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Conversion efficiency enhancement of AlGaAs quantum well solar cells
ARTICLE IN PRESS
Microelectronics Journal 38 (2007) 513–518 www.elsevier.com/locate/mejo
Conversion efficiency enhancement of AlGaAs quantum well solar cells J.C. Rimadaa,, L. Herna´ndezb, J.P. Connollyc, K.W.J. Barnhamc a
Laboratorio de Celdas Solares, Instituto de Ciencia y Tecnologı´a de los Materiales (IMRE), Universidad de La Habana, Colina Universitaria, 10400 La Habana, Cuba b Facultad de Fı´sica, Universidad de La Habana, Colina Universitaria, 10400 La Habana, Cuba c Blackett Laboratory, Physics Department, Imperial College London, London SW7 2BW, UK Received 10 January 2007; accepted 3 March 2007 Available online 16 April 2007
Abstract The quantum efficiency and photocurrent for AlGaAs quantum well solar cell is calculated and compared with experimental results obtaining good agreement. The conversion efficiency as a function of Al composition in barriers and wells is presented showing that there is a wide range of Al composition barrier and Al composition well where the QWSC efficiency is always higher than corresponding homogeneous p–i–n cell without quantum wells. We also show that for up to 15 wells in the intrinsic region an efficiency enhancement for the QWSC over the baseline cell is obtained. r 2007 Elsevier Ltd. All rights reserved. PACS: 68.65.Fg; 73.21.Fg; 73.63.Hs; 84.60.Jt; 85.30.De; 85.35.Be Keywords: Quantum well; Solar cell; AlGaAs; Conversion efficiency; Modelling
1. Introduction The quantum well solar cells (QWSC) are a different approach to multi-band gap solar cells where a multiquantum well system is grown in the intrinsic region of a p–i–n structure [1,2]. The gap of the wells must be narrower than the gap of the baseline p–i–n solar cells in order to insert additional energy levels between valence and conduction bands in the host semiconductor and hence to absorb photons with energies lower than the energy gap of the host material. The output voltage of the QWSC is dominated by the wider band-gap barrier material, the recombination in the wells and in the barrier-well interfaces. The absorption edge, the spectral response and hence short-circuit current are determined by the width and depth of the quantum well. Longer wavelength light can be absorbed by making the quantum well deeper and generating a higher photocurrent.
Corresponding author.
E-mail addresses: jcrimada@fisica.uh.cu (J.C. Rimada), luisman@fisica.uh.cu (L. Herna´ndez). 0026-2692/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.03.007
However, deeper wells increase the recombination rate and consequently a decrease of the output voltage is expected. Thus, the depth of the wells gives rise to a tradeoff between the photocurrent and the output voltage. The effect of quantum well thickness is less clear. Widening the well, the absorption edge decreases and more energy levels are allowed, so enlarging the absorption per well, but a smaller number of wells fit in the intrinsic region. On the other hand, a narrower well absorbs less photons but a greater number of wells can be contained in the intrinsic region. Furthermore, increasing the quantum well width increases the recombination rate and consequently the open-circuit voltage decreases. On increasing the number of the wells, the absorption and photocurrent increase but the recombination in the interfaces also increases leading to a decrease in the output voltage. For these reasons, it is necessary to optimize the cell design in order to achieve enhancements of conversion efficiencies over single band-gap cells. Different theoretical models have been developed to understand the performance of the QWSC and reviews of the theory have been published by Nelson [3], Barnham et al. [4] and Anderson [5]. At present, a controversy exists
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J.C. Rimada et al. / Microelectronics Journal 38 (2007) 513–518
whether the QWSC conversion efficiency would reach beyond that of the baseline bulk device of optimum band gap in the case of radiative recombination dominance. Arau´jo and Martı´ , using thermodynamic arguments, established that the ideal QWSC cannot exceed the efficiency of ideal ordinary solar cell [6]. However, Barnham et al. [2,4] reported experimental results showing enhancement in the QWSC efficiency and they argued that their observations of quasi-Fermi level variation suggest that the assumption made by Arau´jo and Martı´ (spatially constant quasi-Fermi level), is not applicable to the solar cells. A model for QWSC, without considering the escape and capture rates from the wells, was developed by Connolly et al. [7], in which the dark current was characterized in terms of the quantum well density of states. Their calculated dark current compared to the experimental results showed that the dark current was systematically overestimated. They proposed that this suggested a decreased quasi-Fermi level separation between electrons and holes in the wells similar to that found in previous work [8]. Again using thermodynamic arguments, Luque et al. [9] concluded that the ideal QWSC cannot exceed the efficiency of ideal