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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 p r e s e n t i n g the  t h i s thesis i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t t h e  B r i t i s h Columbia, I agree t h a t the a v a i l a b l e f o r reference  and  study.  University  of  L i b r a r y s h a l l make i t f r e e l y I f u r t h e r agree t h a t  permission  f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may g r a n t e d by  the  Head o f my  I t i s understood t h a t f i n a n c i a l gain  Department o f  s h a l l not  be a l l o w e d w i t h o u t my  Physics  2£ September  representatives.  c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada. Date  Department o r by h i s  be  Columbia,  1961.  written  permission.  ii  ABSTRACT  An i n v e s t i g a t i o n has been made i n t o the p o s s i b i l i t y o f observing o p t i c a l t r a n s i t i o n s  ( i n t h e 100-micron r e g i o n )  between the ground s t a t e of a donor i m p u r i t y In s i l i c o n and the remaining  f i v e s t a t e s o f the  Kohn and L u t t i n g e r .  l l s ^ s e t i n t r o d u c e d by  While such t r a n s i t i o n s a r e f o r b i d d e n i n  the u s u a l e f f e c t i v e mass approximation,  i t i s found  that  a p p l i c a t i o n o f c o r r e c t i o n s t o the e f f e c t i v e - m a s s wave f u n c t i o n s l e a d s t o an enhanced t r a n s i t i o n  probability.  Under- the assumption o f a simple c u b i c l a t t i c e of impurities, the c a l c u l a t e d absorption c o e f f i c i e n t i s of the o r d e r o f 10 cm"* a t an i m p u r i t y c o n c e n t r a t i o n o f 1 x 1 0 1  1 8  cm"* , 3  and f a l l s o f f e x p o n e n t i a l l y w i t h d e c r e a s i n g i m p u r i t y concentration.  An upper l i m i t i s p l a c e d on t h e r e g i o n i n which t h e  t r a n s i t i o n s h o u l d be o b s e r v a b l e by the broadening i m p u r i t y band. 18 than 5 x 10 The  I t i s estimated  o f t h e 2s-2p  that f o r concentrations greater  _<i cm ° t h e t r a n s i t i o n o f i n t e r e s t w i l l be obscured.  c a l c u l a t e d values of the absorption c o e f f i c i e n t are  probably o n l y a c c u r a t e t o within, one,  o r even two, o r d e r s of  magnitude, because o f t h e approximations  involved.  However,  t h e r e would appear t o be no f i r m t h e o r e t i c a l reason why t h e ls >-^ls< > ( 0  5  t r a n s i t i o n s h o u l d not be observed.  iii  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 CHAPTER I  i i i  Intr6duction  CHAPTER II  1  The Theory of an Isolated.Donor in Silicon.  Impurity 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.  2. Integrals involved in the secular equation.  27  3. Evaluation of integrals.  30  4. Evaluation of matrix elements.  33  5. The secular equation:  35  Solution.  6. Energy-band broadening. C.  22  Optical Matrix Elements. 1. Simplification of the matrix elements. 2. Evaluation of integrals in the sphericalpotential approximation.  38 39 39 41  3. Corrections to the spherical-potential approximation.  42  4. Evaluation of integrals i n 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.  56  Specialization to a r i g i d lattice.  3. The coefficient of absorption. 4. Specialization to centres i n a dielectric medium*  60  Evaluation of the Absorption Coefficient.  62  Simplifications and approximations.  62  2. The absorption coefficient for the ls<°>—*ls< > transition.  64  B. 1.  5  C.  58  Discussion of Results.  69  1.  Is—*2p  69  2.  ls(°>-*!s< ) transition.  70  APPENDIX A  Properties of the Coefficients D ^ C k ) .  72  APPENDIX B  Properties of the Three-Dimensional  0  transition. 5  Kronecker 6.  75  APPENDIX C  Evaluation of the Integral (IV:36).  78  APPENDIX D  The Absorption Coefficient for the Is —*2p Transition. 0  BIBLIOGRAPHY  83 89  vi  LIST OF ILLUSTRATIONS  FIGURE 1  C o r r e c t e d Ground-State Wave F u n c t i o n .  92  FIGURE 2  L o c a t i o n o f Hydrogen-like  93  FIGURE 3a  Values of A ( x ) and Ag(x).  94  FIGURE 3b  Values o f BjCx) and B ( x ) .  95  1  3  FIGURE 4 FIGURE 5  FIGURE 6  as a F u n c t i o n of Impurity  Lattice  Spacing.  X as a F u n c t i o n o f the Energy o f the I s Level.  9S  2  The  Integral:  Function of FIGURE 7  Energy L e v e l s .  C04 % + c o i y  a ^  97  as a  \.  98  T i g h t - b i n d i n g Energy S u r f a c e s o f (111:58) f o r cos k a = 0. Q  99  FIGURE 8  Comparison o f C o r r e c t e d and Approximate Energy S u r f a c e s .  100  FIGURE 9  The Is ( 0 ) »ls (5) A b s o r p t i o n C o e f f i c i e n t as a F u n c t i o n of Impurity L a t t i c e Spacing.  101  FIGURE 10  X  FIGURE 11  The  a ° P  ls*  a  s  a  F  u  n  c  t  i  o  n  o  f  Impurity  Lattice  Spacing.  102  A b s o r p t i o n C r o s s - S e c t i o n f o r the 0  )  —*2p^  Transition.  103  CHAPTER I  Introduction  The quantum-mechanical theory of crystalline solids depends, to a large extent, upon the consequences of the periodic arrangement of atoms In a crystal.  It i s 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 r i g i d lattice of identical nuclei. one-electron approximation (Reitz,  1955), the only forces  influencing the electron motion are those action with the nuclei.  In this adiabatic,  due to Coulomb inter-  The corresponding one-electron  Schxoedinger equation i s :  where V (r) i s a potential function with the same periodicity 2rr,  as the crystal lattice, h i s Planck's constant divided by and m i s the mass of an electron. 1928;  Wilson, 1953,  p. 21)  The Bloch theorem (Bloch,  extended to three dimensions shows  that the eigensolutions of equation (1) cp„_Cr) = e  i k r  have the form;  u„_(c)  (  where u ^(r) has the same periodicity as V ( r ) . n  p  may he shown (Wannier, 1959,  Chapter 5)  i s a many-valued function £ (_£)  o f  n  t n e  I  :  2  )  From this i t  that the eigenvalue  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 f i r s t Brillouin zone (Jones, 1960, p. 37). The foregoing remarks apply to a completely periodic array of atoms.  If this periodicity i s disturbed slightly in  some way, equation (1) must be replaced by:  m - [K. + U H  - E+  (l!3)  where U represents a perturbing potential due to the departure from s t r i c t 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 w i l l 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 i s that in which one (or more) of the atoms in the lattice i s replaced by an atom of a different type, referred to as a "substitutional impurity".  Insofar as i t s atomic number permits, the impurity  atom w i l l 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 f i e l d requirements.  -3-  nucleus has a smaller charge than the o r i g i n a l nucleus, there w i l l be a deficiency of electrons i n the l a t t i c e due to a number of u n f i l l e d bonds;  while i f the impurity nucleus  a greater charge, there w i l l be an electron excess.  has  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 i n 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. type i s said to be "monovalent".)  (An impurity of t h i s  The nature of the i n t e r -  -  ;  action of the impurity nucleus with the extra electron w i l l depend on whether or not i t i s energetically favourable f o r the electron to occupy an o r b i t close to the nucleus.  If such  an o r b i t i s favoured, what i s known as a "deep" impurity state i s produced, and the electron-nucleus i n t e r a c t i o n i s highly complicated.  However, i n the case of a "shallow" impurity  state, characterized by an o r b i t of large dimensions, only the excess charge on the impurity nucleus w i l l be of importance i n the i n t e r a c t i o n , and the impurity system w i l l resemble a hydrogen-like atom imbedded i n the l a t t i c e . If i t i s assumed that the impurity i n question i s i n a shallow state, the hydrogen-atom analogy mentioned in the l a s t paragraph may be used to give a p a r t i c u l a r form to the perturbation U:  U(C)  s  -  "KF  <I:4)  * A l l numerical r e s u l t s i n the present thesis w i l l apply to t h i s p a r t i c u l a r case.  -4-  (e i s the electronic charge, and K i s the static dielectric constant of the host lattice).  The use of the static  dielectric constant i s justified by the fact that the orbital frequency of the extra electron about the impurity nucleus i s 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 w i l l therefore s t i l l be valid.  It i s this polarization which gives rise to  the static dielectric constant. There i s 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 w i l l have a symmetry determined by the physical arrangement of the atoms surrounding the impurity nucleus.  In  the case of phosphorus in s i l i c o n , 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 a l . , 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  D ""(_) « U C t )  (1:6)  C  m J  (5) may be replaced by the equation: {£-(_)-E}D°"\tO  +  I  U_'D '' (_')<m_|U . .(t)|m'_> <n  (  ,  T  s  I s 7  )  = o The Fourier transform of equation ( 7 ) i s then: [ - - ( - I V ) - E]  +  F \rJ lm  J * * D " ' (_)e <mk|U . .Cr)|m'l<'> c  n  ,  li5r  T  s  =  O  (1:8)  where F ^ C r ) i s 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 i s : + 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  i s 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 solutions of ( 5 ) .  D^Qs), and hence the  Clearly any extra degeneracies caused by  the use of an incorrect potential would be carried over into this case as well.  -6-  U n f o r t u n a t e l y , because the exact form of U not known, the above method of s o l v i n g equation practicable.  T  g_(r) i s ( 5 ) i s not  I t i s necessary t o s o l v e the problem u s i n g 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.  C o r r e c t i o n s t o t h i s approximation may then be s t u d i e d by the i n t r o d u c t i o n of 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 , o r by some o t h e r l e s s exact  procedure.  I t has been shown (Koster and S l a t e r , 1954 b; 1952) t h a t t h e energy e i g e n v a l u e s o f equation to  those of e q u a t i o n  energy  Slater,  (3) a r e s i m i l a r  ( 1 ) , w i t h the e x c e p t i o n t h a t allowed  l e v e l s may now occur i n the " f o r b i d d e n " r e g i o n s between  the bands.  F o r a s h a l l o w i m p u r i t y s t a t e , these  l e v e l s a r e c l o s e t o the parent band;  "split-off"  hence i t i s t o be  expected t h a t the wave f u n c t i o n s c o r r e s p o n d i n g t o these  levels  are c l o s e l y r e l a t e d t o the B l o c h f u n c t i o n s a t the band edge. I f i t i s assumed t h a t the band i n q u e s t i o n i s of s t a n d a r d  form  (Wilson, 1953, p. 4 2 ) , w i t h i t s minimum i n k-space a t k =» 0, then i t may be shown (Kohn, 1957, S e c t i o n 5a) t h a t the e q u a t i o n : {jr.-  £ 1 +  •  E +  (  I  :  i o )  has s o l u t i o n s :  V \& m  ~- F'"*(£> <p (_) mj0  (  _.  