Barrier Tunneling — Question 1

Question 1

Exact barrier transmission is determined by both decay and boundary matching

An electron of energy $E=2.00\,\mathrm{eV}$ encounters a barrier with $V_0=5.00\,\mathrm{eV}$ and $a=.300\,\mathrm{nm}$. 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 $T=[1+V_0^2\sinh^2(\kappa a)/(4E(V_0-E))]^{-1}$. For electrons use $\hbar^2/(2m_e)=.0381\,\mathrm{eV\,nm^2}$.

1. Find $\kappa$ and the dimensionless barrier thickness.

2. Calculate the exact transmission and reflection probabilities.

Question 1 - Solution

Exact barrier transmission is determined by both decay and boundary matching

(a) Method and calculation

$$\boxed{\kappa=\sqrt{\frac{3.00}{.0381}}\,\mathrm{nm}^{-1}=8.87357\,\mathrm{nm}^{-1}}.$$$$\boxed{\kappa a=2.66207}.$$

(b) Interpretation and further calculation

$$T=\left[1+\frac{25\sinh^2(2.6620695)}{24}\right]^{-1}\quad\Rightarrow\quad\boxed{T=.0185439}.$$$$\boxed{R=1-T=.981456}.$$

Physical check

Transmission is a ratio of probability currents. For identical exterior regions it equals the squared transmitted amplitude relative to unit incident amplitude.