Dimensional Crossover of the Dephasing Time in Disordered Mesoscopic Rings

We study dephasing by electron interactions in a small disordered quasi-one dimensional (1D) ring weakly coupled to leads. We use an influence functional for quantum Nyquist noise to describe the crossover for the dephasing time $\Tph (T)$ from diffusive or ergodic 1D ($ \Tph^{-1} \propto T^{2/3}, T^{1}$) to 0D behavior ($\Tph^{-1} \propto T^{2}$) as $T$ drops below the Thouless energy. The crossover to 0D, predicted earlier for 2D and 3D systems, has so far eluded experimental observation. The ring geometry holds promise of meeting this longstanding challenge, since the crossover manifests itself not only in the smooth part of the magnetoconductivity but also in the amplitude of Altshuler-Aronov-Spivak oscillations. This allows signatures of dephasing in the ring to be cleanly extracted by filtering out those of the leads.

We study dephasing by electron interactions in a small disordered quasi-one dimensional (1D) ring weakly coupled to leads. We use an influence functional for quantum Nyquist noise to describe the crossover for the dephasing time τϕ(T ) from diffusive or ergodic 1D (τ −1 ϕ ∝ T 2/3 , T 1 ) to 0D behavior (τ −1 ϕ ∝ T 2 ) as T drops below the Thouless energy. The crossover to 0D, predicted earlier for 2D and 3D systems, has so far eluded experimental observation. The ring geometry holds promise of meeting this longstanding challenge, since the crossover manifests itself not only in the smooth part of the magnetoconductivity but also in the amplitude of Altshuler-Aronov-Spivak oscillations. This allows signatures of dephasing in the ring to be cleanly extracted by filtering out those of the leads. Over the last twenty years numerous theoretical [1,2,3,4,5,6,7,8] and experimental [10,11,12,13,14,15] works have studied the mechanism of dephasing in electronic transport and its dependence on temperature T and dimensionality in disordered condensed matter systems. At low temperatures dephasing is mainly due to electron interactions, with the dephasing time τ ϕ (T ) increasing as T −a when T → 0.
The dephasing time controls the scale of a negative weak localization (WL) correction to the magnetoconductivity, and (under certain conditions) the magnitude of universal conductance fluctuations (UCFs). If T is so low that τ ϕ exceeds τ Th = /E Th , the time required for an electron to cross (diffusively or ballistically) a mesoscopic sample (E Th is the Thouless energy), UCFs become T -independent. This leaves WL as the only tool to measure the T -dependence of dephasing in mesoscopic wires or quantum dots at very low T . For quantum dots, a dimensional crossover was predicted [4] from τ ϕ ∝ T −1 , typical for a 2D electron gas [1], to τ ϕ ∝ T −2 when the temperature is lowered into the 0D regime, where the coherence length and the thermal length are both larger than the system size, independent of geometry and real dimensionality of the sample. Although the τ ϕ ∝ T −2 behavior is quite generic, arising from the fermionic statistics of conduction electrons, experimental efforts [13] to observe it have so far been unsuccessful. The reasons for this are unclear. Conceivably dephasing mechanisms other than electron interactions were dominant, or the regime of validity of the 0D description had not been reached. In any case, other ways of testing the dimensional crossover for τ ϕ are desirable.
Here we study dephasing in a quasi-1D mesoscopic ring weakly coupled to two well-conducting leads through narrow point contacts. We find a dimensional crossover for τ ϕ (T ) from diffusive or ergodic 1D (∝ T −2/3 , T −1 ) to 0D (∝ T −2 ) behavior as T is decreased below E Th , and pro-pose a detailed experimental scenario for observing this behavior. It reveals itself not only via the WL corrections to the smooth part of magnetoconductivity, but also via the amplitude of the Altshuler-Aronov-Spivak (AAS) oscillations [16] that result from closed trajectories with a non-zero winding number acquiring the Aharonov-Bohm phase. Under suitable conditions, discussed below, the magnitude of AAS oscillations will be independent of dephasing in the leads. Thus, the ring geometry provides a more promising setup for the observation of the dimensional crossover. than 2D or 3D settings.
