aSi Final Report

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UPMC Universite Paris VI

Master (M2) in Fluid Mechanics

Internship at E.T.S.I.A., Universidad Politecnica de Madrid

Electrical characterization of amorphoussilicon solar cells

Author:

Pablo Penas

Supervisor:

Dr. Miguel Hermanns

27th August 2013


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Technical abstract

This work begins with an introduction to the fundamentals of operation of an amorphous siliconp-i-n cell in order to qualitatively explain its electrical behaviour. Other important features

such as carrier recombination and the shunt-leakage current are discussed.A procedure to electrically characterize the current-voltage (I V) behaviour of amorphoussilicon p-i-n cells is then presented. The electrical behaviour of the cell is described analytic-ally by an equivalent circuit model, alongside its limitations and underlying assumptions. Thecharacterization process essentially consists in experimentally determining the unknown circuitelements. A Variable Illumination Method is employed, where the experimental data is strictlylimited to various I V curves under specific illumination conditions. The simplicity of theexperimental set up, consisting purely of an appropriate light source and an I V measuringstation, is a main advantage of this particular characterization method.

A single-junction cell is first characterized. All parameters except the effective -product andbuilt-in voltage could be extracted from aI V curve in the dark. The effective -product, an

indicator of the rate of carrier recombination and state of degradation of the cell, has been shownto depend on the illumination spectrum and intensity. The I V curves predicted by the modelwere observed to perform poorly under large reverse voltage biasing. This has been attributedto a non-Ohmic voltage dependence of the shunt leakage current in this region. Therefore, anextended model that takes into account shunt leakage current non-linearities is also presented.

A double-junction a-Si:H/a-Si:H cell is next characterized. The tandem cell was exposed to aparticular IR and UV bias light in order to excite each subcell differently. It was not possible toachieve single subcell excitation. Nevertheless, theoretical relations have been derived for suchscenario. Instead, it was concluded that UV light induced a considerably greater photocurrentin one particular subcell than in the other, while IR light had the opposite effect. The charac-terization process was then adapted to match these particular circumstances. The end resultwas a set of analytical expressions relating graphically-obtained variables (open-circuit voltageand resistance, short-circuit current and resistance) to the circuit parameters. These expressionswere seen to be supported by experiment. The characterization proved useful to determine thestate of degradation of one subcell relative to the other.

Further work is required to model the J V curves for the double-junction cell. This may bedone by first achieving single subcell excitation, therefore making use of the theoretical relationsderived for this scenario.

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Acknowledgments

The contribution of the project supervisor, Dr. Miguel Hermanns, of the Universidad Politecnicade Madrid (UPM), merits special acknowledgment for his excellent guidance, reviews and sug-

gestions that have made this work possible.The author wishes to thank the members of Instituto de Energa Solar (IES) of the UPM,especially Yedileth Contreras for her great help with the apparatus set-up and experimentalmeasurements; and Ignacio Rey-Stolle for kindly providing the electrical characterization facil-ities.

The author gratefully acknowledges the vital contribution of Alonso Pardo in the design andassembly of the circuit and LED boards.

Finally, the author would like to express his gratitude towards Javier Izard from the companySoliker, for providing the test cell and for sharing valuable insight on the electrical behaviour ofamorphous silicon solar cells.

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Contents

1 Introduction 6

2 Electrical behaviour ofa-Si:H solar cells 72.1 Fundamentals of operation of a p-i-n junction . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Diffusion and voltage biasing . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Carrier recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Single-cell electrical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 General equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Limitations of the model and underlying assumptions . . . . . . . . . . . 14

2.3 Extended model for reverse voltage biasing . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Shunt leakage current and shunt-busting . . . . . . . . . . . . . . . . . . . 16

3 Test cell and experimental technique 17

3.1 Test cell description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Apparatus description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Electrical characterization of a single-junction cell 19

4.1 Measurements in dark conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.1 Reverse voltage biasing model . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Measurements under illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 Model performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3.1 Cell response to changes in illumination intensity and spectrum . . . . . . 314.4 Sources of error and uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Electrical characterization of a double-junction cell 37

5.1 Model and general equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3 Measurements in dark conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.4 Theoretical derivations for single subcell excitation . . . . . . . . . . . . . . . . . 41

5.4.1 SC current OC voltage relation . . . . . . . . . . . . . . . . . . . . . . . . 425.4.2 SC and OC resistance property . . . . . . . . . . . . . . . . . . . . . . . . 43

5.5 Measurements under illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.5.1 Baseline assumptions at SC and OC operating conditions . . . . . . . . . 46

5.5.2 SC resistance-current relation . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.5.3 The V-R-J relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.6 Summary of findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6 Conclusions 54

6.1 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


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1 Introduction

Thin-film hydrogenated amorphous silicon (a-Si:H) p-i-nsolar cells are extensively used in a widerange of applications. They are commonly used in power sources for electronic devices such as

calculators and watches, batteries, photosensors and building integrated photovoltaics such assemi-transparent building facades or window glazing. Amorphous silicon solar cells constituteone of the most promising options for low-cost, large scale applications in photovoltaics.

The main advantage of thin-film a-Si:H cells over crystalline silicon cells is their thinness ( 300nm). This does not only imply a-Si:H cells having lower material costs, but it also makes themmore aesthetically attractive and flexible in their applications. Furthermore, manufacturingcosts are potentially low since there is a greater cost reduction potential than in c-Si cells dueto the significantly lower amount of silicon material required in its manufacture.

On the other hand, they suffer from limited efficiency ( 5 10%) partly due to light-induceddegradation (Staebler-Wronski effect) that manifests in the form of relatively high carrier re-combination losses.

A popular solution to improve the cell conversion efficiency is to stack cells of different band gapsto form a multi-junction or tandem cell. It is of uttermost importance to be able to properlycharacterize single-junction and tandem cells to allow for future improved designs.

This leads to the overall purpose of this work: to provide an electrical characterization methodfor a-Si:H cells. Particularly, a double-junction a-Si:H/a-Si:H cell shall be characterized, treatingit first as a single-junction cell and then as a double-junction cell.

The cell may be described analytically by an equivalent circuit. Equivalent circuits are a con-venient way of characterizing and modelling cells. They provide insight on the physical processesthat take place within the cell and evaluation of the circuit parameters gives a clear picture ofthe cells properties.

The characterization process essentially consists in experimentally determining the unknownparameters of the equivalent circuit via a Variable Illumination Method. The experimental datais strictly limited to various I V curves under specific illumination conditions. No knowledgeof the exact irradiance power, spectral response or QE of the individual subcells is required.While for the single-junction cell characterization white light is best suited for the task, forthe tandem cell characterization, bias UV and IR illumination shall be employed to excite thesubcells differently.

The main advantage of this characterization method over other methods is the simplicity of theexperimental set-up, consisting purely of an appropriate light source and an I V measuringstation.

This report is structured as follows. Section 2 covers the fundamentals of operation of p-i-njunctions found in a-Si:H cells in order to understand the shape of their characteristic I Vcurves. Next, the analytical, equivalent circuit model for a single-junction cell is introducedalongside its limitations and underlying assumptions. The physical characteristics of the testtandem cell, together with the measuring apparatus and light sources, are briefly described inSection 3. Section 4 is devoted to the characterization using the single-junction cell model. Themodel performance is evaluated, and the common sources of error are exposed. In Section 5, theequivalent circuit model shall be extended to describe a double-junction cell. The characteriz-ation method for such then follows. Lastly, the concluding remarks followed by possible futurework are presented in Section 6.

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2 Electrical behaviour ofa-Si:H solar cells

This section qualitatively describes the physics of p-i-n junctions and other aspects such as car-rier recombination in order to gain valuable insight on the electrical behaviour of hydrogenated

amorphous silicon cells. An equivalent circuit able to model such behaviour is then presented.

2.1 Fundamentals of operation of a p-i-n junction

The photodiode inside an a-Si:H based cell has a p-i-n structure. The first layer consists in a thin(usually 10 40 nm), p-type doped layer. Since it is negatively charged, holes are the majoritycarriers. The second layer is refered to as the intrinsic (i) layer. Typically, it has a thicknessd = 200 600 nm. The third layer is a thin, n-type doped layer (of similar thickness to the p-layer). Since it is positively charged, electrons are the majority carriers. In thermal equilibrium,electrons are donated from the n-layer to the p-layer. This generates an approximately uniformbuilt-in electric field Eb acting in the direction shown in Figure 1. A built-in voltage, Vb, is also

generated across the p-i-n junction. The following expression may be used to relate the two:

|Eb| Vbd

(1)

Figure 1: Schematic of a typical a-Si:H p-i-n cell. The TCO (transparent conducting oxide) andAZO (aluminium zinc oxide) layers act as the front and back electrical contacts respectively.

