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
that acts on particles. We call it chemical because it tends to change particle numbers.F_{chem} |
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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 | |||||||||||||||||||||

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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 Quasi-Fermi energies, and E_{F}^{
e}, given by
E_{F}^{h} | |||||||||||||||||||||

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We thus have for the chemical forces | |||||||||||||||||||||

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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 , i.e. x and E_{C}
= E_{C}(x). What then determines
the numerical value of the band edge energy (for some defined zero point of the energy)? There are two factors:E_{V} = E_{V}(x) |
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The particular kind of semiconductor or crystal considered - this defines the band structure in general.
We call this part , 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 E_{Cryst} . Examples are materials
with compositions that change gradually (e.g. xSi-Ge alloys, or GaAlAs with "sliding" Ge or
Al concentration, respectively). | |||||||||||||||||||||

External or internal electrical field due to the electrostatic potential E_{x}
= – dV(x)/d x that
must be superimposed on the band energies as V( x)– |e| with V|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 in pink
here, to avoid confusion with the various energies_{x} | |||||||||||||||||||||

We thus can write | |||||||||||||||||||||

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This yields for the force | |||||||||||||||||||||

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

For an electrical field
in E_{x}-direction, we had x< v average velocity due to the electrical
force _{elect}> =. F_{elect} = e · E_{x} |
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<v must be a constant and we defined _{elect}
>< v mobility _{elect}>/E_{x} =µ, or |
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Since the scattering processes that caused < v to be constant _{elect}>are the same for all forces, the proportionality
constant between force and average velocity must be the same, too. We thus can write | |||||||||||||||||||||

< v or (dropping indexes for convenience again):_{chem}>/F
_{chem} = µ/e | |||||||||||||||||||||

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The electrical current carried by this velocity is | |||||||||||||||||||||

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

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

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We thus can always equate | |||||||||||||||||||||

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And this is the Einstein-Smoluchowski relation. | |||||||||||||||||||||

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

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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
is not constant and we will come back to this when discussing heterojunctions or graded
semiconductors.E_{Cryst} | |||||||||||||||||||||

Ohm's Law and Materials Properties

© H. Föll (Semiconductors - Script)