n  )  where F * ( r ) i s a g a i n t h e F o u r i e r t r a n s f o r m o f the c o e f f i c i e n t < m  D ( k ) d e f i n e d by ( 6 ) . F ""*<_) i s a s o l u t i o n o f the e q u a t i o n : w  U^C-tv) - ^ ]  F^t)  *  EF'-'C-}  (1:12)  -7-  and t h e form o f E ^ - i v ) i s , t o second o r d e r :  'E-M.).  (1:13)  ( £ ?* i s t h e energy a t the bottom o f the band, and m^., my, c  and m_. a r e c o n s t a n t s w i t h t h e dimension o f mass).  E q u a t i o n (13)  may be s o l v e d approximately by s e t t i n g m^, = my « ra «= m . In z  t h i s case the f u n c t i o n s F "°(r_) a r e s i m p l y m o d i f i e d c  hydrogen  wave-functions, and t h e allowed v a l u e s of (E - E'^ ) form a 0  hydrogen-like In  spectrum.  t h e c o n s i d e r a t i o n o f energy l e v e l s s p l i t o f f from the  c o n d u c t i o n band of s i l i c o n ,  allowance must be made f o r t h e  f a c t that the band i s not o f s t a n d a r d form, but has s i x e q u i v a l e n t minima, at ( £ k , 0 , 0 ) , ( 0 , ± k , 0 ) , and ( 0 , 0 J k ) i n the o  o  0  f i r s t B r i l l o u i n zone (Herman, 1954 and 1955). estimate of k  A tentative  has been made by Kohn (1957, S e c t i o n 7 c ) , who  Q  g i v e s a v a l u e o f 0.7 k j j ^ .  (^max  i  s  t t i e  magnitude of k a t t h e  zone boundary i n one of t h e s i x a x i a l d i r e c t i o n s ) . conduction-band  energy near one o f these minima i s g i v e n by  an e x p r e s s i o n l i k e :  MhO  =  (Kohn and L u t t i n g e r , 1955a)  £o + ^  (k.-O*  + ^  o  (1:14)  U J+k*)  u s i n g the minimum a t (k ,0,0) as an example. masses m^ and  A l s o , the  The e f f e c t i v e  a r e g i v e n by (R.N. Dexter, e t a l . , m= l  m  t  1954):  0.98 m « Cl9m  <  I : 1 5  >  In t h e approximation o f a s p h e r i c a l l y symmetric p e r t u r b i n g  -8-  p o t e n t i a l j t h e s o l u t i o n o f t h e many-minimum problem f o l l o w s a l o n g r o u g h l y t h e same l i n e s as t h a t f o r a s i n g l e minimum.* The p r o b l e m i s f i r s t s o l v e d f o r a s i n g l e minimum a t one o f the  s i x e q 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  t h a t by v i r t u e o f the s p h e r i c a l symmetry o f t h e 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 o b t a i n e d i f any one o f t h e o t h e r f i v e minima had been used.  I t therefore  f o l l o w s t h a t i f a l l s i x minima a r e p r e s e n t a t t h e same t i m e , the  single-minimum h y d r o g e n - l i k e l e v e l s w i l l each a c q u i r e 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 to  t h e h a m i l t o n i a n of e q u a t i o n ( 1 0 ) , i n t h e c a s e o f many minima,  s t a t i o n a r y p e r t u r b a t i o n t h e o r y ( S c h i f f , 1955, p. 155) shows t h a t t h e z e r o - o r d e r wave f u n c t i o n s f o r t h e s i x degenerate s t a t e s b e l o n g i n g t o a p a r t i c u l a r energy l e v e l a r e g i v e n 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 t h e o r y ( H e i n e , 1960, p. 107) t h e n  i n d i c a t e s what t h e c o r r e c t l i n e a r c o m b i n a t i o n s a r e . of  The  purpose  f o r m i n g t h e s e c o m b i n a t i o n s i s t o e l i m i n a t e a l l non-zero  elements o f t h e p e r t u r b a t i o n between d e g e n e r a t e 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)  t h 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 o n l y p a r t i a l l y remove t h e g r o u n d - s t a t e degeneracy caused by t h e s p h e r i c a l p o t e n t i a l approximation —  t h e 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 t w o - f o l d d e g e n e r a t e l e v e l , * See C h a p t e r I I .  and  -9-  a three-fold degenerate l e v e l .  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 l e v e l s i s the one which i s non-degenerate.  As s p l i t t i n g  may occur, then, i t should be p r o f i t a b l e to examine the p o s s i b i l i t y of radiation-induced  t r a n s i t i o n s between the  ground state and the remaining f i v e Is states. It w i l l be shown that i f the effective-mass wave functions derived on the basis of a s p h e r i c a l l y symmetric  perturbation  are used, the o p t i c a l matrix elements f o r Is-Is t r a n s i t i o n s are very small compared with those f o r t r a n s i t i o n s between other pairs of l e v e l s .  Furthermore, they remain small even  after a correction has been applied to the non-degenerate ground state wave function. phosphorus impurity  Consequently, i n the case of an i s o l a t e d i n s i l i c o n , i t should not be f e a s i b l e to  observe the f i n e structure of the (is] states experimentally. If the impurity concentration  i s increased,  however, the  s e l e c t i o n r u l e governing the t r a n s i t i o n s of interest breaks down.  It may now be possible f o r an electron i n the non-  degenerate ground state on one impurity  atom to make the  t r a n s i t i o n to an excited Is state on another  such atom.  Unfortunately, treatment of t h i s many-impurity problem i s complicated by the random d i s t r i b u t i o n of impurities.  It i s  possible to approximate the actual s i t u a t i o n , however, by assuming that the impurity atoms form a regular l a t t i c e which i s superimposed on the l a t t i c e of the host c r y s t a l .  In t h i s  approximation the problem reduces to that of " s o l i d 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 m a t r i x elements f o r the many-impurity case  a l s o t u r n out t o be very s m a l l i f the s p h e r i c a l - p o t e n t i a l e f f e c t i v e - m a s s wave f u n c t i o n s a r e used.  However, u s i n g  c o r r e c t i o n s t o the e f f e c t i v e mass theory based on those i n t r o d u c e d by Kohn and L u t t i n g e r (1955 a ) , i t w i l l be shown that the matrix elements may be o f a s i z e which w i l l  permit  o b s e r v a t i o n of the t r a n s i t i o n s .  The a c t u a l s t a t e of a f f a i r s  depends upon the a p p r o p r i a t e n e s s  of the c o r r e c t e d wave  f u n c t i o n s used.  I t i s probable  t h a t the  Kohn-Luttinger  approach leads t o r e s u l t s which a r e o n l y good t o w i t h i n a f a c t o r o f two o r t h r e e .  However, there would appear t o be  no f i r m t h e o r e t i c a l reason why the f i n e s t r u c t u r e o f the ( i s ] s t a t e s should not be observed.  * See Chapter I I I .  CHAPTER I I  The Theory of an I s o l a t e d Donor Impurity  in Silicon.  A. E f f e c t i v e Mass Theorem f o r an I s o l a t e d Impurity. Consider t h e problem of an i s 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 .  L e t the Schroedinger  e q u a t i o n f o r t h e pure s i l i c o n l a t t i c e be:  i  ft (p _(r) 0  =  m  (11:1)  U(_)<P«_(e)  where:  A*  •  -  _ £  V  *  +  and V ( r ) i s the p e r i o d i c c r y s t a l AT  V  ( H : 2 )  ^  potential.  By the B l o c h  ' ' "  Theorem, the e i g e n s o l u t i o n s of (1) have the form: f - k ^  «  e  u„_(c)  (11:3)  u _ ( _ l = u^wCr+RO  (11:4)  i b r  where: ,  m  for  any v e c t o r R_ of the pure s i l i c o n  satisfy  lattice.  I f the u_-_(r)  the n o r m a l i z a t i o n :  ( |u-,K(_)i* d.c .) • (An IT CCll  = SL.  where Q i s t h e volume of a u n i t c e l l of the s i l i c o n  (11:5) lattice,  then the B l o c h f u n c t i o n s <P__(r) may be shown t o form a complete  -12-  orthonormal set normalized over the whole crystal. If the effect of introducing an impurity atom into the pure crystal i s considered as a small perturbation, the Schroedinger equation for the isolated-impurity problem may be written: flfCt)  = E+Cc)  (11:6)  - Uo+Vin)  (11:7)  where: il  and U(r) i s the perturbing potential due to the extra charge on the impurity nucleus.*  If the extra electron i s assumed  to move in a spherically symmetric potential, then at large distances from the Impurity nucleus: UCO  S  (11:8)  where e i s the electronic charge, and K i s the static dielectric constant of s i l i c o n . 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 i s over a l l the energy bands of equation ( 1 ) *  For the purposes of this derivation, i t w i l l be assumed that shallow-state theory i s applicable.  •*  See Chapter I for a more detailed discussion.  -13  and the integration i s over the f i r s t Brillouin zone of the s i l i c o n lattice.  For simplicity i t w i l l be assumed that the  wave functions belonging to energy levels s p l i t off from the conduction band of the unperturbed lattice, under a small perturbation U(r), may be written i n terms of conduction-band Bloch functions alone. *(_)  3  Hence:  (dk  (11:10)  D ( _ ) cf>c_(r)  where the integration i s again over the f i 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£ (U>E]D(k) t  + {*_'  U(k,_)&(_')  =  (11:11)  O  where: Itfk.k') = \dr cp J(_) U(r) <(>_'(r) = <cklUC_)|ck'> e  c  (11:12)  Now the conduction band of s i l i c o n has six equivalent minima in the f i r s t Brillouin zone, at (±k ,0,0), (0>k ,0), and (0,0jk ). Q  Q  Q  If i t i s 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 i s taken, in the main, from Luttinger and Kohn, 1955, Appendix A.  -14.  symmetry of the unperturbed l a t t i c e , and the summation i s over the s i x conduction-band minima.  Stationary perturbation theory  (Schiff, 1955, p. 155) shows that there are s i x allowed l i n e a r combinations of the form (13); they w i l l be distinguished by a superscript i : D'Ck) =  _,*) Dj(k) j  (11:14)  Following Kohn and Luttinger (1955 c) the c o e f f i c i e n t s 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.  (11:15)  0,0)  J L ( o , o , 1,-1, 0 , 0 ) _L. ( 0 , 0 . 0 , 0 ,  l.-O  In the s p h e r i c a l - p o t e n t i a l approximation, these s i x states a l l have the same energy. For the (ts^ set, the c o e f f i c i e n t s Dj(k) of (14) may be taken to s a t i s f y : {£i(k)-E}Dj(_) + Sdk'U(_,k')Dj(k') where  =  o  (11:16)  £*(_) i s a second-order expansion about k j of the  conduction-band energy t ( k ) : c  £ (k) = J  c  £0 + Z  l i / i Ck -k X Vy-k^) x  ix  (11:17)  -15-  Here: j  _  9 £c(k) z  £o =» energy a t the conduction-band  minimum.  L u t t i n g e r and Kohn (1955, Appendix A) show t h a t p r o v i d e d U(r) i s a "gentle" p o t e n t i a l , UCk.k') so t h a t  _  UCOe ^'-^":  U(k-_).  1  B  (11:18)  (16) becomes: -E] DJCO  Then, m u l t i p l y i n g  +  f * - ' XU.-fc'} Dj(k')  (19) by e ^-*"—3^— 1  =  O  (11:19)  and i n t e g r a t i n g 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 ^ " ^ - U ' D : ( k - ) fd_'UC,.)e Ctnr j i j r  +  ilk  J  Hence t o the approximation  >^'  =  o  ( I I : 2 0  >  that:  J <Ak e--='> * F»**St 6r'illo_>'«  =  i ( r  (2n^  3  &(r- 0 r  Zone  it  t l  follows  that:  \\^^tT^  + ^  ) i ^ F  (  =  F  i  (  In t h i s case, the f u n c t i o n F j ( r ) must be d e f i n e d  Pitt) =  \  e  l  t  ^ -  r  D_(_)  ^  (ii 21) :  as:  (n.22)  _> BrillovuVi Zone T  * For complete accuracy, t h i s i n t e g r a t i o n s h o u l d be taken over a l l k-space.  