Photoluminescence image evaluation of solar cells based on impliedvoltage distribution

Otwin Breitenstein a,n, Hannes Hof̈fler b, Jonas Haunschild b

a Max Planck Institut of Microstructure Physics, Weinberg 2, 06120 Halle, Germanyb Fraunhofer Institute of Solar Energy Research (ISE), Heidenhofstraße 2, 79110 Freiburg, Germany

a r t i c l e i n f o

Article history:Received 1 April 2014Received in revised form13 May 2014Accepted 19 May 2014

Keywords:Photoluminescence imagingQuantitative evaluationModelingLocal analysisSeries resistance imagingSaturation current density imaging

a b s t r a c t

All previous methods for quantitatively evaluating photoluminescence (PL) images of solar cells assumeda laterally constant short circuit current density Jsc. Moreover, they had to subtract a Jsc PL image from allother PL images for considering the diffusion-limited carriers. Here a more realistic PL evaluationmethod is introduced, which is based on a recently published alternative model of the illuminated solarcell. In this model an analytic expression is derived by considering the illuminated current as a diffusionprocess between bulk and the pn-junction and linking the implied voltage in the bulk with the local pn-junction voltage under illumination. This model does not assume a laterally constant Jsc but a constantlight absorption rate, and it leads to a prediction of the Jsc distribution solely based on PL imaging results.Moreover, it regards the shadowing of the cell by the busbars and grid lines. This model is applied to thequantitative evaluation of PL images of an industrial multicrystalline silicon solar cell. The resultingseries resistance and saturation current density images are compared with that of an established PLevaluation method, and the resulting distribution of Jsc is compared with LBIC results. The results of thenew method appear slightly more realistic than that of the old one, since they consider theinhomogeneity of Jsc.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

In the last decade photoluminescence (PL) imaging of solar cellshas been established as a successful and fast method for quantita-tively imaging the distribution of the local series resistance [1] andof the local saturation current density, see e.g. [2–4]. All thesemethods are based on the model of independent diodes, eachbeing connected with the terminals via an independent seriesresistance, which is expressed in this model area-related in unitsof Ω cm2. Moreover, most of the previous methods assume aninjection intensity-independent lifetime by assuming an idealityfactor of unity for the diode current. Only in [4] this ideality factormay be assumed to be larger than unity for the whole area, whichenables a fit of the simulation results to the global illuminatedcharacteristic.

All previous PL evaluation methods for solar cells have anotherpoint in common: they need to subtract a PL image taken undershort circuit (Jsc) condition from all other PL images for obtaining a‘net’ PL image for further evaluation. This procedure goes back to aproposal of Trupke et al. [1] and was justified theoretically byGlatthaar et al. [3], who showed that the carrier concentration in

the bulk under illumination and current extraction is the sum ofone voltage-dependent but illumination-independent and onevoltage-independent but illumination-dependent part, the latterone dominating under short circuit condition. The physical reasonfor this procedure is the well-known fact that, for the samepn-junction voltage below the open circuit voltage Voc, the carrierconcentration in the bulk is higher under illumination and carrierextraction than that in the dark. These additional carriers, whichhave to diffuse to the pn-junction for generating the illuminatedcell current, are called “diffusion-limited carriers” [1]. By subtract-ing the Jsc PL image from the other PL images, the illuminated caseis reduced to the un-illuminated case, as it would hold e.g. for anelectroluminescence (EL) investigation. After this correctionthe net luminescence intensity depends exponentially on thepn-junction voltage, as in the EL case.

This approach works fine as long as the lifetime remainsindependent of the excitation intensity. In reality the lumines-cence intensity is caused by the excess carrier concentration in thebulk, hence it depends exponentially on the so-called impliedvoltage Vimpl in the bulk, which we here define as the separationbetween the electron and the hole quasi-Fermi level in the middleof the bulk, divided by the electron charge q. In all previous PLevaluation methods the implied voltage did not appear at all. AnyPL evaluation method, which wants to consider an injectionintensity-dependent lifetime, must be based on the implied

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/solmat

Solar Energy Materials & Solar Cells

http://dx.doi.org/10.1016/j.solmat.2014.05.0400927-0248/& 2014 Elsevier B.V. All rights reserved.

