handout16 - 1 ECE 407 Spring 2009 Farhan Rana Cornell...

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Unformatted text preview: 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 , . , = 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 2 . 2 = + + + 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 2 . 2 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 2 . 2 (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 = 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 , , * = ( ) ( ) 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 a Energy x k a a 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 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 x k a a Energy x k a a Energy 3 ECE 407 Spring 2009 Farhan Rana Cornell University Current Density for a Completely Filled or Empty Bands x k a a Energy x k a a 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...
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This note was uploaded on 04/04/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell University (Engineering School).

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handout16 - 1 ECE 407 Spring 2009 Farhan Rana Cornell...

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