Page 221 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
q j
–
for elec-
We use (6.71) to calculate the electric current density i =
n
trons from the form given in (6.11) and assume that the material is homo-
geneous, i.e., ∇µ = 0 . In this way we obtain three contributions
q 4
2
3
–
1
i = q K E + q K B × ( M E) + --------K EB( )MB (6.72)
1 2 3
M
where the components of the tensors K are given by [6.2]
i
1 τ i f ∂ 0
3 ∫
3
K = – --------- --------------------------------------------v ⊗ vd k (6.73)
i
2 2
4π 1 + q τ ∂ E
----------BMB
M
V k
The rank two tensors K are called the generalized transport coefficients.
i
They reflect the fully anisotropic character of constitutive equations in a
crystal. To simplify the discussion, we now write (6.72) by defining a
generalized constitutive equation for the current density
i = σ∗ E (6.74)
where the effective conductivity σ∗ is defined by
q 4
3
2
σ∗ = q K I + q K A + --------K A (6.75)
1 2 2 3 3
M
and where is the identity matrix. The matrices A 2 and A 3 are given by
I
0 – B m B m
3 2 2 3
A = – B m 0 – B m (6.76a)
2 3 1 1 3
– B m B m 0
2 1 1 2
2
B m 1 B B m B B m 1
1
1
1
1
2
3
A = B B m B m B B m (6.76b)
2
3
2 1 2 2 2 2 3 2
2
B B m B B m B m
3 1 3 3 2 3 3 3
Note that these equations are given in coordinates of the crystal direc-
tions. Thus, for the effective masses we may chose the two different
218 Semiconductors for Micro and Nanosystem Technology