Hello, everyone.
I have tried to determine carrier motilities of some materials, by Density Functional Theory, using Quantum ESPRESSO. There are a few methods to do it, like a package called BoltzTrap2, and a QE code called EPW. I have not mastered these yet, but there is another method, which requires to determine the elastic constants of the structure and apply the Boltzmann Transport Theory equation, shown in the image attached.
I’m trying to use this equation, but there is something wrong with my calculation, because I can’t get reasonable results. I tried to replicate the results described in papers like this one:
https://www.researchgate.net/publication/365822924_Atom-to-Device_Simulation_of_MoO3Si_Heterojunction_Solar_Cell
His calculation of mobility, and the one I get by applying the same formula with the same parameters, are completely different. I would be very glad if someone could elucidate this question.
You may need to get a qualitative understanding of the mechanism of transport in your system to identify an appropriate method of calculation.
In general, one of the central tasks in the theoretical prediction of carrier mobility is determining effective mass because carrier mobility varies inversely with carrier effective mass. The effective mass is proportional to the reciprocal of the second derivative of the band energy with respect to wave vector.
m* = (hbar)^2 (d^2E/dk^2)^(-1)
Carrier mass is therefore determined in theoretical calculations by finding the curvature of the highest energy valance band (hole effective mass) or lowest energy conduction band (electron effective mass) in k-space. This can be done numerically by computing the electronic structure with any of the numerous available band structure codes and fitting several points around the VBM/CBM to a parabola. Alternatively, some electronic structure codes include the option to compute the effective mass tensor directly.
Mobility is just the ratio of drift velocity to the applied electric field. To get from carrier mass to mobility requires the mean free time (average time between scattering events). A quick web search turned up this page: http://britneyspears.ac/physics/mobility/mobility.htm
In contrast, a hopping theory approach makes sense for cases where the bands are relatively dispersionless. For this reason, in a molecular crystal, hopping theory is often a reasonable strategy. It was developed by Marcus, [R.A. Marcus and N. Sutin, Biochimica et Biophysica Acta 811 p265-322 (1985); R. A. Marcus, Faraday Discuss. Chem. SOC., 74, p7- 15 (1982)]. (Marcus actually published long before this, but these papers are more assessable.) A relatively convenient protocol has been presented by Deng and Goddard. [Deng and Goddard, J. Phys. Chem. B, 108, 8614-8621 (2004)] The basic idea is to compute hop rates from a representative monomer to its neighbors and take a weighted average of these to determine the diffusion coefficient. Related approaches have been developed by Miller-Abrahams [A. Miller and E. Abrahams, Phys. Rev., vol. 120, no. 3, pp. 745-755, 1960] and Mott [N. F. Mott, Journal of Non-Crystalline Solids 1 (1), 1 (1968); N. F. Mott and E. A. Davis, Electronic processes in Non-crystalline Materials, 2nd ed. (Clarendon Press, Oxford, 1979).]
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