InxGa1-xN 禁带计算公式

Solar Energy Materials & Solar Cells 87 (2005) 595–603 starting from the energy gap of GaN, we fixed the gap of each junction that gives the same Keywords: Solar cell; Simulation; Modeling; InGaN ARTICLE IN PRESS www.elsevier.com/locate/solmat � 0927-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.solmat.2004.08.020 Corresponding author. E-mail address: ahmed.bouazzi@enit.rnu.tn (A.S. Bouazzi). amount of photocurrent density in the structure. Then we calculated the current density accurately and optimized the thicknesses of p and n layers of each junction to make it give the same output current density. The evaluation of ni: the intrinsic concentration permitted to calculate the saturation current density and the open-circuit voltage of each junction. Assuming an overall fill factor of 80%, we divided the output peak power by the incident solar power and obtained the efficiency of each structure. The numerical values for InxGa1�xN were taken from the relevant literature. The calculated efficiency goes from 27.49% for the two-junction tandem structure to 40.35% for a six- junction structure. The six-junction InxGa1�xN tandem structure has an open-circuit voltage of about 5.34V and a short circuit current density of 9.1mA/cm2. r 2004 Elsevier B.V. All rights reserved. Theoretical possibilities of InxGa1�xN tandem PV structures Hasna Hamzaoui, Ahmed S. Bouazzi�, Bahri Rezig PV & Semicond. Mat. Lab. – ENIT, University of Tunis El Manar, P.O.Box 37, Tunis-Belvedere 1012, Tunisia Received 15 May 2004; received in revised form 17 August 2004; accepted 20 August 2004 Available online 13 December 2004 Abstract We designed a model of InxGa1�xN tandem structure made of N successive p–n junctions going from two junctions for the less sophisticated structure to six junctions for the most sophisticated. We simulated the photocurrent density and the open-circuit voltage of each structure under AM 1.5 illumination in goal to optimize the number of successive junctions forming one structure. For each value of N, we assumed that each junction absorbs the photons that are not absorbed by the preceding one. From the repartition of photons in the solar spectrum and 2.1. The identification of the semiconductors ARTICLE IN PRESS H. Hamzaoui et al. / Solar Energy Materials & Solar Cells 87 (2005) 595–603596 Approximate calculations were performed in order to identify the energy gaps of the InxGa1�xN alloys that should be used for the tandem cells. These approximate calculations were done assuming a perfect quantum response of the materials and equal photocurrent densities for all the junctions of the tandem cell, which represents the electricity rule that governs tandem cells. The indium fraction was calculated using the relation given in ref. [9] between the energy gap and the indium fraction EgðxÞ ¼ ð1� xÞEgðGaNÞ þ xEgðInNÞ � bxð1� xÞ; (1) where Eg(GaN) ¼ 3.4 eV, Eg(InN) ¼ 0.7 eV. b is called the bowing parameter. In the InxGa1�xN system the bowing is, however, not constant but it is composition dependent. b is roughly expressed as bðxÞ ¼ ð1� xÞð11:4� 19:4xÞ: (2) 2.2. Computation of the tandem cell short-circuit current density In a tandem cell, the top junction absorbs the photons with energy greater or equal to its energy gap Eg1, which produce a photocurrent density Jph1 and transmits the remaining photons to the junction directly below, and so on until the bottom 1. Introduction It has been shown that, theoretically, the efficiency of tandem photovoltaic cells increases as it incorporates more and more junctions [1]. However, practically, there is a very little range of materials that could be used to make these cells. In fact, the used materials should have some similar properties like the thermal expansion coefficient, the electron affinity and the lattice mismatch. Thus, only few semiconductors have been used for tandem cells; mainly ternary, quaternary and pentanary alloys. As for the number of junctions included in tandem structures, a five-junction tandem cell has just been realized in 2003 showing an open-circuit voltage of 4.1V, and is still under experiments, in order to measure its efficiency [2]. Recently, indium gallium nitride alloys InxGa1�xN are becoming familiar in electronics, they have energy gaps lying between 0.7 and 4.2 eV [3]. Thus, if used for photovoltaic applications, these alloys can lead to realize tandem cells with a greater number of junctions and could so eventually have interesting potentials. In order to evaluate the possibilities of these alloys, we tried, in this work, to model and simulate tandem cells made of two, three, four, five and six InxGa1�xN junctions. The calculations done in this work were done for AM1.5 illumination. 