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The fast and computationally inexpensive Modified Transfer Matrix Method (MTM) is employed to simulate the optical response of kesterite Cu2ZnSnSe4 solar cells. This method can partially take into account the scattering effects due to roughness at the interfaces between the layers of the stack. We analyzed the optical behavior of the whole cell structure by varying the thickness of the TCO layer (iZnO + ITO) between 50 and 1200 nm and the buffer CdS layer between 0 and 100 nm. We propose optimal combinations of the TCO/CdS thicknesses that can locally maximize the device photocurrent. We provide experimental data that qualitatively confirm our theoretical predictions.
© 2016 Optical Society of America
1. Introduction
Kesterite-based solar cells attracted significant attention during the last years as a possible alternative to Cu(In,Ga)Se2 (CIGS) solar cells. High efficiency CIGS solar cells with η = 22.3% have been demonstrated [1] but their large-scale industrial production could be limited because of indium scarcity and growth in its price (high demand from the display industry) [2]. Copper–zinc–tin–sulfide/selenide absorbers (kesterite) could overcome this issue thanks to their cheap and earth-abundant composition. The record device using a pure selenide Cu2ZnSnSe4 absorber (CZTSe, band gap Eg ≈1 eV) showed an efficiency η = 11.6%, open circuit voltage VOC = 0.423V and short circuit current JSC = 40.6mA/cm2. The best kesterite solar cell, obtained with a Cu2ZnSn(S,Se)4 absorber (Eg = 1.13eV), has η = 12.6%, VOC = 0.513V and JSC = 35.2mA/cm2 [2,3].
The performances of kesterite solar cells today are still low to start rivaling the more mature CIGS technology. The low VOC remains the main issue of this technology, with values that can be around 150-200 mV lower than the ones of CIGS cells with equivalent Eg [2,4]. But also sub-optimal values of JSC can furtherly lower the device efficiency: Winkler et al. stated that 75% of the possible JSC-deficit in kesterite cells is due to optical losses related to the parasitic absorption of the buffer/TCO layers and the reflection of photons from the top surface [5]. The optimization of the thickness of the top semi-transparent layers is a possible way to minimize the optical losses due to photons reflection which lead to improvements of the device JSC.
In this work we consider a Molybdenum/MoSe2/CZTSe/CdS/iZnO/ITO structure and we simulate its optical response (reflectance and photocurrent) as a function of the thickness of the TCO (iZnO + ITO) and the buffer (CdS) layers. From the analysis of the simulated device photocurrent we can identify local maxima (optimal points) and minima in the space of the TCO/CdS thicknesses: this phenomenon is due to the constructive/destructive interference effects generated by the multiple reflections of photons inside the top transparent layers [6]. Interference fringes can be experimentally observed in the Reflectance and External Quantum Efficiency (EQE) spectra of CIGS and Kesterite solar cells.
Previous similar works [5,7] proposed ultra-thin thicknesses for the TCO/CdS layers (≤ 60nm) such that a large improvement of the photocurrent generation was mainly due to the minimization of the parasitic absorption of these layers. This strategy is an absolute optimum from the optical transparency point of view but it is not ideal in terms of higher lateral series resistance [8,9] and other possible technological issues (e.g. pinholes [10]) that can compromise the device efficiency. This partially explains why a wide range of TCO/CdS thicknesses has been proposed for Kesterite solar cells in literature: TCO from 60 to ~500 nm and CdS from 25 to ~100 nm at the research scale [11]. At the module scale even thicker thicknesses for the TCO layer could be required: even though kesterite industrial modules have yet to be reported, the standard TCO thickness for CIGS industrial modules is about 1 μm [12].
We investigated all the range of cases by performing simulations of the whole kesterite solar cells varying the thickness of the TCO from 50 to 1200 nm and the CdS from 0 to 100 nm. We employed 1D numerical optical models to calculate the percentage of photons lost by reflection and the percentage of photons absorbed in the CZTSe, as a function of the TCO/CdS layers thicknesses. The software SCAPS [13] was also employed to compute the EQE of the simulated devices. The theoretical results are compared to experimental measurements of devices having the same structure of our model and variable thicknesses for the TCO layer between 400 and 110 nm and fixed thickness for the CdS layer equal to ~50 nm.
