Silicon: Lattice spacing <111><100> Thermodynamics and Bloch theorem

[andrewr]

Hi,
I had a question that maybe someone might know, and that although I have been researching it I am not finding enough information on the web that would solve the issue. (it's the end of the month, too... and being broke and in a hurry is a problem too...)

The project is aimed at saving lives of our USA military (not killing anyone) -- and is guaranteed to do that -- but it's overall usefulness and deploy/logistics issues need to be improved or it might not get used effectively -- and, as I am not in this for the money or with money -- I have some final hurdles to cross before I can improve the item to a level I am *confident* in..

Size and heat are two issues that can be improved, but I lack some specific information to design a robust compensation circuit for unusually hot and cold temperatures (antartica anyone?)

It all boils down to accurate thermodynamics of Silicon... (not exotic new semiconductors...hybrids)

The effective mass of electrons in semiconductor material show a directional dependency; in most texts there are two orientations listed out for for E(k) diagrams. The directions are <111> and <100>
From common knowledge, it's evident that the lattice (atomic) spacing is different depending on which way one observes the crystal. The lattice spacing produces the band changes and affects the energy gap (Bloch theorem too); so it is ideal for use in a simplistic model to attempt to design a better version and look for flaws in the models I have.

I found the "lattice spacing" for silicon (nearly pure) from 0K to melting online in a table -- but it is only for one direction in the crystal and I don't know if it is in vacuum or not. (I'm still happy with it...!) At 300K the chart reads 0.5431092 angstrom... Looking at the chart, I noticed that near 2K, there is a reversal in the contraction rate much like water at 4C begins to expand again before forming ice. This suggests to me that there is a dynamic motion of of the molecular bonds which packs silicon more tightly depending on relative angle (not just vibration length) of the bonds at different temperatures. That makes me uncomfortable with just assuming that the lattice size changes uniformly (scalar) in both directions -- I can (and will) model with that assumption; but I would prefer to KNOW that is a safe assumption by reviewing some studies...

So, I am wondering if anyone has come across the same kind of thermodynamic data (preferably as spacing and not as temperature coefficient) for a different direction and knows where it might be buried on the web. I am interested in graphs (as long as they are big enough to make reasonable estimates from), tables (I love those!), curve fits -- (less trustworthy) -- for silicon that show the nonlinear expansion effects in other crystal directions OR Information for doped silicon in the same or other direction when the doping concentration is given. I can sort out the dopant later...but knowing the approximate dopant level (1 order of magnitude is OK) is important...

Most industry doping is off +-100% from manufacturer to manufacturer anyway. And multiple manufacturers of parts is a good thing for sustainability... (I'm not going to be making the parts...)

Thanks in advance.

 

[DrDu]

Silicon crystallizes in the cubic system. The effective mass is a tensor in general, however, in the cubic system it reduces to an isotropic scalar, i.e., it is not direction dependent.
Furthermore, Silicon melts at 1414 degrees Celsius, so at 2K, it will be an isotropic liquid. So there is no lattice and no different lattice directions anymore.

 

[andrewr]

Silicon crystallizes in the cubic system. The effective mass is a tensor in general, however, in the cubic system it reduces to an isotropic scalar, i.e., it is not direction dependent.
Furthermore, Silicon melts at 1414 degrees Celsius, so at 2K, it will be an isotropic liquid. So there is no lattice and no different lattice directions anymore.

<edited heavily...less violence potential... :) >

OK, and how about at two Kelvin? The bump ends at absolute zero. I'm sorry if my capital K confused you.
AFAIK -- silicon would be solid, unless I missed something. (I'm trying to be clearer this time, as I probably confused others too.)
When applying the bloch theorem in general, there is directional dependency. The effective mass computed for an isotropic *approximation* is a combination of effects from at least two orientations of the crystal.

Are you giving me a "theoretical" answer -- or do you have some actual data to back up what you are saying? The non-linearity of lattice spacing change as a function of temperature is a fact -- and so empirical data is the only way I know to get a guaranteed answer to the question; cf: <will edit when I find link again>

Google Docs at the moment has Robert R Reeber, Kai Wang, Materials Chemistry and Physics. 46 (1996) with an extremely useful lattice spacing table in addition to many historical studies' data-points graphed and curve fitted on pp.262
The data shown below is not lattice spacing but a thermal coefficient of expansion, and it looks like...http://img837.imageshack.us/img837/9563/sitherm.jpg Uploaded with ImageShack.us

You can see the expansion coefficient change direction at around 75K and begin expanding again.
That is what is similar to water at 4C (above freezing), whereas silicon being a solid (that gas laws are often applied to by analogy) at around 75K shows signs similar to fluid with non-orthogonal bond angles being near a "phase" change. It would be more correct for me to say that the 2K point is secondary reversal or endpoint of the reversal by appearances.

