Page 259 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
static approximation (see Section 4.1.2), in which case one of the two
terms drop from the formalism.
• α is the rank two tensor of thermal expansion coefficients (see
EH
Section 2.4.2)
• κ is the rank two tensor of dielectric permittivities.
σHT
• p is the vector of pyroelectric coefficients.
σH
• µ is the rank two tensor of magnetic permeabilities.
σET
• C is the scalar heat capacity (see Section 2.4.2).
σEH
We see that reversible thermodynamics provides us with the form of the
possible functional relationships. For the content of the tensors we have
to turn to detailed theories, as has been done in the text for a number of
the coefficients above. For example, if we include only the symmetry
properties of the crystalline materials under consideration, we can
already greatly reduce the number of possible nonzero entries in the
above tensor coefficients. For a detailed example considering the elastic
stiffness tensor, see Section 2.3.

When analyzing piezoelectric transducers it is usual to greatly reduce the
above constructive relationship. In particular, since the velocity of sound
is so much smaller than the velocity of light, regardless of the medium,
we may assume that as far as the mechanical deformation field is con-
cerned the electromagnetic field changes almost instantaneously. Further-
more, unless parasitically dominant or required as an effect, we assume
that the piezoelectric phenomena is operated under isothermal condi-
tions. Note here that these assumptions are merely a matter of computa-
tional convenience. In fact, nowadays many computer programs exist that
implement the full theory and so allow the designer a considerable
amount of more detailed investigative possibilities. Sticking to our sim-
plifications, we obtain
ε = s :σ + d ⋅ E (7.58a)
EHT HT
D = d HT :σ + k σHT ⋅ E (7.58b)

256 Semiconductors for Micro and Nanosystem Technology