Progress In Electromagnetics Research B, Vol. 11, 265–279, 2009
DESIGN AND MODELING FOR ENHANCEMENT OF
LIGHT EXTRACTION IN LIGHT-EMITTING DIODES
WITH ARCHIMEDEAN LATTICE PHOTONIC
CRYSTALS
J.-Y. Chen
Center for General Education
Hsing Kuo University of Management
No. 600, Sec. 3, Taijing Blvd., Annan District
Tainan 709, Taiwan (R.O.C.)
Y.-G. Li
Institute of Manufacturing Engineering
National Cheng Kung University
No. 1, University Road, Tainan City 701, Taiwan (R.O.C.)
J.-Y. Yeh
Department of Management Information Science
Chung Hwa University of Medical Technology
No. 89, Wen-Hwa 1st ST. Jen-Te Hsiang
Tainan Hsien 717, Taiwan (R.O.C.)
L.-W. Chen
Department of Mechanical Engineering
National Cheng Kung University
No. 1, University Road, Tainan City 701, Taiwan (R.O.C.)
C.-C. Wang
Institute of Manufacturing Engineering
National Cheng Kung University
No. 1, University Road, Tainan City 701, Taiwan (R.O.C.)
266 Chen et al.
Abstract—Light extraction efficiency of light-emitting diodes (LEDs)
based on various photonic crystal (PhC) structures is investigated in
this study. By using the plane wave method and the finite element
method, the influence of several factors on the enhancement of light
extraction is discussed, including lattice type, the density of states
from a photonic band diagram, the ratio of cylinder radius and lattice
constant, and the thickness of a PHC pattern. Some rules are given
for the practical implementation of an optimized PhC-based LED
with higher light extraction efficiency. In the simulation results, the
maximum enhancement of the extraction efficiency could be by a factor
of approximately 2.8 for the optimized Archimedean tiling pattern in
the LED device emitting at the center wavelength of 530 nm. However,
with the use of optimized PhC structures, the square and triangular
lattices reveal enhancements of ∼ 1.7 and ∼ 1.5, respectively. The
extraction efficiency of the Archimedean PhC LED is much greater
than that of the regular lattice PhC LED.
1. INTRODUCTION
Photonic crystals (PhCs) allow one to enhance, attenuate, or suppress
spontaneous emission properties of materials placed inside them [1].
They can control the spatial distribution of radiation power and
redistribute the spectrum of the emitted light into useful forms.
The ability to manipulate light emission has potentials on many
optoelectronics devices [2].
Light-emitting diodes (LEDs), which spontaneously emit radiation
from a p-n junction, have attracted much interest for a wide range of
applications in solid-state lighting, displays, optical communications,
and optical interconnects in computers. The light outcoupling
efficiency of LEDs has important consequences on such applications.
Unfortunately, most of the light emitted from conventional surface-
emitting LEDs is mainly limited by the total internal reflection within
the high dielectric material, leading to a poor external efficiency. The
need for the improvement of light extraction efficiency is exceptionally
significant. Much effort has been exerted to overcome this limit,
including surface texturing, the use of microlenses, reshaping the light
escape cone, distributed feedback structures, and the diffraction by a
two-dimensional (2D) PhC [3–6]. In particular, high light-extraction
efficiency is expected for the integration of 2D PhCs, which allows the
control of photonic behavior in a predictable manner.
PhCs have a periodic dielectric modulation with a spatial scale
on the order of the optical wavelength. Design and optimization of
Progress In Electromagnetics Research B, Vol. 11, 2009 267
the PhCs are not simple due to the multi-parameter problem of a
patterned texture. There are many factors related to the combinations
of intermixing materials, lattice symmetry, lattice constant, filling
factor, shape of the scattering object, and thickness of a PhC layer. It
is not easy to say which one provides the most efficient PhC structure in
the improvement of light extraction efficiency in LEDs. There are two
main options to improve light extraction by means of PhCs: the first
one is the use of the photonic band-gaps (PBGs) to enhance emission in
useful directions and to inhibit it in others; the other is to redirect the
emission from guided modes into radiative modes [7]. Nevertheless, the
first effect generally leads to nonradiative losses and the requirement of
a sufficiently large refractive index contrast to open a full band-gap [8].
Hence, the second option is considered here for the advantage of being
compatible with present material processes and amenable to treatment
in real components.
