- Email: [email protected]
AlGaAs multiple quantum well solar cells
Solar Energy Materials & Solar Cells 70 (2001) 49}69
Temperature dependence of photocurrent components on enhanced performance GaAs/AlGaAs multiple quantum well solar cells E. Aperathitis *, A.C. Varonides, C.G. Scott, D. Sand, V. Foukaraki , M. Androulidaki , Z. Hatzopoulos , P. Panayotatos Microelectronics Research Group, Institute of Electronic Structure & Laser, Foundation For Research and Technology-Hellas, P.O. Box 1527, Heraklion 71110, Crete, Greece Physics Department, University of Crete, P.O. Box 2208, Heraklion 71003, Crete, Greece Physics and Electrical Engineering Department, University of Scranton, Scranton, PA 18510-4642, USA Department of Applied Physics, Hull University, Hull HU67RX, UK Rutgers, The State University of New Jersey, Department of Electrical and Computer Engineering, 94 Brett Rd., Piscataway, New Jersey 08854-8058, USA Received 27 August 1999; received in revised form 6 June 2000; accepted 29 September 2000
Abstract The performance of Al Ga As p/i/n solar cells with multiple quantum wells (MQW) of GaAs/Al Ga As in the i-region has been investigated at various temperatures, ranging from !103C to 1003C, and compared with that of conventional solar cells composed of either the quantum well material (GaAs) or the barrier material (Al Ga As) alone. The dark currents of the MQW cells were found to lie between those of the conventional cells. The increase of dark current with temperature was accompanied by a slight decrease of the diode ideality factor. A linear dependence of open-circuit voltage (< ) on temperature was observed for all cells when illuminated with a 100W halogen lamp. < for the MQW cells was found to be independent of the number of wells, lying between the < 's for the two conventional cells. The MQW cells exhibited performance improvement with temperature when compared to the conventional cells and there was a signi"cant enhancement in the short-circuit current with temperature of those MQW cells that exhibited poorer performance at lower temperatures. Theoretical calculations have quanti"ed the contribution of the tunneling current component
* Corresponding author. Tel.: #30-81-394-105; fax: #30-81-394-106. E-mail address: [email protected] (E. Aperathitis). 0927-0248/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 4 8 ( 0 0 ) 0 0 4 1 1 - 6
50
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
to the total observed photocurrent at the various temperatures examined. It was found that tunneling currents are present at all temperatures and can be the dominant component in MQW cells of thinner wells at low temperatures. These results suggest that GaAs/Al Ga As MQW structures, of good-quality material, when processed as conven tional solar cells with antire#ective coatings should deliver more output power under intense illumination than conventional solar cells composed of the quantum well material alone. 2001 Elsevier Science B.V. All rights reserved. Keywords: Multiple quantum wells; GaAs/AlGaAs; Tunneling
1. Introduction The high cost of the state-of-the-art single-junction GaAs-based solar cells has made their use for one-sun terrestrial applications nonviable, despite the fact that the e$ciency achieved in practice for these devices is now close to their theoretical limit. The alternative approach of multi-junction tandem solar cells has yet to overcome its main disadvantage of increased technological complexity. The use of GaAs solar cells might be more economically attractive for terrestrial applications by utilizing concentrated light [1]. Further cell cost reduction and/or e$ciency improvement, however, is still needed for this approach. Attempts at fabricating cost-e$cient solar cells have already been reported by making use of epitaxial lift-o! techniques [2,3] or growth of GaAs solar cell structures on foreign substrates such as Si [4] and Ge [5]. E$ciency improvement of single-junction solar cell requires new device designs [6,7]. One of these approaches is that of multiple quantum well (MQW) solar cells, in which a number of quantum wells are incorporated in the cell [6]. Fig. 1 illustrates such an approach, along with parameters which will be used in the modeling section. In such an arrangement, the i-region of the p}i}n solar cell structure consists of a number of QWs from material of energy gap E , lower than the barrier material, E . Thus, in addition to the normal photogeneration of electrons and holes by the action of those photons that have energy above the energy gap, E of the p- and n-region material, it is possible for photons with energy between E and E to contribute to the photogenerated current, provided that the charge carriers generated in the wells can escape from these wells before they recombine. Previous studies on AlGaAs cells with GaAs wells [8,9] and on GaAs cells with InGaAs wells [10], as well as on CdMnTe devices incorporating a variety of CdTe single and multiple quantum wells [11], have revealed that increase in the shortcircuit current (I ) can, indeed, be achieved as a result of photogeneration of carriers within the quantum wells and, equally important, the subsequent release of these carriers from the well. Unfortunately, the e$ciency improvement of such devices has been found to be limited by a reduction in the open-circuit voltage, (< ), which accompanies the incorporation of the wells. It is clear, however, that this reduction also depends on factors other than the band gap of the well material. There has been some controversy over the reasons for MQW solar cell e$ciency enhancement,
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
51
Fig. 1. A p/i(MQW)/n solar cell structure. Shown are all the parameters used in the modeling section.
particularly on whether it is associated with the nature and dimension of the well material or it is solely due to enhanced I [12}14]. In all investigations, the material quality of the grown MQW structure has been found to be of paramount importance. Furthermore, photoconductivity measurements have demonstrated [15] that the mechanism for the escape of carriers from the QWs does have a thermal component in forward bias. Taking into account that a signi"cant forward bias voltage is developed in an operating solar cell and the fact that the operating temperature of cells under concentrated light, is signi"cantly above room temperature [16], the e$ciency of solar cells with properly designed MQWs is expected to be enhanced in concentrator applications by the greater probability of carrier escape from the wells. Some theoretical calculations as well as results on the temperature behavior of MQW solar cells [1,17,18] or numerical prediction of performance under concentration [19] have previously been reported and we have also given a detailed account of an experimental study [20] The reader is cautioned that the correct level of illumination is 3.3 mW/cm (corrected for re#ection) on the e!ects of temperature on the output performance of p/i/n Al Ga As/GaAs MQW solar cells with di!erent number of wells and the same i-region widths. The aim of the present work is to examine the e!ects of temperature and illumination on several p/i/n Al Ga As/GaAs MQW solar cells of di!erent well widths and i-region thick nesses, and compare them with standard p/i/n Al Ga As and GaAs solar cells
52
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
Table 1 Details of the i-region structure of p/i/n solar cells. Widths: ¸ for the GaAs well, ¸ for the Al Ga As barrier, and ¸ for the i-region Sample
Well period
¸ (nm)
¸ (nm)
GB68 GB69 GB71 GB66 GB73
23 9.0 6.0 40 5.4 3.6 39 15.0 10.0 Conventional p/i/n Al Ga sample Conventional p/i/n GaAs sample (with AR coating)
¸ (m) 0.35 0.36 0.97 0.90 0.80
without quantum wells in the i-region. In addition the contribution of di!erent current components was modeled and investigated at di!erent operating temperatures. Such modeling is necessary in order to provide guidance in cell design with respect to well, barrier and i-region thickness for optimal operation over speci"ed temperature ranges. Some preliminary results were included in a conference presentation [21].
2. Experimental procedure The p/i/n solar cell structures of this investigation were grown in a VG80 H MBE system, on epi-ready 2 n> type (1 0 0) GaAs substrates. The bu!er layer was n>GaAs, Si doped at 2;10 cm\ and 1 m thick. The n-type (Si-doped) and the p-type (Be-doped) Al Ga As regions (1.8;10 cm\) were 150 and 300 nm thick, respectively. High-quality interface between the n-type Al Ga As and the GaAs bu!er layer was achieved by growing a superlattice, consisting of 20 periods of 2.8 nm Al Ga As and 2.7 nm GaAs (Si-doped 2;10 cm\) on the bu!er layer. Two di!erent i-region thicknesses were examined of approximately 0.35 and 0.97 m widths. Details of the i-region of the structures grown for this study are given in Table 1. The ratio of well width to barrier width was kept constant at 1.5 for all MQW samples. For all MQW samples a 30 nm thick Al Ga As spacer was grown on either side of the i-region. Growth was completed with a 40 nm thick p>Al Ga As, 1;10 cm\, window layer followed by the cap layer (50 nm thick p>GaAs, 1;10 cm\). Sample GB66, which was the conventional Al Ga As sample, had an i-region of 0.90 m and no MQWs. The samples were processed, using standard photolithographic techniques, as mesa-isolated diodes of 720 m diameter, resulting in a junction area of 4.1;10\ cm. The top contact metallization, Pt/Ti/Au, had the ring geometry of 520 m inner diameter. The back ohmic contact, (Ge/Au)/Ni/Au, was evaporated on the n>GaAs substrate. Both contacts were annealed initially at 4103C for 20 s and subsequently at 4003C for 2 min. It has been shown, by transmission line
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
53
measurements, that this sequential annealing procedure improves the ohmic properties of both n- and p-type ohmic contacts. No antire#ective coating was applied on the surface of the samples for the purposes of this study. Observation of the as-grown material surface with an optical microscope revealed the presence of oval defects with densities of around 1000 cm\ for all samples, except for sample GB71 which had 2000 defects/cm\. For comparison purposes, an e$cient p/i/n GaAs solar cell (sample GB73), with a 0.8 m thick i-region , 0.3 m thick GaAs p-region (Be-doped, 3;10 cm\) and 2.6 m thick GaAs n-region (Si-doped, 1;10 cm\) was also examined. It was processed in exactly the same way as the other samples but it also received the application of a ZnS/MgF antire#ective coating. A device of the same structure, processed as a 1 cm;1 cm solar cell (with top contact grid and ZnS/MgF antire#ec tive coating) had exhibited 19.4% e$ciency under 100 mW/cm (approximately AM1 spectrum). One should, however, exercise caution in inferring e$ciencies for the other structures of this study from this, because of the much smaller size of the devices of this study. Both dark and illuminated current}voltage characteristics of the diodes were taken, in rough vacuum (10\ mbar), in a temperature-controlled Biorad DL4600 Polaron system interfaced with an HP916 computer. The Biorad DL4600 Polaron system uses heating elements and liquid nitrogen for controlling the temperature via a Pt 100 element for temperature reading ($13C). A 100 W quartz tungsten halogen lamp (colour temperature 3300 K) was used for illumination. The intensity of the light entering the samples was calculated to be 3.3 mW/cm, after correcting for the re#ectance of the uncoated GaAs [22]. The above value is also the illumination level for Ref. [20]. The open-circuit voltage spectral response of the samples was monitored using a /4 double grating computer controlled monochromator (SPEX 0.22 mm) and standard lock-in techniques.
3. Experimental results and discussion 3.1. Temperature dependence of dark I}< characteristics The dark current}voltage (I}<) characteristics of the samples were taken at temperatures of !103C, 03C, 203C, 503C and 1003C. The I}< curves at the two extreme temperatures of !103C and 1003C have been plotted in Fig. 2(a) and (b), respectively. It is observed from these "gures that the dark current for each of the MQW cells lies well below the dark current of the conventional cell GB73 which consists of the well material (GaAs) alone, and above the control cell GB66 that is composed of the barrier material (Al Ga As) alone. This is as expected and is observed not only at lower temperatures as found previously [7,8,13,23}28] but also in the whole range of temperatures examined in this work. Furthermore, the spread in the I}< curves [24] of the MQW cells observed at !103C disappears at elevated temperatures. The increased series resistance for these samples observed at high forward bias is partially due to the fact that these samples
54
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
Fig. 2. Dark I}< characteristics of p/i/n solar cells at the two extreme temperatures of this investigation: (a) !103C and (b) 1003C.
