Landau-Fokker-Planck equation for hot electrons in semiconductors

In this paper we derive the Landau-Fokker-Planck (LFP) equation for the energy distribution function of a nondegenerate hot electrons in bulk semiconductors with strong electron-electron () interaction by the screened Coulomb potential described by a static dielectric function (the effect of dynamic screening is also analyzed qualitatively). The scattering rates are calculated quantum-mechanically using the first Born approximation. The LFP equation was derived employing two different approaches: (i) a direct general transformation of the integral Boltzmann scattering operator; (ii) using the Chapman-Kolmogorov integral equation (CKE) for conditional transition probability densities. The screened potential allows to avoid any ad hoc cutoff procedures for tackling the problem of the Coulomb divergence of the matrix element in the scattering probability rates. However, the screening at the same time brings to the theory another problem, namely, that increase of the screening strength changes the character of the scattering when it eventually becomes strongly inelastic. This corresponds to large energy transfer between the scattering electrons, which in turn, prohibits in principle considering the electron scattering process as a diffusion in the energy space (the necessary condition for the LFP approximation). We show that the screening effect must be explicitly included into the kinetic theory to correctly address the above two problems, and the cases of a weak and a strong screening need to be analyzed in detail. The LFP approximation is physically justified only for a weak screening. The obtained screening function coincides with the Brooks-Herring screening function. We have established general relationships between the kinetic coefficients in the LFP operator obtained by the above two approaches. The explicit expressions for the dynamic friction and the diffusion coefficients in the energy space are also obtained. Although, both approaches lead to the same solutions, we demonstrate that the CKE method is more universal approach, and the obtained here general relationships can be applied for any other scattering mechanism, provided that the required conditions are satisfied. We discuss the necessary validity conditions for the LFP equation, and show that for typical physical situations, e.g., for high-energy photoexcitation of the carriers, the LFP equation can be used up to the densities <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mo>∼</a:mo><a:msup><a:mrow><a:mn>10</a:mn></a:mrow><a:mn>18</a:mn></a:msup><a:mspace width="0.28em"/><a:mi mathvariant="normal">c</a:mi><a:msup><a:mrow><a:mi mathvariant="normal">m</a:mi></a:mrow><a:mrow><a:mo>−</a:mo><a:mn>3</a:mn></a:mrow></a:msup></a:mrow></a:math>. However, for higher densities the screening energy becomes too large, which results in an increase of the minimum energy transfer in the scattering events, and the required for the LFP conditions are violated. In this case, one must use the integral scattering operator, which we obtained in the form suitable for numerical solution. Dynamic screening extends the LFP validity range. As an example of the application of the LFP equation, we discuss the perovskite solar cells, which are characterized by relatively low and comparable energies of the optical and acoustic phonons, and the generated carrier densities of the order of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msup><e:mrow><e:mn>10</e:mn></e:mrow><e:mn>18</e:mn></e:msup><e:mspace width="0.16em"/><e:mi mathvariant="normal">c</e:mi><e:msup><e:mrow><e:mi mathvariant="normal">m</e:mi></e:mrow><e:mrow><e:mo>−</e:mo><e:mn>3</e:mn></e:mrow></e:msup></e:mrow></e:math>. In this case, the LFP equation can be applied for each of these interactions (electron-phonon and the scattering). This allows to find correct distribution function of the hot electrons, and calculate the electron power loss, to which the scattering makes the direct contribution through the corresponding energy relaxation channel in the kinetic equation.