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  IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 45, NO. 2, FEBRUARY 1998 423 Improved Equivalent Circuit and Analytical Modelfor Amorphous Silicon Solar Cells and Modules J. Merten, J. M. Asensi, C. Voz, A. V. Shah, R. Platz, and J. Andreu  Abstract— An improved equivalent circuit for hydrogenatedamorphous silicon ( a  -Si:H) solar cells and modules is presented. Itis based on the classic combination of a diode with an exponentialcurrent–voltage characteristic, of a photocurrent source plus anew term representing additional recombination losses in the i-layer of the device. This model/equivalent circuit matches the I  ( V  ) curves of  a  -Si:H cells over an illumination range of sixorders of magnitude. The model clearly separates effects relatedto the technology of the device (series and parallel resistance) andeffects related to the physics of the pin-junction (recombinationlosses). It also allows an effective   product in the i-layer of thedevice to be determined, characterizing its state of degradation.  Index Terms— Amorphous silicon solar cells and modules,analytical model, I  ( V  ) characteristics, outdoor measurements,recombination,   -product degradation. I. I NTRODUCTION T HE use of equivalent circuits is a convenient and commonway to describe the electrical behavior of electronicdevices. Generally, an equivalent circuit offers three mainadvantages: it is easy to use within electrical circuits; itallows the device’s properties to be described in a standardizedand abbreviated manner using a simple analytical model; itprovides insights into the complex physical processes that takeplace within the device.The equivalent circuit generally used for photovoltaic solarcells is shown in Fig. 1 (ignoring the dashed section): itessentially consists of a current source shunted by a diode.These two elements correspond to generation and loss of photocurrent in the device. The resistancesand canbe considered to be “parasitic” circuit elements, introduced todescribe the behavior of real solar cells with their technicallimitations. We shall come back to these later.That one may simply superpose a photocurrent sourceon the characteristics of the dark diode is, at first sight,physically surprising—in fact, the photo-induced generation of holes and electrons within the solar cell will change the carrierconcentration at every point, requiring, thus, a new solutionfor the drift-diffusion differential equations throughout the Manuscript received October 10, 1996; revised July 10, 1997. The review of this paper was arranged by Editor P. N. Panayotatos. This work was supportedby the EU Project JOU2-CT94-0403, Project EF-REN (93)032 of the SwissConfederation, by the Spanish Goverment, and the Generalitat de Catalunya.J. Merten, J. M. Asensi, C. Voz, and J. Andreu are with Universidad deBarcelona, Departament de F´ısica Aplicada i Eletr`onica, E-08028 Barcelona,Spain.A. V. Shah and R. Platz are with Institut de Microtechnique, Rue Breguet,Switzerland.Publisher Item Identifier S 0018-9383(98)00969-1.Fig. 1. Equivalent circuit for photovoltaic solar cells and modules. Thecurrent sink (dashed lines) takes into account the current losses due torecombination in the i-layer of the device. whole device. However, as postulated in [1], and as shownin [2], such a simple superposition of a dark diode and aphotocurrent source is indeed valid and can be theoretically justified for crystalline solar cells, consisting of pn-diodes.The theoretical justification [2] is based on the linear form of the drift-diffusion differential equations for minority carriers,within the p- or n-type bulk regions of the pn-diode.It is, however, well known that amorphous silicon solarcells, which are pin-diodes and in which the main part of the photovoltaic generation occurs in the intrinsic i-layer,behave differently. As a striking example, the curvesfor different illumination levels usually all meet at a singlepoint in the first quadrant [3], a fact that can onlybe reconciled with the simple equivalent circuit of Fig. 1 if anadditional loss term, which increases strongly with the forwardvoltage , is introduced. Such a loss term has to take intoaccount the recombination losses in the intrinsic layer of thedevice.Generally speaking, recombination is relatively intensewithin amorphous silicon cells because of the presence of dangling bonds that act as recombination centers—this isespecially true for cells in the degraded state. It is thereforeintuitively “reasonable” to describe amorphous silicon cellsby introducing an additional recombination loss term intothe equivalent circuit, a