Fig. 3. Temperature dependence of ideality factor of p/i/n solar cells as extracted by "tting the I}< characteristics, such as the ones of Fig. 2, of all temperatures of this investigation.
had not been speci"cally processed to minimize series resistance and partially due to defects associated with the quality of the grown material. The linear part of the dark current curves, which corresponds to voltages of about the same magnitude as the < of the individual diodes, has been "tted by the exponential voltage dependence I"I exp(q
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
55
Fig. 4. Temperature dependence of saturation current of p/i/n/ solar cells as extracted by "tting the I}< characteristics, such as the ones of Fig. 2, of all temperatures of this investigation.
increasing temperature, from n"1.95}2.30 at !103C to n"1.75}1.80 at 1003C. The rate of decrease with temperature for the MQW sample GB69 and conventional sample GB73 is much slower than for the MQW sample GB68. Diode GB71 exhibits a very high ideality factor at low temperatures which reduces to 2 at the highest tested temperature. The high n value at lower temperatures must be attributed to the high density of thermally activated defects observed for these diodes [30}32]. However, it is clear that the diode quality improves with temperature and this improved behavior leads to enhanced solar cell e$ciency, despite the increased dark current, as seen below. Both reverse saturation and recombination currents increase with decreasing semiconductor band gap [7,23]. Thus, the currents of the MQW diodes are higher than those of the conventional sample GB66 which has i-region material consisting only of the barrier material of the MQW diodes (Figs. 2 and 4) and lower than the conventional sample GB73 which consists of the well material alone. Furthermore, as it can be seen in Fig. 4, the rate of increase of I with temperature is similar for all diodes, except for diode GB71. The latter diode exhibited the highest I value at low temperatures and the lowest rate of increase with temperature of all diodes examined. It is speculated that this behavior is due to the higher concentration of growth defects associated with this diode, as previously discussed. 3.2. Temperature dependence of I}< characteristics under illumination An indication of the quality of the quantum wells was obtained by examining the photovoltage at room temperature as a function of wavelength. Fig. 5 shows the open-circuit photovoltage of the MQW samples as a function of photon energy. It should be noted that no bias illumination was used for these measurements. The characteristic excitonic peaks arising from absorption of photons within the quantum
56
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
Fig. 5. Photovoltage as a function of wavelength of p/i/n MQW solar cells.
Fig. 6. Temperature dependence of open-circuit voltage of p/i/n solar cells.
wells is easily discerned. As expected, the MQW samples exhibited excitonic peaks at di!erent photon energies since these samples have wells of di!erent width, thus di!erent level distribution, with the thinner the wells in the sample, the higher the energy of the peak. The e!ect of temperature on the illuminated I}< characteristics of the diodes was monitored in the range of temperatures between !103C and 1003C and the dependence of the open-circuit voltage < , under 3.3 mW/cm illumination, with temper ature is depicted in Fig. 6. It is seen that < for GaAs/Al Ga As MQWs is
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
57
Fig. 7. Photo I}< characteristics of p/i/n solar cells at (a) !103C and (b) 1003C.
lower than for the conventional sample (GB66) which is composed from the barrier material alone but higher than for the conventional sample GB73 which is composed from the well material alone, in the whole range of temperature examined. Similar observations have been reported for room temperature MQW solar cells having 20 wells of GaAs (9 nm thick) with 6 nm thick Al Ga As barriers [24]. Furthermore, a linear dependence of < with temperature can be seen in Fig. 6 for both the conventional samples [16,33], as well as for the MQW devices [34]. The spread among < values for the MQW cells reduces with temperature. This is attributed to the observed (Fig. 3) convergence of the ideality factors of the MQW diodes with temperature. It should be noted that the MQW samples GB68, GB69 and the conventional sample GB73 exhibited similar rates of < decrease with temper ature (1.99}2.10 mV/C), whereas the slowest rate, 1.66 mV/C, was observed for sample GB71. The importance of incorporating the MQWs in the i-region of the p}i}n structures can be further observed in Fig. 7 where the full I}< curves, (and consequently the power delivered by these devices), are depicted, again at the two extreme temperatures of this investigation. It should be noted again that all samples, except GB73, possess no antire#ective coating. Under the temperature and illumination conditions used here, the MQW solar cell GB69 has exhibited comparable short-circuit current I with the control sample GB73 consisting of the well material alone, at all of the temperatures examined in this work. The other two MQW solar cells, which had thicker wells than GB69 and showed poor performance at low temperatures, exhibited a remarkable enhancement in I at elevated temperatures. These observations are an indication that tunneling of the carriers through the barriers dominates the escape process from the wells at lower temperatures [35]. At elevated temperatures, the escape contribution dominates over tunneling giving rise to the observed improvement of I . A theoretical analysis and
58
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
comparison of the various current contributions to the observed photocurrent at di!erent temperatures is given in the next section. Furthermore, the very low "ll factor of sample GB71 at lower temperatures is certainly associated with the high ideality factor this diode exhibited (Fig. 3). Indeed, the improvement of the diode ideality factor with temperature resulted in a similarly improved performance of the device under illumination at higher temperatures, albeit lower than the performance of GB69. The remarkable improvement in the performance of GB68 and particularly GB71 with temperature strongly suggests that the higher the operating temperature of MQW solar cells having thick wells in the i-region, the higher the output performance. Clearly, good-quality material and entire control of processing procedures [14,24,25,30}32] are of paramount importance for utilizing the full potential improvement for these devices. The temperature dependence of a MQW structure similar to sample GB68 of this work, was previously reported [17]. The authors of this particular reference, however, ascribed all improvements to the temperature dependence to the dark current. We speculate that this is due to the di!erence in Al content of the AlGaAs layer in our respective studies. The mole fraction x was )0.3 that was used in Ref. [17] produced shallower well from which carriers would be able to escape with probability of more than 90% even at room temperature. Thus, since the authors of Ref. [17] concentrated on temperatures at and above room temperature they would not, and did not, observe additional escaping with rising temperature. In addition, optimized design of such structures involves several width and compositional parameters. Since tunneling is directly a!ected by the width of the barrier layer, absorption is a!ected by the width of the i-region, and thermionic emission depends on the Al mole fraction, the relative contribution of the various current components need to be delineated at di!erent temperatures. An initial calculation was performed towards this end for the MQW solar cells examined in this work.
4. Modeling 4.1. Calculation of tunneling currents For the quantum well (QW) thickness involved, these devices, because of their rather extensive number of repeat distances, are superlattice-based and thus perpendicular transport may take place via mini-bands. In other words, electrons channel through the superlattice (i.e. with large number of repeat distances) region more easily than the `heaviera holes, thus resulting in e!ective mass "ltering [36]. In addition, quantization of the energy in the wells leads to widening of the useful spectrum of incident wavelengths (Fig. 1). It is demonstrated here that non-zero tunneling currents do exist, as a result of carrier con"nement in sub-bands (in disagreement with Ref. [37]), that these currents are indeed present (in agreement with Refs. [38,39]) at both temperature extremes and that they dominate over thermionic emission at low temperatures [29], while at high temperatures the roles reverse. A "rst-principles calculation of the tunneling current density component J is performed, of the 23
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
59
short-circuit current-density component J , that is due solely to tunneling carriers along the growth direction of the device (as long as the tunneling probability is non-zero), and via the two existing channels (i.e. eigen-states) of the GaAs}AlGaAs quantum well system. The starting point is the general current-density relationship for current densities, which is true for any carrier transport settings J"nqv,
(1)
where n is the concentration of carriers with enough energy to overcome the potential barriers (typically in cm\), v(E) is the velocity of the carriers and q is the single electron charge. The currents sought are due to tunneling processes, through potential barriers, of carriers trapped in quantum wells, hence the factor n of the general expression (1) should be composed of (i) (ii) (iii) (iv)
The trapped carriers' non-zero tunneling probability P(E) The Fermi}Dirac occupation probability f (E) The availability of the two-dimensional electron density of states (DOS) g(E), and The net (after absorption) photo-generation rate, namely G ()e\?HV over the extend of the intrinsic region, where () is the absorption coe$cient of the absorbing material involved.
To investigate the transport properties, once the above factors are placed in the right context, one must evaluate the net current #ow in the i-region. The quantum wells play the role of electron reservoirs, and in such systems one has to consider the net electron #ow from left to right and from right to the left. The di!erential form of Eq. (1) reads now as follows:
J"q v(E) dn,
(1)
where dn is the carrier concentration given below in terms of the availability factor g(E) f (E): dn"dEg(E) f (E).
(1)
In this picture, the current injected from the left-hand side reservoir is
A J "q g(E)P(E) f (E!E )v(E), (2) $ ¸ where A is the cross-section of the device, (), are the absorption coe$cient and the carrier transit time. The absorption coe$cient is generally a function of the incident wavelength, but for GaAs}AlGaAs solar cell systems is constant at &10 cm\). Special attention should be given at the g(E) energy function: carriers trapped in quantum wells essentially comprise two-dimensional systems, and for such systems the density of states (eV\ cm\) is a well-known superlattice DOS: (m*"0.067m is the carriers' e!ective mass, with m the electronic rest mass) m g(E)" * . (3)
60
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
The tunneling probability function P(E) [40] is:
16E(!E) E P(E)" exp !k (1! ,
(4)
where E is the energy of a trapped carrier, is the conduction band discontinuity, and k is the potential barrier unit-less factor (2m* k "2¸ (nm) (nm\) (5)
(with ¸ the barrier width in nm). As seen from Eq. (2), the factor g(E)P(E) f (E!E ) ensures carrier availability in the $ quantum wells, while the occupation Fermi}Dirac factor is given by the following expression: 1 f (E!E )" , (6) $ 1#exp(E!E /k¹) $ where ¹ is the temperature, k is the Boltzmann constant, and E is the quasi-Fermi $ level at the left side of the device (see Fig. 1). In a similar fashion one may construct (mirroring Eq. (2)) the current #ow from the right side of the device
A J "q dx dEG e\?Vg(E)P(E) f (E!E )v(E), (7) $ ¸ where the involved quasi-Fermi level E is shifted relative to E by an amount $ $ equivalent to the open-circuit voltage q< . The net current (density) #ow is then J "J !J 23 A "q dx dEG e\?Vg(E)P(E)[ f (E!E )!f (E!E )]v(E), (8) $ $ ¸ where the velocity of the carriers is just:
v(E)"
2(!E) . m*
(9)
(thermionic theory postulates the energy to be the kinetic energy of the mobile carriers) [see for example Ref. [29], and references there in]. Expression (8) is rewritten as follows:
A * J "q dxG (e\?V) dEg(E)P(E)[ f (E)!f (E#q< )v(E)], (10) 23 ¸ where the Fermi}Dirac probability distribution re#ects the fact that there is a clear energy distance between the two quasi-Fermi levels, equaling the open-circuit voltage developed in the device during illumination. The "rst integration, in Eq. (10), sweeps the intrinsic region ¸ (lowest and upper limits at x"0 and ¸ , respectively), while the
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
61
second integration allows for possible trapped carrier energies in the quantum wells, as it is dictated by the mini-bands (Fig. 1). The Fermi factor is of course to be found from the di!erence 1 1 f (E)!f (E#q< )" ! . (11) 1#exp(E!E /k¹) 1#exp(E#q< !E /k¹) $ $ Leading to sinh(q< /2k¹) f (E)!f (E#q< )" . (12) 2cosh(E!E /2k¹) cosh(E!E #q< /2k¹) $ $ In both Eqs. (11) and (12), the Fermi level E is the quasi-Fermi level E taken as $ $ reference (zero-energy level, see below), since E !E "q< . Substitution of Eqs. $ $ (3), (4), (9), (12) into Eq. (10) leads to a general expression of tunneling currents in GaAs}AlGaAs systems (note that the absorption factor is just a fraction (1!e\?* ) of the initial G )
A q< (2m* J "J "8q G (1!e\?* ) sinh 23 ¸ 2k¹ #
(E/)(1!E/) exp(!k (1!E/ . (13) cosh(E!E /2k¹) cosh(E!E #q< /2k¹) $ $ # with < , E and E (low and upper limits corresponding to minimum and
maximum allowed energies in the quantum wells) are measured relative to quasi-Fermi level E "E . $ ;
dE
Making a change in variables E "y.