Most of the photovoltaic generation of the electron-hole pairs takes part in the undoped i-layer. Electron-hole pairs are created through photon absorption as depicted in Figure 2. Aphoton with the right amount of energy may transfer such energy to an electron in the valence

band of the semiconductor material (a-Si:H), where it is tightly bound in a covalent bondbetween neighbouring atoms. The electron becomes excited and jumps over to the higherenergy conduction band. An empty covalent bond is formed, referred to as a hole. This processhas been graphically represented in Figure 3. Electrons in the conduction band (and similarlyholes in the valence band) are free to move and therefore contribute to the electric current. Freeholes and electrons are referred to as carriers.

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Figure 2: Schematic of the generation of electron-hole pairs in a p-i-n cell. Note that theTCO forms the window layer, while the AZO layer acts as a reflector to maximise the captureprobabiliy of the cell.

Figure 3: Band diagram representation of carrier photo-generation. The inicident photon isrepresented by the curly orange arrow, whose energy is transferred to an electron in the semi-conductor valence band. The excited electron jumps over to the conduction band, and a hole isthus created in the valence band.

The electric field drives the photo-generated free electrons and holes from the i-layer to then-type and p-type layers respectively. This is referred to as drift. Diffusion (due to gradients incarrier concentrations) drives carriers in the opposite senses. This has been illustrated in Figure4. Drift and diffusion forces are always in competition with each other.

The following current sign criterion shall be adopted. For the top p-i-n junction schematic inFigure 4, the current driven by drift is larger than the current driven by diffusion. Hence the netoutput current (density) J shall be regarded as positive. For the bottom schematic, diffusiondominates over drift, generating a current in the opposite sense and consequently regarded as

negative. The dominant driving force is determined by the the external voltage V applied overthe cell, together with the illumination conditions. V directly affects the electric field |E| asfollows:

|E| Vb V

d(2)

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Figure 4: Schematics of a p-i-n junction. The length of the arrows symbolises the magnitudeof the carrier fluxes due to drift or diffusion. Top: Case when drift dominates over diffusion(strong |E|). Operating point (A) in Figure 5 belongs to this regime. The net output currentis taken to be positive: J > 0. Bottom: Case when diffusion dominates over drift (weak |E|).Operating point (C) in Figure 5 belongs to this regime. The net output current is taken to benegative: J < 0.

Consider a cell operating under intermediate illuminations. Its characteristic J V curve issketched in Figure 5. At operating point (A), the cell is operating in short-circuit conditions(SC) since there is no external voltage applied (V = 0). At this particular point, E = Eb. Theelectric field is strong, hence the current due to drift (Jdrift) is much bigger than the current dueto diffusion (Jdiff). As a result, the net current J is largely positive. In short, point (A) belongsto the operating regime where Jdrift Jdiff, portrayed in the top p-i-n schematic in Figure 4.

When V < 0 (reverse voltage biasing), the cell operates deeper in the Jdrift Jdiff regime. Thisis because |E| becomes even stronger, and so diffusion effects in the i-layer become even morenegligible. The drift current is relatively insensitive to the applied electric field [5] since it islimited by the number of minority carriers rather than by the electric field not being strongenough. This is observed in Figure 5, where J is shown to increase minimally as V becomes

more negative.

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Figure 5: Sketch of typical J V curves for an a-Si:H cell under illumination (solid line) and inthe dark (dashed line), labelled with the different main regimes of operation. MPP stands forthe Maximum Power Point of the cell.

The diffusion current, on the other hand, is very sensitive to |E|; the mechanism relating thetwo will be explained next in Section 2.1.1. When V > 0 (forward voltage biasing), |E| weakensand the diffusion current increases, eventually becoming comparable to the drift current as Vapproaches the open-circuit voltage VOC. This happens often past the maximum power operatingpoint (MPP), where drift still dominates. Point (B) corresponds to open-circuit conditions (OC),

where V = VOC and J = 0. Therefore, this point belongs to the Jdrift = Jdiff regime. If V isincreased further, |E| weakens even more and Jdiff overcomes Jdrift, resulting in J < 0. Point (C)belongs to this regime where Jdrift Jdiff, portrayed in the bottom p-i-n schematic in Figure 4.

A cell operating in the dark behaves in the same way. However, at a given V, the diffusioncurrent is always stronger than for illuminated conditions. This is because there are no electron-hole pairs being photo-generated in the i-layer, hence the minority carrier diffusion gradientsare steeper. As a result Jdrift dominates over Jdiff only when V < 0. Note that at point (B),V = 0 and J = 0, meaning the cell is essentially in thermal equilibrium.

2.1.1 Diffusion and voltage biasing

Electrons diffusing from the n-side to the p-side have to overcome an electrostatic potentialbarrier of energy EB , where:

EB = EC,p EC,n = q(Vb V) (3)

EC,p is the conduction band energy at the p-layer, EC,n is the conduction band energy at the n-layer, q is the carrier elementary charge. Clearly, EB decreases (increases) with forward (reverse)voltage biasing. Note that the energy of the barrier is proportional to the electric field |E| and tothe difference in the electrostatic potential (n p) across the p-i-n junction. This is picturedin Figure 6.

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Figure 6: (Electron) Energy band diagrams and their corresponding p-i-n junction operatingconditions. EC and EV denote the conduction and valence energy band respectively. Left: Casewhere cell operates at short-circuit conditions (V = 0). The net current density will be J = 0 indark conditions, else J > 0 under illumination. Right: Case where cell operates under forwardvoltage biasing (V > 0). Note that J > 0 if V < VOC, else J < 0 if V > VOC.

With no applied external voltage, the Fermi Level (EF) of the electrons at n and p is the same.When V > 0, the energy input from the power source raises overall the energy of the electronsat n, and the Fermi Level at n is shifted to a higher energy closer to EC,p. This means thatthe probability of an electron in the conduction band at n having energy E > EC,p is greater.Likewise, an electron photo-generated in the i-layer is more likely to have sufficient energy toovercome the now smaller energy barrier. Forward voltage biasing similarly reduces the energybarrier in the valence band, (which of course faces the opposite direction), thus encouraging

hole diffusion. The end result is the increase in Jdiff with forward voltage biasing. Note thatthe opposite happens with reverse voltage biasing, the energy barrier becomes larger, hence Jdiffdecreases.

2.1.2 Carrier recombination

Hydrogenated amorphous silicon contains an amphoteric dangling bond defect that can be neut-ral, positively charged, or negatively charged. The lattice structure of a-Si:H is sketched andcompared to that of c-Si in Figure 7. Dangling bonds are empty covalent bonds, depicted asdashed lines in Figure 7b.

Dangling bonds act as the main recombination centres for carriers. Recombination refers to

electrons in the conduction band losing energy and dropping down to the valence band, wherethey are again bound in a covalent bond. High recombination rates result in small output

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currents and low cell efficiencies. It is obvious from Figure 7a that c-Si based solar cells areabsent of dangling bonds due to the organized, crystalline atomic arrangement of the semicon-ductor material. This explains why c-Si cells offer small recombination current losses and higherefficiencies.

(a) c-Si structure (b) a-Si:H structure

Figure 7: Lattice structure of c-Si and a-Si:H. Dangling bonds are represented by dashed lines.

In a-Si:H cells, most of the bulk recombination in the i-layer occurs due to neutral danglingbonds. However, near the p-i and n-i interfaces the dangling bonds may be charged. As theillumination is reduced to low levels, the neutral zone in the i-layer shrinks. In the regionsnear the p and n doped zones, dangling bonds are hence charged, locally weakening the electricfield and increasing recombination [2]. This is known as interface recombination. In the dopedlayers, the dopant atoms introduce many dangling bonds and do not contribute free electrons.

The recombination rate in these layers is so high that photons absorbed in doped layers do notcontribute to the power generated by solar cells [8]. The mere function of the doped layers isto induce the built-in electric field. a-Si:H cells suffer from light induced degradation, known asthe Staebler-Wronski effect. This causes the cell efficiency to significantly decrease (by 15-30%)during the first few hundred hours of operation. It will be seen that the state of degradation(hence the magnitude of recombination current losses) may be quantified in terms of the effectivemobility-lifetime () product of the photocarriers.

2.2 Single-cell electrical model

The electrical behaviour of an a-Si:H cell may be accurately described by the equivalent electrical

circuit model proposed in [1]. In this section, the equations of the model are first presented andthe range of validity and underlying assumptions of the model are then discussed.

2.2.1 General equations

The equivalent circuit for a non-ideal a-Si:H cell is shown in Figure 8. Here, JL represents theloss-free illumination current (or photocurrent) density generated by the cell. JR is the currentdensity lost due to carrier recombination in the i-layer. JD represents the ideal diode currentdensity and JP is the shunt leakage current density lost across the parallel or shunt resistanceRp. Rs represents the series resistance of the non-ideal cell.

J is the net output current density of the cell, while V corresponds to the voltage across the cell

terminals. It is important to note that V is taken as positive (corresponding to forward voltagebiasing) when the n-terminal is at a higher external voltage potential than the p-terminal.

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Likewise, J is defined to be positive when the current leaves the p-terminal, i.e. when driftdominates over diffussion. This means that this model takes the short-cicuit current (JSC) ofa cell under illumination to be positive. This is consistent with the voltage and current signcriteria established back in Section 2.1. Note that the useful power P generated by the cell is

simply given by P = JV [W/m

2

].

