MathSource Reviews

Diamonds and Gallium Arsenide

Edited by Matthew M. Thomas

This issue of Mathematica in Education and Research (né Mathematica in Education) marks the fifth anniversary of this journal. Those inclined to send gifts to TELOS in commemoration of this event are reminded that wood (traditional) and silverware (modern) are apropos for fifth anniversaries. In seventy years, the appropriate gift will be not wood but diamond, for subscribers to both traditional and modern standards. MathSource offers little by way of gift guides, but, vis-à-vis wood and diamond, it does contain a Material Science directory (pub/Applications/MaterialScience, to be exact). Through March 1997, items in that directory involved not wood and diamond, but gallium arsenide and diamond. Let us now examine the former.

BACKGROUND

Through March 1997, there were two items in the MathSource MaterialScience directory: 0200-316, Crystal Structure Graphics by Wolfram Research Inc., and 0202-059, Gallium Arsenide Crystal Graphics by Alastair McLean of Queen’s College in Kingston, Ontario, Canada. The WRI contribution, accompanied by surprisingly little documentation for a WRI contribution to MathSource, “provides a nice picture of the crystal structure of diamond,” to cite a phrase from the contribution’s introduction. But that work in diamond structure partially inspired McLean’s efforts in gallium arsenide (GaAs) crystal structure. In the 4-kB GaAs-README.txt file that resides in directory 0202-059, McLean lauds the clarity with which CrystalStructure.m (3 kB, in residence in 0200-316) yields a PostScript-rendering of the diamond lattice. His own MathSource item uses Mathematica to render the (110) plane of the GaAs structure.

Why be concerned with the GaAs crystal structure? Because GaAs is one of the most versatile semiconductors currently in use. A compound semiconductor, GaAs comprises elements gallium (Ga) from Group III-A and arsenic (As) from Group V-A of the Periodic Table. Other compound semiconductors formed from elements in these Groups include indium antimonide (InSb) and aluminum phosphide (AlP). Compound semiconductors can offer advantages not found in their pure element counterparts such as silicon (Si). GaAs, to wit, is a much more efficient X-ray detector than is Si (important for radiography applications). Also, GaAs--but not Si--is able to emit light.

Elaboration upon the above information, as well as a colorful illustration of the GaAs zinc-blende structure (a superstructure of the diamond structures), can be had from MIT web page www-dmse.mit.edu/wom//CompoundSemiconductors.html. Just note that this page uses European standard Periodic Table labeling, thus placing Ga, In, and Al in Group III-B, and As, Sb, and P in Group V-B. National Center for Computational Electronics page www-ncce.ceg.uiuc.edu/tutorials/crystal_structures/gaas.html also illustrates the GaAs crystal structure, with related views available to those with 3-D web browsers. In addition, NIH page ntp-server.niehs.nih.gov/htdocs/Results_status/ ResstatG/11192-E.html provides chemical health and safety information (toxicity, carcinogenicity, et al) on GaAs. The interested reader is urged to compare the works just cited with that to be discussed.

ITEM 0202-059

For the Gallium Arsenide Crystal Graphics item, let us first return to the file GaAs-README.txt. The more README files I encounter, the more their usefulness seems to be on the wane. Alastair McLean’s README file proved a pleasant reversal in that sad trend. His affiliation (Queen’s University Physics Department) is cited, as are his postal and email addresses. The work is dated 13 February 1992. In an overview, McLean first pays homage to the aforementioned CrystalStructure.m item from WRI, and describes how it partially inspired his efforts to render the zinc-blende (110) surface of GaAs in Mathematica. He then alludes to his own research in semiconductor surface/interface electronic structures, and explains his need for models of epitaxial antimony and bismuth overlayers on GaAs (110). In the absence of such models, he created his own, corroborated by electron diffraction test results for atom positioning. Thus this MathSource item. McLean concludes his README file with short but informative descriptions of the packages and notebook his item comprises.

McLean’s MathSource item consists of said README text file, seven packages (entitled GaAs1.m through GaAs7.m in the order suggested), and a 70-kB Mathematica notebook originally created for use with the Macintosh. While the packages create lattice model graphics, the notebook unifies the item packages (using the <<filename.m command) with explanatory text and useful journal references. As he wrote in the README file, so McLean writes in the notebook introduction of his motivation for creating this item: The need to illustrate the epitaxial adlayer structure formed by depositing an antimony or bismuth monolayer onto the GaAs (110) plane, and the publication of these illustrations in technical journals.