� Corresponding author. Tel.: þ49 345 5582740; fax: þ49 345 5511223.E-mail address: breiten@mpi-halle.mpg.de (O. Breitenstein).

Solar Energy Materials & Solar Cells 128 (2014) 296–299

voltage. Moreover, as mentioned above, all previous PL evaluationmethods assumed a homogeneous value of Jsc, even below thebusbars and grid lines, which is a strong simplification.

The PL evaluation method introduced here, which we name “PLevaluation based on implied voltage distribution (PL-imp)”, alsouses the model of independent diodes, but it is explicitly based onthe implied voltage distribution, and it does not assume a constantJsc. Instead, it assumes a constant rate of generated carriers perarea in the non-shadowed regions, which appears to be morerealistic. From the Jsc PL signal a measure of the bulk recombina-tion rate at short circuit condition is derived, which allows topredict the Jsc distribution solely from the evaluation of PL images.Moreover our method does not rely on the assumption of nolateral voltage gradients at low injection conditions, which canlead to errors in the final results as shown in [5]. In thiscontribution, the lifetime is still assumed to be independent ofthe excitation intensity, but this method provides the base for afuture extension to regard an injection-dependent lifetime.

2. Physical model

The PL evaluation method introduced here is based on analternative one-diode model for illuminated solar cells publishedin detail elsewhere [6]. Therefore, only the basic ideas and resultsof this theory will be reported here. The main content of thiscontribution is the application of this theory to PL image evalua-tion. The diode theory is described here without any seriesresistances, but for the application of the results to PL evaluationa local series resistance will be considered. As in [6] onlycontributions of the dark and illuminated currents of the p-doped bulk are considered here.

As mentioned above, the alternative one-diode model assumesthat the generation current density Jgen, which describes the totalamount of light-generated carriers per unit area and time, islocally constant and independent of the local pn-junction biasVpn. Provided that the texturing is sufficiently homogeneous, thisis certainly a better approach than assuming Jsc to be homoge-neous, in particular for multicrystalline silicon cells. Under shortcircuit condition, this current density splits into the extractedshort circuit current density Jsc and a current density describingthe bulk recombination under short circuit, which is called here asJrec,0:

Jgen ¼ Jscþ Jrec;0 ð1Þ

The magnitude Jrec,0 is a new diode parameter introduced forthis theory. The alternative diode theory [6] assumes that theexcess carrier concentration in the largest part of the bulk isconstant across the depth. If some current is extracted underillumination, only in a so-called drift region close to the pn-junction a carrier concentration gradient exists, which leads tothe illuminated current flow. According to PC1D simulations of atypical crystalline solar cell, this drift region has an extension ofabout 20 mm [6]. As in the previous theories, the excess carrierconcentration at the pn-junction is given by the local voltagethere. The basic idea of the new model is that under illuminationthe extracted current is proportional to the difference between theexcess carrier concentration in the bulk nbulk and that at the pn-junction npn and can be described formally by an effectivediffusion coefficient Deff (ni¼ intrinsic carrier concentration,NA¼effective bulk acceptor concentration, VT¼thermal voltage):

J ¼ qDeff ðnbulk�npnÞ ¼ qDeffni2

NAexp

V impl

VT

� ��exp

Vpn

VT

� �� �ð2Þ

In contrast to the usual definition of a diffusion coefficient, Deff hasthe unit of cm/s. Thus, it also could be named a carrier extraction

velocity. On the other hand, the extracted cell current is thedifference between Jgen and the bias-dependent recombinationrate in the bulk, which may be expressed by the saturation currentdensity of the bulk recombination J01:

