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A theoretical investigation of optical absorption by donor impurities in silicon Gilliland, John Michael 1961

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A THEORETICAL INVESTIGATION OF OPTICAL ABSORPTION BY DONOR IMPURITIES IN SILICON by JOHN MICHAEL GILLILAND B.Sc, The University of British Columbia, 1960. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1961. In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Physics  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date 2£ September 1961. i i ABSTRACT An investigation has been made into the p o s s i b i l i t y of observing o p t i c a l t r a n s i t i o n s (in the 100-micron region) between the ground state of a donor impurity In s i l i c o n and the remaining f i v e states of the l l s ^ set introduced by Kohn and Luttinger. While such t r a n s i t i o n s are forbidden i n the usual e f f e c t i v e mass approximation, i t i s found that application of corrections to the effective-mass wave functions leads to an enhanced t r a n s i t i o n p r o b a b i l i t y . Under- the assumption of a simple cubic l a t t i c e of impurities, the calculated absorption c o e f f i c i e n t i s of the order of 10 cm"*1 at an impurity concentration of 1 x 1 0 1 8 cm"*3, and f a l l s o f f exponentially with decreasing impurity concen-t r a t i o n . An upper l i m i t i s placed on the region i n which the t r a n s i t i o n should be observable by the broadening of the 2s-2p impurity band. It i s estimated that f o r concentrations greater 18 _<i than 5 x 10 cm ° the t r a n s i t i o n of intere s t w i l l be obscured. The calculated values of the absorption c o e f f i c i e n t are probably only accurate to within, one, or even two, orders of magnitude, because of the approximations involved. However, there would appear to be no f i r m t h e o r e t i c a l reason why the ls ( 0>-^ls< 5> t r a n s i t i o n should not be observed. i i i ACKNOWLEDGEMENT The writer gratefully acknowledges the assistance and encouragement of Dr. Robert Barrie, who suggested the problem and supervised the writing of this thesis. Thanks are also due to Dr. J. Bichard and Dr. J.C. Giles of this department for several illuminating conversations, and for permission to use their unpublished experimental results. iv TABLE OF CONTENTS ABSTRACT i i ACKNOWLEDGEMENT i i i CHAPTER I Intr6duction 1 CHAPTER II The Theory of an Isolated.Donor Impurity in Silicon. 11 A. Effective Mass Theorem for an Isolated Impurity. 11 B. Optical Matrix Elements. 17 CHAPTER III The Theory of Donor Impurities in Silicon for Finite Impurity Concentrations. 22 A. Introduction. 22 B. Tight-Binding (LCAO) Approximation. 22 1. The secular equation; Derivation. 22 2. Integrals involved in the secular equation. 27 3. Evaluation of integrals. 30 4. Evaluation of matrix elements. 33 5. The secular equation: Solution. 35 6. Energy-band broadening. 38 C. Optical Matrix Elements. 39 1. Simplification of the matrix elements. 39 2. Evaluation of integrals in the spherical-potential approximation. 41 3. Corrections to the spherical-potential approximation. 42 4. Evaluation of integrals in the corrected spherical-potential approximation. 47 5. Evaluation of the matrix elements. 51 V CHAPTER IV Optical Absorption by Transitions between Impurity Bands in Phosphorus-Doped Silicon. 54 A. Derivation of the Absorption Coefficient. 54 1. Transition probabilities. 54 2. Specialization to a rigid lattice. 56 3. The coefficient of absorption. 58 4. Specialization to centres in a dielectric medium* 60 B. Evaluation of the Absorption Coefficient. 62 1. Simplifications and approximations. 62 2. The absorption coefficient for the ls<°>—*ls<5> transition. 64 C. Discussion of Results. 69 1. Is—*2p 0 transition. 69 2. ls(°>-*!s<5) transition. 70 APPENDIX A Properties of the Coefficients D ^ C k ) . 72 APPENDIX B Properties of the Three-Dimensional Kronecker 6. 75 APPENDIX C Evaluation of the Integral (IV:36). 78 APPENDIX D The Absorption Coefficient for the Is —*2p 0 Transition. 83 BIBLIOGRAPHY 89 v i LIST OF ILLUSTRATIONS FIGURE 1 Corrected Ground-State Wave Function. 92 FIGURE 2 Location of Hydrogen-like Energy Levels. 93 FIGURE 3a Values of A 1(x) and Ag(x). 94 FIGURE 3b Values of BjCx) and B 3 ( x ) . 95 FIGURE 4 as a Function of Impurity L a t t i c e Spacing. 9S FIGURE 5 X 2 as a Function of the Energy of the Is Level. 97 FIGURE 6 FIGURE 7 FIGURE 8 FIGURE 9 FIGURE 10 FIGURE 11 C04 % + c o i y a ^ as a The Integral: Function of \ . Tight-binding Energy Surfaces of (111:58) fo r cos k Q a = 0. Comparison of Corrected and Approximate Energy Surfaces. ( 0 ) The Is »ls (5) Absorption C o e f f i c i e n t as a Function of Impurity L a t t i c e Spacing. X a P ° a s a F u n c t i o n o f Impurity L a t t i c e Spacing. The Absorption Cross-Section for the l s * 0 ) — * 2 p ^ Transition. 98 99 100 101 102 103 CHAPTER I Introduction The quantum-mechanical theory of crystalline solids depends, to a large extent, upon the consequences of the periodic arrange-ment of atoms In a crystal. It is the purpose of this thesis to discuss some of the effects of deviations from this periodicity. To a f i r s t approximation, an electron in a pure elemental solid may be considered to move independently of other electrons through a rigid lattice of identical nuclei. In this adiabatic, one-electron approximation (Reitz, 1955), the only forces influencing the electron motion are those due to Coulomb inter-action with the nuclei. The corresponding one-electron Schxoedinger equation i s : where V (r) is a potential function with the same periodicity as the crystal lattice, h is Planck's constant divided by 2rr, and m is the mass of an electron. The Bloch theorem (Bloch, 1928; Wilson, 1953, p. 21) extended to three dimensions shows that the eigensolutions of equation (1) have the form; cp„_Cr) = e i k r u„_(c) ( I : 2 ) where u n^(r) has the same periodicity as V p(r). From this i t may he shown (Wannier, 1959, Chapter 5) that the eigenvalue £ is a many-valued function £n(_£) o f t n e electron wave-vector k, -2-each subscript n labelling a different branch, or "energy band", of the function.* For uniqueness, the allowed values of k are confined to a region of momentum space known as the fi r s t Brillouin zone (Jones, 1960, p. 37). The foregoing remarks apply to a completely periodic array of atoms. If this periodicity is disturbed slightly in some way, equation (1) must be replaced by: m - [K. + UH - E+ ( l ! 3 ) where U represents a perturbing potential due to the departure from strict periodicity. Equation (3) has been studied with the aid of Wannier functions by several authors (Wannier, 1937; Koster and Slater, 1954 a and b; Slater, 1956). For the particular application to be studied in this thesis, however, a technique will be used which involves only the Bloch functions <p_jj.(r) of the unperturbed lattice (Luttinger and Kohn, 1955, Appendix A; Kohn, 1957). The specific problem to be considered is that in which one (or more) of the atoms in the lattice is replaced by an atom of a different type, referred to as a "substitutional impurity". Insofar as its atomic number permits, the impurity atom will take over the electronic bonds left unfilled by the removal of the original atom. Consequently, i f the impurity Note that the branches may be degenerate for some values of k. (Wannier, 1959, p. 145.) It should be noted that for complete accuracy U must satisfy self-consistent field requirements. -3-nucleus has a smaller charge than the original nucleus, there w i l l be a deficiency of electrons in the lattice due to a number of unfilled bonds; while i f the impurity nucleus has a greater charge, there w i l l be an electron excess. Impurities of the f i r s t type are known as "acceptors", while those of the second type are known as "donors". For definiteness, consider the case of a donor impurity, * such as phosphorus in s i l i c o n , whose atomic number i s one greater than that of the l a t t i c e atoms. (An impurity of this type i s said to be "monovalent".) The nature of the inter- -; action of the impurity nucleus with the extra electron w i l l depend on whether or not i t is energetically favourable for the electron to occupy an orbit close to the nucleus. If such an orbit i s favoured, what i s known as a "deep" impurity state is produced, and the electron-nucleus interaction i s highly complicated. However, in the case of a "shallow" impurity state, characterized by an orbit of large dimensions, only the excess charge on the impurity nucleus w i l l be of importance in the interaction, and the impurity system w i l l resemble a hydrogen-like atom imbedded in the lattice. If i t i s assumed that the impurity in question i s in a shallow state, the hydrogen-atom analogy mentioned in the last paragraph may be used to give a particular form to the perturbation U: U ( C ) s - " K F <I:4) * A l l numerical results in the present thesis w i l l apply to this particular case. -4-(e is the electronic charge, and K is the static dielectric constant of the host lattice). The use of the static dielectric constant is justified by the fact that the orbital frequency of the extra electron about the impurity nucleus is sufficiently low to be neglected with respect to the orbital frequencies of other electrons in the crystal. The concept of polarization of the host material by means of relative displacements of nuclei and electron shells will therefore s t i l l be valid. It is this polarization which gives rise to the static dielectric constant. There is a major assumption implicit in the foregoing argument — that of a spherically symmetric perturbing potential. The true hamiltonian governing the motion of the extra electron will have a symmetry determined by the physical arrangement of the atoms surrounding the impurity nucleus. In the case of phosphorus in silicon, for example, U(r) must have tetrahedral symmetry (denoted by T.S.), so that equation (3) becomes: (1:5) Group theory (Heine, 1960, Section 6) then shows that the wave functions + belonging to each value of E in (5) must generate an irreducible representation of the tetrahedral group T d (Eyr.ing et al., 1944). If the wave functions + are expanded in terms of the Bloch functions of the unperturbed lattice as follows: * See Chapter II for details of this derivation in the approximation of a spherically symmetric perturbing potential. - 5 -+(_)* Z Uk DC""(_) «UCt) (1 :6) m J (5) may be replaced by the equation: {£-(_)-E}D°"\tO + I U_'D<n'',(_')<m_|UT.s.(t)|m'_> ( I s 7 ) = o The Fourier transform of equation ( 7 ) is then: [ - - ( - I V ) - E] F lm\rJ + J * * Dc"n',(_)eli5r<mk|UT.s.Cr)|m'l<'> = O (1:8) where F^Cr) is the Fourier transform of D(n°(__). Equation (8) may be simplified by neglecting interband terms, and by using a suitable approximation with regard to the "gentleness" (but not the symmetry) of t*T,g# (_,)• The result of this simplification is: + U M . ( E ) } F^Cc) = E F ^ ( r ) ( I . 9 ) Here again the wave functions corresponding to each E must generate an irreducible representation of the group T d. It is obviously to be expected that replacing tt_.s.