handout15 - 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 15 Dynamics of Electrons in Energy Bands In this lecture you will learn: The behavior of electrons in energy bands subjected to uniform electric fields The dynamical equation for the crystal momentum The effective mass tensor and inertia of electrons in energy bands Examples Magnetic fields ECE 407 Spring 2009 Farhan Rana Cornell University Electron Dynamics in Energy Bands 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 , , = 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 , . , = + where the wavevector is confined to the FBZ and n is the band index k r Now we ask the following question: if an external potential is added to the crystal Hamiltonian, then what happens? How do the electrons behave? How do we find the new energies and eigenstates? The external potential could represent, for example, an applied E-field or an applied B-field, or an electromagnetic wave (like light) ( ) t r U H , r + 2 ECE 407 Spring 2009 Farhan Rana Cornell University Recall from homework that the energy bands are lattice-periodic in the reciprocal space, When a function in real space is lattice-periodic, we can expand it in a Fourier series, When a function is lattice-periodic in reciprocal space, we can also expand it in a Fourier series of the form, ( ) ( ) k E G k E n n r r r = + ( ) ( ) ( ) ( ) r G i j j j e G V r V r V R r V r r r r r r r . = = + ( ) ( ) ( ) ( ) = = + j k R i j n n n n j e R E k E k E G k E r r r r r r r . Periodicity of Energy Bands Fourier representation of energy bands ECE 407 Spring 2009 Farhan Rana Cornell University Consider the following mathematical identity (Taylor expansion): ( ) ( ) ( ) ( ) ( ) x f e a x f a x f x f a x f dx d a = + + + = + . .......... ' ' 2 1 ' 2 Generalize to 3 dimensions: ( ) ( ) r f e a r f a r r r r = + . Now go back to the relation: and consider the operator: ( ) ( ) ( ) ( ) = = + j k R i j n n n n j e R E k E k E G k E r r r r r r r . ( ) ( ) = j R j n n j e R E i E . r r We apply this operator to a Bloch function from the same band (i.e. the n-th band) and see what happens: ( ) ( ) ( ) ( ) ? , . , = = r e R E r i E k n j R j n k n n j r r r r r r A New Operator - I 3 ECE 407 Spring 2009 Farhan Rana Cornell University ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r k E r e R E R r R E r e R E r i E k n n j k n R k i j n j k n j j n k n j R j n k n n j j r r r r r r r r r r r r r r r r r r , , ....
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handout15 - 1 ECE 407 Spring 2009 Farhan Rana Cornell...

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