2. Modeling junction of the tandem cell. � ARTICLE IN PRESS H. Hamzaoui et al. / Solar Energy Materials & Solar Cells 87 (2005) 595–603 597 � The surface and the rear recombination velocities were taken to be equal to 103 cm/s. � The electronic properties and the effective masses of the identified InxGa1�xN alloys, are assumed to be the same in all junctions and are supposed to be equal to those of the GaN, which were taken from Refs. [5,6]. For GaN: mnd* ¼ 0.2 m0, mpd* ¼ 0.8 m0, m0 is the electron rest mass, Ln ¼ 125� 10�6 cm, Lp ¼ 79� 10�6 cm, Dp ¼ 9 cm2/s, Dn ¼ 25 cm2/s. We assumed that the absorption coefficients curves of the InxGa1�xN alloys are similar to GaNs [7], which increases linearly from 9.5� 103 cm�1 around the energy gap to 1.3� 104 cm�1 for the most energetic photon of the solar spectrum (hn ¼ 4 eV). In this way, for a InxGa1�xN alloy with energy gap Eg(i), the absorption coefficient is calculated as follows aðhn; EgðxÞÞ ¼ A � hnþ B; (4) where A and B are constants whose values are such that the absorption coefficient is equal to 9.5� 103 cm�1 around the energy gap and equal to 1.3� 104 cm�1 for the most energetic photon of the solar spectrum (hnE4 eV). The junctions and the n-side thicknesses (l and d, respectively) were used as adjusting parameters in order to match the produced photocurrent de The majority carrier concentration was taken equal to 1018 cm�3 on each side of the junction. The properties of the junctions used in the calculations are as follows: 2.3 The short-circuit current density of a tandem cell Jsc is given by the least of the photocurrent densities produced by the junctions of the tandem cell. Jsc ¼ Mini(Jphi), i ¼ 1yn, n is the number of junctions incorporated in the tandem cell. Theoretically, the current mismatch between the junctions’ photocurrent densities should not exceed 5%. In this work, the current mismatch was taken less then 3%. The photocurrent density Jph of an n–p junction with energy gap Eg(i) and receiving light by the n side is taken equal to Jphi ¼ X AM1:5 JpiðhnÞ þ JniðhnÞ; EgðiÞphn; (3) JpiðhnÞ and JniðhnÞ are respectively the holes and electrons current densities produced by the photons of energy hnXEg(i). For the calculations done in this work, the photocurrent produced in the depletion region has been neglected. JpiðhnÞ and JniðhnÞ were calculated using the theoretical conventional equations [4]. . The properties of the InxGa1�xN junctions nsities. ARTICLE IN PRESS H. Hamzaoui et al. / Solar Energy Materials & Solar Cells 87 (2005) 595–603598 2.4. Computation of the open-circuit voltage and the efficiency of a tandem cell The open-circuit voltage was calculated for all the tandem cells being modeled. The open-circuit voltage of a tandem cell is taken to be equal to the sum of the open- circuit voltages of the tandem junctions. Voc ¼ SVocðiÞ; (5) i ¼ 1yn, n is the number of junctions incorporated in the tandem cell. The open-circuit voltage of an n–p junction is given by Voc ¼ kT q ln JL J0 þ 1 � � ; (6) where JL is the junction photocurrent density, J0 is the saturation current density, k is the Boltzmann constant, T is the temperature which was taken equal to 300K and q is the absolute electric charge of electrons. The saturation current density J0 was calculated for all the InxGa1�xN alloys. J0 ¼ qn2i Dnj LnjNA þ Dpj LpjND � � ; j ¼ 1 . . . n; (7) where J is the number of the jth junction. The intrinsic carrier concentration was also computed for all the identified alloys. According to Ref. [10], the intrinsic carrier concentration is expressed as n2i ¼ 2:31� 1031 mndmpd m20 � �3=2 � T3 exp � Eg kT � � : (8) For InxGa1�xN alloys, the saturation current density is given by J0ðInxGa1�xNÞ ¼ 2480945; 33347 expf�EgðxÞ=kTg: (9) The maximum output power that can be produced by a tandem