PEŁNY TEKST/FULL TEXT

JUSTYNA GARUS, SEBASTIAN GARUS, KONRAD GRUSZKA, KATARZYNA BŁOCH,
MARCIN NABIAŁEK
The creation and shifts of band gaps
in binary superlattices
INTRODUCTION
Superlattices are intensively tested materials [1÷24], the filtering
capabilities are particularly attractive [21]. Multilayers are
characterized by the presence of the photonic band gap, so that
electromagnetic waves (EMW) at the specific frequencies does not
propagate them.
Using emulation properties of multilayer systems allows pretesting of the structure and the design to get the specific
characteristics of the electromagnetic wave transmission bands.
This reduces the number of samples performed and reduces the
cost of examinations.
Transmission superlattices are calculated using the matrix
method [18÷21] and by the method of finite-difference in the time
domain for one-dimensional structures (1D FDTD) [22, 25].
while for S type of polarization they take the form:
2n j cos j
t j , j 1 
n j cos j  n j 1 cos j 1
(4)
rj , j 1 
1D FDTD
Using Maxwell's laws, we can determine the equations [23, 24]:


D
  H
t
 
D
   0 r  E

1
H

 E
0
t
MATRIX METHOD
The transmission of the multilayer structure is calculated from the
following equation:
2
n cos out 1
T  out
(1)
nin cosin  11
where: nin, nout – respectively the refractive index of the
electromagnetic wave falling on the multilayer structure and which
goes after passing, in – angle of falling of the EMW for
superlattice,  out – the angle at which EMW leaves multilayers,
 11 – the first word of the characteristic matrix diagonal 
superlattice described the relation:
rin , j 1 
1  1

 
r
1 
tin , j 1  in , j 1
  id j n j 2 cos j
 (2)


rj , j 1  
0
 J e
 1  1
  


2
1  
 id j n j
cos  j t j , j 1  r j , j 1
 j 1 


0
e

 

where: nj – refractive index of the layer j, dj – thickness of the
layer j,  j – determined from Snell's law angle of falling of EMW
for the layer j,  – the wavelength of falling wave. Depending on
the type of polarization t and r determine the amplitude Fresnel
ratios respectively for transmittance and reflectance [20]. For the
polarization P are defined as:
2n j cos j
t j , j 1 
n j cos j 1  n j 1 cos j
,
rj , j 1 
(3)
n j cos j 1  n j 1 cos j
n j cos j 1  n j 1 cos j
Mgr inż. Justyna Garus, mgr inż. Sebastian Garus ([email protected]), dr inż.
Konrad Gruszka, dr Katarzyna Błoch, dr hab. Marcin Nabiałek prof. PCz. –
Instytute of Physics, Technical University in Częstochowa
n j cos j  n j 1 cos j 1
n j cos j  n j 1 cos j 1
(5)
which involve together the electric field E, magnetic field H and
the electric induction vector D.  0 ,  0 are electric and magnetic
permeability in a vacuum, and  r is a relative permittivity of the
medium.
In order to simplify calculation of the electric field and the
electric induction vector are normalized according to the relation:
~
E
~
0
1
 E, D 
D
0
 0  0
(6)
Another transformation allows to obtain necessary for
analyzing the propagation of EMW the system of equations in the
formalism of the FDTD method:
n
n 
~ n 1 ~ n 1
x 
1
1 Hy
1
Dx
Hy

