On accurate simulations of thin-film solar cells with a thick glass superstrate

Klaus Jäger; Grit Köppel; Martin Hammerschmidt; Sven Burger; Christiane Becker

doi:10.1364/OE.26.000A99

1. Introduction

Numerical Maxwell solvers are well suited to simulate layer stacks of periodically structured thin optical films, which for example are used in thin-film solar cells. Because the computational cost scales with the size of the computational domain, the maximal volume of unit cells used for these simulations is limited.

In practice, such thin-film stacks often are created on glass superstrates with a thickness in the order of millimetres. A superstrate is a substrate, through which the incident light reaches the thin-film stack. Such thick glass layers can always be treated incoherently, because sunlight has a spatial coherence length of about 27 µm [1–3] and a coherence time of about 3 fs [4, 5] at 500 nm wavelength, which corresponds to a disc of about 27 µm radius and 1 µm height.

In recent years, several approaches were discussed to treat optical systems with thick incoherent layers [6–9]. With the finite element method (FEM) [10], such thick layers can be taken into account directly for example with a domain decomposition approach [11]. However, then the layer is treated as a coherent layer, which leads to reflectance/absorptance spectra with a very narrow interference pattern. This contradicts experimental observations. If the sample is not strongly scattering, this interference can be eliminated by combining simulations with well-selected different glass thicknesses [12]. But if the sample scatters strongly, eliminating the interference in the different diffraction orders becomes very cumbersome if not impossible.

A common way to take the incoherent superstrate into account in FEM is to represent the glass layer by a thin layer of a few hundreds of nanometres thickness, which is covered by an infinite half space filled with the same material (glass), as illustrated in Fig. 1(a). The effect of the air-glass interface, which is far off the computational domain, then is applied to the simulation results a posteriori.

Fig. 1 (a) Cross section through a periodic unit cell (enclosed by the green box) with a sinusoidal interface [14] and the glass and silicon half spaces above and below the unit cell. The a posteriori corrections described in this paper account for the interaction of light, which is reflected from the unit cell back into the glass half space, with the glass-air interface, depicted in red. The intensities are named as in Section 2: I^0,^a and I⁰ are the incident intensities in air and glass, respectively, and R^g is the reflectivity of the air-glass interface; all three variables must be treated for both polarizations. $I_{out}^{j}$ is the vector of all intensities emitted from the unit cell in the j^th correction order. The glass-air interface reflects the intensities $I_{in}^{j}$ back towards the unit cell. ρ^j is the contribution of the j^th correction order to the total reflectivity. (b) The sinusoidal texture used in the simulations, which is generated according to Eq. (14). (c) Atomic force microscopy (AFM) measurements of a sinusoidally nanotextured sol-gel layer on glass with 750 nm pitch and about 150 nm texture height, hence an aspect ratio of about a = 20%.

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In the zeroth-order correction only the initial reflection of the incident light beam at the air-glass interface is taken into account [13, 14]. Interactions of light, which is reflected from the computational domain back into the glass half space, with the glass-air interface are neglected. Higher-order corrections at the glass-air interface may become important, if a significant fraction of the back-reflected light is directed into angles, which are comparable to or larger as the critical angle [15]. Recently we discussed a first-order correction, which takes the first interaction of the back-reflected light with the glass-air interface into account [16, 17]. The first-order correction agrees much better with experimental data than the zeroth-order correction.

In this manuscript we calculate the scattering matrix, which allows to evaluate not only the first order but an arbitrary number of correction orders at negligible computational cost. The number of the correction order denotes, how often light, which is reflected from the computational domain back into the glass superstrate, interacts with the glass-air interface. We discuss the difference between the zeroth and higher-order corrections in dependence of the wavelength. Further, we confirm the validity of this approach with experimental data. The findings presented in this contribution are important for improving the accuracy in optical simulations of nanostructured solar cells on thick glass superstrates.

