
In a naive (and wrong) view, enough negatively charged carriers in
the material would move to the surface to screen the field completely, i.e.
prevent its penetration into the material. "Enough", to be more
precise, means just the right number so that every field line originating from
some charge in the positively charged plate ends on a negatively charged
carrier inside the material. 


But that would mean that the concentration of
carriers at the surface would be pretty much a d function, or at least a function with a very steep
slope. That does not seem to be physically sensible. We certainly would expect
that the concentration varies smoothly within a certain distance, and this
distance we call Debye length right away.



As you might know, the Debye length is a crucial
material parameter not only in all questions concerning ionic conducitvity (the
field of "Ionics"), but whenever the
carrier concentration is not extremely large (i.e. comparable to the
concenetration of atoms, i.e in metals). 

We will now derive a simple formula
for the Debye length. We start from the
"naive" view given above and consider its
ramifications: 


If all (necessarily mobile) carriers would pile
up at the interface, we would have a large concentration gradient and
Ficks
law would induce a
very large particle current away from the interface, and, since the particles
are charged, an electrical current at the
same time! Since this electrical
diffusion
current j_{el, Diff} is proportional
to the concentration gradient
–grad (c(x)), we have: 


j_{el, Diff}(x) = –
q · D · grad (c(x)) 




With D = diffusion coefficient. Be
clear about the fact that whenever you have a concentration gradient of mobile
carriers, you will always have an electrical current by necessity. You may not
notice that current because it might be cancelled by some other current, but it
exists nevertheless. 

The electrical field E(x), that caused the concentration
gradient in the first place, however, will also induce an electrical
field
current (also called drift current)
j_{field}(x), obeying Ohms law in the most
simple case, which flows in the opposite
direction of the electrical diffusion current. We have: 


j_{field}(x) = q
· c · µ · E(x) 




With
µ =
mobility, q = charge of the particle (usually a multiple of the
elementary charge e of either sign); q · c ·
µ, of course, is just
the conductivity s 


The total
electrical current will then be the sum of
the electrical field and diffusion current. 

In equilibrium, both electrical currents obviously must
be identical in magnitude and opposite in sign for every x, leading
for one dimension to 


q · c(x) · µ
· E(x) = q
· D · 
dc(x)
dx 



Great, but too many unknowns. But, as
we know (????), there is a relation between the diffusion coefficient
D and the mobility µ that we can use; it is the
EinsteinSmoluchowski
relation (the link leads you to the semiconductor Hyperscript). 





We also can substitute the electrical Field
E(x) by –
dU(x)/dx, with U(x) = potential
(or, if you like, voltage) across the system. After some reshuffling we
obtain 


– e 
dU(x)
dx 
= 
kT
c(x) 
· 
dc(x)
dx 
= kT · 
d [lnc(x)]
dx 




We used the simple relation that d
(lnc(x)) / dx = 1/c(x) ·
dc(x)/dx. This little trick makes clear, why we always
find relations between a voltage and the logarithm of a concentration. 


This is a kind of basic property of ionic
devices. It results from the difference of the driving forces for the two
opposing currents as noted before:
The diffusion current is proportional to the gradient of the concentration whereas the field
current is directly proportional to the concentration. 

Integrating this simple differential
equation once gives 


U(x) + 
kT
e 
· ln c(x) = const.





Quite interesting: the sum of two functions of
x must be constant for any x and for any functions
conceivable; the above sum is obviously a kind of conserved quantity. 


That's why we give it a name and call it the
electrochemical potential
V_{ec} (after muliplying with e so we have energy
dimensions). While its two factors will be functions of the coordinates, its
total value for any (x,y,z) coordinate in equilibrium is a
constant (the three dimensional
generalization is trivial). In other words we have 


V_{ec} 
= 
V(x) + 
kT 
· ln c(x) 




with V(x) = e ·
U(x) = electrostatic potential energy. 


The electrochemical potential thus is a real
energy like the potential energy or kinetic energy. 

Obviously, in equilibrium (which means that nowhere in the
material do we have a net current flow) the
electrochemical potential must have the same value
anywhere in the material. 


This reminds us of the Fermi energy.
In fact, the electrochemical potential is nothing but the Fermi energy and the Fermi distribution in
disguise. 


However, since we are considering classical particles here, we get the classical
approximation to the Fermi distribution which is, of course, the
Boltzmann
distribution for
E_{F} or V_{ec}, respectively,
defining the zero point of the energy scale. 

This is easy to see: Just rewriting
the equation from above for c(x) yields 


c(x) = exp – 
(Vx) – V_{ec}
kT 




What we have is the simple Boltzmann distribution for classical particles with
the energy (Vx) –
V_{electrochem}. 


