Abstract. In this paper we propose a designs of solar cells based on III-V semiconductors which can deliver more efficient, less expensive solar cells. Recently proposed approaches to enhance the efficiency utilize the novel III-V nanomaterials containing quantum dots, and new concepts like intermediate bandgap quantum dot based solar cells promise the efficiency over 50

Alexander I Fedoseyev

Frantis ∨ Ek C ∨ Ajko

CiteSeerX

 
 
European Association for the 
Development of Renewable Energies, 
Environment and Power Quality (EA4EPQ) 
 
 
International Conference on Renewable Energies and Power Quality 
(ICREPQ’11) 
Las Palmas de Gran Canaria (Spain), 13th to 15th April, 2011 
 
 
 
Photoabsorption Efficiency Improvement for Photovoltaic Solar Cells by Using the 
Honeycomb Nanostructures 
 
Alexander I. Fedoseyev1  and Frantis∨ek C∨ajko2,  
 
1 CFD Research Corporation,  
215 Wynn Dr., Huntsville, Alabama 35805 (USA) 
Phone/Fax number:+1 256 7264928, e-mail: aif@cfdrc.com 
 
2 FMRI Lab., University of Michigan,  
Ann Arbor, Michigan 48109 (USA) 
 
 
 
Abstract. In this paper we propose a designs of solar cells 
based on III-V semiconductors which can deliver more efficient, 
less expensive solar cells. Recently proposed approaches to 
enhance the efficiency utilize the novel III-V nanomaterials 
containing quantum dots, and new concepts like intermediate 
bandgap quantum dot based solar cells promise the efficiency 
over 50% [1]. Needs for thin flexible space solar cells are 
supposed to be fulfilled with Inverted Metamorphic 
Multijunction (IMM) solar [2]. In this paper we consider the 
alternative design of flexible solar cells, which is based on a 
honeycomb nanostructure consisting of embedded semiconductor 
pillars in a flexible film [3]. Such nanostructures are flexible and 
solar cells therefore are suitable for wider range of applications 
compared to ones with solid panel design, and are less expensive 
due to the smaller fraction of the active semiconductor material. 
However this kind of design may reduce the efficiency of the 
solar cell due to the volume filling factor.. 
 
Key words 
 
Nanostructured, photovoltaic cell, flexible, 
photoabsorption.. 
 
1. Introduction 
 
In this paper we propose a designs of solar cells based on 
III-V semiconductors which can deliver more efficient, 
less expensive solar cells. Recently proposed approaches 
to enhance the efficiency utilize the novel III-V 
nanomaterials containing quantum dots, and new concepts 
like intermediate bandgap quantum dot based solar cells 
promise the efficiency over 50% [1]. Needs for thin 
flexible space solar cells are supposed to be fulfilled with 
Inverted Metamorphic Multijunction (IMM) solar [2]. In 
this paper we consider the alternative design of flexible 
solar cells which is based on a honeycomb nanostructure 
consisting of embedded semiconductor pillars in a flexible 
film [3]. Such nanostructures are flexible and solar cells 
therefore are suitable for wider range of applications 
compared to ones with solid panel design, and are less 
expensive due to the smaller fraction of the active 
semiconductor material. However this kind of design 
may reduce the efficiency of the solar cell due to the 
volume filling factor. 
 
 2. Proposed Approach 
 
We intend to show that the proposed nanostructured solar 
cell have the same efficiency as monolithic cells, while 
contain the smaller fraction of the active semiconductor 
material. 
 
We present the models and simulation results that show 
that the filling factor does not limit the efficiency of such 
solar cell, in contradiction with the common sense 
approach, is not valid for such nanostructures. We 
provide the detailed analysis of such multiscale models 
(nanostructure feature size is much less than light 
wavelength), show that the absorption efficiency is more 
than 90%, and that such nanostructures also work as 
concentrators, comparable to Goetzberg et al[4]. 
There are several parameters that describe the efficiency 
of a solar cell. Usually a recombination rate and a carrier 
transport have to be calculated in order to obtain a power 
efficiency of the device or external quantum efficiency. 
We are primarily interested in the absorption of photons 
and showing that the efficiency of photoabsorption is not 
necessarily proportional to the volume filling factor. We 
will compare a quantum efficiency of the bulk 
semiconductor to the efficiency of composite structure. 
       
