Page 176 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Systems and Ensembles
Γ E() =
which defines the density of states DE()δE (5.4)
⁄
Density of DE() = ∂ VE() ∂ E (5.5)
States
This rather abstract definition becomes lucid when applying to an elec-
tron in a cube with a cubic crystal structure with lattice constant . The
a
⋅
edge of the cube has a length L = n a . Applying periodic boundary
conditions the k-vector comes in discrete portions
⁄
k= ( 2πn 2πn 2πn, , ) L as shown in (3.44). Given the energy we
E
x y z
calculate the number of states that are within the range of 0 E,( ) , i.e., the
number of combinations of n n n,( , ) that results each in an energy
x y z
lower than E , or in other words the sum over all possible k. We have
⁄
n = ( k L) ( 2π) etc.We write the sum over all k as the integral over k-
x x
E
space over all states with energies lower than in spherical coordinates
kE()
3
3
L
L


∑ →   2π ∫ dk dk dk =   2π ∫ 4πk k (5.6)
2
------
------
d
x
y
z
k ( 0 E) 0
,
(5.6) divided by the sample volume L 3 gives the volume Vk E()( ) the
⁄
states occupy in k-space. The energy is given by E = ( h k ) ( 2m) and
2 2
⁄
thus we have k = ( 2mE) h 2 , where k is the absolute value of the vec-
⁄
⋅
2
tor k. Furthermore we have dk = m dE ( h k) . We insert this into
(5.6) in order to calculate the volume with respect to a given energy E
E
/
1  2m  32
h  ∫
d
VE() = --------- ------- E' E' (5.7)
4π 2
0
With the expansion procedure as shown in (5.3) we obtain for the density
of states
/
1  2m  32
DE() = --------- ------- E (5.8)
4π 2 h 
Semiconductors for Micro and Nanosystem Technology 173