The introduction of periodic patterning within the LED device
causes the dispersion curves of Bloch modes to become folded at
the Brillouin zone boundary. Thus, some of the wave-guided modes
(trapped modes lying below the light line) will be shifted to the
diffracted modes lying above the light line and escape from the
device [2]. Photonic quasi-crystals (PhQs) are similar to PhCs but rely
on a quasi-crystal arrangement of scattering objects [5, 9–11]. PhQs
possess high symmetry orders not achievable in nature and offer many
desirable features, such as isotropic PBGs, operation in low refractive
index materials, diffraction not arising from the nearest neighbor
dielectric rod interactions, and flat dispersion bands. However, it
is difficult to numerically establish the dispersion properties of most
PhQs. Archimedean lattices are a category of PhQs, consisting of
regular convex polygons which are not necessarily identical and can fill
the whole plane without gaps [11]. There are 11 Archimedean tilings,
including the familiar traditional Bravais lattices: square, triangular,
and honeycomb structures. The benefits of Archimedean lattices are
a higher order of local rotational symmetry than the regular lattices
and the strict periodicity which allows the calculation of the dispersion
relations by numerical methods. Besides, the isotropic behaviors can
be transposed to light-diffracting PhQs, and the lattice constants can
reach the wavelength of visible light.
In this work, we present a different approach to the optimization
of light extraction efficiency in the LEDs through PhCs (PhQs). A
kind of simple 2D PhC structure constructed with a small portion of
PhQs is proposed for the quasi omnidirectionality of light extraction.
In addition, this numerical study is targeted to examine the light
extraction efficiency of the LEDs based on different kinds of basic PhC
268 Chen et al.
structures and the proposed PhQ one, and to find the most effective
design under optimized parameters for further discussion. Taking
a typical organic LED device as an illustration, the introduction of
PhCs (PhQs) to the improvement of extraction efficiency is discussed
in detail. The relevance between the predictions of the plane wave
method (PWM) [1] and the consequences of the three dimensional
finite element method (3D FEM) calculation is demonstrated for highly
efficient light-extracting structures. The simulation results reveal that
the PhQ-based LEDs show higher light output than the traditional
PhC-based ones.
2. SIMULATION MODEL AND METHOD
The theoretical study of photonic structures is based on the three
geometrical configurations shown in Fig. 1, i.e., square, triangular,
and Archimedean (4, 8
2
) arrays of circular cylinders. The notation
to categorize the Archimedean lattices is a set of shape and number
of polygons (n
a
1
1
,n
a
2
2
,n
a
3
3
,...), denoting a tiling of a vertex type in
the way that n
1
-gon, n
2
-gon, and n
3
-gon, ..., meet clockwise on each
vertex, and the superscript a
i
refers to the number of these polygons
adjacent to each other [12]. The symbol (4, 8
2
) means a tiling in which
a square and two octangles gather edge-to-edge around a vertex. The
(4, 8
2
) lattice has been also called the “bathroom tile” lattice, and its
photonic structure is obtained by the placement of dielectric cylinders
at the vertices. Four parameters are involved in the simulation: lattice
constant a, emitting center wavelength λ, thickness d of a pattern
array, and ratio (f = r/a) of radius and lattice constant, where r is
the radius of the circular cylinders.
As can be seen from Fig. 1(c), such a periodic structure can be
also considered as clusters of 4 atoms organized in a square Bravais
lattice. This indicates that Wigner cells can be defined and all
the theoretical and numerical methods can be applied to model the
multiple wave scattering. Photonic band structures can be calculated
by defining periodically supercells. The primitive cell is formed
through two primitive translation vectors a
1
=(1+1/
√
2)a(1, 1) and
a
2
= (1+1/
√
2)a(−1, 1). The positions of the cylinders in the unit
supercell with respect to the coordinate origin are u
1
, u
2
, u
3
and
u
4
with u
1
= a(1/2, (1/2+1/
√
2)), u
2
= a(1/2, −(1/2+1/
√
2)),
u
3
= a(−1/2, −(1/2+1/
√
2)), and u
4
= a(−1/2, (1/2+1/
√
2)),
respectively. For any integers l
1,2
, R = l
1
a
1
+ l
2
a
2
defines the Bravais
lattice with the periodic dielectric constants ε(r + R)=ε(r), where r
is the position vector. A standard PWM can be used to calculate the
band diagrams.
Progress In Electromagnetics Research B, Vol. 11, 2009 269
There are several positions in which a PhC (PhQ) can be inserted
in to an LED device [4]. For practical applications, the introduction of
a PhC (PhQ) in the anode layer should be the most effective, as it does
not lead to a modification in the electronic properties of the device.
Besides, the light emitted from the LED device can be classified into the
following modes: the radiation mode, the surface plasmon mode, the
waveguide mode, and the substrate mode [4, 13]. From the theoretical
(a) (b)
(c)
Figure 1. Various PhC/PhQ lattice tiling schemes and their
corresponding unit cells in reciprocal space: (a) square lattice; (b)
triangular lattice; (c) (4, 8
2
) Archimedean lattice. These photonic
structures with lattice constant a are composed of cylinders with a
refractive index of n
H
=1.95 and a radius of r = fa. The substrate
refractive index is n
L
=1.48. The primitive cell in (c) including four
cylinders is defined by two primitive translation vectors a
1
and a
2
.
Four cylinders surrounding the coordinate origin are at positions u
1
,
u
2
, u
3
, and u
4
, respectively.