Diffusion model light small-position modulator drift

Photocoupler

According to the Boltzmann transport equation, the current density in the carrier continuity equation can be modeled using a drift-diffusion approach. In this model, the current density is expressed in terms of the Fermi levels, denoted as $ \psi_n $ and $ \psi_p $, which represent the energy levels for electrons and holes, respectively.

The mobility of electrons ($ \mu_n $) and holes ($ \mu_p $) plays a key role in determining how carriers move under the influence of an electric field. The quasi-Fermi levels are related to the carrier concentration and potential through two Boltzmann approximations, as shown below:

In these equations, $ n_i $ represents the effective intrinsic carrier concentration, while $ T_L $ is the lattice temperature. These expressions can then be rearranged to define the quasi-Fermi levels more explicitly:

By substituting these into the current density expression, we obtain:

The final term accounts for the gradient of the effective intrinsic carrier concentration, incorporating the bandgap narrowing effect. The effective electric field is typically given by:

Note that the derivation of the drift-diffusion model assumes the Einstein relation. For Boltzmann statistics, this relationship is expressed as:

Here, $ a $ takes values of $ \pm \frac{1}{2} $, representing the a-point integrals of the Fermi-Dirac distribution, and $ \epsilon_{Fn} $ is defined as $ -qQ_n $.

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