Simulating Nanoscale Optics in Photovoltaics with the S-Matrix Method. Dalton Chaffee, Xufeng Wang, Peter Bermel

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1 ABSTRACT Simulating Nanoscale Optics in Photovoltaics with the S-Matrix Method Dalton Chaffee, Xufeng Wang, Peter Bermel In the push to build high-efficiency solar cells with less materials, thin-film solar cells have attracted an increasing amount of interest. Thin films are particularly attractive if they could exhibit light trapping and photon recycling capabilities exceeding those of traditional wafer-based cells. Recent work by Alta Devices demonstrating a record single-junction efficiency of 28.8% with a gallium arsenide thin film cell shows the potential. However, most existing simulation tools do not handle these properties well particularly photon recycling. In this work, we develop an improved solar cell simulation tool to accurately predict thin-film performance. It is based on a fast layered wave-optical module coupled to a drift-diffusion electronic model. The S-matrix method was used to solve for light absorption at any point in a solar cell given the depths and refractive indices of each layer; these results were then used to calculate initial and recycled photon generation profiles, and coupled self-consistently to an existing solar cell simulator, ADEPT 2.0, available on nanohub.org an open-access science gateway for cloud-based simulation tools and resources in nanoscale science and technology. In general, this improved simulation technique produced more accurate carrier than predicted by standard models. Preliminary results are presented that indicate this approach is also capable of accurately modeling the effects of anti-reflection coatings (ARCs) and various back reflectors. This new capability will be made available through a revised version of ADEPT 2.0. Keywords: Photovoltaics, thin film, light trapping, antireflection coatings, transfer matrix method, S-matrix method INTRODUCTION While solar cell efficiencies have been increasing, they are still not high enough to make solar energy cost-effective without subsidies in all energy markets. In the search for cheaper and higher-efficiency solar cell technologies, a new type of solar cell has recently emerged: the thin-film solar cell, which is built with nanoscale feature sizes, and has a total cell thickness below 50 µm, typically about 1-40 µm. The advantage of these thin-film cells lies in the reduced volume in which dark current can recombine, leading to a higher open-circuit voltage [1], as well as the potential for reduced costs, due to much lower material usage. This principle has been demonstrated by Alta Devices thin-film GaAs cell with an efficiency of 28.8% under the standard AM 1.5 spectrum and 1 sun illumination [2]. This efficiency currently stands as the world record under such conditions for a single junction solar cell of any thickness [2]. Even so, there remains room for improvement in such devices, as they remain significantly below the theoretical Shockley-Quiesser limit of 33.5% [3]. Over the last few decades, theoretical modeling has been used to identify areas of improvement for current designs (e.g., Ref. 1 correctly predicted the need for the development of thin-film technology). Many existing theoretical simulations today focus on the electrical properties that dominate the performance of traditional, wafer-based solar cells. For instance, ADEPT 2.0, a tool published on nanohub.org and available for public use, 1

