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# Virtual Refinement of Semiconductor Structures

IP.com Disclosure Number: IPCOM000084250D
Original Publication Date: 1975-Oct-01
Included in the Prior Art Database: 2005-Mar-02
Document File: 2 page(s) / 42K

IBM

## Related People

Hachtel, GD: AUTHOR [+2]

## Abstract

Semiconductor structures usually contain relatively small regions in which the doping or carrier concentrations vary rapidly over several orders of magnitude. Conventional finite difference methods introduce a grid of solution points, approximate derivatives by differences between function values at the solution points, and solve the resulting set of nonlinear algebraic equations by one of several conventional techniques. Hence a large number of solution points will be needed to resolve the regions, in which the doping or carrier distributions change rapidly.

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Virtual Refinement of Semiconductor Structures

Semiconductor structures usually contain relatively small regions in which the doping or carrier concentrations vary rapidly over several orders of magnitude. Conventional finite difference methods introduce a grid of solution points, approximate derivatives by differences between function values at the solution points, and solve the resulting set of nonlinear algebraic equations by one of several conventional techniques. Hence a large number of solution points will be needed to resolve the regions, in which the doping or carrier distributions change rapidly.

The more recently developed finite element method divides the device into pieces or elements, in each of which the solution is approximated by a function (usually polynomial) of the values of the solution obtained at solution points for each element. Instead of solving a differential equation, in the finite element method an appropriate integral is minimized over all of the elements. The integral is chosen to give the same solution as the differential equation.

This integration is usually carried out numerically over each element, using a number of integration points which need not be the same as the solution points. Information about the structure can be incorporated at these integration points, even though the problem need not be solved at these points. The usual emphasis is to use the minimum number of integration points in each element that will assure the convergence of the finite element method.