homework-semiconductors-II.md

homework-semiconductors-II.md

Problem 8

The ${200}$ set of wavefunctions are $\phi = \exp(i 2\pi/a (\pm 2x))$, $\exp(i 2\pi/a (\pm 2y))$, and $\exp(i 2\pi/a (\pm 2z))$. They can be represented as

$$ \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} \begin{pmatrix} -1 \ 0 \ 0 \end{pmatrix} \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \begin{pmatrix} 0 \ -1 \ 0 \end{pmatrix} \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix} \begin{pmatrix} 0 \ 0 \ -1 \end{pmatrix}, $$ treating the magnitude of each as 1.

The classes are ${E}, {3C_2}, {6S_4}, {6\sigma}, {8C_3}$.

Consider ${6\sigma}$. The typical symmetry operation $(xyz) \rightarrow (yxz)$ from reflection on $(1\bar{1}0)$ in its 3-dim Cartesian form is $\begin{bmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix}$, and it only two wavefunctions remain unchanged, namely $\begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix}$ and $\begin{pmatrix} 0 \ 0 \ -1 \end{pmatrix}$. So the character of this class for the 6-dim representation of the ${200}$ set of wavefunctions is 2.

${3C_2}$ has the form $\begin{bmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & -1 \end{bmatrix}$, and two wavefunctions remain the unchanged, $\begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} \begin{pmatrix} -1 \ 0 \ 0 \end{pmatrix}$. The character is 2.

${6S_4}$ and ${8C_3}$ both give 0. Obviously ${E}$ gives 6.

So the characters for the 6-dim representation of the ${200}$ set of wavefunctions are: $$ \begin{array}{c|ccccc} \textrm{Class} & {E} & {3C_2} & {6S_4} & {6\sigma} & {8C_3} \ \hline \textrm{Characters} & 6 & 2 & 0 & 2 & 0 \end{array} $$

Problem 9 (changed to ${200}$)

Instead of using orthogonality relations to pry out the linear coefficients of the irreducible representations one by one, we can use linear algebra to systematically solve for them. Here is the character table: $$ \begin{array}{c|ccccc} & {E} & {3C_2} & {6S_4} & {6\sigma} & {8C_3} \ \hline A_1 (\Gamma_1) & 1 & 1 & 1 & 1 & 1 \ A_2 (\Gamma_2) & 1 & 1 & -1 & -1 & 1 \ E (\Gamma_3) & 2 & 2 & 0 & 0 & -1 \ T_1 (\Gamma_5) & 3 & -1 & 1 & -1 & 0 \ T_2 (\Gamma_4) & 3 & -1 & -1 & 1 & 0 \ \hline R & 6 & 2 & 0 & 2 & 0 \ \end{array} $$

We are solving for $R = \sum_{j=1}^5 m_j \Gamma_j$. Therefore a typical $\mathbf{Ax} = \mathbf{b}$ form can be constructed. $$ \begin{bmatrix} 1 & 1 & 2 & 3 & 3 \ 1 & 1 & 2 & -1 & -1 \ 1 & -1 & 0 & 1 & -1 \ 1 & -1 & 0 & -1 & 1 \ 1 & 1 & -1 & 0 & 0 \end{bmatrix} \begin{bmatrix} m_1 \ m_2 \ m_3 \ m_4 \ m_5 \end{bmatrix} = \begin{bmatrix} 6 \ 2 \ 0 \ 2 \ 0 \end{bmatrix} $$

As it turns out, $[m_1, m_2, m_3, m_4, m_5] = [1, 0, 1, 0, 1]$, and $R = \Gamma_1 + \Gamma_3 + \Gamma_4$.

Problem 10

We've already been over the two $\Gamma_1$ and first set of three $\Gamma_4$ wavefunctions. The last set of three $\Gamma_4$ wavefunctions are $\sqrt 8 \times$ $$ \phi_1 = \sin(2\pi x/a) \cos(2\pi y/a) \cos(2\pi z/a); \ \phi_2 = \cos(2\pi x/a) \sin(2\pi y/a) \cos(2\pi z/a); \ \phi_3 = \cos(2\pi x/a) \cos(2\pi y/a) \sin(2\pi z/a). $$

The classes are ${E}, {3C_2}, {6S_4}, {6\sigma}, {8C_3}$.

Obviously the identity operation ${E}$ leads to a character of 3.

Consider ${3C_2}$. A typical operation is $(xyz) \rightarrow (x\bar{y}\bar{z})$ from rotation ccw about $[100]$. It leaves $\phi_1$ unchanged, but $\phi_2$ and $\phi_3$ changing sign. The character is $1 + (-1) + (-1) = -1$.

Consider ${6S_4}$. A typical operation is $(xyz) \rightarrow (\bar{x}z\bar{y})$ from four-fold ccw rotation with reflection from $(100)$. It leaves $\phi_1$ changing sign, $\phi_2$ becoming $phi_3$, and $\phi_3$ becoming $-\phi_2$. The character is $-1$.

Consider ${6\sigma}$. A typical symmetry operation is $(xyz) \rightarrow (yxz)$ from reflection on $(1\bar{1}0)$. It leaves $\phi_1$ becoming $\phi_2$, $\phi_2$ becoming $\phi_1$, and $\phi_3$ unchanged. So the character is $1$.

Consider ${8C_3}$. A typical operation is $(xyz) \rightarrow (yzx)$ from three-fold ccw rotation about $[111]$. It results in a cycle of $\phi_1$, $\phi_2$ and $\phi_3$. So the character is $0$.

So the characters for the 6-dim representation of the ${200}$ set of wavefunctions are: $$ \begin{array}{c|ccccc} \textrm{Class} & {E} & {3C_2} & {6S_4} & {6\sigma} & {8C_3} \ \hline \textrm{Characters} & 3 & -1 & -1 & 1 & 0 \end{array} $$ as expected.