2 solves Poisson s equation coupled with the hole and electron continuity equations in one spatial dimension in compositionally nonuniform semiconductors [4]. The effects of optical parameters such as front-surface reflection, depth of absorption, and photon recycling, though, are not precisely calculated: front-surface reflection is estimated as a flat, wavelength-independent value, absorption is estimated using Beer s Law, and the effects of photon recycling are ignored. While these approximations may hold for thick, wafer-based solar cells, evidence from the literature suggests that in order to fully capture the behavior of thin-film cells in particular, and thus improve the performance of these promising devices, the effects of the above optical properties must be considered. While other simulations have been developed for treating the optical properties separately, unifying their treatment in a single tool would be most desirable, in terms of both accuracy and ease of use. Solar cell light trapping keeps unabsorbed photons inside for as long as possible before they escape out the back of the cell as transmission or through the front-surface escape cone; this allows for more photons to be absorbed, and thus higher efficiencies to be reached. It is well-known from early theoretical work on PV cells that ideal ray-optics design of the front surface can scatter light into all possible angles inside the active layer, yielding an enhancement of effective path-length up to 4n 2 (where n is the depth of the device) [5]. However, it has been shown that the efficiency of wave optics-based approaches can in principle exceed that value, using resonances to strongly enhance absorption in critical wavelength ranges [6]. Thin-film cells rarely see such high light trapping, since strongly light-scattering structures must usually be much thicker, yet the importance of this effect is inherently greater in such devices, due to the risk of incomplete light-trapping in thinner layers. Design aspects affecting light trapping include the anti-reflection coating, the material choice for the active region, the thickness of the cell, and the backside mirror reflectivity. The calculation of electric field, and ultimately photon absorption, at any depth in a solar cell has been accomplished previously using the finite difference time domain method [7] as well as the transfer matrix method (TMM) [8, 9]. The latter is a general mathematical model used to solve problems of waves in layered media (such as light waves in a solar cell), and can have important advantages in speed for certain structures found in the PV literature. This method, however, can become numerically unstable in the case of absorptive media. Thus, a more stable approach needs to be adapted into solar cell simulations in order to model the light trapping physics crucial to thin-film cells. Photon recycling describes the effect of spontaneously emitted photons that is, those that are emitted within the cell due to radiative recombination being reabsorbed and used to generate current, rather than exiting the cell as wasted energy [10]. This effect does not occur in wafer-based cells is because the active layer in such cells is generally thinner than the absorption depth of spontaneously emitted photons, meaning that most of these photons are lost in the substrate layer [1]. In fact, the photon recycling factor, or the increase in radiative recombination lifetime due to self-absorption [10], is generally immaterial for many solar cells, due to high non-radiative recombination. However, thin films made from high-quality materials without an absorbing substrate have photon lifetimes potentially much greater than in traditional wafer-based cells for a photon recycling factor on the order of 10, as suggested by Refs. 10 and 11. A higher photon recycling factor leads to a lower saturation current density, and an increase from 1 to 10 can lead to an overall solar cell efficiency increase on the order of 2% [1]. However, simulating the exact value of the photon recycling factor given the 2

3 parameters of a solar cell, rather than assigning an estimated value, has proved difficult. Recently, the value of the photon recycling factor has been reconciled with the Roosbroeck-Shockley equation in a thermodynamically consistent way [12]. This has led to design studies examining the effects of solar cell parameters on photon recycling and overall efficiency, including one incorporating the electronic simulation framework of the previously mentioned ADEPT 2.0 model [12, 13]. Additional research is needed to fully understand the impact of photon recycling on thin-film cells and how this effect can be fully exploited. In this work, we develop a TMM model that uses the S-matrix method to simulate light trapping in a solar cell in a more numerically stable, or less computationally expensive, fashion than the original T-matrix method. This model is then incorporated into the existing electronic framework of ADEPT 2.0 to more accurately model the effects of reflection, absorption, and transmission in any given layered cell. This new capability allows for optimization of parameters such as the thickness and complex refractive index of the antireflection coating, as well as other layers in the solar cell. The simulation also provides a basis for the additional incorporation of photon recycling into the model. Importantly, this improved tool will be made available on nanohub.org, making it globally accessible [14]; to the best of the authors knowledge, there is currently no open-access simulation tool that addresses both the optical and electronic model of a solar cell, and thus many leading experimental research groups go without the benefits of such a model. Ultimately, the capabilities of this improved model will be available to everyone, leading to further optimization and increased efficiencies in thin-film solar cells. METHODOLOGY ADEPT 2.0 currently takes as input a wide variety of parameters for any specific solar cell that the user wants to simulate, such as layer thickness, band gap, dielectric constant, doping strength, absorption spectra, incident sunlight spectrum, and others; these parameters can be inputted either as a flexible control file or in a user-friendly graphical interface, illustrated infigure 1. The model uses the iterative Newton Method to rigorously solve for the behavior of electrical carriers within the solar cell [4]. To take the first step and solve for the initial generation of carriers, ADEPT uses Beer s Law in the general form: ( )= ( ) ( ), (1) where ( ) is the final intensity at wavelength, ( ) is the initial intensity, α( ) is the wavelength-dependent absorption coefficient, and k is the thickness of the layer. This equation holds true for single-layer devices, but does not completely describe the attenuation of light that occurs in multilayer structures. For instance, in the current ADEPT 2.0 input deck, the user must input a single value for front-surface reflection, when in fact this value varies by wavelength in accordance to the Fresnel reflection coefficient [15]. Other discrepancies from the Beer s Law approximation occur when multiple interfaces appear within the solar cell. In order to more precisely calculate the absorption of light in such a device, we implemented the transfer matrix method and integrated this with the existing capabilities of ADEPT 2.0. The transfer matrix method (TMM) is a general mathematical model describing the reflection, transmission, and absorption of waves in layered media. The only inputs needed 3

