Matching of Silicon Thin-Film Tandem Solar Cells for MaximumPower Output

We present a meaningful characterization method for tandem solar cells. e experimental method allows for optimizing the output power instead of the current. Furthermore, it enables the extraction of the approximate AM1.5g efficiency when working with noncalibrated spectra. Current matching of tandem solar cells under short-circuit conditionmaximizes the output current but is disadvantageous for the overall �ll factor and as a consequence does not imply an optimization of the output power of the device. We apply the matching condition to the maximum power output; that is, a stack of solar cells is power matched if the power output of each subcell is maximal at equal subcell currents. e new measurement procedure uses additional light-emitting diodes as bias light in the JJJJJJJJ characterization of tandem solar cells. Using a characterized reference tandem solar cell, such as a hydrogenated amorphous/microcrystalline silicon tandem, it is possible to extract the AM1.5g efficiency from tandems of the same technology also under noncalibrated spectra.


Introduction
e electrical series connection in tandem solar cells requires a design of the electrical and optical properties of each subcell that minimizes the losses arising from an electrical mismatch.Mismatching losses are eclipsed if each subcell delivers its maximum power output at the same current.Unfortunately, this power matching condition is difficult to access experimentally.Usually, the matching (or mismatch) of tandem stacks is investigated in terms of the current matching condition [1], which is achieved if the short-circuit currents of the subcells are equal.is condition is accessible by investigation of the spectral response or quantum efficiency of the subcells.
However, current and power matching conditions may differ signi�cantly especially if the �ll factors of the subcells are markedly different, like in hydrogenated amorphous/microcrystalline silicon (a-Si:H/c-Si:H) thin-�lm tandem cells.is fact is illustrated by Figure 1 displaying simulated power density versus current density curves for two different a-Si:H/c-Si:H tandem stacks.e gray lines correspond to the respective subcell and overall  curves of a stack under current matching condition where the current density at zero voltage is maximized for the superposed  curve.e black lines correspond to a powermatched tandem cell.At the maximum power point, the power of the current-matched cell is outperformed by the power-matched tandem cell.Power matching implies a loss in total short-circuit current, that is, however, overcompensated at the maximum power point by the advantageous shape (advantageous overall �ll factor) of the total  characteristics.
It is not possible to probe experimentally the single cell components of a tandem cell independently, neither electrically nor optically.To reveal some information on the single cell components of a tandem stack, an experimental method has been proposed that consists of recording the current density voltage-()-characteristic of a sample using additional light sources.e method uses an approach similar to the light biasing in quantum efficiency measurement, however, in contrast to the wavelength resolved quantum efficiency measurement it is capable of determining the solar cell power under the changed illumination conditions, whereas the quantum efficiency measurement gives only information on the current matching.F 1: Simulated current-power characteristics of the subcells (dash-dotted: top cell, dashed line: bottom cell) and of the combined hydrogenated amorphous/microcrystalline tandem solar cell (straight line).e output power is maximized to 11.6 mWcm −2 instead of 11.3 mWcm −2 , corresponding to a gain of 2.6% when the currents of the subcells are power matched; that is, the currents are matched at the maximum power point (black) instead of the common current matching at zero voltage (gray).e total thickness of the simulated tandems was 1.5 m.
e method was proposed by Burdick and Glatfelter in 1986 [2]. e additional bias light is chosen to affect mainly either the top or the bottom cell of the tandem.e supplementary bias light increases the fraction of absorbed illumination in the respective subcell similar to an increased thickness of the absorber layer.
"Spectrometric characterization" methods have been applied to various types of solar cells, for example, crystalline (with a light-emitting diode (LED) solar simulator [3]), multijunction (with �ltered and intensity-adapted sun simulator light [4,5]), and concentrator cells (adjusting the intensity of independent light sources [6] or using �lters for a single-source sun simulator [7]).Burdick and Glatfelter used a high-intensity �lament lamp, a �enon-arc lamp, and a monochromator.LEDs offer more opportunities to easily shape further interesting experiments (see, e.g., [3]).
As the amorphous top layer of thin-�lm silicon tandem solar cells is known to degrade under light soaking, the optimization of tandem solar cells is speci�cally difficult and time consuming.e correlation of matching conditions and degradation has already been investigated by Repmann et al. [8], addressing also the effect on outdoor yield.e experimental study was, however, based on a series of tandem solar cells with various  -layer thicknesses as well as modeling and did not use the characterization method presented in this paper.e method is very helpful in such an analysis.e use of the method in the optimization of tandem solar cells minimizes the need to fabricate solar cells of different thicknesses.
In this paper, we present an experimental method based on the use of a white light source supplemented by additional blue and infrared LED arrays.is method allows for the determination of the power mismatch of tandem solar cells, that is, the required change in layer thickness to reach maximum power.We refer to matching with respect to a maximum overall power instead of a maximum current under short-circuit conditions.Power matching is achieved for a stack of solar cells if the power output of each cell is maximal at equal cell currents.e experimental method serves to �nd the needed absorber layer thicknesses by changing the spectrum that can be made equivalent to changing the layer thickness and, thus, slightly shiing the single-cell  curves.We demonstrate the method using a reference tandem solar cell that is bottom limited in terms of the current matching conditions, however, is top limited in terms of power matching conditions.Additionally, and speci�cally to this method using tunable LEDs as light sources complementing the sun simulator spectrum, the method can be used to identify the AM1.5g performance of a tandem solar cell under a noncalibrated illumination spectrum.A preliminary report on the method has been published earlier [9].

