Question 28.12: Transmission Coefficient for an Electron A 30-eV electron is...

A:

Let us assume that the probability of transmission is low so that we can use the approximation in Equation 28.37. For the given barrier height and electron energy, the quantity U – E has the value

U-E=(40 eV -30 eV )=10 eV =1.6 \times 10^{-18} J

T \approx e^{-2 C L}                       [28.37]

Using Equation 28.38, the quantity 2CL is

\begin{aligned}2 C L &=2 \frac{\sqrt{2\left(9.11 \times 10^{-31} kg \right)\left(1.6 \times 10^{-18} J \right)}}{1.054 \times 10^{-34} J \cdot s }\left(1.0 \times 10^{-9} m \right) \\&=32.4\end{aligned}

C=\frac{\sqrt{2 m(U-E)}}{\hbar}                   [28.38]

Therefore, the probability of tunneling through the bar

T \approx e^{-2 C L}=e^{-32.4}=8.5 \times 10^{-15}

That is, the electron has only about 1 chance in 10^{14} to tunnel through the 1.0-nm-wide barrier.

B:

For L = 0.10 nm, we find 2CL = 3.24, and

T \approx e^{-2 C L}=e^{-3.24}=0.039

This result shows that the electron has a relatively high probability, about 4%, compared with 10^{-12} % in part A, of penetrating the 0.10-nm barrier. Notice an important behavior that leads to effective practical applications for tunneling: that reducing the width of the barrier by only one order of magnitude increases the probability of tunneling by about 12 orders of magnitude!

Physics Now™ Investigate the tunneling of particles through barriers by logging into PhysicsNow at www.pop4e.com and going to Interactive Example 28.12.