TITLE: Epitaxy
NAME: Dan Connelly
COUNTRY: United States
EMAIL: djconnel@flash.net
WEBPAGE: http://www.flash.net/~djconnel/
TOPIC: Elements
COPYRIGHT: I SUBMIT TO THE STANDARD RAYTRACING COMPETITION COPYRIGHT.
JPGFILE: epitaxy.jpg
ZIPFILE: epitaxy.zip
RENDERER USED:
POVRay Windows 3.0
TOOLS USED:
Paint Shop Pro 5.01, JPEG Optimizer, ImageMagick
RENDER TIME:
4 CPU-days
HARDWARE USED:
PII 266 MHz, 128 MB memory (primary)
HP Worksation, 100 MHz, 128 MB memory (secondary)
IMAGE DESCRIPTION:
see below
DESCRIPTION OF HOW THIS IMAGE WAS CREATED:
see below
tools used detail:
POV-Ray :
rendering and modeling
Paint Shop Pro 5:
signature
JPEG Optimizer:
JPEG conversion
ImageMagick:
combine
======================================================================
NOTES
theme notes:
introduction
-------------
The ruling class of the periodic table is column IV. Carbon, the
lightest member, is the basis for life in its incredible diversity of
molecular forms. As a crystal, it mechanical and aesthetic properties
make it among the most values substances throughout human history.
But as a crystal, it places second to the next member of the IV family :
silicon. Developments of silicon technology have revolutionalized human
society during the past 40 years.
This image shows the interior of a chemical vapor deposition system
for silicon epitaxy -- the thermally activated growth of crystalling
material from hydrides. However, what is being formed isn't exactly
silcon, but something newer : a zincblend germanium silicon
crystal. GeSi as an alloy has received considerable attention over
the past 15 years due to its superior electronic transport properties
and due its application in "band engineering". However, as an alloy,
it presents some difficulties. An ordered crystal would solve
some of these, and would provide a clear advantage in CMOS
technology. Yet its formation is too difficult -- entropy's pull
is too strong. So it remains an idea.... a concept.... a goal.
chamber
-------
The distant background is dominated by the quartz reaction chamber.
It must be transparent to allow penetration of the light from the
lamp array above and below. This light is the heat source which controls
the reaction rate. It is perhaps ironic that quartz is itself
a compound of silicon -- silicon begets silicon.
gas
---
The reaction is generated from SiH4, GeH4, and trace amounts of B2H4
in an ambient of H2. Atoms are colored according to a scale based
in spectral theory : red for the less tightly bound germanium valence
electrons, and blue for the more strongly bound silicon atoms. Hydrogen,
with its relatively neutral role, is white. Boron, being metallic, is
also colored white (slightly transparent due to its reduced valence).
In the foreground the rather large tetrahedral SiH4 and GeH4 molecules
are clearly visible. Due to its lesser reactivity, more SiH4 must be used
than GeH4, and thus these are shown in 2:1 molecular concentration ratio.
The H2, despite being more numerous still, are more difficult to spot
due to their small size. The B2H4 is considered too rare (its high
reactivity and trace concentration in the crystal means very little
is needed) to be seen and thus is not rendered.
As the nanometers of the foreground give way to the decimeter-distances
to the chamber, the gas loses its discrete visibility and becomes
a gaseous continuum. The red and blues of the Ge and Si average into a
purplish haze which pervades the scene.
crystal
-------
This, clearly, is the focus of the image. The zincblend lattice
(an extension of the diamond lattice), despite the simplicity
of its specification, is a wonderful mix of cubic and hexagonal
symmetries which the raytracer is excellently suited to reveal.
Here is a viewpoint quite close (approx 13 nm) from the edge of
the wafer. To the right is visible a step on the surface -- it
is via such steps that epitaxy typically occurs, due to the
catalytic effect of the corners. The corner of the step is visible
in the distance -- it hasn't yet reached the wafer edge. The step
in this case is a double one -- quite rare in practice, but this
is our lucky day :).
The structure is formed from sp3 hybrid orbitals -- four ellipsoids
per atom. The ellipsoids from adjacent atoms merge in a covalent
bond with two electron states of complementary spin. Layers of
germanium and silicon atoms alternate -- again the same colors are
used : red for germanium and blue for silicon. The reaction is limited
by hydrogen atoms terminating dangling surface bonds. The surface layer
is here silicon, with approximately 800f the surface bonds H-tied.
Note the crystal surface is highly simplified -- in reality there
are complex surface reconstructions which reduce the number of
free states per atom to one from the two shown here. Additionally,
in current alloy deposition technology, there are other complex
surface reactions taking place which are neglected.
The perfection of the lattice is broken, however, by the presence
of an occasional boron atom (here represented as white). With
only three available valence electrons, boron creates a net
electron deficiency which manifests itself as extended quantum states
called "holes". These act as the conduction quanta in "p-type" material,
shown here.
The hole gas is shown as a green cloud permeating the crystal,
fading exponentially away from the surface. Even though the
doping level in this crystal is quite high, the hole gas is
still quite dilute relative to the free carrier gas found in metals.
It is the control of this carrier gas, the ability to attract and
repel it, that makes it so useful in semiconductor switches.
* * * * * * * * * * * * * * * * * * * * * * * *
technical notes:
First, the boring stuff... The chamber is CSG from superellipsoids
textured with a modified quartz texture. An array of six lights placed
at the top surface of the chamber interior lights the scene. They
are visibly represented by toroids at over the top of the chamber...
they affect the scene only in reflections.