J ¼ Jgen� J01expV impl

VT

� �ð3Þ

For Vpn¼0 (short circuit) Jrec,0¼ J � Jgen holds. Then Eq. (3) allowsto calculate Vimpl under short circuit condition:

V impl;0 ¼ VT lnJrec;0J01

� �ð4Þ

This allows to calculate Deff after Eq. (2):

Deff ¼Jsc

q ni2 Jrec;0NA J01

�ni2

NA

� �� JscJ01NA

qni2Jrec;0

ð5Þ

The latter relation in Eq. (5) holds due to Jrec,0 » J01, since Jrec,0 istypically in the lowmA/cm2 region, see Section 4. Hence, by Eq. (5)the formally introduced effective diffusion coefficient Deff can beexpressed by measurable diode parameters. If Eq. (5) is insertedinto Eq. (2), this leads together with Eq. (3) to the major result ofthis theory, which expresses Vimpl by Vpn under illumination:

V implðVpnÞ ¼ VT lnJrec;0J01

þ JscJgen

expVpn

VT

� � !ð6Þ

Here the material parameters ni and NA cancel and only measur-able diode parameters remain. If Eq. (6) is inserted into Eq. (3) forcalculating the illuminated current density, the result is basicallythe same as for the conventional diode model:

JðVpnÞ ¼ Jsc�JscJgen

J01expVpn

VT

� �ð7Þ

The factor (Jsc/Jgen) in Eq. (7) is due to the fact that, in the newmodel, the open circuit condition is established by balancing Jgento the dark current, instead of Jsc in the conventional diode model[6]. Thus, the alternative one-diode model does not change thesolar cell physics significantly, it only provides a deeper under-standing of the pn-junction physics and provides a simple way toexpress Vimpl by Vpn analytically.

3. PL evaluation method

All previous luminescence imaging methods are based on theequation:

Φi ¼ CiexpVpn;i

VT

� �ð8Þ

here Ci is the so-called luminescence scaling parameter, whichdepends e.g. on the local surface conditions and recombinationproperties, and i is the position index. Eq. (8) holds directly for un-illuminated (EL) measurements, since there VimplEVpn holds. Forprevious PL evaluation methods, however, Eq. (8) only holds forthe ‘net’ PL signal after Jsc correction [1]. In our contribution Vimpl

and Vpn are two separate variables, which are connected by Eq. (6).Therefore our PL evaluation is based on:

Φi ¼ CiexpV impl;i

VT

� �ð9Þ

The second basic equation, which is also used in all other PLevaluation methods, is the calculation of Vpn in the modelof independent diodes regarding a local series resistance Rs.Regarding Eq. (7) this equation reads here:

Vpn;i ¼ VþRs;iJi ¼ VþRs;i Jsc;i�Jsc;iJgen

J01;iexpVpn;i

VT

� � !ð10Þ

O. Breitenstein et al. / Solar Energy Materials & Solar Cells 128 (2014) 296–299 297

A major difference between this and previous PL evaluationmethods is the new local diode parameter Jrec,0,i, which describesthe local bulk recombination rate under short circuit condition.This means that a PL signal Φi

(sc) measured under short circuitcondition provides information to Jrec,0,i. Combining Eq.(3) for theshort circuit case with Eq. (9) leads to:

Jrec0;i ¼ J01;iΦi

ðscÞ

Cið11Þ

Since the global currents of the cell are the sum of all pixelcurrents, the global generation current is the global sum of Isc andthe sum of the Irec,0 of all pixels. Here the influence of the lightshadowing by the busbars and grid lines has to be considered,otherwise the simulated local short circuit current densitybecomes too low. Note that Isc of the cell always refers to thewhole cell area, but locally (between the grid lines) the averagedJsc is larger than Isc/A (A¼cell area). This influence is consideredhere by dividing Isc/A by a factor of (1 � fmet), with fmet being thefraction of metallized area. Expressed as current densities, thisleads with X and Y being the dimensions of the image to:

Jgen ¼Isc

Að1� fmetÞþ 1XY

∑iJrec0;i ð12Þ

This equation allows to calculate the value of Jgen, assumed here tobe homogeneous between the grid lines, if Isc is known and Jrec,0,i ismeasured after Eq. (11).

