Enhanced efficiency of Schottky-barrier solar cell with periodically nonhomogeneous indium gallium nitride layer

Abstract. A two-dimensional finite-element model was developed to simulate the optoelectronic performance of a Schottky-barrier solar cell. The heart of this solar cell is a junction between a metal and a layer of n-doped indium gallium nitride (InξGa1−ξN) alloy sandwiched between a reflection-reducing front window and a periodically corrugated metallic back reflector. The bandgap of the InξGa1−ξN layer was varied periodically in the thickness direction by varying the parameter ξ∈(0,1). First, the frequency-domain Maxwell postulates were solved to determine the spatial profile of photon absorption and, thus, the generation of electron–hole pairs. The AM1.5G solar spectrum was taken to represent the incident solar flux. Next, the drift-diffusion equations were solved for the steady-state electron and hole densities. Numerical results indicate that a corrugated back reflector of a period of 600 nm is optimal for photon absorption when the InξGa1−ξN layer is homogeneous. The efficiency of a solar cell with a periodically nonhomogeneous InξGa1−ξN layer may be higher by as much as 26.8% compared to the analogous solar cell with a homogeneous InξGa1−ξN layer.


Introduction
A variety of light-trapping strategies capitalizing on structures engineered on the order of the wavelength of solar light have been examined both experimentally and theoretically to enhance the efficiencies of solar cells. 1 These strategies include texturing the front surface of the solar cell, 2,3 incorporating a periodically corrugated metallic back reflector, [4][5][6] coating the front surface with an "antireflection" (AR) layer, 7,8 embedding nanoparticles inside the light-absorbing layer (s), [9][10][11] and using light concentrators. [12][13][14][15] An attractive approach for boosting the solar-cell efficiency requires the use of photonabsorbing component materials whose electromagnetic properties are periodically nonhomogeneous along the thickness direction. Recently, it was demonstrated that the incorporation of a periodically nonhomogeneous intrinsic layer (i.e., i layer), along with a periodically corrugated back reflector, in amorphous silicon p-i-n junction solar cells can improve overall efficiency by up to 17%. 16 This improvement is likely due to i. the periodic corrugation of the metallic back reflector [4][5][6]17 facilitating the excitation of surface-plasmon-polariton waves [18][19][20] and waveguide modes 21 to intensify the electric field inside the semiconductor region, leading to an increase in the electron-hole pair (EHP) generation rate; ii. the periodic nonhomogeneity of the i-layer that may facilitate the excitation of multiple surface-plasmon-polariton waves 22 and waveguide modes, 17 thereby further boosting the EHP generation rate; and iii. the accompanying spatial gradient in the bandgap that may also aid charge separation and reduce the EHP recombination rate. 23,24 The study described herein concerns an especially simple type of solar cell that has received scant attention from researchers to date: a Schottky-barrier solar cell. 25 The Schottky barrier is provided by a metal-semiconductor junction. 26,27 Recent theoretical studies, 28,29 along with an earlier experimental study, 30 have suggested that Schottky-barrier solar cells may be a particularly promising proposition if the semiconductor layer (i.e., the absorbing layer) was made from alloys of indium gallium nitride (In ξ Ga 1−ξ N), since the bandgap for these alloys can closely match the range of energies of photons across the entire solar spectrum (i.e., 0.70 to 3.42 eV) by varying the relative proportions of indium and gallium through the parameter ξ ∈ ð0;1Þ. 31 Specifically, indium nitride (i.e., ξ ¼ 1) has a bandgap of 0.7 eV 32,33 and absorbs efficiently across the infrared regime in the solar spectrum, while gallium nitride (i.e., ξ ¼ 0) has a bandgap of 3.42 eV and absorbs efficiently across the near-ultraviolet portion of the solar spectrum.
With current technologies, there are significant challenges to overcome in the production of In ξ Ga 1−ξ N alloys for all values of ξ, particularly for smaller-bandgap alloys. When the proportion of indium is large (i.e., ξ ≳ 0.3), poor crystal growth plagues the realization of solar-cell applications. Such poor growth results in decreased carrier transport, background n doping due to Fermi pinning above the conduction band edge, 34 and a bandgap that is greater than expected. 35,36 By using a Schottky-barrier junction with n-doped In ξ Ga 1−ξ N, as opposed to a p-i-n junction, the difficulty of p doping the material is avoided.
