Role of Short-Range Order in Producing An Energy Gapin Amorphous Covalent SemiconductorsBY J. KLIMA *, T. C. MCGILL -f- AND J. M. ZIMANH. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TLReceived 16th June, 1970The role of short-range order in determining the density of states of disordered covalent semi-conductors is investigated using the multiple scattering formalism. Detailed calculations of anapproximate density of states for a model consisting of clusters of carbon atoms in configurationsfound in amorphous and crystalline germanium are reported. These calculations suggest that theenergy gap in amorphous covalent semi-conductors is a result of the short-range order.1. INTRODUCTIONExperimental studies of amorphous silicon and germanium have yielded manyof the properties of the materials.The pioneering work of Ritcher and Breitlingon silicon and germanium, and the detailed studies of Girgorovici and colleagueson germanium indicate that the structures of these amorphous materials may bedescribed in the following way: long-range order is absent; however, almost allsites have four nearest neighbours which are arranged like the nearest neighboursin the perfect crystal. Measurements of the optical properties 3-8 show that thesematerials possess an optical gap which is qualitatively similar to that in the crystallinematerial.The existence of an optical gap in amorphous silicon and germanium posesan interesting theoretical question. In the standard theory of solids, heavy emphasisis placed upon the role of long range order in producing a band gap.The simul-taneous lack of long-range order and existence of an optical gap in these materialsforce one to ask whether any other factor can produce an energy gap.In this paper the authors address themselves to this question. The role of short-range order in determining the density of states is investigated using the multiplescattering f~rmalism.~~ lo The results of a detailed quantitative study of a modelconsisting of carbon atoms with short-range order like that in amorphous germaniumis reported. In 92, a short theoretical discussion of the relevant results from themultiple scattering formalism is presented; in §3, the exact details of the model andthe numerical results are presented and discussed.Conclusions are presented in$4. Throughout this calculation units in which = 1 , me = 3 and energy is measuredin rydbergs are used.2. THEORETICAL DISCUSSIONMultiple scattering formalism 9 s lo deals with the properties of an ensemble ofscattering centres. The potential about each of these centres is assumed to bespherically symmetric and not to overlap that of its neighbours (muffin tin approxi-* on leave of absence from Charles University, Prague.f NATO Postdoctoral Fellow23. KLIMA, T. C. MCGILL AND J . ha. ZIMAN 21mation). For a perfect crystal this formalism reduces to that obtained in the KKRmethod of calculating band structure. 9In this formalism the density of states may be expressed analytically in termsof the scattering properties of each centre and their spatial arrangement.Lloyd l 3has shown that the integrated density of states (the number of states with energy lessthat E ) is given byE+ 23n2 ni.2 N(E) = - - - Im Tr In (BLLIBUr - G&,(l- ll)kLI(l*, E)). (2.1)The first term is the integrated free particle density of states. The last term is theFriedel sum for the solid. In this term, f,l is the volume of the solid, L( = (1,m))refers to a real spherical harmonic, YL(Q), and 1 is the location of a scattering centre.GtL,(l- 1') is given by the expressionwhereCfiI = 1 dQYLII(~)YL(Q)YLI(Q)and K = ,/E. kL(l,E), the single site k-matrix for site 1, is related to the phase shiftfor spherical harmonic L, BL(Z,E), by the equationThe trace is taken over L and 1.To show the importance of short-range order in determining the density of states,eqn (2.1) may be rearranged so that the scattering properties of small clusters of sitesappear in place of the single-site scattering properties.The clusters of sites ofinterest here are defined in the following manner. First, a set of locations for thecentres of the clusters is selected. These locations will be labeled rj where the jrefers to thejth cluster. It is not necessary that the centre of a cluster coincide withone of the scattering centres. Secondly, each scattering centre is identified with thecluster whose centre is closest to the site. If a site belongs to cluster j , its locationwill be denoted by lj.Using the definition of a cluster given above, the last term in the integrated densityof states may be rearranged to givekL(I,E) = -tan 6,O,E)/u (2.3)E* 2N(E) = --- Im Tr 111 ( a L u 6 i j - C G,tII(ri -rj)KLIxLI(rj)) -3n2 ni.2 L I I2nQ - c j Im Tr In (8LLdljlj1 - G&r(ij - l;)kLI(l;)), (2.4)where KLrrLr(rJ), the K-matrix for cluster j , is given byKLnLI(rj) = ALIILrII(rj - l j ) ~ ~ I r ( I J .) M E I ~ ~ , ; ~ ~ v ~ ~ A L ~ ~ ~ ~ O f - ri), (2.5)L!