Dos (k-space)

Density of States (k-space) (DOS): D(E): States per unit volume per unit energy about energy E. DOS controls electronic properties of solids like Electronic transport, Optical absorption, Magnetic behavior etc. D(E)dE is number of states in energy interval E to E+dE To calculate the DOS, we need to know: (i) Dimensionality of the system (ii) E versus k relation that the particles obey Particle of interest is electron since in most applied problems we are dealing with electrons. h2 k 2 E-k relation is parabolic for free electron E = 2m

Volume in k-space that each energy state occupies is 8π 3 V Number of electronic states in volume Ω of k-space:

ΩV 8π 3

3-D System: k-space volume in k to k+dk is 4πk2dk The k-space volume per electron 3 state is 2π L

(

)

The electron states between k and k+dk is

4π k dk V 8π 2

2

3

=

k 2 dk V 2 2π

Denoting the energy and energy interval corresponding to k and k+dk as E and E+dE. Number of electron states between E and E+dE per unit V is k 2 dk D ( E ) dE = 2π 2

Using the E versus k relation for the free electron, we have 3/ 2 1/ 2 0

2 m E dE k dk = 3 h 2

m03 / 2 ( E )1/ 2 dE D ( E ) dE = 2 π 2 h3 since

h2 k 2 E= 2m

2 h and dE = k dk m

Taking spin of e– into account

2 m03 / 2 E 1/ 2 dE D( E ) = π 2 h3