16-  The required normalization of the Fj(r) i s : fdrlFjWl*  =  (2T,)  (II:22«)  3  where the integration i s 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') j d r c ^ J c c ) s  cp -6f) ck  , Jdk I POO I*  while: |<*rlFC )| - (dkjdk< D*(k)DCk') j d r e C  l  = jdk (in) (DC!.)! 3  1  =  l ( k  --^  E  Un) jdr J  e  1  ^ * .  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).  -17-  B. Optical Matrix Elements, In the dipole approximation, the matrix elements of interest are: (11:24)  C L J M P J L ' O  where: |U>  s  (11:25)  +;(_)  It may be shown that: pp. =  <r-component  of momentum  =  - ~  ft]  (11:26)  # = # + U(c)  where:  c  Hence the matrix elements ( 2 4 ) become: l  «  *  '  (11:27)  J2- ( E J ' - E i ) < i J t l * c | l ' i ' >  Now by equation (10): <Ulx„|i'£'> =  <cklx-lc_'>  =  £,*i*ocy  [dLk [d_*D„(k)Dj^(!<*)<c_l^lck'>  [dr *, e a * ~ b V c u*_( C )u e _.(_^  * See Chapter IV.  (11:26)  -18-  ut_ ~ £  has the periodicity of the unperturbed l a t t i c e .  so that: c k  TkT  "  £  s  (11:30)  f c V  where the summation i s over the entire unperturbed reciprocal lattice, and:  Substituting (30) into (29):  =  6(_'-k^ t 3kL  I  __B'jtJ_.(11:32) tr * t < r  Now k and k' both l i e i n the f i r 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) i s for K_ =• 0, so that: *(_.'-_> - ^ B ^ i C f c ' - k ^  <ckla.Uk'> -  ( I I ; 3 3 )  Now: w  =  ~k ( * ^ d  uxit  cell  ^7  (11:34)  Adams (1952) shows that i f the phases of the Ugj^r) are properly chosen, the integral in (34) vanishes.  It w i l l be assumed that  -19-  t h i s c h o i c e of phase has been made. KL  -  Hence:  0  (11:35)  Thus (33) becomes: «• o Kg-  T  »R. l S J > ' . T  J  i e  ^,  i,_  ( I l l 3 e  ,  where (11:37) Substituting  (36) i n t o ( 2 8 ) :  -  _ V ^ - J ' - S - < J * . « , U V >  <  I  I  :  3  8  )  where: \n>  *  v a . * *  (  I  I  :  3  9  )  The m a t r i x elements <jil x, lj'£'> may be w r i t t e n :  <ill*clj O f  For  =  j^tfcrt^Crte^'^dc  (11:40)  t h e t r a n s i t i o n t o be c o n s i d e r e d , from the I s ground  (characterized  state  by a^°h t o another I s s t a t e , t h e s p h e r i c a l -  p o t e n t i a l approximations t o both F j ^ ( r ) and F j t ^ i C r ) a r e even f u n c t i o n s o f x, y, and z.  In t h i s case (40) reduces t o :  -20  <j/l*<rljr> = <•  (11:41)  Fj^Ct) Fj'isCc) sin (kj'-hjVE d£  the real part of the expression having vanished because i t s integrand i s an odd function of x.  Clearly i f j j * , the 08  matrix element (41) i s identically zero.  If j / j , the matrix f  element i s not zero, but as the Fj__Cr) are slowly-varying compared to the lattice spacing (Luttinger and Kohn, 1955, Section III), i t i s reasonable to assume that the presence of an oscillating term in the Integrand w i l l 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  (11:42)  «^F -_(c)F . (E) sin kj-r cosfe r J  i  u  r  and (11:43)  Jdr % F * ( t ) Fj' (_) sinRj'-r »$ kj.r r  l$  u  I* J - J', both integrals are of the order of 5 x 10~ a*, 4  where a* i s the approximate "Bohr radius" for the F ( r ) . It l s  is to be expected that when j ^ j ' these integrals w i l l 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 i s the value of the matrix element <ji.l-x^-l j  1'>  for the  ls-*2p  0  transition when spherical-  potential wave functions are used. * See (111:37) for examples which bear out this contention.  -21-  In Chapter I I I , c o r r e c t i o n s t o t h e s p h e r i c a l - p o t e n t i a l approximation w i l l be considered.  A n t i c i p a t i n g the r e s u l t s ,  i t may be s t a t e d here t h a t , r e p l a c i n g the ground s t a t e wave f u n c t i o n by i t s c o r r e c t e d v a l u e and u s i n g the e s t i m a t i o n —3  *  procedure of the l a s t paragraph, v a l u e s o f l e s s than 1 x 10 " a * are o b t a i n e d f o r (41).  As the Is  2p  Q  m a t r i x element i s  decreased by o n l y a f a c t o r o f two when the c o r r e c t e d  functions  are used, the I s - * - I s m a t r i x elements a r e again n e g l i g i b l e by comparison. of the  I t i s t h e r e f o r e u n l i k e l y t h a t the f i n e s t r u c t u r e  { i s ] s t a t e s w i l l be e x p e r i m e n t a l l y  the i m p u r i t i e s a r e t o o f a r apart  o b s e r v a b l e when  to interact.  -22-  CHAPTER I I I  The Theory of Donor I m p u r i t i e s i n S i l i c o n f o r F i n i t e Impurity  Concentrations.  A. I n t r o d u c t i o n . The case i n which there a r e many i m p u r i t i e s present i n a crystal lattice i s difficult  t o t r e a t because o f the random  nature of the i m p u r i t y d i s t r i b u t i o n .  I f , however, the  i m p u r i t i e s a r e assumed t o l i e on a r e g u l a r l a t t i c e w i t h i n 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 a t t a c k e d by means o f the u s u a l approximations o f s o l i d s t a t e p h y s i c s .  For  s i m p l i c i t y , t h e r e f o r e , i t w i l l be assumed t h a t the i m p u r i t i e s l i e on a simple  c u b i c l a t t i c e , so t h a t the problem of i m p u r i t y -  l e v e l broadening may be t r e a t e d i n the B l o c h scheme.  B. T i g h t - B i n d i n g  (LCAO) Approximation.  1. The s e c u l a r e q u a t i o n : The Schroedinger  Derivation.  equation  f o r the many-impurity problem  is:  K* (_) s  *  =  E (_)*_(_)  <  I I I : 1  >  F o r the treatment of a one-dimensional l a t t i c e c o n t a i n i n g random i m p u r i t i e s , see Lax and P h i l l i p s (1958).  ** Conwell (1956) d i s c u s s e s the i m p l i c a t i o n s of the assumption of a r e g u l a r i m p u r i t y l a t t i c e . See a l s o B a l t e n s p e r g e r (1953).  -23-  where: K  and V  •  « . +  Y  r  t o  {  ,  m  a  )  ( r ) i s a p e r i o d i c p o t e n t i a l w i t h t h e p e r i o d i c i t y o f the Mr  impurity l a t t i c e .  F o l l o w i n g S l a t e r and K o s t e r  (1954)j  V (^ 5 H U(r:-a )  (111:3)  ,  P  5  _  where the summation i s over a l l s i t e s <* o f the i m p u r i t y n  lattice. In the LCAO approximation,  the wave f u n c t i o n s "f_0_)  may  be w r i t t e n :  * *  W  =  ^  -t  d  * 2  +* ^ C c - a J  <IXI:4)  where N i s the number o f i m p u r i t i e s p r e s e n t , and the summation over I and i i n c l u d e s c o n t r i b u t i o n s from a l l e i g e n s t a t e s o f the i s o l a t e d - i m p u r i t y problem.  m  The energy:  )  (  I  I  I  :  5  )  where: |K> =  ' must then be minimized d_£.  (111:5*)  by the proper c h o i c e o f the c o e f f i c i e n t s  Hence:  ddct'  =  — 7 — \4T, < - l ^ ' - > <i_l_> 13<*iy  Now by e q u a t i o n  -  - J  4-,<_l!s>] adi'e'  =  o  , (111:6) T  T  T  (4) t h e m a t r i x elements i n (6) a r e :  <-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'  =  a,a'  e,i  N  Z  e^-fts <ijUOTfl_>  Z  (111:10) ( s h i f t i n g the o r i g i n t o a  n  and d e f i n i n g a  m  « <i .-a ). n  n  Similarly:  J-  <K\H\K'>  Substituting Zd<;{ Z e  1  =  Z  dj Z  e'*-** <£jM «li'i' .> s  (111:11)  (10) and (11) i n t o (6) l e a d s t o the e q u a t i o n :  ^ <iil«U'l  , f t B  ,>  - E C K ^ Z e - < i t ) |i'ra >"i i 5  3 B  =  = f o r each p a i r o f v a l u e s ( i ' , jL ' ) .  (HI 12) :  o  Hence the s e c u l a r determinant  must vanish, g i v i n g :  J Now  by (2) and ( 3 ) :  (111:13)  -25-  n *_  + X <UI OCr-i_U'^'a_>  Ei'.<ilU'i a*.> * ,  (111:14)  Then, substituting (14) into (13): \ Z e  J  - -  Z  B  <iilO(r-ft^U'jfa„>  (111:15)  Using the expression (11:23) for U£s_>t <LlU'£'a_> = X <<*«j/[clkUk' jj'  DjCk^  1  Dj'^'dc') <c_lc_'_-,>  -- ^.ai**!' J<*!ij*- t>J(_^ 1 > J V ^  - TTTJ C  e"  i ! s  (^(dk'Dj^^DjVf-') Ji  '  J  '-  a  oC_-k")  <_lk*g > =  (111:16)  J  where:  (111:17)  Similarly:  =  "Z  UfeU-' ^ ( ^ D ( k ' ) jV  e  l t ,  "*  B  <ckluc -a_Mck'> C  <IH:18)  Now Luttinger and Kohn (1955, Appendix A) have shown that under the assumption that U(r) i s a "gentle*' potential, < c k | U C ^ l  -  c -k " >  J L _  IXnY  < k l U C » « U k ' > .  -  -  -  (111:19)  -26-  Clearly U(r - a ) also satisfies the requirement of gentleness, n  so that (19) may be extended to give: < c k l U ( E - a ) | ck'>  S  a  <V< I U(c-a Mk'>  (111:20)  s  Hence, substituting (20) into (18)<eel UCc-anU i i ' a > ,  2!  * _ i _ s ^ o ( i * « i ' j d k j d k ' D/J(^Dji'(Js') <wiucr:-g ,>ik'a ,> !  ( . i) m  s  2  Then, making the substitution:  f^t)  *  f*fcD (0e J I  U ! l  - « |  )  c  (111:22)  in (16) and ( 2 1 ) , the following expressions are obtained: < i l U'Jt'a >  =  a  J L - ,  <i£lUCc-fl-)|i iJ flB> ,  ,  =  2j«S*«J'' <i*\*'*'&*>  (111:23)  ^SVi' <J*lU(c-a MjTa»>  TJTO  (111:24)  a  where: .jla >B  F  j  t  (  C  -  (111:25)  ^  Finally, substituting (23) and (24) into ( 1 5 ) , the secular equation becomes:  i'.i* L  a  *  (111:26)  b  =  o  For the sake of simplicity, the range of t in (26) i s 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 c o n s i d e r a t i o n , i . w i l l be taken as Is o n l y . f o l d degenerate* and considered For  separately  The  five-  the non-degenerate Is bands w i l l then i n the l i g h t of equation  the f i v e - f o l d degenerate Is band, the  be  (26). secular  equation (26) i s :  ^  (111:27)  w h i l e f o r the non-degenerate Is band: i v i Z c ^ - a -  Il«J*4<jisl  ,  2  0Cc-a.)lj'i$ »> f t  ^  -  (111:28)  u-  x  <J i*> J'IS  z « ; v  —  j j  Equations (27) 2.  and  (28) must now  Integrals involved I t may  be shown  have the p r o p e r t y  be  i n the s e c u l a r  >  TO  o  solved. equation.  t h a t the c o e f f i c i e n t s D j ^ k )  i n (11:23)  that:  t>j i»CI^ =  D./ W 1S  -  (111:29)  In c o n s i d e r i n g the exact s o l u t i o n s of the i s o l a t e d - i m p u r i t y problem, Kohn and L u t t i n g e r (1955 c) note t h a t a l l Is s t a t e s other than the non-degenerate ground s t a t e w i l l have r o u g h l y the same energy. Thus, i n s p i t e of the f a c t t h a t the exact s o l u t i o n w i l l i n v o l v e s e p a r a t e two-fold and t h r e e - f o l d d e g e n e r a c i e s , 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 a l s o Chapter I of t h i s t h e s i s ) . See Appendix A. The s u b s c r i p t " - j " i n (29) denotes the c o e f f i c i e n t a s s o c i a t e d w i t h the conduction-band minimum at -k< .  28-  S u b s t i t u t i o n of (29) i n t o  (22) leads immediately  that F j ( r ) i s the complex conjugate l s  phases o f the F ^ C r )  of F _ j ( r ) . l s  F  (r*)e  j u  Hence i f the  a r e so chosen t h a t Fj^(O) i s r e a l and  p o s i t i v e f o r a l l j , i t f o l l o w s that F j equal.  t o the r e s u l t  ( r ) and F „ j i ( r )  l s  we  s  Therefore: i ! y  -  +F-  j l s  (r) e" l  j r  =  Ij U > + | - j l s >  = 2 F J ( E ) cos k U  i  r  (111:30) F (r*)e ilfc  Jlt  r  -F- (r*)e-  i!5i,r  jl4  lju>  =  - l - J l s > = 2iF i u (cn.v. kj-c  Then, d e f i n i n g : , i  --  >  Cr*T * "J* ^  =  v  0  , < j1  * = a  (111:31)  >  and u s i n g t h e v a l u e s of ctj g i v e n i n (11:15), | F , cosk, .