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# handout10 - 1 ECE 407 – Spring 2009 – Farhan Rana –...

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Unformatted text preview: 1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 10 The Tight Binding Method (Contd…) In this lecture you will learn: • The tight binding method (contd…) • The π-bands in conjugated hydrocarbons Energy k a π a π − s E σ ss V 4 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Tight Binding for a Square Lattice with a Two-Atom Basis Consider a 2D square lattice with a two-atom basis: y a d x a d y a a x a a ˆ 2 ˆ 2 ˆ ˆ 2 1 2 1 = = = = r r r r • The primitive vectors and basis vectors are as follows: • Each basis atom contributes one s-orbital that participates in bonding ⇒ Each primitive cell contributes two s-orbitals that participate in bonding a a 1 d r 1 a r 2 a r 2 d r A B ( ) ( ) ( ) ( ) ( ) [ ] ∑ − − + − − = m m SB d k i SB m SA d k i SA R k i k d R r e k c d R r e k c N e r m 2 . 1 . . 2 1 r r r r r r r r r r r r r r r v φ φ ψ One can write the trial tight-binding solution for wavevector as: k r ( ) ( ) SB SB SA SA E r E r ↔ ↔ r r φ φ 2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Analysis of the Tight Binding Solution a a 1 d r 1 a r 2 a r 2 d r A B ( ) ( ) ( ) ( ) ( ) [ ] ∑ − − + − − = m m SB d k i SB m SA d k i SA R k i k d R r e k c d R r e k c N e r m 2 . 1 . . 2 1 r r r r r r r r r r r r r r r v φ φ ψ (1) Summation over all primitive cells (2) Common phase factor for each primitive cell (4) A phase factor for each orbital that is related to the position of the orbital within the primitive cell w.r.t. lattice point (3) Summation over all orbitals within a primitive cell with undetermined coefficients ECE 407 – Spring 2009 – Farhan Rana – Cornell University Tight Binding Solution a a 1 d r 1 a r 2 a r 2 d r A B Plug the solution into the Schrodinger equation: ( ) ( ) ( ) r k E r H k k r...
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handout10 - 1 ECE 407 – Spring 2009 – Farhan Rana –...

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