Optical design of GaN/InxGa1-xN/cSi tandem solar cells with triangular diffraction grating Leo Jyun-Hong Lin1 and Yih-Peng Chiou1,2,* 1
Graduate Institute of Photonics and Optoelectronics, National Taiwan University, Taipei 10617, Taiwan 2 Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan * [email protected] (Y.-P. Chiou)
Abstract: Optical design in enhancing optical absorption of group-IIInitride- and multiple quantum well-based GaN/InxGa1-xN/cSi dual-junction tandem solar cells with triangular diffraction grating is simulated and optimized by using combined two-dimensional rigorous coupled wave analysis and transfer matrix methods. This paper thoroughly examines these phenomena of optical absorption affected by antireflection coatings, multiple thin-film layers and diffraction gratings with the integrated perspectives of semiconductor physics and electromagnetic theory for the first time. An improvement of 58% in absorption compared to the prototype SC is obtained which means more than 80% of incoming light (hυ > EgSi) can be harvested in this thin-film (< 4 μm in total) design. ©2015 Optical Society of America OCIS codes: (230.1950) Diffraction gratings; (230.5590) Quantum-well, -wire and -dot devices; (310.1210) Antireflection coatings; (310.4165) Multilayer design; (350.6050) Solar energy.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
S. Nakamura, M. Senoh, N. Iwasa, and S. I. Nagahama, “High power InGaN single quantum well structure blue and violet light emitting diodes,” Appl. Phys. Lett. 67(13), 1868–1870 (1995). S. Nakamura, M. Senoh, S. I. Nagahama, N. Iwasa, T. Yamada, T. Matsushita, H. Kiyoku, Y. Sugimoto, T. Kozaki, H. Umemoto, M. Sano, and K. Chocho, “Continuous-wave operation of InGaN/GaN/AlGaN-based laser diodes grown on GaN substrates,” Appl. Phys. Lett. 72(16), 2014–2016 (1998). O. Jani, I. Ferguson, C. Honsberg, and S. Kurtz, “Design and characterization of GaN/InGaN solar cells,” Appl. Phys. Lett. 91(13), 132117 (2007). K. T. Chen, W. C. Huang, T. H. Hsieh, C. H. Hsieh, and C. F. Lin, “InGaN light emitting solar cells with a roughened N-face GaN surface through a laser decomposition process,” Opt. Express 18(22), 23406–23412 (2010). L. J. H. Lin and Y.-P. Chiou, “Improving thin-film crystalline silicon solar cell efficiency with back surface field layer and blaze diffractive grating,” Sol. Energy 86(5), 1485–1490 (2012). S. P. Bremner, M. Y. Levy, and C. B. Honsberg, “Analysis of tandem solar cell efficiencies under AM1.5G spectrum using a rapid flux calculation method,” Prog. Photovolt. Res. Appl. 16(3), 225–233 (2008). L. A. Reichertz, I. Gherasoiu, K. M. Yu, V. M. Kao, W. Walukiewicz, and J. W. Ager III, “Demonstration of a III–Nitride/Silicon Tandem Solar Cell,” Appl. Phys. Express 2(12), 122202 (2009). L. Hsu and W. Walukiewicz, “Modeling of InGaN/Si tandem solar cells,” J. Appl. Phys. 104(2), 024507 (2008). M. Nawaz and A. Ahmad, “A TCAD-based modeling of GaN/InGaN/Si solar cells,” Semicond. Sci. Technol. 27(3), 035019 (2012). C. J. Neufeld, Design, Fabrication and Characterization of III-N Based Solar Cells, Dissertation, UCSB. C. J. Neufeld, S. C. Cruz, R. M. Farrell, M. Iza, J. R. Lang, S. Keller, S. Nakamura, S. P. DenBaars, J. S. Speck, and U. K. Mishra, “Effect of doping and polarization on carrier collection in InGaN quantum well solar cells,” Appl. Phys. Lett. 98(24), 243507 (2011). C. J. Neufeld, N. G. Toledo, S. C. Cruz, M. Iza, S. P. DenBaars, and U. K. Mishra, “High quantum efficiency InGaN/GaN solar cells with 2.95 eV band gap,” Appl. Phys. Lett. 93(14), 143502 (2008). M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71(7), 811–818 (1981). M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986). L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13(5), 1024–1035 (1996). M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovolt. Res. Appl. 3(3), 189–192 (1995).
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A614
17. M. A. Green, Solar Cells: Operating Principles, Technology and System Applications (Prentice-Hall, Englewood Cliffs, New Jersey, 1982). 18. Filmetrics, Inc., “Refractive Index Database,” http://www.filmetrics.com/. 19. M. Polyanskiy, “Refractive Index Database,” http://refractiveindex.info/. 20. M. Levinstein, “Semiconductors on NSM,” http://www.ioffe.ru/SVA/NSM/Semicond/. 21. S. Adachi, The Handbook on Optical Constants of Semiconductors (World Scientific Publishing, 2012). 22. S. Adachi, Optical Constants of Crystalline and Amorphous Semiconductors (Springer, 1999). 23. G. F. Brown, J. W. Ager III, W. Walukiewicz, and J. Wu, “Finite element simulations of compositionally graded InGaN solar cells,” Sol. Energy Mater. Sol. Cells 94(3), 478–483 (2010). 24. M. J. Bergmann and H. C. Casey, “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84(3), 1196–1203 (1998). 25. M. Leung, A. Djurisic, and E. Li, “Refractive index of ingan/gan quantum well,” J. Appl. Phys. 84(11), 6312– 6317 (1998). 26. O. D. Miller, E. Yablonovitch, and S. R. Kurtz, “Intense internal and external fluorescence as solar cells approach the Shockley-Queisser efficiency limit,” IEEE J. Photovoltaics 2, 303–311 (2012). 27. J. L. Gray, A. Luque, and S. Hegedus, Handbook of Photovoltaic Science and Engineering, 2nd ed., (Wiley, West Sussex, UK, 2011). 28. ASTM, “Reference Solar Spectral Irradiance: Air Mass 1.5 Spectra,” http://rredc.nrel.gov/solar/spectra/. 29. S.-Y. Lien, D. Wuu, W. Yeh, and J. Liu, “Tri-layer antireflection coatings (SiO2/SiO2-TiO2/TiO2) for silicon solar cells using a sol-gel technique,” Sol. Energy Mater. Sol. Cells 90(16), 2710–2719 (2006). 30. D. Holec, P. M. F. J. Costa, M. J. Kappers, and C. J. Humphreys, “Critical thickness calculations for InGaN/GaN,” J. Cryst. Growth 303(1), 314–317 (2007).