2. Methodology
The optical constants of the simulated stack (see Fig. 1) were selected from literature [14–18] with the exception of the CdS layer, whose optical constants (n-k coefficients) were measured by Spectroscopic Ellipsometry (SE). For the CZTSe layer we chose optical constants computed by ab-initio simulation (by Persson [16]) because the available experimental works from literature reporting n-k coefficients of kesterites are often inaccurate in the wavelengths range near the fundamental absorption edge (λ~1250nm for CZTSe) because of issues related to the high surface roughness of the polycrystalline kesterite absorbers [5]. Anyhow, with the optical constants from Persson [16], the simulated EQE underestimates the absorption in the wavelengths between 1100 and 1300 nm compared to the experimental one (see Fig. 2): this is expected to have some limited quantitative impacts on our presented results. The longer absorption edge visible in the experimental EQE measurements has been explained to be related to several possible non-idealities of the kesterite absorbers, like tail states and inhomogeneities [20].
Fig. 1 (a) Refractive indices n and (b) extinction coefficients k (reported in logarithmic scale) of the materials considered in the optical simulations: ITO [14], iZnO [15], CdS (SE), CZTSe [16], MoSe2 [17], Mo [18].
The electrical parameters for our model were discussed in detail in previous works [11]. In Table 1 we have summarized the most relevant among them. The Molybdenum (Mo) back contact was modeled with a work function Wf = 4.95eV [23]. Specific defect states and tunneling at the interfaces between the layers were not implemented.
Table 1. Electrical properties of simulated materials
The software SCAPS, which was employed for the computation of EQE, natively performs its optical simulations considering a user-defined constant reflection (between 0 and 100%) for the first and the last interfaces of the stack and it applies the Beer-Lambert law for the estimation of photons absorption inside the layers of the stack. Alternatively the user can input files representing the spectral reflectance for the first/last interfaces, obtained by other optical simulation methods or also by experimental measurements.
We carried out 1D optical simulations using two different methods: the standard Transfer Matrix Method (TMM) and the “Modified Transfer Matrix” (MTM) method that was previously employed to model CuInSe2 and CuGaSe2 solar cells by Yin et al. [24]. The TMM is an exact analytical method to compute the optical response of multilayer stacks with perfectly flat interfaces [6]: it only requires the optical constants n and k and the thickness of each layer. The incoming electromagnetic radiation is modeled as plane waves in normal incidence. As we anticipated above, kesterite absorbers present significant surface roughness that can be up to a hundred of nanometers [5]. Also the CdS and TCO layers introduce roughness features at smaller length scales. These interface roughness components can lead to scattering within the films and results in substantial intensity reduction of reflectance (R) especially at short wavelengths [24]. For TMM calculations we employed a code written in Python by S. Byrnes [25]. The MTM method is identical to the TMM method except for the substitution of the Fresnel coefficients ra,b and ta,b (complex reflection and transmission coefficients) at the interface from the layer a to the layer b, with the following quantities [24]:
where na and nb are the refractive indexes (real part) of the layers a and b respectively, and σ is the surface roughness (unit nm) in terms of root mean square (RMS) between the layers a and b. The MTM can partially consider the effects of interfaces roughness, when σ/λ<<1 [24]. Both the TMM and MTM are relatively fast and computationally inexpensive compared to other more complex and time consuming optical methods (e.g. the FDTD, Finite-difference time-domain, methods) [26].Our baseline model was based on a device with a CZTSe absorber (Eg ≈1.02 eV) fabricated by reactive thermal annealing of metallic precursor stacks deposited by DC sputtering (Alliance Concepts AC450), and described more in detail in Neuschitzer et al. [19] and Giraldo et al. [27]. CdS was deposited using chemical bath deposition, while the TCO was deposited using DC-pulsed sputtering (Alliance Concepts CT100). In order to provide an experimental validation of our theoretical results, four more solar cells samples with variable ITO thicknesses (denoted as ITO1-4) were fabricated. Several individual solar cells with an area of 3 × 3mm2 were obtained on each of the ITO1-4 samples, using a manual microdiamond scriber with a scribed line width of 20μm. When dealing with these four samples in the following paragraphs, we will consider only the best-efficiency cell present on each of them.