(14)
Expression (13) is reformed into the following:
A (2m* q< J "J "8q G (1!e\?* ) sinh 23 ¸ 2k¹
y(1!y) exp(!k (1!y) (15) cosh(y/2k¹) cosh(y/2k¹#q< /2k¹) (from y "E / to y "E /, as shown by Eq. (14), see also Fig. 1).
The second term of Eq. (15) is just a numerical factor I(y) that depends on the y-variable and its possible (if any) numerical instabilities. Thus Eq. (15) simplixes into the following general formula for tunneling currents: ; dy
A (2m* q< . J "8I(y)q G (1!e\?* ) sinh 23 ¸ 2k¹
(16)
62
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
Tunneling current densities comprise in fact one of the components of the total short circuit current density J of the devices under study, as it can be seen from Table 2. Note that for any solar cell, the J}< characteristic is given by the standard expression J"J (eO4 I2!1)!J , (17) where J is the reverse saturation current density, and J is the current component under short-circuit conditions. At open-circuit conditions (i.e. when <"< ) Eq. (17) dictates that 0"J (eO4 I2!1)!J or that J J eO4-! I2. (18) Short-circuit current densities as described by Eq. (18) include all possible contributions under illumination and at open-circuit conditions. The rest of the components (see also Table 2) that contribute to the total J are to be derived in the next sections. 4.2. Calculation of thermionic currents According to thermionic emission theory ([29], and references there in), a thermal current of carriers escaping from the quantum wells can be established by following the scheme of current densities imposed by expression (1) or (1) via (1)
A J "q G (1!e\?* ) d Ev(E)g(E) f (E#q< )(1!P(E)), (19) 2& ¸ where the last term in the integration ensures non-tunneling (note that P(E) is the tunneling probability used earlier), and where the integration will include all the available energy values from the lowest possible to the conduction band (hence practically to in"nity). The lowest available energy state of the trapped carriers is the ground state E measured from the bottom of the well. Since the energies of the trapped carriers in the wells are many k¹s away from the quasi-Fermi level E , one may replace the Fermi}Dirac with a Maxwell}Boltzmann distribution, $ so that expression (19) successively becomes as follows (with P(E) 1, i.e. neglecting numerical contributions from the second integral in Eq. (19) that includes the P(E) term):
A 2 m* G (1!e\?* ) dE (E!E J "q 2& ¸ m * # E#q< !E $ (1!P(E))exp ! k¹
"q
A (2m* G (1!e\?* ) e\O4-! ># \#$ I2 dE(E!E e\#\# I2 ¸ #
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
A (2m* E!E $ G (1!e\?* ) e\O4-! I2 dE(E!E exp ! ¸ k¹ #
q
63
A (2m* G (1!e\?* ) e\O4-! I2 ¸
"q
dE(E!E e\#\# I2e\# \#$ I2.
Changing variables: x"[(E!E )/k¹]:
(2m* A J "q G (1!e\?* ) e\O4-! ># \#$ I2 dx(xe\V(k¹) 2& ¸
"q
A (2m*(k¹) G (1!e\?* ) e\O4-! ># \#$ I2 dx(xe\V ¸
"q
A (2m*(k¹) G (1!e\?* ) e\O4-! ># \#$ I2(1/2), ¸
(20)
where the integral at the end of Eq. (20) is just the gamma function ()" . Expression (20) may now be written as
A (2m*(k¹) J "q G (1!e\?*G ) e\O4-! ># \#$ I2. 2& ¸
(21)
The energy di!erence E !E can be evaluated in a variety of ways: based on $ Fig. 1 and measuring from E , this di!erence may be re-written as follows: $ E !E "E #(E !E )!, $ $
(22)
where E is the energy miniband in the well measured from the bottom of the well, is the conduction band discontinuity and equal to E !E . As before, E refers to wide gap material (in this case AlGaAs: E "E !E "1.88 eV) and E to the conduction edge of the narrow-gap semiconductor (GaAs: E "E !E "1.42 eV). The energy di!erence E !E is easily (i.e. by standard tech $ niques, see for instance Ref. [29]) determined from the fact that
N E !E "k¹ ln $ p
(23)
[N (cm\) is the e!ective density of states for GaAs/AlGaAs systems, and p(cm\) is the doping of the p-regions as described in Section 2] Hence E !E "E !(E !E )"E !k¹ ln(N /p). $ $
(24)
64
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
substituting Eq. (24) into Eq. (21), by virtue of Eq. (22)
A (2m*(k¹) J "q G (1!e\?* ) e\O4-! >B# \ I2e\#! \#$ I2 2& ¸
"q
A (2m*(k¹) N G (1!e\?* ) e\O4-! >B# \ I2 e\# I2 ¸ p
"q
A (2m*(k¹) N e\O4-! >B# \ I2e\# I2 G (1!e\?* ) ¸ p
A (2m*k N ¹ e\# >O4 >B# \ I2 G (1!e\?* ) ¸ p "(A*)¹ e\# >O4 >B# \ I2,
"q
(25)
where the coe$cient A* is
A (2m*k N . G (1!e\?* ) (26) A*"q ¸ p from Eq. (25) it is concluded that thermionic currents of multiple quantum well (MQW's) solar cells include a ¹ temperature factor, as opposed to a ¹-dependence for the bulk case (i.e. no MQWs, see also Ref. [29]). 4.3. Excess carriers in the n-region An estimate of any excess carriers in the n-region may be pursued via the di!usion of minority carriers (holes) in the n-region. The current density due to minority hole concentration (in the n-AlGaAs region, ignoring general minority carriers in the very thin p-region) can be evaluated [41] from "rst principles namely, by evaluating the minority holes p as a function of distance in the n-region and then by evaluating the hole-di!usion current. These steps are as follows: First, one has to solve the di!usion equation, past the intrinsic region ¸ : dp p ! #G e\?* >V"0, D (27) dx where the "rst term indicates the di!usion of excess holes as a function of x, while the next two terms are the carriers: generated minus recombined/absorbed. Following standard techniques, the solution of Eq. (27) is the sum of the `homogeneousa and the `particulara solution ¸G e\?* >V, p "AeV * #e\V * # (28) D (1!¸) where A, B are constants to be found from the boundary conditions: dp zero at the farthest end of the n-region, thus A"0. On the other hand, p (0)"0, with x"0
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
65
taken at the very end of the intrinsic region, thus determining B: ¸G e\?* . B"! D (1!¸ ) Application of the boundary conditions "nalizes Eq. (28) to ¸ G e\?* (e\?V!e\V * ). p " D (1!¸)
(29)
From which the hole-di!usion current is
¸ e\?* , "qG (30) 1#¸ V where ¸ is the di!usion length of the generated holes in the bulk n-region of the p/i/n device, estimated in terms of the di!usion coe$cient D , and hole relaxation time , by means of the Einstein's relation. For a standard hole mobility value at 400 cm/V s, and with "10\ s (minority lifetimes in GaAs/AlGaAs systems), the di!usion length is found to be 3.6 m at 1003C and 2.96 m at }103C, respectively. dp J "qD dx
4.4. Ewects of recombination Treating the source of photon in#ux as a `black-body radiationa event, one may connect the incident photon #ux and the corresponding frequencies via Planck's formula, and at any frequency f, via standard techniques (see for instance Ref. [42]): 8f G ( f )" . c(eFD I2!1)
(31)
In thermal equilibrium, recombination rates are equal to generation rates: thus, the corresponding recombination rates may be found from G given above, and by integrating over all frequencies (in reality frequencies or wavelengths re#ecting energies of the order of the optical gap of the device and beyond: by the term optical gap, the energy di!erence between miniband E valence band is hinted). By setting y"hf/k¹, and assuming no frequency dependence of the dielectric constant , and absorption coe$cient :
8(k¹) y R "G " dy . (32) c eW!1 # I2 When thermal equilibrium is disturbed, direct (electron}hole) recombination rates may always be expressed as R"Cnp
66
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
while in equilibrium these rates are simply R "Cn p "Cn (where: C is a constant factor, and where the intrinsic carrier concentration n is connected to the equilibrium values: n p "n). Therefore, comparison of the two rates at non-equilibrium and at equilibrium conditions respectively, leads to the following correspondence: pn R"R , n
(33)
where p and n include excess carriers on top of the equilibrium values. R is found to be of the order of 10\ cm\ s\ in agreement with Ref. [12], while the excess carriers in the quantum wells are [43]:
m (k¹) E !E n" * ln 1#exp $ k¹
10 cm\
(34)
(at E !E "0.72 eV), where the Fermi level used is the quasi-Fermi level after $ splitting has taken place due to illumination, and where E is the ith miniband in the quantum wells. By means of Eq. (30), the recombination rate R (via Eq. (29)) is estimated to be of the order of 10}10 cm\ s\, and hence much less than the incident photon #ux G ("2.6;10 cm\ s\). 4.5. Fitting the experimental results The total short-circuit current density is calculated by summing J , J , and 23 2& J from Eq. (30). Results of the calculations for each sample at the two temperature extremes are summarized in Table 2. Experimental values of open-circuit voltage were used in Eqs. (16) and (25). The values of tunneling and thermal current densities depend on the transit time which is used as a "tting parameter consistent for all device designs depicted in this communication: of the order of ps in tunneling [43] and ms for thermionic emission [44]. Final J values seem to be remarkably close to the experimental ones reported here, thus making a strong case for the existence of quantum size ewects in MQW solar cells. Finally, by comparing thermionic emission current values, one observes that they are dominant and comparable at 1003C but di!er greatly at }103C, where the deeper energy levels of the wider wells result in reduced thermionic current contribution. By contrast, it is particularly striking the fact that tunneling currents behave almost `by the booka (and in agreement with textbook quantum mechanics): (a) they are non-negligible at any temperature; (b) they dominate at }103C and as expected (less scattering events); (c) they contribute a higher current density in the case of thin barriers than in the case of thick ones.
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
67
Table 2 Current density components at 1003C and !103C for samples GB68 (23 wells), GB69 (40 wells) and GB71 (39 wells) Current density (mA/cm)
GB68 (thick wells)
GB69 (thin wells)
GB71 (thick wells, thick i-region)
#1003C
!103C
#1003C
!103C
#1003C
J 2& J 23 J .
51.28 3.50 2.28
5.85 21.23 2.19
52.78 6.92 2.26
14.20 42.94 2.17
52.41 4.20 1.23
1.80 7.60 1.18
Total Experimental
57.06 56.09
29.27 31.71
61.96 63.41
59.31 54.87
57.84 59.26
10.58 9.75
!103C
Thickness of barrier is proportional to well thickness. J is the thermionic short-circuit current density, 2& J is the tunneling short-circuit current density, J is the excess carriers short-circuit current density, Total 23 . is the total theoretical short-circuit current density (J #J #J ) and Experimental is the measured 2& 23 . short-circuit current density.
