5.3 Interfacial Dielectric Constant ε
117
P x
E x
r
E z
P z
r
(c)
(d)
P x
E x
E z
P z
(a)
(b)
r
r
ext
ext
ext
ext
Fig. 5.1 Lorentz models in the bulk (a, b) and at the interface (c, d). (a, c) External electric field
E ext
x parallel to the interface, (b, d) E ext
z perpendicular to the interface
We notice that ε corresponds to the ratio of the local field factors, ε = f x /f z .
The dielectric constant acts as a screening factor of the electric field normal to a
surface by induced surface charges. In a similar manner, Zhuang et al. [16] applied
the Lorentz model to describe the dielectric property at the interface, and proposed
ε
=
ε(ε + 5)
2(2ε + 1)
.
(5.33)
[Problem 5.1] Let us derive ε in Eq. (5.33) on the basis of the Lorentz model
illustrated in Fig. 5.1a–d.
(a, b) Bulk First, suppose a spherical cavity of radius r embedded in the bulk of a
slab with a dielectric constant ε, and put the system in an electric field E ext
x or E ext
z .
Show that the local fields inside the cavity in the two cases are
E
loc
x = f x E
ext
x =
ε + 2
3
E
ext
x
and E
loc
z = f z E
ext
z =
ε + 2
3ε
E
ext
z .
(5.32)