Question #43899

In a solid-state device, minority charge carriers are modeled as a 1D classical ideal gas in thermal equilibrium at temperature $T$. The probability density function for a carrier's kinetic energy $E$ is $f(E) = \frac{1}{k_B T} e^{-E/k_B T}$ for $E \ge 0$, where $k_B$ is the Boltzmann constant.

A carrier successfully tunnels through a junction if its kinetic energy $E$ exceeds a dynamic potential barrier $U(\theta)$. The barrier height depends on the instantaneous bending angle $\theta$ of a lattice bond, given by $U(\theta) = U_0 + \Gamma(1 - \cos\theta)$, where $U_0$ and $\Gamma$ are positive constants with units of energy.

The lattice bond itself is a classical system in thermal equilibrium at the same temperature $T$, with its own potential energy given by $V(\theta) = \Gamma(1 - \cos\theta)$.

Given that $\Gamma \gg k_B T$, find the overall probability $P$ that a randomly selected carrier has enough energy to overcome the barrier at a random instant in time.

(Assume the small angle approximation is valid for the thermal distribution of $\theta$, and use $\int_{-\infty}^{\infty} e^{-a x^2} dx = \sqrt{\frac{\pi}{a}}$).