cell is given by Pm ¼ FF� Jph � Voc; (10) where FF is the fill factor, which was taken as constant and equal to 80%. In this theoretical work, we supposed that the junctions are perfect with negligible series resistance and infinite shunt resistance. Jph is the photocurrent density of the tandem cell. The efficiency of the tandem cell is given by Z ¼ JphVocFF F0 ; (11) F0 is the incident irradiance per unit area in mW/cm 2. For the considered solar 2 spectrum [8], F0 ¼ 96:366mW=cm : 3. Results and discussions 3.1. Simulations for a six-junction tandem cell InxGa1�xN tandem cells comprising two, three, four, five and six junctions were simulated. Hereafter, the results computed for a InxGa1�xN tandem structure comprising six junctions are given. The following table shows the energy gaps of the identified materials, the thicknesses of the junctions Li and that of the n-side di as well as the indium fraction x for the identified InxGa1�xN alloys. We can see from Table 1 that the energy gaps of the junctions decrease from the top to the bottom of the tandem cell. Table 2 gives the computations of the photocurrent densities, the open-circuit voltages and the output peak power for a six-junction InxGa1�xN tandem cell. Simulations show that the six-junctions InxGa1�xN tandem cell could reach an efficiency of more than 40% with a short-circuit current density of 9.1mA/cm2 and an open-circuit voltage of 5.3V. We can notice from Table 2 and Fig. 1 that the open-circuit voltages produced by the junctions of the tandem cell decrease almost linearly from the top to the bottom ARTICLE IN PRESS Table 1 Energy gaps and thicknesses for a six-junction tandem cell H. Hamzaoui et al. / Solar Energy Materials & Solar Cells 87 (2005) 595–603 599 Band gap (eV), indium fraction for InxGa1�xN alloys Junction thickness L (mm) n-side thickness di (mm) Eg1 ¼ 2.25, x ¼ 0.11718 L1 ¼ 2 d1 ¼ 0.60 Eg2 ¼ 1.79, x ¼ 0.613 L2 ¼ 2 d2 ¼ 0.28 Eg3 ¼ 1.475, x ¼ 0.766659 L3 ¼ 2 d3 ¼ 0.65 Eg4 ¼ 1.19, x ¼ 0,8535 L4 ¼ 2 d4 ¼ 0.30 Eg5 ¼ 0.95, x ¼ 0,921 L5 ¼ 2 d5 ¼ 0.25 Eg6 ¼ 0.7, x ¼ 1 L6 ¼ 2 d6 ¼ 0.30 Table 2 Simulation results for a six-junctions tandem cell Junction N1i Jph(i) (mA/cm 2) Voc(i) (V) Pm (mW/cm 2) 1 9.1 1.74741 38.88 2 9.2 1.28769 3 9.2 0.97269 4 9.1 0.68741 5 9.1 0.44741 Efficiency (%) 6 9.2 0.19798 40.346 Voc (V) 5.34062 ARTICLE IN PRESS H. Hamzaoui et al. / Solar Energy Materials & Solar Cells 87 (2005) 595–603600 0.5 1 1.5 2 en c irc ui t v ol ta ge (V ) of the tandem cell; the open-circuit voltage of the tandem cell is mainly produced by the first three junctions. This remark was as true for all the simulated tandem cells. The logarithm of the saturation current density increases almost linearly from the top to the bottom of the tandem cell (Fig. 2). This increase is mainly due to the decrease of the energy gaps of the junctions as we move from the top to the bottom junction of the tandem cell. Given relation (7) between the open-circuit voltage and the saturation current density of an n–p junction, we conclude that the decrease of the open-circuit voltage from the top to the bottom junction of the tandem cell (Table 2 and Fig. 1) is due to 0 1 2 3 4 5 6 Number of the junction O p Fig. 1. Open-circuit voltage versus the junction number for a six-junction tandem structure. Semi-logarithmic representation of the saturation current 1.00E-29 1.00E-24 1.00E-19 1.00E-14 1.00E-09 1.00E-04 1.00E+01 1 2 3 4 5 6 The junction number Lo g (Jo ) Fig. 2. Saturation current density versus the junction number for a six-junction tandem structure. the increase of the saturation current density, as the photocurrent density is constant in a tandem cell. 3.2. Potentials of InxGa1�xN tandem cells For the simulated InxGa1�xN tandem cells, Table 3 gives the short-circuit current densities, the open-circuit voltages the output peak powers and the efficiency. Fig. 3 and Table 3 show simultaneously the variation of the short-circuit current density, the open-circuit voltage and the output power as a function of the number of junctions included in the cell. It seems that the output power saturates when the number of junctions increases. More than six junctions would not give more power proportional to the complication of adding more cells. ARTICLE IN PRESS