c_(r) (11:10) where the integration is again over the fi r s t Brillouin zone, and the subscript c refers to the conduction band. Substituting (10) into (6)*, multiplying by the complex conjugate of ^ ^ . ( r ) , and integrating over the entire crystal leads to the equation: i \u00a3 t ( U > E ] D ( k ) + {*_' U(k,_)&(_') = O (11:11) where: Itfk.k') = \\dr cpeJ(_) U(r) <(>c_'(r) = * = <\u2022 Fj^Ct) Fj'isCc) sin (kj'-hjVE d\u00a3 (11:41) the real part of the expression having vanished because its integrand is an odd function of x. Clearly i f j 0 8 j * , the matrix element (41) is identically zero. If j \/ j f , the matrix element is not zero, but as the Fj__Cr) are slowly-varying compared to the lattice spacing (Luttinger and Kohn, 1955, Section III), i t is reasonable to assume that the presence of an oscillating term in the Integrand will tend to make the integral very small. A more quantitative estimate of the non-zero matrix elements may be obtained from consideration of the Integrals: Jdr \u00ab^F J-_(c)F i. u(E) sin kj-r cos fer r (11:42) and Jdr %r F* l $ ( t ) Fj ' u (_) sinRj'-r \u00bb$ kj.r (11:43) I* J - J', both integrals are of the order of 5 x 10~4a*, where a* is the approximate \"Bohr radius\" for the F l s ( r ) . It is to be expected that when j ^ j ' these integrals will have even smaller values.* (41), which 1st equal to the difference of (42) and (43), should therefore be negligible compared with 2 * 1.9 x 10 a , which is the value of the matrix element for the ls-*2p0 transition when spherical-potential wave functions are used. * See (111:37) for examples which bear out this contention. -21-In Chapter I I I , corrections to the sphe r i c a l - p o t e n t i a l approximation w i l l be considered. Anticipating the r e s u l t s , i t may be stated here that, replacing the ground state wave function by i t s corrected value and using the estimation \u20143 * procedure of the l a s t paragraph, values of less than 1 x 10 \"a* are obtained f o r (41). As the Is 2p Q matrix element i s decreased by only a factor of two when the corrected functions are used, the Is-*-Is matrix elements are again n e g l i g i b l e by comparison. It i s therefore u n l i k e l y that the f i n e structure of the {is] states w i l l be experimentally observable when the impurities are too far apart to inte r a c t . - 2 2 -CHAPTER III The Theory of Donor Impurities i n S i l i c o n f o r F i n i t e Impurity Concentrations. A. Introduction. The case i n which there are many impurities present i n a c r y s t a l l a t t i c e i s d i f f i c u l t to treat because of the random nature of the impurity d i s t r i b u t i o n . I f , however, the impurities are assumed to l i e on a regular l a t t i c e within the c r y s t a l , the problem of t h e i r i n t e r a c t i o n may be attacked by means of the usual approximations of s o l i d state physics. For s i m p l i c i t y , therefore, i t w i l l be assumed that the impurities l i e on a simple cubic l a t t i c e , so that the problem of impurity-l e v e l broadening may be treated i n the Bloch scheme. B. Tight-Binding (LCAO) Approximation. 1. The secular equation: Derivation. The Schroedinger equation f o r the many-impurity problem i s : K * s ( _ ) = E (_)*_(_) < I I I : 1> * For the treatment of a one-dimensional l a t t i c e containing random impurities, see Lax and P h i l l i p s (1958). ** Conwell (1956) discusses the implications of the assumption of a regular impurity l a t t i c e . See also Baltensperger (1953). -23-where: K \u2022 \u00ab . + Y r t o { m , a ) and V (r) i s a periodic p o t e n t i a l with the p e r i o d i c i t y of the Mr impurity l a t t i c e . Following Slater and Koster (1954)j V P(^ 5 H U(r:-a5,) (111:3) _ where the summation i s over a l l s i t e s <*n of the impurity l a t t i c e . In the LCAO approximation, the wave functions \"f_0_) may be written: * * W = -t ^ d * 2 + * ^ C c - a J *