handout13 - Handout 13 Properties of Electrons in Energy...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 13 Properties of Electrons in Energy Bands In this lecture you will learn: • Properties of Bloch functions • Average momentum and velocity of electrons in energy bands • Energy band dispersion near band extrema • Effective mass tensor • Crystal momentum ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bloch Functions: A Review 1) The quantum states of an electron in a crystal are given by Bloch functions that obey the Schrodinger equation: ( ) ( ) ( ) r k E r H k n n k n r r r r r , , ˆ ψ = 2) Under a lattice translation, Bloch functions obey the relation: ( ) () r e R r k n R k i k n r r r r r r r , . , = + 3) Bloch functions can be written as the product of a plane wave times a lattice periodic function: () () r u e r k n r k i k n r r r r r r , . , = where the wavevector is confined to the FBZ and “ n ” is the band index k r () ( ) ( ) = + j r G k i j k n k n j e V G c r r r r r r r r . , , 1 4) Bloch function of wavevector can be written as a superposition of plane waves with wavevectors that differ from by reciprocal lattice vectors: k r k r
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Orthogonality: Bloch functions are eigenstates of a Hermitian operator and therefore must be orthogonal. In “ d ” dimensions: () ( ) () m n d d m n k k k m k n d k k V r r r d , , ' , ' , , * ' 2 δ π ψ r r r r r r r r r = = Bloch Functions: Orthogonality and Completeness Both expression valid depending upon context Completeness: Bloch functions for ALL wavevectors in the FBZ and for ALL energy band satisfy the following completeness relation in “ d ” dimensions: () ( ) ( ) ' ' 2 ' FBZ , * , FBZ in , * , r r r r k d V r r d k n k n d d k n k n k n n r r r r r r r r r r r r = ∑∑ = ECE 407 – Spring 2009 – Farhan Rana – Cornell University Another Schrodinger-like Equation for Bloch Functions The periodic part of a Bloch function satisfies a Schrodinger-like equation: () () () () ( ) ( ) r u k E r u r V m k k m P m P r u k E r u r V m k P r u e k E r u r V m k P e r k E r r V m P r k E r H k n n k n k n n k n k n r k i n k n r k i k n n k n k n n k n r r r r h r h r r r r r r r h r r r r r r h r r r r r r r r r r r r r r r r r r r r r r r , , 2 2 2 , , 2 , . , 2 . , , 2 , , ˆ 2 . ˆ 2 ˆ ˆ 2 ˆ ˆ 2 ˆ ˆ 2 ˆ ˆ = + + + = + + = + + = + = Where the following two relations have been used: () ( )() () ( ) () r f k P e r f e P r f k P e r f e P r k i r k i r k i r k i r r h r r r r r h r r r r r r r r r r r 2 . . 2 . . ˆ ˆ ˆ ˆ + = + = Result:
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bloch Functions and Electron Momentum • For an electron with wavefunction given by a plane wave: the quantity is the momentum of the electron () r k i k e V r r r r r .
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handout13 - Handout 13 Properties of Electrons in Energy...

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