&vwit22 ENERGY GAP IN AMORPHOUS COVALENT SEMICONDUCTORSand(see ref. (lo), (14)).The last term in eqn (2.4) contributes to the expression only when the K-matrix for thecluster or the single site k-matrix are not defined. Suitably defining the logarithmin the second term at these points allows us to discard this term.Eqn (2.4), without the last term, is the result desired since the presence of theK-matrix for the cluster makes it possible to study the influence of the presence ofshort-range order.However, even when the K-matrices for all the clusters are known,eqn (2.4) is still a difficult equation to evaluate exactly. For disordered structures,correlations in the position and orientation of nearby clusters must be known tomake an exact evaluation of the average of eqn (2.4) possible. For a perfect crystal,a complete band structure calculation is required to evaluate it exactly. To circumventthis difficulty and to show the importance of short-range order, eqn (2.4) is approxi-mated by neglecting the influence of multiple scattering between clusters.In thisapproximation, eqn (2.4) reduces towhere the sum o n j spans all the clusters in the solid. Eqn (2.9) shows that thisapproximation is equivalent to summing the Friedel sum for each of the clusters inthe solid. It has the particular advantage of not requiring that one specifies eitherthe relative angular orientations or the correlations in position of the clusters.However, this approximation is a potentially serious one. Intuitively, it would seemthat the error is related to the size of the cluster considered. Since the use of verylarge clusters in calculating the K-matrices in eqn (2.9) should lead to a density of stateswhich deviates only slightly from the exact density of states, one would expect thecorrection terms to eqn (2.9) to decrease with increasing size of the clusters used.The authors hope to give a more detailed discussion of this point in a futurepublication.3.CALCULATIONS AND RESULTSTo evaluate eqn (2.9), the geometry of the clusters and the single site k-matrixare required. In this calculation, the two types of clusters of eight carbon atomsshown in fig. 1 are used. The configuration on the left, the staggered configuration,2has exactly the same structure as the cubic cell of the diamond lattice ; the diamondlattice may be built up from these structures. The configuration on the right, theeclipsed configuration,2 is not found in the diamond lattice. However, it is thoughtto be present in amorphous gerrnaniwm2 Throughout this calculation, the inter-atomic distance is taken to have the value 6.74 ao, the crystalline value.Determination of the single site k-matrix requires a detailed knowledge of themuffin tin potential about each site.This potential is assumed to be identical foreach site in the material and its value is taken to be that appropriate for the perfectcrystal. Unfortunately, the muffin tin approximation is poor for covalently bondedsemiconductors. Consequently, finding a potential and phase shifts which generatea realistic band structure presents some difficulty. To avoid this difficulty (which iJ . KLIMA, T. C. MCGILL AND J . M. ZIMAN 23outside the range of interest of this paper). carbon was chosen as a model materialsince the large hand gap of diamond makes the existence of a gay, rather inseiisitiveto these problems.Following Mertens,' the crystal potential is approximatedsimply as the sum of the Herman Skillman potentials o f the two nearest neighboursalong their connecting line. The muffin tin zero was taken to be -2.207ry.16Integration of the radial Schriidinger equation for this potential yields the phasesshifts used in this calculation. As shown in fig. 2, the resulting p-phase shift exhibitsa weak resonance which is important in determining the character of the scatteringproperties of the cluster. The s-phase shift has little structure. Only s- andp-singlesite phase shifts were used in this calculation. When a KKR calculation was carriedout for diamond using these phase shifts, a qualitatively correct band structure wasobtained.(4 (6)FIG.1.-The staggered (a) and eclipsed (b) clusters used in the calculation of the density of states.The K-matrix for each type of cluster was evaluated numerically and the resultingK-matrix was substituted into the derivative of eqn (2.9) with respect to energy toproduce the density of states. The energy range 0-1.4 ry above the muffin tin zerowas investigated since this is the range where the KKR calculation indicated thatthe top of the valence band and the bottom of the conduction band are located.The densities of states as a function of energy obtained from these calculationsare presented in fig. 3. The values of the energy at the r $ s level, the top of the valenceband, and at the r15 level, the lowest level with r-symmetry in the conduction band,are also indicated along the energy axis in fig.3. These values were obtained froma KKR calculation with the phase shifts used in the calculation of the density ofstates.These results show three facts. First, the energy dependence of the density ofstates for a solid made up of these clusters can be divided into three ranges : from0.0 to 0.54 ry, from 0.54 to 1.05 ry, and above 1.05 ry. In the first and third ranges24 ENERGY GAP IN AMORPHOUS COVALENT SEMICONDUCTORS'T 2n9 1;iiit:-5 sa - '."4- 2-3'c4. --energy (rY)FIG. 2.-Single-site phase shifts for carbon. The broken and full lines correspond to the s- andp-phase shifts, respectively. The zero of energy is at the muffin tin zero.FIG.3.