Optimization of Light-Trapping in Thin-Film Solar Cells Enhanced with Plasmonic Nanoparticles Wael Itani

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Wael Itani. Optimization of Light-Trapping in Thin-Film Solar Cells Enhanced with Plasmonic Nanoparticles. 2021. ￿hal-03129380v3￿

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Wael Itani Department of Electrical & Computer Engineering Technical University of Munich Munich, Germany [email protected] electronics-grade, pure, crystalline silicon [8], is Abstract—In this paper, we motivate the need for used to ensure long99 lifetime.999% of carriers for their collection. photovoltaic technology and its improvement before we overview the contributions of Akimov and Koh’s 2010 paper to Solar technology could be broadly classified by material plasmonic thin film cells. The paper results are complemented of active region [9]. Traditional, crystalline Silicon, solar cells, by the necessary theoretical description and formulation. We which current dominate the market could only be made so then conclude by recalling the limitations of the technology thin. Although the diminishing size improves the open circuit under review, and providing an outlook. The empirical nature voltage, it decreases the absorbance and corresponding of Akimov and Koh’s paper has led to focus on providing a photocurrent [9]. This puts a hard cap on their development, thorough theoretical background for readers, with a rich as nearly half of their production costs come from the choice appendix for comprehensiveness. of material [9]. Dye sensitized cells on the other hand, while they offer a cheap alternative, suffer from poor electrical Keywords—plasmonics, nanoparticles, thin film, solar cells, properties, and, thus, overall poor effectiveness. photovoltaics II. SUMMARY I. INTRODUCTION Akimov and Koh [1] study forward scattering spherical In the past decade, Akimov and Koh presented a seminal metallic nanoparticles embedded in the top transparent series of papers which have since taken plasmonic thin-film conductive layer ( ), in a square lattice arrangement, for solar cells mainstream [1]–[6], with thin films capturing 20푛푚 20% an amorphous silicon cell ( ) with the aim of achieving of the photovoltaic market share less than five years later [7]. maximal broadband light240-trapping푛푚 for optimized overall In this paper, we revisit their work which investigated the power absorption, and indirectly photoelectric carrier optimal material for the plasmonic nanoparticles, and put it in generation. They optimize the material, size (radius) and perspective with recent developments [1]. After motivating surface coverage for scattering in the dipolar and multipolar the need for plasmonic-enhanced thin-film solar cells, we regime in comparison with an ideal, non-dissipative non- review their underlying theory, its applications, and the dispersive (real negative frequency-independent permittivity), challenges that remain before concluding with an outlook. metal of and a control cell. The problem is modelled in A. The Landscape of the Energy Sector COMSOL with 3D Maxwell equations for a normal incident solar profile plane wave with periodic and perfectly As the global energy consumption is set to nearly double matched퐼퐴푀1.5 boundary conditions, and refractive indices in 2050 from its current figure of 17 TW in 2020 [8], and the interpolated from SOPRA. Results show that a perfect bells continues to toll warning us of climate change happening conductor, with infinitely negative permittivity is needed to as you read this paper, it is time to tap our alternative energy improve scattering, and to avoid parasitic absorption by sources. Solar energy an abundant clean source able to supply particles due to surface plasmon resonance. This corresponds ten thousand times our current needs with 176,000 TW to an infinitely high surface plasmon resonance frequency. radiation striking earth [8], [9]. That is covering the Venzuela Two maximas are obtained for the surface coverage and alone with 10% efficient panels would meet the globe’s radius, corresponding to dipolar and multipolar regimes. The demands. However, this goes against a fundamental former is preferred, whereas the latter reduces parasitic advantage of solar energy as a distributed resource, for which absorption in lower modes of resonance. Amongst the metals the geographic location, landuse, the weather, the season, and considered, aluminum comes closes to the perfect conductor the climate need to be considered [8] given the lifetime of with a surface plasmon frequency in the ultraviolet range, solar projects typically spanning several decades. In reality, away from the optimal frequency determined for solar energy “reserves”, are less than a third of the identified 2푒푉 resources, at 50,000 TW [8] which is still enough to meet a maximum absorption of 푊 , as opposed to the visible 430 2 hundred times our current demand. range for other metals. With푒푉 푚 frequency being geometry dependent, aluminum, of permittivity, achieves B. The Solar Challenge optimal enhancement of −53 with푒푉 radius and Solar energy is already capturing an increasing share of the surface coverage, compared23% to 21푛푚 , 59% energy mix around the globe, with decreasing prices and enhancement, with radius−10~ and− 15 푒푣 5 − ,12% increasing competitiveness. However, to compete with fossil surface coverage for 23silver,− 30 gol푛푚d and copper in20 dipolar− 30% regime.3% fuels, and motivate the uprooting of existing power infrastructure for more renewable alternatives, the prices need III. THEORETICAL DESCRIPTION to be brought further down. While the material contributes to nearly of the cost of a crystalline silicon cell, a A. Principles of Photovoltaics significant50% portion of the bulk material does not contribute to Solar or photovoltaic cells convert light energy into efficient absorption [8]. On the contrary, it is required for electricity by