Barrier Tunneling — Question 8

Question 8

Above-barrier reflection vanishes at discrete transmission resonances

For a rectangular barrier with $E>V_0$, use $q=\sqrt{2m(E-V_0)}/\hbar$ and $T=[1+V_0^2\sin^2(qa)/(4E(E-V_0))]^{-1}$. An electron encounters $V_0=5.00\,\mathrm{eV}$ and $a=.300\,\mathrm{nm}$. For electrons use $\hbar^2/(2m_e)=.0381\,\mathrm{eV\,nm^2}$.

1. Derive the energies of unity transmission with positive integer n.

2. Calculate the lowest such energy and explain why the formal choice $n=0$ must be excluded.

Question 8 - Solution

Above-barrier reflection vanishes at discrete transmission resonances

(a) Method and calculation

$$\sin(qa)=0,\quad qa=n\pi\quad\Rightarrow\quad\boxed{E_n=V_0+\frac{n^2\pi^2\hbar^2}{2ma^2}},\quad n=1,2,\ldots.$$$$\boxed{T(E_n)=1}\quad\text{for every positive integer }n.$$

(b) Interpretation and further calculation

$$\boxed{E_1=5+\frac{\pi^2(.0381)}{(.300)^2}=9.17813\,\mathrm{eV}}.$$$$n=0:\ E=V_0,\quad\boxed{T=\left(1+\frac{mV_0a^2}{2\hbar^2}\right)^{-1}<1}.$$

Physical check

At the barrier top both the sine factor and its energy denominator vanish, so direct substitution as though the denominator were finite is invalid.