5. Conclusions The behavior of p/i/n Al Ga As solar cells with GaAs/Al Ga As MQWs in the i-region has been studied with temperature under illumination and compared to that of a p/i/n GaAs and AlGaAs solar cell. The dark currents of the MQW diodes were found to lie between those of conventional cells formed from the well material and from the barrier material alone. It was found that the increase in the dark current of the diodes with temperature was accompanied by an improvement of diode ideality factor, even for the one diode which exhibited poor dark current characteristics at room temperature. Under illumination and at elevated temperatures, the MQW solar cells exhibited enhanced I , which was comparable to the photocurrent of the control (p/i/n GaAs) sample. In addition they exhibited higher < . The MQW solar cells with thin barrier width exhibited enhanced output performance in the whole range of temperatures examined, due to contribution of tunneling current to the total observed photocurrent, whereas the MQW cell with thicker barrier width showed remarkable performance only at elevated temperatures. It was con"rmed by modeling that non-zero tunneling currents exist at all temperatures but is more signi"cant in thin barriers as expected. It can readily be inferred from these results that p/i/n GaAs/AlGaAs MQW structures of high-quality material should exhibit enhanced performance over conventional p/i/n GaAs solar cells under intense illumination when processed speci"cally for application as solar cells. The relative contribution of tunneling and thermionic emission currents as a function of temperature was determined by modeling and "tting and should provide guidance for the optimized design of MQW solar cells tailored for operation in speci"c temperature ranges.
68
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
Acknowledgements The authors wish to thank the British Council and the Greek Ministry for Development for "nancial support through a bilateral program between Greece and the UK. In addition, one of the samples reported here was developed under a bilateral program between Greece and Germany. P. Panayotatos acknowledges the support of Rutgers University through the FASP program.
References [1] K. Barnham, I. Ballard, J. Barnes, J. Connolly, P. Gri$n, B. Kluftinger, J. Nelson, E. Tsui, A. Zachariou, Appl. Surf. Sci. 133/114 (1997) 722. [2] G.B. Lush, M.P. Patkar, M.P. Young, M.R. Melloch, M.S. Lundstrom, S.M. Vernon, E.D. Gagnon, L.M. Geo!roy, M.M. Sanfacon, Thin-"lm GaAs solar cells by epitaxial lift-o!, Proceedings of the 23rd IEEE Photovoltaic Specialists Conference, 1993, p. 1343. [3] P.R. Hageman, G.J. Bauhuis, A. Van Geelen, P.C. Van Rijsingen, J.J. Schermer, L.J. Giling, Large area, thin "lm epitaxial lift-o! III/V solar cells, Proceedings of the 25th Photovoltaic Specialists Conference, 1996, p. 57. [4] M. Yang, T. Soga, T. Jimbo, M. Umeno, Jpn. J. Appl. Phys. 33 (1994) 6605. [5] C. Flores, B. Bollani, F. Campesato, F. Paletta, D. Passoni, G. Timo, A. Tosoni, Sol. Energy Mater. 23 (1991) 356. [6] K.W.J. Barnham, G. Duggan, J. Appl. Phys. 67 (1990) 3490. [7] F.W. Ragay, E.W.M. Ruigrok, J.H. Wolter, GaAs-AlGaAs heterojunction solar cells with increased open-circuit voltage, Proceedings of the First World Conference on Photovoltaic Energy Conversion, 1994. [8] J. Nelson, K.W.J. Barnham, J.P. Connolly, G. Haarpaintner, C. Button, J. Roberts, Quantum well solar cell dark current, Proceedings of the 12th European Photovoltaic Solar Energy Conference, 1994, p. 1370. [9] As . Aperathitis, Z. Hatzopoulos, P. Panayotatos, D. Sands, C.G. Scott, M. Yousaf, PIN solar cell e$ciency enhancement by the use of MQW structures in the I-region, Presented at 14th European Photovoltaic Solar Energy Conference, Barcelona, Spain, 30 June}4 July, 1997. [10] J. Barnes, T. Ali, K.W.J. Barnham, J. Nelson, E.S.M. Tsui, J. Roberts, M.A. Pate, S.S. Dosanjh, Gallium Arsenide/Indium Gallium Arsenide multi- quantum well solar cells, Proceedings of the 12th European Photovoltaic Solar Energy Conference, 1994, p. 1374. [11] D.E. Ashenford, A.W. Dweydari, D. Sands, C.G. Scott, M. Yousaf, E. Aperathitis, Z. Hatzopoulos, P. Panayotatos, J. Crystal Growth 159 (1996) 920. [12] R. Corkish, M. Green, Recombination of carriers in quantum well solar cells, Proc. 23rd IEEE Photovoltaic Specialists Conference, 1993, p. 675. [13] G.L. Araujo, A. Marti, F.W. Ragay, J.H. Wolter, E$ciency of multiple quantum well solar cells, Proceedings of the 12th European Photovoltaic Solar Energy Conference, 1994, p. 1481. [14] N.G. Anderson, J. Appl. Phys. 78 (1995) 1850. [15] K.W.J. Barnham, J.M. Barnes, B. Braun, J. P. Connolly, G. Haarpaintner, J. A. Nelson, M. Paxman, C. Button, J.S. Roberts, C.T. Foxon, A novel approach to higher e$ciency-the quantum well solar cell, Proceedings of the 11th European Photovoltaic Solar Energy Conference, 1992, p. 146. [16] A. Luque, Solar Cells and Optics for Photovoltaic Concentration, Adam Hilger, Bristol, 1989. [17] I. Ballard, K.W.J. Barnham, J. Nelson, J.P. Connolly, C. Roberts, J.S. Roberts, M.A. Pate, The e!ect of Temperature on the e$ciency of multi-quantum well solar cells, Proceedings of the 2nd World Conference and Exhibition on Photovoltaic Solar Energy Conversion, Vienna, Austria, 6}10 July 1998, p. 3624
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
69
[18] P.R. Gri$n, J. Barnes, K.W.J. Barnham, I. Ballard, M. Cabodi, M. Mazzer, J.S. Roberts, R. Grey, G. Hill, M.A. Pate, A. Marti, J.C. Maroto, R.F. Reyna, L. Cuadra, M. Yang, and M. Yamaguchi, A method for optimising strained GaAs/InGaAs quantum well solar cells and preliminary results under concentration, Presented at 14th European Photovoltaic Solar Energy Conference, Barcelona, Spain, 30 June}4 July 1997. [19] H. Ohtsuka, T. Kitatani, Y. Yazawa, T. Warabisako, Sol. Energy Mater. Sol. Cells 50 (1998) 251}257. [20] E. Aperathitis, C.G. Scott, D. Sands, V. Foukaraki, Z. Hatzopoulos, P. Panayotatos, Mater. Sci. Eng. B 51 (1998) 85. [21] A.C. Varonides, E. Aperathitis, P. Panayotatos, C.G. Scott, Signi"cance of tunneling and thermionic emission currents in performance enhancement of multiple quantum well GaAs/AlGaAs p}I}n solar cells, Proceedings of the Second World Conference and Exhibition on Photovoltaic Solar Energy Conversion, Vienna, Austria, 6}10 July 1998, p. 66 [22] E. Aperathitis, Z. Hatzopoulos, M. Androulidaki, V. Foukaraki, A. Kondilis, C.G. Scott, D. Sands, P. Panayotatos, Sol. Energy Mater. Sol. Cells 45 (1997) 161. [23] K.J.W. Barnham, J. Barnes, G. Haarpaintner, J. Neslon, M. Paxman, T. Foxon, J. Roberts, Quantum well solar cells, MRS Bulletin, October 1993, pp.51}55. [24] F.W. Ragay, J.H. Wolter, A. Marti, G.L. Araujo, Experimental analysis of the e$ciency of MQW solar cells, Proceedings of the 12th European Photovoltaic Solar Energy Conference, 1994, p. 1429. [25] F.W. Ragay, P.J. van Hall, M.R. Leys, E.A.E. Zwaal, J.H. Wolter, GaAs}AlGaAs p}i}n heterostructures for high-e$ciency solar cells, Proceedings of the 11th European Photovoltaic Solar Energy Conference, 1992, p. 359. [26] G. Haarpaintner, J. Barnes, K.J.W. Barnham, J. P. Connolly, S.S. Dosanjh, J. Neslon, C.Roberts, C. Button, G. Hill, M.A. Pate, and J.S. Roberts, Voltage performance of quantum well solar cells in the Al Ga As/GaAs and the GaAs/In Ga As material systems, Proceedings of the First World V \V W \W Conference on Photovoltaic Energy Conversion, 1994, p. 1783. [27] K. Barnham, J. Connolly, P. Gri$n, G. Haarpaintner, J. Nelson, E. Tsui, A. Zachariou, J. Osborne, C. Button, G. Hill, M. Hopkinson, M. Pate, J. Roberts, T. Foxon, J. Appl. Phys. 80 (1996) 1201. [28] F.W. Ragay, A. Marti, G.L. Araujo, J.H. Wolter, Sol. Energy Mater. Sol. Cells 40 (1996) 5. [29] S.M. Sze, Physics of Semiconductor Devices, Wiley, New York, 1981. [30] S.P. Tobin, S.M. Vernon, C. Bajgar, L.M. Geo!roy, C.J. Keavney, M.M. Sanfacon, V.E. Haven, Sol. Cells 24 (1988) 103. [31] S.P. Tobin, S.M. Vernon, C. Bajgar, S.J. Wojtczuk, M.R. Melloch, A. Keshavarzi, T.B. Stellwag, S. Venkatensan, M.S. Lundstrom, K.A. Emery, IEEE Trans. Electron Dev. 37 (1990) 469. [32] M.R. Melloch, Sol. Cells 30 (1991) 313. [33] H.J. Hovel, Solar cells, in: R.K. Willardson, A.C. Beer (Eds.), Semiconductors and Semimetals, Vol. 11, Academic Press, New York, 1975, pp. 166}180. [34] I. Ballard, Temperature e!ects on the e$ciency of MQW solar cells, M.Sc. Thesis, September 1995, Imperial College of Science and Medicine, London. [35] J. Nelson, M. Paxman, K.W.J. Barnham, J.S. Roberts, C. Button, IEEE Trans. Quantum Electron. 29 (1993) 1460. [36] F. Capasso, K. Mohammed, A.Y. Cho, R. Hull, A.L. Hutchinson, Appl. Phys. Lett. 47 (1985) 420. [37] N. Bernhard, G.H. Bauer, W.H. Bloss, Progress in Photovoltaics: Res. Appl. 3 (1995) 149. [38] M. Hirose, S. Miyazaki, J. Non-Cryst. Sol. 9798 (1987) 23. [39] I. Pereyra, M.N.P. Carreno, R.K. Omnori, C.A. Sassaki, A.M. Andrade, F. Alvarez, J. Non-Cryst. Sol. 9798 (1987) 871. [40] L.I. Schi!, Quantum Mechanics, McGraw-Hill, New York, 1968. [41] E.S. Yang, Microelectronic Devices, McGraw-Hill, New York, 1988. [42] D.K. Ferry, Semiconductors, Macmillan, New York, 1991. [43] M. Shur, GaAs Devices and Circuits, Plenum Press, New York, 1987. [44] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Halsted Press, New York, 1988.