2. Theory

In numerical simulations, periodically structured thin-film layer stacks are usually treated with periodic boundary conditions on the side faces of the unit cell. Because of the perfect periodicity, far-field reflection and transmission into the top and bottom infinite half spaces, which cover the layer stack, only happens into discrete and well-defined diffraction orders. The reflectivity R of the layer stack in the glass halfspace can be calculated with

R_{p} (λ) = \frac{1}{{| E_{i, p}^{g} (λ) |}^{2} \cos θ_{i}^{g} (λ)} \sum_{n} {| E_{n, p}^{g} (λ) |}^{2} \cos θ_{n}^{g} (λ),

for p polarization and similar for s polarization, where the E_i_,_p and

E_{n, p}^{g}

are replaced by E_i_,_s and

E_{n, s}^{g}

, respectively. In Eq. (1) the electromagnetic field components

E_{n}^{g}

and the angles

θ_{n}^{g}

are output from a Fourier transform postprocess of the Maxwell solver. The subscript i denotes the incident wave, the superscript g denotes fields and angles in glass, and λ is the vacuum wavelength. The sum is taken over all directions into which the structure reflects.

The zeroth-order correction for reflection accounts only for the initial reflection of the air-glass interface. The reflectivity in air R⁰ is calculated using

R_{p}^{0} (λ) = R_{p} (λ) [1 - R_{p}^{g} (λ)] + R_{p}^{g} (λ),

for p polarization and similar for s polarization. Here, the superscript 0 denotes the zeroth-order correction,

R_{p}^{g}

and

R_{s}^{g}

are the reflectivities of the air-glass interface for the two polarizations (about 4% at normal incidence), as illustrated in Fig. 1(a) [13, 14].

We obtain higher-order corrections for reflection with the scattering matrix $S$ . Initially, light is incident onto the nanotexture with a k-vector k_i with absolute value k_i = 2πn_g(λ)/λ, where n_g(λ) is the refractive index of glass. The hexagonal nanotexture reflects light into several diffractions orders with directions $k_{n}^{out}$ , where n ∈ {1, 2 … N}. As illustrated in [14], N depends on the wavelength, the period of the nanotexture, its shape (square or hexagonal), the direction of incidence, and the refractive index of the material into which the light is diffracted.

The scattering matrix connects all outgoing directions $k_{n}^{out}$ to the directions $k_{n}^{in}$ , from which light impinges onto the nanotexture after it is reflected from the glass-air interface back towards the computational domain. They are connected to each other via $k_{n, ∥}^{out} = k_{n, ∥}^{in}$ and $k_{n, ⊥}^{out} = - k_{n, ⊥}^{in}$ , where ‖ denotes the component parallel to the lattice plane of the nanotexture (i.e. in the x y-plane) and ⊥ denotes the perpendicular component, i.e. in z-direction. Note that light incident onto the nanotexture from one of the incident directions $k_{m}^{in}$ with m ∈ {1, 2 … N) leads to the reflection in all directions $k_{n}^{out}$ with n ∈ {1, 2 … N}. Because both p and s polarizations have to be taken into account, the dimension of the scattering matrix is 2N ×2N, $S \in ℝ^{2 N \times 2 N}$ . In other programs, such as OPTOS [9] or GenPro4 [18], the scattering matrices are defined for non-periodic structures. Their size corresponds to the angular discretization and can be used to steer the accuracy of the calculations.

The incident k-vectors can be calculated using the reciprocal grid, which we illustrated in [14]; its gridpoints G_pq are given by

G_{p q} = p g_{1} + p g_{2},

where p, q ∈ ℤ are integers and g₁ and g₂ are the primitive vectors of the reciprocal grid. For a hexagonal grid with period (pitch) P they are given by

g_{1} = \frac{4 π}{\sqrt{3} P} (\frac{1}{2}, - \frac{\sqrt{3}}{2}), g_{2} = \frac{4 π}{\sqrt{3} P} (\frac{1}{2}, + \frac{\sqrt{3}}{2}) .

Now we find all the possible k-vectors via the condition that their parallel components, are shorter or of equal length than the k-vector of the incident light k_i. The parallel components are equivalent to the vectors connecting the origin O of the reciprocal grid to the gridpoints G_pq plus the parallel component of the incident direction, k_i_‖. Hence, the condition reads

| k_{p q, ∥}^{in} | = | G_{p q} - O + k_{i ∥} | \leq k_{i} .