Regarding the lattice shape, sure it's not only cubic system, but 'diamond'. It is not a rigid cube lattice like rock salt. There are spaces which line up non-uniformly depending on which way the crystal is turned. That's why I mentioned <100> and <111> which are the two commonly graphed electrical orientation of interest -- although I think there may be a third commonly considered orientation; there are L,Γ,Χ endpoints on a typical E(k) diagram. And I can't remember off the top of my head how crystallographers denote the remaining portion of the diagram as I don't see it commonly. (and when I do it's blurry...)

[URL]http://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Silicon-unit-cell-3D-balls.png/220px-Silicon-unit-cell-3D-balls.png[/URL]

What I am concerned about is the correlation between lattice spacing when viewed from different angles; as I said, I'm going to use the working assumption that no matter which way one views the crystal -- that the lattice constants will scale together -- but I'd like to verify that because the nonlinear distortions in coefficient of expansion suggest (to me) that there is a twisting of some kind over temperature, which in turn makes me think that the twisting may or not be uniform or perhaps has periodic tendencies that are out of phase from one direction to another so that the average would be correlated but there would be distinct changes (periodic?) in the correlation as temperature was decreased from 400K.

As an over simplified example (non quantum), one might take a straight piece of spring steel (a needle with flattened ends) and compress it axially -- at some point it will *usually* bow along some random radial direction. The compression was uniform along the axis, but the distortion went into the empty space around the spring. If a true cube lattice were made, and compressed along all axii simultaneously -- the same type of effect could be expected to happen -- and some prediction could be made as to the *net* effect by symmetry and continuity (non quantization) of macroscopic springs -- however in a microscopic diamond pattern (obviously having more complications than a non-quantum spring) -- I am uncertain what would happen and 1D calculations based on say the <111> "!" orientation while assuming (falsely) that the <100> orientation has the exact same E(K) relationship produces large enough errors that I can spot them in physical experiments that don't go much below room temperature...

Thanks for your reply in any event -- I'll take it as a vote in favor of the simplest model.

Oh, and on second thought -- another question -- are you saying that no one except Intel has bothered to try the experiment, and there are no publications to find?

 

[andrewr]

Well, Memorial day is coming to an end -- so most of us can go back to forgetting about people who made an ultimate sacrifice to protect our way of life. It was good to think about those who suffered for us, at least for one day;

I'll close by just posting a sample E(k) diagram that is missing something, but I couldn't make out exactly what. Perhaps tomorrow, someone who spent today remembering -- will be free to make a difference.

http://img837.imageshack.us/img837/5950/siliconek.png

 

[DrDu]

I have no idea why the expansion behaves as shown by you at deep temperatures, however, I am only a layman as far as silicon is concerned. But I doubt that there is any feature in the behaviour of silicon which hasn't been investigated both experimentally and theoretically in utmost detail given it's tremendous importance in industry. So maybe someone else can answer this question (would be interested to know the reason, too).
Sorry that I misunderstood you writing about 2K (I thought you meant K for kilo).
However, I was not talking about an isotropic approximation. Effects of anisotropy in the cubic system are only seen when going beyond tensors of rank 2 (which includes the tensor of effective mass). This does not mean that anisotropy isn't important in silicon, only that it will not show up in effective mass.
Btw. this is an international forum on physics. So while your interest in US military is certainly honourable, it is off topic.

 

[andrewr]

However, I was not talking about an isotropic approximation. Effects of anisotropy in the cubic system are only seen when going beyond tensors of rank 2 (which includes the tensor of effective mass). This does not mean that anisotropy isn't important in silicon, only that it will not show up in effective mass.

Sure, I understand what you meant now.

Btw. this is an international forum on physics. So while your interest in US military is certainly honourable, it is off topic.

No offense to the international community of physicists -- and I didn't specify an exclusive military that the effort was for; nor even for the allies, though your assumption is understandable for I clearly don't exclude the US.

It is inevitable that I will have to cite the national systems of information to reference documents; even (for example) old documents of Werner Heisenberg which are not of U.S. origin. I have only found one reference to a study on silicon, that I haven't access to, that I would like to see. Being as it is from a US company -- and the US was on holiday yesterday (except for perhaps Intel in China) -- my comment was simply a hope that the search engines will point some relevant people (US intel ex-employees?) in this general direction after the get off holiday. As the forums don't appear to even enforce their own rules, and I saw no specific rule to the effect of revealing identifying information of which country a physics experiment or engineering feat was done, I see no reason to worry about the comment. People will choose to be angered or not regardless of my intentions.

I, too, hope in having my question answered in an OPEN forum where all are free to see and use the information. I wouldn't post on the internet information I don't want to share. If there is some rule I missed, I'd be happy to at least read it. Cheers.

 

[andrewr]

And so the question remains...

I'll just collects bits and pieces until I have enough to figure out some solution (even if not the right one...).

