Phys560Notes-7 - Band Structure Calculations using a plane...

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1 Band Structure Calculations – using a plane wave basis set       2 2 2 U m kk k k rr r r Fourier transform:   i UU e Kr K K r       iii ii eu e c e e kr k - K k + K KK       2 2 2 i i ce U e c e c e m kKr kK r k-K K k K K kK . 2 nd term  i cU e kK K r K Let  K . 2 nd term i cUe K K K,K . ..... ..... 0 i ee e . Each Fourier coefficient   ... must be zero.    2 2 0 2 c m - K K K k K K 2 An infinite set of linear homogeneous equations Nontrivial c ’s determinant = 0 secular equation of order. Often use a truncated set diagolization of H approx. n k and c Accuracy depends on truncation. NFE (nearly free electron approximation) Assume U small, use perturbation theory. Unperturbed states: 1 i e V k ; 22 0 2 k m k in extended zone. Perturbation theory:      2 03 00 U UO U k k k , assuming it is convergent.      33 , 11 i i K VV Ue U e e d r U e d r U Kk kr k r K k k K K K 3 ,0 0 1 V d r V K r = average crystal potential, independent of k ; set to zero.
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3          2 03 00 0 U OU K K kk K    02 valid if     U K K K . Band structure is free-electron-like except where ~ U K K ; for those regions in k space, the expansion is not valid. Need degenerate perturbation theory. Define   kkK Worst case: exact degeneracy     , or .  & Laue condition electron Bragg diffraction (just like x-rays) k and k states strongly coupled by the crystal potential. Assume just two states k & kK are strongly coupled: U K Κ . Consider the subspace spanned by k & :    1,2 ii ec c e kr Kr K r Ignore coupling to all other states,   2 . Set 0 0 U . 4 From     2 2 0 2 cU c m - K K K k K K , one gets a pair of equations 2 2 0 2 c m Κ k k  2 2 0 2 kc U c m k K k * UU KK because U is real. With 22 0 2 k m k ,  0* 0 c k k -K K k   0 0 c k K k K Determinant = 0 2 2 1 4 2 U k K k k K K i e , i e kKr , and - K  i i ce c e K , and k . 0 c k k . This and the normalization condition determine the coefficients.
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Phys560Notes-7 - Band Structure Calculations using a plane...

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