Figure 8: Equivalent electrical circuit for a single-junction a-Si:H cell.

This circuit may be described mathematically by applying Kirchhoffs first law, which statesthat the sum of currents around any node in an electrical circuit must be zero:

J = JL JR JD JP (4)

The current term JD may be replaced by a more detailed expression describing the diode beha-viour. Similarly, JR may be described through a recombination model proposed in [1], and JPmay be expressed in terms of Rs and Rp. Equation (4) may therefore be fully written as:

J = JL JLd2

()eff[Vb (V + JRs)] J0

eV+JRsnVTe 1

V + JRsRp

(5)

The symbols J0, n, ()eff correspond to the diode saturation current density, diode idealityfactor and effective carrier mobility and lifetime product respectively. It is also recalled thatd and Vb are the i-layer thickness and built-in voltage. VTe is the thermal voltage, defined asfollows:

VTe =kBT

q(6)

where kB is Boltzmanns constant, T the absolute temperature and q the elementary charge.

Equation (5) will be referred to as the characteristic equation of the cell, which essentiallydescribes the relationship between cell current density J and cell voltage V. In the cell char-acterization process, it is often more convenient to work in terms of the non-dimensional ideal

voltage of the cell, V

, and non-dimensional current density J

, rather than in terms of V andJ. V and J are defined as:

V =V + JRs

VTe(7)

J =JRsVTe

(8)

Note that V represents the voltage drop across Rp or the diode element, normalised by VTe.J represents the voltage drop across Rs, normalised by VTe. J0 and JL shall be likewise non-dimensionalised as follows:

J0 =J0RsVTe

(9)

JL =JLRsVTe

(10)

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In terms of V and J, the characteristic equation of the cell (5) may be rewritten as follows:

J = JL

1

d2

()effVb [1 V(VTe/Vb)]

J0

eV

/n 1

RsRp

V (11)

This will be referred to as the non-dimensionalised characteristic equation of the cell.At this point, it is worth stating the well used term R denoting the cells characteristic resistancein the V J plane:

R = V

J(12)

R is usually evaluated at short-circuit and open-circuit conditions, giving the short-circuit res-istance RSC and open-circuit resistance ROC of the cell respectively. Useful information can beextracted from these quantitities. It is therefore pertinent to introduce the term R, defined asthe cells dimensionless resistance in the V J plane:

R = V

J

(13)

R is obtained by differentiating (11) with respect to J. This gives:

R =

JL

d2

()effVb [1 V(VTe/Vb)]2

VTeVb

+ J01

neV

/n +RsRp

1(14)

Unfortunately, R still depends on JL, a quantity which is a-priori unknown and difficult todetermine experimentally. It is therefore logical to eliminate JL from (14) through direct sub-stitution. This is done first by solving for JL in (11):

JL =J + J0

eV

/n 1

+ RsRpV

1

d2

()effVb[1V

(VTe/Vb)]

(15)

Substituting (15) into (14), an useful expression for R is obtained:

R =

J + J0

eV

/n 1

+ RsRp V

1 d2

()effVb[1V(VTe/Vb)]

d2

()effVb [1 V(VTe/Vb)]2

VTeVb

+J0n

eV/n +

RsRp

1

(16)

Note that in the absence of illumination, (JL = 0) the non-dimensional characteristic equationand R reduce to:

J = J0 eV

/n 1

+

RsRp

V (17)

R =

J0n

eV/n +

RsRp

1

(18)

2.2.2 Limitations of the model and underlying assumptions

This model differs from the typical equivalent circuit of P-N junction solar cells in the inclusionof a current loss term JR, that strongly increases with forward voltage. This term takes intoaccount the recombination losses in the i-layer of the cell previously discussed in Section 2.1. Itis a function of the effective product (combines the lifetime and mobility of free electronsand holes), that determines the state of degradation. This recombination function is taken fromthe neutral dangling-bond recombination model described in [4].

The electrical model, or rather, the recombination current term has been developed under thefollowing assumptions:

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constant, strong electric field |E| within the i-layer, diffusion effects, which are supposed much weaker than drift effects, are neglected, bulk recombination in the i-layer is determined by the neutral dangling bonds.

The first two assumptions are applicable to illuminated p-i-n cells under small positive or neg-

ative voltage biasing, with thin i-layers and relatively low defect densities therein. For largeforward voltages the model loses its validity since diffusion can no longer be neglected, as ithas already been seen. Looking back at Figure 5, the recombination model is most valid forthe Jdrift Jdiff regime. Its accuracy near point (B), i.e. near OC conditions, will still beacceptable even though the theoretical assumptions are no longer fulfilled.

The third assumption is only valid for sufficiently strong illuminations such that JSC 1 A/m2

[2]. It is recalled that at lower illumination levels than this, most of the recombination occursin the p-i and n-i interfaces. This is because at the end regions in the i-layer near the p andn doped zones, dangling bonds are charged, locally weakening the electric field (which mayno longer be approximated as uniform), thus increasing recombination. As the illuminationintensity is increased, charge defects are neutralized, the field becomes more uniform and inter-

face recombination decreases. It has been found that the effective -product may vary undersignificant changes in irradiance, usually increasing as the illumination level increases [2].

Finally, a very important assumption is that the model takes the shunt-leakage current of thecell to be Ohmic, i.e. always linearly dependent on the external voltage. This is often untruewhen the cell is operating under large reverse voltage biasing, where the model is thus no longervalid. This shall be explained and discussed in the next section below.

2.3 Extended model for reverse voltage biasing

A cell will normally operate near its maximum power point (MPP) under significant forwardvoltage biasing. However, in a photovoltaic module, heavy shading of a particular cell mayforce that cell to operate at large negative voltages. At large reverse voltage biasing, the modeldescribed by (5) will show significant deviations from the experimental JV curves. The reasonis that the model does not take into account the full physics of the reverse bias characteristicof the cell. Two important effects are the diode avalanche breakdown at large negative voltagesand the non-linearity of the shunt leakage current.

The excess variable dark leakage current observed at low voltage biasing is commonly referredto in the literature as shunt leakage current (ISH). In the equivalent circuit model given byEquations (4) and (5), the shunt leakage current is assumed to be Ohmic (linearly dependent onvoltage). The term JP has been used to denote the strictly Ohmic shunt leakage current density.Consequently, it has been represented as the current across a parallel or shunt resistance ( Rp),

as pictured in Figure 8. This model provides a satisfactory fit for forward voltage biasing.However, at high enough reverse voltage biases, the leakage current shows a non-linear voltagedependence, where ISH |V|

, with 2 3 [7]. The non-linearity of ISH at large negativevoltages is clearly observed in Figure 18, where the measured current almost purely consists ofthe shunt leakage current.

An extension term has been developed [11] in the form of an additional current term JB whichdescribes the diode avalanche breakdown and the shape of the reverse bias characteristic of thecell. When the voltage reaches the breakdown voltage, Vbr, the cell will allow large reversecurrents to flow through it. The additional current term JB may be modelled as follows:

JB = V + JRs

Rp1 V + JRs

Vbr

(19)

Here and are positive correction coefficients, obtained experimentally. The full model hence

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becomes:

J = JL JR JD (JP + JB) (20)

The term JP + JB = JSH is the effective shunt leakage current density. This model is especially

suited for Vbr < V < 0, and its accuracy has been validated, for example, in [3]. For V 0 thenew current JB may be neglected without any loss of accuracy.

There are some other recent models that explain this non-linear behaviour, such as the space-charge-limited current model proposed in [7].

2.3.1 Shunt leakage current and shunt-busting

In a-Si:H p-i-n cells, the physical origin of the shunt conduction paths, along which the shuntleakage current flows, has been attributed to lateral drift currents arising from differently sizedelectric contacts [9] or to local non-uniformities.

The p-type and n-type layers are very thin ( 10 nm). Therefore, doping inhomogeneities,surface roughness or metal/contact material diffusion can create possible shunt paths. Accordingto [7], the most likely way is through an aluminium incursion from the AZO contact layer tothe n-type layer. During the deposition of the AZO layer, Al can diffuse into the a-Si:H cell toform an Al filament that destroys part of the n-i junction. This is sketched in Figure 9. Theresult is a localized p-i-metal structure along which the shunt leakeage current flows.

Figure 9: Schematic of a typical a-Si:H p-i-n cell layout with a shunt structure due to (alumin-

imum) contact diffusion into the a-Si:H layer. The TCO layer forms the front electric contact,while the AZO layer forms the back contact. The dashed red lines represent the paths of theshunt leakage current (JSH). It flows in parallel to the ideal exponential diode current (JD),whose paths are represented by the solid blue lines. At high forward voltage biases, JD JSH,while at small forward voltages or negative voltages JSH JD.

