energybands_3_4

energybands_3_4 - A and perturbation v from neighboring...

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Introduction to solid state physics WS 2005/06 M. Wolf sheet 7.3 Quasi-free electron dispersion: Reduced zone scheme Consequences of the Bloch-theorem: (1) Reduced zone scheme: Band structure can be reduced to the 1. Brillouin zone: ) G k ( ) k ( n n v v v + ψ = ψ ) G k ( E ) k ( E n n v v v + = the index n counts the Eigenstates E n of electrons in the periodic potential. (2) The crystal momentum k v h can be translated by reciprocal lattice vectors and thus the wave vector k v does not represent the real electron momentum. (3) Wave functions in the periodic potential: Bloch waves in contrast to free electrons. (4) An electron indexed by n and k v has a non- vanishing mean velocity h v v v v ) k ( E ) k ( v n k n = . In a periodic potential stationary states exist where electrons move with constant velocity, i.e. vanishing resistivity. K=2 π /a

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Introduction to solid state physics WS 2005/06 M. Wolf sheet 7.4 Tight-binding approximation For more localized electrons in atomic orbitals i ϕ a description by quasi free electrons is inadequate: linear combination of atomic orbitals (LCAO) Hamiltonian consists of contribution of free atoms H
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Unformatted text preview: A and perturbation v from neighboring atoms: ) r r ( v ) r r ( V m 2 v n n A 2 A v v v v h − + − + ∆ − = + = H H with ∑ − = − ≠ n m n A n ) r r ( V ) r r ( v v v v v The wave functions k Ψ are derived from free atom orbitas i ϕ and form Bloch waves: ∑ − ϕ ⋅ = Ψ n n i r k i k ) r r ( e n v v v v For overlapping wave functions between nearest neighbour atoms n,m: ( ) ∑ − − ≈ − m r r k i i m n e B A E ) k ( E v v v v with ) r r ( ) r r ( v ) r r ( r d A n i n n * i 3 v v v v v v − ϕ ⋅ − ⋅ − ϕ ∫ − = and ) r r ( ) r r ( v ) r r ( r d B n i n m * i 3 v v v v v v − ϕ ⋅ − ⋅ − ϕ ∫ − = Example: simple cubic lattice ⇒ ) a , , ( ); , a , ( ); , , a ( r r m n ± ± ± = − v v ( ) ( ) ( ) ( ) a k cos a k cos a k cos B 2 A E ) k ( E z y x i + + − − ≈ v expansion near the center of the 1. BZ: 2 2 i k Ba B 6 A E ) k ( E + − − ≈ v ⇒ E(k) ∝ k 2 like for free electrons....
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energybands_3_4 - A and perturbation v from neighboring...

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