Two-dimensional quantum confinement

The considerations in the previous section, similarly apply when quantum confinement is predominant in the $y-z$ plane. The two-dimensional Schrödinger equation reads,

$${-\left[\frac{\hbar^2}{2}\frac{\partial}{\partial y}\frac{1}{m_y}... ...al z}\frac{1}{m_z}\frac{\partial}{\partial z}\right]\chi_i} = E_{i,k_x} \chi_i.$$

while the three-dimensional eigenfunctions can be expressed as

$$\Psi_{k_x,i}=\frac{e^{jk_xx}}{\sqrt{L_x}}\chi_i(y,z)$$ (29)

where $\chi_i(y,z)$ are the eigenfunctions associated to the energy level $\epsilon_i(x)$ and the total energy reads

$$E_{i,k_x}=\epsilon_i(x)+\frac{\hbar^2k_x^2}{2m_x}$$ (30)

The electron concentration is

$$n(x,y,z)=\sum_i \vert\chi_i(x,y,z)\vert^2 \alpha_i$$ (31)

where

$$\alpha_i=\sum_{k_x}\frac{f_0(E_{i,k_x})}{L_x}$$ (32)

Using (4),

$$\alpha_i=2\frac{L_x}{2\pi}\int_{-\infty}^{+\infty} \frac{f_... ...rac{2}{\pi}\int_{0}^{+\infty} \frac{f_0(E_{i,k_x})}{L_x} dk_x$$ (33)

Expressing $k_x$ as

$$k_x=\frac{\sqrt{2m_x(E-\epsilon_i)}}{\hbar}$$ (34)

and deriving with respect to $E$ we obtain

$$dk_x=\frac{\sqrt{2m_x}}{2\hbar \sqrt{E-\epsilon_i}}dE$$ (35)

Substituting (37) in (35), we obtain

$$\alpha_i=\frac{2m_x}{\pi \hbar}\int_{\epsilon_i}^\infty \frac{1}{\sqrt{(E-\epsilon_i)}} \frac{1}{1+e^\frac{E-E_F}{K_BT}}dE$$ (36)

and multiplying and dividing by $\sqrt{K_BT}$, we obtain the final expression for the electron density

$$n(y,z)=\frac{1}{\pi}\left(\frac{2m_xK_BT}{\hbar^2} \right)^{... ...F_{-\frac{1}{2}} \left(\frac{E_F-\epsilon_i(x)} {K_BT} \right)$$ (37)

where $F_{-\frac{1}{2}}$ is the Fermi integral of order $-\frac{1}{2}$ :.

$$F_{-\frac{1}{2}}(y)=\int_0^\infty \frac{x^{-\frac{1}{2}}}{1+e^{x-y}}dx$$ (38)