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5 Semiconductor Modeling
80

Fig. 5.3. Intel Celeron Processor (downside)

A ‘closed’ system of partial differential equations is obtained by coupling the
current relations/continuity equations to the Poisson equation for the electric
potential:
2
λ ΔV = n − p − C(x).
This equation is a direct consequence of the Maxwell field equations, when
magnetic and relativistic effects are neglected. Its right hand side represents the
(negative and scaled) space charge density. The parameter λ, whose square mul-
tiplies the Laplace operator in the Poisson equation, is the scaled Debye length of
the doped semiconductor device, determining the radius of electrical influence
of an impurity ion in the semiconductor crystal. In many real life applications
it is a small parameter. It is often assumed that diffusivities and mobilities are