1. Introduction

Silicon is a promising element for energy conversion and harvest [1–3]. One method for such conversion is to trap most incoming solar light with engineered surfaces to convert the photon energy into useful heat [4,5]. Another method for such conversion is to generate electricity using photovoltaic effects thanks to the match between the silicon band gap (1.12 eV) and majority energy of solar radiation [6,7]. While the two working principles are essentially different, their success commonly relies on the low reflectance (R) from silicon surfaces [8]. Various ways have been attempted to reduce R, such as adding metallic islands [9], fabricating random or periodic structures [10–13], modifying lattice interfaces [14], coating anti-reflection films [15], and generating a porous or oxidized layer [16,17]. Besides them, physically roughening the surface profile has been viewed as a promising choice because of its simplicity and freedom from other materials involvement [18]. Roughened crystalline silicon surfaces, similar to other commonly-seen rough surfaces, were usually assumed to be Gaussian random surfaces (Gauss) [19,20] or exponential random surfaces (Exponent) [21].

However, their optical response consistently showed up differences between the experimental results and those from modeling [22–24]. Neither a Gauss surface nor an Exponent surface was able to illustrate the anisotropic roughness of crystalline silicon [25]. The perfect crystal periodicity and lattice orientation actually made rough silicon surfaces very unique. Their optical responses accordingly differ from those of other rough surfaces. The objective of this work is therefore to quantitatively investigate the uniqueness in both profile and optical responses of roughened crystalline silicon surfaces. Optical responses here are scattering patterns and the hemispherical R from surfaces. For the profile investigation, a numerical model will be proposed to differentiate fabricated samples from other well-known surface types. For scattering patterns, they will be simulated and measured from our samples as a consistency check. Modeling results from a planar surface and a Gauss surface will be plotted together for demonstrating the uniqueness. The R from all surfaces will also be calculated considering the potential for energy-related applications. At the same time, the mass production feasibility and roughness flexibility of crystalline silicon rough surfaces will be demonstrated.

The first step of our work is to develop a surface maker, which is able to generate rough surfaces automatically. Two samples of different roughness will be fabricated to assure the roughness flexibility. The second step is to obtain surface profiles of samples using an atomic force microscopy (AFM). The uniformity, randomness, and other profile statistics will then be calculated. The commonness and distinction in profile statistics between samples and other rough surfaces will be quantitatively illustrated. The third step is to check the consistency of scattering patterns from measurements and modeling based on partial sample profiles. Scattering patterns from samples will be measured with a custom-designed scatterometer [26]. Their scattering patterns and R will also be acquired using the finite-difference-time-domain (FDTD) algorithm [27]. The fourth step is to plot scattering patterns from a planar surface, Gauss, and our samples together. The R from all surfaces will be listed for comparison.

2. Surface maker and fabricated samples

Figure 1 shows the sketch of our proposed randomly rough surface maker, which fabricates samples purely through scratching. Its static and dynamic analysis has been conducted with the finite element method software (ANSYS 13.0). The safety factor of maximum stress is 2.0 to assure robustness. The current size of samples is 40 mm × 40 mm for measurement convenience, but the maker can be enlarged to fabricate bigger samples easily. The maker is mainly composed of a Scotch yoke [28,29] and Roberval balance [30] besides a driving motor. Both the yoke and balance are made of aluminum 6061, whose material properties can be found in Ref. 31. The Scotch yoke turns rotary motion of the motor into linearly back-and-forth motion of its platform. The bottom of the platform is attached to a piece of sandpaper for scratching. The linear motion generates a rough profile consistent along one direction. The Roberval balance has two platforms in contrast to the single platform of the yoke. The upper platform supports a silicon sample facing the sandpaper. The lower one holds a weight defining the applied pressure on the sample. Though a constant loading (2 kg) is not uniformly applied on the lower platform, the Roberval balance gives uniform pressure on the contact area.

 

Fig. 1 The randomly rough surface maker and its two main components, Scotch yoke and Roberval balance. Arrowheads in the full view mark moving directions of components. A sample is placed on the upper platform of Roberval balance as specified with a square.

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Samples were fabricated using a 508-μm-thick and polished silicon wafer, which was diced into small squares. A square was mounted on the upper platform of the balance carefully such that the crystal orientation (100) was parallel to the direction of linear motion. Each sample was a square scratched for fifty minutes. The roughness of two samples was determined by SiC particles stuck on the sandpaper. The average diameters and their corresponding sandpaper number (No.) were 35 ± 1.5 μm (No. 400) and 15.3 ± 1 μm (No. 1200) following the FEPA (Federation of European Products of Abrasives) standard. Each sandpaper was renewed during the scratching process, assisting the roughness uniformity. After scratching, samples were cleaned with de-ionized water and purged with nitrogen.

The profile statistics and roughness uniformity of samples were investigated using profiles measured by an AFM (Veeco D5000) with a tip of radius 7 nm. The AFM stage resolution along lateral and vertical direction was 1.5 nm and 0.1 nm, respectively. Each sample was scanned nine times, but the location was different. A scanning area was 10 μm × 10 μm, and the height h(x, y) along x- or y-direction was specified using 512 data points. For the comparison consistency, the standard deviation σ and correlation length ζ were obtained following the definition from a Gauss surface [32]. Their uniformity among nine scanned areas was confirmed, but only the average of all scanned areas were illustrated here for conciseness. The height standard deviation $({\sigma }_{\overline{x},\overline{y}})$ and correlation length $({\text{ζ}}_{\overline{x},\overline{y}})$ averaged over nine areas of Sample A (using sandpaper No. 400) were 58 nm and 605 nm, respectively. Subscripts $\overline{x}$ and $\overline{y}$ represent the average along the x- and y-direction, respectively. In contrast, ${\sigma }_{\overline{x},\overline{y}}$ and ${\text{ζ}}_{\overline{x},\overline{y}}$ of Sample B (using sandpaper No. 1200) were 35 nm and 566 nm, respectively. Sample A had larger height variation clearly, but its profile was more correlated than that of Sample B.

Figure 2 shows the average surface profile along the x-direction $h_{\overline{y}}(x)$ for three scanned areas of each sample. An inset plots the original height h(x, y) of the sampling area measured by the AFM. The other inset specifies ${\sigma }_{\overline{x},\overline{y}}$ and ${\text{ζ}}_{\overline{x},\overline{y}}$ averaged over all scanned areas. $h_{\overline{y}}$ is the middle curve in every sub-figure, and its data point represents the average of heights along the y-direction of the scanned area. ${\sigma }_{\overline{y}}$ is the average height standard deviation similarly defined. ${\sigma }_{\overline{y}}$ is sometimes large due to particles, which cause height variation in a small area. As a result, the variation of $h_{\overline{y}}$ enlarges with ${\sigma }_{\overline{y}}$ occasionally. For example, the variation of $h_{\overline{y}}$ becomes larger and ${\sigma }_{\overline{y}}$ gets bigger at the right part than the left part of first sub-figure in Fig. 2(a). But the rough surface profile along the x-direction is mainly contributed from scratching. One proof is the usual independence between $h_{\overline{y}}$ variation and ${\sigma }_{\overline{y}}$ value. Another is the stripe patterns in insets, showing trivial ${\sigma }_{\overline{y}}$ variation. Three curves ($h_{\overline{y}}$, $h_{\overline{y}}-{\sigma }_{\overline{y}}$ and $h_{\overline{y}}+{\sigma }_{\overline{y}}$) are usually close. Therefore, we consider h as a function of only x, and the subscript $\overline{y}$ will not be specified hereafter.

 

Fig. 2 The average height along y-direction $(h_{\overline{y}})$ for three 10 μm × 10 μm scanned areas of: (a) Sample A; (b) Sample B. The inset in each sub-figure shows the original contour measured.

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Figure 3 shows the normal cumulative distribution function (CDF) of two samples to validate the “height randomness.” The normal CDF of a profile is a function of h as [32]:

 

Fig. 3 (a) The CDF of Sample A and the ideal CDF obtained with ${\sigma }_{\overline{x},\overline{y}}$ = 58 nm. (b) The CDF of Sample B and the ideal CDF obtained with ${\sigma }_{\overline{x},\overline{y}}$ = 35 nm.

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3. Types of randomly rough surfaces

The identification of a rough surface type needs an auto-correlation function (ACF) ρ, which illustrates the height correlation at locations x₁ and x₂ [35]:

 

Fig. 4 (a) The ACF of Sample A, a Gauss surface, an Exponent surface, and numerically generated surfaces (Bessel_25 and Bessel_50). All surfaces have the same correlation length ${\text{ζ}}_{\overline{x},\overline{y}}$ = 605 nm. (b) The ACF of Sample B, the Gauss surface, the Exponent surface, Bessel_25, and Bessel_50. All surfaces share ${\text{ζ}}_{\overline{x},\overline{y}}$ = 566 nm.

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The profile distinction of our samples from other randomly rough surfaces can be determined from its ACF (ρ_S), where the subscript S means sample. Figure 4(a) and 4(b) show the ρ_S of Sample A and Sample B, respectively. Each ρ_S varies largely with τ and becomes negative occasionally. The ρ_S variation of Sample A is a little stronger than that of Sample B. For fitting ρ_S of significant variations, we propose a curve ρ_B based on the Fourier-Bessel series [37]:

The functions of correlation length ζ for a Gauss surface and an Exponent surface cannot be fully applicable to that for Bessel’s rough surfaces. There are multiple lengths such that ρ_B = e⁻¹. Furthermore, there exists no ζ of ρ_B making h(x₁) and h(x₂) uncorrelated. The function invalidity of ζ results from intrinsic connections built on the crystal periodicity and lattice orientation preference. As a result, ζ of ρ_B can still be defined as the distance making ρ_B(ζ) = e⁻¹, but it is the shortest distance among multiple choices. The surface heights can still be strongly correlated even though locations are far apart.

4. Obtaining scattering patterns

The studied scattering patterns are the product of BRDF and cosθ_r, where BRDF and θ_r are the bi-directional reflectance distribution function and polar angle of reflectance, respectively. The BRDF is defined as [38]:

In this work, the incident light is either transverse electric (TE) waves or transverse magnetic (TM) waves. The wavelength (λ) is fixed at 405 nm considering the potential applications for solar energy. Three angles of incidence (θ_i = 15°, 30°, and 60°) will be studied. The material for our samples is lightly-doped p-type crystalline silicon, whose refractive index n and extinction coefficient κ are set to n = 5.32 and κ = 0.32 [39] during modeling.

Programs were developed to generate surfaces and obtain their BRDF*cosθ_r [27]. The Gauss surface was generated using either set of ${\sigma }_{\overline{x},\overline{y}}$ and ${\text{ζ}}_{\overline{x},\overline{y}}$ from the two samples. A total of thirty surfaces were generated for each (σ, ζ), and their ACFs characteristics of a Gauss surface were validated. The BRDF*cosθ_r were calculated for each surface and then averaged for plotting. On the other hand, the nine measured profiles of each sample could be input into programs for surface generation. Each BRDF*cosθ_r was calculated and averaged. Hence, the BRDF*cosθ_r and R from Gauss and Bessel’s rough surfaces of identical (σ, ζ) could be compared.

A three-axis automated scatterometer (TAAS) was employed for BRDF measurements [26]. It determined angles of three axes automatically, and the sample on its holder could be rotated manually. BRDF allows for the four degrees of freedom, describing incidence reflected light, but only three are needed for the in-plane BRDF measurement here. The calibration of TAAS was conducted because the wavelength λ is now 405 nm. Figure 5(a) shows R measured from a double side polished silicon wafer, which is the same raw material for our samples. Numerical results from the Fresnel’s equations were also plotted for comparison [38]. The R showed excellent agreement at 1° ≤ θ_i ≤ 84° between numerical and experimental results for both incident TE and TM waves. The standard deviation was less than 1°, and the Brewster angle occurred at θ_i = 80° as predicted. Such agreement confirmed not only the capability of TAAS but also the accuracy of silicon optical constants. Each R measurement was conducted three times. The repeatability of TAAS was illustrated by short error bars. We further validated the capability of TAAS using a microstructured surface as shown in Fig. 5(b). A silicon binary grating was designed and fabricated. Its profile was defined with the trench depth d, ridge width w, and period Λ. Dimensions in the unit of micrometers were d = 0.59, w = 3.72, and Λ = 6.00. The 0th order diffraction efficiency η₀ from the grating was measured every 5° of θ_i. Deviations due to imperfection of the sample profile were small. Most measurements agreed with numerical ones, assuring capability of TAAS for scattering patterns from structured surfaces at various θ_i.

 

Fig. 5 Calibration results of the TAAS at λ = 405 nm for both TE and TM waves: (a) The reflectance R from a crystalline silicon substrate; (b) The zeroth order reflected diffraction efficiency η₀ from a silicon grating.

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5. Scattering patterns

Figure 6 shows the BRDF*cosθ_r from numerically generated Bessel’s rough surfaces (Bessel_50), Gauss, and our samples at θ_i = 15°. The incident light in Fig. 6(a) and 6(b) is TE waves, while that in Fig. 6(c) and 6(d) is TM waves. On the other hand, (σ, ζ) = (58, 605) in the unit of nanometers is shared among surfaces in Fig. 6(a) and 6(c). (σ, ζ) = (35, 566) is for surfaces in Fig. 6(b) and 6(d). The inset in a sub-figure shows the value of BRDF*cosθ_r higher than 2.0. In Fig. 6(a), the BRDF*cosθ_r from both Bessel_50 and Sample A shows a sharp peak at θ_r = 15°. The common existence of this peak supports the consistency of Bessel’s rough surfaces from both experiments and modeling. The curve from a Gauss surface does not have a sharp peak at θ_r = 15°; instead, it shows a shallow valley between two nearby peaks. The valley and two side peaks from the Gauss surface clearly differentiate optical responses between Bessel’s rough surfaces and a Gauss surface. Another unique scattering pattern of Bessel’s surfaces is the confinement of reflected energy. In general, the reflection of light by a rough surface composes a peak around the direction of specular reflection, an off-specular lobe, and a diffuse component. But the latter two of Bessel’s rough surfaces are quite trivial in contrast to those of a Gauss surface.

 

Fig. 6 The BRDF*cosθ_r of samples and that of a Gauss surface and Bessel_50 at the oblique (θ_i = 15°) incidence of linearly polarized light. Results at TE wave incidence are (a) and (b), while those at TM wave incidence are (c) and (d). ${\sigma }_{\overline{x},\overline{y}}$ = 58 nm and ${\text{ζ}}_{\overline{x},\overline{y}}$ = 605 nm are shared in (a) and (c); while ${\sigma }_{\overline{x},\overline{y}}$ = 35 nm and ${\text{ζ}}_{\overline{x},\overline{y}}$ = 566 nm are shared in (b) and (d).

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Figure 6(b) shows the BRDF*cosθ_r from surfaces of smaller σ and ζ than those in Fig. 6(a). The curve from Bessel_50 is still closer to that from Sample B than Gauss. All three curves show a peak at θ_r = 30°; however, off-specular peaks can be observed in curves from the Gauss surface and Bessel_50. The Gauss surface has two at θ_r ≈-5° and θ_r ≈ + 30°, while the Bessel_50 has only one at θ_r ≈ + 30°. Characteristics at the incidence of TE waves above can also be found at the TM wave incidence as shown in Fig. 6(c) and 6(d). Results from Bessel_50 and our samples are consistent. But their scattering patterns are distinct from those of the Gauss surface, even though they three share identical (σ, ζ).

Figure 7 shows counterparts of Fig. 6, except θ_i is enlarged to θ_i = 30°. Besides similar observations in Fig. 6, Fig. 7 shows additional interesting results. First, the specular peak at θ_r = 30° can only be found from curves of Bessel’s rough surfaces. There is a small valley at θ_r = 30°sandwiched between two shoulders in the curve from the Gauss surface. Second, the area below associated curves is larger in Fig. 7(a) than Fig. 7(c). The area is larger in Fig. 7(b) than Fig. 7(d) as well. Such area difference comes from the incidence polarization. The reflectance is higher at the incidence of TM waves than TE waves, regardless of a silicon plane shown in Fig. 5(a) or a rough surface here.

 

Fig. 7 The BRDF*cosθ_r of samples and that of a Gauss surface and Bessel_50 at the oblique (θ_i = 30°) incidence of linearly polarized light.

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Figure 8 shows counterparts of Fig. 6 and Fig. 7, but θ_i is enlarged to θ_i = 60°. Four new and interesting results are found as follows. First, the peak at θ_r = 60° exists in all curves because rough surfaces become smooth ones at a larger θ_i. All scattering patterns approximate to those from a smooth plane, and the difference among rough surfaces is weakened. Second, the curve from a Gauss surface is sometimes closer to that from samples than Bessel_50. An example is the peaks in Fig. 8(c) and 8(d). One reason for such confusion is still the reduced influence of roughness at large θ_i. The other is the weak reflectance at the oblique incidence of TM waves. Third, only an off-specular lobe or shoulder shows up in curves from the Gauss surface at θ_r ≈40°. The other lobe cannot show up at θ_r ≥ 60° because it is too weak as θ_r is close to 90°. Fourth, the area below a curve depends stronger on the incidence polarization than that in previous two figures. The area (hemispherical) difference between two polarizations gets large, and it will be quantitatively discussed below.

