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Impurity optical absorption and magnetic susceptibility in silicon and germanium Macek, Vilko 1971

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IMPURITY OPTICAL ABSORPTION AND MACNETIC SUSCEPTIBILITY IN SILICON AND GERMANIUM by V i l k o Macek D i p l . Ing. , U n i v e r s i t y c f Ljubljana, 1966 (Yugoslavia) A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1971 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Depa rtment The University of Brit ish Columbia Vancouver 8, Canada Date 21 r i i ABSTRACT Macek, V i l k o ; Ph.D. Thesis, U n i v e r s i t y of B.C., November 1971 IMPURITY OPTICAL ABSORPTION AND MAGNETIC SUSCEPTIBILITY  IN SILICON AND GERMANIUM Professor: Dr. Robert Barrie The eigenvalue problem for the two-electron system of a donor pa i r i n germanium and s i l i c o n was solved. Using the Heitler-London method the l s - l s e l e c t r o n energy spectra were c a l c u l a t e d as a function of the separation between the two donors i n the c r y s t a l . Also the widths of bands of c l o s e l y spaced ls-2p donor molecular l e v e l s were calculated as a i u n c t i o n or mtercionor separation. i n L i n t ; si-uuy Lue m a i i y — v a l l c y nature of the conduction band as well as the d i s c r e t e structure of the c r y s t a l medium of the host semiconductor were taken i n t o account. I t was observed that a l l donor p a i r s i n a germanium c r y s t a l can be divided i n t o three classes; any two donor p a i r s belonging to the same cl a s s have i d e n t i c a l e l e c t r o n energy spectra i f t h e i r interdonor separa-tions are equal. As an a p p l i c a t i o n of t h i s study of the donor-donor i n t e r a c t i o n , the linewidths of B and E i n f r a - r e d absorption l i n e s i n s i l i c o n and ger-manium were estimated as a function of donor concentration and compared with experimental r e s u l t s . A random d i s t r i b u t i o n of donors i n the c r y s t a l was assumed. It was supposed that o p t i c a l absorption takes place through e x c i t a t i o n of the e l e c t r o n system of a donor p a i r from an occupied state to a higher l y i n g excited state. The agreement between theory and experiment was very good. In another a p p l i c a t i o n of the r e s u l t s of the study of the donor-donor i n t e r a c t i o n , the magnetic s u s c e p t i b i l i t y of weakly i n t e r a c t i n g donors 16 3 i n germanium was studied as a function of donor concentration (10 cm 17 -3 £ £ 10 cm ) and temperature (1°K £ T £ 5°K). The system of i n t e r a c t -ing donors was treated as a system of i s o l a t e d donor p a i r s . A random d i s t r i b u t i o n of donors i n the c r y s t a l was assumed. The spin paramagnetic and o r b i t a l diamagnetic s u s c e p t i b i l i t i e s were ca l c u l a t e d and an upper l i m i t to the Van Vleck paramagnetic s u s c e p t i b i l i t y was c a l c u l a t e d . The r e s u l t s were compared to r e s u l t s of experimental measurements, and again a good agreement was observed. Special a t t e n t i o n was paid to the behaviour of antimony-doped germanium f o r which case the Sonder and Schweinler theory breaks down. iv TABLE OF CONTENTS Abstract i i Figure Captions v Acknowledgements , v i i I. Introduction 1 II. Eigenvalue Problem for a Donor Pair in Ge and Si 1. Electron States of a Single Donor .7 2. Electron States of a Donor Pair 19 a) The Role of Interference Coefficients ...38 b) Intermediate Interdonor Separations 44 c) Large Interdonor Separations 50 III Cci7CC7itr -*"Lz~ d~~cc ~f T T ? A%or>r<•>••-> i n fi«»fKf>') and Si(P) 62 IV. The Magnetic Susceptibility of Weakly Interacting Donors in Ge 1. Introduction 77 2. The Magnetic Susceptibility of Impure Semiconductors 81 3. The Spin Paramagnetic Susceptibility 90 4. The Orbital and Van Vleck Magnetic Susceptibilities 100 V. Summary •• 109 References v H3 Appendices A) Interference Coefficients for a Donor Pair in Ge 115 B) Calculation of — -( 4 T T ) 2 dR dH<*(l,2)\TC (1,2)|*(1,2)> 123 m V FIGURE CAPTIONS Page Fig. 2-1: The Is- and 2p-like shallow donor levels in Ge(Sb) and Si(P). 14 Fig. 2-2: Donor pair; A and B donor cores, 1 and 2 electrons. 20 Fig. 2-3: A schematic i l l u s t r a t i o n of the effect of the interference coefficient on the donor-molecular spectrum ( l s ^ l - l s ^ l states in S i ) . 32 Fig. 2-4: Simple face-centered cubic l a t t i c e structure. 39 Fig. 2-5: A schematic i l l u s t r a t i o n of the effect of the interference coefficients on the donor-molecular spectrum ( l s - l s states in Ge). 41 Fig. 2-6: The l s - l s spectrum of Sb donor molecule i n Ge: The envelopes of energy levels of the three molecular classes shown collec-tively as a function of the absolute interdonor separation. 47 Fig. 2-7: The l s - l s spectrum of Sb donor molecule in Ge: The gl and u^ energy levels of the three molecular classes shown separately. 48 Fig. 3-1: The E and B optical transitions in a single donor in Se and Ge. 63 Fig. 3-2: The E and B optical transitions in a donor pair i n Si and Ge. 64 Fig. 3-3: The broadening of the absorption line resulting from the tran-si t i o n from a ground l s - l s donor molecular state to a particular ls-2p excited state due to the random distribution of donors. 68 Fig. 3-4: The estimated concentration dependence of the widths of the E and B absorption lines in Si(P), together with experimental results. 70 Fig. 3-5: The estimated concentration dependence of the widths of the . E and B absorption lines in Ge(Sb), together with experimental results. 74 Fig. 4-1: The observed temperature and donor concentration dependence of the spin paramagnetic susceptibility of n-type semiconductor. (Schematic diagram). 78 Fig. 4-2: The relative size of kT at 4°K i n comparison with the l s - l s energy levels of a Sb donor pair in Ge. 85 Fig. 4-3: The spin paramagnetic susceptibility of Sb-doped Ge as a function of temperature and donor concentration. 93 v i Fig. 4-4: The spin paramagnetic s u s c e p t i b i l i t y of P-doped Ge as a function of temperature and donor concentration. 94 Fig. 4-5: Schematic i l l u s t r a t i o n of the temperature dependence of the f a c t o r ^ which gives r i s e to deviations from the Curie law. 97 Fig. 4-6: The energy l e v e l s relevant to the spin paramagnetic s u s c e p t i -b i l i t y , f o r two d i f f e r e n t dopants. 99 Fig. 4-7: The sum of the o r b i t a l diamagnetic and Van Vleck paramagnetic s u s c e p t i b i l i t i e s as a function of donor concentration, f or Sb- and As-doped Ge, together with the values extracted from r e s u l t s of experimental measurements of the t o t a l magnetic s u s c e p t i b i l i t y . 107 Fig. 4-8: The estimated t o t a l magnetic s u s c e p t i b i l i t y of Sb-doped Ge as a function of temperature, together with experimental r e s u l t s . 107 v i i ACKNOWLEDGEMENTS I wish to express my gratitude to Professor R. Ba r r i e for h i s valuable guidance and assistance during the course of t h i s work, and for hi s support i n obtaining f i n a n c i a l assistance for my studies. I am indebted to Dr. J . Marko for h i s h e l p f u l discussions and suggestions. Thanks are also due to Dr. I.W. Sharpe f o r discussions and encouragement. I wish to thank Dr. R. Kuwahara for providing me with the r e s u l t s of his experimental studies of o p t i c a l absorption i n s i l i c o n . I wish to acknowledge f i n a n c i a l support provided by the U n i v e r s i t y of B r i t i s h Columbia through the award of three Graduate Fellowships. I also thank Miss Rose Chabluk for typing t h i s d i f f i c u l t manuscript, - 1 -I. INTRODUCTION When a group V element, l i k e antimony or phosphorus, replaces a host atom at a l a t t i c e s i t e i n a germanium or s i l i c o n c r y s t a l , i t becomes a donor giving up an e l e c t r o n to the conduction band of the c r y s t a l . Experi-mental studies have shown that the p h y s i c a l properties of n-type semiconduc-tors strongly depend upon the concentration of donors. I t i s convenient to c l a s s i f y the impure semiconductors into three groups regarding t h e i r degree of doping, the low, intermediate, and high concentration of i m p u r i t i e s . B o t h extremes, the low and the high concen-t r a t i o n regions, have received considerable t h e o r e t i c a l treatment, and the experimental r e s u l t s have been reasonably well explained. The same cannot be said of the intermediate concentration region. The work presented i n t h i s t h e s i s i s intended to be a c o n t r i b u t i o n towards a better understanding of the behaviour of impure semiconductors i n the intermediate impurity concentration range. In the low impurity concentration range the behaviour of n-type semiconductors i s well understood i n terms of s i n g l e i s o l a t e d donors. At s u f f i c i e n t l y low temperatures the donor e l e c t r o n i s l o o s e l y bound to the p o s i t i v e donor core, thus g i v i n g r i s e to energy l e v e l s s l i g h t l y below the conduction band, w i t h i n the forbidden gap. These are the s o - c a l l e d shallow (2) donor l e v e l s . (3 4) An e f f e c t i v e mass theory has been developed by Kohn and Luttinger ' f o r the shallow donor st a t e s . In t h i s theory a donor i s treated as a hydrogen-like atom where the Coulomb a t t r a c t i o n between the donor core and - 2 -the bound e l e c t r o n i s reduced by the d i e l e c t r i c constant of the semiconductor host, and the e l e c t r o n i s assumed to move with the e f f e c t i v e mass of a conduction band e l e c t r o n . Thus the shallow donor l e v e l s are described by wave functions that are a c t u a l l y the conduction band Bloch functions modu-la t e d by the hydrogen-like s o - c a l l e d envelope functions. Since i n both s i l i c o n and germanium the conduction band has several equivalent minima, shallow donor states are degenerate. This degeneracy i s p a r t l y removed by i n t e r a c t i o n of the donor e l e c t r o n with the c r y s t a l l a t t i c e and the donor core i n the immediate v i c i n i t y of the l a t t e r , (the s o - c a l l e d c e n t r a l c e l l i n t e r a c t i o n ) . This i n t e r a c t i o n i s p a r t i c u l a r l y strong when the electron i s i n the l s - l i k e donor state that belongs to the i d e n t i t y representation of the c r y s t a l symmetry group ( i . e . the T^ group for s i l i c o n and germanium). As a r e s u l t t n x s state l i e s u u n s i u e i a u l j b e x u w Lu«= u u i c i l o - l i k c o t o L i J o i » i the energy spectrum. The energy d i f f e r e n c e between t h i s state and the other l s - l i k e states i s c a l l e d the v a l l e y - o r b i t s p l i t t i n g . I t v a r i e s s i g n i f i c a n t l y from dopant to dopant. The e f f e c t i v e mass theory s u c c e s s f u l l y accounts for the experimen-t a l l y observed low-temperature behaviour of n-type semiconductors at low donor 15 —3 16 —3 concentrations, < 10 cm i n germanium, and < 10 cm i n s i l i c o n (orders of magnitude), i . e . i n the region where no i n t e r a c t i o n s between donors take place. In the intermediate region, i n t e r a c t i o n s between donors s t a r t to take place, and become stronger and stronger with increasing donor concentra-17 —3 18 —3 t i o n . At high concentrations, > 10 cm i n germanium, > 10 cm i n s i l i c o n (orders of magnitude) these i n t e r a c t i o n s are so strong that they - 3 -give r i s e to the s o - c a l l e d impurity bands. Even at low temperatures donor electrons are not l o c a l i z e d any longer, at these high concentrations. This concentration region has received considerable t h e o r e t i c a l treatment i n (5 6) terms of a band theory. ' The intermediate concentration region i s characterized by the presence of donor-donor i n t e r a c t i o n s that are weak enough that donor electrons are l o c a l i z e d on t h e i r donor cores, yet strong enough that donors cannot be treated as i s o l a t e d . This donor i n t e r a c t i o n i s r e f l e c t e d i n the behaviour of the semiconductor i n t h i s concentration range. An example i s the magnetic s u s c e p t i b i l i t y . The molecular i n t e r a c t i o n between donors gives r i s e to spin alignment and a corresponding departure from the Curie law which i s obeyed ( 7 8") very w e l l at low donor concentrations. v ' ' There has been no t h e o r e t i c a l study of donor molecular states done so f a r , which would take into account the m u l t i v a l l e y nature of the conduction band, although the idea of donor molecules i s not new. Sonder and Schweinler have explained the observed deviations from the Curie law of donor magnetic (9) s u s c e p t i b i l i t y of n-type s i l i c o n on the basis of a molecular model. In t h e i r theory they assume a simple s p h e r i c a l l y symmetric conduction band with an e f f e c t i v e mass that i s an appropriate average of the a c t u a l transverse and l o n g i t u d i n a l masses of a conduction band e l e c t r o n . This amounts to employ-ing the r e s u l t s for a hydrogen molecule and s c a l i n g them so as to account for the d i f f e r e n c e i n mass and f o r the reduced Coulomb a t t r a c t i o n due to the d i e l e c t r i c constant of the host c r y s t a l . Attempting to apply the Sonder and Schweinler theory to donors i n germanium one notes that, while i t accounts reasonably w e l l f o r the observed deviations from the Curie law for phosphorus - 4 -and arsenic donors, i t breaks down i n the case of antimony donors. ^°' Our a s s e r t i o n i s that t h i s f a i l u r e i s due to the use of a s i m p l i f i e d conduction band and the r e s u l t i n g neglect of the higher l y i n g Is-type states. Namely, the v a l l e y - o r b i t s p l i t t i n g i n antimony i s so small that even at l i q u i d He temperatures some other states besides the lowest l y i n g one are thermally occupied. (For a l l dopants i n s i l i c o n , as w e l l as f o r phosphorus and arsenic i n germanium, the v a l l e y - o r b i t s p l i t t i n g i s large compared to kT at l i q u i d He temperatures, which j u s t i f i e s the neglect of higher l y i n g states.) This suggests the need for a d e t a i l e d study of the donor-molecular spectrum i n which the actual m u l t i v a l l e y nature of the conduction band i s taken into account. The a v a i l a b l e experimental data on the concentration dependence of the f a r i n f r a - r e d absorption l i n e w i d t h s 1 , 1 " 1 " ' s u g g e s t too that a study of donor-molecular spectra i s needed. One would expect that the increased broadening of the s i n g l e donor l e v e l s with increasing concentration would c e r t a i n l y r e s u l t i n an increased broadening of the absorption l i n e s . Therefore the object of t h i s thesis i s to study the molecular donor-donor i n t e r a c t i o n taking i n t o account the ac t u a l m u l t i v a l l e y nature of the conduction band and the anisotropy of the donor e l e c t r o n mass, and then apply the r e s u l t s of the study to the o p t i c a l absorption and magnetic s u s c e p t i b i l i t y of i n t e r a c t i n g donors. The work i s presented i n the following way. Chapter II i s devoted to the study of i n t e r a c t i o n between two donors. In section I I - l we review the e f f e c t i v e mass theory for i s o l a t e d donors, and then i n section II-2 we solve the eigenvalue problem f o r the donor p a i r , (12) following the Heitler-London method. ' This s e c t i o n contains three subsections. In II-2d (and Appendix) we discuss i n d e t a i l a s p e c i a l feature of donor molecules which i s absent i n ordinary molecules, namely the interference c o e f f i c i e n t s . They are a r e f l e c t i o n of the d i s c r e t e nature of the c r y s t a l medium which the donor molecule i s part of, as w e l l as of the m u l t i v a l l e y nature of the conduction band of the host semiconductor. In II-2b we present the r e s u l t s of computation of the l s - l s donor-molecular energy spectrum as a function of interdonor separation i n the molecule. The computation i s done f o r two d i f f e r e n t dopants i n germanium, phosphorus and antimony. Although the treatment of molecular states i n Ch. II-2 i s a p p l i -cable to n-type s i l i c o n as w e l l , the numerical computation of i t s energy spectrum i s not performed since i n ap p l i c a t i o n s i t i s not needed. (The model employing a s i n g l e conduction band i s quite s u f f i c i e n t for n-type s i l i c o n , due to the large value of the v a l l e y - o r b i t s p l i t t i n g . ) In Ch. II-2c we present an alternate approach that i s s u i t a b l e when the interdonor separations are large compared to the average separation of a donor electron from the donor core which i t i s l o c a l i z e d on. We use t h i s approach to c a l -culate the widths of the molecular "bands" of energy l e v e l s as a function of interdonor separation. In Chapter I I I we apply the r e s u l t s of Ch. I I to the case of o p t i c a l absorption at l i q u i d He temperatures, assuming that o p t i c a l absorp-t i o n takes place through e x c i t a t i o n of the e l e c t r o n system from a thermally occupied molecular l e v e l to an excited one. We take i t that most of the i n t e r a c t i o n between donors i s taken into account b y t r e a t i n g the assembly - 6 -of i n t e r a c t i n g donors as an assembly of i s o l a t e d donor p a i r s . We suppose that donors are d i s t r i b u t e d randomly throughout the c r y s t a l , and that the d i s t r i b u t i o n of i n t e r a c t i n g donors can be w e l l represented by Chandrasekhar 1s nearest neighbour p r o b a b i l i t y d i s t r i b u t i o n function (which i s a binary Poisson function, mathematically). In t h i s way we estimate the widths of absorption l i n e s f o r both s i l i c o n and germanium, and compare them with the experimental r e s u l t s . In Chapter IV we c a l c u l a t e the magnetic s u s c e p t i b i l i t y of i n t e r -a c t i n g donors i n germanium at l i q u i d He temperatures, on the basis of the r e s u l t s of Chapter I I , assuming again - as i n Chapter I I I - that most of the donor i n t e r a c t i o n i s taken into account by allowing each donor to i n t e r a c t with i t s nearest neighbour. Again we describe the d i s t r i b u t i o n of donor p a i r s i n the crystal, by C i i a u u i a s e t i i i c i i . & ZaacLiou. Tlic p ^ i t u u i ' ^ oLJt»cLivc cf t h i s chapter i s to c a l c u l a t e the spin parmagnetic s u s c e p t i b i l i t y of i n t e r -a c t i n g donors since i t i s very s e n s i t i v e to the d i s t r i b u t i o n of l e v e l s i n the donor e l e c t r o n energy spectrum. Since at present the only a v a i l a b l e experimental data on the spin paramagnetic s u s c e p t i b i l i t y are those extracted from the r e s u l t s of measurement of the t o t a l s u s c e p t i b i l i t y which includes also the o r b i t a l and Van Vleck s u s c e p t i b i l i t i e s , we include a c a l c u l a t i o n of these as w e l l , i n Chapter IV, and then make a comparison with experimen-t a l r e s u l t s . Chapter V contains a summary of r e s u l t s and observations of Chapters I I , I I I , and IV. - 7 -I I . EIGENVALUE PROBLEM FOR A DONOR PAIR IN Ge AND S i 1. E l e c t r o n States of a Single Donor When a group V element l i k e Sb or P replaces a host atom at a l a t t i c e point i n a germanium or s i l i c o n c r y s t a l , i t becomes a donor el e c t r o n up an e l e c t r o n to the c r y s t a l . At s u f f i c i e n t l y low temperatures t h i s donor i s l o o s e l y bound to the p o s i t i v e donor core, thus g i v i n g r i s e to energy l e v e l s s l i g h t l y below the conduction band, within the forbidden gap. These (2) are c a l l e d shallow donor l e v e l s . ' An effective-mass model of these donor states has been developed by Kohn and L u t t i n g e r . ^ In t h i s model the l o o s e l y bound donor e l e c t r o n and ion core are treated as a hydrogen-like atom where the Coulomb a t t r a c -ductor host, and the e l e c t r o n i s assumed to move with the e f f e c t i v e mass of a conduction band e l e c t r o n . Thus i n the effective-mass model one writes the wave function of a l o c a l i z e d donor e l e c t r o n as a product of the conduction band Bloch wave function and the eigenfunction of the Schrodinger equation f o r the hydrogen-like atom formed by the donor core and i t s l o o s e l y bound el e c t r o n . O u t l i n i n g the d e r i v a t i o n of Kohn and Lu t t i n g e r , we write the Hamiltonian f o r a donor e l e c t r o n i n the form ~K = 3K + U ( r ) , (2.1) where TCQ i s the Hamiltonian f o r an e l e c t r o n i n a pe r f e c t c r y s t a l , * 2 ? ^ = - T - V + V ( r ) , (2.2) o zm — V(r) being an e f f e c t i v e p e r i o d i c p o t e n t i a l . The eigenfunctions of 3c*o are the w e l l known Bloch functions <j> (k , r ) , n The second term i n the expression (2.1), U(_r), i s the a d d i t i o n a l p o t e n t i a l due to the replacement of a host atom i n the c r y s t a l l a t t i c e by a donor core. For distances r = |r| large compared to the l a t t i c e constant, U(r) i s of the form of a Coulomb p o t e n t i a l shielded by the p o l a r i z a t i o n of the c r y s t a l , 2 U(r) = , (2.4) where K denotes the s t a t i c d i e l e c t r i c constant of the c r y s t a l . Ho (-raat; TJ (y V i r i (9 .1 ") a n e r t i i r h a f i n n . We e.Trnand the eieen— functions iK_) of (2.1), 7^^(r) = E<Kr) (2.5) i n terms of the Bloch functions $ (k»_.) > defined by (2.3), tp(r) = 1 A (k) <J) (k,r) , (2.6) — n — n  n,k where n s p e c i f i e s the energy band to which the state of the ele c t r o n with the wave vector k belongs. The equations which A (k) s a t i s f y are given by n — (E (k) - E) A (k) + Z <nk|u|n'k'> A ,(k') = 0 (2.7) n - n , , - n -where - 9 -<nk|u|n'k'> = <j>n*(k,r)U ( r ) ^ , (k' ,r) dr . (2.8) One can show supposing that the p o t e n t i a l U(r) i s weak and slowly varying with p o s i t i o n i n the l a t t i c e , that the system (2.7) has solutions i n which A^OO i s n e g l i g i b l e unless n = 0 and the value of k i s near the p o s i t i o n of a conduction band minimum, k^. So as an approximation we set W k ) = / n ~ I 0 A Q ( j ) ( k ) ^ 0 i f n = 0 and k « k^, (2.9) otherwise and replace E o ( k ) and <0k|u|0k'> i n (2.7) by t h e i r expressions f o r k a k_. and k' c k., 2 Eo Q 0 -\ ( i - ^ ) 2 (2.10) . 