The two possible ways to evaluate PL images are either toevaluate the same number of images as the number of local diodeparameters (which are usually J01, Rs, and C), plus at least one Jsc PLimage for performing the above mentioned PL data correction[2,3]. The second way is to evaluate a larger number of PL imagesand to perform a linear regression procedure for obtaining thelocal diode parameters [4]. Here the first way will be used. Sinceour local diode parameters are J01,i, Jrec,0,i, Rs,i, and Ci, for ourprocedure we have to evaluate four PL images. For obtaining thesediode parameters, we are looking for a simultaneous solution of

Eqs. (9) – (12) regarding Eq. (6) applied to these PL images. Ourstrategy is to optimize the imaging conditions for these PLmeasurements so that each of them reacts most sensitively toone of the diode parameters. Then an iterative loop procedure isused for fitting Eq. (9) – (12) with Eq. (6) simultaneously to all PLresults. We propose to perform the following PL measurements:

Φi(1) measured under Voc condition at low intensity I(1) (e.g.

0.1 sun) used for fitting Ci after Eq. (9)Φi

(2) measured under Jsc condition at high intensity I(2) (e.g.1 sun) used for fitting Jrec,0,i after Eq. (11) and Jgen after Eq.(12)

Φi(3) measured with weak load (e.g. 25% of Isc) at high intensity

I(3) (e.g. 1 sun) used for fitting J01,i after Eq. (10)Φi

(4) measured with stronger load (e.g. 75% of Isc) at highintensity I(4) (e.g. 1 sun) used for fitting Rs,i after Eq. (10)

We have developed a software package performing this itera-tive fitting procedure named “PL-imp”, since this procedure isbased on the analysis of the implied voltage. On a usual PC theevaluation of four 1024�1024 pixel sized PL images using 20iterations takes about 4 s.

4. Results

This PL evaluation method was applied to a typical industrialmulticrystalline silicon solar cell with a standard size of156�156 mm2. For comparison, the same PL images were eval-uated by a “Coupled Voltage Calibration” (C-VC) method [3], whichalso regards the voltage drop at series resistances for the calibra-tion measurement, but assumes a homogeneous Jsc. Also here theshadowing effect was considered by assuming a slightly increasedJsc, see Eq. (12). Fig. 1 shows the resulting images of C, J01, and Rsfor both methods. This comparison shows that the results of bothmethods regarding these parameters are very close to each other.

Fig. 1. Images of the luminescence calibration constant C (a, d), the saturation current density J01 (b, e), and the local series resistance Rs (c, f) after the newly proposedmethod (a–c, top) and the C-VC method [3] (d–f, bottom).

O. Breitenstein et al. / Solar Energy Materials & Solar Cells 128 (2014) 296–299298

This proves that the iterative fitting procedure is generally work-ing as expected. Looking more into the details we see that themaximum values of J01 in the poor crystal quality regions areslightly higher in the C-VC method than in the PL-imp method,here they differ by about 9%. This is due to an overestimation of Jscin these regions by C-VC. Altogether, though the differencesbetween the C-VC and the PL-imp results are weak, the latterprocedure seems to be more realistic from a physical point of view.