Temperature dependence of photocurrent components on enhanced performance GaAs/AlGaAs multiple quantum well solar cells E. Aperathitis *, A.C. Varonides, C.G. Scott, D. Sand, V. Foukaraki , M. Androulidaki , Z. Hatzopoulos , P. Panayotatos Microelectronics Research Group, Institute of Electronic Structure & Laser, Foundation For Research and Technology-Hellas, P.O. Box 1527, Heraklion 71110, Crete, Greece Physics Department, University of Crete, P.O. Box 2208, Heraklion 71003, Crete, Greece Physics and Electrical Engineering Department, University of Scranton, Scranton, PA 18510-4642, USA Department of Applied Physics, Hull University, Hull HU67RX, UK Rutgers, The State University of New Jersey, Department of Electrical and Computer Engineering, 94 Brett Rd., Piscataway, New Jersey 08854-8058, USA Received 27 August 1999; received in revised form 6 June 2000; accepted 29 September 2000
Abstract The performance of Al Ga As p/i/n solar cells with multiple quantum wells (MQW) of GaAs/Al Ga As in the i-region has been investigated at various temperatures, ranging from !103C to 1003C, and compared with that of conventional solar cells composed of either the quantum well material (GaAs) or the barrier material (Al Ga As) alone. The dark currents of the MQW cells were found to lie between those of the conventional cells. The increase of dark current with temperature was accompanied by a slight decrease of the diode ideality factor. A linear dependence of open-circuit voltage (< ) on temperature was observed for all cells when illuminated with a 100W halogen lamp. < for the MQW cells was found to be independent of the number of wells, lying between the < 's for the two conventional cells. The MQW cells exhibited performance improvement with temperature when compared to the conventional cells and there was a signi"cant enhancement in the short-circuit current with temperature of those MQW cells that exhibited poorer performance at lower temperatures. Theoretical calculations have quanti"ed the contribution of the tunneling current component
* Corresponding author. Tel.: #30-81-394-105; fax: #30-81-394-106. E-mail address: [email protected] (E. Aperathitis). 0927-0248/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 4 8 ( 0 0 ) 0 0 4 1 1 - 6
50
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
to the total observed photocurrent at the various temperatures examined. It was found that tunneling currents are present at all temperatures and can be the dominant component in MQW cells of thinner wells at low temperatures. These results suggest that GaAs/Al Ga As MQW structures, of good-quality material, when processed as conven tional solar cells with antire#ective coatings should deliver more output power under intense illumination than conventional solar cells composed of the quantum well material alone. 2001 Elsevier Science B.V. All rights reserved. Keywords: Multiple quantum wells; GaAs/AlGaAs; Tunneling
1. Introduction The high cost of the state-of-the-art single-junction GaAs-based solar cells has made their use for one-sun terrestrial applications nonviable, despite the fact that the e$ciency achieved in practice for these devices is now close to their theoretical limit. The alternative approach of multi-junction tandem solar cells has yet to overcome its main disadvantage of increased technological complexity. The use of GaAs solar cells might be more economically attractive for terrestrial applications by utilizing concentrated light [1]. Further cell cost reduction and/or e$ciency improvement, however, is still needed for this approach. Attempts at fabricating cost-e$cient solar cells have already been reported by making use of epitaxial lift-o! techniques [2,3] or growth of GaAs solar cell structures on foreign substrates such as Si [4] and Ge [5]. E$ciency improvement of single-junction solar cell requires new device designs [6,7]. One of these approaches is that of multiple quantum well (MQW) solar cells, in which a number of quantum wells are incorporated in the cell [6]. Fig. 1 illustrates such an approach, along with parameters which will be used in the modeling section. In such an arrangement, the i-region of the p}i}n solar cell structure consists of a number of QWs from material of energy gap E , lower than the barrier material, E . Thus, in addition to the normal photogeneration of electrons and holes by the action of those photons that have energy above the energy gap, E of the p- and n-region material, it is possible for photons with energy between E and E to contribute to the photogenerated current, provided that the charge carriers generated in the wells can escape from these wells before they recombine. Previous studies on AlGaAs cells with GaAs wells [8,9] and on GaAs cells with InGaAs wells [10], as well as on CdMnTe devices incorporating a variety of CdTe single and multiple quantum wells [11], have revealed that increase in the shortcircuit current (I ) can, indeed, be achieved as a result of photogeneration of carriers within the quantum wells and, equally important, the subsequent release of these carriers from the well. Unfortunately, the e$ciency improvement of such devices has been found to be limited by a reduction in the open-circuit voltage, (< ), which accompanies the incorporation of the wells. It is clear, however, that this reduction also depends on factors other than the band gap of the well material. There has been some controversy over the reasons for MQW solar cell e$ciency enhancement,
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
51
Fig. 1. A p/i(MQW)/n solar cell structure. Shown are all the parameters used in the modeling section.
particularly on whether it is associated with the nature and dimension of the well material or it is solely due to enhanced I [12}14]. In all investigations, the material quality of the grown MQW structure has been found to be of paramount importance. Furthermore, photoconductivity measurements have demonstrated [15] that the mechanism for the escape of carriers from the QWs does have a thermal component in forward bias. Taking into account that a signi"cant forward bias voltage is developed in an operating solar cell and the fact that the operating temperature of cells under concentrated light, is signi"cantly above room temperature [16], the e$ciency of solar cells with properly designed MQWs is expected to be enhanced in concentrator applications by the greater probability of carrier escape from the wells. Some theoretical calculations as well as results on the temperature behavior of MQW solar cells [1,17,18] or numerical prediction of performance under concentration [19] have previously been reported and we have also given a detailed account of an experimental study [20] The reader is cautioned that the correct level of illumination is 3.3 mW/cm (corrected for re#ection) on the e!ects of temperature on the output performance of p/i/n Al Ga As/GaAs MQW solar cells with di!erent number of wells and the same i-region widths. The aim of the present work is to examine the e!ects of temperature and illumination on several p/i/n Al Ga As/GaAs MQW solar cells of di!erent well widths and i-region thick nesses, and compare them with standard p/i/n Al Ga As and GaAs solar cells
52
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
Table 1 Details of the i-region structure of p/i/n solar cells. Widths: ¸ for the GaAs well, ¸ for the Al Ga As barrier, and ¸ for the i-region Sample
Well period
¸ (nm)
¸ (nm)
GB68 GB69 GB71 GB66 GB73
23 9.0 6.0 40 5.4 3.6 39 15.0 10.0 Conventional p/i/n Al Ga sample Conventional p/i/n GaAs sample (with AR coating)
¸ (m) 0.35 0.36 0.97 0.90 0.80
without quantum wells in the i-region. In addition the contribution of di!erent current components was modeled and investigated at di!erent operating temperatures. Such modeling is necessary in order to provide guidance in cell design with respect to well, barrier and i-region thickness for optimal operation over speci"ed temperature ranges. Some preliminary results were included in a conference presentation [21].
2. Experimental procedure The p/i/n solar cell structures of this investigation were grown in a VG80 H MBE system, on epi-ready 2 n> type (1 0 0) GaAs substrates. The bu!er layer was n>GaAs, Si doped at 2;10 cm\ and 1 m thick. The n-type (Si-doped) and the p-type (Be-doped) Al Ga As regions (1.8;10 cm\) were 150 and 300 nm thick, respectively. High-quality interface between the n-type Al Ga As and the GaAs bu!er layer was achieved by growing a superlattice, consisting of 20 periods of 2.8 nm Al Ga As and 2.7 nm GaAs (Si-doped 2;10 cm\) on the bu!er layer. Two di!erent i-region thicknesses were examined of approximately 0.35 and 0.97 m widths. Details of the i-region of the structures grown for this study are given in Table 1. The ratio of well width to barrier width was kept constant at 1.5 for all MQW samples. For all MQW samples a 30 nm thick Al Ga As spacer was grown on either side of the i-region. Growth was completed with a 40 nm thick p>Al Ga As, 1;10 cm\, window layer followed by the cap layer (50 nm thick p>GaAs, 1;10 cm\). Sample GB66, which was the conventional Al Ga As sample, had an i-region of 0.90 m and no MQWs. The samples were processed, using standard photolithographic techniques, as mesa-isolated diodes of 720 m diameter, resulting in a junction area of 4.1;10\ cm. The top contact metallization, Pt/Ti/Au, had the ring geometry of 520 m inner diameter. The back ohmic contact, (Ge/Au)/Ni/Au, was evaporated on the n>GaAs substrate. Both contacts were annealed initially at 4103C for 20 s and subsequently at 4003C for 2 min. It has been shown, by transmission line
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
53
measurements, that this sequential annealing procedure improves the ohmic properties of both n- and p-type ohmic contacts. No antire#ective coating was applied on the surface of the samples for the purposes of this study. Observation of the as-grown material surface with an optical microscope revealed the presence of oval defects with densities of around 1000 cm\ for all samples, except for sample GB71 which had 2000 defects/cm\. For comparison purposes, an e$cient p/i/n GaAs solar cell (sample GB73), with a 0.8 m thick i-region , 0.3 m thick GaAs p-region (Be-doped, 3;10 cm\) and 2.6 m thick GaAs n-region (Si-doped, 1;10 cm\) was also examined. It was processed in exactly the same way as the other samples but it also received the application of a ZnS/MgF antire#ective coating. A device of the same structure, processed as a 1 cm;1 cm solar cell (with top contact grid and ZnS/MgF antire#ec tive coating) had exhibited 19.4% e$ciency under 100 mW/cm (approximately AM1 spectrum). One should, however, exercise caution in inferring e$ciencies for the other structures of this study from this, because of the much smaller size of the devices of this study. Both dark and illuminated current}voltage characteristics of the diodes were taken, in rough vacuum (10\ mbar), in a temperature-controlled Biorad DL4600 Polaron system interfaced with an HP916 computer. The Biorad DL4600 Polaron system uses heating elements and liquid nitrogen for controlling the temperature via a Pt 100 element for temperature reading ($13C). A 100 W quartz tungsten halogen lamp (colour temperature 3300 K) was used for illumination. The intensity of the light entering the samples was calculated to be 3.3 mW/cm, after correcting for the re#ectance of the uncoated GaAs [22]. The above value is also the illumination level for Ref. [20]. The open-circuit voltage spectral response of the samples was monitored using a /4 double grating computer controlled monochromator (SPEX 0.22 mm) and standard lock-in techniques.
3. Experimental results and discussion 3.1. Temperature dependence of dark I}< characteristics The dark current}voltage (I}<) characteristics of the samples were taken at temperatures of !103C, 03C, 203C, 503C and 1003C. The I}< curves at the two extreme temperatures of !103C and 1003C have been plotted in Fig. 2(a) and (b), respectively. It is observed from these "gures that the dark current for each of the MQW cells lies well below the dark current of the conventional cell GB73 which consists of the well material (GaAs) alone, and above the control cell GB66 that is composed of the barrier material (Al Ga As) alone. This is as expected and is observed not only at lower temperatures as found previously [7,8,13,23}28] but also in the whole range of temperatures examined in this work. Furthermore, the spread in the I}< curves [24] of the MQW cells observed at !103C disappears at elevated temperatures. The increased series resistance for these samples observed at high forward bias is partially due to the fact that these samples
54
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
Fig. 2. Dark I}< characteristics of p/i/n solar cells at the two extreme temperatures of this investigation: (a) !103C and (b) 1003C.