-The approximate density of states as a function of energy. The full and broken lines arethe densities of states for a solid made up of eight-atom clusters with staggered and eclipsed configura-tions, respectively. The dotted line is the density of states for a solid of clusters consisting of asingle site. The range of the forbidden gap in crystalline carbon is indicated by the arrows on theenergy axisJ . KLIMA, T. C. MCGlLL AND J . M. ZIMAN 25the density of states is approximately an order of magnitude greater than that foundin the second range.Secondly, the sharp transition region between range one and range two occurs atalmost exactly the value of the energy for the ri5 level ; the transition between rangetwo and range three occurs at approximately the value of the energy for the Tlslevel.Thus, the density of states obtained here are similar to what would be foundfor a perfect crystal. The ranges of energy giving large density of states wouldcorrespond to the valence and conduction bands, and the range of energies giving asmall density of states would correspond to the band gap.Thirdly, the similarities between the densities of states obtained for the twotypes of clusters indicate that the presence of a range of energies where the density ofstates is small bounded by ranges where the density of states is large is insensitiveto the differences in the short-range order which are found in disordered germanium.For comparison, the density of states resulting when clusters consisting of a singlesite are used is also shown in fig.3. The graph shows that the single-site clustersfail to produce a rapid increase in the density of states for energies greater that 0.5 ry.The differences between the density of states for the single-site clusters and eight-site clusters may be understood in the following way. The peak in the density ofstates for the single-site clusters results from the weak p-resonance which appears inthe single-site phase shifts. For comparison, the density of states resuIting whenclusters consisting of a single site are used is also shown in fig. 3. The graph showsthat whilst the single-site clusters produce a peak on the density of states in the firstrange, they do not produce as small a density of states in the second range as themulti-site clusters and fail to produce a peak in the third range.The differences inthis density of states and those obtained from the multi-site clusters may be explainedin the following way. In the first case the weak p-resonance produces only one peakin the density of states. However, in the multi-site cluster the s- andp-waves scatteredfrom the various sites interact to produce a number of resonances in the total scatteringproperties. These resonances fall into two distinct groups producing the double-peaked structure separated by a small density of states region shown in fig. 3. Itis the explicit inclusion of the short-range order in the multi-site cluster which givesthis result.The small-but finite-density of states in the gap region is the result of theincomplete cancellation of the free electron density of states (the first term in the eqn(2.9)) by the contribution from the multiple scattering properties (the second termin eqn (2.9)), and is probably due to the rather drastic approximation of neglectingmultiple scattering between clusters. Similar calculations are currently in progressfor silicon and germanium.While the results are not complete, it appears thatsimilar results will be obtained for these materials. The p-phase shift of both siliconand germanium have a weak p-resonance ; and preliminary calculations on siliconindicate that this weakp-resonance is split in a way analogous to that for carbon.4. CONCLUSIONThe scattering properties of clusters of eight carbon atoms have an energydependence which produces an approximate density of states with clear indicationof an energy gap.Even though this calculation is only an approximate one for adisordered material, it does suggest that the presence of an energy gap in the spectrumof amorphous covalent semi-conductors is closely related to the presence of short-range order. The qualitative similarity between this gap and that found in theperfect crystal is also explained by the relative insensitivity of the size of the gap to thedifference in short-range order found in these two materials26 ENERGY GAP IN AMORPHOUS COVALENT SEMICONDUCTORSThe authors acknowledge helpful discussions with G. J. Morgan. They areparticularly indebted to him for suggesting the possibility of a weak p-resonance inthese materials.Helpful correspondence with J. Treusch is also acknowledged.H. Richter and G. Breitling, 2. Nuturforsch., 1958, 13a, 988.R. Grigorovici and R. Manaila, J. Non-Cryst. Solids, 1969, 1, 371.J. Tauc, R. Grigorovici and A. Vancu, Phys. Status Solidi, 1966, $5, 627.A. H. Clark, Phys. Rev., 1967, 154,750.J . Wales, G. J. Lovitt and R. A. Hill, Thin Solid Films, 1967, 1, 137.T. M. Donovan, W. E. Spicer and J. M. Bennett, Phys. Reu. Letters, 1969, 22, 1058.R. Grigorovici and A. Vancu, Thin Solid Films, 1968,2, 105.D. Beaglehole and M. Zavetova, J. Non-Cryst. Solids, 1970, 4, 272.J. L. Beeby, Proc. Roy. SOC. A, 1964,279,82.lo P. Lloyd, Electrons in Metals and MuZtipZe Scattering Theory, unpublished.l1 J . Korringa, Physica, 1947, 13,392.l2 W. Kohn and N. Rostoker, Phys. Rev., 1954, 94, 1111.l3 P. Lloyd, Proc. Phys. SOC., 1967,90,207.l4 T. McGill and J. Klima, in preparation.0. Madelung and J. Treusch, Proc. 9th Int. ConJ Physics of Semiconductors, (Nauka, Lenin-grad, 1968), vol. 1, p. 38.l6 K. Mertens, Diss. (Univ. Marburg, 1967).l7 F. Herman and S. Skillman, Atomic Structure Calculations, (Prentice Hall, Inc., EnglewoodCliffs, New Jersey, 1963).l 8 Mertens l6 includes a d-phase shift in his band structure calculation. The omission of thed-phase shift in our calculation produces slight differences between the band structure obtainedby Mertens and our results