THEORY OF AN ELECTREC -CURRENT-FLOW liNDUCED POLARIZATEQN EFFECT ON THE OPTICAL «ABSO-RPTEON OF A (SEMICONDUCTOR Thesis for the Degree of Ph. D. ' _ MICHIGAN STATE UNIVERSITY ’ (CARL A. BAUMGARDNER 1957 C nus-~- LIBRARY THESlé E ‘ ‘.’ ‘Ffl" 'fi.‘ '~ «Ix Cf . I .m l _ lee-M: q‘ 1 ‘3'. % . l 1’ 'x 'n' ‘1”! I'f 1" ’ .'. I, ' This is to certify that the thesis entitled THEORY OF AN ELECTRIC-CURRENT—FLOW INDUCED POLARIZATION EFFECT ON THE OPTICAL ABSORPTION OF A SEMICONDUCTOR presented by Carl A. Baumgardner has been accepted towards fulfillment of the requirements for Ph.D. degree in Physics ma' Major professor DMe August 1, 1967 0-169. ABSTRACT THEORY OF AN ELECTRICfCURRENT-FLOW INDUCED POLARIZATION EFFECT ON THE OPTICAL ABSORPTION OF A SEMICONDUCTOR By Carl A. Baumgardner A calculation is made of the direct optical absorp- tion in a current-carrying semiconductor. It is assumed that the only effect of the d.c. electric field which pro- duces the current, is to shift the distribution of carriers in 5 space. It is found that the absorption depends on the relative orientations of the electric field and the pol— arization vector of the light. This polarization effect is found to be proportional to the square of the carrier drift. A physical picture of the effect is presented. Curves of the effect versus photon energy are calculated for both fundamental and inter-valence—band absorption in indium antimonide, for both the intrinsic and extrinsic semiconductor, and for various field strengths and temp- eratures. The results are analyzed and found to be con- sistent with the physical picture of the effect. Sug- gestions are made for using the effect to investigate the distribution functions and wave functions of a semi- conductor'and a comparison is made With a similar effect measured in p-type germanium. THEORY OF AN ELECTRIC-CURRENT—FLOW INDUCED POLARIZATION EFFECT ON THE OPTICAL ABSORPTION OF A SEMICONDUCTOR By hdfid ‘l’ Carl AYIBaumgardner A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1967 ACKNOWLEDGMENT I wish to thank Professor Truman O. Woodruff for his concern, guidance and assistance throughout the course of this work, and for his critical reading of the manuscript. I also wish to thank the members of the Solid State Theory group of the Michigan State University Physics Department for their many valuable discussions. This work was supported by a grant from the National Science Foundation. ii TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION ................................. 1 II. THE ABSORPTION COEFFICIENT ................... A III' THE POLARIZATION EFFECT IN THE LIMIT OF SMALL CARRIER WAVE VECTOR ...................... ... 13 l. The Band Structure ....................... 15 2. The Matrix Elements ...................... l9 3. The Distribution Functions For Electrons and Holes ................................ 22 A. Calculation of the Polarization Effect ... 2A IV. THE PHYSICAL EXPLANATION OF THE POLARIZATION EFFECT ...................................... 29 V. THE POLARIZATION EFFECT FOR ARBITRARY WAVE VECTOR ...................................... 3A VI. DISCUSSION OF RESULTS ....................... A3 REFERENCES ......................................... 53 APPENDIX, Listing of computer Program For Calculation of the Polarization Effect ... 56 iii LIST OF FIGURES FIGURE PAGE 1. Direct Optical transitions ................. 5 2. Shift of electrons in k-space under the in- fluence of an electric field .. ....... ...... 1A 3. The energy bands in indium antimonide ...... 35 A. Wave function coefficients of the conduction band ...... .... .................. ........... 35 5. Wave function coefficients of the second va- lence band ........ ...... . .......... ........ 35 6. Wave function coefficients of the third va— lence band... ............................... 35 7. The polarization effect for the fundamental absorption ............................... .. A0 8. The polarization effect for the inter-valence band absorption at 3000K. .............. ..... Al 9. The polarization effect for the inter-valence band extrinsic absorption at 770K ........... A2 10. The absorption for valence band two to va- lence band one transitions in indium anti— monide...... ................................ A9 11. Measured absorption curves for p-type germanium, after Pinson. and Bray --------- 50 iv LIST OF TABLES TABLE PAGE 1. The matrix elements between the conduction and valence bands in InSb .................... 22 2. Parameters used in calculations for InSb ..... 38 I. INTRODUCTION If a large external electric field is applied to a semi- conductor in which there are free carriers, the most im- mediate result is a change in the energy distribution of those carriers. Many physical properties of semiconductors depend on the energy distribution of the carriers and it is to be expected that these properties will change with the electric field. Since the 1930's many such changes have been observed and studied; for example changes in the mobility and the Hall effect. New effects, such as nega- tive resistance and anisotropies in the conductivity, have also been found to be associated with carrier streaming. It is evident that an understanding of the energy distri- bution of the carriers is fundamental to an understanding of these changes of physical properties which have come to be known as "hot-electron effects". The fact that our present-day knowledge of the distribution function as a function of the electric field