Page 210 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Local Equilibrium Description
j
n N 11 N 12 X 1
= – (6.35)
j Q N 21 N 22 X 2
X j represents the respective thermodynamic force, which in this special
case is
∂ η
X = ------ (6.36)
1 ∂ x

the gradient of the electrochemical potential and the thermal driving
force

∂
1 T
X = --------- (6.37)
2
∂
T x
which is the temperature gradient per unit temperature. N , the so–
ij
called transport coefficients, connect fluxes and forces. In real anisotropic
materials they are found to be tensors. The first coefficient to be calcu-
lated relates the particle current density to the applied electrochemical
potential

f ∂ 0
∂ ∫
3
N 11 = τ – -------- v ⊗( v)d v (6.38)
E
V
The next two coefficients are symmetric and given by
f ∂ 0
∂ ∫
(
3
N = τ – -------- E –( µ) v ⊗ v)d v = N (6.39)
12 E 21
V
They tell us, in principle, what part of the particle current flows due to the
temperature gradient applied, or what part of the heat current results from
an applied electrochemical potential gradient. In reality they are not
directly observed, and we shall discuss this later. The last coefficient
gives the heat current due to a thermal driving force

f ∂ 0
∂ ∫
3
2
N = τ – -------- E –( µ) v ⊗( v)d v (6.40)
22 E
V
Semiconductors for Micro and Nanosystem Technology 207