Stochastic current switching in bistable resonant tunneling systems

Current-voltage characteristics of resonant-tunneling structures often exhibit intrinsic bistabilities. In the bistable region of the $I\text{−}V$ curve one of the two current states is metastable. The system switches from the metastable state to the stable one at a random moment in time. The mean switching time $\tau$ depends exponentially on the bias measured from the boundary of the bistable region $V_{\mathrm{th}}$. We find full expressions for $\tau$ (including prefactors) as functions of bias, sample geometry, and in-plane conductivity. Our results take universal form upon appropriate renormalization of the threshold voltage $V_{\mathrm{th}}$. We also show that in large samples the switching initiates inside, at the edge, or at a corner of the sample depending on the parameters of the system.

Figure 1

The $I\text{−}V$ curve of the DBRTS. The bistable region is present in the range of bias between ${\tilde{V}}_{th}$ and $V_{th}$. The bold dashed line corresponds to the unstable current state.

Figure 2

Schematic potential profile of the double-barrier resonant tunneling structure. The structure consists of a quantum well separated from two leads by tunneling barriers. The electrons with three-dimensional wave vectors $\mathbf{k}$ and $\mathbf{p}$ fill all the states up to the Fermi energies $E_F$ in the left and right leads, respectively. In the quantum well the motion of electrons in the $z$ direction is quantized, and the electrons with two-dimensional transverse wave vectors $\mathbf{q}$ occupy all the states up to the Fermi energy $E_F^w$. The inset shows the potential profile at zero bias.

Figure 3

(a) Generic behavior of $U\left(N\right)$ at different values of bias. Outside the bistable region $U\left(N\right)$ has one minimum (top curve). Inside the bistable region the function $U\left(N\right)$ has two minima and a maximum, which correspond to the locally stable current branches and the unstable branch, respectively (middle and bottom curves). (b) The sketch of $U\left(N\right)$ for the model of Fig. 1. Solid line corresponds to a bias slightly below $V_{th}$, whereas dashed line depicts $U\left(N\right)$ for the bias slightly above ${\tilde{V}}_{th}$.

Figure 4

The sketch of the density profile at the saddle point $z_s\left(\mathbf{\rho }\right)$ corresponding to the solution of Eq. (53) with the boundary conditions (54). The precise radial dependence $z_s\left(\rho \right)$ obtained by solving Eq. (55) numerically is shown in the inset.

Figure 5

Schematic dependence of the logarithm of the mean switching time $\tau$ on voltage. (a) In round samples at $L⪡d$, close to the threshold there is a region of $3∕2$-power law dependence of $\ln \tau$, followed by the region of linear dependence corresponding to the switching at the edge in the regime of the large sample. In samples with pronounced corners the region of linear dependence corresponds to the switching at the sharpest corner (dashed line). (b) At $L⪢d$, two regions of different linear behavior corresponding to the switching in the interior and at the edge of the large round sample are present. In samples with pronounced corners these two regions are followed by an additional region of linear voltage dependence corresponding to the switching at the corner of smallest angle $\theta$ shown by dashed line. The slope of this linear dependence is smaller by a factor of $\pi ∕\theta$ than that of edge switching.