is to a large extent only qualitative, is reflected in the large number of hot—elec— tron effects for which there is not yet a good quantita- tive explanation.1 The purpose of this paper is to extend the theoretical understanding of some recent experiments aimed at obtaining quantitative information about the distribution function under current flow conditions, and to propose some new experiments with the same objective. Much of the earliest work on the effects of large electric fields on solids was directed toward understanding 2,3,4 the energy distribution function. Only recently, how- ever have there been any successes in describing the dis- 5’6 The most notable of 7 tribution function quantitatively. these has been the work of Pinson and Bray on the experi- mental determination of the distribution function in p—type germanium by means of "free—electron" absorption experi- ments. The facts that a unique correlation between the absorption frequency and the electronic energy of the ab— sorbing state is possible, and that the amount of absorp— tion at this frequency is a function of the number of car- riers available to absorb the light, allowed the number of carriers as a function of the energy to be roughly determined. It was noted by Pinson and Bray at that time,8 as well as by others working independently,9 that the absorp- tion was dependent on the direction of polarization of the light relative to the electric field, and that this effect yields some additional details concerning the angular dependence of the hot—carrier distribution func- tion. The complexity of the wave functions for p-type germanium prevented a complete quantitative analysis and thus impeded the theoretical explanation of the ef- fect and the description of the distribution function. In this paper we predict a similar polarization effect for indium antimonide, and using the relatively simple wave functions given by Kane10 for this semiconductor, we analyze the effect in detail. Consideration of the results yields a clear physical picture of the causes of the effect and shows that measurements of the effect not only can be a tool for understanding the distribution function, but also can yield information about the wave functions of the semiconductor. In Chapter II of this paper the optical absorption co— efficient is developed in a form particularly suited to our problem. In Chapter III the polarization effect is calculated in the approximation that the electronic wave vector, k, is small. A physical explanation of the effect is given in Chapter IV. The restriction to small wave— vectors is removed in Chapter V and theoretical curves of the polarization effect versus photon energy for various distribution functi0n8,temperatures and electric field strengths are computed; a discussion of the results is presented in Chapter VI. II. THE ABSORPTION COEFFICIENT The amount of light of a certain frequency,w, absorbed by a crystal upon which it is incident, is described by the absorption coefficient,11 defined by a(w) = power absorbed per unit volume . (2.1) incident flux The power absorbed per unit volume is (2.2) “wrind where h is Planck's constant divided by 2n. rind is the net rate of transitions induced in the crystal by the light, per unit volume, per unit time, and is a function of the frequency. The incident flux is the magnitude of the time averaged Poynting vector, S = 0 EL X'H . (2.3) AR EL and H are the electric and magnetic field strengths of the incident light. In Chapters III, IV, and V we shall be interested in calculating the absorption due to direct electron transitions between energy bands in a semiconductor (see figUre 1). These can either be transitions bet- ween two valence bands (dotted lines), or between a valence band and the conduction band (solid lines). The former is sometimes called free-electron absorp— tion, because it lies in the same frequency range A t Energy Conduction Band E Energy Gap Valence Bands \ 7 /I 2 Fig. 1. Direct Optical transitions U‘I as true free—electron absorption, to distinguish it from the latter which is called fundamental absorption. in this pa— per we call them inter—valence band absorption and funda- mental absorption, respectively. The electric field of the light causing the transi— tions ( not to be confused with the d.c. electric field, to be introduced later to modulate the absorption ), can be described by a vector potential, A<;,t> = A. eXp[i(§°:-wt)] + A: eXpE-i(§'§-wt], <2.u) where A0 =|Ao|exp(i6)ao is complex, a. is a unit vector in the direction of A0, s is the wave vector and w the frequen— cy of the electromagnetic wave. Our first step in calculat- ing the rate of induced transitions will be to find the interaction energy between this field and the electrons in the bands. The interaction energy, H', is found from the classi— cal Hamiltonian, H, for a particle of charge e and mass m in an electromagnetic fieldl2: H=%—m(p_—%A)2+e. (2.5) p is the momentum, c the speed of light in a vacuum and T is the scalar potential of the electric field. Multiplying out the squared term in Eq.(2.5), and using the relation f(§)px — pr<3> = 1% r<;), (2.6) 3. 