Page 212 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Local Equilibrium Description
thermal gradient may nevertheless leave us with an electrical current
induced heat flow, i.e.
∂ η N 21 N 21
j = N ------ = --------- j = – ------------i = Πi (6.44)
Q 21 ∂ x N 11 n qN 11
The last factor of proportionality between the thermal current density
j and the electrical current density is termed the Peltier coeffi-
i
Q
cient
N 21
Peltier Π = – ------------ (6.45)
qN
Coefficient 11
Since thermal currents are not directly observable but rather their gra-
dients, we recall the energy balance equation (6.22) in the form
∂ ux t) = Fj x t,( ) ∇ ( j ( x t) + µj x t)) (6.46)
(
,
,
,
(
–
t ∂ n x Q n
In terms of the electrical current density, using the definition of the
electrochemical potential (6.31), we therefore have
∂ ( , 1 2
σ
t ∂ ux t) = – i∇Π + ---i (6.47)
The second term in (6.47) we identify as the Joule heating that a cur-
rent produces. Note that it is quadratic in the current density and its
coefficient is the inverse of the electrical conductivity as given by
(6.42). After the discussion of Bloch oscillations it became clear that
electrical conductivity is due to a coupling of electrons to a new dissi-
pation channel, namely the phonon system. This coupling was intro-
duced phenomenologically by means of the relaxation time
approximation. The first term in (6.47) shows that no effect will be
observed in a homogeneous material, since with Π = const ,
∇Π = 0 . No prediction about Π is possible. Therefore, we look at
B
the interface or two materials and with different Peltier coeffi-
A
cients Π A and Π B , and two different electrical conductivities σ A
and σ .
B

Semiconductors for Micro and Nanosystem Technology 209