(_) by a potential of higher symmetry would introduce additional degeneracies into the energy level scheme of (9). After the solutions of (9) have been obtained, they may be Fourier transformed to give the D^Qs), and hence the solutions of ( 5 ) . Clearly any extra degeneracies caused by the use of an incorrect potential would be carried over into this case as well. -6-Unfortunately, because the exact form of U T g_(r) i s not known, the above method of solving equation ( 5 ) i s not practicable. It i s necessary to solve the problem using the s p h e r i c a l l y symmetric p o t e n t i a l (4) as a f i r s t approximation. Corrections to t h i s approximation may then be studied by the introduction of a tetrahedrally symmetric perturbation, or by some other l e s s exact procedure. It has been shown (Koster and Slater, 1954 b; Slate r , 1952) that the energy eigenvalues of equation (3) are s i m i l a r to those of equation (1), with the exception that allowed energy l e v e l s may now occur i n the "forbidden" regions between the bands. For a shallow impurity state, these " s p l i t - o f f " l e v e l s are close to the parent band; hence i t i s to be expected that the wave functions corresponding to these l e v e l s are c l o s e l y r e l a t e d to the Bloch functions at the band edge. If i t i s assumed that the band i n question i s of standard form (Wilson, 1953, p. 42), with i t s minimum i n k-space at k =» 0, then i t may be shown (Kohn, 1957, Section 5a) that the equation: { j r . - £ 1 + • E + ( I : i o ) has solutions: V m\& ~- F'"*(£> <pmj0(_) ( _ . n ) where F < m*(r) i s again the Fourier transform of the c o e f f i c i e n t D w ( k ) defined by (6). F ""*<_) i s a solut i o n of the equation: U^C-tv) - ^ ] F ^ t ) * EF'-'C-} (1:12) - 7 -and the form of E ^ - i v ) i s , to second order: 'E-M.). (1:13) ( £c?* i s the energy at the bottom of the band, and m^., my, and m_. are constants with the dimension of mass). Equation (13) may be solved approximately by s e t t i n g m^, = my « raz «= m . In t h i s case the functions Fc"°(r_) are simply modified hydrogen wave-functions, and the allowed values of (E - E'^0) form a hydrogen-like spectrum. In the consideration of energy l e v e l s s p l i t off from the conduction band of s i l i c o n , allowance must be made for the fact that the band i s not of standard form, but has s i x equivalent minima, at (£k o,0,0), (0,±k o,0), and (0,0Jk 0) i n the f i r s t B r i l l o u i n zone (Herman, 1954 and 1955). A tentative estimate of k Q has been made by Kohn (1957, Section 7c), who gives a value of 0.7 k j j ^ . (^ max i s t t i e magnitude of k at the zone boundary i n one of the s i x a x i a l d i r e c t i o n s ) . Also, the conduction-band energy near one of these minima i s given by an expression l i k e : (Kohn and Luttinger, 1955a) MhO = £o + ^ ( k . - O * + ^ U J + k * ) (1:14) using the minimum at (k o,0,0) as an example. The e f f e c t i v e masses m^  and are given by (R.N. Dexter, et a l . , 1954): ml= 0.98 m mt « C l 9 m < I : 1 5> In the approximation of a s p h e r i c a l l y symmetric perturbing - 8 -p o t e n t i a l j the s o l u t i o n of the many-minimum problem f o l l o w s along roughly the same l i n e s as that f o r a s i n g l e minimum.* The problem i s f i r s t s o l v e d f o r a s i n g l e minimum at one of the s i x eq u i v a l e n t p o s i t i o n s l i s t e d above. Then i t i s argued that by v i r t u e of the s p h e r i c a l symmetry of the p e r t u r b i n g p o t e n t i a l , a s i m i l a r s o l u t i o n would have been obtained i f any one of the other f i v e minima had been used. I t t h e r e f o r e f o l l o w s that i f a l l s i x minima are present at the same time, the single-minimum hydrogen-like l e v e l s w i l l each acquire a s i x - f o l d degeneracy i n a d d i t i o n t o i t s s p i n degeneracy. I f now a t e t r a h e d r a l l y symmetric p e r t u r b a t i o n i s a p p l i e d t o the hamiltonian of equation (10), i n the case of many minima, s t a t i o n a r y p e r t u r b a t i o n theory ( S c h i f f , 1955, p. 155) shows that the zero-order wave f u n c t i o n s f o r the s i x degenerate s t a t e s belonging t o a p a r t i c u l a r energy l e v e l are given by l i n e a r combinations of the s i x corresponding individual-minimum wave f u n c t i o n s . Group theory (Heine, 1960, p. 107) then i n d i c a t e s what the c o r r e c t l i n e a r combinations are. The purpose of forming these combinations i s t o e l i m i n a t e a l l non-zero elements of the p e r t u r b a t i o n between degenerate s t a t e s . For the case of the conduction-band i m p u r i t y l e v e l s of phosphorus i n s i l i c o n , i t may be shown (Kohn, 1957, S e c t i o n 5b) th a t a t e t r a h e d r a l l y symmetric p e r t u r b a t i o n can onl y p a r t i a l l y remove the ground-state degeneracy caused by the s p h e r i c a l -p o t e n t i a l approximation — the maximum p o s s i b l e s p l i t t i n g i s i n t o a non-degenerate l e v e l , a two-fold degenerate l e v e l , and * See Chapter I I . - 9 -a three-fold degenerate level. Experimental studies of the hyperfine structure of the ground state (Fletcher et a l . , 1954 a and b) indicate conclusively that the lowest of these three levels i s the one which i s non-degenerate. As sp l i t t i n g may occur, then, i t should be profitable to examine the possibility of radiation-induced transitions between the ground state and the remaining five Is states. It w i l l be shown that i f the effective-mass wave functions derived on the basis of a spherically symmetric perturbation are used, the optical matrix elements for Is-Is transitions are very small compared with those for transitions between other pairs of levels. Furthermore, they remain small even after a correction has been applied to the non-degenerate ground state wave function. Consequently, in the case of an isolated phosphorus impurity in s i l i c o n , i t should not be feasible to observe the fine structure of the (is] states experimentally. If the impurity concentration is increased, however, the selection rule governing the transitions of interest breaks down. It may now be possible for an electron in the non-degenerate ground state on one impurity atom to make the transition to an excited Is state on another such atom. Unfortunately, treatment of this many-impurity problem i s complicated by the random distribution of impurities. It i s possible to approximate the actual situation, however, by assuming that the impurity atoms form a regular l a t t i c e which is superimposed on the la t t i c e of the host crystal. In this approximation the problem reduces to that of "solid hydrogen" -10-lmbedded i n a d i e l e c t r i c medium.* The o p t i c a l matrix elements for the many-impurity case also turn out to be very small i f the sphe r i c a l - p o t e n t i a l effective-mass wave functions are used. However, using corrections to the e f f e c t i v e mass theory based on those introduced by Kohn and Luttinger (1955 a), i t w i l l be shown that the matrix elements may be of a s i z e which w i l l permit observation of the t r a n s i t i o n s . The actual state of a f f a i r s depends upon the appropriateness of the corrected wave functions used. It i s probable that the Kohn-Luttinger approach leads to r e s u l t s which are only good to within a factor of two or three. However, there would appear to be no firm t h e o r e t i c a l reason why the f i n e structure of the (is] states should not be observed. * See Chapter I I I . CHAPTER II The Theory of an Isolated Donor Impurity i n S i l i c o n . A. E f f e c t i v e Mass Theorem for an Isolated Impurity. Consider the problem of an is o l a t e d monovalent donor impurity i n a c r y s t a l of s i l i c o n . Let the Schroedinger equation for the pure s i l i c o n l a t t i c e be: i ft0(pm_(r) = U(_)<P«_(e) (11:1) where: A * • - _ £ V * + V ^ ( H : 2 ) and V (r) i s the periodic c r y s t a l p o t e n t i a l . By the Bloch AT ' ' " Theorem, the eigensolutions of (1) have the form: f - k ^ « e i b r u „ _ ( c ) (11:3) where: u m_(_ ,l = u^wCr+RO (11:4) for any vector R_ of the pure s i l i c o n l a t t i c e . If the u_-_(r) s a t i s f y the normalization: ( |u-,K(_)i* d.c = SL. .) • (11:5) ( A n I T CCll where Q i s the volume of a unit c e l l of the s i l i c o n l a t t i c e , then the Bloch functions <P__(r) may be shown to form a complete -12-orthonormal set normalized over the whole crystal. If the effect of introducing an impurity atom into the pure crystal is considered as a small perturbation, the Schroedinger equation for the isolated-impurity problem may be written: flfCt) = E+Cc) (11:6) where: il - Uo+Vin) (11:7) and U(r) is the perturbing potential due to the extra charge on the impurity nucleus.* If the extra electron is assumed to move in a spherically symmetric potential, then at large distances from the Impurity nucleus: UCO S (11:8) where e is the electronic charge, and K is the static dielectric constant of silicon. Since the ^__.(_) form a complete set, the solutions of equation (6) may be written: = Z \dik W"\k) <JU(r} ( I I . 9 ) where the summation is over a l l the energy bands of equation ( 1 ) * For the purposes of this derivation, i t will be assumed that shallow-state theory is applicable. •* See Chapter I for a more detailed discussion. -13 and the integration is over the fir s t Brillouin zone of the silicon lattice. For simplicity i t will be assumed that the wave functions belonging to energy levels split off from the conduction band of the unperturbed lattice, under a small perturbation U(r), may be written in terms of conduction-band Bloch functions alone. Hence: * ( _ ) 3 (dk D ( _ ) cf>c_(r) (11:10) where the integration is again over the fi r s t Brillouin zone, and the subscript c refers to the conduction band. Substituting (10) into (6)*, multiplying by the complex conjugate of ^ ^ . ( r ) , and integrating over the entire crystal leads to the equation: i £ t ( U > E ] D ( k ) + {*_' U(k,_)&(_') = O (11:11) where: Itfk.k') = \dr cpeJ(_) U(r) <(>c_'(r) = <cklUC_)|ck'> (11:12) Now the conduction band of silicon has six equivalent minima in the fi r s t Brillouin zone, at (±kQ,0,0), (0>kQ,0), and (0,0jk Q). If i t is assumed that the coefficients D(k) corresponding to k near different minima are very weakly coupled, the solutions of (11) may be written approximately: D(i0 = H «j DjCls) (11:13) where the {a£ are numerical coefficients determined by the * The following argument is taken, in the main, from Luttinger and Kohn, 1955, Appendix A. - 1 4 . symmetry of the unperturbed latt i c e , and the summation i s over the six conduction-band minima. Stationary perturbation theory (Schiff, 1955, p. 155) shows that there are six allowed linear combinations of the form (13); they w i l l be distinguished by a superscript i : D'Ck) = _,*) Dj(k) (11:14) j Following Kohn and Luttinger (1955 c) the coefficients required for the (is] set of solutions are: ^ ( 1 , 1 . 1 , 1 , 1 . 1 ) < - - ^ 0 , 1 , - 1 , - 1 , 0 . 0 ) ± ( l , l . O , 0 , - l . - l ) Z <Xj - J _ ( l , - l , 0 , 0. 0,0) J L ( o , o , 1,-1, 0, 0 ) _L. ( 0 , 0 . 0 , 0 , l . - O (11:15) In the spherical-potential approximation, these six states a l l have the same energy. For the (ts^ set, the coefficients Dj(k) of (14) may be taken to satisfy: {£i(k)-E}Dj(_) + Sdk'U(_,k')Dj(k') = o (11:16) where £*(_) is a second-order expansion about kj of the conduction-band energy t c ( k ) : £Jc(k) = £0 + Z l i / i Ckx-kixX Vy-k^) (11:17) -15-Here: j _ 9z£c(k) £o =» energy at the conduction-band minimum. Luttinger and Kohn (1955, Appendix A) show that provided U(r) i s a "gentle" p o t e n t i a l , UCk.k') _ UCOe1^'-^": B U ( k - _ ) . (11:18) so that (16) becomes: -E] D J C O + f * - ' XU.-fc'} Dj(k') = O (11:19) Then, multiplying (19) by e1^-*"—3^— and integrating over the f i r s t B r i l l o u i n zone: I, ^ ( ^ J i ^ ) - E ) ( A k T ^ e ^ - ^ + J _ . fdLfe e ^ " ^ - r U ' D:(k-) fd_'UC,.)e i l k>^' = o ( I I : 2 0 > Ctnr J j i j Hence to the approximation that: J <Ak ei(r--='> * = (2n^3 & ( r- r0 F»**St 6r'illo_>'« Zone i t follows that: t l \\^^tT^ + ^ ) F i ( ^ = F i ( ^ ( i i : 2 1 ) In t h i s case, the function F j ( r ) must be defined as: Pitt) = \ e l t ^ - r D_(_) (n.22) _>T BrillovuVi Zone * For complete accuracy, t h i s integration should be taken over a l l k-space. 