modeling of downconverter based on pr3 yb3 codoped fluoride glasses to improve sc si solar cells efficiency

3+ 3+ Modeling of downconverter based on Pr -Yb codoped fluoride glasses to improve sc-Si solar cells efficiency P Song and C Jiang Citation: AIP Advances 2, 042130 (2012); doi: 10.1063/1.4766187 View online: http://dx.doi.org/10.1063/1.4766187 View Table of Contents: http://aip.scitation.org/toc/adv/2/4 Published by the American Institute of Physics AIP ADVANCES 2, 042130 (2012) Modeling of downconverter based on Pr3+ -Yb3+ codoped fluoride glasses to improve sc-Si solar cells efficiency P Song and C Jianga State key Laboratory of Advanced Optical Communication Systems and Network, Shanghai Jiao Tong University, Shanghai 200240, China (Received 14 June 2012; accepted 22 October 2012; published online November 2012) Quantum cutting via a two-step resonant energy transfer in a spectral downconverter of Pr3+ -Yb3+ codoped fluoride glass is investigated numerically by proposing up and solving the theoretical model of rate equations and power propagation equations Based on the optimal Pr3+ -Yb3+ concentration and the thickness of the spectral downconverter, the total power conversion efficiency of 175% and total quantum conversion efficiency of 186% are obtained The performance of a sc-Si solar cell covered with a spectral downconverter is evaluated with the photovoltaic simulation programme PC1D For sc-Si solar cells, the energy conversion efficiency of 14.90% for the modified AM1.5G compared to a 12.25% energy conversion efficiency for the standard AM1.5G has been obtained, and the simulated relative energy conversion efficiency for the sc-Si solar cell approaches up to 1.21 Our results show that the use of a spectral downconverter yields better sc-Si solar cell performance compared to the standard AM1.5G irradiation The paper also provides a framework for investigating and optimizing the rare-earth doped spectral downconverter, potentially enabling a sc-Si solar cell with an efficiency improvement Copyright 2012 Author(s) This article is distributed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4766187] I INTRODUCTION The main energy loss in the conversion of solar energy to electricity is related to the spectral mismatch Photons with energy lower than the band gap cannot be absorbed by solar cells While for photons with energy larger than the band gap, the excess energy is lost by thermalization of hot charge carriers The mismatch between the solar spectrum and the band gap energy of silicon semiconductor limits the energy conversion efficiency (ECE) of crystalline-silicon (c-Si) based solar cells.1 Spectral downconverters on top of c-Si solar cells are simple and cheap applications of spectral downconversion (DC) for enhancing performance of c-Si solar cells Spectral downconverters convert the wavelengths where the spectral response is low to wavelengths where the spectral response is high.2–5 A promising method can help single-crystalline Si (sc-Si) solar cells capture more energy from the solar spectrum via spectral DC of lanthanide rare earth (RE) ions near-infrared (NIR) quantum cutting (QC): Cutting an ultraviolet (or a blue photon) into two or more NIR photons, which can be well absorbed by sc-Si solar cells, will reduce the energy loss related to thermalization of hot charge carriers and then improve the ECE of sc-Si solar cells.6 The potential applications of NIR QC in sc-Si solar cells have attracted a lot of recent research attentions.7 NIR QC was achieved firstly in YPO4 : Tb3+ -Yb3+ where a visible photon is converted into two NIR photons through energy transfer (ET) from Tb3+ to Yb3+ and the sc-Si solar cell shows an excellent spectral response.8 The ET mechanism also has been established in Tb3+ -Yb3+ couple: the ET occurs through a cooperative dipole-dipole mechanism.1, Following the work, similar phenomena of