44
Compact Models for Integrated Circuit Design
where the minority carrier hole lifetime in an n-type semiconductor is given by
τ
σ
p
th p t
v N
=
1
(2.52)
In an n-type material, lots of electrons are available for capture. Therefore,
Equation 2.51 shows that the minority carrier hole lifetime τ p is the limiting
factor in recombination process in an n-type material.
Similarly, for a p-type semiconductor, we can show from Equation 2.50 that
the net recombination rate for electrons is given by
U
n
n
=
∆
τ
(2.53)
where
τ
σ
n
th n t
v
N
=
1
(2.54)
is the minority carrier electron lifetime. Thus, for a p-type semiconductor the
minority carrier electron lifetime is the limiting factor in the recombination
process.
The other recombination process in silicon that does not depend on deep
level impurities and that sets an upper limit on lifetime is Auger recombination. In this process, the electrons and holes recombine without trap levels and the released energy (of the order of energy gap) is transferred to
another majority carrier (a hole in a p-type and electron in an n-type silicon).
Usually, Auger recombination is important when the carrier concentration
is very high (>5 × 10 18 cm –3 ) as a result of high doping or high-level injection.
2.2.7 Basic Semiconductor Equations
2.2.7.1 Poisson’s Equation
Poisson’s equation is a very general differential equation governing the operation of IC devices and is based on Maxwell’s field equation that relates the
charge density to the electric field potential. Conventionally, the electrostatic
potential, f in a semiconductor is defined in terms of the intrinsic Fermi level
(E i ) such that
φ = −
E
q
i
(2.55)
The negative sign in Equation 2.55 is due to the fact that E i is defined as
the electron energy whereas f is defined for a positive charge. The electric
field E, which is defined as the electrostatic force per unit charge, is equal to
the negative gradient of f, such that
Compact Models for Integrated Circuit Design
where the minority carrier hole lifetime in an n-type semiconductor is given by
τ
σ
p
th p t
v N
=
1
(2.52)
In an n-type material, lots of electrons are available for capture. Therefore,
Equation 2.51 shows that the minority carrier hole lifetime τ p is the limiting
factor in recombination process in an n-type material.
Similarly, for a p-type semiconductor, we can show from Equation 2.50 that
the net recombination rate for electrons is given by
U
n
n
=
∆
τ
(2.53)
where
τ
σ
n
th n t
v
N
=
1
(2.54)
is the minority carrier electron lifetime. Thus, for a p-type semiconductor the
minority carrier electron lifetime is the limiting factor in the recombination
process.
The other recombination process in silicon that does not depend on deep
level impurities and that sets an upper limit on lifetime is Auger recombination. In this process, the electrons and holes recombine without trap levels and the released energy (of the order of energy gap) is transferred to
another majority carrier (a hole in a p-type and electron in an n-type silicon).
Usually, Auger recombination is important when the carrier concentration
is very high (>5 × 10 18 cm –3 ) as a result of high doping or high-level injection.
2.2.7 Basic Semiconductor Equations
2.2.7.1 Poisson’s Equation
Poisson’s equation is a very general differential equation governing the operation of IC devices and is based on Maxwell’s field equation that relates the
charge density to the electric field potential. Conventionally, the electrostatic
potential, f in a semiconductor is defined in terms of the intrinsic Fermi level
(E i ) such that
φ = −
E
q
i
(2.55)
The negative sign in Equation 2.55 is due to the fact that E i is defined as
the electron energy whereas f is defined for a positive charge. The electric
field E, which is defined as the electrostatic force per unit charge, is equal to
the negative gradient of f, such that