16-The required normalization of the Fj(r) is: fdrlFjWl* = (2T,) 3 (II : 2 2 « ) where the integration is taken over the entire crystal. This follows from the requirement that the ^ (r) defined by (10) be normalized to unity over the entire crystal, since: jdr |+(_}!' = jdk jdk' D*(k)D(k') jdrc ^ J c c ) cpck-6f) s, Jdk I POO I* while: |<*rlFCC)|l - (dkjdk< D*(k)DCk') j d r e l ( k - - ^ E e 1 ^ * = jdk (in) 3 (DC!.)!1 = Un) J jdr . Substitution of (14) into (10) leads to the equation: *i<>^ = Z otj Jdk Dj/k) fck(_) (11:23) = S OC'- F^(C) ^cjsjte). (II:23') where the subscript i labels the eigenvalues of equation (21). - 1 7 -B. Optical Matrix Elements, In the dipole approximation, the matrix elements of interest are: C L J M P J L ' O ( 1 1 : 2 4 ) where: |U> s +;(_) ( 1 1 : 2 5 ) It may be shown that: pp. = <r-component of momentum = - ~ ft] ( 1 1 : 2 6 ) where: # = # c + U(c) Hence the matrix elements ( 24 ) become: l * ' ( 1 1 : 2 7 ) « J2- (EJ ' - E i)<iJtl* c|l'i'> Now by equation (10): < U l x „ | i ' £ ' > = £ , * i * o c y [dLk [d_*D„(k)Dj^(!<*)<c_l^lck'> ( 1 1 : 2 6 ) < c k l x - l c _ ' > = [dr *, e a * ~ b V c u*_(C)ue_.(_^ * See Chapter IV. - 1 8 -ut_ ~ £ has the periodicity of the unperturbed lattice. so that: c k T k T " £ s f c V (11:30) where the summation is over the entire unperturbed reciprocal lattice, and: Substituting (30) into (29): = 6(_'-k^ _ _ B ' j t J _ . ( 1 1 : 3 2 ) t 3kL I t r * t < r Now k and k' both l i e in the fir s t Brillouin zone, by equation (10). Their difference can therefore never become as great as a non-zero vector of the reciprocal lattice. Hence the only non-zero contribution of the summation in (32) is for K_ =• 0, so that: <ckla.Uk'> - *(_.'-_> - ^ B ^ i C f c ' - k ^ ( I I ; 3 3 ) Now: w = ~k ( d * ^ ^7 (11:34) uxit cell Adams (1952) shows that i f the phases of the Ugj^r) are properly chosen, the integral in (34) vanishes. It will be assumed that - 1 9 -th i s choice of phase has been made. Hence: KL - 0 (11:35) Thus (33) becomes: «• o Kg-T »R. l S J > ' . T i e J ^, i,_ ( I l l 3 e , where (11:37) Substituting (36) into (28): - _ V ^ - J ' - S - < J * . « , U V > < I I : 3 8 ) where: \n> * v a . * * ( I I : 3 9 ) The matrix elements <jil x, lj'£'> may be written: <i l l*clj f O = j ^ t f c r t ^ C r t e ^ ' ^ d c (11:40) For the t r a n s i t i o n to be considered, from the Is ground state (characterized by a^°h to another Is state, the s p h e r i c a l -p o t e n t i a l approximations to both F j ^ ( r ) and F j t ^ i C r ) are even functions of x, y, and z. In t h i s case (40) reduces to: -20 <j/l*<rljr> = <• Fj^Ct) Fj'isCc) sin (kj'-hjVE d£ (11:41) the real part of the expression having vanished because its integrand is an odd function of x. Clearly i f j 0 8 j * , the matrix element (41) is identically zero. If j / j f , the matrix element is not zero, but as the Fj__Cr) are slowly-varying compared to the lattice spacing (Luttinger and Kohn, 1955, Section III), i t is reasonable to assume that the presence of an oscillating term in the Integrand will tend to make the integral very small. A more quantitative estimate of the non-zero matrix elements may be obtained from consideration of the Integrals: Jdr «^F J-_(c)F i. u(E) sin kj-r cos fer r (11:42) and Jdr %r F* l $ ( t ) Fj ' u (_) sinRj'-r »$ kj.r (11:43) I* J - J', both integrals are of the order of 5 x 10~4a*, where a* is the approximate "Bohr radius" for the F l s ( r ) . It is to be expected that when j ^  j ' these integrals will have even smaller values.* (41), which 1st equal to the difference of (42) and (43), should therefore be negligible compared with 2 * 1.9 x 10 a , which is the value of the matrix element <ji.l-x^-l j 1'> for the ls-*2p0 transition when spherical-potential wave functions are used. * See (111:37) for examples which bear out this contention. -21-In Chapter I I I , corrections to the sphe r i c a l - p o t e n t i a l approximation w i l l be considered. Anticipating the r e s u l t s , i t may be stated here that, replacing the ground state wave function by i t s corrected value and using the estimation —3 * procedure of the l a s t paragraph, values of less than 1 x 10 "a* are obtained f o r (41). As the Is 2p Q matrix element i s decreased by only a factor of two when the corrected functions are used, the Is-*-Is matrix elements are again n e g l i g i b l e by comparison. It i s therefore u n l i k e l y that the f i n e structure of the {is] states w i l l be experimentally observable when the impurities are too far apart to inte r a c t . - 2 2 -CHAPTER III The Theory of Donor Impurities i n S i l i c o n  f o r F i n i t e Impurity Concentrations. A. Introduction. The case i n which there are many impurities present i n a c r y s t a l l a t t i c e i s d i f f i c u l t to treat because of the random nature of the impurity d i s t r i b u t i o n . I f , however, the impurities are assumed to l i e on a regular l a t t i c e within the c r y s t a l , the problem of t h e i r i n t e r a c t i o n may be attacked by means of the usual approximations of s o l i d state physics. For s i m p l i c i t y , therefore, i t w i l l be assumed that the impurities l i e on a simple cubic l a t t i c e , so that the problem of impurity-l e v e l broadening may be treated i n the Bloch scheme. B. Tight-Binding (LCAO) Approximation. 1. The secular equation: Derivation. The Schroedinger equation f o r the many-impurity problem i s : K * s ( _ ) = E (_)*_(_) < I I I : 1> * For the treatment of a one-dimensional l a t t i c e containing random impurities, see Lax and P h i l l i p s (1958). ** Conwell (1956) discusses the implications of the assumption of a regular impurity l a t t i c e . See also Baltensperger (1953). -23-where: K • « . + Y r t o { m , a ) and V (r) i s a periodic p o t e n t i a l with the p e r i o d i c i t y of the Mr impurity l a t t i c e . Following Slater and Koster (1954)j V P(^ 5 H U(r:-a5,) (111:3) _ where the summation i s over a l l s i t e s <*n of the impurity l a t t i c e . In the LCAO approximation, the wave functions "f_0_) may be written: * * W = -t ^ d * 2 + * ^ C c - a J <IXI:4) where N i s the number of impurities present, and the summation over I and i includes contributions from a l l eigenstates of the isolated-impurity problem. The energy: m ) ( I I I : 5 ) where: |K> = ' (111:5*) must then be minimized by the proper choice of the c o e f f i c i e n t s d_£. Hence: = — 7 — \4T, < - l ^ ' - > - 4-,<_l!s>] = o , T T T ddct' <i_l_> 13<*iy adi'e' - J (111:6) Now by equation (4) the matrix elements i n (6) are: < - i * > = i Z L < * ; * v z e ^ ^ - " - ^ < ^ , s , i r w > ( 1 I I s 7 ) -24-and: < K i « l K > = jjTLZ^U^-r * • < * » " - < i £ f l o l « U - r f t a « > (IH:8) where: (111:9) Thus: Bdtfe' N e,i a,a' = Z Z e^-fts <ijUOTfl_> (111:10) ( s h i f t i n g the o r i g i n to a n and defining a m « <i n.-a n). S i m i l a r l y : J- <K\H\K'> = Z d j Z e'*-** <£jM «li'i's.> (111:11) Substituting (10) and (11) into (6) leads to the equation: Z d < ; { Z e 1 ^ < i i l « U ' l , f t B , > - E C K ^ Z e i 5- 3 B<it) |i'ra=>"i ( H I : 1 2 ) = o for each pair of values ( i ' , jL ' ) . Hence the secular determinant must vanish, giving: J (111:13) Now by (2) and (3): -25-n * _ Ei'.<ilU'i,a*.> + X <UI OCr-i_U'^'a_> * (111:14) Then, substituting (14) into (13): \ Z e J - - B Z <iilO(r-ft^U'jfa„> (111:15) Using the expression (11:23) for U£s_>t <LlU'£'a_> = X <<*«j/[clkUk' D j C k ^ Dj'^ 'dc') <c_lc_'_-,> j j ' 1 -  ^.ai**!' J<*!ij*- t>J(_^ 1 > J V ^ e " i ! s ' - a oC_-k") - TTTJ ( ^ ( d k ' D j ^ ^ D j V f - ' ) < _ l k * g = > C Ji' J J (111:16) where: (111:17) Similarly: = "Z UfeU-' ^ (^D j V(k') e l t , " * B <cklucC-a_Mck'> <IH:18) Now Luttinger and Kohn (1955, Appendix A) have shown that under the assumption that U(r) is a "gentle*' potential, < c k | U C ^ l c k " > J L _ < k l U C » « U k ' > . - - IXnY - - - (111:19) -26-Clearly U(r - a n) also satisfies the requirement of gentleness, so that (19) may be extended to give: <cklU(E-a a)| ck'> S <V< I U(c-a sMk'> (111:20) Hence, substituting (20) into (18)-<eel UCc-anU i , i 'a 2 ! > * _i_ s^o ( i *« i ' jdkjdk' D/J(^Dji'(Js') <wiucr:-g!,>ik'as,> ( m . 2 i ) Then, making the substitution: f^t) * f * f c D J I ( 0 e U ! l - | « ) - c (111:22) in (16) and (21), the following expressions are obtained: < i l U'Jt'aa> = J L - , 2j«S*«J'' <i*\*'*'&*> (111:23) <i£lUCc-fl-)|i,iJ,flB> = TJTO ^SVi' <J*lU(c-a a MjTa»> (111:24) where: . j l a B > - F j t ( C - ^ (111:25) Finally, substituting (23) and (24) into (15), the secular equation becomes: i'.i* L a * b (111:26) = o For the sake of simplicity, the range of t in (26) is normally restricted to a small number of values, under the assumption that wave functions belonging to widely separated energy levels do not interact appreciably. For the problem -27-under consideration, i . w i l l be taken as Is only. The f i v e -f o l d degenerate* and the non-degenerate Is bands w i l l then be considered separately i n the l i g h t of equation (26). For the f i v e - f o l d degenerate Is band, the secular equation (26) i s : ^ (111:27) while for the non-degenerate Is band: i v i Z c ^ - a - I l « J * 4 < j i s l 2 0 C c - a . ) l j ' i $ f t » > , ^ (111:28) - u - x z « ; v <J i*> J 'IS TO> — j j - o Equations (27) and (28) must now be solved. 2. Integrals involved i n the secular equation. It may be shown that the c o e f f i c i e n t s D j ^ k ) i n (11:23) have the property that: t>j i»CI^  = D./1SW - (111:29) In considering the exact solutions of the isolated-impurity problem, Kohn and Luttinger (1955 c) note that a l l Is states other than the non-degenerate ground state w i l l have roughly the same energy. Thus, i n s p i t e of the fact that the exact solut i o n w i l l involve separate two-fold and three-fold degeneracies, the s i t u a t i o n may be approximated by a s i n g l e f i v e - f o l d degeneracy. (See also Chapter I of t h i s t h e s i s ) . See Appendix A. The subscript " - j " i n (29) denotes the c o e f f i c i e n t associated with the conduction-band minimum at -k< . 28-Substitution of (29) into (22) leads immediately to the r e s u l t that F j l s ( r ) i s the complex conjugate of F _ j l s ( r ) . Hence i f the phases of the F ^ C r ) are so chosen that Fj^(O) i s r e a l and pos i t i v e for a l l j , i t follows that F j l s (r) and F„ji s(r) we equal. Therefore: F j u ( r * ) e i ! y - + F - j l s ( r ) e " l - j r = Ij U > + | - j l s > = 2 FJ U(E ) cos k i r (111:30) F J l t ( r*)e l f c i - r -F - j l 4 (r*)e- i ! 5 i , r= l j u > - l - J l s > = 2iF iu(cn.v. kj-c Then, defining: , i - - > = Cr*Tv* "J* 0^ ,<j 1* a= >- (111:31) and using the values of ctj given i n (11:15), i t follows that: |0,a =,> = | F , cosk, .(r-a-?") + F , cos k j ( r-a B^ + F s c o s ks CC-SH,)} 11, O,M> = — z \ F, cos k,C*:-a3') - ^ C o s k j C r - a . „ ) ] C2TT)V* L J U , « H > = jj^jfa |^F,cos k.-Cc-fte*) - F 5 c o s kj.Cc-aw^ (111:32) I 4 > a = ? > = _ i _ & \ FjSiV, k 5 .(c-a»^ ( Z t n v * «. J where the argument of F^ i s (r - a m ) i n each case. The integrals involved i n evaluating < i I i'^m^ are: - 2 9 -gjjy JF/Cs*»Fi'(C.-*a"* <**kj-r cos ky(c-aa) A r Jjj^ I F*(t) Fj'CC-So*) Sinker Cos Wy.(r-So) dr jJJ-y | Fj*Cr*)Fi'(£-a,=,l sin K»-r s i n kj.(r-S^ An - «i'*TIy= " j»J'B If!' a (111:33) where: (111:34) and: = c o s i&j'. a 3 ( i f * ? ' | Fj" Fj'(£-W) Cos k j . r C o s k j ' r (aw)1! [p*(C*l Fj'fe-Sm*) cos k j . r Sin kj/.r dr T !»' i J Fj*C;"l Fi-'(c-oB*j SiV» k , . r c o s k j ' j : 4r 1 (in*) 1 {FfCc) Fj'Cc-Oa.*) Sin k j - r S in Isj'-rT d r (111:35) S i m i l a r l y , the integ r a l s involved i n evaluating <x I U(r-a a^| Va** "> are combinations of integrals Ujj»mn> ujj*mn> u/j»m n» uL'mn of the form: = (F/CJ !