Both the baseline and the ITO1-4 samples have a 1800 nm thick CZTSe layer and about 380 nm of MoSe2, estimated from Scanning Electron Microscopy (SEM) cross section images. The nominal thicknesses of the ITO, iZnO and CdS layers of our cell are equal to 350, 50 and 50 nm respectively [19], whereas the ITO1-4 samples differ from the baseline for the ITO thickness (as a result of shorter deposition time). These “nominal” thicknesses are values obtained from the deposition systems calibrations over flat reference samples, like glass or silicon [19]: when deposited to different substrates and rough surfaces, as it is the case of kesterite films, these thicknesses are expected to slightly vary. But after growth over the kesterite, it is difficult to experimentally determine their actual thickness by analysis of SEM cross sections because of the rugosity of the underlying CZTSe layer and a non-perfect thickness homogeneity over the whole sample.
For these reasons we refined the estimation of the thickness of the ITO, iZnO and CdS layers using a DOE (design of experiments) fit method based on TMM and MTM simulations, looking for the best matching of the simulated/measured spectral reflectance in terms of position of the local maxima and minima points. The considered parameter space for the fitting procedure was the thickness of the three top layers (ITO/iZnO/CdS), because the position of the interference fringes in the spectral reflectance is closely correlated to the thickness of these layers as well as their optical constants. The thickness of the considered layers was varied over a range of ± 50 nm with respect to the known nominal values. The best fitting combinations obtained with this procedure are summarized in Table 2 and no other similar results could be obtained with considerably different combinations inside the considered ranges.
Table 2. Thickness (in nm) of the top layers estimated by TMM and MTM
In Table 3 we summarize the values of interfaces roughnesses (σ) employed for the MTM method: these values were estimated with a similar procedure as the one explained above. The estimated σ for the Air/ITO/iZnO/CdS interfaces are well inside the range of standard values that are typically reported in literature for these layers (σ from 5 to 25 nm) [28–30]. Surface roughness σ for kesterite films can typically be in the order of magnitude from tens to a hundred of nm [3,5], which are values close to the limit of validity of the MTM method (σ/λ<<1) for the wavelength range considered in this work (λ ∈ [280,1300] nm). We estimated a value of σ = 35 nm for the CdS/CZTSe interface that, for this technology, represents a relatively moderate amount of roughness.
Table 3. Surface roughness (σ) values, in terms of RMS for the MTM method
EQE measurements were made using a Bentham PVE300 system. Total reflectance measurements R were made with a Perkin Elmer R LAMBDA 950 UV/Vis Spectrophotometer (integrating sphere). The JSC of all the solar cells present on the ITO1-4 samples was measured using a Sun 3000 class AAA solar simulator (Abet Technologies Inc.).
3. Results
In Fig. 2, spectra of simulated reflectance (R) and EQE obtained with four different approaches are compared to the measured EQE/R of the baseline experimental cell [19]. In Fig. 2(a) we can observe the results obtained with the TMM and MTM methods. In Fig. 2(b) we show the EQE simulations obtained with a constant reflection value (R = 10%) and a so called Hybrid model, such that the experimental R of the baseline cell (red solid line) is input in our model in order to compute the Hybrid EQE (blue dots).
The simulation with a constant reflectance (Fig. 2(b)) represents the basic result that can be obtained with SCAPS only, which obviously cannot reproduce the interference fringes. The TMM model, visible in Fig. 2(a), is able to simulate the interference fringes in the Reflectance spectrum but it strongly overestimates their amplitudes with respect to the experimental one. Finally the MTM method produces better results than the TMM, in terms of reduction of the amplitude of the interference oscillations of the simulated reflectance: as anticipated in the previous section, this is due to the implementation of the scattering effects due to the interfaces roughness.
