2.12: The 2-D Density of States - Quantum Wells Confined in 1-D
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We showed above that the energy of electrons in a quantum well is
\[ E = \frac{\hbar^{2}\pi^{2}}{2mL_{z}^{2}}n^{2}+\frac{\hbar^{2}(k_{x}^{2}+k_{y}^{2})}{2m},\ n=1,2, \dots \nonumber \]
For the DOS calculation, the specifics of the confining potential are irrelevant; we note only that the electron is unconfined in two dimensions. If the quantum well has area \(L_{x}\times L_{y}\) then each allowed value of k-space occupies an area of \(2\pi/L_{x} \times 2\pi/L_{y}\).
It is convenient to convert to cylindrical coordinates \((k,\phi,z)\) where k is the magnitude of the k-vector in the x-y plane. The number of states within a ring of thickness dk is then
\[ n_{s}(k)dk=2 \times \frac{1}{4\pi^{2}/A}\times 2\pi kdk \nonumber \]
where \(A=L_{x} \times L_{y}\), and again we have multiplied by two to account for the electron spin.
Now k is related to the energy by
\[ E-E_{n}=\frac{\hbar ^{2}k^{2}}{2m},\ E\geq E_{n} \nonumber \]
Thus, from Equation 2.12.3,
\[ g(E)dE = \frac{Am}{\pi \hbar^{2}}\sum_{n}u(E-E_{n})dE , \nonumber \]
where u is the unit step function. The DOS is plotted in Figure 2.12.2.