Fig. 3. Temperature dependence of ideality factor of p/i/n solar cells as extracted by "tting the I}< characteristics, such as the ones of Fig. 2, of all temperatures of this investigation.
had not been speci"cally processed to minimize series resistance and partially due to defects associated with the quality of the grown material. The linear part of the dark current curves, which corresponds to voltages of about the same magnitude as the < of the individual diodes, has been "tted by the exponential voltage dependence I"I exp(q
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
55
Fig. 4. Temperature dependence of saturation current of p/i/n/ solar cells as extracted by "tting the I}< characteristics, such as the ones of Fig. 2, of all temperatures of this investigation.
increasing temperature, from n"1.95}2.30 at !103C to n"1.75}1.80 at 1003C. The rate of decrease with temperature for the MQW sample GB69 and conventional sample GB73 is much slower than for the MQW sample GB68. Diode GB71 exhibits a very high ideality factor at low temperatures which reduces to 2 at the highest tested temperature. The high n value at lower temperatures must be attributed to the high density of thermally activated defects observed for these diodes [30}32]. However, it is clear that the diode quality improves with temperature and this improved behavior leads to enhanced solar cell e$ciency, despite the increased dark current, as seen below. Both reverse saturation and recombination currents increase with decreasing semiconductor band gap [7,23]. Thus, the currents of the MQW diodes are higher than those of the conventional sample GB66 which has i-region material consisting only of the barrier material of the MQW diodes (Figs. 2 and 4) and lower than the conventional sample GB73 which consists of the well material alone. Furthermore, as it can be seen in Fig. 4, the rate of increase of I with temperature is similar for all diodes, except for diode GB71. The latter diode exhibited the highest I value at low temperatures and the lowest rate of increase with temperature of all diodes examined. It is speculated that this behavior is due to the higher concentration of growth defects associated with this diode, as previously discussed. 3.2. Temperature dependence of I}< characteristics under illumination An indication of the quality of the quantum wells was obtained by examining the photovoltage at room temperature as a function of wavelength. Fig. 5 shows the open-circuit photovoltage of the MQW samples as a function of photon energy. It should be noted that no bias illumination was used for these measurements. The characteristic excitonic peaks arising from absorption of photons within the quantum
56
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
Fig. 5. Photovoltage as a function of wavelength of p/i/n MQW solar cells.
Fig. 6. Temperature dependence of open-circuit voltage of p/i/n solar cells.
wells is easily discerned. As expected, the MQW samples exhibited excitonic peaks at di!erent photon energies since these samples have wells of di!erent width, thus di!erent level distribution, with the thinner the wells in the sample, the higher the energy of the peak. The e!ect of temperature on the illuminated I}< characteristics of the diodes was monitored in the range of temperatures between !103C and 1003C and the dependence of the open-circuit voltage < , under 3.3 mW/cm illumination, with temper ature is depicted in Fig. 6. It is seen that < for GaAs/Al Ga As MQWs is
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
57
Fig. 7. Photo I}< characteristics of p/i/n solar cells at (a) !103C and (b) 1003C.
lower than for the conventional sample (GB66) which is composed from the barrier material alone but higher than for the conventional sample GB73 which is composed from the well material alone, in the whole range of temperature examined. Similar observations have been reported for room temperature MQW solar cells having 20 wells of GaAs (9 nm thick) with 6 nm thick Al Ga As barriers [24]. Furthermore, a linear dependence of < with temperature can be seen in Fig. 6 for both the conventional samples [16,33], as well as for the MQW devices [34]. The spread among < values for the MQW cells reduces with temperature. This is attributed to the observed (Fig. 3) convergence of the ideality factors of the MQW diodes with temperature. It should be noted that the MQW samples GB68, GB69 and the conventional sample GB73 exhibited similar rates of < decrease with temper ature (1.99}2.10 mV/C), whereas the slowest rate, 1.66 mV/C, was observed for sample GB71. The importance of incorporating the MQWs in the i-region of the p}i}n structures can be further observed in Fig. 7 where the full I}< curves, (and consequently the power delivered by these devices), are depicted, again at the two extreme temperatures of this investigation. It should be noted again that all samples, except GB73, possess no antire#ective coating. Under the temperature and illumination conditions used here, the MQW solar cell GB69 has exhibited comparable short-circuit current I with the control sample GB73 consisting of the well material alone, at all of the temperatures examined in this work. The other two MQW solar cells, which had thicker wells than GB69 and showed poor performance at low temperatures, exhibited a remarkable enhancement in I at elevated temperatures. These observations are an indication that tunneling of the carriers through the barriers dominates the escape process from the wells at lower temperatures [35]. At elevated temperatures, the escape contribution dominates over tunneling giving rise to the observed improvement of I . A theoretical analysis and
58
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
comparison of the various current contributions to the observed photocurrent at di!erent temperatures is given in the next section. Furthermore, the very low "ll factor of sample GB71 at lower temperatures is certainly associated with the high ideality factor this diode exhibited (Fig. 3). Indeed, the improvement of the diode ideality factor with temperature resulted in a similarly improved performance of the device under illumination at higher temperatures, albeit lower than the performance of GB69. The remarkable improvement in the performance of GB68 and particularly GB71 with temperature strongly suggests that the higher the operating temperature of MQW solar cells having thick wells in the i-region, the higher the output performance. Clearly, good-quality material and entire control of processing procedures [14,24,25,30}32] are of paramount importance for utilizing the full potential improvement for these devices. The temperature dependence of a MQW structure similar to sample GB68 of this work, was previously reported [17]. The authors of this particular reference, however, ascribed all improvements to the temperature dependence to the dark current. We speculate that this is due to the di!erence in Al content of the AlGaAs layer in our respective studies. The mole fraction x was )0.3 that was used in Ref. [17] produced shallower well from which carriers would be able to escape with probability of more than 90% even at room temperature. Thus, since the authors of Ref. [17] concentrated on temperatures at and above room temperature they would not, and did not, observe additional escaping with rising temperature. In addition, optimized design of such structures involves several width and compositional parameters. Since tunneling is directly a!ected by the width of the barrier layer, absorption is a!ected by the width of the i-region, and thermionic emission depends on the Al mole fraction, the relative contribution of the various current components need to be delineated at di!erent temperatures. An initial calculation was performed towards this end for the MQW solar cells examined in this work.
4. Modeling 4.1. Calculation of tunneling currents For the quantum well (QW) thickness involved, these devices, because of their rather extensive number of repeat distances, are superlattice-based and thus perpendicular transport may take place via mini-bands. In other words, electrons channel through the superlattice (i.e. with large number of repeat distances) region more easily than the `heaviera holes, thus resulting in e!ective mass "ltering [36]. In addition, quantization of the energy in the wells leads to widening of the useful spectrum of incident wavelengths (Fig. 1). It is demonstrated here that non-zero tunneling currents do exist, as a result of carrier con"nement in sub-bands (in disagreement with Ref. [37]), that these currents are indeed present (in agreement with Refs. [38,39]) at both temperature extremes and that they dominate over thermionic emission at low temperatures [29], while at high temperatures the roles reverse. A "rst-principles calculation of the tunneling current density component J is performed, of the 23
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
59
short-circuit current-density component J , that is due solely to tunneling carriers along the growth direction of the device (as long as the tunneling probability is non-zero), and via the two existing channels (i.e. eigen-states) of the GaAs}AlGaAs quantum well system. The starting point is the general current-density relationship for current densities, which is true for any carrier transport settings J"nqv,
(1)
where n is the concentration of carriers with enough energy to overcome the potential barriers (typically in cm\), v(E) is the velocity of the carriers and q is the single electron charge. The currents sought are due to tunneling processes, through potential barriers, of carriers trapped in quantum wells, hence the factor n of the general expression (1) should be composed of (i) (ii) (iii) (iv)
The trapped carriers' non-zero tunneling probability P(E) The Fermi}Dirac occupation probability f (E) The availability of the two-dimensional electron density of states (DOS) g(E), and The net (after absorption) photo-generation rate, namely G ()e\?HV over the extend of the intrinsic region, where () is the absorption coe$cient of the absorbing material involved.
To investigate the transport properties, once the above factors are placed in the right context, one must evaluate the net current #ow in the i-region. The quantum wells play the role of electron reservoirs, and in such systems one has to consider the net electron #ow from left to right and from right to the left. The di!erential form of Eq. (1) reads now as follows:
J"q v(E) dn,
(1)
where dn is the carrier concentration given below in terms of the availability factor g(E) f (E): dn"dEg(E) f (E).
(1)
In this picture, the current injected from the left-hand side reservoir is
A J "q g(E)P(E) f (E!E )v(E), (2) $ ¸ where A is the cross-section of the device, (), are the absorption coe$cient and the carrier transit time. The absorption coe$cient is generally a function of the incident wavelength, but for GaAs}AlGaAs solar cell systems is constant at &10 cm\). Special attention should be given at the g(E) energy function: carriers trapped in quantum wells essentially comprise two-dimensional systems, and for such systems the density of states (eV\ cm\) is a well-known superlattice DOS: (m*"0.067m is the carriers' e!ective mass, with m the electronic rest mass) m g(E)" * . (3)
60
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
The tunneling probability function P(E) [40] is:
16E(!E) E P(E)" exp !k (1! ,
(4)
where E is the energy of a trapped carrier, is the conduction band discontinuity, and k is the potential barrier unit-less factor (2m* k "2¸ (nm) (nm\) (5)
(with ¸ the barrier width in nm). As seen from Eq. (2), the factor g(E)P(E) f (E!E ) ensures carrier availability in the $ quantum wells, while the occupation Fermi}Dirac factor is given by the following expression: 1 f (E!E )" , (6) $ 1#exp(E!E /k¹) $ where ¹ is the temperature, k is the Boltzmann constant, and E is the quasi-Fermi $ level at the left side of the device (see Fig. 1). In a similar fashion one may construct (mirroring Eq. (2)) the current #ow from the right side of the device
A J "q dx dEG e\?Vg(E)P(E) f (E!E )v(E), (7) $ ¸ where the involved quasi-Fermi level E is shifted relative to E by an amount $ $ equivalent to the open-circuit voltage q< . The net current (density) #ow is then J "J !J 23 A "q dx dEG e\?Vg(E)P(E)[ f (E!E )!f (E!E )]v(E), (8) $ $ ¸ where the velocity of the carriers is just:
v(E)"
2(!E) . m*
(9)
(thermionic theory postulates the energy to be the kinetic energy of the mobile carriers) [see for example Ref. [29], and references there in]. Expression (8) is rewritten as follows:
A * J "q dxG (e\?V) dEg(E)P(E)[ f (E)!f (E#q< )v(E)], (10) 23 ¸ where the Fermi}Dirac probability distribution re#ects the fact that there is a clear energy distance between the two quasi-Fermi levels, equaling the open-circuit voltage developed in the device during illumination. The "rst integration, in Eq. (10), sweeps the intrinsic region ¸ (lowest and upper limits at x"0 and ¸ , respectively), while the
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
61
second integration allows for possible trapped carrier energies in the quantum wells, as it is dictated by the mini-bands (Fig. 1). The Fermi factor is of course to be found from the di!erence 1 1 f (E)!f (E#q< )" ! . (11) 1#exp(E!E /k¹) 1#exp(E#q< !E /k¹) $ $ Leading to sinh(q< /2k¹) f (E)!f (E#q< )" . (12) 2cosh(E!E /2k¹) cosh(E!E #q< /2k¹) $ $ In both Eqs. (11) and (12), the Fermi level E is the quasi-Fermi level E taken as $ $ reference (zero-energy level, see below), since E !E "q< . Substitution of Eqs. $ $ (3), (4), (9), (12) into Eq. (10) leads to a general expression of tunneling currents in GaAs}AlGaAs systems (note that the absorption factor is just a fraction (1!e\?* ) of the initial G )
A q< (2m* J "J "8q G (1!e\?* ) sinh 23 ¸ 2k¹ #
(E/)(1!E/) exp(!k (1!E/ . (13) cosh(E!E /2k¹) cosh(E!E #q< /2k¹) $ $ # with < , E and E (low and upper limits corresponding to minimum and
maximum allowed energies in the quantum wells) are measured relative to quasi-Fermi level E "E . $ ;
dE
Making a change in variables E "y.
