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At usual diffusion temperatures the dopants are often ionized and the
so released electrons cause an electric field which operates as an
additional drift term to the normal diffusion flux.
=  D^{ . }grad (C) + ^{ . }C 


(2.4) 
The velocity of the charged particles can be described
with
= ^{ . }
=  ^{ . }grad



(2.5) 
where
is the electric field and
the electrostatic potential. The mobility
is related to the
diffusion coefficient by Einstein's relation
= D^{ . } 


(2.6) 
where q denotes the elementary charge and k is known as the
Boltzmann constant.
Introducing (2.6) and (2.5) into
(2.4) and taking more than one dopant into account Fick's
law can be extended to
=  D_{A}^{ . }grad (C_{A}) + z_{A}^{ . }^{ . }C_{A}^{ . }grad () 


(2.7) 
where z_{A} denotes the charge state of the belonging dopant (+1
for singly charged acceptors and 1 for singly charged donors).
The electrostatic potential
is determined by the Poisson equation
div^{ . }grad ()
= q^{ . }(n  p  C_{net}) 


(2.8) 
with
C_{net} =  z_{A}^{ . }C_{A} 


(2.9) 
where the quantity C_{net} represents the net concentration of all ionized
dopants and n, p the concentration of electrons and holes, respectively.
Under the assumption of thermodynamic equilibrium the carrier
concentrations n and p can be obtained with
n^{ . }p = n_{i}^{2} 


(2.10) 
n = n_{i}^{ . }e^{} 


(2.11) 
p = n_{i}^{ . }e^{ } 


(2.12) 
In case of global charge neutrality
n  p  C_{net} = 0 


(2.13) 
and by means of the equilibrium carrier concentrations
(2.11)(2.12) the electrostatic potential can explicitly
be calculated by
With the explicit form of the gradient of ,
grad () =  ^{ . }^{ . }z_{i}^{ . }C_{i} 


(2.15) 
substituting into (2.7) the flux
for dopant C_{A} now depends also on the gradients of the concentrations C_{i}
of all other dopants.
=  D_{A}^{ . }1 +
^{ . }grad (C_{A})  



D_{A}^{ . }^{ . }z_{i}^{ . }grad (C_{i}) 


(2.16) 
If only one dopant is present (2.16) simplifies to
=  D_{A}^{ . }1 +
^{ . }grad (C_{A}) 


(2.17) 
Comparing Fick's first law (2.1) and
(2.17) an effective diffusion coefficient with the field enhancement factor
can be extracted
D_{eff} = D_{A}^{ . }1 +
= ^{ . }D_{A} 


(2.18) 
For intrinsic conditions (
C_{A}/n_{i}
1) this factor has a value
close to one and for high concentrations (
C_{A}/n_{i}
1) a value close to two.
Next: 2.1.2 PairDiffusion Mechanism
Up: 2.1 Diffusion
Previous: 2.1 Diffusion
Mustafa Radi
19981211