the photoelectric effect, whereby a valence mechanical support as the silicon cells need to be cut with a electron is excited into the conduction band if the incoming minimum thickness of from wafers [8]. Moreover, photon provides it with enough energy to overcome the 200휇푚 bandgap. The process produces an electron-hole pair which Nanotechnology for Energy Systems ©2021 Wael Itani must be separated and collected at opposite electrodes. In a p- The hydrogenation of a-Si serves a dual purpose. For one, n junction, this separation occurs with diffusion, whereas a p- it improves electron mobility, and reduces recombination i-n junction allows an electric field to drive the drift of the losses by strengthening the bonds of the material [8]. Second, carriers. As is evident, there are many losses that could occur it resolves the Staebler-Wronski effects of degradation with through the process, from the photon not being absorbed due exposure to sunlight [8], by stabilizing the structure at shorter to very short or long wavelength, basically transmitted, or due exposure times, and, thus, higher performance. This is to reflection, to the charge carriers recombining before otherwise resolved by restorative thermal annealing, electric collection. The second figure in [10] shows the relative stressing [26] or introducing nanocrystalline Silicon in the significance of the losses. The losses are typically amorphous matrix. parametrized by the two quantum efficiencies, the external and internal ones, described in Appendix A. Thin film cells offer the flexibility of exploring new materials and architecture [27] to make the best use of B. Thin Film Solar Cells plasmonics, as the overall bulk material, and, thus, thickness Thin film cells, on the other hand, offer a middle ground is reduced. This advantage is typically characterized by the between cost of cell and performance. They benefit from mass-power ratio [9]: plasmonics to make the latter more competitive, especially improving light absorbance which deteriorates in the range 푚 푡휌 () near their bandgap, due the mismatch between optical and 푃 = 퐼퐴휂 carrier diffusion lengths. While the paper They are typically a where refers to the module, the power it provides, , or film, compared to typical of 푚 푃 푡 1 2 휇푚 200 − 300 휇푚 , , , its thickness, density, area, and efficiency crystalline silicon cells [9], deposited on a cheap substrate, respectively,휌 퐴 휂 and the incident solar flux. The decreased including but not limited to glass or plastic [16]. With thickness directly 퐼 contributes to improving collection [9]. enhanced properties, they could be brought down all the way Ideally, the thickness should be comparable to minority carrier to the order of . The reduced thickness in turn 100 푛푚 [8] diffusion length [9]. With the enhancement provided by the reduced recombination losses due to abridged carrier diffusion inclusion of plasmonic nanoparticles for absorption, active lengths [5]. It is this balance which make thin film cells an regions with thickness less than could be achieved attractive application of plasmonics, even though they have [9]. At this scale traditional surface100 texturing푛푚 is simply too been proven effective with crystalline, organic [17] and dye large, optimizing recombination losses at the surface rather sensitized [18], [19] as well. than absorption [16]. Another advantage of thin films cells is Various semiconducting material could be used for the that some modules even show enhanced performance with film itself, such as Silicon (Si), Cadium Telluride (CdTe), elevated temperatures [27], which could benefit from the Copper Inidium Diselenide and Copper Inidium Gallium relaxation of plasmonic-generated hot carriers [28]. Selenide, with the former’s efficiency reaching [9]. Ray optics dictate an upper limit on absorption Their fabrication method allow for higher integration20% with enhancement of where is the refractive index of the other systems, such as building envelopes [8], [9]. Amorphous 2 absorbing material4푛 [25]. It 푛 has been theorized that with Silicon (a-Si) [20], the second and last remain the most subwavelength confinement, the limit, known as Lambertian commercialized with efficiencies around at 10% $0.76/푊 limit, could be exceeded times. Ideally plasmonic [9], to less than comparable silicon cells [21]. Like 12 50% 80% nanoparticles should be able to achieve that. On the other Si, CdTe requires high-temperautre processing [22]. hand, this enhancement has been challenged as angle-specific, However, unlike Si cells, they do not benefit from an abundant such that overall absorption enhancement remains within limit material. A detailed discussion on the matter could be found [29]. in [9]. C. Introduction of Plasmonic Nanoparticles Due to the aforementioned reason, and the industry’s familiarity with Si, Si thin film cells were the first to be The interaction of light with nanoparticles has been th commercialized with a similar range of efficiency [9]. Due to employed for stained glass since 7 century [30]. This the cost of processing Si into its crystalline or even interaction has been studied since the days of Faraday [16]. microcrystalline state, a-Si has become a target of research, However, renewed interest in their understanding and with its base efficiency as much as half that of others application has accompanied the work of Mie which brought mentioned [9]. The first a-Si cell to use plasmonics was likely about scattering over six orders of magnitude stronger than introduced in 2006 [23]. a-Si cells, studied by Akimov and Raman scattering through the use of nanostructures of size Koh, have absorption coefficients higher than comparable or larger than the wavelength of light [16]. crystalline silicon, at the cost of their conversion100 × efficiencies, Surface plasmons were experimentally observed 6 years prior, as carrier diffusion length is greatly reduced in the absence of and then extended to films 50 