Density of states 2 including spin the total number

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online