ordinary solar cells unless carriers in the well are pumped by the absorption of a second photon. Recently, Ramey and Khoie [10] showed that incorporation of multiple quantum wells in the intrinsic region of p–i–n solar cell can improve the conversion efficiency and they found out that Al0:1 Ga0:9 As=GaAs cell with multiple quantum wells of 15 nm is more efficient than to corresponding cell without quantum wells. A theoretical model developed by Rimada and Hernandez [11,12] showed that the insertion of multi-quantum wells into the depletion region of a p–i (MQW)–n AlX Ga1X As solar cell can enhance the conversion efficiencies. Open-circuit voltages short-circuit current densities, I–V curves and conversion efficiencies were calculated as functions of the well and barrier band gaps, width and depth of the wells, number of wells in the intrinsic region and the recombination rate in the interfaces. Subsequently, Lade and Zahedi [13] revised and extended the ideal model and found that the absorption was overestimated in the original model of Rimada and Hernandez, although also they found a good agreement with previous experimental results for the opencircuit voltage of AlGaAs/GaAs QWSCs [14]. The present article extends the improved theoretical ideal model reported in Ref. [15] for QWSCs where both barrier and well materials are made of AlGaAs. We show that the insertion of multi-quantum wells into the depletion region of a p–i (MQW)–n AlGaAs solar cell can enhance the conversion efficiency. The cell materials are AlX Ga1X As for the host cell and barriers and AlY Ga1Y As for the wells, where 0pypxp0:35 in order to ensure minor gap for the well material and direct gap. The quantum efficiency for AlGaAs QWSC was calculated and compared with available data from the
group at Imperial College London, obtaining good agreement. The photocurrent then is calculated from the quantum efficiency spectrum and compared with experimental results. We study the conversion efficiency as a function of Al composition in barriers and wells as reported below. From this dependence, it can be determined that for wide Al composition range the conversion efficiencies of the QWSC are higher than the corresponding homogeneous p–i–n solar cell. This work also shows that up to 15 wells in the intrinsic region an efficiency enhancement for the QWSC over the baseline cell is obtained. 2. Model A QWSC with N W wells each of length LW in the intrinsic region of length W with barrier band gap EgB and well band gap EgW is studied. The p–n regions are uniformly doped. Under these conditions, the current– voltage relation of the MQW cell is given by [11]: qV J MQW ¼ J 0 ð1 þ rR bÞ exp 1 kT qV þ ðarNR þ J S Þ exp ð1Þ 1 J PH , 2kT where rR and rNR are the ‘‘radiative enhancement ratio’’ and ‘‘non-radiative enhancement ratio’’, respectively, and represent the increment in the net intrinsic region radiative and non-radiative recombinations due to the insertion of the quantum wells, a and b are parameters also defined by Anderson [16]. The J S is the surface recombination current and J PH is the photocurrent. The term J S was introduced to represent the well-barrier interface recombination, which is characterized by the recombination velocity nS : DE J S ¼ 2N W qniB gDOS nS exp , (2) kT where DE ¼ EgB EgW , and gDOS ¼ gW =gB is the density of states enhancement factor, with gW and gB as the effective volume densities of states for the wells and barriers and niB is the equilibrium intrinsic carrier concentration for the baseline cell material. The photocurrent J PH is calculated from the external quantum efficiency of the cell (QE). The p-region and n-region contribution to QE was classically evaluated solving the carrier transport equations at room temperature within the minority carrier and depletion approximations [17]. The contribution of photo-generated carriers in the intrinsic region to QE values is calculated by the expression n X QEðlÞ ¼ ½1 RðlÞ exp ai z i ½1 expðaB W NaW Þ , ð3Þ where RðlÞ is the surface reflectivity spectrum, the first exponential factor is due to the attenuation of light in the precedent layers of the cell, ai and zi are the absorption
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coefficient and the width of the precedent layers, respectively, the aB is the absorption coefficient of the bulk barrier material, N is the number of wells and aW is the non-dimensional quantum well absorption coefficient, used for energies below the barrier band gap. Following Bastard [18], we calculate the density of states for the single quantum well within the envelope function approximation. When mixing between light and heavy valence sub-bands is neglected, the absorption coefficient can be calculated as follows: aW ðEÞ ¼ aW L, X X aen hhm ðEÞ þ aen lhm ðEÞ, aW ðEÞ ¼ n;m
ð4Þ ð5Þ
n;m
