
So far we implicitly defined
(thermal) equilibrium as a total
equilibrium involving three components if
you think about it: 


1. Equilibrium of the electrons in the
conductance band. This means their concentration was given (within
the usual approximations) by 


n_{e} = N_{eff}
^{e} · exp – 
E_{C} – E_{F}
kT_{ } 




2. Equilibrium of the holes in the valence
band. This means their concentration was given within the usual
approximations by 


n_{h} = N_{eff}
^{h} · exp – 
E_{F} – E_{V}
kT_{ } 




3. Equilibrium between the electrons and
holes, i.e. between the bands. This means
that the Fermi energy is the same for both
bands (and positioned somewhere in the band gap). 

If an equilibrium is disturbed, it
takes a certain time before it is restored again; this is described by the
kinetics of the processes taking place. In
a strict sense of speaking, the Fermi energy is not defined without
equilibrium, but only after it has been restored. This restoring process occurs
in the bands and between the bands: 


In the bands,
local equilibrium (in
kspace) between the carriers will be obtained after there was
time for some collisions, i.e. after some multiples of the
scattering
time. This process is known as thermalization and occurs typically in picoseconds. 


Between the
bands, equilibrium will be restored by generation and recombination
events and this takes a few multiples of the carrier
life time, i.e. at least nanoseconds if not milliseconds. 

This means that we can have a
partial (or local) equilibrium in the bands
long before we have equilibrium between the bands. This
local equilibrium implies that: 


Nonequilibrium means something changes in time.
Changes in the properties of the particle ensemble considered (i.e. electrons
and holes) in local equilibrium are only
due to "traffic" between the
bands while the properties of the particles in the band do not change anymore.
The term "local" of course, does not refer to a coordinate, but to a
band. 


The carrier concentrations
therefore do not have their total or global equilibrium value as given, e.g.,
by the mass action law, but their
local equilibrium concentration can still be given in terms of the equilibrium
distribution by 


n_{e} = N_{eff}
^{e} · exp – 
E_{C} –
E_{F}^{e}
kT_{ } 
n_{h} = N_{eff}
^{h} · exp – 
E_{F}^{h} –
E_{V}
kT_{ } 




with the only difference that the Fermi energy now is different for the electrons and
holes. Instead of one Fermi
energy E_{F} for the whole system, we now have
two Quasi
Fermi energies, E_{F}^{e} and
E_{F}^{h}. 

For the product of
the carrier densities we now obtain a somewhat modified massaction law 


n_{e} · n_{h} =
N_{eff}^{e} ·
N_{eff}^{h} · exp – 
(E_{C} –
E_{V}) + (E_{F}^{e} –
E_{F}^{h})
kT_{ } 
= 
n_{i}^{2} · exp
– 
E_{F}^{e} –
E_{F}^{h}
kT_{ } 




For this we used the
by now basic relation 


N_{eff} ^{e} ·
N_{eff}^{h} · exp – 
E_{C} –
E_{V}
kT_{ } 
=
n_{i}^{2} 



The name "Quasi Fermi energy" is maybe not so good,
there is nothing "quasi" about
it. Still, that's the name we and everybody else will use. Sometimes it is also
called "Imref" (Fermi backwards), but that
doesn't help much either. 


Rewriting the equations from
above gives a kind of definition for the Quasi Fermi energies: 


E_{F}^{e} =
E_{C} – kT · ln 
N_{eff} ^{e}
n^{e} 
E_{F}^{h}
= E_{V} + kT · ln 
N_{eff} ^{h}
n^{h} 



Quasi Fermi energies are extremely
helpful for the common situation where we do have nonequilibrium, but only
between the bands  and that covers most of
semiconductor devices under conditions of current flow. We will make frequent
use of Quasi Fermi energies. 

