local and near x ¼ Æ1 μm they have already stabilized. Near the rod ends at far away
with x ¼ ÆL, there exist field concentrations which cannot be seen in the figure.
When θ ¼ 0 K, the curves are either symmetric or antisymmetric about the origin.
When θ is nonzero, the curves lose their symmetry or antisymmetry. Specifically,
(a) and (b) show the usual built-in electric potential and field. When plotting the
electric potential, x ¼ À1 μm is the reference where the potential vanishes, i.e.,
a ¼ À1 μm in Eq. (7.58). It can be seen that although the built-in voltage determined
by the potential difference at x ¼ Æ1 μm in (a) is essentially unaffected by the
temperature change, the slope of the potential field or the built-in electric field in
(b) is. When θ ¼ 0.5 K, the potential in (a) is continuous but not smooth at the origin.
This is related to that the corresponding electric field in (b) in fact has a jump
discontinuity at the origin which has been connected by a vertical line in the figure.
The total charge ρ
t in (c) is directly responsible for the behavior of the electric field in
(b).
The difference between the homogenous and heterogeneous junctions is caused
by the reversal of the c-axis in the left half of the rod, which leads to the fundamentally different polarizations shown in Fig. 7.10a, b for the homogeneous and
heterogeneous junctions, respectively. Away from the junction and on its both
sides, the P in (a) is the same but that in (b) has opposite signs. The different
behaviors of P in (a) and (b) have implications on the effective polarization charge
density ρ
P . For the homogeneous junction in (a), the corresponding ρ
P is essentially
antisymmetric without any net charge and shows no visible temperature effect.
Therefore it is not shown. For the heterogeneous junction in (b), the corresponding
ρ
P is shown in (c) which produces some net positive charge near the junction. This
affects the total charge density in Fig. 7.9c.
In addition to the volume effective polarization charge density described by ρ
P ,
there also exists surface effective polarization charge density σ
P on the interface of
the PN junction due to the discontinuity of P there. For our junction interface, σ
P can
be calculated from
σ
P
¼ P
À
À P
þ ,
P
À
¼ P 0
À
ð Þ, P
þ
¼ P 0
þ
ð Þ:
ð7:70Þ
We list σ
P for the same three temperature changes in Table 7.1. For a homogeneous
junction or when θ ¼ 0 K, we have σ
P ¼ 0. For a heterogeneous junction, σ
P
increases with θ. σ
P is positive and contributes further to the net positive charge at
the junction produced by ρ
P in Fig. 7.10c.
We now consider the case when the two ends of the rod at x ¼ ÆL are under an
applied voltage 2V with the following boundary conditions:
φ ÀL
ð Þ ¼ V, φ L
ð Þ ¼ ÀV:
ð7:71Þ
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
7.3 Temperature Effects on PN Junctions
193
with x ¼ ÆL, there exist field concentrations which cannot be seen in the figure.
When θ ¼ 0 K, the curves are either symmetric or antisymmetric about the origin.
When θ is nonzero, the curves lose their symmetry or antisymmetry. Specifically,
(a) and (b) show the usual built-in electric potential and field. When plotting the
electric potential, x ¼ À1 μm is the reference where the potential vanishes, i.e.,
a ¼ À1 μm in Eq. (7.58). It can be seen that although the built-in voltage determined
by the potential difference at x ¼ Æ1 μm in (a) is essentially unaffected by the
temperature change, the slope of the potential field or the built-in electric field in
(b) is. When θ ¼ 0.5 K, the potential in (a) is continuous but not smooth at the origin.
This is related to that the corresponding electric field in (b) in fact has a jump
discontinuity at the origin which has been connected by a vertical line in the figure.
The total charge ρ
t in (c) is directly responsible for the behavior of the electric field in
(b).
The difference between the homogenous and heterogeneous junctions is caused
by the reversal of the c-axis in the left half of the rod, which leads to the fundamentally different polarizations shown in Fig. 7.10a, b for the homogeneous and
heterogeneous junctions, respectively. Away from the junction and on its both
sides, the P in (a) is the same but that in (b) has opposite signs. The different
behaviors of P in (a) and (b) have implications on the effective polarization charge
density ρ
P . For the homogeneous junction in (a), the corresponding ρ
P is essentially
antisymmetric without any net charge and shows no visible temperature effect.
Therefore it is not shown. For the heterogeneous junction in (b), the corresponding
ρ
P is shown in (c) which produces some net positive charge near the junction. This
affects the total charge density in Fig. 7.9c.
In addition to the volume effective polarization charge density described by ρ
P ,
there also exists surface effective polarization charge density σ
P on the interface of
the PN junction due to the discontinuity of P there. For our junction interface, σ
P can
be calculated from
σ
P
¼ P
À
À P
þ ,
P
À
¼ P 0
À
ð Þ, P
þ
¼ P 0
þ
ð Þ:
ð7:70Þ
We list σ
P for the same three temperature changes in Table 7.1. For a homogeneous
junction or when θ ¼ 0 K, we have σ
P ¼ 0. For a heterogeneous junction, σ
P
increases with θ. σ
P is positive and contributes further to the net positive charge at
the junction produced by ρ
P in Fig. 7.10c.
We now consider the case when the two ends of the rod at x ¼ ÆL are under an
applied voltage 2V with the following boundary conditions:
φ ÀL
ð Þ ¼ V, φ L
ð Þ ¼ ÀV:
ð7:71Þ
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
7.3 Temperature Effects on PN Junctions
193