# Effective mass as a consequence of energy band structure

The Wikipedia article on effective mass defines it as follows:

In solid state physics, a particle's effective mass (often denoted $${\textstyle m^{*}}$$) is the mass that it seems to have when responding to forces, or the mass that it seems to have when interacting with other identical particles in a thermal distribution.

In semiconductor physics, due to the presence of, and, therefore, interactions with, the atoms of a semiconductor, electrons in a semiconductor are unable to move as freely as they can in a vacuum. To account for this decreased mobility, we say that the electron has an effective mass.

I have then seen it said that effective mass is a consequence of energy band structure: (1) It is determined by the curvature of the energy band, (2) it depends on the material, (3) it depends on the band. I also found the following related diagram:

What I'm struggling with is interpreting/understanding this graph in terms of the description of effective mass as a consequence of energy band structure. I would greatly appreciate it if people would please take the time to explain this graph in terms of the description that I gave of effective mass as a consequence of energy band structure.

• Effective mass is defined as $(m^*)^{-1}= \frac{\partial^2 E}{\partial k^2}$ so this equation explains the relation of the effective mass to the curvature of the parabolic band structure! – Simon Jan 17 at 15:24
• @Simon what does that PDE have to do with parabolas? (I’m kind-of aware that PDEs can have solutions that are parabolas, but my PDE knowledge is not very deep, so I’d appreciate a more elaborate explanation, so that I can explore this further.) – The Pointer Jan 17 at 15:47

So what is illustrated in this figure is the energy band structure, these represent the dispersion, i.e. the energy $$E$$ in function of the wave vector $$k$$. Now, you can find in standard textbooks on solid state physics that the effective mass is defined as $$(m^*)^{-1} =\frac{1}{\hbar^2}\frac{\partial^2 E}{\partial k^2}$$ So the curvature of these (approximate) parabola in the band structure determine the effective mass. Also note that the concave parabola will have a corresponding negative effective mass, these correspond to holes. Furthermore, the curvature of the parabola is inversely proportional to the effective mass, hence, these strongly curved parabola will correspond to a low effective mass, that's why they call it 'light hole', and the other way around.
• The energy has a $k$-dependence which gives rise to a parabolic band structure. The curvature of this parabola (in more general the curvature of any curve) is determined by the second derivative of the energy (curve/function) w.r.t. $k$? – Simon Jan 17 at 16:07
• @ThePointer it is not a partial differential equation, but an expression. We assume that $E = p^2/2m^*$ and from it want to get $m$, so about a point $k_0$ we want to see how the energy changes when we change the momenta. We take the second derivative of the energy at that point $\frac{1}{m^*} = \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k^2}|_{k=k_0}$. We don't have to solve anything, just carry out the derivative. – yu-v Jan 17 at 16:09