the efficient a Author to whom correspondence should be addressed; Electronic email: cjiang@sjtu.edu.cn 2158-3226/2012/2(4)/042130/10 2, 042130-1 C Author(s) 2012 042130-2 P Song and C Jiang AIP Advances 2, 042130 (2012) FIG Energy level scheme and the DC mechanism for Pr3+ -Yb3+ couple showing the QC process involved a two-step consecutive resonant ET DC for sc-Si solar cells were reported for a variety of other RE3+ -Yb3+ (RE = Yb, Tm, Ce, etc.) codoped phosphors and glasses.10–12 The majority of studies for QC in Pr3+ -Yb3+ couple focussed on single crystals.13–31 However, QC has been seldom studied in the fluoride glass which has been demonstrated to be a valid alternative host to support an effective QC process.32 In this work, a promising two-step QC of Pr3+ -Yb3+ codoped fluoride glass is investigated numerically Theoretical models of the Pr3+ -Yb3+ system are founded and simulated numerically by using MATLAB in order to obtain the power conversion efficiency (PCE) and the quantum conversion efficiency (QCE) The performance of a sc-Si solar cell by employing the spectral downconverter containing Pr3+ -Yb3+ codoped fluoride glass is evaluated by using the PC1D software.33 II THEORY The diagram of Pr3+ and Yb3+ energy levels, the relevant absorption and emission transitions, spontaneous emission, and the ET process between Pr3+ and Yb3+ is shown in Fig Praseodymium ion, when excited by blue photons into the PJ (J = 0, 1, 2), I6 levels, relax nonradiatively to the P0 level and then can interact with Yb3+ ions through a two-step consecutive ET The QC processes are depicted schematically: Resonant ET transfers from the level of the Pr3+ donor, and sequential transferred to two neighboring Yb3+ acceptors, Pr3+ (3 P0 ) + Yb3+ (2 F7/2 ) → Pr3+ (1 G4 ) + Yb3+ (2 F5/2 ) and Pr3+ (1 G4 ) + Yb3+ (2 F7/2 ) → Pr3+ (3 H4 ) + Yb3+ (2 F5/2 ) Referring to the energy level system of Pr3+ -Yb3+ shown in Fig 1, the QC system of Pr3+ -Yb3+ is equivalent to a system of pump, excitation and transition We propose the population-rate and power propagation equations model of Pr3+ -Yb3+ The group of rate and power propagation equations can be written as follows ∂ N1 = −W14 N1 + W21 N2 + A21 N2 + Ccr N2 N5 ∂t (1) ∂ N2 = W32 N3 + A32 N3 − W21 N2 − A21 N2 + Ccr N3 N5 − Ccr N2 N5 ∂t (2) ∂ N3 = A43 N4 − W32 N3 − A32 N3 − Ccr N3 N5 ∂t (3) 042130-3 P Song and C Jiang AIP Advances 2, 042130 (2012) ∂ N4 = W14 N1 − A43 N4 ∂t (4) ∂ N5 = −W56 N5 + W65 N6 + A65 N6 − Ccr N3 N5 − Ccr N2 N5 ∂t (5) ∂ N6 = −W65 N6 − A65 N6 + W56 N5 + Ccr N3 N5 + Ccr N2 N5 ∂t (6) d Pp (z, λ) = [σ32 (λ)N3 + σ21 (λ)N2 − σ14 (λ)N1 − α p ]Pp (z, λ) dz (7) d Ps (z, λ) = [σ65 (λ)N6 − σ56 (λ)N5 − αs ]Ps (z, λ) dz (8) where N1 (3 H4 ), N2 (1 G4 ), N3 (3 P0 ), N4 (3 P2 ), N5 (2 F7/2 ) and N6 (2 F5/2 ) are the population densities of relevant energy levels of Pr3+ and Yb3+ Aij (i, j = 1∼6) is the spontaneous transition rate and non-radiation transition rate between the energy levels i and j σ ij (λ) (i, j = 1∼6) is the absorption and emission cross section of the transition between the energy levels i and j Pp (z, λ) and Ps (z, λ) are the corresponding input solar power and output light power, where z is the thickness of the ion doping layer α p and α s are scattering losses and are assumed as frequency-independent constants so as to simplify the analysis The concentration-dependent Ccr1 and Ccr2 are the ET cross-relaxation co-efficient describing the interaction between Pr3+ and Yb3+ , and they are linearly-increasing functions of the Pr3+ concentration according to the resonant ET theory.34 Ccr = 2.0 × 10−21 + 8.00 × 10−48 (N Pr 3+ − 1.0 × 1025 ) (9) Ccr = 2.0 × 10−21 + 8.00 × 10−48 (N Pr 3+ − 1.0 × 1025 ) (10) Wij (i, j = 1∼6) is the transition rate between energy levels i and j, and can be expressed as Wi j (z, λ) = σi j (λ)P(z, λ) hνi j Ae f f (11) where h is Plank constant, vij (i, j = 1∼6) is