3x which is true for any function that can be expressed in a power series in x, y, and z, we have H = E: ’S;A°E + ieh V-A + e , A + e9. (2.7) ‘ m 2mc 2m c _ x 3 B 3 Here 2 — 1 5x + j 5 + R a where i, 5, R, are unit vectors in the x,y,and z direc— tions. We choose the transverse (Coulomb) gauge, so that V° A = O T = O We also drop the term in A2 as much smaller than the remaining terms. Adding to Eq. (2.7) the potential V (3) of the electron in the periodic lattice and sub- stituting p = -i My, we arrive at the Hamiltonian for an electron in a periodic lattice interacting with an electric field. To first order H ; Ho+ H' (2.9) with H =1"-2 v2+ V(r) (2 10) 9 2m - ’ ° and HI: -2 A02 (2 11) mct-‘, ' ' H0 is the unperturbed Hamiltonian leading to the un- perturbed band structure of the semiconductor, and H is the interaction energy we have been seeking. Using time dependent perturbation theory we can 13 the probability that an electron in the calculate valence-band state CV at time t=O will be induced by the perturbation H' to change to the state CC in a higher band (perhaps but not necessarily the conduction band) at the later time t. This transition probability is given by T0 (t w) = |<¢ |H'|¢ >|2 x S(E) (2 12) CV ’ c T v ' where A Sin2[(EC(k) — Ev(k) - Mw)t/2Ml S(E) = 2 ’ (2.l2b) [Ec(k) - Ev(k) - Mm] <> indicates the quantum mechanical expectation, and ' _ e . 2 HT - ‘53 exp(is°r)AO°p. ( .13) Ec(kc) and EV(kV) are the electron energies in states ¢c and ¢v respectively, and are functions of k the wave vector of the electron. Tgv is not the total transition probability. We must multiply it by an occupation factor (distribution func- tion), f(EV), which is the probability that the state ¢v is occupied, and also by the occupation factor [1 - f(Ec)] which is the probability that the state Cc is not occupied. Thus the total probability of a transition from state CV to state Cc is TCV = |<¢C|HT|¢V>|ZS(E){f(EV)[l - f(EC)]} (2.1M) The probability that an electron in state CO at time i t = 0 will be induced by H to change to the state CV at time t, can also be found by time-dependent perturbation theory. It is T30 = |<¢.V1H.'f'|¢c>|2 x S(E), (2.15) where $'= %% exp (—i§°r)A:°g. (2.16) It can be shownlu I<¢CIH§|¢V>|2 = |<¢V|Hyl¢c>|2 (2.17) therefore T30 = T‘gv (2.18) The occupation factor for TV is f(EC)[l — f(EV)]. There— c fore the net transition probability (the probability for transitions upward minus the probability for transitions downward) is THV = ch - Tvc = IclHTl¢v>l2 8(E) [f(Ev) — f030)] (2.19) The rate of transitions is found by summing the transition probabilities for all possible combinations of ¢ summation is performed by multiplying by the density of c and ¢V and dividing by t. In the most general case the . l . . states per unit volume, (€573, and Integrating over the two bands 0 and v. However conservation of momentum re- quires that the momentum of the electron in the initial state plus the momentum of the photon must equal the momentum of the electron in the final state; i.e. lO KEV + Mg = Mk0. (2.20) But s is so much smaller than kv or kc that essentially k = 5 —v c' This establishes a one—to-one correspondence be— tween the two states ¢v and OC which allows all possible combinations of ¢v and ¢c to be included simply by integrat- ing once over all k space. Therefore _ l_ ' 2 rind _ E Il<¢c|HT|¢v>l 8(E) x (2.22) k2 [f(EV) — f(EC)] Sine d6d¢dk. (2M3 To find the magnitude of S we use - g: 9A=-2_UJ. c A EL ‘c 8 CIAOI Sin (s r - wt + 6)ao, H = I x A = —2§ x aOIAOI sin (s-E — wt +5), (2.23) - 2 _ 1 Sin (s-r - wt + 6) — 2’ and kc . . —$ = n = index of refraction. (2.2A) Then using Eq. (2.3) we have, ISI = flu :2 <2 25> 2N0 o ' ' From Eqs. (2.25), (2.22) and (2.1) we arrive at the absorption coefficient: 11 ezh _ A . 2 Anznmzcwt f|<¢clao E|¢V>| S(E) d(w) = (2.26) [f(EV) — f(EC)]k2 sine dedb. We have used the fact that s is essentially zero to drop the exp (is-r) in the matrix element. The integral over k in Eq. (2.26) can be done im- mediately even though we do not have explicit expressions for the k- dependence of the energy or distribution func— tion. This is possible because in the limit of large time, t, we have for S(E) (Eq.(2.l2b)): A Sin[{E (k)—E (k)—hw}t/2h] 2nt Lim C V = ——— 6[Ec(k)—Ev(k)-hw] M t+°° [Ec(k)—EV(E)-bwl2 (2.27) Taking this limit makes sense physically if the light has been turned on for a time long compared to the inverse of its frequency, which is generally true experimentally. Using _ l 6[f(x)] - Qfi 6(x - x0) (2.28) dx Eq. (2.26) then becomes 0(w) = _ezk2 lsbclaoiglbv>|2 sine [f(EV) — f(EC)]ded¢ 2wmzcnw C1(EC — Ev) , dk (2.29) 12 where k is now determined by Ec(k) - Ev(k) — Mm = O (2.30) If there are more than two bands involved in the transitions we must sum Eq. (2.29) over all possible com— binations of bands. The total absorption coefficient is then _ ezkz |<¢c|ao ' El¢v>|2[f(EV)—f(Ec)l sin ededq; C(w) - Z Z d (E _ E ) c v 2nm2cnw c v dk (2.31) To proceed any further in a calculation of the ab- sorption coefficient we need explicit expressions for the matrix element, the energies and the distribution functions. III. THE POLARIZATION EFFECT IN THE LIMIT OF SMALL CARRIER WAVE-VECTOR In this chapter we calculate the absorption coef- ficient for intrinsic indium antimonide with a d.c. electric field, E (which shall always appear with a vector sign to distinguish it from the energy), applied perpendicular to the direction of propagation of the light (See Figure 2). We take the electric field to be in the z-direction and consider two cases. In one case the vector potential, A, is directed parallel to E, and in the other A is directed perpendicular to E. The difference in the absorption for these two cases is the polarization effect. We consider in this chapter only the absorption due to transitions be- tween a valence band and the conduction band; i.e. the fundamental absorption, and this is calculated in the ap- proximation of small k (electron wave-vector magnitude). We assume that the d.c. electric field does not change the band structure of the crystal. This assumption is thought to be valid because many collisions take place during the characteristic time of a polarization effect ex— periment, which washes out changes that would otherwise 15 require consideration. We have chosen indium antimonide as the model semiconductor for our calculation because of its high mobility, which permits a large shift in its 13 Relative orientations of E and A. , Light— ] A pol rized either T or —-+ Fig. 2. Shift of electrons in g-space under the influence of an electric field. 