Apparent low-temperature saturation of the electron dephasing time from trapping in non-uniform disordered conductors: solving and testing the virtual-electron-trap mechanism

In many disordered conductors the dephasing time $\tau_\varphi$ extracted from weak-localization magnetoresistance saturates below roughly 0.1-1 K, although standard theories of electron-electron and electron-phonon dephasing predict $1/\tau_\varphi\to 0$ as $T\to 0$. One long-standing proposal is that this saturation is only apparent: if the sample is spatially non-uniform, with small low-diffusion regions acting as electron traps, then fitting the magnetoresistance with uniform-sample theory can yield a temperature-independent effective dephasing time even while the microscopic one keeps growing. Germanenko, Minkov, and Rut demonstrated this inhomogeneity effect by simulation (Phys. Rev. B 64, 165404 (2001)), and Chang and Wu gave an analytic realization, the virtual-electron-trap-scattering (VETS) model (Physica B 406, 3036 (2011)), deriving a closed-form effective dephasing rate with explicit regimes in which the extracted $\tau_\varphi$ saturates at the trapping-free dwell time and later de-saturates at the lowest temperatures. That derivation, however, holds only in the small-island limit $Q_H L_b\ll 1$ with configurational averaging, has never been independently checked even numerically, and the trapping mechanism has never been tested against the competing extrinsic mechanisms (dilute magnetic impurities, dynamical two-level systems, external noise, heating). The problem posed here is to (i) solve the coherent-backscattering (Cooperon) problem for non-uniform conductors beyond the small-island, configuration-averaged limit, establishing the effective dephasing time that uniform-system analysis would extract, and (ii) determine, by testable criteria, whether trapping-induced apparent saturation actually accounts for the residual saturation observed in real conductors.

Background

In a disordered metal or semiconductor at low temperature, the conductivity receives a quantum correction from weak localization (WL): electrons that traverse closed diffusive loops interfere with their own time-reversed paths, and the sign of the interference shifts the resistance. The correction is measured through the low-field magnetoresistance/magnetoconductance (MR/MC), the change of resistance/conductance with a weak perpendicular magnetic field, which destroys the interference once a flux quantum threads the loop area. The interference is also suppressed by dephasing — the loss of phase coherence due to inelastic scattering — after a time $\tau_\varphi$, so the MR/MC shape is routinely inverted to extract $\tau_\varphi$. Theoretically the interference is described by the Cooperon, the propagator of the diffusion-mode pair formed by an electron and its time-reversed partner; for a spatially uniform conductor with diffusion constant $D$ it is P(\mathbf{Q},\omega)=1/(-i\omega+DQ^{2}+1/\tau_\varphi}), and electron-electron and electron-phonon processes give $1/\tau_\varphi\propto T^{p}$, vanishing as $T\to 0$.

Since the observation of Mohanty, Jariwala, and Webb (1997) that the extracted $\tau_\varphi$ saturates at low temperature in some mesoscopic wires, the origin of this saturation has been contested. Golubev and Zaikin argued for an intrinsic dephasing by zero-point fluctuations that survives at $T=0$, while detailed diagrammatic treatments (Cohen and Imry 1999; Marquardt, von Delft, Smith, and Ambegaokar 2007) conclude that in equilibrium and linear transport the dephasing rate must vanish as $T\to 0$; on this mainstream view, observed saturation must be extrinsic. Specific extrinsic mechanisms have been verified in particular systems: extremely dilute magnetic impurities whose spin-flip scattering dephases while remaining hidden in the resistivity (Pierre and Birge 2002; Pierre et al. 2003; the Kondo-scaling experiments of Mallet et al. 2006), externally coupled microwave noise (Wei, Pereverzev, and Gershenson 2006), non-linear-transport and heating artifacts (Imry, Ovadyahu, and Schiller 2003), and dynamical structural defects behaving as tunneling two-level systems (TLS), for which Galperin, Kozub, and Vinokur (2004) predict a residual dephasing time scaling as $\tau_\varphi^{0}\propto 1/D$, matching the systematic observations in polycrystalline metals and alloys (Lin and Kao 2001). Reviews by Lin and Bird (2002) and continuing work through the 2010s-2020s show that in many systems — indium tin oxide, topological insulators, 2D hole gases, granular films — a residual, weakly temperature-dependent dephasing term persists whose microscopic origin is still unidentified.