According to "Materials Science and Engineering" (An Introduction) -- one of those shallow courses, by William D. Callister, Jr. -- He says: "The Cv is zero at 0K, but it rises rapidly with temperature; this corresponds to an increased ability of the lattice waves to enhance their average energy with ascending temperature." (pp. 661)

What he is getting at is that the phonon (or quantized thermal/acoustical) vibrations have a larger number (not necessarily density) of states as the temperature rises. eg: the molecular bonds (covalent) in a crystal lattice have a bond energy vs. radii of separation; and this effectively makes a quantum well; so that the NUMBER of discrete energy levels a particle "in" the well could have increase with temperature.

Eg:

Given a potential energy diagram of bond energy, at 0K the energy must be that at the bottom of the potential energy trough.

This being a 1D quantum mechanical well (bonds are more/less linear) the energy states that the bond may "have" are going to follow the energy distribution of a simple harmonic oscillator. Eg: constant energy increase between each energy level.

So, at the low temperature limit -- Cv=AT**3 -- all crystals behave the same qualitatively with respect to temperature and heat capacity. At higher energies, I suppose, the linearity of the bonds becomes questionable for there will be transverse forces as well as linear spring (Young's modulus) due to random vibrations. eg: the density of states of vibration modes increases with crystal dimensions, and therefore with temperature too -- for crystals generally expand with heat.

The author of that book also makes an astute observation: fig 20.3 on pp. 664; "For a symmetric potential energy-versus-interatomic distance curve, there is no increase in interatomic separation with rising temperature."

So, if the well were symmetrical the average energy point would be at the same location, X. However, in real crystals -- the bond potential energy has a steeper slope in compression than in expansion, so the crystals favor expansion with increased temperature.

It would seem, then, on a bond by bond basis that the spring constant (Young's modulus/elasticity) would scale with the average *slope* of potential energy vs. an average potential energy; whereas the crystal *expansion* (lattice) would scale with the average POSITION of the well width for a given energy. The two measurements taken together could be used to make a crude characterization of silicon bond lengths.

Most treatments of heat capacity use what is known as a debye temperature (θD) as an empirical value characterizing a material's (overall) heat capacity. There are excellent resources right here in the UK on that: see: "PHONON HEAT CAPACITY", Lecture 11, A.H. Harker -- Physics and Astronomy
UCL; By Google search. I'm not sure I agree with his statement that the cutoff frequency is "fudge factor", it's more than a "fudge" factor; it has a physical significance with respect to packet velocity of phonons.

But, here's where my question starts to get interesting; I think it possible to model the spring constants and all of silicon bonds, and get an idea of what thermal stresses there are in silicon. Silicon, with respect to Young's modulus, is an ANISOTROPIC crystal. Therefore, it will expand differently and contract differently with respect to applied direction of stress with respect to the crystal lattice and with compression or tension.

Some papers indicate up to six different Young's moduli for silicon; and this is only at room temperature. So, I am pretty sure anisotropic effects in bond lengths with respect to <100><111> etc, are going to occur over temperature as I originally suspected. And remember, computation of effective isotropic mass is accomplished by lattice parameters; so the two are coupled.

Does anyone know of experiments where the mechanical elasticity constant vs. temperature are graphed for silicon, or doped silicon, (Google searchable?).

 

[DrDu]

Yes, you can certainly get some information on the anharmonicity of the bond potential from Youngs modulus and thermal expansion as a function of temperature.
I would try "M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Oxford UP, who discusses these topics at length.
You are also right that the anisotropic effects can be seen in the tesor of Young's moduli, which is a 4th rank tensor. If silicon were completely isotropic, this tensor would only have 2 independent components, so the remaining 4 are due to anisotropic effects.

 

[andrewr]

Dr.Du !
Thank you. Your recommendation is confirmed, the book is absolutely superb!
In chapter 3, the main consideration that I came to by a different route -- namely, the nearest neighbor and second neighbor effects are sufficient for calculation of basic localized lattice interactions -- corresponds to the orbital distance of a trapped electron not thermally excited in silicon semiconductor at an impurity site.

In chapter 4, the same approach that I wanted to take -- Simple Harmonic Analysis -- is quite obviously going to be enabling for me to finish my project. It is in this analysis, that a discussion of the 3 dimensional effects of vibrating oscillators begins. In silicon, although the thermal expansion of the whole cube would expected to be isotropic -- the compressibility of silicon is not; and therefore one can expect different nominal SHO frequencies for differing directions.

I never thought of using the energy of the cohesion as a way to work the oscillator problem sort of "backward" without defining a "box" size that the nucleii are trapped in. His approach is sure to be more accurate than mine.

I'll see how well it agrees with experimental results, and focus on that expansion change at 75K in silicon to see if I ought to have been concerned about it or not. I'll keep you posted!