Honeycomb photovoltaic nanostructures, consisting of 
embedded semiconductor rods in a flexible film, are 
proposed for flexible solar cell design to make the 
devices resistant against thermal and mechanical stresses, 
therefore suitable for wider range of applications 
compared to ones with solid panel design.. However this 
may reduce the efficiency of the solar cell due to the 
https://doi.org/10.24084/repqj09.429 706 RE&PQJ, Vol.1, No.9, May 2011
volume filling factor. Here we present the multiscale 
models for such systems (nanostructure feature size is 
much less than light wavelength), and simulation results 
that show that the filling factor does not limit the 
efficiency. Therefore we show that the common sense 
approach is not valid for such nanostructures. 
An attenuation of a light intensity in the structure is 
characterized by its penetration depth. Its value is in a 
range of values of materials used that span over four orders 
of magnitude. Furthermore, the intensity has terms that 
decays 100-times faster than the main exponential term. 
These fast decaying components create a boundary layer 
with a comparable contribution to energy transmitted into 
the structure. Hence, the traditional methods have 
difficulties due to a multiscale nature of the problem. In 
our approach we solve for the fields on top boundary and 
then use analytical expressions for the light propagation 
modes. 
 
3. Problem Formulation and Solution Method 
 
Photoabsorption is a process of capturing a photon that 
yields a transition of a valence electron to the conductive 
band. The probability that the captured photon contributes 
an electron-hole pair (e-h) generation rather than an 
increase of heat is called a quantum efficiency η and is 
defined as a ratio of e-h generation rate G to photon 
absorption rate Nph  
 G
N ph
η =  (1) 
A capture of every photon decreases the magnitude of the 
electromagnetic field inside the structure. The depth where 
the intensity of the field drops down to 1/e is called a 
penetration depth δ and its inverse an attenuation constant 
α. Both quantities describe how fast the field in the 
structure is attenuated. 
 
The quantum efficiency as defined above requires 
switching from a classical description of an 
electromagnetic field to a quantum one. By introducing a 
photoabsorbed power PG that contributes the e-h 
generation and the total absorbed power P0 we can rewrite 
(1) into form  
 h= 
PG
P0
 (2) 
 
If P0 is an incident power rather than an absorbed power 
and a portion of the energy of light escapes from the 
structure, then (1) and (2) are not equivalent but the 
efficiency η represents the effectiveness of the light 
capture in the honeycomb structure. Therefore the second 
definition will be preferred. 
 
A design of a solar cell allows a current and therefore an 
equilibrium can be developed. However, the generated DC 
current doesn’t interact with the light and is not considered 
in our analysis. All relevant charges and currents at the 
considered frequency are implicitly included in the 
complex permittivity ε of the material. Assuming the 
time factor e-iωt, the electromagnetic fields E, H are 
described by Maxwell equations  
 
 0iωµ∇× =E H  (3) 
 iωε∇× = −H E  (4) 
 
where µ0 is a permeability of vacuum, ω is an angular 
frequency of the light and i is the imaginary unit. An 
equivalent way of describing the material properties is to 
use a reflection coefficient n and an extinction coefficient 
k. Then, the absorption coefficient can be calculated from 
the extinction coefficient k and the wavelength in vacuum 
λ0=2πc/ω  
 α= 
4πk
λ0
 (5) 
The absorption constant α corresponds to the total 
absorbed power and the density of the absorbed power 
can be calculated according to  
 
 2
0
1 | |
2ab
np E
Z
α= ⋅  (6) 
 
where the intrinsic impedance of vacuum Z0=377 Ω and 
E is magnitude of the electric field at given point in 
space. 
 
As a good performance was expected in the 
homogenization limit, the geometry was simplified to the 
extent possible by considering only a two dimensional 
problem. This reduces complexity of the calculation and 
let us concentrate on the mechanism of absorption in the 
structure to demonstrate that absorbed energy is not 
simply proportional to a filling factor of GaAs in 
alumina.  
We consider a periodic structure consisting of vertical 
slabs of GaAs and Al2O3 depicted in Fig. 1. The slabs are 
oriented in y-z plane and periodically repeat along the x-
direction. A volume filling factor of GaAs is   
 
 FF=b/d  (7) 
 
This structure fills out the lower semi-space. There is a 
single-layer antireflection coating to maximize the energy 
flow into the structure. The incident plane wave of given 
free space wavelength λ0 and irradiance I0 impinges 
perpendicularly from the top. 
 
A.  Maxwell equations governing the problem 
The problem is two dimensional because neither the 
material properties nor the driving field depends on the y-
coordinate. The solution can be decomposed into two 
https://doi.org/10.24084/repqj09.429 707 RE&PQJ, Vol.1, No.9, May 2011
polarizations that don’t interact with each other: E-mode 
and H-mode  
 
Figure 1:  Periodic structure of alumina (thickness a) and GaAs 
(thickness b) slabs with a period d and an antireflection coating 
of height h on top of it. An incident light impinges from the top. 
 
 mod ˆE e yyE− =E  (8)  
 mod ˆ ˆE e x zxH zH− = +H  (9) 
 mod ˆ ˆH e x zxE zE− = +E  (10) 
 mod ˆH e yyH− =H  (11) 
 
where x, y and z are unit vectors of the canonical base. 
After substituting into Maxwell equations (3,4), the 
equations for the Ey and Hy components can be derived  
 