In the following sections, a two-dimensional (2-D) numerical simulation is described for a Schottky-barrier solar cell made with In ξ Ga 1−ξ N. It is essential that the coupled processes of optical absorption and electrical-current generation are simultaneously accommodated in the simulation. This is because a focus on the computation of only the optical (i.e., at full quantum efficiency) short-circuit current density, but not of the open-circuit voltage, overplays the importance of the EHP generation rate by not taking the EHP recombination rate into account. 16 In our simulation, particular attention is paid to the role of periodic nonhomogeneity of the absorbing material. Thus, the effect of periodically varying the fractional composition parameter ξ of In ξ Ga 1−ξ N in the direction perpendicular to the mean plane of the back reflector is explored, the back reflector being periodically corrugated in one direction. The model is briefly described in Secs. 2.1 and 2.2, with further details being available elsewhere. 16,37 Numerical results are presented in Secs. 3.1 and 3.2. Closing remarks are presented in Sec. 4.

Physical Model
A schematic diagram of the simulated Schottky-barrier solar cell is shown in Fig. 1. The back reflector is corrugated along the x-axis, and the z axis is normal to the plane of the Schottky contact and the mean plane of the back reflector. In the remainder of this paper, the term "width" refers to extent in the x-direction, while "thickness" refers to extent in the z-direction. Insolation is provided via an excitation port at z ¼ −L Air − L w − L c . A planar AR window made of an insulating material occupies the region −L w − L c < z < −L c . To align with our earlier work, the optical permittivity of this layer was taken to be identical to that of aluminum-doped zinc oxide. 38 The region 0 < z < L z is occupied by n-doped In ξ Ga 1−ξ N, forming a Schottky junction and two ohmic junctions with a metal in the region 0 < z < −L c . For calculations, the metal is assumed to be silver. 39 The Schottky contact of width L s is centered at The two regions between the metal contacts for 0 < z < L c are occupied with the same material as the AR window that occupies −L w − L c < z < −L c .
A silver back reflector, covered with an insulating window (made of the same material as the window that occupies −L w − L c < z < −L c ), occupies the region L z < z < L z þ L r . This back reflector is periodically corrugated in the x-direction with period L x . The region L z < z < gðxÞ is filled with the window material, while the region gðxÞ < z < L z þ L r is filled with silver. The corrugation in the unit cell is specified by the function E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 1 6 ; 3 6 3 where L x is the corrugation period, L g ≤ d a is the corrugation height, and ζ ∈ ð0;1 is the duty cycle. As the corrugation is invariant along the y-axis, our model is 2-D. Absorption of the normally incident solar flux with AM1.5G spectrum 40 is calculated by solving the frequency-domain Maxwell postulates. The semiconductor charge-carrier drift-diffusion equations model the electron and hole density spatial distributions. 41,42 Because of the nonhomogeneity of the semiconductor (i.e., In ξ Ga 1−ξ N), the effective dc electric field acting on (a) electrons includes a contribution from gradients in the electron affinity and (b) holes includes contributions from gradients in both the electron affinity and the bandgap. Direct, mid-gap Shockley-Read-Hall, and Auger recombination are all included in our simulation. The current density J, which is averaged over the Schottky contact (or, identically, both of the ohmic contacts), is calculated for a range of external biasing voltages V ext .
By varying the proportion of indium relative to that of gallium, the bandgap of In ξ Ga 1−ξ N can be engineered to take any value from E InN g ¼ 0.7 eV (i.e., for InN when ξ ¼ 1) continuously through to E GaN g ¼ 3.42 eV (i.e., for GaN when ξ ¼ 0). Thus, allowing for nonhomogeneity in the z-direction, the In ξ Ga 1−ξ N alloy has bandgap given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 1 1 6 ; 1 3 3 where the bowing parameter b ¼ 1.43 eV. 43 To estimate ξ for a material with a specific bandgap E g0 , Eq. (2) may be solved as The electron affinity χ 0 for In ξ Ga 1−ξ N is modeled in an analogous manner to the bandgap. Hence E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 1 1 6 ; 6 6 9 where χ InN and χ GaN are the electron affinities of InN and GaN, respectively. For all other material parameters of In ξ Ga 1−ξ N, as described in the first column of Table 1, Vegard's law of linear interpolation 44 is assumed to apply, i.e., E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 1 1 6 ; 6 0 2 where τ ∈ fN C ; N V ; C n ; C p ; C rad ; α oc g. Data for τ InN and τ GaN are provided in Table 1.