2 <0k|u|0k'> -> — — (2.11) K|k - k'| Z * where the volume of the c r y s t a l i s set equal to un i t y . In (2.10) m denotes the e f f e c t i v e mass of a conduction band e l e c t r o n . Substituting (2.10) and (2.11) into (2.7) we get 9 •, 9 A ^ ^ I c M ° ~ K a l l k' ]k - k'| 2 •k 2m which i s the Schrodinger equation i n momentum space of an electron of mass m moving i n the a t t r a c t i v e Coulomb p o t e n t i a l (2.4). In order to convert t h i s equation into one i n coordinate space we introduce the so- c a l l e d envelope function F (r) as the Fourier transform of A o ^ ( k ) , F (r) = 1 A ( j ) ( k ) e 1 - - . (2.13) J ~ a l l k ° ~ - 10 -The equation (2.12) now becomes = E F.(r) (2.14) which i s a hydrogen-like, s o - c a l l e d e f f e c t i v e mass equation. Knowing the d i e l e c t r i c constant K of the semiconductor host c r y s t a l and the e f f e c t i v e mass tensor m of the conduction electron we can solve the equation (2.14) to get the spectrum of the shallow donor l e v e l s . which we wanted to dei"ive. I f we solve the Schrodinger equation (2.14), i t s eigenfunctions F^(r) turn out to extend over many l a t t i c e s i t e s i n the c r y s t a l . This implies that l o c a l l y a donor el e c t r o n described by the wave function (2.15) moves as i f i t were i n the Bloch state <J>o (k_ ,_r); i f we follow the el e c t r o n over a distance of a number of l a t t i c e s i t e s , however, i t s wave function i s slowly modulated by the hydrogen-like function F^(r) as a r e s u l t of the weak Coulomb a t t r a c t i o n of the donor core. This i s why we c a l l the function F^ . (r) an envelope function. Yet we must not forget that t h i s i s only the f i r s t approximation; the a c t u a l donor electron wave function i s a wave packet composed of Bloch waves a r i s i n g from the v i c i n i t y of the conduction band minimum [see eq. (2.13)]. The conduction band i n both germanium and s i l i c o n has several equivalent minima. Therefore the t o t a l effective-mass-theory wave function of a donor el e c t r o n i s a l i n e a r combination of functions of the form (2.15), Combining (2.13) with (2.9) and (2.6) we obtain <Kr) = r J'(r) = F. (r)6 (k ,r) j • o '-(2.15) * 00 = Z u — J = (V) _)* 0CJSj,_), y = 1, . • •, n v (2.16) - 11 -where the sum over i is a summation over a l l n conduction band minima. In v germanium, n^ = 4, and in s i l i c o n n^ = 6. The coefficients a^ . (u) in (2.16), describing the relative contribution from the j-th minimum of the conduction band to the wave function ^ (r), also provide that I|J (r_) belongs to an irreducible representation of the point group of the host crystal (i.e. T^ group in the case of germanium and s i l i c o n ) . Cyclotron resonance studies have shown that the effective mass of a conduction electron in s i l i c o n and germanium is anisotropic. In si l i c o n , the conduction band minima l i e at about 85% the way between k = 0 and the zone boundaries on the (0,0,1) axis and the equivalent a x e s . ^ ^ The energy surfaces near these points are ellipsoids whose axes are oriented in the direction of the corresponding conduction band minima, 2 2 E (k) = ^ - ( k - k ) 2 + (k 2 + k 2) , (2.17) o — 2m^  z o 2mfc x y where energy is measured from the bottom of the conduction band, where k = (k , k , k ) — v x' y' z and (2.18) h. = (0, 0, k o) Here k.^  i s any of the conduction band minima. The longitudinal and trans-verse masses, m and m , appearing i n (2.17) have been found to have the following values: m = 0.98 m 16 (2.19) mfc = 0.19 m m being the mass of a free electron. - 12 -In germanium the conduction band minima occur at the boundaries of the zone i n the d i r e c t i o n (1,1,1) and the equivalent ones. Again, the energy i n the v i c i n i t y of a conduction band minimum i s given by an expression of the form (2.17), with the l o n g i t u d i n a l and transverse masses having the following values: m = 1.60 m 1 (2.20) mt = 0.08 m In terms of m^ and mt the Schrodinger equation (2.14) i s expressed i n the form 2 .2 Jl 7 ,2 2 2m7 7~~2 " 2m~ ( ~ 2 " 2 J ~ ^r" F j <-> = * e f f *.,<_> , ( 2 . 2 1 ) ii 3z. t dx. 9y. J J J 9y. where we. wrote E - . i n s t e a d of E to s t r e s s that ( 2 . 2 1 ) i s an e f f e c t i v e mass e n equation. For the l s - l i k e s o l u t i o n of t h i s equation Kohn and L u t t i n g e r v ' have used a t r i a l f u n ction of the form F (r) = 1 e ~ \ A + y L + f L , (2.22) a 2 • b 2 /rra^b and determined the i o n i z a t i o n energy E i n equation (2.21) and the e f f e c -t i v e transverse and l o n g i t u d i n a l Bohr r a d i i , a and b, appearing i n (2.22), using a v a r i a t i o n a l method. For germanium, using the values (2.20) and the d i e l e c t r i c constant K = 16, the following values were found: a = 65.4 A b = 22.7 A (2.23) E c c = 9.2 meV ef f - 13 -The r e s u l t i s the same for a l l four v a l l e y s , of course. Therefore the ground s t a t e as given by the e f f e c t i v e mass theory i s f o u r f o l d degenerate, and i s described by the wave functions (2.16), the envelope functions (r_) being given by (2.22). A s i m i l a r r e s u l t i s obtained f o r the case of s i l i c o n . The ground state i s s i x - f o l d degenerate, and i s described by the wave functions (2.16), where the envelope functions have the form (2.22) again. The values (2.19) for the l o n g i t u d i n a l and transverse masses and K = 12 f o r the d i e l e c t r i c constant, y i e l d the following: a = 25.0 A b = 14.2 A (2.24) E _ = 29.0 meV We note that the i o n i z a t i o n energy as given by the e f f e c t i v e mass theory i s determined by the properties of the conduction band of the host semiconductor, and does not depend on the p a r t i c u l a r dopant. Experimental observations, however, have shown that i o n i z a t i o n energies do vary from dopant to dopant. Moreover, the l s - l i k e states, described by the wave functions (2.16) have been found to be s p l i t i n energy. In germanium, ^ --^ there are two l s - l i k e donor l e v e l s , a nondegenerate ground l e v e l , c a l l e d s i n g l e t , and a thr e e f o l d degenerate excited l s - l i k e l e v e l , c a l l e d t r i p l e t . In s i l i c o n , there are three l s - l i k e donor l e v e l s , a s i n g l e t , a doublet, and a t r i p l e t . These l e v e l s are shown i n the diagram F i g . 2.1. In the diagram, each l e v e l i s l a b e l l e d by the i r r e d u c i b l e representation of the group T^ (the point group of the germanium and s i l i c o n c r y s t a l ) , according Ge(Sb) 0.+ ///////////////// c.B. (8) 2p" (E + T± + T 2) (4) 5. 2p u (A x + T x) 10. 4-(3) (1) Is (Tj_) Is (Aj) -v if [meV] 0. 4-S i CP) /////////////// C.B. 20. 40. (12) (6) (2) (3) (1) -E T [meV] 2p _(2T 1 + 2T 2) 2p°(A 1 + E + T j ] Is (E) Is (T) i s (A x) F i g . 2-1. The I s - and 2p-like shallow donor l e v e l s i n G;(Sb) and S i ( P ) . - 15 -to which the wave functions describing the p a r t i c u l a r l e v e l transform. In terms of the c o e f f i c i e n t s a.(y)'appearing i n (2.16) these l s - l i k e states are described by the following: y a .(y) 1 Irrep. 1 1 /6 a, i , i , i , i , D A l 2 1 Jl2 a, 1, 1, 1, -2, -2) E 3 1 2 (-i , -1, 1, 1, 0, 0) 4 . 1 Jl a, -1, 0, 0, 0, 0) 5 1 /2 (0, 0, 1, -1, 0, 0) T l 6 1 /2 • (0, 0, 0, 0, 1, -1) Ge: y a.(y) Irrep. l 1 2 (1, 1, 1, 1) A l 2 1 2 ( i , -1, -1, 1) 3 1 2 (1, -1, 1, -1) T i : 4 1 2. (1, 1, -1, -1) (2.26) Two notes: 1) The two rows belonging to E are d i f f e r e n t from those given by Kohn and Luttinger. The l a t t e r are not orthogonalized. 2) Some authors, e.g. Tinkham, use notation T2 i n place of our T-^ . - 16 -For a l l donors the energy of the doublet and t r i p l e t l e v e l s ( i f they e x i s t ) l i e reasonably close to that predicted by the e f f e c t i v e mass theory, whereas the s i n g l e t l e v e l l i e s considerably below i t . The magni-tude of separation of t h i s l e v e l from the others (the s o - c a l l e d v a l l e y -o r b i t s p l i t t i n g ) v a r i e s from dopant to dopant. In germanium i t ranges from 0.32 meV (Sb) to 4.2 meV (As), and i n s i l i c o n from 10 meV (Sb) to 21 meV (As). The reason for f a i l u r e of the e f f e c t i v e mass theory to reproduce t h i s dopant-dependent s p l i t t i n g i s the f a c t that the Hamiltonian i n (2.21) i s only an approximation. While i t i s a good approximation at large d i s -o tances from the donor core ( i . e . r >> a, , which i s equal to 5.66 A l a t t i c e ' n o for germanium, and to 5.42 A f o r s i l i c o n ) , i t i s wrong at small distances. The concept of d i e l e c t r i c constant appearing i n (2.21) i s meaningless at small distances. Also the s u b s t i t u t i o n a l donor core may d i s t o r t the l a t t i c e s i g n i f i c a n t l y i n i t s immediate v i c i n i t y . Besides, i f r £ a. . xattxce strong i n t e r a c t i o n s between the donor el e c t r o n and the electrons of the donor core take place. For these reasons the a c t u a l energy l e v e l s l i e lower than predicted by the e f f e c t i v e mass theory. The s i n g l e t l e v e l i s a f f e c t e d much more than the others because the s i n g l e t wave function i s nonvanishing at the s i t e of the donor core whereas the t r i p l e t and doublet wavefunctions do vanish, as one can see from (2.25) and (2.26). (2) To account for t h i s , Kohn introduced the s o - c a l l e d c e n t r a l c e l l c o r r e c t i o n f a c t o r 3 into the envelope functions (2.22), defined as - 17 -where i s the observed energy of the state described by i/V (r_). The corrected envelope functions become, for large r ( i . e . r >> a n . ), j_at txce x 2 + y 2 . z 2 F ( y ) ( £ ) _ c o n s t x e 6 y V ^ a2 j + (2.28) This i s the asymptotic s o l u t i o n of the e f f e c t i v e mass Schrodinger equation (2.21) with E e j £ replaced by the observed energy E ^ \ -2 .2 .^ 2 ..2 ..2 2 Ti 3 h 3 3 e 2m„ 2 2ni _ 2 2 «r £ 3z. t 3x. 3y. 3 3 3 F ( y ) _ ) = E ( y ) F ( v ) < r ) . (2.29) Since E ^ i s not an eigenvalue of the Hamiltonian appearing i n (2.29) we had to drop a boundary condition, F (0) = f i n i t e , i n order to get (2 .28) , so that we cannot use the equation (2.29) f o r small r . This does not pre-sent any a d d i t i o n a l problem since the Hamiltonian i n (2.29) i s wrong at small r anyway, as mentioned above. In a l l our ap p l i c a t i o n s l a t e r on of the envelope functions, r w i l l be much la r g e r than the c r y s t a l l a t t i c e spacing, i n the region of our i n t e r e s t , so that we can pretend that F ^ ^ ( r ) has the form (2.28) f o r a l l r , and normalize i t over the whole space, x 2 + y 2 z? In terms of these corrected envelope functions the donor el e c t r o n wave functions (2.16) assume the form E V a i - y ) F , ( v , ) C r ) ^ ( K , , r ) , u =» 1, • • • > % . (2.31) - 18 -We can treat the 2p-like states i n the same way as the l s - l i k e s t a t e s . Again they are described by, the wave functions of the form (2.31), The corresponding envelope functions are the 2p-like s o l u t i o n s of the e f f e c t i v e mass Schrodinger equation (2.29). Since t h i s equation i s c y l i n -d r i c a l l y symmetric one can define the magnetic quantum number m. States corresponding to d i f f e r e n t |m| are associated with d i f f e r e n t l e v e l s , but states corresponding to m and -m are s t i l l degenerate. So we l a b e l the 2p-like states by o f o r m= 0, and by ± for m = ±1. J 1 iTa b J ' ira^b a/2 where 6 and 6, are the c o r r e c t i o n f a c t o r s analogous to 3 defined by (2.27), o ± u Since the functions (2.32) and (2.33) vanish at the o r i g i n , the c e n t r a l c e l l i n t e r a c t i o n s are very small. In f a c t , within the accuracy of the /-i £\ experimental measurements of energy of the 2p-like donor l e v e l s the (2) f a c t o r s 6 and <$, are equal to 1 , w o ± The c o e f f i c i e n t s a f or these 2p-like states are d i f f e r e n t j from those f o r the l s - l i k e states, i n general. They can be determined by elementary group theory. In F i g . 2.1 the 2p-like l e v e l s are shown for antimony donor i n germanium and f o r phosphorus donor i n s i l i c o n . They are l a b e l l e d by the i r r e d u c i b l e representations of the group T^, i n the same fashion as the l s -l i k e ones. - 19 -2. E l e c t r o n States of a Donor F a i r In the previous section we reviewed the e l e c t r o n i c states of a s i n g l e donor i n a germanium or s i l i c o n host c r y s t a l . Let us create now an i d e n t i c a l donor at a separation R from the f i r s t one thus forming a two-donor "molecule", for now each donor e l e c t r o n i n t e r a c t s with both cores as w e l l as with the other e l e c t r o n . The e s s e n t i a l d i f f e r e n c e between an ordinary molecule and ours i s i n the f a c t that the l a t t e r i s part of a c r y s t a l . The interdonor sepa-r a t i o n R has to comply with the c r y s t a l l a t t i c e s t r u c t u r e ; i t can only assume d i s c r e t e values. In a d d i t i o n to the i n t e r a c t i o n with the donor cores and the other e l e c t r o n , each electron also interacts, with the p e r i o d i c f i e l d of the c r y s t a l l a t t i c e . Therefore the Hamiltoaaian 7C(ltZ) for the electrons of our molecule has the following form: *<1,2) - - i_ <vj + V*) + V ( l ) + V(2) " ^ ~ 2 2 2 - — — + , (2.34) K r l B K r 2 A K r 1 2 where 1 and 2 r e f e r to the coordinates of the two e l e c t r o n s , and V denotes the e f f e c t i v e p e r i o d i c p o t e n t i a l of the c r y s t a l l a t t i c e , as i n eq. (2.2). In the Coulomb terms i n (2.34) A and B denote the two donor cores so that r, » i s the distance from the f i r s t e l e ctron to the donor core A, etc. (see 1A F i g . 2.2); r ^ i s the distance between the electrons.. The Coulomb i n t e r -a c t i o n between the donor cores i s omitted i n the Haiailtonian (2.34) since i t does not a f f e c t the e l e c t r o n states of our molecule. The p o s i t i o n of 20 ~ T i g . 2-2. Donor p a i r ; A and B donor cores, 1 and 2 electrons. the donor cores i s f i x e d i n the c r y s t a l l a t t i c e so that they cannot adjust t h e i r separation so as to y i e l d the minimum i n the t o t a l energy of the molecule. Arranging the terms i n (2.34) we can express #(1,2) i n terms of the single, donor Hamiltonians #(1) and 7t'(2), 2 2 2 *(1,2) = Xf(l) + fc(2) - — — + 6 K r 2 B K r 2 A K r 1 2 (2.35) with - i ! V J + v ( 1 ) . _ £ 1A (2.36) and * ( 2 ) = - ^ V 2 + V(2) 2B (2.37) - 21 -The s i n g l e donor Hamiltonian has been discussed i n the previous section of t h i s chapter. We know i t s energy eigenvalues and eigenfunctions. 7 ( 1 ) ^ ( 1 ) = E ( y ) ^ ( l ) (2.38) • 7 5 f ( 2 ) ^ ( 2 ) = E ( y ) ^ ( 2 ) (2.39) The superscripts A and B on the donor electron wave function mean that the corresponding functions are centered on the donor cores A and B, respec-t i v e l y . In terms of the donor el e c t r o n wave functions (2.31) they are expressed as follows: Ka) bV±IA> = ^ ( ^ i + f ) ( 2 - * 0 ) 0 2 > 5 = * P < r 2 - f ) . ^ (2.41) where r_ and are coordinates of the electrons 1 and 2, r e s p e c t i v e l y , measured from the midpoint between the two donor cores. Let us proceed now to solve the eigenvalue problem f o r our donor p a i r system whose Hamiltonian i s given by (2.34). There i s a v a r i e t y of methods a v a i l a b l e f o r t h i s purpose. We s h a l l follow the Heitler-London method. The reason we choose t h i s p a r t i c u l a r one i s that apart from being the simplest one i t i s known that i t gives reasonably good r e s u l t s -e s p e c i a l l y concerning the spin s i n g l e t - t r i p l e t energy differences - for hydrogenic molecules with internuclear separations |R| i n the i n t e r v a l R n < IRI < R , where R„ ^ 1.6 and R ~ 50 i n units of the Bohr r a d i u s . ( 1 7 ^ £ ' — 1 u £ u In our a p p l i c a t i o n of the r e s u l t s of t h i s c a l c u l a t i o n , the interdonor separations w i l l be well within the above-mentioned i n t e r v a l . Besides we - 22 -already have the s i n g l e donor electron wave functions that are needed i n the Heitler-London treatment of a donor pair system. The basic p r i n c i p l e of the Heitler-London method i s to b u i l d up the two-electron molecular basis functions of proper symmetries using the (one-center) one-electron wave functions, and then f i n d l i n e a r combina-tions of these basis functions such that they minimize the Hamiltonian of the system. - • There i s one d i f f i c u l t y , however, i n applying the Heitler-London method to the case of our donor molecule. I t i s obvious that the Hamilton-ian (2.34) for our donor electron p a i r system lacks ( i n general) the center of i n v e r s i o n symmetry as well as the r e f l e c t i o n symmetry plane. This i s a consequence oZ the e r ^ s t & l c t r u c t u r c o f the r.cdit!TT. 5" ch the H o n o r molecule i s embedded. When we write the Hamiltonian i n the form (2.35) a l l the depen-dence on the p e r i o d i c f i e l d of the c r y s t a l l a t t i c e comes into the si n g l e donor Hamiltonians. Therefore by using the eigenfunctions of the s i n g l e donor Hamiltonian to b u i l d up the Heitler-London molecular basis functions a great deal of the e f f e c t of the c r y s t a l l a t t i c e i s being taken into account. The s i n g l e donor e l e c t r o n wave functions r e f l e c t the c r y s t a l l a t t i c e structure both i n t h e i r composition [see eq. (2.15), (2.31)] and i n t h e i r energy eigenvalues [eqs. (2.14), (2.29)]. This of course does not mean that now we can forget that the Hamiltonian (2.35) l a c k s , i n general, the center of in v e r s i o n symmetry and the r e f l e c t i o n symmetry plane; the proper molecular eigenfunctions are, i n general, symmetric neither under - 23 -inversion nor under reflection through a plane. But for the purpose of energy calculation i t certainly w i l l be a good approximation to treat the Hamiltonian (2.35) formally as though i t had the symmetries of an ordinary two-center homonuclear molecule, not forgetting - when calculating the matrix elements between the Heitler-London basis functions and the Hamil-tonian - that the simple donor wave functions composing the Heitler-London basis functions are eigenfunctions of ^(1) and #t2) [see eqs. (2.35), (2.38), (2.39)]. In other words, in our model of a donor molecule we take i t that the molecule i s embedded in a continuous medium (thus possessing the center of inversion symmetry and the reflection symmetry plane) while at the same time i t s electrons move local l y in the periodic potential f i e l d due to the discrete crystal l a t t i c e as well as in the Coulomb potential f i e l d due to the donor cores and the other electron. (Mathematically, one can calculate the energy spectrum of the donor molecule as a function of a continuous interdonor separation R, although the result has a physical meaning only at discrete values of _.) This results in some undue degeneracies in the energy spectrum. Yet we expect the splittings due to the difference between the real Hamiltonian and ours to be small, especially at larger interdonor separations. The larger the distance between the two donors (as compared to the l a t t i c e parameter, a ) the better one would .Lattice expect our approximation to be. Let us consider now the case of a two-donor molecule whose two electrons are in the same, nondegenerate single donor state, the other states being far enough away that we can neglect their mixing in. (Such a case arises with donors in s i l i c o n where due to the - 24 -large value of the v a l l e y - o r b i t s p l i t t i n g we can take into account only the lowest l s - l i k e donor state i f we are interes t e d i n the lowest l y i n g molecular l e v e l s . ) If we denote the sing l e donor wave function by ^ and the molecu-l a r wave function by T' ', then the l a t t e r can be expressed as follows, according to the Heitler-London method, ^ ( 1 , 2 ) = - ^ ( D ^ ( 2 ) , (2.42) where 1 and 2 r e f e r to coordinates of the two electrons, and A and B denote the donor cores. on which the p a r t i c u l a r s i n g l e donor wave functions are centered. Since the Hamiltonian (2.35) i s i n v a r i a n t under interchanges of both the electrons and the donor cores, the appropriate basis functions have to possess the corresponding symmetries as w e l l . Such basis functions can be formed by properly symmetrizing the fu n c t i o n (2.42). Formally the f K) symmetrizing operators 0 are given by the following: o ( 1 ) = ( i + P 1 2 ) (1 + J ) o ( 2 ) = ( i - P 1 2 ) (1 - J ) (2.43) where P^ 2 represents a space-coordinate permutation operator, and J i s an inv e r s i o n operator. p i 2 V fcr ^ = \v fe2'ri> C2.44) J ¥uy - f c l ' £2> = \ v (-*L» - 25 -The o r i g i n of r_ i s taken to be at the midpoint of R, the separation of the two donors. Applying the operators (2.43) on the function (2.42), we get two Heitler-London basis functions, / K ) ( l , 2 ) = 0 ( K ) Y (1,2) , K = 1, 2 , (2.45) uy uu or, i f we write them out e x p l i c i t l y , ^ ( 1 , 2 ) = ^ ( D ^ ( 2 ) + ^ ( 1 ) ^ ( 2 ) (2.46) V™ (1,2) = ^ ( 1 ) ^ ( 2 ) - ^ ( 1 ) ^ ( 2 ) . (2.47) Since we are only interested i n c a l c u l a t i n g the energy spectrum i n the absence of a magnetic f i e l d , we do not have to include spin wave functions e x p l i c i t l y into our basis functions. However, the function (2.46) describes a spin s i n g l e t state, and the function (2.47) a spin t r i p l e t s t ate. Regard-ing t h e i r symmetry with respect to inversion, the state described by the function (2.46) i s c a l l e d the gerade ( i . e . "even" i n German) state, and the one described by (2.47) the ungerade ("odd") st a t e . As discussed above, we can l a b e l the states by these names i n view of our approximation only, i n general. The functions (2.46) and (2.47) are not normalized. They are, however, orthogonal to each other, owing to t h e i r symmetry properties. Therefore, the energy of the states described by these functions i s deter-mined by the equations (1,2)| tf(l,2)|/^(l,2)> - £ < ^ ) ( 1 , 2 ) | ^ ) ( 1 , 2 ) > = 0 , K = 1,2,(2.48) - 26 -or « F ( K ) ( 1 , 2 ) | #(l,2)|f w ( l , 2 ) > c. yy 1 1 yy £ K ,(K) < / K ) ( l 2) ,2)> ,00. , K = 1,2. C2.49) If we i n s e r t the expression (2.31) f or the s i n g l e donor wave function into (2.46) and (2.47), and then use these to c a l c u l a t e the matrix elements appearing i n (2.48) we get the following < / K ) (1,2) I #(1,2) 1 ^ ( 1 , 2 ) > = yy 1 yy ,00 i j k l 1 J k 1 J * I F k V ) ^ * o f e k ^ l A > F l P ) ^ 2 B > * o ^ l ^ 2 B > l d * l ^ 2 - • I T 1 J k 1 i j k l  ; [ ^ j ) ( r 1 B ) * 0 ( k . , r 1 B ) F ^ > ( r 2 A ) * o ( k . , r 2 A ) ] * #(1,2) [ F - ^ r ^ ^ ^ . r ^ F ^ ^ r ^ ) ^ ^ , ^ ) ] ^ ± L a. a. a, a, . ., , x 1 k 1 i j k l J J ^ • ^ ^ I A H O ^ ^ A ^ ^ ^ B ^ O ^ ^ B ^ * ^1 ' 2 ) * ^ ^ o K ^ F i y ) C £ 2 A > * o ^ l ^ 2 A > l d ^ l ^ 2 i j k l [ ^ ( r U ( k . , r 1 R ) F ^ ( r 9 A H < ^ , r 9 A ) ] * * ( l , 2 ) x i " j u k u l J L i v - l B / Y o v ^ i ' - l B y i j v - 2 A / Y o ^ j ' - 2 A ' ,00 and an analogous expression f o r < ¥ " " ^ ( 1 > 2 ) l ^ " " ^ ( 1 , 2 ) > . In (2.50) the uu ? 