Page 256 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Electron-Phonon
Box 7.2. Basic thermodynamic relations [7.11].
Extensive and Intensive Parameters. The exten- The Thermodynamic Potentials. By the Legen-
sive parameters are denoted by dre transformation we can replace any extensive
,
,
{ SV N , N … N } ≡ parameter in (B 7.2.3) by an intensive parameter,
, ,
r
1
2
(B 7.2.1) yielding a new fundamental relation that is closer
{ X X X … X, 1 , 2 , , t }
0
to laboratory conditions. The most familiar of
for the volume and the N the mole numbers
V
k these “potentials” are the Helmholtz free energy
of the constituent chemical species. The intensive
(replacing with )
S
T
parameters are denoted by
F = U – TS , (B 7.2.11)
{ TP η η …η } ≡ the Enthalpy (replacing with )
,
,,
,
,
V
r
2
1
P
(B 7.2.2)
{ P P P … P, 1 , 2 , , t }
0
H = U + PV , (B 7.2.12)
P
T
Here, is the temperature, is the pressure, and
the Gibbs free energy (combining the above)
the η are the electrochemical potentials of the
k
constituent species. G = U – TS + PV , (B 7.2.13)
and the Grand canonical potential (replacing S
The Fundamental Equations. The fundamental
µ
T
with , and N with )
energy equation of a simple system is
K = U – TS – ηN . (B 7.2.14)
,
,
,
,
(
U = U X X X … X ) (B 7.2.3)
0 1 2 t The Maxwell Relations. We have that
In its differential form, the fundamental thermody-
∂X ⁄ ∂P = ∂X ⁄ ∂P (B 7.2.15)
namic equation for the energy is j k k j
∂X ⁄ ∂X = – ∂P ⁄ ∂P j (B 7.2.16)
j
k
k
dU = TdS + ∑ t k = 1 P dX k
k
(B 7.2.4) ∂P ⁄ ∂X = ∂P ⁄ ∂X j (B 7.2.17)
k
k
j
= ∑ t k = 0 P dX k These define the material properties in convenient
k
The intensive parameters P k are defined by forms. Also called reciprocal relations.
⁄
P = ∂U ∂X (B 7.2.5) Affinities and Fluxes. In a discrete system, an
k k
extensive parameter flux is defined by
The fundamental entropic relation is
J = dX ⁄ dt (B 7.2.18)
S = S X X X … X,( , , , ) (B 7.2.6) k k
0 1 2 t
Taking the time derivative of the entropic funda-
In its differential form, the fundamental thermody-
mental relation we obtain
namic equation for the entropy is
1 t dS dX k
⁄
dS = ---dU + ∑ F dX dS dt = ∑ -------------------- = ∑ F J (B 7.2.19)
k k
T k = 1 k k dX dt
k
(B 7.2.7) k k
= ∑ t 0 F dX k where we have defined the extensive parameter’s
k
k =
⁄
associated affinity F = dS dX . For continu-
The intensive parameters F k are defined by k k
ous systems we have that
⁄
F = ∂S ∂X (B 7.2.8)
k k
The Euler Relation. U = ∑ t k = 0 P X k (B 7.2.9) s˙ = ∑ ∇ F • j k (B 7.2.20)
k
k
k
The Gibbs-Duheim Relation. In both instances the rate of entropy production is
the sum of the products of affinities with their
∑ t k = 0 X dP = 0 (B 7.2.10) respective fluxes.
k
k
Semiconductors for Micro and Nanosystem Technology 253