Question

Exercise 39 gives a condition for resonant tunneling through two barriers separated by a space width of$\mathbf{2}\mathbf{s}$, expressed I terms of factor$\mathbf{\beta }$given in exercise 30. Show that in the limit in which barrier width$\mathbf{L}\mathbf{\to }\mathbf{\infty }$, this condition becomes exactly energy quantization condition (5.22) for finite well. Thus, resonant tunneling occurs at the quantized energies of intervening well.

Short answer

In the limit in which barrier width $\mathrm{L}\to \mathrm{\infty }$, this condition becomes exactly energy quantization condition (5-22) for finite well. Thus, resonant tunneling occurs at the quantized energies of intervening well.

Step-by-step solution

Concepts involved

Resonant tunneling is a phenomenon in which an electron enters from one side of a double barrier structure, and travels across it at or near the metastable levels in the quantum well.

Given parameters

The condition for resonant tunneling from the Exercise 39 is given by,

$2\mathrm{s}=\mathrm{\beta }/\mathrm{k}$

Where, $\mathrm{\beta }={\tan }^{-1}\left(\frac{2\mathrm{\alpha k}}{{\mathrm{k}}^2-{\mathrm{\alpha }}^2}\coth \left(\mathrm{\alpha L}\right)\right)$.

$k$= wave number

$L$ = barrier width

$\alpha$ =constant

Show that in the limit in which barrier width L→∞, this condition becomes exactly energy quantization condition

In the limit $\mathrm{L}\to \mathrm{\infty }$$\coth \to 1$ and so:

$\begin{array}{c}\mathrm{\beta }={\tan }^{-1}\left(\frac{2\mathrm{\alpha k}}{{\mathrm{k}}^2-{\mathrm{\alpha }}^2}\right)\\ 2\mathrm{sk}={\tan }^{-1}\left(\frac{2\mathrm{\alpha k}}{{\mathrm{k}}^2-{\mathrm{\alpha }}^2}\right)\\ \tan \left(2\mathrm{sk}\right)=\left(\frac{2\mathrm{\alpha k}}{{\mathrm{k}}^2-{\mathrm{\alpha }}^2}\right)\end{array}$

Rearranging, $2\cot \left(\mathrm{k}.2\mathrm{s}\right)=\frac{\mathrm{k}}{\mathrm{\alpha }}-\frac{\mathrm{\alpha }}{\mathrm{k}}$

2s is the distance between the barriers; this is same as the expression (5.22) which is the energy quantization condition.

Here, the condition of resonant tunneling is proved to be equal to the energy quantization condition, when$\mathrm{L}\to \mathrm{\infty }$ . Thus, resonant tunneling occurs at the quantized energies of intervening well.