1. Introduction Ternary compound Indium Gallium Nitride (InGaN) semiconductor material made of a mix of gallium nitride (GaN) and indium nitride (InN) is prevalently used in various optoelectronic devices including high-brightness violet-, blue- and green- light-emitting diodes (LEDs) [1], laser diodes (LDs) [2], solar cells (SCs) [3] and light-emitting solar cells (LESCs) [4]. InxGa1-xN alloys have broad bandgap (Eg) values, which can be engineered to vary In content x, ranging from 0.7 eV (i.e. InN) to 3.42 eV (i.e. GaN), which covers most of the solar spectrum from ultraviolet (UV) to infrared (IR) light. Crystalline silicon (cSi) is an indirect Eg semiconductor, i.e. absorption is phonon-assisted at near- Eg [5], where Eg is 1.12 eV at a cell temperature of 300 K. cSi’s absorption coefficient (α, abbreviated as AC) varies over many orders of magnitude with varying wavelengths (λs). They are suitable for highefficiency dual-junction (DJ) tandem SCs which requires subcell Eg energies of 1.1 eV and 1.7 eV for an ideal two terminal tandem SC under the condition of current-matching [6]. cSi Eg is close to 1.1 eV and we can engineer the In content of InxGa1-xN ternary alloy to be 1.7 eV. The fundamental structure of an InGaN SC is similar to that of an InGaN LED and its single chip InGaN-based SCs can even be operated as LEDs and therefore are called LESCs [4]. Growth of GaN SCs (or InGaN SCs with similar manufacturing process) on n-type cSi(111) substrate [7] needs a thin low-temperature AlN buffer layer (usually 10’s of nm) to accommodate the lattice mismatch between Si and GaN (or InGaN) and the AlN buffer layer serves as p-type doping layer for cSi through Al in-diffusion. Previous analytical calculations have predicted a conversion efficiency of 31% [8] for pn-InGaN/pn-cSi SCs. TCAD semiconductor simulation [9] for a DJ GaN/In0.5Ga0.5N (1 μm)/cSi (200 μm) tandem design predicted a conversion efficiency of 29%. The advantage of this InGaN/cSi tandem structure is that the low resistance tunnel junction [8] at the n+-InxGa1-xN/p+-cSi interface can be formed for certain InxGa1-xN alloy fractions (x ~0.45-0.5) depending upon how AlN buffer layer is incorporated and the actual alignment of n+-InxGa1-xN conduction band and p+-cSi valence band (at an alloy composition of x = 0.46, perfect alignment can be obtained). In c-Si SCs, a majority of the incident light is absorbed in the neutral regions of the device, and minority carriers must diffuse to the depletion region to be separated by the built-in field of the junction. This design relies on long minority carrier diffusion lengths (MCDLs) around 100’s of μm, especially for cSi with relatively low ACs where carriers are generated far from the junction. P-n junction based SC designs are typically implemented in cSi where the absorption length (10’s of μm) is greater than the depletion region width (~1 μm) but less than the MCDL [10]. cSi SC operates as a diffusion-dominant device. On the other hand, in InxGa1-xN SC where the absorption length (300 nm) is comparable to the depletion region
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A615
(400 ~500 nm) and MCDL is short (< 60 nm), a SC design based on a p-i-n junction is preferable. InGaN SC operates as a drift-dominant device. 2. Structure and simulation methods Figure 1 shows the whole tandem SC which consists of a tri-layer anti-reflection coating (ARC) stack, a DJ GaN/InxGa1-xN/cSi tandem SC design with InxGa1-xN (Quantum Well: QW)/GaN (Quantum Barrier: QB) MQW structure to counteract threading dislocations due to the large lattice mismatch, a symmetrically Al nano-structured triangle grating (TG), and a Al back metal reflector. Since the back metal reflector with TG also serves as a back contact, we should limit TG design with the grating angle as small as possible for better electrical contact and as large as possible to increase the optical path length (OPL) of light absorption (a tradeoff). The back surface field (BSF) layer in terms of improving minority carrier collection is implicitly assumed on top of the grating and can be neglected in this optical simulation due to negligible refractive index change induced by heavy doping. The advantage of the BSF is to establish an additional drift field for minority carriers that assists minority carrier collection and reduces the effective back-surface recombination velocity for the minority carriers approaching the back electrode [5]. The MQW InGaN junction with QW/QB pairs up to 100 periods and In fraction up to 0.27 was demonstrated without significant material degradation [10]. We have extended this concept further to 200 periods of QW/QB pair and In fraction up to 0.5 for the ideal DJ tandem SC. Due to the depletion region width comparable to the absorption length, we simulate the MQW InGaN SC with thicknesses less than 500 nm in total. Due to the very low MCDL, we tend to make the p-GaN layer as thin as possible (< 150 nm) in order to reduce the carrier recombination before they can diffuse.
Fig. 1. MQW GaN/InxGa1-xN/cSi dual-junction (DJ) tandem solar cell (SC).
The MQW InGaN top junction is a Ga-polar (0001) oriented design with a p-side-up geometry, the polarization-induced dipole is in the opposite direction of the depletion field [11,12], resulting in a reduced or even reversed electric field in the MQW InGaN region. Instead of promoting charge separation, the electric field in the MQW InGaN layer inhibits carrier collection by forcing electrons (e-) towards the p-GaN side of the junction and holes (h+) towards the n-GaN side of the junction. The polarization induced dipole effectively sets up a potential barrier that carriers must overcome to be collected. Without the large depletion field to separate carriers generated in the MQW InGaN region, carriers would simply recombine and not contribute to the photocurrent. If the doping is sufficient to effectively screen the polarization charges, the electric field in i-region can be reversed back to the correct direction for carrier collection. If the doping is high enough on the n-GaN side to screen the positive polarization sheet charge at that interface and pin the conduction band near the Fermi level, and as the doping on the p-side is increased to screen the negative polarization sheet charge at that interface, the magnitude of the electric field increases from flat band where the conduction band is at the Fermi level on both sides of the i-region to a
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A616
nearly constant electric field-bearing band across the whole depletion region. The effect of the increasing doping density is to screen the polarization sheet charges. As the In fraction is increased, the piezoelectric polarization sheet charges will increase due to increased strain, and the doping density required to screen the polarization charge will need to be increased as well. Note that it is impractical to dope the layers with arbitrarily high concentrations which would instead damage the layers and make them dead layers. Dead layers have an extremely low minority lifetime and thus is prone to recombine most of the short-λ photogenerated electrons (e-) and holes (h+) [5]. As UV photons with increasingly higher energies penetrate less and less deep into the InGaN material, it appears that a thinner MQW InGaN pn-junction located closer to the surface would always be beneficial for carrier collection; thus leading ultimately to a higher quantum efficiency of the InGaN top cell. We have assumed the sufficient doping concentrations for both sides to overcome the polarization sheet charges so that the e- and h+ carriers can be effectively collected by the recovered electric field within the MQW InGaN region without causing the dead layers. In this tandem design, the use of III-nitride on cSi provides a promising low cost and highly efficient SC structure. This stack structure will not only provide enhanced spectral coverage but also introduce a natural tunnel junction between the p-cSi and n-InGaN, thereby eliminating the need for highly doped regions in the conventional DJ SCs [8]. Since InGaN can be grown on cSi [7], realization of highly efficient and cheap tandem cell presents a new possibility for photovoltaics. In this electromagnetic simulation, polarization dependent twodimensional (2D) numerical simulations based on rigorous coupled wave analysis (RCWA: TE and TM modes) [13,14] is implemented for the design and optimization of optical absorption of the SC structure. We use transfer matrix method (TMM) [15] to effectively calculate reflectivity and transmissivity of optical stratified films. We combined these two methods coded in Matlab (Mathworks, Inc.) in our lab and verified with the Finite Element Method (FEM) [5] to calculate the optical absorptance of the whole MQW GaN/InxGa1-xN/cSi DJ tandem SC. In other words, if αλ, Rλ, and Tλ represent the optical absorptance, reflectance, and transmittance, respectively, then the conservation of energy implies that αλ + Rλ + Tλ = 1, that is, the amount of incident energy is equal to the sum of the absorbed, reflected, and transmitted energy. Both TE and TM reflectances, transmittances and absorptances are calculated and averaged as unpolarized light. The solar spectrum spans a broad range of λs and the spectral range of interest is 300 nm – 1200 nm [5]. The refractive index dispersion effect of each layer is taken into consideration. To enable more incident light to enter the SC, the sun’s spectral characteristics should be considered during the design of anti-reflection coatings (ARCs). The weighted average reflectance Rave within the spectrum under consideration can be calculated by the integral of incident photon flux F(λ) multiplying the reflectance of monochromatic light R(λ) normalized by the integral of F(λ) with the assumption of constant internal quantum efficiency (IQE) for simplicity.
Rave =
λ2
λ2
λ1
λ1
F ( λ ) ⋅ R ( λ ) ⋅ IQE ( λ ) d λ λ2
F ( λ ) ⋅ IQE ( λ ) d λ
λ1
=
F (λ )⋅ R (λ ) dλ λ2
.
