The fundamental problem of treating light incoherence in photovoltaics and its practical consequences

The incoherence of sunlight has long been suspected to have an impact on solar cell energy conversion efficiency, although the extent of this is unclear. Existing computational methods used to optimize solar cell efficiency under incoherent light are based on multiple time-consuming runs and statistical averaging. These indirect methods show limitations related to the complexity of the solar cell structure. As a consequence, complex corrugated cells, which exploit light trapping for enhancing the efficiency, have not yet been accessible for optimization under incoherent light. To overcome this bottleneck, we developed an original direct method which has the key advantage that the treatment of incoherence can be totally decoupled from the complexity of the cell. As an illustration, surface corrugated GaAs and c-Si thin-films are considered. The spectrally integrated absorption in these devices is found to depend strongly on the degree of light coherence and, accordingly, the maximum achievable photocurrent can be higher under incoherent light than under coherent light. These results show the importance of taking into account sunlight incoherence in solar cell optimization and point out the ability of our direct method in dealing with complex solar cells structures.

Optimization of the efficiency of solar cells is a major challenge for renewable energies. Using a rigorous theoretical approach, we show that the photocurrent generated in a solar cell depends strongly on the degree of coherence of the incident light. In accordance with Heisenberg uncertainty time-energy, incoherent light at photons of carrier energy lower than the active material bandgap can be absorbed whereas coherent light at the same carrier energy cannot. We identify cases where incoherence does enhance efficiency. This result has a dramatical impact on the way solar cells must be optimized regarding sunlight. As an illustration, surface-corrugated GaAs and c-Si thin-film solar cells are considered. 1 These authors contributed equally to this work. Renewable energies, and especially solar energy, are of great interest in the quest for sustainable energy sources. In this context, any improvement in the ability to convert solar light energy into electric energy is important. Among many methods available for this purpose, the optimization of light-trapping structures at the front and/or back-side(s) of solar cells is a promising approach [1][2][3][4][5][6]. Most researches focus on finding the optimal structure geometry [7][8][9][10][11][12] that increases the absorption inside surface-corrugated ultrathin slabs via quasi guided modes [13][14][15]. The aim is to find the fundamental upper bound limit on absorption (efficiency) [15]. Many theoretical works have contributed to substantial progresses along this line [15][16][17][18][19]. Nevertheless, at present, an important issue remains quite unexplored: the impact of solar light incoherence on solar cell efficiency. Indeed, it is well known that the response of an optical device depends on the degree of coherence of the incident light [20]. Until now, the lack of investigations in this area was due to the complexity of the numerical methods dealing with incoherence. Calculation methods exist but are time consuming [21][22][23][24][25][26][27][28], limiting thereby the capacity to model the effects of solar light incoherence on solar cell efficiency. As a consequence, major works done in optimizing the absorption (hence the photocurrent) consider coherent incident light. In a recent article, we developed a rigorous theory accounting for the effects of temporal incoherent behavior of light on absorption and photocurrent [29]. Our method is not only more accurate and simpler than previous one but it offers a large gain in computational time, paving the route for extensive studies of incoherent versus coherent effects. In the present Letter, * e-mail: aline.herman@unamur.be † e-mail: michael.sarrazin@unamur.be ‡ e-mail: olivier.deparis@unamur.be we show that the degree of sunlight coherence has a dramatical, unsuspected impact on the way solar cells must be optimized. Especially, we show that the photocurrent produced by a thin-film solar cell strongly depends on the coherence time of the incident light. This result is explained in terms of the time-energy uncertainty relation (i.e. Heisenberg uncertainty principle) which allows to break the energy bandgap limitation in solar cell absorption. To prove our assertion, the photocurrent generated in two types of uncoated semi-conductor slabs, namely crystalline silicon (c-Si) and Gallium Arsenide (GaAs), under exposure to incoherent light is numerically investigated. The slabs have their top surfaces corrugated with wavelength-scale arrays of square or cylindrical holes for light-trapping purpose.