1A 15 distribution function with relatively small electric fields, and because accurate, relatively simple analytic expressions for its wave functions are available. 1. The Band Structure The wave functions and energies for indium antimonide have been calculated by Kane.lO He added to the classical Hamiltonian Eq. (2.9), several spin dependent terms and applied the resulting Schroedinger equation to the Bloch function, exp (ik-r)uK(r), to arrive at the Schroedinger equation for the cell-periodic function uK(r), which we write as 2 2 E35 + V<£> + Mk-E + Mk 5— 2m + SlJuK = EKu.K(;>. (3.1) The first four terms result from the classical Hamiltonian and S1 is a spin-dependent term added by Kane. We have dropped a second spin-dependent term because Kane found it very small. The solutions of Eq. (3.1) are found in terms of the solutions of [P— + VJU. = EiU. (3.2) These solutions are known from group theory to be St and 8+ for the conduction band, and Xt, X+, Yt, Y+, 2+, and 2+ for the valence bands. S signifies a function with the symmetry properties of an atomic s orbital under the operations 16 of the tetrahedral group, and X, Y, and Z signify wave func- tions with symmetry properties of the x, y, 2 atomic p- functions. The 5 functions have energy ES and the p func- tions all have energy Ep For k in the z direction, Kane solved the Hamiltonian matrix corresponding to Eq. (3.1) for its eigenvalues which he found to be given by the equations E = o (3.3) E'(E' - EG)(E' + A) - k232(E' + 2A/3) = 0. (3.A) Here EG = Es’ A = -3Ep, E' = EK — (h2/2m)k2, (3.5) .P_= "' %, (3-6) and m is the free electron mass. EG is the energy gap and A is the energy separation of the "split-off band"; i.e. a valence band which is lower in energy than the other two bands which are degenerate at k = 0. Both parameters can be evaluated experimentally. If k2 is very small Eq. (3.A) is approximately equal to I t i E (E — EG)(E + A) = 0, (3.7) which has the solutions ! t t E 2 O, E 2 E E 2 -A. (3.8) 17 We can find an approximate solution of Eq. (3.A) corresponding to each of these three values by writing it in the form (13'- EO);= r(E'), (3.9) where E0 is one of the values Eq. (3.8). I Then substituting this same EO for the E s on the right we have _ MZKZ P2k2 2 l _ hzkz Ec _ EG + 2m + 3 E_ + E + A _ EG + 2m G G (3.10) — Mzkz szz __M2k2 E2 ' 2m ' 2‘3Eé ‘ 2m2 (3'11) — h2k2 P2k2 MZKZ E3 ‘ "A + ‘25— ' 3(EG + A) "A ‘ ‘25; (3°12) The highest valence band is the solution of Eq. (3.3), which is 2 2 E1 = M2§ . (3.13) This is inaccurate, as explained by Kane, so we use instead __M2k2 El - 2m 3 (3.14) l where ml is given experimentally. EC is the conduction band, E1 is the heavy mass band, E2 is the light mass band and E is the split-off band. 3 18 The wave functionstflmfizresult from the diangonali- zation of the Hamiltonian corresponding to Eq. (3.1) are: ¢ia = eitis+]' + bi[(X — iY)+//2]' + ci[z+]' . . (3.15) ¢iB = ai[i8+] + bi[-(X + iY)+//2] + Ci[Z+] ¢la = [(X + iY)+J'//2 (3.16) $16 = [(X - iY)+l'//§ The index i takes on the values c, 2, 3 referring to the conduction band, the second valence band and the third valence band. The 1 refers to the first valence band. Wave functions with first subscript the same but different in the second subscript (d or B) are degenerate in energy but differ in spin. The primes in Eqs. (3.15) and (3.16) indicate that for directions of k other than the z—direction, I i t I i l X , Y , Z , S , t and + are obtained by the transformations: F.1 '- 7F.) t exp (-i¢/2) cos (6/2) exp (i¢/2) sin (6/2) t l L+ L-exp (—i¢/2) sin (6/2) exp (i¢/2) cos (6/2) L+, (3.17) {‘9‘} '3 17 X Cos 6 cos ¢ cos 6 sin ¢ -sin 6 X} Y. = -sin 9 cos O O Y _Z'J Lsin 6 cos 9 sin 6 sin 9 cos 6 _[Z_ (3.18) S = S (3.19) l9 6 is the angle k makes with the z-axis and O is the angle k makes with the x—axis. The ai, bi’ and c1 are real coefficients given by: ' ai = kP(Ei + 2A/3)/N /2 ' 20) V l Ci - (E1 - EG)(Ei + 2A/3)/N l where N = (ai + bi + c:?L)l/2 and the E1 are the solutions of Eq. (3.A). For small k Eqs. (3.20) become: aC = l, bC = C0 = o (3.21) _ _ 1/2 _ 1/2 a2 — 0, b2 - (1/3) , c2 - (2/3) a3 = 0, b3 = (2/3)l/2, c3 = -(1/3>1/2 2. The Matrix Elements Including spin differences, there are six valence bands and two conduction bands. Therefore in the summation over the conduction and valence bands in Eq. (2.31) we will have twelve terms. We must calculate the matrix element for each of these twelve terms and for each of the two di— rections of polarizations of the vector potential A. Thus there are twenty-four matrix elements to be calculated. In the small 5 limit many of the coefficients of the wave functions, as given in Eq. (3.21), are zero. However we 20 shall derive the matrix