term which is symbolized by thecurrent sink (dashed lines) in Fig. 1. Recombination losseswithin the i-layer are in a first approximation proportional tothe carrier concentrations observed therein, and thus to thephotogenerated current This has been shown theoreticallyin [4] and is used in the formulation of the loss currentThe aims of this article are fourfold:1) to show empirically that the equivalent circuit in Fig.1 describes quite precisely the experimentally measuredelectrical behavior of illuminated solar cells and is ableto do so for illumination levels varying over six ordersof magnitude; 0018–9383/98$10.00 © 1998 IEEE Authorized licensed use limited to: Universitat de Barcelona. Downloaded on February 16, 2009 at 04:39 from IEEE Xplore. Restrictions apply.  424 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 45, NO. 2, FEBRUARY 1998 Fig. 2. Comparing the experimental I  ( V  ) -data of the laboratory cellshown in Fig. 3 with the results of the analytical model. The decrease of  R  sc =( @V=@I  ) V  =0 of the cell in the degraded state is attributed to enhancedrecombination in the i-layer of the device. Note that the VIM-method doesnot require calibrated illumination levels, these may therefor differ for themeasurement of the cell in the degraded and initial state. 2) to present an experimental method that can be used todetermine systematically the elements of this equiva-lent circuit, including the new recombination loss term(dashed symbol in Fig. 1);3) to provide a link between the equivalent circuit of Fig. 1,especially between the newly introduced recombinationloss term, and theoretical treatments of recombination inamorphous silicon cells, such as [4];4) and to illustrate the use of the new, complete equiva-lent circuit to investigate the long-term behavior of acommercial module during outdoor exposition.II. E XPERIMENTAL P ROCEDURE The authors have used the standard characterization proce-dure for solar cells that consists of measuring the curveat a given illumination level as it shown in Fig. 2. The resultingdata may be condensed into the six characteristic parameterswhich are:1) the short circuit current ;2) the open circuit voltage ;3) the fill factor ;4) the efficiency ;5) the “open circuit resistance” , whichmay be related to the series resistance ;6) and the “short circuit resistance” ,which may be related to the parallel resistanceThe latter two parameters and are key parametersfor the present treatment; they are reciprocal slopes of thecurve.The basic idea is to measure the curve over a widerange of illumination levels, rather than at a fixed illuminationlevel of say 1000 W/m Thereby, additional information aboutthe device can be gained. We call this method the VariableIllumination Measurement (VIM) method.The experimental results are plotted as a function of theshort circuit currentor open circuit voltage avoidingthe need of calibrated illumination levels. This has been donewith the data obtained from a typical laboratory solar cell[5] (single junction, glass/SnO /pin-structure with an i-layerthickness of 0.35- m, 10.7% initial efficiency) which areare indicated by the symbols in Fig. 3.The light source may be either a laboratory lamp or, alternal-tively, sunlight for outdoor measurements. In the laboratory,variation of the illumination level over six orders of magnitudeis obtained by varying the distance between the lamp andthe sample and by using neutral (grey) filters. The outdoorVIM-method makes use of the natural variation of the solarirradiance, the spectral variations having only a slight influenceon the results in the case of single junction cells.III. T HE A NALYTICAL M ODEL The single exponential model represented by the equivalentcircuit in Fig. 1 ignoring the dashed section is known tomatch well the curves of crystalline solar cells [1],[2]. This model predicts that the short-circuit resistanceshould be equal to the parallel resistance of the deviceover a large range of illumination levels [dot-dashed linein Fig. 3(b)]. The experimental -data show that thisis not the case for amorphous silicon cells. This constitutesa clear empirical motivation for the introduction of a newcurrent loss term into the equivalent circuit (represented by thedashed section in Fig. 1): a term which explicitly takes intoaccount the recombination losses in the i-layer of the device.A simple expression for this current can be deduced assumingthe electrical field to be constant within the i-layer, andto be strong enough to mask the effects of the diffusion of the carriers [4]. This very crude assumption is expected tobe valid only for small or negative external voltages forcells with