applications and analogies Conducting Midshipmen-A Classroom Activity Modeling Extended Bonding in Solids Joseph F. Lomax U.S. Naval Academy Annapolis, MD 21402 The solid is recognized a s one of the primary states of matter, but it is often treated a s a n outcast in the general chemistry courses taught in high school and college. This is often due to the perceived difficulty in explaining the concepts of extended bonding and structure in solids. These concepts are needed to understand the properties of . . insulators, s&niconductors, and metals. The familiar two-electron two-center bond that works no well in isolated molecules does little to explain the behavior of many of the most common solids in our technical world. For example, in a computer, the power cord is a plastic-covered copper wire, which is connected to a computer whose major component is the silicon chip. So we have a n insulator protecting a metallic conductor, which brings current to a semiconductor. Part of the difficulty for teachers of chemistry is that solid state science typically has been considered the province of the physics community. The success of the free electron and band models of conduction, however, has been a t the expense of the local view of electron sharing with which chemists are so comfortable. Conductivity Conductivitv in a solid has one of the largest dvnamic ranges of' any physical meilsurement. ~monshc.el&nents it w r i e s from 5 x 10.'" S n n for sulfur to 6 x 10" S cm for silver ( 1 ) .Despite this wide range, all compounds and elements can be placed into one of three classes: insulators, semiconductors, and metals. These conductivitv classes are unified. in the most microscopic form, by the necessity of charge movement from atom to atom. Because the most basic form of charge is the electron and because electrons reside in orbitals around an atom, this is the most logical place to start the analogy. Modeling the Solid A model is a n analogy that helps us visualize, often in a simplified way, something that cannot be directly observed (2). I n its most rudimentary form, a n orbital is a place where a n electron might reside, so a model for this could be a chair in which a person might or might not be sitting.' If this analogy is used in a classroom, the person would be a 'Strictly speaking, there should be two electrons per orbital with different spins. This could could be described by an S-shaped Love Seat. This analogy ignores this complication because this type of seat is uncommon in classrooms. Nevettheless, this description can be a useful aside in discussing orbitals. ZThisanalogy ignores the existence of impurity states between the valence and conduction bands in extrinsic semiconductors because they are difficult to model in a classroom. One can assume that the model describes semiconductors in which the impurity states have energies that overlap those of the host bands.
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Journal of Chemical Education
edited by
RON DELORENZO Middle Georgia College Cochran, GA31014
student-and, in my case a t the Naval Academy, a midshipman. J u s t as a n electron is confined to a distinct orbital, the options for sitting in a chair are restricted. For example, you cannot sit a foot above the chair or half-way through the chair: You must either lower your potential energy and fall to the chair, or raise your potential enough to sit in the chair. I t seems reasonable to describe a chair a s a simple model for a n isolated valence orbital. A two-centered bond would be modelled by two people sitting in separate chairs bolding hands. However, we must now expand our concept of a n orbital to create a physical model of a n extended solid. There will now be less energy between the orbital states and easier movement between them. An ensemble of orbitals in a n extended solid can be compared to a room full of chairs. The change in potential energy for movement between chairs must be depicted as quite small. Thus, the chairs must overlap so that eventually, for a one-dimensional case, we have a bench-r, in physics parlance, a band. The Electron-Hopping Model Although it is easy to build a conceptual bench out of chairs, i t is not practical to build a real bench in a classroom. This forces us away from a model based strictly on band theory (people on a bench) to a n electron-hopping model (people moving between chain).
