Theoretical Studies of Plasmonic Effects in Organic Solar Cells - Organic Solar Cells: Materials and Device Physics - page 180

consisting of two or more separated regions. It is based on expanding the fields and
matching them at the boundaries of different regions, and thus lends itself naturally
to the analysis of multilayer optical devices. The most representative of mode-
matching methods involves rigorous coupled-wave analysis [ 66 ], scattering-matrix
method [ 67 ], and plane wave expansion method [ 68 ]. Using cheap computer
resources, these methods are specially useful in characterizing the optical response
of periodic OSC devices. The zero-order and high-order transmittance and
reflectance can be easily obtained with a program. However, it is not well suited
for characterizing plasmonic effects, because a large quantity of modes are
required to describe the plasmon coupling and hybridization. For mode-matching
methods, computing the optical absorption of active materials for the metal-
incorporated OSCs is nontrivial. The trivial absorbance cannot truly represents the
optical absorption of OSCs, and electric fields should be postprocessed for
excluding the metallic absorption. Additionally, cascading many layer media with
repeated matrix multiplications will cause the stability and accuracy issues, which
always happens in OSC modeling with the curved geometries.

7.3.2 Finite-Difference Frequency-Domain Method

Considering a 2-dimensional OSC structure, the Maxwell's equations can be
decoupled into a TE and TM modes. The wave equations for TE and TM modes
are respectively formulated as [ 59 ]

1
e r

o
ox

1
l r

o E z
ox

þ 1
e r

o
oy

1
l r

o E z
oy

þ k 0 E z ¼ 0 ð 7 : 1 Þ

1
l r

o
ox

1
e r

o H z
ox

þ 1
l r

o
oy

1
e r

o H z
oy

þ k 0 H z ¼ 0 ð 7 : 2 Þ

where k 0 is the wave number of incident light, and e r and l r are the relative per-
mittivities and permeabilities, respectively. Regarding nonmagnetic optical mate-
rials, l r ¼ 1, e r ¼ n c , and n c is the complex refractive index of optical materials.
With the Yee lattice [ 69 ], the 2D finite-difference frequency-domain (FDFD)
method is utilized to characterize the optical properties of OSCs. As shown in
Fig. 7.1 , the five-point stencil is adopted for the FDFD method. The discretized
forms for the TE and TM wave equations are respectively of the form

! U 0

e k 0 U 0 U 1 þ U 3

U 2 þ U 4
eD y

1
D x

þ 1
D y

2 ¼ 0 ; U ¼ E z ð 7 : 3 Þ

eD x