Rapid thermal evaporation for cadmium selenide thin-film solar cells

Kanghua LI, Xuetian LIN, Boxiang SONG, Rokas KONDROTAS, Chong WANG, Yue LU, Xuke YANG, Chao CHEN, Jiang TANG

Front. Optoelectron. ›› 2021, Vol. 14 ›› Issue (4) : 482-490.

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Front. Optoelectron. ›› 2021, Vol. 14 ›› Issue (4) : 482-490. DOI: 10.1007/s12200-021-1217-1
RESEARCH ARTICLE
RESEARCH ARTICLE

Rapid thermal evaporation for cadmium selenide thin-film solar cells

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Abstract

Cadmium selenide (CdSe) belongs to the binary II-VI group semiconductor with a direct bandgap of ~1.7 eV. The suitable bandgap, high stability, and low manufacturing cost make CdSe an extraordinary candidate as the top cell material in silicon-based tandem solar cells. However, only a few studies have focused on CdSe thin-film solar cells in the past decades. With the advantages of a high deposition rate (~2 µm/min) and high uniformity, rapid thermal evaporation (RTE) was used to maximize the use efficiency of CdSe source material. A stable and pure hexagonal phase CdSe thin film with a large grain size was achieved. The CdSe film demonstrated a 1.72 eV bandgap, narrow photoluminescence peak, and fast photoresponse. With the optimal device structure and film thickness, we finally achieved a preliminary efficiency of 1.88% for CdSe thin-film solar cells, suggesting the applicability of CdSe thin-film solar cells.

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cadmium selenide (CdSe) / rapid thermal evaporation (RTE) / solar cells / thin film

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Kanghua LI, Xuetian LIN, Boxiang SONG, Rokas KONDROTAS, Chong WANG, Yue LU, Xuke YANG, Chao CHEN, Jiang TANG. Rapid thermal evaporation for cadmium selenide thin-film solar cells. Front. Optoelectron., 2021, 14(4): 482‒490 https://doi.org/10.1007/s12200-021-1217-1

1 Introduction

Recent progress in chiral photonics, i.e., bianisotropic metamaterials [17] and bianisotropic metasurfaces [815], has significantly advanced our understanding of light transport in complex photonic structures and spurred numerous applications. Indeed, bianisotropic metamaterials have been used to realize novel resonators [1,2,6] and construct photonic topological insulators [4,5,7]. Bianisotropic metasurfaces have been used to manipulate the polarization of light [8,9] and to achieve exotic refraction and transmission of light [1014]. In metamaterials or metasurfaces, it is common to use optically complicated structures inside each unit cell together with a very large number of inclusions to obtain certain functionalities. Thus the complete multiple scattering of light among all the inclusions is hardly traceable. Therefore, under proper approximation, the chiral metamaterials [1520] or chiral metasurfaces [21,22] can be described using effective constitutive parameters (permittivity and permeability), which in principle accounts for the bianisotropic response of the complicated unit cell. Notably, the grading effect in a wide range of metasurfaces, e.g., with a rotated orientation of each unit cell structure, can also be included in the spatial dependent bianisotropic constitutive parameters in the effective medium description.
However, the available means of calculating photonic bianisotropic medium is limited within analytical methods, which are usually complicated and only apply to simple structures, i.e., interfaces or slab structures [2326]. A generic numerical approach that calculates the electromagnetic properties for bianisotropic structure with arbitrary shapes is lacking. In this paper, we intend to fill this gap and propose a finite element method (FEM) to simulate the optical properties of bianisotropic media.
The paper is organized as follows: In Section 2, we derive the balanced formulation of the weak form for the finite element implementation. In Sections 3−5, we illustrate and validate our numerical approach via three examples, i.e., bianisotropic slab structure, bianisotropic photonic crystal, and bianisotropic ring resonator respectively. Finally, Section 6 concludes the paper.