(1)
F (λ ) dλ
λ1
We adopt a tri-layer ARC for broad spectral range with low reflectance. In order to proceed with the RCWA and TMM simulations, we need to get dispersive refractive complex indices for InGaN, GaN, InN, AlN, cSi, Al, SiO2, TiO2 and Ta2O5 and ITO (current spreading layer or transparent conducting layer which can reduce contact resistance and enhance photogenerated carrier collection). Moreover, the real and imaginary parts of the refraction index of c-Si are obtained from [16]. The imaginary part is obtained from the AC via α = 4πk/λ [17]. We can get λ-dependent complex refractive indices for cSi, Al, SiO2, TiO2 and Ta2O5 from the websites [18–20] and for AlN (buffer layer), GaN and InN from the handbooks [21,22]. But for such dispersion relations describing n(E) and k(E) of the wide Eg III-nitride are usually unavailable, we need an indirect method to generate the λ-dependent #236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A617
complex refractive indices in the RCWA and TMM simulations. First, we adopt the λdependent complex refractive indices for GaN, InN from two handbooks [21,22]. Then we start from AC fitting formula [23] and get the λ-dependent imaginary part k from it and we estimate the λ-dependent real part n by shifting the energy scale of the refractive index of GaN [24]. Quantum-confinement induced refractive index change is ignored due to the lack of experimental data compared to the theoretical calculation [25]. 4-terminal tandem SC which can facilitate LESC module can be formed by increasing the thickness of AlN buffer layer in terms of its insulator-like properties. If we can design a LESC with good light extraction, then we can obtain good light trapping for the solar cell function and good light extraction for the light emitting function in one device. Therefore, an optical design is required that can accommodate these dual functions in one device. To this end, we look to the reciprocity theorem that connects these two optical devices. This idea is profound, and we extend it to the design of InGaN/cSi tandem solar cell in this paper. According to Owen and Eli Yablonovitch [26], a great solar cell also needs to be a great light emitting diode due to their reciprocity relation if high efficiency is expected to be achieved. They have shown that high efficiency solar cell will require complete optical design such that the solar cell achieves optimal light extraction at open-circuit condition and optimal light absorption at short-circuit condition. At open-circuit condition, the carriers have no current loop to go and they build up in density at terminals and, ideally, they recombine radiatively and emit external fluorescence (represented by external fluorescence efficiency ηext which is related to the light extraction) in exact proportion to the incoming sunlight. If the optical design is done poorly, the resulting open-circuit voltage would be greatly reduced (Voc = Voc-ideal – kT/q | ln ηext |) due to very low external fluorescence efficiency ηext. This result is further buttressed by the rigorous thermodynamic theory literature. It is clear that top surface texturing is not helpful in InGaN MQW SC due to lots of defect-induced recombination centers which can annihilate useful electrons and holes. Also any additional non-radiative recombination centers impair the carrier density buildup, limiting both the open-circuit voltage and short-circuit current. Therefore, external fluorescence efficiency depends not only on optical design but also the quality of material. We combine anti-reflection coatings and triangular diffraction grating to achieve better optical design for the purpose of obtaining higher Voc and Jsc in GaN/InxGa1-xN/cSi tandem solar cell. If the MQW InxGa1-xN junction with different In content can be manufactured ideally with less recombination centers, then the open-circuit voltage approaching Voc-ideal lies mainly with optical design. 3. Results and discussion For light incident at the front of the solar cell, the optical generation rate Gj in the absorber layer j which is either InGaN multiple quantum well or silicon p-n junction takes the form [27] Gj =
λ < hc / E g
P (λ ) d λ, 1 − R ( λ ) − T ( λ ) ⋅ Aj ( λ , α ) ⋅ hc / λ
where F ( λ ) =
P (λ ) hc / λ
,
Aj ( λ , α ) = 1 − exp −α j ( λ ) x j exp − α i ( λ ) xi , i< j
(
(2)
)
(3)
where Aj(λ,α) is absorption which is related to the layer structure of the solar cell and P(λ) is the spectral power density (W/m2/μm). The structure of the GaN/InxGa1-xN/cSi tandem solar cell has hundreds of layers in this optical simulation. In the case of 200 pairs of MQW, the total structure can be mounted to more than 400 layers in this TMM/RCWA simulation. Essentially, only photons with energy larger than Eg contribute to the generation rate.
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A618
Therefore, absorbed sunlight in the absorber layers produces charge carriers including electrons and holes. In the dual-junction tandem solar cell, the generation rates for top junction (G1) and bottom junction (G2) can be described for the first order analysis as follows [27]: λ1 P (λ ) G1 = 1 − R ( λ ) ⋅ 1 − exp −α1 ( λ ) x1 ⋅ d λ, hc / λ 0
(
G2 =
)
( ) 1 − R ( λ ) ⋅ (1 − exp −α ( λ ) x ) exp −α ( λ ) x ⋅ hc / λ d λ.
λ2
P λ
2
2
1
1
(4)
(5)
0
While the four-terminal module design is adopted, both generation rates lead to short-circuit currents and can be harvested as current output. While the two-terminal module design is adopted, current-matching condition Jsc1(qG1) = Jsc2(qG2) is required so that harvested charge carriers wouldn’t be wasted. We adopt thin-film structure for both top and bottom junctions in order to maximize the carrier collection. The AM 1.5G solar spectrum (1000 W/m2) is used in our TMM/RCWA simulations which has a maximum possible short-circuit currents (Jsc,ideal) of 45.48 mA/cm2 (> 1.12 eV) in cSi SC [5,28]. The short-circuit current, Jsc, contributed by the subcell is given by λ
J sc =
q 2 λ ⋅ EQE ( λ ) ⋅ P ( λ ) d λ , hc λ1
(6)
where EQE(λ) is the external quantum efficiency of the subcell and P(λ) is the spectral power density (W/m2/μm). The external quantum efficiency takes into account the effects of reflectance Rλ, transmittance Tλ = 0 (due to the Al back metal reflector) and absorption A. EQE ( λ ) = 1 − R ( λ ) − T ( λ ) ⋅ IQE ( A ( λ , α ) , λ ) ,
(7)
where α is the absorption coefficient. The maximum ideal short-circuit current, Jsc,ideal, occurs when EQE(λ) = 100% at λ < λG, where λG is the wavelength corresponding to the subcell material’s bandgap. λ
J sc ,ideal =
q 2 λ ⋅ P ( λ ) d λ. hc λ1
(8)
And Jsc can be written as two terms with the assumption of IQE(A(λ,α),λ) = 100% for simplicity λ
J sc =
q 2 λ ⋅ EQE ( λ ) ⋅ P ( λ ) d λ hc λ1 λ
=
q 2 λ ⋅ 1 − R ( λ ) ⋅ IQE ( A ( λ , α ) , λ ) ⋅ P ( λ ) d λ hc λ1
=
q 2 λ ⋅ 1 − R ( λ ) ⋅ P ( λ ) d λ hc λ1
=
q 2 q 2 P d λ ⋅ λ λ − λ ⋅ R (λ ) ⋅ P (λ ) dλ ( ) hc λ1 hc λ1
λ
λ
(9)
λ
= J sc ,ideal − ΔJ sc , reflection loss .
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A619
In order to achieve lower reflection loss of the SC, and a tri-layer (Ta2O5/SiO2-TiO2/SiO2) ARC is adopted instead of a double-layer (Ta2O5/SiO2) ARC. The refractive index of SiO2TiO2, which can be formed by, for example, sol–gel method [29], is between the Ta2O5’s and SiO2’s. Therefore, we propose this tri-layer ARC to reduce the reflection loss. The minimum weighted average reflectance Rave = 1.6% with a tri-layer AR stack is achieved. The thicknesses of Ta2O5, SiO2, and SiO2-TiO2 are 53.5 nm, 92 nm and 51 nm, respectively (Figs. 2(a) and (b)). Almost further 20% reduction of reflection loss is achieved with the tri-layer ARC compared to the double-layer Ta2O5/SiO2 ARC as shown in Fig. 2(c).
Fig. 2. (a) Reflection loss of a tri-layer ARC. (b) Contour plot of the weighted average reflectance Rave of a tri-layer ARC (c) Comparison of reflectance between a tri-layer ARC and a double-layer ARC. (d) Optical absorptance of MQW GaN/InxGa1-xN/cSi DJ tandem SC without diffraction grating.
The advantage of using a tri-layer ARC is to further reduce the reflection loss in the optical spectrum ranging between 300 nm and 400 nm as shown in Fig. 2(c). 60% further reduction of the reflection loss in the UV spectrum is obtained. A tri-layer ARC can achieve broader spectrum than a double-layer ARC for much lower reflectance in the UV range. The critical thickness of an InxGa1-xN layer grown on GaN decreases significantly as the In content x of the layer is increased. The theoretical critical thickness of an InGaN layer coherently grown to GaN calculated by Sisher et al. [30] is smaller than 9 nm with the In content x larger than 0.3. If the energy balance model is calculated, then the critical thickness is much smaller. It is less than 4 nm with In content larger than 0.2. As for the DJ SC, the In content of the top junction is around 0.5, only QW thicknesses smaller than 4 nm are considered in our simulation. Figure 2(d) shows the Fabry-Perot (FP) modulation effect on the absorption curves. Incorporation of the MQW structure as the active region increases the optical absorption from 350 nm to 700 nm because of the modifications in both the amplitude and phase of the optical wave modes in the active region. A MQW structure sandwiched between air-ARCs-GaN and GaN-Si interfaces operates as a better absorber of light. FP
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A620
oscillations according to thickness and refractive index of InGaN/GaN film stack are obviously observed in Fig. 2(d). After adding tunneling junction, cSi junction and Al reflector, we can see greater absorption enhancement in the optical spectrum ranging from 400 to 700 nm whose absorptances reach above 0.9. The Al reflector reflects incoming light back to the InxGa1-xN MQW active region for 2nd absorption and the OPL of absorption in the InGaN doubles (> 400 nm). According to the Beer-Lambert law, 98% of the incident light can be absorbed. This explains why the absorption of this visible spectrum (from 400 to 700 nm) is very high. The thicker Si junction can slightly enhance further absorption in the visible spectrum (from 500 to 700 nm). The spectrum from 700 to 1100 nm has poor absorption mainly contributed by the bottom Si junction.