4 are the amplitude and wavelength of the incoming waves (in this case, the incident sunlight spectrum), and the layer thicknesses and complex refractive indices; in other words, all parameters that were previously already entered into ADEPT 2.0, except for the refractive indices, which are well-known for most photovoltaic materials [16]. In the case of solar cell simulations, the TMM calculates the intensity of light at any point in the device, which can be then used to determine the absorption of photons at each of these points. This is the same parameter calculated using Beer s Law in the current ADEPT 2.0 model. Figure 1: The current graphical user interface of ADEPT 2.0 When implementing the TMM, the most straightforward, intuitive approach is the T-matrix method, which has been used for solar cell applications previously [9]. One challenge arises with this method, however, and that is that matrix elements may grow exponentially large, making the calculation numerically unstable, or at least computationally expensive, for absorptive materials [17]. Thus, another method is desired to ensure numerical stability; the solution to this is found in a reformulation of the T-matrix method known as the S-matrix method [17]. The model described here was built from scratch; the T-matrix method was developed first, then the S-matrix method was developed using the the T-matrix method in the numerically stable regime, and cross-checking new S-matrix results. Here, we will describe the formulation of both methods and the reasons for the S-matrix method s superior stability. T-matrix and S-matrix formulation 4

5 The T-matrix method uses a layer-by-layer approach to the wave optics problem, where a distinct matrix is constructed for every layer in the cell, which will henceforth be referred to as the layer matrix. Each layer matrix is then subdivided once more into a matrix describing the interface between one layer and the next and a matrix describing the flow of light within the material in question; these will furthermore be referred to as the interface and propagation matrices, respectively. The device is bounded on either side by a semi-infinite region, which in most relevant applications is air for the front side. The interface matrix is described by the Fresnel reflection and transmission coefficients as follows [18]: =[ 1 1 ] /, (2) where is the interface matrix and and are the Fresnel reflection and transmission coefficients of the interface, respectively. Next, the propagation matrix is constructed as follows [17]: =[ ( ) 0 0 ( ) ], (3) where is the propagation matrix, ( ) is the wavelength-dependent impedance of the layer, and k is the depth of the layer. The interface and propagation matrices are simply multiplied to obtain the layer matrix [17]: =. (4) This process is repeated for each layer until all layer matrices are found, and then they are simply multiplied together to calculate the matrix describing the entire structure, known as the stack matrix [17]: =, (5) where the structure is described as shown in Figure 2: Figure 2: The layered cell (adapted from Ref. 17) where u (0) is the incident light and d (n+1) is set to zero, making d (0) the light reflected and u (n+1) the light transmitted. The reflection and transmission can then be calculated by substitution into the characteristic equation for the stack matrix [17]: 5