Experimental Details
Figure 2 shows the experimental setup consisting of a sun simulator with two additional LED sources (with emissions at wavelengths  ∼  nm and 850 nm, resp.) to tune the spectrum applied to the device under test.e sample is homogeneously illuminated by a sun simulator acting as the wide spectral light source and the LEDs.
e method comprises the following steps: a tandem cell with known quantum efficiency is exposed to the sun simulator spectrum supplemented by different fractions of infrared and blue LED light.Under each illumination condition, an  curve and the illumination spectra are acquired.In the second step, by acquiring data for another test tandem cell the previously characterized tandem cell can be used as a reference so that the acquisition of the spectra becomes obsolete.
Figure 3(a) displays the external quantum efficiencies (EQE) of the top and bottom cells of an a-Si:H/c-Si:H tandem solar cell.e sample under investigation is deposited on an Asahi U-type substrate and consists of two - - junctions followed by a ��� silver rear re�ector layer [10].corresponding to the AM1.5g spectrum [11], to the spectrum of the sun simulator, and to illumination by the sun simulator with additional light bias provided by the two LED sources, respectively.e total thickness of the tandem cell is 2.4 m.It is seen from Figures 3(a) and 3(b) that the wavelengths range of the LED light sources is chosen to affect either the top or the bottom cell exclusively.Furthermore, the output power of the sun simulator is slightly attenuated in order to keep the total light power incident on the cell equal to the standard value of 1000 W/m 2 when the LEDs are added.erefor the power of the sun simulator light sources was reduced (�lters could have been used as well).e LED bias illumination intensity is modi�ed from 100% infrared (IR) to 100% blue in equidistant power steps, increasing the power at the IR sources while decreasing the blue sources.e total illumination intensity is kept constant.