The metal diffusion hypothesis is reinforced by the shunt-busting phenomenon observed in a-Si:H cells. Shunt-busting involves applying a reverse voltage bias to the cell for a certain periodof time (without exceeding its breakdown voltage), which causes the shunt leakage current todecrease to a lower value. In the equivalent circuit model, this corresponds to a drastic increase

in the value of Rp. It is likely that reverse voltage biasing forces the Al filaments out of thea-Si:H layer through oxidation or evaporation, eliminating thus the parasitic shunt path andimproving the cells electric yield.

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3 Test cell and experimental technique

3.1 Test cell description

The test cell on which the electrical characterization was performed consists of a thin-film,double-junction a:Si:H cell. It has a total area of approximately 40 mm 40 mm. It is shownin Figure 10.

Figure 10: Photograph of the test a-Si:H tandem cell (front view).

A diagram detailing its physical, layered structure is represented in Figure 11. It contains twoa-Si:H subcells of p-i-n structure for improved efficiency. The top subcell is designed to absorbhigh energy photons, while the bottom subcell mainly absorbs lower energy photons. This meansthat the i-layer of the bottom subcell must have a lower band gap. Note that the i-layer of thetop subcell is somewhat thinner than that of the bottom subcell.

Figure 11: Diagram of the test cell portraying its layer arrangement. The approximate layerthicknesses are indicated. Adapted from [10].

The tandem cell only has two electrical contacts. The TCO layer is a transparent (anti-reflective)coating that acts as the front contact, while the AZO back reflector (chemically composed

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of ZnO:Al) acts as the back contact. The front (high-emmisivity) glass and back glass aresupporting layers that essentially add structural rigidness and protect the cell.

The tandem cell has an an active cell area A (where photo-generation takes place) of dimensions15 mm 30 mm. Therefore A = 4.5 104 m2. To avoid confusion, it should be clarified that

although the measuring apparatus obviously reads the total current (I), the current density (J)is used instead in the characterization process. It is simply given by:

J =I

A[A/m2] (21)

Consequently, this means that the cells series and parallel resistances will be automaticallynormalised by the cell area, with units [m2].

3.2 Apparatus description

Measuring Station

All measurements were taken at the Instituto de Energa Solar (IES). The I V curves wereobtained through a 4-point measuring station, equipped with a variable resistor and linked toa computer via the Agilent VEE Pro software for data acquisition and processing. It enabledto perform automatic I or V sweeps between specified minimum and maximum values, for aspecified number of measuring points.

The test cell was always placed on top of a thermoelectric support (Peltier cooler), designed tokeep the test cell at the desired temperature.

Solar Simulator

The solar simulator essentially consists of a 1000 W Xenon lamp of adjustable height. It providedthe illumination used in the characterization of the single-junction cell described in Section 4.

LED board

For the characterization of the tandem cell described in Section 5, the illumination was providedby a board containing 36 LEDs (in a 6 6 arrangement). Half of these were focused infrared(IR) LEDs ( = 830 nm, = 5 mm, 70 mW/sr) while the other half were focused ultraviolet(UV) LEDs ( = 405 nm, = 3 mm, 10 mW/sr). A photograph displaying the IR and UVLED arrangement is shown in Figure 12.

Figure 12: Photograph of the LED board used as the IR and UV light source.

The LED board was connected to a circuit board. The LED intensities could be varied bymanually tuning trimming-type potentiometers. The UV LED branches were independendent

of the IR LED branches. This allowed to turn on just the IR LEDs or UV LEDs on their own(partial operation), or all LEDs simultaneously. Note that the intensities of both UV and IRLEDs were controlled individually via two separate potentiomenters.

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4 Electrical characterization of a single-junction cell

The aim of this section is to provide an experimental method that can be used to systematicallydetermine the unknown elements of the equivalent circuit for a single-junction cell previously

shown in Figure 8, described mathematically by (11). The methods were applied to the testtandem cell, but in this case it was treated as a single cell in all regards.

The circuit parameters Rp, Rs, J0, and n may be determined from a J V curve in the dark.The remaining parameters, ()eff and Vb, may be readily evaluated from a set of J V curvesunder sufficiently strong illumination. A variable illumination method (VIM) is employed, whichessentially consists in obtaining a set of J V curves under different illumination intensities.

It is important to note that all these parameters are temperature dependent. In this work, thethermal variation of these parameters is not dealt with. All measurements were taken at 298 K,which results in a constant thermal voltage, VTe = 25.7 mV.

4.1 Measurements in dark conditions

The non-dimensional characteristic equation for the cell operating in the dark, previouosly givenin Equation (17), is recalled:

J = J0

eV

/n 1

+RsRp

V (22)

When V 1, given that RpJ

0/Rs 1, the J

0-dependent term in Equation (22) is negligiblewith respect to the last term. Physically, this means that the leakage current is much largerthan the diode current. The characteristic equation simplifies to:

J

=

Rs

Rp V

(23)

Therefore, an experimental (J) V plot in the V 1 region will be a straight line of slopem = Rs/Rp. This slope can be directly obtained through a linear regression fit in the V

1regime.

When V 1, the exponential term in (22) is dominant. Physically, this means that theexponential diode current overwhelms the shunt leakage current. It is found that:

J = J0eV/n ln (J) =

1

nV + ln

J0

(24)

Therefore, a plot of ln (J) V can be approximated as a straight line of slope m = 1/n and

y-intercept c = ln (J0).

To represent the experimental curves in terms of V and J, the series resistance Rs must beknown. Rs can be found from the exponential region in the J V curve. This region isdescribed by (24) and the condition V 1 applies. Note that V can be approximated asV /VTe for voltages up to the start of the exponential region (since JRs V). Therefore at theexponential region the condition V /VTe 1 must also apply.

An educated guess for Rs must first be taken. It is known that for the correct value of Rs, theln(J) V curve (at the V 1 region) will be a straight line. If, on the other hand, theguessed value is far from the correct one, the ln(J) V curve will show a strong curvature.A linear regression fit is then performed for V /VTe 1 and the correlation coefficient between

the linear fit and the experimental curve is recorded. Then, Rs is systematically varied and itscorresponding correlation coefficient recorded. For the correct Rs, the correlation coefficient willbe a maximum.

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It is important to note that the results depend on the minimum value of V /VTe that is consideredto be much greater than 1, which marks the lower bound of the linear fit. The V /VTe lower limitmust be chosen carefully. At its optimal value of Rs the greatest possible correlation coefficientmust be attained and at the same time the largest possible portion of the exponential region of

the dark J V curve should be captured, all while ensuring that V /VTe 1 is satisfied.

Experimental results

Figure 13 shows the experimentally obtained J V dark characteristic curve of the cell. It isclearly linear up to V /VTe 20. It is then followed by an exponential region that blows upafter V /VTe 50. It can be seen that after V /VTe 50 the exponential term in (22) must bedominant, meaning that this portion of the curve lies in the V 1 region indeed.

Figure 13: J V experimental curve in dark conditions. Left: J V curve in the dark for smallforward voltages. A linear region up to V /VTe 20 is observed. Right: J V curve in the darkfor large forward voltages. This portion of curve belongs to the exponential-dominated regime.

The correct value for Rs was found to be 0.0032 m2. This result shall be verified shortly.

Using this value, experimental plots of J V were constructed at the V 1 region (Figure 14)and at the V 1 region (Figure 15). Rp, J0 and n could then be easily obtained. It may beobserved in Figure 15 that for the chosen value of Rs, the ln(J

) V curve indeed becomesa straight line. The red solid line marks the linear regression fit performed in the top graph in

order to model Equation (24). The fit is again displayed in the bottom graph of Figure 15 toverify that it lies and matches well the exponential region of the dark J V curve.

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Figure 14: J

V

plot with a linear fit in the V

1 region in order to get Rp from its slope.

Figure 15: J V plots and linear fit for the V 1 region at the chosen value of Rs. Lowerlimit of fit: V /VTe 52. Top: Semilog plot on which linear fit is performed, enabling J0 andn to be found. Bottom: Plot in a log-free scale to verify that the exponential part of curve isexclusively captured by the fit.

The V /VTe lower limit of the fit was set to 52. Figure 16 shows the correlation coefficientbetween the experimental points and the fit for a range of values of Rs. Clearly, the correlationis maximised at Rs = 0.0032 m

2.

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Figure 16: Correlation coefficient of the linear fit performed in the V 1 region, as a functionof Rs. Lower limit of fit: V /VTe 52.

Lastly, the choice for the V /VTe lower limit being set to 52 is justified in Figure 17. It showsthat the maximum correlation is greatest when the linear fit is set to start at V /VTe 50 60.For V /VTe > 65, the correlation coefficient falls since the number of useful experimental pointsstart to run out. One can also observe the strong dependence of Rs with the V /VTe lower limit.In this case, V /VTe = 52 was seen as a proper lower limit according to the criteria previouslystated.

Figure 17: Dependance of the maximum correlation coefficient and corresponding Rs on theV /VTe lower limit. Top: Maximum correlation coefficient as a function of the chosen V /VTe lowerlimit. Bottom: Correct Rs corresponding to the maximum correlation coefficient according tothe chosen V /VTe lower limit.

