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Point defects generally are mobile -
at least at high temperatures. They are the vehicles that make the atoms of the
crystal mobile - point defects are the cause of solid state diffusion. |
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Many products of modern technology
depend on solid state diffusion and thus on point defects. Some examples are:
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Microelectronics and Optoelectronics. |
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Solid state Sensors, e.g. the oxygen sensor regulating the
emissions of your car. |
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Solid state batteries,accumulators and fuel
cells. |
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High strength
materials. |
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The concentration of point defects,
their specific kind (including impurity atoms), their migration parameters,
equilibrium or non-equilibrium conditions and the atomic mechanisms of
diffusion determine what you get in a specific solid state experiment involving
diffusion. |
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Small wonder that many diffusion phenomena are
not yet totally clear! |
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If we use the term
"diffusion", we always and exclusively mean that something moves around in a random fashion,. i.e.
does a random walk. |
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The "something" in the context of this lecture is an
atom moving around randomly in a crystal.
Generally, it can be any particle or even
quasi-particle we like - atoms, molecules, electrons, positrons, photons,
phonond, excitons, ... |
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Lets quickly go over the basic laws
of diffusion which were discovered by Adolf
Fick on a phenomenological base long before
point defects were known. The starting point is
Ficks 1.
Law, stating: |
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The flux
j of diffusing
particles (not necessarily atoms) is
proportional to the gradient of their
concentration, or |
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The index i refers to the particular
particle with number i observed; D is the diffusion
coefficient of that particle. |
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Note that even this purely
phenomenological description applies to everything - e.g. liquids - as long as
we discuss diffusion and not, e.g., some
kind of flow. |
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This means that the underlying dynamics of the
particles on an atomic scale is essentially
random
walk. |
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The derivation of the simple continuum equation
above from the primary events of random scattering (causing random walk) of
many discrete particles takes a lot of averaging. If you don't know how it's
done (or forgot), do consult the
proper
(english) modul of "Introduction to Materials Science II" (and
the links from this modul). |
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If there are several interacting
particles, the formulation of Ficks 1. law must be more general, we
have |
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With µ =
chem. potential;
M = mechanical mobility. |
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Since the gradient of the chemical potential may
be different from zero even for constant concentrations, special effects as,
e.g., uphill diffusion are contained
within this formulation. |
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The next basic equation is the
continuity
equation. It states: |
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Changes of
the particle concentration within a volume element must express the difference of what goes in to what goes out - we
have conservation of the particle number
here. In mathematical terms this means |
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This is if course only true as long as no
particles are generated or annihilated (as, e.g., in the case of electrons or
holes in an illuminated semiconductor). |
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Combining the two equations from
above we obtain Ficks 2. law: |
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The temporal
change in concentration at a given point is proportional to the
2nd derivative of the concentration,
or |
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¶c
¶t
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= div (D · Ñc) = |
D · Dc |
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With the final equation being valid only for
D = const. |
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Ficks equations look innocent enough,
but solutions of the rather simple differential equations forming Ficks laws
are, in general not all that simple! They do follow some general rules,
however: |
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They involve almost always statistical functions,
as well they should, considering that diffusion is a totally statistical
process at the atomic level. |
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The solutions to heat conducting problems are
quite similar, as well they should, because the conduction of heat can be
treated as a diffusion phenomena. (it actually
is a diffusion phenomena). |
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This link gives some
more information about Fick's laws and standard solutions. |
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Many diffusion phenomena can be dealt
with on the phenomenological base of Ficks laws. All that is required, is to
know the diffusion coefficient and its dependence on temperature and possibly
other variables - you do not have to know anything about the atomic mechanisms involving point defects to solve
diffusion problems. |
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It turns out, however, that complex diffusion
problems - e.g. the simultaneous diffusion of B and P in
Si can not be modeled adequately without knowing the atomic mechanisms
and their interaction. This explains the impetus behind major efforts to
unravel the precise mechanisms of diffusion in Si and other
semiconductors. |
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All we have to know
about random walk is the
general relation between the average distance <r2> covered by a randomly "walking" object
(which we often also call diffusion
length L) , the number
N of steps made, and the (average) distance
r0 covered in one step. |
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For three-dimensional random walk we have quite
generally |
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We now must link the phenomenological
description of diffusion (that only works on averages and thus only if many
particles are considered) with the basic diffusion event, the single jump of a
single atom or defect. |
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We describe the
net flux of particles as the difference in
the number of particle jumps to the left and to the right. With the jump
frequency
n we obtain Ficks 1. law with an expression
for the diffusion coefficient (for cubic crystals), a
detailed derivation is given in
the link. |
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With a = lattice
constant, n = jump frequency, i.e. the number
of jumps per second from one position to a
neighboring one . |
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g is the
geometry factor of the lattice type
considered. It takes into account that considering all jumps that are possible
in the given lattice, only some have a component in the
x-direction. Its definition is |
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g is always about 1 as you
will find out doing the exercise, so we will not consider it any more. |
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The jump frequency
n = N/t (= number of jumps
N per second)
is given by |
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with Gm =
free enthalpy for the jump or for the migration of the atom or defect. n0 is the frequency of "attempts"
to overcome the enthalpy barrier for a jump; it is, of course, the vibration
frequency of the lattice atoms, i.e. around 1013 Hz. |
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This gives us the
second important parameter set describing a property of a point defect, namely
its migration energy and
entropy |
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All we have to do is to express
Gm = HM –
TSM, with HM = migration
enthalpy (or -energy), and SM = migration entropy.
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The magnitude of the migration
entropy will be comparable to the formation entropy because it has the same
roots. It is thus around 1 k for "normal" crystals. |
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Combining everything, we obtain an
expression for the diffusion coefficient D in terms of the
migration energy: |
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Where all constant (or nearly
constant) factors have been included in D0. Some
numerical values are given
in the link. |
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These formulas relate the atomic
properties of defects to the diffusion coefficient from Ficks laws. |
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There is one more
expression of prime importance when it comes to diffusion. It brings together
statistical considerations from looking at random walk (which is exactly what a
vacancy does) as given above with the diffusion coefficient.
All we have to do is to express N by the diffusion
coefficient. |
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What we get is the famous
Einstein -
Smoluchowski
relation (for 3-dimensional diffusion). |
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With L = mean square displacement or
diffusion length, t = time since start of the diffusion (or, if the
particle "dies", e.g. by recombination in the case of minority
carriers in semiconductors, its lifetime). |
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Einstein
derived this in 1905 in a slightly more general form: |
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With r = vector
between "start" and "stop" of the diffusing particle for
the time t; <r2> is thus the average of the square of the mean
displacement (this is something different
from the square of the average!), and g* is some factor "in the
order of 1", i.e. 2, or 6, depending if the diffusion
is 1 -, 2 - or 3 -dimensional and what kind of symmetry
(cubic, etc.) is involved. |
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Now let's do an exercises: |
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© H. Föll
