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In a typical diffusion experiment, some impurity atoms are introduced into a host
by first putting them (ideally) with d - distribution at the surface. |
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After annealing for a specified time at a specified temperatures, some diffusion
of the impurity atoms will have produced a diffusion profile, i.e. a smooth curve
of the concentration
c vs. depth x in the sample (usually plotted as lg(c) - x curve).
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Some experimental care is necessary. Simply depositing the impurity atoms on the surface of
the host crystal may not lead to any results, because e.g. an impenetrable oxide layer may prevent any diffusion of the
impurity atoms into the crystal. "Shooting" the impurity atoms into a surface-near area via ion implantation will
overcome that problem, but may create its own problems by generating point defects which change the regular diffusion behavior. |
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There are some well established standard methods for measuring the diffusion profiles after
a successful diffusion experiment (see below). Fitting the profile to the applicable solution of Ficks law will provide
two results:
- The numerical value of the diffusion coefficient D for the set of parameters considered.
- The validity of Ficks law for this case as evidenced by the quality of the fit.
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Of course, any "macroscopic" method for measuring profiles relies on having a profile
on a, lets say, 10 µm scale in the first place, i.e. each impurity atom must have made many
individual jumps. |
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This will in most cases only happen at sufficiently high temperatures. Waiting a long time
is not very effective; this is immediately clear if looking at diffusion phenomena in a slightly
different way. |
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How then do we get experimental data at small concentrations or small numbers
of jumps? The answer is: use radioactive tracer atoms as the diffusing atoms, that can
be found and identified in extreme small concentrations! |
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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 g-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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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, but here I won't go into that. |
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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(Theo)
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 – |
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 (See also chapter 4.2.1) |
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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 w,
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. |
© H. Föll (Defects - Script)
