Page 255 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
Actually, the charge interactions are long-range. Since the crystal is neu-
tral, we assume that the remaining crystal screens off the rest of the
charges, and the local disturbance is only “felt” by the direct neighbors of
the moving charge. Clearly, this is a simplistic model used only to show
the features of the piezoelectric interaction.

The restriction to next-neighbor interactions and identical interatomic
force constants yields the following equations of motion when the
expressions (7.47) and (7.48) are inserted into the Lagrange equations

E
--- 2u
mu˙˙ im = – ( im – u iM – u ( i 1)M ) (7.50a)
–
a
E
--- 2u
Mu˙˙ iM = – ( iM – u ( i + 1)m – u ) (7.50b)
im
a
Reversible For a more general model, we first consider a general thermodynamic
Thermo- phenomenological model of piezoelectricity (see e.g. [7.5] or [7.12]).
dynamic
The internal energy of the crystal, U , is defined as (for the general idea,
Derivation
see Box 7.1 and Box 7.2)
Box 7.1. The thermodynamic postulates [7.11].
The Four Callen Postulates. Thermodynamics is The values assumed by the extensive parameters
founded upon a set of “laws”. Modern theorists in the absence of a constraint are those that maxi-
have reformulated these laws as a set of postu- mize the entropy over the manifold of constrained
lates. We closely follow [7.11]: equilibrium states.
I. There exist particular states (called equilibrium III. The entropy of a composite system is additive
states) that, macroscopically, are characterized over the constituent subsystems (whence the
completely by the specification of the internal entropy of each subsystem is a homogeneous first-
energy U , and a set of extensive parameters order function of the extensive parameters). The
X X … X, , , . entropy is continuous and differentiable and is a
1 2 r
S
II. There exists a function (called the entropy ) monotonically increasing function of the energy.
of the extensive parameters, defined for all equi- IV. The entropy of any system vanishes in the state
⁄
librium states, and having the following property: for which T ≡ ( ∂U ∂S) = 0 .
X 1 X 2 …
,
,
⋅
⋅
dU = σ:dε + E dD + H dB + TdS (7.51)

252 Semiconductors for Micro and Nanosystem Technology