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# lecture9 - 2.57 Nano-to-Macro Transport Processes Fall 2004...

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2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 9 3.4 Density of states (1) Electron in a quantum well = U U=0 x ENERGY AND n= 1 n= 2 n= 3 WAVEFUNCTION For electrons in a quantum well, the energy has discrete levels as 2 h n 2 E = (n=1,2,…) 8 m D 2 For wavefunction Ψ , we have degeneracy g(n)=2 due to the spin. , n s (2) Harmonic oscillator The energy is 1 K E = h ν ( n + 1/ 2); ν = (n=0,1,2…) n 2 π m The wavefunction is Ψ , and the degeneracy is g(n)=1. n (3) Rigid rotation The energy eigenvalues are 2 = ( ( E l = A A + 1) = hB A A + 1) (for |m| A , A =0,1,2, …). I 2 For wavefunction Ψ nlm , the degeneracy is g( l )=2 l +1. (4) Hydrogen atom E n el = − Mc 1 2 = − 13.6 eV ( n , 1 n A + 1 and m A , A =0, 1, 2, …) 2 2 2 = n n 2 The wavefunction Ψ nlms corresponds to degeneracy g(n)=2n 2 . Now let us consider electrons in a solid. The parabolic approximation at the band edge gives 2 2 ) = 2 2 = 2 ( k + k y 2 + k k x z E E = = c 2 m * 2 m * , 2.57 Fall 2004 – Lecture 9 1

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2 π na x , , k = ± . In different directions, n values can be different. For where k k y z L wavefunction Ψ ( , k k y , k ) , we have three quantum numbers. x z 15 10 Debye k ω π /a Normalized Electron Energy E/Eo 5 0 -1 0 1 k / ( π /a) In the Debye approximation, we have energy dispersion as ω = vk = v 2 2 2 x y z k k k + + ; E n = h ν ( n + 1 2 ) , where v is sound velocity. Density of (quantum mechanical) states (DOS): (a) Electron Volume of L π 2 0 L π 4 L π 6 L π 2 L π 4 dk k+dk One k x k y k k x k y k z Unit Cell 2.57 Fall 2004 – Lecture 9 2
3 In above figure, we can find the volume of one state is V = (2 π / L ) . In the above sphere, 1 the number of states within k and k+dk is 2 4 π k k Vk k 2 N = = V 1 2 π 2 , in which V=L 3 is the crystal volume.

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