years after Mie’s work [24]. The the crystalline structure to around [8], [24]. They breakthrough moment came with the work of Stuart et al. absorb over of the incident light100 with 푛푚 less than of the before the turn of the millennium, as they have shown a silicon. How90%ever, their overall efficiencies have 1% staggered narrowband 18 fold increase in the photocurrent of a around for years, unable to overcome recombination photodetector using silver particles, and 30% overall [31]. losses in12% heavily “defective” active region [20]. A detailed Gold particles have subsequently been employed for a comparison of carrier diffusion in both materials is found in narrowband eighty percent enhancement of solar cell [8]. absorption [32]. Despite being less energy intensive, the production of Plasmonic refers to the coherent excitation of conduction amorphous silicon remains capital intensive [25]. We need to band electrons at metal-dielectric interfaces by the incident consider the energy payback period for our cells, around 1 or electromagnetic radiation [8], [28]. These oscillations are 2 years for silicon cells with 80% of energy embedded due to damped by interband transitions [33], and radiative and processing [8]. nonradiative decay [28]. We say that the phenomenon is restricted to the interface [8] since it is related to the “free” electrons in metals and the induced surface charge oscillations [5], and reduces exponentially away from them [5]. The construction and working of a heat exchanger in practical advantage of this interaction is that “subwavelength” terms. Symmetrically, the effect have been used for enhancing structures are able to “capture” light beyond their scale, electroluminescence, with a factor of seven achieved for making it suitable for a range of application, including silicon light-emitting diodes (LEDs) [16]. While the spectroscopy, sensing [34], nanolithography [28], even cancer nanoparticles in use with photovoltaics exhibit localized therapy [9] and, of course, the one under consideration, solar surface plasmons, the interaction is applicable to all metal cells. interfaces, including flat surfaces where charge compression waves are observed [9], yielding another metaphor with the We note that the historical development has shaped the study of thermo-fluids. unfolding of the field, including the widespread use of the dipole model. Mie theory continues to make an appearance in The third mechanism, which is a bit more involved, modelling plasmonic-enhanced solar cells, and precious enhances the absorption in the active region through coupling metals with their narrowband enhancement continue to be the light to the waveguide modes introduced by the studied [20], [35]–[37]. While described by the same nanostructuring. This is called surface plasmon polariton equations, the difference in the underlying mechanisms has (SPP), and is commonly done by a corrugated metal film on caused the adoption of new findings to lag. However, the rear side of the active region [28]. It could be termed as conclusions drawn from the paper under consideration [1] propagating, because surface might be understood to mean have started to capture the attention they deserve. We see that that of any metal-dielectric interface which could otherwise the focus has been shifted to tuning the frequency response of exhibit the localized surface plasmonic effect described the plasmonics, and its duration [28]. These are indirectly above, including the not necessarily planar films. The controlled through the choice of material, whether for the waveguide modes could either be surface plasmon polaritons dielectric or the metal, the shape, arrangement [38], or volume at the metal-semiconductor interface, or photonic modes fraction [5] of the latter, and the geometric dimensions of propagating in-plane. This propagation could reach to either. By embedding the nanoparticles in a transparent before the energy is absorbed completely, compared10 to conductive oxides, the distance of the particles to the active the100 20 휇 to 30 nm scale of interaction for the localized ones [28], region could also be optimized to excite the gap modes [24] in [47]. Poursafar et al. have demonstrated a combined the farfield [36]. configuration by introducing the silver nanostripes, subtracted from the back contact metal film to make corrugated, into the We now focus our attention to the principle of operations, active region achieving light absorption enhancement, or the mechanisms, through which plasmonic enhancement and almost short-1circuit.5 × current enhancement [48]. takes place. To point out, first, plasmonic-enhancement does Waveguide modes2 × alone are suggested to be a loss mechanism not need nanoparticles per se to be achieved. A range of other afterall [29]. metallic structures could be used, whether holes, gratings [39], [40], nanowires [41], nanocone [42] or continuous film. These We note that other mechanisms exist, which are starting to could be placed in front or behind the active region or gain momentum in the research community, such as hot embedded inside it. Combinations of different shapes and carrier, from the decay of plasmonics forming a nanostructure- configurations could also be used. Sun and Wang report at semiconductor Schottky junction [28], and multiple exciton least improvement by using a combination of frontal generation for up and down shifting [20], [28]. Another nanocyl20%inders and backward nanospheres than any single mechanism is the cavity mode resulting from the standing shape and configuration [43]. The frontal configuration used waves due to coupling of plasmonics [21]. Plasmoelectric in [1] remains the most popular [27]. Metal particles were first mechanisms are still in their infancy, waiting to be utilized. laid on top of solar cells as an “anti-reflection” coating [15]. These include occupation electron excitation by electric field Whether placed on front or back of the active