The band dispersion of electrons in a two dimensional square lattice (lattice constant a) is given by,
\(E\left(k_x, k_y\right)=-2\left(t_x \cos k_x a+t_y \cos k_y a\right) \)
where tx, ty > 0. The effective mass tensor \(m^*=\left(\begin{array}{ll}m_{x x} & m_{x y} \\ m_{y x} & m_{y y}\end{array}\right)\) of electrons at \(\vec{k}=\left(\frac{\pi}{a}, \frac{\pi}{a}\right)\) is
1
\(\left(\begin{array}{cc}0 & \frac{\hbar^2}{2 a^2 \sqrt{t_x t_y}} \\ \frac{\hbar^2}{2 a^2 \sqrt{t_x t_y}} & 0\end{array}\right)\)
2
\(\left(\begin{array}{cc}\frac{\hbar^2}{2 a^2 t_x} & 0 \\ 0 & \frac{\hbar^2}{2 a^2 t_y}\end{array}\right)\)
3
\(\left(\begin{array}{cc}-\frac{\hbar^2}{2 a^2 t_x} & 0 \\ 0 & -\frac{\hbar^2}{2 a^2 t_y}\end{array}\right)\)
4
\(\left(\begin{array}{cc} 0 & -\frac{\hbar^2}{2 a^2\left(t_x+t_y\right)} \\ -\frac{\hbar^2}{2 a^2\left(t_x+t_y\right)} & 0 \end{array}\right)\)