The narrowing of the bandgap associated with doping was incorporated through the Slotboom model. 45 While this model was developed for silicon, similar narrowing behavior under heavy-doping conditions has been observed in GaN. 46 When doped, the bandgap of the semiconductor narrows to E g ¼ E g0 − ΔE g , while the electron affinity reduces to Table 1 Electronic data used for GaN and InN. The composition of In ξ Ga 1−ξ N was estimated using Eq. (3), with linear interpolation used to estimate data for the semiconductor-filled region 0 < z < L z with bandgaps not presented here in all cases, except for the electron affinity χ 0 which uses Eq. (4).
with the conduction-band fraction α oc being a material-specific parameter.
Here, the bandgap narrowing is estimated as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 1 1 6 ; 7 1 1 where N d and N a are the doping densities of the donor and the acceptor, respectively, and N ref and E ref are the empirically determined reference doping density and reference energy, respectively. The real part of the optical refractive index of In ξ Ga 1−ξ N is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 7 ; 1 1 6 ; 6 1 2 Refng ¼ Re ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi per the Adachi model, 29 where A A and B A are parameters given in Table 1. The photon energy is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi where the constants come from interpolation of parameters given by Brown et al. 47 An empirical low-field mobility model-called either the Caughey-Thomas 29 or the Arora 48 mobility model-describes the variations of the electron mobility μ n and the hole mobility μ p with temperature and doping. Thus E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 1 ; 1 1 6 ; 3 1 4 where l ¼ n for electrons and l ¼ p for holes. In the foregoing equation, μ l is the maximum value of the carrier mobility μ l when the material is undoped, μ ð1Þ l is the minimum value of μ l when the material is heavily doped, N crit l is the critical doping density of the In ξ Ga 1−ξ N alloy, T is the lattice temperature and T ref ¼ 300 K is the reference temperature, and α ct;l , β l , γ l , and δ l are empirically determined parameters. The solar cell was taken to be operating at T ¼ T ref ¼ 300 K; therefore, it should be noted that the choices of α ct;l , β l , and γ l have no effect on the results. Following Hamady et al., 29 in lieu of experimental data, we assumed that α ct;l ¼ β l ¼ γ l ¼ 1 for all simulations reported here. With these assumptions, the carrier mobilities simplify to E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 2 ; 1 1 6 ; 1 3 5 l ∈ fn; pg; whereÑ l ¼ ðN d þ N a Þ∕N ref l is the ratio of doping density to the critical doping density. The electronic data used for In ξ Ga 1−ξ N presented in Table 1 were provided by Hamady et al., 29 who also demonstrated that a relatively large metal work-function Φ for the Schottky-barrier contact improves efficiency. Accordingly, the relatively large value of Φ ¼ 6 eV was chosen here.
The bandgap profile of In ξ Ga 1−ξ N for the solar cells simulated in this study is given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 3 ; 1 1 6 ; 6 9 9 where E Ã g is the baseline (maximum) bandgap, A is the amplitude, L p ¼ L z ∕κ is the period with κ > 0, ϕ is a phase shift, and α is a shaping parameter, which governs the profile's gradient. Three example bandgap profiles are shown in Fig. 2 for α ∈ f0.2; 1;10g, when κ ¼ 3 and ϕ ¼ 0.75.

Computational Model
A 2-D finite-element optoelectronic model was implemented in the COMSOL Multiphysics (V5.1) software package 48 in two major steps, as described now. Let it be noted that terms in block capitals are COMSOL Multiphysics 48 terms.
