Buffer Layer between a Planar Optical Concentrator and a Solar Cell

The effect of inserting a buffer layer between a periodically multilayered isotropic dielectric (PMLID) material acting as a planar optical concentrator and a photovoltaic solar cell was theoretically investigated. The substitution of the photovoltaic material by a cheaper dielectric material in a large area of the structure could reduce the fabrication costs without significantly reducing the efficiency of the solar cell. Both crystalline silicon (c-Si) and gallium arsenide (GaAs) were considered as the photovoltaic material. We found that the buffer layer can act as an antireflection coating at the interface of the PMLID and the photovoltaic materials, and the structure increases the spectrally averaged electron-hole pair density by 36% for c-Si and 38% for GaAs compared to the structure without buffer layer. Numerical evidence indicates that the optimal structure is robust with respect to small changes in the grating profile. C 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.


I. INTRODUCTION
2][3] The substitution of the photovoltaic material by a cheaper dielectric material in a large area of the light-harvesting structure could reduce the fabrication costs without significantly reducing the efficiency of conversion of photonic energy into electrical energy.The large-area fabrication of such composite structures using patterning techniques such as soft lithography and particle assembly [4][5][6] is now inexpensive enough that the concentrators and the photovoltaic components can have sub-mm dimensions.These micro-cell designs are forgiving with respect to the incorporation of optically lossy materials such as metals into the POC.They are also advantageous for heat dissipation, especially at the modest concentration ratios that are appropriate with relatively low-cost single-junction crystalline silicon (c-Si) and thin-film compound-semiconductor photovoltaic solar cells.
A POC comprising a periodic multilayered isotropic dielectric (PMLID) material backed by a metallic surface-relief grating was proposed and optimized earlier for c-Si solar cells. 2 The geometrical parameters and the refractive indexes of the materials in the POC were selected in order to maximize the solar-spectrum-integrated power-flux density inside the PMLID material.Consequently, the spectrally averaged electron-hole pair (EHP) density generated in the solar cells was also maximized.Since there is a mismatch between the optical permittivities of the PMLID constituents and the solar cell, reflection occurs at the PMLID/c-Si interfaces.If this reflection could be reduced, the conversion of photonic energy to electrical energy would be further enhanced.
Therefore, we decided to investigate the effect of an isotropic dielectric buffer layer inserted between the PMLID material and the solar cell in an effort to maximize the transfer of optical energy from the PMLID material to the solar cell.Based upon the optical permittivities of the layers in PMLID material and of the semiconductor, this buffer layer should act as an antireflection coating.
Figure 1 presents the schematic of one unit cell of the light-harvesting structure in the xz plane, the structure being invariant along the y axis.At the bottom is a metal grating of period L along the x axis.Atop one period of the metal grating sits a solar cell, atop N c consecutive periods of the grating sits a PMLID material with N d periods in the vertical direction, and one period each on both sides of the solar cell are shared by the PMLID material and the buffer layer.The topmost layer is indium-tin oxide (ITO) functioning as an optically transparent electrode.Using data on an optimized PMLID material, 2 our objective here is to optimize the thickness L b and the refractive index n b of the buffer layer, when the solar cell is made of either c-Si or gallium arsenide (GaAs).
The plan of this paper is as follows.Section II provides a brief description of the theory used to compute the specular and non-specular reflectances and transmittances of the light-harvesting structure shown in Fig. 1, when it is illuminated by a linearly polarized plane wave whose propagation vector lies wholly in the xz plane.An exp(−iωt) dependence on time t is implicit, with ω as the angular frequency and i = √ −1.The spectrally averaged EHP density is calculated using the computed solution.Section III presents numerical results, when the solar cell is made of either c-Si or GaAs.The paper concludes with a few remarks in Sec.IV.Each period is of thickness 2Ω = L d /N d and comprises N layers of equal thicknesses d = 2Ω/N.The relative permittivity ε r j > 1 of the j-th layer, j ∈ [1, N], is taken to be a real-valued function of the free-space wavelength λ 0 .Our interest lies in the spectral regime λ 0 ∈ [λ 0 mi n , λ 0 ma x ] dictated by the solar spectrum.