1 uu • upper sign applies when K = 1, and the lower ones when K = 2. To f a c i l i t a t e the c a l c u l a t i o n of the i n t e g r a l s i n (2.50) we make use of the f a c t that the envelope functions F(r) ~vary slowly as compared to the Bloch functions <j>0(k>r_). We trea t the envelope functions i n i n t e g r a l s i n (2.50) as being constant over a un i t c e l l , and break up the i n t e g r a l i n v o l v i n g both envelope and Bloch functions into an i n t e g r a l over the whole space i n v o l v i n g envelope functions, and an i n t e -g r a l over a unit c e l l i n v o l v i n g Bloch functions. In the c a l c u l a t i o n of the l a t t e r i n t e g r a l , we make use of the f a c t that Bloch functions are orthogonal and normalized i n a un i t c e l l , J ° u n i t c e l l and p e r i o d i c i n the sense r) <f> (k.,r)dr = 6... , , 0 J C J - ->-J v -> <f>0(k,r_ ± R) = <j>o(k,r) e ± : L—*— , (2.52) where R i s a vector connecting any two l a t t i c e s i t e s i n the c r y s t a l . Having c a r r i e d out these intermediate steps we can write (2.50) i n the following form < V ™ (1,2) | *(1,2) | ¥ ^ ) ( 1 > 2 ) > = = 2 I a ^ a ^ a W ^ <AV | | AV> (2.53) ± j 1 1 3 J y y 1 1 u y ± 2 Z a < y ) a < y V P ) a < y ) <AVMBV> COS [ (k. - k. )-R] i j i i j j y y 1 1 y y - I - J -- 28 -where <A1BJ' I TC U V > y y 1 1 y y ,(y) ,(y) t ^ y ( r 1 A ) F ] - ( r 2 B ) ] " 7 i r ( l , 2 ) [ F ^ ( r 1 A ) F ^ ( r ^ ) ] d r _ d r 2 , (2.54) (y) and <A 1 B ^ |7C|B J A |> i s defined analogously. y y 1 1 y y We use the notation a) B A H F * ^B> = F i ' f e - f *> C2.55) We get a s i m i l a r expression f o r the overlap i n t e g r a l , r(K) <y V i V /.(l,2) k ( K ) ( l , 2 ) > = yy 1 yy ' 2 S a ^ W ^ a ^ A V u V * ± i 1 i J J y y ' y y C2.56) ± 2 S O W ^ W ^ V . B V : * cos[(k. - k.)-R] i j 1 1 J 3 M M{ \x U - l - j y -If now we i n s e r t the expression (2.35) for the Hamiltonian i n t o (2.54), we obtain the follow i n g : < A V | 7 ? | A V > = 2 E ( P ) - <Ai\-^- \Al> - <B J |-^— + < A V 1 - ^ - 1 A V > y' K r _ B - y y ' K r 2 A ' y C2.57) 12 y y and s i m i l a r l y , <A I B J |^|B 1 AJ> = 2 E ( Y ) S 2 - ^ M H B S S - <BJ |-^MA J > S y y ' y y v y K r ] _ g y y y K R 2 A Y 2 < R 2 A ' Y Y + < A V | - ^ — | B V > , y y 1 < r 1 2 1 y y ' (2.58) <A1B:3 I A^^ y y ' y y = 1 , (2.59) - 29 -<AV = S 2 . (2.60) y y y u y Here we took i n t o account the fac t that the envelope functions are nor-' malized over the whole space, <Ai|A1> = <Bj|Bj> = 1 , (2.61) y 1 y y y ' and we defined the overlap of the envelope functions S as S H <Ai|B1> = <Aj|Bj> . (2.62) y y1 y y1 y In order to s i m p l i f y the c a l c u l a t i o n of matrix elements of Coulomb terms between envelope functions (which are a n i s o t r o p i c , see eq. (2.30)), entering the expressions (2.57) and (2.58), we introduce the * s o - c a l l e d mean e f f e c t i v e Bohr radius a , that i s , we make the following replacement i n the envelope functions: 2 ± 2 2 2 ^ 2 ^ 2 0 x. + y. z. x. + y. + z. 2 2 ,2 *2 *2 U.bJ) a b a a No e s s e n t i a l feature i s l o s t by such an approximation i n the matrix e l e -ments since the important information about the many-valley nature of the energy surface i s already contained i n the c o e f f i c i e n t s a . ^ . Numeri-J * o c a l l y , the mean Bohr radius for donors i n germanium i s equal to a = 45.0 A, A ° + and i n s i l i c o n a = 20.4 A. In t h i s approximation we can evaluate a l l the matrix elements i n (2.57) and (2.58) using standard formulae and tables of molecular i n t e g r a l s . Now we can write (2.57) and (2.58) i n the form <A±BJ 17^  |AV> = 2 E ( y ) + ~ ( 2 ' 6 4 ) y y1 y y c ^ The numerical values f o r a* are. chosen i n such a way that e 2/x a* i s equal to the i o n i z a t i o n energy obtained by the e f f e c t i v e mass theory. 30 -<A± B ^ l B 1 AJ> = 2 E ( y ) S 2 + H^? , (2.65) y y 1 1 u Vi ]i E where the quantity H ^ can be referred to as the Coulomb i n t e g r a l , and • Li t L g ^ as the exchange i n t e g r a l , i n analogy with the standard Heitler-London treatment of molecules. How they are defined i n terms of the matrix elements of the Coulomb p o t e n t i a l terms between envelope functions, i s evident by comparing (2.64) with (2.57), and (2.65) with (2.58). We note that they depend on the p a r t i c u l a r s i n g l e donor state y, but because of the approximation (2.63) they do not depend on p a r t i c u l a r v a l l e y s i and j any longer. Thus when i n s e r t i n g (2.64) and (2.65) into (2.53) we can take them out of the summation and put them i n front of the summation sign. < T ( K ) (1,2)1 #(1,2)|Y ( K )(1,2)> = 2 { ( 2 E ( y ) + H< y )) E a <y V y ) a < y )a < y ) yy yy c . ' i i j j ± ( 2 E ( y ) S 2 + H< y )) E a ^ a ^ a ? 0 ^ cos[(k. - k.)-R]) (2.66) y E ± ^  i i 3 3 -x -3 -I f now we take into account that the c o e f f i c i e n t s are normalized, J Z a f y V y ) - 1 , (2.67) i 3 2 and we define a c o e f f i c i e n t . I ( y ) by the following expression, I(y) = Z a f y ) a f p ) a f y V y ) cos [ (k. - k.)-R] , (2.68) ± j i i J J - l - j - ' we can write the matrix element (2.50) i n the f i n a l form < ^ ) ( 1 , 2 ) | ^ ( 1 , 2 ) | ^ ) ( 1 , 2 ) > = = 2 { ( 2 E ( y ) + H c ( y ) ) ± ( 2 E ( y ) S 2 + H E ( y ) ) I(y)} . (2.69) - 3 1 -In the same way we obtain the following expression f o r the molecular overlap i n t e g r a l : ( 1 , 2 ) 1 ^ ( 1 , 2 ) > = 2 { 1 ± S 2 I O J ) } . ( 2 . 7 0 ) Therefore, the energy of the molecular states £ [see eq. ( 2 . 4 9 ) ] described by the wave functions ( 2 . 4 6 ) and ( 2 . 4 7 ) i s given by the expression (u) H C ( y ) ± H E ( y ) * ™ £ K = 2 E ( Y ) + - ^ ^ , K = 1 , 2 . ( 2 . 7 1 ) 1 ± S Z I ( y ) y Here, as w e l l as i n ( 2 . 6 6 ) , ( 2 . 6 9 ) and ( 2 . 7 0 ) the upper sign applies for K = 1 , and the lower one for K = 2 . E ^ i s the energy of the s i n g l e donor states into which the molecular states described by ( 2 . 4 6 ) and ( 2 . 4 7 ) d i s s o c i a t e i n the l i m i t of very large interdonor separations, jkj -> °°. We n o t i c e that the expression ( 2 . 7 1 ) i s very s i m i l a r to the f a m i l i a r r e s u l t of the Heitler-London treatment of hydrogenic molecules. The d i f f e r e n c e i s i n the presence of an extra f a c t o r , I ( y ) . This factor - we s h a l l r e f e r to i t as the interference c o e f f i c i e n t - w i l l be studied i n more d e t a i l and i n more general terms l a t e r i n t h i s chapter. From the above d e r i v a t i o n i t i s obvious that the presence of the interference co-e f f i c i e n t i n the expression for the energy of a donor-molecular state i s a d i r e c t consequence of the many-valley nature of the conduction band of the host semiconductor. From i t s d e f i n i t i o n , eq. ( 2 . 6 8 ) , we see that the interference c o e f f i c i e n t can assume values between 0 and 1 , depending upon the i n t e r -- 32 -donor separation _ of the -molecule, and upon the p a r t i c u l a r s i n g l e donor states into which the molecular state d i s s o c i a t e s at |_| -> 0 0. When I(y) = 1 the expression (2.71) i s equal to the standard r e s u l t of the Heitler-London treatment of hydrogenic molecules. The following diagram i l l u s t r a t e s the e f f e c t the interference c o e f f i c i e n t I(y) has on the p o s i t i o n of energy l e v e l s i n the spectrum of a donor molecule. The diagram shows the energy l e v e l s £ K , K = 1,2 i . e . the spin s i n g l e t and spin t r i p l e t l e v e l s , r e s p e c t i v e l y , as a function of the interdonor separation R, such that — / R = const. On the R axis the l a t t i c e s i t e s that a donor can occupy are i n d i c a t e d . (The other donor i s at the o r i g i n (0,0,0).) The diagram i s purely schematic; the energy units are a r b i t r a r y . F i g . 2-3. A schematic i l l u s t r a t i o n of the e f f e c t of the interference c o e f f i c i e n t on the donor-molecular spectrum ( l s ^ l - l s ^ l states i n S i ) . - 33 -The wayy curves i n the above diagram represent the functions £ - ^ ( R ) a n d Siven by ( 2 . 7 1 ) . Mathematically they are continuous functions of R , but they have a p h y s i c a l meaning only at d i s c r e t e values of R , as shown i n the diagram by x's and o's for tine p a r t i c u l a r d i r e c t i o n —/ . We see that a l l spin s i n g l e t l e v e l s l i e between two l i m i t i n g energies, the l i m i t s being determined by I(y) = 1 , i . e . the hydrogenic case, and I(u) = 0 . The same applies to spin t r i p l e t l e v e l s as w e l l . Therefore, the exchange energy, i . e . the d i f f e r e n c e i n energy between the spin t r i p l e t and the spin s i n g l e t state too, v a r i e s between the value given by the hydrogenic model ( i . e . associated with I(y) = 1 ) and the value 0 . This i s the most important feature of a donor-molecule energy spectrum as compared to the one of an ordinary molecule. Let us consider now a more general case, mamely the case of Heitler-London molecular states which are composed o f several d i f f e r e n t s i n g l e donor sta t e s . Examples of such molecular states are the l s - l s states i n germanium, where due to the weakness of the v a l l e y - o r b i t s p l i t t i n g and the degeneracy of the upper l s - l i k e s i n g l e donor states, configuration mixing between a l l four s i n g l e donor states takes place. The following treatment of these states can be applied also to otiher donor-molecular s t a t e s . If we are i n t e r e s t e d i n the excited l s - l s s t a t e s i n s i l i c o n , or ° ± the ls-2p and ls-2p states i n either s i l i c o n or germanium, the configura-t i o n mixing has to be taken into account owing to tine degeneracy of the s i n g l e donor states involved. - 34 -If each electron of a two-donor molecule can be i n any of the n 2 s i n g l e donor states , then we can form n molecular wave functions V it' • -v yv analogous to (2.42), * y v ( l , 2 ) = 4 A ( D ^ ( 2 ) , y,v = 1, n v , (2.72) where the notation i s the same as the one used i n (2.42) and l a t e r on. Now (K) we can form four non-combining sets of basis functions » K = 1,2,3,4, (K) by applying symmetrizing operators 0 on the functions (2.72), ^ ( 1 , 2 ) = 0 ( K ) ¥ y v ( l , 2 ) , K = 1, 4; y,v = 1 n y (2.73) (K) where the operators 0 are expressed i n terms of the space-coordinate per-mutation operator P^ 2 a*id the inve r s i o n operator J [defined by eq. (2.44)], as follows: o(1) = (i + P 1 2 ) (1 + J ) 0 ( 2 ) = (1 - P ) (1 - J ) (3) ( 2 ' 7 4 ) o u ; = ( i + P 1 2 ) (1 - J ) o(4) = ( i - P 1 2 ) (1 + J ) . Owing to t h e i r p a r t i c u l a r symmetry properties, the wave functions (2.73) with K = 1 and 3 describe spin s i n g l e t states, and those with K = 2 and 4 spin t r i p l e t s t a t e s . Regarding t h e i r symmetry with respect to i n v e r s i o n about the midpoint of the molecule, the states described by the wave functions (2.73) with K = 1 and 4 are c a l l e d the gerade states, and those with K = 2,3 the ungerade ones. - 35 -As was discussed at the beginning of t h i s section the true Hamiltonian of our system does not have ( i n general) a center of inversion symmetry; the above wave functions are only approximate eigenfunctions which we expect to y i e l d an energy spectrum that w i l l be a good approximation to the a c t u a l one. We can write (2.73) e x p l i c i t l y i n terms of the s i n g l e donor wave functions as follows: ^ ( 1 , 2 ) = s < K ) * A ( 1 ) * J ( 2 ) + s < K ) ^ ( 1 ) ^ ( 2 ) + s 3 ( K ) ^ ( 1 ) * A ( 2 ) + s ^ K ) ^ ( 1 ) ^ ( 2 ) , K = 1, 4; y,v = 1, n v , (2.75) (K) where the c o e f f i c i e n t s s. have the following values: s ( 1 ) = (1, 1, 1, 1) s ( 2 ) = (1, -1, -1, 1) m (2.76) s U ; = (1, -1, 1, -1) s ( 4 ) = (1, 1, -1, -1) 2 Not a l l of the 4n^ functions (2.75) are l i n e a r l y independent from each other. Since ¥ ( K ) ( 1 , 2 ) = vy f + (1,2) fc (2.77) i o r K = 1, 2 yv 1., ( K ) ( 1 , 2 ) f or K = 3, 4 yv ' ' and y ( K ) ( l , 2 ) = 0 for K = 3, 4 ' (2.78) yy - 36 -2 i t follows that there are only 2n^ l i n e a r l y independent basis functions: / ^ 1 , 2 ) , u < v , y,v = 1, n v ; K = 1,2 no ( 2 ' 7 9 ) r ; v ' ( l , 2 ) , y < v , y,v = 1 n y ; K = 3, 4 Proceeding now to minimizing the Hamiltonian ^ with respect to a l i n e a r combination of a l l the l i n e a r l y independent basis functions of a giyen symmetry, we get four determinantal equations, det || < ¥ ^ } (1,2)| ff(l,2)|^, (1,2)> - £ < ^ } ( 1 , 2 ) 1 ^ , ( 1 , 2 ) > || = 0 , (2.80) K = 1, 4 which are analogs of the equations (2.48) . The roots of these secular equations represent the ele c t r o n energy spectrum of the molecule. A c a l c u l a t i o n e n t i r e l y p a r a l l e l to the one leading to (2.69) and (2.70) gives the following expressions f o r the matrix elements appearing i n the secular equation (2.80): <¥ (1,2 ) | Ti (1,2 ) | V™ ,(1,2) > = = S l ( K ) [ E ( y ) + E ( V ) + H P ( P V ) ] 6 , 6 , 1 C yy' vv + s < K > [ ( E ( y ) + E ( V ) ) S<W"> S ( V V , ) + H ( ^ ' V V , ) ] I ( y , y \ v , v ' ) 2 E + s 3 ( K ) [ ( E ( y ) + E ( V ) ) S ( V I J , ) S ( y V , ) + H E ( v y ' y V , ) ] I(v,y',v,v') + s f K ) [ E ( y ) + E ( V ) + H p ( V y ) ] 6 , 6 , , n . . . 4 C vy yv (2.81) and <,(K) ( ) | f ( K ) = (K) , + (K) s ( y y ' ) (vv') , , yv 1 y v 1 yy vv 2 » ' + s < K ) s ( v y , ) s ( y v , ) i( Hy;y,V) + s, ( K ) 6 , 6 , (2.82) 3 ' i r , r ' 4 vy' y v' - 3 7 -where we took into account the orthonormality of the c o e f f i c i e n t s a . ^ , 3 j J J y y The Coulomb i n t e g r a l H^, exchange i n t e g r a l H^ ,, and the s i n g l e donor overlap i n t e g r a l S i n (2.81) and (2.82) are defined as follows: (2.83) (yv) ^ l A ^ ^ l " Fv^2B> % / i 2 + F 2 ( r , , ) F 2 ( r J - — d r . d r - , y —1A' v —2B < r 1 2 -1 -2 (2.84) H. (yy'vv') E F ( r . . ) F , ( r 1 B ) dr. y —1A' y —IB K r ^ g —1 F y ^ l A > V ^ l B > F v ^ 2 B > F v ^ 2 A > ^ ~ d ^ 2 (2.85) ,(yy') FP^1A> V^IB* ^ 1 (2.86) The interference c o e f f i c i e n t I i s defined as I(y,y',v,v') = Ea i j (y) (y') (v) (v«) a. a cos[(k. - k.)-R] , (2.87) and i s a g e n e r a l i z a t i o n of the interference c o e f f i c i e n t defined by (2.68). I t i s now straightforward to c a l c u l a t e the interference c o e f f i c i e n t s (2.87) fo r a l l possible interdonor separations R, evaluate the i n t e g r a l s (2.84), (2.85), and (2.86), i n s e r t them i n t o (2.81) and (2.82), and then solve the secular equations (2.80) to get the energy spectrum of the molecule as a function of R_. However, as we s h a l l see i n the next s e c t i o n we can get the same r e s u l t with much l e s s computing, i n the case of donor molecules i n germanium. - 38 -a) The Role of Interference C o e f f i c i e n t s The presence of interference c o e f f i c i e n t s i n the secular equation provides the e s s e n t i a l d i f f e r e n c e between a hydrogenic molecule and a donor molecule. The interference c o e f f i c i e n t s are a d i r e c t consequence - and manifestation - of the many-valley nature of the donor s t a t e s . As we have seen e a r l i e r i n a simple example the f a c t that the s i n g l e donor states are composed of Bloch states that a r i s e from several minima of the conduction band, gives r i s e to a number of d i s t i n c t molecular energy l e v e l s . These l e v e l s l i e i n between the l e v e l s that would have been obtained by using a s i n g l e conduction band with an i s o t o p i c mass equal to the appropriate average of m^  and m . The presence of the interference terms means that such a simple s i n g l e s p h e r i c a l conduction band model would be quite inappropriate f o r c e r t a i n problems. From (2.87) we can c a l c u l a t e the set of interference c o e f f i c i e n t s of a l l p o s s i b l e combinations of s i n g l e donor wave functions f o r each p a r t i -c u lar (discrete) value of the interdonor separation R. This has been done i n the Appendix i n a systematic way. There i t has been shown that due to the p a r t i c u l a r structure of the germanium conduction band there are only, three d i s t i n c t , non-equivalent sets of interference c o e f f i c i e n t s p o s s i b l e . (We c a l l two sets of interference c o e f f i c i e n t s equivalent i f they y i e l d i d e n t i c a l energy spectra provided the absolute interdonor distance R = | R | i s the same i n both cases.) In other words, i f we consider a l l donor molecules such that t h e i r absolute interdonor separation R = | R | l i e s i n the i n t e r v a l [R, R + dR], there are only three d i s t i n c t l e v e l structures p o s s i b l e , depending on the r e l a t i v e l o c a t i o n of the donors of the molecule i n the c r y s t a l l a t t i c e . - 39 -The germanium c r y s t a l l a t t i c e consists of two i n t e r l o c k i n g face-centered cubic l a t t i c e s . We can di v i d e a l l l a t t i c e s i t e s of each l a t t i c e i n t o two c l a s s e s : simple cubic s i t e s and face-centered s i t e s , as i l l u s t r a t e d by the following f i g u r e ; F i g . 2-4. Simple face-centered cubic l a t t i c e s t r u c t u r e . The second l a t t i c e i s displaced along the body diagonal of the cube i n the above f i g u r e by one quarter of i t s length. The above-mentioned three d i s t i n c t cases occur i n the following s i t u a t i o n s : a) Both donors that c o n s t i t u t e the molecule belong to e i t h e r simple cubic or face-centered p o s i t i o n s i n the same l a t t i c e ; b) One donor occupies a simple cubic s i t e while the other i s on a face-centered s i t e i n the same l a t t i c e ; c) The two donors occupy s i t e s that do not belong to the same l a t t i c e . This c l a s s i f i c a t i o n of a l l donor molecules i n a germanium c r y s t a l i n t o three classes i s not only a r e f l e c t i o n of the nature of the c r y s t a l - AO -l a t t i c e but i s also a consequence of the p a r t i c u l a r structure of the con-duction band of germanium, as we have already mentioned. More s p e c i f i c -a l l y , i t i s a consequence of the f a c t that the conduction band minima i n germanium l i e at the edges of the R r i l l o u i n zone. (Such i s not the case with s i l i c o n . ) Due to t h i s c l a s s i f i c a t i o n of donor molecules i t i s s u f f i c i e n t f o r us to c a l c u l a t e the energy spectra of three molecules only, each belonging to a d i f f e r e n t c l a s s , as a function of a continuous interdonor absolute distance R. (Since the interference c o e f f i c i e n t s do not depend on R, the only dependence on R comes into the secular equations through the Coulomb, exchange and overlap i n t e g r a l s , a l l of which are continuous functions of R.) Having done t h i s we have the knowledge of the energy spectra of a l l possible two-donor molecules i n a germanium c r y s t a l . This i s i l l u s t r a t e d i n the following schematic diagram, F i g . 2-5. The curves i n t h i s diagram represent the (continuous) solutions of the secular equations f o r the three classes of molecules. Each energy l e v e l i s l a b e l l e d by the c l a s s which i t i s associated with. The diagram i s purely schematical; neither the number of l e v e l s nor t h e i r d i s t r i b u t i o n r e f e r to any a c t u a l case. Supposing one donor i s kept fixed at a simple cubic s i t e at (0,0,0), then the p o s s i b l e p o s i t i o n s of the second donor c o n s t i t u t i n g the molecule are indicated on the R axis (0 denotes simple cubic l a t t i c e s i t e s , x face-centered s i t e s , and A s i t e s on the displaced l a t t i c e ) , for three d i f f e r e n t d i r e c t i o n s i n the c r y s t a l , [1,0,0], [1,1,0], [1,1,1]. The energy spectra associated with the above (discrete) interdonor separations R are shown by' points on the c o n t i n -uous curves. The energy l e v e l s of a l l molecules such that t h e i r second 4> R , fill [1,0,0] R ,fill [1,1,0] R ,R||[U,I] F i g . 2-5. A schematic i l l u s t r a t i o n of the e f f e c t of the interference, c o e f f i c i e n t s on the donor-molecular spectrum (Ls-ls states in.Ge). - 42 -donor occupies a simple cubic s i t e ( i . e . molecules of cla s s a) l i e on the curves l a b e l l e d a, energy l e v e l s of.molecules of c l a s s b on curve b and those of molecules of c l a s s c on curves c. Thus with the help of the above diagram we can determine the energy spectrum f o r any molecule i n the c r y s t a l . had c a l c u l a t e d energy spectra of donor molecules as a function of (discrete) R i n a straightforward way (see page 37), and p l o t t e d them i n a £ vs. R diagram (to get a set of points, as i n the above diagram), we would have noticed that a l l the points tend to l i e on several d i s t i n c t smooth curves. Drawing i n t e r p o l a t i o n curves through the computed points we would have gotten the continuous curves shown i n the above diagram. molecule to belong to a p a r t i c u l a r c l a s s L. The r a t i o between the number of simple cubic l a t t i c e s i t e s and the number of face-centered cubic s i t e s i s 1:3 (see F i g . 