region, the amplified locally by the plasmon [28]. nanoparticles could scatter light, at high angles as a subwavelength structure, to increase its effective path length, With that being said, we can classify the nanoparticles increasing exponential decay of incident light in the material introduced to enhance the performance of a solar cell as resonant and non-resonant, with the rule of thumb being described by Lambert’s law −훼푙 , where the exponents refer to the absorption퐼푎푏푠 coefficient= 퐼푖푛푐푒 and optical path resonant nanoparticles are considered those which have their length respectively, and promoting total internal reflection resonance frequency in the visible range [2]. Those are [35] allowing the cell to absorb more [28]. This scattering typically the “shiny”, precious metals, silver and gold [49], as process is the first mechanism for plasmonic-enhancement, well as copper, with transition wavelengths of 327, 517 and and it is the one studied in the paper under review [1]. The 590 nm respectively. It is typically resonant nanoparticles that nanostructures reduce contact resistance [44], open-circuit are tuned [16]. The most common non-resonant particle used voltage and fill factor, thereby exhibiting competing electrical is aluminum with a transition wavelength of 827 nm, outside properties [24] at the interface of the active region, the visible range [33]. challenging the advantages of reduced wafer size [45]. Apart from the cost, and broadband effect, aluminum is The second mechanism refers to the plasmonic interaction also preferred due to its oxidative properties, as aluminum amplifying the electromagnetic field locally, in the near field, oxide has a low dielectric permittivity insignificantly affecting within the semiconductor, thereby enhancing absorption its performance as a plasmonic material [2]. Furthermore, its which is proportional to the intensity at a given point. The is nanoparticles have been shown to benefit from introduction of sometimes referred to as the “antenna” effect, or localized functional groups by wet chemical synthesis for enhanced surface plasmon resonance, especially when the nanoparticle stability [19]. Alternatively, gold, alloy has been shown to have performance10% within 90% that of aluminum, is excited near resonance. Localized resonance requires 1% surfaces curves or kinks [40]. In solar cells, this involves the while providing an oxidative protection [50]. Synthesis of plasmonic material being embedded inside the active region. such alloys has been demonstrated for other solar applications When embedded inside the active region of the cell, the [51], [52]. plasmonic resonance is typically used to improve absorption The sphere is typically used as an ideal shape in studies by enhancing the electromagnetic field locally. While due to its isotropic nature, and the availability of analytical governed by other physical phenomena, this is similar to the solutions through Mie theory. In addition, the understanding 1 2 () of the underlying mechanisms become more complex with 푃푎푏푠 = 2 휔휀0퐼푚[휀(휔)]|퐸(휔)| other shapes, such as nanoshells which introduce optical We have skipped over the derivation of the above term from vortexing [16]. Shape optimization is critical, as it might the Poynting vector, the cross product of the electrical and lengthen the optical thickness by times. Apart from, hard magnetic fields [81]. Implicit here is the spatiotemporal to manufacture, idealized spheres30 and ellipsoids, nanocubes dependencies of the functions. The carrier generation rate, [8] have shown to be the optimal shape for practical assuming each absorbed photon corresponds to an electron- applications. It might take a bit, however, for their adoption, hole pair generated, could then be calculated [10]: with recent studies still citing shape optimization as future work [50], just as it had to accept broadband enhancement with aluminum as a material of choice. 푃푎푏푠 () 푔 = ℎ휔 IV. MODEL OF PLASMONIC NANOPARTICLES IN SOLAR CELLS where is Planck’s constant. The above could be integrated ℎ A. Maxwell’s Equations over frequency, or wavelength if expressed otherwise, to obtain the volumetric carrier generation rate, and then over the While the phenomena considered are quantum mechanical volume for the total carrier generation rate. in nature, such as the strong interaction between the plasmons and band electrons of the substrate, rather than a quantized D. Dimensionless Parameters representation [53], they are typically modelled with To better quantify the study of nanoparticle-enhanced electromagnetic waves governed by Maxwell equations, solar cells, we must introduce a few parameters, including the detailed in Appendix B. It has been previously speculated that size parameter , defined as the ratio of the “effective” size of this led to overestimation of possible enhancements, as losses the particle to the푥 wavelength, and the refractive index ratio: are not properly accounted for in the classical model [54]. Recently, it has been reported that experimental observations deviate from the results of classic model in inverse proportion 2휋푛푚푟 () 푥 = = 푘푛푚푟 to the size of nanoparticles [28], by nearly a factor of half [36]. 휆 This is expected as experimental and modelling efforts in the 푛푝 () research community appear to be isolated, with few modelling 푚 = 푛푚 papers reviewed [55]–[57] validating against experimental where is the medium refractive index, that of the results [38], [56], [58]–[60]. The challenges of reproducing 푛푚 푛푝 idealized models remain a challenge, starting with that particle, the particle’s radius, and the wavelength of light. 