PH VS ICAL REVIEW VOLUME 1 75, NUMBER 3 15 NOVEMBER 1968 Single-Site Approximations in the Electronic Theory of Simple Binary Alloys* B. VzLrcz1, t S. Krzzpsrzrcz, sm H. ErrzzNzzrcrr Disisium uf Emgieesrimg used A pphed Physics, Pursurd University, Cuerhridge, hgusrushusstir 02138 (Received 26 June 1968) A single-band model Hamiltonian is used to describe the electronic structure of a three-dimensional disordered binary alloy. Several common theories based on the single-site approximation in a multiple- scattering description are compared with exact results for this Hamiltonian. The coherent-potential theory of Soven and others is shown to be the best of these. Within the appropriate limits, it exhibits dilute-alloy, virtual-crystal, and well separated impurity-band behavior. Hubbard and Onodera's and Toyozawa's simple model density of states is employed in numerical calculations for a wide variety of concentrations and scattering-potential strengths. Explicit results are exhibited for the total density of states, the partial density contributed by each component, and such k-dependent properties as the Bloch-wave spectral density and the distribution function. These illustrate the general conclusions as well as the limitations of the quasiparticle description, I. DTTRODUCTION i 'HIS pRpcr ls coDcclncd with two Rspccts of tbc single-particle theory of the electronic structure of disordered binary alloys. It presents a systematic deriva- tion of the so-called coherent-potential (CP) theory' of such systcIQs, clariGcs its meaDlng and limitations, and discusses numerical results for a moderately realistic single-band model corresponding to R three-dimensional system. In addition it presents a number of exact re- sults for this model. These are useful as a basis for comparison with approximate calculations, Rnd also when thc two constltucnts give risc to t%'0 well-scpa" ra, ted. Sub-bands, R situation in which thc Quctuations of the random potential are so large that the CP theory is not expected to be valid. The Cp concept has generally been developed within the framework of the multiple scattering description2 of disordered systems. ' — 5 In this approach the propa- gation of an electron or lattice wa, ve in an alloy is re- gaxded as a succession of elementary scatterings on the random atomic scattcrcrs& which Rlc thcD Rvcx'Rgcd ovcl Rll con6gurations of atoms. Taylor' and Soven, ' de~.»rig respectively with the case of the lattice vibration and electron-cxcitation spectrum in an alloy, returned to the Ewald-Lax theory' of multiple scattering. They viewed a given scatterer as being embedded in an effective medium whose choice was open and could be made self-consistently. This choice in turn determined an ef- fective Hamiltonian called the coherent-potential. *Supported in part by grant No. GP-8019 of the National Science Foundation and the Advanced Research Projects Agency. t Permanent address: Institute of Sohd State Physics, Czech~ slovak Academy of Sciences, Prague. r P. Soven, Phys. Rev. 156, 809 (19N). ~ M. Lax, Rev. Mod. Phys. 23, 287 (1951). 3 S. F. Edwards, Phil. Nag. 6, 617 (1961). . L. Beeby and S. F. Edwards, Proc. Roy. Soc. (London) A2 4, 39$ (1963). ~ A most useful recent source of references for the problem of liquid metals is the I'roceedirjgs of tIte I&ereeQogal Conference ow tIIe Properties of I rquid Metals, Brookhaeee, EN6, given in Advan. Phy . 16, 14' g. 9N'). D. %'. Taylor, Phys, Rev. 156, 1017 (19@'). Harniltonian. The physical condition corresponding to this choice is simply that a single scatterer embedded lD this effective IncdluIQ should ploducc Qo fux'ther SCRt- tering on thc average. The C6ectivc Hamiltonian in question is to be regarded as an unknown of the problem, and in contrast to the known "unperturbed" Hamilton- ian that forms the usual starting points for multiple scattering theories, is not impmved further by consider- ing scattering corrections of ever increasing order. In this sense, the self-consistent choice of Hami1. tonian is optimal among all single-site approximations, which neglect the scattering from clusters of atoms. Although the clustering effects may bc important under certain circumstances, thc single-site Rpproxlmatlon renders the problem tractable. It may be said to play the same role in alloys as the molecular-6eld theory in magnetism. McMillan and Anderson~ used a similar approachs in their treatment of liquid iron. Crudely speaking, their model binary alloy consisted of iron atoms and vacancies. A quite di6erent and important application of thc same formalism %'as xQRdc very x'cccDtly by Onodcra and Toyozawa. 9 They described Frenkel exci- tons m mixed lonlc crystals using a simple three- dimensional single-exciton band model which permits a detailed solution of the problem. In addition, quantities other than the density of states, namely thc spectral density describing the optical absorption, werc calcu- lated and discussed for the Grst time. The CP approxi- mation vras there rightly regarded as a scheme which interpolates between properly described limits corre- sponding to the entire range of impurity concentrations and strong and weak scattering. A band model similar to that used by these authors, and previously intro- duced by Hubbard, " will also be employed here to ob- r P. W. Anderson and W. L. McMillan, in Prozeed&sgr uf ths Igterea&orhal School of Physics "Set'ico Fernsi, " Course 37, edited by W. Marshall (Academic Press Inc. , New York, 1967). 8 For a different approach to dilute liquid alloys, see E. A. Stern, Phys. Rev. 168, 730 (1968). ' Y, Onodera and Y. Toyozawa, J. Phys. Soc. Japan 24, 341 (1968). '&' J. Hubbard, Proc. Roy. Soc. (London) A276, 238 (1963). 747