Journal of Applied Mathematics and Physics, 2014, 2, 41-46
Published Online January 2014 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2014.21007
OPEN ACCESS JAMP
Electronic Modeling and Optical Properties of
CuIn0.5Ga0.5Se2 Thin Film Solar Cell
Rongzhen Chen1*, Clas Persson1,2
1Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm, Sweden
2Department of Physics, University of Oslo, Oslo, Norway
Received October 2013
ABSTRACT
In this work, the band structure and optical-related properties of CuIn0.5Ga0.5Se2 thin film are presented. The
calculation is performed by the full-potential linearized augmented plane wave (FPLAPW) method. The spin-
orbit coupling is considered. The result for the dielectric function is in good agreement with earlier experimental
measurements and simulations. Based on the complex dielectric function, the dielectric constant, the absorption
coefficient, the complex refractive index and the reflectivity at normal incidence are explored. We found that
they are comparable with the earlier results.
KEYWORDS
Thin Film; CuIn0.5Ga0.5Se2; Band Structure; Dielectric Function; Dielectric Constant; Absorption Coefficient;
Complex Refractive Index; Reflectivity; Spin-Orbit Coupling
1. Introduction
The chalcopyrite CuIn1-xGaxSe2 (CIGS) alloy is one of the most promising thin film absorber materials in thin
film photovoltaic technology [1,2]. Currently, the best efficiency of solar cells based on CIGS contains around
30% Ga. However, optimum Ga content is theoretically 50% ~ 60%. Therefore, in this work, we will study band
structure and optical-related properties of CuIn0.5Ga0.5Se2 (50% Ga).
In this work, the band structure is calculated without and with the spin-orbit coupling (NonSOC and SOC),
respectively. The SOC is important for the calculation of band structure. It affects the curvature of energy band
strongly, especially for the valence bands (VBs) near the Γ point. The complex dielectric function is obtained
based on the calculation of band structure. Therefore, the other optical properties are possible to be investigated
using the complex dielectric function, such as the dielectric constant, the absorption coefficient and many others.
The result based on our calculation method is reported few, although it is investigated by some experimental
measurements and simulations [3-7].
2. Theoretical Details
2.1. Compuational Details
The program package Wien2k [8] is utilized to perform all the calculations. It based on the full-potential linea-
rized augmented plane wave (FPLAPW) plus local orbitals method. The relativistic effects and SOC are consi-
dered in our calculation. The generalized gradient approximation (GGA) plus an onsite Coulomb interaction U
of the Cu d states is treated as exchange correlation potential, which will improve the energy gap. The k mesh is
8 × 8 × 8 to sample the Brillouin zone (BZ). The CuIn0.5 Ga0.5Se2 has chalcopyrite structure with tetrahedral
bonding character. Eight atoms are contained in primitive cell.
2.2. Optical Properties
The calculation of optical properties does not go beyond the Kohn-Sham eigenstates in Wien2k. The complex
*Corresponding author.

R. Z. CHEN, C. PERSSON
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42
dielectric function can be written:
12
() ()()i εωε ωεω = +
, (1)
where
1()
εω
is the real part and
2
() εω
is the imaginary part. Both of them are tensor. The imaginary part in
atomic units is calculated as follows [9]:
; ,',; ,',
2,'
0 ,0',',,
1
Im( )π
(()()) (),
ijinn jnn
nn
nn nn
pp
ff
εω
εε δεεω
=
− −−
∑∫
kκ
k
kk kk
(2)
where
; ,',inn k
p
is the ith momentum matrix element for the band index
n
and
'n
with crystal momentum k.
The real part of dielectric function is derived by the imaginary part tensor:
22
0
Re( )',
Im(( '))
2'
π(' )
ij
d
ij ij
εδ ω
εω
ωωω
∞
= +
−
∫
P
(3)
where P is the principal value.
()
ii
n ω
(refractive index) and
()
ii
k ω
(extinction coefficient) are the real and
imaginary part of refractive index, respectively. They can be expressed by the complex dielectric function:
(){||Re()} 2,
iiii ii
n ωε ε = +
(){||Re()} 2.
iiiiii
k ωε ε = −
(4)
The absorption coefficient is obtained by:
2() c.
ii
Ak ωω =
(5)
The reflectivity at normal incidence can be calculated by this complex refractive index.
22 22
(){(1) }{(1) }
iiiiii iiii
Rn kn k ω = −+++
. (6)
The loss function can be calculated by:
() Im(1()),
ij ij
L ω εω = −
(7)
and the real part of the optical conductivity is:
Re(())(4)Im(( ).
ij ij
σω ωεω = π
(8)
3. Results and Discussion
3.1. Band Structure
The electronic band structure of CuIn0.5Ga0.5Se2 is presented without and with spin-orbit coupling in Figure 1. It
is plotted in the energy range from −3 eV to −4 eV (upper panel), which is the visible sunlight spectrum. It de-
monstrates that it has direct band gap and the band gap for NonSOC case is 0.81 eV, and it is around 0.75 eV for
the SOC case. However, the plot has been corrected the band gap to 1.33 eV.
Figure 1 demonstrates that the energy band dispersion is strongly affected by SOC, especially for the valence
bands (VBs) near the Γ point (lower panel), where the SOC split is ~0.2 eV. However, the conduction bands are
only affected slightly by the SOC. Therefore, it is important to consider the SOC when calculating energy band
dispersion for semiconductor, because it will directly impact on the calculation of effective mass and carrier
concentration. It is consistent with earlier results [10,11].
3.2. Optical Properties
The average of complex dielectric function is presented in Figure 2. The band gap is corrected to 1.33 eV for
the calculations of optical properties. The main shape of this complex dielectric function is similar for both of
NonSOC and SOC cases. However, the main peak position is slightly different. It is around 2.81 eV and 2.85 eV
for NonSOC and SOC cases, respectively. The average high frequency dielectric constant is
Re( (0)) εω =
=
5.74 for the NonSOC case, and it is around 5.70 for the SOC case. It has good agreement with earlier experi-
mental measurements and simulations results [12-15].
In Figure 3, the tensor of this complex dielectric function is demonstrated. There is a difference in the main

