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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 electronhole 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 Schrodingerlike 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 nth band can be written in a manner similar to the freeelectron 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).
 Spring '08
 RANA

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