According to the Boltzmann transport equation, the current density in the carrier continuity equation can be described using a drift-diffusion model. In this model, the current density is expressed in terms of the Fermi levels for electrons (¢n) and holes (¢p).
Here, μn and μp represent the mobility of electrons and holes, respectively. The quasi-Fermi level is related to the carrier concentration and potential through two Boltzmann approximations, as shown below:
In these equations, nie denotes the effective intrinsic carrier concentration, and TL is the lattice temperature. These expressions can be rearranged to define the quasi-Fermi levels, which are given by:
Substituting these into the current density expression yields:
The last term represents the gradient of the effective intrinsic carrier concentration, taking into account the bandgap narrowing effect. The effective electric field is generally expressed as:
Note that the derivation of the drift-diffusion model assumes the Einstein relationship. For Boltzmann statistics, this relationship is given by:
Where (a takes values of 1/2 or -1/2) refers to the a-point integral of the Fermi-Dirac distribution, and &epSILon; Fn is defined as -qQn.
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