ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR QUARTERLY PROGRESS REPORT NO. V THEORY OF CERTAIN ENERGY SURFACES AND BRILJOUIN ZONES By Et. KATZ Professor of Physics G. B. SPENCE Research Assistant Project 2158 AIR RESEADRCH AND DEVELOPMENT COMMAND CONTRACT AF 18(600)-750, PROJECT NO. R-355-40-10 September, 1953

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN THEORY OF CERTAIN ENERGY SURFACES AND BRILLOUIN ZONES INTRODUCTION The general purpose of this project is the investigation of the relations between the shapes of energy surfaces and of Brillouin zones in solids. There are three distinct phases of the project. The first phase is an investigation of the shape of the higher Brillouin zones for the more important crystal structures. The principles of the construction of such zones have long been known, but the forms of all but a few of the simplest zones have not been studied or described. The major contribution of this phase of the work will be the construction of models and suitable figures to aid in visualizing the shapes of the zones and the tabulation of data pertinent to these zones. The problem of the occurrence of higher-order zones of simple shape is of interest here. The second phase is an investigation-of the shape of energy surfaces in reciprocal space (k-space). In the past, such investigations have been carried out, with varying degrees of success, through extensive numerical calculations which have had to be performed separately for each substance. It is the purpose of the present study to keep the work as general as possible by investigating certain critical and near-critical surfaces by approximate analytical methods, avoiding elaborate numerical work on individual substances. These critical surfaces are characterized by having singular points on the zone boundaries either in a plane on an edge or at a corner. The third phase is an application of the results of the first two phases to certain problems in the theory of metals and alloys and to the appear ance of energy gaps due to faint superstructures. For example, Jones' explanation of the Hume-Rothery rules for simple binary substitutional alloys is based on a study of one type of critical energy surface. This surface was determined by the approximation of nearly free electrons. It is intended tu carry this approximation to a higher degree. Another application arises from _______________-________ 1 _________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN a remark of Slater's that a superstructure will cause a splitting of a band. E. Katz has pointed out that a splitting into nonoverlapping bands will take place only if the superstructure gives rise to new zone boundaries which almost coincide with an original energy surface. Thus, by correlating critical energy surfaces with zone boundaries of higher zones we can predict when a superstructure is likely to lead to nonoverlapping bands. If the second phase can be completed with sufficient accuracy, then several applications can be made to problems which depend on electron overlap into a higher zone. Two such problems are the behavior of certain elastic constants in aluminum alloys and the c/a ratio of magnesium alloys. The second phase is of course much harder than the first, and so it was thought best to work first on the second phase. This first quarterly report summarizes preliminary work done on the shape of energy surfaces near zone boundaries. SURVEY OF POSSIBLE METHODS The applications mentioned in the introduction depend mainly on knowing the shape of the energy surfaces and the size of the energy gaps in the region near the Brillouin zone boundaries. However, it is in just this region that the energy is determined least accurately by existing methods. Clearly, methods based on spherical symmetry such as that of Wigner and Seitz are not applicable. Methods such as Slater's modification of the Wigner-Seitz method and Herring s method of orthogonalized plane waves are good, but require considerable numerical work for each case considered. It is felt that there is a need for a simpler, even though perhaps less accurate, method by which a large number of elements and alloys could be surveyed. Two possibilities are the tight-binding approximation and the nearly-free-electron approximation. The tight-binding approximation is receiving considerable attention at present. However, the author is unaware of any instance where the nearly-free-electron approximation has been worked out with the same thoroughness as other methods. In fact, very little fundamental work has been done along this line beyond that published by Sommerfeld and Bethel, 1933. Therefore it seems worthwhile to extend the nearly-free-electron approximation. This has the additional advantage