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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. |
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But that would mean that the concentration of carriers at the surface would be
pretty much a δ- 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. |
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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). |
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We will now derive a simple formula for the Debye length.
We start from the "naive" view given above and consider its ramifications: |
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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 jel, Diff is proportional
to the concentration gradient –grad (c(x)), we have: |
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jel, Diff(x) = – q · D · grad (c(x))
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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. |
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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) jfield(x), obeying Ohms law
in the most simple case, which flows in the opposite direction of the electrical diffusion
current. We have: |
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jfield(x) = q · c · µ ·
E(x) |
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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
σ |
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The total electrical current will then be the sum of the electrical field and diffusion current. |
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In equilibrium, both electrical currents obviously
must be identical in magnitude and opposite in sign
for every x, leading for one dimension to |
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q · c(x) · µ · E(x)
= q · D · |
dc(x) dx |
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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 Einstein-Smoluchowski relation (the link leads you
to the semiconductor Hyperscript). |
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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 |
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– e |
dU(x)
dx | = |
kT c(x) |
· |
dc(x) dx |
= kT · |
d [lnc(x)] dx |
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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. |
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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. |
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Integrating this simple differential equation once gives |
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U(x) + | kT e
| · ln c(x) = const. |
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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. |
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That's why we give it a name and call it the electrochemical
potential
Vec (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 |
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Vec | = |
V(x) + | kT |
· ln c(x) |
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with V(x) = e · U(x) = electrostatic potential energy. |
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The electrochemical potential thus is a real energy like the potential energy or kinetic energy. |
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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. |
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This reminds us of the Fermi
energy. In fact, the electrochemical potential is nothing but
the Fermi energy and the Fermi distribution in disguise. |
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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 EF or Vec,
respectively, defining the zero point of the energy scale. |
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This is easy to see: Just rewriting the equation from above for c(x)
yields |
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c(x) = exp – |
(Vx) – Vec kT |
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What we have is the simple Boltzmann distribution for
classical particles with the energy (Vx) – Velectrochem. |
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