In the first major step, the ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN module was used to calculate the 2-D EHP generation rate as a function of x ∈ ð−L x ∕2; L x ∕2 and z ∈ ð0; L z Þ. Normally incident monochromatic light sampled at 5-nm intervals on the λ 0 -scale across the AM1.5G spectrum, 50% s polarized and 50% p polarized, was activated by the PERIODIC PORTS option at z ¼ −L c − L w − L Air . DIFFRACTION ORDER ports for diffraction orders f−3; −2; −1;1; 2;3g were added. The inclusion of diffraction ports for even higher orders has little impact on the resulting EHP generation rate, predominantly due to strong absorption of shorter wavelength photons by In ξ Ga 1−ξ N, the majority of which are absorbed close to the surface of the device before they can be scattered. The boundaries parallel to the z-axis have FLOQUET PERIODICITY with the wavevector provided by the periodic ports. The region z > L z þ L þ f behind the back reflector was taken to be a PERFECTLY MATCHED LAYER. The semiconductor region was first meshed with a MAPPED mesh with a 10-nm DISTRIBUTION in both the z-and x-directions and then split into to a triangular distribution using the insert center points conversion. The back-reflector region was covered with an EXTRA FINE, DELAUNAY, and FREE TRIANGULAR mesh. Data for a spatial map of the computed EHP generation rate were stored in an external file.
The EHP generation rate can be used to compute the optical short-circuit current density J Opt SC , assuming that every absorbed photon creates an EHP in the In ξ Ga 1−ξ N layer and that no recombination takes place. Neglect of recombination implies that J Opt SC is necessarily larger than the short-circuit current density J SC , which is the electronically simulated current density that flows when the solar cell is illuminated and no external bias is applied (i.e., when V ext ¼ 0).
In the second major step, the SEMICONDUCTOR module was used to calculate the electron and hole densities in the semiconductor region, and thereby the current densities. Due to the symmetry of the unit cell, only its right half (i.e., 0 ≤ x ≤ L x ∕2) needs to be electrically simulated. FERMI-DIRAC CARRIER STATISTICS, along with CONTINUOUS QUASI-FERMI LEVELS at any internal boundary, were employed. FINITE VOLUME (CONSTANT SHAPE FUNCTION) discretization was employed as this inherently conserves current throughout the solar cell. 48 COMSOL uses a Scharfetter-Gummel upwinding scheme for solving the charge carrier transport equations. THE FREE TRIANGULAR, DELAUNAY mesh has a maximum element size of 15 nm. A potential difference of V ext was applied between the ohmic and IDEAL SCHOTTKY contacts. THERMIONIC CURRENTS, with standard Richardson coefficients of A Ã n ¼ 110 A K −2 cm −2 , and A Ã p ¼ 90 A K −2 cm −2 were applied at the Schottky barrier. 29,48 INSULATOR INTERFACES were placed at the remaining external electrical boundaries. External Mathematica™ or MATLAB™ codes were used to calculate the USER-DEFINED GENERATION from the output of the first major step. Recombination was incorporated via AUGER, DIRECT and TRAP-ASSISTED (MIDGAP SHOCKLEY-READ-HALL) pathways, with parameters as provided in Table 1. To facilitate convergence, the nonhomogeneity, EHP generation, and EHP recombination physics were slowly activated as the solver progressed by use of a CONTINUATION PARAMETER. AN ANALYTIC DOPING MODEL was used to set the donor concentration to N d . THE FREE TRIANGULAR, DELAUNAY mesh used has a maximum element size of 15 nm.
3 Numerical Simulations of In ξ Ga 1−ξ N Schottky-Barrier Solar Cells

Design of Periodically Corrugated Back Reflector
The numerical results presented here are for devices with parameters listed in Table 2. A preliminary study was carried out to ascertain reasonable values for the dimensions of the device. Figure 3 presents J Opt SC as a function of the back-reflector period L x ∈ ð300;1000Þ nm for a Schottky-barrier solar cell with a homogeneous In ξ Ga 1−ξ N layer of thickness Anderson, Mackay, and Lakhtakia: Enhanced efficiency of Schottky-barrier solar cell. . . L z ∈ f200;400;600;860;1000g nm. For this set of simulations, L g ¼ 130 nm, d a ¼ 135 nm, and ζ ¼ 0.5 were determined as optimal for a solar cell with a homogeneous In ξ Ga 1−ξ N layer characterized by E g0 ¼ 1.6 eV, L x ¼ 600 nm, and L z ¼ 860 nm. While this optimization is far from exhaustive, significant optical variation was seen to occur for thinner In ξ Ga 1−ξ N solar cells: a 6% variation in J Opt SC was calculated for a solar cell with L z ¼ 200 nm, with a maximum J Opt SC of 24.9 mA cm −2 when L x ¼ 480 nm. Importantly, for solar cells with In ξ Ga 1−ξ N layers thicker than about 600 nm, J Opt SC saturates as L x increases beyond a threshold value. Therefore, L x ¼ 600 nm was chosen to be optimal. While the peak value of J SC of a Schottky-barrier solar cell with a 600-nm-thick In ξ Ga 1−ξ N layer is obtained for larger periods than this (L x ≈ 680 nm), J SC falls more rapidly for devices with back reflectors of overly large periods than for those with periods that are too small.