2 Model

The constitute relations of the generic bianisotropic materials are given by D=ϵ¯E+χ¯ehH, B=μ¯H+χ¯heE, where D and B are electric displacement vector and the magnetic induction intensity respectively, E and H are electric and magnetic field respectively, ε¯=ε0ε¯r and μ¯=μ0μ¯r are permittivity and permeability, respectively, χ¯eh=ε0μ0χ¯ehr and χ¯he=ε0μ0χ¯her are coupling constants. As for the source-free Maxwell’s equations with time-harmonic dependence of eiωt, i.e., ×E=iωB, ×H=iωD, i2=1, one reformulated Maxwell’s equations using the normalized field, i.e., e(r)=E(r) and h(r)=μ0ε0H(r), as follows
(×+ik0χ¯her)e(r)+ik0μ¯rh(r)=0,(×ik0χ¯ehr)h(r)ik0ε¯re(r)=0.
Eliminating the magnetic field h(r), one arrives at the vector wave equation given as follows
(×ik0χ¯ehr)[1μ¯r(×+ik0χ¯her)]e(r)k02ε¯re(r)=0.
Once the electric e field known, the magnetic field h is given by h(r)=1ik0μ¯r1×e(r)μ¯r1χ¯here(r). As for Eq. (2), the vectorial wave equation is essentially determined by operator L, i.e., L=(×ik0χ¯ehr)[1μ¯r(×+ik0χ¯her)]k02ε¯r. Under scaler inner product, we show that the operator L is indeed self-adjoint, which is described as follows
(F,LE)=(LF,E),
see proof in Supplementary Material A. Importantly, we can further proof that
dV([1μ¯r(×+ik0χ¯her)]F[×ik0χ¯ehr]TEk02ε¯rFE)=(F,LE)=(LF,E)=dV([1μ¯r(×+ik0χ¯her)E][×ik0(χ¯ehr)T]Fk02ε¯rEF),
which is coined as the balanced weak form. This balanced weak form contains a few implications that deserves further discussion: (1) In practical implementation of FEM, the Galakin procedures is usually adopted. In the Galakin procedures, the test function F is by default selected as the basis functions that are used to interpolate the electric field. Thus, it is essential to keep the test function space and the expansion function spaces undergone the same operations, i.e., transformation in exactly the same manner; (2) Applying standard FEM procedure leads to the unbalanced formulation of weak form in bianisotropic media as used in reference [27], which gives rise to artificial numerical errors due to the spatial differentiation of constitutive parameters, i.e., giving rise to the unusual imaginary part of the propagation constant of bianisotropic waveguides; (3) In this paper, the balanced formulation is adopted, it turns to be extremely important in the finite element modeling in bianisotropic medium and overcome the problem in unbalanced formulation from the comparison between the standard weak form and balanced weak form formulation that is not shown in the paper.

3 Reflection and transmission of light in layered bianisotropic medium

First, we benchmark our finite element model of light reflection and transmission against the analytical calculations. We consider a slab geometry in which light is incident in normal direction, as shown in Fig. 1(a). The slab in dark gray surrounded with air in light gray has εr=4, μr=1 and the magnetoelectric coupling constants
χ¯her=(χ¯eh)T=(00000000iχ33).
The slab has a finite thickness L=1m in horizontal direction, and the incident light contains in-plane (p) polarization and out-of-plane (s) polarization. The practical implementation is realized by modifying the original weak form in COMSOL Multiphysics [28] into our balanced weak form (the right term in Eq. (4)) for the bianisotropic medium.
Fig.1 (a) Schematic diagram of simulating reflection and transmission spectrums of bianisotropic slab with normal incident light. A bianisotropic slab (dark gray) is surrounded by air (light gray), the boundaries are vertical with y-axis and light propagates along y direction. Horizontal (p) and vertical (s) polarized light are excited (received) at port 1 (2) and 3 (4) respectively. (b)−(e) Reflection (r) and transmission (t) spectrum of horizontal (p) and vertical (s) polarized light. The results from FEM and semi-analysis method are represented by blue circles and red lines respectively

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The numerical calculation of the χ33 dependent reflection/transmission spectrum at vacuum wavelength λ=1m from our reformulated weak form are perfectly coincides with semi-analysis method (see details in Supplementary Material B) displayed in Figs. 1(b)−1(e). As a side remark, there is no difference between p and s polarization for normal incidence at χ33=0. As for nonzero χ33, the reflection/transmission of p and s polarization differ from each other. This is reasonable since, inside the bianisotropic slab, the p and s polarization are not eigen-polarization and gets mixed due to χ33, and the true eigen-polarization essentially has different effective refractive index. As a result, the incident p and s polarization can be decomposed into the two eigen-polarizations that has different refractive index depending on χ33, which leads to the variation of reflection and transmission against χ33, as confirmed both by our numerical simulation and analytical calculations.

4 Band structure of bianisotropic photonic crystals

Bianisotropic metamaterials and metasurfaces have been used to realize photonic topological insulator or photonic valley Hall effect. Dong and his coworkers [29] explored the photonic valley in honeycomb photonic crystals, in which the inversion symmetry is broken via the material bianisotropy displayed in Fig. 2(a). The unit cell in this honeycomb lattice has two different rods with radii r in purple and blue, respectively, and lattice constant a. These two rods have uniaxial permittivity and permeability εr=μr=diag(ε//,ε//,ε), and nonzero bianisotropic tensor
χ¯ehr=χ¯her=(0iκε//0iκε//00000),
where the parameter κ in rods 1 and 2 have opposite sign. With the modified weak form, we calculated band structure of this photonic lattice in Fig. 2(b), where the dispersion curves for two pseudo-spins (spin-up and spin-down) of light on the high symmetry line of Brillouin zone are shown and have perfect agreement with the data from Ref. [29]. The two spins have almost the same band structure in low frequency, but the spin-up state has a smaller bandgap near K and a larger bandgap around K' than that of the spin-down state. As a result, in the range of frequency from 0.375 to 0.435, the bandgap vanishes at K valley for the spin-up state; the same is true for spin-down state at K' valley. Thus, the two spin states propagate in different directions determined by the associated valley index, as illustrated with two arrows in Fig. 2(a).
Fig.2 (a) Bianisotropic honeycomb photonic crystal. The blue and purple circles represent two kinds of rods, and a unit cell is extracted as the green hexagon with r=0.25a. ε//=8, εprep=1, κ=0.5 in rod 1 but κ=0.5 in rod 2. (b) Band structure on the high symmetry line. Red and blue points represent spin-down and spin-up sampled form literature, and blue circles is the result from COMSOL with modified weak form. The inset shows the Brillouin zone where G1 and G2 are reciprocal lattice vectors. K and K' are two kinds of vertex