A. Geometry
The regions {x ∈ (0, are occupied by a metallic surface-relief grating with its troughs filled by a dielectric material of relative permittivity ε r 1 .The metal/dielectric interface has a periodic rectangular profile with a period L along the x axis, corrugation height L g , and duty cycle ζ ∈ [0, 1].The λ 0 -dependent relative permittivity of the metal is denoted by ε m with Re(ε m ) < 0 and Im(ε m ) > 0.
The regions {x are occupied by a semiconductor of λ 0 -dependent relative permittivity ε s with Re(ε s ) > 0 and Im(ε s ) > 0.
The regions {x which is taken to be non-dissipative and non-dispersive in the spectral regime of interest.
The remainder of the unit cell is occupied by a metal of relative permittivity ε m .The half space z > L t is vacuous.

B. Plane-wave response
Suppose that the light-harvesting structure is illuminated by an obliquely incident plane wave whose electric field phasor is given by Here and hereafter, θ is the angle of incidence with respect to the z axis, a s is the amplitude of the s-polarized component, a p is the amplitude of the p-polarized component, k 0 = ω/c 0 = 2π/λ 0 is the free-space wavenumber, c 0 = 1/ √ ε 0 µ 0 is the speed of light in free space, η 0 =  µ 0 /ε 0 is the intrinsic impedance of free space, ε 0 is the permittivity and µ 0 is the permeability of free space, and ûx , ûy , ûz is the triplet of Cartesian unit vectors.
As depolarization cannot occur in this problem, the electric field phasors of the reflected and the transmitted fields can be stated as 7,8 E r e f (x, z, and respectively, where Z ≡ {0, ±1, ±2, . ..}, and The λ 0 -dependent reflection coefficients of order n are denoted by r (n) ss and r (n) p p , and the corresponding transmission coefficients by t (n) ss and t (n) p p .Thus, the reflected and the transmitted field phasors comprise specular components identified by n = 0 and non-specular components identified by n 0. Provided that L m is sufficiently thick, the transmitted field will transport virtually no energy in the +z direction.
In the region z ∈ (−L o , L t ), the electric and magnetic field phasors must satisfy the frequencydomain Maxwell curl equations We used the finite-element method (FEM) 9,10 to solve Eqs.(6).After decoupling the s-and p-polarization states and exploiting the independence of the field phasors from y, we reduced Eqs.( 6) to two scalar Helmholtz equations, one for each of the two linear polarization states.The boundary conditions on the sides x = 0 and x = N c L + 3L of the unit cell have to be quasi-periodic.Standard transmission conditions hold across the top boundary z = −L o and bottom boundary z = L t , whose satisfaction involves Eqs. ( 1)-( 2).We used a special-purpose finite-element meshing function based on the TRIANGLE library. 11Inside each triangle of the mesh, the field phasors were approximated by cubic polynomials.The summations in the expressions of the reflected and transmitted field phasors were truncated to n ∈ {−M t , M t }.

C. Spectrally averaged EHP density
After obtaining the reflection and transmission coefficients for any λ 0 ∈ [λ 0 min , λ 0 max ], the electric field phasor E(x, z, λ 0 ) can be obtained at any location inside the unit cell, for an unpolarized incident plane wave (a s = a p = E o / √ 2) with an electric field of magnitude E o = 1 V m −1 .Thereafter, the spectrally averaged EHP density can be computed as 2 where is the reduced Planck constant, S(λ 0 ) is the AM1.5 solar spectrum, 12 λ 0 min = 400 nm, and λ 0 max = 1000 nm.

III. NUMERICAL RESULTS
For all calculations, we set L o = 100 nm, a practical value.Also, we chose the metal to be silver and L m = 40 nm, and we used measured data for ε m as a function of λ 0 available elsewhere. 13s optimization of the POC alone had previously 2 provided the relative permittivities of all N = 9 layers in every period of the PMLID material, we used the same method here as well.The measured values of the relative permittivity of silicon oxynitrides of 9 different compositions 13 for λ 0 ∈ [400, 1000] nm were chosen for ε r j , j ∈ [1, N].Finally, we fixed L = 400 nm, consistently with the exploitation of Floquet theory to excite surface-plasmon-polariton waves [14][15][16] and waveguide modes. 16,17

A. Optimization of POC without buffer layer
We began by assuming that the indium-tin-oxide layer, the buffer layer, and the solar cell are absent.The PMLID material was taken to have N d = 3 periods along the z axis, and that the relative permittivities ε r j from Ref. 13 were changed to (1 + δ) 2 ε r j , δ ∈ [−0.25, 0.25] for all j ∈ [1, N].We then calculated the solar-spectrum-integrated power-flux density in the plane x = 0, as a function of θ, d, ζ, L g , and δ, when an unpolarized plane wave is incident on the structure.Here, P x (x, z; λ 0 ) is the x-directed component of the monochromatic time-averaged Poynting vector P(x, z; λ 0 ).Our experience was that optimization of q 0 in any other plane parallel to x = 0 is equally effective. 2aximization of q 0 was carried out using the differential equation algorithm, 18 while ensuring that a variation of up to 5% in any parameter does not result in a change in q 0 by more than 10%.This modified DEA is designed to avoid any sharp resonances that are either of numerical origin or are physical but would require high-precision manufacturing steps.The following practical design of the POC alone emerged: θ = 18 deg, d = 130 nm, ζ = 0.75, L g = 40 nm, and δ = 0.25.This design delivered q 0 = 4.17 avg increases with  N c , but  N c cannot be too large in a practical situation.As N (2) avg is double of N (0) avg but N (3) avg is not triple of N (0) avg ,  N c = 2 is a practical choice.Significantly, the replacement of two thirds of a solar cell by the POC (  N c = 2) results in only about ∼ 25% loss of efficiency.
Similar results were found when the solar cell is made of GaAs, 20 as Table II shows.However, in this case, the replacement of two thirds of a solar cell by the POC results in only about ∼ 10% loss of efficiency.

B. Optimization of buffer layer
Next, in order to determine the optimal buffer layer, we fixed θ = 18 deg, d = 130 nm, ζ = 0.75, L g = 40 nm, δ = 0.25, N d = 3, L = 400 nm and L o = 100 nm.Our objective was to      determine the combination {L b , n b } that maximizes N avg when the solar cell is made of either c-Si or GaAs. 20We also fixed N c = 2 in order to always have at least two periods of POC in all the simulations.Hence, the POC's continuous spans along the x axis would decrease from 4L to 2L as L b would increase from 0 to L. the transmitted and reflected waves would be negligible in comparison to the error inherent in partitioning space into triangles, as required by the FEM.

