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Radioactive tracer atoms can be easily detected
whenever they decay, emitting some high energy radiation. |
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If the half-life time of the tracer
used is relatively small (but still large enough to allow an experiment before
the tracer has vanished), a large percentage of the tracer atoms can be
detected by their decay products - typically a, b, or γ-rays. We thus may have an extremely high
detection efficiency, many orders of magnitude below the detection limits of
standard methods. |
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Lets consider the general way a tracer experiment
is done: |
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Deposit a thin layer of the atoms that are to
diffuse on the (very clean) host crystal. Some of those atoms should be a
suitable radioactive isotope of the species investigated. Use any deposition
technique that works for you (evaporation, sputtering techniques, sol-gel
techniques ("painting it on")...), but make sure that the deposition
technique does not alter your substrate (sputtering, e.g., may produce point
defects) and that you have no "barrier layer" between the substrate
and the thin layer. |
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Anneal for a suitable time at a specific
temperature. |
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Remove thin layers from the surface (ideally one
atomic layer after the other) by, e.g. sputtering techniques, anodic oxidation
and chemical stripping, ultramicrotomes, chemical dissolution, ...). |
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Measure the radioactivity of each layer. |
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With the known half-life of the tracer and the
time since the deposition of the layer, calculate how many tracer atoms are in
your layer. From the measurement of many layers a concentration profile of the
tracer atoms results. |
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The rest is conventional: Fit the
profile against a standard solution of Ficks law or against your own solution
and extract the diffusion coefficient for the one temperature used. This gives one data point. And then: |
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Repeat the experiment for several other
temperatures, collecting data points for different temperatures. |
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From an Arrhenius representation of the measured
diffusion coefficients you obtain D0 and an activation energy for the tracer diffusion if
your data are on a (halfway) straight line. |
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If this sounds tedious, it's because
it is! You appreciate why students doing a master or PhD thesis are so
essential to research. Still, nothing beats tracer experiments when it comes to
sensitivity and accuracy. |
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There is, however, a basic
problem that we have to discuss if you want to extract information about the
vehicle of the tracer diffusion, i.e. about the vacancies or, in some cases,
interstitials from a tracer experiment. This is always the case when dealing
with self-diffusion. |
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The diffusion coefficient of the
tracer atom is not necessarily identical with the diffusion coefficient for
self-diffusion as defined for the vehicles - usually vacancies. |
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The reason for this is that the tracer is a
specific atom, while we look at many vacancies that help it along - and we must not
confuse the vehicle wih the diffusing impurity (or tracer) atom,
as noted before. In particular, the
jumps of the tracer atom may be correlated
with the jumps of the individual vacancy coming by. |
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In other words, whereas a particular vacancy may
(and usually does) jump around in a perfect random
walk pattern (i.e. each jump contributes to the mean square
displacement of the vacancy), the tracer
atom may not move randomly! |
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Lets look at a simple example for a
two-dimensional vacancy diffusion mechanism. |
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The tracer atom is marked in light blue, it has a vacancy as a
neighbor, a jump is possible. |
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Vacancy and tracer atom have exchanged their positions. |
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Next, the vacancy will jump again - with equal
probability on one of the 6 surrounding atom sites - so it is truly
doing a random walk. And one of those
jumps goes back to position 7, with exactly the same probability as to
the other available sites. |
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The "viewpoint" of the
tracer atom, however, is different. It will jump back to site 6 with a
higher probability than to the sites 1 -
5 because a vacancy is available on
6, whereas for the other sites the passing of some other vacancy must be awaited. There is a correlation between jump 1 and jump 2
- there is no random walk. The jumps back
will lead to wrong values of the mean square displacement, because this
combination does not add anything and occurs more frequently as it would for a
truly random walk. |
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The correlation effects between individual jumps
of the tracer atom and the random jumps of vacancies can be calculated by a
rigid theory of diffusion by individual jumps - an outline is given in the
advanced section. |
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As a result, these
correlation effects (in
all dimensions and for all lattice types) can be dealt with by defining a
correlation factor f that must
be introduced into the equations coupling the tracer diffusion to the vacancy
diffusion. |
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We define a correlation coefficient
f1V that allows to correlate the diffusion coefficient
for the (vacancy driven) self-diffusion,
DSD(T), as measured by
a tracer experiment, to the diffusion coefficient for
self-diffusion, DSd as given
by theory via the equation |
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As an example for a real correlation factor we
look at f1V(cub), the correlation factor for
self-diffusion mediated by single vacancies in a cubic lattice. It is given in
a good approximation by |
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f1V(cub) |
» 1
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2
z |
= |
5/6 fcc
3/4 bcc |
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With z = number of nearest
neighbors. |
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To illustrate the correlation phenomena, suppose
that f = 0. In this case, even for wildly moving vacancies
(DSD >> 0), the tracer atoms would not move - we
would not observe any diffusion. |
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This case is fully realized for one-dimensional diffusion, where it is also easy to
see what happens - just consider a chain of atoms with one vacancy: |
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The vacancy may move back and forth the chain like
crazy - the tracer atom (light blue) at most moves between two position,
because on the average there will be just as many vacancies coming from the
right (tracer jumps to the left) than from the left (tracer jumps to the
right). |
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Correlation coefficients can be calculated - as long as the diffusion mechanism and
the lattice structure are known. They are, however, very difficult to measure which is unfortunate, because they contain
rather direct information about the mechanism of the diffusion. The
calculations, however, are not necessarily easy. |
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Impurity atoms, which may have some interaction
with a vacancy, may show complicated correlation effects because in this case
the vacancy, too, does no longer diffuse totally randomly, but shows some
correlation to whatever the impurity atom does. |
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If a kick-out mechanism is active, the tracer atom
might quickly be found immobile on a lattice site, whereas another atom - which
however will not be detected because it is not radioactive - now diffuses
through the lattice. The correlation factor is very small. |
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Some examples for correlation coefficients are
given in the table for a simple vacancy mechanism (after Seeger). The correct value
from extended calculations is contrasted to the value from the simple formula
given above. |
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Lattice type |
coordination
number z |
f1V »
1 2/z |
f1V (correct) |
One dim. lattice |
Chain |
2 |
0 |
0 |
Two dim.
