Direct Energy, 2018a

2 CAPACITORS AND PIEZOELECTRIC DEVICES 33
If an electric eld is also applied, stress and strain are related by
(
)
1
strain =
· stress + −→ E · d (2.17)
Young's elastic modulus
where d is the piezoelectric strain constant.
The energy stored in a piezoelectric device under stress −→ ς is given by
E = | −→ ς |·A · l · η eff (2.18)
where A is the cross sectional area of a device in m 2 , l is the deformation in
m, and η eff is the eciency. Devices which are bigger, are deformed more,
or are made from materials with larger piezoelectric constants store more
energy.
According to Eq. 2.14, the material polarization of an insulating crystal
is linearly proportional to the applied stress. While this accurately
describes many materials, it is a poor description of other materials. For
other piezoelectric crystals, the material polarization is proportional to the
square of the applied stress
∣ −→ P
∣ =
∣ −→ D
∣ ∣∣ −→
∣ ∣∣
∣ − ɛ 0 E + d |
−→ ς | + dquad | −→ ς | 2 (2.19)
where d quad is another piezoelectric strain constant. To model the material
polarization in other materials, terms involving higher powers of the stress
are needed.
2.3.2 Piezoelectricity in Crystalline Materials
To understand which materials are piezoelectric, we need to introduce some
terminology for describing crystals. Crystalline materials may be composed
of elements, such as Si, or compounds, such as NaCl. By denition, atoms
in crystals are arranged periodically. Two components are specied to
describe the arrangement of atoms in a crystal: a lattice and a basis [25,
p. 4]. A lattice is a periodic array of points in space. An n-dimensional
lattice is specied by n lattice vectors for integer n. We can get from one
lattice point to every other lattice point by traveling an integer number of
lattice vectors. Three vectors, −→ a 1 , −→ a 2 , and −→ a 3 , are used to describe physical
lattices in three-space. The choice of lattice vectors is not unique. Lattice
vectors which are as short as possible are called primitive lattice vectors. A
cell of a lattice is the area (2D) or volume (3D) formed by lattice vectors.
A primitive cell is the area or volume formed by primitive lattice vectors,
and it is the smallest possible repeating unit which describes a lattice.