of immediately extending Jones' work on the Hume-Rothery rules. Sommerfeld, A., and Bethe, H., Handb. d. Physik, 2nd ed. vol. 24, pt. 2, 1933, p. 333. _________________-_ 2 _______________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN NEARLY-FREE-ELECTRON APPROXIMATION Summary of Known Results2 The procedure of the nearly-free-electron approximation is to expand the Block wave function and the periodic potential as Fourier series'\/~ (r) = e2iktPr A e2 (1) p V (r) =.v e2 iiir (2) P On inserting these into Schr'ddinger's equation an infinite set of homogeneous linear equations for the unknown amplitude coefficients AA in terms of the energy E and the known coefficients Vf is obtained; the m th equation is i IL -v+ (2-hlk + mI2 - E)& m]A = 0. (5) p P The energy is determined by the condition that the infinite determinant of the coefficients.must vanish. If j k | k + ml for all m for which VmA~O i.e. for points that do not lie on a zone boundary, generally speaking, the energy is given approximately by the usual perturbation-theory result, h2 2m V2 E 2m., k t+iF: (2 E =- 2 k2+VOO + T k + 4 (4) P However, for points on a plane forming a Brillouin zone boundary across which there is an energy gap (V]mO) but not near another such plane, Iki = k + mi for one value m of p. The degeneracy of two unperturbed plane waves must be taken into account. This is done by considering the two equations h2 k2' (Voo0 + k2 - E) Aooo + V AA = -i A -a k0 E) A0 0 + - A - AO00 P and A VAm A000 + (V00 + h2-k.+ Am - E) Am =- VAmp AA P where the summation on p excludes the vectors 0 and m. For such points only two wave function coefficients AO and Am will be large. This means that the coefficients on the right-hand side can be neglected. There results a quadratic equation for the energy with solutions E = V0+0-l0 2 (k2 + Ij k + [-Ji -^(k2- k+-m I2)]2+1m1-2. (5) 2 2m- + 1 2m+ It follows from this that the energy gap at any point on the plane m (the plane which perpendicularly bisects the vector m) is gi3ven by 2See for example Wilson, A. H., The Theory of Metals, Cambridge University Press, Cambridge, 1956. _______________________- 5 ____3

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN AE = 2IM1|. (6) These results have formed the basis of the applications of the nearly-freeelectron approximation to the theory of metals. Jones has shown how the Fourier coefficients VI can be calculated from X-ray data. However, the calculated energy gap of 2 IV is well known to be much larger than the experimentally determined energy gap. This poor agreement is due partly to the calculation of Vm and partly to the fact that only one boundary plane was taken into account. All other boundary planes were assumed tobe far enough away from the center of the plane studied to have no effect on the value of the energy. In all cases that we are aware of, this neglect of other planes seems hardly justified. One objective of the present study is to take account of the other boundary planes properly. When this is done, a correction to the energy gap formula 21Vml I is obtained which turns out to be large compared with the gap itself. This study will also make it possible to discuss the shape of the energy surfaces and the size of the energy gaps near and at the intersection of two or more faces of the zone boundary. A new comparison between experiment and these new results of the nearly-freeelectron approximation will be required. Some New Results3 Secular Equation for Two Planes. For a first extension of the nearly free-electron approximation we are investigating the energy near the intersection of two faces of the zone boundary, say the perpendicular bisectors of the vectors -m and -n. Along such an edge |k = 1 k + ml = k + nl and we must consider the corresponding three equations for the amplitude coefficients. The secular equation for the coefficients of AQo0, Am, An is 2 Em - ( kl) I2i+ I + m 2 + k + 2] E 2 |2+ M)2(k 21k+m12+k2 n 12- + 121, )-(Vm2+!vi_2+!^22)] E - (4)3 k2 St+mI 2 k+2+( ) ( V n m1 2k2+ VA2 2 k+m 12+lVM 2 1k+n 2) - 2Re (V V- Vn-m) = 0, (7) where the zero of energy is taken such that V000 = 0 and Re denotes the real part (the Fourier coefficients of noncubi': structures can be complex numbers). 3Complete derivations will be given in a technical report ____________________________________ 14 ___________________4