Dephasing in weak localization. The WL correction to the conductivity is governed by coherent back-scattering of the electrons from static disorder and, to the lowest order, is due to the enhancement of the return probability caused by constructive interference of two time reversed trajectories, described by the so-called Cooperon C [17,18]. In this order, the WL correction to the conductivity, in units of the Drude conductivity σ 0 , is given by [19]: Here ν is the electron density of states per spin at the Fermi surface, and = 1 henceforward. Dephasing limits the scale of this contribution and effectively results in the suppression of the Cooperon at long times: We consider here low temperatures where the phonon contribution to dephasing is negligible and three main sources contribute to the Cooperon decay with time: an applied magnetic field H, characterized by the time scale τ H [20]; the leakage of particles from the ring, characterized by the dwell time τ dw [21]; and electron interactions, whose effects can be described in terms of the decay function F(t) [1,3], which grows with time and may be used to define a dephasing time via F(τ ϕ ) = 1. F(t) can be obtained using the influence functional approach [6,7], which gives results for the magnetoconduc-arXiv:0905.1213v3 [cond-mat.mes-hall] 4 Dec 2009 tivity that are practically equivalent to those originally obtained in [1]. Roughly speaking, an electron traversing a random walk trajectory x(t 1 ) of duration t acquires a random phase ϕ t = t 0 dt 1 V (x(t 1 ), t 1 ) due to the random potential V describing the Nyquist noise originating from electron interactions; the variance of this phase, averaged over all closed random walks (crw), gives the decay function, F(t) = 1 2 δϕ 2 t crw . A careful treatment [6,7] gives where The noise is assumed Gaussian, with correlation function Here A is the wire's cross-sectional area, the diffuson Q(x) is the time-averaged solution of the diffusion equation and δ T (t) is a broadened δ-function of width τ T 1/T and height T , given by [7,8] δ T (t) = πT w(π T t) , w(y) = y coth(y) − 1 sinh 2 (y) .
This form takes into account the Pauli principle in a quantum description of Nyquist noise and reproduces the results [5] of leading order perturbation theory in the interaction for ∆g. The broadening of δ T (t) is the central difference between quantum noise and the classical noise considered in previous treatments [1,6], which used a sharp δ(t) function instead. Note that Eq. (4) is free from IR singularities, because the x-independent part of V V (the diffuson "zero mode") does not contribute to F. Qualitative picture. We begin with a qualitative discussion of dephasing in an isolated quasi-1D system of size L. Since Nyquist electric field fluctuations are white noise in space, the x-dependence of V behaves like a random walk in space (∼ |x|), so that Q(x) ∼ |x|. For τ T τ Th , the potential seen during one traversal of the system is also white noise in time, i.e. δ T (t) → δ(t).
In the diffusive regime (τ T t τ Th ), a random walk x(t 1 ) of duration t does not feel the boundaries, In the ergodic regime (τ T τ Th t), the trajectory fully explores the whole system, thus |x(t 1 )| ∼ L instead, which reproduces F(t) ∝ T Lt and τ ϕ ∝ T −1 [6].
We are primarily interested in the 0D regime reached at T E Th (τ Th τ T t). In contrast to the previous two regimes, a typical trajectory visits the vicinity of any point x in the interval [0, L] several times during the time τ T (see Fig. 1). On time scales shorter than this time τ T the potential is effectively frozen, so that the broadened δ-function in Eq. (5) saturates at its maximum, δ T (t) → T , and the variance of V is of order T 2 |x|. The phase picked up during τ T becomes Adding up the contributions from t/τ T independent time intervals (t τ T ), we find F(t) ∝ L 3 T 2 t, implying τ ϕ ∼ T −2 , characteristic of 0D systems [4]. Thus, when τ Th becomes the smallest time scale, a dimensional crossover occurs and the system becomes effectively 0D.
The qualitative behavior of τ ϕ in all three regimes also follows upon extracting τ ϕ selfconsistently from the standard perturbative expression for the Cooperon selfenergy [2,7,8]. Inserting the usual cutoffs T and 1/τ ϕ for the frequency transfered between the diffusive electrons and their Nyquist noise environment and excluding the diffuson 0th mode via a cutoff at 1/L of the transfered momentum, we have (omitting numerical prefactors): where g 1 is the 1D dimensionless conductance, D = v F l the 1D diffusion constant, v F the Fermi velocity and the mean free path. Writing E Th = D/L 2 this yields τ ϕ ∝ (g 1 / √ E Th T ) 2/3 , g 1 /T or E Th g 1 /T 2 for the diffusive (τ T τ ϕ τ Th ), ergodic (τ T τ Th τ ϕ ) or 0D (τ Th τ T τ ϕ ) regimes, respectively, as above (with dimensionful parameters reinstated). Eq. (7) illustrates succinctly that the modes dominating dephasing lie near the infrared cutoff (ω τ −1 ϕ or E Th ) for the diffusive or ergodic regimes, but near the ultraviolet cutoff ω T for the 0D regime (which is why, in the latter, the broadening of δ T (t) becomes important).
