 | Nernst law is a special answer to the general
and important question: |
| |
How do charged and mobile particles redistribute themselves in an electrical
potential if there are some restrictions to the obvious solution that they all move to one or the
other pole of the field? |
|
|  | It is the answer to this question that
governs not only pn-junctions, but also batteries, fuel cells, or gas sensors, and, if you like, simply all junctions. |
 | Let us consider a material
that essentially contains mobile carriers of only one kind, i.e. a metal
(electrons) , a (doped) semiconductor (electrons or holes, depending on doping), or
a suitable ionic conductor (one kind of mobile ion). |
|  | We imagine that we hold a positively charged plate at some (small) distance to the surface of a material
having mobile negative charges (a metal, a suitable ionic conductor, a n-doped semiconductor, ...). In other
words, the positively charged plate and the material are insulated, and no currents
of any kind can flow between the two. However, there will be an electrical field, with field lines starting at the
positive charges on the plate and ending on the negative charges inside the material. We have the following
situation: |
| |
|
 | 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 jel, Diff is proportional to the concentration gradient –grad (c(x)), we have: |
| |
jel, 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)
jfield(x), obeying Ohms law in the most simple case, which flows in the opposite direction of the electrical diffusion current. We have: |
| |
jfield(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 Einstein-Smoluchowski 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 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 |
| |
Vec | = | 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 EF or Vec, 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)
– Vec kT |
|
|
|  | What we have is the simple Boltzmann distribution for classical particles with the energy (Vx) –
Velectrochem. |
| | |
| |
 | We may thus assume within a very good approximation that the carrier density at any point is
given by the constant volume density c0 of the field free material, plus a rather small space dependent addition c1(x);
i.e. |
| |
|
|  | Obviously, only
c1(x) is important for Poissons equation. |
 | From Boltzmanns distribution we know that |
| |
c(x)
c0 | = 1 + | c1(x) c0
| = exp | æ ç è | - | D(energy) kT | ö ÷ ø | = exp |
æ ç è | – | V(x) kT | ö ÷ ø |
|
|
|  | because the difference in energy of a
carrier in the field free volume (i.e. where we have c0) is simply the electrostatic energy
associated with the electrical field. |
|  | Since we assumed c1 << c0, we may with impunity express the exponential function as a Taylor series
of which we only retain the first term, obtaining: |
| |
1 + |
c1(x) c0 | » 1 + | V(x) kT |
|
|
 | This is a simple trick, but important.
Feeding the result back into Poissons equation yields: |
| |
d2 [c1(x)] dx2 | = | e2 · c0 · c1(x) e · e0 ·
kT |
|
|
 | For a simple one-dimensional case with
a surface at x = 0 we obtain the final solution |
| |
c1(x) =
c1(x = 0) · exp – | x d |
|
|
 | The quantity d is the
Debye length we were after, it is obviously given by |
| |
d = Debye
length = |
æ ç è | e ·
e0 · kT e2 ·
c0 | ö ÷
ø | 1/2 | |
|
|
 | The Debye
length is sometimes also called Debye-Hückel
length (which is historically correct and just). |
|  | c1(x = 0), of course, is given by the boundary condition, which for our
simple case is: |
| |
c1 (x = 0) = c0
· | V (x = 0) kT |
|
|
| What is the meaning of the Debye length? Well, generalizing a bit, we
look at the general case of a material having some surplus charge at a definite
position somewhere in a material |
|  | Consider for
example the phase boundary of a (charged) precipitate, a charged grain boundary in some crystal, or simply a (point)
charge somehow held at a fixed position somewhere in some material. The treatment would be quite similar to the one-dimensional case given
here. |
 | What we know now is quite
important: |
|  | If you are some Debye lengths away from
these fixed charges, you will not "see" them anymore; their effect on the equilibrium carrier distribution
then is vanishingly small. |
|  | The Debye length
resulting in any one of these situations thus is nothing but the typical distance needed for screening the surplus
charge by the mobile carriers present in the material. |
|  | In other words, after you moved about one Debye length away from the surplus charge, its effects on the
mobile charges of the material are no longer felt. |
 | More about the
Debye length can be found in the
Hyperscript "Semiconductors". |
| |
|
© H. Föll (Electronic Materials - Script)