2-4). I t follows, assuming that each donor can occupy any l a t t i c e s i t e with equal p r o b a b i l i t y , that the r a t i o between the number of p o s s i b l e molecules belonging to c l a s s a, and the number of those belong-ing to c l a s s b, and the number of those belonging to c l a s s c, i s We can i n t e r p r e t the above diagram i n another way, too. If we Let us see now what i s the p r o b a b i l i t y , p. for an a r b i t r a r y N a b N = 5 3: 8 (2.88) c or 0.3125 i f L = a 0.1875 i f L = b (2.89) 0.5000 i f L = c - 43 -A d e t a i l e d study of the interference c o e f f i c i e n t s helps us to reduce the amount of computation necessary to acquire the knowledge of the energy spectra of a l l the molecules i n a c r y s t a l even more. A close look at the interference c o e f f i c i e n t (see Appendix) with respect to the matrix elements (2.81) and (2.82) reveals that i t i s not necessary to solve the secular equation (2.80) for a l l four symmetries of the Heitler-London wave functions ( i . e . a l l four values of K) f o r a l l three c l a s s e s of molecules. Namely, f o r molecules belonging to c l a s s a the s p i n s i n g l e t gerade and ungerade states are degenerate, as well as the spin t r i p l e t gerade and ungerade ones, and f o r the molecules belonging to c l a s s b the spin s i n g l e t and t r i p l e t gerade l e v e l s are degenerate, as well as those described by the s p i n s i n g l e t and t r i p l e t ungerade wave functions. Thus only for the mole-cules that belong to c l a s s c we have to solve the secular equations for a l l four values of K since each of them y i e l d s a d i f f e r e n t pattern of energy l e v e l s . From the r e s u l t s of t h i s subsection follows the following very u s e f u l summation r u l e : If a p h y s i c a l quantity Q depends on R through the energy eigen-values £(R) of a two-donor molecule, representing the whole set of eigen-values, then T E P(R) Q(£(R)) = E p E P(R) Q (R) , (2.90) a l l R L a l l R where the summation over R goes over a l l possible R i n a c r y s t a l , P(R) i s a (discrete) p r o b a b i l i t y d i s t r i b u t i o n function, and Q L(R) i s a continuous fu n c t i o n such that - 44 -Q( £(R)) = Q (R) i f |R| = R and R e # L , (2.91) where i s the set of a l l interdonor separations R of molecules belonging to class L. Thanks to the summation rule of eq. (2.90) we do not have to com-pute the quantity Q for a l l possible molecular positions R in the crystal; i t is sufficient to compute i t for three molecules only, each belonging to a different class L, as a function of continuous R = |RJ. If we can replace P(R) by an isotropic, continuous function P(R), the eq. (2.90) can be written in the following form: P(R) QL(R) dR . (2.92) J £ P(R) Q( t (R)) = E P. a l l R L " o b) Intermediate Interdonor Separations The preceding calculation of energy spectra of the l s - l s - l i k e electron states of two-donor molecules in germanium was performed numerically using a computer. The interdonor separations were in the o o range ^ 80 A < R < 400 A, i.e. in the so-called intermediate range. The computation has been done in this particular range for two reasons. This i s the range of interdonor separations that i s important for our application of the result of the present calculation of energy spectra to magnetic suscep-o t i b i l i t y . Apart from that, at R < 80 A the Heitler-London approach is not suitable (because i t does not take into account the donor-ionic contributions to the electronic energy of the molecule that are non-negligible at this * ° ° small separation; (a = 45 A)), and at R > 400 A we can make use of an approx-imation that greatly simplifies the calculation (see Chapter II - 2c). - 45 -The computer program was organized i n such a way that one donor was being kept f i x e d at a simple cubic s i t e d e f i n i n g the o r i g i n (0,0,0) while the other donor c o n s t i t u t i n g the molecule was running over a l l simple cubic p o s i t i o n s i n the above range of distances i n the d i r e c t i o n [0,0,1], a l l face-centered p o s i t i o n s i n the same distance range i n the d i r e c t i o n [1,1,0], and s i m i l a r l y over a l l displaced l a t t i c e s i t e s i n the d i r e c t i o n [1,1,1]. At each simple cubic and each face-centered l a t t i c e s i t e the secular equation for the spin s i n g l e t gerade states was solved, i . e . the c o e f f i c i e n t s i n the l i n e a r combination of the basis functions (2.79) and the corresponding energy eigenvalues were found, and then, using the same c o e f f i c i e n t s i n the l i n e a r combination of the spin t r i p l e t ungerade basis functions, the energy eigenvalues of these states were c a l c u l a t e d . At the displaced l a t t i c e s i t e s eigenfunctions and energy eigenvalues of the spin s i n g l e t , gerade and ungerade, states were computed by s o l v i n g the corres-ponding secular equations, and then the energy l e v e l s of the spin t r i p l e t , gerade and ungerade, states were found using the same c o e f f i c i e n t s i n the l i n e a r combination of the corresponding basis functions. In t h i s way, by a minimum amount of matrix d i a g o n a l i z i n g (which i s a computer time consuming operation) the eigenfunctions and energy spectra of a l l molecules i n the d i r e c t i o n [0,0,1] i . e . molecules of cl a s s a, a l l molecules of c l a s s b i n the d i r e c t i o n [1,1,0], and a l l molecules of c l a s s c i n the d i r e c t i o n [1,1,1] having t h e i r interdonor separations i n the range o o 80 A < R < 400 A were obtained. The spacings between l a t t i c e s i t e s of the o same type were small enough (maximum spacings were ^ 8 A) that we can use these r e s u l t s to obtain the energy spectrum of any possible molecule i n the c r y s t a l by i n t e r p o l a t i o n (see F i g . 2-5). - 4 6 ^ The numerical input data f o r t h i s computation were: energies of the l s - l i k e s i n g l e donor l e v e l s the -effective mass i o n i z a t i o n energy * E £ £ , the mean e f f e c t i v e Bohr radius a ( i t was used as the basic u n i t of ef f ' length throughout the computation), the l a t t i c e parameter l a t t i c e ' t* i e s t a t i c d i e l e c t r i c constant K, and the i n t e r f e r e n c e c o e f f i c i e n t s I. In (K) L the output there were the molecular energy l e v e l s £ ^ and the cor r e s -(K.) L ponding eigenfunctions $ n ' (1,2) expressed as l i n e a r combinations of the basis functions ¥ v '(1,2), yv ' (K) # ( 1 , 2 ) * < K ) » L ( 1 , 2 ) = £ ^ K ) ' L $ ( K > > L C L , 2 ) , • ( 2 . 9 3 ) $ ( K ) > L ( 1 , 2 ) = E a ( K ) ' L T ( K ) ( 1 , 2 ) , ( 2 . 9 4 ) n ' yv n,yv yv ' ' where # ( 1 , 2 ) i s the Hamiltonian ( 2 . 3 4 ) , and 1 ^ ( 1 , 2 ) the functions ( 2 . 7 9 ) . The Coulomb, exchange, and s i n g l e donor overlap i n t e g r a l s , ( 2 . 8 4 -8 6 ) were c a l c u l a t e d u t i l i z i n g formulas and tables of molecular i n t e g r a l s . A f t e r the matrix of the secular equation ( 2 . 8 0 ) was brought into the standard form by m u l t i p l y i n g i t by the inverse of the molecular overlap matrix, i t was diagonalized. To t h i s end i t was f i r s t reduced to a symmet-r i c t r i d i a g o n a l matrix using Householder's method, and then i t s eigenvalues were found by a b i s e c t i n g method. The corresponding eigenvectors were c a l -culated using inverse i t e r a t i o n technique. The whole c a l c u l a t i o n was done for two d i f f e r e n t donors, f o r a n t i -mony and for phosphorus donor molecules i n germanium. A l l the computer out-put data were put into the computer storage for future use and reference. F i g . 2-6. The l s - l s spectrum of Sb donor molecule i n Ge: The envelopes of energy l e v e l s of the three molecular classes shown c o l l e c t i v e l y as a function of the absolute interdonor separation. 22.5 23.0 + [meV] 23.5 R = 36 R = .51 * ~ a u S3 R = 81 x H a 4 F i g . 2-7. The l s - l s spectrum of Sb donor molecule i n Ge: Tae g and u energy l e v e l s of the three molecular classes shown separately. - 49 -The diagrams i n F i g . 2-6 and 2-7 i l l u s t r a t e the r e s u l t s . Numer-i c a l data r e f e r to antimony donor molecules. In the f i r s t of these diagrams, F i g . 2-6, envelopes of energy l e v e l s of molecules of a l l three c l a s s e s c o l l e c t i v e l y are shown as a f u n c t i o n of inte r d o n o r s e p a r a t i o n R. We n o t i c e how the sharp s i n g l e donor l e v e l s (at R -* °°) "broaden" i n t o "bands" * as the in t e r d o n o r s e p a r a t i o n decreases. At the s e p a r a t i o n R ^  4.25 a o (= 190 A) these "bands" merge i n t o one another. This means that a t i n t e r -donor separations R ^  4.25 a an e l e c t r o n i n a s i n g l e donor s t a t e y may come -through molecular i n t e r a c t i o n s w i t h another e l e c t r o n - lower i n energy than an e l e c t r o n i n the s i n g l e donor s t a t e y' although at separations R -* °° ( i . e . i n the s i n g l e donor spectrum) the s t a t e y l i e s higher than the s t a t e y'. * T-, A-, [For i n s t a n c e , a t R -t, 4.25 a the energy of a Is - I s molecular s t a t e may ^ A, CV <V U C i U W C l I- i i c i l i L L ICL U U J . C*. -i_ U 0.0 o L . < - t - i - ' ^ } u x |_ J. x \ J U ^ , J_< x-i » j j . i i .J.L> can only occur i n cases where the maximum of the exchange energy (exchange energy i s a f u n c t i o n of i n t e r f e r e n c e c o e f f i c i e n t s ) i s of the same order of magnitude as the v a l l e y - o r b i t s p l i t t i n g . T his i s a s p e c i a l f e a t u r e of donor molecules and has important e f f e c t s on the magnetic p r o p e r t i e s of i m p u r i t y -doped semiconductors, si n c e due to molecular i n t e r a c t i o n s between donors the st a t e s w i t h a s s o c i a t e d l a r g e r overlaps and exchange energies may be more populated than one would expect on the b a s i s of energy c o n s i d e r a t i o n of the s i n g l e donor s t a t e s i n t o which they d i s s o c i a t e a t R -> °°. The sma l l e r the magnitude of v a l l e y - o r b i t s p l i t t i n g the more pronounced the e f f e c t i s . The above mentioned "bands" of molecular s t a t e s are of course not bands i n the us u a l meaning of the word; i f we blew up the energy s c a l e of the diagram F i g . 2-6 we would see that these "bands" are merely set s of - 50 -r e l a t i v e l y c l o s e l y spaced energy l e v e l s . This i s i l l u s t r a t e d by the diagram F i g . 2-7. This diagram shows the s i n g l e t gerade and t r i p l e t ungerade Is - Is energy l e v e l s of a two-donor molecule for three d i f f e r e n t p o s i t i o n s i n the c r y s t a l . These p o s i t i o n s are chosen i n such a way that each of them i s associated with a d i f f e r e n t c l a s s of molecule (a, b, and c ) . The absolute interdonor separations are approximately equal i n a l l three cases. We know that a simple conduction band would give r i s e to a Is - Is molecular spectrum that has the same pattern of l e v e l s (for a l l molecular p o s i t i o n s ) as the s i n g l e t gerade and t r i p l e t ungerade Is - Is spectrum of a molecule of cl a s s a i n the actual case, i . e . i n the case of a m u l t i v a l l e y conduction band. I t i s evident from F i g . 2-7 that the a c t u a l donor molecu-l a r spectrum e x h i b i t s a number of energy l e v e l s that are absent i n the model employing a simple conduction band, apart from the l e v e l s described by the s i n g l e t ungerade and t r i p l e t gerade states (not shown i n F i g . 2-7) whose existence i s conditioned by the m u l t i v a l l e y nature of the conduction band. c) Large Interdonor Separations In t h i s subsection we s h a l l consider Heitler-London states of a donor pair with interdonor separations R = |R| large compared to the mean distance of an e l e c t r o n from the donor core i t i s l o c a l i z e d on, i . e . Ir I = Ir I << R '-1A1 2B 1 (2.95) l£1 Bl = | £ 2 A | * R (For notation see F i g . 2-2). At these large interdonor separations i t i s - 51 -convenient to write the Hamiltonian (2.35) i n the form *(1,2) = *(1) + 7C(2) - | + V ( 1 , 2 ) , (2.96) where V(l,2) + . (2.97) R r i B r2A r i 2 2 In (2.96) and (2.97) the energy i s measured i n un i t s of e / *' , and the Ka A length i n un i t s of the mean Bohr radius a . When the condition (2.95) i s s a t i s f i e d we can expand (2.97) into a Taylor s e r i e s i n terms of r and r , —X A —2. D V(l,2) = t ( £ 1 A ' £ 2 B ) " 3 ( - l A ' ^ (^2B*^ ) ] ••• ' ( 2 , 9 8 ) R since ^1B - I l A " * • —2A = —2B + — ( 2 ' 9 9 ) -12 = - " -1A + -2B * In (2.98) R denotes the u n i t vector i n the d i r e c t i o n of R. The advantage of (2.98) over the exact expression (2.97) i s obvious. I t contains no two-—3 3 — p a r t i c l e operator. To the order of r /R (r being an average value of | r - . | ^ |_r | ) i t i s a l i n e a r combination of the ele c t r o n p o s i t i o n vectors, which w i l l make the c a l c u l a t i o n of i t s matrix elements between Heitler-London wave functions very simple. - 52 -Is - Is States In order to obtain the energy spectrum of the Is - Is states, we can use the functions (2.75) again to set up the secular equations and solve them. However, i f we use the approximation (2.98) i n the Hamiltonian (2.96), we can express the basis functions (2.75) i n terms of r . and r O T ) — J L A —La as w e l l , *£>(1,2) = s f \ < r u ) * v < r 2 B ) + s f \ ( r u - R) * v ( r f f l - R) + + S 3 K ^ v ^ l A " *> V^2B + & + S4(K)^V^1A>V(-2B) > • ( 2 ' 1 0 0 ) (K) and then expand the s i n g l e donor wave functions associated with and (K) . „, , s^ into a Taylor s e r i e s , (r . — R) = xp (-R) + r 1 A - V ^ (r) + ... y — l A y —1A — y — 1 * r - -R r = R (2.101) It follows that the s i n g l e donor overlap i n t e g r a l s are n e g l i g i b l y small as compared to uni t y . Also, when the accuracy of the f i r s t term i n the expan-sion (2.98) i s s u f f i c i e n t , the c o n t r i b u t i o n from the terms associated with (K) (K) s 2 and s^ i n (2.100) to the matrix element of (2.98) between the H e i t l e r -London basis functions i s zero [see eq. (2.81)]. This means that the secular equations are t r i v i a l ; t h e i r matrices are diagonal. Their eigen-values are the following (K) (u) (v) 1 < ^ ) d , 2 ) | v ( l , 2 ) | ^ ) ( l , 2 ) > ? * V ) - E ( y ) + E ( V ) - | + m (2.102) - 53 -Ins e r t i n g (2.98) and (2.100) into the l a s t term i n (2.102) we get <V^ )(1,2)|V(1,2)|¥^ )(1,2)> = 0 , (2.103) since f o r a l l y and v, % ( £ 1 A ) l l . 1 A k v ( £ l A ) > = [0.0.0] > (2.104) because the s i n g l e donor wave functions involved are l s - l i k e . Therefore, :-lA\ " ^ 2 B to order of r /R (r being an average value of |r_ | ^  |r_ | ) , the energy of the Is - Is l e v e l s i s given by ^ } ( R ) = E ( M ) + E ( V ) - | . (2.105) Is - 2p States In the same way as above we can c a l c u l a t e the energy spectra of the Is - 2p state s , i . e . the molecular states that d i s s o c i a t e i n t o a l s - l i k e and a 2p-like s i n g l e donor state i n the l i m i t R -> °°. In analogy to (2.100) we write the Heitler-London basis functions d e s c r i b i n g these states as ^ p = S l K \ s felA>*2p fe2B> + S 2 K \ s Q-lA - 5 ) * 2 P (^2B " y v y v y v + S 3 K ^ 2 p k l A " ^ I s (^2B + *> +sfVs ^lA>^2p ^2B> > ( 2 - 1 0 6 > v y v . u o ± where 2p^ stands for ei t h e r 2p^ or 2p^ s i n g l e donor s t a t e . Again we can expand these basis functions i n t o Taylor s e r i e s to f i n d that the secular equations are t r i v i a l again. The energy eigenvalues are given by an expression analogous to (2.102), ^ l s } 2 (*> = E ( 1 S y ) + e ( 2 ? v ) " R + £V<® > (2'107)  S p p v y V - 54 -< y i ? 2 P ( 1 , 2 ) ^ ( 1 , 2 ) 1 ^ (1,2)> e^> (R) s ~ y " (2.108) P <<T K ) (1 2 ) k ( K ) (1 2)> Is 2p U ' Z ; | T l s 2p ^  y v y r v (K) (K) Again contributions from terms i n (2.106) associated with s^ and s^ to the matrix element £ i s n e g l i g i b l e . Besides yv ° ° ^ l s felA>liu'*l8 ^ 1 A ) > " [ 0 ' ° ' ° ] y y and (2.109) <*2P C I 2 B )^2B I*2 P ^ B ^ = [ ° > ° ' 0 ] v v because we can express these two matrix elements as l i n e a r combinations of matrix elements of _r between two l s - l i k e and two 2p-like envelope functions, r e s p e c t i v e l y , which are equal to zero. I t follows that £ y v } = 1 ^ i s ( ^ 2 P fe2B)|va,2)|* ( r 1 A ) ^ s ( r ^ ) , (2.110) y v *v y where the + sign applies when K = 1,2, and - sign when K = 3,4, [see eq. (2.76)]. I n s e r t i n g (2.98) into (2.110) we get the following expression tor £ : yv - +~ { ( * l s felA>lilAl*2p felA> ' ^ ^ 1 ^ 1 * 1 8 ( ^ 2 B ^ " y v v y (2.111) - 3 ( < * l s ^1A^2AK ( - l A } > ' ^ H < % ^ B ^ B ^ l s 3 ' *v v y R We note that i n t h i s expression the s i n g l e donor wave functions as well as the operators i n each of the one - p a r t i c l e matrix elements are associated - 55 -with the same donor core, so that we can suppress the l a b e l s 1A and 2B i n (2.111), £ ( K )=±'{(<!()- | r | K > • <i|; |r|ih >-yv Is '—1 2p 2p '—1 Is V v v y - 3 ( < * l g | r | * > • R ) (<* | r | * > • £ ) } . (2.112) y *v v y R I f we define a vector A by —yv J A = <i|»_ |r|^„ > = <iK |r|i|). > , "(2.113) —yv Is '—1 2p 2p '—1 Is y v v y we can write (2.112) i n the form £( K ) ( R ) = ± {(A -A ) - 3(A -R)(A - R ) } \ . (2.114) c yv — —yv —yv —yv — —yv — _3 K The e i n g l c doner T T?"? f u r c f i o n s appearing i " thp. matrix element (2.113) are the complete wave functions as defined i n Ch. I I - l . The l a b e l s y and v s p e c i f y the p a r t i c u l a r i r r e d u c i b l e representation of the point group T^ according to which the functions and transform under the symmetry y p v operations of T^. Under these operations the operator r_ transforms accord-ing to the i r r e d u c i b l e representation T^. (See note 2 on page 15). Thus we can use group t h e o r e t i c a l considerations to determine which of the matrix elements (2.113) are non-vanishing. According to a well-known group t h e o r e t i c a l theorem the matrix element <' r ,^ s | r j ^ > m a y be nonvanishing i f the d i r e c t product T x T^ y v contains T , where V and T denote the i r r e d u c i b l e representations to v y v which the functions i ( ; ^ g and , r e s p e c t i v e l y , belong; i . e . y v - 56 -A + 0 i f r x T, contains T —yv y 1 V character table for the group E 8 C 3 3C 2 6a , a 6S. q A l 1 1 1 1 1 A "2 1 1 1 -1 -1 E 2 -1 2 0 0 T l 3 0 -1 1 -1 T 3 0 -1 -1 -1 2 (2.115) (2.116) we f i n d the following decompositions of the relevant d i r e c t products (see diagram F i g . 2-1), A l X T l " T l T l X T l = A l + E + + T 2 E x T = T x + T 2 (2.117) Therefore, f or donors i n germanium the following matrix elements are equal to zero: N l 8 111* 2 po>l - 0 i f r u = A 1 a n d r v = A 1 2p y v (2.118) r U . ±> = 0 i f r = A, and T = E or T 1 Is '—1 2p 1 y 1 v 2 y v (2.119) S i m i l a r l y f o r donors i n s i l i c o n , - 57 -l<* l 8 Ir|*2 o>| - 0 ( if r " = A, and r = 1 o r E y • 1 v l (2.120) i f T = E and r = A . o r E y v 1 \r\Tn ±> = 0 i f T = A. and T = T_ . (2.121) 1 T l s '—1 2p ' y 1 v 2 y r v From (2.118 - 121) we can conclude which of the Is - 2p H e i t l e r -London states are not a f f e c t e d by the i n t e r a c t i o n p o t e n t i a l V(l,2) to the 3 3 rg\ order of r / R . In order to f i n d the s p l i t t i n g s £ of the states that b yv are a f f e c t e d by V ( l , 2 ) , we would have to c a l c u l a t e the c o e f f i c i e n t s d e s c r i b -ing the r e l a t i v e contributions from each i n d i v i d u a l v a l l e y of the conduc-t i o n band to the s i n g l e donor 2p-like wave functions (analogous to ^ f o r l s - l i k e states given by eqs. (2.25) and (2.76)), then c a l c u l a t e the non-vanisnmg matrix elements A leq. (./.n/)j, and r i n a i l y evaluate ',^v' a s a f u n c t i o n of R. [eq. (2.112)]. In t h i s way we would get the energy spectra of the Is - 2p Heitler-London states f o r large | R | . From eqs. (2.107) and (2.112) we can conclude that they consist of a number of r e l a t i v e l y c l o s e l y spaced energy l e v e l s . The width of these "bands" i s determined by the (K) extremes of £ ( R ) . yv — The c a l c u l a t i o n of a l l the d i s t i n c t energy l e v e l s as outlined above i s very simple and straightforward, but tedious because of the many degenerate s i n g l e donor states that one has to take into account and because the r e s u l t i n g molecular l e v e l s depend on the o r i e n t a t i o n of the molecule R i n such a way that we cannot make any systematic study as we were able to do with the Is - Is s t a t e s . (At intermediate interdonor separations the Is - Is states give r i s e to "bands" of energy l e v e l s due to the interference - 58 -c o e f f i c i e n t s ; the matrix elements involved were independent of R. This i s the reason that at large interdonor separations, where the terms asso-c i a t e d with the interference c o e f f i c i e n t s become n e g l i g i b l e , we do not get any Is - Is "bands". On the other hand, matrix elements involved i n the expressions for energy of the Is - 2p states do depend on R, so that even at separations large enough that the matrix elements associated with the i n t e r f e r e n c e c o e f f i c i e n t s are n e g l i g i b l e , they give r i s e to "bands".) Besides we are not going to need the energies of the d i s t i n c t l e v e l s i n our a p p l i c a t i o n l a t e r on. Therefore, l e t us estimate the widths of these "bands" only. (K) From (2.114) i t follows that the extremes of £' (R) occur when R_ i s p a r a l l e l (or a n t i p a r a l l e l ) with and when i t i s perpendicular to A • So we can write (2.114) i n the form £ ( K ) ( R ) = ± p (R) A 2 ^ , (2.122) yv — " y V x — yv R 3 ' A. where p (R) i s a function that can assume values between -2 and +1 I yv — A\ depending on the o r i e n t a t i o n of R. with respect to A ^ . From (2.113) we obtain, using the same assumption as the one leading to eq. (2.53), |A | = | Z c f 1 S y ) c ( f 2 p v ) <F 1 Sy (r) | r | F 2 p (r)> | . (2.123) The