Semiconductor equations, Electrical Engineering

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Semiconductor Equations 

The semiconductor equations that are relating these variables are shown below:

Carrier density:

n = ni exp (EFN - Ei / KT)        (1)

p = ni exp (Ei - EFP / KT)        (2)

In which EFN is the electron quasi Fermi level and EFP is the hole quasi Fermi level. These above 2 equations lead to 

Np = n­i2 exp (EFN - EFP/ KT)   (3)

In equilibrium EFN = EFP = Constant

Current:

There are two mechanism of current; electron current density and hole current density. There are various mechanisms of current flow:

  1. Drift
  2. Diffusion
  3. Thermionic emission
  4. Tunnelling

The final two mechanisms are significant frequently only at the interface of two different materials like a metal-semiconductor junction or a semiconductor-semiconductor junction where the two semiconductors are of dissimilar materials. Tunneling is as well significant in the case of PN junctions in which both sides are heavily doped.

The dominant conduction mechanisms include drift and diffusion in the bulk of semiconductor. The current densities because of these two mechanisms can be written as

JN = qnμNε + qDN dn/dx   (4)

JP = qnμPε + qDP dP/dx   (5)

In which μN and μP are electron and hole mobilities correspondingly and DN, DP are their diffusion constants.

Potential:

The potential and electric field in a semiconductor can be described in the following ways:

  1. Ψ = - EC /q + constant ; ε =  (1/q) (dEc / dX)
  2. Ψ = - EV /q + constant ; ε =  (1/q) (dEV / dX)
  3. Ψ = - Ei /q + constant ; ε =  (1/q) (dEi / dX)
  4. Ψ = - EO /q + constant ; ε =  (1/q) (dEO / dX)

All these definitions are equal and one or the other may be selected on the basis of convenience. The potential is connected to the carrier densities through the Poisson equation: -

2 Ψ / ∂X2 = - q/ε (p-n+ N+D - N-A)      (6)

In which the last two terms present the ionized donor and acceptor density.

 


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