 2 2 0( , ) ( ) ( , ) 0y yE x z x E x zω µ ε∇ + =  (12)  
 2 0
1 ( , ) ( , ) 0
( ) y y
H x z H x z
x
ω µ
ε
∇ ∇ + =i  (13)  
 
The general solution in terms of variable u (Ey in the E-
mode and Hy in the H-mode) has form of a periodic 
function uper(x)  with the period d modified by an 
exponential factor  
 
 u(x,z)=uper(x,z)e
ikBx  (14) 
 
To obtain a solution one usually chooses one period in the 
x-direction as a computational domain (the z-coordinate 
bounds depend on the problem). So called Bloch boundary 
conditions flow directly from (14)  
 
 u(x0,z)=u(x0+d,z)e
ikBx  (15) 
 
where the Bloch wave number kB is to be found. Since in 
our case a normal incidence is considered, there is no 
preferred direction in x-y plane due to the translational 
symmetry. Therefore Bloch boundary conditions can be 
reduced to periodic conditions on the y-z sides of the 
computational domain  
 
 u(x0,z)=u(x0+d,z)  (16) 
B.  Plane Wave Expansion 
The plane wave expansion (PWE) method is applied to 
solve the problem, i.e. the general solution (14) for kB=0 
is expressed as a superposition of plane waves  
 
For our analysis we chose GaAs as a prospective 
candidate for a photoabsorbing material with a cut-off 
wavelength of 871 nm. Its internal quantum efficiency 
was estimated to be 99%. The word internal reflects the 
fact that recombination was not taken into account and 
needs to be included during an optimization of the 
structure height. Alumina (Al2O3) was chosen for its 
good mechanical properties and a low heat generation.  
 
However, it is not a photovoltaic material and thus 
degrades the performance of the structure. The results 
below give a quantitative estimate of this degradation. 
Properties of both materials at 827 nm were estimated 
from various sources and are summarized in Table (not 
shown). 
2.  Antireflection layer 
Because of the field matching procedure described above 
and uniqueness of the expansion there will be a plane 
wave component in the reflected field. For the cell size d 
sufficiently smaller than the wavelength λ0 one can 
expect that the plane wave component will be prevailing. 
As the εr in the air is 1 the plane wave component decays 
very slowly and carries energy away. Therefore its 
reflection has to be eliminated by an antireflection layer.  
 
The antireflection layer (Fig. 1) was calculated for the 
plane wave component and its properties were 
determined to be h=λ0/4n, air effn n n= ⋅   with 
1airn =  and effn  being the real part of eigenvalue β for 
the longest propagating mode. 
 
The higher harmonics have different equivalent refraction 
index β, so they will be reflected even if the plane wave 
component is totally eliminated. Fortunately, by a good 
choice of the cell size d it is possible to make the 
refraction index imaginary, meaning that the higher 
harmonics will exponentially decay with a distance from 
the reflecting surface. Then, the energy conservation law 
implies that these modes cannot carry energy away 
https://doi.org/10.24084/repqj09.429 708 RE&PQJ, Vol.1, No.9, May 2011
because there is no absorption. In order to make β 
imaginary one needs to request  
 
 εr- ⎝⎜
⎛
⎠⎟
⎞ md/λ0
2
<0 (17) 
 
for every value of m except m=0. Hence for air (εr=1), the 
cell size must be smaller than wavelength  
 
 d<λ0 (18) 
 
The smaller the cell size d is, the more confined to the 
surface the modes are. 
 
In our simulations the observed total reflected power was 
less than 2% for cell sizes less than 0.3 λ0 for both 
polarizations and all filling factors and it was decreasing 
with the cell size. These results include the contribution of 
higher harmonics.  
 
The related eigenproblems were solved by a MatlabTM 
eigenvalue solver. The value β was calculated as a square 
root of the eigenvalue with a sign chosen so that the 
imaginary part is positive. A typical Fourier spectrum is 
depicted in Fig. 2. 
 
  
Figure 2:  Fourier spatial harmonics of modes in the periodic 
structure excited by a plane wave incident in the normal 
direction. Normalized Fourier component amplitude (y-axis) vs. 
the component number m (x-axis). 
3.  Discussion of Results  
A convergence of the method was checked for d/λ0=0.12 
and filling factors 0.1 and 0.5. It was found that small and 
large filling factors demand more harmonics to be used. 
128 harmonics expansion was found to be satisfactory to 
achieve about 1% accuracy. 
 