P P where n;m aen hhm ðEÞ and n;m aen lhm ðEÞ are the absorption coefficients due to electron-heavy hole and electronlight hole transitions to conduction band, respectively; aW is the well absorption coefficient and L is the ‘‘quantum thickness of the heterostructure’’ [18]. To match accurately with experimental data in the long wavelength region the exciton absorption is considered in the theoretical calculation. The exciton binding energies are analytically evaluated in the framework of fractional–dimensional space [19]. Once the total QE is calculated by means of the AM1.5, [20] solar spectrum F ðlÞ the photocurrent is determined by integration: Z J PH ¼ q F ðlÞQETOTAL ðlÞdl (6) and then, Eq. (1) is completely determined. 3. Results and discussion As a test of our model we compare the QE calculated with the experimental values of G951 ðAl0:33 Ga0:67 As= GaAsÞ QWSC sample from the research group at Imperial College London. Table 1 displays the pertinent features of several solar cells that were used to compare our model with experimental parameters. The absorption spectrum of bulk AlX Ga1X As was generated from the GaAs spectrum aðlÞ, for which comprehensive data are available, using the same nonlinear shift of the energy axis reported by Paxman et al. [21]. The expressions for the generation of the bulk absorption spectrum and the values of AlGaAs parameters used in the calculation are shown in Table 2. The internal quantum efficiency for G951 QWSC is calculated as
515
function of energy and is matched up to experimental curve in Fig. 1. A good fit is achieved between experimental and modelled spectra where only the QWSC growth parameters were used, without any fitting parameter. Although the fundamental contribution is due to p-region, the deconvolved spectra clearly show that the existence of quantum wells in the intrinsic region increases the QE values in the short wavelength region and consequently the short-circuit current should increase. Good agreement between modelled and experimental QE spectra was observed for all solar cells reported in Table 3. We determined the photocurrent by integration of Eq. (6) with modelled and experimental QE and the results are shown in Table 3. For the studied solar cells, the difference between both photocurrents did not overcome 10%. In a previous work [15], we compared the calculated open-circuit voltages with experimental values obtaining also good agreement showing good agreement. The dependence of conversion efficiency on quantum well and baseline band gaps are examined as a function of Al composition of barrier (xB ) and well (xW ) in the AlGaAs QWSC. The calculations were performed varying Al fraction well from GaAs to AlGaAs composition corresponding to a homogeneous p–i–n solar cell. Unfortunately, we do not have available experimental data for conversion efficiencies in order to compare the absolute values obtained by our model, then we use the normalized efficiency Z defined as the ratio between QWSC efficiency and its equivalent homogeneous p–i–n solar cell efficiency, in order to investigate the relative behaviour of the conversion efficiency. The effects of the barrier and well band gap energies upon the normalized efficiency Z are shown in Fig. 2. These values were calculated for QWSC with well width of 15 nm and 15 wells where the efficiency reaches a maximum as it was shown in a previous work [15]. The greatest Z is attained for Al0:1 Ga0:9 As=GaAs QWSC. This agrees with the results reported in Ref. [10]. An approximately 20% of efficiency enhancement is obtained between this QWSC structure and its equivalent baseline cell. It can also be observed from Fig. 2 that for high Al composition in the barrier, the efficiency decreases with the increase of well depth. However, note that there is a wide range of Al composition barrier (0:05oxB o0:3) and Al composition well (0oxW oxB ) where the QWSC efficiency is always
Table 1 Details of cell structure Cell
Cap (mm)
p layer (mm)
p doping (cm3 )
n layer (mm)
n doping (cm3 )
i layer (mm)
Number of wells
Well width (nm)
G946 QT76 G951 QT468A QT229
0.017 0.02 0.02 0.04 0.045
0.15 0.3 0.15 0.15 0.5
1:3 1018 7 1017 1:3 1018 9 1017 2 1018
0.46 0.6 0.46 0.6 0.5
1:3 1018 3 1017 1:3 1018 2:5 1017 6 1018
0.51 0.48 0.81 0.48 0.8
50 30 50 30 50
8.5 8.7 8.5 8.4 10
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516
Table 2 Material data and references used for modelling of AlGaAs/AlGaAs QWSCs Parameter
Analytic expression
Reference
Absorption (aðlÞ) coefficient
aX ðlÞ ¼ aGaAs ðl0 Þ E ¼ hc=l E EgðxÞ ¼ E 0 Egð0Þ 0:62x½E Egð0Þ0:5
[21]
Dielectric constant ()
¼ ð13:1 2:2xÞ0
[22]
Mobility
log10 ðme Þ ¼ ½1:5545 þ 0:0016 þ ð0:735 þ 0:0013xÞlog10 ðNaÞ 2 300 cm ð0:0253 þ 0:0052xÞlog10 ðNaÞ2 T Vs
[22]
(me ; mh )
log10 ðmh Þ ¼ ½9:723 þ 0:0095 þ ð1:576 þ 0:0012xÞlog10 ðNdÞ 3=4 2 300 cm ð0:0507 þ 0:0034xÞlog10 ðNdÞ2 T Vs
[22]
Minority carrier
Le ¼
Diffusion length
Lh ¼
me ðNa; xÞ 1=2 expð9:72xÞ½210:06 þ 27:254 log10 ðNaÞ me ðNa; 0Þ 0:87 T 0:850 log10 ðNaÞ2 mm 300 mh ðNd; xÞ 1=2 expð9:72xÞ½116:92 þ 14:466 log10 ðNdÞ mh ðNd; 0Þ T mm 0:438 log10 ðNdÞ2 300
[22]
[22]
(Le ; Lh ) Effective masses (me ; mlh ; mhh )
me ¼ ð0:0632 þ 0:0856x þ 0:0231x2 Þm0 mlh ¼ ð0:088 þ 0:0372x þ 0:0163x2 Þm0 mhh ¼ ð0:50 þ 0:2xÞm0
Interface recombination rate (nS )
30 cm/s
Incident spectrum
AM1.5
Internal Quantum Efficiency
[20]
Experiment intrinsic region base region emitter region Theory
0.6 0.5 0.4 0.3 0.2 0.1 0 2
[23]
3
Energy (eV) Fig. 1. Experimental and modelled IQE for G951 QWSC, the contributions for p, i and n layers are shown separately.