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4.1.1 Reverse voltage biasing model

Should the extended model with the breakdown current term be required, a JV experimentalplot in the dark for the V < 0 region may be used to estimate the coefficients and presentedin (19). In the absence of illumination, at the reverse voltage bias region (Vbr < V < 0), theJ0-dependent terms in (22) are negligible, which suggests that the output current is purely madeup of the effective leakage shunt current: J = JP + JB. The extended characteristic equationin terms of V becomes:

J =RsRp

1 +

1 V

VTeVbr

V (25)

The graph in Figure 18 shows that the bare model described by (22) will always underestimateJ at sufficiently large negative voltages since it does not take into account the non-linearity ofthe shunt leakage current nor the diode avalanche breakdown at high negative voltages. Typicalbreakdown voltages for a-Si:H cells lie between -6 V and -8 V [6]. Hence, for this double-junctioncell in question a rough estimate for its Vbr is 16 < Vbr < 12 V, which is quite large.

The parameter may be determined considering the curve in the 100 < V < 0 region, whereV(VTe/Vbr) 1 and (25) may be approximated as:

J =RsRp

[1 + ]V (26)

Therefore, a (J) V plot in such region may be approximated as a straight line of slopem = (Rs/Rp)[1 + ]. The parameter was then determined by systematically varying it untila good model-experiment matching was obtained.

Figure 18: Comparison between the experimental and modelled JV curves for reverse voltagebiasing. The extended model is given by (25) with Vbr = 16 V, = 4.2 and = 0.5.

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4.2 Measurements under illumination

As it will be shown, the remaining unknown parameters, ()eff and Vb, are linked to the short-circuit dimensionless resistance RSC of illuminated J V curves, defined as:

RSC = V

J

SC

(27)

RSC may be obtained from the gradient evaluated at the short-circuit operating point (SC) inan experimental J V plot. It is recalled that when the cell is operating at SC, J = JSC andV = 0. This results in VSC = J

SC = JSCRs/VTe. In order to determine ()eff and Vb, the cellmust be exposed to intensities of illumination (irradiances) that induce such current densitiesJSC that the next two conditions are satisfied:

J0eVSC/n

RsRp

(28a)

J

0eV

SC/n J

SC= V

SC(28b)

An expression for RSC may be obtained through direct evaluation of Equation (16) at SC, notingthat the J0-dependent terms in the denominator are negligible in magnitude in comparison tothe rest of the terms if (28a) and (28b) are satisfied. The result is:

RSC = V

J

SC

=

JSC + RsRp VSC

1 d2

()effVb[1VSC(VTe/Vb)]

d2

()effVb

1 VSC(VTe/Vb)2 VTeVb +

RsRp

1

(29)

It is algebraically convenient to introduce the following dimensionless parameter and variable

:

=d2

()effVb(30)

= (VSC) = 1 V

SC(VTe/Vb) (31)

Here , where 0 < < 1, is a measure of the subcells recombination current magnitude relativeto the illumination current JL. Typically, decreases with illumination strength. incorporatesthe effect of irradiance (through VSC) on the recombination current. It should be noted that is always constant (provided ()eff remains invariant). In these new terms, the expression forRSC reads:

RSC =

1 + RsRp

(1 )( )

+ RsRp

1(32)

and , which contain the remaining unknown circuit parameters, may be grouped togetherand solved for in the equation above:

(1 )

( )= Fexp =

[RSC]1 RsRp

1 + RsRp

1

RSC(33)

Fexp implies that the grouped parameters form a function F (dependent on R

SC) which may beevaluated directly from experiment. Note that the approximation in (33) is only valid if:

RsRp

[RSC]1 1 (34)

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which may not always be the case. If condition (34) is not satisfied, the full expression for Fexpshould be used, rather than just [RSC]

1.

A method to determine ()eff and Vb is proposed. More precisely, this method may be employedto obtain a representative value of ()eff, mainly valid for irradiances of the same order of

magnitude as the experimental curves used in its finding. This is because ()eff depends onthe intensity of the illumination and it cannot be assumed to be invariant across illuminationsof different orders of magnitude. Vb must not necessarily be known beforehand.

The procedure involves systematically varying Vb over a sensible range of values. For eachvalue of Vb, the corresponding mean ()eff is then computed from the experimental data, thusobtaining a set of ()effVb pairs. The optimal ()effVb pair must be the one that gives thebest matching of the model against the experimental J V curves. If Vb is too high or too low,the modelled curves will not adjust themselves well to the experimental J V curves especiallynear the open-circuit region. In particular VOC will be overestimated (or underestimated) if Vbis too high (or too low).

Assuming a value for Vb, ()eff can be computed at a particular intensity of illumination by

substituting (30) into (33):

()eff =d2

Vb

2Fexp

1+Fexp

(35)Note that and of course Fexp are determined experimentally.

A simplified expression from which ()eff can be quickly estimated can be attained. Recom-bination losses must be assumed to be small ( 1) and the irradiance is taken to be weakenough so that is close to 1 (i.e. VSC(Vb/VTe) 1). Furthermore, the condition in (34) isassumed to hold. In this case Equation (33) simplifies to:

(1 ) =1

RSC(36)

After substituting (30) and (31) into (36) and rearranging, the simplified, approximate expres-sion for ()eff introduced in [2] is obtained:

()eff, approx =d2

Vb(1 )RSC =

d

Vb

2JSCRsR

SC=

d

Vb

2JSCRSC (37)


The parameter ()eff was estimated by employing both the full approach given by (35) andthe simplified approach found and used in the literature, given by (37). This was done in orderto assess the accuracy of the simplified approach with respect to the full approach. The fullapproach later proved to yield better and consistent estimates for the ()eff Vb pairs.

The cell was exposed to six different illuminations of white light produced by the solar simulator(briefly described in Section 3.2) ranging approximately from 0.3 to 1.8 suns. The experimentalJ V curves are plotted in Figure 19, from which VSC and R

SC were obtained. The inducedJSC lies between 20 and 120 A/m

2 in magnitude. After non-dimensionalisation, JSC was seento comply with conditions (28a) and (28b). RSC also complied with condition (34).

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Figure 19: Experimental J V curves for different illuminations (approx. 0.3 1.8 suns). Foreach curve, the short-circuit operating point (SC) is shown, along with the fitted tangent withgradient m (where m = 1/RSC).

Typically, the built-in voltage of a simple a-Si:H cell lies between 0.8 and 1 V. For a tandem cell,the equivalent Vb will approximately be the sum of each subcells Vb. The minimum possible

value of Vb must always exceed VOC attained at strong irradiances, in this case Vb > 1.75 V.Hence, for this particular tandem cell the range of Vb considered was from 1.75 V to 2.1 V.

Figure 20 shows the variation of RSC and ()eff with J

SC (which is fairly proportional tothe irradiance) for Vb = 1.85 V. The decaying trend in R

SC with J

SC is expected, and itmay be inferred from (29). On the other hand, ()eff can be seen to remain constant atthis narrow range of irradiances considered. This is because the full approach is more preciseas it will be next seen. The difference in ()eff calculated by both approaches widens withirradiance since (0 < < 1) moves away from 1 with increasing VSC (or irradiance). For bothapproaches, an average value of ()eff was computed by taking the mean ()eff over the 6different illuminations at a given Vb. The average ()eff vs Vb is plotted in Figure 21. Notethat ()eff 1/Vb

2 according to the simplified expression given in (37).

Next, the validity of using the average ()eff Vb pairs to characterize the cell in this particularexperimental illumination range was evaluated. The validity of the pairs obtained by bothapproaches were examined, i.e. by the full approach given in Equation (33) and the simplifiedapproach in (36).

In order to do this, the following functions were defined:

Fth =(1 )

( )(38a)

Fth, approx = (1 ) (38b)

Fexp =1

R

SC

(38c)

Fth represents the grouped expression of and used in the full approach while Fth, approxrepresents that of the simplified approach.

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Figure 20: Variation of RSC and ()eff with J

SC (irradiance) for Vb = 1.85 V, computed fromthe J V curves in Figure 19. Top: Experimentally determined variation of RSC with J

SC.Bottom: Computed ()eff vs J

SC using both approaches.

Figure 21: Variation of the average ()eff with Vb obtained by the full and the simplifiedapproaches.

Using any pair of values for the average ()eff and Vb from Figure 21, the parameter andvariable (VSC) were computed, the latter spanning a sensible V

SC range. The theoreticalFth J

SC and Fth,approx V

SC curves were constructed and compared against the experimentalFexp V

SC points. They are shown in Figure 22.

It is thus corroborated that both theoretical functions F are essentially invariant regardless ofthe ()eff Vb pair used. It is clearly observed that Fth correlates well with the experimentalfunction Fexp for the whole range of illuminations considered. As a result, all illuminationswithin this range will give a similar ()eff value by this approach. This was seen on Figure20. It may be concluded that the average ()eff value is indeed representative over the entireexperimental illumination range.

However, there is a noticeable worse correlation between Fth, approx and Fexp. The average()eff value for the simplified approach is observed to be only accurate for the lower half of the

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illumination range. Note that as the illumination strength is lowered, the difference bewteensimplified and full approaches decreases, and F VSC becomes a straight line. It is expectedthat the validity of using the simplifed approach increases.