(r-a-?") + F , cos k j ( r - a ^  |0,a ,>  =  11, O,M>  = — z C2TT) *  \ F, cos L  = jj^jfa  |^F,cos k.-Cc-fte*) -  =  B  V  U , « H >  k,C*:-a ') 3  i t follows that: + F c o s k CC-SH,)} s  s  ^CoskjCr-a.„)] J F cos 5  kj.Cc-aw^  (111:32)  I4 a > >  = ?  = _ i _  (Ztn * v  &\  FjSiV, «.  k .(c-a»^ 5  J  where the argument o f F^ i s ( r - a ) i n each case. m  The  integrals involved i n evaluating  < i I i'^m^ a r e :  -  2  9  -  gjjy JF/Cs*»Fi'(C.-*a"* <** j-r c o s ky(c-aa) A r k  J j j ^ I F*(t) Fj'CC-So*) Sinker Cos Wy.(r-So) d r  jJJ-y | Fj*Cr*)F'(£-a,l sin K»-r s i n kj.(r-S^ An ,  i  =  - «i'* Iy= " j»J' If!'  (111:33)  T  B  a  where: =  c o s i&j'. a  3  (111:34) (if*  and:  | Fj"  ?'  [p*(C*l (aw) ! 1  J Fj*C;"l  i  T !»'  1 (in*)  Similarly, are  Fj'(£-W) Cos k j . r C o s k j ' r Fj'fe-Sm*) c o s k j . r  Sin kj/.r d r  F -'(c-o *j SiV» k , . r c o s k j ' j : i  (111:35)  4r  B  {FfCc) Fj'Cc-Oa.*) S i n k j - r S i n Isj'-rT d r 1  the i n t e g r a l s  involved i n evaluating  combinations of i n t e g r a l s  Ujj»mn>  u  jj*mn>  <x I U(r-a ^| Va** "> a  u  / j » m n»  u  L'mn  of the form: =  (F/CJ!*) Fj/fc-Sm) UC£-ft c ) C o s k j . * :  coskj'r.  dr (111:36)  etc.  -30-  3. E v a l u a t i o n of i n t e g r a l s . The i n t e g r a l s of (35) and  (36) may  be shown t o have the  following  approximate v a l u e s (using t h e s p h e r i c a l l y  potential  (1:4) as an approximation t o U ( r ) ) :  f Tr  iio  l  - ± . ± 4  2.  symmetric  Cl +• *<.*8»)»  1 (111:37) -a«/  X  '  B  I jj m  4 4  I.vo  I*. TT. _ J-4JS -  1  it  =  o  wi+Vi  [All  the  m = O  case.  jj']  jj m L  (111:38) O  ii'a  •1-1 a  oology  O  Wy  analogy  wttb  "Hie  <X- i n t e g r a l s .  Jje  L j * j ' , all S3 1 (111:39)  c  ijjia =  U  IjJB  --§1 f-L  JJ°  Xa« I 2  +  !  "I  2(i + K*B*)J  (111:40) Uj*,«  ^  Ujjm  analogy  wVHi  "Hie  wj = O  case,  -31-  u  <  jj'o  "  "if.-  O  '-'jjm =  <• *  u». -  Uft  v  l)5  tall  (111:41) by  <  B  analogy  Willi  +V»e « -  =  e*  Uj-jo  <  (111:42)  I n t e g r a l s Ujj» m nj are  f  o  r  which a  d i f f i c u l t to evaluate.  n  i s neither equal to  ?m nor  zero,  However, I t i s t o be expected that  they w i l l be s m a l l e r than the two-centre i n t e g 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 i g n o r e d , w i l l be used. In  (37) t o (42), the n o t a t i o n i s as f o l l o w s : m  the  magnitude of a .  *b  the  magnitude of k j ,  e  the  e l e c t r o n i c charge.  K  the  static  A  the t r a n s v e r s e "Bohr r a d i u s " f o r F j in silicon.  a  B n a  *  m  dielectric  (111:43) constant of s i l i c o n .  l s  (r)  » the l o n g i t u d i n a l "Bohr r a d i u s " f o r F ^ . - ( r ) in silicon. ~* J  =? an a p p r o p r i a t e average of A and B.  The f u n c t i o n a l form of F j  l s  (111:44)  (r) used i s that g i v e n by Kohn (1957,  * See S l a t e r and K o s t e r , 1954, S e c t i o n I I I .  32-  equation  6.4): F =  S ^ l  exp  (111:45)  where the z-axis i s taken i n the d i r e c t i o n of k . i  For s i l i c o n ,  Kohn gives the values: A - 25.0 x 10~  8  cm.  B - 14.2 x 10"  8  cm.  (111:46)  The necessity of using a* a r i s e s from the occurrence of the term 1/r i n the integrands of (40) to (42).  S i m p l i f i c a t i o n of  the exponential terms i n these integrands leads to a complicated expression f o r the remaining r-dependence unless an average Bohr radius i s used. of  a  From (46) i t may be seen that the value  w i l l be approximately 20 x 10  equation 5.10).  (Kohn,  1957,  Use of t h i s value leads to:  -§1^ = fc.O By Chapter I , k  Q  X  10"* eY.  8  S  (111:47)  silicon]  lattice  and by (46) B i s approximately  Thus the value of k ^ 1 + k. B*  [for  ~ — i i . , where d i s the s i l i c o n 2d  spacing 5.42 x 10"~ cm.; to 2.5 d.  cm.  - i -  =  equal  i s approximately 12, so that: 6.9 x l o "  3  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 Chapter I gives k s | k ^ . Q  is d  -33  T*  -  - JL  7*  I-integrals  all other  all other  U-integrals  =  O  (111:48)  =  O  (111:49)  If i t i s further assumed that:  o unless a  m  (111:50)  i s one of (±a,0,0), (0,±a,0), (0,0,+a), or zero  (where a i s 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: <olo>  =  1  <ll  =  1  < 1 U >  =  <2ll>  =  <il3>  »  <1|4>  [ <Z'Z>  = =  ;<3i-i> [<3I4>  i>  i  «  z <H5>  =  O  I <ZI4>  »  <2|5>  =  o  =  <4I4>  =  <5l5>  =  1  -  <3l5>  «  <4!5>  =  o  -34-  <OlU  1  O  cj„>  '<11UIlfl >  a  <llUUaj,>  a  <2.| U | l a >  .<i 1 uI ia >  s  O  C i = 3.4.y ]  O  Ci= 3,4,53.  a  s  a  ' <z\v 1 z O  r  <3lU 1 3* >  (111:52)  s  2  | <4IU | 4 a >  a  ] <5l U ( 5 a >  »  a  5  [<3l U | 4 a >  <31UI5ftg>  s  =  <4|UlSa >  =  a  Then, multiplying the matrix elements (52) by e ' - ' - s  , and  summing over nearest neighbours:  le --' <OlUlOa;,>  =  | U  Z,e -*a<l|0|la >  =  2 0 \ ( i + c o s k . i X cos K a + cos K a ) + 2c<>s K»a}  Zc -' -<2|Ul 2.4 >  a  Z U { ( i + c o s k „ a X c o s K«a + cos k » a ^ +  =  zu{ k»*  cosk«a  £ e ' - * = » < 3 | U|3 9. >  =  4U J c o $ k . a  cos  <4|Ul4a >  =  4U^cosKx*  £e/K » » < s i U | Sa >  =  4U  1  i  i  £e  5  ft  i K f t  a  =»<lIUUa > s  =?  £,C  iSfta  =  5  \ 2+co3k.a]^cosk,«  +• c o s K a + coa K , a ] y  x  tos  {cos  y  x  co» K a ^  ••-cos K a  cos K » a ]  Y  + co k.4co»K 4  K*a +  Y  + cos K * + y  Ka  zcosK a3  S  toj  v  K,a  + +  t  co K»a} $  + cosk.aco$  K»a]  O  -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 a cos K^a + cos K a + cos K^a. Q  y  M «• cos K^a + cos k a cos K^a + cos K a. Q  N  35  z  (111:55)  cos K^a + cos K a + cos k a cos K^a. y  Q  (111:56) Then, making use of the results of (51) and (53), equation (27) becomes: (L+M}-£  L- t/  2  O  O  O O  O  O  Z L - t  O  O  O  o  O  2.M-1  O  ZN-E  =  o  (111:57)  The roots of equation (57) are:  (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): d „ d * O.  A l l others  S3  d,,dx4  A l l others  ea  0.  t  (  £,:  O.  0.  d» =  1  .  A l l others  S3  0.  d««  1  .  A l l others  SO  0.  d =i. y  A l l others  -  (111:59)  (111:60)  0.  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 i s only approximate.  There are actually two sets of degenerate states:  one ( +*,  ) corresponding to the group B, and the other  l  (  , + \ 4<" <4  ) corresponding to the group T  Luttinger, 1955 c).  g  (Kohn and  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 sphericalpotential effective-mass approximation as: E-E', , 0  1  = i u \ z+ co*k.a^{ cot. K»a +  cos K «  +• cos  t  }  (m 61) :  For a simple comparison of the line broadenings described by (58) and (61), consider the case when cos k a - 1. Q  In this  -37-  instance, a l l the energies reduce to: E.-£  l s  = 4 U \ cos K a + cos kr a x  y  + c o » K 4 *j  (III-62)  B  The implication of ( 6 2 ) i s that the degenerate and nondegenerate Is levels broaden at approximately the same rate. Clearly this behaviour i s due to the fact that sphericalpotential 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, 1 9 5 5 c), the results of ( 5 8 ) w i l l be taken to apply i n this case without alteration.  The non-  degenerate Is level, on the other hand, i s considerably lower in the exact formulation of the problem than i t i s in the spherical-potential approximation (Kohn and Luttinger, 1 9 5 5 a and c). In this case, therefore, a corrected wave-function should have been used.  It i s 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 i s 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 i t s spherical-potential counterpart, so that Is functions centered on neighbouring donor atoms would overlap much less than the corresponding l s ' functions. N  6. Energy-band broadening. Baltensperger (1953) has calculated the broadening of the Is and 2s-2p bands of equation (1) assuming a singleminimum conduction band and using purely hydrogenic effectivemass wave functions. He finds that the lower edge of the 2p band i s depressed: AE ° =  o i 5 _SL  2p  *  5"x io" ev.  (111:63)  1  below the Isolated-impurity level when r » 3a*. s  ^  ^  Since: (111:64)  the corresponding value of the impurity lattice spacing i s a = 5a*. This spacing gives an impurity concentration, in 18 3 s i l i c o n , 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 i s raised above the isolated-impurity level by at most: = -iaU  2  i.3*  io"* «v.  (111:65)  =. - 4 0  s  o . l x io- eV..  (111:66)  and at least: AE*.*  J  -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 2p ^ ^ levels i s : 6  Q  & E. touted  =  (3.2,x.io-*  -  II  KIO"  )  1  E V  .  2.1  =  x  lo" ev. 1  (111:67)  By (63), (65), and (67), the separation at a - 5a* i s at least: a  &E.4 « a  -a - o . l « i o ~ ) « v . (2-1 * to" - O-S" X icT* 1  IS  x  to  l  - 1  (111:68)  Hence at a = 5a , the separation has dropped to about twothirds of i t s 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 i s found that the Is and 2s-2p bands overlap near a = 3ex 4 x 10 /cm ), 18  3  (impurity concentration  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 w i l l not be obscured by that for the I s — * 2 p  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 N  i d * Y,^-- '^-S  i=l  f  l  3  'lp.lol*a > n  n' n  a . ±. f , a r f e  s  <ils  S  e  l  e  l  B  f  t  9  <i l s a ' l 3  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 i s also a vector of the unperturbed lattice,  so that: = Ej; Uia >  U\ila >  (111:70)  a  s  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 '|x,|o i s a > 5  5  »  Hence (71) may be written:  <xa ' l « , | o a > 2  a  (111:72)  -41  n'n  l  di* T. e'--=  x X  <iU«rtoa„>  • S,e"^'>»--' (shifting the origin to a «  (  , and defining a  Q  - a  m  n  I  I  I  :  7  3  )  - a »). n  By (73), therefore: <5K'lp»-loK> = where:  Es^CH')  e.,(tf.  Effw-  5 .=  J- X e  S K . K ' Z * *  £e -" <alx,loa > ( m 7 4 ) lK  B  =!  ESCIO  :  (  I  N  :  7  5  )  and B>K  .  i  (  (  ^^'  (111:75')  c i s  OK,K*  *  Kronecker o in three dimensions.  a  Its value i s  one whenever K «• K ' , and zero whenever K / K * . 2. Evaluation of integrals in the spherical-potential approximation. The integrals involved in evaluating (for  i  03  <i U Io a > f  B  1,2,3,4,5) are similar to those defined in (35) and  (36), and w i l l be denoted by Xjj» , X.fy , Xjy * m  m  m  a n d  *Jym  m  For example: X  j"Via  =  ^ »  (<rFi^)Fi'Cr-a Uosbi-£  * See Appendix B.  B  cosk/.r  <*£  (111*76)  -42  If spherical-potential effective-mass wave functions are used for the evaluation of the integrals (76), i t i s found that j j » x  a  n  d  m  cases of interest.  ^J'm  v a n i s h  identically in almost a l l  Furthermore, the only non-zero integrals  of this type (certain of the X?Jt for which j f j * and m  ra ?