A penetration of the light into the material is given by the 
most slowly decaying mode. Figure 3 shows a 
homogenization limit for cell sizes d<0.1λ0, i.e. the 
penetration is independent on the cell size and varies only 
with the filling factor FF. The variation is mild except for 
FF<0.05 where it abruptly increases (Fig.3).  
 
  
Figure 3:  Top: Dependence of the normalized penetration 
depth δ/λ0 of 827nm light on a cell size d of the periodic 
structure for three filling factors FF. Theoretical penetrations 
for pure GaAs and pure alumina are shown. E-polarization. 
Bottom: Penetration vs. filling factor in the homogenization 
limit for cell size d=50nm=0.06λ0. Comparison between E- and 
H- modes. 
 
The opposite limit (the cell size d is several times larger 
than λ0 – not shown) exhibits very large penetration, i.e. 
a slow absorption. The longer penetration requires a 
thicker active area to collect the solar power. That in turn 
means increased recombination, and hence, a lost of 
efficiency. Fortunately, for cell sizes d comparable to the 
wavelength λ0 is possible to achieve a penetration depth 
of several wavelengths. The power efficiency η defined 
in (2) is almost not affected (Fig. 4), meaning that 99% of 
the absorbed power contributes e-h generation. However, 
the current response will be smaller because of increased 
recombination. 
https://doi.org/10.24084/repqj09.429 709 RE&PQJ, Vol.1, No.9, May 2011
  
  
Figure 4:  Power efficiency of photoabsorption. An unpolarized 
incident light is assumed. 
When the height of the periodic structure is not big 
enough, the light won’t be absorbed before reaching the 
boundary and a part of the energy escapes from the 
structure without being absorbed. To minimize an energy 
loss and an excessive recombination the height of the 
active area must be optimized. In the homogenization limit 
(for d<0.1λ0) there is only one dominant mode. The 
second longest mode has about 100 times shorter 
penetration depth. This allows to use the dominant 
exponential decay with an amplitude properly calculated 
from the boundary condition evewhere but in the vicinity 
of the boundary. 
4.  Conclusion 
A numerical analysis of a periodical structure consisting of 
alumina and GaAs slabs driven by an infrared light 
propagating along the slabs was presented. A significant 
portion of the work deals with eigenmodes of this 
structure. It was argued that the analysis of fields can be 
performed for two independent polarizations thoughout 
this paper referred to as E- and H-modes. Since the most 
practical case is when the spatial period d of the structure, 
referred to as a cell size, is smaller than the wavelength an 
efficiency of photoabsorption was calculated in this region. 
Equal contributions of both polarizations in the incident 
light was assumed. 
 
A photoabsorption efficiency was calculated as a ratio of 
photoabsorbed power to the power entering the periodic 
structure. It was shown that the resulting efficiency is 
nearly the one of GaAs. From the energy flux into GaAs 
was concluded that a diffraction is an important factor 
increasing the amount of absorbed power at the expense 
of larger penetration depth. 
 
The absorption constant of the structure is important for 
calculation of the height of the structure. Although, the 
light intensity doesn’t decay always exponentially (e.g. 
when two excited modes have similar attenuation 
constants), it was observed that the mode with the largest 
attenuation constant is dominant because of its large 
plane wave component. This allows at least a quantitative 
estimate of the optimal height of the structure (too high 
structures has a higher recombination rate so finding a 
trade-off is necessary). 
 
There are some results that can be extended for GaAs 
rods or bars buried in alumina. The most important is that 
when the cross sectional size of alumina is less than or 
comparable to a wavelength in alumina, the energy from 
alumina is diffracted into GaAs, and thus, substantially 
increasing the efficiency of photoabsorption. As a side 
effect the light attenuation constant is close to the one of 
GaAs. 
 
References 
 
[1] Shao Q., A. Balandin, A. Fedoseyev, M. Turowski, 
Intermediate-band solar cells based on quantum dot 
supracrystals", App. Phys. Let. 91, No.16, 2007.  
[2] "http://www.solartoday-digital.org/ solartoday/200903/? 
pg=26#pg26"  
[3] Fan Z., H. Razavi, J. Do, A. Moriwaki et al.,"Three-
dimensional nanopillar-array photovoltaics on low- cost 
and flexible substrates", Nature Materials 8, 648, 2009.  
[4] Goetzberger A.,J.C.Goldschmidt,M.Peters, P.Loper, " 
Light trapping, a new approach to spectrum splitting", 
Solar EnergyMaterials & Solar Cells 92, 1570–1578, 2008. 
https://doi.org/10.24084/repqj09.429 710 RE&PQJ, Vol.1, No.9, May 2011


Photoabsorption Efficiency Improvement for Photovoltaic Solar Cells by Using the Honeycomb Nanostructures

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Photoabsorption Efficiency Improvement for Photovoltaic Solar Cells by Using the Honeycomb Nanostructures

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