bandwidth, and Aeff is the effective cross-section area The conservation laws are given by Ni − N Pr 3+ = (12) Ni − NY b3+ = (13) i=1 i=5 where NPr3+ and NYb3+ are total Pr3+ and Yb3+ concentrations, respectively III RESULTS AND DISCUSSION A The theoretical model calculation We consider an initial steady state The above model can be solved numerically in MATLAB, where the population rate equation groups (1)–(6) can be solved by Newton’s Iterative Method, while the propagation rate equation groups (7) and (8) form a system of coupled differential equations which can be solved via fourth-order Runge-Kutta methods and some boundary conditions The modified solar spectrum Ps (z, λ) is optimized and obtained For simulation, we select fluoride glasses as host materials for spectral downconverters The spectroscopic parameters used in calculations are chosen carefully from the literature, as listed in Table I We choose 482 nm and 980 nm as the Pr3+ center excitation wavelength and the system emission wavelength, respectively Incident solar spectrum is normalized for the case of the complete absorption of incident solar emission in the spectral region corresponding to 482 nm absorption band 042130-4 P Song and C Jiang AIP Advances 2, 042130 (2012) TABLE I Primary parameters in the theoretical model Parameter Scattering loss coefficient 482nm absorption cross section (Pr3+ ) 923nm radiation cross section (Pr3+ ) 1350nm radiation cross section (Pr3+ ) Pr3+ ion spontaneous emission rate Pr3+ ion spontaneous emission rate 980nm absorption cross section (Yb3+ ) 980nm radiation cross section (Yb3+ ) Yb3+ ion spontaneous emission rate Initial effective pump power \ initial signal power Symbol Unit Value References α p, α s σ 14 σ 32 σ 21 A32 A21 σ 56 σ 65 A65 Pp0 \ Ps0 db/m m2 m2 m2 s-1 s-1 m2 m2 s-1 W 0.1 2.8×10-24 3.88×10-24 0.15×10-24 1653 4680 1.32×10-24 1.40×10-24 10 0.191 \ 0.418 35 36, 37 36, 37 36, 37 36, 37 36, 37 35 35 35 — FIG (a) Pr3+ absorption band and incident solar spectrum for the calculation of overlap coefficient (b) Normalized spectrum of the input solar light, and the inset: normalized solar energy spectrum of Pr3+ ions In order to calculate the effective absorption of input solar spectrum accurately, the overlap coefficient δ of the Pr3+ absorption spectrum and the solar spectrum is defined as δ= λ Iab (λ)dλ λ Isolar (λ)dλ (14) where Iab (λ) and Isolar (λ) are the Pr3+ absorption spectrum around 482 nm and the solar spectrum around 482 nm, respectively The calculated δ is 0.3817, as shown in Fig 2(a) It means that 38.17% of incident solar energy can be absorbed effectively in the system under 482 nm excitation process In our simulation, we set 500 mW to be initial solar power incident at 482 nm, and initial effective solar power Pp0 is calculated as 190.8 mW We also calculate initial 980 nm output solar light power Ps0 of 148 mW via the normalized solar power spectrum, as shown in the inset of Fig 2(b) Initial Pr3+ concentration, Yb3+ concentration and the thickness of the spectral downconverter are set to be 1.0×1026 ions/m3 , 1.0×1026 ions/m3 and mm, respectively In order to depict the effect of spectral DC based on 482 nm excitation and 980 nm emission, PCE (ηPCE ) and QCE (ηQCE ) are defined as 1500 η PC E = λ=200 1500 λ=200 Ps (z, λ)dλ (15) Pin (z, λ)dλ 042130-5 P Song and C Jiang AIP Advances 2, 042130 (2012) 1500 λ=200 1500 η QC E = λ=200 Ns (z, λ)dλ (16) Nin (z, λ)dλ The total PCE (η PCE ) is defined as the ratio of the total output light power to the total input light power and the total QCE (η QCE ) based on the whole solar spectrum in the range of 200 nm-1500 nm is defined as the ratio of the total output photon number to the total input photon number 1500 η PC E = λ=200 1500 = λ=200 1500 q QC E = λ=200 1500 λ=200 1500 Pout (z, λ)dλ λ=200 1500 Pin (z, λ)dλ λ=200 1500 Nout (z, λ)dλ = Nin (z, λ)dλ [Pin (z, λ) + Ps (z, λ) − Pab (z, λ)]dλ λ=200 (17) Pin (z, λ)dλ [Nin (z, λ) + Ns (z, λ) − Nab (z, λ)]dλ 1500 λ=200 (18) Nin (z, λ)dλ where Pout (λ) and Nout (λ) are the spectrum of total output light power and the spectrum of total output photon number, respectively Pin (λ), Pab (λ) and Ps (λ) are the spectrum of input light power, the spectrum of Pr3+ absorption power at 482 nm and the spectrum of output light power at 980 nm, respectively Nin (λ), Nab (λ) and Ns (λ) are the corresponding spectrum of the photon number B The effect of Pr3+ -Yb3+ concentrations on PCE and QCE The effect of Pr3+ and Yb3+ concentrations on PCE and QCE is shown in Fig Here the thickness of the spectral downconverter is fixed at mm As Pr3+ concentration increases from 2.6×1026 ions/m3 to 4.0×1026 ions/m3 , PCE and QCE decrease slightly, as shown in Fig 3(a) Meanwhile, PCE and QCE increases clearly with increased Yb3+ concentration from 1.2×1026 ions/m3 to 3.8×1026 ions/m3 , as shown in Fig 3(b) Figure indicates that in the definite range of Pr3+ -Yb3+ concentration, the optimal PCE and QCE could be much better for lower Pr3+ concentration and higher Yb3+ concentration, and moreover, variations of PCE and QCE are less sensitive to a variation of Pr3+ concentration than to a variation of Yb3+ concentration In our simulation, PCE and QCE reach maximums only when Pr3+ and Yb3+ concentrations reach the optimal values of 1.7×1025 ions/m3 and 3.6×1026 ions/m3 , respectively, ascribed to the concentration quenching of Pr3+ and Yb3+ , is observed C The effect of the spectral downconverter thickness on PCE and QCE When investigating the thickness effect of the spectral downconverter on PCE (the total PCE) and QCE (the total QCE), both Pr3+ and Yb3+ concentrations are fixed at 1×1026 ions/m3 , and Figure shows their relationships As the thickness increases from mm to 10 mm (0.5 mm to mm), PCE and QCE (the total PCE and QCE) increase clearly It should be pointed out that PCE and QCE (the total PCE and QCE) will no longer increase when the thickness exceeds the maximum (the optimal value), ascribed to absorption losses and side scatting losses within the spectral downconverter.38 In our simulation, for a thickness of the spectral downconverter z = mm, we obtain the optimal total PCE of 175% and the total QCE of 186% when optimal Pr3+ and Yb3+ concentrations are 1.7×1025 ions/m3 and 3.6×1026 ions/m3 , respectively It is shown that the maximum PCE and the corresponding QCE in our simulation are in good coincident with experimental results in measured literatures,14, 17, 29 verifying feasibility and effectiveness of our simulated QC system of Pr3+ -Yb3+ We can calculate accurately the efficiency 042130-6 P Song and C Jiang AIP Advances 2, 042130 (2012) FIG The thickness of the spectral downconverter is fixed at 3mm (a) Variation of PCE and QCE with Pr3+ concentration from 2.6×1026 ions/m3 to 4.0×1026 ions/m3 (b) Variation of PCE and QCE with Yb3+ concentration from 1.2×1026 ions/m3 to 3.8×1026 ions/m3 FIG Pr3+ and Yb3+ concentration are fixed at 1×1026 ions/m3 (a) Variation of PCE and QCE with the thickness of the spectral downconverter from 1mm to 10mm (b) Variation of the total PCE and the total QCE with the thickness of the spectral downconverter from 0.5mm to 3mm of spectral DC and acquire optimal system parameters: ion concentrations, thickness of the spectral downconverter and NIC QC host materials D The modified solar spectrum The spectrum of total output light power Pout and the spectrum of total output photon number Nout are shown in Fig The obvious absorption peak around 488 nm and emission peak around 980 nm is observed The spectrum curves directly indicate that effective processes of QC in Pr3+ Yb3+ couple and the energy in the excited P2 level can be transferred efficiently to codoping Yb3+ ions Figure also illustrates that a sc-Si solar cell may obtain more energies and photons from the modified