6 =. (6) Total absorption can be calculated from =1. To find absorption at any particular depth in the device, the field amplitude must be calculated as follows [9]: (1,1) ( ) ( ) + (2,1) ( ) ( ) = (1,1) (1,1) ( ) + (1,2) (2,1) ( ), (7) where is the stack matrix considering each layer behind the layer in question, is the stack matrix considering each layer in front of the layer in question, and x is the depth at which the amplitude is to be calculated, relative to the layer in which it lies. The S-matrix approach shares similarities with the T-matrix approach, but there are several key differences as well. To begin, if the T-matrix interface is represented as =, then the S-interface matrix can be calculated as: / / = [ / 1/ ] (8) Next, the S-layer matrix is calculated comparably to the T-layer matrix in (3) and (4): =[ ( ) ] * * [ ( ) ]. (9) The S-stack matrix, in contrast to the T-stack matrix, is calculated iteratively, layer by layer, with each subsequent stack matrix being calculated using the previous stack matrix ( ) and the current layer matrix, as follows [17]: (10) where = ( ) ( ) ( ) ( ), = and = ( ) ( ) ( ) ( ). ( ) ( ) ( ) ( ), = , This S-stack matrix can then be tied back to Figure 2 via the following formula [17]: =. (11) 6

7 Like (7), the field amplitude at any depth in the device using the S formulation is: = (, ) ( ( ) ( ) (, ) ( ) ( ) ) (, ) (, ) ( ). (12) Numerical Stability The T-matrix method becomes unstable for highly absorptive media, because certain terms in the propagation matrix become exponentially large. This is due to the impedance, β, as used in (3), which is proportional to the imaginary part of the refractive index and thus absorption. For example, when modeling a silicon wafer, the T-matrix method could not always produce a finite result for shorter wavelengths where the absorption of silicon is high. In contrast, the S-matrix recursive formulation, Eq. (10), inverts these exponentially large terms, leading to a less computationally expensive calculation and a more reliable results. During the exploration of this work, no case in which the S-matrix method became numerically unstable was discovered. Integration with ADEPT 2.0 Equation (12) yields the field amplitudes at any depth, and can be used to calculate the corresponding absorption to feed back into ADEPT 2.0 s electrical calculations. The absorption rate for every photon in the spectrum may be found as follows: = ( ) ( ), [ ( )] (13) where ( ) is the real part of the refractive index of the layer in question and Re[ ( )] is the real part of the refractive index of the front semi-infinite region (1 for air). To find the total absorption at any depth in the device, the absorption rate of every photon was integrated across all respective wavelengths using Riemann sums. This yields the number of photons absorbed per cubic centimeter per second, the units used by ADEPT 2.0 in its electrical calculations. The code was modified to read in a standard ADEPT 2.0 control file with the additional option of having an optical layer (such as an anti-reflection coating) that influences the optical but not electrical calculations. Using this control file, the appropriate wavelength-dependent refractive indices, and the light spectrum, this model calculates the photons absorbed per cubic centimeter per second at any depth in the device; these results can then be used by a slightly modified version of ADEPT 2.0 to calculate useful performance information about the device. RESULTS AND DISCUSSION Two key examples drawn from the literature, wafer-based and thin-film crystalline silicon solar cells were studied using both the S-matrix method optical generation calculation and ADEPT 2.0 s original Beer s Law calculation. 7