푟 휆 spherical shape, and there appears to be a range of We note that the size parameter is sometimes defined as 푘푟 recommended parameters optimized for experimental solely without the refractive index correction [5]. Several reproducibility [5]. However, it has been shown that this other parameters could be defined as well, but most would approach, as well as the Mie approach introduced below, come into use for empirically-driven modelling approach, could be adjusted for empirical findings [61], and corrected such as the thickness-normalized effective optical path length for finite size using the random phase approximation derived [57]. in a dual quantum mechanical analytical-empirical approach E. Point Dipole Model [62]. Moreover, coupled optical-electrical-thermal simulations have been done [10] to better account for the For a sufficiently small particle, typically taken as , it could be modelled as a point dipole, where the particles푥 ≪ losses. While thermal effects could be accounted for with 0.1 respective heat transfer equations discussed in [10], electro- are assumed to be noninteracting [82]. The scattering and optical losses are sufficiently described by the Maxwell absorption cross sections, and , are taken as [16]: 휎푠푐푎 휎푎푏푠 equations, and the carrier continuity and drift-diffusion equations reproduced in Appendix C. Overall, the problem of 1 4 2 () isolated disciplines tackling the problem has been identified 휎푠푐푎 = 6휋 푘 |훼| with attempts to resolve it by comprehensive reviews [20]. () 휎푎푏푠 = 푘퐼푚[훼] B. Boundary Conditions with the wavenumber : The boundary conditions used [1], the periodic boundary 푘 [5], [8], [10], [28], [29], [34], [38], [39], [42], [43], [45]–[47], [53], [55], [63]–[77], and the perfectly matched boundary 2휋 휔 () layer [2], [5], [10], [10], [12], [15], [25], [29]–[31], [35]–[37], 푘 = 휆 = 푐 [46]–[56] have become almost standard for this type of where the frequency of the light, and its speed. Moreover, simulation. The former allows to reduce the computational the polarizability휔 is defined as [15]: 푐 domain to one unit of a periodic geometry, allowing the 훼 coupling of diffrenent components. This is applied to the sides 휀푝−휀푚 () of the domain containing the nanoparticle. The latter 훼 = 3푉( ) represents a total absorption layer [10] which is also used to 휀푝+2휀푚 reduce the domain in perpendicular to the interface without with the wavelength of light, and the particle volume, introducing artificial (numerical artifact) reflection or typically휆 taken to be the volume of a sphere.푉 and are the interfering waves. dielectric functions of the particle and embedding휀푝 휀 푚 medium C. Power Absorption & Carrier Generation respectively [16]. They are essentially the complex-valued wavelength-dependent permittivity. We also note that the In full wave simulations, the electric field distribution is expressions are for the farfield, applicable to the forward- obtained allowing us to calculate the absorbed power per unit scattering configuration of [1], whereas the nearfield volume [2]: description could be found in [70]. We also note that the above formulation summons the quasi-static assumption which

becomes invalid with larger particle sizes, as dynamic () depolarization and radiation damping and other mechanisms 휎푠푐푎 = Σ푛휎푠푐푎,푛 become significant and need to be considered in our () understanding of the plasmons [16]. Dynamic depolarization 휎푎푏푠 = Σ푛휎푎푏푠,푛 occurring for a finite-size particle, with its electrons or simply the summation of the respective cross section oscillating out of phase with a reduced restoring force, contributed by each mode. This is in contrast to the point explaining the red-shifted, broadened resonance observed for dipole model which accounts for only the first mode larger particle sizes. In other words, the oscillation frequency interaction for sufficiently small particles it considers [35]. of all modes decreases with size, less so for higher modes [5]. However, higher modes have been shown to have significant The optimum nanoparticle size, for a forward-scattering contributions to the overall plasmonic enhancement [5]. configuration as considered here, has been identified as Plasmons could then be understood as the eigemodes of the between 30 and 60 nm [21], whereas it is less than 30 nm for metals, yielding cross sections surpassing the geometrical near-field enhancement [47]. The broadening is described as ones. silver nanoparticles, for example, exhibit an advantageous for solar applications for the same reasons albedo100 of 푛푚. The resonant frequency of each mode decreases Akimov and Koh recommended aluminum to be used. The with the size0.9 of the particle, with higher modes showing less size is then tuned to optimize resonant interaction of higher sensitivity. The cross sections for the modes display a gamma modes while decreasing absorption in lower modes in the distribution with an early peak and long-tailed decay away peaks of the solar profile [5]. from the optimum value [5]. From the formulation above, it is mathematically This is attributed to the higher modes featuring “the straightforward to see how resonance behavior arises from the distance between any two adjoining poles” [5]. It presents the denominator approaching nought. The matter is not as simple basis of introducing the particles with a “dimer” arrangement as optimizing scattering and minimizing absorbance, for it is [50], or in pairs [12], [73], at each site, as further modes are seen that forward and backward scattering cross sections are introduced due to the Coulumb interaction between their correlated. Whether the particles are placed at the front or back surface charges. The resulting modes could be classified as , of the active region, maximal scattering also means a large , and modes according to the value their effective퐿 portion could be lost in the wrong direction. We also see that resonance푀 푇 dielectric permittivity approaches, , , and the dependence on the volume of the particle appears in both otherwise respectively [12]. The wavefunction−∞ remains0 expressions, albeit quadratic in the scattering cross section. unbounded for and , and bounded, with strong local Thus, optimizing size would also need to account for parasitic enhancement, for퐿 . 