Alternative Derivation of the Einstein Relation

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 Fchem 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 one-dimensional case we thus have
Fchem, x  =  F  =  –   chem
dx
 =  –   dEF
dx
Of course, we will never confuse µchem, the chemical potential, with µ, the carrier mobility!
Looking at the most general case with only local equilibrium in the bands, we use the Quasi-Fermi energies, EFe and EFh, given by
EFh   =    EC  –  kT · ln  Neeff
ne
       
EFh   =    EC  +  kT · ln  Nheff
nh
We thus have for the chemical forces
Fe  =  –   dEFe
dx
 =  –  dEC
dx
 +  kT 
ne
 ·  dne
dx

Fh  =  –   dEFh
dx
 =  –  dEA
dx
 +  kT 
nh
 ·  dnh
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. EC = EC(x) and EV = EV(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 ECryst, and note, while ECryst 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. Si-Ge alloys, or GaAlAs with "sliding" Ge or Al concentration, respectively).
External or internal electrical field Ex = – dV(x)/dx due to the electrostatic potential V(x) that must be superimposed on the band energies as – |e|V 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 Ex in pink here, to avoid confusion with the various energies).
We thus can write
dEC
dx 
 =  dECryst
dx 
 +  e · Ex
This yields for the force
Fchem  =  –  dECryst
dx 
 –  e  · Ex  –   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
<vchem> = average velocity due to the chemical force = const. · Fchem.
For an electrical field Ex in x-direction, we had
<velect> = average velocity due to the electrical force Felect = e · Ex.
<velect> must be a constant and we defined <velect>/Ex = mobility µ, or
<velect>
Felect
 =  µ
e
Since the scattering processes that caused <velect> 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
<vchem>/Fchem = µ/e or (dropping indexes for convenience again):
<vchem>  =  v  =   µ
e
 · Fchem
The electrical current carried by this velocity is
j  = e · n · <vchem>  =  n · µ ·   dE(x)Cryst
dx 
  +  e  · n · µ ·Ex(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 · Ex  –  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
j  =  –  e · D · dn
dx
We thus can always equate

D  =  µ · kT
e

And this is the Einstein-Smoluchowski relation.
 
 
 
Substituting this in the equation above, we get exactly the same equation as in the first derivation
n · µ · Ex  =  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 ECryst is not constant and we will come back to this when discussing heterojunctions or graded semiconductors.
 

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