* 0), as well as the integrals X j j i  m  and X^y , are m  extremely small, being of the order of 1 x IO"* B. Con6  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 i s incorrect. It i s 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  l e s s exact techniques of c o r r e c t i o n . Kohn and L u t t i n g e r ( 1 9 5 5 a) have proposed  technique. of  They argue t h a t except  such a  i n the immediate v i c i n i t y  the i m p u r i t y nucleus, the s p h e r i c a l - p o t e n t i a l  w i l l be v e r y n e a r l y v a l i d .  Consequently,  approximation  i f the s p h e r i c a l -  p o t e n t i a l e f f e c t i v e - m a s s e q u a t i o n i s s o l v e d i n an  "exterior"  r e g i o n which excludes a s m a l l volume s u r r o u n d i n g the donor atom, and the r e s u l t i n g wave f u n c t i o n i s j o i n e d t o an exact s o l u t i o n of energy,  i n the " i n t e r i o r " r e g i o n t o determine  (11:6)  an improved theory s h o u l d r e s u l t .  the d i f f i c u l t i e s  the  In order t o a v o i d  i n h e r e n t i n the second s t e p of  this  procedure, Kohn and L u t t i n g e r have made use of the a l l y observed ground s t a t e i o n i z a t i o n energy  experiment-  t o determine  the  e x t e r i o r wave f u n c t i o n , and then have made a rough e s t i m a t e of the s o l u t i o n f,or the i n t e r i o r r e g i o n . t h e i r r e s u l t s are approximately  They have found t h a t  c o n s i s t e n t w i t h the r e q u i r e -  ment t h a t the two s o l u t i o n s j o i n smoothly. The e x p e r i m e n t a l i o n i z a t i o n of a phosphorus donor i n silicon  i s (Morin e t a l . , 1 9 5 4 ) E- £  0  =  - oo44  ev-  (111:77)  Kohn and L u t t i n g e r ( 1 9 5 5 a) show t h a t i f t h i s v a l u e of ( E - £ o ) i s s u b s t i t u t e d i n t o the e f f e c t i v e mass e q u a t i o n  (11:27),  the  s o l u t i o n o f the e q u a t i o n i s ; (111:78)  -44-  where: x  *» r/cx*  n - 0.81  (111:79)  C •= normalization constant W » the Whittaker function. The function (78) has the limiting behaviour: FC*}  FCxi  = Ce""  /n  =  c  -0-«Un  Zx  4 lj  e" "l7r' x/  Thus F(x) diverges near the origin.  a  as  s  4r  x  o.  * •  (111:80)  f 00  (This behaviour i s 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.  ii.  For large values of x the approximation to F(x)/C given by (i) was found to be f i t t e d very closely by the expression:  -45 0.65 exp (-1.25 x) iii.  (111:81)  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 i s greater than 1.25, so that the second exponential decays more rapidly than the f i r s t . iv.  The value of p was determined by requiring the expression (82) to go to 2.30 at x » 0. According to ( i ) , this value i s slightly greater than the value of F(x)/C at the Wigner-Seitz sphere boundary.)  v.  The value of q was determined by requiring that (82) have the same normalization as the function plotted i n ( i ) .  The result was:  F(x)/C - 0.65 exp (-1.25 x)  (111:83)  + 1.65 exp (-3.00 x) (83) i s compared graphically with the result of (i) in Figure 1. vi.  The value of C was determined by integrating the curve of (i) numerically, and requiring that: (111:84)  -46-  The result was: (111:85) vii.  (85) was substituted into (83) to give for the corrected ground state effective-mass wave function the expression: 2-24 ,  er ] ir/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 i t s 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 i n 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 i s not known accurately, however, the corrections to be used in this case are d i f f i c u l t to determine. * See Chapter I.  Kohn and Luttinger (1955 c) have  -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 i s  so close to the effective-mass value (-0.029 ev.), i t w i l l be assumed that the correction i s small enough for the effective-mass functions to be used without alteration. An estimate of the error introduced by this assumption w i l l be made in conjunction with the evaluation of optical matrix elements. 4. Evaluation of integrals in the corrected sphericalpotential approximation. Using the corrected wave functions introduced in the last section, the integrals (76) may be evaluated approximately.  Hence, i lra° 0j or i f m ft 0, but a  m  i s perpendicular to  the <r-axis, X  Jj=  =  (111:87)  °  since the integrand i s an odd function.  If o  m  i s parallel  to the cr-axis, on the other hand,  "= cS*  x  =  f { (^FjCOFTCt-^as +  (ao-FjCcVF/Ct-ar-'Kos *t»sjr  * see Section C.4 of this chapter.  (111:88) j  -48-  The second i n t e g r a l i n ( 8 8 ) cannot be evaluated a n a l y t i c a l l y . By the arguments of Chapter I I , Section C, however, t h i s Integral should be small.  (  FtoPV-a^ dr  ^»  *  Numerically, for|a |« 4a j m  F ^ F ° ( r - a , ) o s 2 k . i dr  ,-Ti  (111:89)  -LIST mo- ) *  C  1  a  ~  -C2-88 x iar») a*  (111:90)  Hence at t h i s value of ja^J, the second i n t e g r a l 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: (111:91)  l i r t * [«••' Fj(cWCc-*«r> c o V k j . c *c  w i l l d i f f e r i n value from ( 8 8 ) . c l e a r l y small;  However, t h i s difference i s  i t vanishes altogether when the Bohr r a d i i  A and B are replaced by t h e i r average, a .  Consequently,  i t w i l l be assumed that a l l non-zero integrals of the form ( 9 1 ) are approximately equal to: Xi = *  -( XK i - 4irr 2  J  P(OF"(C-&m> * E  =  Lo.o.al)'  -49-  wher©:  B Cy^ = ( i V ^ t  (111:93)  t  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.  i s plotted as a function of a in  Figure 4. Other integrals of interest are: Xii*  = ^ » | < Fi( ^ FrCc-ft») eoskj-r  s  r  t  S.«  l y - r dr  (111:94) (ZTrt  For m  s  x  )  i  ~  1  "*  0 the integrals (94) would be expected to have values  of the same order of magnitude as those in (90). |S<r  Xjj  In any case,  fler  m  i s less than or equal to X J J Q .  non-zero and easy to evaluate. X*f  0  ^  The latter integral i s  It i s found that:  + 0-sr * i o - M  (111:95)  Comparing this value with (89), i t i s seen that XJ^ w i l l be at least a factor of ten less than X^, though the two integrals w i l l not necessarily have the same sign. Finally, XJTB  = ^  =  [ * , F j ( t f F?Cr-a > S ^ k j . r <*r B  _L__ . _L | J<,F (r') F"(r-ft ^ dr i  a  {*, F/r*) F^Cr-a^ cos^hiC * r ]  -50-  Clearly, If m  or i f a  m  i s perpendicular to the cr-axis,  o If m / 0 and  is parallel to the  (111:96) -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") w i l l be neglected.  By analogy with (36), i t i s  expected that these Integrals w i l l 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 terms of X^.  The dependence of this quantity on the five-fold  degenerate level energy i s shown in Figure 5.  Clearly the  difference between values for E ^ f - £ o - -0.029 ev. and E  (5) l 5  - E o - -0.032 ev. is sufficiently small for the use of  the 0,029 ev. energy to be a reasonable approximation. 5. Evaluation of the matrix elements. The i n t e g r a l s i n (87) to (97) may now be used to evaluate the matrix elements then be substituted into j <llx |oa >  *  t < i l a , | O a >  ~  r  B  5  s  O  l = !  -  Xa  jy^ ( « , , , - * ) Xa 5=  (5| Xftj  o a > «v  7f P = XL.  c  , < 5 | a,|oa > 5  , which may  (75).  as  '<JI v l o ^ > < <4 1  <i lx<r\oa >  a  =  (111:98)  =  3  £  (111:99)  Substituting (98) and (99) into (75),  r e s t r i c t i n g the  summation to nearest neighbours, and noting that:  - -- i -  ( <A* Ay \ c  It i s found that:  (111:100)  -52-  <iK'lp,|0K>  =  -  E (K) Sfi  - d\$L „•„ k„a cos\K a] XI  (III: 101)  x  - d* i l S . n W.a  COS  {Kyd} X3 j  _ 0L5 ^ s.vi k.a cos iK*a^  For simplicity, consider:  * j z^^T o  (111:102)  - « , ^ * . n K,a  - df ±L sinlua cosK»a "l V? J  By (59) and (60), d  and d  2  5  are mutually exclusive. Hence  only one of the bracketted terras i n (102) w i l l be non-zero. Expressions similar to (102) w i l l arise for other choices of  a , the main difference being that d * w i l l be replaced by 2  d^*, or by -(d^* + d * ) . 2  Since dj and d both depend on K, 2  and thus lead to cumbersome expressions, i t w i l l 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  <5K'IP^IOK>  s  %^ xl  - r i z y f (1-cos k.O  a. ±. sin k,* E™t«) i  K t f  x Z cos  sm K,a  K^a.  (111:103)  (111:104)  -53  Both o f these m a t r i x elements v a n i s h when cos k a =° 1 , Q  but t h e i r sum i s l a r g e when cos k a » 0 or -1. Q  Consequently  the combined e f f e c t o f these m a t r i x elements w i l l be r e l a t i v e l y important, except i n the neighbourhood of i m p u r i t y l a t t i c e s p a c l n g s f o r which k a i s an even i n t e g r a l m u l t i p l e Q  of TT.  ( C l e a r l y t h i s behaviour w i l l not be o b s e r v a b l e , s i n c e  i n r e a l i t y the i m p u r i t y l a t t i c e s p a c i n g has o n l y the s i g n i f i c a n c e o f an average d i s t a n c e between i m p u r i t i e s —• the randomness of the i m p u r i t y d i s t r i b u t i o n : , removes e f f e c t s which a r e dependent  upon the assumption o f r e g u l a r i t y . )  -54-  CHAPTER IV  Optical Absorption by Transitions between Impurity Bands i n 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 i s the semi-classical approximation. This approximation, in which the radiation f i e l d i s treated classically and the atomic system i s treated quantum-mechanically, leads to essentially correct results, and has the great advantage of simplicity over more accurate treatments involving quantization of the f i e l d .  Consequently the semiclassical theory w i l l  be used in this thesis in obtaining an expression for the coefficient of absorption. Consider an atomic system in state  U> upon which i s  incident a polarized beam of photons, of intensity angular frequency range (<>>, oo + d«>).  I(w)d<J  in the  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, i s : xicS) |M .(5J| U  (IV:1)  -55where: e  =• e l e c t r o n i c  charge  m  =» e l e c t r o n mass  c  » velocity of light  QLw =• photon wave-number v e c t o r , of magnitude o3  UJl  »  (where Eu and E a r e the e n e r g i e s o f iu> and u> respectively) A  «  M (3J  -  V  ^<r  ^ .  <ule  l 3  (IV:  2  )  (IV:3)  - v U> r  r  =* component o f grad i n the d i r e c t i o n o f polarization.  S i n c e the f u n c t i o n  i s s h a r p l y peaked a t «o -  equation  P  (1)  =  and:  may be w r i t t e n approximately as:  cLco -a>  J  S  f  on  If-*-.  ICCO) I M ^ C ^  \  \  z  bi^-cS)  m ccO* z  (IV:4)  so t h a t :  (IV:5) I f !(*«>) i s approximately constant a t the f r e q u e n c i e s  -56.  c o n s i d e r e d , i t may  be r e p l a c e d i n (5) by I«o , i t s average  over u n i t frequency range at co.  Pto• ^  then becomes the  p r o b a b i l i t y per u n i t time o f a t r a n s i t i o n between K>  ic> and  due t o the a b s o r p t i o n of a photon of frequency  from  a beam of i n t e n s i t y Io per u n i t frequency range. The i n t e n s i t y may Ifco^ oLco  a l s o be =  written: <A*o.  wCto'i ticJ  (IV:  6)  where n(<j) d^o i s the number of photons i n (", co+ d<>>) c r o s s i n g u n i t a r e a per u n i t time.  