Fig. 3. (a) TM 1st order DE of TG. (b) Maximum short-circuit current minus reflection loss of MQW GaN/InxGa1-xN/cSi DJ tandem SC with diffraction grating. (c) Unpolarized optical absorptance.
Metal blaze gratings are proved to be an ideal reflector, light would be reflected completely and scattered in all directions to increase its OPL. The goal is to enlarge the angles of scattered lights in the diffractive orders 1 and −1 [5]. The diffractive orders are defined by the grating equation:
nout sin (θ m ) + ninc sin (θ inc ) =
mλ , d
(10)
where ninc and nout are the refractive indices of material on the incoming and outgoing sides of the grating, respectively, m is the diffractive order, θinc is the incident angle, d is the grating groove spacing, λ is the wavelength in vacuum, andθm is the diffractive angle of the mth diffractive order. We adopt the symmetrically TGs to enhance the OPL. We design TG with large 1st-order diffraction efficiency (DE) and large diffracted angle which can enhance the
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A621
OPL of absorption. In the TMM/RCWA simulation, DER,m,n and DET,m,n are diffraction efficiencies for reflective and transmissive diffraction modes and are defined by the conservation of energy.
DE
R,m,n
+ DET , m , n = 1, n = TE or TM .
(11)
m
Fig. 4. Optical absorption of MQW GaN/InxGa1-xN/cSi DJ tandem SC with diffraction grating with varying (a) p-GaN thicknesses, or (b) Si thicknesses, or (c) total number of MQW pairs.
We start from the TM mode due to its sensitivity to varying grating parameters. We target the grating design at λ = 0.8 μm. While Λ = 0.25 μm, it has the largest DE shown in Fig. 3(a) and its diffracted angle is around 60。 . This diffracted angle is much larger than the critical angle, which is around 45。 , with respect to the normal to the interface of Si and In0.5Ga0.5N. The total internal reflection occurs while the 1st- and minus 1st-order waves strike the interface of Si and In0.5Ga0.5N and can be further absorbed by the Si material with much longer OPLs. Then we change the grating thickness to increase the 1st-order DE. And we can obtain that around 75 nm the grating has the largest DE at grating period 250 nm shown in Fig. 3(a). After analyzing the maximum 1st-order diffraction efficiency versus grating thickness, we found the optimum grating thickness is 63 nm in TM case. In TE case, the optimized grating period is 432 nm. After considering TM and TE cases, the grating period should be located in the range between 250 to 432 nm. The OPL difference is about 3 times difference (between 60 and 30。 ). In the Fig. 3(b) we limit our grating thickness less than 100 nm, which would result in grating angles smaller than 40。 , for reliable manufacturing purpose. The optimum 1st-order diffraction efficiency for TM is at Λ = 250 nm and thickness = 80 nm. The larger the grating angle, the larger it contributes to the TM light absorption. The
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A622
absorption is increasing with the increasing grating thickness and increasing period. Conversely, the larger the grating angle and the smaller it contributes to the TE optical absorptance. Therefore, it is a trade-off between TE and TM optimization grating angles. For the unpolarized case, the case with period 250 nm and grating thickness 80 nm has the largest absorptance. In this case the grating angle is around 32.6。 . Due to the trade-off between the TE and TM optimization, we got the optimum grating angle ~25。 for unpolarized light. Figure 3(c) shows the case with period 432 nm and grating thickness 100 nm has the largest absorptance. In this case the grating angle is 24.8。 . From the optical absorptance spectrum shown in Fig. 3(c), we can see improvement for the whole spectrum. Figure 4(a) shows that decreasing the p-GaN thickness can affect the spectrum from 700 to 1100 nm. The total harvested currents (InGaN SC + cSi SC) are 35.86, 35.89 and 36.12 mA/cm2 for p-GaN thicknesses 150 nm, 100 nm and 50 nm, respectively. Due to this small blue-shift in the spectrum, a small improvement 0.72% is obtained. Note that decreasing pGaN thickness can greatly increase the IQE in the UV range smaller than 364 nm. Due to the fact that InGaN SC operates as a drift-dominant device, thinner p-GaN can greatly enhance carrier collection in order not to block electric field-included depletion region carrier collection. The benefit is twofold. Figure 4(b) shows that increasing Si thickness can increase the optical absorption, especially the case of 2 μm Si thickness with 3.54% increment. The simulated thicknesses of silicon p-n junction are much smaller than minority carrier diffusion lengths which are around 100’s of μm and it means any generated charge carriers can be effectively collected. The challenging part is to increase its absorption as large as possible due to its indirect band gap weak absorption. Therefore, increasing optical path length of light absorption in silicon is the core part of optimization. Figure 4(c) shows that more pairs of MQW (with total InGaN thickness 200 nm) don’t contribute to more optical absorption with the QB thickness 8 nm chosen to avoid the formation of intermediate minibands. The coupling of adjacent QWs due to tunneling effect (QB thickness > 5 nm) can be neglected. It should be noted that the increasing QB thickness and MQW pairs would increase series resistance. The optical absorption decreases a little bit but the overall optical absorption doesn’t change that much, only ~0.2%. Therefore the number of MQW pairs depends upon the lattice-mismatch induced critical thickness for growth which can provide the maximum Jsc with least series resistance and current-matching while the two-terminal module design is adopted. 4. Conclusion
We present a thorough electromagnetic simulation TMM/RCWA study with semiconductor physics perspectives on the absorptance enhancement design of a MQW GaN/InxGa1-xN/cSi DJ tandem SC with triangular diffraction grating over a broad spectrum of λs from 300 to 1200 nm for the first time. We combine anti-reflection coatings and triangular diffraction grating to achieve optimal optical design for the purpose of obtaining higher Voc and Jsc in MQW GaN/InxGa1-xN/cSi tandem solar cell. Engineering photon dynamics within the solar cell structure is important to achieve high short-circuit current and high open-circuit voltage and especially important to design a novel either two-terminal or four-terminal MQW InGaN/cSi light emitting solar cells which definitely require good light extraction. A tri-layer (Ta2O5/SiO2-TiO2/SiO2) ARC can achieve broader spectrum than a double-layer ARC for lower reflectance within the whole considered spectrum and surprisingly much lower reflectance in the UV range. Almost further 20% reduction of reflection loss is achieved with the tri-layer ARC compared to the double-layer Ta2O5/SiO2 ARC. Due to the trade-off between the TE and TM light trapping optimization in triangular diffraction grating, we got the optimum grating angle ~25。 for maximized absorption of unpolarized light. Adjusting pGaN thickness and MQW pairs can greatly enhance InGaN IQE and optical absorption which can serve as another degree of freedom to fulfill the requirement of current-matching at the two-terminal module design or to maximize the optical absorption at the four-terminal module design. In the case of 200 pairs of MQW, the total structure can be mounted to more than 400 layers in this TMM/RCWA simulation. More than 80% of incoming light can be
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A623
harvested with the total structure thickness less than 4 μm including enhanced optical absorption path length in silicon p-n junction and optimized MQW InGaN p-i-n junction with extra degree of freedom to do ether current-matching or maximized short-circuit current for different module configurations. An improvement of 58% is obtained for the whole structure as compared with the structure without the ARC and diffractive triangular grating. The superiority of enhancement is twofold due to the optical maximized absorptance and electrically better carrier collection designs resulting in larger Jsc and open-circuit voltage Voc. It is imperative for SC to acquire maximum absorptance in the balance between optical and electrical designs. The results presented here support an efficient approach to optimize optical design of InGaN/cSi tandem SC. If the MQW InxGa1-xN junction can be manufactured ideally with much less recombination centers with different In content in the near future, then the open-circuit voltage approaching Voc-ideal will mainly lie with optical design which we integrate it in the beginning of the light-emitting solar cell design. Acknowledgments
The authors gratefully acknowledge the financial support for this research from the National Science Council of Taiwan under grant No. NSC102-2221-E-002-174 and No. NSC1022120-M-110-005 and from the Excellent Research Projects of National Taiwan University under Grant 104R89085. The authors also would like to give thanks to Kelly Marissa Ramos Lin for her patiently technical proofreading help.