magnitude of the matrix element <F^ Sy |_r l-^j^ > > does not depend on the p a r t i c u l a r v a l l e y of the conduction band s p e c i f i e d by j ; i t only depends on the p a r t i c u l a r l s - l i k e s i n g l e donor state s p e c i f i e d by y (because of the c e n t r a l c e l l c o r r e c t i o n f a c t o r contained i n the envelope functions; a l l - 59 -2p-like states are degenerate), and on the magnetic quantum number of the 2p-like s t a t e s . Since a l l the c o e f f i c i e n t s a . ^ ^ are normalized to unity, we 3 estimate that for Is - 2p° states o u v A... I - { (2.124) - 2P* y v 1 (y) ± A ^ y f o r Is, - 2p t states , where A^^ and A ^ are defined as: o + • • . A o ( y ) ••= |<Fj Sy(r)|r|F 2P°(r)>| (2.125) = l < F f y ( r ) | r | F 2 P " ( r ) > | From (2.3), (2.32), and (2.33) we obtain the following expressions f or (2.125), A ( y ) = b j ( y ) o o A ^ = a (2.126) + + where >M 3 3M 3 2 / e u 6 c (2.127) J (y) y ± ( B y + « ± ) 5 In (2.126) a and b are the transverse and l o n g i t u d i n a l Bohr r a d i i . I t - 60 -follows that and - 2 b 2 > > 2 4 < € f> o(R) < bV y>2 4 , K - 1,2 o „3 Is 2p — o R ' ' R y *v - b 2 j (y)2 1 < £ ( K ) o ( R ) < 2 b 2 j ( y ) 2 1 = R 3 1 S y 2 p v ~ ° R - 2 a 2 d f > 2 4 < ±(R) < a 2 J + ( y ) 2 4 , K - 1,2 R y P v _ R _ a 2 a ( y ) _ l < , 0 0 ± ( R ) < 2 ^ ( 1 1 ) 2 1 ? R = 3 > 4 _ R J - L S y Z P V - R J R~ (2.128) (2.129) Now we can write (2.107) i n the form £<f 2 o(R) = E ( l s y ) + E^O - | ± p b 2 J o ( p ) 2 4 (2.130) and S ™ * . * ® " E & V + e ( 2 p ^ " I * P a 2 j i " ) 2 "5 • < 2 - 1 3 1 > y v R where p i s a parameter that can assume values between -2 and +1 depending upon the o r i e n t a t i o n of the molecule and on the p a r t i c u l a r l s - l i k e and 2p-like s i n g l e donor states involved. The upper sign i n (2.130) and (2.131) applies f or K = 1,2 and the lower for K = 3,4. o ± We note that the widths of the Is - 2p and Is - 2p "bands" of energy l e v e l s are proportional to the square of the l o n g i t u d i n a l and trans-verse Bohr radius, r e s p e c t i v e l y , which i s a s p e c i a l feature of the donor-molecular spectra. Had we been using i s o t r o p i c hydrogenic wave functions - 61 -as an approximation to the s i n g l e donor wave functions ( i . e . a simple s p h e r i c a l conduction band model) we would not have obtained any d i f f e r e n c e i n the widths of the Is - 2p° and Is - 2p~ "bands". - 62 -I I I . CONCENTRATION DEPENDENCE OF FAR IR ABSORPTION IN Ge(Sb) and Si(P) Experimental studies of f a r i n f r a - r e d absorption i n group V doped Ge and S i ^ " ^ at low temperatures ( l i q u i d He temperature) and low donor concentrations (5 x 10^ < N d £ l O 1 ^ f o r Ge, 5 x l O 1 ^ < N d 1 10"*"7 for S i) have y i e l d e d a concentration dependence of the absorption l i n e width. Various processes that contribute to the broadening of absorption l i n e s have been considered by other authors, l i k e electron-phonon i n t e r -(19,20,21) . . . a c t i o n , d i r e c t overlap of neighbouring impurities forming an (22) (23) "impurity band" , ionized-impurity broadening v ; , e t c , yet none of these processes gives r i s e to the observed concentration dependence of the l i n e width i n the above mentioned temperature and concentration range. Recently^ P ^ e r a n t z s u ^ ^ ^ t e d (24) t-T-,aj- s t r a i n s caused bv impur-i t i e s i n the l a t t i c e are responsible f o r the concentration broadening, but l a t e r Stoneham showed(25) t j i a t pomerantz's mechanism was too weak to account f o r the observed concentration dependence of the linewidths. In t h i s chapter we s h a l l attempt to account f o r the observed concentration dependence of the absorption l i n e width on the basis of the molecular model of i n t e r a c t i n g donors i n a c r y s t a l , developed i n Chapter n . At very low donor concentrations when the o p t i c a l properties of donors i n a c r y s t a l are well represented by i s o l a t e d donors, the o p t i c a l absorption takes place through the e x c i t a t i o n of a donor e l e c t r o n from i t s ground state to an excited s t a t e . The diagrams of F i g . 3-1 show schemati-c a l l y the observed t r a n s i t i o n s with t h e i r standard notation. T 63 -S i ( P ) Ge(Sb) 2p" 2p l s ( E ) I S C T - L ) l s ( A 1 ) 2 P' 2 P C ls(Tx) lsCA 1) E l E 3 B l B 3 F i g . 3-1. The E and K o p t i c a l t r a n s i t i o n s i n a s i n g l e donor i n S i and Ge. T At l i q u i d helium temperatures no t r a n s i t i o n s o r i g i n a t i n g from the Is 1 and Is l e v e l s i n s i l i c o n are observed, since these states are not populated at these low temperatures, due to the large v a i i e y - o r b i t s p l i t t i n g i n s i l i c o n . With increasing donor concentration, i n t e r a c t i o n s between donors s t a r t to take place, r e s u l t i n g i n a broadening of the s i n g l e donor l e v e l s i n t o molecular "bands" of l e v e l s , as observed i n Chapter I I . Now the o p t i c a l absorption takes place through e x c i t a t i o n of the el e c t r o n system of the i n t e r a c t i n g donors to excited molecular sta t e s . The dependence of the width of molecular "bands" on the separations between the i n t e r a c t i n g donors, and the random d i s t r i b u t i o n of donors throughout the c r y s t a l give r i s e to a concentration dependence of the width of t r a n s i t i o n l i n e s i n the absorption spectrum. The range of donor concentrations of our i n t e r e s t i s low enough that we can assume that each donor i n t e r a c t s with only one other donor at - 64 -a time. So we can t r e a t the c r y s t a l containing donors per cm as an 3 assembly of i s o l a t e d two-donor molecules per cm . O p t i c a l absorption of photons r e s u l t s i n t r a n s i t i o n of the e l e c t r o n system to an excited two-donor molecular s t a t e . These states were the object of study of Chapter 16 —3 II-2. The high l i m i t of the concentration range of our i n t e r e s t (10 cm 17 -3 f o r Ge, and 10 cm for Si) i s low enough that we can apply the r e s u l t s of the s e c t i o n II-2c, i . e . the molecular states at large interdonor separa-t i o n s , to the present study. F i g . 3-2 shows schematically molecular t r a n s i t i o n s corresponding to the i s o l a t e d donor ones shown i n F i g . 3-1. (48) (24) (2) Si(P) Is 2p" i 1 0 o Is 2p A A Is Is C96) (32) (48) (16) (18) (12) Ge(Sb) T I (2) - 1 -E l E 3 • T. . Is "L2p" A l + Is 2p Is 2p Is 12P° T T i 1-, 1 - Is Is A l T l . - Is Is - A l A l - Is Is F i g . 3-2. The E and K o p t i c a l t r a n s i t i o n s i n a donor p a i r i n S i and Ge. In the above f i g u r e , each "band" of l e v e l s i s l a b e l l e d by the i s o l a t e d donor states into which the p a r t i c u l a r molecular states d i s s o c i a t e i n the l i m i t R -> °°. In the concentration range of our i n t e r e s t the Is - Is l e v e l s are not broadened, whereas the Is - 2p° and Is - 2p _ are, [see eqs. (2.105), - 65 -(2.130), (2.131)]. The Is - 2p~ l e v e l s were observed to broaden more than the Is - 2p° ones, so we expect the B t r a n s i t i o n l i n e s to be broader than the E ones. We note that i n a l l t r a n s i t i o n s indicated i n F i g . 3-2, only one donor e l e c t r o n i s a c t u a l l y being excited; the other e l e c t r o n does not change i t s s i n g l e donor state during the molecular t r a n s i t i o n . I t follows that i n order that a t r a n s i t i o n be allowed both the molecular and the s i n g l e donor t r a n s i t i o n must be allowed. For molecular t r a n s i t i o n s the s e l e c t i o n r u l e s are 1 ^ 1 u g 3 3 (3.1) u -e* g and f o r s i n g l e donor t r a n s i t i o n s A l ~ T l T l " T l (3.2) T 2 ->T 2 T -*-»• T These l a t t e r s e l e c t i o n r u l e s follow from the f a c t that the e l e c t r i c dipole momentum operator transforms according to the i r r e d u c i b l e representation T^ under the symmetry operations of the point group T^ [see (2.116)]. The spin m u l t i p l i c i t y and "geradeness" of the molecular l e v e l s i n F i g . 3-2 are not i n d i c a t e d . In Chapter I I we observed that there are no spin s i n g l e t A A T T ungerade and spin t r i p l e t gerade Is 1 Is 1 and I s ^ l molecular states; a l l other molecular states of F i g . 3-2 e x i s t , [see eq. (2.79 ) ] . The numbers i n parentheses on the l e f t hand side i n d i c a t e the t o t a l number of - 66 -molecular states i n each p a r t i c u l a r "band" of l e v e l s . I f we denote by £ n(j0 the energy of the t r a n s i t i o n from a ground Is - Is state to the n-th state i n a p a r t i c u l a r Is - 2p "band" of a mole-cule with the interdonor separation R, measured r e l a t i v e l y to the t r a n s i t i o n energy i n the l i m i t R -»- °°, then the energy d i s t r i b u t i o n S(E) of the t r a n s i -t i o n l i n e i s determined by the expression S(E) = | I I 6(E - e (R)) P(R), ' (3.3) n R where P(R) i s the p r o b a b i l i t y of f i n d i n g two i n t e r a c t i n g donors at a separation R_ i n the c r y s t a l at a c e r t a i n donor concentration. The sum over n i n (3.3) goes over a l l N l e v e l s that contribute to the p a r t i c u l a r t r a n s i -t i o n l i n e , and the sum over R i s a summation over a l l l a t t i c e s i t e s i n the c r y s t a l . The function E (R) i n (3.3) i s i d e n t i c a l with <f (R) which we n — yv — were dealing with i n Chapter II-2c. I t i s a rather-complicated function of R. To compute i t f o r a l l molecular states and a l l R p o s s i b l e i n the c r y s t a l l a t t i c e would involve an enormous amount of work, which would not be worth doing since we are only i n t e r e s t e d i n the width of the energy d i s t r i b u t i o n function (3.3). Besides a l l experimental data on the concentration broaden-ing line-widths are bound to contain considerable errors since experimentally the t o t a l l i n e width i s measured which i s a r e s u l t of combined a c t i o n of several d i f f e r e n t processes. Therefore we s h a l l only estimate the l i n e width which i s due to the concentration broadening. We s h a l l make the estimate on the basis of the upper and lower bounds of the band involved i n the t r a n s i t i o n . - 67 -In Ch. II-2c we found that the extremes of a "band" are given by (see page 60) £ (R) = P \ u u D 3 and (3.4) £ 1 ( R ) = P l 3 ' K where and i n eq. (3.4) can assume values +2, +1, -1, -2, depending upon the symmetries of the states involved, and A i s a constant which depends on the magnetic quantum number of the s i n g l e donor 2p-like state and on the l s - l i k e donor states into which the molecular states taking part i n the tran-s i t i o n d i s s o c i a t e at R -> •». The subscripts u and 1 i n eq. (3.4) r e f e r to the upper and lower bounds of the "band". To s i m p l i f y the c a l c u l a t i o n we s h a l l suppose that the d i s c r e t e probah"" 1 i t y f u n r » M n n P'(R) i n p o . (3.3) can be w e l l approximated by an i s o -t r o p i c , continuous p r o b a b i l i t y density function P ( R ) . In t h i s way we can convert the sum over R. i n eq. (3.3) into an i n t e g r a l over R . Let us consider the energy d i s t r i b u t i o n function S n(E) of the absorption l i n e associated with the t r a n s i t i o n from a ground l s - l s state to the n-th excited ls-2p state. The energy of t h i s t r a n s i t i o n i s given by an expression analogous to eqs. (3.4), E ( R ) = P n K . (3.5) n R 3 Here the value of p l i e s between p, and p . Apart from the normalization n 1 u constant, s n ( E ) i s the n-th term i n the sum over n i n eq. (3.3), S J E ) - S(E-e (R))P(R)dR . (3.6) n Assuming that the p r o b a b i l i t y density function P(R) i s well described by (13) Chandrasekhar's function , defined as - 68 -R_ 3 P(R)dR = e" ( R s } d (|-) 3 , s (3.7) where R i s r e l a t e d to the donor concentration N by the r e l a t i o n s a 4nR we get from eqs. (3.5) and (3.6) the expression ,(n) • • / i A n J \ ~ ,(r S C E ) E n s ,(n) In eq. (3.9) the energy E i s defined as n R 3 (3.8) C 3 . 9 ) (3.10) S(E)E S> I-4 e - 2 f \ \ 1 i 2 e ~ 2 1 i - - / - - 4- -/ 1 / I / 1 \ i , ! E 0 0.5 1 131 1.5 F i g . 3-3. The broadening of the absorption l i n e r e s u l t i n g from the t r a n s i t i o n from a ground l s - l s donor molecular state to a p a r t i c u l a r l s - 2 p excited state due to the random d i s t r i b u t i o n of donors. F i g . (3-3) shows a graph of the energy d i s t r i b u t i o n f unction given by eq. (3.9), From the graph we see that the function S r ( E ) reaches i t s half-maximum value at E ( / } = 1.31 E ( N ) 2 S ' (3.11) - 69 -The energy d i s t r i b u t i o n function S ( E ) f o r an actu a l absorption l i n e , which r e s u l t s from t r a n s i t i o n s from the ground l s - l s l e v e l to the "band" of ls-2p l e v e l s , i s given by eq. (3.3). In terms of s n ( E ) > eq. (3.9), i t i s expressed as , . (n) . E ( n ) f s S ( E ) = i I - ~ - e E . (3.12) E This i s a superposition of l i n e s of the form shown i n F i g . 3-3. The outermost l i n e s c o n t r i b u t i n g to S ( E ) are those determined by £ U ( R ) end" e ^ ( R ) , eq. (3.4), i . e . S ^ ( E ) and S ^ ( E ) , where u and 1 r e f e r to the upper and lower bounds of the ls-20 "band". I f we define the width E of S ( E ) as the d i f f e r e n c e between the w energies at which S ^ ( E ) and S ^ ( E ) reach t h e i r half-maximum values, then i t follows from eqs. (3.10), (3.4) and (3.11) that E i s given by w Ew = 1-31 ( p u - pl } I ' ( 3 ' 1 3 ) where p = +1 and p.. = -2 for K = 1, 2, and p = +2 and p . = -1 f o r K = 3,4 u 1 u 1 (see page 60). In the following we s h a l l evaluate E ^ and compare i t with the exper-imentally measured halfwidths E i of absorption l i n e s . However, without a d e t a i l e d c a l c u l a t i o n of the ls-2p energy l e v e l s l y i n g i n between the upper and lower bounds, we can consider E ^ only a rough upper l i m i t of E J . S i l i c o n A A In s i l i c o n a l l the observed t r a n s i t i o n s o r i g i n a t e from the Is 1 Is 1 1 3 l e v e l , where there are only g and u molecular states ( i . e . K = 1,2). There-fore i t follows from (2.130) and (3.13) that the width of the E l i n e (see F i g . 3-2) i s given by E ( E ) = 1 . 3 1 - 3 . b 2 J ( 1 ) 2 - i r , (3.14) o R 3 s and the width of the B l i n e by •(B) . , ,,.,.,,2,(1)2 _1 R E W = 1 . 3 1 - 3 - a V ^ , (3.15) w o _ 3 s * 2 * where lengths are expressed i n units of a , and energy i n units of e /<a . I t i s to be noted that the e f f e c t i v e Bohr radius a does not appear i n any of the - 71 -ft above expressions e x p l i c i t l y . Therefore the results are independent of a . ft In the present study a is used merely as a unit of length. Using the following numerical values for phosphorus doped si l i c o n , a = 25.0 A b = 14.2 A J . o a" - 20.4 A A E( l s 1 ) = 45.1 . meV E ( ^ } = 29 meV ef f K = 12 5 = 6 = 1 o ± ' we obtain from (2.127) and (3.14), (3.15) the result: E(E) = 8 5 ^ [ n ] e V j ( 3 > 1 6 ) w R 3 s , E(B) = 265^ [ m e V ] _ ( 3 a ? ) w „ J Fig. 3-4 shows the plot of the estimated line widths, together with the experimental results obtained by Kuwahara.^*^ These results were obtained b y separating the concentration dependent part of the line half-widths from the total halfwidth. Germanium In germanium the situation i s more complicated since there are several l s l s levels that may be occupied also at low temperatures, (see Fig. 3-2). Various transitions may have different relative probabilities to occur due to differences in population and number of states available. Since we are only estimating the line widths, l e t us estimate the maximum effect that the differences in relative transition probabilities - 72 -can have on the l i n e widths. This w i l l occur at temperatures where the A i T i T i A i A i h t r a n s i t i o n Is Is Is 2p dominates over the t r a n s i t i o n Is Is Is 2p (due to a greater number of a v a i l a b l e i n i t i a l s tates i n s p i t e of the Boltzmann facto r, for the former t r a n s i t i o n ) i n the E- and B l i n e s , and A, T, A, T„ T, T, 1 1 1 1 1 1 where the t r a n s i t i o n Is Is ->• Is 2p dominates over Is Is -> Is 2p Cdue to the more favourable Boltzmann f a c t o r , f o r the former t r a n s i t i o n ) i n the E^ and B^ l i n e s . Under these conditions the widths of these.lines are given by the following expressions: E ( l ) = 1.31 -4 . b 2 J ( 3 ) 2 - ~ (3.18) ° R 3 s (E ) (E ) / \ 2 E. 3 = E . 1 • / - 4 ^ ) (3.19) o (B ) (E.) 2 E = E (3.20) w w \ b / (B ) (B ) / J \ 2 E „ 3 " E w " ' (73)) < 3 - 2 1 ) o Using the following data for the antimony-doped germanium, a = 64.5 A b = 22.7 A * 0 a = 45.0 A E ( l s = 1 0 . 2 meV - 73 -T 3 E C l S } = 9.9 meV E « 9.2 mey e f f K = 16 6 =6^ =1 o ± we obtain the following numerical -values f o r the l i n e widths ( V 3 E = 121./R w s E( V = 112./R3 w (3.22) ( V 3 E = 15.2/R w s (B ) E 6 = 14.0/R"5 w s F i g . 3-5 shows p l o t s of our estimates for the widths of absorption l i n e s i n antimony-doped germanium. Experimental points are the r e s u l t s given by Nishida and H o r i i from which we subtracted the concentration (±) independent widths h Q due to phonon broadening; h^ = 0.10 meV, h^ 0^ = 0.04 meV. These values were obtained by extrapolating Nishida and H o r i i ' s r e s u l t s to N^ = 0, and are therefore bound to contain large errors since the accuracy of the experimental r e s u l t s themselves i s not very high. The experimental r e s u l t s are p a r t i c u l a r l y u n r e l i a b l e for the E l i n e s since as stated by t h e i r authors the instrumental broadening may not have been f u l l y accounted f o r , so that the r e s u l t s given are the upper l i m i t to the l i n e widths due to the concentration broadening. Keeping t h i s i n mind when comparing our estimates and experimental r e s u l t s , we can say that agreement i s good. - 74 -F i g . 3-5. The estimated concentration dependence of the widths of the E and B absorption l i n e s i n Ge(Sb), together with experimental r e s u l t s . - 75 -The agreement between our estimates and experimental r e s u l t s f o r phosphorus-doped s i l i c o n which are much more accurate, (see F i g . 3-4), i s e x c e l l e n t . Therefore we can conclude that, the observed concentration-dependent width of absorption l i n e s i s a r e s u l t of molecular i n t e r a c t i o n s between donors. Our r e s u l t s for antimony-doped germanium are i n agreement with the experimental observation that the widths of B^ and l i n e s are l a r g e r than the widths of B^ and E^ l i n e s , r e s p e c t i v e l y , too. However, the act u a l d i f f e r e n c e s i n widths seems to be much l a r g e r than those yielded by our model. This suggests that there must be another concentration dependent broadening mechanism present. I t must be much weaker than mole-cular i n t e r a c t i o n s , but i t s e f f e c t must strongly depend upon the symmetries of the e l e c t r o n states involved. Most probable source of such a broadening (?4) seems to be the s t r a i n s , as suggested by Pomerantz. It i s to be noted that the same matrix element [eq. (2.113)] that i s responsible f o r the broadening of the (ls2p) "bands" determines the pro-b a b i l i t i e s for the e l e c t r i c dipole t r a n s i t i o n s Is -> 2p i n i s o l a t e d donors. ± o One would expect therefore that the r a t i o s of the (ls2p ) and (ls2p ) "band" widths i s approximately equal to the r a t i o s of i n t e n s i t i e s of ± o Is •> 2p and Is -> 2p t r a n s i t i o n s i n s i n g l e donors. These r a t i o s are not exactly equal since the 2p - and 2p° l e v e l s are not degenerate, i n s i n g l e donors. - 76 -In s i l i c o n , + .E~ . . 2 = * 3.05 (3.23) E° b 2 w and .E..OT.E. _?2 iS_ V 2.65 (3.24) J l s This l a t t e r r e s u l t i s i n agreement with the experimental r e s u l t s obtained by Kuwahara. IV. THE MAGNETIC SUSCEPTIBILITY OF WEAKLY INTERACTING DONORS IN Ge 1. Introduction The magnetic properties of impure semiconductors at low temper-atures very strongly depend upon the degree of doping. At very low impurity concentrations, such that overlap between im p u r i t i e s i s n e g l i g i b l e , the magnetic behaviour of impure semiconductors i s w e l l represented by the magnetism of i s o l a t e d i m p u r i t i e s . I t has been observed that at very low donor concentrations the magnetic s u s c e p t i b i l i t y of n-type semiconductors i s a l i n e a r function of the concentration of donors and obeys the Curie / T O N law. ' ' At very high impurity concentrations, on the other hand, high enough that merging of the conduction band and impurity l e v e l s has taken place, the magnetism of impurities i s completely determined by the band magnetism of the merged band.^''' Both of these extreme cases are reason-ably simple from a t h e o r e t i c a l point of view, and very good agreement between theory and experiment has been obtained. For intermediate donor concentrations, however, the behaviour of semiconductors i s more complicated. Experimental studies i n s i l i c o n ^ ^ and germanium have shown that as the concentration of donors increases the spin paramagnetic s u s c e p t i b i l i t y deviates from that expected f o r i s o l a t e d donors more and more. The slope of the curve of s u s c e p t i b i l i t y vs. inverse temperature decreases with decreasing temperature. This i s schematically i l l u s t r a t e d i n F i g . 4-1. -. 