higher than corresponding homogeneous p–i–n cell without quantum wells. Fig. 3 shows the dependence of the conversion efficiency on the number of wells in the intrinsic region of the p–i(MQW)–n solar cells. In this figure, the Al compositions of wells and barriers are xW ¼ 0 and xB ¼ 0:1, respectively, the width of wells is LW ¼ 15 nm and the width of the intrinsic region is W ¼ 0:5 mm, for up to 15 wells in order to avoid tunnelling transitions. For these parameter values and with used material properties, Fig. 3 shows an efficiency enhancement overall for the QWSC over the baseline cell in contrast to results reported in the work of Lade and Zahedi [13] where the QWSC efficiency never overcomes the baseline solar cell efficiency for any value of the number of wells. These authors, as stated above, revised and extended the first version of our model, and found an expression for the radiative recombination coefficient for the quantum wells. In our opinion by using
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4. Summary
Table 3 Experimental and calculated photocurrent of AlGaAs QWSCs Photocurrent (A=m2 )
Sample
G946 QT76 G951 QT468A QT229
Calculated
Experimental
87.8 76.0 112.0 76.8 18.9
82.2 81.5 132.8 77.0 20.6
0.325
We
ll (A
0.2
l fr
act
0.1
ion )
1.2
1
0.1
0.2
0.8 0.025
Normalized Efficiency
0
0.3
517
n) rrier (Al fractio
Ba
Fig. 2. QWSC normalized efficiency as function of the aluminium composition for barrier and well for N W ¼ 15 wells, LW ¼ 15 nm and W ¼ 0:5 mm.
Extending and improving the theoretical ideal model reported in Refs. [11,12] we have shown that the insertion of multiple quantum wells into the depletion region of a p–i(MQW)–n AlX Ga1X As solar cell can enhance the conversion efficiency compared with the baseline cell. The quantum efficiency and the photocurrent for the AlGaAs QWSC was calculated and compared with experimental results obtaining good agreement. These results together with previous comparisons of the calculated open-circuit voltage with experimental values confirm the reliability of the improved model presented in this work. We have studied the conversion efficiency behaviour as a function of Al composition in barriers and wells as reported in Fig. 2, compared with the efficiency of the baseline solar cells without quantum wells. From this dependence, it can be determined that for QWSC of AlGaAs, there is a wide range of Al composition barrier (0:05oxB o0:3) and Al composition well (0oxW oxB ) where the QWSC efficiency is always higher to that of the corresponding homogeneous p–i–n cell without quantum wells. We have also shown that at least until 15 wells in the intrinsic region an efficiency enhancement for the QWSC over the baseline cell is obtained. The current study has only examined QWSCs of AlGaAs, therefore it would be interesting to apply the model to other QWSC material systems and structures. For example, the more recent strain-balanced QWSC of InGaAsP, which have a gap close to the optimal for the AM1.5 spectrum and structures where the quantum wells interact to form superlattices.
References
Fig. 3. Normalized conversion efficiency as function of number of wells. The Al composition of wells and barriers are xW ¼ 0 and xB ¼ 0:1, respectively, the width of wells is LW ¼ 15 nm and the width of intrinsic region is W ¼ 0:5 mm.
that expression they overestimate the radiative recombination in the quantum wells thereby obtaining lower values in the conversion efficiency.
[1] K.W.J. Barnham, C. Duggan, A new approach to high-efficiency multi-bandgap solar cells, J. Appl. Phys. 67 (7) (1990) 3490–3493. [2] K.W.J. Barnham, J.P. Connolly, N. Ekins-Daukes, B. Kluftinger, J. Nelson, C. Rohr, Recent results on quantum well solar cells, J. Mater. Sci. Mater. Electron. 21 (2000) 531–536. [3] J. Nelson, Quantum-well structures for photovoltaic energy conversion, in: Thin Films, vol. 21, Academic Press, New York, 1995, pp. 311–368. [4] K.W.J. Barnham, I. Ballard, J.P. Connolly, N.J. Ekins-Daukes, B.G. Kluftinger, J. Nelson, C. Rohr, Quantum well solar cells, Physica E 14 (2002) 27–36. [5] N.G. Anderson, On quantum well solar cell efficiencies, Physica E 14 (2003) 126–131. [6] G.L. Arau´jo, A. Martı´ , Absolute limiting efficiencies for photovoltaic energy conversion, Solar Energy Mater. Solar Cells 33 (1994) 213–240. [7] J.P. Connolly, J. Nelson, K.W.J. Barnham, I. Ballard, C. Roberts, J.S. Roberts, C.T. Foxon, Simulating multiple quantum well solar cells, in: Conference Records of the 28th IEEE Photovoltaic Specialists, NJ, 2000. [8] J. Nelson, J. Barnes, E. Tsui, K.W.J. Barnham, T. Foxon, T. Cheng, J.S. Roberts, Observation of suppressed radiative recombination in single quantum well p–i–n photodiodes, J. Appl. Phys. 82 (12) (1997) 6240–6246.
