Unformatted text preview: ype Intrinsic pType 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.
 Spring '11
 Leburton

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