78 -N. (4) N (3) d (2) I (I) JL T N d ( l ) < N d ^ < N d ^ ) < N d ( 4 ) Curie Law Fig. 4-1. The observed temperature and donor concentration dependence of the spin paramagnetic susceptibility of n-type semiconductor. (Schematic diagram), In the diagram of Fig. 4-1 the straight dashed lines represent the Curie law as expected for isolated donors at four different donor concentrations N,. Concentration belongs to the range of low donor concentrations; d d here the spin paramagnetic susceptibility i s s t i l l well represented by the susceptibility of isolated donors, i.e. the Curie law. We see how the donor-donor interactions give rise to deviations from the Curie law at higher concentrations. The magnetic susceptibility may actually, become (4) independent of temperature at certain concentrations, (N^ in Fig. 4-1). ™ (4) > e. 1^16 -3 (8) In antimony-doped germanium J N , o x 1U cm . v ' Sonder and Schweinler^) have ca l c u l a t e d the spin paramagnetic s u s c e p t i b i l i t y of weakly i n t e r a c t i n g donors using a model i n which they tre a t the i n t e r a c t i n g donors as an assembly of i s o l a t e d donor p a i r s . In ft t h e i r model they use a s i n g l e conduction band with an i s o t r o p i c mass m equal to an appropriate average of the l o n g i t u d i n a l and transverse masses. They take into consideration only the spin s i n g l e t and t r i p l e t l s - l s Heitler-London molecular states that d i s s o c i a t e into the lowest l y i n g l s -l i k e s i n g l e donor s t a t e . The other molecular states are not taken into account i n t h e i r model. Assuming a random d i s t r i b u t i o n of donors, they get the following r e s u l t s f o r the magnetic s u s c e p t i b i l i t y per gram Ap ^1-a where C i s given by the expression and a i s defined by N, * 3 B(m y (4.3) In eq. (4.2) p, 8 and k are the s p e c i f i c weight of the c r y s t a l , Bohr mag-neton and Boltzmann constant, r e s p e c t i v e l y . A and B i n (4.2) and (4.3) are constants r e l a t e d to the spin s i n g l e t - s p i n t r i p l e t energy s p l i t t i n g of the hydrogen molecule. The Sonder and Schweinler theory s u c c e s s f u l l y accounts f o r the deviations from Curie law that were observed experimentally for donors i n s i l i c o n . Attempting to apply t h e i r theory to the case of donors i n - 80 -germanium one gets reasonably good agreement with r e s u l t s of measurements for phosphorus donors. In the case of antimony donors however, the Sonder and Schweinler theory breaks down. } I t does not account for the magnetic s u s c e p t i b i l i t y becoming independent of temperature at 16 —3 \ 6 x 10 cm as has been observed experimentally. (See F i g . 4-1.) In t h i s chapter we s h a l l attempt to improve the Sonder-Schweinler theory. Our a s s e r t i o n i s that the f a i l u r e of the theory i n the case of antimony donors i n germanium i s due to neglect of the higher l y i n g l s ( T ^ ) donor s t a t e s . While i n s i l i c o n the p r o b a b i l i t y for these states to be occupied at l i q u i d He temperatures i s n e g l i g i b l e (due to the large value of the v a l l e y - o r b i t s p l i t t i n g ) , t h i s i s no longer true i n germanium where donors e x h i b i t a much smaller v a l l e y - o r b i t s p l i t t i n g (see Ch. I I - l ) . One i s led to the conclusion that the higher l y i n g donor states are important by the experimentally observed f a c t that deviations from Curie law increase with decreasing value of the v a l l e y - o r b i t s p l i t t i n g , at a fi x e d temperature and donor concentration. Among a l l the species of donors i n germanium that have been studied experimentally, the antimony donors have the smallest v a l l e y - o r b i t s p l i t t i n g . I t s value i s 0.32 meV, which i s of the same order of magnitude as the thermal energy kT at l i q u i d helium temperatures. Thus one would expect the e f f e c t of the higher l y i n g donor states to be e s p e c i a l l y pronounced i n the case of antimony donors. Therefore we are going to use an improved model i n order to c a l -c u late the spin paramagnetic s u s c e p t i b i l i t y of donors i n germanium. In t h i s model we s h a l l take into account a l l the l s - l i k e s i n g l e donor sta t e s . - 81 -The concentration range of our i n t e r e s t w i l l be 10" cm a, a, 10 cm , 1. e. the s o - c a l l e d intermediate concentration range. I t i s to be noted that we s h a l l now be working at a higher concentration than with the o p t i c a l study of germanium i n the previous chapter. Our assumption of r e l a t i v e l y large interdonor separations i s no longer applicable here and interference terms play an important r o l e . We s h a l l take into account the m u l t i v a l l e y nature of the conduction band which gives r i s e to the interference terms, (see Ch. II-2a). At present, the only a v a i l a b l e experimental data on the spin para-magnetic s u s c e p t i b i l i t y are those obtained from the r e s u l t s of measurement of the t o t a l magnetic s u s c e p t i b i l i t y which includes also o r b i t a l and Van Vleck s u s c e p t i b i l i t i e s . One separates the spin paramagnetic s u s c e p t i b i l i t y f r o m I-VIOTTI by p s c i i m i n p t h a t - i t gives the only t e m D e r a t u r e - d e p e n d e n t c o n t r i -bution to the t o t a l s u s c e p t i b i l i t y . In our model t h i s assumption may not be true any longer (due to the small magnitude of the v a l l e y - o r b i t s p l i t t i n g ) . Thus we s h a l l include a c a l c u l a t i o n of o r b i t a l and Van Vleck magnetic suscep-t i b i l i t i e s as w e l l , i n t h i s chapter. 2. Magnetic S u s c e p t i b i l i t y of Impure Semiconductors Accepting the Busch-Mooser'^7) basic model of the magnetic s u s c e p t i b i l i t y of a semiconductor c r y s t a l , we consider the s u s c e p t i b i l i t y 3 of a germanium c r y s t a l , X c ry S t a]_» doped with N^ , group V donors per cm to d i f f e r from the s u s c e p t i b i l i t y of the pure germanium c r y s t a l , x » only through the c o n t r i b u t i o n from the donor electrons, i . e . X ^ I = X + X J (4.4) ^ c r y s t a l A g * > - 82 -where x 1 S t n e s u s c e p t i b i l i t y of the donor e l e c t r o n s . This i s only an approximation to the actual case, of course, since time s u b s t i t u t i o n of host atoms by donors c e r t a i n l y does a f f e c t the l a t t i c e , and through i t the l a t t i c e s u s c e p t i b i l i t y X • However, the Buscn-Mooser model, with minor modifications by Stevens et a l . (28) ^ a s a n o w e c j l f o r g 0 0 d i n t e r p r e t -ations of the behaviour of many semiconductors(29)^ w f i i c h would imply that the e f f e c t of donor cores on the l a t t i c e s u s c e p t i b i l i t y x 1 S small. We are int e r e s t e d i n the donor magnetic s u s c e p t i b i l i t y i n the concentration range 10"^ £ 10"^ donors per cm3 a t l i q u i d He tempera-tures. This concentration range i s characterized by an i n t e r a c t i o n between donors that i s strong enough that donors cannot be considered i s o l a t e d , yet weak enough that we can regard donor electrons l o c a l i z e d . Measurements 17 -3 upper l i m i t f o r the l a t t e r assumption to be v a l i d , f o r antimony donors; (^ 17 -3 fo r phosphorus donors the l i m i t i s 3 x 10 cm In our treatment of i n t e r a c t i n g donors we s h a l l assume that i n the concentration range of our i n t e r e s t the most s i g n i f i c a n t c o n t r i b u t i o n to the donor magnetic s u s c e p t i b i l i t y x a r i s e s from tlie pairwise i n t e r a c t i n g donors. Thus we s h a l l assume that the s u s c e p t i b i l i t y of i n t e r a c t i n g donors i s we l l represented by the s u s c e p t i b i l i t y of H^/2 i s o l a t e d donor p a i r s . We also assume that the d i s t r i b u t i o n of donors i s random throughout the c r y s t a l . Our donor p a i r s are assumed to be fonnmed by the nearest neighbours. Thus the donor magnetic s u s c e p t i b i l i t y i s given by the expression X = - f E P(R) x ( £ ) > (A.5) a l l l a t t i c e s i t e s - 83 -where x(R) i s the magnetic s u s c e p t i b i l i t y of a donor p a i r with the i n t e r -donor separation R, and P(R) i s the p r o b a b i l i t y of f i n d i n g such a donor p a i r i n the c r y s t a l at a given concentration of donors. Since the d i s t r i -bution of donors i s random, we can f i x the o r i g i n of R at an a r b i t r a r y l a t t i c e s i t e ; the sum i n (4.5) then represents a summation over a l l the pos s i b l e p o s i t i o n s of the second donor i n the c r y s t a l . In the concentration range of our i n t e r e s t the average interdonor separations are reasonably large compared to the l a t t i c e spacings (|R| va r i e s between ^20 and ^ 50 a, . ). So we can replace the sum over a l l l a t t i c e l a t t i c e s i t e s of the c r y s t a l by an i n t e g r a l over the absolute value of interdonor separation R - | R | , and a sum over a l l l a t t i c e s i t e s w i t h i n a s p h e r i c a l s h e l l of radius R and thickness dR. In Chapter I I , we observed that i n the absence of a magnetic f i e l d a l l donor p a i r s which have the second donor within the same s p h e r i c a l s h e l l can be divided into three c l a s s e s ; a l l pa i r s belonging to the same c l a s s have the same energy spectrum. (They a l l have the same set of values for the interference c o e f f i c i e n t s . ) In the presence of a magnetic f i e l d t h i s i s no longer true. As we s h a l l see l a t e r , the magnetic s u s c e p t i b i l i t y i s determined by the energy spectrum of our el e c t r o n system i n the presence of a magnetic f i e l d . In the l i m i t of a very weak magnetic f i e l d one would expect the di f f e r e n c e s i n s u s c e p t i b i l i t i e s to be much smaller among donor p a i r s belonging to the same c l a s s than among the pa i r s belonging to d i f f e r e n t c l a s s e s . Therefore, we carry out an averaging over R, the unit vector i n the d i r e c t i o n of the o r i e n t a t i o n of the donor p a i r , separately f o r each of the three classes L. Eq. (4.5) now becomes N d • ' L=a,b,c P(R) X L(R)dR , (4.6) - 84 -where P(R)dR i s the p r o b a b i l i t y of f i n d i n g the nearest neighbour i n the s p h e r i c a l s h e l l [R, R + dR], and p i s the p r o b a b i l i t y that a donor p a i r l-i belongs to the c l a s s L. The p r o b a b i l i t i e s p were given i n Ch. I I , eq. (2.89). X^(R) ^ n e9« (4.5) i s the average magnetic s u s c e p t i b i l i t y of a molecule of class L, i n the s h e l l [R, R + dR], X ( R ) = 47 X L(R)dR , ( 4 . 7 ) where R i s the un i t vector along R. Here we assumed that the d i s t r i b u t i o n of donors i s i s o t r o p i c . Quantum mechanically, the magnetic s u s c e p t i b i l i t y of a molecule of c l a s s L and interdonor separation R i s given by the following expression: X L(R) = ~ 3 ! In Z L(R) . (4.8) Here k i s the Boltzmann constant, T the temperature, H the magnetic f i e l d and Z^'(R) the p a r t i t i o n function defined by Z L(R) = E exp [- £ ( K^' L(R,H)/kT] . (4.9) n,K,M n ' W (K)' L In t h i s expression £ ' (R,H) i s the e l e c t r o n i c energy of the donor pair i n the presence of the magnetic f i e l d H. The subscript n s p e c i f i e s the p a r t i c u l a r Heitler-London combination of the o r b i t a l s i n g l e donor states, K denotes the spin m u l t i p l i c i t y of the p a r t i c u l a r molecular state, and M stands f or the component of the spin i n the d i r e c t i o n of the magnetic f i e l d ; M = 0 f o r K = 1,3 ( i . e . spin s i n g l e t s ) and M = 0, ± 1 for K = 2,4 ( i . e . spin t r i p l e t s ) . We s h a l l take into account a l l the l s - l s molecular sta t e s . The c o n t r i b u t i o n from the higher excited molecular states to the sum (4.9) - 85 -can be neglected since t h e i r energies are large compared to kT at l i q u i d He temperatures. The diagram i n F i g . 4-2 shows schematically the l s - l s energy l e v e l s and the r e l a t i v e s i z e of kT. 1 0.6 --spin s i n g l e t spin t r i p l e t 0.2 0. kT R ^ 40 a T ^ 4°K F i g . 4-2. The r e l a t i v e s i z e of kT at 4°K i n comparison with the l s - l s energy l e v e l s of a Sb donor p a i r i n Ge. The energy appearing i n (4.9) i s an eigenvalue of the Hamiltonian •#? u(l,2) f o r the electrons of the donor p a i r system i n the presence of a H magnetic f i e l d , # H ( 1 , 2 ) = #(1,2) + ^ m a g d , 2 ) (4.10) - 86 -Here TC(JL,T) i s the Hamiltonian i n the absence of a magnetic f i e l d , and •ft (1,2) i s the Hamiltonian describing the i n t e r a c t i o n of the electrons e x mag ' b with the magnetic f i e l d . The Hamiltonian " ^ ( l ^ ) has been discussed i n d e t a i l [see eq. (2.34)] and i t s eigenvalue problem solved i n Chapter I I . '#(1,2) * (5 ) , L(1,2) = £ ( K ) > L ( R ) *(5)jL (4.1D nM ' n - nM 3» (^ ) , L(1,2) = [Z a ( K ) ' L T ( K ) ( 1 , 2 ) ] ? < K ) (4.12) nM n,yv yv M yv (K) In (4.12) ? M i s the spin wave function; f o r K = 1,3 i t describes a spin s i n g l e t state, M = 0, and f o r K = 2,4 a spin t r i p l e t s t a t e , M = 0, ± 1 . The magnetic Hamiltonian K (1,2) defined by (4.10) i s given mag by the following expression: W 1 ' 2 ) • k - ^ i - t f v a ) - f A d ) ] + f i V 2 ( D 2 + 2m" [T^(2) " fA(2)]'[fv(2) - | A ( 2 ) ] + | - v 2 ( 2 ) + 3g (S(l) + S(2))-H . (4.13) Here m i s the (free) e l e c t r o n mass, 6 the Bohr magneton, and g the gyro-magnetic r a t i o of a donor e l e c t r o n . (Experimental studies of the g-factor have shown that i t i s p r a c t i c a l l y independent of the donor concentration , which j u s t i f i e s the use of a single-donor g-factor i n the Hamiltonian des c r i b i n g a system of i n t e r a c t i n g donors.) S_(l) and S_(2) are spins of the electrons 1 and 2. A ( l ) i n (4.13) i s the vector p o t e n t i a l of the f i e l d H - 87 -as f e l t by the e l e c t r o n 1, and A(2) the p o t e n t i a l of the same f i e l d at the p o s i t i o n of e l e c t r o n 2, A ( l ) = | (H x r 1 ) A(2) = | (H x r 2 ) (4.14) In the l i m i t of a very low magnetic f i e l d we can employ perturba-(K) L t i o n theory i n order to c a l c u l a t e the energies (R»H)» using the (K) L eigenfunctions $ ^ * of the unperturbed Hamiltonian 7C(1,2), eq. (4.11). nM Perturbation theory y i e l d s the energy expressed i n the form of a power se r i e s i n terms of H.. A f t e r performing an averaging over a l l d i r e c t i o n s of the magnetic f i e l d H, f ^ ' ^ H ) ^ J d £ ^ > L (R,H) , 14 . 1 b , we obtain the following r e s u l t : c^),L02»H) = ' ^ K ) , L ( R ) +D^ K )' L(R) x H 2 , K = 1,3 (4.16) fo r s i n g l e t states, and /f (5),L(R.H) = £ ( K ) ' L ( R ) + e g MH + D ( K ) ' L ( R ) x H 2, K = 2,3 (4.17) *"nM —' n — n — fo r t r i p l e t s t a t e s . In eq. (4.15) we took an average over a l l d i r e c t i o n s of the magnetic f i e l d not only i n order to s i m p l i f y the c a l c u l a t i o n (by reducing the number of v a r i a b l e s ) but also i n order to bring theory closer to experimental conditions. Namely, i n experimental studies of the magnetic s u s c e p t i b i l i t y one usually uses powdered samples. Apart from s t r a i n s (that are randomized i n a powdered sample) there i s t h e o r e t i c a l l y no d i f f e r e n c e between a powdered sample i n a magnetic f i e l d of f i x e d d i r e c t i o n , and a s i n g l e c r y s t a l i n a magnetic f i e l d whose d i r e c t i o n i s changing randomly during the measurement so that the r e s u l t of the measurement i s an average over a l l d i r e c t i o n s of the magnetic f i e l d . OO L In eq. (4.16) and (4.17) the quantity (R) i s given by the following expression: D ( K ) ' L ( R ) = n ~ 4TTH2 dH [ < 0 ^ ) ' L ( l , 2 ) | 7 i r o r b ( l , 2 ) | ^ K I ) ' L ( l , 2 ) > + z t<*g )' La » 2 ) i y v v(i , 2 ) + y B(i , 2 ) i»g>» L(i > 2 ) > i 2 ] ^ ^ n7n I f ^ ' V ) - £ ( K ) ' L ( R ) M* n n Here the summation over n' and M' goes over a l l the states of the donor p a i r , i n c l u d i n g the excited and continuum states, except the state s p e c i -f i e d by n. 7C K ( l , 2 ) i n (4.18) i s the part of ~tf (1,2) that contains orb mag the magnetic f i e l d q u a d r a t i c a l l y , i . e . 2 7 ? o r b ( l , 2 ) = [A(1).A(1) + A(2)-A(2)] . (4.19) 2mc . 78T (1,2) and 7^(1,2) are associated with the terms containing H l i n e a r l y ; they are defined as 7C (1,2) = - ~ - [ V ( l ) - A ( l ) + A ( l ) - V ( l ) + V(2)-A(2) .+ A(2 ) -V(2)} (4.20) vv 2mic — — — — __ _ and - 89 -7^(1,2) = 3g (1(1) + S(2))-H . (4.21) The operator 7 £ (1,2) gives r i s e to the so-called o r b i t a l diamagnetism, and 34^(1,2) gives r i s e to Van Vleck parama.gnetirm. Upon i n s e r t i n g (4.16) and (4.17) into (4.9), i t follows from (4.8) that i n the l i m i t of a very low magnetic f i e l d , BgH << kT, the magnetic T s u s c e p t i b i l i t y X (R) i s given by the expression 2 »V f r £ " K ) ' L ( - i x ( 1 ) = 7 ^ ^ „ ! K E X I > 1 — s - 1 K=2,4 E n ( K ) , L ( R ) e x p J _ n - j _ ( A > 2 2 ) Z L(R) n,K K n ~ ^ k T Here n has the following values: K / 1 f o r K = 1,3 ( i . e . s i n g l e t s ) n = (4.23) 3 for K = 2,4 ( i . e . t r i p l e t s ) Z L(R) appearing i n eq. (4.22) i s the p a r t i t i o n function defined by eq. (4.9) taken i n the l i m i t of a very low magnetic f i e l d , i . e . Z L(R) = Z % exp [-£ ( K )' L(R)/kT] . (4.24) n,K n — - 90 -3. The Spin Paramagnetic S u s c e p t i b i l i t y The f i r s t term i n eq. (4.22) represents the spin paramagnetic s u s c e p t i b i l i t y . P h y s i c a l l y i t a r i s e s from the di f f e r e n c e s i n population of e l e c t r o n energy l e v e l s associated with d i f f e r e n t components of spin i n the d i r e c t i o n of the magnetic f i e l d . (V\ T The energy £ ^ ' (R) i n eqs. (4.22) and (4.24) i s the energy i n the absence of a magnetic f i e l d . As observed i n Chapter II i t depends only on the absolute value R of the interdonor separation R f o r a given c l a s s L. Therefore the spin paramagnetic part of the expression (4.22) i s independent of the o r i e n t a t i o n of the molecule, i . e . X^W = X^(R) = - T 2 — E ' e x p [ - ^ K ) ' L ( R ) / k T ] , (4.25) Z (K) "" n,K K=2,4 where Z L(R) = S ^ exp [ - ^ K ) ' L ( R ) / k T ] . (4.26) n,K The expression (4.25) f o r the spin paramagnetic s u s c e p t i b i l i t y of a donor p a i r can now be wri t t e n i n the following form: 2 2 X£(R) = 2 x -§J^ x G L(R,T) , (4.27) where the function G^(R,T) i s defined as j ? ( K ) > L ( R ) , s n ( K )' L(R) r 1 G L(R,T) = 4 x V exp [- ] x j E n K exp [- ] } . (4.28) n,K n,K K=2,4 - 91 -2 2 If i t i s r e c a l l e d that 6 g /4kT i s the spin paramagnetic s u s c e p t i b i l i t y f o r an i s o l a t e d donor ( i . e . the Curie law), i t i s evident from (4.27) that the value of the spin paramagnetic s u s c e p t i b i l i t y for a pa i r of i n t e r a c t i n g donors i s given by that f o r two i s o l a t e d donors m u l t i p l i e d by the c o r r e c t i o n f a c t o r G L. This c o r r e c t i o n f a c t o r , eq. (4.28), i s a fun c t i o n of both the temperature and the distance between the two donors. It has the correct asymptotic behaviour; i n the l i m i t T -> 0, R ->• °°, the spin s i n g l e t and spin t r i p l e t l e v e l s coincide, the higher l y i n g states are unpopulated and G L •* 1. Inserting (4.27) into (4.6) we obtain the fol l o w i n g expression for the spin paramagnetic s u s c e p t i b i l i t y of the whole c r y s t a l , „2 2 = N x A / . V T J ( N , , T ) , (4.29) where ?(N d,T) = E p L L=a,b,c P(R) G L(R,T)dK . (4.30) X as given above was evaluated numerically on a computer. The c a l c u l a t i o n (K) L i s straightforward. In Ch. I I we computed the eigenvalues ^ ' ( R ) as a function of a d i s c r e t e R for a l l three classes L, (see page 39). The co r r e c t i o n f a c t o r G' I j(R , T) defined by (4.28) was computed at the same values of d i s c r e t e R as a function of 1/ T i n the range 1/T = [0.2, 1.0]. Using the Aitken quadratic i n t e r p o l a t i o n formula, we them joined the d i s c r e t e values of G L ( R,T) i n t o a continuous function. The i n t e g r a l appearing i n (4.30) was consequently c a r r i e d out by the Gauss quadratic i n t e g r a t i o n - 92 -method. For the pa i r d i s t r i b u t i o n function P(R) we employed Chandrasekhar's fu n c t i o n . I t i s given by d (^) - e" ( P T ) 3 d ( ^ ) 3 , (4.31) s "s s where R i s r e l a t e d to the donor concentration N, by the r e l a t i o n s d 4TTR 3 §L = J L . (4.32) 3 N. d The computation was performed for two d i f f e r e n t dopants i n germanium, antimony and phosphorus. In the case of antimony donors, the gyromagnetic r a t i o g i s a n i s o t r o p i c . Experimental studies have shown that there are four g tensors; they are e l l i p s o i d s of r e v o l u t i o n with symmetry axes pointing i n [1,1,1] directions.(~*0) Q n e w r i t e s ? 2 . 2 _ . 2 2 „ //, oo\ g = gj^ S i l l O - r fe| | C u o O , V'-'--) where gj^ and g|j are the p r i n c i p a l g values perpendicular and p a r a l l e l to the symmetry axes, r e s p e c t i v e l y , and 9 i s the angle between the magnetic 2 f i e l d and the symmetry a x i s . In an actual case the value of g depends not only on the angle 9 but also on s t r a i n s i n the sample. In a powdered sample these s t r a i n s are randomized. Thus i t seems reasonable to use an average 2 value f o r g i n eq. (4.29). We take g 2 = i (g| + 2 g 2 j ) . (4.34) Numerical values of gj^ and g|| are 1.917 and 0.842. ^ 3 0 \ The.g-factor associated with phosphorus donors i n germanium i s i s o t r o p i c and has the value 1.56. ( 1 5 ) F i g . 4-3. _ The spin paramagnetic s u s c e p t i b i l i t y of Sb-doped Ge as a function of temperature and donor concentration. F i g . 4-4. The spin paramagnetic s u s c e p t i b i l i t y of P-doped Ge as a function of temperature and donor concentration. - 95 -Results of the c a l c u l a t i o n s are presented i n Figures 4-3 and 4-4. The quantity A appearing i n the figures i s the magnitude of the v a l l e y - o r b i t s p l i t t i n g . The only a v a i l a b l e experimental data on the magnetic s u s c e p t i -b i l i t y of n-type germanium are r e s u l t s obtained by the Faraday method. This method gives the value of the t o t a l magnetic s u s c e p t i b i l i t y of the sample, X c r y s t a l ° ^ e sV^-n P a ramagnetic c o n t r i b u t i o n to the t o t a l suscep-t i b i l i t y i s very small; the order of magnitude of ^ s t a l "*"S ^ ^ e r a u g \ -9 -1 whereas X ^ 10 ' emu g . The accuracy of the most accurate experimental (8) measurements i n n-type germanium a v a i l a b l e i s 0.2%. Supposing that the spin paramagnetic s u s c e p t i b i l i t y i s responsible for the temperature dependence of the r e s u l t s , one can separate i t from the t o t a l s u s c e p t i b i l i t y . — - - ... .... " :. • . . . ' • • 1 ....... .. i. _• _ 4. J T. _• 1 _• 4-,. 