Let us first recall some first principles as well as the framework of our method. The maximum achievable photocurrent J supplied by a solar cell is given by: where A(λ) is the active layer absorption spectrum, S(λ) is the global power spectral density of the solar radiation and J(λ) is the maximum achievable photocurrent spectrum. In order to take into account the incoherent nature of sunlight, the absorption A(λ) must be the effective incoherent absorption A incoh (λ) undergone by the solar cell. In numerous works previously published [8-10, 13, 31], A(λ) is actually the coherent absorption A coh (λ) which is computed using numerical methods that propagate the coherent electromagnetic field. In some works, numerical methods were proposed in order to compute A incoh (λ) [21][22][23][24][25][26][27][28]. However, those methods often rely on multiple numerical runs, each one being performed for a coherent incident wave which is dephased with respect to the previous one. The final result is then obtained from a statistical analysis. In a recent work [29], we have shown that the incoherent absorption spectrum A incoh (ω), where ω = 2πc/λ is the angular frequency, can be simply obtained from the convolution product between the coherent absorption spectrum A coh (ω) and an incoherence function I(ω) [29]: where ⋆ denotes the convolution product. The incoherence function is defined by the Gaussian distribution [29]: with a Full Width at Half Maximum ∆ω = 2π/τ c inversely related to the coherence time τ c . Physically, I(ω) describes the stochastic behavior of each spectral line (at frequency ω) composing the whole solar spectrum. This simple formula is easy to use in practice and allows to reduce the computational time cost and the algorithm complexity. Full rigorous demonstration of formula (2) was given in ref. [29]. Nevertheless, this expression can be easily justified. Upon incoherent illumination, the solar cell undergoes many incident wave trains randomly dephased with respect to each other. These wave trains have the same frequency ω and an average duration equal to the coherence time τ c . As a consequence, the solar cell feels an incoherent wave which is essentially a coherent wave that is filtered in time, i.e. the function f incoh (t) which describes the time-dependent incoherent signal is given by the coherent counterpart f coh (t) multiplied by a random temporal window function m(t). Then, in analogy with signal processing theory [30], the effective solar cell response A incoh (ω) in the frequency domain is given by a convolution product, i.e. Eq.(2). In the present case, it can be proved [29] that Let us now clarify the effect of finite temporal coherence of sunlight on solar cell efficiency. Under coherent illumination, the type of front-side corrugation in thinfilm solar cells has a strong influence on the photocurrent [9,10,31]. Therefore, it is important to properly optimize the corrugation (we do not consider here additional improvements brought by conformal antireflection coatings). In order to investigate the role of the coherence time on such an optimization, we studied two different corrugations, cylindrical holes and square holes (Fig. 1). Both corrugated slabs have a fixed thickness (t = 1µm) and a fixed height of the hole (h = 500 nm). Slab thickness and hole depth are typical of ultrathin solar cells designs where photonic light trapping effects are exploited [10,15]. The ratio between hole size (diameter or side) and the period of the hole array has a fixed value equal to 0.9 (D/p = 0.9 for the cylindrical hole and a/p = 0.9 for the square hole). These values resulted from previous optimization [10]. The period (p) is the only varying parameter (from 250 nm to 1250 nm). Two type of materials were investigated: crystalline silicon (c-Si) and gallium arsenide (GaAs). The holes were supposed to be filled with air (i.e. incidence medium). The aim here is not to find again the best corrugation shape (like in [10]) but to highlight the effect of the coherence time on different shapes.
The photocurrent (Eq. 1) under incoherent illumination was calculated for various coherence times. The integration was carried out from λ = 200 nm to 2500 nm. Under incoherent illumination, the absorption spectrum A(λ) was A incoh(λ) which was deduced from A coh (λ) using the convolution formula (Eq. 2). The coherent absorption spectrum was numerically calculated using the Rigorous Coupled Wave Analysis (RCWA) method [32,33]. The incident light was supposed to be unpolarized and impinging under normal incidence. Strictly speaking, the convolution product involves an integration from −∞ to +∞. However, the numerical integration range is limited by the knowledge of the complex permittivity ǫ(ω) of the material. Therefore, the integration (convolution product) was performed from λ = 10 nm to 5 µm.