푇 absorbance of the nanoparticles themselves. 푀 H. Effective Medium Theory F. Mie Theory For practical application, short of numerical modelling, we In his work, Mie introduced an analytical solution for the do not wish to consider a singular particle as in Mie theory. interaction of plane electromagnetic waves with an ideal, Instead, the effective medium theory is employed [33], homogeneous and isotropic, spheres [33]. The embedding allowing the calculation of the effective permittivity, and other medium is assumed to be non-absorbing which limits the macroscopic properties. A shortcoming of such a treatment, substrates for which it could be used [5], [83]. The results for however, is that is assumes specular transmission whereas the amorphous silicon, however, are within the accuracy of the wavevectors are typically wide-spread [53]. The most mo del. The solution is expressed as the summation of infinite commonly used effective medium methods are Maxwell- series in terms of the modes , from which the total extinction, Garnett, and Bruggeman, an extension of the former that helps scattering and backscattering푛 efficiencies, , , and 푄푒푥푡 푄푠푐푎 푄푏 us account for polydisperse particles [25]. Arinze et al. have are expressed. We note that the extinction efficiency is the modified the Maxwell-Garnett method to specifically model summation of that of scattering and absorption, closely related plasmonics in thin films which they have validated to the concept of albedo [82], typically used to normalize the experimentally [25]. They introduced effective absorption scattering efficiency [33]. The latter ratio is sometimes coefficient and film thickness: referred to as radiative efficiency [53]. The efficiencies and the coefficients are defined in 휎푝 () Appendix D. We, thus, see that the relative size of the particle 훼푒푓푓 = 훼푚 (1 + 푑2 (푄푠 − 푄푎)) with respect to the wavelength of light is necessary to determine their interaction. Moreover, the efficiencies above 휎푝 () tend to zero as approaches unity, meaning as the particle 푡푒푓푓 = 푡√1 − 2 refractive index 푚approaches that of the dielectric. For , 푑 the near-field is enhanced [66]. 푚 > 1 where is the interparticle separation, and the film 푑 푡 Mie theory is considered applicable for , thickness. For backscattering configuration, one approach to where the particle and the wavelength are “comparable”.푥 ∈ [0.1,100 For] effective parameters is summarized in [57]. Other [42], and , or more explicitly, , Rayleigh scattering, or the more thorough effective medium theory, i.e. spectral density, elastic푥 ≪ 1 scattering of light by푥 < particles0.1 of “negligible” size is exist [84], but we have chosen the above as its expression is considered [8], and eventually quantum effects, such as more suggestive, and amenable to insight. Landau damping [28] must be accounted for. I. Dielectric Function Model G. Higher Resonance Modes As it should have become obvious through the prior Mie theory is said to be a “monochromatic” theory, discussion, we need a model of the dielectric function of the dismissing the coupling between the different wavelengths material for a full description of the problem. A description of [83]. However, in reality, it considers the interaction of the the most common Doyle [63], Lorentz and Drude models [66] could be found in the respective references. The form of the incoming light with the normal modes of the plasmonic dielectric function itself could be a design parameter [47]. We nanostructure [5], such that푛 the overall cross-sections are present here the most applicable, Drude, model [70]. defined as :

2 VI. OUTLOOK 휔푝 () 푅푒(휀(휔)) = 1 − 2 1 휔 +휏 As a start, quick gains are waiting to be made with arrangement optimization. Second generation devices are 2 defined as those which have benefitted for low-cost 푁푒푒 () 푝 fabrication methods. The third generation is that which 휔 = √휀0푚푒 combines low-cost fabrication with low-cost enhancement, where is the plasma frequency of the electron “gas” of such as the introduction plasmonic nanoparticles. Novel the metal, 휔푝 the relaxation time of the electron, and its devices, whether by material, configuration, otherwise, are set effective mass.휏 푚푒 to benefit more from enhancement techniques. For one, efficiency enhancements by light trapping in organic devices J. Transfer Matrix Method have been shown to be superior than those of silicon cells [15]. After obtaining the “effective” properties of the layer with Based on the review completed for this paper, and others [9], nanoparticles, it is then possible to consider them as part of a [28], [52], [54], [76], [78][70]–[81], we see that up and stack of layers, or multilayer thin-film. Analytically, this is coming materials, including quantum dot, organic, and done by the transfer matrix method introduced in Appendix E perovskite cells are to be the focus of plasmonic [33]. Despite the limitations of the Mie-Effective Medium- enhancements, especially with their favourable roll-to-roll Transfer Matrix method, it is included in detail here due to its fabrication [25], with plasmonic configuration outside the practicality for swift evaluations of different particle materials active region being preferred [25] to simplify manufacturing, and sizes. After all, numerical simulations popular in the field and avoid introducing defects [47]. could only be so accurate given their shortcomings previously discussed. The materials of nanoparticles itself would also be open to more flexible choices, with dielectric particles as Mie V. LIMITATIONS scatterers [20], [38], nanocrystallines [55], and even dielectric shells for metallic nanoparticles [71] being introduced for Nanoparticles in plasmonic-enhanced thin film cells are reduced losses [85]. The field of photovoltaics could also learn still limited by their sensitivity to the angle of incidence [85], lessons for nearby fields, leveraging lessons from solar filtering effect, apart from the backscattering configuration thermal