2.4.2.1 Effective Mass Tensor

If the extremum of the band $n$ is located at the point $\vec{p}_{0}^{n}=\hbar\vec{k}_{0}^{n}$, the function $\epsilon_{n}(\vec{p})$ can be expanded into the Taylor series as follows^2.22

$$\epsilon_{n}(\vec{p})=\epsilon_{n}(\vec{p}_{0}^{n})+\frac{1}{2}m_{\alpha\beta}^{-1}(p_{\alpha}-p_{\alpha 0}^{n})(p_{\beta}-p_{\beta 0}^{n})+\cdots.$$ (2.73)

Here the linear terms vanish due to the definition of the point $\vec{p}_{0}^{n}$. The quantities $m_{\alpha\beta}^{-1}$ have the dimension of an inverse mass. They represent the second derivatives of the energy

$$m_{\alpha\beta}^{-1}=\frac{\partial^{2}\epsilon_{n}(\vec{p})}{\partial p_{\alpha}\partial p_{\beta}}\bigg\vert _{\vec{p}=\vec{p}_{0}^{n}}.$$ (2.74)

As the value of a second derivative does not depend on the differentiation order, quantities $m_{\alpha\beta}^{-1}$ represent a symmetric tensor of the second rank^2.23 called the inverse effective-mass tensor. The components of this tensor depend on the coordinate system in quasi-momentum space. In particular the coordinate system can be chosen so that the non-diagonal components vanish, that is, $m_{\alpha\beta}^{-1}=0$ for $\alpha\ne\beta$. This coordinate system is called a principle coordinate system. In the principle coordinate system (2.73) can be rewritten as:

$$\epsilon_{n}(\vec{p})=\epsilon_{n}(\vec{p}_{0}^{n})+\frac{1}{2}m_{\alpha}^{-1}(p_{\alpha}-p_{\alpha 0}^{n})^{2}.$$ (2.75)

The equation of constant surface is obtained from the condition $\epsilon_{n}(\vec{p})=$const using (2.75):

$$\frac{(p_{x}-p_{x,0})^{2}}{2m_{x}}+\frac{(p_{y}-p_{y,0})^{2}}{2m_{y}}+\frac{(p_{z}-p_{z,0})^{2}}{2m_{z}}= .$$ (2.76)

Thus the constant energy surface near a non-degenerate extremum represents an ellipsoid with the half-axes being proportional to $\sqrt{\vert m_{x}\vert}$, $\sqrt{\vert m_{y}\vert}$ and $\sqrt{\vert m_{z}\vert}$ and the center being placed at $\vec{p}=\vec{p}_{0}$. This means that the coordinate system chosen to diagonalize the tensor $m_{\alpha\beta}^{-1}$ coincides with the principle axes of an ellipsoid. S. Smirnov: