Light trapping in thin silicon solar cells: A review on fundamentals and technologies

Thin, flexible, and efficient silicon solar cells would revolutionize the photovoltaic market and open up new opportunities for PV integration. However, as an indirect semiconductor, silicon exhibits weak absorption for infrared photons and the efficient absorption of the full above bandgap solar spectrum requires careful photon management. This review paper provides an overview on the fundamental physics of light trapping and explains known theoretical limits. Technologies that have been developed to improve light trapping will be discussed, and limitations will be addressed.


| INTRODUCTION
Forty years after Eli Yablonovitch submitted his seminal work on the statistics of light trapping in silicon, 1 the topic has remained on the forefront of solar cell research due to the prevalence of silicon in the photovoltaic (PV) industry since its beginnings in the 1970s. 2,3 Despite the rise of a plethora of alternative technologies, more than 90% of newly installed PV plants are still based on silicon. 3 The reasons for this are multifaceted; certainly, silicon PV is the most mature and reliable technology with the first modern solar cell ever made dating back to 1954. 4 Coincidentally, silicon does not only encompass excellent material properties-with respect to both performance and processability-it also happens to be the second most abundant material in earth's crust. 5 The cost of silicon has seen some volatility in the past due to rapid demand increases with limited manufacturing capabilities, 6 while the silicon wafer contributes an average of 14% to total module production costs. 7 Therefore, it is of economic interest to reduce the amount of silicon used in a PV cell by reducing the absorber thickness. 8 It will be further detailed below that thickness reductions can also lead to efficiency enhancements and to mechanical flexibility. However, crystalline silicon is an indirect semiconductor and therefore exhibits poor absorption for low-energy photons. 9 To ensure complete absorption of the above bandgap solar spectrum, the optical path length of light within the silicon must be increased. Strategies that increase the optical path length with the goal to mitigate photon escape are referred to as light-trapping methods. 10 In this paper, the fundamentals of light trapping in crystalline silicon will be discussed and a review is presented on existing light-trapping strategies. First, the optical properties of silicon and the benefits of thin silicon solar cells will be addressed. Subsequently, known theoretical concepts will be derived and discussed. The fifth part of the paper presents examples of existing light-trapping strategies. The sixth part discusses two practical issues arising from advanced light-trapping technologies: parasitic absorption and surface recombination. In summary, this review provides a comprehensive overview on the topic of light-trapping in silicon with the goal to facilitate the development and benchmarking of emerging strategies.

| OPTICAL PROPERTIES OF SILICON
Silicon is an indirect semiconductor with a bandgap of 1.124 eV at room temperature, 11 which corresponds to a vacuum wavelength of 1103 nm. The band structure of silicon as determined by Chelikowsky and Cohen is presented in Figure 1A. 12 The band structure relates the electron momentum within the crystal, expressed by its wave vector k, with the energy of the electron. The relation depends on the direction of the momentum relative to the crystal directions. The crystal directions in reciprocal or momentum space are displayed on the x axis in Figure 1A. Regions of the band diagram in which no possible k vector exist for a certain energy are called an energetic bandgap. 13 The states above the bandgap belong to the conduction band, and the states below the bandgap belong to the valence band. The numerical value of the bandgap is defined as the energetic difference between the highest point of the valence band and the lowest point of the conduction band. 13 Through absorption of photons and phonons (i.e., crystal lattice vibrations), electrons can be excited from the valence into the conduction band.
This process is called an interband transition. Photons barely carry momentum, while phonons barely carry energy. For an electron to significantly change its energy, it needs to absorb a photon, and to change its momentum, it needs to absorb a phonon. Silicon is an indirect semiconductor, which means that the maximum of the valence band and the minimum of the conduction band occur for different k vectors. Accordingly, for an interband transition to occur, either the photon energy has to be significantly larger than E gap as shown by the blue arrow in Figure 1B or the transition has to be performed by simultaneous absorption of a photon and a phonon 13 as shown by the red arrows in Figure 1B. As the latter process involves three particles rather than only two, it is less likely than a direct transition. Therefore, indirect semiconductors such as silicon absorb less strongly for photon energies right above their bandgap. Figure 1C shows the wavelength-dependent absorption coefficient of silicon 9 and GaAs. 14 The direct bandgap in GaAs leads to a steep onset of its absorption coefficient near the bandgap, whereas the absorption coefficient of silicon increases slowly with decreasing photon wavelength. Thus, for the absorption of the full above bandgap solar spectrum, GaAs solar cells can be made thin, typically in the order of a few hundred nanometers to a couple of micrometers, 15 while silicon solar cells have to be significantly thicker, typically in the range of 100-500 μm. 16

| PROPERTIES OF THIN SILICON
While the economic benefits of decreasing material usage were briefly discussed in Section 1, this section focusses on the physical properties of thin silicon. The section is divided into the mechanical and electrical properties of thin silicon.

| Mechanical properties
When thinning a material, the obvious advantage that comes to mind is a reduction in weight. However, silicon solar cells have to be carefully encapsulated and packaged to avoid degradation and mechanical failure such as cracking. The encapsulation is usually performed with several hundreds of micrometers of EVA and a few millimeter-thick glass. 17 Compared to the approximately 100 μm of silicon, the weight of the EVA and glass dominate the overall module weight. Furthermore, when thinning silicon from a few 100 μm to 50 μm, silicon becomes brittle and fragile and handling such thin wafers becomes increasingly difficult. 16 However, when decreasing thickness even further, eventually silicon stops being brittle and instead starts becoming flexible. 18,19 Figure 2 shows pictures of silicon with a thickness of 10 μm being either bent 20 or cut with scissors. 21 Therefore, a high performing silicon solar cell with thickness of 10 μm or less would be F I G U R E 1 (A) Band diagram of silicon according to Chelikowsky and Cohen. 12 (B) Schematics of direct and phonon-assisted optical transitions. (C) Wavelength-dependent absorption coefficient of silicon and GaAs. Data from Green and Aspnes and Studna 9,14 highly desirable for mechanical reasons and would allow easy integration of silicon solar cells with any surfaces. Very thin and flexible high efficiency solar cells have been demonstrated based on GaAs. 22,23 Contrary to silicon, GaAs is a direct bandgap semiconductor, and hence, less than a couple micrometer thickness is sufficient for complete solar spectrum absorption. While very desirable for economic and ease of use reasons, obtaining thin, flexible, and yet highly efficient silicon solar cells is only possible through very effective light trapping. Yet first flexible crystalline solar cells have been demonstrated 19,24 ; for example, Xue et al. reported an efficiency of 12.3% with only 2.7-μm silicon thickness. 25

| Electrical properties
A solar cell is an optoelectronic device, and while efficient photon collection is crucial, optimizing the electrical properties is equally important. The thickness influences the electrical properties in many different ways. In this section, the influence of thickness on shortcircuit current density, open-circuit voltage, and fill factor will be discussed. The short-circuit current density j sc of a solar cell is determined by the external quantum efficiency EQE(λ) and the solar irradiance flux Φ(λ) via where λ gap is the wavelength that corresponds to the bandgap energy.
The external quantum efficiency describes how many electron hole pairs are created out of incoming photons with a specific wavelength and can in turn be described by the absorption within the absorber material A(λ) and by the charge carrier collection efficiency, also called internal quantum efficiency IQE(λ): Evidently, the absorption is the main topic of this paper and will be discussed further below.  which the short-circuit current density changes: Consequently, any improvements on the short-circuit current will also improve the open-circuit voltage.
The fill factor of a solar cell is related to the open-circuit voltage 36 and to the series and shunt resistances. 37 A thinner cell can be beneficial for the fill factor due to the higher open-circuit voltage, 36 but a strong correlation is not always observed. 33 Thinning the wafer thickness decreases the vertical series resistance but increases the lateral series resistance.
All in all, thin silicon solar cells are desirable for several reasons as long as full light absorption can be achieved.

| THE FUNDAMENTALS OF LIGHT TRAPPING
In this section, the fundamentals of light trapping will be discussed.
First, the concept of path length enhancement will be introduced, then the Lambertian limit, also called Yablonovitch or 4n 2 limit, will be explained. It will be shown how the local optical density of states (LDOS) relates to this limit and how through manipulation of the LDOS the Lambertian limit can be beaten.

| Path length enhancement
The Lambert-Beer law states that the extinction of light within absorptive media is described by where I is the intensity of light having traveled a distance x and I 0 is the intensity at x = 0. α is the absorption coefficient, which is connected to the imaginary part of the refractive index n k of the material and to the vacuum wavelength λ by α = 4πn k /λ. Hence, the amount of light that is absorbed in a medium depends on the distance x traveled through the medium, also called the path length, and on the absorption coefficient α, which in turn depends on the wavelength as shown in Figure 1C. Figure 3 provides an understanding on how long of a distance light with a spectral distribution according to the standard AM 1.5 spectrum has to travel through silicon in order to be fully absorbed. The graph presents the corresponding short-circuit current density that is lost due to transmission through the silicon dependent on the thickness calculated via For the thickness and wavelength-dependent absorption A(λ, t), it was assumed that no other losses (such as reflection) occur, and hence, the absorption is equal to 1 minus transmission: The maximum current j sc max that could be generated from the AM 1.5 spectrum with a material with 1.12-eV bandgap, if all above bandgap photons were absorbed, is 44 mA/cm 2 . Figure 3 shows that to almost fully absorb the AM 1.5 spectrum, 1 mm of silicon would have to be used, which is significantly thicker than even conventional solar cells, currently ranging at around 180-μm thickness. Increasing the absorption while maintaining or even decreasing the silicon thickness requires light-trapping strategies that enhance the path length x of the light. One way to avoid the escape of light on the rear is by placing a mirror, which immediately doubles the path length and is done in almost all monofacial solar cells. But even doubling of the path length is not enough to achieve loss-free absorption, and rather, it is desirable to have light traveling under an angle to the surface. This can, for example, be achieved by texturing the front and rear surface as will be explained further below.
In the next section, the theoretical limits and alternative ways to look at path length enhancement will be discussed. Guidelines on how to measure the path length enhancement can be found in this review paper by Mokkapati and Catchpole. 38

| Lambertian limit
The Lambertian limit is also referred to as the 4n 2 limit or as the Yablonovitch limit acknowledging its first publication in 1982. 1 There are different ways of deriving this limit in a ray optical picture. As the original derivation by Yablonovitch is widely available in literature (e.g., Fonash 10 ), here, a different derivation is presented, which is only briefly mentioned by Yablonovitch as the geometric series approach. 1 Let us imagine a slab of silicon in air-thick enough so that wave optical effects do not occur-and light incident with energy close to the band edge, such that we obtain weakly absorbing conditions. In the case of a smooth front and rear surface ( Figure 4A), the light ray will enter the material under an angle φ to the surface normal, whereas φ depends on the incoming angle and the refractive index according to Snell's law. The absorption path length x through the silicon will be where t is the thickness of the slab. For weakly absorbing conditions, most of the light will exit the slab at the rear as depicted in Figure 4A.
Escape on the rear can easily be avoided by placing a mirror on the rear as shown in Figure 4B. An ideal mirror enhances the path length by a factor of 2, but for a weakly absorbing medium, there will still be escape on the top surface. If instead of an ideal specular mirror, we use an ideal Lambertian reflector, the light will be randomized in its F I G U R E 3 Equivalent short-circuit current density lost due to transmission depending on the thickness of silicon calculated with the Lambert-Beer law for the AM 1.5 spectrum direction upon reflection on the rear. The Lambertian light-trapping limit assumes perfect randomization of the light within the silicon, whereas it does not matter how this randomization is achieved. In the following, we are discussing the case of an ideal Lambertian rear reflector, but a textured front surface could achieve the same randomization. Depending on its angle, light reaching the front surface will either escape ( Figure 4C, arrow 1) or encounter total internal reflection (TIR) ( Figure 4C, arrow 2). The critical angle φ c at which the transition between escape and TIR occurs is given by And with small-angle approximation: This defines a three-dimensional cone within which light can escape as shown in Figure 4C. For silicon at a wavelength of 1000 nm, the refractive index amounts to 3.572, 9 and hence, φ c = 16.30. Note that on the air side, light can be accepted from the full hemisphere and will also escape into angles within the full hemisphere due to reciprocity. Let us now derive the probability for randomized light to escape through the escape cone. To calculate this probability, we need to divide the number of rays within the escape cone by the total number of rays. Due to the perfect randomization, we can assume an isotropic distribution of rays, and hence, the number of rays within the escape cone is proportional to the solid angle of the sphere element defined by 2φ c . In small-angle approximation, this solid angle becomes The total number of rays is proportional to the surface of the full hemisphere. This leads to a probability of escape P e : From this result, there are two different strategies to derive the intensity within the silicon slab: (1) a simple detailed balance consideration and (2) summing up over all possible reflections via a geometric series. In both cases, it is assumed that no photons are lost other than through escape within the front surface escape cone. In other words, no photons are absorbed in the bulk or at the surfaces. In this case, detailed balance dictates that the number of incoming photons # inc is equal to the number of escaping photons # esc . As P e ¼ 1 2n 2 , and the number of photons # S1 reaching the front surface must be 2n 2 # inc in order to ensure # esc = P e # S1 = # inc . The same result is reached when adding up all reflections that occur at the front surface: Hence, we derive that the intensity of rays within the silicon slab going towards the upper hemisphere is 2n 2 greater than the intensity of incoming photons. For isotropy reasons, the same is true for the intensity of photons going towards the bottom surface. Therefore, in total, we have increased the number of rays by a factor of 4n 2 as compared to the incoming number of rays, and these rays are isotopically F I G U R E 4 Schematic of light with energy close to the bandgap passing through a thin silicon slab in air (A) without any light trapping, (B) with ideal specular mirror, and (C) with ideal Lambertian reflector on the rear side. In (C), the escape cone of α c = 16.30 within the silicon and the air escape/acceptance cone that corresponds to the full hemisphere are shown distributed over the whole solid angle within the silicon. Again, from here, there are multiple ways to determine the effective path length enhancement. One way is to determine the absorption within a volume element as described by Yablonovitch and as schematically presented in Figure 5A. First, we define B as the internal intensity per unit internal solid angle, which is an isotropic function with constant value of Absorption A within a volume element dV = tdA i is then given by This equation yields the famous 4n 2 absorption enhancement.
Again, it is assuming weak absorption, so no change in intensity across the thickness of the silicon slab and perfect light randomization, that is, isotropic light conditions within the slab. For a more general derivation of the Lambertian limit that does not require weak absorption, the author refers to work by Green. 39 An alternative way to arrive at the 4n 2 absorption enhancement is depicted in Figure 5B. One can assume each point of the surface to reflect light in an ideal Lambertian way, that is, distributed over the solid angle of a hemisphere with intensity dropping with cos(φ). At the same time, each reflected ray will have a path length of t/cos(φ).
Therefore, the average length t of a ray weighted with its intensity amounts to the slab thickness t: In Figure 5B, there are three different rays shown, which despite their different length, all have the same intensity weighted length t .
There are 4n 2 # inc rays inside the slab as derived above, and hence, the path length enhancement amounts to 4n 2 .
In this section, the Lambertian light-trapping limit was derived from a ray optical perspective, with a slightly different but yet equivalent path to the original work by Yablonovitch. 1 In the following, this result will be connected with statistical mechanics, and it will be shown that the Lambertian limit is not actually a hard limit in the thermodynamic sense but can be overcome through design of the optical environment.

| Statistics and density of states
In the following section, we will derive the Lambertian limit from a statistics point of view. Again, the emphasis is on motivating the derivation conceptionally, while approaching the topic from a slightly different angle than other literature sources. First, let us take a look at the LDOS in a medium, which shall be homogeneous and isotropic throughout the section. The LDOS is defined as the number of existing states per unit volume at a certain energy and can be derived from the dispersion relation, that is, from the relation between momentum and energy. The dispersion relation is shown in Figure 6 for light in vacuum (blue curve) and light in a homogenous, isotropic, and nondispersive medium with refractive index n (red curve). The This means that in a 3D homogenous, isotropic, and nondispersive medium with refractive index n, the LDOS is n 2 higher than in vacuum. Now, we imagine the (isotropic and homogenous) silicon slab sitting in vacuum and within isotropic irradiance. Furthermore, we consider an ergodic system, that is, a system in which all states with the same energy are equally likely to be populated. In this case, the available states within and outside of the silicon slab will be populated according to Bose-Einstein distribution statistics. As the LDOS within the silicon is n 2 higher than outside, we will have an n 2 higher light intensity within the silicon. Once we place a mirror, the system is no longer ergodic, as there is no exchange between all states possible anymore. In this case, we can use the same arguments as explained in Concretely, there are two different routes to achieve this goal: (1) decreasing the LDOS outside of the silicon or (2) increasing the LDOS within the silicon. We will see below how such tasks can be performed, for example, with gratings or photonic crystals. It is important to note that, without such alterations of the LDOS, the 4n 2 limit cannot be overcome 38 ! Naturally, LDOS modification also exhibits fundamental limits. The smallest LDOS possible in air would mean that light can only escape from the silicon into air under one specific angle.
Due to reciprocity, this means that light would also only be allowed to enter from one certain direction. Therefore, strategies that decrease the air LDOS inherently also decrease the light acceptance angle 38 and hence can never work properly without sun tracking or under diffuse light conditions. 40

| LIGHT-TRAPPING TECHNOLOGIES
This section presents an overview of different light-trapping methods and technologies that have been used in production or have otherwise been presented in literature.

| Pyramids
Pyramidal textures are the most common method to enhance the path length in silicon by randomizing the light direction. Their success is based on two major reasons: (1) industrial-scale texturing is feasible using anisotropic alkaline etching 49 and (2) close to perfect randomization can be achieved with pyramidal textures. 50 Figure 7A shows a scanning electron microscopy image of a silicon heterojunction solar cell with random pyramid texture captured under 45 viewing angle. In Figure 7B, the cross section of the same solar cell as in Figure 7A 59 Interestingly, it has been shown that random pyramids perform better than periodic arrays as they offer closer to Lambertian properties. 54,55 Another variant of the inclined surface approach, which can also be fabricated with anisotropic etching, is triangular cross-sectional grooves. 45,60 The performance of such structures with respect to antireflection and light-trapping properties has been extensively This shows that for optimal light management, not only light trapping but also avoiding reflection and parasitic absorption is crucial. 27,68 If the thickness of the silicon absorber is reduced to only a few micrometers and is therefore similar to the pyramid size, (random) pyramids are no longer viable as Lambertian reflectors. The following sections will present light-trapping strategies that also work with very thin silicon. Note that most of these strategies rely on LDOS manipulation within or outside of the silicon and therefore have either limited acceptance angle or limited wavelength bandwidth.

| Gratings and photonic crystals
In this chapter, we will discuss light-trapping strategies based on periodic structures with periodicity being in the same order of magnitude as the wavelengths contained are the solar spectrum. Contrary to the micrometer-sized pyramids introduced in the previous chapter, a ray optical treatment is not possible for such structures. The mathematical description requires to use wave optical physics as dictated by Maxwell's equations. 69 In this section, nanostructures with periodic features will be discussed. Depending on the community and framework, such structures are referred to either as gratings or as photonic crystals. Because the term photonic crystal implies a periodic structure, we will use this term in the following. However, many of the literature examples that are cited refer to their structures as gratings. 56,70,71 A photonic crystal is a material with periodic modulation of the refractive index 69 resulting in an energetic band structure in reciprocal space equivalent to the electronic band structure in crystalline solids. 73 Similar to semiconductors, photonic crystals feature energetic bandgaps in which propagation of light is prohibited due to the absence of allowed states. 73 The dispersion relation or also called band diagram of a one-dimensional photonic crystal with reciprocal lattice constant g is shown in Figure 8A and compared with light in free space. 69 From such a (photonic) band structure, the LDOS can be derived as explained in Figure 6. photonic crystal compared to the LDOS in free space is shown in Figure 8B. 72 Within the energetic bandgap area, the LDOS disappears; no propagation of light with the respective frequencies is allowed.
Consequently, light incident on a photonic crystal within this frequency regime is perfectly reflected. The bandgap usually corresponds to a narrow frequency regime, and therefore, the observed reflection has a specific color. Furthermore, the frequency regime depends on the propagation direction, that is, a photonic crystal observed from different angles will appear in different colors. We can observe such that is, LDOS, that results from sub-and near-wavelength periodic structures. 70,77 They found an enhancement of around 12n 2 to be possible, however, at a narrow periodicity size to wavelength ratio. 70 Mokkapati and Catchpole summarized the limits of nanophotonic light trapping in their 2012 review paper and also point out that enhancements beyond the Lambertian limit in bulk structures must have a limited acceptance angle due to étendue conservation. 38  A popular class of two-dimensional photonic crystals are periodic nanowire and microwire arrays. [80][81][82][83][84] As one example, Garnett and Yang reported a factor 73 of path length enhancement. 81 The inverse geometry, nanohole arrays, has also been explored for light-trapping silicon solar cells. 85

| Nonperiodic nanostructures
In this chapter, we will discuss light-trapping strategies based on nonperiodic structures with structure size similar to the light wavelengths involved. Above, we saw for the ray optical regime that random pyra- To this end, promising results were obtained through modulated surfaces with random features at different length scales, including nanocones and micropyramids. 88,89 What is to date the record absorption within 1-μm-thick crystalline silicon and would correspond to an equivalent short-circuit density of 26.3 mA/cm 2 was achieved with a hyperuniform texture. 90 A further approach to achieve enhanced absorption with nonperiodic nanostructures is the creation of a Fabry-Pérot cavity with the absorber material lying within this cavity and either metals or dielectric thin films effectively acting as mirrors. 91,92 This approach is based on the principle of multi-beam interference and therefore has inherently narrow wavelength response. 93  F I G U R E 9 Schematics of (A) 3D-printed concentrator arrays for external light trapping 94   It is important to distinguish between absorption enhancement within the silicon absorber layer and absorption within other components, also known as parasitic absorption. Parasitic absorption usually does not contribute to the photocurrent and is instead dissipated into heat. This loss mechanism is particularly pronounced in structures in which considerable electromagnetic field enhancement occurs within materials other than the crystalline silicon such as it is the case in plasmonic structures. Researchers working on light-trapping strategies should carefully evaluate their results with respect to parasitic absorption and angle acceptance. If absorption enhancement beyond the Lambertian limit is observed, the structure needs to provide either LDOS enhancement within the absorber or LDOS reduction (and thereby, acceptance angle reduction) outside the absorber. If none of these conditions are present, then absorption enhancement beyond the Lambertian limit is with today's knowledge not possible.