Table 3 Potentials of InxGa1�xN tandem cells Number of junctions in the cell Isc (mA/cm 2) Voc (V) Output peak power (mW/cm2) Efficiency (%) 2 27 1.22 26.48 27.485 3 18.2 2.22 32.42 33.642 4 13.5 3.28 35.48 36.825 5 11 4.28 37.68 39.105 6 9.1 5.34 38.88 40.346 25 30 35 40 Photocurrent density (mA/cm2) Open circuit voltage (V) H. Hamzaoui et al. / Solar Energy Materials & Solar Cells 87 (2005) 595–603 601 2 3 4 5 6 0 5 10 15 20 Output power (mW/cm2) Number of junctions Fig. 3. Variation of the open-circuit voltage, the short-circuit current density and the output maximum power versus the number of junctions in the cell. ARTICLE IN PRESS 30 %) H. Hamzaoui et al. / Solar Energy Materials & Solar Cells 87 (2005) 595–603602 We notice from Table 2 and Fig. 1 that the achievable short-circuit current density 2 3 4 5 6 0 5 10 15 20 25 Ef fic ie nc y ( Number of junctions in the tandem structure Fig. 4. Variation of the efficiency with the number of junctions included in the tandem cell. 35 40 45 AM 1.5 efficiency (%) decreases as the tandem cell.contains more layers However, the increase of the open-circuit voltage as a function of the number of junctions is almost linear; hence, it compensates the decrease of the short-circuit current density, which is a function of the inverse of the same variable; this explains the increase of the output maximum power and the cell efficiency with the number of junctions. The highest efficiency (40.3%) was reached for the six-junction cell (Fig. 4) with a short-circuit current density of 9.1mA/cm2 and an open circuit voltage of about 5.3V. InxGa1�xN tandem cells comprised of two and three junctions have also interesting potentials with an efficiency of 27.4% for the two-junctions tandem cell and of 33.6% for the three-junctions one. It is noticeable that the increase of the efficiency is more important when we move from a two-junction to a three-junction InxGa1�xN tandem cell. 4. Conclusion Simulation of tandem cells comprising, respectively, two, three, four, five and six junctions exclusively made of InxGa1�xN alloys showed that the InxGa1�xN alloys have interesting possibilities for tandem cells applications if compared with the tandem cells that were realized until now. In fact, an efficiency of about 40% is achievable for a six-junctions InxGa1�xN tandem cell with a photocurrent density of 9.1mA/cm2 and an open-circuit voltage of 5.3V. InxGa1�xN tandem cells comprised of two and three junctions have also interesting potentials with efficiency of 27.4% for the two-junction tandem cell and of 33.6% for the three-junction cell. InxGa1�xN tandem cells have an additional advantage as they can be produced with a simpler technology than the ones used to produce tandem junctions made of [10] P. Kireev, La physique des semi-conducteurs, MIR, MOSCOU, (translated from Russian by ARTICLE IN PRESS H. Hamzaoui et al. / Solar Energy Materials & Solar Cells 87 (2005) 595–603 603 S. Medvedev. 1975). different materials. In fact, the InxGa1�xN alloys have similar properties, which make their deposition in successive films easier. References [1] M.A. Green, Third generation photovoltaics: advanced structures capable of high efficiency at low cost, http://www.pv.unsw.edu.au/16thEPSEC/green1.html. 01/04/2003. [2] F. Dimroth, C. Baur, M. Meusel, S. van Riesen, A.W. Bett, 5-junctions III-V solar cells for space applications, WCPEC-3, Osaka, Japan, May 11–18, 2003. [3] F. Bechstedt, J. Furthmueller, M. Ferhat, L.K. Teles, L.M.R. Scolfaro, J.R. Leite, V.Yu. Davydov, O. Ambacher, R. Goldhahn, Energy gap and optical properties of InxGa1�xN, Phys. Status. Solide. (a) 195 (3) (2003) 628–633. 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Davydov, O. Ambacher, R. Goldhahn, Energy gap and optical properties of InxGa1�xN, Phys. Status. Solid. (a) 195 (3) (2003) 628–633. Theoretical possibilities of InxGa1minusxN tandem PV structures Introduction Modeling The identification of the semiconductors Computation of the tandem cell short-circuit current density The properties of the InxGa1minusxN junctions Computation of the open-circuit voltage and the efficiency of a tandem cell Results and discussions Simulations for a six-junction tandem cell Potentials of InxGa1minusxN tandem cells Conclusion References