And so this item provides these illustrations. McLean describes how the zinc blende structure of GaAs can be created using the diamond structure as a template. One need only substitute equal numbers of anions and cations into the diamond lattice, so as to create a zigzag anion-cation chain parallel to the (110) plane. Figure 1,

Figure 1 - Anion-Cation Chains Parallel to GaAs (110) Plane.

created using GaAs1.m, illustrates that chain. And when you add another (110) section, as GaAs2.m does, Figure 2 results (compare it to Figure 1).

Figure 2 - Addition of Another (110) Section to GaAs Structure. 

Note--as McLean does--that this GaAs structure also represents that of compound semiconductors InSb and AlP; they also crystallize into the zinc-blende crystal structure. In these two and subsequent pictures, cation Ga is rendered gray, while anion As is black.

As one would expect, Figure 3 is obtained from GaAs3.m via the Show[ ] command. Figure 3 is just Figure 2, but with interplanar bonds connecting the first and second layers.

Figure 3 - Interplanar Bonds Connect GaAs Layers. 

By changing ViewPoint on Figure 3, McLean illustrates that one can examine the Mathematica-rendered lattice from a number of perspectives. Package GaAs4.m is then used to illustrate relaxed (110) surfaces, in which bond angle rotation forces As top-layer atoms upward, and forces Ga top-layer atoms downward.

Recall that the item’s aim is to illustrate an epitaxial adlayer formed through using the GaAs (110) plane as a substrate. In specific preparation for that, GaAs5.m adds a third (110) section to the structure shown in Figure 2, with all interplanar bonds connected as in Figure 3. Figure 4 illustrates GaAs5.m output rendered through Show[],

Figure 4 - Addition of a Third (110) Section to GaAs Structure.

with atomic positions unrelaxed. By altering the shading of Figure 4’s top layer, and by adding the relaxation of the top and second layers, one creates an acceptable illustration of bismuth/antimony overlayers on the GaAs (110) surface. GaAs6.m provides that illustration -- compare the shading of the top-level atoms in Figure 5

Figure 5 - Bismuth/Antimony Overlayers on the GaAs (110) Surface.

with that in Figure 4. Finally, Figure 6 through GaAs7.m illustrates relaxation in the top two levels, but using relaxation parameters from a paper evidently absent from the notebook bibliography.

Figure 6 - Relaxation in the Bismuth/Antimony Overlayers. 

The illustrations in this notebook have made their way into articles published in Physical Review B, Journal of Vacuum Science and Technology, and Applied Surface Science; the notebook bibliography contains the details.

SUMMARY

“Notebook” has long been the metaphor for graphics-enabled files created in Mathematica. As regards McLean’s notebook, this metaphor is most apt. A technical notebook tends to be rife with information, though not necessarily in the most meticulously organized manner. There are indeed passages within GaAsNotebook.ma that are a bit incongruous: Some references appearing in the explanatory text fail to find their way into the notebook bibliography; some surface science-related discussion is put forward without the background information an outsider to surface science might welcome. But then, is such not the hallmark of a thorough technical notebook--compiled not for straight reading but to supply information when the compiler wants it? What you get in GaAsNotebook.ma ... what is created in GaAs1.m through GaAs7.m ... has found roost in the pages of the technical journals for which the material was originated. Kudos to Alastair McLean for sharing his notebook through MathSource.

McLean does more for MathSource than just share his Mathematica notebook. He promotes the spirit in which MathSource came to be, by calling for others to share their information through this means. To quote from the introduction to GaAsNotebook.ma: “It would be convenient for example to have a library of the common crystallographic bulk and surface structures available.” As McLean has done for GaAs and its cousins, so he invites others to do for other compounds: Render models of them through Mathematica, and make them available through MathSource. It’s sad when a well-motivated plea falls on deaf ears—the date of the Gallium Arsenide Crystal Graphics item is February 1992, yet it remains the newest member of the pub/Applications/MaterialScience directory. Then again, however, the rendered models called for by McLean might yet make their way onto the World-Wide Web, which is, after all, the larger repository.

ABOUT THE EDITOR

Matthew M. Thomas took his D.Sc. in chemical engineering from
Washington University in St. Louis in December 1995. He 
serves as the University Liaison for the R&D arm of a large
aerospace manufacturer in the midwestern United States.