In general, when all of the terms in Eq. (3.187) are considered, we write the
solution using the following trigonometric series:
b n ¼
X M
m¼1
a m cos mψ þ b m sin mψ,
ð3:190Þ
where a m and b m are undetermined constants. We substitute Eq. (3.190) into
Eq. (3.187), write both sides of the resulting equation in terms of cosmψ and
sinmψ using trigonometric identities, and set their corresponding coefficients
equal. This yields a hierarchy of linear equations for a m and b m . These equations
are truncated and solved on a computer.
As a numerical example, consider a rod with a cross-sectional dimension characterized by some diameter d ¼ 1 mm. n ¼ 10
13
=m
3 . For a wave with wavelength
λ ¼ 10 mm which is much larger than the cross-sectional dimension, the wave
number ξ ¼ 2π/λ ¼ 628.3 1/m. Then, from Eq. (3.183), the wave frequency is
ω ¼ 3.49 Â 10
6 1/s. The wave amplitude A will be varied in the computation. In the
case when A ¼ 3 Â 10
À8 m, the corresponding amplitude of the axial strain S 3 ¼ u 3,3
is Aξ ¼ 18.85 Â 10
À6 . We calculate b n for increasing values of A in the range from
10
À9 m to 3 Â 10
À8 m with five terms in the series in Eq. (3.190), i.e., M ¼ 5 which
has been verified numerically to have sufficient accuracy for examining the qualitative behaviors of the waves. The results are summarized in Fig. 3.25. In (a), when
A is small, the wave is essentially sinusoidal. As A increases, deviation from a
sinusoidal wave becomes visible. The height of the crests is larger than the depth of
the troughs, and the crests are narrower than the troughs. As A increases further in
(b), the wave deviates from a sinusoidal one severely. These are believed to be due to
the electrical nonlinearity and are to be verified numerically by, e.g., COMSOL.
3.14 Extension of a PN Junction
In this section we study the extension of a piezoelectric semiconductor rod with a PN
junction (see Fig. 3.26). The rod has a unit cross-sectional area and is under an axial
force F. It is electrically open at both ends. Both homogeneous and heterogeneous
junctions will be considered. For a homogeneous junction, the material is uniform
along the entire rod, but the two halves are doped differently. The left half is
dominated by holes and the right half by electrons. For a heterogeneous junction,
the materials of the two halves are different. We consider the case of a relatively
simple heterogeneous junction obtained by reversing the c-axis of the material of the
left half. When the c-axis is reversed, the relevant piezoelectric constants change
signs. The problem is static. Below is a combination of a linear theoretical analysis
and a nonlinear numerical analysis by COMSOL [11]. A nonlinear analysis using
both the perturbation method and COMSOL can be found in [12].
3.14 Extension of a PN Junction
75
solution using the following trigonometric series:
b n ¼
X M
m¼1
a m cos mψ þ b m sin mψ,
ð3:190Þ
where a m and b m are undetermined constants. We substitute Eq. (3.190) into
Eq. (3.187), write both sides of the resulting equation in terms of cosmψ and
sinmψ using trigonometric identities, and set their corresponding coefficients
equal. This yields a hierarchy of linear equations for a m and b m . These equations
are truncated and solved on a computer.
As a numerical example, consider a rod with a cross-sectional dimension characterized by some diameter d ¼ 1 mm. n ¼ 10
13
=m
3 . For a wave with wavelength
λ ¼ 10 mm which is much larger than the cross-sectional dimension, the wave
number ξ ¼ 2π/λ ¼ 628.3 1/m. Then, from Eq. (3.183), the wave frequency is
ω ¼ 3.49 Â 10
6 1/s. The wave amplitude A will be varied in the computation. In the
case when A ¼ 3 Â 10
À8 m, the corresponding amplitude of the axial strain S 3 ¼ u 3,3
is Aξ ¼ 18.85 Â 10
À6 . We calculate b n for increasing values of A in the range from
10
À9 m to 3 Â 10
À8 m with five terms in the series in Eq. (3.190), i.e., M ¼ 5 which
has been verified numerically to have sufficient accuracy for examining the qualitative behaviors of the waves. The results are summarized in Fig. 3.25. In (a), when
A is small, the wave is essentially sinusoidal. As A increases, deviation from a
sinusoidal wave becomes visible. The height of the crests is larger than the depth of
the troughs, and the crests are narrower than the troughs. As A increases further in
(b), the wave deviates from a sinusoidal one severely. These are believed to be due to
the electrical nonlinearity and are to be verified numerically by, e.g., COMSOL.
3.14 Extension of a PN Junction
In this section we study the extension of a piezoelectric semiconductor rod with a PN
junction (see Fig. 3.26). The rod has a unit cross-sectional area and is under an axial
force F. It is electrically open at both ends. Both homogeneous and heterogeneous
junctions will be considered. For a homogeneous junction, the material is uniform
along the entire rod, but the two halves are doped differently. The left half is
dominated by holes and the right half by electrons. For a heterogeneous junction,
the materials of the two halves are different. We consider the case of a relatively
simple heterogeneous junction obtained by reversing the c-axis of the material of the
left half. When the c-axis is reversed, the relevant piezoelectric constants change
signs. The problem is static. Below is a combination of a linear theoretical analysis
and a nonlinear numerical analysis by COMSOL [11]. A nonlinear analysis using
both the perturbation method and COMSOL can be found in [12].
3.14 Extension of a PN Junction
75