(14)
Expression (13) is reformed into the following:
A (2m* q< J "J "8q G (1!e\?* ) sinh 23 ¸ 2k¹
y(1!y) exp(!k (1!y) (15) cosh(y/2k¹) cosh(y/2k¹#q< /2k¹) (from y "E / to y "E /, as shown by Eq. (14), see also Fig. 1).
The second term of Eq. (15) is just a numerical factor I(y) that depends on the y-variable and its possible (if any) numerical instabilities. Thus Eq. (15) simplixes into the following general formula for tunneling currents: ; dy
A (2m* q< . J "8I(y)q G (1!e\?* ) sinh 23 ¸ 2k¹
(16)
62
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
Tunneling current densities comprise in fact one of the components of the total short circuit current density J of the devices under study, as it can be seen from Table 2. Note that for any solar cell, the J}< characteristic is given by the standard expression J"J (eO4 I2!1)!J , (17) where J is the reverse saturation current density, and J is the current component under short-circuit conditions. At open-circuit conditions (i.e. when <"< ) Eq. (17) dictates that 0"J (eO4 I2!1)!J or that J J eO4-! I2. (18) Short-circuit current densities as described by Eq. (18) include all possible contributions under illumination and at open-circuit conditions. The rest of the components (see also Table 2) that contribute to the total J are to be derived in the next sections. 4.2. Calculation of thermionic currents According to thermionic emission theory ([29], and references there in), a thermal current of carriers escaping from the quantum wells can be established by following the scheme of current densities imposed by expression (1) or (1) via (1)
A J "q G (1!e\?* ) d Ev(E)g(E) f (E#q< )(1!P(E)), (19) 2& ¸ where the last term in the integration ensures non-tunneling (note that P(E) is the tunneling probability used earlier), and where the integration will include all the available energy values from the lowest possible to the conduction band (hence practically to in"nity). The lowest available energy state of the trapped carriers is the ground state E measured from the bottom of the well. Since the energies of the trapped carriers in the wells are many k¹s away from the quasi-Fermi level E , one may replace the Fermi}Dirac with a Maxwell}Boltzmann distribution, $ so that expression (19) successively becomes as follows (with P(E) 1, i.e. neglecting numerical contributions from the second integral in Eq. (19) that includes the P(E) term):
A 2 m* G (1!e\?* ) dE (E!E J "q 2& ¸ m * # E#q< !E $ (1!P(E))exp ! k¹
"q
A (2m* G (1!e\?* ) e\O4-! ># \#$ I2 dE(E!E e\#\# I2 ¸ #
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
A (2m* E!E $ G (1!e\?* ) e\O4-! I2 dE(E!E exp ! ¸ k¹ #
q
63
A (2m* G (1!e\?* ) e\O4-! I2 ¸
"q
dE(E!E e\#\# I2e\# \#$ I2.
Changing variables: x"[(E!E )/k¹]:
(2m* A J "q G (1!e\?* ) e\O4-! ># \#$ I2 dx(xe\V(k¹) 2& ¸
"q
A (2m*(k¹) G (1!e\?* ) e\O4-! ># \#$ I2 dx(xe\V ¸
"q
A (2m*(k¹) G (1!e\?* ) e\O4-! ># \#$ I2(1/2), ¸
(20)
where the integral at the end of Eq. (20) is just the gamma function ()" . Expression (20) may now be written as
A (2m*(k¹) J "q G (1!e\?*G ) e\O4-! ># \#$ I2. 2& ¸
(21)
The energy di!erence E !E can be evaluated in a variety of ways: based on $ Fig. 1 and measuring from E , this di!erence may be re-written as follows: $ E !E "E #(E !E )!, $ $
(22)
where E is the energy miniband in the well measured from the bottom of the well, is the conduction band discontinuity and equal to E !E . As before, E refers to wide gap material (in this case AlGaAs: E "E !E "1.88 eV) and E to the conduction edge of the narrow-gap semiconductor (GaAs: E "E !E "1.42 eV). The energy di!erence E !E is easily (i.e. by standard tech $ niques, see for instance Ref. [29]) determined from the fact that
N E !E "k¹ ln $ p
(23)
[N (cm\) is the e!ective density of states for GaAs/AlGaAs systems, and p(cm\) is the doping of the p-regions as described in Section 2] Hence E !E "E !(E !E )"E !k¹ ln(N /p). $ $
(24)
64
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
substituting Eq. (24) into Eq. (21), by virtue of Eq. (22)
A (2m*(k¹) J "q G (1!e\?* ) e\O4-! >B# \ I2e\#! \#$ I2 2& ¸
"q
A (2m*(k¹) N G (1!e\?* ) e\O4-! >B# \ I2 e\# I2 ¸ p
"q
A (2m*(k¹) N e\O4-! >B# \ I2e\# I2 G (1!e\?* ) ¸ p
A (2m*k N ¹ e\# >O4 >B# \ I2 G (1!e\?* ) ¸ p "(A*)¹ e\# >O4 >B# \ I2,
"q
(25)
where the coe$cient A* is
A (2m*k N . G (1!e\?* ) (26) A*"q ¸ p from Eq. (25) it is concluded that thermionic currents of multiple quantum well (MQW's) solar cells include a ¹ temperature factor, as opposed to a ¹-dependence for the bulk case (i.e. no MQWs, see also Ref. [29]). 4.3. Excess carriers in the n-region An estimate of any excess carriers in the n-region may be pursued via the di!usion of minority carriers (holes) in the n-region. The current density due to minority hole concentration (in the n-AlGaAs region, ignoring general minority carriers in the very thin p-region) can be evaluated [41] from "rst principles namely, by evaluating the minority holes p as a function of distance in the n-region and then by evaluating the hole-di!usion current. These steps are as follows: First, one has to solve the di!usion equation, past the intrinsic region ¸ : dp p ! #G e\?* >V"0, D (27) dx where the "rst term indicates the di!usion of excess holes as a function of x, while the next two terms are the carriers: generated minus recombined/absorbed. Following standard techniques, the solution of Eq. (27) is the sum of the `homogeneousa and the `particulara solution ¸G e\?* >V, p "AeV * #e\V * # (28) D (1!¸) where A, B are constants to be found from the boundary conditions: dp zero at the farthest end of the n-region, thus A"0. On the other hand, p (0)"0, with x"0
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
65
taken at the very end of the intrinsic region, thus determining B: ¸G e\?* . B"! D (1!¸ ) Application of the boundary conditions "nalizes Eq. (28) to ¸ G e\?* (e\?V!e\V * ). p " D (1!¸)
(29)
From which the hole-di!usion current is
¸ e\?* , "qG (30) 1#¸ V where ¸ is the di!usion length of the generated holes in the bulk n-region of the p/i/n device, estimated in terms of the di!usion coe$cient D , and hole relaxation time , by means of the Einstein's relation. For a standard hole mobility value at 400 cm/V s, and with "10\ s (minority lifetimes in GaAs/AlGaAs systems), the di!usion length is found to be 3.6 m at 1003C and 2.96 m at }103C, respectively. dp J "qD dx
4.4. Ewects of recombination Treating the source of photon in#ux as a `black-body radiationa event, one may connect the incident photon #ux and the corresponding frequencies via Planck's formula, and at any frequency f, via standard techniques (see for instance Ref. [42]): 8f G ( f )" . c(eFD I2!1)
(31)
In thermal equilibrium, recombination rates are equal to generation rates: thus, the corresponding recombination rates may be found from G given above, and by integrating over all frequencies (in reality frequencies or wavelengths re#ecting energies of the order of the optical gap of the device and beyond: by the term optical gap, the energy di!erence between miniband E valence band is hinted). By setting y"hf/k¹, and assuming no frequency dependence of the dielectric constant , and absorption coe$cient :
8(k¹) y R "G " dy . (32) c eW!1 # I2 When thermal equilibrium is disturbed, direct (electron}hole) recombination rates may always be expressed as R"Cnp
66
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
while in equilibrium these rates are simply R "Cn p "Cn (where: C is a constant factor, and where the intrinsic carrier concentration n is connected to the equilibrium values: n p "n). Therefore, comparison of the two rates at non-equilibrium and at equilibrium conditions respectively, leads to the following correspondence: pn R"R , n
(33)
where p and n include excess carriers on top of the equilibrium values. R is found to be of the order of 10\ cm\ s\ in agreement with Ref. [12], while the excess carriers in the quantum wells are [43]:
m (k¹) E !E n" * ln 1#exp $ k¹
10 cm\
(34)
(at E !E "0.72 eV), where the Fermi level used is the quasi-Fermi level after $ splitting has taken place due to illumination, and where E is the ith miniband in the quantum wells. By means of Eq. (30), the recombination rate R (via Eq. (29)) is estimated to be of the order of 10}10 cm\ s\, and hence much less than the incident photon #ux G ("2.6;10 cm\ s\). 4.5. Fitting the experimental results The total short-circuit current density is calculated by summing J , J , and 23 2& J from Eq. (30). Results of the calculations for each sample at the two temperature extremes are summarized in Table 2. Experimental values of open-circuit voltage were used in Eqs. (16) and (25). The values of tunneling and thermal current densities depend on the transit time which is used as a "tting parameter consistent for all device designs depicted in this communication: of the order of ps in tunneling [43] and ms for thermionic emission [44]. Final J values seem to be remarkably close to the experimental ones reported here, thus making a strong case for the existence of quantum size ewects in MQW solar cells. Finally, by comparing thermionic emission current values, one observes that they are dominant and comparable at 1003C but di!er greatly at }103C, where the deeper energy levels of the wider wells result in reduced thermionic current contribution. By contrast, it is particularly striking the fact that tunneling currents behave almost `by the booka (and in agreement with textbook quantum mechanics): (a) they are non-negligible at any temperature; (b) they dominate at }103C and as expected (less scattering events); (c) they contribute a higher current density in the case of thin barriers than in the case of thick ones.
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
67
Table 2 Current density components at 1003C and !103C for samples GB68 (23 wells), GB69 (40 wells) and GB71 (39 wells) Current density (mA/cm)
GB68 (thick wells)
GB69 (thin wells)
GB71 (thick wells, thick i-region)
#1003C
!103C
#1003C
!103C
#1003C
J 2& J 23 J .
51.28 3.50 2.28
5.85 21.23 2.19
52.78 6.92 2.26
14.20 42.94 2.17
52.41 4.20 1.23
1.80 7.60 1.18
Total Experimental
57.06 56.09
29.27 31.71
61.96 63.41
59.31 54.87
57.84 59.26
10.58 9.75
!103C
Thickness of barrier is proportional to well thickness. J is the thermionic short-circuit current density, 2& J is the tunneling short-circuit current density, J is the excess carriers short-circuit current density, Total 23 . is the total theoretical short-circuit current density (J #J #J ) and Experimental is the measured 2& 23 . short-circuit current density.