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A624
Graduate Institute of Photonics and Optoelectronics, National Taiwan University, Taipei 10617, Taiwan 2 Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan * [email protected] (Y.-P. Chiou)
Abstract: Optical design in enhancing optical absorption of group-IIInitride- and multiple quantum well-based GaN/InxGa1-xN/cSi dual-junction tandem solar cells with triangular diffraction grating is simulated and optimized by using combined two-dimensional rigorous coupled wave analysis and transfer matrix methods. This paper thoroughly examines these phenomena of optical absorption affected by antireflection coatings, multiple thin-film layers and diffraction gratings with the integrated perspectives of semiconductor physics and electromagnetic theory for the first time. An improvement of 58% in absorption compared to the prototype SC is obtained which means more than 80% of incoming light (hυ > EgSi) can be harvested in this thin-film (< 4 μm in total) design. ©2015 Optical Society of America OCIS codes: (230.1950) Diffraction gratings; (230.5590) Quantum-well, -wire and -dot devices; (310.1210) Antireflection coatings; (310.4165) Multilayer design; (350.6050) Solar energy.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
S. Nakamura, M. Senoh, N. Iwasa, and S. I. Nagahama, “High power InGaN single quantum well structure blue and violet light emitting diodes,” Appl. Phys. Lett. 67(13), 1868–1870 (1995). S. Nakamura, M. Senoh, S. I. Nagahama, N. Iwasa, T. Yamada, T. Matsushita, H. Kiyoku, Y. Sugimoto, T. Kozaki, H. Umemoto, M. Sano, and K. Chocho, “Continuous-wave operation of InGaN/GaN/AlGaN-based laser diodes grown on GaN substrates,” Appl. Phys. Lett. 72(16), 2014–2016 (1998). O. Jani, I. Ferguson, C. Honsberg, and S. Kurtz, “Design and characterization of GaN/InGaN solar cells,” Appl. Phys. Lett. 91(13), 132117 (2007). K. T. Chen, W. C. Huang, T. H. Hsieh, C. H. Hsieh, and C. F. Lin, “InGaN light emitting solar cells with a roughened N-face GaN surface through a laser decomposition process,” Opt. Express 18(22), 23406–23412 (2010). L. J. H. Lin and Y.-P. Chiou, “Improving thin-film crystalline silicon solar cell efficiency with back surface field layer and blaze diffractive grating,” Sol. Energy 86(5), 1485–1490 (2012). S. P. Bremner, M. Y. Levy, and C. B. Honsberg, “Analysis of tandem solar cell efficiencies under AM1.5G spectrum using a rapid flux calculation method,” Prog. Photovolt. Res. Appl. 16(3), 225–233 (2008). L. A. Reichertz, I. Gherasoiu, K. M. Yu, V. M. Kao, W. Walukiewicz, and J. W. Ager III, “Demonstration of a III–Nitride/Silicon Tandem Solar Cell,” Appl. Phys. Express 2(12), 122202 (2009). L. Hsu and W. Walukiewicz, “Modeling of InGaN/Si tandem solar cells,” J. Appl. Phys. 104(2), 024507 (2008). M. Nawaz and A. Ahmad, “A TCAD-based modeling of GaN/InGaN/Si solar cells,” Semicond. Sci. Technol. 27(3), 035019 (2012). C. J. Neufeld, Design, Fabrication and Characterization of III-N Based Solar Cells, Dissertation, UCSB. C. J. Neufeld, S. C. Cruz, R. M. Farrell, M. Iza, J. R. Lang, S. Keller, S. Nakamura, S. P. DenBaars, J. S. Speck, and U. K. Mishra, “Effect of doping and polarization on carrier collection in InGaN quantum well solar cells,” Appl. Phys. Lett. 98(24), 243507 (2011). C. J. Neufeld, N. G. Toledo, S. C. Cruz, M. Iza, S. P. DenBaars, and U. K. Mishra, “High quantum efficiency InGaN/GaN solar cells with 2.95 eV band gap,” Appl. Phys. Lett. 93(14), 143502 (2008). M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71(7), 811–818 (1981). M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986). L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13(5), 1024–1035 (1996). M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovolt. Res. Appl. 3(3), 189–192 (1995).
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A614
17. M. A. Green, Solar Cells: Operating Principles, Technology and System Applications (Prentice-Hall, Englewood Cliffs, New Jersey, 1982). 18. Filmetrics, Inc., “Refractive Index Database,” http://www.filmetrics.com/. 19. M. Polyanskiy, “Refractive Index Database,” http://refractiveindex.info/. 20. M. Levinstein, “Semiconductors on NSM,” http://www.ioffe.ru/SVA/NSM/Semicond/. 21. S. Adachi, The Handbook on Optical Constants of Semiconductors (World Scientific Publishing, 2012). 22. S. Adachi, Optical Constants of Crystalline and Amorphous Semiconductors (Springer, 1999). 23. G. F. Brown, J. W. Ager III, W. Walukiewicz, and J. Wu, “Finite element simulations of compositionally graded InGaN solar cells,” Sol. Energy Mater. Sol. Cells 94(3), 478–483 (2010). 24. M. J. Bergmann and H. C. Casey, “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84(3), 1196–1203 (1998). 25. M. Leung, A. Djurisic, and E. Li, “Refractive index of ingan/gan quantum well,” J. Appl. Phys. 84(11), 6312– 6317 (1998). 26. O. D. Miller, E. Yablonovitch, and S. R. Kurtz, “Intense internal and external fluorescence as solar cells approach the Shockley-Queisser efficiency limit,” IEEE J. Photovoltaics 2, 303–311 (2012). 27. J. L. Gray, A. Luque, and S. Hegedus, Handbook of Photovoltaic Science and Engineering, 2nd ed., (Wiley, West Sussex, UK, 2011). 28. ASTM, “Reference Solar Spectral Irradiance: Air Mass 1.5 Spectra,” http://rredc.nrel.gov/solar/spectra/. 29. S.-Y. Lien, D. Wuu, W. Yeh, and J. Liu, “Tri-layer antireflection coatings (SiO2/SiO2-TiO2/TiO2) for silicon solar cells using a sol-gel technique,” Sol. Energy Mater. Sol. Cells 90(16), 2710–2719 (2006). 30. D. Holec, P. M. F. J. Costa, M. J. Kappers, and C. J. Humphreys, “Critical thickness calculations for InGaN/GaN,” J. Cryst. Growth 303(1), 314–317 (2007).