In terms of averages over u n i t  frequency range: I co where " n  w  -  n^<^  (IV:  i s the number of photons c r o s s i n g  time per u n i t frequency range about co. (7) i n t o between  7)  u n i t a r e a per u n i t  Then,  substituting  ( 5 ) , the p r o b a b i l i t y per u n i t time of a t r a n s i t i o n IA> and  K> due t o the a b s o r p t i o n of a photon of  frequency to from an i n c i d e n t beam of n"^ p a r t i c l e s per u n i t a r e a per u n i t time per u n i t frequency range about Jla-.ujL  =  ^  !  2. S p e c i a l i z a t i o n t o a r i g i d  I M . J ^ J I  2  co  i s found t o be:  oCo^-co).  (IV:8)  lattice.  I f the r e s u l t s of s e c t i o n  1 are applied  a r r a y of atoms, the wave-vectors  i£> and  both the c o o r d i n a t e s of the e l e c t r o n s  to a c r y s t a l l i n e  iu> w i l l depend upon  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 d i f f i c u l t i e s , since the lattice coordinates change as the atoms vibrate about their equilibrium positions.  These  d i f f i c u l t i e s 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 r i g i d 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 *  (IV:9)  b(co*'« - co)  where now: (IV: 10) and: (IV:11)  In effect this approximation involves consideration of the crystal at the absolute zero of temperature, with a l l zeropoint energies of vibrational states being ignored.  -583. The coefficient of absorption. In a cubic crystal of volume V, the number of electronic states whose K-vector l i e s in the range (K, K + dK) i s (Dekker, 1957, p. 256): V<*K  considering one spin orientation only.  Thus i f f (K*) denotes u  the Fermi distribution function for the energy band labelled by u, the total number of empty states of each spin in the range (K', K  1  + dK') in this band i s :  [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 i s : ^•u,G^,iO  =  J 7«ilM(•*-.!$•.LI-^CK')]^  where the integration i s over the f i r s t Brillouin zone.  ( I V  ,  1 3 )  It i s  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 s m a l l volume i s :  Adx  Each of these s t a t e s may multiplying  dK  a c t as an "absorbing u n i t " .  by two t o account f o r both s p i n  Then,  orientations,  the t o t a l number of absorbing u n i t s i n (K, K + dK) i n the volume Adx i s : Adx  o £ r *\ Z  4  >  dK  «  C  (IV:14)  Thus the p r o b a b i l i t y per u n i t time of a t r a n s i t i o n from band Z i n t o band u due t o the a b s o r p t i o n 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  u n i t time per u n i t a r e a per u n i t frequency range about co i s : f ^ - . u , (nco^ AbLx  =  f ^. , u  where the i n t e g r a t i o n  ( n « , i O "Z*>C*0  Ad*  (IV:15)  i s a g a i n 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 g i v e s the decrease per u n i t  time  of the number of photons i n the beam due t o a b s o r p t i o n i n Adx at the frequency of i n t e r e s t . mean energy f l u x W«,» ( dWc Now  =  ra  -  The corresponding change i n the  H o At»co) i s :  (JxJ) Act* tuo  the a b s o r p t i o n c o e f f i c i e n t  ^ ^ ( c o )  (IV:16) i s d e f i n e d by the  d i f f e r e n t i a l equation: OLWUJ  dLx  -/x^Cco}  ( I V : 17)  -60-  Hence, by  (16) and  .  (17):  J * * I * K ' ^ i ^ i ^ . p  W  B  - S «  (IT,18)  l  where both i n t e g r a t i o n s 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 The  r e s u l t (18)  i n a d i e l e c t r i c medium.  a p p l i e s to the  i n t e r a c t i o n of r a d i a t i o n  w i t h a c r y s t a l l i n e a r r a y of atoms imbedded i n f r e e space. the a r r a y i s now  considered  t o be  imbedded i n a d i e l e c t r i c  medium, s e v e r a l c o r r e c t i o n s must be a p p l i e d 1958,  Sections The  2 and  (D.L.  Dexter,  4).  f i r s t c o r r e c t i o n i s concerned w i t h m o d i f i c a t i o n  the magnitude of the r a d i a t i o n f i e l d . 35.11  and  ;Mil  c o e f f i c i e n t , depends on  tttf  hence a l s o  f i r s t pointed  general  out by Lax  the  , where e,« i s the  magnitude of the f i e l d which a c t s at the atoms of the I t was  of  S c h i f f (1955, e q u a t i o n s  35.14) shows t h a t T^ (*w,Ktc'), and  absorption  If  (1952) that  €«  equal to the average macroscopic f i e l d  crystal.  i s not  f  e  in  i n the  surrounding medium.  Consequently, (18) must be m u l t i p l i e d by  a factor  .  The  { /e]* €e(t  effective field ratio  accurately. overlap,  €eff  /«  is difficult  to  evaluate  For a v e r y t i g h t - b i n d i n g approximation, i n which  exchange e f f e c t s , and m u l t i p o l e  interactions  higher  61than dipole-dipole may be neglected, D.L. Dexter (1956) derives the expression: T "  =  3  (IV; 19)  where n i s 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 f i e l d 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/.  by the energy flux W  (jn«>)  w  . In  a dielectric medium (D.L. Dexter, 1956), Wo> i s given by i t s free-space value multiplied by ~  , where K i s the static  dielectric constant of the medium, and & i s the energy velocity c/n.  For photon energies at which the medium i s effectively  transparent, K i s 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 i s s t i l l correct when the medium i s not transparent.  In this case n i s  the real part of the index of refraction.) On applying the above corrections to (18), the absorption coefficient of interest for a r i g i d crystalline array imbedded in a dielectric medium of refractive index n i s found to be: r-o*  - TTC m  & fe-? i *  s  i • ^CK^ Lt-fwCK'O  i*  (iv,  $(co . fi s  C J )  20)  -62-  B. E v a l u a t i o n e f the A b s o r p t i o n 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 s t a n d , the i n t e g r a t i o n s i n (20) a r e i n t r a c t a b l e . Consequently,  s e v e r a l approximations must be Introduced, as  follows: i.  S i n c e the a b s o r p t i o n p r o c e s s I s t o be c o n s i d e r e d o n l y i n the neighbourhood temperature,  o f the a b s o l u t e z e r o o f  the Fermi f u n c t i o n s f ( K ) and f ( K ' ) A  assume a s i m p l e form.  u  I t i s t o be expected t h a t  when T i s near 0, a l l the e l e c t r o n s w i l l be i n the ground-state band, l a b e l l e d by & .  fu.(K') =  Hence:  o  (IV:21)  The form o f f ^ ( K ) when T i s e x a c t l y z e r o must i n v o l v e a d i s c o n t i n u i t y a t the mid-point of the i-band.  T h i s behaviour r e s u l t s from the f a c t t h a t  w h i l e t h e r e a r e as many i m p u r i t y e l e c t r o n s i n the c r y s t a l as there a r e i m p u r i t y n u c l e i ,  spin-degeneracy  ensures t h a t t h e r e w i l l be twice t h i s number of e l e c t r o n i c s t a t e s a v a i l a b l e i n the lowest i m p u r i t y band.  Thus when T =» 0, o n l y the lower h a l f of band  JL w i l l be o c c u p i e d . As T i n c r e a s e s s l i g h t l y from z e r o , however, the d i s c o n t i n u i t y i n the Fermi f u n c t i o n f (K) a c q u i r e s A  a f i n i t e spread. the  F o r a v e r y narrow band, such as  i - b a n d i s expected t o be, t h i s spread w i l l  * See Chapter III, S e c t i o n 5.  -63-  p r o b a b l y equal o r exceed  the band width, so t h a t  the e l e c t r o n s 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 ,  assumed t h a t i n t h i s  i t w i l l t h e r e f o r e be  case,  = i ii.  (IV: 22)  By ( 3 ) , the m a t r i x elements  M^CK'K  ico v") a r e : 5  (IV:23) "to  In the e l e c t r i c d i p o l e approximation,  i t i s assumed  t h a t the photon wave-number v e c t o r i s o f n e g l i g i b l e magnitude, so t h a t the e x p o n e n t i a l i n (23) may be approximated  by the f i r s t  e  tl-.'a-r  fi  term i n i t s expansion: (IV:24)  t  Thus, i n t h e cases o f i n t e r e s t , t h e m a t r i x elements are: MS.O^'K)  =  i  <iK'lp,loK'>  (IV:25)  The e x p r e s s i o n s (25) may then be e v a l u a t e d w i t h the a i d of equations iii.  (111:103) and (111:104).  In order t h a t the i n t e g r a t i o n s i n (20) may be c a r r i e d out, the t h r e e - d i m e n s i o n a l Kronecker  6,  *  d e f i n e d i n (111:75*) must be r e p l a c e d by a D i r a c  -64-  delta-function (Dirac, 1958, p. 58).  It i s shown  in Appendix B that with the correct normalization: (6KS0Z  =  * K K - = - ^ J p l SCK-K").  (IV:26)  where V i s the volume of the impurity-lattice "crystal". iv. By (ll),  the function ^jf^K* i s defined to be: WKK'  =  ^ I E (K">- E , ( K > | u  Thus, in the cases of interest, the tight-binding expressions E^CK').  (111:58) may be used for the energies  A S stated in the discussion of (111:62),  the broadening of the lower band i s expected to be negligible, so that E (K) may be set equal to the X  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 w i l l be used in explicit calculations. 2. The absorption coefficient for the ls ^-» l s ^ transition. Co  (  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  -65the ground state and the d band. 5  s  M^,o(K'Ka« < ) a  5  smk,a Xi  i  E *1»U} cos K a S ' ( 5  (IV:27)  ss  m  Hence:  |ru  $0  Ci<'K^ .^l K  '  a  *  ~  j£  Cxi) U'f.Wiof  i± s.v,*k.a 3  H  =  2  0  3  . cos *,* 1  (ZTTV*  S K  K  '  (IV:28)  CxiViEV'ic^r  S,V,^ A "to*  cos*K,a  SCK-K').  Then, substituting (28) into (20) and using the deltafunction &(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 i t s maximum value when c o s k a = 0. For 0  simplicity, intermediate cases w i l l be ignored. the  Then, under  condition that cos k a » 0, (111:55), (111:56), and Q  (111:58) show that: E /o(£) C  = - 4 l U l (cos K,a + coi K z] y  +  A. E5  0  (IV:30)  denoting: ^E .o s  =  [ E f f - EJV ] ,' .iaW  ,  s  Hence, substituting (30) into (29) and taking  T V  ( K ^ J out-  side the integral sign as tfco , by virtue of the delta1  . „. .  1  —66—  function: a . r u l  s  •.i - L { l 4 « \  ^-  (eXD*  /•" * ( d K n/  ,  t * d K  4/  iTr  i*f* d K »  v/  4IUI •'-•n  y  i-«y  / a  cos K a l  a  a  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 ^ )  IVCwt  d t dpOOS X + Co&g)  +.oo«|p I  Vsin'x  (IV:33)  3"  where d £ i s a line element of constant (cos x + cos y). Hence, defining: AE^O  -  CO  4IUI  (IV:34)  the absorption coefficient (32) becomes:  n c U  S * a».ui <°  \ U i ^ . z i ^ 3  _| cot  (IV:35)  X 4-Co  =1 for % between +2 and -2. In Appendix C the integration in (35) i s carried out, The results are shown in Figure 6, where the integral:  dt  (IV:36)  >Js«n*x + Sin**}' 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 coefficient which i s divergent at the centre of the band. However, i t may be shown* that the divergence i s sufficiently mild for the area under the absorption-coefficient curve to remain f i n i t e . The divergent behaviour of (36) i s a direct consequence of the tight-binding approximation, correcting the energy surfaces.  