thin i-layers (small value of and for low defectdensities therein. In this case, a homogenous generation of carriers leads to linearily varying profiles for the free electronsand holes in the i-layer [4]. The recombination functionis taken from [6](1)where and are the capture times of the electronsand holes by the neutral dangling bonds. Inserting the linearcarrier profiles mentioned above, this recombination functionbecomes(2)where is the position in the i-layer measured as the distancefrom the p-layer,and the band mobilities of the freecarriers. Remember that this expression has been obtainedneglecting diffusion currents and is only valid for strong fieldsin the i-layer, thin cells and low defect densities in the i-layer.The expression for the recombinationin (2) maybe integrated over the whole i-layer (fromto toobtain the total current loss due to recombination withinthe i-layer; we thereby obtain(3) Authorized licensed use limited to: Universitat de Barcelona. Downloaded on February 16, 2009 at 04:39 from IEEE Xplore. Restrictions apply.  MERTEN et al .: IMPROVED EQUIVALENT CIRCUIT AND ANALYTICAL MODEL 425 (a)(b)(c)(d)Fig. 3. Illumination level dependence of the I  ( V  ) parameters. The symbolsindicate the experimental data of a typical single junction laboratory cell,the lines the predictions of the model. Omitting the recombination current(dot-dashed lines) affects only the predictions for R  sc and FF; but notthose for V  oc and R  oc : The marks on the x  -axis denote I  sc under one-sunillumination, and the dotted line ( @V=@I  dark  ) as a function of  I  dark  : Theoscillations of  R  oc are a numerical effect. with(4)It should be noted that equals the generation currentmultiplied by the ratio of the cell thickness over theeffective driftlength (Schubweg) in the i-layer. Theelectrical field in the i-layer (assumed to be constant) isset to the difference between the built-in voltage and thevoltage over the junctionThe effective -product TABLE IP ARAMETER V ALUES U SED TO F IT THE M ODEL TO THE E XPERIMENTAL D ATA OFA T YPICAL L ABORATORY C ELL IN THE D EGRADED (B) AND I NITIAL (A) S TATE suitably combines the -products of electrons andholes 1 , as resulting from the integration of (2).Introducing this recombination current into the equivalentcircuit (dashed symbol in Fig. 1) leads to the followinganalytical expression for the curve of amorphous siliconsolar cells and modules(5)The photogenerated current is reduced by the losscurrents due to recombination in the i-layer, by the diode withits saturation current and its quality factor denoteshere the elementary charge, Boltzmanns’ constant andthe absolute temperature of the device), and by the parallelresistanceNote that we expect this model to be valid for small forwardvoltages only, where the crude assumptions mentioned abovecan be considered to be fulfilled. This is the case in the shortcircuit region [4] and we find from (5) that the slope isdetermined by the recombination term of the model(6)Here the effect of and have been neglected, which iscorrect for intermediate illumination levels (see Section IV).The measurement of provides direct information aboutthe effective -product within the i-layer of the device.Numerical simulation of amorphous silicon solar cells usingthe program described in [8] with a standard defect modeldemonstrates that the recombination current within the i-layeris the loss current showing the strongest variation with theexternal voltage [9]. Other loss currents (for example thosedue to recombination at the interfaces between the intrinsic andthe doped layers) were found to have a very weak dependenceon the external voltage leading thus to a neglible influence on[9].The built-in voltage in amorphous silicon solar cells wasdetermined by [10] to be V and we will use thisvalue for the following calculations. Assuming that doesnot change during degradation, monitoring allows directmeasurement of the degradation state of the i-layer of thedevice. 1 Note that recent investigations on a  -Si:H films lead to the empiricalconclusion that the   -products are approximately equal in the case of compensated or strongly degraded a  -Si:H [7], i.e., (   ) e      0 n    0 n      0 p    0 p  : If, in fact   0 n    0 n  =    0 p    0 p  holds, a more precise calculation of recombinationlosses within the i-layer of the pin cell can be employed for our model andleads to a final result that is identical to that given above in (3) and (4). Authorized licensed use limited to: Universitat de Barcelona. Downloaded on February 16, 2009 at 04:39 from IEEE Xplore. Restrictions apply.
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