ECE340_L8_S14_Distribution

# Density of states 2 including spin the total number

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Unformatted text preview: ype Intrinsic p-Type Addi`onal Informa`on Conduc`on Band Density of States Consider a cube of material having sides of length "L" A free electron in this material has a wavefunction given by: j k x + k y+ k z ψ = Ae ( x y z ) where 2π 2π 2π , k y = ny , k z = nz L L L We also know: 2D k x = nx p2 2 k 2 2π 2 2 2 2 E= = = * 2 nx + ny + nz2 * * 2m 2m mL The constant energy surface is therefore the ( ) the surface of a sphere of radius "n" where: ( n= n +n +n 2 x 2 y ) 1 22 z 3D The total number of states within the sphere is: 4 N ′ = π n3 3 Note: Other variants of this derivation get the same result 11 Conduc`on Band Density of States (2) Including spin, the total number of states up to a given energy is therefore: 4 8 N ′(n ) = 2 × π n 3 = π n 3 3 3 Rewriting the expression for E: 2π 2 2 2 L m*E E = * 2 n so n = and: 2 mL π 2 8 3 ⎛ m*E ⎞ N′(E) = 2 L ⎜ 2 ⎟ 3π ⎝ 2 ⎠ dN ′ ( E ) L3 ⎛ 2 m * ⎞ = dE 2π 2 ⎜ 2 ⎟ ⎝ ⎠ 3/2 3/2 E The Density of States is defined as the number of states per incremental unit of energy near E per unit volume: dN ′ ( E ) 1 1 ⎛ 2 m* ⎞ N (E) = = 3 dE L 2π 2 ⎜ 2 ⎟ ⎝ ⎠ 3/2 E 12 Other Cases •  Valence Band –  Need to include both light holes and heavy holes in the density of states •  Near the Direct- Indirect Crossover –  Need to include both valleys •  Indirect Materials –  Mul...
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## This note was uploaded on 03/06/2014 for the course ECE 340 taught by Professor Leburton during the Spring '11 term at University of Illinois, Urbana Champaign.

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