*) Fj/fc-Sm) UC£-ft c ) C o s k j . * : c o s k j ' r . dr (111:36) etc. -30-3. Evaluation of in t e g r a l s . The integrals of (35) and (36) may be shown to have the following approximate values (using t h e s p h e r i c a l l y symmetric po t e n t i a l (1:4) as an approximation to U ( r ) ) : l f T-i i o r - ± . ± 2. 4 Cl +• *<.*8»)» 1 - a « / B X 4 4 I j j m o o l o g y w i+Vi t h e m = O c a s e . I.vo = o [ A l l j j ' ] ' I*. it TT. _ J-4JS - 1 jj m Lii'a O Wy analogy w t t b "Hie <X- in tegrals . • 1 - 1 Jje a O L j * j ' , all S3 1 c ijjia = IjJB (111:37) (111:38) (111:39) UJJ° --§1 f-L + ! "I Xa« I 2 2(i + K*B*)J Uj*,« ^ U j j m analogy wVHi "Hie wj = O c a s e , (111:40) -31-u j j ' o " " i f . - O t a l l < '-'jjm = <• * (111:41) u». -v l ) 5 = U f t B < b y analogy W i l l i +V»e « -e* Uj-jo < (111:42) Integrals Ujj» m nj f o r which a n i s neither equal to ?m nor zero, are d i f f i c u l t to evaluate. However, It i s to be expected that they w i l l be smaller than the two-centre integ r a l s of (40, (41), and (42). Consequently a two-centre approximation, i n which these i n t e g r a l s are ignored, w i l l be used. In (37) to (42), the notation i s as follows: am *b e K the magnitude of a m . the magnitude of k j , the e l e c t r o n i c charge. the s t a t i c d i e l e c t r i c constant of s i l i c o n . (111:43) A the transverse "Bohr radius" for F j l s (r) i n s i l i c o n . B » the longitudinal "Bohr radius" for F^.-(r) i n s i l i c o n . J ~* n * (111:44) a =? an appropriate average of A and B. The functional form of F j l s (r) used i s that given by Kohn (1957, * See Slater and Koster, 1954, Section I I I . 32-equation 6.4): F = S ^ l exp (111:45) where the z-axis is taken in the direction of k i. For s i l i c o n , Kohn gives the values: A - 25.0 x 10~8 cm. (111:46) B - 14.2 x 10"8 cm. The necessity of using a* arises from the occurrence of the term 1/r in the integrands of (40) to (42). Simplification of the exponential terms in these integrands leads to a complicated expression for the remaining r-dependence unless an average Bohr radius i s used. From (46) i t may be seen that the value of a w i l l be approximately 20 x 10 cm. (Kohn, 1957, equation 5.10). Use of this value leads to: -§1^  = fc.O X 10"* eY. [ f o r s i l i c o n ] (111:47) By Chapter I , k Q ~ — i i . , where d i s the s i l i c o n l a t t i c e 2d spacing 5.42 x 10"~8 cm.; and by (46) B i s approximately equal to 2.5 d. Thus the value of k ^ i s approximately 12, so that: S - i - = 6.9 x lo" 3 1 + k. B* 14? To a reasonable approximation, therefore, the integrals (37) to (42) may be taken as: * Jones (1960, p. 121) shows that the value of k^x i s Chapter I gives k Q s | k ^ . d -33 T* - 7* - JL all other I -integrals = O (111:48) al l other U-integrals = O (111:49) If i t is further assumed that: o (111:50) unless a m is one of (±a,0,0), (0,±a,0), (0,0,+a), or zero (where a is the impurity lattice spacing), then the secular equation (27) may be easily solved. (This assumption constitutes a nearest-neighbour approximation.) 4. Evaluation of matrix elements. Using the results of (47), (48), and (49), the matrix elements of interest reduce to: < o l o > = 1 < l l i > = 1 < 1 U > = < 2 l l > = i z < i l 3 > » <1|4> « < H 5 > = O [ < Z ' Z > = I = < Z I 4 > » <2|5> = o ; < 3 i - i > = < 4 I 4 > = < 5 l 5 > = 1 [ < 3 I 4 > - < 3 l 5 > « < 4 ! 5 > = o -34-< O l U 1 O cj„> ' <11UIl f l a > a < l l U U a j , > a <2.| U | l a a > . < i 1 u I i a s > s O Ci= 3.4.y ] ' <z\v 1 z O O Ci= 3,4,53. r <3lU 1 3* 2> s (111:52) | <4IU |4a a> a ] <5l U ( 5a 5 > » [<3l U |4a s > <31UI5ftg> = <4|UlSa a> = O Then, multiplying the matrix elements (52) by e'- '-s , and summing over nearest neighbours: le 1--' <OlUlOa;,> = | U \ 2 + c o 3 k . a ] ^ c o s k , « +• c o s K y a + coa K , a ] Z,e i - *a< l |0|la 5 > = 2 0 \ ( i + c o s k . i X cos K x a + cos K y a ) + 2c<>s K»a} Zci-'ft-<2|Ul 2.4a> a Z U { ( i + cosk„aXcos K«a + cos k » a ^ + z cosK Y a 3 £ e i K f t = » < l I U U a s > = zu{ t o s k » * c o s k « a + cos K y * + co» K t a ^ £e ' - * = » < 3 | U|3 9.=?> = 4U J c o $ k . a cos K xa ••-cos K Y a + cos K » a ] £,C i S f t a <4|Ul4a = > = 4 U ^ c o s K x * + c o S k . 4 c o » K v 4 + c o $ K » a } £e/K » » < s i U | Sa 5> = 4U { c o s K*a + t o j K,a + c o s k . a c o $ K»a] -35-a l l others - 0. where: U - Ujjm evaluated at a m - (a,0,0). (111:54), 5. The secular equation: Solution. For brevity denote: L a cos k Qa cos K^a + cos Kya + cos K^a. M «• cos K^a + cos k Qa cos K^a + cos Kza. N 35 cos K^a + cos K ya + cos k Qa cos K^a. (111:55) (111:56) Then, making use of the results of (51) and (53), equation (27) becomes: ( L + M } - £ L - t / 2 O O O O Z L - t O O o O 2.M-1 O O O O Z N - E = o The roots of equation (57) are: (111:57) (111:58) Then, making use of these results in conjunction with the -36-requirement that i g(r) be normalized over the entire crystal, the corresponding d-coefficients may be obtained from equation (12): (111:59) d „ d t * O . A l l others S 3 0. d , , d x 4 O. A l l others e a 0. ( £ , : d» = 1 . A l l others S 3 0. d « « 1 . All others S O 0. d y = i . A l l others - 0. (111:60) In the isolated-impurity problem, as noted in Section B.l of this chapter, group-theoretical considerations show that the five-fold degeneracy of the upper Is state is only approximate. There are actually two sets of degenerate states: one ( +l*, ) corresponding to the group B, and the other ( , + < 4\ 4<" ) corresponding to the group T g (Kohn and Luttinger, 1955 c). In the present approximation, therefore, the results of (59 and (60) indicate that the states corresponding to T 3 form bands independently of each other, while the states corresponding to E mix. Substitution of (51) and (53) into equation (28) gives the energy for the non-degenerate Is state In the spherical-potential effective-mass approximation as: E-E',0,1 = i u \ z+ co*k.a^{ cot. K»a + cos K t« +• cos } ( m : 6 1 ) For a simple comparison of the line broadenings described by (58) and (61), consider the case when cos k Qa - 1. In this - 3 7 -instance, a l l the energies reduce to: E . - £ l s = 4 U \ cos K x a + cos krya + co» K B 4 *j (III - 6 2 ) The implication of ( 6 2 ) is that the degenerate and non-degenerate Is levels broaden at approximately the same rate. Clearly this behaviour is due to the fact that spherical-potential wave functions were used throughout the calculation. However, i t was assumed in the derivation of ( 2 7 ) and ( 2 8 ) that the non-degenerate and five-fold degenerate levels were sufficiently far apart to broaden independently. For this assumption to be valid, corrections to the spherical-potential approximation should have been taken into account. As the corrected degenerate Is levels are displaced relatively l i t t l e from their spherical-potential values (Kohn and Luttinger, 1955 c), the results of ( 5 8 ) will be taken to apply in this case without alteration. The non-degenerate Is level, on the other hand, is considerably lower in the exact formulation of the problem than i t is in the spherical-potential approximation (Kohn and Luttinger, 1955 a and c). In this case, therefore, a corrected wave-function should have been used. It is expected that i f the correction were carried out in detail, the broadening of the non-degenerate level would be negligible for the impurity lattice spacings considered. This contention is based on the fact that the See Section C.3 of this chapter. 38-correct l s ^ ^ wave function would be much less extensive than its spherical-potential counterpart, so that Is functions centered on neighbouring donor atoms would over-lap much less than the corresponding l s N ' functions. 6. Energy-band broadening. Baltensperger (1953) has calculated the broadening of the Is and 2s-2p bands of equation (1) assuming a single-minimum conduction band and using purely hydrogenic effective-mass wave functions. He finds that the lower edge of the 2p band is depressed: AE 2 p ° = o i 5 _SL * 5"x io"1 ev. (111:63) below the Isolated-impurity level when r s » 3a*. Since: ^ ^ (111:64) the corresponding value of the impurity lattice spacing is a = 5a*. This spacing gives an impurity concentration, in 18 3 silicon, of 1 x 10 /cm . At a «• 5 a*, the results of (58) indicate that the upper edge of the five-fold degenerate Is band is raised above the isolated-impurity level by at most: = - i a U 2 i . 3 * io"* «v. (111:65) and at least: A E * . * = . - 4 0 s o . l x io- JeV.. (111:66) -39-Using the values of the isolated-impurity level energies listed by Kohn and Luttinger (1955 c, Table VII), the separation of the l s ^ and 2pQ^6^ levels is: & E. touted = (3.2 ,x.io-* - II K I O " 1 ) E V . = 2.1 x lo" 1 ev. (111:67) By (63), (65), and (67), the separation at a - 5a* is at least: &E.4a« a (2-1 * to" 1 - O-S" X icT* - o . l « i o ~ l ) «v . I S x t o - 1 -a (111:68) Hence at a = 5a , the separation has dropped to about two-thirds of its value in the isolated-impurity case. Since Baltensperger's results are based on the cellular method, they are more applicable at small values of a than are the tight-binding results of (58). Using Baltensperger's figures as a rough guide, i t is found that the Is and 2s-2p bands overlap near a = 3ex (impurity concentration 4 x 1018/cm3), Hence any investigation of the ls*°—*ls*5* transition must be carried out for an impurity lattice spacing greater than 3a*, so that the line will not be obscured by that for the Is—*2p Q transition. C. Optical Matrix Elements. 1. Simplification of the matrix elements. In the electric-dipole approximation, the optical matrix * elements of interest are: * See Chapter IV. -40-= - i - i d * Y,^--S'^--S < i l s f l 3 ' l p . l o l * a n > N i=l n' n s a . ±. f, a r f e e l e l B f t 9 < i l s a 3 ' l I O l S &n>] (111:69) making use of (11:26). Under the assumption that a l l impurities are substitutional, so that each vector of the impurity lattice is also a vector of the unperturbed lattice, so that: U\ilas> = Ej; Uia a> (111:70) It then follows from (69) and (70) that: <SK'lp.tOK> . - S l E u ^ - E ^ U (111:71) Now by a simple extension of the result of (11:36), < i i s a 5'|x,|o is a 5 > » <xa 2' l « , | o a a > (111:72) Hence (71) may be written: -41 l n'n x X di* T. e'--= <iU«rtoa„> • S , e " ^ ' > » - - ' ( I I I : 7 3 ) (shifting the origin to a Q« , and defining a m - a n - a n»). By (73), therefore: <5K'lp»-loK> = Es^CH') S K . K ' Z * * £e l K-" B<alx,loa = !> ( m :74) where: e.,(tf. E f f w - ESCIO ( I N : 7 5 ) and 5 B > K.= J- X ( e i ( ^ ^ ' (111:75') . c * O K , K * i s a Kronecker o in three dimensions. Its value is one whenever K «• K ' , and zero whenever K / K * . 2. Evaluation of integrals in the spherical-potential  approximation. The integrals involved in evaluating < i U f I o a B > (for i 03 1,2,3,4,5) are similar to those defined in (35) and (36), and will be denoted by Xjj»m, X.fym, Xjym* a n d *Jymm For example: Xj"Via = ^ » ( < r F i ^ ) F i ' C r - a B U o s b i - £ c o s k / . r <*£ (111*76) * See Appendix B. - 4 2 If spherical-potential effective-mass wave functions are used for the evaluation of the integrals (76), i t is found that xjj» m a n d ^J'm v a n i s h identically in almost a l l cases of interest. Furthermore, the only non-zero integrals of this type (certain of the X?Jt m for which j f j* and ra ?* 0), as well as the integrals X j j i m and X^ym, are extremely small, being of the order of 1 x IO"*6 B. Con-sequently the matrix elements are very much smaller than those for Is—*2p transitions. As was pointed out in Chapter I, however, the assumption of a spherically symmetric perturbing potential is incorrect. It is therefore not a valid approximation to make use of the procedure outlined in the last paragraph. Corrections to the spherical-potential approximation must be considered. 