8.12 Continuity Equation
251
D n n = −G n + U n
(8.65a)
D p p = −G p + U p .
(8.65b)
8.13 Heat Conduction
We consider here the heat transport [814] due to a temperature gradient. The heat flow q, i.e. energy
per unit area per time in the direction ˆ
q, is proportional to the local gradient of temperature. The
proportionality constant κ is called, heat conductivity,
q = −κ ∇T .
(8.66)
In crystals, the heat conductivity can depend on the direction and thus κ is generally a tensor of rank 2.
In the following, κ will be considered as a scalar quantity. The quite generally valid Wiedemann–Franz
law connects the thermal and electrical conductivities
κ =
π
2
3
k
e
2
T σ .
(8.67)
The balance (continuity) equation for the heat energy Q is
∇ · q = −
∂ Q
∂t
= − ρ C
∂T
∂t
+ A ,
(8.68)
where ρ denotes the density of the solid and C the heat capacity. A denotes a source or drain of heat,
e.g. an external excitation. Combining (8.66) and (8.68), we obtain the equation for heat conductivity
T =
ρ C
κ
∂T
∂t
−
A
κ
,
(8.69)
which simply reads T = 0 for a stationary situation without sources.
The random mixture of various atoms in natural elements represents a perturbation of the perfectly
periodic lattice. Since the mass of the nuclei varies, in particular lattice vibrations will be perturbed.
Thus we expect an effect on the heat conductivity. In Fig. 8.28, the thermal conductivity of crystals
from natural Ge and enriched
74 Ge are compared [815], the latter having, as expected, the higher heat
conductivity, i.e. less scattering. The T
3 -dependence of the heat conductivity at low temperature has
been attributed to scattering of phonons at the sample boundary [816]. The thermal conductivity of
isotopically pure
28 Si thin films has been measured to be 60% greater than natural silicon at room
temperature and at least 40% greater at 100
◦ C, a typical chip operating temperature [817, 818].
8.14 Coupled Heat and Charge Transport
The standard effect of coupled charge and heat transport is that a current heats its conductor via Joule
heating. However, more intricate use of thermoelectric effects can also be employed to cool certain
areas of a device. For further details see [819, 820].
For the analysis of coupled charge and heat transport we first sum the electric field and the concentration gradient to a new field ˆ
E = E + ∇ E F /e. Then, the heat flow and charge current are
251
D n n = −G n + U n
(8.65a)
D p p = −G p + U p .
(8.65b)
8.13 Heat Conduction
We consider here the heat transport [814] due to a temperature gradient. The heat flow q, i.e. energy
per unit area per time in the direction ˆ
q, is proportional to the local gradient of temperature. The
proportionality constant κ is called, heat conductivity,
q = −κ ∇T .
(8.66)
In crystals, the heat conductivity can depend on the direction and thus κ is generally a tensor of rank 2.
In the following, κ will be considered as a scalar quantity. The quite generally valid Wiedemann–Franz
law connects the thermal and electrical conductivities
κ =
π
2
3
k
e
2
T σ .
(8.67)
The balance (continuity) equation for the heat energy Q is
∇ · q = −
∂ Q
∂t
= − ρ C
∂T
∂t
+ A ,
(8.68)
where ρ denotes the density of the solid and C the heat capacity. A denotes a source or drain of heat,
e.g. an external excitation. Combining (8.66) and (8.68), we obtain the equation for heat conductivity
T =
ρ C
κ
∂T
∂t
−
A
κ
,
(8.69)
which simply reads T = 0 for a stationary situation without sources.
The random mixture of various atoms in natural elements represents a perturbation of the perfectly
periodic lattice. Since the mass of the nuclei varies, in particular lattice vibrations will be perturbed.
Thus we expect an effect on the heat conductivity. In Fig. 8.28, the thermal conductivity of crystals
from natural Ge and enriched
74 Ge are compared [815], the latter having, as expected, the higher heat
conductivity, i.e. less scattering. The T
3 -dependence of the heat conductivity at low temperature has
been attributed to scattering of phonons at the sample boundary [816]. The thermal conductivity of
isotopically pure
28 Si thin films has been measured to be 60% greater than natural silicon at room
temperature and at least 40% greater at 100
◦ C, a typical chip operating temperature [817, 818].
8.14 Coupled Heat and Charge Transport
The standard effect of coupled charge and heat transport is that a current heats its conductor via Joule
heating. However, more intricate use of thermoelectric effects can also be employed to cool certain
areas of a device. For further details see [819, 820].
For the analysis of coupled charge and heat transport we first sum the electric field and the concentration gradient to a new field ˆ
E = E + ∇ E F /e. Then, the heat flow and charge current are