elements for the general case, where the coefficients are non-zero, since we shall need them in that form in a later chapter. The following properties of the wave function will be useful in Calculating the matrix elements: a) In the absence of spin-dependent terms in the perturbing Hamiltonian ' <[exp(i¢/2) cos(6/2)t + exp(-i¢/2) sin(6/2)+]| l[-exp(-i¢/2) sin(6/2)+ + exp(i¢/2) cos(6/2)+]> = O (3.22) and likewise ' l i l ' <+ |+> = o; <+ |+ > = 1, <+ |+ > = l. (3.23) b) Electric dipole selection rules, group theory, and Eq. (3.6) yield: — -111: — _ M (3.2A) _ =_imP — -F_ (3.25) J¢i = o; J = X,Y,Z; i = x,y,z (3.26) = 0; i = x,y.z; J = X.Y.z; K = X.Y.z (3.27) We now calculate, as an example, the matrix element <¢Ca|aoop|¢28> for A, (1.e., a0) 1n the x-direction. <¢ca|px|¢28> = [Using Eqs. (3.22) and (3.23)] [Using Eqs. (3.18) and (3.19)] [Using Eqs. (3.26) and (3.27)] [Using Eqs. (3.2A) and (3.25)] + + + + , a b2 i 0 (cos 9 cos ¢—i sin ¢) + a X bca2 + 1 (cos 6 cos o-i sin ¢), @ X mP/(h/2)(cos 6 cos O-i sin ¢)(bca2-acb2). The twenty-four matrix elements calculated in this way are listed in Table I. In this table i = 2,3, D = mP/M, G = (acci + ccai), and Gi— = (acbi - bcai)° 1+ 22 TABLE I.—-The matrix elements between the conduction and valence bands in In Sb. A in z-direction ¢ia ¢is ¢1o d’18 ¢ca Dcos6Gi+ (2—)sin6Gi- O -(2—)acsin6 /2 /2 D D . ¢CB —(;E)sin6Gi— Dcos6Gi+ -(;E)a081n6 O A in x-direction ¢io ¢iB ¢1o. $18 p Dsinecos¢G + -2—(cos6cos¢- O Dac ca i ;:—(cos6cos¢+ 2 Sin‘bmi' i sin¢) D Dac A08 ——(cos6cos¢+ Dsin cos¢Gi+ ———(cos6cos¢- O /2 /2 i sin¢)Gi- i sino) 3. The Distribution Functions for Electrons and Holes We assume that the effect of the electric field is to shift the distribution of carriers in k-space by the amount k This is shown in Figure 2 for a Fermi-sphere. d’ For non-zero temperatures the sphere of course will not be sharply defined but will be "blurred" near the edge. At zero electric field, the carriers will have the distribu- tion given by the dotted circle. When the electric field is applied in the positive z-direction the distribution shifts in the negative z—direction by the amount kd. Thus carriers 23 at A with a certain energy, E(A) will have an occupation probability corresponding to A + g of the unshifted distri- bution. d is a positive number which is equal to ‘kd: the size of the shift in k due to the electric field. kd has the same sign for the conduction and valence bands although in our problem the sign is not important since the polariza- tion effect is independent of this sign. For the small k calCulation we take the distribution functions to be: exp {[EF — EC(A + g)l/KT} (3.29) for the conduction band, and l - eXp {[Ev(5 + g) - EFJ/KT} (3.30) for the valence bands. EF is the Fermi energy, T is the temperature, and K is Boltzmann's constant. These are Fermi-Dirac distribution functions in the limit IE — EF|>>KT. (3.31) Eq. (3.31) is valid for intrinsic In Sb at all but very high temperatures, except near the conduction-band edge where it is good only up to about 300°K. Eqs. (3.29) and (3.30) are not necessarily the actual distribution functions for In Sb. More likely the experimental crystal will be an extrinsic semiconductor with some form of Maxwell-Boltzmann distribution function with the normalization depending on the concentration of impurities. However, these distribu— tion functions do have the essential Boltzmann factor, and 2A furthermore the simplicity of the intrinsic situation will make the physical interpretation of the theory clearer. A. Calculation of the Polarization Effect We now have explicit expressions for all the factors in Eq. (2.31). We calculate the absorption due to transi- tions between valence band one and the conduction band for the case when A is polarized in the x-direction, perpendi- cular to the d.c. electric field, as an example. Substituting Eqs. (3.10) and (3.1A) in Eqs. (3.29) and (3.30) we have: M2 _ 2 2 f(EC) — exp{[EF-EG-ch(kl+dc+2kldc cos 6)]/KT} (3.32) 2 f(EV) = 1-exp{[%%:(ki+di+2kldl cos 6)-EF]/KT} (3.33) We have used (5 + g)-(g + g) = k2 + d2 + 2kd cos a, (3.3A) and we have written k with a subscript because k is different for transitions between different bands according to Eq. (2.30). Substituting Eqs. (3.32), (3.33), (3.10) and (3.14) in (2.31) and summing over the four matrix elements involved (two of which are zero), we have: di(w) = ggaif(cosz6cos2¢+sin2¢)sin6[1-Allexp(Vllcose) -A exp(-VC c1 cos 6)]d6d¢, l 25 where we have written 222 1 4e aCP |k1| a = (3.36) O 3canl+ l— + £— m m c l A- - ex {FEE-(k2 + d2) E J/KT} (3 37) 11 ' p 2ml 1 1 ‘ F ‘ A - {[E’ E M2(k2 + d2)/KT} (3 38) cl ‘ eXp F 5 G ' 2R; 1 c ' hzk d __ 1 1 V11 - _5;HT_ (3.39) hzkldc VCl = _mc—KT_ . (3.140) a: is the absorption in the limit T = O, E = O, for trans- actions between valence band 1 and the conduction band. The integral over 9 gives a factor of R. The inte- gral over 6 can be simplified with the change of variable cos 6 = x which implies —sin 6d6 = dx. Eq. (3.35) then becomes: V -x -V x l_3ll 2 _11 Cl Oi - 8&0 {l (x + 1)(l - Alle - Acle )dx, (3.41) which integrates to a1 = %Gl{£ - A [(e ll_e ll) - (e ll+e‘ 11) + (e 11.65 11)] L o 3 11 V11 V11 V11 (3.42) V -V V —V V -V -A [(e Cl-e 01) _ (e Cl+e C1) + (e Cl-e Cl)]} cl V V 2 V 27 Cl cl cl On expanding the exponentials for I Vij |<< l (3-A3) we find A v2 A v2 oi = aO{l-Acl—All - 01 c1 —;%~ll } Eq. (3.A3) is valid if k.d. M2—%—i << KT (3.44) 1 Since we have already assumed k is small this is good if di is small: i.e. the drift velocity in band i must be much less than the thermal velocity, which is true in InSb for fields of lOOv/cm or less. Similarily we find: 1 _ 1 all — do[l Acl All] 2 _ 2 _ _ _ _; 2 2 “L ‘ O‘oEl Ac2 A22 12(Ac2Vc2 + A22V22)J (3.45) 2 _ 2 _ _ _ 1 2 2 0‘ll ' O‘o':l Ac2 A22 3(Ac2vc2 + A22V22)] 3 3 = 3 _ _ _ i 2 2 0‘II “1 0‘o[1 Ac3 A33 0(Ac3vc3 + A33V33)] 27 O A A.. V . V i = 2, 3 are given by Eqs. where a. .. 