Barrier Tunneling — Question 7

Question 7

Transmission remains finite when incident energy reaches the barrier top

Consider the limit $E\to V_0^-$ for a rectangular barrier with fixed finite width a. A rectangular barrier has $V=V_0$ for $0<x<a$ and zero outside, with equal mass on both sides. For $0<E<V_0$, use $\kappa=\sqrt{2m(V_0-E)}/\hbar$ and $T=[1+V_0^2\sinh^2(\kappa a)/(4E(V_0-E))]^{-1}$. Use $\sinh z\simeq z$ as $z\to0$. For electrons use $\hbar^2/(2m_e)=.0381\,\mathrm{eV\,nm^2}$.

1. Derive the limiting transmission at $E=V_0$.

2. Evaluate it for an electron with $V_0=5.00\,\mathrm{eV}$ and $a=.300\,\mathrm{nm}$.

Question 7 - Solution

Transmission remains finite when incident energy reaches the barrier top

(a) Method and calculation

$$\frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\longrightarrow\frac{V_0^2}{4E(V_0-E)}\frac{2m(V_0-E)a^2}{\hbar^2}.$$$$\boxed{T(E=V_0)=\left(1+\frac{mV_0a^2}{2\hbar^2}\right)^{-1}}.$$

(b) Interpretation and further calculation

$$\frac{m_eV_0a^2}{2\hbar^2}=\frac{V_0a^2}{4[\hbar^2/(2m_e)]}=\frac{5(.300)^2}{4(.0381)}.$$$$\boxed{T=.252988}.$$

Physical check

Zero decay constant does not make a finite barrier invisible. Boundary mismatch still produces reflection; taking the limit before canceling the vanishing factors would miss this result.