7.6 The quantum tunneling of particles through potential barriers  (Page 4/22)

The two types of solutions in the three regions are illustrated in [link] .

Now we use the boundary conditions to find equations for the unknown constants. [link] and [link] are substituted into [link] to give

[link] and [link] are substituted into [link] to give

Similarly, we substitute [link] and [link] into [link] , differentiate, and obtain

Similarly, the boundary condition [link] reads explicitly

We now have four equations for five unknown constants. However, because the quantity we are after is the transmission coefficient, defined in [link] by the fraction F / A , the number of equations is exactly right because when we divide each of the above equations by A , we end up having only four unknown fractions: B / A , C / A , D / A , and F / A , three of which can be eliminated to find F / A . The actual algebra that leads to expression for F / A is pretty lengthy, but it can be done either by hand or with a help of computer software. The end result is

In deriving [link] , to avoid the clutter, we use the substitutions $\gamma \equiv \beta \text{/}k-k\text{/}\beta$ ,

We substitute [link] into [link] and obtain the exact expression for the transmission coefficient for the barrier,

or

where

For a wide and high barrier that transmits poorly, [link] can be approximated by

Whether it is the exact expression [link] or the approximate expression [link] , we see that the tunneling effect very strongly depends on the width L of the potential barrier. In the laboratory, we can adjust both the potential height $U_0$ and the width L to design nano-devices with desirable transmission coefficients.

Transmission coefficient

Strategy

Solution

For a lower-energy electron with $E_1=7.00\text{eV}$ :