Quantum Tunneling Question

A 50-eV electron approaches a square barrier 70 eV high and (a) 1.0 nm thick, (b) 0.1 nm thick. What is the probability that the electron will tunnel through?

SOLUTION (a) Inside the barrier $U_0-E=(70\mathrm{eV}-50\mathrm{eV})\left(1.6\times 10^{-19}\mathrm{J}/\mathrm{eV}\right)=3.2\times 10^{-18}\mathrm{J}$. Then, using Eqs. 38-17, we have $2G\ell =2\sqrt{\frac{2\left(9.11\times 10^{-31}\mathrm{kg}\right)\left(3.2\times 10^{-18}\mathrm{J}\right)}{{\left(1.055\times 10^{-34}\mathrm{J}\cdot \mathrm{s}\right)}^2}}\left(1.0\times 10^{-9}\mathrm{m}\right)=46$ and $T=e^{-2Gt}=e^{-46}\approx 1\times 10^{-20},$ which is extremely small. (b) For $\ell =0.10\mathrm{nm},2G\ell =4.6$ and $T=e^{-4.6}=\mathrm{0.010.}$ Thus the electron has a $1\%$ chance of penetrating a 0.1 -nm-thick barrier, but only 1 chance in $10^{20}$ to penetrate a $1-\mathrm{nm}$ barrier. By reducing the barrier thickness by a factor of 10 , the probability of tunneling through increases $10^{18}$ times! Clearly the transmission coefficient is extremely sensitive to the values of $\ell$, $U_0-E$, and $m$.