handout7 - Handout 7 Properties of Bloch Functions and...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University x k y k Energy FBZ Handout 7 Properties of Bloch Functions and Electron Statistics in Energy Bands In this lecture you will learn: • Properties of Bloch functions • Periodic boundary conditions for Bloch functions • Density of states in k-space • Electron occupation statistics in energy bands ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bloch Functions - Summary • Electron energies and solutions are written as ( is restricted to the first BZ): • The solutions satisfy the Bloch’s theorem: and can be written as a superposition of plane waves, as shown below for 3D: • Any lattice vector and reciprocal lattice vector can be written as: • Volume of the direct lattice primitive cell and the reciprocal lattice first BZ are: ( ) r k n r r , ψ and ( ) k E n r ( ) ( ) r e R r k n R k i k n r r r r r r r , . , ψ ψ = + ( ) ( ) ( ) + = + j r G k i j n k n j e V G k c r r r r r r r r . , 1 ψ k r 3 3 2 2 1 1 b m b m b m G r r r r + + = ( ) 3 2 1 3 . b b b r r r × = Π 3 3 2 2 1 1 a n a n a n R r r r r + + = ( ) 3 2 1 3 . a a a r r r × =
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bloch Function – Product Form Expression A Bloch function corresponding to the wavevector and energy band “ n ” can always be written as superposition over plane waves in the form: k r ( ) ( ) ( ) + = + j r G k i j n k n j e V G k c r r r r r r r r . , 1 ψ The above expression can be re-written as follows: ( ) ( ) ( ) ( ) r u e e V G c e e V G k c e r k n r k i j r G i j k n r k i j r G i j n r k i k n j j r r r r r r r r r r r r r r r r r r , . . , . . . , 1 1 = = + = ψ Where the function is lattice periodic: ( ) r u k n r r , ( ) ( ) ( ) ( ) ( ) r u e V G c e V G c R r u k n j r G i j k n j R r G i j k n k n j j r r r v r r r r r r r r r r , . , . , , 1 1 = = = + + satisfies Bloch’s theorem ( ) ( ) r e R r k n R k i k n r r r r r r r , . , ψ ψ = + ( ) ( ) r u e r k n r k i k n r r r r r r , . , = ψ Note that: ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bloch Functions – Periodic Boundary Conditions • Any vector in the first BZ can be written as: k r 3 3 2 2 1 1 b b b k r r r r α α α + + = where α 1 , α 2 , and α 3 range from -1/2 to +1/2 Reciprocal lattice for a 2D lattice Consider a crystal made up of N 1 primitive cells in the direction, N 2 primitive cells in the direction and N 3 primitive cells in the direction Direct lattice 1 b r 1 a r 2 b r 2 a r 1 a r 2 a r 3 a r Assuming periodic boundary conditions in all three directions we must have: ( ) ( ) ( ) r r e a N r a N k i r r r r r r ψ ψ ψ = = + 1 1 . 1 1 ( ) ( ) ( ) r r e a N r a N k i r r r r r r ψ ψ ψ = = + 2 2 . 2 2 ( ) ( ) ( ) r r e a N r a N k i r r r r r r ψ ψ ψ = = + 3 3 . 3 3 Volume of the entire crystal is: ( ) 3 3 2 1 3 3 2 2 1 1 . = × = N N N a N a N a N V r r r
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bloch Functions – Periodic Boundary Conditions The periodic boundary condition implies: ( ) ( ) { { 1 1 1 1 1 1 1 1 1 1 .
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