conversion, on say nickel for broadband absorption parasitic absorption [25], exciton quenching, chemical and [72], to minimize losses. These advances further solidify the physical compatibility with the solvent Apart from the leading stance of maneuvering around the bandgap limitation limitation of parasitic absorption, the Fano effect, resulting altogether , rather than tuning resonance frequencies [27]. from the destructive interference of incident and plasmonic- More explicitly, we need to optimize for the energy source, scattered light needs to be managed. the Sun, first and foremost, rather than the nitty gritty details. I do not doubt the creativity of the nanotechnology It may seem that there is an overwhelming number of community in coming up with ways to fabricate parameters to tune. However, let us recall that the cell, at the nanostructures for solar devices, based on Appendix F. end of the day, is of finite size, and there is a finite number of However, interdisciplinary, inter-sectorial, work is needed to abundant materials, say carbon [98], to choose from. Mix-and- push the fabrication methods out of the lab, at industrial scale, match optimizations at the particle level have not yielder into the real world where costs matter [7], [20]. This is considerable improvement [29]. Rather than proceeding with thoroughly discussed in [20], [90]. Nanoparticles already have the heuristic optimization manually [47], the design space cost-effective production methods that integrate well with could be explored computationally. This could be done with photovoltaic product [20]. These methods must be considered. present-day machine-learning technology. The identification Alternatively, we note that parasitic absorption by metallic of complete competing effect pairs, such as the enhancement nanoparticles remains a challenge. We mention this with by closely placing particles [50] and separating them widely fabrication, as the “roughness” or “kinks” due to tolerances [81], makes this a perfect problem for optimization. could be detrimental, which is nanoparticle aggregates are disappointing [29], [96]. The cold silicon processing The advent of quantum computing appears promising for technique recently introduced might revive interest in the the same reason of numerical optimization, and two others. material and in commercial photovoltaic ventures as well [20]. The second is that it would present a more capable platform for modelling of phenomena quantum mechanical in nature. A significant obstacle to this is establishing a unified, Third, it has pushed “quantum” into the popular culture, accessible, representation for parameters under consideration breaking down the hesitance of utilizing the theory for and obtained results. This is a problem in any academic field, practical applications. This has started with analytical models however, our understanding of plasmonic nanoparticle in thin being corrected [62], and quantum effects more effectively film cells has matured enough [25], [85], and is ready for such leveraged, to make silicon’s bandgap effectively direct by a step to accelerate its commercialization. This could simply quantum confinement for example [14]. be done by devoting the same zealot of reviews to database design. Such a database would be one step towards This should motivate more fundamental representation, translatable optimization of the wealth of plasmonc-enhanced allowing to model the complexity of the cell and its underlying cell parameters currently shackled by simplified models [27]. mechanism to achieve overall efficiency, otherwise currently It would also help develop current models to account for more limited [25]. It is likely that specialized software packages mechanisms and their coupling based on their relative would be developed, akin to ASA[99], such that the choice of significance reported, rather than starting with the most a cell’s operating mechanism(s) would become modular simplified models. In addition, such efforts would help compared to generic scientific simulation software currently produce cells that operate well under different real used (COMSOL, Luminar, …). environmental conditions, such as the irradiance and zenith The ideal remains to construct an optically-thick, angle geographical variation [97]. physically-thin device with broadband efficiency and angle isotropy. Far from the ideal, we already see a push to commercialize neonatal third generation technology, with vol. 152, pp. 285–290, Jun. 2018, doi: patents getting filed [88]. 10.1016/j.vacuum.2018.03.026. [13] M. Aerts et al., “Free Charges Produced by Carrier VII. 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Hylton et al., “Loss mitigation in plasmonic solar 푉푚푝 퐽푚푝 () cells: aluminium nanoparticles for broadband 퐹퐹 = 푉표푐 퐽푠푐 photocurrent enhancements in GaAs photodiodes,” Sci Rep, vol. 3, no. 1, p. 2874, Dec. 2013, doi: The maximum device efficiency could then be written in 10.1038/srep02874. terms of the above: [94] M. Kriisa et al., “Growth and properties of ZnO films on polymeric substrate by spray pyrolysis method,” Thin 푃푚푝 푉푚푝퐽푚푝 퐹퐹푉표푐퐽푠푐 () 휂 = = = Solid Films, vol. 555, pp. 87–92, Mar. 2014, doi: 푃푖푛 푃푖푛 푃푖푛 10.1016/j.tsf.2013.05.150. The input power could be taken from the solar constant [95] M. Kriisa, M. Krunks, I. Oja Acik, E. Kärber, and V. 푃푖푛 or some other standard value, such as the 푘푊 . Mikli, “The effect of tartaric acid in the deposition of 푃퐴푀1.5 = 1 푚2 Sb2S3 films by chemical spray pyrolysis,” Materials Further discussion on this could be found in [9]. The open- circuit voltage itself could be defined as [9]: Science in Semiconductor Processing, vol. 40, pp. 867– 872, Dec. 2015, doi: 10.1016/j.mssp.2015.07.049. [96] S. Morawiec, M. J. Mendes, F. Priolo, and I. Crupi, 휅푇 퐽퐿 () 푉표푐 = ln ( + 1) “Plasmonic nanostructures for light trapping in thin-film 푞 퐽0 solar cells,” Materials Science in Semiconductor where is Boltzmann constant, the temperature, and the Processing, vol. 92, pp. 10–18, Mar. 2019, doi: carrier 휅 charge together defining푇 the thermal voltage.