Effect of Periodic Nonhomogeneity of In ξ Ga 1−ξ N Layer
The essential effects of including periodic nonhomogeneity in the thickness direction in the In ξ Ga 1−ξ N layer may be inferred from Fig. 4. For a fixed bandgap-nonhomogeneity amplitude A ∈ f0; 0.2; 0.4; 0.6; 0.8; 1g, the baseline bandgap E Ã g is shown to dramatically affect the simulated efficiency Fig. 3 Optical short-circuit current density J Opt SC as a function of the back-reflector period L x for a Schottky-barrier solar cell with In ξ Ga 1−ξ N layer of thickness L z ∈ f200;400;600;860;1000g nm. The absorbing layer of In ξ Ga 1−ξ N is homogeneous with ξ ¼ 0.6, i.e., the bandgap E g0 ¼ 1.45 eV. Thin solid lines show J Opt SC for the equivalent solar cells with a flat back-reflector. Fig. 4 Efficiency η as a function of baseline bandgap E Ã g for bandgap-nonhomogeneity amplitude A ∈ f0; 0.2; 0.4; 0.6; 0.8; 1g eV, when κ ¼ 3, α ¼ 2, and ϕ ¼ 0.75.
which equals the maximum power produced by the solar cell as a function of V ext , divided by the incident AM1.5G solar power flux of 1000 W∕m 2 . The solar spectrum was discretized at 5-nm λ 0 -intervals; with the minimum λ 0 chosen to be 300 nm; the maximum λ 0 chosen as min½2400;1240∕ðE Ã g − AÞ nm; the other bandgap-nonhomogeneity parameters fixed at κ ¼ 3, α ¼ 2, and ϕ ¼ 0.75. At the minimum bandgap for an In ξ Ga 1−ξ N alloy, i.e., E g ¼ 0.7 V which arises at ξ ¼ 1, there is a minimum permissible baseline bandgap E Ã g for each value of A. Irrespective of the choice of value of A ∈ ð0; 0.8Þ eV, a baseline bandgap of E Ã g ¼ 1.5 eV turns out to be optimal. From Fig. 4, it may be inferred that the difference in the efficiencies with a homogeneous In ξ Ga 1−ξ N layer and a periodically nonhomogeneous In ξ Ga 1−ξ N layer is greatest when a. the bandgap-nonhomogeneity amplitude A is the greatest and b. when the maximum bandgap E Ã g ≈ 1.5 eV. Indeed, at E Ã g ¼ 1.5 eV, the efficiency is 13.26% for A ¼ 0 and 16.81% for A ¼ 0.8 eV, i.e., the relative increase in efficiency attributable to the periodic nonhomogeneity of the In ξ Ga 1−ξ N layer is 26.8%, which is this paper's most significant result. For A ¼ 1 eV, E Ã g ¼ 1.7 eV is optimal, yielding an efficiency of 15.8%.
The importance of the periodicity in the bandgap variation when seeking maximal efficiency may be inferred from Figs. 5 and 6. In Fig. 5, the thickness of the In ξ Ga 1−ξ N layer is fixed at L z ¼ 600 nm while the ratio κ varies from 0.5 to 4. The efficiency of the solar cell is seen to reach a maximum value of about 17% at κ ¼ 1.4 and κ ¼ 3. Between these two values of κ, only a slight decrease in η was found. Significantly, the efficiency for all values of κ > 0 is greater than it is when the In ξ Ga 1−ξ N layer is homogeneous. Figure 6 illustrates the effect of varying the thickness L z of the In ξ Ga 1−ξ N layer when the nonhomogeneity period L p is fixed. Irrespective of the period, the solar cell with the nonhomogeneous In ξ Ga 1−ξ N layer performs better than the one with a homogeneous In ξ Ga 1−ξ N layer. Furthermore, the solar cell with the smaller period (L p ¼ 200 nm) is found to perform more efficiently than the one with larger period (L p ¼ 400 nm). Distinct maximums can be seen at κ ¼ 1.5 and κ ¼ 3, which correspond to L z ¼ 300 nm and L z ¼ 600 nm, respectively. The efficiencies at these points, respectively, are 17.28% and 16.88%, which correspond to relative increases in efficiency of 16.7% and 28.5% as compared to the analogous solar cell with a homogeneous In ξ Ga 1−ξ N layer.
