Page 224 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

From Global Balance to Local Non-Equilibrium
∂n
t ∂ 
x
t ∂ + ∇ nv( a ) =  ∂n   C (6.78)
We use the force term F = – qE and performing the time derivative of
j = nv a taking (6.78) into account the momentum balance equation
n
will read
∂ nv a q  ∂v a 
----------- – v ∇ nv( ) + ----En + ∇ ΠΠ ΠΠ = n (6.79)
t ∂ a x a m x t ∂   C
(
(
(
We write – v ∇ nv ) = – ∇ nv ⊗ v ) + nv ∇ ⊗ v ) and then
x
x
x
a
a
a
a
a
a
obtain
v ∂ a q 1  ∂v a 
n------- + nv ∇ ⊗( x v ) + ----En + ----∇ nk T = n (6.80)
t ∂ a a m m x B t ∂   C
where we have replaced the momentum current tensor ΠΠ ΠΠ by the temper-
T
ature tensor . This is done by performing the integral in (6.15) over the
differences between the microscopic group velocities and the average
velocities
m
--------- (
(
3
,,
–
–
nk T = 4π ∫ vv ) ⊗ ( vv ) f kx t)d k (6.81)
B 3 a a
Ω k
We interpret the r.h.s. of (6.81) as this part of the average energy that
remains when we subtract the average kinetic energy in harmonic
⁄
approximation E = 12m∗ v 2 from the total average energy. This is a
kin a
term that shows random motion with vanishing average velocity, namely
the temperature. This means that we can write the average energy consis-
tent of two parts asw = m 2⁄ v 2 + 12⁄ Tr k T( ) , where the trace opera-
a B
tor on a tensor means Tr k T( ) = k B∑ T . With the same arguments
B i ii
we obtain
∂ w 1  ∂w 
(
n------ + ∇ Q + nv ∇ w) + qEnv + ---∇ nk T = n (6.82)
x
x
t ∂ a a n x B t ∂   C
where we set the heat flow Q as
Semiconductors for Micro and Nanosystem Technology 221