The Hybrid model (in Fig. 2(b)) ideally should produce an EQE identical to the experimental one. The differences between the experimental EQE (black line) and Hybrid model (blue circles), visible in Fig. 2(b) for some wavelengths ranges, can be explained as follow:
- • the differences in the [1100-1300] nm range are due to the optical constants of the CZTSe layer, as we have already explained in the previous section;
- • the differences in the [750-1000] nm range can be due to the optical constants of the ITO layer (underestimation of the long-wavelengths absorption) as well as to defect interfacial states at the back contact (not implemented);
- • the differences in the [400-500] nm range can be due to the optical constants of the CdS layer (underestimation of absorption) as well as to defect interfacial states between the CdS-CZTSe layers (not implemented).
Even though the Hybrid model can reproduce the experimental EQE better than all the other considered models, it cannot be employed to perform predictions or investigations of device optimizations because it requires experimental reflectance spectra and it does not provide information of the layers internal absorption required to compute the device photocurrent. From the comparison of the TMM and the MTM models we show that the MTM produces a closer agreement between the simulated and experimental EQE/R spectra (Fig. 2(a)), thus it has been selected as the preferential method to perform the optimization study that will be described in the followings.
In Fig. 3 we report the results of multiple optical simulations performed using the MTM method.
Fig. 3 Results of MTM optical simulations varying the thicknesses of the TCO (iZnO + ITO) and CdS layers: (a) percentage of reflected photons R% considering the wavelengths range [400-1100] nm and (b) photocurrent Jph generated in the CZTSe absorber. The iZnO thickness is constant and equal to 50nm.
Figure 3(a) represents the reflected photons percentage (R%) and Fig. 3(b) represents the photocurrent generated in the CZTSe layer (Jph). Both graphs share the same X and Y-axis: on the X-axis we varied the thickness of the TCO layer from 50 to 1200 nm and on the Y-axis we varied the thickness of the CdS layer from 0 to 100 nm. Actually, the TCO layer is composed by a thin intrinsic Zinc Oxide (iZnO) layer of constant thickness equal to 50 nm and an ITO layer with variable thicknesses between 0 and 1150 nm. The photocurrent Jph has been computed as follow:
where q is the electron charge, Nph-AM1.5(λ) is the photon flux in the AM1.5 solar spectrum and ACZTSe(λ) is the spectral absorption of photons inside the CZTSe layer computed by a MTM simulation. Jph represents an upper bound of the maximum achievable JSC of the device [5] thus the qualitative trends observed in Fig. 3(b) related to Jph are expected to be valid also for JSC. The wavelength integration range between 280 and 1300 nm takes into account the onset of the AM1.5 spectrum (280nm) and the limit of absorption of the CZTSe absorber in our structure (below 1300nm, as seen in Fig. 1(b)). The Reflected Photons percentage R% has been computed as follow:where R(λ) is the spectral reflectance at the top interface of the device, computed by MTM. The integration interval for R% is between [400-1100] nm of wavelength because, for the considered materials, the minimization of reflectance plays an important role mainly in this range: for λ<400nm the absorption of the CdS/TCO layers is dominant and for λ>1100nm the absorption of the CZTSe layer is weak (see Figs. 1(b) and 2(b)).The dots plotted in Figs. 3(a) and 3(b) represent the simulated values of R%/Jph for the thicknesses of CdS/TCO corresponding to the experimental samples (see also Table 2, MTM columns). In Fig. 3(b) we can see that the samples ITO1 and ITO3 are inside two different non-optimal regions. The sample ITO2 is inside an optimal region and the sample ITO4 seems to be at the border between a non-optimal and an optimal region.