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5 Bianisotropic ring resonator

We consider a bianisotropic slab waveguide as shown in Fig. 3(a), which extends infinitely along x and z direction and is sandwiched by air. The bianisotropic slab has a relative permittivity εr=1, relative permeability μr=1, and magnetoelectric coupling tensor
χehr=(χher)T=(00000000iχ33).
Light propagates along x, the effective refractive index of the fundamental mode increases for larger χ33, as can be seen from the simplified wave equation in the presence of χ¯ehr in Supplementary Material C. The fundamental mode is linearly polarized for vanished χ33, and is elliptically polarized for nonzero χ33. By time reversal symmetry, the polarization of light will be converted into its time-reversal partner, i.e., the helicity flipping sign, for opposite propagating direction. As the slab waveguide is rolled to form a ring resonator shown in Fig. 3(b), the x, y and z components of an electric field in an ordinary slab waveguide corresponds to the azimuthal (Eθ), radial (Er) and z components. Therefore, light propagates along the azimuthal direction in the ring resonator is the same as it transports along x in the slab waveguide. Thus, light propagates clockwise or counter-clockwise, the polarization of light will have opposite helicity. Consequently, the ring resonator could function as a polarization depended insulator as shown in Figs. 3(c)−3(f). The clockwise and counter-clockwise modes in the ring resonator have opposite helicity of polarization in electric field basis (Ez1.72iEθ) at vacuum wavelength 1.0615μm. A slab waveguide with the same permittivity and permeability supports transverse electric and transverse magnetic modes, which span the degenerate basis for the polarization. In Figs. 3(c) and 3(d), the polarized optical beam Ez+1.72iEy is excited at the left and the right ports of the slab waveguide respectively, it couples with counter-clockwise mode Ez+1.72iEθ in the ring resonator with positive direction propagation (from left to right) while it propagates through the slab without any coupling with the ring from right to left. In contrast, its time-reversal partner, i.e., Ez1.72iEy, propagates without coupling in a positive direction but couples with clockwise mode Ez1.72iEθ in negative propagation as shown in Figs. 4(e) and 4(f). Therefore, the polarized beams Ez±1.72iEy are insulated in positive and negative directions, respectively. Furthermore, the polarization ellipticity of the modes in the ring can be manipulated by changing χ33, this structure provides a platform to isolate or pick out any ellipse polarized beam in a simple way. Evidently, our numerical results are fully consistent with the earlier polarization-dependent optical insulation based on the bianisotropic ring resonator.
Fig.3 Bianisotropic ring resonator behaves as polarization depended insulator. Bianisotropic slab waveguide (a) and bianisotropic ring resonator formed by rolling slab waveguide (b). Optical beam with polarization Ez+1.72iEy propagates in the slab couples with ring from the left to the right (c) but propagates without coupling from the right to the left (d). Beam with polarization Ez1.72iEy goes through the slab in positive direction (e) but is insulated in reversal direction (f). The width of slab and ring is 0.15μm, the radius of ring is 2.3272μm. εr=μr=1 and χ11=2.7442 in bianisotropic ring, εr=μr=2.366 in slab. Vacuum wavelength is 1.0615μm and background medium is air

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6 Conclusion

We propose a numerical method based on FEM approach to study the optical properties of complex bianisotropic structures. We first benchmark our method in reflection and transmission of a bianisotropic slab, and further validate our numerical calculation of the band structure in a bianisotropic photonic crystal that plays a relevant role in topological photonics. Our results have perfectly coincided either with the analytical results or the literature data. Finally, we illustrate that our method can be used to design a bianisotropic ring resonator, which functions as a polarization-dependent optical insulator.

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61725401, 61904058, and 62050039), the National Key R&D Program of China (No. 2016YFA0204000), the Innovation Fund of WNLO, National Postdoctoral Program for Innovative Talent (No. BX20190127), the Graduates’ Innovation Fund of Huazhong University of Science and Technology (No. 2020yjsCXCY003), and China Postdoctoral Science Foundation Project (Nos. 2019M662623 and 2020M680101). The authors thank the Analytical and Testing Center of HUST and the facility support of the Center for Nanoscale Characterization and Devices (CNCD), WNLO-HUST.

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