E. C. van Reufh U. S. N a v y Marine Engineering Laboratory Annapolis, Maryland
An Analogy for the Band Theory of Metals
I n attempting to introduce the Band Theoly of Metal4 to those who are unfamiliar with this topic, the author has discovered an analogy which has proved rather useful.' Quantum mechanics views the "free" electrons of a metal as being restricted to discrete energy levels. These levels can be liicened to the rungs of a ladder. As contrasted to an inched plane, where any vertical height is attainable, a ladder permits its user only those heights corresponding to its rung positions. Likewise, the energy values of the electrons are restricted to certain levels (ladder), rather than being continuous (inclined plane) as in the classical mechanics viewpoint. The opinions or assertions made in this paper are those of the author and are not to he construed as official or reflecting the views of the Department of the Navy or the naval service at large. 'The reader mdamiliar with the fundamental theory should A. H., consult authoritative presentations such as COTTRELL, "Theoretical Structurd Netrtllurgy," 2nd ed., St. Martin's P r m , Inc., New Tork, 1959, p. 44-73, or ZIMAN,J. M., "Principles of the Theory of Solids," Cambridge University Prws London, 1964, p. 1-125.
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Journal of Chemicol Edumfion
The Analogy
The reader is asked to imagine that he is observing a fire which is on the 10th floor of a building. The fire must he fought by a bucket brigade with two firemen on each rung of the ladder. The man on the right side of each rung passes a filled bucket of water from the man below him to the man above him. Theman on the right side of the top rung throws the water into the burning window. Each empty bucket is then passed down the left side of the ladder. Notice that every rung is occupied by two men each of whom is performing a function opposite to that of his partner. This situation is very similar to the electron picture in metals where there are many energy levels (rungs), each of which is occupied by only two electrons (firemen). As each bucketful of water rises up the ladder, it is attaining more and more energy and hence requires less and less energy to he thrown into the fire. The two electrons (firemen) on each level have opposite spin; one with spin up (full bucket) and one with spin down (empty bucket). Notice also that every rung must be occupied for the fire crew to perform its function. This restriction is the analogy to the Pauli exclusion principle.
Now, the reader is asked to imagine that there are only three fixed-length ladders available, each a different length. One reaches up to the eighth floor, another to the nint,h floor and another to the tenth floor. (See Fig. 1) The fireman on the top rung of the first ladder finds that his buckets of water will not reach the fire even when he throws them as high as possible. The fireman on the top rung Figure I of the second ladder gets a fraction of each bucketful into the tenth floor window when he throws as hard as he can. But the top man on the third ladder is able to throw all of every bucketful of water into the fiery tenth floor window. The above three cases can be likened to the three basic types of solids: an insulator, a semi-conductor, and a metal. The ladders correspond to the so-called valence euergy bands and are of different heights (different energies) in the three cases (see Fig. 2). The tenth floor of the building can be likened to the conduction band in solids. The first case is analogous to a tvnical ". insulator. Here, the valence band-width (ladder length) is such that the electrons' energies at the top of the valence band (top of the ladder) are so far below the conductionband (10thfloor) that they cannot be accelerated into the conduction band by an applied potential (fireman throwing water). The second case is similar to a typical intrinsic semi-conductor. Here, the electrons Figure 2 at the top of the valence band are close enough in energy to the conduction band that some of them can be accelerated into it (i.e., some water gets into the 10th floor window). Xotice here that the quantity of water that actually gets into the fire is a function of how far the top fireman is below the fire and the strength of the top fireman. (The current flow in a semi-conductor is proportional to the energy gap and the applied voltage.) The third case is analogous to most metals. Here, the electrons at the top of the valence band have suffcient energy to be within the conduction band and are capable of participating in the electrical conductivity process (all of the water at the top of the ladder goes into the fire). I n attempting to visualize the situation in real solids, one must recognize the limitations of the preceding analogy. It is valid when one considers the discrete energy levels available to the electrons. It is also valid to the extent that it demonstrates that only those electrons whose energies correspond to the top of the band are capable of extensive mobility. However, the reader must not think that the electrons have to travel
"up" the band to the top as do the buckets of water. I t is here that the analogy is inapplicable since present-day band theory tells us nothing about such details. L i k e wise, the occupation of each rung by two firemen merely represents two electrons; one with spin "up" and the other with spin "down." Finally, for further amplification, let us say that the top rung on the ladder corresponds to the term which is defined as the Fenni level. This is the highest energy level which is occupied by electrons. Thus, we can see that metals are conductors, because their Fermi energy is high enough so that the electrons at that energy level can enter the conduction band. Since most solids do have directionality properties (anisotropy), we must think also in terms of the Fermi energy variations with direction. The Fermi energy for an electron traveling in the X-direction may be quite different from that in the Y- or 2-directions. Thus, the Fermi energy can be visualized in three dimensional spare roordinates as t,hc so-called Fermi surface. Themost difficult task for those just being introdured to the theory of solids is to learn to think in terms of energy rather than in terms of our usual space coordinate system. We must learn to think of the "location" of an electron with reference to the lo~vestenergy level in its band, not in terms of its physical location among the atoms of the solid. When one talks of atoms, one discusses their location coordinates. However, in diseussing the electrons we must refer to their energy coordinates instead, since an electron's energy will determine its "location" within a band.
Solution to puzzle on p. 445 of the August issue.
Volume 43, Number 9, Sepfember 1966
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