By W Shockley
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Additional info for Electrons and Holes in Semiconductors with Applications to Transistor Electronics
0) (0) new representation, which is the direct product r 1 x r J . This representation can be decomposed into irreducible components r(k). In the following, 26 we shall label their basis functions fr(k) ,m . 37) of V in the stress tensor components we obtain < ~ r (i) ,m IVI~ r (i) ,R. J ,Cl Gr(j),Cl < ~ r (i) ,m IVr (j) ,Cl I~ r (i) ,R. 47) ... each matrix element is a linear combination of r ,n terms like (1. 48) with [f r (k) ,n IVr (j) ,Cl ] = J fr (k) ,n Vr (j) ,Cld, (1. 50) This allows the reduction of all matrix elements in terms of a minimal number of parameters, the [fIV]r(k).
However, instead of working directly with the ~nk' we shall again take their Fourier transforms as we have done previously for the ~nk. We thus define modified Wannier functions vn(r - ~i) as vn(r - ~i) = ;, YI~ L exp(-i~. 39) - The set of vn(r-~i) defined in this way is also orthonormal since the quantities (llYN) exp(-~~i) are the coefficients of a unitary transformation. These modified Wannier functions have also a localized character since they are Fourier transforms of delocalized functions.
18 for a T2 state of the group Td . defect orientational stre .. orientation splitting spliHing 'I'~ ,'I'.. //;,"" _ - - - '1'. -:: '" '" ====:1(",,- --~-------- 't. '1'•. '1'. , 'fL 0> ~-------- 'I'~ '1',. p~ (00') ......... ~ ______- '1'. 18. 81) where D is the dipole moment of the system. For one-electron states ~i and ~j' the intensity is simply the product ~ • r, where ~ is the unit vector of the direction of polarization and r the position vector of the electron. In this case, the calculation of this intensity requires knowledge of the matrix elements of x,y,z, the components of r which belong to the representation T2 , between the basis functions of the levels ~i and ~j' The procedure for calculating these elements is completely analogous to the calculation of the matrix elements of the perturbation produced by a stress aT2 ,i' 35 Consider the case of the Td group developed in the preceeding section.