4-1, „_ J-ll U 1 I J . & W d y c u e : U U L a x n o \ac*.L-tx u u L U C 41*^-1-1.1 y *. u u 1 u 5 . 1 L •_ ^ ^ ^ w r -L ^ — . -j .. error of ^ 20%. The error i s a c t u a l l y even larger since the o r b i t a l and Van Vleck s u s c e p t i b i l i t i e s are not exactly independent of temperature, as we s h a l l see i n the next section of t h i s chapter. This large error i n the experimental data makes i t meaningless to compare them q u a n t i t a t i v e l y with the r e s u l t s of our c a l c u l a t i o n . However, we can make a q u a l i t a t i v e compar-is o n . Comparing our r e s u l t s for antimony doped germanium with those for the case of phosphorus donors, we notice that at low donor concentra-tions the s u s c e p t i b i l i t y obeys Curie law very w e l l , i n both cases. As the concentration increases the deviations from the Curie law become larger and l a r g e r ; the Curie law i s breaking down f a s t e r f o r antimony than for phosphorus donors. The low temperature s u s c e p t i b i l i t y becomes a c t u a l l y - 96 -temperature independent f o r antimony donors at concentrations where i t i s s t i l l strongly temperature dependent i n the case of ptiosphorus donors. Damon and G e r r i t s e n ^ ^ give 6 x 1 0 ^ cm 3 as the concentration at which the s u s c e p t i b i l i t y of an antimony-doped sample becomes independent of temperature. We can conclude that the q u a l i t a t i v e agreement between experimental i n d i c a t i o n s and our c a l c u l a t i o n i s very good. How does our model explain these features? The spin paramagnetic s u s c e p t i b i l i t y of an impure semiconductor i s expresseel as a product of two terms, according to our model [see eq. (4.2.9)]. The f i r s t of these, 2 2 N^g g /4kT represents the Curie law; i t i s the same expression as would a r i s e from the d i f f e r e n c e i n thermal population between the single-donor spin — states with spin p a r a l l e l and a n t i p a r a l l e l witlh the magnetic f i e l d . But i i i c r ;;cd;l ~~z c2.2.c"cJ- ~ci**c ~ f ^-"-^cr ~1 ->'*»*-,**r»T* tr- fr*^-™> f n n n _ magnetic) spin s i n g l e t and (magnetically active) spdun. t r i p l e t states, the source of t h i s f i r s t term i s a c t u a l l y the d i f f e r e n c e i n population of the molecular t r i p l e t states associated with d i f f e r e n t rarientations of spin with respect'to the magnetic f i e l d . The second term, ^ ( N ^ j T ) , r e f l e c t s the presence of the molecular spin s i n g l e t s t a t e s . I t represents e s s e n t i a l l y the average r e l a t i v e population of the t r i p l e t l e v e l s . I t i s normalized i n such a way that i t i s equal to unity when the average thermal population f a c t o r s are equal for s i n g l e t and t r i p l e t l e v e l s , ((i.e. when the actual populations of s i n g l e t and t r i p l e t l e v e l s are i n the r a t i o 1:3.) At low donor concentrations the average imterdonor separations are large, and the spin s i n g l e t - t r i p l e t s p l i t t i n g i s small compared to kT (at l i q u i d He temperatures). There i s therefore IMB d i f f e r e n c e i n the - 97 -thermal population p r o b a b i l i t i e s between the spin s i n g l e t and spin t r i p l e t s t a t e s . I t follows that jjr(Nd,T) = 1, and the spin paramagnetic s u s c e p t i -b i l i t y i s given by that of the i s o l a t e d donors, i . e . the Curie law. With increasing donor concentration the average interdonor sepa-r a t i o n decreases and the i n t e r a c t i o n between donors becomes stronger and stronger. This r e s u l t s i n l a r g e r spin s i n g l e t - t r i p l e t s p l i t t i n g s . In dopants that have a v a l l e y - o r b i t s p l i t t i n g large compared to kT, there are only two l e v e l s , a spin s i n g l e t and a s p i n t r i p l e t one, that are relevant to the temperature dependence of the f a c t o r ^ . This case i s i l l u s t r a t e d i n F i g . 4-5. In (a) we divided the temperature range of our i n t e r e s t into three regions, regarding the r e l a t i v e magnitude of kT compared to the spin s i n g l e t - t r i p l e t s p l i t t i n g e.. Graph (b) shows the factor ^ i n the corresponding regions. In region a, kT i s l a r g e compared to e, the population p r o b a b i l i t y of the t r i p l e t l e v e l i s comparable to the s i n g l e t l e v e l , and ^ - 1. In region y, kT i s small compared to E; the population p r o b a b i l i t y of the t r i p l e t l e v e l i s small compared to the s i n g l e t F i g . 4-5. Schematic i l l u s t r a t i o n of the temperature dependence of the f a c t o r which gives r i s e to deviations from the Curie law. - 98 -l e v e l , and ^ - 0. In between, i n the region denoted by 8 i n F i g . 4-5a, kT ^ e; both l e v e l s are considerably populated. With decreasing temper-ature the average r e l a t i v e population of the t r i p l e t l e v e l decreases i n favour of the non-magnetic s i n g l e t l e v e l . Therefore the f a c t o r decreases as w e l l . In a c e r t a i n temperature region t h i s decrease i s roughly a l i n e a r f u n c t i o n of T, which compensates the T ^ dependence of the Curie law and y i e l d s a temperature independent spin paramagnetic s u s c e p t i b i l i t y . An example of the above case i s phosphorus-doped germanium (A = 2.9 meV). However, for intermediate donor concentrations the temper-atures i n the range of our i n t e r e s t ( i . e . T = 1°K to 5°K) belong to the region a [see F i g . 4-5]. The region 8, i . e . the region characterized by a temperature independent spin paramagnetic s u s c e p t i b i l i t y , belongs to temperatures below 1°K, for phosphorus-doped germanium. (See the r e s u l t s of our c a l c u l a t i o n , F i g . 4-4.) The s i t u a t i o n may be d i f f e r e n t from the case above i f the dopant has a v a l l e y - o r b i t s p l i t t i n g which i s of the same order of magnitude as kT. If i n a d d i t i o n to t h i s , the number of higher l y i n g populated states i s large enough to compensate for the smaller thermal population p r o b a b i l i t y f a c t o r , then higher l y i n g states w i l l play the same r o l e as the "ground" s i n g l e t and t r i p l e t states did i n the previous case. This w i l l r a i s e the temperature at which the spin paramagnetic s u s c e p t i b i l i t y becomes tempera-ture independent, as i l l u s t r a t e d i n F i g . 4-6, schematically. - 99 -kT << A kT ^ A (a) (b) Fig. 4-6. The energy levels, relevant to the spin paramagnetic susceptibility, for two different dopants. Fig. 4-6 shows the relevant energy levels for two different dopants at same donor concentration and same temperature. Diagram (a) corresponds to the case of a dopant with a large valley-orbit splitting (A >> kT), and (b) to that of a dopant with a small valley-orbit s p l i t t i n g (A ^ kT). It i s evident that while the temperature T in the case (a) belongs to region a [see Fig. 4-5], the same temperature T belongs to region 3 in the case (b) i.e. the region which features a temperature independent spin paramagnetic susceptibility. The fact that in case (b) there are several distinct spin singlet and t r i p l e t levels with energies ^ kT makes the transition region g even wider than i t is in case (a) i.e. in the case of only two levels. An example of the above case (b) is antimony-doped germanium. Here the valley-orbit s p l i t t i n g is 0.32 meV, and there are 6 spin singlet A l T l A l A l and 6 spin t r i p l e t Is Is states above the "ground" Is Is spin singlet and spin t r i p l e t state. The above discussion explains why anti-mony donors exhibit a temperature independent spin paramagnetic suscepti-b i l i t y at concentrations and temperatures at which the susceptibility of - 100 -phosphorus donors i s s t i l l strongly dependent on temperature. [See F i g . 4-3 for the r e s u l t s of our c a l c u l a t i o n of spin paramagnetic s u s c e p t i -b i l i t y of antimony donors.] This i s a demonstration of the m u l t i v a l l e y nature, of the s i n g l e donor states as w e l l as of the interference i n t e r -actions between donors. 4. The O r b i t a l and Van Vleck Magnetic S u s c e p t i b i l i t i e s Let us proceed now to c a l c u l a t e the o r b i t a l and Van Vleck magnetic s u s c e p t i b i l i t i e s . For a donor p a i r of separation R t h e i r sum i s given by OO L the second term of eq. (4.23). The quantity ' (R) i s given by eq. (4.18). 00 L The operators appearing i n the expression for ' (R) are sums of o n e - p a r t i c l e operators. Therefore the two-particle matrix elements of eq. (4.18) can be broken down into one-particle matrix elements, which i n turn can be expressed as a l i n e a r combination of matrix elements associated with one v a l l e y only (assuming that the Bloch functions vary r a p i d l y enough that a matrix element i n v o l v i n g both the envelope functions and the Bloch functions can be w r i t t e n as a product of a matrix element i n v o l v i n g the envelope functions only, and another one i n v o l v i n g the Bloch functions o n l y ) . Considering the operators TV r b (1» 2) and 7*^(1,2) involved i n these matrix elements we also note that they are composed of l i n e a r momen-tum operators, those i n the absence of a magnetic f i e l d and those i n i t s presence, [see eq. (4.13)]. R e c a l l i n g that the s i n g l e donor states are a c t u a l l y wave packets composed of Bloch waves a r i s i n g from the v i c i n i t y - 101 -of the conduction band minima [see eq. (2.6)], we replace the free e lectron mass m i n operators " 3^^(1,2) and 7£ v v(l,2) by the appropriate component of the e f f e c t i v e mass tensor m i n the one-valley matrix elements. Since we are using unperturbed wave functions i n our c a l c u l a t i o n ( i . e . wave functions that do not depend on the magnetic f i e l d JO we can carry out the averaging over H. i n eq. (4.18) i n the operators alone, and having done t h i s c a l c u l a t e the matrix elements of these averaged operators, instead of f i r s t c a l c u l a t i n g the matrix elements e x p l i c i t l y i n terms of H and l a t e r performing the averaging over _H. This greatly s i m p l i f i e s the c a l c u l a t i o n . A l l terms i n eq. (4.18) containing any component of H l i n e a r l y , drop out i n the average. (K) L Now the c a l c u l a t i o n of ' (R) i s straightforward, but very tedious. P a r t i c u l a r l y the second term i n (4.18) i s very d i f f i c u l t . It involves an i n f i n i t e s e r i e s that cannot be summed up a n a l y t i c a l l y . If we (K) L can replace c^? (R) i n the denominator by an e f f e c t i v e , constant energy E then we can make use of the closure r e l a t i o n to evaluate the remaining c sum. We can write i i<*(S>'Lr* I * ° 2 , ; l > I 2 = , , 1 nM 1 v v s 1 n M ' n r n M' (4.35) z \ < * V ' L \ * + * I * ° 2 ; L > I 2 - Z|<* <$ )' L|* I * ( M ! , , 1 nM 1 w s' n M' 1 , 1 nM 1 w s 1 nM' (K),L,2 n'M' M Since spin i s conserved the contributions from TC cancel on the r i g h t s hand side of (4.35). This leaves only K i n (4.35). Applying now the - 102 -closure r e l a t i o n we get * i < * ( ! ? ' L i * I * % L > I 2 1 nM 1 w 1 n M 1 n f n M' nM 1 w' nM 1 nM 1 w 1 nM 1 from which i t follows that dH r i<$ (^' L|X |$ (^; L>| 2 = <$ (^' L| dH 75-2|$(5)>L> , ( 4 . 3 7 ) — , , ' nM 1 w 1 n M' 1 nM 1 — w 1 nM n p i J M since the second term on the righthand side vanishes, being a matrix element of 7(^(1,2) between the l s - l s molecular wave functions. Now eq. ( 4 . 1 8 ) can be wr i t t e n as D ^ ' V ) =4 < $ i ( K ) ' L ( l , 2 ) | 4 f d £ ^ r b ( l , 2 ) | ^ K ) ' L ( l , 2 ) > n — R 2 n '4ITJ < * ( K ) ' L ( 1 , 2 ) | 4 n 1 4TT d H ^ ( l , 2 ) l ^ > L ( l , 2 ) > ^ ( 4 > 3 8 ) E c(R) - ^ K ) , L ( R ) In ( 4 . 3 8 ) we retained only the o r b i t a l part of the molecular wavefunctions [see 4 . 1 2 0 ] , since H , and K. do not depend on spin. orb w Ultima t e l y we are int e r e s t e d i n the magnetic s u s c e p t i b i l i t y of the whole c r y s t a l . In view of the dis c u s s i o n at the beginning of t h i s Chapter [see eq. ( 4 . 7 ) ] we have to take an average over a l l p o s s i b l e o r i e n t a t i o n s R of the donor p a i r . We can carry i t out at t h i s point. Namely from ( 4 . 2 2 ) i t follows that - 103 -W b (R) + X _ ( R ) = " vv Z (R) n,K 'K v 1 v 4TT dR D^K )' L(R)_ exp L-g ( K ) , L ( R ) kT •], (4.39) since a l l donor p a i r s belonging to the same c l a s s L that are located w i t h i n a t h i n s p h e r i c a l s h e l l have the same energy spectrum (they a l l have the same set of values f o r the interference c o e f f i c i e n t s ) . What we a c t u a l l y need therefore i s an average value of the expression (4.38), the average being taken over a l l o r i e n t a t i o n s R, , dR D ( K ) ' L ( R ) = 4 — L o d R < ^ K ) ' L ( l , 2 ) dH It ,(1,2) U ( K ) , L ( 1 , 2 ) > — orb 1 n d R < * ( K ) , L ( l , 2 ) I d H # 2 (1,2)|$ ( K )' L(1,2) — n . 1 — vv 1 n ^ » ' E c(R) - g ^ K ) ' L ( R ) ( 4 . 4 0 ) Now the expression (4.40) i s a function of the absolute i n t e r -donor separation R only. I t s evaluation i s straightforward. The c a l c u l a -t i o n of the molecular matrix elements involved i n (4.40) i s discussed i n some d e t a i l i n the Appendix B. F i r s t we express the above molecular matrix elements i n terms of one-valley matrix elements. In the operators of each of these one-v a l l e y matrix elements we replace the free e l e c t r o n mass m by the appropriate * components of the e f f e c t i v e mass tensor m , i . e . m^_ and m^ , and then take an average over the d i r e c t i o n s of the magnetic f i e l d . We get the following expressions, 2 4 ^ H 2 J dH WorbCL,2) 2mc 2 [ p 2 ( l ) + p 2 ( l ) + p 2 ( 2 ) + p 2 ( 2 ) ] (4.41) 4TTH 2 J 2 2 dH t f 2 (1,2) - 6 * -— w ' , 2 2 4m c [ A 2 ( l ) + A 2(2) + A ( l ) A(2) + A (2) A ( l ) ] , (4.42) - 104 -where (4.43) A 2 j + ( x 2 + z 2) - i 9y _9_ 3y - x-9x - 2xy-9x9y ,m .2 r 2 .2. 9 2 (—) [(x + y ) —5-j m . Z I 9t 12 m m [-_9_ 9x - 2yz-9y9 (4.44) The expressions for the operators A(1) A.(2) and A(2) A(1) are not given above since the matrix element or A between the I s - i i k e envelope functions vanishes. The above expressions, (4.43) and (4.44), are given i n the same coordinate system as the envelope functions between which the corresponding operator appears i n a p a r t i c u l a r one-valley matrix element, i . e . the z-axis i s oriented i n the d i r e c t i o n of the minimum of the v a l l e y . The d i f f e r e n -2 t i a t i o n s involved i n A are c a r r i e d out e x p l i c i t l y before performing the averaging over R. In the process of averaging over R we encounter i n t e g r a l s that cannot be c a r r i e d out a n a l y t i c a l l y due to the anisotropy of the envelope functions. To s i m p l i f y them we introduce the e f f e c t i v e Bohr radius a (as we did i n Ch. I I ) . The r e s u l t i n g expressions were evaluated numerically on a computer, using the double Gauss i n t e g r a t i o n method i n e l l i p t i c coordinates. The r e s u l t s of t h i s c a l c u l a t i o n turn out to depend strongly - 105 -* upon the choice of the e f f e c t i v e Bohr radius a . Therefore at the moment * we t r e a t a as a parameter rather than t r y i n g to j u s t i f y t h e o r e t i c a l l y any p a r t i c u l a r numerical value f o r i t . In t h i s way we obtained the numerical values of the two-el e c t r o n matrix elements appearing i n eq. (4.40) as a function of R. Now we have to choose a s u i t a b l e e f f e c t i v e energy i n eq. (4.40) i n order to evaluate the sum of the o r b i t a l and Van Vleck magnetic s u s c e p t i b i l i t i e s as given by eq. (4.39). The choice i s not t r i v i a l . We know the low l i m i t to E ; i t i s given by the energy of the lowest molecular state that (K) L i s connected with $^ (1,2) through the operator *^ v v« This i s a ls-2p s t a t e . But no upper l i m i t can be given f o r E^ due to the continuum states, The c o n t r i b u t i o n from these states may be s i g n i f i c a n t , i n general, i n sums of the type appearing i n eq. (4.18), An estimate for E^ for an analogous sum i n the hydrogen molecule showed that E c might be associated with the continuum sta t e s . (32) There are no experimental data on the Van Vleck magnetic suscep-t i b i l i t y alone a v a i l a b l e at the present. However, experimental studies of the t o t a l magnetic s u s c e p t i b i l i t y have i n d i c a t e d ^ ^ that the contribu-t i o n from the Van Vleck paramagnetism may be s i g n i f i c a n t , at c e r t a i n donor concentrations i t may even overwhelm the o r b i t a l diamagnetism. We s h a l l therefore be s a t i s f i e d with an estimate of the upper l i m i t to the Van Vleck paramagnetic contributions to the t o t a l magnetic s u s c e p t i b i l i t y . Using the low l i m i t for E^ we thus evaluate the expression (4.39) as a function of R. We perform t h i s c a l c u l a t i o n for two cases, f o r antimony and for arsenic doped germanium. (The energies of the antimony - 106 -donor molecular l e v e l s had been calculated i n Chapter I I . For the purpose of t h i s present c a l c u l a t i o n we c a l c u l a t e the energy l e v e l s of the arsenic donor molecule taking into account the lowest l y i n g l s - l i k e s i n g l e donor state only.) Having done t h i s we c a l c u l a t e the sum of the o r b i t a l and the Van Vleck magnetic s u s c e p t i b i l i t i e s f o r the whole c r y s t a l , according to eqs. (4.6) and (4.31), as a function of both the donor concentration and the temperature. In F i g . 4-7 the r e s u l t s of the c a l c u l a t i o n of the sum of the o r b i t a l and Van Vleck magnetic s u s c e p t i b i l i t i e s are presented for antimony and f o r arsenic doped germanium. The "experimental" data are those extracted from r e s u l t s of the measurement of the t o t a l magnetic s u s c e p t i -(8) b i l i t y , performed by Damon and Ge r r i t s e n . In order to obtain these data we supposed that the Curie law i s obeyed i n the case of arsenic donors, and used the r e s u l t s of the Ch. IV.3 f o r the spin paramagnetic s u s c e p t i b i l i t y i n the case of antimony donors. As mentioned e a r l i e r , we treated a as a parameter i n the c a l c u l a t i o n of the sum of the o r b i t a l and Van Vleck s u s c e p t i b i l i t i e s . The r e s u l t s shown i n F i g . 4-7 are those obtained * f o r the same values of a as used by Damon and Ge r r i t s e n i n the i n t e r p r e t -* ° * ° a t i o n of t h e i r r e s u l t s , i . e . a = 45 A for antimony donors and a = 35 A for arsenic donors. F i g . 4-7 exh i b i t s a good agreement between our estimates and the experimental ones. From t h i s agreement one would conclude that our estimate of the upper l i m i t to the Van Vleck magnetic s u s c e p t i b i l i t y i s close to i t s a c t u a l value, at a given donor concentration and temperature. This i n d i c a t e s that the f i r s t term of the se r i e s i n eq. (4.18) represents the - 107 b + X w ) A C I O " 9 emu g _ I : 4 .0 J G e (Sb) I G e ( A s ) ^ C l O ^ c m " 3 : F i g . 4-7. The sum of the o r b i t a l diamagnetic and Van Vleck paramagnetic s u s c e p t i b i l i t i e s as a function of donor concentration, for Sb- and As-doped Ge, together with the values extracted from r e s u l t s of experimental measurements of the t o t a l magnetic s u s c e p t i b i l i t y . X T A C I O " 7 emu g" 1 3 1.14 G e ( S b ) Nd = 5.9 x IO 1 6 c m " 3 N d =3.1 x | 0 l 6 c m - 3 F i g . 