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[9] A. Luque, A. Mart, L. Cuadra, Thermodynamic consistency of subbandgap absorbing solar cell proposals, IEEE Trans. Electron Devices 48 (2001) 2118–2124. [10] S.M. Ramey, R. Khoie, Modelling of multiple quantum well solar cells including capture, escape, and recombination of photo-excited carriers in quantum wells, IEEE Trans. Electron Devices 50 (2001) 1179–1188. [11] J.C. Rimada, L. Hernandez, Modeling of ideal AlGaAs quantum well solar cells, Microelectron. J. 32 (2001) 719–723. [12] J.C. Rimada, L. Hernandez, A new approach to ideal AlGaAs MQW solar cells, Mod. Phys. Lett. B 15 (17) (2001) 778–781. [13] S.J. Lade, A. Zahedi, A revised ideal model for AlGaAs/GaAs quantum well solar cells, Microelectron. J. 35 (5) (2004) 401–410. [14] E. Aperathitis, Z. Hatzopoulos, M. Kayambaki, V. Foukaraki, M. Ruzinsky, V. Saly, P. Sirotny, A.C. Varonides, P. Panayotatos, 1 cm 1 cm GaAs/AlGaAs MQW solar cells under one sun and concentrated sunlight, in: Conference Record of the Twenty-eighth IEEE Photovoltaic Specialists Conference, 2000. [15] J.C. Rimada, L. Hernandez, K.W.J. Barnham, J.P. Connolly, Quantum and conversion efficiencies calculation of AlGaAs/GaAs multiple quantum well solar cells, Phys. Status Solidi B 242 (9) (2005) 1842–1845. [16] N.G. Anderson, Ideal theory of quantum well solar cells, J. Appl. Phys. 78 (3) (1995) 1850–1861.
[17] H.J. Hovel, Semiconductor and Semimetals: Solar Cells, vol. 11, Academic Press, New York, 1975. [18] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Editions de Physique, Paris, 1988. [19] H. Mathieu, P. Lefebre, P. Christol, Simple analytical method for calculating exciton binding energies in semiconductor quantum wells, Phys. Rev. B 46 (1) (1992) 4092–4101. [20] ASTM G173-03 Standard Tables for Reference Solar Spectral Irradiances: Direct Normal and Hemispherical on 37 C Tilted Surface, ASTM International. For referenced ASTM standards, visit the ASTM website hwww.astm.orgi, or contact ASTM Customer Service at [email protected]. For Annual Book of ASTM Standards volume information, refer to the standard’s Document Summary page on the ASTM website. [21] M. Paxman, J. Nelson, B. Braun, J.P. Connolly, K.W.J. Barnham, C.T. Foxon, J.S. Roberts, Modeling the spectral response of the quantum well solar cell, J. Appl. Phys. 74 (1) (1993) 614–621. [22] H.C. Hamaker, Computer modeling study of the effects of inhomogeneous doping and/or composition in GaAs solar cell devices, J. Appl. Phys. 58 (6) (1985) 2344–2351. [23] E.H. Li, Material parameters of InGaAsP and InAlGaAs systems for use in quantum well structures at low and room temperatures, Physica E 5 (2000) 215–273.
Microelectronics Journal 38 (2007) 513–518 www.elsevier.com/locate/mejo
Conversion efficiency enhancement of AlGaAs quantum well solar cells J.C. Rimadaa,, L. Herna´ndezb, J.P. Connollyc, K.W.J. Barnhamc a
Laboratorio de Celdas Solares, Instituto de Ciencia y Tecnologı´a de los Materiales (IMRE), Universidad de La Habana, Colina Universitaria, 10400 La Habana, Cuba b Facultad de Fı´sica, Universidad de La Habana, Colina Universitaria, 10400 La Habana, Cuba c Blackett Laboratory, Physics Department, Imperial College London, London SW7 2BW, UK Received 10 January 2007; accepted 3 March 2007 Available online 16 April 2007
Abstract The quantum efficiency and photocurrent for AlGaAs quantum well solar cell is calculated and compared with experimental results obtaining good agreement. The conversion efficiency as a function of Al composition in barriers and wells is presented showing that there is a wide range of Al composition barrier and Al composition well where the QWSC efficiency is always higher than corresponding homogeneous p–i–n cell without quantum wells. We also show that for up to 15 wells in the intrinsic region an efficiency enhancement for the QWSC over the baseline cell is obtained. r 2007 Elsevier Ltd. All rights reserved. PACS: 68.65.Fg; 73.21.Fg; 73.63.Hs; 84.60.Jt; 85.30.De; 85.35.Be Keywords: Quantum well; Solar cell; AlGaAs; Conversion efficiency; Modelling
1. Introduction The quantum well solar cells (QWSC) are a different approach to multi-band gap solar cells where a multiquantum well system is grown in the intrinsic region of a p–i–n structure [1,2]. The gap of the wells must be narrower than the gap of the baseline p–i–n solar cells in order to insert additional energy levels between valence and conduction bands in the host semiconductor and hence to absorb photons with energies lower than the energy gap of the host material. The output voltage of the QWSC is dominated by the wider band-gap barrier material, the recombination in the wells and in the barrier-well interfaces. The absorption edge, the spectral response and hence short-circuit current are determined by the width and depth of the quantum well. Longer wavelength light can be absorbed by making the quantum well deeper and generating a higher photocurrent.
Corresponding author.
