 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}. 
  