# Where does the lowered effective mass of electrons come from?

It is a known fact that electrons in the conduction band of a semiconductor can (in certain scenario's) be described as having an approximate parabolic dispersion relation of the form $E_c(k) = E_c + \frac{\hbar^2k^2}{2m^*}$ where $m^*$ is the so called effective mass, which increases with the size of the band gap. This effective mass is often measured as a fraction of the standard electron mass $m_e$ and it can be much smaller: for example, in GaAs we have that $m^* = 0.067m_e$.

Now, the way I was taught, this was just a result of standard $\vec{k}\cdot{\vec{p}}$ perturbation theory, which somehow involves the crystal lattice structure and related periodicity to look at band structure near band extrema. This formulation turns out to be effective, and thus it is used.

But for me, the origin of this effective mass was never explained. Because isn't this amazing? Why does an electron suddenly behave as if it is much, much lighter, when placed into a lattice? I suppose it is a quantum effect, maybe having to do with interference? That is just a guess though. I would be very grateful if someone could help me gain some insight into how this effect comes about.

• Its like the refractive index of a medium: because of the periodic disposition of the atoms, the light seems to move slower. This effect can be explained classically (standard optics result), by considering the polarisability of the media (Lorentz-Lorenz). In the case of electrons, you can think of the periodicity of the lattice makes (Bloch) the electrons have an effective momentum, i.e, a different mass. – AccidentalFourierTransform Jan 13 '16 at 22:20
• @AccidentalFourierTransform I am not entirely sure I follow. The way I interpret that is as follows: in typical (classical) explanations of the refractive index an EM wave's phase velocity is slowed because the field disturbs the charges of the atoms proportional to the susceptibility, shaking them, which causes them to emit a wave at the same frequency with some delay. The superposition of all these waves is then the slowed down EM wave. I suppose this is similar to your story, because polarisability is related to susceptibility by Clausius-Mossoti. But how do I apply this story here? – user129412 Jan 13 '16 at 22:38
• @AccidentalFourierTransform Hm, perhaps to follow up on the above, are you arguing that the electron interacts with the various atoms in the periodic lattice in such a way that the superposition of each of these interactions leads to the overall effect, which is the lowered mass in this case? In that case I suppose I'm trying to find a picture in which I can understand how these interactions that are summed up (effectively) lower the mass of the electron – user129412 Jan 13 '16 at 22:45
• note that scattering by a potential does induce a phase shift in the electron's wave-function (e.g., see Phase shifts in scattering theory). Anyway, we shouldnt take the refractive index analogy too far. We are talking about QM after all... – AccidentalFourierTransform Jan 13 '16 at 22:49
• There is a price to pay for a low effective mass of charge carriers: it usually seems to come with a high dielectric constant of the material. My intuition may be completely off here, but I wonder if one can attribute some of the effect to a classical mean field approximation over the dielectric behavior of the lattice? – CuriousOne Jan 13 '16 at 22:51

If you're looking for a strict derivation of the effective mass equation, check out

What he does is take the full Schrödinger equation with the periodic potential, and write it in the Bloch state basis. He then writes the effective mass equation in the plane wave basis. By comparing the matrix elements of both equations, he reaches the set of approximations necessary for them to be equivalent.

• Thanks for the source! I'm afraid that a strict derivation isn't exactly what I'm trying to find though, more of an intuitive picture of how the electrons can behave as if they are so light, rather than that it follows from the mathematics. – user129412 Jan 13 '16 at 23:47
• Very basically, the result of all the complex interactions between electron and lattice looks like a lighter electron. – ignacio Jan 13 '16 at 23:59
• Right, and it is related to the periodicity, as well as the atoms themselves. But is it an interference effect? – user129412 Jan 14 '16 at 0:10
• Yes, you can see it as the interference of the scattering against each lattice site. – ignacio Jan 14 '16 at 0:56
• Okay, yes, I think I can follow now. The idea is that (choosing one of the two approaches) you express your states in the Bloch or the plane wave basis, thus taking into account interactions with many lattice sites at once, for an interference effect. This interference then leads to the lighter (looking) electron. Perhaps I'd like to delve a little deeper into how these complex interactions then contribute to a lower effective mass individually though; this seems unintuitive for me but perhaps it is not a very reasonable thing to ask, seeing as this is simply an approximation. – user129412 Jan 14 '16 at 14:57