Why is effective mass a tensor? So I came across the effective mass concept for solids the other day. It was mentioned that the effective mass is a tensor and may have different values in different directions. However, this is stark contrast to ordinary mass which is direction independent (as far as I know). So how do we physically explain this direction dependence of electrons (holes) mass inside a solid? Is it called "mass" just because it has the dimensions of mass? Or is it just a mathematical tool without any physical significance?
 A: Effective mass $m_{\ast}$ is just a constant that shows up in the dispersion relation $\epsilon(\mathbf{k})$ of an energy band. Consider a one-dimensional band with dispersion $\epsilon(k)$. Expanding near a minimum of $\epsilon$
\begin{align}
\epsilon(k) \approx \epsilon_0 + \frac{\hbar^2 k^2}{2m_{\ast}}
\end{align}
It's defined by analogy to the quantum mechanical energy of a free particle of mass $m$,
\begin{align}
E(k) = \frac{\hbar^2 k^2}{2m}.
\end{align}
For example, consider a one-dimensional tight-binding model with dispersion
\begin{align}
\epsilon(k) = -2 t\cos(ka).
\end{align}
Taylor expanding $\epsilon$ near $k=0$ we get
\begin{align}
\epsilon(k)\approx -2t + tk^2 a^2.
\end{align}
so that the effective mass is
\begin{align}
m_{\ast}= \frac{\hbar^2}{2ta^2}.
\end{align}
In higher dimensions, the dispersion relation can be more complicated. If the dispersion is isotropic (the same in every direction), then
\begin{align}
\epsilon(\mathbf{k}) = \epsilon_0 + \frac{\hbar^2 |\mathbf{k}|^2}{2m_{\ast}}.
\end{align}
However, you could have a dispersion where the constants in front of different components of $\mathbf{k}$ are different, e.g.
\begin{align}
\epsilon(\mathbf{k}) = \frac{\hbar^2}{2}\left(\frac{k_x^2}{m_x} + \frac{k_y^2}{m_y}\right)
\end{align}
Such a dispersion could arise from, e.g., a tight-binding model in which the hopping parameters or lattice constants in each direction are different. You could even have a dispersion with cross-terms between the components of $\mathbf{k}$, e.g.
\begin{align}
\epsilon(\mathbf{k}) = \frac{\hbar^2}{2}\left(\frac{k_x^2}{m_1} + \frac{k_x k_y}{m_2} + \frac{k_y^2}{m_3}\right)
\end{align}
In general, the dispersion (to quadratic order, and dropping the constant) will be
\begin{align}
\epsilon(\mathbf{k}) = \frac{\hbar^2}{2}\sum_{a,b}h_{ab} k_a k_b
\end{align}
for a tensor $h_{ab}$ — the inverse of the effective mass tensor.
A: Your surprise for a tensor mass is originating from a misconception: to assume that the usual point particle mechanics is the "natural" mechanics and everything less usual (i.e. beyond the introductory physics level) requires a special explanation in terms of it.
Actually things can be stated in a different, more general way. Assuming that a basic tenet of Newtonian mechanics is a linear relation between force and acceleration, taking into account the vector character of both quantities, the most general relation among them should be a tensor relation. It is an experimental fact that for point particles at low energy such a tensor is diagonal and only one parameter is enough. However, even without quantum mechanics, there are cases where the necessity of a more general linear relation shows up. 
One case is that of constrained systems. It is well known that one of the most effective way to deal with them is to use the constraint equations to reduce the umber of degrees of freedom and this is efficiently done within the Lagrangian formalism. The passage from the usual descritpion in  terms of three-dimensional position vectors ${\bf r_i}$ to generalized coordinates $q_{\alpha}$ implies that in general the kinetic energy term has the form 
$$
T = \sum_{\alpha \beta} m_{\alpha \beta}\dot q_{\alpha} \dot q_{\beta}
$$
where $m_{\alpha \beta}$ is a  positive definite matrix.
Also in relativistic mechanics, the relation between acceleration and force may be written as a tensor relation, provided a mass tensor depending on velocity is introduced. 
The common feature of such different systems is the non-alignment between acceleration and force.
It is then not completely weird that the kinetic energy term of a quantum mechanics system could show a similar anisotropy.