E-mail addresses: jcrimada@fisica.uh.cu (J.C. Rimada), luisman@fisica.uh.cu (L. Herna´ndez). 0026-2692/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.03.007
However, deeper wells increase the recombination rate and consequently a decrease of the output voltage is expected. Thus, the depth of the wells gives rise to a tradeoff between the photocurrent and the output voltage. The effect of quantum well thickness is less clear. Widening the well, the absorption edge decreases and more energy levels are allowed, so enlarging the absorption per well, but a smaller number of wells fit in the intrinsic region. On the other hand, a narrower well absorbs less photons but a greater number of wells can be contained in the intrinsic region. Furthermore, increasing the quantum well width increases the recombination rate and consequently the open-circuit voltage decreases. On increasing the number of the wells, the absorption and photocurrent increase but the recombination in the interfaces also increases leading to a decrease in the output voltage. For these reasons, it is necessary to optimize the cell design in order to achieve enhancements of conversion efficiencies over single band-gap cells. Different theoretical models have been developed to understand the performance of the QWSC and reviews of the theory have been published by Nelson [3], Barnham et al. [4] and Anderson [5]. At present, a controversy exists
ARTICLE IN PRESS 514
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whether the QWSC conversion efficiency would reach beyond that of the baseline bulk device of optimum band gap in the case of radiative recombination dominance. Arau´jo and Martı´ , using thermodynamic arguments, established that the ideal QWSC cannot exceed the efficiency of ideal ordinary solar cell [6]. However, Barnham et al. [2,4] reported experimental results showing enhancement in the QWSC efficiency and they argued that their observations of quasi-Fermi level variation suggest that the assumption made by Arau´jo and Martı´ (spatially constant quasi-Fermi level), is not applicable to the solar cells. A model for QWSC, without considering the escape and capture rates from the wells, was developed by Connolly et al. [7], in which the dark current was characterized in terms of the quantum well density of states. Their calculated dark current compared to the experimental results showed that the dark current was systematically overestimated. They proposed that this suggested a decreased quasi-Fermi level separation between electrons and holes in the wells similar to that found in previous work [8]. Again using thermodynamic arguments, Luque et al. [9] concluded that the ideal QWSC cannot exceed the efficiency of ideal ordinary solar cells unless carriers in the well are pumped by the absorption of a second photon. Recently, Ramey and Khoie [10] showed that incorporation of multiple quantum wells in the intrinsic region of p–i–n solar cell can improve the conversion efficiency and they found out that Al0:1 Ga0:9 As=GaAs cell with multiple quantum wells of 15 nm is more efficient than to corresponding cell without quantum wells. A theoretical model developed by Rimada and Hernandez [11,12] showed that the insertion of multi-quantum wells into the depletion region of a p–i (MQW)–n AlX Ga1X As solar cell can enhance the conversion efficiencies. Open-circuit voltages short-circuit current densities, I–V curves and conversion efficiencies were calculated as functions of the well and barrier band gaps, width and depth of the wells, number of wells in the intrinsic region and the recombination rate in the interfaces. Subsequently, Lade and Zahedi [13] revised and extended the ideal model and found that the absorption was overestimated in the original model of Rimada and Hernandez, although also they found a good agreement with previous experimental results for the opencircuit voltage of AlGaAs/GaAs QWSCs [14]. The present article extends the improved theoretical ideal model reported in Ref. [15] for QWSCs where both barrier and well materials are made of AlGaAs. We show that the insertion of multi-quantum wells into the depletion region of a p–i (MQW)–n AlGaAs solar cell can enhance the conversion efficiency. The cell materials are AlX Ga1X As for the host cell and barriers and AlY Ga1Y As for the wells, where 0pypxp0:35 in order to ensure minor gap for the well material and direct gap. The quantum efficiency for AlGaAs QWSC was calculated and compared with available data from the
group at Imperial College London, obtaining good agreement. The photocurrent then is calculated from the quantum efficiency spectrum and compared with experimental results. We study the conversion efficiency as a function of Al composition in barriers and wells as reported below. From this dependence, it can be determined that for wide Al composition range the conversion efficiencies of the QWSC are higher than the corresponding homogeneous p–i–n solar cell. This work also shows that up to 15 wells in the intrinsic region an efficiency enhancement for the QWSC over the baseline cell is obtained. 