8 The first example is a single-layer, 200 µm-thick crystalline silicon wafer. The structure is also given a front semi-infinite region of silicon rather than air to prevent reflection at the front interface. With no reflection and little transmission, the two models were expected to perform very similarly, and indeed this can be seen to be the case in Figure 3. Optical Generation in a 200 Micron Si Wafer with Reflection = 0 Optical Generation (photons absorbed/cm 3 /s) Max Current: TMM: 37.6 ma/cm 2 ADEPT: 37.2 ma/cm 2 Depth (microns) Figure 3: For the one-layer case, the old and new results are similar In this calculation, the Max Current values are simply the area under the curve of the photons absorbed at each point up to the bandgap wavelength (estimated using the ASTM AM1.5G solar spectrum, and integrated using Riemann sums). It was found that the traditional Beer s Law approach in ADEPT yields a max current of 37.2 ma/cm 2, while introducing a TMM approach yielded a slightly higher max current of 37.6 ma/cm 2. Thus, the two methods match to within 1%. It should also be noted that not all of these photons will lead to an electron-hole pair being collected by the p-n junction, thus the actual would be something less than the Max Current value. Additionally, photons with wavelengths above the bandgap wavelength of 1107 nm are discounted from the above figure because they do not have enough energy to have a significant chance to excite silicon electrons from the valence to the conduction band. The next case considered was the case of a thin-film, 2 µm-thick c-si cell. The front semiinfinite region was allowed to be air rather than silicon, bringing reflection into consideration. The TMM calculation assumed a 100 nm silicon nitride anti-reflection coating (ARC); because the old ADEPT 2.0 input cannot accept optical layers or wavelength-dependent reflections, a fixed reflection value was assigned in lieu of the ARC (see Figure 4). With the difference in reflection induced by the ARC and the increased probability of transmission due to the thin active layer, the results between the two methods were expected to strongly differ; this proved to be the case, as seen in Figure 5. 8

9 Optical Generation in a 2 Micron Thin-Film Si Cell with Reflection > 0 Optical Generation (photons absorbed/cm 3 /s) Max Current: TMM: 11.4 ma/cm 2 ADEPT: 16.2 ma/cm 2 Depth (microns) Figure 4: TMM s improved reflection Figure 5: TMM s more precise optical calculation leads to a different result A reflection value of was assigned to the ADEPT 2.0 calculation, by taking the average reflection value of each photon of appropriate energy reaching the device (as calculated by the TMM code) in the sunlight spectrum being considered. In practice, an experimentalist may estimate this value by either measuring the total power reflection of a standardized solar source from the sample, or measuring its reflection spectrum with a spectrophotometer and taking a spectral average using literature data. This approach leads to substantially different Max Current values for the two techniques: while 16.2 ma/cm 2 was estimated before in ADEPT, a more accurate value is revealed by TMM to be 11.4 ma/cm 2 (a difference of 29.6%), as shown in Figure 5. The different treatments also lead to a different rate of change of optical generation over depth: the ADEPT 2.0 values decrease more quickly, as they overweight short wavelength photons which have small absorption depths. Because the ARC allows more photons of higher wavelengths ( nm) and thus higher absorption depths into the cell, more photons are found to be absorbed towards the back of the cell. This suggests that the junction depth needed for optimal current collection will be found to be deeper with the TMM method than estimated using ADEPT s previous optical model. CONCLUSIONS & FUTURE WORK The improved light trapping calculation of the transfer matrix method module describes optical generation in solar cells more accurately. Furthermore, the S-matrix method has proved stable given the computation power available on nanohub.org. Using these improvements, design parameters could be investigated that are specific to such calculations, 9

10 such as the design of ARCs and the effects of a backside mirror. Another step needs to be taken, however, before the optical properties of thin-film cells can be fully be modeled: performing a detailed calculation of photon recycling, the extent to which radiatively recombined photons can be reabsorbed. Previous work has been done to model photon recycling by tracking the location of radiative emission and collection inside a solar cell [12]. Without a full optical model, though, the precise effects of photon recycling upon a cell cannot be fully anticipated; fortunately, the transfer matrix method model provides the framework for such a calculation. The TMM module here currently is formulated to only calculate the effects of solar photons incident on the front of the device to incorporate photon recycling, one would simply need to reformulate the method to be capable of calculating absorption given the emission of photons within the device. This capability would open the door to advanced design studies that will precisely optimize parameters such as material choice, thickness, doping strength, and backside mirror reflectivity; optimizing nanostructured photovoltaics is a major topic of current research [13]. Currently, the light trapping TMM module has not been incorporated into the official version of ADEPT that is currently available on nanohub.org. Ultimately, these capabilities will be implemented into both the control file and graphical user interface options of the online tool for global accessibility and ease-of-use. 10

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