and may be modified by  (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 » 0, K y  z  « 0, and the faces of the f i r s t  Brillouin zone, at right angles.  Clearly this requirement i s  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 w i l l correspond to a surface whose cross-section i s shown schematically in Figure 8, and the corresponding absorption coefficient w i l l l i e between i n f i n i t y 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 i t s value at % 0 for a l l values of \ considered. m  -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 i s probable that the correct maximum of the absorption-coefficient curve may be obtained by rounding off the curve of Figure 6.  If this i s  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 isolatedimpurity energy gap between the  i*  C  o  >  and  is  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 f i e l d ratio, /e  €<fe  • Since at the wave lengths of interest (about 100  microns), silicon i s effectively transparent (Bichard and * Giles), i t should be a good approximation to write: n = /J?  (IV:38)  where K i s the dielectric constant quoted by Kohn (1957, equation 5.8).  To this approximation, the value of n i s 3.46.  Also, since centres in s i l i c o n are f a i r l y diffuse, having a * See Section A.4 of this chapter.  69-  Bohr r a d i u s o f about h a l f t h a t i n germanium (Kohn, equation 5.10), the e f f e c t i v e f i e l d r a t i o s h o u l d approximately  1957,  be  1, a c c o r d i n g t o the arguments of Lax  (1952).  The r o u n d i n g - o f f of the t i g h t - b i n d i n g a b s o r p t i o n coefficient  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 c o n s i s t e n c y might be o b t a i n e d by  cutting  o f f the curve of F i g u r e 6 at the r e s o l u t i o n of the s p e c t r o meter used, such a method o f f e r s no guarantee of  accuracy.  In the c a l c u l a t i o n s performed i n t h i s t h e s i s , the c u t - o f f value  (d  max  w i l l be taken as  five.  S u b s t i t u t i n g the approximations  of the l a s t two  para-  graphs i n t o (37), the e x p r e s s i o n f o r the a b s o r p t i o n coe f f i c i e n t becomes:  (Vatfll+*/«'J  c  (IV:39)  v a l u e s of t h i s f u n c t i o n are p l o t t e d i n F i g u r e 9.  C. D i s c u s s i o n of R e s u l t s . 1. Is —• 2 p  Q  transition.  In Appendix D, the method used f o r the  ls^-^-ls^ ) 5  t r a n s i t i o n i s extended t o g i v e an approximate e x p r e s s i o n f o r the ls^ -l-».2p ^ ^ a b s o r p t i o n c o e f f i c i e n t . 0  6  o  t r a n s i t i o n has been observed  As the  latter  e x p e r i m e n t a l l y ( B i c h a r d and  Giles),  i t s h o u l d be p o s s i b l e t o 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 * See S e c t i o n A.4  of t h i s  chapter.  as  -70-  a rough check on the method i n g e n e r a l . it  In the Appendix  i s shown t h a t at c o n c e n t r a t i o n s f o r which the  of t i g h t - b i n d i n g s h o u l d be a good approximation, calculated absorption c o e f f i c i e n t agreement w i t h experiment.  i s in quite  assumption the  reasonable  From t h i s i t would appear t h a t  there are no gross e r r o r s i n the c a l c u l a t i o n , and  that  the  method used might be expected to g i v e a reasonable d e s c r i p t i o n of the l s * ^ — . . I s * * 5  2.  ls  (  Q  ^— ls*  The  5 )  transition.  transition.  r e s u l t s p l o t t e d i n F i g u r e 9 i n d i c a t e t h a t at  i m p u r i t y c o n c e n t r a t i o n s of about 10*®  per cnr*, the  a b s o r p t i o n c o e f f i c i e n t i s of the order of 10 cm" .  l s ^ — * l s ^ ^ As t r a n -  1  s i t i o n s w i t h a b s o r p t i o n c o e f f i c i e n t s i n t h i s range are s e r v a b l e by e x i s t i n g techniques  ( B i c h a r d and G i l e s ) ,  ob-  the  theory would appear t o i n d i c a t e t h a t the f i n e s t r u c t u r e of Is  s t a t e s w i l l be i d e n t i f i a b l e .  the  There a r e , however, s e v e r a l  sources of e r r o r i n the c a l c u l a t i o n which might tend t o i n v a l i d a t e the r e s u l t s The  first  obtained.  p o s s i b i l i t y of major e r r o r l i e s i n the  of wave f u n c t i o n s f o r the two  l s - s t a t e s of i n t e r e s t .  L u t t i n g e r approximation used i n t h i s p a r t of the may  o n l y be good t o w i t h i n a f a c t o r of two  matrix element.  choice The Kohn-  calculation  or t h r e e , f o r the  A l s o , as i n d i c a t e d i n F i g u r e 5, i f the  l e v e l i s depressed below i t s u n c o r r e c t e d  Is* * 5  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 u n c e r t a i n t y i n the c a l c u l a t i o n i s the  -71-  assumption of a r e g u l a r random nature of the  l a t t i c e of i m p u r i t i e s .  a c t u a l d i s t r i b u t i o n w i l l remove  pendence on such q u a n t i t i e s that  as cos k a .  the  de-  I t i s t o be  Q  expected  t h i s "smoothing-out" e f f e c t w i l l be accompanied by  o v e r a l l decrease i n the magnitude of the T h i s c o r r e c t i o n may  of the  (111:103) and  f a c t o r of t h r e e or f o u r D  a  n  d  s  i s  *°  the are  instance. f o r example,  s h o u l d be m u l t i p l i e d by  i f the e f f e c t of t r a n s i t i o n s t o  other p o i n t s i n the d e r i v a t i o n  might be a p p l i e d .  e r r o r s have a l r e a d y been d i s c u s s e d . s h o u l d be noted here, however:  the  at  In most cases the One  incident  alteration is  u n p o l a r i z e d , the a b s o r p t i o n c o e f f i c i e n t s h o u l d be reduced  to  (D.L.  i f the  further  possible  radiation  one-third  of i t s v a l u e f o r a completely p o l a r i z e d beam,  Dexter, 1958). There i s no guarantee, t h e r e f o r e ,  F i g u r e 9 are a c c u r a t e t o w i t h i n of magnitude.  1  See,  the r e s u l t s  l e s s than one  or two  ls^°^  -ls^ ^ 5  of to  transition  unobservable. f o r example, equations (111:103) and  of  orders  can be o b t a i n e d , t h e r e would appear  be no f i r m t h e o r e t i c a l reason f o r the t o be  that  However, i n view of the f a c t that r e s u l t s  the order of 10 cm""  a  included.  D e  There are s e v e r a l which c o r r e c t i o n s  given  (111:104) i n d i c a t e s ,  the a b s o r p t i o n c o e f f i c i e n t (39)  **1 " **2  absorption c o e f f i c i e n t .  f i v e degenerate Is s t a t e s  a v a i l a b l e f o r o p t i c a l t r a n s i t i o n s i n any Comparison of  an  be p a r t i a l l y compensated, however, by  f a c t that more than one  that  Clearly  (111:104).  -72-  APPENDIX A  P r o p e r t i e s of the C o e f f i c i e n t s D.  By e q u a t i o n (11:16) the c o e f f i c i e n t s D  i  (k)  (k) and  D_ . (k) s a t i s f y : (U(k.k') Dj lV) dk'=  {£!(=)-E]T> lk)  +  {£"i(kV-E]D. (!s >  + jUCk.k'V D.^Ck'Wk' =  jf  t  o  (  A  :  1  )  and: ,  Jl  o  (A 2) :  r e s p e c t i v e l y , where the s u b s c r i p t " - j "  denotes the c o n d u c t i o n  band minimum a s s o c i a t e d w i t h - k j .  by (11:17)  Now  (A:3) where: 3k> 9k  A  k= -ti  3(-i<03C-kp  (A:4)  8k ak^ x  Hence: (A:5) A l s o , by (11:12): (A:6)  -73-  The p e r i o d i c p a r t s , B l o c h f u n c t i o n s may to  ij(£) $ °* the  u c  conduction-band  be shown (Jones, 1960,  equation  2.25)  satisfy:  vV k(rJ> +  u w  C  ck  U c C ^ - l s l k l - V C ^ UckCO = o  % -  +  P  (A:7)  Taking the complex conjugate of ( 7 ) :  vV  e  k  - Z i k .v uck  + ^  - V p ( ^ ] u*w  { t Ck) c  = o  By the symmetry of the c r y s t a l , £ (k) «- £. (-k), so t h a t c  may  c  (A:8)  (8)  be w r i t t e n :  V ^ i  - lik.v u.1  +  ^ 1 J c-u>> - V±£L £c  ^  =  o  ( A : 9 )  Hence, p r o v i d e d t h a t the c o n d u c t i o n band i s non-degenerate: utkC^  =  "-c-isCr.^  (A:  10)  and: (A:ll)  Therefore, s u b s t i t u t i n g  <Pck(c) <Pck'(rt de.  U*(k,se^ = = =  (11) i n t o ( 6 ) :  J UC«^  ip *-k Cr>> <p_, (r> dr c  11 C-k,-ls')  c  f e  (A:12)  -74-  From (2) i t f o l l o w s It'iC-W-E^D.uC-ft J  Then, a p p l y i n g  that:  + (ui-k.k'^p , (w ) dk' = * 1  J  o (A:13)  J  the r e s u l t s of (5) and (12) t o (13):  { e ( ^ - E ] D. (-^ J  c  jft  U c ( k V E } D-  jt  4- ju(-w,-k') D. C-«£') Aw j4  (-k^ + j u*(k,k'>  b.j.C-k-) dk'  ^ o o (A:14)  UiCk^-E] D . j , ^  Thus, provided  +  j Utk.k'lp.^C-k-i  dk' = o  that t h e t h energy l e v e l of equation (1) i s  non-degenerate: =  &-£H^  ( A : 1 5 )  In the case of the I s l e v e l the non-degeneracy c o n d i t i o n i s c l e a r l y s a t i s f i e d , so t h a t : (A:16)  APPENDIX B i  P r o p e r t i e s of the Three-Dimensional  a.  Equation  Kronecker 6.  (111:75') defines a function 6 ^ i k  such  that:  (B:l) where N i s the number of atoms i n a simple c u b i c " 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 Expanding the s c a l a r  crystal.  product o c c u r r i n g i n the e x p o n e n t i a l  term of ( 1 ) , and making use of the c u b i c symmetry of the c r y s t a l being c o n s i d e r e d :  •  ^  '  '  -  ^  Now the allowed v a l u e s of a  <  B  :  2  >  a r e na, where a i s the l a t t i c e  n  A  I/.  s p a c i n g and n runs from 0 t o N - 1.  ,  <  A - 1  , (Svmilarl^  y>  (of ^ and. z.")  Hence:  (B:3)  -76-  by the u s u a l r u l e s f o r summing a geometric Now  series.  by the p e r i o d i c boundary c o n d i t i o n used to  the allowed v a l u e s of K i n the f i r s t B r i l l o u i n zone 1960,  p.  determine (Jones,  36), iKxN' »a  _  /  e  (B:4) f o r each allowed v e c t o r K. a vector.  By d e f i n i t i o n ,  i s such  Hence: AkKkw'  Thus i f k /  =  e  = 1  (B:D)  k*: 5 '= KW  — \  ](  ^^-1  M  *-*  " TT ?  l U k j - i Jl k r  (6) and  " =  e  -  \\  I f , on the other hand, k = k',  Hence, by  (k - k*)  IT  A  k  ,  k  i  - i  o  J  (B:6)  then by ( 1 ) : =  (B:7)  1  (7): 'O  if  k*  k'  Okk'  (B:8) 1  T h i s behaviour dimensional b.  if  k = k'  i s analogous t o t h a t of the u s u a l  one-  Kronecker  The f u n c t i o n 6j ^j t may E  c  e a s i l y be approximated  a D i r a c d e l t a f u n c t i o n , as f o l l o w s :  by  -77-  N  MCI r»  vi  crystal  (B:9)  where V i s the volume of the c r y s t a l . Clearly:  (B:10) Hence: V  1  •———•  t  •  (B:ll)  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  coefficients.  -78-  APPENDIX C  E v a l u a t i o n of the I n t e g r a l  In Chapter  IV, the  (IV:36).  integral:  I  Coix + co* = ^  (C:l)  0 4 X,^ $ TT  o c c u r r e d i n connection w i t h the c a l c u l a t i o n of the absorption c o e f f i c i e n t  i n the t i g h t - b i n d i n g  ld°-—*ls  approximation.  