NNSE508_EM-L9-bands-tight-binding

NNSE508_EM-L9-bands-tight-binding - 1 Lecture contents...

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NNSE 508 EM Lecture #9 1 Lecture contents Bloch theorem k-vector Brillouin zone Tight-binding model Almost free-electron model Bands Effective mass Holes
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NNSE 508 EM Lecture #9 2 Translational symmetry: Bloch theorem ) ( ) ( R r V r V 3 3 2 2 1 1 a m a m a m R   ) ( ) ( 2 2 r E r r V m p If V(r) is a periodic function: One-electron Schrödinger equation (each state can accommodate up to 2 electrons): ) ( ) ( r u e r k ikr k The solution is : ) ( ) ( R r u r u k k where u k (r) is a periodic function: From: Linearity of the Schrödinger equation Fourier theorem 2 2 ) ( ) ( R r r k k   Quasi-wavevector k is analogous to a wavevector for free electrons (V=const) u k might be not a single valence electron function but is close to linear combination of valence electron wavefunctions Important :
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NNSE 508 EM Lecture #9 3 Introduced k -vector quantum number for periodic potential (to enumerate states) Momentum is not conserved (not a quantum number), however quasi-momentum is conserved k -vector can be considered to lie in the first Brillouin zone Solution with periodic boundary conditions gives eigen-functions u n,k for a given k which forms orthogonal basis (compare with Fourier expansion) n –values enumerate bands Electron occupying level with wavevector k in the band n has velocity (compare to group velocity) Bloch theorem: consequences ) ( ) ( R r u r u k k ) ( ) ( ) ( 1 2 2 2 r u E r u r V k i m k k k ) ( ) ( r u e r k ikr k   ) ( ) ( 2 2 r E r r V m p k n k n E r u , , ), ( ) ( 1 ) ( k E k v n k n 2 2 2 2 E k n ma    
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NNSE 508 EM Lecture #9 4 Reciprocal space (1D) ) ( ) ( r u e r k ikr k Wavefunction of an electron in crystal : 1D reciprocal lattice vector : m k E 2 2 2 m a b 0 2 1D free electrons “band structure” is: k’ k’-b   ' ' ' ' 2, ( ) ( ) ( ) ( ) ik r ikr ibr ikr k k k k r e u r e e u r e u r periodic function First Brillouin zone: 0 0 a k a 2 nd band
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NNSE 508 EM Lecture #9 5 Reciprocal space: 3D 3 3 2 2 1 1 b m b m b m b     23 1 2 3 1 2 3 2 , , ...
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NNSE508_EM-L9-bands-tight-binding - 1 Lecture contents...

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