Phys560Notes-10 - Electron Dynamics (of Semiconductors)...

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Electron Dynamics (of Semiconductors) Basic assumption: independent electron approximation (single-particle picture) Question: what happens to an electron under an external field (transport properties)? Bloch state   i n eu kr kk r         i nn ie u k pp k r n k is not a momentum eigenfunction. k p method : calculating  0 n from 0 n k by perturbation n Hu u k k  22 2 pk HU mm m k kp r  00 0 0 HH H H m k k kp k H = perturbation      2 3 2 n n n uH u k u O k m k Taylor expansion             2 3 000 0 2 3 0 1 2 1 2 nnn n ni n i j n ii j j Ok k O k k k k k k k k Comparing terms ,   0 0 0 n n n n uu k k k p ν k . Let 0 to simplify notation: Expectation value of v = group velocity 1 n   vk k k Comparing terms 2 , 2 2 0 0 1 ij n j i j k m mk k    k  2 2 n i in n j jn n ij n n n up k u up ku i j m m    k k k Theorem : if iji j j ij ij kk f kk g then SS ij ij f g (symmetric part). SA ij ij ij f ff , where ij ji f f (symmetric part) and A A ij ji f f   (antisymmetric part).
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Definition:    2 1 2 1 n n ij ij kk k M = inverse effective mass tensor for band n at k Making use of   00 0 0 0 nn n n uu k k pk p         1 2 11 ni n nj n j ij n p pi j mm k k M For a free electron gas: 22 2 n k m k and all matrix elements vanish.    1 1 1 1 n m m m M , as expected. In general, 1 n M = real & symmetric; diagonalizable with a real rotation of axes (principal axes transformation). 1 1 2 3 1 1 1 ij m m m M ; m 1,2,3 = effective masses      0 2 i n i i k m k k ν k At critical points (van Hove singularities), such as band edges (CBM and VBM),  1 0 ν 2 i i i k m k The system is like a free electron gas, but with different masses along different directions. Referring to the principal axes,  2 2 1 n n ni n n p m Semiclassical Model for Electron Dynamics Semiclassical regime: wavepackets with   ave k and   ave r .  1 kr (uncertainty principle).  1 kB Z a (reasonably well defined k in BZ). ra (large size relative to individual atoms).
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(wavelength of applied field) r , and so the wave packet is particle-like.
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This note was uploaded on 03/21/2012 for the course PHYS 560 taught by Professor Flynn during the Spring '08 term at University of Illinois, Urbana Champaign.

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Phys560Notes-10 - Electron Dynamics (of Semiconductors)...

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