\phantom{\rule{0.16em}{0ex}}{E}_{\mathrm{Th}}^{\ensuremath{-}1}$ (and $\ensuremath{\langle}\mathcal{G}\ensuremath{\rangle}=0)$, where ${E}_{\mathrm{Th}}=\ensuremath{\hbar}D/{L}²$ is the Thouless energy of the wire and $D$ the diffusion constant; the small dimensionless factor results from screening, i.e., cannot be obtained within a simple theory for noninteracting electrons. Electronic interactions are also responsible for an asymmetry under magnetic field reversal; the antisymmetric part of the nonlinear conductance (at high magnetic field) being much smaller than the symmetric one, $\mathrm{rms}\left({\mathcal{G}}_a\right)\ensuremath{\simeq}0.001\phantom{\rule{0.16em}{0ex}}{(g{E}_{\mathrm{Th}})}^{\ensuremath{-}1}$, where $g\ensuremath{\gg}1$ is the dimensionless (linear) conductance of the wire. In a weakly coherent wire (i.e., ${L}_{\ensuremath{\varphi}}\ensuremath{\ll}L$, where ${L}_{\ensuremath{\varphi}}$ is the phase coherence length), the nonlinear conductance is of the same order as the result ${\mathcal{G}}₀$ of a free electron calculation (although screening again strongly reduces the dimensionless prefactor); we get $\mathcal{G}\ensuremath{\sim}{\mathcal{G}}₀\ensuremath{\sim}{({L}_{\ensuremath{\varphi}}/L)}^7/2{E}_{\mathrm{Th}}^{\ensuremath{-}1}$, while the antisymmetric part (at high magnetic field) now behaves as ${\mathcal{G}}_a\ensuremath{\sim}{({L}_{\ensuremath{\varphi}}/L)}^11/2{(g{E}_{\mathrm{Th}})}^{\ensuremath{-}1}\ensuremath{\ll}\mathcal{G}$. The effect of thermal fluctuations is studied: when the thermal length ${L}_T=\sqrt{\ensuremath{\hbar}D/{k}_BT}$ is the smallest length scale, ${L}_T\ensuremath{\ll}{L}_{\ensuremath{\varphi}}\ensuremath{\ll}L$, the free electron result ${\mathcal{G}}₀\ensuremath{\sim}{({L}_T/L)}³{({L}_{\ensuremath{\varphi}}/L)}^1/2{E}_{\mathrm{Th}}^{\ensuremath{-}1}$ is negligible and the dominant contribution is provided by screening, $\mathcal{G}\ensuremath{\sim}({L}_T/L){({L}_{\ensuremath{\varphi}}/L)}^7/2{E}_{\mathrm{Th}}^{\ensuremath{-}1}$; in this regime, the antisymmetric part is ${\mathcal{G}}_a\ensuremath{\sim}{({L}_T/L)}²{({L}_{\ensuremath{\varphi}}/L)}^7/2{(g{E}_{\mathrm{Th}})}^{\ensuremath{-}1}$. All the precise dimensionless prefactors are obtained. Crossovers from zero to strong magnetic field regimes are also analyzed.