3. Corrections to the spherical-potential approximation. In Chapter I i t was pointed out that the most appropriate method of correcting the spherical-potential shallow-impurity approximation would be to apply a tetrahedrally symmetric perturbation to the hamiltonian of equation (11:7). This perturbation would have to account for: i . Deviations from the approximation of large impurity-electron orbits. i i . Departures from spherical symmetry in general. Unfortunately, the form of the required perturbing potential is not knownj i t therefore becomes necessary to make use * See Chapter II, Section B. - 4 3 of less exact techniques of correction. Kohn and Luttinger ( 1 9 5 5 a) have proposed such a technique. They argue that except i n the immediate v i c i n i t y of the impurity nucleus, the s p h e r i c a l - p o t e n t i a l approximation w i l l be very nearly v a l i d . Consequently, i f the s p h e r i c a l -p o t e n t i a l effective-mass equation i s solved i n an " e x t e r i o r " region which excludes a small volume surrounding the donor atom, and the r e s u l t i n g wave function i s joined to an exact solut i o n of ( 1 1 : 6 ) i n the " i n t e r i o r " region to determine the energy, an improved theory should r e s u l t . In order to avoid the d i f f i c u l t i e s inherent i n the second step of t h i s procedure, Kohn and Luttinger have made use of the experiment-a l l y observed ground state i o n i z a t i o n energy to determine the ex t e r i o r wave function, and then have made a rough estimate of the solu t i o n f,or the i n t e r i o r region. They have found that t h e i r r e s u l t s are approximately consistent with the require-ment that the two solutions j o i n smoothly. The experimental i o n i z a t i o n of a phosphorus donor i n s i l i c o n i s (Morin et a l . , 1 9 5 4 ) Kohn and Luttinger ( 1 9 5 5 a) show that i f t h i s value of ( E - £ o ) i s substituted into the e f f e c t i v e mass equation (11:27), the solution of the equation i s ; E - £ 0 = - o o 4 4 e v - (111:77) (111:78) - 4 4 -where: x *» r/cx* n - 0.81 (111:79) C •= normalization constant W » the Whittaker function. The function (78) has the limiting behaviour: FC*} = Ce""/n - 0 - « U n Zx 4 l j a s x 4r o. (111:80) FCxi = ce"x/"l7r' as * f 00 • Thus F(x) diverges near the origin. (This behaviour is due to the inapplicability of the potential —~- close to the donor nucleus.) Kohn and Luttinger therefore round off the solution within the Wigner-Seitz unit sphere of radius 0.08a* enclosing the impurity atom. The following procedure was used to obtain a usable approximation to the function F(x): i . The approximations (80), divided by the normalization constant C, were plotted, and were found to be close together near x =• 4.5. Accordingly, the two functions were joined at this value to give F(x)/C approximately for a l l x. i i . For large values of x the approximation to F(x)/C given by (i) was found to be fitted very closely by the expression: -45 0.65 exp (-1.25 x) (111:81) i i i . iv. v. vi. For small values of x, F(x)/C was assumed to be approximately of the form: 0.65 exp (-1.25 x) + p exp (-qx) (111:82) where q is greater than 1.25, so that the second exponential decays more rapidly than the f i r s t . The value of p was determined by requiring the expression (82) to go to 2.30 at x » 0. According to (i ) , this value is slightly greater than the value of F(x)/C at the Wigner-Seitz sphere boundary.) The value of q was determined by requiring that (82) have the same normalization as the function plotted in (i). The result was: F(x)/C - 0.65 exp (-1.25 x) (111:83) + 1.65 exp (-3.00 x) (83) is compared graphically with the result of (i) in Figure 1. The value of C was determined by integrating the curve of (i) numerically, and requiring that: (111:84) -46-The result was: (111:85) v i i . (85) was substituted into (83) to give for the corrected ground state effective-mass wave function the expression: 2-2,4 erir/of] (111:86) v i i i . As a very rough check on the accuracy of (86) with respect to energy, the radius for which the radial probability distribution (Pauling and Wilson, 1935) has its maximum was plotted as a function of energy for the Is, 2s, and 3s states of a hydrogen-like atom (Figure 2). The probability distribution maximum of (86) was found to be consistent with an energy of -0.044 ev. in this scheme. If the requirements of tetrahedral symmetry are to be satisfied , the effective mass wave functions used in connection with the five-fold degenerate Is states must be different from those used with the non-degenerate ground state. As the energy of the five-fold degenerate level is not known accurately, however, the corrections to be used in this case are difficult to determine. Kohn and Luttinger (1955 c) have * See Chapter I. -47-estimated that th© level corresponds to an energy of about -0.032 ev. Using this value, and applying the procedure used above for the ground state, an approximate wave function similar to (86) should be obtained. As the energy used is so close to the effective-mass value (-0.029 ev.), i t will be assumed that the correction is small enough for the effective-mass functions to be used without alteration. An estimate of the error introduced by this assumption will be made in conjunction with the evaluation of optical matrix elements. 4. Evaluation of integrals in the corrected spherical- potential approximation. Using the corrected wave functions introduced in the last section, the integrals (76) may be evaluated approximately. Hence, i l ra ° 0j or i f m ft 0, but a m is perpendicular to the <r-axis, XJj= = ° (111:87) since the integrand is an odd function. If o m is parallel to the cr -axis, on the other hand, x"= = cS* f { (^FjCOFTCt-^as (111:88) + (ao-FjCcVF/Ct-ar-'Kos *t»sjr j * see Section C.4 of this chapter. - 4 8 -The second integral in (88) cannot be evaluated analytically. By the arguments of Chapter II, Section C, however, this Integral should be small. Numerically, for|am|« 4a j ( ^ » FtoPV-a^ d r * -LIST mo-1) a* ( 1 1 1 : 8 9 ) , - T i F ^ F ° ( r - a , ) C o s 2 k . i d r ~ -C2-88 x iar») a* ( 1 1 1 : 9 0 ) Hence at this value of ja^ J, the second integral i s smaller than the f i r s t by approximately a factor of ten, and has the same sign. It i s also to be expected that integrals of the form: l i r t * [«••' Fj(cWCc-*«r> coVkj.c *c ( 1 1 1 : 9 1 ) w i l l differ in value from ( 8 8 ) . However, this difference i s clearly small; i t vanishes altogether when the Bohr r a d i i A and B are replaced by their average, a . Consequently, i t w i l l be assumed that a l l non-zero integrals of the form (91) are approximately equal to: X i = - K i - 4- P ( O F " ( C - & m > * E = Lo.o.al) * ( X i r r 2 J ' - 4 9 -wher©: BtCy^ = ( i V ^ t (111:93) Values of these functions have been tabulated by Rosen (1931) for arguments greater than 1.5. For the values used here, see Figures 3a and 3b. is plotted as a function of a in Figure 4. Other integrals of interest are: X i i * s = ^ » | < r Fi( t ^ FrCc-ft») eoskj-r S.« l y-r dr ( Z T r t s x ) i 1 ~ "* (111:94) For m 0 the integrals (94) would be expected to have values of the same order of magnitude as those in (90). In any case, |S<r fler X j j m is less than or equal to X J J Q . The latter integral is non-zero and easy to evaluate. It is found that: X*f0 ^ + 0-sr * i o - M (111:95) Comparing this value with (89), i t is seen that XJ^ will be at least a factor of ten less than X^ , though the two integrals will not necessarily have the same sign. Finally, X J T B = ^ [ * , F j(tf F?Cr-aB> S ^ k j . r <*r = _L__ . _L | J<,Fi(r') F"(r-ft a^ dr - {*, F/r*) F^Cr-a^ cos^hiC *r ] -50-Clearly, If m or i f a m is perpendicular to the cr-axis, o (111:96) If m / 0 and is parallel to the -axis, (111:97) As in the development leading to (47) and (48), a l l integrals involving two different conduction-band minima (i.e. j 5* j") will be neglected. By analogy with (36), i t is expected that these Integrals will be considerably smaller than those listed above. As was pointed out in Section C.3, corrections should be applied to the effective-mass functions associated with the five-fold degenerate Is level as well as to those associated with the ground state. Approximately, the corrected functions should have the same form as (86), namely: where the values of P,Q,S, and T are determined by the energy of the five-fold degenerate level. If these functions are used, corresponding corrections must be applied to the integrals in (87) to (97), a l l of which may be expressed in degenerate level energy is shown in Figure 5. Clearly the difference between values for E ^ f - £ o - -0.029 ev. and (5) E l 5 - E o - -0.032 ev. is sufficiently small for the use of terms of X^ . The dependence of this quantity on the five-fold the 0,029 ev. energy to be a reasonable approximation. 5. Evaluation of the matrix elements. The integrals in (87) to (97) may now be used to evaluate the matrix elements <i lx<r\oas> , which may then be substituted into (75). j < l l x r | o a B > * O l = ! - Xa a t < i l a , | O a 5 > ~ jy^ (« , , , - * 5 =) Xa= (111:98) Substituting (98) and (99) into (75), restricting the summation to nearest neighbours, and noting that: < , < 5 | a,|oa 5> £ ' < J I v l o ^ > as <4 1 o a c> «v (5|= Xftj 7f P3= XL. (111:99) - - - i - ( <A* Ay \ c (111:100) It i s found that: - 5 2 -< i K ' l p , | 0 K > = - ESfi(K) - d\$L „•„ k„a cos\Kxa] XI (III: 101) - d* i l S . n W.a COS {Kyd} X3 _ 0L5 ^  s.vi k.a cos iK*a^ j For simplicity, consider: * j z^^T o - « , ^ * . n K,a (111:102) - df ±L sinlua cosK»a "l V? J By (59) and (60), d 2 and d 5 are mutually exclusive. Hence only one of the bracketted terras in (102) will be non-zero. Expressions similar to (102) will arise for other choices of a , the main difference being that d 2* will be replaced by d^*, or by -(d^* + d 2*). Since dj and d 2 both depend on K, and thus lead to cumbersome expressions, i t will be assumed for simplicity that each of the coefficients involving these terms has a magnitude equal to one. Hence, in the calculation of the absorption coefficient, two matrix elements are of importance: <*K'tp,rloK> g - r i zyf (1-cos k.O %^ xl sm K,a (111:103) < 5 K ' I P ^ I O K> s a. ±. sin k,* E™t«) i K t f x Z cos K^ a. (111:104) -53 Both of these matrix elements vanish when cos k Q a =° 1 , but t h e i r sum i s large when cos k Q a » 0 or -1. Consequently the combined e f f e c t of these matrix elements w i l l be r e l a t i v e l y important, except i n the neighbourhood of impurity l a t t i c e spaclngs for which k Q a i s an even i n t e g r a l multiple of TT. (Clearly t h i s behaviour w i l l not be observable, since i n r e a l i t y the impurity l a t t i c e spacing has only the si g n i f i c a n c e of an average distance between impurities —• the randomness of the impurity distribution:, removes e f f e c t s which are dependent upon the assumption of r e g u l a r i t y . ) -54-CHAPTER IV Optical Absorption by Transitions between Impurity Bands in Phosphorus-doped Silicon. A. Derivation of the Absorption Coefficient. 1. Transition probabilities. In treating the phenomena of absorption and induced emission of electromagnetic radiation by atomic systems, the most commonly used approach is the semi-classical approximation. This approximation, in which the radiation field is treated classically and the atomic system is treated quantum-mechanically, leads to essentially correct results, and has the great advantage of simplicity over more accurate treatments involving quanti-zation of the field. Consequently the semiclassical theory will be used in this thesis in obtaining an expression for the coefficient of absorption. Consider an atomic system in state U> upon which is incident a polarized beam of photons, of intensity I ( w ) d < J in the angular frequency range (<>>, oo + d«>). Using first-order time-dependent perturbation theory, Schiff (1955, Chapter X, Section 35) has shown that the probability per unit time of a transition from \H> to another state lu> , due to absorption of a photon from, the incident beam, is: xicS) |MU.(5J| (IV:1) -55-where: e =• e l e c t r o n i c charge m =» electron mass c » v e l o c i t y of l i g h t QLw =• photon wave-number vector, of magnitude ^ . o3 U J l » (where Eu and E A are the (IV: 2 ) « energies of iu> and u> respectively) M V ( 3 J - < u l e l 3 - r v r U > (IV :3) ^ < r =* component of grad i n the d i r e c t i o n of po l a r i z a t i o n . Since the function i s sharply peaked at «o - f and: equation (1) may be written approximately as: P = cLco If-*-. ICCO) I M ^ C ^ \ \z bi^-cS) J-a> mzccO* S on so that: ( I V : 4 ) If !(*«>) i s approximately constant at the frequencies (IV:5) -56. considered, i t may be replaced i n (5) by I«o , i t s average over unit frequency range at co. Pto• ^  then becomes the p r o b a b i l i t y per unit time of a t r a n s i t i o n between ic> and K> due to the absorption of a photon of frequency from a beam of i n t e n s i t y Io per unit frequency range. The i n t e n s i t y may also be written: Ifco^ oLco = wCto'i ticJ <A*o. (IV: 6 ) where n(<j) d^ o i s the number of photons i n (", co+ d<>>) crossing unit area per unit time. In terms of averages over unit frequency range: I co - n^<^ (IV: 7) where "n w i s the number of photons crossing unit area per unit time per unit frequency range about co. Then, su b s t i t u t i n g (7) into (5), the p r o b a b i l i t y per unit time of a t r a n s i t i o n between IA> and K> due to the absorption of a photon of frequency to from an incident beam of n"^  p a r t i c l e s per unit area per unit time per unit frequency range about co i s found to be: Jla-.ujL = ^ ! I M . J ^ J I 2 o C o ^ - c o ) . ( I V : 8 ) 2. S p e c i a l i z a t i o n to a r i g i d l a t t i c e . If the r e s u l t s of section 1 are applied to a c r y s t a l l i n e array of atoms, the wave-vectors i£> and iu> w i l l depend upon both the coordinates of the electrons within the i n d i v i d u a l -57-atoms, and the coordinates of the atoms themselves within the lattice. In this form solution of the problem presents formidable difficulties, since the lattice coordinates change as the atoms vibrate about their equilibrium positions. These difficulties may be reduced (D.L. Dexter, 1958, Chapter II, Section 5) by using the adiabatic approximation to partially separate the two types of coordinate dependence, and the Condon approximation to average over the dependence on the coordinates of the lattice. However, further simplification is to be desired. Such simplification may be achieved by assuming that the lattice in question consists of a completely rigid array of atoms. In this case, IO and l»0 refer solely to electronic states which may be denoted by i*'i> and U K ' X . The probability (8) then becomes: Pco-u* ( « « , K . j O = -Jj-TT* i M ^ K . K ' . n ^ O l * b(co*'« - co) where now: (IV:9) and: (IV: 10) (IV:11) In effect this approximation involves consideration of the crystal at the absolute zero of temperature, with a l l zero-point energies of vibrational states being ignored. -58-3. The coefficient of absorption. In a cubic crystal of volume V, the number of electronic states whose K-vector lies in the range (K, K + dK) is (Dekker, 1957, p. 256): V<*K considering one spin orientation only. Thus i f f u(K*) denotes the Fermi distribution function for the energy band labelled by u, the total number of empty states of each spin in the range (K', K1 + dK') in this band is: [1- f.(s">] ^  ( 1 V : 1 2 ) It then follows from (9) and (12) that the probability per unit time of a transition from the state I£K> into band u, due to the absorption of a photon of frequency to from an incident beam of n"a» particles per unit time per unit area per unit frequency range about to is: ^ • u , G ^ , i O = J 7 « i l M ( • * - . ! $ • . L I - ^ C K ' ) ] ^ ( I V , 1 3 ) where the integration is over the fir s t Brillouin zone. It is unnecessary to include both spin orientations, since the spin remains unaltered by the transition. Consider a small cylindrical volume of the crystal, of cross-section A and length dx parallel to the Incident photon beam. By an argument similar to that given above, the number of f i l l e d band-jL states of each spin in the range (K, K + dK) -59-i n t h i s small volume i s : Adx d K Each of these states may act as an "absorbing unit". Then, multiplying by two to account for both spin orientations, the t o t a l number of absorbing units i n (K, K + dK) i n the volume Adx i s : o £ r *\ A d x d K Z 4 > C « (IV:14) Thus the p r o b a b i l i t y per unit time of a t r a n s i t i o n from band Z into band u due to the absorption i n volume Adx of a photon of angular frequency from an incident beam of "n",,* p a r t i c l e s per unit time per unit area per unit frequency range about co i s : f ^ - . u , (nco^ AbLx = f ^ . u , (n«, iO "Z*>C*0 A d * (IV:15) where the integration i s again over the f i r s t B r i l l o u i n zone. Equation (15) e f f e c t i v e l y gives the decrease per unit time of the number of photons i n the beam due to absorption i n Adx at the frequency of i n t e r e s t . The corresponding change i n the mean energy f l u x W«,» ( ra H o At»co) i s : dWc = - (JxJ) Act* tuo (IV:16) Now the absorption c o e f f i c i e n t ^ ^ ( c o ) i s defined by the d i f f e r e n t i a l equation: OLWU J dLx - / x^Cco} ( I V : 17) -60-Hence, by (16) and (17): . J * * I * K ' ^ i ^ i ^ . p W B - l S « ( I T , 1 8 ) where both integrations are over the f i r s t B r i l l o u i n zone. 4. S p e c i a l i z a t i o n to centres i n a d i e l e c t r i c medium. The r e s u l t (18) applies to the interaction of r a d i a t i o n with a c r y s t a l l i n e array of atoms imbedded i n free space. If the array i s now considered to be imbedded i n a d i e l e c t r i c medium, several corrections must be applied (D.L. Dexter, 1958, Sections 2 and 4). The f i r s t correction i s concerned with modification of the magnitude of the r a d i a t i o n f i e l d . S c h i f f (1955, equations 35.11 and 35.14) shows that T^;Mil(*w,Ktc'), and hence also the absorption c o e f f i c i e n t , depends on tttf , where e,« i s the magnitude of the f i e l d which acts at the atoms of the c r y s t a l . It was f i r s t pointed out by Lax (1952) that € f« i s not i n general equal to the average macroscopic f i e l d e i n the surrounding medium. Consequently, (18) must be m u l t i p l i e d by a factor {€e(t/e]* . The e f f e c t i v e f i e l d r a t i o € e f f/« i s d i f f i c u l t to evaluate accurately. For a very tight-binding approximation, i n which overlap, exchange e f f e c t s , and multipole interactions higher 61-than dipole-dipole may be neglected, D.L. Dexter (1956) derives the expression: T " = 3 (IV; 19) where n is the refractive index of the surrounding medium. However, for the case of diffuse centres such as impurities in germanium, Lax (1954) has shown that the effective field ratio must be taken as 1. The second correction to be applied to (18) involves modification of photon energies and velocities by the dielectric medium. Essentially, (18) was derived by dividing the tran-sition probability IRo-u/. (jn«>) by the energy flux Ww . In a dielectric medium (D.L. Dexter, 1956), Wo> is given by its free-space value multiplied by ~ , where K is the static dielectric constant of the medium, and & is the energy velocity c/n. For photon energies at which the medium is effectively transparent, K is approximately equal to n . Thus in these regions the multiplying factor may be replaced by 1/n. (Dexter, 1958, Section 2 points out that multiplication by 1/n is s t i l l correct when the medium is not transparent. In this case n is the real part of the index of refraction.) On applying the above corrections to (18), the absorption coefficient of interest for a rigid crystalline array imbedded in a dielectric medium of refractive index n is found to be: r-o* - TTC m & fe-? i * s i i* (iv, 20) • ^CK^ Lt-fwCK 'O $(cofi.s - C J ) - 6 2 -B. Evaluation ef the Absorption C o e f f i c i e n t . 1. S i m p l i f i c a t i o n s and approximations. As they stand, the integrations i n (20) are intractable. Consequently, several approximations must be Introduced, as follows: i . Since the absorption process Is to be considered only i n the neighbourhood of the absolute zero of temperature, the Fermi functions f A ( K ) and f u ( K ' ) assume a simple form. It i s to be expected that when T i s near 0, a l l the electrons w i l l be i n the ground-state band, l a b e l l e d by & . Hence: fu.(K') = o (IV:21) The form of f^(K) when T i s exactly zero must involve a dis c o n t i n u i t y at the mid-point of the i-band. This behaviour r e s u l t s from the fac t that while there are as many impurity electrons i n the c r y s t a l as there are impurity nu c l e i , spin-degeneracy ensures that there w i l l be twice t h i s number of el e c t r o n i c states av a i l a b l e i n the lowest impurity band. Thus when T =» 0, only the lower half of band JL w i l l be occupied. As T increases s l i g h t l y from zero, however, the disc o n t i n u i t y i n the Fermi function f A (K) acquires a f i n i t e spread. For a very narrow band, such as the i-band i s expected to be, t h i s spread w i l l * See Chapter III, Section 5. -63-probably equal or exceed the band width, so that the electrons w i l l be d i s t r i b u t e d throughout the band. For s i m p l i f i c a t i o n , i t w i l l therefore be assumed that i n t h i s case, = i (IV: 22) i i . By (3), the matrix elements M ^ C K ' K ico5v") are: (IV:23) "to In the e l e c t r i c dipole approximation, i t i s assumed that the photon wave-number vector i s of n e g l i g i b l e magnitude, so that the exponential i n (23) may be approximated by the f i r s t term i n i t s expansion: e t l - . ' a - r fi t (IV:24) Thus, i n the cases of in t e r e s t , the matrix elements are: M S .O ^ ' K ) = i <iK'lp,loK'> (IV:25) The expressions (25) may then be evaluated with the aid of equations (111:103) and (111:104). i i i . In order that the integrations i n (20) may be c a r r i e d out, the three-dimensional Kronecker 6, * defined i n (111:75*) must be replaced by a Dirac -64-delta-function (Dirac, 1958, p. 58). It is shown in Appendix B that with the correct normalization: ( 6 K S 0 Z = * K K - = - ^ J p l SCK-K"). (IV:26) where V is the volume of the impurity-lattice "crystal". iv. By (ll), the function j^f^ K* is defined to be: W K K ' = ^ I E u (K">- E , ( K > | Thus, in the cases of interest, the tight-binding expressions (111:58) may be used for the energies E^CK'). AS stated in the discussion of (111:62), the broadening of the lower band is expected to be negligible, so that EX(K) may be set equal to the isolated-impurity value of the ground-state energy. Examination of (111:58) shows that the expressions for the energy bands associated with the group E are rather cumbersome. Consequently, only the simpler expressions for the energies associated with the group Tg will be used in explicit calculations. 2. The absorption coefficient for the lsCo^-» l s ( ^ transition. Using the approximations outlined in the last section, substitution of (111:104) into (25) leads to an expression for the matrix element associated with the transition between -65-the ground state and the d 5 band. M^,o(K'Ka« a< 5) s i smk,a Xi E(5*1»U} cos Kma Sss' (IV:27) Hence: |ru $ 0 C i < ' K ^ K . ^ l a * j £ i± s.v,*k.a Cxi)2 U'f.Wiof cos*K,a S K K ' ' ~ H 3 = S,V,^0A C x i V i E V ' i c ^ r ( I V : 2 8 ) "to* 3 . c o s 1 * , * (ZTTV* S C K - K ' ) . Then, substituting (28) into (20) and using the delta-function &(K* - K) to carry out the integration over K': " isM&a ( I V : 2 9 ) i (*!