 

Fig. 8 The BRDF*cosθ_r of samples and that of a Gauss surface and Bessel_50 at the oblique (θ_i = 60°) incidence of linearly polarized light.

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6. Hemispherical reflectance R

Table 1 lists the hemispherical R from a generated Bessel’s rough surface, a Gauss surface, a planar surface, and Sample A. The three rough surfaces have the same σ and ζ, and their R is numerically obtained via the integration of BRDF*cosθ_r. That is, each R is the area below a curve in sub-figure (a) or (c) of Fig. 6, 7, or 8. The specular R from a plane shown in Fig. 5(a) is also the hemispherical R, assuming the plane is perfectly smooth. The listed R in table is quite informative and interesting. First, the R from Sample A is always the lowest at the same polarized incidence and θ_r. The reduction in R is the out-of-plane scattering caused by imperfection of real samples. Second, the R from a planar surface is often the largest among four surfaces. The larger R from a planar surface than rough surfaces is intuitive because the surface (plane) is mirror-like. The only exception at the TM wave incidence of θ_i = 60° is due to the nearby Brewster angle. The incident light can penetrate completely into a smooth interface between free space and silicon, while surface roughness ruins the access easiness. Third, the R from Bessel_50 is lower than those from a Gauss surface and a planar surface usually. Reduced R of Bessel’s rough surfaces exhibits promising potentials for absorbing solar energy. Besides crystalline silicon, other materials can also have a Bessel’s rough surface profile by coating a layer on a roughened silicon surface. As long as the top profile remains parallel to the bottom, it becomes a Bessel’s rough surface made of other materials. Fourth, the R at the TM wave incidence is less than at the TE wave. The only exception caused by out-of-plane scattering occurs at θ_i = 15° from Sample A.

Table 1. The hemispherical R from a generated Bessel’s rough surfaces (Bessel_50), a Gauss surface, a planar surface, and Sample A. The ${\sigma }_{\overline{x},\overline{y}}$ = 58 nm and ${\text{ζ}}_{\overline{x},\overline{y}}$ = 605 nm are shared by Bessel_50, the Gauss surface, and Sample A.

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Table 2 lists the hemispherical R from another set of rough surfaces, which are also marked Bessel_50, Gauss, and Sample B. Their σ and ζ are smaller than those of rough surfaces in Table 1. Findings in Table 1 consistently show up, for example, the lowest R among the same column is from Sample B. Bessel’s rough surfaces favor more light trapping than Gauss surfaces.

Table 2. The hemispherical R from a generated Bessel’s rough surface, a Gauss surface, and Sample B. They all have ${\sigma }_{\overline{x},\overline{y}}$ = 35 nm and ${\text{ζ}}_{\overline{x},\overline{y}}$ = 566 nm.

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On the other hand, the comparison between two tables gives two interesting messages. First, the R from Bessel_50 in Table 2 is between that from a planar surface and Bessel_50 in Table 1 at the incidence of both TE and TM waves for θ_i = 60°. The R from the Gauss surface also exhibits such a relation. But the order is inconsistent at θ_i = 30°, i.e., the R in Table 1 is between that from a planar surface and that in Table 2. The R clearly cannot be determined by either σ or ζ, even for the same type of rough surfaces. Second, while the R from Bessel_50 is usually lower than that from a Gauss surface, the difference does not vary much with σ and ζ. Moreover, the average R of Bessel_50 considering both polarizations and three θ_i are about the same in each table. No absorption superiority shows up from the two sets of σ and ζ for Bessel’s rough surfaces here. Individual impacts from σ and ζ on optical responses of Bessel’s rough surfaces needs an in-depth study and more samples.

7. Conclusion

This work successfully demonstrated the uniqueness in both surface profile and optical responses of roughened crystalline silicon surfaces. Their ACFs could only be fit with Bessel’s functions. Their BRDF*cosθ_r usually have no off-specular lobe(s) like a Gauss surface at smaller θ_i. Samples of a large area can be mass fabricated; moreover, the surface material can be extended to metals and other dielectrics. Reduced R from surfaces at various incidence orientations and polarizations indeed facilitates absorbing light and gives hope to solar cells and thermal collectors. Since two samples are insufficient, a future work will be followed to investigate impacts of σ and ζ individually on scattering patterns and R.