1’ ci’ 11’ 01’ 11’ (3.36) through (3.AO) with obvious changes in the sub— scripts. II and 1 indicate the relative orientations of E and A. The polarization effect is given by i = i _ d1 . = . up all IL , 1 1,2,3 Thus we have cl: o1 [A v2 + A V2 J (3 A6) ' 0 cl cl 11 11 ° Q2F+dl [A V2 + A V2 l (3 A7) p 1 0 c2 c2 22 22 ’ 3= 4 up 0 (3. 8) The polarization effect (Eqs. (3.A6) through (3.A8) is roughly proportional to E2 through the variables Vij' The factors in the A's which are proportional to E (d: in Eq. (3.37) for example) are not important, Since d is very small, which makes exp (-d) nearly 1 and slowly varying. The effect goes to zero with A as expected, however the difference in signs of the first two equations and the zero result for the last is noteworthy. This last result becomes even more interesting when the integrals analogous to Eq. (3.35) are written out for these transitions. One finds the effect is identically zero no matter what distribution function is used. This points to the wave functions as the 28 cause of the peculiarites and as we shall see in the next chapter, this inference is correct. IV. THE PHYSICAL EXPLANATION OF THE POLARIZATION EFFECT In Chapter III we suggested connections between cer- tain features of the wave functions and certain peculiar- ities of the polarization effect. To clarify these connections we investigate the matrix elements for the different polarizations and different directions of electronic motion given by A, for an idealized set of wave functions. Consider transitions between a conduction band with a purely S—like wave function and a valence band with a Z— like wave function, when A is in the z-direction. For A also polarized in the z-direction, the matrix element is = M . (4.1) If we now take A in the x-direction, 8+8 and 2+X by transformations (3.18) and (3.19). Thus the matrix element is <§|pZ|X> = 0. (A.2) Now take A to be polarized in the x-direction. For A in the z-direction the matrix element is = O, (4.3) and for A in the x-direction it is = M. (A.A) Next we calculate the absorption associated with 29 30 the two directions of A. (Picture two electrons doing the absorbing, one moving in the z-direction and one moving in the x-direction with the same speed; ice.'the same magnitude of A.) Because the energy is a spher- ically symmetric function of A we have dz: [|M|2+ O] D IMIZD: (14.5) ax: [O+|N|’] D = 1MIZD, (4.6) where dzand uxare the absorptions for the light polar- ized in the z-iand x-directions respectively and D is some constant which depends on the magnitude of A. We see the sum total of dzand axis the same for these directions. We call the z and x directions correspond- ing directions because what one contributes to uxthe other contributes to dz, and vice versa. When we consider the absorption due to electrons moving in other directions, we find, because of the spheriCal symmetry of the transformation equations and the energy, that for each direction Bz there is a corresponding direction Bx thatigives the same contri- bution to ax as Bz gives to dz and vice versa. This is a one-to- one and single valued correspondence. When we integrate over all angles as prescribed by Eq. (2.31), we find that dz = dx just as in Eqs. (A.5) and (A.6). 31 If we now insert in the absorption expression an angularly dependent distribution factor, —exp(Cose ), like the one due to an electric field in the z-direction, then, for A also innthe z—direction a = D [-exp(l)] — |M|2 De a = D [-exp(l)] O. xv x For A in the x-direction a = D [-exp(0)] = O a = D [ -exp(0)] =-IMI2 D Adding up the total absorption for the two direc- tions of A we have d = [—lMl2 De+O] = —IM|2 De, [ o — |M|2 D ] = — IMIZD, Q II and therefdre az- aX = all - al = — IMIZD (e-l) Thus there is a polarization effect because the absorption contribution from each of the corresponding directions x and z is multiplied by a different occupa- tion factor. LikewiSee when we integrate over all angles there will be a polarization effect for the same reason. The field acts to lowero the absorption because of the negative sign of the distribution factor. We can now understand the reason for the sign difference between Eqs. (3.A6) and (3.A7). The wave 32 functions for small A given by Eqs. (3.15) and (3.16) with coefficients (3.21) show that for A in the z-di— rection valence band number one has no Z component. Thus it contributes little or nothing to aZEall when the dis- tribution factor is large. At the same time it contri— butes most to aXE a i when the distribution factor is llarge. Thus the action of the distribution factor is larger for ai than all, and because the sign of this action is negative, the resulting polarization effect is positive. Valence band two, however, has a much larger contri- bution from the Z-like wave than the X-like wave. Thus it will tend to contribute more to all and less to ai when the distribution factor is large. Therefore the action on allis larger than the action on ai and the sign of the distribution factor makes Eq. (3.A7) neg- ative. The cause of the zero result for the third valence band is also to be found in the coefficients (3.21.). Because the relative probability of an electron, in this band, being in the state X, Y, or Z is given by the square of the respective coefficients of these functions, we see from Eqs. (3.21) and (3.15) that the probability of an electron's being in X + i Y is twice the proba- bility of its being in Z. Dividing the probability of being in X and Y equally between the two, we find the probabilities of being in X,Y, or Z are all equal. 