The respective perpendicular components are given by

k_{p q, ⊥}^{in} = \sqrt{k_{i}^{2} - {| k_{p q, ∥}^{in} |}^{2}}

Since we are interested in the incoherent propagation of light in the glass superstrate, we have to look at the intensities. Hence, the elements of the scattering matrix are given by

S_{b a} = {| E_{b a}^{g} |}^{2} \frac{\cos θ_{b}}{\cos θ_{a}},

where E_ba is the b^th outgoing field originating from an incident field distribution where only the a^th incident field is different from zero with strength 1. The indices a, b ∈ {1, 2 … 2N} denote the possible direction and polarization combinations. We further need two vectors that account for reflection and transmission at the glass-air interface: R ∈ ℝ²^N and T ∈ ℝ²^N. Their components are calculated using the Fresnel equations.

The j^th-order correction for the reflection corresponds to the j^th interaction of the light, which is reflected from the computational domain into the glass superstrate with the glass-air interface, as illustrated in Fig. 1(a). It is calculated as follows: let $I_{in}^{j - 1} \in ℝ^{2 N}$ be the intensity distribution incident onto the nanotexture in the (j − 1)^th-order correction for reflection. The intensities emitted from the nanotexture are given by

I_{out}^{j} = S I_{in}^{j - 1} .

The contribution of the j^th-order to the reflection is given by

ρ^{j} = I_{out}^{j} \cdot T,

where · denotes the scalar product of two vectors. The intensity distribution incident onto the nanotexture in the j^th order

I_{in}^{j}

is given by component-wise multiplication of

I_{out}^{j}

with R,

I_{in, a}^{j} = I_{out, a}^{j} R_{a} .

We start the iterative process with $I_{in}^{0}$ , where only the two components corresponding to the k_i direction are different from zero. These two components are the incident intensities in p- and s-polarization. They are $I_{p}^{0} = 0.5 [1 - R_{p}^{g} (λ)]$ and $I_{s}^{0} = 0.5 [1 - R_{s}^{g} (λ)]$ , where interface. $R_{p}^{g}$ and $R_{s}^{g}$ are the initial reflectivites of the air-glass interface.

The reflectivity of the structure in the ℓ^th-order correction is given by

R^{ℓ} (λ) = \sum_{j = 1}^{ℓ} ρ^{j} (λ) + \frac{1}{2} [R_{p}^{g} (λ) + R_{s}^{g} (λ)] .

In this manuscript, we sum iteratively until convergence is reached, i.e. when

\frac{ρ^{ℓ}}{R^{ℓ - 1}} < τ,

with a convergence threshold τ = 10⁻⁴.

Note that the zeroth-order correction R⁰—defined in Eq. (2)—cannot be derived from Eq. (11). R¹ with ℓ = 1 is equivalent to the first-order correction, which also can be calculated directly without using the scattering matrix [16, 17],

R^{1} = \frac{1}{{| E_{i}^{a} |}^{2} \cos θ_{i}^{a}} \sum_{n} [{(t_{n}^{s} E_{s, n}^{g})}^{2} + {(t_{n}^{p} E_{p, n}^{g})}^{2}] \cos θ_{n}^{a} + R^{g} .

Here, the

E_{n}^{g}

vectors are decomposed into s- and p-polarized components before multiplying them with the Fresnel coefficients

t_{n}^{s}

and

t_{n}^{p}

, respectively, which describe transmission from glass into air. Note that

E_{i}^{a}

is connected to

E_{i}^{g}

, which is used in Eq. (1), via transmission from air into glass. For incoherent sunlight, Equation (13) has to be applied to incident s- and p-polarized light, independently.

3. Simulation and experimental details

To test the approach described above, we used the sinusoidal nanotexture denoted as negative cosine texture in earlier work [14, 19], which is illustrated in Fig. 1(b). Mathematically, it is—up to vertical and horizontal scaling—described with

f (x, y) = - \cos (x) \cos [\frac{1}{2} (x + \sqrt{3} y)] \cos [\frac{1}{2} (x - \sqrt{3} y)] .

