φ ¼
eλ þ pc
ð
Þ θ
k e 2 þ cε
ð
Þ
sinh ka exp Àkx
ð
Þ,
E ¼
eλ þ pc
ð
Þ θ
e 2 þ cε
sinh ka exp Àkx
ð
Þ,
D ¼
eλ
c
þ p
θ sinh ka exp Àkx
ð
Þ,
P ¼
eλ
c
þ p
θ sinh ka 1 À
cε 0
e 2 þ cε
exp Àkx
ð
Þ,
ρ
P
¼ k
eλ
c
þ p
θ sinh ka 1 À
cε 0
e 2 þ cε
exp Àkx
ð
Þ,
ð7:50Þ
Δn ¼
k
q
eλ
c
þ p
θ sinh ka exp Àkx
ð
Þ,
ρ
P
À qΔn ¼ Àk
eλ
c
þ p
θ sinh ka
cε 0
e 2 þ cε
exp Àkx
ð
Þ:
ð7:51Þ
For a numerical example, consider a ZnO rod with 2a ¼ 1.2 μm, n 0 ¼ 10
21 /m
3 , and
θ ¼ 0.1 K. As indicated by the above equations, a local temperature change produces
many electromechanical fields. The situation is similar to those produced by local
stresses. Our main interest is the local potential barrier and well shown in Fig. 7.5a
which is caused by the thermally induced local polarization in (b). The polarization
is under the screening effect of the electrons. As a linear solution, the potential
barrier and well exhibit an antisymmetry about the origin.
The above potential barrier and well predicted by a linear analysis neglect the
inherent nonlinearity in the drift currents. As a comparison, for the same rod with a
finite length of 2L ¼ 8.4 μm, we perform a nonlinear analysis using COMSOL with
larger values of θ. The results are shown in Fig. 7.6. In (a), when θ ¼ 1 K, the
nonlinear result looks qualitatively similar to the linear result. However, when
θ ¼ 5 K, fundamentally different from the linear result, the potential barrier height
and well depth become clearly different, losing the antisymmetry of the linear
solution about the origin. This is typical for a nonlinear solution. If θ is increased
further, (b) shows that the well becomes very deep and the barrier is barely visible.
Next we consider the case when the two ends of a finite rod with length 2L is
under an applied voltage 2V with the following boundary conditions:
φ ÀL
ð Þ ¼ V, φ L
ð Þ ¼ ÀV:
ð7:52Þ
Our main interest is the current-voltage relation or I-V curve. We use the nonlinear
version of the constitutive relations for currents in Eq. (7.32) and perform a numerical analysis using COMSOL. Specifically, consider a rod with 2L ¼ 8.4 μm,
2a ¼ 1.2 μm, and n 0 ¼ 10
21 /m
3 . The I-V curves of interest are shown in Fig. 7.7a
for different θ, where J
n
¼ I. For a specific curve corresponding to a given θ, when
the applied voltage is low, no currents can flow through the rod in either direction.
When the applied voltage is high and above a critical value that is temperature
186
7 Thermal Effects
eλ þ pc
ð
Þ θ
k e 2 þ cε
ð
Þ
sinh ka exp Àkx
ð
Þ,
E ¼
eλ þ pc
ð
Þ θ
e 2 þ cε
sinh ka exp Àkx
ð
Þ,
D ¼
eλ
c
þ p
θ sinh ka exp Àkx
ð
Þ,
P ¼
eλ
c
þ p
θ sinh ka 1 À
cε 0
e 2 þ cε
exp Àkx
ð
Þ,
ρ
P
¼ k
eλ
c
þ p
θ sinh ka 1 À
cε 0
e 2 þ cε
exp Àkx
ð
Þ,
ð7:50Þ
Δn ¼
k
q
eλ
c
þ p
θ sinh ka exp Àkx
ð
Þ,
ρ
P
À qΔn ¼ Àk
eλ
c
þ p
θ sinh ka
cε 0
e 2 þ cε
exp Àkx
ð
Þ:
ð7:51Þ
For a numerical example, consider a ZnO rod with 2a ¼ 1.2 μm, n 0 ¼ 10
21 /m
3 , and
θ ¼ 0.1 K. As indicated by the above equations, a local temperature change produces
many electromechanical fields. The situation is similar to those produced by local
stresses. Our main interest is the local potential barrier and well shown in Fig. 7.5a
which is caused by the thermally induced local polarization in (b). The polarization
is under the screening effect of the electrons. As a linear solution, the potential
barrier and well exhibit an antisymmetry about the origin.
The above potential barrier and well predicted by a linear analysis neglect the
inherent nonlinearity in the drift currents. As a comparison, for the same rod with a
finite length of 2L ¼ 8.4 μm, we perform a nonlinear analysis using COMSOL with
larger values of θ. The results are shown in Fig. 7.6. In (a), when θ ¼ 1 K, the
nonlinear result looks qualitatively similar to the linear result. However, when
θ ¼ 5 K, fundamentally different from the linear result, the potential barrier height
and well depth become clearly different, losing the antisymmetry of the linear
solution about the origin. This is typical for a nonlinear solution. If θ is increased
further, (b) shows that the well becomes very deep and the barrier is barely visible.
Next we consider the case when the two ends of a finite rod with length 2L is
under an applied voltage 2V with the following boundary conditions:
φ ÀL
ð Þ ¼ V, φ L
ð Þ ¼ ÀV:
ð7:52Þ
Our main interest is the current-voltage relation or I-V curve. We use the nonlinear
version of the constitutive relations for currents in Eq. (7.32) and perform a numerical analysis using COMSOL. Specifically, consider a rod with 2L ¼ 8.4 μm,
2a ¼ 1.2 μm, and n 0 ¼ 10
21 /m
3 . The I-V curves of interest are shown in Fig. 7.7a
for different θ, where J
n
¼ I. For a specific curve corresponding to a given θ, when
the applied voltage is low, no currents can flow through the rod in either direction.
When the applied voltage is high and above a critical value that is temperature
186
7 Thermal Effects