1. Introduction Ternary compound Indium Gallium Nitride (InGaN) semiconductor material made of a mix of gallium nitride (GaN) and indium nitride (InN) is prevalently used in various optoelectronic devices including high-brightness violet-, blue- and green- light-emitting diodes (LEDs) [1], laser diodes (LDs) [2], solar cells (SCs) [3] and light-emitting solar cells (LESCs) [4]. InxGa1-xN alloys have broad bandgap (Eg) values, which can be engineered to vary In content x, ranging from 0.7 eV (i.e. InN) to 3.42 eV (i.e. GaN), which covers most of the solar spectrum from ultraviolet (UV) to infrared (IR) light. Crystalline silicon (cSi) is an indirect Eg semiconductor, i.e. absorption is phonon-assisted at near- Eg [5], where Eg is 1.12 eV at a cell temperature of 300 K. cSi’s absorption coefficient (α, abbreviated as AC) varies over many orders of magnitude with varying wavelengths (λs). They are suitable for highefficiency dual-junction (DJ) tandem SCs which requires subcell Eg energies of 1.1 eV and 1.7 eV for an ideal two terminal tandem SC under the condition of current-matching [6]. cSi Eg is close to 1.1 eV and we can engineer the In content of InxGa1-xN ternary alloy to be 1.7 eV. The fundamental structure of an InGaN SC is similar to that of an InGaN LED and its single chip InGaN-based SCs can even be operated as LEDs and therefore are called LESCs [4]. Growth of GaN SCs (or InGaN SCs with similar manufacturing process) on n-type cSi(111) substrate [7] needs a thin low-temperature AlN buffer layer (usually 10’s of nm) to accommodate the lattice mismatch between Si and GaN (or InGaN) and the AlN buffer layer serves as p-type doping layer for cSi through Al in-diffusion. Previous analytical calculations have predicted a conversion efficiency of 31% [8] for pn-InGaN/pn-cSi SCs. TCAD semiconductor simulation [9] for a DJ GaN/In0.5Ga0.5N (1 μm)/cSi (200 μm) tandem design predicted a conversion efficiency of 29%. The advantage of this InGaN/cSi tandem structure is that the low resistance tunnel junction [8] at the n+-InxGa1-xN/p+-cSi interface can be formed for certain InxGa1-xN alloy fractions (x ~0.45-0.5) depending upon how AlN buffer layer is incorporated and the actual alignment of n+-InxGa1-xN conduction band and p+-cSi valence band (at an alloy composition of x = 0.46, perfect alignment can be obtained). In c-Si SCs, a majority of the incident light is absorbed in the neutral regions of the device, and minority carriers must diffuse to the depletion region to be separated by the built-in field of the junction. This design relies on long minority carrier diffusion lengths (MCDLs) around 100’s of μm, especially for cSi with relatively low ACs where carriers are generated far from the junction. P-n junction based SC designs are typically implemented in cSi where the absorption length (10’s of μm) is greater than the depletion region width (~1 μm) but less than the MCDL [10]. cSi SC operates as a diffusion-dominant device. On the other hand, in InxGa1-xN SC where the absorption length (300 nm) is comparable to the depletion region
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A615
(400 ~500 nm) and MCDL is short (< 60 nm), a SC design based on a p-i-n junction is preferable. InGaN SC operates as a drift-dominant device. 2. Structure and simulation methods Figure 1 shows the whole tandem SC which consists of a tri-layer anti-reflection coating (ARC) stack, a DJ GaN/InxGa1-xN/cSi tandem SC design with InxGa1-xN (Quantum Well: QW)/GaN (Quantum Barrier: QB) MQW structure to counteract threading dislocations due to the large lattice mismatch, a symmetrically Al nano-structured triangle grating (TG), and a Al back metal reflector. Since the back metal reflector with TG also serves as a back contact, we should limit TG design with the grating angle as small as possible for better electrical contact and as large as possible to increase the optical path length (OPL) of light absorption (a tradeoff). The back surface field (BSF) layer in terms of improving minority carrier collection is implicitly assumed on top of the grating and can be neglected in this optical simulation due to negligible refractive index change induced by heavy doping. The advantage of the BSF is to establish an additional drift field for minority carriers that assists minority carrier collection and reduces the effective back-surface recombination velocity for the minority carriers approaching the back electrode [5]. The MQW InGaN junction with QW/QB pairs up to 100 periods and In fraction up to 0.27 was demonstrated without significant material degradation [10]. We have extended this concept further to 200 periods of QW/QB pair and In fraction up to 0.5 for the ideal DJ tandem SC. Due to the depletion region width comparable to the absorption length, we simulate the MQW InGaN SC with thicknesses less than 500 nm in total. Due to the very low MCDL, we tend to make the p-GaN layer as thin as possible (< 150 nm) in order to reduce the carrier recombination before they can diffuse.
Fig. 1. MQW GaN/InxGa1-xN/cSi dual-junction (DJ) tandem solar cell (SC).
The MQW InGaN top junction is a Ga-polar (0001) oriented design with a p-side-up geometry, the polarization-induced dipole is in the opposite direction of the depletion field [11,12], resulting in a reduced or even reversed electric field in the MQW InGaN region. Instead of promoting charge separation, the electric field in the MQW InGaN layer inhibits carrier collection by forcing electrons (e-) towards the p-GaN side of the junction and holes (h+) towards the n-GaN side of the junction. The polarization induced dipole effectively sets up a potential barrier that carriers must overcome to be collected. Without the large depletion field to separate carriers generated in the MQW InGaN region, carriers would simply recombine and not contribute to the photocurrent. If the doping is sufficient to effectively screen the polarization charges, the electric field in i-region can be reversed back to the correct direction for carrier collection. If the doping is high enough on the n-GaN side to screen the positive polarization sheet charge at that interface and pin the conduction band near the Fermi level, and as the doping on the p-side is increased to screen the negative polarization sheet charge at that interface, the magnitude of the electric field increases from flat band where the conduction band is at the Fermi level on both sides of the i-region to a
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A616
nearly constant electric field-bearing band across the whole depletion region. The effect of the increasing doping density is to screen the polarization sheet charges. As the In fraction is increased, the piezoelectric polarization sheet charges will increase due to increased strain, and the doping density required to screen the polarization charge will need to be increased as well. Note that it is impractical to dope the layers with arbitrarily high concentrations which would instead damage the layers and make them dead layers. Dead layers have an extremely low minority lifetime and thus is prone to recombine most of the short-λ photogenerated electrons (e-) and holes (h+) [5]. As UV photons with increasingly higher energies penetrate less and less deep into the InGaN material, it appears that a thinner MQW InGaN pn-junction located closer to the surface would always be beneficial for carrier collection; thus leading ultimately to a higher quantum efficiency of the InGaN top cell. We have assumed the sufficient doping concentrations for both sides to overcome the polarization sheet charges so that the e- and h+ carriers can be effectively collected by the recovered electric field within the MQW InGaN region without causing the dead layers. In this tandem design, the use of III-nitride on cSi provides a promising low cost and highly efficient SC structure. This stack structure will not only provide enhanced spectral coverage but also introduce a natural tunnel junction between the p-cSi and n-InGaN, thereby eliminating the need for highly doped regions in the conventional DJ SCs [8]. Since InGaN can be grown on cSi [7], realization of highly efficient and cheap tandem cell presents a new possibility for photovoltaics. In this electromagnetic simulation, polarization dependent twodimensional (2D) numerical simulations based on rigorous coupled wave analysis (RCWA: TE and TM modes) [13,14] is implemented for the design and optimization of optical absorption of the SC structure. We use transfer matrix method (TMM) [15] to effectively calculate reflectivity and transmissivity of optical stratified films. We combined these two methods coded in Matlab (Mathworks, Inc.) in our lab and verified with the Finite Element Method (FEM) [5] to calculate the optical absorptance of the whole MQW GaN/InxGa1-xN/cSi DJ tandem SC. In other words, if αλ, Rλ, and Tλ represent the optical absorptance, reflectance, and transmittance, respectively, then the conservation of energy implies that αλ + Rλ + Tλ = 1, that is, the amount of incident energy is equal to the sum of the absorbed, reflected, and transmitted energy. Both TE and TM reflectances, transmittances and absorptances are calculated and averaged as unpolarized light. The solar spectrum spans a broad range of λs and the spectral range of interest is 300 nm – 1200 nm [5]. The refractive index dispersion effect of each layer is taken into consideration. To enable more incident light to enter the SC, the sun’s spectral characteristics should be considered during the design of anti-reflection coatings (ARCs). The weighted average reflectance Rave within the spectrum under consideration can be calculated by the integral of incident photon flux F(λ) multiplying the reflectance of monochromatic light R(λ) normalized by the integral of F(λ) with the assumption of constant internal quantum efficiency (IQE) for simplicity.
Rave =
λ2
λ2
λ1
λ1
F ( λ ) ⋅ R ( λ ) ⋅ IQE ( λ ) d λ λ2
F ( λ ) ⋅ IQE ( λ ) d λ
λ1
=
F (λ )⋅ R (λ ) dλ λ2
.