33 This is true for all directions of A because of the spherical symmetry of the transformations (3.18). Thus for any given direction, all = a i; i.e. the correspond— ing directions are the same, and the distribution func— tion can make no difference. The deScription of the polarization effect for small A can be made in the simple terms used above because the conduction band is completely S-like in this limit. Away from A = O the conduction band rapidly becomes a mixture of all four orbitals, and the valence bands take on some S—like character and depart from the simple combinations of X, Y, and Z orbitals that made the above analysis so unambiguous. No matter how complicated the admixtures become, however, an analysis similar to that above can be carried out with the additional re- quirement that transitions proceeding downwards from the conduction band to a valence band (or from a higher valence band to a lower valence band ), are negative additions to the absorption. V. THE POLARIZATION EFFECT FOR ARBITRARY WAVE VECTOR The approximate determination of the polarization effect given in Chapter III cannot be used to plot a curve of absorption versus frequency for a comparison with experiment because the wave functions for small A rapidly lose accuracy as we move just a small interval in fre- quency from the absorption edge (see figs. A,5,6). Therefore in this chapter we calculate the polarization effect using the wave functions for arbitrary A given by Eqs. (3.15) and (3.16) with coefficients (3.20). We calculate first for the fundamental absorption and then for the inter—valence band absorption. Summing Eq. (2.31) over the conduction band and valence bands, and using the matrix elements in Table I,we find Mt» al|(k) =e2 p2 k2 x Hon i=1 B—w. d(E -E.3 1 ——aE——r—l (5.1) R 2 . 2 2 .§ lo{2(acci+ccai)SIn6+[(acbi-bcai)-2(acci+ccai) ]S1 6} x {f(Ei)-f(EC)}d6, and 3 91: _e_2_ '5 92 k2 X (5.2) hcn 1=l h mi d(EC—Ei)| dk n 2 _ _ 2 . 3 lo{[(acci+ccai) (bcai acbi) ] Sin 6 + 2 +2(bcai-acbi)2 Sine} {f(Ei)-f(EC)} d6 34 Fig. 3. Energy Bands Fig. A. Conduction Band 1.0- 1.0 E a //’ C O 8. /’ E O. t O / Evl ev \\ E I “-2. /’ v2 0 O. I l..._J [O m <: UK) I I l I _ U 1* I I I 1 I* i _ 5 10 15 20x10 “ 2 4 6 ,8 10 12 lelO “ k2 in atomic units k2 in atomic units 1.0l l 3 0.8 b 0.6 C 0.4 O.% —a l l l l l T i _ i'i’ I I i I r _ O 2 4 6 8 10 12 le10 1 O 4 6 8 10 12 lelO “ 2 k2 in atomic units k2 in atomic units Fig. 5. Valence band two Fig. 6. Valence band three Fig. 3. The energy bands for m1 = .5Am. Figs.A,5,6. Wave function coefficients for the energy bands of indium antimonide after Kane.lO 35 36 where hwi = Ec(k>-Ei. (5.3) and it is understood that f1, 1-2, 2.8 while for the third term we use the prescription,c-£. The re- sulting equations foral_and allare solved numerically as before. The parameters used in calculating the polarization effect for InSb are shown in Table II. 38 TABLE II Parameters used in calculations for InSb. — = _ ‘5 ..._.. Eg (.23 9.6 x 10 x T) ev mC Ol3mEg/.23 m1 = .5Am m2 = .Ol5m m3 = .12m A = .9ev P2: .AA atomic units _ a -1.5 2 = e -2.1 “n - 7x 10 x T cm /v.sec up 1.1 x 10 x T cmyv.sec The band parameters are as given by Kane 16 except Eg, 17 m0 and m which are as given by Ehrenreich . The values I of m determined by various experiments are not consistent. 1 We have chosen the value tabulated in preference to others because it is most defensible, and it also produces theo- retical absorption versus photon energy curves closer to the experimental resultS. The mobilities vary with temp- erature and hole concentration and are as given in ref— erence 18. The results of the computations are given in terms of a dimensionless function we shall call the polari- zation effect coefficient, defined by PE=OL.|__|_:.O:_£ ’ (5 7) 0L ZEI‘O where aZ is the absorption at the temperature at which ero all - ai is measured,for zero electric field. Figure 7 shows the polarization effect coefficient for the fundamental absorption for intrinsic InSb,as a 39 function of the photon energy, for T = 300°K and Aé100V/cm. The exact Fermi - Dirac distribution function was used in this case, not its Maxwellian limit as in Chapter III. Fig.53 shows the intra—valence band absorption at T=300°K and §?100 V/cm’ which turns out to be much the same whether we use the intrinsic Fermi-Dirac distribu- tion or go to the extrinSic case for a hole concentra- tion of A x lOls/Cmg and a Maxwell Boltzmann distribu— tion given by Aflaphfexp(E/KT) ' NE) (5.8) (2HKT) 3/2(m13/2+ m2 3/2) where we have averaged over the highest valence band masses and p is the concentration of holes. Fig. 9 shows PE for inter—valence band absorption in extrinsic InSb at 77A) DO 5‘J321‘ EGAPAD(J)'00 SOB!L(J)-91.1Eta/(Tot(2.1)) VMASS(19=.013*EfiAP/EGAPZER0 VMASS(2)3e.015 VMASS(3)=9.12 VHASS¢4)=:.54 CKANE=2.*(1./VMASS(4)!1.) BOLTZC=8.616~5 TK=TrBOLTZC EFERH1= EGAPT/2.* 3.