Observation of geometry-dependent conductivity in two-dimensional electron systems

Abstract

We report electrical conductivity σ measurements on a range of two-dimensional electron gases (2DEGs) of varying linear extent. Intriguingly, at low temperatures (T ) and low carrier density (ns) we find the behavior to be consistent with σ ∼ L α , where L is the length of the 2DEG along the direction of transport. Importantly, such scale-dependent behavior is precisely in accordance with the scaling hypothesis of localization [Abrahams et al., Phys. Rev. Lett. 42, 673 (1979)] which dictates that in systems where the electronic wave function ξ is localized, σ is not a material-specific parameter, but depends on the system dimensions. From our data we are able to construct the "βfunction" ≡ (h/e 2 )d ln σ/d ln L and show this to be strongly consistent with theoretically predicted limiting values. These results suggest, remarkably, that the electrons in the studied 2DEGs preserve phase coherence over lengths ∼ 10 µm. This suggests the utility of the 2DEGs studied towards applications in quantum information as well as towards fundamental investigations into many-body localized phases.The scaling hypothesis of localization [1], formulated over thirty years ago, is a statement that the electrical conductivity σ is lengthscale-dependent in finite systems where the conduction electrons are short-ranged or localized. This can be understood by considering electronic states with localization length ξ in systems of different spatial extents: As depicted in Fig. 1(a), if ξ is greater than the linear extent of the system, then electrons are able to communicate across the system ends and there will be a finite conductance G even at T = 0 K. However, this conductance will decrease as the system size increases, ultimately vanishing for infinitely large systems. On the other hand, if the electronic states are extended, ξ → ∞, then even in the infinite system-size limit, G = 0. This intuitive picture is at the very heart of the scaling hypothesis which distinguishes between metallic and insulating states on the basis of the range of ξ: If the electronic states at the chemical potential µ are extended, then the system is a metal, but if they have a finite extent, the system is an insulator. In other words, the metallic state is defined by σ independent of system dimensions, whereas the insulating state is characterized by σ decaying with increasing system dimensions. This underlies the Anderson metal-to-insulator transition in which a "mobility-edge" in wave vector k-space demarcates short-ranged and long-ranged states [2].However, since the scaling hypothesis was put forward, to our knowledge there have been no experimental reports of length-dependent σ. In this paper, working with mesoscopic GaAs-based 2DEGs of varying linear extent L, we provide the experimental demonstration of σ-scaling consistent with the scaling hypothesis. We continuously tune ξ in the 2DEGs by applying a top-gate voltage V G and observe a crossover from a regime in which the electrical resistivity ρ ≡ 1/σ is independent of L to one where it is strongly dep...

An electronic avalanche model for metal–insulator transition in two dimensional electron gas

“…For the latter case however, the picture can be semi-classical, since quantum fluctuations in this scale become weak, so that one can use the classical Boltzmann transport equation and particle transport can be treated semi-classically. This picture is compatible with experimental observations in [19] in which it was proposed that the transport is different for scales L l φ and L l φ to explain the power-law behavior σ ∼ L α L with some α L exponent. In this approach one subdivides the system into many cells with the linear sizes ∆L ∼ l φ .…”

“…In the first scale the electrons retain their quantum phase, whereas for the latter case, the picture can be semi-classical, since quantum fluctuations in this scale do not play a vital role and one can use the classical Boltzmann transport equation [18]. This approach has been proved to be useful in many situations and physical interpretation of some phenomenon, like the interpretation of finite-size power-law conductivity of 2DEG [19], the self-averaging [20], and the percolation prescription of 2DEG [21] each of which considers the linear size ∆L ∼ l φ as an important spatial scale. We have treated the electron gas inside these cells purely quantum mechanically, but for the transport of the particles to the neighboring cells some semi-classical rules have been developed.…”

Flicker Noise in Two-Dimensional Electron Gas

“…the quantum fluctuations do not play a vital role. This approach has been proved to be very useful for the interpretation of finite-size power-law conductivity of 2DEG [39], the self-averaging [40] and percolation prescription of 2DEG [41], each of which considers the linear size ∆L ∼ ζ φ as an important spatial scale [3].…”

Some Properties of Sandpile Models as Prototype of Self-Organized Critical Systems

“…In the first scale the electrons retain their quantum phase, whereas, for the latter case, the picture can be semi-classical, since quantum fluctuations in this scale do not play a vital role and one can use the classical Boltzmann transport equation [191]. This approach has been proved to be useful in many situations and physical interpretation of some phenomena, like the interpretation of finite-size power-law conductivity of 2DEG [192], the self-averaging [193], and the percolation prescription of 2DEG [194] each of which considers the linear size ∆L ∼ l φ as an important spatial scale. We have treated the electron gas inside these cells purely quantum mechanically, but for the transport of the particles to the neighboring cells some semi-classical rules have been developed.…”