Pho.rs: Semiconductor band gap

A1 ^?? Substitute $\Psi(x) = e^{ikx/a} u(x/a)$ into the Schrödinger equation and get the equation
\[ u'' + 2ik u'+\left[ \frac{2ma^2}{\hbar^2} \left( E- U(x) \right) - k^2 \right] u = 0\]on $u(y)$.

A2 ^?? Consider a linear differential equation with constant coefficients $\alpha$ and $\beta$:
\[ u'' + 2i\alpha u' +(\beta-\alpha^2) u = 0.\]Find the values of $\lambda_{1,2}$ for which the function $u_{1,2}(y) = e^{\lambda_{1,2} y}$ is a solution to this equation. Check that for any constants $A$ and $B$, the function $u(y) = A e^{\lambda_1 y} + B e^{\lambda_2 y}$ is also a solution to this equation.

Ответ: \[\lambda_1 = -i\alpha + \sqrt{-\beta}, \quad \lambda_2 = -i\alpha - \sqrt{-\beta}\]

A3 ^?? Write a system of equations for the coefficients $A_1$, $A_2$, $B_1$ and $B_2$ corresponding to the requirements on the functions $u_I(x)$ and $u_II(x)$, in matrix form
\[ M \begin{pmatrix}
A_1 \\
A_2 \\
B_1 \\
B_2
\end{pmatrix} =
\begin{pmatrix}
M_{11} & M_{12} & M_{13} & M_{14} \\
M_{21} & M_{22} & M_{23} & M_{24} \\
M_{31} & M_{32} & M_{33} & M_{34} \\
M_{41} & M_{42} & M_{43} & M_{44}
\end{pmatrix}
\begin{pmatrix}
A_1 \\
A_2 \\
B_1 \\
B_2
\end{pmatrix} = 0.\]
Write the coefficients of the matrix $M_{ij}$ by $\lambda_{11}$, $\lambda_{12}$, $\lambda_{21}$, $\lambda_{22}$, $a$, and $b$.

Ответ: \[M=\begin{pmatrix}
1 & -1& 1 & -1\\
\lambda_{11} & -\lambda_{21} & \lambda_{12} & -\lambda_{22} \\
e^{-\lambda_{11}b/a} & -e^{\lambda_{21}(1-b/a)} & e^{-\lambda_{12}b/a} & -e^{\lambda_{22}(1-b/a)} \\
\lambda_{11}e^{-\lambda_{11}b/a} & -\lambda_{21}e^{\lambda_{21}(1-b/a)} & \lambda_{12}e^{-\lambda_{12}b/a} & -\lambda_{22}e^{\lambda_{22}(1-b/a)} \\
\end{pmatrix}
\]

A5 ^?? Show that the de Broglie wave $\Psi(x) = e ^{ikx/a}$ has the energy $E=\frac{\hbar^2 k^2}{2m a^2}$.

A6 ^?? Express $E_F$ in terms of $L$, $N$, $m$, and $\hbar$.

Ответ: \[k_\text{max}=\pi N a / 2L\]\[ E_F = \frac{\pi^2 \hbar^2 N^2}{8m L^2} \]

A7 ^?? Use the program energy.py to obtain the dependence of the band gap $E_g$, where the Fermi level is located, on the parameter $k_0$ at $b/a=0.1$. Consider $a=0.5~\text{nm}$, electron mass $m=9.1 \cdot 10^{-31}~\text{kg}$, Planck constant $\hbar=1.05 \cdot 10^{-34}~\text{J} \cdot \text{s}$, elementary charge $e=1.60 \cdot 10^{-19}~\text{C}$.
Fill in the table.txt table and use the program bandgap.py to draw a graph of this relationship. In table.txt table, the separator between columns is tab.

A8 ^?? What $k_0$ corresponds to the forbidden zone of silicon $E_{\rm Si} = 1.12~\text{eV}$? Do all further calculations for this $k_0$.

\[k_0=1.9\]

A9 ^?? Use the program energy.py to find the values of $E_c-E_F$, $E_F-E_v$, $\alpha$, and $\beta$. Check for yourself that $E_c - E_v = E_{\rm Si}$. Also find the values of $(E_c - E_F)/k_\text{B}T$ and $(E_F - E_v)/k_\text{B}T$ for $T=300~\text{K}$.

Ответ: \[ E_c = E_F + 0.71~\text{eV}, \quad \alpha = 1.99~\text{eV}, \quad \frac{E_c - E_F}{k_\text{B}T}=27\]\[ E_v = E_F - 0.48~\text{eV}, \quad \beta = 1.68~\text{eV}, \quad \frac{E_F - E_v}{k_\text{B}T}=18\]

A10 ^?? Show that the total number $p$ of NOT-filled states in the valence band is
\[p = \frac{L}{a} e^{\frac{E_v - E_F}{k_\text{B}T}} \sqrt{\frac{k_B T}{\pi\beta}}.\]
This number $p$ is the number of holes involved in the transfer processes.

Show that $p \cdot n$ does not depend on the position of the Fermi level $E_F$. Find the value of $n_i= \sqrt{p \cdot n} \cdot a / L$ for silicon at temperature $T=300~\text{K}$.

Ответ: \[pn = \frac{L^2}{a^2} e^{-\frac{E_g}{k_\text{B}T}} \frac{k_\text{B} T}{\pi\sqrt{\alpha \beta}} \]\[ n_i = \sqrt{pn} \cdot a/L = 2.7 \cdot 10^{-11}\]

A11 ^?? What should the Fermi level $E_F$ be so that the crystal is electrically neutral, i.e., the number of holes $p$ is equal to the number of electrons $n$?

Ответ: \[E_F = \frac{E_v + E_c}{2} - \frac{k_\text{B}T}{2} \ln \sqrt{\frac{\beta}{\alpha}} \]

A12 ^?? What fraction $c$ of silicon $\rm Si$ atoms must be replaced by aluminum $\rm Al$ to increase the ratio of holes to electrons $p/n$ to $q=10^6$ with respect to pure silicon? Consider the temperature to be $300~\text{K}$.

In pure silicon $p=n$, therefore in silicon with impurities $p = qn$. Changing of aroms only shifts the Fermi level, so $\sqrt{pn} =n_i$ and
\[ p =n_i \sqrt{q}, \quad n = \frac{n_i}{\sqrt{q}}.\]The condition of electroneutrality is
\[ p - \frac{cL}{a} = n.\]Then, answer is
\[ c = \left( \sqrt{q} + \frac{1}{\sqrt{q}} \right) \cdot 2.6 \cdot 10^{-11} \simeq 2.6 \cdot 10^{-8}. \]

Semiconductor band gap

Решение