2. Model A QWSC with N W wells each of length LW in the intrinsic region of length W with barrier band gap EgB and well band gap EgW is studied. The p–n regions are uniformly doped. Under these conditions, the current– voltage relation of the MQW cell is given by [11]: qV J MQW ¼ J 0 ð1 þ rR bÞ exp 1 kT qV þ ðarNR þ J S Þ exp ð1Þ 1 J PH , 2kT where rR and rNR are the ‘‘radiative enhancement ratio’’ and ‘‘non-radiative enhancement ratio’’, respectively, and represent the increment in the net intrinsic region radiative and non-radiative recombinations due to the insertion of the quantum wells, a and b are parameters also defined by Anderson [16]. The J S is the surface recombination current and J PH is the photocurrent. The term J S was introduced to represent the well-barrier interface recombination, which is characterized by the recombination velocity nS : DE J S ¼ 2N W qniB gDOS nS exp , (2) kT where DE ¼ EgB EgW , and gDOS ¼ gW =gB is the density of states enhancement factor, with gW and gB as the effective volume densities of states for the wells and barriers and niB is the equilibrium intrinsic carrier concentration for the baseline cell material. The photocurrent J PH is calculated from the external quantum efficiency of the cell (QE). The p-region and n-region contribution to QE was classically evaluated solving the carrier transport equations at room temperature within the minority carrier and depletion approximations [17]. The contribution of photo-generated carriers in the intrinsic region to QE values is calculated by the expression n X QEðlÞ ¼ ½1 RðlÞ exp ai z i ½1 expðaB W NaW Þ , ð3Þ where RðlÞ is the surface reflectivity spectrum, the first exponential factor is due to the attenuation of light in the precedent layers of the cell, ai and zi are the absorption
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coefficient and the width of the precedent layers, respectively, the aB is the absorption coefficient of the bulk barrier material, N is the number of wells and aW is the non-dimensional quantum well absorption coefficient, used for energies below the barrier band gap. Following Bastard [18], we calculate the density of states for the single quantum well within the envelope function approximation. When mixing between light and heavy valence sub-bands is neglected, the absorption coefficient can be calculated as follows: aW ðEÞ ¼ aW L, X X aen hhm ðEÞ þ aen lhm ðEÞ, aW ðEÞ ¼ n;m
ð4Þ ð5Þ
n;m
P P where n;m aen hhm ðEÞ and n;m aen lhm ðEÞ are the absorption coefficients due to electron-heavy hole and electronlight hole transitions to conduction band, respectively; aW is the well absorption coefficient and L is the ‘‘quantum thickness of the heterostructure’’ [18]. To match accurately with experimental data in the long wavelength region the exciton absorption is considered in the theoretical calculation. The exciton binding energies are analytically evaluated in the framework of fractional–dimensional space [19]. Once the total QE is calculated by means of the AM1.5, [20] solar spectrum F ðlÞ the photocurrent is determined by integration: Z J PH ¼ q F ðlÞQETOTAL ðlÞdl (6) and then, Eq. (1) is completely determined. 3. Results and discussion As a test of our model we compare the QE calculated with the experimental values of G951 ðAl0:33 Ga0:67 As= GaAsÞ QWSC sample from the research group at Imperial College London. Table 1 displays the pertinent features of several solar cells that were used to compare our model with experimental parameters. The absorption spectrum of bulk AlX Ga1X As was generated from the GaAs spectrum aðlÞ, for which comprehensive data are available, using the same nonlinear shift of the energy axis reported by Paxman et al. [21]. The expressions for the generation of the bulk absorption spectrum and the values of AlGaAs parameters used in the calculation are shown in Table 2. The internal quantum efficiency for G951 QWSC is calculated as
515
function of energy and is matched up to experimental curve in Fig. 1. A good fit is achieved between experimental and modelled spectra where only the QWSC growth parameters were used, without any fitting parameter. Although the fundamental contribution is due to p-region, the deconvolved spectra clearly show that the existence of quantum wells in the intrinsic region increases the QE values in the short wavelength region and consequently the short-circuit current should increase. Good agreement between modelled and experimental QE spectra was observed for all solar cells reported in Table 3. We determined the photocurrent by integration of Eq. (6) with modelled and experimental QE and the results are shown in Table 3. For the studied solar cells, the difference between both photocurrents did not overcome 10%. In a previous work [15], we compared the calculated open-circuit voltages with experimental values obtaining also good agreement showing good agreement. The dependence of conversion efficiency on quantum well and baseline band gaps are examined as a function of Al composition of barrier (xB ) and well (xW ) in the AlGaAs QWSC. The calculations were performed varying Al fraction well from GaAs to AlGaAs composition corresponding to a homogeneous p–i–n solar cell. Unfortunately, we do not have available experimental data for conversion efficiencies in order to compare the absolute values obtained by our model, then we use the normalized efficiency Z defined as the ratio between QWSC efficiency and its equivalent homogeneous p–i–n solar cell efficiency, in order to investigate the relative behaviour of the conversion efficiency. The effects of the barrier and well band gap energies upon the normalized efficiency Z are shown in Fig. 2. These values were calculated for QWSC with well width of 15 nm and 15 wells where the efficiency reaches a maximum as it was shown in a previous work [15]. The greatest Z is attained for Al0:1 Ga0:9 As=GaAs QWSC. This agrees with the results reported in Ref. [10]. An approximately 20% of efficiency enhancement is obtained between this QWSC structure and its equivalent baseline cell. It can also be observed from Fig. 2 that for high Al composition in the barrier, the efficiency decreases with the increase of well depth. However, note that there is a wide range of Al composition barrier (0:05oxB o0:3) and Al composition well (0oxW oxB ) where the QWSC efficiency is always