Filtering out leads. For simplicity, above we did not model the leads explicitly. In real experiments, however, ∆g is affected by dephasing in the leads, which might mask the signatures of dephasing in the confined region (the ring). This concern also applies to quantum dots connected to leads (cf. the τ ϕ ∝ T −1 -behavior observed in [13]), or finite-size effects in a network of disordered wires [15], where paths encircling a given unit cell might spend significant time in neighboring unit cells as well (cf. T −1/3 -behavior observed in [15,25] at τ ϕ /τ Th ≥ 1). The effect of leads can be filtered out [15] by constructing from |∆g(T, φ)| its nonoscillatory envelope |∆g en (T, φ)|, obtained by setting θ = 0 in Eq. (11) while retaining τ H = 0, and studying the difference ∆g(T, φ) = |∆g en (T, φ)| − |∆g(T, φ)| ; (12) this procedure is illustrated in Fig. 2. ∆g is dominated by paths with winding numbers n ≥ 1 which belong to the ring. Contributions to ∆g from Cooperons extending over both the ring and a lead will be subleading for wellconducting leads with a small contact-lead-contact return probability. Concretely, for N -channel point contacts with conductance g cont = N T cont , this requires leads with dimensionless conductance g lead N [26]. Suggested Experiments. To observe the predicted 1Dto-0D crossover experimentally, several conditions need to be satisfied. Our theory assumes (i) L L W λ F (λ F is the Fermi wavelength). Ensuring that we stay in the WL regime requires (ii) a large dimensionless conductance, g 1 ∝ ( /L)(L W L H /λ 2 F ) 1, and (iii) a finite τ dw to limit the growth of ∆g with decreasing T ; choosing the limit, somewhat arbitrarily, as ∆g l=2µm, g 1 =25, τ dw /τ Th =3: this implies 8 g cont , and thus the absence of Coulomb blockade. (iv) We also need τ Th τ dw , or g cont g 1 , to ensure that trajectories with |n| ≥ 1, responsible for AAS oscillations, remain relevant. (v) To maximize the WL signal, the transmission per channel should be maximal, thus we suggest T cont 1 and N 10. (vi) The relevant temperature range, [T dil , T ph ], is limited from below by dilution refrigeration (T dil 10mK) and from above by our neglect of phonons (T ph 5K). (vii) The ring should be small enough that c 2 E Th T dil . (viii) The interaction-induced dephasing rate τ −1 ϕ , though decreasing with decreasing T , should for T T dil not yet be negligible compared to the T -independent rates τ −1 H and τ −1 dw . These constraints can be met, e.g., with rings prepared from a 2D GaAs/AlGaAs heterostructure with λ F ≈ 30nm, v F ≈ 2.5·10 5 m/s, and g 1 = 4πL W l/λ F L, by adjusting g 1 and E Th by suitably choosing L and L W .
To illustrate this, numerical results for |∆g| and ∆g, obtained from Eq.(2) using experimentally realizable parameters [9,10,15,27], are shown in Figs. 2 and 3 for several combinations of , L and L W . The regime where ∆g exhibits diffusive T −1/3 behavior (7g 1 E Th T T ph ) is visible only for our smallest choices of both g 1 and E Th (Fig. 3a, heavy dashed line). AAS oscillations in |∆g| and ∆g (Fig. 2), which require τ Th τ ϕ , first emerge at the crossover from the diffusive to the ergodic regime. They increase in magnitude with decreasing T , showing ergodic T −1 behavior for 30E Th T 7g 1 E Th (Figs. 3a,b), and eventually saturate towards their T = 0 values, with ∆g(0, φ) − ∆g(T, φ) showing the predicted 0D behavior, ∝ T 2 , for T 5E Th , see Fig. 3c (there τ ϕ τ Th , i.e. dephasing is weak).
Conclusions. The AAS oscillations of a quasi-1D ring weakly coupled to leads can be exploited to filter out the effects of dephasing in the leads, thus offering a way to finally observe, for T 5E Th , the elusive but fundamental 0D behavior τ ϕ ∼ T −2 . This would allow quantitative experimental tests of the role of temperature as ultraviolet frequency cutoff in the theory of dephasing.