Maps of the photocurrent were computed for both slabs defined in Fig. 1 with c-Si or GaAs as active material, according to various corrugations, periods and coherence times (Fig. 2). The permittivities of materials were taken from the literature [34]. Since the coherence time of the sunlight is about τ c = 3 fs [35] and since the photocurrent almost does not vary when the coherence time is longer than 10 fs (Fig. 2), the abscissa values in Fig. 2 are plotted using a logarithmic scale, in order to better highlight the influence of τ c on J.
Two optimal periods are found for the c-Si slab corrugated with cylindrical holes and illuminated under coherent light: p = 450 nm and 750 nm (Fig. 2(a)). If we only think in terms of coherent light, we could use both optima since they lead to the same photocurrent. However, when the coherence time decreases, we notice that J depends strongly on τ c . We also notice that, depending on the degree of coherence, a structure could be optimized under coherent light (i.e. high values of τ c ) while remaining optimal or being even better under incoherent light. Therefore, the choice of the optimal corrugation (period and hole shape) strongly depends on the coherence time of the incident light. An optimal structure under coherent light is not necessarily the optimal one under incoherent light. At the contrary, a non optimal structure under coherent light could become the optimal one under incoherent light. The choice of the optimal corrugation also depends on the material used in the active layer (e.g. Fig. 2(a) vs. Fig. 2(b)). We also notice that, when the shape of the holes changes from square to cylinder, the optima shift to smaller coherence times for both materials (Fig. 2(a) vs. Fig. 2(c) and Fig. 2(b) vs. Fig. 2(d)). Since the photocurrent is strongly influenced not only by the shape of the hole [10] but also by the coherence time, it turns out to be necessary to optimize the structure taking also into account the coherence time of the solar radiation, which is not currently done in literature.
In order to better understand the influence of the coherence time on J, we plotted a cross-section of the maps of Fig. 2 for a period equal to 450 nm (Fig. 3). This period of 450 nm is not the optimal one for the four studied structures. However, it is a compromise since J is high for the four structures under coherent light. Fig. 3 shows that the photocurrent is quite constant at high coherence times. As τ c decreases, however, J increases, reaches a maximum, decreases and then increases again.
Let us now explain why the photocurrent increases when the coherence time decreases from τ c = 100 fs to τ c = 1 − 2 fs. Due to their finite coherence time, each photon from the solar radiation cannot be defined with a definite energy E 0 (or wavelength λ 0 = hc/E 0 ). According to the time-energy uncertainty relation, i.e. Heisenberg uncertainty principle: Each photon is characterized by a spectral width ∆E with a time uncertainty related to the coherence time, i.e. ∆t ≈ τ c . From Eq. 4, we must consider that a photon with a carrier wavelength λ 0 occupies a spec- tral domain roughly defined by D s ∼ [λ 0 − ∆λ, λ 0 + ∆λ] with ∆λ ≈ λ 2 0 /(4πcτ c ). That means that the photon does not feel a single value of the complex refractive index n(λ 0 ) + ik(λ 0 ), but a range of values n(λ) + ik(λ) with λ ∈ D s . As a consequence, even if k(λ 0 ) is almost equal to zero, typically above the bandgap wavelength (λ g ≈ 1.1µm for c-Si) the photon can be yet absorbed provided that k(λ) = 0 on D s . To illustrate this point, Fig. 4 shows the normalized incoherence function I N (λ) (panel (a)), the absorption spectrum A(λ) (panel (b)) and the photocurrent spectrum J(λ) (panel (c)) for the case of cylindrical holes (p = 450 nm) in c-Si. Real and imaginary parts of the c-Si refractive index, n(λ) + ik(λ), are also shown (panel (a)). Note that the incoherence function is normalized in order to help the comparison with the c-Si refractive index data. Since the incoherence function