L i n e s t y p i c a l of those over which the i n t e g r a t i o n must be taken are sketched i n F i g u r e 7. Now,  by elementary  calculus, (C:2)  Along the l i n e  (cos x +• cos y) « 1 , dx and dy are r e l a t e d  by:  Sinx  T h e r e f o r e , by  dx  (C:3)  (2) and ( 3 ) :  (C:4)  -79-  Hence the i n t e g r a n d i n (1) becomes: _  V  + JinN  ^ S m ^ x  <Lx I-  (C:5)  C ^ - c o s x ^  with x v a r y i n g between 0 and c o s " ^ ^ -1) whenever f i s g r e a t e r than z e r o , and between cos""*(^+ X  i s l e s s than zero.  1) and v whenever  By ( 5 ) , the i n t e g r a l i n (1) may  thus  be w r i t t e n : ,cos'a-0  J  >/1-  J t os  -. ^ c  Vi-CI-art»^  0  r  (c 6) :  (C:6»)  Changing the v a r i a b l e i n ( 6 ) t o :  and  defining: 2^5  the i n t e g r a l  ( 6 ) becomes:  Z  (''  1-1  \  a  x  y ^ C i - ^ L ? +• XCI-XU'  (C:9)  -80-  Similarly, 2  (6*) becomes: (*  dx  (C:9'>  where: (C:10) C l e a r l y , t a k i n g i n t o account the allowed v a l u e s of \ i n each case, i n t e g r a t i o n of  (9) and  \ » 0.  which i s symmetric about considered  Hence o n l y  (9) need be  in detail.  In the l i m i t as (9) may  (9') w i l l l e a d t o a f u n c t i o n  \ approaches 2,  be approximated  £ becomes l a r g e ,  and  by:  (C:ll)  Comparison of the v a l u e s of range of X c o n s i d e r e d  $x(i-")0  i n d i c a t e s that  and  X = 0,  1  i n the  (11) w i l l be a v a l i d  approximation f o r a l l ? g r e a t e r than When  LXCI-TO"]  0.5.  (9) becomes:  (C:12)  As t h i s i n t e g r a l d i v e r g e s ,  the l i m i t i n g behaviour of  % goes t o zero must be examined i n more d e t a i l . The  i n t e g r a l (9) may  be r e w r i t t e n  as:  (9)  as  [  4X  l-xd-x^n-  Vxu-xV  (C:13)  Clearly, for values of X much less than 5 , the expression under the square root In (13) w i l l be dominated by i t s second term, while for values of X much greater than the f i r s t term w i l l dominate.  ,  Thus in the limit of small  £ , which corresponds to the limit of small 1 , (13) may be approximated by:  -i- (  z _  s(.>rr  , JTea.-'{iS.  +  cos- j 1 - 1 3 - 1  ft  >J i+r$ +  «  }  /TUT J  (C:14)  - i - W i H i i  4  Neglecting the f i r s t term, which goes to two as  goes to  zero, (14) i s seen to go to i n f i n i t y as:  (C:15)  -82-  As the n a t u r a l l o g a r i t h m o f 2/3|  goes t o i n f i n i t y  more  s l o w l y than any power o f 2/3"? , i t i s c l e a r from (15) t h a t the area under the a b s o r p t i o n - c o e f f i c i e n t curve w i l l  remain  f i n i t e i n s p i t e of the d i v e r g e n c e o f (12). Values o f the i n t e g r a l F i g u r e 6.  (1) a r e p l o t t e d a g a i n s t  f in  -83-  APPENDIX D  The Absorption Coefficient for the Is  *>2p Transition. 0  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 ls  ( 0  * — • 2p  ( 6 ) 0  transition.  Rather than carry out a complicated energy-band calculation for the broadening of the 2s-2p impurity band, a tight-binding expression w i l l be assumed for the 2p  Q  energy,  mixing with the other n = 2 states being ignored.  The  parameters in this tight-binding expression w i l l then be determined from the results of Kohn and Luttinger (1955 c), and of Baltensperger (1953). l  f c  -  The function so obtained i s :  U [cos K*d + cos K a 3  •+- cos  K,a]  (D:l) where: 1 2p  *"  D  «= 1/6 of the 2p band-width given by  E  t h e  f  i s o l a t e d  ~^Purity P 2  0  —2 ev.  energy, -1.1 x 10  Baltensperger .  (D:2)  In the case of high concentration, E* must be modified Po 2  according to Baltensperger's calculations. Corrections to the effective mass theory w i l l be ignored for the 2p  Q  wave functions, so that:  -84-  *  V r T ^ W**  fc  *  ( D : 3  )  Using (3) and the corrected ground-state wave function (111:86), the integral:  x I° 2  =  - i - ,  ( F * Cr^ * Fis C c - S ) d r z  may be calculated.  o  (D:4)  B  Rough values for this integral are shown  in Figure 10. By (1), (4), and (111:74), the optical matrix element of interest i s , under the assumption that the six degenerate 2p  Q  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.GO  -  E\rCK^  (D:6)  By (5), (IV:20), and (IV:23), the absorption coefficient for the  Is^L»- 2p * *transition i s : 6  Q  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 i s included in (7) to account for the six-fold degeneracy of the 2p  Q  band.  The expression (7) may be simpli-  fied by the procedure which led to equation (IV:25) for the l s * ° ^ l s * ) case.  The result of this simplification i s :  5  • ^{"W^ « l * ? - H X t \ I  (D.9)  where:  4  dS J i n  1  *  S>"n'«i 4- S i V i a ,  +  l  (D:10)  cosx +<o$^ +• cos i - t  The integral (10) may be evaluated approximately by the methods of Appendix C. <* =  The element of surface area i s :  i  S  C  ± l-Ci^-cosx -conj">*  dx dy  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-integral  i  i n (11) may be e v a l u a t e d by means o f ( C : l l ) :  >/a<n-c*s*V  7 1 7 ^ ) ^=r/5(.-x^-tt-Tt\»  =  d  (D:14)  v/here: *  (D:15)  =  I t may e a s i l y be seen that  (12) corresponds t o r e s t r i c t i o n o f  x, y, and z t o v a l u e s near the bottom of t h e band. If ^  i  assumed l a r g e ,  s  the band, the i n t e g r a l  as i t w i l l be near the bottom o f  i n (14) may be e v a l u a t e d approximately.  The f i n a l r e s u l t i s :  * ~ JaR^iT )„ S(<-%V* The r e s u l t sideration  ~ /a*  ^  (16) may a l s o be o b t a i n e d from d i r e c t  o f the i n t e g r a l  i n (7).  ( D j l 6 )  con-  At the bottom of the band,  when x, y, and z a r e near z e r o , the c o s i n e terms i n (7) may be expanded  i n a Taylor series  Cos x + cos  + cos i  ~  about the o r i g i n : 3 - -1 r *  (D:17)  Hence the i n t e g r a l becomes:  o  (  TA  TI/  ./fir  (D:18)  -87-  Using th© d e l t a f u n c t i o n , (18) becomes: 4=  i  ^5=7  (D:19)  which i s the same as the r e s u l t o b t a i n e d i n (16). Near the c e n t r e of the band, the i n t e g r a l difficult  to evaluate.  (9) i s  Hence, as a f i r s t approximation,  the  e x p r e s s i o n o b t a i n e d f o r the bottom of the band w i l l be assumed to h o l d f o r a l l ( x , y , z ) .  The e x p r e s s i o n f o r the a b s o r p t i o n co-  e f f i c i e n t at the c e n t r e of the band then becomes:  since  I * 0 at the band c e n t r e .  The  absorption cross-section  i s then found by m u l t i p l y i n g (20) by a CT-  In F i g u r e  2 11  0  to obtain:  •^^°)f T,{ X »7a-] 4  (D:21)  u  the a b s o r p t i o n c r o s s - s e c t i o n (21) i s  p l o t t e d as a f u n c t i o n of a/a . that  3  (21) i s approximately  I t may  be seen from the graph  constant between a - 8a*  ik  a =» 14a. , w i t h a v a l u e of about 3 x 10  —15  and  2  cm  .  This value  i s i n q u i t e good agreement w i t h the experimental r e s u l t s of B i c h a r d and G i l e s , who  o b t a i n an a b s o r p t i o n c r o s s - s e c t i o n —15 2 v a l u e between 2 and 4 x 10 cm . The non-constancy of the c a l c u l a t e d c r o s s - s e c t i o n beyond  -88-  a = 14 a: i s p r o b a b l y due width 6U'  mainly t o i n a c c u r a c i e s i n the band-  as read from B a l t e n s p e r g e r ' s  from the n a t u r a l l i n e - w i d t h should be The  graph.  A contribution  included.  c l o s e agreement between the observed and  calculated  c r o s s - s e c t i o n s i s probably f o r t u i t o u s , s i n c e the use of expression justified.  (16)  f o r x,y,  the  and z away from the o r i g i n i s un-  I t i s expected t h a t c o r r e c t i o n of t h i s e r r o r  would i n c r e a s e the a b s o r p t i o n  coefficient  (20).  However,  t h i s c o r r e c t i o n should be at l e a s t p a r t i a l l y c a n c e l l e d  by  t a k i n g i n t o account the random d i s t r i b u t i o n of the i m p u r i t i e s .  -89-  BIBLIOGRAPHY Adams, E.N.  II. Phys. Rev. 85, 41 (1952).  Baltensperger, W.  Phil. Mag.  Bichard, J. and J.C. Giles. unpublished results. Bloch, F. Z. Physik Conwell, E.M.  7  , 44, 1355 (1953).  Private communication of  52, 555 (1928).  Phys. Rev. 103, 51 (1956).  Dekker, A.J. Cliffs,  Solid State Physics, Prentice-Hall, Englewood 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. Rev. 96, 222 (1954).  Phys.  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 S t a t e P h y s i c s 5_, 257  (1957).  Kohn, W.  and J.M.  Luttinger.  Phys. Rev. 97, 883  (1955).  Kohn, W.  and J.M.  Luttinger.  Phys. Rev.  97, 1721  Kohn, W.  and J.M.  Luttinger.  Phys. Rev.  98, 915  (1955). (1955).  K o s t e r , G.F.  and J.C. S l a t e r .  Phys. Rev. 95, 1167  (1954).  K o s t e r , G.F.  and J.C. S l a t e r .  Phys. Rev. 96, 1208  (1954).  Lampert, M.A.  Phys. Rev. 97, 352  (1955).  Lax, M.  J . Chem. Phys. 20, 1752  Lax, M.  "Proceedings o f the A t l a n t i c C i t y Conference on P h o t o c o n d u c t i v i t y , 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 . Luttinger,  J.M.  (1952).  Phys. Rev.  and W. Kohn.  110, 41  Phys. Rev.  (1958).  97, 869  Morin, F . J . , J.P. Maita, R.G. Shulman, and N.B. Phys. Rev. 96, 833(A) (1954).  (1955). Hannay.  P a u l i n g , L. and E.B. Wilson. I n t r o d u c t i o n to Quantum Mechanics. McGraw-Hill, New York, 1935. Reitz,  J.R.  Rosen, N.  S o l i d S t a t e P h y s i c s JL, 1 (1955). Phys. Rev.  38, 255  (1931).  S c h i f f , L . I . Quantum Mechanics. New York, 1955.  Second E d i t i o n ,  S e i t z , F. The Modern Theory of S o l i d s . York, YMU~. Slater,  J.C.  Phys. Rev.  87, 807  McGraw-Hill,  Wannier, G.H.  Koster.  Phys. Rev.  Phys. Rev.  52, 191  New  (1952).  S l a t e r , J.C. " E n c y c l o p e d i a of P h y s i c s " , ed. S. S p r i n g e r , B e r l i n , V o l . XIX, p. 1, 1956. S l a t e r , J.C. and G.F.  McGraw-Hill,  Flttgge,  94, 1498  (1954).  (1937).  Wannier, G.H. Elements of S o l i d S t a t e Theory. Cambridge U n i v e r s i t y P r e s s , 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  energy  r  (units o-P  - */xa" ) e  FIGURE 2 L o c a t i o n of Hydrogen-like Energy L e v e l s .  -941 x 10"',-  5K lo"*|  For higher values see Rosen (i93l).  of X ,  5 x ld*4-  £-5x10  1 x 10 5x lo" i5  Z-5 x »0*4-  Ix  10  FIGURE 3 a V a l u e s o f A-^x) and A g ( x )  -95-  O  o2  o.4  06  0.8  X FIGURE 3 b Values of Bj(x) and B ( x ) . 3  1.0  (4.6 x l o ' \ r n ) J  (20  x I0"crr.- )  ( l o x \o"cm-*)  3  (imparity concentrations in  (58  parentheses)  FIGURE 4 X? as a Function of Impurity Lattice Spacing.  x lo'^m ) 1  -97-  11  1-2  ls  < 5 >  1-3  energy  (units o f  1-4 -  Yxa* )  c  FIGURE 5 X* as a F u n c t i o n of the Energy of the l s  v  ' Level.  IS  FIGURE 6 The Integral: <J($) = as a Function of  [ < t J C J V s * * + sin*y i n  Cos K * co$  y «  ^  FIGURE 7 Tight-binding Energy Surfaces of (111:58) for cos k a Q  - 1 0 0 -  FIGURE 8 Comparison of Corrected and Approximate Energy Surfaces.  -101-  +  a  /a«  FIGURE 9 The l s * ° — > l s * ^ 5  Absorption C o e f f i c i e n t  a F u n c t i o n of Impurity L a t t i c e  Spacing.  as  -103-  H  1  1  h  1  1  H  h  i  5"x.o l ,5  I * id"*  5  25  x IO'I  KId  IKIO-'I  H  &  1  7  8  9  1  1  1  io  il  \Z  1  13  (14  FIGURE 11 The Absorption Cross-Section for the ls  ( 0  L,2p  0 ) o  Transition.  H  

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