(1)
F (λ ) dλ
λ1
We adopt a tri-layer ARC for broad spectral range with low reflectance. In order to proceed with the RCWA and TMM simulations, we need to get dispersive refractive complex indices for InGaN, GaN, InN, AlN, cSi, Al, SiO2, TiO2 and Ta2O5 and ITO (current spreading layer or transparent conducting layer which can reduce contact resistance and enhance photogenerated carrier collection). Moreover, the real and imaginary parts of the refraction index of c-Si are obtained from [16]. The imaginary part is obtained from the AC via α = 4πk/λ [17]. We can get λ-dependent complex refractive indices for cSi, Al, SiO2, TiO2 and Ta2O5 from the websites [18–20] and for AlN (buffer layer), GaN and InN from the handbooks [21,22]. But for such dispersion relations describing n(E) and k(E) of the wide Eg III-nitride are usually unavailable, we need an indirect method to generate the λ-dependent #236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A617
complex refractive indices in the RCWA and TMM simulations. First, we adopt the λdependent complex refractive indices for GaN, InN from two handbooks [21,22]. Then we start from AC fitting formula [23] and get the λ-dependent imaginary part k from it and we estimate the λ-dependent real part n by shifting the energy scale of the refractive index of GaN [24]. Quantum-confinement induced refractive index change is ignored due to the lack of experimental data compared to the theoretical calculation [25]. 4-terminal tandem SC which can facilitate LESC module can be formed by increasing the thickness of AlN buffer layer in terms of its insulator-like properties. If we can design a LESC with good light extraction, then we can obtain good light trapping for the solar cell function and good light extraction for the light emitting function in one device. Therefore, an optical design is required that can accommodate these dual functions in one device. To this end, we look to the reciprocity theorem that connects these two optical devices. This idea is profound, and we extend it to the design of InGaN/cSi tandem solar cell in this paper. According to Owen and Eli Yablonovitch [26], a great solar cell also needs to be a great light emitting diode due to their reciprocity relation if high efficiency is expected to be achieved. They have shown that high efficiency solar cell will require complete optical design such that the solar cell achieves optimal light extraction at open-circuit condition and optimal light absorption at short-circuit condition. At open-circuit condition, the carriers have no current loop to go and they build up in density at terminals and, ideally, they recombine radiatively and emit external fluorescence (represented by external fluorescence efficiency ηext which is related to the light extraction) in exact proportion to the incoming sunlight. If the optical design is done poorly, the resulting open-circuit voltage would be greatly reduced (Voc = Voc-ideal – kT/q | ln ηext |) due to very low external fluorescence efficiency ηext. This result is further buttressed by the rigorous thermodynamic theory literature. It is clear that top surface texturing is not helpful in InGaN MQW SC due to lots of defect-induced recombination centers which can annihilate useful electrons and holes. Also any additional non-radiative recombination centers impair the carrier density buildup, limiting both the open-circuit voltage and short-circuit current. Therefore, external fluorescence efficiency depends not only on optical design but also the quality of material. We combine anti-reflection coatings and triangular diffraction grating to achieve better optical design for the purpose of obtaining higher Voc and Jsc in GaN/InxGa1-xN/cSi tandem solar cell. If the MQW InxGa1-xN junction with different In content can be manufactured ideally with less recombination centers, then the open-circuit voltage approaching Voc-ideal lies mainly with optical design. 3. Results and discussion For light incident at the front of the solar cell, the optical generation rate Gj in the absorber layer j which is either InGaN multiple quantum well or silicon p-n junction takes the form [27] Gj =
λ < hc / E g
P (λ ) d λ, 1 − R ( λ ) − T ( λ ) ⋅ Aj ( λ , α ) ⋅ hc / λ
where F ( λ ) =
P (λ ) hc / λ
,
Aj ( λ , α ) = 1 − exp −α j ( λ ) x j exp − α i ( λ ) xi , i< j
(
(2)
)
(3)
where Aj(λ,α) is absorption which is related to the layer structure of the solar cell and P(λ) is the spectral power density (W/m2/μm). The structure of the GaN/InxGa1-xN/cSi tandem solar cell has hundreds of layers in this optical simulation. In the case of 200 pairs of MQW, the total structure can be mounted to more than 400 layers in this TMM/RCWA simulation. Essentially, only photons with energy larger than Eg contribute to the generation rate.
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A618
Therefore, absorbed sunlight in the absorber layers produces charge carriers including electrons and holes. In the dual-junction tandem solar cell, the generation rates for top junction (G1) and bottom junction (G2) can be described for the first order analysis as follows [27]: λ1 P (λ ) G1 = 1 − R ( λ ) ⋅ 1 − exp −α1 ( λ ) x1 ⋅ d λ, hc / λ 0
(
G2 =
)
( ) 1 − R ( λ ) ⋅ (1 − exp −α ( λ ) x ) exp −α ( λ ) x ⋅ hc / λ d λ.
λ2
P λ
2
2
1
1
(4)
(5)
0
While the four-terminal module design is adopted, both generation rates lead to short-circuit currents and can be harvested as current output. While the two-terminal module design is adopted, current-matching condition Jsc1(qG1) = Jsc2(qG2) is required so that harvested charge carriers wouldn’t be wasted. We adopt thin-film structure for both top and bottom junctions in order to maximize the carrier collection. The AM 1.5G solar spectrum (1000 W/m2) is used in our TMM/RCWA simulations which has a maximum possible short-circuit currents (Jsc,ideal) of 45.48 mA/cm2 (> 1.12 eV) in cSi SC [5,28]. The short-circuit current, Jsc, contributed by the subcell is given by λ
J sc =
q 2 λ ⋅ EQE ( λ ) ⋅ P ( λ ) d λ , hc λ1
(6)
where EQE(λ) is the external quantum efficiency of the subcell and P(λ) is the spectral power density (W/m2/μm). The external quantum efficiency takes into account the effects of reflectance Rλ, transmittance Tλ = 0 (due to the Al back metal reflector) and absorption A. EQE ( λ ) = 1 − R ( λ ) − T ( λ ) ⋅ IQE ( A ( λ , α ) , λ ) ,
(7)
where α is the absorption coefficient. The maximum ideal short-circuit current, Jsc,ideal, occurs when EQE(λ) = 100% at λ < λG, where λG is the wavelength corresponding to the subcell material’s bandgap. λ
J sc ,ideal =
q 2 λ ⋅ P ( λ ) d λ. hc λ1
(8)
And Jsc can be written as two terms with the assumption of IQE(A(λ,α),λ) = 100% for simplicity λ
J sc =
q 2 λ ⋅ EQE ( λ ) ⋅ P ( λ ) d λ hc λ1 λ
=
q 2 λ ⋅ 1 − R ( λ ) ⋅ IQE ( A ( λ , α ) , λ ) ⋅ P ( λ ) d λ hc λ1
=
q 2 λ ⋅ 1 − R ( λ ) ⋅ P ( λ ) d λ hc λ1
=
q 2 q 2 P d λ ⋅ λ λ − λ ⋅ R (λ ) ⋅ P (λ ) dλ ( ) hc λ1 hc λ1
λ
λ
(9)
λ
= J sc ,ideal − ΔJ sc , reflection loss .
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A619
In order to achieve lower reflection loss of the SC, and a tri-layer (Ta2O5/SiO2-TiO2/SiO2) ARC is adopted instead of a double-layer (Ta2O5/SiO2) ARC. The refractive index of SiO2TiO2, which can be formed by, for example, sol–gel method [29], is between the Ta2O5’s and SiO2’s. Therefore, we propose this tri-layer ARC to reduce the reflection loss. The minimum weighted average reflectance Rave = 1.6% with a tri-layer AR stack is achieved. The thicknesses of Ta2O5, SiO2, and SiO2-TiO2 are 53.5 nm, 92 nm and 51 nm, respectively (Figs. 2(a) and (b)). Almost further 20% reduction of reflection loss is achieved with the tri-layer ARC compared to the double-layer Ta2O5/SiO2 ARC as shown in Fig. 2(c).
Fig. 2. (a) Reflection loss of a tri-layer ARC. (b) Contour plot of the weighted average reflectance Rave of a tri-layer ARC (c) Comparison of reflectance between a tri-layer ARC and a double-layer ARC. (d) Optical absorptance of MQW GaN/InxGa1-xN/cSi DJ tandem SC without diffraction grating.
The advantage of using a tri-layer ARC is to further reduce the reflection loss in the optical spectrum ranging between 300 nm and 400 nm as shown in Fig. 2(c). 60% further reduction of the reflection loss in the UV spectrum is obtained. A tri-layer ARC can achieve broader spectrum than a double-layer ARC for much lower reflectance in the UV range. The critical thickness of an InxGa1-xN layer grown on GaN decreases significantly as the In content x of the layer is increased. The theoretical critical thickness of an InGaN layer coherently grown to GaN calculated by Sisher et al. [30] is smaller than 9 nm with the In content x larger than 0.3. If the energy balance model is calculated, then the critical thickness is much smaller. It is less than 4 nm with In content larger than 0.2. As for the DJ SC, the In content of the top junction is around 0.5, only QW thicknesses smaller than 4 nm are considered in our simulation. Figure 2(d) shows the Fabry-Perot (FP) modulation effect on the absorption curves. Incorporation of the MQW structure as the active region increases the optical absorption from 350 nm to 700 nm because of the modifications in both the amplitude and phase of the optical wave modes in the active region. A MQW structure sandwiched between air-ARCs-GaN and GaN-Si interfaces operates as a better absorber of light. FP
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A620
oscillations according to thickness and refractive index of InGaN/GaN film stack are obviously observed in Fig. 2(d). After adding tunneling junction, cSi junction and Al reflector, we can see greater absorption enhancement in the optical spectrum ranging from 400 to 700 nm whose absorptances reach above 0.9. The Al reflector reflects incoming light back to the InxGa1-xN MQW active region for 2nd absorption and the OPL of absorption in the InGaN doubles (> 400 nm). According to the Beer-Lambert law, 98% of the incident light can be absorbed. This explains why the absorption of this visible spectrum (from 400 to 700 nm) is very high. The thicker Si junction can slightly enhance further absorption in the visible spectrum (from 500 to 700 nm). The spectrum from 700 to 1100 nm has poor absorption mainly contributed by the bottom Si junction.