< lE'&Ortj'cos'K.a 6{E ' ? , « > -M CO J ' Clearly the absorption coefficient (29) vanishes whenever o sin koa «• 0 and has its maximum value when cosk 0 a = 0. For simplicity, intermediate cases will be ignored. Then, under the condition that cos k Qa » 0, (111:55), (111:56), and (111:58) show that: E C / o ( £ ) = - 4 l U l (cos K,a + coi Kyz] + A. E 5 0 (IV:30) denoting: ^E s .o = [ Eff - EJV ] ,'s.iaW , T V . „ . . Hence, substituting (30) into (29) and taking ( K ^ J 1 out-side the integral sign as tfco1 , by virtue of the delta-—66— function: • . i /•"n/* tv/* i*f* a . r u l s - L { l 4 « \ (eXD* ^- ( d K , d K y d K » c o s l K a a 4IUI •'-•n/4 i T r / a i-«ya 41 (IV:32) 41UI One of the integrations in (32) may be carried out immediately (Kahn, 1955, equations 3 and 4), by setting: ©l£ d Ccol X *• C e i ^ ) d t dpOOS X + Co&g) I V C w t +.oo«|p I Vsin'x 3" (IV:33) where d£ is a line element of constant (cos x + cos y). Hence, defining: A E ^ O - CO 4IUI (IV:34) the absorption coefficient (32) becomes: 3 _| cot X 4-Co = 1 n c U S * a».ui <° \ U i ^ . z i ^ (IV:35) for % between +2 and -2. In Appendix C the integration in (35) is carried out, The results are shown in Figure 6, where the integral: dt >Js«n*x + Sin**}' (IV:36) COS X +• COS «j = ^ i IT -67-is plotted as a function of | . As may be seen from the graph, use of (36) leads to an expression for the absorption co-efficient which is divergent at the centre of the band. How-ever, i t may be shown* that the divergence is sufficiently mild for the area under the absorption-coefficient curve to remain finite. The divergent behaviour of (36) is a direct consequence of the tight-binding approximation, and may be modified by correcting the energy surfaces. (The tight-binding surfaces are shown schematically in Figure 7.) Jones (1960, pp. 44-46) shows that for a simple cubic lattice, such as that considered here, the surfaces of constant energy must intersect the planes - 0, K y » 0, K z « 0, and the faces of the fi r s t Brillouin zone, at right angles. Clearly this requirement is not satisfied by the surfaces of Figure 7. The corrected surfaces are shown by Wilson (1953, p. 42, Figure 11.10). In this case, the mid-point of the band will correspond to a surface whose cross-section is shown schematically in Figure 8, and the corresponding absorption coefficient will l i e between infinity and the free-electron value obtained by approximating this surface by a right circular cylinder.** See Appendix C. Free-electron energy surfaces are described by Dekker (1957, p. 262, Figure 10-10b). The free-electron absorption coefficient may be determined by setting (36) equal to its value at % m 0 for a l l values of \ considered. -68-As the electrons In question should be described much more accurately by the tight-binding approximation than by the free-electron approximation, i t is probable that the correct maximum of the absorption-coefficient curve may be obtained by rounding off the curve of Figure 6. If this is done, the maximum absorption coefficient may be written as a function of the impurity lattice spacing as follows: on substituting the expression (III;49) for U in (35). denotes the round-off value from Figure 6, while c O j % denotes the frequency corresponding to the isolated-impurity energy gap between the i * C o > and i s s > levels.) In order to be able to make use of (37) to calculate the absorption coefficient, values must be determined for the index of refraction, n , and for the effective field ratio, €<fe/e • Since at the wave lengths of interest (about 100 microns), silicon is effectively transparent (Bichard and * Giles), i t should be a good approximation to write: n = /J? (IV:38) where K is the dielectric constant quoted by Kohn (1957, equation 5.8). To this approximation, the value of n is 3.46. Also, since centres in silicon are fairly diffuse, having a * See Section A.4 of this chapter. 69-Bohr radius of about half that i n germanium (Kohn, 1957, equation 5.10), the e f f e c t i v e f i e l d r a t i o should be approximately 1, according to the arguments of Lax (1952). The rounding-off of the tight-binding absorption co-e f f i c i e n t c a r r i e d out i n (37) i s a somewhat a r b i t r a r y procedure. While consistency might be obtained by cutting o f f the curve of Figure 6 at the resolution of the spectro-meter used, such a method o f f e r s no guarantee of accuracy. In the ca l c u l a t i o n s performed i n t h i s thesis, the cut-off value (d m a x w i l l be taken as f i v e . Substituting the approximations of the l a s t two para-graphs into (37), the expression f o r the absorption co-e f f i c i e n t becomes: c (Vatfll+*/«'J (IV:39) values of t h i s function are plotted i n Figure 9. C. Discussion of Results. 1. Is —• 2p Q t r a n s i t i o n . In Appendix D, the method used for the l s ^ - ^ - l s ^ 5 ) t r a n s i t i o n i s extended to give an approximate expression f o r the ls^0-l-».2po^6^ absorption c o e f f i c i e n t . As the l a t t e r t r a n s i t i o n has been observed experimentally (Bichard and G i l e s ) , i t should be possible to use the r e s u l t s of t h i s c a l c u l a t i o n as * See Section A.4 of t h i s chapter. -70-a rough check on the method i n general. In the Appendix i t i s shown that at concentrations f o r which the assumption of tight-binding should be a good approximation, the calculated absorption c o e f f i c i e n t i s in quite reasonable agreement with experiment. From t h i s i t would appear that there are no gross errors i n the c a l c u l a t i o n , and that the method used might be expected to give a reasonable description of the l s * ^ — . . I s * 5 * t r a n s i t i o n . 2. l s ( Q ^ — l s * 5 ) t r a n s i t i o n . The r e s u l t s plotted i n Figure 9 indicate that at impurity concentrations of about 10*® per cnr*, the l s ^ — * l s ^ ^ absorption c o e f f i c i e n t i s of the order of 10 cm"1. As tran-s i t i o n s with absorption c o e f f i c i e n t s i n t h i s range are ob-servable by e x i s t i n g techniques (Bichard and G i l e s ) , the theory would appear to indicate that the f i n e structure of the Is states w i l l be i d e n t i f i a b l e . There are, however, several sources of error i n the c a l c u l a t i o n which might tend to i n -validate the r e s u l t s obtained. The f i r s t p o s s i b i l i t y of major error l i e s i n the choice of wave functions for the two l s - s t a t e s of i n t e r e s t . The Kohn-Luttinger approximation used i n t h i s part of the c a l c u l a t i o n may only be good to within a factor of two or three, for the matrix element. Also, as indicated i n Figure 5, i f the I s * 5 * l e v e l i s depressed below i t s uncorrected effective-mass p o s i t i o n , the matrix element f o r the t r a n s i t i o n w i l l be reduced. A second source of uncertainty i n the c a l c u l a t i o n i s the -71-assumption of a regular l a t t i c e of impurities. C l e a r l y the random nature of the actual d i s t r i b u t i o n w i l l remove de-pendence on such quantities as cos k Qa. It i s to be expected that t h i s "smoothing-out" e f f e c t w i l l be accompanied by an o v e r a l l decrease i n the magnitude of the absorption c o e f f i c i e n t . This correction may be p a r t i a l l y compensated, however, by the f a c t that more than one of the f i v e degenerate Is states are a v a i l a b l e for o p t i c a l t r a n s i t i o n s i n any given instance. Comparison of (111:103) and (111:104) indicates, for example, that the absorption c o e f f i c i e n t (39) should be m u l t i p l i e d by a factor of three or four i f the e f f e c t of t r a n s i t i o n s to the **1 " **2 D a n d s i s *° D e included. There are several other points i n the derivation at which corrections might be applied. In most cases the possible errors have already been discussed. One further a l t e r a t i o n should be noted here, however: i f the incident r a d i a t i o n i s unpolarized, the absorption c o e f f i c i e n t should be reduced to one-third of i t s value for a completely polarized beam, (D.L. Dexter, 1958). There i s no guarantee, therefore, that the r e s u l t s of Figure 9 are accurate to within less than one or two orders of magnitude. However, i n view of the f a c t that r e s u l t s of the order of 10 cm""1 can be obtained, there would appear to be no f i r m t h e o r e t i c a l reason fo r the ls^°^ - l s ^ 5 ^ t r a n s i t i o n to be unobservable. See, f o r example, equations (111:103) and (111:104). -72-APPENDIX A Properties of the C o e f f i c i e n t s D. (k) By equation (11:16) the c o e f f i c i e n t s D i (k) and D_ . (k) s a t i s f y : {£!(=)-E]T>jflk) + (U(k.k') Dj tlV) dk'= o ( A : 1 ) and: {£"i(kV-E]D.Jl(!s ,> + jUCk.k'V D.^ Ck'Wk' = o (A :2) respectively, where the subscript " - j " denotes the conduction band minimum associated with - k j . Now by (11:17) (A:3) where: 3k> 9kA 3(-i<03C-kp k= -ti 8kx ak^ (A:4) Hence: (A:5) Also, by (11:12): (A:6) -73-The periodic parts, u cij(£) $ °* the conduction-band Bloch functions may be shown (Jones, 1960, equation 2.25) to s a t i s f y : vVCk(rJ> + uckw + % - U c C ^ - l s l k l - V P C ^ UckCO = o (A:7) Taking the complex conjugate of (7): v V e k - Zik .v uck + ^ { t cCk) - - Vp(^] u*w = o (A:8) By the symmetry of the c r y s t a l , £ c(k) «- £.c(-k), so that (8) may be written: V ^ i - lik . v u.1 + ^ 1 J £c -u>> - V±£L ^ = o ( A : 9 ) Hence, provided that the conduction band i s non-degenerate: u t k C ^ = "-c-isCr.^ ( A : 10) and: Therefore, s u b s t i t u t i n g (11) into (6): U*(k,se^ = <Pck(c) <Pck'(rt de. = J UC«^ ipc*-k Cr>> <pc_fe, (r> dr = 11 C-k,-ls') ( A : l l ) ( A :12) -74-From (2) i t follows that: It'iC-W-E^D.uC-ft + (ui-k.k'^p , (w1) dk' = o J J J* (A:13) Then, applying the r e s u l t s of (5) and (12) to (13): {e J c(^-E ] D.jft(-^ 4- ju(-w,-k') D.j4C-«£') Aw ^ o U c(kVE} D-jt (-k^  + j u*(k,k'> b.j.C-k-) dk' o (A:14) UiCk^-E] D . j , ^ + j Utk.k'lp.