# Non-parabolic correction to effective electron mass in III-V semiconductors

My question came about when reading Resolving Ambiguities in Nanowire field-effect transistor characterization by Heedt et al. 2015. This paper deals with the modeling of the electrostatics of an InAs nanowire (I care about InSb but the same model should hold with different parameters). Now, specifically, at the bottom of page 3 they write that they take the nonparabolic correction to the electron effective mass $$m^*_e(E)$$ into account via the energy-dependent Kane model in the context of $$k\cdot p$$ perturbation theory, which accounts for the coupling of the conduction band to the valence bands and remote bands.

My question is quite simple; how do they take this into account? Clearly they do not use a static value for $$m_e^*$$ but instead use an energy dependent one. This energy dependence however is not given, and I am wondering what the explicit form is; how would I rewrite my parabolic dispersion $$\frac{\hbar^2 k^2}{2m_e^*}$$?

The authors of the paper refer to the paper Band Structure of Indium Antimonide by Kane 1957, to Band Parameters of Semiconductors with Zincblende, Wurtzite, and Germanium Structure by Cordona 1963, and to Band parameters for III–V compound semiconductors and their alloys by Vurgaftman et al 2001. These papers essentially all list the same type of expression (where the final one tabulates experimentally determined values), namely that for the conduction band effective mass $$m_e^*$$ we have

$$$$\frac{m_e}{m_e^*} = \left(1+2F\right) + \frac{E_p\left(E_g+2\Delta_{SO}/3\right)}{E_g(E_g+\Delta_{SO})}$$$$

where $$F$$ is due to coupling to other bands, $$E_g$$ is the gap and $$\Delta_{SO}$$ is the spin-orbit coupling (Vurgaftman equation 2.15).

Now, I can see that this is a term that redefines the effective mass by taking all of this into account. What I do not see is how this is energy dependent; all of the parameters given above are tabulated as constants in the paper (with a temperature dependence). How then does one obtain an expression where $$m_e^*(E)$$ is energy or $$k$$-dependent? If I plug them into the expression above, I just get the tabulated value one tends to use.

• A tough question. On the other hand, you are saying that the tabulated values seem to take it into account already. A different way to look at it would be to see what an estimate of $m^{*}$ would be from published band diagrams vs the tabulated values. – Jon Custer Jun 9 '17 at 0:32
• @JonCuster Indeed, the tabulated values do take this into account. But in that case it is not clear to me why the original paper lists an energy dependent effective mass $m_e^*(E)$, as it is not energy dependent in this model. Nor in any of the cited sources, as far as I can deduce. – user129412 Jun 9 '17 at 17:47

Answering this question requires understanding effective mass theory and how Evan Kane's $$k \cdot p$$ model is employed for modeling semiconductor band structure.

Let's think about the conduction band in a semiconductor. The band structure is simply energy as a function of $$k$$ or $$E(k)$$. Now $$E(k)$$ can be expanded as a Taylor series about any given $$k$$. The lowest point in the conduction band is important since this is where the electron carriers will be (in their lowest energy states). Since the band structure is continuous and we are looking at the lowest energy point, a Taylor series expansion about the minimum will not have a term linear in $$k$$, so we can model $$E = E_g + a k^2 + ...$$ quite generally, where $$E_g$$ is the band gap. Next, quantum mechanics tells us that a free particle's kinetic energy is $$\frac{\hbar^2 k^2}{2 m}$$. If we simply re-write that Taylor expansion coefficient $$a$$ as $$\frac{\hbar^2}{2 m^*}$$ it is still just a simple parabolic expansion but we are going to call $$m^*$$ the "effective mass". There are lots of nifty things you can do with this effective mass theory, and one of them is to calculate the density of states at any given energy above the band gap because you know how dense states are as a function of $$k$$ and you now have a mapping of energy to $$k$$.

Next we move on to the Kane model, which is an application of $$k \cdot p$$ perturbation theory. https://en.wikipedia.org/wiki/K%C2%B7p_perturbation_theory The parameters in a Kane model produce a non-parabolic band structure $$E(k)$$, however they can still be used to calculate the parabolic term right at the band gap, which is the effective mass, and the formula you quote in your question is exactly that. And you are right - it is not a function of the energy!

So, what do the authors mean when they say "The nonparabolic correction to the electron effective mass of the density of states ..."? In the paper the authors are seeking to find a carrier charge density as a function of energy and this requires them to have a band structure to determine how many states are available below any give energy level. They are using Kane's non-parabolic band structure as a mapping from $$k$$ to $$E$$ because they expect there to be so many carriers that the parabolic approximation is not quite good enough.

So, what do they mean when they use the function $$m^*_n(E)$$? Remember how we simply fit a parabola to a band structure and just used the $$k^2$$ coefficient to simply "define" an effective mass? Well, what the authors of your paper are doing is using Kane's model band structure to calculate a density of states below any given energy $$E$$. Then they are saying "at any given $$E$$ what purely parabolic band structure would have resulted in the density of states that we have just calculated with Kane's model?" And they call that $$m^*_n(E)$$. Quite close to the band gap it will be the effective mass you quote, but as the band begins to fill it will get "heavier" since in Kane's model bands begin to "flatten out" as $$k$$ gets further from the minimum (the gamma point).