*YKtLOGF(eVMASSC4)/VHASS(1)>14. REF!ND=3.96 CALL-ENERGY CALL ABSORBZ EFIELD=1on. NSR=80 _ FORMAT (5Xa4HNSR=.IS) PRINT 7.NSR CALL ABPARPER END 56 (30¢) 11 12 13 18 20 22 45 55 57 suaaourtue ENERGY THIS SUBRDUTINEFINDS THE-E sue K OPEEQUATIONt3.5) BY SOLVING 50. (3.4). THEN ADDING THE FREE ELECTRON ENERGY (EFREELEC) AND CORRECTIONS FOR HIGHER BANDS (HPERT). COMMON/aLocx1/EaAE.VMASS(4).DELTA ‘COHMON/ENABSISHV‘loz)aA(4¢162)23(40102)90(4.102)aEKANE(4:102)oS(10 12) GOMMON/MAINENSISGE.cKANEbEGAPAD¢4> COMMON/ENERGYSIHHozu.5P5.00.09.08 DIMENSION~E¢3).MI<31 HHOZMt*E(J)+2.¢DD*E(J)eTA DELTAE=FEIDFE E(J)=E(J)EDELTAE 1F (EPSEABSFthLTAE))45.55.55 CONTINUE TF=E(J)§DB TG=E(J)IEGAP AN=¢S(K)Et.5)tTr BN:DB*TG/1.414214 cN=TG¢TP FNORMI(ANOANtBNtBN+fiNECN)**.5 A(J.K>=AN/FNORM 65 58 acJ.K)=RN/F~0RM C(J.K)=CN/FNORM HPERT'(B‘JoK)*3(J:K)/2v+C(J0K)'C(JnK))*CKANE¢EFREELEC EKANE‘JEK)=E(J)tEFREELEC’HPERTOEGAPAD(J) EKANE‘4pK)=EFREELEC/VMASS(4)fiEGAPADC4) A¢4,K)=g. 8(41K’=1. C(4.K)=O. ‘FORMAT (61303.4F14000317) PRINT 69:3”V‘K)aEKAME‘laK,nEKANE‘4OK)DEKANE(20K)JEKANE(SOK)D 1MI‘1’QMI(?)RHI(3) PRINT a1 FORMAT ¢.n K SOUAREn AC ac cc AV2 BV2 1 cvz AVS Rvs CV3.) no as 1g2.102 FORMAT (E13.3.9F9.35 PRINT 83.5wv<1).A¢1.1).e(1.1).c<1.1).A(2.1>.ec2.I).C(2.1).A(3.1).a 1‘30!)AC(331) RETURN END COCO 6O SUBROUTINE ABPARPER THIS'SUBROUTINE‘CALgULATES'YHE ABSORPTION FOR A PARALLEL T0 E (ABSPAR) AND FOR A PERPENDICULAR 70 E (ABSPER). USING THESE AND THE RESULTS 0F!SU8ROUTINE ABSORBZ IT THEN CALOULATBS VARlOUS CURVES OF INTEREST: COMMON/BLOCKi/EGARiVMASS(4).DELTA COMMON/RAINABSIT.TK.EFERMI . COMMON/MAINAaR/ErrELo.soexL(4).Nsn COMMON/ENABSISHV(102)1A(41102)38(40102)aC(4a102)EEKANE(4:102):S(10 12, COMMON/ABSORPS/ONSpNV‘lOZ):DENOH(4E102)EGAC4:102)cGB(4n102):EPH0T0 1N(4}102,oALPHA0(4:102):ABZERO‘41102) COMMON/ABRENGY/EZERO(3).ENTHETA(4’ DIMENSION DRIFTCA)3HRPLUSK(4)aDISFUN(2.4).SUN83(2a4).SUM81(2.4).AR 1§As3(4)pAREA31(4)EPOLFEC(43102’ECHGPARC4)1CHGPERC4):XCZ):CO(2):SI( 22’1FK(234)1ABSPARC4;102):ABSRER(4:102):POLFOAC4o102) RMASSE=9.108E82R HBAR=(1.0544/5o2917$*1uE'18 DO 199 13134 199 DRlFTCI’:VMASS(I)QRMASSE*SOBIL‘!)REFIELD/HBAR 200 FORMAT (07H TEMP=nF690:9H EPERMI‘oF8.5:9H EFIELD=:F6.039H EMOB 11L=.E12.4.9u RM091L=.E12.43 PRINT ZOOETEEFERMIoEFIELDoSOBXL‘l)ESOBILNZ’ 201 FORHAY (07H CMASSB$F995oSX:7HV1MASS'JF9osa3X:7HV2HA5331F9oSE3X:7HV 13NA533F9.5:3Xo9HDRIFTKC)3:512o5p3X;10HDRIFTCV1)81512o5} PRINT 2O1:VHASSC1,3VMASS(4)nVHASSCZ)aVMASS(3)oDRlFT(1)gDRIFT(4) 204 FORMAT (tn KSQUARED FK(ODD:C) FK(EVENnC’ FK(ODDnV1) FK‘O 1091V2’ DISFUN(E:C) DISFU(E:V1) DISFU(E:V2) ENTHETACC) ENTHET 2A‘V?)')‘ PR‘NT 204 EZERO(1)=EGAP EZERO¢2,'RO EZER0(3)3!DELTA DO R80 K=3.102 DO 210 :3194 210 DRPLUSK(!)=SHV‘K)*DRIFT(I)*DR1FT(I) H33.1415927/(2c*NSR3 DO 215 N311? Do 214 J3204 SUMSStN.J)=o. 214 SUMSi‘N.J)=O. 215 CONTINUE Do 760 1A811NSR X(1,=(2.*KSR¢1oI2.*IA)'H X(23=l2.*k$R92.tIA)oH C0(1)'COSF(X(1)’ COC?)3COSF(X(2)) S!‘1)'S!NE(X(1)) S!‘?)!SIN'(X(2’, DO 250 J3114 D0 240 N313? FK‘NtJ)IDRPLUSKCJ’*Qo'HVCK)*DRIFT(J,¢CO(N’ CALL ENGY(JOFK‘NlJ), 61 D 221 DISFUN(N1J)I 1./ (1. * EXPF((ENTHETA(J) - EFERHI)/TK)) IF (J'1)240.240.224 224 suns3tN.JI=5un$5=PRon-<(nAIJ.KI~GBIJ.KI/2.>~AREAS$IJ>+GB) 27o POLEEC(J.K)=ABSPAR(J:K’sABSPER(J:K) 279 FORMAT (E11.3.9El3.5) 280 PRINT 279.SNVIKI.PKT1.II.FKI2.1I.FKI1.4>.th1.2).OISFUN(2.1).OISFU 1N(2.4I.OISFUNI2.2I.ENTHETA(1).ENTHETAI2I 281 FORMAT («a KSOUARED IN ATOMIC UNITS. EPHOTON IN EV. ABSORPTION Is IN INVERSE CENTINFTERSRI PRINT 281 282 FORMAT I-n K SQUAREO EPHOTON(V1> ABSPAR-ABSPERl EPHOTON(V2) 1ARSPAR-ABSRER2 EPHOTON(V3) ABSPARnABSPERSt) PRINT 282 D0 284 K=a.102 253 FORMAT (511.3.3IFI4.7.516.7I) 284 PRINT 283.8NVIK)aEPHOTON(4.K)oPOLFEC(4.K).EPNOTON(2.K).POLFEC(2.K) 1.EPHOYON(3.K).POLFEC(3oK) 265 FORMAT (In EPHOYON(V1) ABSPAR(V1) ARSPER(V1) EPHOTON(V2) 1 ABSPAR(V2) ABSPER(V2) ERHOTON(V3) ABSPAR(V3) AB ZSRER(V3)t) PRINT 285 D0 790 K=3o102 289 FORMAT (3(F12.7p2516.7)) 290 PRINT ZaqlEpHoTnN<4gK)IABSPAR<4IK)DABSPERt4JK,IEPHOTON‘QJK)DABSPAR 1(21K)oARSRERCZIK)IEPH0TON(3IK)nABSPAR‘SaK):ABSPER(3:K) 292 FORMAT ('6 EPHOYON(V1) CHANGERAR‘Vl’ CHANGEPER(V1) EPHOTON(V2) 1 CHANGEPAR‘V2) CHANGEPERCVZ) EPHOTON‘V3, CHANGEPAR(V3) CHAN ZGEPERIV3)t) PRINT 292 DO 299 K=31101 D0 994 J=2,4 POLFOA(JIK)IPOLPEC(JJK)/ABZERO(JIK) CHGRAR(J)=ABSPAR(J:K)BAQZERO‘JaK) 294 CHGRER(J)IABSPER(J:K)9ABZER0(J0K1 295 FORMAY (3(F12-7.2516.6)) 299 PRINT ZQSIEPHOTnN(4.K’:CHGPAR“)ICH69ER‘4,oEPHOTON(2-K):CHGPAR(2): 1CHGRER(?)nEPHOTON(3,K)oCHGPAR(3)pCHGPER‘S) 300 FORMAT (in K SQUARE” EPHOTON‘Vl) PAR'PEROABZEROl EPHOTON(V2) 1PARQPEROABZER02 EPHOYON(V33 PAR'PEROABZEROS') PRINT 3ND D0 30‘ K330101 303 FORMAT (511.3:3(F14.71516-7)) 62 304 PRINT SOSISWV(KInEPH0T0N(4:K)TROLF0A(4oK).EPHOTON(2:K).POLF0A(2nK) 1.EPHOTON(3:K)nPOLFOA(3nK) RETURN END I CO 32 O 325 345 355 365 63 SUBROUTINE ENGYQInFK) THIS SUBROUTINE CALCULATES VALUES 0F E(K*D>n(ENTHETA):CALLED FOR BY SUBROUTINE ABPARPER. COMMON/BLOCKI/EOAP.VMASSI4).DELTA COMMON/ENERGYSIHHoau.EPS.OD.DP.OB COMMON/ABRENsv/EZERN(3I.ENTHETAI4I COMMON/MAINENSISQP.CKANE:EGAPAD(4) EFREELEC=HH02MfiFK !F(4-I)32n.520p325 ENTHETAI4)=EFREELEC/VMASS(4INEGAPADI4) GO To 365 SBSOPfiFKt27o21'27.21 TA=OPNS TRBOBNS DO 345 L=I.2o FE=FZERO(I)vi3*ODtE7ERO(I)NEZEROII)-TAREZERO(I)-TB DFE=3.EEZERO(I)0EZEROII)*2.ODDtEZERO(I)9TA DELYAE=FEIDFE EZEROIII=EZEROIII~DELYAE IF(EPS-ABSF(DELTAE)I345.355.355 CONTINUE TF=EZERO(I)NDB TO:EZEROIIIEEGAP AN=IS-t.5)tTF BN=OB*TG/I.414214 cN=YGRTF FNORMI(AN.AN¢BN~BN¢CN*CN)0'.5 BaBN/FNORN c:CN/FNORP HPEPT=(HRS/2.NCOCINCKANE'EFREELEC ENTHETAIII=EZERn(I).EEREELEC~HPERT+EGAPAOIII RETURN END TIN MW 82 8622 I E II” v ”0 m “Il'i3 U"0 1293 IIHINMWJIII