NNSE618-L6-eff-mass-approx-impurities

# NNSE618-L6-eff-mass-approx-impurities - 1 Lecture contents...

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NNSE 618 Lecture #6 1 Lecture contents Effective mass approximation Impurities

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NNSE 618 Lecture #6 2 Effective mass approximation   ) ( ) ( ) ( 2 0 2 r E r r V r V m p i i One-electron Schrödinger equation with weak and slow varying perturbation V i : ) ( ) ( ) ( ) ( * 2 2 r F E E r F r V m p C i It is possible to substitute the effect of periodic potential into the effective Hamiltonian: With F(r) - slow varying envelope function: r e r V i 2 ) ( Packet localized in the k - space (close to minimum) Let’s introduce small perturbation of periodicity And as usual build a solution as a wave packet of Bloch wavefunctions : ) ( ) ( ) ( , r u e k c r nk ikr k n n Weak and slow varying perturbation, such as screened Coulomb potential: Closest band considered only (conduction band) ) ( ) ( ) ( ) ( ) ( 0 0 r u r F r u e k c r ikr k For non-degenerate, isotropic, parabolic minimum at k = 0
NNSE 618 Lecture #6 3 Effective mass approximation Bloch wave packet: With dimensions in real and k-space 0 1 a k r  ) ( ) ( ) ( 0 r u r F r Depending on sign of the perturbation, the top-most or bottom-most state splits from the band : i V

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NNSE618-L6-eff-mass-approx-impurities - 1 Lecture contents...

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