Decay of metastable current states in one-dimensional resonant tunneling devices

Current switching in a double-barrier resonant tunneling structure is studied in the regime where the current-voltage characteristic exhibits intrinsic bistability, so that in a certain range of bias two different steady states of current are possible. Near the upper boundary $V_{\mathit{th}}$ of the bistable region the upper current state is metastable, and because of the shot noise it eventually decays to the stable lower current state. We find the time of this switching process in strip-shaped devices, with the width small compared to the length. As the bias $V$ is tuned away from the boundary value $V_{\mathit{th}}$ of the bistable region, the mean switching time $\tau$ increases exponentially. We show that in long strips $\ln \tau \propto {(V_{\mathit{th}}-V)}^{5∕4}$, whereas in short strips $\ln \tau \propto {(V_{\mathit{th}}-V)}^{3∕2}$. The one-dimensional geometry of the problem enables us to obtain analytically exact expressions for both the exponential and the prefactor of $\tau$. Furthermore, we show that, depending on the parameters of the system, the switching can be initiated either inside the strip, or at its ends.

Figure 1

Current-voltage characteristic of the DBRTS. The bistable region is present in the range of bias between ${\tilde{V}}_{\mathit{th}}$ and $V_{\mathit{th}}$. The inset shows a sketch of generic potential $u\left(n\right)$ in the bistable region. The points $\mathit{A}$ and $\mathit{C}$ on the $I\text{‐}V$ curve correspond to the local and global minima of $u\left(n\right)$, respectively. The point $\mathit{B}$ on the unstable current branch corresponds to the maximum of $u$.

Figure 2

(a) Saddle point solution $z_s\left(\xi \right)$ for strips of length $\Lambda =\pi$, 3.2, 3.5, 4, 5, and ∞. The allowed motion of a classical particle in the potential $u\left(z\right)=-z^2∕2+z^3∕3$ is shown in the inset. (b) The dependence $I\left(\Lambda \right)$ defined by Eq. (13). At $\Lambda <\pi$ it is linear: $I\left(\Lambda \right)=\Lambda ∕6$; while for $\Lambda >\pi$ it is described by Eq. (14). At $\Lambda \to \infty$ the saddle point $z_s\left(\xi \right)$ is given by Eq. (11), and $I=3∕5$.

Figure 3

Prefactor ${\tau }_s^{*}$ of the mean switching time in units ${\tau }_0^{*}$ vs the dimensionless length of the strip $\Lambda$. At $\Lambda <\pi$ the prefactor is given by Eq. (37), whereas above $\pi$ it is calculated numerically. At $\Lambda \to \infty$ the prefactor approaches $\sqrt{2}∕5$ in agreement with Eq. (36).

Figure 4

Prefactor ${\tau }_r^{*}$ of the mean switching time in units ${\tau }_0^{*}∕\sqrt{U_1}$ vs the dimensionless circumference of the ring $2\Lambda$ above $\Lambda =\pi$. The dotted line shows the asymptote $\sqrt{\pi }∕15\Lambda$ of ${\tau }_r^{*}$ at $\Lambda \to \infty$, see Eq. (45).

Figure 5

(a) Logarithm of the mean switching time $\tau$ vs voltage at $L⪡d$. Near the threshold there is a region of $3∕2$-power law dependence of $\ln \tau$ (regime of short strip), then follows the region of $5∕4$-dependence corresponding to the switching at the ends in the regime of long strip. (b) Logarithm of the switching time $\tau$ vs voltage at $L⪢d$. Close to the threshold one first observes $5∕4$-power law behavior corresponding to the interior switching, then at voltage below $V_1^{*}$ follows the region of $5∕4$-dependence corresponding to switching at the ends.