The simulations required to determine the scattering matrix were performed with the Maxwell solver JCMsuite, which utilizes the finite element method (FEM) [20]. The optical system studied for the simulations in this work is sketched in Fig. 1(a). The optimal dimensions of the tetrahedra and prisms constituting the three-dimensional mesh for the FEM simulations were found to be maximally 50 nm for the sinusoidal interface and 100 nm for the volume elements, as discussed earlier [14]. One advantage of JCMsuite is that all 2N incident electric fields (N directions and two polarizations) can be treated in one simulation, which makes these calculations very efficient.

Experimentally, Köppel and co-workers successfully demonstrated the use of sinusoidal nanotextures as antireflective scheme for thin-film solar cells with a liquid-phase crystallized silicon (LPC-Si) absorber [21, 22]. The nanotextures are fabricated with nanoimprint lithography [8, 23] using high-temperature stable sol-gels [24], which are suited for the LPC process; an atomic force microscopy (AFM) picture of such a texture is shown in Fig. 1(c). Onto the nanotexture an about 10 µm thick nanocrystalline silicon layer is deposited, which subsequently is molten and re-crystallized with a continuous-wave (cw) line laser [25, 26].

To calculate the aspect ratio we divided the texture height, derived from AFM measurements, by the pitch; the error of the aspect ratio was deduced from the AFM error of ±20 nm. The reflectance spectra were obtained with an integrating sphere of 150 nm diameter, which is attached to a PerkinElmer LAMBDA 1050 spectrophotometer, at θ_in = 8° angle of incidence.

The experimental sample has an LPC-Si layer thickness of about 10 µm, while in the simulations the silicon layer is considered to be infinitely thick. For wavelength shorter than about 600 nm, all light that reaches the 10 µm-thick silicon layer will be absorbed. Hence, we compare simulated and measured reflectance spectra in a 350–600 nm range.

4. Results

Figure 2 shows the angles of the different diffraction orders in glass and air for light, which is diffracted by hexagonal periodic structures with 500 nm and 750 nm pitch at normal incidence. The zeroth diffraction order (θ_g = θ_a ≡ 0) is not shown. In glass, the different diffraction orders are present until much longer wavelength than in air. The zeroth order correction [Eqs. (1) and (2)] takes all diffraction orders in glass into account, even if they cannot propagate in air, leading to an overestimated reflectivity.

Fig. 2 Angles of the diffraction orders into which a hexagonal periodic structure with a pitch of (a) P = 500 nm and (b) P = 750 nm scatters light at normal incidence. The figure shows the angles in glass and in air. The zeroth diffraction order (θ_g = θ_a ≡ 0) is not shown. Experimental data is available in the wavelength range (350–600 nm) – this range is marked by the white boxes. In glass, the diffraction orders are present up to much longer wavelength than in air, which is also illustrated in (c). For P = 750 nm, also the 4^th and 5^th diffraction orders are present in glass at short wavelengths, but in (b) they are omitted for clarity.

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Figure 3(a) shows numerical and experimental 1 − R spectra for a nanotextured sample with 500 nm pitch. Large differences are seen between the 0^th-order correction [Eq. (2)] and the other two curves. The converged data was obtained iteratively with the scattering matrix approach until the threshold of τ = 10⁻⁴ was reached [Eq. (12)]. It is a bit lower than the 1^st-order correction [Eq. (13)], which constitutes an upper bound to 1 − R. Both the 1^st-order correction and the converged curve agree well with the measurement, in contrast to the 0^th-order correction. Also for a pitch of 750 nm, shown in Fig. 3(b), the 0^th-order correction shows much worse agreement with the experimental data than the other two numerical curves.

Fig. 3 Numerical and experimental 1 − R spectra for the nanotextured layer stacks illustrated in Fig. 1(a) with (a) 500 nm pitch and (b) 750 nm pitch. Numerical results were calculated with the 0^th-order correction [Eq. (2)], the 1^st-order correction [Eq. (13)], and iterated using the scattering matrix, until convergence was reached for the threshold τ = 10⁻⁴ [Eq. (12)]. The 0^th-order correction differs strongly from the other curves because not all diffraction orders that are present in glass can propagate into air [see Fig. 2]. Simulation and experimental results are shown for θ_in = 8°.

