Calculation of effective mass from bandstructure

Question

The effective mass is defined as $$ \frac{1}{m_{ij}^*} = \frac{1}{\hbar^2} \frac{\partial^2\epsilon}{\partial k_i \partial k_j} $$

where, $m_{ij}^*$ is the effective mass, $\hbar$ is the Planck's constant, $\epsilon$ is the energy and $k_i,\ k_j$ are reciprocal latttice vectors.

Now let us consider that we have the values of energy corresponding to the points in a line connecting the center of Brillioun zone and a point in the reciprocal space (say $k_x = 0.5$, $k_y = 0.5$, $k_z = 0.5$). Now I would like to know if it is possible to get value of effective mass from this data. If yes how can I do that?

Answer 1

From your data, it seems like you can obtain the longitudinal effective mass in that direction. By the way, that line in the reciprocal space, the [111] direction, is often called the $\Lambda$ axis.

I suppose that if you represent the energy as a function of $|\vec{k}|$ in that direction there will be a minimum or a maximum. You could try then to fit the data around the minimum or maximum to a parabola $\varepsilon(k)=Ak^2+Bk+C$ and from there $$\frac{1}{m^*_l}=\frac{1}{\hbar^2}\frac{d^2\varepsilon}{dk^2}=\frac{2A}{\hbar^2}$$