c-Si solar cell
Let us first present results for the solar cell made of c-Si, whose relative permittivity as a function of λ 0 is available in the literature. 19n Fig. 3 the computed values of N avg for 2L b ∈ {0, 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 650, 700, 750, 800} nm and n b ∈ {1.8, 2.2, 2.6, 3.0, 3.4} are displayed.The set of values of 2L b was chosen so that the outer edges of the buffer layer always coincide with the mesh lines on the grid.
The spectrally averaged EHP density is always higher if a buffer layer with refractive index larger than 2.2 is present, indicating a reduction in the reflectance at the POC/solar-cell interface.The photon flux in sunlight is maximum at λ 0 ≃ 680 nm, 21 and the average volume-averaged refractive index of the PMLID material (with refractive indices ) at the same wavelength equals 2.1.Since the real part of the refractive index of c-Si is 3.8 at the same wavelength, and the refractive indices of all materials involved are higher for λ 0 < 680 nm but are weakly dependent on λ 0 for λ 0 > 680 nm, having n b 2.1 is unfruitful for enhancing N avg .
In addition, N avg achieves a maximum at some 2L b ∈ [500, 800] nm and does not seem to significantly change with further increase in 2L b in Fig. 3. Taking the highest value of N avg from this figure, we concluded that the optimal parameters of the buffer layer are: 2L opt b = 700 nm and n opt b = 3.4.Then, N opt avg = 2.37 × 10 27 m −3 s −1 which is 36% higher than when there is no buffer layer (i.e., 2L b = 0).
Let us denote by F opt and F ref the values of F when the buffer layer is optimal (i.e., 2L b = 700 nm and n b = 3.4) and absent (i.e., 2L b = 0), respectively.Figure 4 shows F opt and F ref as functions of λ 0 .In much of the spectral regime of interest, F opt > F ref .The maximum value of the ratio F opt /F ref is 3.50, which occurs for λ 0 = 909.15nm.
In Fig. 5, we present the spatial distributions of |E(x, z, λ 0 )| 2 in the entire structure for 2L b ∈ {0, 200, 450, 800} nm, when λ 0 = 909.15nm and n b = 3.4.A band of hot spots occurs in the POC, the buffer layer, and the solar cell when 2L b = 700 nm.This band explains the enhancement of F by the insertion of the optimal buffer layer.We attribute the occurrence of such bands to the excitation of (i) surface-plasmon-polariton (SPP) waves whose electric fields are maximal some distance away from the metal/dielectric interface 8 and (ii) waveguide modes. 17Parenthetically, when a metal and a homogeneous dielectric material form a planar interface, only one SPP wave can propagate at any given λ 0 . 22Its fields are maximal at the interface and decay exponentially with distance from the interface.When the dielectric material is periodically nonhomogeneous normal to the interface, more than one SPP wave can propagate at any given λ 0 . 8These SPP waves differ in phase speed, attenuate rate, degree of localization to the interface, and field profiles.On the dielectric side, some SPP waves have maximal fields at the interface, but others can have maximal fields some distance away from the interface.The significance of multiple SPP waves and multiple waveguide modes excitable at the same value of λ 0 has been experimentally established for the periodically corrugated interface of a metal and a PMLID material. 16n a per-unit-area basis, replacement of 80% of the solar cell by the the optimal POC will lead to a 49% loss in efficiency.However, replacement of a part of the POC by the optimal buffer layer reduces the efficiency loss to 30%.The use of very cheap dielectric materials for the POC and the buffer layer could offset this reduction.

GaAs solar cell
The relative permittivity of GaAs as a function of λ 0 was retrieved from a standard resource. 20et us note here that GaAs has zero conductivity for λ 0 ∈ [950, 1000] nm.
Similarly to Fig. 3, the computed values of N avg versus 2L b and n b are displayed in Fig. 6.Taking the highest value of N avg from this figure, we concluded that the optimal parameters are: 2L opt b = 800 nm and n opt b = 2.6.Then, N opt avg = 3.77 × 10 27 m −3 s −1 which is 38% higher than when there is no buffer layer (i.e., 2L b = 0).Moreover, in Fig. 7  In Fig. 8, we present the spatial distributions of |E(x, z, λ 0 )| 2 in the entire structure for 2L b ∈ {0, 600, 800} nm, when λ 0 = 909.15nm and n b = 2.6.The intensity is higher in the solar cell when the buffer layer is present.
On a per-unit-area basis, replacement of 80% of the solar cell by the combination of the optimal POC and the optimal buffer layer will result in efficiency loss of just 10%.The use of very cheap dielectric materials for the POC and the buffer layer is very likely to offset this reduction.