Figure 22: Fth, Fth,approx as functions of V

SC, compared against experimental Fth,exp V

SC

points. The variable thickness in the full model curve accounts for the fact the curve is composedof several overlapping Fth V

SC subcurves, by taking several ()eff Vb pairs from Figure 21over the whole Vb range.

The final step involved picking the optimal ()eff Vb pair from Figure 21 through direct

modelling of the experimental J V curves. For this particular illumination range, it was foundthat the pair comprised by Vb = 1.85 V and ()eff = 4.6 10

12 m2/V gave the best matchingbetween the modelled and experimental J V curves.

There is a good chance that even when modelling single-junction cells, the value of Vb that givesthe best fit is not the same as the actual true value of the aSi:H cell. As previously mentionedin Section 2.2.2, VOC will very likely be underestimed by the model when the true value of Vbis used. In this case, it must be kept in mind that the cell is actually a double-junction tandemcell. Here, Vb loses part of its physical meaning since it is in fact the equivalent built-in voltageof the combination of both subcells, and therefore must be treated as a tuning parameter whosevalue should be adjusted to minimise the mismatch of the modelled and experimental J Vcurves.

4.3 Model performance

The cell parameter values are summarised in Table 1.

With all elements the equivalent circuit now known, the J V curves modelled by (5) wereconstructed. Since a closed-form exact solution of equation (5) or (11) is not available, a Newton-Raphson iterative method was employed. Note that the photocurrent JL is required. It wascomputed by directly specifying the desired JSC. Rearranging Equation (5) evaluated at SCgives:

JL =

JSC

1 + RsRp + J0eJSCRsnVTe 1

1 d2

()eff(VbJSCRs)

(39)

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Parameter Value Units

Starter values

VTe 25.7 103 V

d 600 nm

Dark measurementsRs 3.20 103 m2

Rp 47.6 m2

J0 8.6 107 A/m2

n 3.78

Illumination measurements

()eff 4.6 1012 m2/V

Vb 1.85 V

Table 1: Parameter values for the single-cell equivalent circuit model.

The performance of the model with these parameters was assessed for forward and reverse voltagebiasing.

Forward voltage biasing

Figure 23 verifies the good agreement between experimental and modelled J V curves with thechosen parameters. However, VOC is slightly overestimated. The VOC predicted by the modelwas found to be largely dependent on the ()effVb pair chosen. This particular pair gave goodagreement for a wide range of positive V at the expense of overestimating VOC. The fact thatthe tandem cell is approximated as a single-junction cell is expected to undermine the modelaccuracy. Furthermore, the underlying assumptions of the recombination model are no longervalid near the OC region as previously stated in Section 2.2.2. Therefore the model is expected

to perform worse in this region.

Figure 23: Experimental and modelled illuminated J V curves for forward voltage biasing.

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Ideally, the cell will mostly operate at voltages near the maximum power point ( MPP). Again,the model accurately describes the P V behaviour of the cell in this region. This is plotted inFigure 24.

Figure 24: Experimental and modelled illuminated P V curves for forward voltage biasing.

Reverse voltage biasing

In the V < 0 regime, Figure 25 shows the bare model fits well the experimental curves up toV = 3 V, quite a large value. For larger negative voltages, the model described by (5) willalways underestimate J since it does not take into account the non-linearity of the shunt leakagecurrent nor the avalanche diode breakdown at high negative voltages. The same model, but nowextended to include the JB term described in Section 2.3, faithfully solves this problem.

Extreme off-design operation is not an issue when dealing with a single cell. However, forphotovoltaic modules composed by strings of many cells in series, it is not unlikely that acertain cell will be subject to some degree of shading, resulting in current mismatch. For heavy

shading, if each cell was tested individually, the output current of the shaded cell would besignificantly lower than that of the rest of the unshaded cells. Note that the output current Jof a set of cells connected in series is dictated by the smallest J that corresponds to that of theshaded cell. The operating point of the rest of the unshaded cells will forcefully move so thatevery cell now produces the same J as that of the shaded cell.

As an example, suppose all unshaded cells are initially operating at the MPP. The reductionin J imposed by the shaded cell will force the unshaded cells to operate at a higher V (henceat a lower J) for a particular J V curve. As it can be seen from Figures 23 and 24 eachunshaded cell now produces less useful power, less J and generates extra (positive) voltage.This extra voltage difference must be cancelled by the shaded cell, assuming the whole moduleis operating under a fixed external voltage. In this case, the shaded cell operating point will

be at a large negative voltage. There is thus a danger of cell damage due to a hot spot, if thepower dissipation through the cell is high enough, and of avalanche diode breakdown of the cell

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if the voltage across it is negative enough. The more cells there are in series, the worse it getsfor the shaded cell. This is corroborated in [6], where it is concluded that the power loss due toshading decreases with increasing number of cells connected in series, but the risk of cell damageincreases.

Figure 25: Experimental and modelled J V curves for reverse voltage biasing at three distinctillumination levels. The coefficients for the extended model are = 4.2 and = 0.5.

4.3.1 Cell response to changes in illumination intensity and spectrum

The validity of the procedure for obtaining ()eff and Vb was confirmed by calculating theseparameters from a new set of weaker illuminations of different spectrum. The induced JSCwas approximately one order of magnitude less than before (2 < JSC < 10 A/m

2). This wasachieved by placing filters on top of the cell. The precise spectrum of the light that reached thecell was not known. In the electrical model used, all parameters are assumed to be independentof the illumination intensity or spectrum except ()eff. Even on tandem cells, changing the

illumination spectrum has little effect on the shape of the J V curves provided all subcellsremain excited at a similar level. This means that all photocurrents (JL) generated by thesubcells are of similar orders of magnitude. A change in the illumination spectrum may causethe photocurrent magnitudes of both subcells to shift by different amounts, and so will therecombination currents. The effect on the J V curve will be efficienctly captured by re-evaluating the value of ()eff under this particular illumination.

The illumination spectrum under which both subcells must be excited may be known from thespectral response of the tandem cell. The spectral response (SR) has units [A/W] and it isdefined by:

SR() =JSC()

Pirr()=

JSC()

E()()(40)

JSC() is the short-circuit current density generated by the cell under monochromatic light of

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wavelength , Pirr() is the illumination intensity, E() is the energy of a photon and () isthe photon flux per unit area and time.

The spectral response for the whole cell is plotted in Figure 26. This SR() portrays thecombined response of both subcells. When SR() > 0 , both subcells are being simultaneously

excited by the incident illumination, hence an output current JSC is generated. Note that thespectral response of a tandem cell is always limited by one of its subcells.

Figure 26: Spectral Response of the tandem cell.

For the test tandem cell, the spectral response of the tandem cell hints that the incident illumin-

ation spectrum determined by the filters must have displayed at least one wavelength between300 and 700 nm since both subcells were excited. The latter was verified experimentally sinceJSC was observed to vary proportionally to the irradiation.

For Vb = 1.85 V, the average ()eff (obtained by the full approach) was 3.46 1012 m2/V. Note

that it is around 25% lower than for the stronger set of illuminations (previously presented inSection 4.2). This agrees with the physical explanation that at lower illuminations, the p-i andi-n interface recombination increases due to the increased number of charged dangling bondsand the fact that charge-assisted capture is much more likely than capture by neutral danglingbonds [4].

Using this value and the rest of values from Table 1, the J V curves at both forward and

reverse voltage biasing in this illumination regime were modelled. They are plotted in Figures27 and 28 respectively. The deviation of the experimental curves from the bare model at largenegative voltages is perhaps even more noticeable as the illumination is weakened. Once again,VOC is overestimated by the model for the same reasons.

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Figure 27: Experimental and modelled illuminated J V curves for the new set of weakerirradiances with an altered spectral distribution.

Figure 28: Experimental and modelled illuminated J V curves for reverse voltage biasing forthe new set of weaker irradiances with an altered spectral distribution. The extended modelincludes the JB term defined in Section 2.3.

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4.4 Sources of error and uncertainties

All the elements in the electrical model are temperature-dependent. For consistent results, it istherefore very important that all measurements are taken with a strict eye on the temperature,with a recommended maximum allowance of 1 C offset. As an example, Figure 29 portrays thestrong effect of temperature on the dark J V curve of the cell. The computed parameters fromthe dark J V curves at 25 C and 28 C are compared in Table 2:

Parameter T = 25 C T = 28 C Units

Rs 0.0032 0.0024 m2

Rp 47.6 55.0 m2

J0 8.6 107 2.3 107 A/m2

n 3.80 3.88

Table 2: Parameters computed from a dark J V curve measured at different temperatures.

Figure 29: Measured dark J V curves at different temperatures.

It is often difficult to maintain the cell at the desired temperature due to its tendency to heatup when it is subject to sufficiently strong illuminations. To avoid this, the cell was placed atopa Peltier cooler designed to maintain the cells temperature at a specified value, thus minimisingany sources of error coming from temperature-related effects.

