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# handout14 - Handout 14 Statistics of Electrons in Energy...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 14 Statistics of Electrons in Energy Bands In this lecture you will learn: ECE 407 – Spring 2009 – Farhan Rana – Cornell University Example: Electron Statistics in GaAs - Conduction Band Consider the conduction band of GaAs near the band bottom at the Γ -point: = e e e m m m M 1 0 0 0 1 0 0 0 1 1 This implies the energy dispersion relation near the band bottom is: () ( ) e c e z y x c c m k E m k k k E k E 2 2 2 2 2 2 2 2 h h r + = + + + = Suppose we want to find the total number of electrons in the conduction band: We can write the following summation: ( ) ( ) × = FBZ in 2 k f c E k E f N r r

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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University () () () + = KT E k E f c f c E k E f r r exp 1 1 ( ) ( ) × = FBZ in 2 k f c E k E f N r r Where the Fermi-Dirac distribution function is: We convert the summation into an integral: () () () () + × = × = KT E k E k d V E k E f N f c k f c r r r r exp 1 1 2 2 2 FBZ 3 3 FBZ in π Then we convert the k-space integral into an integral over energy: () () () () = + × = ? ? FBZ 3 3 exp 1 1 2 2 f c f c E E f E g dE KT E k E k d V N r r Need to find the function g ( E ) and need to find the limits of integration Example: Electron Statistics in GaAs - Conduction Band ECE 407 – Spring 2009 – Farhan Rana – Cornell University Density of States in Energy Bands Energy x k a a s E σ ss V 4 ( ) ( ) a k V E k E x ss s x cos 2 = ss s V E 2 + ss s V E 2 Consider the 1D energy band that results from tight binding: We need to find the density of states function g 1D ( E ): () dE E g L dE dE dk L dk L dk L ss s ss s ss s ss s x V E V E D V E V E x a x a a x k × × × × + + 2 2 1 2 2 0 FBZ in 2 2 4 2 2 2 () a k aV dk dE x ss x sin 2 = () () ( ) 2 2 1 2 1 2 s ss D E E V a E g = E ( ) E g D 1 s E ss s V E 2 ss s V E 2 +
3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University () () () () = + × = ? ? FBZ 3 3 exp 1 1 2 2 f c f c E E f E g dE KT E k E k d V N r r π Since the electrons are likely present near the band bottom, we can limit the integral over the entire FBZ to an integral in a spherical region right close to the Γ -point: Since the Fermi-Dirac distribution will be non-zero only for small values of k , one can safely extend the upper limit of the integration to infinity: () () () () () × × Γ point 3 2 FBZ 3 3 8 4 2 2 2 f c f c E k E f dk k V E k E f k d V r r () () () () () × × 0 3 2 FBZ 3 3 8 4 2 2 2 f c f c E k E f dk k V E k E f k d V r Example: Electron Statistics in GaAs - Conduction Band

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handout14 - Handout 14 Statistics of Electrons in Energy...

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