I n a room full of chairs with several students, not even one student will be able to move about the room if it is totally crowded or, ironically, totally empty. For convenience, this situation can be simulated by a full row in a partially filled room. Similarly, in a n extended solid, when equivalent orbitals on all atoms are either all full or all empty, nocharge can move among the atoms through thew orbitals. Tlui is analogous to a n insulator. Semiconductors Providing a mechanism to model the charge movement is quite easy. All you need is empty chairs. One can do this in three ways. Displace same students to an adjacent roam. 'Involve fewer students. Involve more students,but put all newcomers in an adjacent and nearly empty room where they have the space to move around. These correspond to a n intrinsic semiconductor, and the p-type and n-type extrinsic semiconductors, respectively. The original and adjacent rooms correspond to the valence and conduction bands, re~pectively.~ Illustrating Movement within a Band Hole Movement The concept of hole movement, which is necessary for understanding intrinsic and p-type extrinsic semiconductors, often confuses students. To remedy this, take a row that is
the room. Without some outside force, students sit in seats in a more or less random manner. When there is a pattern, it usually occurs because of some repulsive force (e.g., away from the instructor and towards the back of the room) or attractive force (e.g., towards a window or door). Any movement they make would be more or less random, too. However, if the instructor requests that a number of students move to one end of the room, this would require work, and there would be an overload of students on one side of the room. This illustrates how a conductor can be polarized. If a conveyance for movement is provided, such as a suggestion from the instructor (closing the circuit), students (coulombs) can work their way (watts) from the filled edge of the room to open seats at the other edge (volts). This illustrates conductivity. The Energy Gap and Conductivity in Semiconductors
The only movement choice for students in a fdled class is out of the mom. When we pretend that the energy necessary to move out of the room (the energy gap, E,,) is unattainable, we are illustrating an insulator. When we pretend that the necessary energy is attainable, we illustrate an intrinsic semiconductor,which has a filled valence band and an empty conduction band. As the necessary energy gets smaller, more students will be able to exit to the adjacent room. Also, as one increases the incentive to move (increasing the temperature or excitation with light), more students will move to the adjacent room. More empty seats will appear, and more students will be in a nearly empty room with plenty of room to move. These are analogous to hole formation in the valence band with free-electrons in the conduction band. The number of these charge carriers dominates conductivity in semiconductors, and both increase greatly as temperature increases. Mobility and Conductivity in Metal Bands
Students in a half-filled room have many alternatives when wishing to move. They move with much greater facility, and there is no incentive to leave the room. Metals tend to have much ereater conductivitx As oooosed to semiconductors, the temperature dependance ofmetals is not dominated bv the number of charee carriers but bv the carriers' mobilit; Movement of lone &ectrons tends b occur with greater facility than the concerted electron movement necessary for hole movement. Thus, one observes a higher magnitude of mobility in metals than in p-type semiconductors. Mobility has many competing factors especially in semiconductors, but discussing them is beyond the range of this article. However, within this analogy, one can imagine that it becomes more difficult to maneuver from chair to chair as the chairs and other people move about with increasing energy. Therefore, metal mobility and conductivity decrease with increasing temperature. Just stating that conductivity is charge movement implies action. Action enlivens. With this analogy, a seemingly dimcult set of conce~tscan be broueht to life and learned in a quick, entertaining way. Finally, witha little effort and imapination. this analow can be extended to exolain other te&inology'and phen&ena included in more' advanced classes.
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Row a is a representation of a one-dimensional p-type semiconductor. Aperson represents an electron, and the line of chairs represents a group of orbitals, all at the same energy level. In rows k,hole movement from left to right through the semiconductor is illustrated. almost filled, and have a student move to an adjacent empty seat (Fig. la). Have his neighbor move to the newly formed empty seat, and so on (Figs. lb-4. As the students are moving in concert in one direction, you can point out to them that the empty seats are "moving" to the other end. Thus, a physical illustration of hole movement has occurred. For humor-and to dramatize your point-you can "order" your holes to move back to their original seats. It should not surprise students that this particular order can only be "obeyed" if they move, in concert, back to their initial positions. The coordination involved helps to drive home the point that hole movement is less facile than electron movement, a point that hecomes more significant later. Electron Movement An illustration of electron movement as seen in most
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Literature Cited 1. CRC H o n d h k olChemistry and Physics, 67th ed.; Weast, R C., Ed.; CRC:Cleue-
Volume 69 Number 10 October 1992
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