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The occurrence of Vn-m as well as Vm and VE1 gives this equation certain limited symmetry properties. Thus the energy at the position k as determined. by planes m and n is the same as the energy at the positions A A A A A A A A k' = k + m or k1'~ = k + n or k"t = -k as determined by the planes ml = -m m" = -n mT = -m or or n' = n-m n = -(n-m) n1' = -n. This means, for example, that the energy must be determined near only one of the two types of edges of the first zone of the face-centered cubic structure. Modification of Perturbation Theory Near an Edge. If the Fourier coefficients are small, then near the origin the, smallest root of the secular equation, equation (7), is given approximately by 2 Vh2 E k: I _~ + ) 2m:h2:2 I +2 _ M k + n2 ~ This is the energy of a free electron plus the contribution to the perturbation theory result of planes m and n. However, this approximate solution breaks down at points near the line of intersection where kl = k + + and the exact solution must be used. It can be seen from equation (4) that the contribution to the perturbation-theory result of the different planes is additive. Thus, the energy in the first band in the region from the origin to the intersection of planes -m and -n is given by the perturbation-theory result, equation (4), omitting the terms p = m, n from the sulmmation and adding the exact lowest root of the secular equation. There is a possible difficulty in this procedure which will be discussed in a technical report. The condition of degeneracy Ikl =1k + ml = | + n. is satisfied for k on the line of intersection of the planes -m and -n (the minus sign is due to the sign convention chosen for the Fourier series). The point of intersection of this line with the plane of the vectors -m and -n is called -n the critical point, and the vector from the origin to this point is denoted by kc. The magnitude of kc is -m ----------- ^ ---------— ~~~~~~~~~'~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN kc =m-2n cos 0 + n~ (8) 2 sin @ In order to discuss the energy near an edge, it is convenient to take out the free electron energy by the substitution E (k)= k2 + (k) (9) 2m and to expand (k) about the critical point by the substitution k = +. (10) The secular equation for (X) is 2 ) 2 2 ~ 2m liv2 -2Ree(V ^nnV n ) = 0. (11) Formulas for the exact solution of this equation have been obtained from the well-known trigonometric solution of the cubic equation. The Energy Gap at an Edge. The energies in the three bands which meet at a line of intersection are given by the roots of 3-2(Iv +) (v + K- n+ l m2 )-2 Re (VVm n-V) = 0, (12) sincFor s for th e of intersection K is perpendicular to m and n and the scalar products vanish. The discussion of equation (12) for noncubic structures must await a tabulation of the complex Fourier coefficients. For the f.c.c. and b.c.c. structures, the Fourier coefficients are real, negative numbers and for the first zone either ( m,12 (1 Vn) 2n Vi - - 2V'n = -m anI d Vnm = arm or Vn-m = VA and Vn = V-m, where 0<V/1:. In this special case the roots of equation (12) are given by 6

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 1 = - 1t ( ) m ~l2 = -|V V 5 =1 (478 +2 -) l1vA. (15) These are plotted in the figure. 2. _3 E2 |, O | j.2.4.6.8 1.0 1, -2 For the first zone of the b.c.c. structure V = 1, and for the f.c.c. structure V-/0.8 for most substanceso Thus for these structures the energy gap between the first and second energy bands is of the order of 315V near an edge and decreases to not less than 2Vjm| in the zone face far from an edge. This means that the Fourier coefficients of the potential must be somewhat smaller than the value predicted from a gap of 21VJml. This rather surprising result tends to make the agreement between theory and experiment even worse. The significance of this result must be investigated further. In order to analyze the energy near the line of intersection it is desirable to introduce a coordinate system. This has been done and an expression of as a function of K obtained. This permits a better discussion of the energy splitting at a zone face and of the shape of the energy surfaces within the zone near an edge. These results will not be given in this report. PLANIS FOR FUTURE WORK At the present time the investigation of the energy near a line of intersection is approaching completion. In order to interpret and apply the results properly, the numerical values of the Fourier coefficients of the potential must be known. Consequently, an investigation of the Fourier coefficients has been undertaken which will include tabulation of the more important coefficients.for some common substances. The main problem.is to remove --------------------- 7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN the discrepancy between the value as calculated by Jones' formula and the smaller value derived from emperical data. When this work has been completed, it is planned to investigate the energy near a zone corner at which three faces meet. This will result in the study of a quartic secular equation. _____1_________________ 8 _______________________