C. Robustness of the optimal structure
As we were interested in determining the sensitivity of the optimal designs with respect to manufacturing errors, we analyzed the effects on N avg of changes in the shape of the grating.
Perfect rectangular profiles are difficult to realize efficiently in any mass-manufacturing scenario at the nanoscale.Hence, we changed the rectangular shape into trapezoidal by reducing the top dimension of the metal in each unit cell in the region This change did not necessitate a significant change in the FEM mesh.The spectrally averaged EHP densities computed with the optimal parameters determined previously in this section are N opt avg = 2.40 × 10 27 m −3 s −1 (c-Si) and N opt avg = 3.80 × 10 27 m −3 s −1 (GaAs).These values for the trapezoidal profile differ only by 1.3% (c-Si) and 0.8% (GaAs) with respect to those of N opt avg for the rectangular profile.
We also determined the variation of N avg with respect to L g ∈ [36, 44] nm.For ±10% variation of L g from its optimal value of 40 nm, we found a variation of less than 0.4% in the spectrally averaged EHP density for both c-Si and GaAs cells.

IV. CONCLUDING REMARKS
Given that dielectric materials used in planar optical concentrators are cheaper than photovoltaic materials in common use, the incorporation of a POC can reduce the cost of fabricating solar-cell modules.Our previous work 2 had shown that a 4:1 POC made of a PMLID material could result in a loss of 48% (on a per-unit-area basis) in efficiency compared to a pure planar solar cell.Our current work improves upon that result through the insertion of a buffer layer between the PMLID material and the solar cell.Our calculations show that the buffer layer can have a wide tolerance in thickness from 250 to 400 nm, which is highly desirable from a manufacturing perspective, and may have a potential optimal thickness in the 350-to-400-nm range.More importantly, the addition of the buffer layer between the PMLID material and the solar cell improves the spectrally averaged electron-hole pair density in a c-Si solar cell by 36% and in a GaAs solar cell by 38% compared to the same structure without the buffer layer for the same solar irradiance conditions.
Relative to a solar-cell module without the POC and the buffer layer, there is an efficiency loss of 30% for c-Si solar cells and 10% for GaAs solar cells accompanied by the combination of the optimal POC and the optimal buffer layer.These losses are amply compensated by the much lower costs of the dielectric materials in comparison to those of the semiconductors.

Figure 1
Figure 1 shows the unit cell of the concentrator {x ∈ (0, N c L + 3L), z ∈ (−L o , L t )}, where L t = L d + L g + L m .The region z < −L o is vacuous, while the top layer {x∈(0, N c L + 3L), z∈ (−L o , 0)} is made of indium-tin oxide.The PMLID material occupies the regions {x ∈ (0, N c L + L − L b ), z ∈ (0, L d )} and {x ∈ (N c L + 2L + L b , N c L + 3L), z ∈ (0, L d )}.This material has continuous spans of (N c + 2)L − 2L b =  N c L − 2L b along the x axis periodically.Furthermore, this material has N d ≥ 1 periods along the z axis.
× 10 3 W m −2 .Next, both the indium-tin-oxide layer and the solar cell made of c-Si 19 were considered along with the POC, the buffer layer still being absent.Since L b = 0, the POC has a period of  N c L along the x axis, where  N c = N c + 2. When  N c = 0 (i.e., N c = −2), the POC is absent and only the solar cell is present in the unit cell.Only the integer  N c was kept variable, and values N (  N c ) avg of the spectrally averaged EHP density N avg were calculated for  N c ∈ [0, 4].The calculated data are presented in Table I.Clearly, N (  N c ) of the spectrally averaged EHP density N a v g in relation to  N c when the buffer layer is absent (i.e., L b = 0), θ = 18 deg, d = 130 nm, ζ = 0.75, L g = 40 nm, δ = 0.25, L = 400 nm, and the solar cell is made of c-Si.

Figure 2
FIG. 2. FEM mesh (dark lines) of the light-harvesting structure.Colors in the background correspond to different regions: Metallic grating (brown), solar cell (red), buffer layer (orange), the PMLID material with N d = 3 (green), and the ITO layer (blue).

TABLE I .
Values N

TABLE II .
Same as TableI, but the solar cell is made of GaAs.
we observe that F opt > F ref for all wavelengths of interest.The maximum value of the ratio F opt /F ref is 2.34, which occurs for λ 0 = 887.19nm.