lattices |
hex. close packed |
6 |
0.6666 |
0,56006 |
square |
4 |
0,5 |
0,46694 |
Three
dim. lattice |
cub. primitive |
6 |
0,6666 |
0,65311 |
Diamond |
4 |
0,5 |
0,5 |
bcc |
8 |
0,75 |
0,72722 |
fcc |
12 |
0,83 |
0,78146 |
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There is a
plethora of methods, some are
treated in other lecture courses. In what follows a few important methods are
just mentioned. |
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Concentration Profile Measurements |
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Secondary Ion Mass
Spectrometry (SIMS) for
direct measurements of atom concentrations. This is the most important method
for measuring diffusion profiles of dopants in Si (and other
semiconductors). |
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Rutherford
Backscattering (RBS) for direct measurements of atom
concentrations. |
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Various methods for measuring the conductivity as a function of depth for
semiconductors which corresponds more or less directly to the concentration of
doping atoms. In particular:
- Capacity as a function of the applied voltage
("C(U)") for
MOS and junction structures)
- Spreading resistance measurements
- Microwave absorption.
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Local growth kinetics of defects, e.g. the
precipitation of an impurity, contain information about the diffusion, e.g.
- Growth of oxidation induced stacking faults
in Si
- Impurity -"free" regions around grain boundaries (because the
impurities diffused into the grain boundary where they are trapped).
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An example for a "diffusion
denuded" zone along grain boundaries can be seen in the
illustration |
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Annealing
experiments |
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These experiments are in a class of their own. In this case
point defects which have been rendered immobile in a large supersaturation, e.g. by rapid cooling from high
temperatures, are made mobile again by controlled annealing at specified
temperatures. Since they tend to disappear - by precipitation or outdiffusion -
measuring a parameter that is sensitive to point defects - e.g. the residual
resistivity - will give kinetic data. |
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A classical experiment produces supersaturated point defects
by irradiation at low temperatures with high-energy electrons (a few
MeV). The energy of the electrons must be large enough to displace single
atoms - Frenkel pairs may be formed - but not large enough to produce extended
damage "cascades". |
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Annealing for a defined time at a specified temperature will
remove some point defects which is monitored by measuring the residual
resistivity - always at the same very low temperature (usually 4K).
Repeating the sequence many times at increasing temperatures gives an annealing
curve. A typical annealing curve may look like this: |
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What "impurities" means in this context is left
open. They may form small complexes, interact with nearby vacancies or
interstitials, or whatever. |
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The interpretation of the steps in the annealing curves as
shown above is not uncontested. The "Stuttgart school" around
A. Seeger has a
completely different interpretation, invoking the
"crowdion",
than the (more or less) rest of the world. |
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Methods
measuring single atomic jumps |
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This ultimate tool can be used if the point defects have
rather low symmetry. The best example is the
dumbbell
configuration of the interstitial or interstitial carbon in Fe |
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In the classical experiment the crystal is uniaxially deformed
at not too low temperatures. The dumbbells will, given enough time, orient
themselves in the direction of tensile deformation (there is more space
available, so the energy is lower) and thus carry some of the strain. We have
more dumbbells in one of the three possible orientations than in the two other
ones (see below) |
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The tensile stress is now suddenly relieved. Besides the
purely and instantaneous elastic relaxation, we will now see a slow and
temperature dependent additional relaxation because the dumbbells will
randomize again. The time constant of this process directly contains the jump
frequency for dumbbells. This effect, which exists in many variants, is called
"Snoek effect". |
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If you do not use a static stress, but a periodic variation
with a certain frequency ω, you have a whole
new world of experimental techniques! |
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Last, there are methods which monitor
the destruction (or generation) of some internal
order in the material. The prime technique is
Nuclear Magnetic Resonance
(NMR), which monitors the
decay of nuclear magnetic moments which were first oriented in a magnetic field
and then disordered by atomic jumps, i.e. diffusion. The Mößbauer
effect may be used in this connection,
too. |
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© H. Föll