4-8. The estimated t o t a l magnetic s u s c e p t i b i l i t y of Sb-doped Ge as a function of temperature, together with experimental r e s u l t s . - 108 -major part of the Van Vleck magnetism. To confirm t h i s one would have to compute t h i s term exactly. -This computation however would involve a great amount of work which does not seem worth doing, e s p e c i a l l y since at present there i s no experimental method a v a i l a b l e to measure the Van Vleck magnetic s u s c e p t i b i l i t y d i r e c t l y . The good agreement between our estimates and the experimental ones also implies a q u a n t i t a t i v e agreement between our theory and the actual spin paramagnetic s u s c e p t i b i l i t y of the antimony-doped germanium. Our r e s u l t s also show that the spin paramagnetic s u s c e p t i b i l i t y i s not the only source of the temperature dependence of the t o t a l magnetic s u s c e p t i b i l i t y of n-type germanium, as was supposed by Damon and G e r r i t s e n . estimates for the t o t a l s u s c e p t i b i l i t y as a function of temperature for two concentrations and also the corresponding experimental r e s u l t s . - 109 -V. SUMMARY The objective of t h i s work was the study of donor-donor i n t e r -a c t i o n . In Chapter II we treated a pair of i n t e r a c t i n g donors as a hydrogen-l i k e molecule, and c a l c u l a t e d i t s electron energy spectrum, taking into account the d i s c r e t e c r y s t a l structure of the medium the donor p a i r i s part of, as well as the m u l t i v a l l e y nature of the conduction band of the host semiconductor. F i r s t we considered the case of n-type s i l i c o n where due to the large magnitude of the v a l l e y - o r b i t s p l i t t i n g one can tre a t the lowest l y i n g A l Is s i n g l e donor state independently from the higher l y i n g l s - l i k e states. A l A l We formed spin s i n g l e t and t r i p l e t Is - Is molecular states following the Heitler-London method, and calculated t h e i r energies. We obtained an expression s i m i l a r to the f a m i l i a r r e s u l t of the Heitler-London treatment of ordinary molecules. The d i f f e r e n c e was i n the presence of an extra f a c t o r - we c a l l e d i t interference c o e f f i c i e n t - associated with the exchange and overlap terms. I t was observed that i t can assume values between 0 and 1, depending upon the interdonor separation of the donor molecule i n the c r y s t a l . A l A l As a r e s u l t the Is - Is spin s i n g l e t - t r i p l e t energy spectrum consists of a number of l e v e l s l y i n g i n between, the l e v e l s that would have been obtained i n a simple conduction band model. As the interdonor separation decreases from R = « to smaller values the s i n g l e donor energy l e v e l "broadens" into a molecular "band" of r e l a t i v e l y c l o s e l y spaced energy l e v e l s . Next we were dealing with a donor molecule i n a germanium c r y s t a l , i n the same way. We c a l c u l a t e d i t s l s - l s energy spectrum. But t h i s time a l l four l s - l i k e s i n g l e donor states were taken i n t o consideration, due to - 1 1 0 -the small value of the v a l l e y - o r b i t s p l i t t i n g . We observed that again the inte r f e r e n c e c o e f f i c i e n t s played an important r o l e i n determining the energy spectrum. Also we found that one can d i v i d e a l l po s s i b l e two-donor molecules i n a germanium c r y s t a l into three classes; molecules belonging to the same c l a s s have the same energy spectrum i f they have the same absolute value of the interdonor separation. This follows from the f a c t that i n germanium a l l molecules belonging to the same clas s have the same set of values f o r the interference c o e f f i c i e n t s . The interference c o e f f i -c i e n t s represent a manifestation of both the d i s c r e t e structure of the c r y s t a l and the m u l t i v a l l e y nature of the conduction band. In Chapter II we. also calculated the width of molecular "bands" ± o of the ls-2p and ls-2p energy l e v e l s , f o r both s i l i c o n and germanium. We observed that at a fi x e d interdonor separation t h e i r widths are propor-t i o n a l to the square of the transverse and l o n g i t u d i n a l Bohr r a d i i , r e s p e c t i v e l y . In Chapter I I I we applied the r e s u l t s of Chapter II to the case of f a r i n f r a - r e d absorption i n n-type s i l i c o n and germanium, at l i q u i d He temperatures. We estimated the width of the B ( i . e . Is ->• 2p~) and E ( i . e . Is -> 2p°) absorption l i n e s , and made a comparison with.the experi-mental r e s u l t s . For s i l i c o n we obtained a very good agreement. We can say that we obtained a good agreement for the r e l a t i v e widths of the B and E l i n e s f o r germanium too, although the a v a i l a b l e experimental data are not as r e l i a b l e as those for s i l i c o n . However, our c a l c u l a t i o n did not account f o r the observed d i f f e r e n c e i n the widths between B^ and B^, and E^ and E^ l i n e s , although i t c o r r e c t l y indicated that the widths of B^ and E^ l i n e s are larger than those of B^ and E^. We a t t r i b u t e d t h i s d i f f e r e n c e to the s t r a i n s i n the c r y s t a l . - I l l -In Chapter IV we applied the r e s u l t s of Chapter II to the case of the magnetic s u s c e p t i b i l i t y of n-type germanium, at l i q u i d He tempera-tures. F i r s t , we cal c u l a t e d the spin paramagnetic s u s c e p t i b i l i t y of antimony and phosphorus donors, i n the concentration range 10"^ to 1 0 ^ 3 donors/cm . We observed how the Curie law that i s v a l i d at low concen-t r a t i o n s breaks down with increasing donor concentration; t h i s breakdown i s much f a s t e r for antimony than f o r phosphorus donors. At concentrations 16 —3 above 6 x 10 cm we found a temperature independent spin paramagnetic s u s c e p t i b i l i t y for antimony-doped germanium. A l l t h i s i s In agreement with experimental observations. In Chapter IV we also c a l c u l a t e d the o r b i t a l diamagnetic su s c e p t i -b i l i t y and estimated the upper l i m i t to the Van Vleck paramagnetic s u s c e p t i -b i l i t y tor antimony-and arsenic-doped germanium. m tne c a l c u l a t i o n we had A a parameter, a . We obtained the best agreement with the experimental estimates for the values of a that were used by Damon and Ge r r i t s e n . The experimental estimates were obtained from the Damon and Ge r r i t s e n r e s u l t s of the measurements of the t o t a l magnetic s u s c e p t i b i l i t y by assuming that the Curie law i s obeyed f o r arsenic-doped germanium, and using the previously c a l c u l a t e d values for the spin paramagnetic s u s c e p t i b i l i t y of the antimony-doped germanium. The comparison showed a good agreement between our estimates and the experimental ones. This suggests that the estimated upper l i m i t to the Van Vleck magnetic s u s c e p t i b i l i t y i s close to i t s ac t u a l value. We confirmed the experimental i n d i c a t i o n that i t represents a s i g n i f i c a n t c o n t r i b u t i o n to the t o t a l magnetic s u s c e p t i b i l i t y , and that i t may even overwhelm the o r b i t a l magnetism i n arsenic-doped germanium at c e r t a i n donor concentrations. - 112 -In conclusion we note that the work presented i n t h i s thesis s u c c e s s f u l l y explained and reproduced a number of experimental observations i n the intermediate donor concentration range, thus f u l f i l l i n g the purpose of t h i s thesis as stated i n the Introduction. There are ways i n which the r e s u l t s presented i n t h i s thesis could be improved. One main l i m i t a t i o n of the model employed i n our work r e s u l t s from the p a i r approximation which c e r t a i n l y s t a r t s to break down at the high l i m i t of our concentration range. Attempting to include i n t e r -actions between three and more donors, however, would mean an enormous increase i n the amount of computation, which would not be worth doing i n view of the accuracy and a v a i l a b i l i t y of experimental r e s u l t s as well as of the agreements obtained by the pair approximation. 113 -References (1 (2 (3 (4 (5 (6 (7 (8 (9 (10 (11 (12 (13 (14 (15 (16 (17 (18 (19 M.N. Alexander and D.F. Holcomb, Revs. Mod. Phys. 4_0, 815 (1968). W. Kohn i n S o l i d State Physics, V.5, (Academic Press, New York 1957), W. Kohn and J.M. Luttinger, Phys. Rev. 97_, 1721 (1955). W. Kohn and J.M. Luttinger, Phys. Rev. 98, 915 (1955). H.M. James and A.S. Ginzberg, J . Phys. Chem. _5J?, 840 (1953). M. Lax and J.G. P h i l l i p s , Phys. Rev. 110, 41 (1958). E. Sonders and D.K. Stevens, Phys. Rev. 110, 1027 (1958). D. H. Damon and A.N. Ger r i t s e n , Phys. Rev. 127, 405 (1962). E. Sonder and H.C. Schweinler, Phys. Rev. 117, 1216 (1960). R. Kuwahara, Ph.D. t h e s i s , U n i v e r s i t y of B r i t i s h Columbia, 1971. Y. Nishida and K. H o r i i , J . Phys. Soc. Japan 26, 388 (1969). J.C. S l a t e r , Quantum Theory of Molecules and S o l i d s , V . l , (McGraw-H i l l Book Co., Inc., New York, 1963). S. Chandrasekhar, Revs. Mod. Phys. 15_, 86 (1943). G. Feher, Phys. Rev. 114, 1219 (1959).. G. Feher, D.K. Wilson, E. Gere, Phys. Rev. L e t t . J3> 25 (1959). J.H. Reuszer and P. Fis h e r , Phys. Rev. 135 A 1125 (1964). C. Herring, Revs. Mod. Phys. 34, 631 (1962). M. Kotani, A. Amemiya, E. Ishiguro,, and T, Kiramra, Tables of  Molecular Integrals (Maruzen Co., Ltd., Tokyo 1955). R. Barrie and K. Nishikawa, Can. J . Phys. 41, 1823 (1963). - 114 -(20) C.Y. Cheung and R. Bar r i e , Can. J . Phys. 45, 1421 (1967). (21) M. Lax and E. Burstein, Phys. Rev. 100, 592 (1955). (22) W. Baltensperg,ir, P h i l Mag. 44, 1355 (1953). (23) R. Ba r r i e and C.Y. Cheng, IX. International Conference on the Physics of Semiconductors, Moscow 1968. (24) M. Pomerantz, J . Phys. Soc. Japan J29, 140 (1970). (25) A.M. Stoneham, J . Phys. Soc. Japan 30, 576 (1971). (26) R. Bowers, Phys. Rev. 108, 683 (1957). (27) G. Busch and E. Mooser, Helv. Phys. Acta 26_, 611 (1953). (28) D.K. Stevens et a l . , Phys. Rev. 100, 1084 (1955). (29) R. Bowers, J . Phys. Chem. S o l i d s , 8_, 206 (1959). (30) R.E. Pontinen, Ph.D. t h e s i s , U n i v e r s i t y of Minnesota, 1962, (31) A. Dalgarno and J.T. Lewis, Proc. Nat. Acad. S c i . 46, 70 (1955). (32) J.H. Van Vleck and A. Frank, Proc. Nat. Acad. S c i . 15, 539 (1929). 115 -APPENDIX A Interference C o e f f i c i e n t s f o r a Donor P a i r i n Ge In Chapter II we defined the interference c o e f f i c i e n t s by eq. (2.87) I ( A 1 5 A 2 > X 3 , A A ) = I a<V a ^ a f c K j W cos [ (k^-k^,) • R] , (2.87) where the c o e f f i c i e n t s are those given by (2.26), ft) a ) 1 1 2 (1, 1, 1, 1) 2 1 2 (1, -1, -1, 1) 3 1 2 (1, -1, 1, -1) 4 1 2 (1, 1, -1, -1) (7. 9f>) The vectors k^ and k^, are the wave vectors associated with the Jl-th and £'-th v a l l e y . In Cartesian coordinates the wave vectors s p e c i f y i n g the four v a l l e y s of the conduction band are = (1, -1, -1) k o k 2 = (-1, 1, -1) k Q (1) ^3 = (1, 1, 1) k *4 (-1, -1, 1) k Q , with k Q being equal to k , o a ' (2) - 116 -where a i s the l a t t i c e parameter. The vector R appearing i n (2.87) i s the separation of the two donors forming the nolecule. I t can only assume di s c r e t e values. Consider-ing that the germanium c r y s t a l l a t t i c e consists of two i n t e r l o c k i n g face-centered cubic l a t t i c e s the vector R has to be expressible either as p " l + n 2 n l + N 3 N 2 + % R = ( 2 ' 2 ' 2 ) A ' ( ' or 1 1 1 n, + n 0 + n.. + n„ + •» n 0 + n„ + ~z R = ( _ 1 2 ^ _1 _3 2 ^ _2 _3 2 ) A ^ ( 4 ) where n^, n 2 , n^ are integers, and a i s the l a t t i c e parameter. If we define the o r i g i n of the coordinate system i n such a way that one donor i s s i t t i n g at the o r i g i n (0,0,0) on a simple cubic s i t e , then the second donor i s occupying: a) a simple cubic s i t e on the same l a t t i c e i f a l l three components of R i n un i t s of a are integers, i . e . - = ( N 1 ' N 2 S N 3 ) A ' Nl» N 2 ' N 3 i n t e S e r ( 5) b) a face-centered s i t e on the same l a t t i c e i f two of the components of II i n units of a are h a l f - i n t e g e r s , R = (N x, N 2, N 3)a, two of N^, N 2 > N 3 h a l f - i n t e g e r (6) (t h i r d integer) c) a s i t e on the displaced l a t t i c e i f a l l three components of R. i n u n i t s of a are quarter-integers, i . e . - 117 -R = (N , N , N 3)a, 1^, N 2, N 3 quarter-integer. (7) In order to s i m p l i f y the evaluation of the interference c o e f f i -c i e n t s l e t us defina • 1 ' 2 ~ I l ~ so that we can write (2.87) i n the form I(X 1,.\ 2,X 3,X 4)-= Re[L(X 1,A 2)-L*(A 3,.X 4)], (9) where L means the complex conjugate, value of L, and Re symbolizes the r e a l part of a complex number. Inserting (2.26) and (1) into (8) we f i n d that L (X^X^) can be ex r , r o c : ? ; : pd ~>? fnlInwS-L(X,X) = ^ (L± + L2 + L3 + L 4 ) , X = 1,. 2, 3, 4 1,(1,2) = * ( L 1 - L 2 - L 3 + L 4 ) L( l , 3 ) = | (L± - L 2 + L 3 - L 4 ) where L( l , 4 ) = | ( L x + L 2 - L 3 - L 4 ) L(2,3) = L (1,4) L(2,4) = L (1,3) L(3,4) = L (1,2) L(X 2,X 1) = L(X 1,X 2) , X 1,X 2 = 1, 2, 3, 4 L, = E •(E •E ) _ 1 1 x v y z L = E »(E -E ) _ 1 2 y x z L = E 'E 'E 3 x y z L. = E »(E «E ) - 1 4 z v x y (10) ( I D - 118 with „ _ i k R E = e o x x _ _ i k R /t rj\ Ey = e o y (12) i k R E = e o z z where R , R , and R are components of R, and k has been defined i n (2). x y z — o From (10) we see that a l l the interference c o e f f i c i e n t s are completely determined by the set of four L(X^,\^), namely <^ = { L ( l , l ) , L ( l , 2 ) , L ( l , 3 ) , L(l,4)} . • (13) If we define that ^ ^ <** means that both <£f^  and <S€^ 8 i v e t n e same int e r f e r e n c e c o e f f i c i e n t s , then c*f<v ±±? = {± L ( l , l ) , ± L ( l , 2 ) , ± L ( l , 3 ) , ± L(l,4)} (14) and <*?^  ± = {± L * ( l , l ) , ± L * ( l , 2 ) , ± L * ( l , 3 ) , ± L*(l,4)} . (15) In view of (5), (6), and (7) there are three cases to be considered, a) I f we define t = (E , E , E } , (16) *- x y z then i t follows from (5) that ^ j- i N . i T iN„iT iN„iT-, , 1 7 , . £ = { e l , e 2 , e 3 } , (I/) or £ = {± 1, ± 1, ± 1} (18) ~ 119 -The equation (18) comprises eight d i f f e r e n t groups of simple cubic l a t t i c e s i t e s . But i t i s easy to see that they a l l would give the same set of interference c o e f f i c i e n t s , from (12), (11), (10), (14). Thus i n order to f i n d the interference c o e f f i c i e n t s i t i s enough to consider only one of the cases (18), e.g. £ = {1,1,1} . (19) From (11) and (10) i t follows {1,0,0,0} , and f i n a l l y t X I I I ( X ^ ,X^ i ^ o *^4) = j 1 i f X, = X^ , and X^ = X^ (20) 0 otherwise We conclude that a l l the simple C U D I C l a t t i c e s i t e s give tne same set ox interference c o e f f i c i e n t s as given by (20). Therefore, a l l molecules con-s i s t i n g of two donors with equal interdonor distance | R | y i e l d the same energy spectrum i f both donors occupy simple cubic s i t e s of the same l a t t i c e , or face-centered s i t e s of the same l a t t i c e , b) Here we have three p o s s i b i l i t i e s : a) £ = {1, i , i} (21) {0, 0, -1, 0} * {0, 0, 1, 0} 1 i f ( X r X 2 ) = (1,3) or (2,4) I ( X 1 } X 2 , X 3 , X 4 ) = I and ( A ^ ) = (1,3) or (2,4) (22) 0 otherwise g) t = { i , 1, i> y = {0, 1, 0, 0} - 120 -i i f a l f A 2 ) (1,2) or (3,4) and (A ,X.) (1,2) or (3,4) (23) 0 otherwise Y) t = U, i , 1} x = {0, 0, 0, 1} i i f (x 1,x 2) (1,4) or (2,3) I(A^,A^,A^,A^) - < and (A„,A.) (1,4) or (2,3) (24) 0 otherwise P h y s i c a l l y the r e s u l t s (22), (23), (24) imply that as we go from one face-centered l a t t i c e , s i t e to another one i n the same l a t t i c e (keeping one donor at (0,0,0)) we perform a c e r t a i n transformation amongst the s i n g l e donor wave functions l a b e l l e d by y = 2, 3, 4 i n eq. (2.16). (For instance, going from a s i t e g i v i n g r i s e to interference c o e f f i c i e n t s (22) to a s i t e y i e l d i n g (23) has the same e f f e c t as interchanging states 2 and 3.) Since these three states are degenerate, the transformation w i l l not have any e f f e c t on the energy spectrum of the molecule. In other words, any two molecules having one donor on a simple cubic l a t t i c e s i t e , and the other one on a face-centered p o s i t i o n i n the same l a t t i c e , w i l l have i d e n t i c a l energy spectra, provided the interdonor separations |R| are the same. c) From (7) i t follows that i n view of (12), (11), and (15), we have four p o s s i b l e cases that give r i s e to d i f f e r e n t sets of values for inte r f e r e n c e c o e f f i c i e n t s . (25) 121 -This r e s u l t s i n the following interference c o e f f i c i e n t s : - | i f (A 1,A 2) or a 3 , A 4 ) = (1,3) or (2,4) I(X 1,A 2,A 3,A 4) = j and ( A ^ ) f .V<y\) (27) + y- otherwise 4 ( fST,.ST, ST ST ST . Si Si ST ) . .«> t- f f + i f * f + ± J T > " ) J <32> v i ST , . /2 / ST . ST x ST . . ST ST . . /2 i , Q O N ^ I^T + 1 4 " ' - ( 2 T + 1 4 " ) ' 4 ~ + 1 4 ~ ' — + 1 4 ~ | ( 3 3 ) Combining (29), (31), (33) with (26), we see that going from one s i t e to another on a displaced l a t t i c e again has the same e f f e c t on the interference c o e f f i c i e n t s as r e l a b e l l i n g the s i n g l e donor wavefunctions, t h i s time a l l four, 1, 2, 3, 4, among themselves. E n e r g e t i c a l l y the state l a b e l l e d by 1 i s d i f f e r e n t from those l a b e l l e d 2, 3, 4, so that s t r i c t l y speaking there are two d i f f e r e n t sets of energy l e v e l s as we move from one s i t e to another on a displaced l a t t i c e . However, these transformations only a f f e c t the terms i n the secular equation which are associated with the overlap i n t e g r a l s , the e f f e c t being to change the sign. Thus the small change i n value (due to the d i f f e r e n c e of energy between the states 1 and 2, 3, 4) has a n e g l i g i b l e e f f e c t on the eigenvalues of the secular equation. So we can say that a l l molecules c o n s i s t i n g of a donor located on one l a t t i c e and another donor on the d i s -placed l a t t i c e y i e l d the same energy spectra, provided t h e i r interdonor separations | R | are the same. - 122 -Let us remark that t h i s systematic study of interference c o e f f i -c i e n t s for germanium and the subsequent c l a s s i f i c a t i o n of donor molecules in t o three classes regarding t h e i r energy spectra was conditioned by the f a c t that i n germanium, the conduction band minima l i e at the edge of the B r i l l o u i n zone. Such i s not the case with s i l i c o n where the conduction band minima l i e at about 85% of the way from the center to the edge of the zone. As a consequence, moving a donor from one s i t e to another i n a s i l i c o n c r y s t a l does not r e s u l t i n any simple systematic changes i n the int e r f e r e n c e c o e f f i c i e n t s . - 123 -APPENDIX B C a l c u l a t i o n of 1 dR dH <*(1,2)| 7C (1,2)|*(1,2)> (1) The operator T€ (1,2) i s either the o r b i t a l or the Van Vleck operator, and i s a l i n e a r combination of one-particle operators. y (1,2) = ei t h e r % ,(1,2) or 7(2 (1,2) m orb w (1) K<l>v - *ma) + ^ m ( 2 ) (2) It follows- that the above matrix element can be broken down into one-electron matrix elements, which i n turn can be expressed as a l i n e a r combination of matrix elements associated with one v a l l e y only (assuming that the Bloch functions vary r a p i d l y enough that a matrix element i n v o l v i n g both envelope functions and Bloch functions can be w r i t t e n as a product of a matrix element i n v o l v i n g envelope functions only, and another one i n v o l v i n g Bloch functions o n l y ) . Therefore, i n order to c a l c u l a t e the above matrix element, we have to evaluate matrix elements of the following form (3) and > (4) where j denotes a p a r t i c u l a r v a l l e y . - 124 -X (1) i n the above matrix elements depends on H and r = r . Since m — — — 1 the envelope functions F do not depend on H, < 3> = Hit dR [<F A|f-— J 4 IT dH X F?>] - m1 j (5) and <4> = h dR [<F j '4TT dH ~K \ T > (6) It i s convenient to express J£ i n the coordinate svstem that i s m oriented i n such a way that the z-axis i s i n the d i r e c t i o n of the j - t h v a l l e y . We write the vector p o t e n t i a l A i n the form A = - H x r (7) where r = (x, y, z) and (8) (0,0,1) || ^ . For instance, It - | — [ A 2 + A 2 ] + 2 - A 2 / orb 2 1m x y m z J mc t J l (9) From (7) and (8). "#orb . 2 , ... 4mc t / — [(zH - yH ) 2 + (xH - zH ) 2 ] + — ( y H - xH )' I m y } z z x m ' x y - 125 -Taking the average over H, a l l terms i n (10) containing any component of II l i n e a r l y drop out, H = H = H = H H = H H = H H = 0 x y z x y x z y z (ID I t follows, 4TTH d H £ o r b ( l , 2 ) 2mc 2 [p£(D + p*(l) + p'(2) + p*(2)] 2 y (12) where we wrote 2 -2 „2 1 „2 H = H" = H = z J and 2 l m . 2 , 2 , . 2 . p t H 1 2 ^ ( x + y + 2 z > (13) 2 1 m , 2 . 2 P. = I 1 2 m o (x + y ) (14) (3) Applying (12) and (14) on F^, we can write 1_ 4TT d i l # „ = 0 (x,y,z,R , R , R ) , — m j m x y z 3 (15) sxnce + R / 2 ) 2 + (y + R / 2 ) 2 ( z + R / 2 ) 2 x y i Zj T — H —^ (16) In (15) 0 denotes a function of el e c t r o n coordinates m r_ = (x,y,z) and molecule p o s i t i o n vector R = (R ,R ,R ). Now, x y z ( 5 ) - t r dR M ^ R ) (17) - 126 -(6) = ~ 1 4rr dR MB A(R) (18) where (R) = 0 (x,y,z,R ,R ,R ) e m x y z - 2 (x+R /2)2+(y+R / 2 ) 2 (z+R / 2 ) 2 a2 + b2 dxdydz, (19) and M M ( R ) = 0 (x,y,z,R ,R m x y z x e - \ R /2) 2+(y-R / 2 ) 2 (z-R / 2 ) 2 a2 + b2 (x+R /2)2+(y+R / 2 ) 2 (z+R / 2 ) 2 x y + z a^ b 2 dxdydz (20) The matrix elements (19) and (20) could, i n p r i n c i p l e , be evaluated on a computer as a function of p o s i t i o n of the molecule i n the c r y s t a l . But t h i s would involve an enormous amount of work. (4) Let us concentrate our att e n t i o n on the matrix element ( 2 0 ) . Introducing new coordinates, x = x i a (21) z = z we get «BV> - 4 a 0'(x1,y',z',R',R',R') e m J x y z ( x ' - R V 2 ) 2 + ( y ' - R ' / 2 ) 2 + ( z , - R , / 2 ) 2 x y z ft 2 - 127 -x e -I (x ,+R'/2) 2+(y'+R'/2) 2f(z'+R ,/2) 2  x J y z * 2 dx'dy'dz' (22) where R' x = R x a R' y y a R' z z b (23) In the above expression a can be any constant, (5) Let us introduce a new coordinate system, such that the new z-axis, z", w i l l point i n the d i r e c t i o n of R' = (R',R',R'). — x y z This i s accomplished by the matrix S , S r' (24) where cosS cost)) - sin<j) sinO cos<{) cos0 sincf) cos<j) sin0 sin<|> - sin0 0' cos6 (25) Inserting (24) i n t o (22), our matrix element becomes M M ( R ) *3 ./ x" 2+y" 2+Cz a'-R'/2) : 0"(x",y",z",R,,R',R') e~ V *2 mv " ' ' x y' z " a I x "2+y" 2+(z"+R'/2) : x e \J *2 d x " % " d z " , (26) - 128 -where ft ft R' = |R'| = (R 2 + R 2) (^-) 2 + R 2 fyZ (27) '— 1 x y a z b Up to t h i s point we haven't made any approximation. The important point i s that i n the exponential f a c t o r s we now have only|R'| appearing; i t i s to be remembered, however, that |R'| i s a function of the d i r e c t i o n of R. BA RA In order to get the average value of M (R), M , which we need i n the c a l c u l a t i o n of the magnetic s u s c e p t i b i l i t y of the whole c r y s t a l , we have to perform the i n t e g r a t i o n MBA 1 " RA dR M (R) . (28) BA This i n t e g r a t i o n cannot be c a r r i e d out a n a l y t i c a l l y with the M (R) func t i o n as i t i s now. Therefore we s h a l l make an approximation. We replace the d i r e c t i o n dependent R' i n eq. (26) by R (which i s independent of the d i r e c t i o n of the donor p a i r ) . This approxi-mation i s r e l a t e d to the choice of the numeric value f o r a . But rather than t r y i n g to t h e o r e t i c a l l y j u s t i f y any p a r t i c u l a r value ft f o r a , we tr e a t i t as a parameter, at t h i s p o i n t . Later on i t turns out that we get the best agreement with experimental data for ft the values of a that were used by Damon and G e r r i t s e n i n the inter-p r e t a t i o n of t h e i r experimental r e s u l t s . BA Now the exponentials i n the expression (26) f o r M (R) are independ-ent of R, so that we can write - 129 -M B . 2, r I x 2+y 2+(z-R/2) 2 BA a b 7-ft, . - »/ 1 — ± z -0 (x,y,z,R) e V «2 e m a *3 J x 2+y 2+(z+R/2) 2 * 2 dxdydz, (29) where o; (x sy,z,R) H - dR 0"(x,y,z,R',R',R') — m x y z (30) The i n t e g r a l (30) can now be c a r r i e d out a n a l y t i c a l l y , and so can (29). But since 0^ turns out to be a rather complicated function, the i n t e g r a t i o n i n (29) i s performed on a computer using the double Gauss method i n e l l i p t i c coordinates. 

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