A distinct, purely structural possibility is that the saturation is an artifact of analyzing a spatially non-uniform conductor with uniform-sample formulas. Germanenko, Minkov, and Rut (Phys. Rev. B 64, 165404 (2001)) showed by numerical simulation that in a macroscopically inhomogeneous two-dimensional system the low-field negative magnetoresistance, fitted with uniform-system WL theory, yields a dephasing time that saturates as $T$ decreases, even though the true phase-breaking length diverges. Chang and Wu (Physica B 406, 3036 (2011)) supplied an analytic realization, the virtual-electron-trap-scattering (VETS) model: the conductor is a percolating, sample-spanning high-diffusion region (H-region, diffusion constant $D_H$) containing many small, nearly insulating islands (L-islands with $D_L\approx 0$) that act as virtual traps. Two dwell times govern the dynamics: $\tau_f$, the mean duration an electron stays in the H-region before being captured by an island, and $\tau_b$, the mean residence time inside an island. Writing region-resolved Cooperons $P_{ij}$ ($i,j\in\{H,L\}$) coupled by Dyson integral equations, and simplifying under the small-island condition $Q_H L_b\ll 1$ (island size $L_b$ small compared with the spatial variation scale $Q_H^{-1}$ of the H-region Cooperon) plus a configurational average over island positions, the dressed H-region Cooperon takes the uniform form with a frequency-dependent effective dephasing rate, which in the DC limit is

$$\frac{1}{\tau_\varphi^{\mathrm{eff}}(0)}=\frac{1}{\tau_\varphi}+\frac{1/\tau_f}{1+\tau_\varphi/\tau_b}.$$

For $\tau_b\gg\tau_f$ this predicts three regimes as $\tau_\varphi$ grows with falling temperature: $\tau_\varphi^{\mathrm{eff}}\approx\tau_\varphi$ when $\tau_\varphi\ll\tau_f$; an intermediate regime $\tau_f\ll\tau_\varphi\ll\tau_b$ in which $\tau_\varphi^{\mathrm{eff}}\approx\tau_f$ is nearly temperature-independent — apparent saturation set by the trapping-free time, requiring no intrinsic saturation mechanism at all; and a lowest-temperature regime $\tau_\varphi\gg\tau_b$ in which $\tau_\varphi^{\mathrm{eff}}\approx(\tau_f/\tau_b)\tau_\varphi$ grows again, so the apparent saturation is transient. The model also predicts that the value extracted from MR/MC depends on the fitting form chosen (absolute versus relative conductance change). Two limitations are essential: the derivation is controlled only in the small-island, configuration-averaged limit — for larger or internally structured islands the interface coordinates of the Dyson equation are entangled and the island contribution to the Cooperon becomes momentum-dependent, and this general case was left unsolved — and the paper provided no independent numerical check of its closed form. The mechanism has remained untested: the ITO study of Wu et al. (2012) found comparable residual dephasing terms in homogeneous and granular films, without measuring the dwell times the model's criteria require.

Problem Statement

Consider a spatially non-uniform disordered conductor: a percolating, sample-spanning high-diffusion region (H-region, diffusion constant $D_H$) contains low-diffusion islands (L-islands with $D_L\approx 0$) acting as virtual electron traps, with mean dwell times $\tau_f$ in the H-region before capture and $\tau_b$ inside an island, and a microscopic inelastic dephasing time $\tau_\varphi(T)$ that grows as the temperature falls. In the known small-island, configuration-averaged limit ($Q_H L_b\ll 1$, where $Q_H^{-1}$ is the spatial variation scale of the H-region Cooperon and $L_b$ the island size), the dephasing time that standard uniform-system weak-localization analysis of the magnetoresistance would extract obeys $1/\tau_\varphi^{\mathrm{eff}}(0)=1/\tau_\varphi+(1/\tau_f)/(1+\tau_\varphi/\tau_b)$, producing a temperature-independent plateau $\tau_\varphi^{\mathrm{eff}}\approx\tau_f$ for $\tau_f\ll\tau_\varphi\ll\tau_b$ — apparent saturation without any intrinsic saturation mechanism — followed by re-growth $\tau_\varphi^{\mathrm{eff}}\approx(\tau_f/\tau_b)\tau_\varphi$ at the lowest temperatures. Two linked questions, which together decide whether disorder inhomogeneity explains the observed residual low-temperature saturation of the extracted dephasing time, remain open.

(1) Solve the coherent-backscattering (Cooperon) problem for such non-uniform conductors beyond the small-island, configuration-averaged limit — for islands of finite size and internal structure, where the interface coordinates of the coupled region-resolved Cooperon (Dyson) equations are entangled and the island contribution to the Cooperon acquires momentum dependence — and determine the effective dephasing time (and its full temperature dependence) that uniform-system MR/MC analysis would extract, including whether the closed-form expression above survives.

(2) Determine whether this trapping-induced apparent saturation actually accounts for the residual saturation observed in real disordered conductors, through the model's distinctive predicted signatures: the pinning of the saturation plateau to the structural dwell time $\tau_f$ rather than to any temperature scale, and the predicted re-growth of the extracted dephasing time below the $\tau_\varphi\sim\tau_b$ crossover. Discrimination from the competing extrinsic mechanisms identified in the literature — dilute magnetic impurities, dynamical two-level systems, external high-frequency noise, and heating — and the correlation (or absence) of the residual saturation scale with independently characterized microstructure and measured dwell times across samples would bear directly on this question.

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