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Diffusion Current Discretization

  Generally, tex2html_wrap_inline4781 is a function of several physical variables defined on two given grid points ij (4.1-5), where tex2html_wrap_inline4803 is the electrostatic potential, tex2html_wrap_inline4805 the net doping, tex2html_wrap_inline4807 the intrinsic carrier density and C the doping concentration at the grid points i and j, respectively.

  equation437

We will discuss now several discretizations of diffusion current models starting with the diffusion flux followed after the first Fickian law (4.1-6).

  equation442

Using the finite-difference approximation the discretized diffusion current at the ij-th grid line becomes to (4.1-7).

  equation447

The average diffusion coefficient tex2html_wrap_inline4817 is defined as the projection of the diffusivities at the grid points i and j, tex2html_wrap_inline4823 and tex2html_wrap_inline4825 , to the mid-edge point of tex2html_wrap_inline4787 (see 4.1-8).

  equation457

The current discretization (4.1-7) allows the easy implementation of a static clustering model. Only the dopant concentrations tex2html_wrap_inline4829 and tex2html_wrap_inline4831 have to be replaced by the mobile species tex2html_wrap_inline4833 and tex2html_wrap_inline4835 , where an algebraic relation (4.3-20) can be used to calculate the active species.

To account for the electric field of the charged dopants the diffusion flux model (4.1-6) has to be extended by a field enhancement term, where Z denotes the charge state of the dopant. Equation (4.1-9) gives the diffusion flux for single negatively charged dopants under thermal equilibrium and by assuming Boltzmann statistics to predict the electrostatic potential tex2html_wrap_inline4839 , where the electron concentration is scaled to the intrinsic carrier density tex2html_wrap_inline4841 .

  equation471

For a general formulation of the field enhancement the electron concentration n has to be replaced by the majority carrier concentration. The discretized diffusion current is then given by (4.1-10).

  equation477

Using the box integration method offers the possibility of using the Scharfetter-Gummel discretization scheme for the diffusion current [Sch69]. Again, the field enhancement flux reads

  equation491

The electrostatic potential is scaled to the thermal voltage tex2html_wrap_inline4845 and the Einstein relation is used to express the mobility tex2html_wrap_inline4847 . Assuming a constant diffusion flux along the discretization interval tex2html_wrap_inline4849 , we obtain a differential equation for the dopant concentration within the discretization interval tex2html_wrap_inline4849 to (4.1-12). The boundary conditions are given to tex2html_wrap_inline4853 and tex2html_wrap_inline4855 .

  equation498

This equation represents an ordinary differential equation for the unknown tex2html_wrap_inline4857 , where the electrostatic field is assumed to be constant between the interval tex2html_wrap_inline4849 . This implies a linear dependence of the electrostatic potential within tex2html_wrap_inline4849 . The electric field is approximated by the finite difference expression (4.1-13).

  equation508

By solving the differential equation (4.1-12) for tex2html_wrap_inline4857 and re-inserting the solution, the discretized current between the grid points ij is given by (4.1-14), where B(x) are the the Bernoulli-coefficients (4.1-15) and tex2html_wrap_inline4869 an auxiliary variable (4.1-16).

    eqnarray521


next up previous contents
Next: Time Discretization Up: 4.1.2 Discretization of the Previous: Box-Integration Method

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