Calibration of the Reference Cell
e experiments described in the following provide a detailed analysis of the matching behavior of an a-Si:H/c-Si:H tandem solar cell.ese experiments include the measurement of the EQE of the top and bottom cells, the measurement of the  characteristics of this cell under systematically varied contributions from the LEDs, and the measurement of the combined illumination spectra corresponding to each of those measurements.is somewhat cumbersome procedure provides an analysis method on its own right but is especially valuable when using the analyzed cell as a reference cell for a simpli�ed analysis of further cells as described in Section 4.
In the �rst step, we measure the EQE of the top and the bottom cells of the reference sample (see Figure 3 Note that we have tuned the illumination steps Δ  LED (, Δ 2  LED (, and the output  sunsim ( of the sun simulator in a way such that the total illumination intensity corresponds to approximately 1000 W/m 2 for each measurement.e original sun simulator spectrum was attenuated to meet the requirement.Yet, experiments with a slightly changed, in terms of spectral shis, sun simulator spectrum lead to the same results (data not shown). By Here , ,  0 , and   are coefficients describing the relationship mathematically.SC resulting from the      measurements using linear spectral variations of the applied illumination.For clarity, the short-circuit current densities calculated for the standard AM1.5g spectrum as well as for the attenuated and not attenuated sun simulator as calculated with the help of (2) are also shown.It can be seen from Figure 4 that by variation of the LED light bias we are able to approximate very well the short-circuit current densities de�ned by the AM1.5g spectrum as well as those de�ned by the unattenuated sun simulator.It is worth noting that the present method enables one to analyze the cell close to standard conditions even if the sun simulator is not as well matched to the AM1.5g spectrum as in the present case.
Figure 5 shows the current density difference Δ SC   top SC −  bot SC as a function of the index  of the measurement and demonstrates that the linear relationship between Δ SC and  as postulated by ( 3) is very well ful�lled.e next step requires the measurement of the  characteristics of the cell under each of the  illumination bias conditions.For each of the curves we determine the maximum output power , the �ll factor FF, and the shortcircuit current density  SC .For our measurement setup, the uncertainties for those quantities are roughly 4%.A normalized set of these data as a function of the measurement index  is shown in Figure 6.Note that due to the linear relation between  and the short-circuit current density difference Δ SC , we may use Δ SC (instead of ) as a scale for the display of the data (top axis in Figure 6).As expected, the maximum of the overall short-circuit current density  SC and the minimum of the �ll factor FF is close to Δ SC  0, that is, at the current matching condition.
F 6: e normalized solar cell parameters are given for each measurement number.As the measurement number  is linearly related to the difference in short-circuit current Δ SC , the second scale is attached above.e sample is slightly mismatched.e total short-circuit current density  SC (green symbols) is maximized when the subcell currents are approximately equal, close to Δ SC  0, which does not correspond to AM1.5g-illumination conditions.�nder the same current matching condition the �ll factor (orange symbols) is minimal.e power (blue symbols) peaks for larger values of Δ SC .A layer thickness adaption, speci�cally an increase of the top cell layer thickness, would result in a power increase.
e reference point given by the AM1.5g-standard spectrum is found at Δ SC (AM.5g) 0.5 mAcm − .us, we have  top SC >  bot SC and the cell is bottom limited in terms of the current matching conditions.However, the maximum of the output power is found at Δ SC ( max )  0.7 mAcm − , that is, Δ SC ( max ) > Δ SC (AM.5g) and therefore the cell is top limited in terms of power matching conditions.e loss induced by the power mismatch under current matching is about 0.9% of the power achieved under optimum power matching conditions.However, the value of Δ SC ( max ) = 0.72 mAcm −2 shows that the different matching conditions are met by considerably different situations and lead to considerably different conclusions.e increase in the �ll factor of the  curve partially compensates current losses due to the mismatch.In order to optimize the present cell design for current matching a thicker bottom cell would be needed, whereas optimization of the power matching (and output power) requires a thicker top cell.