Comparing the Fig. 3(b) with the Fig. 3(a) we can clearly see an inverse dependency between the local maxima of Jph with the local minima of R%. This can be also observed in Fig. 4(a) where we plotted the simulated Jph and R% for a constant thickness of the CdS layer equal to 50 nm (black and red solid lines respectively). Considering this specific case, five local maxima are present in the Jph curve for TCO thicknesses equal to 1100, 845, 570, 280 nm and for a thickness lower than 100 nm. The grey line in Fig. 4(a), representing the linear regression of the simulated Jph curve, highlights the lowering of Jph itself that is directly due to the increasing parasitic absorption of the ITO layer with its increasing thickness.
Fig. 4 (a) Simulated Jph and R% (solid line) and experimental JSC and R% (dashed lines). The simulations consider a varying ITO thickness and constant thickness for the CdS and iZnO layers, both equal to 50nm. Figure (b) is a zoom-in of figure (a) in the [50-450] nm range, showing only Jph and JSC: the empty circles represent the MTM simulations considering the exact values of CdS-iZnO-ITO thickness corresponding to the experimental samples (Table 2) while the black solid line is the same as the one plotted in figure (a).
In Fig. 4(b) our theoretical results in terms of Jph (solid line) are compared to the experimental JSC measurements related to the solar cells of the samples ITO1-4 (blue crosses). Qualitatively the average values of the experimental JSC (dashed line) seem to confirm the expected trends of simulated Jph obtained by MTM. In Fig. 4(a) we can see that also the simulated trends in terms of R% (red solid line) are confirmed by the experimental R% (red dashed line) computed applying the Eq. (4) on the measured spectral reflectances of the samples ITO1-4 visible in Fig. 5 (red lines). The simulations show some bigger discrepancies for the case of sample ITO4, having the thinner value of ITO thickness equal to 70 nm (iZnO + ITO = 110 nm). The experimental ITO4 exhibits the best average JSC whereas in our model the best Jph is obtained in the case of ITO2 (290 nm thick TCO). From the analysis of Figs. 4(a) and 5 we can see that the MTM simulations exceed the estimation of the reflectance for the thinner thicknesses of the TCO, motivating the underestimation of Jph for the simulated ITO4. Besides this specific case, a lower mismatch between the simulated/measured EQE/R spectra are observed for all the other samples (see Fig. 5).
Fig. 5 Measured and simulated EQE/R spectra of the four samples ITO1-4 (in this order from (a) to (d)). The black/red solid lines represent the experimental EQE/R. The green/brown dashed lines represent the simulated EQE/R obtained by MTM.
4. Conclusions
In this work we employed the MTM numerical method to investigate the optical behavior of a CZTSe solar cell over the variation of the thickness of its TCO and CdS layers. Considering a TCO thickness from 50 to 1200 nm and a CdS thickness from 0 to 100 nm we individuated the combinations of the TCO/CdS thicknesses that can provide improvements on the device photocurrent due to local minimizations of the Reflected Photons percentage (see Figs. 3(a)-3(b)). Comparisons with experimental data of devices with variable TCO thicknesses (between 400 and 110 nm) and constant CdS thickness (~50 nm) provided evidences of the consistency of our calculations in terms of expected trends, even though we could not obtain an exact quantitative estimation of the improvements because of some limitations of the model itself. In general the choice of a thinner TCO thickness is preferable to obtain record cells at the research scale (parasitic absorption minimization) whereas a thicker TCO is required for scaling up the devices to the sub-module stage (sheet resistance minimization and better reliability). From a methodology point of view we discussed the reasons for the selection of the MTM over the pure TMM: the MTM improves the matching between the simulated and experimental EQE/R spectra thanks to its capability of partially consider the effects of scattering due to the interfaces roughness in the CZTSe solar cell. Although other more complex methods (e.g. FDTD) could be employed to furtherly improve the accuracy of the optical simulations, they would not guarantee the same performances of the MTM method, in terms of high speed and low computational cost, that were required to perform the proposed optimizations.
Acknowledgments
This research was supported by the Framework 7 program under the project KESTCELLS (FP7-PEOPLE-2012-ITN316488). S.G. thanks the Government of Spain for the FPI fellowship (BES-2014-068533), and E.S. for the “Ramón y Cajal” fellowship (RYC-2011-09212).
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