mse 486, what a challenging cause. Such hard working can only I succeed! Anyway, I would continue to try.
Prof. Afromowitz argues with me
You raise some interesting issues.
With regard to the Boltzmann-Matano technique, the equation I gave in class
is directly applicable to the case of pre-dep diffusion, but it must be
modified to apply to drive-in diffusion. Take a look at the article at
http://doc.tms.org/ezMerchant/prodtms.nsf/ProductLookupItemID/MMTA-9910-2605
/$FILE/MMTA-9910-2605F.pdf?OpenElement
Actually, only the first page of this article is displayed at that URL, but
the essence of the method is described. It says that the point x=0 must
occur at the "Matano interface", which is the plane at which the amount of
impurity diffusing out of the left-hand region is equal to the amount of
impurity diffusing into the right-hand region. For a pre-dep diffusion,
this plane is at the surface of the wafer. But for a drive-in, the Matano
interface is below the wafer surface (within the silicon wafer), and
although it is hard to locate exactly, it eliminates the problem you point
out at the surface of a gaussian distribution, where the derivative of N(x)
goes to zero.
With regard to the digital integration of the diffusion equation with
non-constant D, there are certainly other complications to consider. One
obvious problem you did not address is what happens if the medium in which
the impurity is diffusing is made up of layers having abrupt interfaces, and
different diffusion coefficients. Then, mathematically, the derivative
dD/dx is a delta function at those interfaces. However, there are boundary
conditions at the interface. The impurities, in equilibrium, must satisfy
certain thermodynamic properties, similar to the segregation coefficient,
ko, that we saw at liquid-solid interfaces. For solid-solid interfaces, it
is usually called m (equal to C1/C2, the ratio of the concentrations on the
two sides of the interface). Also, one must guarantee that the flux on the
two sides are equal, so that there is no continuing build-up of
concentration at the interface. Thus, these conditions need to be
incorporated into the simulation of diffusion. (The problem is similar to
what happens when an electric field impinges on an interface between two
materials having a different dielectric constant. The electric flux normal
to the interface is continuous.) If the diffusivities vary slowly with x or
with N, then the extra terms that you suggest are needed would be second
order corrections. You might try to do this type of problem using your
computer model, one with the extra terms and one without, and compare the
results and see which makes sense.
Thank you for your thoughts on this problem.
Prof. Afromowitz
Prof. Cao’s comments about my project
Wen Hao,
It is great that you already succeeded in measuring the impedance and simulating
the circuit. It is a very good start.
The data is reasonable, and stranger. We experienced many weired results and
more than often we had to re-do the measurements several times. However, simple
repetition does not warrant a better result. We need to consider several issues:
(1) the electrodes we put on the samples. Sometime the electrodes de-attached
from the samples. Other times, silver penetrated inside the samples. Making
things more complicated is that we hardly can tell what really happened with the
electrodes. In most cases, we can rely on our guess. (2) the capacitance from
the electrode wires sometime can play an importance role as well. We had such
problems in our early studies and we had to physically space two electrode wires
far apart. (3) temperature equilibrium may not be reached inside the samples if
we raise the temperature too fast and don’t wait for sufficient time to let the
sample reach an equilibrium temperature distribution. Inhomogeneity in
temperature is one of the leading causes for stranger impedance data.
More specifically to your own data, even the three sets of the more reasonable
results are questionable. In general, resistance for both grain boundaries and
grains decreases drastically with an increasing temperature (follow Arrhenius
equation), but the interface (between the electrode and sample) resistance would
increase slightly with an increasing temperature. Your data does not follow this
trend.
Please check your experiment procedures and the sample to see what are the
possible explanations.
But you have a very good start and it is a bit surprise to me that you have gone
that far already.
Cao
Best Regards!
flash teaching
After having spending a whole afternoon, I have finally finished teaching these 2 guys flash. It is fun to be a good teacher. But my throat is somewhat drying.
wonderful day!
Haha, I have eventually successed in learning the horrible software MatLab.
I would be the master of software
in MSE.
Hehe
Thankgoodness
Haha, today I have found such a great place where I can place all my photos on.