Fig. 3. (a) TM 1st order DE of TG. (b) Maximum short-circuit current minus reflection loss of MQW GaN/InxGa1-xN/cSi DJ tandem SC with diffraction grating. (c) Unpolarized optical absorptance.
Metal blaze gratings are proved to be an ideal reflector, light would be reflected completely and scattered in all directions to increase its OPL. The goal is to enlarge the angles of scattered lights in the diffractive orders 1 and −1 [5]. The diffractive orders are defined by the grating equation:
nout sin (θ m ) + ninc sin (θ inc ) =
mλ , d
(10)
where ninc and nout are the refractive indices of material on the incoming and outgoing sides of the grating, respectively, m is the diffractive order, θinc is the incident angle, d is the grating groove spacing, λ is the wavelength in vacuum, andθm is the diffractive angle of the mth diffractive order. We adopt the symmetrically TGs to enhance the OPL. We design TG with large 1st-order diffraction efficiency (DE) and large diffracted angle which can enhance the
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A621
OPL of absorption. In the TMM/RCWA simulation, DER,m,n and DET,m,n are diffraction efficiencies for reflective and transmissive diffraction modes and are defined by the conservation of energy.
DE
R,m,n
+ DET , m , n = 1, n = TE or TM .
(11)
m
Fig. 4. Optical absorption of MQW GaN/InxGa1-xN/cSi DJ tandem SC with diffraction grating with varying (a) p-GaN thicknesses, or (b) Si thicknesses, or (c) total number of MQW pairs.
We start from the TM mode due to its sensitivity to varying grating parameters. We target the grating design at λ = 0.8 μm. While Λ = 0.25 μm, it has the largest DE shown in Fig. 3(a) and its diffracted angle is around 60。 . This diffracted angle is much larger than the critical angle, which is around 45。 , with respect to the normal to the interface of Si and In0.5Ga0.5N. The total internal reflection occurs while the 1st- and minus 1st-order waves strike the interface of Si and In0.5Ga0.5N and can be further absorbed by the Si material with much longer OPLs. Then we change the grating thickness to increase the 1st-order DE. And we can obtain that around 75 nm the grating has the largest DE at grating period 250 nm shown in Fig. 3(a). After analyzing the maximum 1st-order diffraction efficiency versus grating thickness, we found the optimum grating thickness is 63 nm in TM case. In TE case, the optimized grating period is 432 nm. After considering TM and TE cases, the grating period should be located in the range between 250 to 432 nm. The OPL difference is about 3 times difference (between 60 and 30。 ). In the Fig. 3(b) we limit our grating thickness less than 100 nm, which would result in grating angles smaller than 40。 , for reliable manufacturing purpose. The optimum 1st-order diffraction efficiency for TM is at Λ = 250 nm and thickness = 80 nm. The larger the grating angle, the larger it contributes to the TM light absorption. The
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A622
absorption is increasing with the increasing grating thickness and increasing period. Conversely, the larger the grating angle and the smaller it contributes to the TE optical absorptance. Therefore, it is a trade-off between TE and TM optimization grating angles. For the unpolarized case, the case with period 250 nm and grating thickness 80 nm has the largest absorptance. In this case the grating angle is around 32.6。 . Due to the trade-off between the TE and TM optimization, we got the optimum grating angle ~25。 for unpolarized light. Figure 3(c) shows the case with period 432 nm and grating thickness 100 nm has the largest absorptance. In this case the grating angle is 24.8。 . From the optical absorptance spectrum shown in Fig. 3(c), we can see improvement for the whole spectrum. Figure 4(a) shows that decreasing the p-GaN thickness can affect the spectrum from 700 to 1100 nm. The total harvested currents (InGaN SC + cSi SC) are 35.86, 35.89 and 36.12 mA/cm2 for p-GaN thicknesses 150 nm, 100 nm and 50 nm, respectively. Due to this small blue-shift in the spectrum, a small improvement 0.72% is obtained. Note that decreasing pGaN thickness can greatly increase the IQE in the UV range smaller than 364 nm. Due to the fact that InGaN SC operates as a drift-dominant device, thinner p-GaN can greatly enhance carrier collection in order not to block electric field-included depletion region carrier collection. The benefit is twofold. Figure 4(b) shows that increasing Si thickness can increase the optical absorption, especially the case of 2 μm Si thickness with 3.54% increment. The simulated thicknesses of silicon p-n junction are much smaller than minority carrier diffusion lengths which are around 100’s of μm and it means any generated charge carriers can be effectively collected. The challenging part is to increase its absorption as large as possible due to its indirect band gap weak absorption. Therefore, increasing optical path length of light absorption in silicon is the core part of optimization. Figure 4(c) shows that more pairs of MQW (with total InGaN thickness 200 nm) don’t contribute to more optical absorption with the QB thickness 8 nm chosen to avoid the formation of intermediate minibands. The coupling of adjacent QWs due to tunneling effect (QB thickness > 5 nm) can be neglected. It should be noted that the increasing QB thickness and MQW pairs would increase series resistance. The optical absorption decreases a little bit but the overall optical absorption doesn’t change that much, only ~0.2%. Therefore the number of MQW pairs depends upon the lattice-mismatch induced critical thickness for growth which can provide the maximum Jsc with least series resistance and current-matching while the two-terminal module design is adopted. 4. Conclusion
We present a thorough electromagnetic simulation TMM/RCWA study with semiconductor physics perspectives on the absorptance enhancement design of a MQW GaN/InxGa1-xN/cSi DJ tandem SC with triangular diffraction grating over a broad spectrum of λs from 300 to 1200 nm for the first time. We combine anti-reflection coatings and triangular diffraction grating to achieve optimal optical design for the purpose of obtaining higher Voc and Jsc in MQW GaN/InxGa1-xN/cSi tandem solar cell. Engineering photon dynamics within the solar cell structure is important to achieve high short-circuit current and high open-circuit voltage and especially important to design a novel either two-terminal or four-terminal MQW InGaN/cSi light emitting solar cells which definitely require good light extraction. A tri-layer (Ta2O5/SiO2-TiO2/SiO2) ARC can achieve broader spectrum than a double-layer ARC for lower reflectance within the whole considered spectrum and surprisingly much lower reflectance in the UV range. Almost further 20% reduction of reflection loss is achieved with the tri-layer ARC compared to the double-layer Ta2O5/SiO2 ARC. Due to the trade-off between the TE and TM light trapping optimization in triangular diffraction grating, we got the optimum grating angle ~25。 for maximized absorption of unpolarized light. Adjusting pGaN thickness and MQW pairs can greatly enhance InGaN IQE and optical absorption which can serve as another degree of freedom to fulfill the requirement of current-matching at the two-terminal module design or to maximize the optical absorption at the four-terminal module design. In the case of 200 pairs of MQW, the total structure can be mounted to more than 400 layers in this TMM/RCWA simulation. More than 80% of incoming light can be
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A623
harvested with the total structure thickness less than 4 μm including enhanced optical absorption path length in silicon p-n junction and optimized MQW InGaN p-i-n junction with extra degree of freedom to do ether current-matching or maximized short-circuit current for different module configurations. An improvement of 58% is obtained for the whole structure as compared with the structure without the ARC and diffractive triangular grating. The superiority of enhancement is twofold due to the optical maximized absorptance and electrically better carrier collection designs resulting in larger Jsc and open-circuit voltage Voc. It is imperative for SC to acquire maximum absorptance in the balance between optical and electrical designs. The results presented here support an efficient approach to optimize optical design of InGaN/cSi tandem SC. If the MQW InxGa1-xN junction can be manufactured ideally with much less recombination centers with different In content in the near future, then the open-circuit voltage approaching Voc-ideal will mainly lie with optical design which we integrate it in the beginning of the light-emitting solar cell design. Acknowledgments
The authors gratefully acknowledge the financial support for this research from the National Science Council of Taiwan under grant No. NSC102-2221-E-002-174 and No. NSC1022120-M-110-005 and from the Excellent Research Projects of National Taiwan University under Grant 104R89085. The authors also would like to give thanks to Kelly Marissa Ramos Lin for her patiently technical proofreading help.
#236065 - $15.00 USD Received 13 Mar 2015; revised 30 Apr 2015; accepted 30 Apr 2015; published 11 May 2015 (C) 2015 OSA 1 Jun 2015 | Vol. 23, No. 11 | DOI:10.1364/OE.23.00A614 | OPTICS EXPRESS A624
