
In this derivation we consider the forces acting on carriers and the currents resulting
from these forces. 


The important point to know is that within the general
framework of mechanics and thermodynamics, forces are generalized and expressed
as the (space) derivatives of thermodynamic potentials. 


In particular, diffusion currents due to concentration
gradients of the diffusing species may be seen as an expression of a
chemical force
F_{chem} that acts on particles. We call it chemical
because it tends to change particle numbers. 


The value of the chemical force is always given by
the derivative of the chemical
potential; looking at a onedimensional case we thus have 


F_{chem, x} = F 
= – 
dµ_{chem}
dx 
= – 
dE_{F}
dx 




Of course, we will never
confuse µ_{chem}, the chemical potential, with
µ, the carrier mobility! 

Looking at the most general case with only
local equilibrium in the bands, we use the
QuasiFermi energies,
E_{F}^{e} and
E_{F}^{h},
given by 


E_{F}^{h} 
= 
E_{C} – kT · ln 
N^{e}_{eff}
n_{e} 




E_{F}^{h} 
= 
E_{C} + kT · ln 
N^{h}_{eff}
n_{h} 




We thus have for the chemical forces 


F^{e} 
= – 
dE_{F}^{e}
dx 
= – 
dE_{C}
dx 
+ 
kT^{ }
n^{e} 
· 
dn^{e}
dx 
F^{h} 
= – 
dE_{F}^{h}
dx 
= – 
dE_{A}
dx 
+ 
kT^{ }
n^{h} 
· 
dn^{h}
dx 




In what follows we drop the indexes
"e" and "h" and write only one set of
equations for the conduction band ( i.e. for electrons). For holes everything
is the same, both equations can be retrieved at the end by proper indexing.


We allow for the band edges to be functions of
x, i.e. E_{C} =
E_{C}(x) and E_{V} =
E_{V}(x). What then determines the numerical value of
the band edge energy (for some defined zero point of the energy)? There are two
factors: 


The particular kind of semiconductor or crystal
considered  this defines the band structure in general. We call this part
E_{Cryst}, and note, while
E_{Cryst} is constant in semiconductors of one kind of
material (and omitted from formulas), it generally may be a function of
x. Examples are materials with compositions that change gradually
(e.g. SiGe alloys, or GaAlAs with "sliding" Ge
or Al concentration, respectively). 


External or internal electrical field E_{x} = –
dV(x)/dx due to the electrostatic potential
V(x) that must be superimposed on the band energies as
– eV with e = magnitude of the elementary charge.
In the following we drop the magnitude signs for the sake of convenience.
(We will write the electrical field E_{x} in pink
here, to avoid confusion with the various energies). 

We thus can write 


dE_{C}
dx_{ } 
= 
dE_{Cryst}
dx_{ } 
+ ^{ }e · E_{x} 




This yields for the force 


F_{chem} 
= – 
dE_{Cryst}
dx_{ } 
– e^{ } · E_{x} – 
kT
n 
· 
dn
dx 



The chemical force will cause a particle movement
exactly as an electrical force (which is now a part of the chemical force). The
result is the same as in the basic treatment of the
electrical conductivity:
There will be a constant average drift velocity in the direction of the force
and we obtain 


<v_{chem}> = average velocity due to the chemical force =
const. · F_{chem}. 


For an electrical field E_{x} in xdirection,
we had
<v_{elect}> = average velocity due to the electrical force
F_{elect} = e · E_{x}.



<v_{elect}> must be a constant and we defined <v_{elect}>/E_{x} = mobility µ,
or 


<v_{elect}>
F_{elect} 
= 
µ
e 



Since the scattering processes that caused
<v_{elect}> to be constant are the
same for all forces, the proportionality constant between force and
average velocity must be the same, too. We thus can write 


<v_{chem}>/F_{chem} = µ/e or (dropping
indexes for convenience again): 


<v_{chem}> 
= v = 
µ
e 
· F_{chem} 




The electrical current carried by this velocity is 


j 
= e · n · <v_{chem}>
= n · µ · 
dE(x)_{Cryst}
dx_{ } 
+ e^{ } · n
· µ ·E_{x}(x) – µ
· kT · 
dn(x)
dx 



If we now consider the usual case of a
semiconductor with E(x)_{Cryst} = const., and a
zero net current (j = 0), we are left with 


e · n · µ · E_{x} – µ ·
kT · 
dn
dx 
= 
0 




The second term is an (electrical) current due to a
concentration gradient which, according to Ficks first law, always can be written as 





We thus can always
equate 





And this is the EinsteinSmoluchowski relation. 







Substituting this in the equation above, we get
exactly the same equation as
in the first derivation 


n · µ · E_{x} 
= D · 
dn
dx 



The consideration of the currents caused by the
chemical force, however, is much more general. The arguments used would also
apply for the case where E_{Cryst} is not constant and we
will come back to this when discussing heterojunctions or graded
semiconductors. 