Measurements Using the Calibrated
Tandem Cell as a Reference e full set of experiments described in the previous section is rather time consuming and not suitable for the investigation of many tandem solar cells.In the following, we describe a procedure that does not require the measurement of the EQE of each device nor the determination of the irradiation spectra.e spectrum delivered by the sun simulator does not need to be identical to the calibration measurement but should be unaltered during one test series.
In the following we use four terms describing the different settings: a reference tandem cell (ref, we use the tandem cell and the data that was presented above) and a test tandem cell (test) are both exposed to a calibration illumination (cal, as was done before) and to an experimental test illumination (exp).As for the calibration procedure, we need a series of  measurements under varied illumination conditions.However, with our automated setup such a measurement series is performed within a few minutes for one sample.We have to perform this measurement under the same conditions for the devices under test as well as for the calibrated reference sample.We apply the method to a top limited (for current as well as power matching) test tandem cell and show only 11 measurement points for the experiment instead of 21 to emphasize the fact that it is not necessary to perform such a large number of measurements in the experiment.Note that the reference sample must not be degraded since the calibration measurement.For the evaluation, we have further to assume that the linearity expressed by ( 1) and ( 3) is valid for the test series.
Figure 7 shows the recorded output power for the device under test as well as for the calibrated reference sample as a function of the measurement index of the test procedure (open symbols).Also shown is the output power  of the calibrated reference sample as it was recorded during the calibration measurement (full symbols).Due to (intentional) change of the sun simulator spectrum, the absolute value and the position of the maxima of the two curves for the reference sample are changed.However, due to the linearity in ( 1) and (3), we are allowed to rescale the  axis of the test measurement by a linear transformation into the Δ cal SC axis of the calibration measurement according to Figure 8 compares the normalized output parameters, maximum output power , �ll factor FF, and short-circuit current density  SC as obtained from the calibration measurement (same data as in Figure 6, full symbols) to those data obtained during the test measurement aer rescaling the  axis of the test measurement to the Δ ref, cal SC axis of the calibration measurement.e coefficients  1 and  0 in (4) are chosen to achieve an optimum match between the respective  data.As can be seen from Figure 8, the linear transformation of the  axis into the Δ ref, cal SC axis reproduces the curves from the calibration measurement very well.e normalized parameters measured under the experimental test illumination and then shied differ by less than 0.1% from those measured under calibration conditions.
In the next step, we use the same transformation that was used to recalculate the values of the calibrated reference tandem solar cell under reference illumination conditions.We apply this transformation to the values measured for the test tandem sample under test illumination.
Figure 9 shows the normalized parameters , FF, and  SC for the test tandem using the scale of the reference sample.e scale of the reference sample under calibration illumination differs from the short-circuit current density difference of the subcells of the test tandem sample measured under test illumination.As axis the short-circuit current difference of the subcell currents of the reference solar cell under calibration illumination is used.e Δ the test cell is current matched.e red line marks the shortcircuit current difference of the reference for illumination conditions closest to AM1.5g.is illumination corresponds also for the DUT to a current generation as under AM1.5g illumination if the DUT has a quantum efficiency similar to that of the reference at the LED wavelengths.Assuming this was true, we can state that the test cell is top limited under AM1.5g.For a maximum output power, the thickness of the top subcell would have to be increased implying an expected increase in power by ∼2%.
Note that in the speci�c case of this tandem solar cell, the power peaks close to short-circuit current matching.is is due to the good �ll factors of the two subcells and implies that current and power matching are very close.
Note also that the proposed method is also interesting for the investigation of multijunction concentrator solar cells, where the �ll factor can be a valuable diagnostic tool �12] and (for all types of cells) in the optimization to speci�c spectra. .e difference in the current values of the  and  axes comes from the difference in the respective experimental measurement spectra and quantum efficiencies of the subcells of the two tandems.If the DUT and the reference had the same quantum efficiencies in the two wavelength ranges of the LEDs used in the characterization method, the slope of the line would be one.A steeper or shallower slope re�ects the difference in the absorption and charge carrier generation in the single subcells with the LED illumination.Keeping this in mind, it is clear that the quantum efficiency of the subcells of the reference tandem cell and of the DUT must meet the sole criterion of being sufficiently high and exclusive (only in one subcell) in the wavelength range of the LEDs.

Control Experiment
To prove the concept we also exposed the device under test to reference illumination conditions.Data is shown in Figure 11, full symbols.e shied values from the F 11: To prove the method, measured data of the tandem device under test under calibration illumination conditions are given (full symbols).e calculated transformation for the experimental test illumination to the calibration conditions is given by open symbols (same data as Figure 9).e fact that the peak position of  SC differs from Δ dut, cal SC = 0 is probably due to measurement uncertainties in the external quantum efficiency.experiment are reproduced very well.e peak in  SC is not found at Δ dut, cal SC = 0. e difference is probably due to uncertainties in the external quantum efficiency that was measured with a �lter wheel device.

Conclusions
e proposed characterization tool allows for the quanti�cation of the power mismatch of tandem solar cells and is thus very helpful in the optimization.e power matching of the currents is applied to the maximum performance output instead of the common current matching under short-circuit conditions.e LED-based method additionally enables the determination of the performance under AM1.5g illumination without time-consuming calibration of the sun simulator spectrum.e  curves measured under IR and blue rich illumination can also be of use in an analysis of the subcell �ll factors.

F 2 :
e experimental setup consists of a sun simulator that illuminates the solar cell.Additional illumination comes from two LED sources emitting at ∼450 nm and 850 nm.