R. Z. CHEN, C. PERSSON
OPEN ACCESS JAMP
43
Figure 1. The electronic energy band structure of CuIn0.5Ga0.5Se2 along the four symmetry directions (100), (100),
(100) and (100) without and with spin-orbit coupling. The notation v1, v2, v3 and c1 refer to a spin-independent
band index. The energies are referred to the VBM. The lower panel is the close-up near the Γ point VBM.
Figure 2. The average of complex of dielectric function is presented (left and right panel). Re(ε) is the real part of
the dielectric function, and Im( ε) is the imaginary part of dielectric function.
and overall shape between the SOC and NonSOC calculations. However, in the lower panel of Figure 3, one
can observe that there are more peaks for the SOC case than for the NonSOC case around the main peak, it
comes from SOC split from band-to-band transitions (v3 → c1, v2 → c1 and v1 → c1) at the Γ point [12] in
Figure 1. Therefore, it is important to consider SOC in the more accurate calculation of the optical respond.
The absorption is shown in Figure 4. One can see that CuIn0.5G0.5Se2 has a broader range of absorption. The
main visible sunlight spectrum could be possible to be absorbed. This result is similar with the earlier measure-
ments [13]. The difference between NonSOC and SOC calculations is subtle. However, the SOC shifts the first
peak around 50 meV. This shift is due to the SOC at the Γ point (see Figure 1). It shows that the absorption in-

R. Z. CHEN, C. PERSSON
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44
Figure 3. The tensor of complex of the dielectric function is presented (left and right panel). The average of the
polarization along the x and y directions is
⊥
ε
and parallel with the z direction is

ε
. The main peak is presented
in the lower panel.
Figure 4. The average absorption coefficient is presented (solid and dashed line), the result is presented in the
range of visible sunlight spectrum.

R. Z. CHEN, C. PERSSON
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45
creases rapidly from the energy range 1.33 eV to 2.80 eV, and it is stable up to 4.60 eV, then it increases rapidly
again.
Figure 5 demonstrates that the average of real (left panel
() (2 ()())/3nn wnw ω
⊥
=+ 
) and imaginary (right
panel
() (2 ()())/3k kk ω ωω
⊥
= +

) part of refraction index. The
( 0)n ω =
is about 2.40 and 2.39 for the Non-
SOC and SOC calculations, respectively. It is in good agreement with other researchers results [14-16]. The
SOC only slightly affects the complex refraction index, which shifts the result little. The peak is shifted by 30
meV for refractive index, and 70 meV for the extinction coefficient.
The optical reflectivity for NonSOC and SOC calculations is shown in Figure 6. The measurement is rare for
the compound CuIn0.5Ga0.5Se2. However, the CuInSe2 is calculated as well using the same method, which has
good agreement with experimental measurements [17]. Therefore, our calculation result is reliable in that sense.
The
( 0)R ω =
is 0.17 for both of NonSOC and SOC calculations. The difference between NonSOC and SOC
is small. However, SOC shifts the result about 70 meV.
4. Conclusion
In this work, the band structure and many optical properties of CuIn0.5Ga0.5Se2 are calculated without and with
considering the spin -orbit coupling. The software program Wien2k is utilized. The band structure is influenced
Figure 5. The average real and imaginary part of refractive index (left panel and right panel) is shown.
Figure 6. The average reflectivity is shown in the energy range of visible sunlight spectrum.

R. Z. CHEN, C. PERSSON
OPEN ACCESS JAMP
46
strongly by the SOC, especially the VBs near the Γ point. The main shape of optical properties such as the di-
electric function, the absorption coefficient and others show the similarity in both NonSOC and SOC calcula-
tions. However, the SOC causes more peaks in the dielectric function, and the shape of most optical properties is
shifted slightly by the effect of SOC. Our result is in good agreement with results from earlier experiments and
simulations.
Acknowledgements
This work was supported by the China Scholarship Council, the Swedish Energy Agency and the Swedish Re-
search Council. We acknowledge access to HPC resources at NSC and HPC2N through SNIC/SNAC and Matter
network.
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