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We take a closer look into the convergence of the scalar scattering matrix approach in Fig. 4, which shows the correction order ℓ required to reach convergence with τ = 10⁻⁴ for a nanotexture with 750 nm pitch and 25% aspect ratio. Further, the figure shows the number of directions N, into which the nanotexture diffracts light in glass. We observe that ℓ always increases towards the edge where the number of directions N decreases. This can be understood as follows: towards such an edge, the angles of the outermost scattered diffraction order become larger and larger, as shown on the right hand side of Fig. 4. Hence, the reflectivites at the glass-silicon and glass-air interfaces increase as well, and less light can leave the glass per correction order – either into air or into silicon, and a larger number of correction orders is required to reach convergence.

Fig. 4 The correction order ℓ required to reach convergence with τ = 10⁻⁴ [red line, see Eq. (12)] for a nanotexture with 750 nm pitch and 25% aspect ratio at normal incidence. Further, the number of directions N is shown, into which the nanotexture diffracts light in glass [cyan dots]. For three wavelengths, these directions are shown on the right. The numbers correspond to the polar angles. More information is given in [14].

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Figure 5 shows the mean 1 − R for numerical and experimental data as a function of the aspect ratio a of the nanotexture and two different pitches. The mean is taken between 350 and 600 nm wavelength. For both pitches, the first-order correction and the converged curve resemble the experimental data much better than the zeroth-order correction. Even for planar structures (a = 0), R⁰ and R¹ differ by several percent. Figure 5 shows that 1 − R stabilizes for aspect ratios a > 0.4. At 500 nm pitch, 1 − R reaches almost 0.92 while for P = 750 nm, only 0.90 are reached. However, aspect ratios exceeding ≈ 0.3 are difficult to realize experimentally [21].

Fig. 5 Simulated and experimental mean 1 − R between 350 and 600 nm wavelength for a nanotexture pitch of (a) P = 500 nm and (b) 750 nm. All results are shown for θ_in = 8°.

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5. Summary and outlook

When simulating the optics of periodically nanotextured layer stacks on thick glass superstrates it is imperative to take the front interface into account adequately. Often, one only corrects for initial reflection losses at the air-glass interface, which are about 30% at normal incidence. We show that it is also very important to take higher-order reflections at the glass-air interface into account, which occur for light that is reflected from the nanotextured layer stack back into glass. The incoherent scattering matrix allows to evaluate an arbitrary number of reflection orders at negligible computational cost. However, for a rough estimate also the first-order correction might be sufficient, as recently shown [17]. The first-order correction is easier to evaluate because it can be calculated without the scattering matrix.

In this manuscript we only discuss the correction for glass superstrates with a flat air-glass interface. The model can be easily expanded to superstrates with periodically textured glass-air interfaces—if their period is the same as that of sinusoidal nanotexture. Then, the vectors R and T, as defined in Section 2 have to be replaced by a scattering matrix for the glass-air interface. If that interface has a different period or is randomly textured, both nanotextured interfaces would need to be described with scattering matrices for non-periodic nanotextures [9, 18].

Currently, the method described in this manuscript can be used to correct reflection spectra. However, it cannot correct the absorptivity in thick absorbing layers, such as silicon, yet. Scattering layer stacks, which are represented as scattering matrices, can be coupled to incoherent layers in matrix-based solvers [9, 18]. In a next step, we will investigate combinations between our FEM solver and such approaches in order to calculate full absorption profiles of layer stacks consisting of thick incoherent layers and thin periodically nanotextured systems.

Funding

German Federal Ministry of Education and Research (BMBF) via Nachwuchswettbewerb NanoMatFutur (No. 03X5520); Einstein Foundation Berlin (ECMath project SE6).

Acknowledgments

The results were obtained at the Berlin Joint Lab for Optical Simulations for Energy Research (BerOSE) of Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Zuse Institute Berlin and Freie Universität Berlin.

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On accurate simulations of thin-film solar cells with a thick glass superstrate

Abstract

1. Introduction

2. Theory

3. Simulation and experimental details

4. Results

5. Summary and outlook

Funding

Acknowledgments

References and links

Cited By

Figures (5)

Equations (14)

Optics Express