F 3 :
Figure2shows the experimental setup consisting of a sun simulator with two additional LED sources (with emissions at wavelengths  ∼  nm and 850 nm, resp.) to tune the spectrum applied to the device under test.e sample is homogeneously illuminated by a sun simulator acting as the wide spectral light source and the LEDs.emethod comprises the following steps: a tandem cell with known quantum efficiency is exposed to the sun simulator spectrum supplemented by different fractions of infrared and blue LED light.Under each illumination condition, an  curve and the illumination spectra are acquired.In the second step, by acquiring data for another test tandem cell the previously characterized tandem cell can be used as a reference so that the acquisition of the spectra becomes obsolete.Figure3(a) displays the external quantum efficiencies (EQE) of the top and bottom cells of an a-Si:H/c-Si:H tandem solar cell.e sample under investigation is deposited on an Asahi U-type substrate and consists of two - - junctions followed by a ��� silver rear re�ector layer[10].Figure3(b) shows the spectral power densities

2 )F 4 :
e measured external quantum efficiency of the reference cell and the illumination spectra are used to calculate the shortcircuit current densities  top SC and  bot SC of the subcells.Additionally, the current densities for the AM1.5g standard spectrum (red star) and for the sun simulator spectrum (violet diamond) are given along with the value for the attenuated sun simulator spectrum (orange diamond).e stepwise changed LED illumination (measurements number 1 to 21) leads to, respectively, increased top and decreased bottom cell current densities (black squares).

Figure 4
Figure 4 depicts the variations of the short-circuit current densities  top SC and  botSC resulting from the      measurements using linear spectral variations of the applied illumination.For clarity, the short-circuit current densities calculated for the standard AM1.5g spectrum as well as for the attenuated and not attenuated sun simulator as calculated with the help of (2) are also shown.It can be seen from Figure4that by variation of the LED light bias we are able to approximate very well the short-circuit current densities de�ned by the AM1.5g spectrum as well as those de�ned by the unattenuated sun simulator.It is worth noting that the present method enables one to analyze the cell close to standard conditions even if the sun simulator is not as well matched to the AM1.5g spectrum as in the present case.Figure5shows the current density difference Δ SC

2 )∆J SC = 0 F 5 :
e current density difference Δ SC   top SC −  bot SC rises linearly as a function of the index  of the measurement as postulated by (3).e gray dashed line is a linear �t.e AM1.5g-illumination conditions are most closely met for measurement number   0, marked in red.e subcell currents are equal, Δ SC  0 mA/cm  , approximately for measurement number   .

F 10 :
With an additional measurement of the external quantum efficiency of the test cell and acquisition of the experimental test spectra, the short-circuit current of the subcells of the test tandem sample under test illumination (dut, exp) can be calculated.e relationship of the differences Δ dut, exp SC to the differences Δ ref, cal SC of the short-circuit currents for the reference under calibration illumination is linear.

Figure 10
Figure 10 proves the linear relationship between the current density differences of the device under test under the experimental test illumination Δ dut, exp SC and of the calibrated reference tandem solar cell under calibration illumination Δ ref, cal SC

2 )
FF dut, cal FF dut, exp P dut, cal , P dut, exp J (a)).e second step comprises number    of  measurements (   2     ) under systematically varied light bias by the LEDs.Here the power provided by each LED is varied by equidistant steps Δ  LED ( and Δ 2 LED (, for LED 1 and 2, respectively.At each step the output of LED 1 is increased whereas that of LED 2 is decreased such that we have for the total light intensity arriving at the sample  (   (   Δ LED (  (     Δ 2 LED (   sunsim ( . using the measured EQEs and the recorded spectra (  we calculate the short-circuit current densities  is the velocity of light,  is the wavelength, and EQE top/bot ( are the external quantum efficiencies as given in Figure3(a).For our measurement setup the uncertainty of for each measurement .In (2),  is the elementary charge, ℎ denotes Planck's constant, SC  (       0 . F 7: e converted power was determined in  measurements under the sun simulator illumination applying additional LED illumination.e reference tandem solar cell was measured under an attenuated sun simulator spectrum (�lled blue symbols) and under a noncalibrated illumination (open blue symbols).e device under test was characterized under the same test spectra (open cyan symbols).e measurement number  labels the different steps in additional LED bias illumination.We show 11 steps for the experiment, 21 in the calibration.e red star marks the illumination conditions closest to AM1.5g.
F 8: e normalized output parameters, maximum output power , �ll factor FF, and short-circuit current density  SC are given as determined for the reference tandem under calibration illumination (ref, cal, same data as Figure6, �lled symbols).esamedata acquired under the experimental test illumination was shied to match the data under calibration conditions(ref, exp, open  symbols).e data overlap.e  axis is the one from the calibration measurement.