solar spectrum than the unmodified (unoptimizable) solar spectrum, and the ECE improvement for a sc-Si solar cell can be expected E The optical configuration of a sc-Si solar cell We imitate an optical configuration of a sc-Si solar cell covered with a spectral downconverter from the analysis of Richards,38, 39 as shown in Fig 6(a) Solar light (AM1.5G) radiates into the top covered spectral downconverter, and some wavelengths of solar light are converted due to the 042130-7 P Song and C Jiang AIP Advances 2, 042130 (2012) FIG Variation of the total output power density and the total output number of photons density with wavelength in the range of 200 nm-1100 nm FIG (a) Schematic diagram of the optical configuration of the top covered spectral downconverter and sc-Si solar cell (b) Schematic diagram of the interface between the top covered spectral downconverter and sc-Si solar cell spectral conversion mechanism Solar light incidents from the surface of the top covered spectral downconverter, and is absorbed by the doping RE ions and re-emits During the process after incident and before absorbing, scattering and escape cone losses weaken the strength of the incident solar light The emission light from the RE luminescent center is unidirectional and the emitted light can arrive to the surface of the sc-Si solar cell, only when the angle of incidence less than the angle of total reflection Pyramid structure of Si3 N4 coated on the surface of the sc-Si solar cell is used to increase the absorbance of the solar light,39 as shown in Fig 6(b) We suppose the incident solar light is vertical to the surface of the top spectral downconverter, considering the incident size of solar light is several orders of magnitude smaller than the size of the sc-Si solar cell We can calculate the angle of total reflection from fluoride glass to air to be 39.8◦ by the equation:sin θ =n air /n fluoride glass , where n is the refractive index, n (air) = 1, n (fluoride glasses) = 1.56 and n (Si3 N4 ) = When the incident angle is less than 39.8◦ , the light can arrive to the surface of the sc-Si solar cell, while the effective absorption is further weakened owing to the scattering occurring on the Si3 N4 surface Detailed calculations of the optical configuration for the sc-Si solar cell covered with the spectral downconverter we analyzed above are executed in PC1D simulations 042130-8 P Song and C Jiang AIP Advances 2, 042130 (2012) TABLE II A sc-Si solar cell primary configuration and calculation parameters for PC1D Parameter Cell area Cell thickness Band gap Temperature P-type background doping Dielectric constant Cell exterior front reflectance Constant intensity Cell front surface texture Cell front surface depth Unit Values m2 m eV K ion/m3 — — W/m2 — m 1×10-2 3×10-4 1.124 300 1.531×1010 11.9 6.77% 1000 Pyramid 7.5×10-8 TABLE III Simulated performance parameters of a sc-Si solar cell in PC1D Voc (V) Jsc (mA/cm2 ) Pm (mW/cm2 ) FF (%) η (%) ηm /ηs Standard AM1.5G spectrum Modified AM1.5G spectrum 0.5918 32.20 12.25 64.28 12.25 0.6002 37.84 14.90 65.60 14.90 1.22 F The simulated performance of a sc-Si solar cell A typical sc-Si solar cell is selected in order to obtain the ECE by using PC1D Several primary configuration parameters of the sc-Si cell are listed in Table II Parameters not mentioned in the table are assigned with the default values.33 The AM1.5G spectrum and modified AM1.5G spectrum Pout are used as the incident spectra in PC1D simulation The performance of the sc-Si solar cell can be evaluated conveniently and primary parameters of the sc-Si solar cell are obtained We define relative ECE as ECEms /ECEs , where ECEms (ηms ) and ECEs (ηs ) are the ECE (η) under the modified AM1.5G spectrum and standard AM1.5G spectrum, respectively Simulated performance parameters of the sc-Si solar cell are listed in Table III For the sc-Si solar cell, the short-circuit (Jsc ) and maximum output power (Pm ) increase effectively while the open-circuit voltage (Voc ) and fill-factor (FF) increase slightly Very little effect of the operation of the spectral downconverter is obtained on FF and Voc because the spectral downconverter changes just the distribution of the incident photon intensity, to connect directly with the generation of carrier densities and lead to the increase of Jsc in the sc-Si solar cell without changing the diode property of the sc-Si solar cell Pm is proportional to Jsc from the relationship Pm = Jsc Voc FF and so, FF and Voc to be nearly uninfluenced under the operation of the spectral downconverter.40 It can be seen that the inclusion of a spectral downconverter based on Pr3+ -Yb3+ codoped fluoride glass allows for a better use of the solar spectrum Since the simulated sc-Si solar cell Eff is improved after incorporating enhancement mechanism of sc-Si solar cells employing spectral DC, the potential benefits for actual sc-Si solar cells are expected We must indicate that in practice the results for performance improvement of sc-Si solar cells may perhaps decrease, due to the fact that our simulations are based on the standard AM1.5G spectrum and normal (perpendicular) incidence of irradiation without taking into account other irradiation conditions and varied angle of incidence of irradiation Further studies need to be done in subsequent work 042130-9 P Song and C Jiang AIP Advances 2, 042130 (2012) The QC system is a promising and simple application of spectral DC for enhancing performance of sc-Si solar cells Such spectral downconverters can be applied to existing sc-Si solar cells without any modifications to sc-Si solar cell configurations Hence, optimization of the spectral downconverter to yield an increase in the ECE of the sc-Si solar cell can be carried out independently of the sc-Si solar cell Our relational work shows that some RE couples have similar capabilities to assist effectively in improving performance of sc-Si solar cells and we will report further work subsequently IV CONCLUSIONS In conclusion, the ET mechanism of NIR QC for Pr3+ -Yb3+ codoped fluoride glass based on the rate equations and power propagation equations is studied The NIR QC model is solved numerically in MATLAB Variation of PCE and QCE with the thickness and Pr3+ -Yb3+ concentrations of the spectral downconverter are investigated The total PCE of 175% and the total QCE of 186% under the optimal Pr3+ -Yb3+ concentrations are obtained In PCID simulation, the relative ECE of 1.22 for a sc-Si solar cell has been obtained for the modified solar spectrum In this work, we set up an effective model for simulating Pr3+ -Yb3+ system, and calculate accurately the efficiency of spectral DC and acquire optimal system parameters The model and technique we propose in this work will be helpful to further investigate and optimize much more efficient NIC QC system containing other RE ion couples based on different hosts for further enhancing the ECE of sc-Si solar cells ACKNOWLEDGMENTS This work was supported by National Natural Science Foundation of China (Grant No 61177056) and sponsored by Shanghai Pujiang Program T Trupke, M A Green, and P Wăurfel, J Appl Phys 92, 1668 (2002) G van Sark, Appl Phys Lett 87, 151117 (2005) W G J H M van Sark, A Meijerink, R E I Schropp, J A M van Roosmalen, and E H Lysen, Sol Energy Mater Sol Cells 87, 395 (2005) W G J H M van Sark, Thin 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(2012) Modeling of downconverter based on Pr3+ -Yb3+ codoped fluoride glasses to improve sc- Si solar cells efficiency P Song and C Jianga State key Laboratory of Advanced Optical Communication Systems... absorption cross section (Pr3+ ) 923nm radiation cross section (Pr3+ ) 1350nm radiation cross section (Pr3+ ) Pr3+ ion spontaneous emission rate Pr3+ ion spontaneous emission rate 980nm absorption