2508 HEINRICH AND JANTSCH ~Work supported by the "Ponds zur F5xderung dex vrissenschaftlichen Forschung, " Austria, and the Lud~ig Boltzmann Gesellschaft. Now at Department of Physics, Lakehead University, Thunder Bay, Ontario, Canada. For example, G. Bauer and H. Kahlert, in Proceed- ings of the Tenth International Conference on Physics of Semiconductors, Cambridge, Mass. , 1970 (unpublished). p. 65. A. Mooradian and A. L. McWhorter, in Hef. 1, p. 380. E. Burstein, Phys. Hev. 93, 632 (1954). M. S. Shur, Phys. Letters 29A, 490 (1969). 5W. Franz, Z. Naturforsch. 13a, 484 (1958). 6 L. V. Keldysh, Zh. Eksperim. i Teor. Fix. 34, 1138 (1958) [Sov. Phys. JETP 7, 788 (1958)]. VH. Heinrich, K. Hess, W. Jantsch, and W. Pfeiler (unpublished) . Jantsch~ and H. Heinrich, Phys. Hev, B ~3 420 (1971). ~V. Roberts and'J. E. Quarrington, J. Electron. 1, 152 (1952). ~C. Y. Liang, J. Appl, Phys. ~39 3866 (1968). ~lH. Piller, J, Appl. Phys. 37, 763 (1966). PHYSICAL RE VIE W B 15 OCTOBER 1971 Electronic Properties of an Amorphous Solid. I. A Simple Tight-Binding Theory D. Weaire and M. F. Thorpe +8eton C8mt8t', F018 Vmi58tsity, N8% HAveg, COgg8etieut 06520 (Received 25 May 1971) Using a simple Hamiltonian of the tight-binding type, rigorous bounds are derived for the density of states of a tetrahedrally bonded solid. These include inner bounds which define a band gap bebveen occupied and unoccupied states. The derivation uses only the assumed per- fect coordination of nearest neighbors, and so it holds for all tetrahedrally bonded crystal structures and random netvrorks of the kind proposed for amorphous Si and Ge. Various other results are obtained for the, fractional 8- and p-like character of vrave functions, the attain- ment of bounds, and other features of the density of states. A band-structure calculation for the diamond cubic structure serves as a test case. I. INTRODUCTION Two broad classes of disordered systems are encountered in solid-state theory (see Fig. 1). In the case of what we shall call quagtjtutjye disorder, one defines a periodic array of potentials which are not identical. They may, for instance, be of two types, randomly distributed, in which case the the- oretical model would be appropriate to a disordered binary alloy. On the other hand, one may define an array of potentials which are identical but not periodically positioned. One might call this po- sitional disorder. Such a model would be appro- priate for, say, a liquid metal. If a positionally disordered system has the same coordination of nearest neighbors everywhere and we describe it with a Hamiltonian which involves only nearest-neighbor coordination, we have the special ease of toPologgccl di8order. The distinc- tion between this ease and that of quantitative dis- order is somewhat clearer. The matrix elements which specify the Hamiltonian are the same every- where throughout the structure. It is the eonnec- tivity of the structure which is disordered. Such a topologically disordered Hamiltonian would seem to be an appropriate starting point for a theory of the electronic properties of amorphous elemental semiconductors, and in the subsequent sections the model will be analyzed in detail. The motivation for this study lies in recent ex- perimental work' 6 on amorphous Si and Ge. From the outset it was evident that these substances were highly disordered, and yet in many x espects their electronic properties are closely similar to those of the corresponding crystals. In particular, a band gap between valence and conduction bands persists in the amorphous state. The extent to which such a gap contains a small density of states tailing off from the two bands is still a subject of debate. Be that as it may, this remains a re- markable experimental result. A model for the structure of these elemental amorphous semiconductors which has gained wide acceptance is the random netmcn k model, in which every atom is almost perfectly tetrahedrally co- ordinated with its nearest neighbors. Distortions of bond lengths and angles from the values in the crystalline state are of the order of 10/~, and yet the distribution of second and further neighbors is highly disordered. 7 It is by no means obvious that such a geometrical arrangement can be con- structed in practice. Ho~ever, this appears to