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Section 11_Methods_for_calculating_band_structure

# Section 11_Methods_for_calculating_band_structure - Physics...

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Unformatted text preview: Physics 927 E.Y.Tsymbal 1 Section 11: Methods for calculating band structure The computational solid state physics is a very fast growing area of research. Modern methods for calculating the electronic band structure of solids allow predicting many important properties of solids. All these methods involve the development of quite complicated computer codes. Nowadays some of such programs are available in the market and can be purchased and used by researchers. Tight-binding approximation Tight-binding method uses atomic orbitals as basis wave functions. Let us see how the energy spectrum gradually changes as atoms are assembled to form the solid. Consider lithium as an example. In a free atom electrons moves in a potential well, as shown in Fig. 1a. The atomic spectrum consists of a series of discrete energy levels, which are denoted by 1s, 2s, 2p, etc. The lithium atom contains three electrons, two of which occupy the 1s shell which is completely full, and the third electron is in the 2s shell. If the two Li atoms are assembled to form the molecule Li 2 , the potential "seen" by electrons is now the double well shown in Fig.1b. Due to a coupling between atoms, each of the atomic levels - that is, the 1s, 2s, 2p, etc. - has split into two closely spaced levels. We may, therefore, speak of the 1s, 2s, 2p, etc., molecular energy levels, recognizing that each of these is, in fact, composed of two sublevels. The amount of splitting depends strongly on the internuclear distance of the two atoms in the molecule: the closer the two nuclei, the stronger the perturbation and the larger the splitting. The splitting also depends on the atomic orbital: The splitting of the 2p level is larger than that of the 2s level, which is larger still than that of the 1s level. The reason is that the radius of the 1s orbital, for instance, is very small, and the orbital is therefore tightly bound to its own nucleus. It is not greatly affected by the perturbation. The same is not true for the 2s and 2p orbitals, which have larger radii and are only loosely bound to their own nuclei. It follows that, generally speaking, the higher the energy, the greater the splitting incurred. Fig.1 The evolution of the energy spectrum of Li from an atom (a), to a molecule (b), to a solid (c). The above considerations may be generalized to a polyatomic Li molecule of an arbitrary number of atoms. Thus in a 3-atom molecule, each atomic level is split into a triplet, in a 4-atom molecule into a quadruplet, and so forth. The Li solid may then be viewed as the limiting case in which the number of atoms has become very large, resulting in a gigantic Li molecule. Each of the atomic levels is split into N closely spaced sublevels, where N is the number of atoms in the solid. Since N is so very large, about 10 23 , the sublevels are so extremely close to each other that they coalesce, and form an energy band. Thus the 1s, 2s, 2p levels give rise, respectively, to the 1s, 2s, and 2p...
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Section 11_Methods_for_calculating_band_structure - Physics...

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