In chapter 2.4 (figure 2.14), the differences between a silicon wafer with a pyramidal texture and an as-cut Si-wafer show the tremendous advantages of the pyramids in increasing the light absorption of Si-wafers. The reason for this will be explained below.
We first consider a planar silicon wafer (see figure 3.8), in which an incident ray strikes the surface and is transmitted into the wafer. Inside the wafer, the ray continues its path and reaches the back side of the wafer, where it escapes from the wafer [13: Yablonovitch 1982].
Fig. 3.8: Silicon with two flat surfaces. Incident light on the front side of the Si-wafer escapes through the back side of the wafer [13: Yablonovitch 1982].
In the case in which both surfaces are not flat, the direction of the incident light will be randomized, and the probability decreases that the incident ray will escape after it has been transmitted into the wafer. It can be said that the ray has been trapped in the wafer by traveling several times from the back side to the inner side of the wafer’s front surface, see figure 3.9.
Fig. 3.9: Silicon wafer with two rough sides. The incoming ray is randomized into the wafer and is thus “trapped” [13: Yablonovitch 1982].
In the following figure 3.10, we see a schematic representation of a silicon wafer with a front pyramidal texture and a planar back side. An incident ray (red line) on the front side of the Si-wafer is also shown [14: McIntosh 2009].
Fig. 3.10: Si-wafer with pyramidal texture on the front side and planar back side enhances light absorption [14: McIntosh 2009].
The incoming ray on the pyramid has two chances to be conducted into the Si-wafer. The first possibility (number 1) is due to the reflection of the incident ray onto a neighboring facet. The second possibility refers to the transmitted ray in the Si-wafer (absorber); the transmitted ray is reflected from the back planar side of the wafer and then is again reflected into the Si-wafer by the facets of the pyramids on the front side. The ray will be reflected only if the angle of incidence is larger than the critical angle. Thus, the chance of the rays being absorbed into the Si-wafer is considerably increased.
In principle, the idea is to give the ray reflected from the back side an incident angle with respect to a normal line with the surface on the front side (into the Si-wafer) greater than the critical angle and thereby to use a phenomenon known as total reflection [15: Goetzberger 1997], as shown in figure 3.11.
In figure 3.11, we see a representation of the critical angle ɸc on a Si-wafer. It has a value of about 17o with respect to a normal line perpendicular to the surface. If the rays reflected from the backside have an angle greater than the critical angle, they will be completely reflected. Thus the path length of the ray is increased, and thus the possibility for the ray to be absorbed into the Si-wafer is also increased. This phenomenon is known as “light trapping” (it should be referred to only as “increased total light reflection”), and it has major significance for wavelengths in the infrared range.
Experimentally, from reflection measurements on the pyramidal texture of Si-wafers, a strong reduction of total light reflection can be observed (figure 3.6). From SEM pictures of textured Si-wafers (figure 3.5), it is obvious that such a reduction of total light reflection is due to the pyramids covering the surface.
Fig. 3.11: Representation of the critical angle ɸc into a silicon wafer with a rough back side and a flat front side. Those rays coming from the back side of the silicon wafer at an angle greater that the critical angle ɸc will be reflected back into the silicon bulk, thus getting a second chance to be absorbed. This phenomenon is known as “light trapping” [15: Goetzberger 1997].
In the case of an optimal KOH-IPA pyramidal texture (figure 3.5 c), pyramids have sizes greater than 5 µm. As silicon has a band gap energy of 1.12 eV, (which corresponds to a wavelength of 1107 nm), light with wavelengths shorter than 1107 nm (1.1 µm) will be absorbed, whereas light with longer wavelengths will not be absorbed.
The great difference between pyramid size and the wavelength of the absorbed light leads us to explain the absorption of light by means of geometrical optics.
However, if the size of a pyramid is smaller than the wavelength, refraction phenomena have to be considered to explain light absorption [16: Llopis 2005].
One of the first theoretical works on light trapping in silicon wafers with neither planar nor spherical surfaces was given by Yablonovitch in 1982 [13: Yablonovitch
1982]. He found that the light intensity inside the wafer is n2 times the incident light intensity, where n is the refraction index of silicon.
Theoretical work on light trapping on pyramidal textures was done by Campbell et al. in 1987 [17: Campbell 1987]. He found that silicon wafers textured on both sides have better light trapping characteristics than silicon wafers textured on only one side. He also found that the randomized pyramidal texture on silicon wafers also has better light trapping properties than wafers with Lambertian surfaces.
By computer modeling he calculated the light remaining in the wafer after a certain number of passes through the wafer, see figure 3.12.
Fig. 3.12: Computer simulation showing the percentage of light remaining as a function of the number of passes through the silicon wafer, with both sides having a pyramidal texture (figure inside right) [17: Campbell 1987].
For silicon solar cells, light trapping has important advantages on the solar cell level, because then more electron-hole pairs will be created, and therefore the electrical current generated by the solar cell will be increased. Campbell also calculated the maximum possible short-circuit current as a function of cell thickness under AM1.5 solar spectrum illumination conditions, see figure 3.13.
From figure 3.13, it can be seen that the maximum possible short-circuit currents for cells with Lambertian and pyramidal surfaces (on both sides) are almost identical.
It is also evident from the figure that the short-circuit current increases with the thickness of the wafer, because then more light can be trapped (absorbed).
A first commercial program, called TEXTURE, to simulate light trapping was developed by Smith et al. in 1990 [18: Smith 1990]. The program used Monte Carlo simulation to calculate the path of the light rays inside the wafer.
Three years later, Brendel [19: Brendel 1993] expanded on Smith’s work.
Brendel’s program is known as SUNRAYS. Like Smith, Brendel also used Monte Carlo simulation to describe the paths of light rays into and through the solar cell structure according to the laws of geometrical optics. The probability of a light ray being reflected, diffracted or absorbed depends on its angle of incidence, its state of polarization, and the optical properties of the interface. In this work, we use the SUNRAYS program to simulate light trapping in our pyramidal texture, see chapter 5.
Fig. 3.13: Maximum possible short-circuit current obtained by solar cells with different textures and thicknesses [17: Campbell 1987].
To use the SUNRAYS program, the texture has to be defined, as well as the dimensions of a single pyramid (unit cell), see figure 3.14.
In our case, we select the following values for our texture to simulate light trapping (see the simulation in figure 5.14):
Texture: Upright / Upright pyramids Dimension of the unit cell:
Thickness of the silicon wafer = 200 µm
Depth top (height of the pyramid) = Depth Bot = 4 µm
Period X (length of the base of the pyramid) = Period Y = 5.6 µm Apex X Top = Apex Y Top = Apex X Bot = Apex Y Bot = 2.8 µm Flat part (region not covered with pyramids) = 0 µm
Fig. 3.14: Unit cell of the Upright pyramid \ Upright pyramid texture [19: Brendel 1993].
A SUNRAYS simulation on upright pyramid \ upright pyramid can be observed in the next figure 3.15.
The SUNRAYS program is well suited for simulation of light trapping for a single cell or for a pyramidal texture with the same pyramid size, and for pyramid sizes whose lengths are considerably bigger than the wavelength of the incoming light.
Fig. 3.15: SUNRAYS simulation of light trapping in a unit cell of the Upright pyramid \ Upright pyramid texture. The simulation of the unit cell can be observed at the leftmost side. From the second unit cell (from left to right) the increasing on light trapping into the unit cell can be observed.
Unfortunately such ideal texture is far away from a random pyramidal texture because not considering the random nature of wet chemical produced textures, and not considering of diffraction phenomena which happen when pyramid sizes have similar lengths to that of the wavelength of the incident light [16: Llopis 2005].
To consider diffraction phenomena and a random pyramidal texture, Llopis et al., simulate light trapping on two dimensional pyramid structures (see figure 3.16). They did not use geometrical optics; they solve the wave equation. For the magnetic field along the z direction (see figure 3.16) the wave equation is written as
whit ko = 2πλ, where λ is the vacuum wavelength. Permitivity values are ε = ε1 = n12/μo in D+ and ε = ε2 = n22/μo in D-. The same equation holds for the electric field along the z direction Ez in TE (electric field in the z direction, normal to the plane of the drawing, see figure 3.16) polarization.
The wave equation is then solved with rigorous electromagnetic methods. The 2-dimensional pyramidal texture is shown in figure 3.16.
Fig. 3.16: Two dimensional pyramidal texture considered by Llopis to simulate light trapping by solving the wave equation by means rigorous electromagnetic methods. Here, diffraction phenomena are under consideration [16: Llopis 2005].
After solving the wave equations by means of numerical approximation and using the coordinate transformation method, simulations on pyramidal texture with different sizes (random pyramidal texture) were performed, see figure 3.17.
Fig. 3.17: Light trapping simulation obtained by solving the wave equation. Ray tracing simulation is also shown for comparison [16: Llopis 2005].
The simulation A corresponds to the consideration of a pyramidal texture with different pyramids sizes (different length of their basis (D), see figure 3.16): 0.15 µm - 0.15 µm - 0.3 µm - 0.45 µm - 0.75 µm - 1.20 µm - 1.95 µm - 3.15 µm - 5.10 µm, whereas for the simulation B, the length of the pyramid basis (D) were: 0.05 µm - 0.05 µm - 0.10 µm - 0.15 µm - 0.25 µm - 0.40 µm - 0.65 µm - 1.05 µm - 1.70 µm. The simulation A considers pyramids whose basis length D is larger than those in simulation B. The simulation using the ray tracing method is also shown in figure 3.15 for comparison.
In figure 3.17 it can be observed that the light trapping properties of the pyramidal texture with random pyramids of simulation A are better than those observed on simulation B and from the ray tracing method. This result is in accordance with experimental results observed on our random pyramidal texture with small pyramid sizes (KOH-HBA texture).
Further publications related to light trapping on textured silicon surfaces also solve the wave equation [20: 2007: Sai] when diffraction phenomena have been taken into account, or they continue using geometrical optics [21: 2012: Baker-Fich] when not.
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