^C-k-i dk' = o Thus, provided that the th energy l e v e l of equation (1) i s non-degenerate: = & - £ H ^ ( A : 1 5 ) In the case of the Is l e v e l the non-degeneracy condition i s c l e a r l y s a t i s f i e d , so that: (A:16) APPENDIX B i Properties of the Three-Dimensional Kronecker 6. a. Equation ( 1 1 1 : 7 5 ' ) defines a function 6 ^ k i such that: (B:l) where N i s the number of atoms i n a simple cubic " c r y s t a l " , and the summation i s over a l l l a t t i c e s i t e s of t h i s c r y s t a l . Expanding the scalar product occurring i n the exponential term of (1), and making use of the cubic symmetry of the c r y s t a l being considered: • ^ ' ' - ^ < B : 2 > Now the allowed values of a n are na, where a i s the l a t t i c e A I/. spacing and n runs from 0 to N - 1. Hence: , y> < (B : 3 ) , A - 1 (Svmilarl^ (of ^ and. z.") -76-by the usual rules for summing a geometric series. Now by the periodic boundary condition used to determine the allowed values of K i n the f i r s t B r i l l o u i n zone (Jones, 1960, p. 36), eiKxN'/»a _ (B:4) for each allowed vector K. By d e f i n i t i o n , (k - k*) i s such a vector. Hence: A k K k w ' = e = 1 ( B : D ) Thus i f k / k*: 5 K W'= — \ ]( * - * \\ - o M ^^-1 l U k r k j - i Jl A k , k i - i J (B:6) I f , on the other hand, k = k', then by (1): " TT ? e " = IT = 1 (B:7) Hence, by (6) and (7): ' O i f k * k' Okk' (B:8) 1 i f k = k' This behaviour i s analogous to that of the usual one-dimensional Kronecker b. The function 6j E^j ct may e a s i l y be approximated by a Dirac delta function, as follows: -77-N vi MCI r» c r y s t a l (B:9) where V i s the volume of the c r y s t a l . C l early: (B:10) Hence: V 1 • — — — • t • ( B : l l ) The r e s u l t (11) i s used i n Chapter IV i n the c a l c u l a t i o n of absorption c o e f f i c i e n t s . -78-APPENDIX C Evaluation of the Integral (IV:36). In Chapter IV, the i n t e g r a l : I Coix + co* = ^  0 4 X,^ $ TT (C:l) occurred i n connection with the c a l c u l a t i o n of the ld°-—*ls absorption c o e f f i c i e n t i n the tight-binding approximation. Lines t y p i c a l of those over which the integration must be taken are sketched i n Figure 7. Now, by elementary calculus, Along the l i n e (cos x +• cos y) « 1 , dx and dy are related (C:2) by: S i n x d x (C:3) Therefore, by (2) and (3): (C:4) - 7 9 -Hence the integrand in (1) becomes: _ <Lx ^ S m ^ x + J i n N V I - C ^ - c o s x ^ (C:5) with x varying between 0 and c o s " ^ ^ -1) whenever f i s greater than zero, and between cos""*(^+ 1) and v whenever X i s less than zero. By (5), the int e g r a l in (1) may thus be written: , c o s ' a - 0 J r >/1- ( c : 6 ) tJos-.c^0 V i - C I - a r t » ^ ( C : 6 » ) Changing the variable i n ( 6 ) to: and defining: 2^ 5 the in t e g r a l ( 6 ) becomes: Z (''x 1 -1 \a y ^ C i - ^ L ? +• XCI-XU' (C:9) -80-S i m i l a r l y , (6*) becomes: 2 (* dx (C:9'> where: (C:10) Clea r l y , taking into account the allowed values of \ i n each case, integration of (9) and (9') w i l l lead to a function which i s symmetric about \ » 0. Hence only (9) need be considered i n d e t a i l . In the l i m i t as \ approaches 2, £ becomes large, and (9) may be approximated by: Comparison of the values of $x(i-")0 and LX C I - T O " ] 1 i n the range of X considered indicates that (11) w i l l be a v a l i d approximation for a l l ? greater than 0.5. When X = 0, (9) becomes: ( C : l l ) (C:12) As t h i s i n t e g r a l diverges, the l i m i t i n g behaviour of (9) as % goes to zero must be examined i n more d e t a i l . The i n t e g r a l (9) may be rewritten as: [ 4X l - x d - x ^ n - V x u - x V (C:13) Clearly, for values of X much less than 5 , the expression under the square root In (13) will be dominated by its second term, while for values of X much greater than , the first term will dominate. Thus in the limit of small £ , which corresponds to the limit of small 1 , (13) may be approximated by: - i - ( + ft « } (C:14) z _ s ( . > r r >J i+r$ / T U T J , JTea.-'{i- + - i - W i H i i S. cos- j 1 - 1 3 - 1 4 Neglecting the fir s t term, which goes to two as goes to zero, (14) is seen to go to infinity as: (C:15) -82-As the natural logarithm of 2/3| goes to i n f i n i t y more slowly than any power of 2/3"? , i t i s clear from (15) that the area under the absorption-coefficient curve w i l l remain f i n i t e i n s p i t e of the divergence of (12). Values of the i n t e g r a l (1) are plotted against f i n Figure 6. -83-APPENDIX D The Absorption Coefficient for the Is *>2p0 Transition. By a method completely analogous to that used for the ls^°-—• l s ^ transition, an approximate expression may be derived for the absorption coefficient associated with the l s ( 0 * — • 2 p 0 ( 6 ) transition. Rather than carry out a complicated energy-band cal-culation for the broadening of the 2s-2p impurity band, a tight-binding expression will be assumed for the 2pQ energy, mixing with the other n = 2 states being ignored. The parameters in this tight-binding expression will then be determined from the results of Kohn and Luttinger (1955 c), and of Baltensperger (1953). The function so obtained is: l f c - U [cos K*d + cos K 3 a •+- cos K,a] (D:l) where: 1 —2 E2p *" t h e i s o l a t e d ~ ^ P u r i t y 2 P 0 energy, -1.1 x 10 ev. Df «= 1/6 of the 2p band-width given by Baltensperger . (D:2) In the case of high concentration, E* must be modified 2Po according to Baltensperger's calculations. Corrections to the effective mass theory will be ignored for the 2pQ wave functions, so that: -84-* f c V r T ^ W** * ( D : 3 ) Using (3) and the corrected ground-state wave function (111:86), the integral: x2I° = - i - , ( F z* oCr^ * Fis Cc-S B) dr (D:4) may be calculated. Rough values for this integral are shown in Figure 10. By (1), (4), and (111:74), the optical matrix element of interest is, under the assumption that the six degenerate 2pQ levels broaden independently: \ X ? ° + 2 X ? ° ( t o s K,a + a * k^a + cos K * a ^ ' ^ where: E i p > i l s (!0 = & * e . G O - E\rCK^ (D:6) By (5), (IV:20), and (IV:23), the absorption coefficient for the Is^L»- 2p Q* 6*transition i s : Tf j j j f l U A f l <ti & (.cos*. +• COS £ 4- COS s - "O -85-where: a- UU-*x-H (D:8) and a factor 6 is included in (7) to account for the six-fold degeneracy of the 2pQ band. The expression (7) may be simpli-fied by the procedure which led to equation (IV:25) for the ls*°^ls* 5) case. The result of this simplification is: • ^{"W^ « l * ? - H X t \ I (D.9) where: dS 4 J i n 1 * + S>"n,'«i 4- S i V i l a cosx +<o$^  +• cos i - t (D:10) The integral (10) may be evaluated approximately by the methods of Appendix C. The element of surface area is: <*S = i ± j dx dy C l-Ci^-cosx - c o n j " > * j (D:ll) Hence ^  reduces to: ^ = J D * ) 4 I — (.M-CoSX-COSnO'1 (D:12) Whenever: *\- cos x £ 2. (D:13) -86-the y - i n t e g r a l i n (11) may be evaluated by means of ( C : l l ) : i >/a<n-c*s*V = 7 1 7 ^ )d^=r/5(.-x^-tt-Tt\» (D:14) v/here: * = (D:15) It may e a s i l y be seen that (12) corresponds to r e s t r i c t i o n of x, y, and z to values near the bottom of the band. If ^ i s assumed large, as i t w i l l be near the bottom of the band, the i n t e g r a l i n (14) may be evaluated approximately. The f i n a l r e s u l t i s : * ~ JaR^iT )„ S(<-%V* ~ /a* ^ ( D j l 6 ) The r e s u l t (16) may also be obtained from d i r e c t con-si d e r a t i o n of the int e g r a l i n (7). At the bottom of the band, when x, y, and z are near zero, the cosine terms i n (7) may be expanded i n a Taylor s e r i e s about the o r i g i n : Cos x + cos + cos i ~ 3 - -1 r* Hence the i n t e g r a l becomes: o (T A TI/ . / f i r (D:17) (D:18) - 8 7 -Using th© delta function, (18) becomes: 4= i ^5=7 (D:19) which i s the same as the r e s u l t obtained i n (16). Near the centre of the band, the i n t e g r a l (9) i s d i f f i c u l t to evaluate. Hence, as a f i r s t approximation, the expression obtained f o r the bottom of the band w i l l be assumed to hold for a l l (x,y,z). The expression f o r the absorption co-e f f i c i e n t at the centre of the band then becomes: since I * 0 at the band centre. The absorption cross-section i s then found by multiplying (20) by a 3 to obtain: CT- 2 0 •^^°)f uT,{X»7a-] 4 (D:21) In Figure 11 the absorption cross-section (21) i s plotted as a function of a/a . It may be seen from the graph that (21) i s approximately constant between a - 8a* and i k — 1 5 2 a =» 14a. , with a value of about 3 x 10 cm . This value i s i n quite good agreement with the experimental r e s u l t s of Bichard and G i l e s , who obtain an absorption cross-section —15 2 value between 2 and 4 x 10 cm . The non-constancy of the calculated cross-section beyond -88-a = 14 a: i s probably due mainly to inaccuracies i n the band-width 6U' as read from Baltensperger's graph. A contribution from the natural line-width should be included. The close agreement between the observed and calculated cross-sections i s probably fortuitous, since the use of the expression (16) for x,y, and z away from the o r i g i n i s un-j u s t i f i e d . It i s expected that correction of t h i s error would increase the absorption c o e f f i c i e n t (20). However, t h i s correction should be at least p a r t i a l l y cancelled by taking into account the random d i s t r i b u t i o n of the impurities. -89-BIBLIOGRAPHY Adams, E.N. II. Phys. Rev. 85, 41 (1952). Baltensperger, W. Phil. Mag. 7 , 44, 1355 (1953). Bichard, J. and J.C. Giles. Private communication of unpublished results. Bloch, F. Z. Physik 52, 555 (1928). Conwell, E.M. Phys. Rev. 103, 51 (1956). Dekker, A.J. Solid State Physics, Prentice-Hall, Englewood Cliffs, 1SS7. Dexter, D.L. Phys. Rev. 101, 48 (1956). Dexter, D.L. Solid State Physics 6, 353 (1958). Dexter, R.N., B. Lax, A.F. Kip, and G. Dresselhaus. Phys. Rev. 96, 222 (1954). Dirac, P.A.M. Quantum Mechanics, Fourth Edition, Oxford University Press, London, 1958. Eagles, D.M. J. Phys. Chem. Solids 16, 76 (1960). Eyring, H., J.E. Walter, and G.E. Kimball. Quantum Chemistry, John Wiley and Sons, Inc., New York, 19441 Fletcher, R.C., W.A. Yager, G.L. Pearson, and F.R. Merritt. Phys. Rev. 95, 844 (1954). Fletcher, R.C, W.A. Yager, G.L. Pearson, A.N. Holden, W.T. Read, and F.R. Merritt. Phys. Rev. 94, 1392 (1954). Herman, F. Physica 20, 801 (1954). Herman, F. Phys. Rev. 95, 847 (1955). Heine, V. Group Theory in Quantum Mechanics, Pergamon Press, New York, London, Oxford, Paris, 1960. Jones, H. The Theory of Brillouin Zones and Electronic States  in Crystals, North-Holland Publishing Co., Amsterdam, 1350. Kahn, A.H. Phys. Rev. 97, 1647 (1955). -90-Kohn, W. S o l i d State Physics 5_, 257 (1957). Kohn, W. and J.M. Luttinger. Phys. Rev. 97, 883 (1955). Kohn, W. and J.M. Luttinger. Phys. Rev. 97, 1721 (1955). Kohn, W. and J.M. Luttinger. Phys. Rev. 98, 915 (1955). Koster, G.F. and J.C. Slater. Phys. Rev. 95, 1167 (1954). Koster, G.F. and J.C. Slater. Phys. Rev. 96, 1208 (1954). Lampert, M.A. Phys. Rev. 97, 352 (1955). Lax, M. J. Chem. Phys. 20, 1752 (1952). Lax, M. "Proceedings of the A t l a n t i c City Conference on Photoconductivity, November, 1954", John Wiley and Sons, Inc., New York, p. I l l , 1955. Lax, M. and J.C. P h i l l i p s . Phys. Rev. 110, 41 (1958). Luttinger, J.M. and W. Kohn. Phys. Rev. 97, 869 (1955). Morin, F.J., J.P. Maita, R.G. Shulman, and N.B. Hannay. Phys. Rev. 96, 833(A) (1954). Pauling, L. and E.B. Wilson. Introduction to Quantum  Mechanics. McGraw-Hill, New York, 1935. Reitz, J.R. S o l i d State Physics JL, 1 (1955). Rosen, N. Phys. Rev. 38, 255 (1931). S c h i f f , L.I. Quantum Mechanics. Second Ed i t i o n , McGraw-Hill, New York, 1955. Seitz, F. The Modern Theory of Solids. McGraw-Hill, New York, YMU~. Slater, J.C. Phys. Rev. 87, 807 (1952). Slater, J.C. "Encyclopedia of Physics", ed. S. Flttgge, Springer, B e r l i n , Vol. XIX, p. 1, 1956. Slater, J.C. and G.F. Koster. Phys. Rev. 94, 1498 (1954). Wannier, G.H. Phys. Rev. 52, 191 (1937). Wannier, G.H. Elements of S o l i d State Theory. Cambridge University Press, London and New York, 1959. -91-Wilson, A.H. The Theory of Metals. Second Edition, Cambridge University Press, London and New York, 1953. -92-FI (JURE 1 Corrected Ground-State Wave Function, -93-4 1 1 1 1 r e n e r g y (units o-P - e */xa" ) FIGURE 2 Location of Hydrogen-like Energy Levels. -94-1 x 10"',-5K lo"*| 5 x ld*4-£-5x10 1 x 10 5x lo"5i-Z-5 x »0*4-I x 10 For h igher values of X , see Rosen ( i93l). FIGURE 3 a Values of A-^x) and Ag(x) - 9 5 -O o2 o.4 0 6 0.8 1.0 X FIGURE 3 b Values of Bj(x) and B 3(x). (4.6 x l o ' \ r n J ) ( 2 0 x I0"crr.- 3) ( l o x \o"cm-*) ( 5 8 x l o ' ^ m 1 ) (imparity concentrations in parentheses) FIGURE 4 X? as a Function of Impurity Lattice Spacing. -97-11 1-2 1-3 1-4 l s < 5 > energy (units o f - cYxa* ) FIGURE 5 X* as a Function of the Energy of the l s v ' Level. IS FIGURE 6 The Integral: <J($) = as a Function of [ <tJC J V s i n * * + sin*y Cos K * co$ y « ^ FIGURE 7 Tight-binding Energy Surfaces of (111:58) for cos k Qa - 1 0 0 -FIGURE 8 Comparison of Corrected and Approximate Energy Surfaces. -101-+ a / a « FIGURE 9 The ls*°—>ls* 5^ Absorption C o e f f i c i e n t as a Function of Impurity L a t t i c e Spacing. -103-5"x.o,5l H 1 1 h I * id"* 5 x IO'I 25 KId I K I O - ' I H h H 1 1 1 1 1 1 1 (-& 7 8 9 io i l \Z 13 14 i H FIGURE 11 The Absorption Cross-Section for the l s ( 0 L , 2 p o 0 ) Transition. 

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