By varying the bandgap-nonhomogeneity shaping parameter α, the spatial profile of the In ξ Ga 1−ξ N nonhomogeneity can be varied. Thus, α ¼ 1 holds for a sinusoidal bandgap profile, Fig. 5 Efficiency η as a function of the ratio κ ¼ L z ∕L p for the In ξ Ga 1−ξ N layer when L z ¼ 600 nm, E Ã g ¼ 1.5 eV, A ¼ 0.8 eV, α ¼ 2, and ϕ ¼ 0.75. The horizontal red line indicates the efficiency when the In ξ Ga 1−ξ N layer is homogeneous.
while smaller and larger values of α yield bandgap profiles with steeper gradients, as illustrated in Fig. 2. For all three values of α in Fig. 7, the efficiency of the solar cell is substantially greater than it is for the corresponding solar cell with a homogeneous In ξ Ga 1−ξ N layer. The maximum increase in efficiency is found for a nearly sinusoidal bandgap profile; indeed, the efficiency increases from 13.2% for a Schottky-barrier solar cell with a homogeneous In ξ Ga 1−ξ N layer to 16.88% for its analog with a nonhomogeneous In ξ Ga 1−ξ N layer with α ¼ 2, which is a relative increase in efficiency of 28.5%.
Let us note here that as α becomes much larger or smaller than unity, the spatial gradients of the bandgap increase in magnitude and the peaks in Fig. 2 become narrower. Accordingly, at extreme values of α, the semiclassical carrier-transport equations implemented in our model become less appropriate as quantum processes become significant. Then, it may be necessary to take into account the quantization of allowed energy states between the peaks of E g0 ðzÞ and tunneling through those peaks, especially for large amplitudes A.

Closing Remarks
A 2-D finite-element model was devised to simulate the combined optical and electrical performances of In ξ Ga 1−ξ N Schottky-barrier solar cells. First, it was found that a periodically Fig. 7 Efficiency η plotted against bandgap-nonhomogeneity shaping parameter α, when corrugated back reflector of a period of 600 nm is optimal for photon absorption in a solar cell containing a homogeneous layer of In ξ Ga 1−ξ N. Second, the effects of periodic nonhomogeneity of the In ξ Ga 1−ξ N layer were elucidated. The nonhomogeneity was directed perpendicular to the mean plane of the periodically corrugated back reflector. For the particular model investigated here, the efficiency of a solar cell with a 600-nm-thick layer of In ξ Ga 1−ξ N was found to increase by 26.8% when suitable periodic nonhomogeneity was incorporated. Thus, the incorporation of a periodically nonhomogeneous In ξ Ga 1−ξ N layer in a Schottky-barrier solar cell can substantially increase the efficiency as compared to the analogous solar cell with a homogeneous In ξ Ga 1−ξ N layer. A comprehensive optimization of study of material and design parameters may well yield even greater nonhomogeneity-induced increases in efficiency. However, such an optimization study-which is justified by the substantial efficiency boosts reported hereinrepresents a major undertaking that lies beyond the scope of this paper. Similarly, it would be of some value to delineate the photon-absorption mechanisms that underpin the boost in the lightto-electricity conversion efficiency that arises following the introduction of periodic nonhomogeneity, but this too is matter for future study.
The feasibility of producing In ξ Ga 1−ξ N alloys with the prescribed spatial variation in ξ to achieve efficiency boosts in Schottky-barrier solar cells is a matter for our experimentalist colleagues to shed light on. On the basis of our numerical study and the relatively huge boosts in efficiency that may be attained, it would be worthwhile for major efforts to be directed toward the production of appropriate In ξ Ga 1−ξ N alloys.
Last, the principal finding of our study is that the introduction of periodic nonhomogeneity can, in principle, substantially boost the efficiency of a Schottky-barrier solar cell. To focus on efficiency, our attention was restricted to only normally incident solar radiation for all cell configurations considered. The influence of the angle of incidence is planned for a future study in which the mechanisms for efficiency boosting will be more fully explored.