Two- and Three-Dimensional Electron Traps — Question 7

Question 7

Strong confinement in one direction freezes its excitation at low energies

In a three-dimensional rectangular infinite well, $E=C(n_x^2/L_x^2+n_y^2/L_y^2+n_z^2/L_z^2)$, with positive integer indices and $C=\pi^2\hbar^2/(2m)$. Ignore spin. Let $L_x=L_y=L$ and $L_z=L/10$. Define $A=C/L^2$. Only excitations at most $20A$ above the ground energy are accessible.

1. Find the ground energy and the lowest excitation gap along each axis.

2. Determine which z mode is accessible and write the effective low-energy two-dimensional spectrum.

Question 7 - Solution

Strong confinement in one direction freezes its excitation at low energies

(a) Method and calculation

$$\boxed{E_{111}=A(1+1+100)=102A}.$$$$\boxed{\Delta E_x=\Delta E_y=3A,\qquad\Delta E_z=300A}.$$

(b) Interpretation and further calculation

$$20A<300A\Rightarrow\boxed{n_z=1\text{ throughout the accessible subspace}}.$$$$\boxed{E_{n_xn_y,1}=100A+A(n_x^2+n_y^2)},\quad n_x^2+n_y^2\le22.$$

Physical check

The frozen direction still contributes a large zero-point energy. Effective two-dimensional motion does not mean that this energy disappears.