Fig. 2 (a) shows an image of the PL-simulated Jsc data, incomparison to a light beam-induced current (LBIC) image(b) scaled to a nominal excitation intensity of 1 sun1 at the sameexcitation wavelength as used for PL imaging of 780 nm. Inaddition, an artificially blurred version of the LBIC image of (b) isshown in (c). It is visible that the PL-based Jsc image (a) shows aclearly worse spatial resolution than the directly measured one(b). This is due to the lateral spreading of the local voltage in asolar cell due to the horizontal balancing currents in the emitterand the grid. Note that this PL evaluation procedure is based onthe model of independent diodes, which neglects any lateralinteraction between neighboring pixels. In reality, the seriesresistance is a distributed one, which is not considered here. Ifthe LBIC image is artificially blurred (c) it looks very similar to thesimulated one (a). Note that the scaling limits are not exactly thesame but close to each other. Thus, it can be judged that our PLevaluation method leads to a good modeling of Jsc. The remainingquantitative differences to the direct measurement may be due tothe very different excitation processes between (a) and (b, c) ordue to natural limitations of this relatively simple physical model,e.g. to the assumption of a depth-independent carrier concentra-tion, an excitation intensity-independent lifetime, or the indepen-dent diode model.

5. Summary and discussion

In this contribution a novel photoluminescence evaluationmethod is introduced (PL-imp), which is based on the evaluationof the local implied voltage instead of the local pn-junctionvoltage. Moreover, this method does not assume a homogeneous

short circuit current density but only a homogeneous density ofabsorbed photons, and it considers the shadowing by metallizedareas. Therefore, in contrast to all previous PL evaluation methods,PL-imp leads to a realistic prediction of a Jsc image for thewavelength used for PL excitation. The comparison of such animage with a conventionally measured LBIC image shows a highdegree of conformance, in spite of the relatively coarse physicalmodel underlying this PL-imp procedure. Except the simulation ofJsc data, the results of PL-imp do not deviate significantly from thatof previous PL evaluation methods, which all had to subtract a JscPL image from the others. However, in contrast to the previous PLevaluation methods, only PL-imp is able to be extended to evaluateintensity-dependent lifetimes. It also appears physically moremeaningful than the previous methods, since it evaluates the PLsignal directly and regards an inhomogeneous Jsc.

It has to be mentioned that PL-imp is more demanding withrespect to light filtering than previous PL evaluation methods. Ifsome amount of reflected excitation light reaches the camera usedfor PL detection, this amount is canceled out by the usualsubtraction of the Jsc PL image. This is not the case for the PL-imp method.

References

[1] T. Trupke, E. Pink, R.A. Bardos, M.D. Abbott, Spatially resolved series resistanceof silicon solar cells obtained from luminescence imaging, Appl. Phys. Lett. 90(2007) 093506.

[2] M. Glatthaar, J. Haunschild, M. Kasemann, J. Giesecke, W. Warta, Spatiallyresolved determination of dark saturation current and series resistance ofsilicon solar cells, Phys. Stat. Sol. RRL 4 (2010) 13–15.

[3] M. Glatthaar, J. Haunschild, R. Zeidler, M. Demant, J. Greulich, B. Michl,W. Warta, S. Rein, R. Preu, Evaluating luminescence based voltage images ofsolar cells, J. Appl. Phys. 108 (2010) 014501.

[4] C. Shen, M.A. Green, O. Breitenstein, T. Trupke, M. Zhang, H. Kampwerth,Improved local efficiency imaging via photoluminescence for silicon solar cells,Sol. Energy Mater. Sol. Cells 123 (2014) 41–46.

[5] H. Hof̈fler, H. Al-Mohtaseb, J. Haunschild, B. Michl, M. Kasemann, Voltagecalibration of luminescence images of silicon solar cells, J. Appl. Phys. 115 (2014)034508.

[6] O. Breitenstein, An alternative one-diode model for illuminated solar cells, IEEEJ. Photovolt. 4 (2014) 899–905.

Fig. 2. Image of Jsc simulated from the PL-imp evaluation (a), LBIC image for 1 sun measured at 780 nm (b) and artificially blurred LBIC image (c).

1 In this work we define a 1 sun equivalent in the context of luminescencemeasurements as an incident photon flux of 2.55�1017 photons/(cm1� s)

O. Breitenstein et al. / Solar Energy Materials & Solar Cells 128 (2014) 296–299 299