Table 1 Details of cell structure Cell
Cap (mm)
p layer (mm)
p doping (cm3 )
n layer (mm)
n doping (cm3 )
i layer (mm)
Number of wells
Well width (nm)
G946 QT76 G951 QT468A QT229
0.017 0.02 0.02 0.04 0.045
0.15 0.3 0.15 0.15 0.5
1:3 1018 7 1017 1:3 1018 9 1017 2 1018
0.46 0.6 0.46 0.6 0.5
1:3 1018 3 1017 1:3 1018 2:5 1017 6 1018
0.51 0.48 0.81 0.48 0.8
50 30 50 30 50
8.5 8.7 8.5 8.4 10
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516
Table 2 Material data and references used for modelling of AlGaAs/AlGaAs QWSCs Parameter
Analytic expression
Reference
Absorption (aðlÞ) coefficient
aX ðlÞ ¼ aGaAs ðl0 Þ E ¼ hc=l E EgðxÞ ¼ E 0 Egð0Þ 0:62x½E Egð0Þ0:5
[21]
Dielectric constant ()
¼ ð13:1 2:2xÞ0
[22]
Mobility
log10 ðme Þ ¼ ½1:5545 þ 0:0016 þ ð0:735 þ 0:0013xÞlog10 ðNaÞ 2 300 cm ð0:0253 þ 0:0052xÞlog10 ðNaÞ2 T Vs
[22]
(me ; mh )
log10 ðmh Þ ¼ ½9:723 þ 0:0095 þ ð1:576 þ 0:0012xÞlog10 ðNdÞ 3=4 2 300 cm ð0:0507 þ 0:0034xÞlog10 ðNdÞ2 T Vs
[22]
Minority carrier
Le ¼
Diffusion length
Lh ¼
me ðNa; xÞ 1=2 expð9:72xÞ½210:06 þ 27:254 log10 ðNaÞ me ðNa; 0Þ 0:87 T 0:850 log10 ðNaÞ2 mm 300 mh ðNd; xÞ 1=2 expð9:72xÞ½116:92 þ 14:466 log10 ðNdÞ mh ðNd; 0Þ T mm 0:438 log10 ðNdÞ2 300
[22]
[22]
(Le ; Lh ) Effective masses (me ; mlh ; mhh )
me ¼ ð0:0632 þ 0:0856x þ 0:0231x2 Þm0 mlh ¼ ð0:088 þ 0:0372x þ 0:0163x2 Þm0 mhh ¼ ð0:50 þ 0:2xÞm0
Interface recombination rate (nS )
30 cm/s
Incident spectrum
AM1.5
Internal Quantum Efficiency
[20]
Experiment intrinsic region base region emitter region Theory
0.6 0.5 0.4 0.3 0.2 0.1 0 2
[23]
3
Energy (eV) Fig. 1. Experimental and modelled IQE for G951 QWSC, the contributions for p, i and n layers are shown separately.
higher than corresponding homogeneous p–i–n cell without quantum wells. Fig. 3 shows the dependence of the conversion efficiency on the number of wells in the intrinsic region of the p–i(MQW)–n solar cells. In this figure, the Al compositions of wells and barriers are xW ¼ 0 and xB ¼ 0:1, respectively, the width of wells is LW ¼ 15 nm and the width of the intrinsic region is W ¼ 0:5 mm, for up to 15 wells in order to avoid tunnelling transitions. For these parameter values and with used material properties, Fig. 3 shows an efficiency enhancement overall for the QWSC over the baseline cell in contrast to results reported in the work of Lade and Zahedi [13] where the QWSC efficiency never overcomes the baseline solar cell efficiency for any value of the number of wells. These authors, as stated above, revised and extended the first version of our model, and found an expression for the radiative recombination coefficient for the quantum wells. In our opinion by using
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4. Summary
Table 3 Experimental and calculated photocurrent of AlGaAs QWSCs Photocurrent (A=m2 )
Sample
G946 QT76 G951 QT468A QT229
Calculated
Experimental
87.8 76.0 112.0 76.8 18.9
82.2 81.5 132.8 77.0 20.6
0.325
We
ll (A
0.2
l fr
act
0.1
ion )
1.2
1
0.1
0.2
0.8 0.025
Normalized Efficiency
0
0.3
517
n) rrier (Al fractio
Ba
Fig. 2. QWSC normalized efficiency as function of the aluminium composition for barrier and well for N W ¼ 15 wells, LW ¼ 15 nm and W ¼ 0:5 mm.
Extending and improving the theoretical ideal model reported in Refs. [11,12] we have shown that the insertion of multiple quantum wells into the depletion region of a p–i(MQW)–n AlX Ga1X As solar cell can enhance the conversion efficiency compared with the baseline cell. The quantum efficiency and the photocurrent for the AlGaAs QWSC was calculated and compared with experimental results obtaining good agreement. These results together with previous comparisons of the calculated open-circuit voltage with experimental values confirm the reliability of the improved model presented in this work. We have studied the conversion efficiency behaviour as a function of Al composition in barriers and wells as reported in Fig. 2, compared with the efficiency of the baseline solar cells without quantum wells. From this dependence, it can be determined that for QWSC of AlGaAs, there is a wide range of Al composition barrier (0:05oxB o0:3) and Al composition well (0oxW oxB ) where the QWSC efficiency is always higher to that of the corresponding homogeneous p–i–n cell without quantum wells. We have also shown that at least until 15 wells in the intrinsic region an efficiency enhancement for the QWSC over the baseline cell is obtained. The current study has only examined QWSCs of AlGaAs, therefore it would be interesting to apply the model to other QWSC material systems and structures. For example, the more recent strain-balanced QWSC of InGaAsP, which have a gap close to the optimal for the AM1.5 spectrum and structures where the quantum wells interact to form superlattices.
References
Fig. 3. Normalized conversion efficiency as function of number of wells. The Al composition of wells and barriers are xW ¼ 0 and xB ¼ 0:1, respectively, the width of wells is LW ¼ 15 nm and the width of intrinsic region is W ¼ 0:5 mm.
that expression they overestimate the radiative recombination in the quantum wells thereby obtaining lower values in the conversion efficiency.
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