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# handout16 - Handout 16 Conductivity of Electrons in Energy...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 16 Conductivity of Electrons in Energy Bands In this lecture you will learn: • Inversion symmetry of energy bands • The conductivity of electrons in energy bands • The electron-hole transformation • The conductivity tensor • Examples • Bloch oscillations ECE 407 – Spring 2009 – Farhan Rana – Cornell University Inversion Symmetry of Energy Bands - I Recall that a Bloch function can be written as: ( ) ( ) r u e r k n r k i k n r r r r r r , . , = ψ Where the periodic part satisfied a Schrodinger-like equation: ( ) ( ) ( ) ( ) r u k E r u r V m k k m P m P k n n k n r r r r h r h r r r r , , 2 2 2 ˆ 2 . ˆ 2 ˆ = + + + Now let go to in the above equation: k r k r ( ) ( ) ( ) ( ) r u k E r u r V m k k m P m P k n n k n r r r r h r h r r r r = + + , , 2 2 2 ˆ 2 . ˆ 2 ˆ And then take the complex conjugate of the whole equation to get: ( ) ( ) ( ) ( ) r u k E r u r V m k k m P m P k n n k n r r r r h r h r r r r = + + + , * , * 2 2 2 ˆ 2 . ˆ 2 ˆ (1) (1) Comparing (1) and (2) we get the results: ( ) ( ) r u r u k n k n r r r r , , * = ( ) ( ) k E k E n n r r =

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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Inversion Symmetry of Energy Bands - II ( ) ( ) r u r u k n k n r r r r , , * = ( ) ( ) k E k E n n r r = One can also write: ( ) ( ) r r k n k n r r r r , , * ψ ψ = ( ) ( ) k E k E n n r r = Recall that: ( ) ( ) k E k v n k n r h r r r = 1 Now let go to in the above equation: k r k r ( ) ( ) ( ) ( ) ( ) ( ) ( ) k v k v k v k E k E k E k v n n n n k n k n k n r r r r r r r h r h r h r r r r r = = = = = 1 1 1 x k a π a π Energy ECE 407 – Spring 2009 – Farhan Rana – Cornell University Current Density for Energy Bands For a free electron gas the current density was given as: ( ) ( ) ( ) ( ) ( ) ( ) k v k f k d e k v k f V e J k r r r r r r r r r × = × = 3 3 all 2 2 2 π In Drude model , the electron current density was given as: ( ) v e n J r r = Now we want to find the current density due to electrons in energy bands The current density due to electrons in the n -th band can be written in a manner similar to the free-electron case: ( ) ( ) ( ) ( ) ( ) ( ) k v k f k d e k v k f V e J n n n k n n r r r r r r r r r × = × = FBZ 3 3 FBZ in 2 2 2 π x k a π a π Energy
3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Current Density for a Completely Filled or Empty Bands x k a π a π Energy • Completely filled bands do not contribute to electrical current or to electrical conductivity where I have used the fact: ( ) ( ) k v k v n n r r r r = E f • Completely empty bands do not contribute to electrical current or to electrical conductivity Only partially filled bands contribute to electrical current and to electrical conductivity Of course, if for all in FBZ: ( ) 0 = k f n r k r ( ) ( ) ( ) 0 2 2 FBZ 3 3 = × = k v k f k d e J n n n r r r r r π Consider a completely filled band for which for all in FBZ: ( ) 1 = k f n r k r ( ) ( ) ( ) ( ) ( ) 0 2 2 2 2 FBZ 3 3 FBZ 3 3 = × = × = k v k d e k v k f k d e J n n n n r r r r r r r r π

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handout16 - Handout 16 Conductivity of Electrons in Energy...

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