The hole gas is done with halos. The main crystal has a single large
halo with an exponential density function at the top surface. The
atoms in the step have a separate halo, each with a simple spherical
density function combined into a single, very expensive, halo field.
Rendering of the middle part of the image is at under 2 lines per
hour as a result of this structure.
The rest is blobs.
To assist with the definition of the structure, TMPOV
(http://twysted.net/) was used for its array support (POV 3.1 Beta also
has arrays, but I chose 3.0 instead due to its UNIX support and for
my greater familiarity with halos at the time). Arrays were used
to define an extended unit cell, which was stepped to create the
entire crystal.
The dimensions of the crystal are 25 x 50 x 10 of these extended
unit cells, each containing four atoms, each containing four sp3 orbitals.
This would be 200k blob components -- way too much for practice.
Thus several optimizations needed to be made:
1. the depth of the crystal was made nonuniform : it increases linearly
from zero to 10, then decreases exponentially to the wafer edge. This
strategy was chosen to make sure there was reasonable depth along
channeled directions (those with long lines of site without obscuring
atoms).... at shallow angles, only atoms near the surface are
visible anyway.
2. The breadth of the crystal was reduced by making the x extent
in proportion to the z distance -- this more efficiently handles
the field of view.
3. It was rendered in two pieces, which were assembled The first phase
rendered the top of the scene, including only the top 1-2 layers of
atoms. The second phase handled the foreground and those gas molecules
close enough to cast shadows -- the full depth of the crystal was
rendered, but it wasn't rendered to full distance. This two-phase
scheme required some care in use of random number streams. The
first phase was done on an HP workstation, while the second was done
on my home PC. Each took close to 48 hours with full CPU usage.
Note this isn't in violation of the IRTC rules. Glenn McCarter
used a similar "layered" technique in the nature round
( http://www.irtc.org/ftp/pub/stills/1998-06-30/dominion.txt ).
The source code also has a "phase 0" for the rendering of the whole
thing in one shot.... 256MB is needed, I suspect.
The gas is handled by blobs for the foreground molecules, spheres for
the more distant molecules, and a fog for the continuum in the far
distance. Atmosphere was tried, here, for greater effect, but it
interacted in undesirable ways with the halos. The fog is much faster,
as well.
Of interest is the strange coloration of the bonds between the silicon
and germanium atoms. These were intended to be a smooth fade from
red to blue, due to the "as advertised" nature of blobs. However, I
suspect the approximations imposed by the POV root solver resulted
in some anomalous coloration. A bug? No -- a feature! These are
but quantum mechanical fluctuations in the valence field :).
* * * * * * * * * * * * * * * * * * * * * * * *
artistic notes:
The composition of this scene actually is quite similar to my
last entry -- http://www.flash.net/~djconnel/POV/baseball.jpg .
A strong foreground with a right focus extends into the distance,
where the background has a left focus. Asymmetry with a touch
of balance...
* * * * * * * * * * * * * * * * * * * * * * * *
footnote on camera :
Assuming a chamber temperature in the range of 600C, the mean
speed of the silane molecules is approximately 860 m/sec. Since
the width of the near field of view is approximately 10^-8 meters,
each pixel is approximately 10^-11 meters in width. This means
the shutter on the camera is open on order 10^-14 seconds. With
light speed at 3 10^10 cm/sec, this means the shutter can move no
more than 1.5 micrometers in each direction. Clearly this camera
doesn't have a mechanical shutter....
=======================================================================
APPENDIX I : source files
camera.inc : generic definitions of camera parameters
camera.pov : generic definition of cameras
epitaxy.txt : this file
final.ini : a file with some settings used in the final rendering
(phase 2, in this case)
header.inc : a generic header file
main.pov : the main scene
=======================================================================
APPENDIX II : rendering stats (phase 2 only)
main.pov Statistics (Partial Image Rendered), Resolution 800 x 600
----------------------------------------------------------------------------
Pixels: 320800 Samples: 1312272 Smpls/Pxl: 4.09
Rays: 1474661 Saved: 1768 Max Level: 5/12
----------------------------------------------------------------------------
Ray->Shape Intersection Tests Succeeded Percentage
----------------------------------------------------------------------------
Blob 17821938 8307319 46.61
Blob Component 1168266263 122375575 10.47
Blob Bound 14463783035 3935823745 27.21
Box 40610483 40610483 100.00
CSG Intersection 46996820 43564302 92.70
CSG Merge 23498410 23498357 100.00
Superellipsoid 93993640 83748302 89.10
Torus 11783144 0 0.00
Torus Bound 11783144 0 0.00
Bounding Box 2008677095 33595489 1.67
Light Buffer 2928364429 2928359137 100.00
Vista Buffer 22068446 21562807 97.71
----------------------------------------------------------------------------
Roots tested: 15145420 eliminated: 0
Calls to Noise: 12902610 Calls to DNoise: 10
----------------------------------------------------------------------------
Halo Samples: 16843657970 Supersamples: 0
Shadow Ray Tests: 25806648 Succeeded: 16861490
Reflected Rays: 312
Refracted Rays: 240
Transmitted Rays: 161909
----------------------------------------------------------------------------
Smallest Alloc: 26 bytes Largest: 9120716
Peak memory used: 114422529 bytes
----------------------------------------------------------------------------
Time For Parse: 0 hours 3 minutes 39.0 seconds (219 seconds)
Time For Trace: 45 hours 23 minutes 35.0 seconds (163415 seconds)
Total Time: 45 hours 27 minutes 14.0 seconds (163634 seconds)