Another main issue in calculating the parameters from the dark J V curve is that it relieson a single curve. Discrepancies between different J V curves at the same temperature were

seen to occur. Errors in the measuring apparatus, which was quite temperamental, were notuncommon, whether they were oscillations in the measurements or severe offsets in the readings.Therefore care must be taken in choosing a dark J V curve that is indeed representative atthe desired temperature by taking several curves. Alternatively, taking the mean of each of theparameter values computed over several J V dark curves is perhaps a better option.

In order to obtain the desired I V curves, the measuring apparatus performed an automaticvoltage sweep between specified minimum and maximum V values, taking an inputted numberof measuring points in between. In order to obtain the smoothest curves possible, the apparatustook an inputted number ofI-readings (usually set as 30 60) at each measuring point, and themean I was recorded as the final value. Even so, small-scale oscillations were often unavoidable.The reason for this is that the current measuring device is accurate to approximately 1 A. This

translates in J being accurate to 2 104. Most oscillations observed had an amplitude of thissame order of magnitude. Oscillations 50 times larger than this were nevertheless registered, as

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clearly seen in Figure 30.

Figure 30: Close-up of the SC region for J V curve (full curve is plotted in Figure 19). SCdenotes the (smoothed) short-circuit operating point. The tangent to gradient line is essentiallya linear regression fit performed between Vmin = 0.25 V and Vmax = 0.25 V.

The oscillations add to the uncertainty of accurately determining the gradients from a J V

plot, e.g. at the SC point of an illuminated curve in order to evaluate ()eff, or at the V 1

region of a dark curve in order to find Rp. The oscillations meant that the gradients wereforcefully determined through a linear regression fit limited by arbitrary minimum and maximumvoltage values in the region of interest. Figure 30 shows the linear fit constructed betweenVmin = 0.25 V and Vmax = 0.25 V used to determine R

SC. R

SC displays a strong dependenceon the minimum and maximum voltage values chosen as shown in Figure 31. The fit is initiallycentered around the SC point, until Vmin hits the first experimental point at V = 0.25 V, afterwhich only Vmax is increased. This plot shows that the computed value of R

SC hits a plateauat Vmax Vmin = 0.3 0.7 V, which can be assumed to be around the true value.

Therefore care must be taken to ensure that the fit indeed covers a sensible V range at theregion of interest.

Figure 31: Estimated R

SC for the J

V

curve in Figure 30 through a linear regression fitlimited by Vmax and Vmin, plotted as a function of the V length of the fit.

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4.5 Conclusions

An experimental method to evaluate the parameters of the electrical model has been presented.The performance of the electrical model for a single-junction cell has then been assessed. Themodel has been seen to accurately represent J V curves for different illumination regimes,provided ()eff (which has been shown to be spectrum and intensity dependent) is reevaluatedaccordingly. Changes in the spectrum and intensity of the illumination will be captured by()eff. This is also true for tandem cells, provided all subcells are excited in a similar way.

For the double-junction cell this is not the case when for example, the bottom subcell is beingexcited (by pure IR light) while the top subcell may be operating the dark. JSC will be verysmall, while VOC will be significantly greater than 0 given that the excited subcell is generatinga positive voltage difference in OC conditions. Modelling this particular J V curve is beyondthe scope of the single-junction cell model.

The next logical step is to introduce a tandem cell equivalent electrical circuit, where each sub-cell is now represented separately. A procedure to experimentally determine the unknown circuit

elements for each subcell is therefore presented in the next section. Note that the procedureassumes the subcells cannot be accessed by electrical contacts separately. The potential advant-ages of the tandem-cell model with respect to the single-cell model are that the former will beable to properly capture the response of a tandem cell to (extreme) changes in the spectrum,and be able to model the J V curves of each subcell individually.

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5 Electrical characterization of a double-junction cell

A multi-junction or tandem cell may be treated as a set of N subcells connected in series.Each subcell may be modelled independently by the single cell model presented in the previous

sections. In this case the characterization process shall be limited to the case of a double-junctioncell (N = 2).

5.1 Model and general equations

The electrical behaviour of an a-Si:H tandem cell may be modelled using the equivalent circuitdrawn in Figure 32. This equivalent circuit specifically describes a double-junction cell. Theequivalent circuit is composed of two subcircuits connected in series. Each subcircuit has adistinct set of elements and represents a single subcell. Note that each subcircuit is identical tothe single-junction cell equivalent circuit.

Figure 32: Equivalent circuit for a double-junction cell. It is treated as two subcells connectedin series. Each subcell is represented by a subcircuit comprised by a distinct set of elements.

Making use of the same notation for the various currents as for the single-junction cell model,this circuit may be described mathematically by again applying Kirchoffs first law. For thegeneral case of N subcells:

J = JL,i JR,i JD,i JP,i i = 1, 2 . . . N (41a)

V =Ni=1

Vi (41b)

where subscript i refers to the ith subcell.

Expanding (41a) as before gives the characteristic J Vi equation for each subcell:

J = JL,i JL,id2i

()eff,i[Vb,i (Vi + JRs,i)] J0,i

eVi+JRs,iniVTe 1

Vi + JRs,iRp,i

(42)

It is again more convenient to work in terms of the dimensionless quantities V

i and J

, defined

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as:

Vi =V + JRs,i

VTe(43)

J

=

JRs

VTe (44)

where Rs denotes the combined series resistance of the whole cell:

Rs =Ni=1

Rs,i (45)

Likewise, the subcell diode saturation current J0,i and subcell photocurrent JL,i may be non-dimensionalised as follows:

J0,i =J0,iRs

VTe(46)

JL,i =JL,iRs

VTe(47)

It is algebraically convenient to introduce the non-dimensional parameter i and variable i:

i =d2i

()eff,iVb,i(48)

i = 1 V

i (VTe/Vb,i) (49)

Here i, where 0 < i < 1, is a measure of the subcells recombination current magnituderelative to the illumination current JL,i. Typically, i decreases with illumination strength. i

models the effect of the external voltage and irradiance on the the recombination current. Itdecreases (increases) with forward (reverse) voltage biasing.

Using these terms, the non-dimensionalised characteristic equations of the tandem cell may bewritten as:

J = JL,i

1

ii

J0,i

eV

i /ni 1

Rs

Rp,iVi i = 1, 2 . . . N (50a)

V =Ni=1

Vi (50b)

V, not unlike J, depends on the combined resistance Rs. From the definitions of V

i in (43),

the combined resistance Rs in (45) and Equation (50b), it follows that:

V =V + JRs

VTe(51)

The dimensionless resistance R of the tandem cell is defined by:

R = V

J=

Ni=1

V iJ

=Ni=1

Ri (52)

where Ri is the non-dimensional resistance of the ith subcell, given by:

Ri = V iJ

=

J + J0,i

eV

i /ni 1

+Rs

Rp,iVi

i(VTe/Vb,i)

i(i i)+

J0,ini

eV

i /ni +Rs

Rp,i

1

(53)

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5.2 Experimental procedure

It has been shown in Section 3.1 that the tandem cell is composed to two subcells. The topsubcell (i = 1) is designed to absorb mostly high energy photons (short wavelength spectrum),while the bottom subcell (i = 2) is optimised for the absorption of lower energy photons (longwavelength spectrum).

As for the single-junction cell case, the aim of the following sections is to determine all unknowncircuit elements. In order to achieve this, the characterization process will once again rely on aVariable Illumination Method. The experimental data available is strictly limited to just a set ofexperimental J V curves obtained for different illumination conditions. From each curve, fourkey quantities may be graphically obtained. These four quantities, vital in the characterizationprocess, are: JSC, R

SC, V

OC and R

OC.

A J V curve under no illumination was initially recorded. It is essential to determine Rs andthus enabling the whole non-dimensionalisation process. The tandem cell was then exposed toIR light at 830 nm (in otherwise dark conditions) in order to induce a much greater photocurrent

in subcell 2 than in subcell 1. The intensity of the IR light was varied and several J V curvesat different irradiances were recorded. Likewise, it was next exposed to UV light at 405 nm ofvarying intensity in order to induce the opposite effect. Lastly, the cell was exposed to IR lightat a fixed intensity in conjunction with variable intensity UV light.

The light source in all cases was the LED board already described in Section 3.2.

5.3 Measurements in dark conditions

The combined series resistance Rs = Rs,1+ Rs,2 may be determined by following the exact sameprocedure previously presented for the single-junction cell analysis. It is impossible to determineRs,1 or Rs,2 independently. This is not a problem since both V

and J depend on Rs and not

just on Rs,i. Furthermore, it will be seen that the plots under dark conditions used for theidentification of parameters are identical to those used for the single-junction cell case.

The characteristic equations for a double-junction cell operating under no illumination are givenby:

+ J = J0,i

eV

i /ni 1

Rs

Rp,iVi i = 1, 2 (54a)

V = V1 + V

2 (54b)

When Vi 1, given that Rp,iJ

0,i/Rs 1, the J

0,i-dependent term in (54a) is negligible withrespect to the last term, leaving:

Rp,iJ

Rs= Vi i = 1, 2 (55)

Substituting (55) into (54b) leads to:

J =Rs

Rp,1 + Rp,2V (56)

Therefore, an experimental (J) V plot in the V 1 region will be a straight line of slopem = Rs/(Rp,1 + Rp,2). This slope can be directly obtained through a linear regression fit in theV 1 regime.