is plotted against wavelength λ, the plotted function is not Gaussian-like as it would be against frequency ω (see Eq. 3). To support our explanation, I N (λ) is centered around a value λ 0 = 1700 nm (vertical gray dashed line in Fig. 4) as justified hereafter. Two cases are considered, i.e. coherent light and incoherent light: In the coherent case, when the perfectly coherent limit is reached (i.e. coherence time tends to infinity), the incoherence function I N (λ) becomes a Dirac function (red line in Fig. 4(a)) centered at λ 0 = 1700 nm. This wavelength was chosen because photons at λ 0 are not absorbed, since k(λ 0 ) ≈ 0 (see Fig. 4(a)). Therefore, at 1700 nm wavelength, the computed coherent absorption is almost equal to zero (red curve in Fig. 4(b)). Accordingly, the coherent photocurrent is also almost equal to zero at that wavelength (red curve in Fig. 4(c)). In the incoherent case, the spectral width of the incoherence function I N (λ) increases as the coherence time decreases (Fig. 4(a)). Therefore, a wider range of wavelengths enters into the calculation, including shorter wavelengths that are absorbed by the material (i.e. k = 0). Physically, in accordance with Heisenberg uncertainty principle, this arises from the fact that a time-windowed sinusoidal signal (∆t ≈ τ c ) becomes a polychromatic signal (with a width ∆E). In other words, while the carrier sinusoidal wave is at λ 0 = 1700 nm, the windowed wave contains shorter wavelengths which can be absorbed. As a consequence, the whole spectral range weightened by the incoherence function must be considered to compute the incoherent absorption A incoh (λ 0 = 1700 nm). This explains why the incoherent absorption increases around 1700 nm as the coherence time decreases (compare e.g. red and green curves in Fig. 4(b) around 1700 nm). Conversely, a wavelength that is strongly absorbed in the coherent case can lead to a lower absorption in the incoherent case (compare e.g. red and green curves in Fig.  4(b) around 500 nm). This is due to the fact that longer wavelengths experiencing k ≈ 0 come to play when determining the incoherent absorption. As a result, the increase or decrease of the absorption at a specific wavelength affects the photocurrent J(λ) as τ c varies. For instance, at λ 0 = 1700 nm, no photocurrent is generated in the coherent case. However, as τ c decreases, A(λ 0 = 1700 nm) and J(λ 0 = 1700 nm) increase. On the other hand, at λ 0 = 500 nm, a high photocurrent is achieved in the coherent case. However, as τ c decreases, A(λ 0 = 500 nm) and J(λ 0 = 500 nm) decrease. Since the total (integrated) photocurrent J = J(λ)dλ is obtained by integrating over a wide range of wavelengths, it can increase or decrease according to the values of the coherence time τ c , in agreement with the trends observed in Fig. 3. Interestingly, in the examples showed in Fig.  3, the coherence time of sunlight (τ c = 3 fs) is such that the photocurrent is higher in the incoherent case than in the coherent one.
In summary, we have proven that the coherence time of the light illuminating a solar cell influences drastically the generated photocurrent. Depending on the shape of the surface corrugation and on the active layer material, the photocurrent may increase or decrease as the coherence time changes. Such a result is fundamentally related to Heisenberg uncertainty principle and shows that incoherent light actually enhances solar cell efficiency by contrast to coherent light. Then, an optimal solar cell structure under coherent illumination is not necessarily an optimal one under incoherent light and vice versa. The optimization of a solar cell must therefore be carried on taking into account light incoherence, and not according to a coherent illumination as it is done usually. It is then necessary to consider a realistic value of the coherence time of the incident light in order to optimize the photocurrent. The present work highlights that further improvements of solar cells should take into account the incoherence of solar light.