푞 In 10.1016/j.mssp.2018.04.035. addition, is the dark current, the current present in the [97] S. A. Choudhury and M. H. Chowdhury, “Use of absence of퐽0 light, due to thermalization, and the full- plasmonic metal nanoparticles to increase the light luminesce current. We note that the dark 퐽퐿 current is absorption efficiency of thin-film solar cells,” in 2016 proportional to the device volume. Thereby, with light- IEEE International Conference on Sustainable Energy trapping, a thin film device benefits from reduced while 퐽0 Technologies (ICSET), Hanoi, Vietnam, Nov. 2016, pp. maintaining to increase . However, surface 퐽퐿 푉표푐 196–201, doi: 10.1109/ICSET.2016.7811781. recombination and lower shunt resistance might dominate [98] S. Paulo, E. Palomares, and E. Martinez-Ferrero, over these improvements. “Graphene and Carbon Quantum Dot-Based Materials in The short circuit current could also be defined as [12]: Photovoltaic Devices: From Synthesis to Applications,” p. 20, 2016. 휆 () [99] M. Zeman, O. Isabella, K. Jäger, P. Babal, S. Solntsev, 퐽푠푐 = 푞 ∫ ℎ푐 퐸푄퐸(휆)푃푖푛(휆)푑휆 and R. Santbergen, “Modeling of Advanced Light WHERE IS THE CHARGE OF AN ELECTRON. Trapping Approaches in Thin-Film Silicon Solar Cells,” 푒 = −푞 MRS Proc., vol. 1321, pp. mrss11-1321-a23-05, 2011, APPENDIX B. MAXWELL EQUATIONS doi: 10.1557/opl.2011.955. The Maxwell equations describe the propagation of electromagnetic waves, through the electric field , magnetic 퐸 APPENDIX A. PRINCIPLES OF PHOTOVOLTAICS field , electric displacement , and magnetic displacement [10]퐻: 퐷 The first efficiency expression, , is the ratio of the 퐵 spectral power absorbed over 퐸푄퐸 the incident power [11], used to characterize the푃푎푏푠 spectral response of the cell 푃[2]푖푛푐. 휕퐵 () The second, , on the other hand, gives the ratio of the ∇ × 퐸 = − 휕푡 number of electrons퐼푄퐸 generated from to the number of incident 휕퐷 () photons[11], [12]. This understanding of assumes all ∇ × 퐻 = 휕푡 + 퐽푒 photons absorbed generate an electron-hole 퐼푄퐸pair. The number is easily obtained by recalling that photons are a quantized () ∇ ∙ 퐷 = 휌푒 () () ∇ ∙ 퐵 = 0 푛1 sin 휃1 = 푛2 sin 휃2 where , , and the magnetic current density where is the refractive index of each medium respectively, 퐷 ,= with휀퐸 the퐵 = magnetic휇퐻 and dielectric permeabilities is the푛 angle of incidence and the angle of refraction. The and퐽푒 = 휎, 푒the퐸 electrical conductivity and the net charge density휇 transfer휃1 matrix method assumes휃2 Fresnel reflection and . These휀 could also be transformed휎푒 to be solved in the Fourier transmission across some layers, with incident plane waves space휌푒 [55]. Computing methods relevant for this set of partial [33]. We label each layer 푙with an index , such that is the 푗 푑푗 differential equations are briefly reviewed [29]. thickness, and the refractive index of the 푡ℎ layer, and assume that the푛 thin푗 -film is preceded by and 푗followed by a APPENDIX C. CARRIER DYNAMICS substrate layer , assumed to be 푎 semi-infinite. The 푠 The continuity equation, derived from the conservation of transmission and propagation matrices, between the 푡ℎ mass, describes the generation and recombination of electrons 푀푗푘 푗 and 푡ℎ layer, and , could then be written as: [10]: 푘 푃푗

휕푁푒 1 () 1 1 푟푗푘 () 푒 푒 푒 푀푗푘 = [ ] 휕푡 = 푒 ∇ ∙ 퐽 + 퐺 − 푅 푡푗푘 푟푗푘 1

휕푁ℎ 1 () 2휋 −푖 푑푗푛푗 cos 휃푗 휕푡 = − 푒 ∇ ∙ 퐽ℎ + 퐺ℎ − 푅ℎ 푒 휆 0 () 푃푗 = [ 2휋 ] 푖 푑푗푛푗 cos 휃푗 Where , , , and are the number density, current density, 0 푒 휆 푁 퐽 퐺 푅 generation rate, and recombination rate for the respective where and are the Fresnel transmission and reflection carrier, electron or hole. We note that the generation and coefficients푡푗푘 at 푟the푗푘 interface of the aforementioned layers, and recombination rates for both carrier types could be considered the angle of refraction in the respective layer [33]. The total equal, or otherwise could be accounted for by a separate transfer휃푗 matrix from the first interface with air, to the last source term, and the collection is embedded into the current −1 interface with the푴 substrate, becomes: term already. The recombination term covers Shockley-Read- Hall, radiative and Auger recombination [10]. In semiconductors, the dynamics of these carriers could be + + + 퐸푎 퐸푠 퐸푠 () described by the Poisson equation: [ −] = 푴 [ ] = 푀푎1푃1푀12푃2 … 푀푙푠 [ ] 퐸푎 0 0 () relating the electric field in air to that in the substrate. This −∇ ∙ 휀∇푉 = 푒(푁ℎ − 푁푒 + 푁푖푖) allows us to define the overall reflectance 2 and 푅 = |푟 | where refers to ionic impurities, and to electrostatic transmittance 푛푠 cos 휃푠 , and calculate the power 푖푖 푉 푇 = potential. The Drift-Diffusion equation is: transferred by a layer푛푎 cos with휃푎 embedded nanoparticles to the substrate for absorption () 푃푤: 퐽푐 = 휇푐(푒푅푐 + 푘∇T)Nc + 휇푐푘푇∇푁푐 ∫ 푇(휆)퐼푖푛푐(휆)푑휆 () where refers either carrier type. 푃푤 = 푐 ∫ 퐼푖푛(휆)푑휆 APPENDIX D. MIE THEORY COEFFICIENTS APPENDIX F. NONEXHAUSTIVE LIST OF NANOSTRUCTURE MANUFACTURING METHODS 2 () • Ion beam milling [64] 푄푒푥푡 = 푥2 Σn(2푛 + 1)푅푒(푎푛 + 푏푛) • “Printable” structures [86] 2 2 2 () • Solution processing [25] 푄푠푐푎 = 2 Σn(2푛 + 1)(푎푛 + 푏푛) 푥 • Soft nanoimprint lithography [87]

1 푛 2 () • Pulse current electrodeposition [27] 푄푏 = 푥2 (Σn(2푛 + 1)(−1) (푎푛 + 푏푛)) • Electron beam lithography and are defined in terms of Riccati-Bessel functions • Gas aggregation [20] [33]푎: 푛 푏푛 • Sputter deposition [88] • Rapid thermal annealing [89] ′ ′ • Soft deposition [90] 푚휓푛(푚푥)휓푛(푥)−휓푛(푥)휓푛(푚푥) () 푎푛 = ′ ′ • Masked deposition [44] 푚휓푛(푚푥)휉푛(푥)−휉푛(푥)휓푛(푚푥) • Close-spaced sublimation [91] ′ ′ 휓푛(푚푥)휓푛(푥)−푚휓푛(푥)휓푛(푚푥) () • Chemical annealing [92] 푏푛 = 푚휓 (푚푥)휉′ (푥)−휉 (푥)휓′ (푚푥) 푛 푛 푛 푛 • Metal-organic vapour phase epitaxy [93] APPENDIX E. TRANSFER MATRIX METHOD • Spray pyrolysis [56], [94], [95] • Chemical vapour deposition. Let us first recall Snell’s law, which dictates the crossing of light rays from one refractive medium to another: