Segregation and Striations in CZ Silicon
|General Remarks, Macrosegregation, and Controlling Crystal Growth|
|Segregation in Silicon, a ultra-high purity material, seems to be an oxymoron. What, exactly, is there to segregate?|
|A lot of atoms other than silicon, And a lot
of vacancies and self-interstitials! Most prominently are the dopant atoms, specified by the
customer, usually either arsenic (As), boron (B) or phosphorous (P), in concentrations between
1014 cm3 to 1019 cm3, which translates
to about 0.5 ppb to 20 ppm. One ppm (= 1/1000 ppb) translates to 0.0001at %, far below anything
interesting in iron and steel. |
Then we have 1 ppm oxygen (O) and roughly 0.1 ppm carbon (C) around, plus the rest of the periodic table at concentrations below ppt or ppqt (look it up).
Not to forget, close to the melting point we have the equilibrium concentration of vacancies and self-interstitials. We don't know exactly how large those concentrations are - but they must be around or just below ppm.
|The customer of silicon wafers specifies the allowed bandwidth of the average doping concentration per wafer (e.g. ± 5 %) and the maximum local deviations he can tolerate within one wafer. Same thing for the oxygen concentration. It should be, for example 6.5 · 1017 cm3 ±10 %.|
|The message is clear: With respect to segregation it is not just the absolute concentration
that counts, but the deviations from some specified value, however small. |
The steps for making a Si crystal are covered in this link. Here we assume the role of the crystal grower. We need to worry about:
|The question we must ask ourselves is: What can we crystal growers do about this?
What kind of "buttons" do we have that we can play with to produce an optimal crystal?
I'm not talking science and theory here. I'm talking about standing in front of a big machine
and setting the controls to get started, and adjusting the controls while the process is running
(for many hours, by the way).
Well, before we talk about fixes to a problem, we first need to understand the problem. We are serious technicians and not politicians, after all.
|As far as macrosegregation is concerned, we only need to look at the effective segregation coefficients to see that we are in serious trouble. Here is the figure from the "Nirvana Si" module once more:|
|Oxygen is not included because its concentration is not constant. The melt slowly dissolves the SiO2 lined crucible and thus gradually increases the oxygen concentration, precluding analysis by the usual theories. Nevertheless, oxygen does show strong segregation effects.|
|The problem is clear: Some effective segregation coefficients
are extremely small, and some change a lot with increasing growth speed. The first property
will lead to strong macrosegregation because in the beginning
of solidification the concentration is the crystal is far lower then the nominal one, whereas
towards the end it must be far larger. |
The second property leads to large concentration changes with fluctuations in the growth velocity, i.e. to strong micro segregation effects.
|Not so good! So what can be done to ameliorate those unwanted segregation effects? What's in the crystal growers tool box? Let's look at the schematic drawing of a Czochralski crystal-puller again , this time with adjustable parameters shown in red.|
|Let's start to look at the parameters, in particular the "easy " ones
that actually do have a button on the control panel. What we have are: |
|You guessed it! The first law of applied science asserts itself once more: There is nothing more practical than a good theory! Fiddling with
any button changes everything
- not just some segregation property. You need a good model or theory that can be cast in
software and then "automatically" adjusts all parameters all the time to values
that are "just right". We have the Goldilock principle at work here, and you, the operator, are not even allowed to touch
the machine as soon as it has been started up and runs on some pre-selected routine. |
The major parameters to work with are the rotation rates and the magnetic field - and this are the difficult parameters, where it is far from obvious what some changes in the settings will produce.
|The industry does have good working theories and
models (that are not much talked about) and can do amazing things with respect to the homogeneity
of its huge crystals. There are limits, however. The article in the link will go into details
For example, the segregation coefficients of gallium (Ga), aluminum (Al), or antimony (Sb) are much smaller (and thus worse) than those of arsenic (As), boron (B) and phosphorous (P). That is one of the reasons why these elements are (almost) never used for doping a crystal.
|All you can do is to go for the best compromise. Rotating crystal and crucible is essential for almost everything concerning the uniformity of the crystal at large - but that also produces the striations and thus specific microsegregation as we shall see in the next section.|
|Microsegregation and Limits|
|This part is heavily connected to the "Segregation Science" module cluster. You should at least browse through these modules before reading on. I cannot dwell at length at all the special words and topics that come up here but have their roots in the segregation science modules.|| |
|What we always find in silicon crystals is a striated distribution of everything we care to analyze. In other words, the maxima and minima of the concentrations found on a planes that follow the shape of the solid-liquid interfaces. Illustrative pictures and more explanations to the geometry can be found here.|
In a simple model the loci of extreme concentrations form
a system of nested bowls that have about the shape of the solid-liquid interface at that point
in time when the respective "bowl" solidified. |
In a less simple but more accurate model, the loci of extreme concentrations are on a tightly wound spiral akin to long drill chips, looking roughly like this (excuse my limited drawing skills):
|Nested bowls or spiral - it maks hardly a difference when you look at a cut in radial direction. Structures like that can only be caused by a rhythmic, quasi-periodic modulation of the average growth speed <v>!|
|Did you realize the conundrum I just posed? No! Then let's quickly recall what we know about casting; we'll need it:|
|The decisive quantity controlling
the solid-liquid interface geometry (planar, cellular, dendritic) for all
solidification processes is the ratio of the temperature gradient G and
the growth velocity v, G/v. Both parameters change a lot during casting
and subsequent cooling. It is obvious that they cannot be controlled
directly and independently - no dials for that on your casting machine. You do get macro-
and microsegregation in cast objects, and microsegregation may appear in the form of striations. |
In this module I discussed striations in some detail and posited that rhythmic fluctuations of the growth speed occur either due to external disturbances of a periodic kind, or because some kind of self-organization of local "stochastic oscillators" at the more or less chaotically moving solid-liquid interface occurs
In stark contrast to casting, when we grow a silicon crystal (or any other crystal) with the Czochralski method we have:
|In short: we have ideal conditions for testing all those theories. I'm not aware that this has been done, however. Part of the reason for this is that big silicon crystals are just far to valuable for sacrificing them to test some theories that are not all that important for the silicon business.|
|Anyway, the conundrum is: How can the growth speed fluctuate in a quasi-periodic "rhythmic" fashion if it and everything else is solidly fixed?|
|The answer is: because of the rotation!
The macroscopic growth speed vmac is indeed fixed to a precise and constant
value but that need not be true for the microscopic growth speed vmic! It
could fluctuate around vmac as long as <vmic> = vmac
. The question is: why should it do that? |
The answer is: We have a well-defined axis of rotation, and we have lateral temperature gradients. The temperature decreases from a maximum value somewhere in the center of the freshly grown crystal to smaller values on the outside. This gradient has a radial symmetry, meaning that the way the temperature goes down from the center to the outside is about the same in all radial directions. That implies that the temperature gradient also has an axis: the loci of the maximum temperatures, running roughly up the center of the crystal.
The rotation axis and the thermal gradient axis are thus almost identical - but never exactly so. It's just a bit hotter on one side of the crucible than on the other one for all kinds of possible reason. What the growing crystal then experiences is comparable to this:
|When we "pull" the
meat for a doener kebab, the radial thermal gradients, as shown above, are far more asymmetric
than during pulling of a Si crystal. It's just much hotter in the back compared to the front.
That means that any point on the doener meat surface experiences a rhythmic up-and-down in
And so does any point on the solid-liquid interface ! All the points on the interface that experience the maximum temperature at the same time are found on a line from the center straight to the outside (a radius in other words).
When we pull the crystal up, the "hot line" screws up into the crystal, forming the kind of spiral shaped surface that I tried to picture above. The distance between subsequent striations is constant and given by the ratio of growth rate and rotation rate.
|Serefe! We have microsegregation in silicon crystals covered!|
|Or do we? Not really. On occasion people grow crystals in some CZ set-up without rotating anything - and find striations! The NASA plus the German Space Agency and probably other lunatics too, grow crystals up there in the spacelab to figure out exactly what causes these nonrotational striations 2). Obviously, looking into this is worth some 50 million $ or more to somebody. In essence, these non-rotational striations are blamed on peculiars of the melt flow, even in small liquid droplets that crystallize. Experimental evidence for that comes from the fact that magnetic fields, that interferes in some tricky but well.understood way with the flow in the liquid, can wipe out striations, induce them, or make irregular once quite regular - depending on what you do with it.|
|Nevertheless, the doener model goes a long way in explaining the main features of striations in CZ-pulled Si crystals correctly. Here are two pictures of striations in silicon single crystals:|
|In the right-hand picture we probably see the effects of atomically dissolved impurities
including the intentionally introduced doping element and the still present impurities. |
The right hand picture shows "big" defects, mostly agglomerates of self-interstitials that nucleated at some segregated impurity (possibly carbon in this case).
|We have, it appears, a relatively clear idea of how striations in (some kinds of) single crystals come about. They exist because their is some rhythmic "distortion" impressed on the system. We do not need really self-organization.|
|Well, yes - but I have a strong feeling that we would get self-organized striations,
too, if we were to stop all that rotation business and just grow (somewhat warped and crooked,
to be sure) crystals straight from the melt. |
I can't prove that, however, so I stop here and must point out that there is a somewhat involved advanced module that gives an example of exactly how self-organization can lead to rhythmic pulsing of an electrochemical system, and what we might be able to learn from that for striations in solidified objects like wootz steel.
|1)||That's how you toast to your Turkish friends, drinking beer and eating a doener kebab (the S should actually have a little hook called cedille on the bottom). Alternatively your can say: Prost, Cheers, Kampai, Skål, or just shut up and enjoy your beer.|
|2)||Helmut Kölker: The Behavior of Nonrotational Striations in Silicon; Journal of Crystal Growth 50 (1980) 852-858|
|Link Hub: Segregation Science|
|Special Modules in Segregations Science:|
|1. Basics of Segregation|
|2. Constitutional Supercooling and Interface Stability|
|3. Supercooling and Microstructure|
|4. Segregation at High and Ambient Temperatures|
|Segregation in Silicon This module|
|Microsegregation and "Current Burst" theory|
6.2.2 Solidification and the Art of Casting
Defect Etching in Silicon
11.5.2 Structure by Dendrites?
Segregation at Room Temperature
Units of Length, Area, and Volume
Segregation in Silicon
Producing "Nirvana" Silicon or Nearly Perfect Silicon Single Crystals
Single Crystal Silicon
The Story of Self-Interstitials in Silicon
1. Basics of Segregation
The Goldilocks Principle
Segregation at Room Temperature
Dislocation Science - 2. The Reality
Microsegregation and "Current Burst" Theory
Constitutional Supercooling and Interface Stability
© H. Föll (Iron, Steel and Swords script)