When V

i 1, the exponential term in (54a) is dominant, leaving:J = J0,ie

Vi /ni i = 1, 2 (57)

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Solving (57) for Vi and substituting it into (54b) leads to:

ln (J) =1

n1 + n2V +

1

n1 + n2ln Kd (58)

where:

Kd =

J0,1n1 J0,2n2 (59)

A plot of ln(J) V in the V 1 region can be approximated as a straight line of slopem = [1/(n1 + n2)] and y-intercept c = [1/(n1 + n2)]ln(Kd).


The dark J V curve used for the current tandem cell characaterization process was taken afew months later than that used in the single-junction cell analysis. This time, value for Rswas found to be 0.00254 m2, which is, as expected, of the same order of magnitude to that

obtained for the single-junction cell analysis. Values for n1 + n2 and Kd could then be easilyobtained from a linear fit in the V 1 region as shown in Figure 33.

Figure 33: J V plots and linear fit for the V 1 region at the chosen value of Rs. Lowerlimit of fit, V /VTe 50. Top: Semilog plot on which the linear fit is performed, enabling n1+ n2

and Kd to be found. Bottom: Plot in a log-free scale to verify that the exponential part of curveis exclusively captured by the fit.

On the other hand, Rp,1+Rp,2, which are the equivalent ofRp in the single-junction cell analysis,came out to be much larger. There is a recorded evolution for Rp in time. Its value was observedto increase from 5 m2 [12] to 50 (by the time of the single-junction cell analysis), then 500 and finally 1000 (tandem cell analysis). Within this period, several reverse voltage biasmeasurements were taken. Therefore, this drastic increase in Rp, (analogous to a big reductionof the shunt leakage current) is attributed to shunt-busting, described back in Section 2.3.1.

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Figure 34: J V plot with a linear fit in the Vi 1 region in order to get Rp,1 + Rp,2 from itsslope.

5.4 Theoretical derivations for single subcell excitation

Before embarking upon the characterization of the cell using measurements under bias IR and UVillumination, it is worth detailing the expected cell behaviour for the case in which the incidentillumination only induces a photocurrent in a single subcell, and the remaining unexcited subcellis forced to operate in dark conditions.

The purpose of this section is to derive useful relations and properties that may be used asindicators for the case of single subcell excitation. As it will seen later in Section 5.5, noneof these properties were satisfied experimentally. This led to the conclusion that single subcellexcitation was not attained, hence an alternative characterization route was then developed.

The case where only subcell 2 is excited while subcell 1 remains in the dark ( JL,1 = 0, JL,2 > 0)will be considered in the following derivations.

Typical operating curves for both subcells and the resultant curve for the whole cell are sketchedin Figure 35. It clearly portrays the case where subcell 2 is excited, while subcell 1 operates inthe dark. Note that when the cell is operating at short-circuit conditions (SC), subcell 1 andsubcell 2 operate at 1, SC and 2, SC respectively.

The first baseline assumption in this case is that at the open-voltage (OC) operating condition,V1,OC = 0 always, hence V

OC = V

2,OC. Secondly, it is evident from Figure 35 that the proximityof points 2, SC and 2, OC = OC implies that V2,SC V

2,OC. The smaller J

SC is, the closer toeach other these two points will be.

A third important baseline assumption is that the shunt leakage current must be Ohmic not onlywhen the subcell operates at forward voltage biases but also at reverse voltage biases (at leastin the range Vi,OC < V

i < 0). This implies that in dark conditions, the subcell will display alinear J V characteristic curve within this negative voltage range.

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Figure 35: J V sketch of the expected operating curves of each subcell when exposed toillumination that excites only subcell 2. Subcell 1 is constrained to operate in the dark.

5.4.1 SC current OC voltage relation

Bearing the first baseline assumption in mind, the characteristic equation for subcell 2 at OCreads:

0 = JL,2

1

2OC

J0,2

eV

OC/n2 1

Rs

Rp,2VOC (60)

where OC = 1 V

OC(VTe/Vb,2).At the SC operating condition, the charactersitic equations for subcells 1 and 2 become:

JSC = J

0,1 Rs

Rp,1V1,SC (61a)

JSC = J

L,2

1

22,SC

J0,2e

V2,SC/n2 Rs

Rp,2V2,SC (61b)

Note that at SC, subcell 1 is operating in reverse voltage biasing (V1,SC < 0). Here (61a) treatsthe J V1 characteristic curve of the subcell in this quadrant (V

1 < 0, J > 0) to be linear

(corresponding to an Ohmic shunt leakage current). It is modelled as a straight line of slope

mrev = Rs/Rp,1, and y-intercept c = J0,1. Furthermore, the magnitude of c is very small, andmay only be taken into consideration if JSC is of the same order of magnitude as J

0,i. Otherwisec may be treated to be 0 if JSC J

0,i.

The second baseline assumption implies that 2,SC OC. It is thus valid to assume that:

1 2

2,SC= 1

2OC

(62)

After premultilying (61a) and (61b) by Rp,i/Rs, adding them together and then substituting in(60) to eliminate JL,2, one gets:

(Rp,1 + Rp,2)JSCRs

= Rp,2J

0,2

Rs

eV

OC/n2 eV

2,SC/n2

+ VOC (V

1,SC + V

2,SC) +Rp,1J

0,1

Rs(63)

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The term (V1,SC + V

2,SC) = V

SC = J

SC may be neglected since it is much smaller than the restof the terms if Rp,i/Rs 1. Using Equation (61a), an expression for V

2,SC may be formulated:

V2,SC = V

SC V

1,SC = J

SC +Rp,1(J

SC J

0,1)

Rs

Rp,1(J

SC J

0,1)

Rs

(64)

Introducing this expression for V2,SC into (63) finally gives:

(Rp,1 + Rp,2)J

SC

Rs=

Rp,2J

0,2

Rs

eV

OC/n2 exp

Rp,1(J

SC J

0,1)

n2Rs

+ VOC +

Rp,1J

0,1

Rs(65)

This is a non-linear equation relating JSC and V

OC. It may be further simplified depending onthe recorded magnitude of JSC with respect to J0,i, which is typically of the order 10

6 A/m2.

For sufficiently small irradiances, JSC may be comparable in magnitude to J

0,i. It may also bethe case that both are always comparable in magnitude, regardless of the irradiance used. This

means that the slope mrev is small, indicating that the cell is well-built and has small parasiticresistances. This implies large values for Rp,i and a small Rs. In any case, when J

SC J

0,i issatisfied, the 2, SC and OC points will practically merge and the approximation V2,SC = V

OC

is valid. Equation (65) then simplifies to a linear form:

(Rp,1 + Rp,2)J

SC

Rs= VOC +

Rp,1J

0,1

Rs(66)

It is thus expected that the magnitude for VOC is such that V

OC V1,SC Rp,1J

0,1/Rs iscomplied. This means that ifJ0,i Rs/Rp,i then V

OC 1. This implies very small illuminationintensities. It should be borne in mind that it is often difficult to obtain sufficiently accurateJ V measurements in the extremely small 0 < V < V

OCregion under very small irradiances.

If on the other hand, JSC J

0,1, the approximation V

2,SC = V

OC is no longer valid, but all theJ0,1-dependent terms in (65) may then be neglected. It keeps a nonlinear form, and may onlybe solved through iteration:

(Rp,1 + Rp,2)J

SC

Rs=

Rp,2J

0,2

Rs

eV

OC/n2 exp

Rp,1J

SC

n2Rs

+ VOC (67)

5.4.2 SC and OC resistance property

The case where subcell 2 is excited while subcell 1 is in the dark is still being considered. This

property states that both R

SC and R

OC should tend to Rp,1/Rs as the irradiance exciting subcell2 is increased.

Evaluating first R1,SC, it is noted that V

1,SC < 0 hence the J

0,1-dependent terms in the definitionfor R1,SC in (53) may be neglected provided J

0,1 Rs/Rp,1. This results in:

R1,SC =Rp,1Rs

(68)

R2,SC shall be evaluated next for the case where the cell is exposed to strong enough irradiancesthat induce VOC V

2,SC 1. The recombination-free exponential term in (53) is dominant.Hence:

R2,SC n2J0,2eV

2,SC/n2 (69)

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With increasing irradiance and V2,SC, R

2,SC becomes more negligible in magnitude with respectto R1,SC and as a result R

SC Rp,1/Rs.

It must be emphasized that this applies only if the J V1 curve in the dark (of subcell 1) atreverse voltage biases (for VOC < V

1,SC < 0) is indeed described by a straight line of slope

mrev = 1/R

1,SC = Rs/Rp,1. This occurs exclusively for Ohmic shunt leakage currents. Thismay not always be the case, since the shunt leakage current of a particular subcell may possess

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