Electron-phonon decoupling in two dimensions

Abstract

In order to observe many-body localisation in electronic systems, decoupling from the lattice phonons is required, which is possible only in out-of-equilibrium systems. We show that such an electron-phonon decoupling may happen in suspended films and it manifests itself via a bistability in the electron temperature. By studying the electron-phonon cooling rate in disordered, suspended films with two-dimensional phonons, we derive the conditions needed for such a bistability, which can be observed experimentally through hysteretic jumps of several orders of magnitude in the nonlinear current-voltage characteristics. We demonstrate that such a regime is achievable in systems with an Arrhenius form of the equilibrium conductivity, while practically unreachable in materials with Mott or Efros-Shklovskii hopping.

Figure 1

(a) The bistability region, where two stable solutions for $t_{\text{el}}$ exist in a certain range of the applied voltage, is shown for $t_{\text{ph}}=0.75t_{\text{ph}}^{\text{cr}}$ for the Arrhenius equilibrium resistance, $\gamma =1$. The blue dots correspond to cold and hot electron states at temperatures $t_{\text{el}}^{<}$ and $t_{\text{el}}^{>}$, respectively, and the red dot to an unstable solution. (b) The dependence of $t_{\text{el}}$ on $v^2$ for different phonon bath temperatures is shown as follows from Eq. (7). Above the critical bath temperature this corresponds to the actual $t_{\text{el}}(v)$ dependence while below $t_{\text{ph}}^{\text{cr}}$ the electronic system will fall either to $t_{\text{el}}^{<}$ or to $t_{\text{el}}^{>}$, making temperatures in between experimentally inaccessible.

Figure 2

Dependence of the bistability boundaries on the phonon temperature for (a) the electron temperature and (b) the source-drain voltage, for $\gamma =1$. The region of electron temperatures inside the curve (a) is experimentally inaccessible as it corresponds to the unstable states.

Figure 3

(a) The S-shape solution to the equation for the non-linear conductance, Eq. (10), for $t_{\text{ph}}=0.75t_{\text{ph}}^{\text{cr}}$. The dotted part corresponds to unstable states, resulting in hysteretic jumps, denoted by the arrows. Note that the jumps do not necessarily occur at the boundaries of the bistability (dashed lines). (b) The numerically predicted I-V characteristics for various lattice temperatures. The jumps here are shown to be at the bistability boundaries, though this may not be the case in reality. The $V>0$ side of the graph illustrates the transition from the cold electron (low conductance) state to the hot electron (high conductance) state, which occurs when the source-drain voltage is increased. The $V<0$ side displays the opposite transition when the voltage is decreased, going from the hot to cold electron states. In both (a) and (b) the voltage is measured in units of $V_0$ and the current is in units such that the resistance is measured in units of $R_0$.

Figure 4

The two lowest-order diagrams that contribute to the collision integral in Eq. (12) due to the interaction of electrons with transverse phonons via impurity scattering: the smaller squares correspond to ${\mathit{g}}_{\mathit{k}}^{\text{imp}}$, the straight lines are the electron Green’s functions, the wavy lines are the phonon Green’s functions and the dashed lines describe the standard averaging over impurities.