2  Dx
2
k
k



 0   0 t 
k
k
2
2

1
1
1
~ n
~ n
1
n 
Ex
  Dx
 Ix
2
2
2
  t 
k
k
k 
r  r

0
1
1
 r  t ~ n  1
n
n
I
Ix


 Ex
x
2
2
2
0
k
k
k

1 ~
1
n 1
n
x  ~ n 
1
n 
1  Hy
1
Hy
Ex

 Ex
2
2
k
k
 0   0 t 
k
k 1 
2
2

(7)
where n is a step in the one-dimensional k-space, I is a auxiliary
matrix, and Δx, Δt describe respectively discretization of
coordinates of the position and time, and are related for the
stability of the simulation by Courant condition [25]
t 
x
N *c0
(8)
where N is the dimension of the simulation, and c0 is the EMW
velocity of the material. Simulation used a soft source of EMW,
the area has been limited by absorbing boundary conditions (ABC),
and a wavelength characteristic was prepared using FFT [25].
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RESEARCH
In this study the behaviour of the electromagnetic wave passing
through the superlattice ABABAB binary structure composed of
lossless materials whose properties do not depend on the
frequency of the electromagnetic wave, using the matrix method
and the FDTD algorithm. Electromagnetic wave fell perpendicular
to the sample surface by what the type of polarization does not
affect the properties of transmission. Binary superlattice symmetry
makes the structure ABABAB transmission is the same as
transmission of the structure BABABA. Due to the use multilayers
as a filters of visible light the calculation performed for the
wavelength range of 300 to 750 nm. The refractive indexes
materials of the layers are nA = 3, nB = 2. And a value of the
thickness varies in the range from 100 to 200 nm.
Influence of the amount of clusters AB
in the transmission properties
Fig. 2. Superposed transmission graph from Figure 1
Rys. 2. Superpozycja transmisji z rysunku 1
Figure 1 shows how the transmission properties of the multilayer
changes with increasing number of pairs of layers AB. Thicknesses of the individual layers are dA = dB = 200 nm what gives
a sample thickness increase 400 nm with each increase of the
parameter nkl determining the number of repetitions of cluster AB.
The structures in Figures 1 and 2 was surrounded by air so that the
refractive index before nin and after nout the structure was equal
to 1.
In Figures 1a and 1b, respectively, for samples with the
structures of AB and ABAB it can be seen changes in transmission
as a function of the size of the wavelength of light, but has not yet
outlines the structure of band gaps. By increasing the number of
consecutive pairs of layers (Fig. 1c÷j) can be seen at wavelengths
near the 340, 500 and 670 nm clearly the formation of photonic
band gaps.
Figure 2 was created by the imposition of all the graphs in
Figure 1. It can be seen that increasing the number of layers
influences on the formation and stabilization of the band gaps for
given, specific to the structure, electromagnetic wavelengths.
Influence of the environment on the superlattice
transmission
The graphs in Figure 3 describe the EMW transmission in wavelength function for the parameters of binary superlattice: nA = 3,
nB = 2, dA = dB = 200 nm, nkl = 10. The following graphs show how
it changes the structure of the transmission according to the
refractive index of the medium in which is placed the multilayer.
The environment has no effect on the structure of band gaps,
but influences of the nature of the interband (Fig. 3).
It should be noted that the most interesting filtration properties
of superlattice have refraction indexes close to the value of
refractive index of the environment where the fluctuations in the
transmission are smaller interband.
Fig. 3. Surrouding material impact on the superlattice transmission
Rys. 3. Wpływ otoczenia materiału na własności transmisyjne supersieci
Influence of layer thickness on the shift
and broadening of band gaps
Figure 4 shows the influence of increasing the thickness of the
cluster on the occurrence of band gaps shift towards higher
wavelengths. Shift was observed for the band gap in Figure 4a
near a wavelength of 340 nm, and the results are summarized in
Figure 5, which shows the linear nature of the change. In Figure 6
can be seen how to change the width of the control band gap as
a function of the thickness of a single layer.
Fig. 1. Figure shows transmission T for binary superlattices depending of the lengthwave λ. Nkl parameter specifies the number of times the
cluster AB. The thickness of a single layer is 200 nm, the refractive index of the material A is 3 and material B is 2
Rys. 1. Wykres przedstawia transmisję T dla supersieci binarnej w zależności od długości fali padającej λ. Parametr Nkl określa liczba powtórzeń
klastera AB. Grubość pojedynczej warstwy wynosi 200 nm, współczynnik załamania materiału A wynosi 3, a materiału B wynosi 2
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Fig. 5. Influence of layer thickness on the 340 nm band gap shift. The
value of λ determines place of occurrence of band gap for the layer
thickness d
Rys. 5. Wpływ grubości warstwy na przesunięcie pasma 340 nm. Wartość
λ determinuje miejsce występowania pasma wzbronionego dla grubości
warstwy d
Fig. 4. Influence of layer thickness on the band gaps shift; nin = nout = 1.
Values of dA and dB are in nm
Rys. 4. Wpływ grubości warstwy na przesunięcie pasm wzbronionych;
nin = nout = 1. Wartości grubości dA i dB są w nm
Fig. 6. The value of x is the width of the band gap for the occurrence
of the layer thickness d
Rys. 6. Wartość x to szerokość przerwy wzbronionej w zależności od
grubości warstwy d
Analysis of the behaviour of electromagnetic waves
in binary superlattice using the FDTD algorithm
In order to examine the behavior of a monochromatic electromagnetic wave having a wavelength equal to the wavelength of
the band gap analysis was performed using the FDTD algorithm,
during which were determined by FFT (Fast Fourier Transform)
wavelength characteristics. Layer thicknesses were equal to
200 nm, and 5 nm was the resolution of simulation (test area
divided into 1300 equal parts). The time step was 8,3391  1018 s
and was selected to fulfill the Courant stability condition. The
refractive indexes of the layers were respectively nA = 3, nB = 2.
The calculations were conducted for 8000 time steps when the
outgoing electromagnetic wave structure has been stabilized [22].
EMW penetrating the superlattice structure, for each boundary
layer is partially reflected and partially passes. Another part of the
electromagnetic wave reflected inside the structure interfere with
each other, which involves a change in the output wavelength and
causes the formation of superlattice band gaps.
Figures 7 and 8 illustrate the characteristics of the wavelength
of the monochromatic electromagnetic wave with wavelengths of
500 and 337 nm corresponding to photonic band gaps. Interference
within the material changes the wavelength distribution and
creates peaks around the gap area.
Figure 9 shows the behaviour of the electromagnetic wave
when it is illuminated by a wave length corresponding to full
transmittance. It is important that at each of the figures from 7 to 9
we can see, all the band gaps.
Fig. 7. The wavelengths characteristic in the FDTD simulation of the
structure illumination of monochromatic light with a wavelength
500 nm
Rys. 7. Charakterystyka długości fali w symulacji FDTD dla struktury
oświetlonej światłem monochromatycznym o długości fali 500 nm
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The use of the FDTD method has allowed to illustrate the
behaviour of electromagnetic waves in areas of high and low
transmission.
REFERENCES
Fig. 8. The wavelengths characteristic in the FDTD simulation of the
structure illumination of monochromatic light with a wavelength 337 nm
Rys. 8. Charakterystyka długości fali w symulacji FDTD dla struktury
oświetlonej światłem monochromatycznym o długości fali 337 nm
Fig. 9. The wavelengths characteristic in the FDTD simulation of the
structure illumination of monochromatic light with a wavelength 640 nm
Rys. 9. Charakterystyka długości fali w symulacji FDTD dla struktury
oświetlonej światłem monochromatycznym o długości fali 640 nm
CONCLUSIONS
Based on the study it can be concluded that the matrix method and
FDTD very well complement the study of photonic properties of
multilayer structures. It has been found that the presence of the
photonic band gap depends strongly on the number of superlattice
layers.
The environment of the multilayer structure does not affect the
existence of band gaps, but it strongly influences the interband
areas.Increasing the thickness of the superlattice layers shifts in
a linear photonic band gaps toward the higher wavelength and the
influences of their size.
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