5. Conclusions The behavior of p/i/n Al Ga As solar cells with GaAs/Al Ga As MQWs in the i-region has been studied with temperature under illumination and compared to that of a p/i/n GaAs and AlGaAs solar cell. The dark currents of the MQW diodes were found to lie between those of conventional cells formed from the well material and from the barrier material alone. It was found that the increase in the dark current of the diodes with temperature was accompanied by an improvement of diode ideality factor, even for the one diode which exhibited poor dark current characteristics at room temperature. Under illumination and at elevated temperatures, the MQW solar cells exhibited enhanced I , which was comparable to the photocurrent of the control (p/i/n GaAs) sample. In addition they exhibited higher < . The MQW solar cells with thin barrier width exhibited enhanced output performance in the whole range of temperatures examined, due to contribution of tunneling current to the total observed photocurrent, whereas the MQW cell with thicker barrier width showed remarkable performance only at elevated temperatures. It was con"rmed by modeling that non-zero tunneling currents exist at all temperatures but is more signi"cant in thin barriers as expected. It can readily be inferred from these results that p/i/n GaAs/AlGaAs MQW structures of high-quality material should exhibit enhanced performance over conventional p/i/n GaAs solar cells under intense illumination when processed speci"cally for application as solar cells. The relative contribution of tunneling and thermionic emission currents as a function of temperature was determined by modeling and "tting and should provide guidance for the optimized design of MQW solar cells tailored for operation in speci"c temperature ranges.
68
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
Acknowledgements The authors wish to thank the British Council and the Greek Ministry for Development for "nancial support through a bilateral program between Greece and the UK. In addition, one of the samples reported here was developed under a bilateral program between Greece and Germany. P. Panayotatos acknowledges the support of Rutgers University through the FASP program.
References [1] K. Barnham, I. Ballard, J. Barnes, J. Connolly, P. Gri$n, B. Kluftinger, J. Nelson, E. Tsui, A. Zachariou, Appl. Surf. Sci. 133/114 (1997) 722. [2] G.B. Lush, M.P. Patkar, M.P. Young, M.R. Melloch, M.S. Lundstrom, S.M. Vernon, E.D. Gagnon, L.M. Geo!roy, M.M. Sanfacon, Thin-"lm GaAs solar cells by epitaxial lift-o!, Proceedings of the 23rd IEEE Photovoltaic Specialists Conference, 1993, p. 1343. [3] P.R. Hageman, G.J. Bauhuis, A. Van Geelen, P.C. Van Rijsingen, J.J. Schermer, L.J. Giling, Large area, thin "lm epitaxial lift-o! III/V solar cells, Proceedings of the 25th Photovoltaic Specialists Conference, 1996, p. 57. [4] M. Yang, T. Soga, T. Jimbo, M. Umeno, Jpn. J. Appl. Phys. 33 (1994) 6605. [5] C. Flores, B. Bollani, F. Campesato, F. Paletta, D. Passoni, G. Timo, A. Tosoni, Sol. Energy Mater. 23 (1991) 356. [6] K.W.J. Barnham, G. Duggan, J. Appl. Phys. 67 (1990) 3490. [7] F.W. Ragay, E.W.M. Ruigrok, J.H. Wolter, GaAs-AlGaAs heterojunction solar cells with increased open-circuit voltage, Proceedings of the First World Conference on Photovoltaic Energy Conversion, 1994. [8] J. Nelson, K.W.J. Barnham, J.P. Connolly, G. Haarpaintner, C. Button, J. Roberts, Quantum well solar cell dark current, Proceedings of the 12th European Photovoltaic Solar Energy Conference, 1994, p. 1370. [9] As . Aperathitis, Z. Hatzopoulos, P. Panayotatos, D. Sands, C.G. Scott, M. Yousaf, PIN solar cell e$ciency enhancement by the use of MQW structures in the I-region, Presented at 14th European Photovoltaic Solar Energy Conference, Barcelona, Spain, 30 June}4 July, 1997. [10] J. Barnes, T. Ali, K.W.J. Barnham, J. Nelson, E.S.M. Tsui, J. Roberts, M.A. Pate, S.S. Dosanjh, Gallium Arsenide/Indium Gallium Arsenide multi- quantum well solar cells, Proceedings of the 12th European Photovoltaic Solar Energy Conference, 1994, p. 1374. [11] D.E. Ashenford, A.W. Dweydari, D. Sands, C.G. Scott, M. Yousaf, E. Aperathitis, Z. Hatzopoulos, P. Panayotatos, J. Crystal Growth 159 (1996) 920. [12] R. Corkish, M. Green, Recombination of carriers in quantum well solar cells, Proc. 23rd IEEE Photovoltaic Specialists Conference, 1993, p. 675. [13] G.L. Araujo, A. Marti, F.W. Ragay, J.H. Wolter, E$ciency of multiple quantum well solar cells, Proceedings of the 12th European Photovoltaic Solar Energy Conference, 1994, p. 1481. [14] N.G. Anderson, J. Appl. Phys. 78 (1995) 1850. [15] K.W.J. Barnham, J.M. Barnes, B. Braun, J. P. Connolly, G. Haarpaintner, J. A. Nelson, M. Paxman, C. Button, J.S. Roberts, C.T. Foxon, A novel approach to higher e$ciency-the quantum well solar cell, Proceedings of the 11th European Photovoltaic Solar Energy Conference, 1992, p. 146. [16] A. Luque, Solar Cells and Optics for Photovoltaic Concentration, Adam Hilger, Bristol, 1989. [17] I. Ballard, K.W.J. Barnham, J. Nelson, J.P. Connolly, C. Roberts, J.S. Roberts, M.A. Pate, The e!ect of Temperature on the e$ciency of multi-quantum well solar cells, Proceedings of the 2nd World Conference and Exhibition on Photovoltaic Solar Energy Conversion, Vienna, Austria, 6}10 July 1998, p. 3624
E. Aperathitis et al. / Solar Energy Materials & Solar Cells 70 (2001) 49}69
69
[18] P.R. Gri$n, J. Barnes, K.W.J. Barnham, I. Ballard, M. Cabodi, M. Mazzer, J.S. Roberts, R. Grey, G. Hill, M.A. Pate, A. Marti, J.C. Maroto, R.F. Reyna, L. Cuadra, M. Yang, and M. Yamaguchi, A method for optimising strained GaAs/InGaAs quantum well solar cells and preliminary results under concentration, Presented at 14th European Photovoltaic Solar Energy Conference, Barcelona, Spain, 30 June}4 July 1997. [19] H. Ohtsuka, T. Kitatani, Y. Yazawa, T. Warabisako, Sol. Energy Mater. Sol. Cells 50 (1998) 251}257. [20] E. Aperathitis, C.G. Scott, D. Sands, V. Foukaraki, Z. Hatzopoulos, P. Panayotatos, Mater. Sci. Eng. B 51 (1998) 85. [21] A.C. Varonides, E. Aperathitis, P. Panayotatos, C.G. Scott, Signi"cance of tunneling and thermionic emission currents in performance enhancement of multiple quantum well GaAs/AlGaAs p}I}n solar cells, Proceedings of the Second World Conference and Exhibition on Photovoltaic Solar Energy Conversion, Vienna, Austria, 6}10 July 1998, p. 66 [22] E. Aperathitis, Z. Hatzopoulos, M. Androulidaki, V. Foukaraki, A. Kondilis, C.G. Scott, D. Sands, P. Panayotatos, Sol. Energy Mater. Sol. Cells 45 (1997) 161. [23] K.J.W. Barnham, J. Barnes, G. Haarpaintner, J. Neslon, M. Paxman, T. Foxon, J. Roberts, Quantum well solar cells, MRS Bulletin, October 1993, pp.51}55. [24] F.W. Ragay, J.H. Wolter, A. Marti, G.L. Araujo, Experimental analysis of the e$ciency of MQW solar cells, Proceedings of the 12th European Photovoltaic Solar Energy Conference, 1994, p. 1429. [25] F.W. Ragay, P.J. van Hall, M.R. Leys, E.A.E. Zwaal, J.H. Wolter, GaAs}AlGaAs p}i}n heterostructures for high-e$ciency solar cells, Proceedings of the 11th European Photovoltaic Solar Energy Conference, 1992, p. 359. [26] G. Haarpaintner, J. Barnes, K.J.W. Barnham, J. P. Connolly, S.S. Dosanjh, J. Neslon, C.Roberts, C. Button, G. Hill, M.A. Pate, and J.S. Roberts, Voltage performance of quantum well solar cells in the Al Ga As/GaAs and the GaAs/In Ga As material systems, Proceedings of the First World V \V W \W Conference on Photovoltaic Energy Conversion, 1994, p. 1783. [27] K. Barnham, J. Connolly, P. Gri$n, G. Haarpaintner, J. Nelson, E. Tsui, A. Zachariou, J. Osborne, C. Button, G. Hill, M. Hopkinson, M. Pate, J. Roberts, T. Foxon, J. Appl. Phys. 80 (1996) 1201. [28] F.W. Ragay, A. Marti, G.L. Araujo, J.H. Wolter, Sol. Energy Mater. Sol. Cells 40 (1996) 5. [29] S.M. Sze, Physics of Semiconductor Devices, Wiley, New York, 1981. [30] S.P. Tobin, S.M. Vernon, C. Bajgar, L.M. Geo!roy, C.J. Keavney, M.M. Sanfacon, V.E. Haven, Sol. Cells 24 (1988) 103. [31] S.P. Tobin, S.M. Vernon, C. Bajgar, S.J. Wojtczuk, M.R. Melloch, A. Keshavarzi, T.B. Stellwag, S. Venkatensan, M.S. Lundstrom, K.A. Emery, IEEE Trans. Electron Dev. 37 (1990) 469. [32] M.R. Melloch, Sol. Cells 30 (1991) 313. [33] H.J. Hovel, Solar cells, in: R.K. Willardson, A.C. Beer (Eds.), Semiconductors and Semimetals, Vol. 11, Academic Press, New York, 1975, pp. 166}180. [34] I. Ballard, Temperature e!ects on the e$ciency of MQW solar cells, M.Sc. Thesis, September 1995, Imperial College of Science and Medicine, London. [35] J. Nelson, M. Paxman, K.W.J. Barnham, J.S. Roberts, C. Button, IEEE Trans. Quantum Electron. 29 (1993) 1460. [36] F. Capasso, K. Mohammed, A.Y. Cho, R. Hull, A.L. Hutchinson, Appl. Phys. Lett. 47 (1985) 420. [37] N. Bernhard, G.H. Bauer, W.H. Bloss, Progress in Photovoltaics: Res. Appl. 3 (1995) 149. [38] M. Hirose, S. Miyazaki, J. Non-Cryst. Sol. 9798 (1987) 23. [39] I. Pereyra, M.N.P. Carreno, R.K. Omnori, C.A. Sassaki, A.M. Andrade, F. Alvarez, J. Non-Cryst. Sol. 9798 (1987) 871. [40] L.I. Schi!, Quantum Mechanics, McGraw-Hill, New York, 1968. [41] E.S. Yang, Microelectronic Devices, McGraw-Hill, New York, 1988. [42] D.K. Ferry, Semiconductors, Macmillan, New York, 1991. [43] M. Shur, GaAs Devices and Circuits, Plenum Press, New York, 1987. [44] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Halsted Press, New York, 1988.