## 1.

## 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 photon-absorbing 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 $p\text{-}i\text{-}n$ 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; andiii. 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 (${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$), 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 $\xi \in (\mathrm{0,1})$.^{31} Specifically, indium nitride (i.e., $\xi =1$) 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., $\xi =0$) 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 ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ alloys for all values of $\xi $, particularly for smaller-bandgap alloys. When the proportion of indium is large (i.e., $\xi \gtrsim 0.3$), 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 ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$, as opposed to a $p\text{-}i\text{-}n$ 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 ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$. 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 $\xi $ of ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ 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.

## 2.

## Summary of the Model

## 2.1.

### 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 $z=-{L}_{\text{Air}}-{L}_{w}-{L}_{c}$. 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 ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$, 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 $x=0$. The two ohmic contacts, each of width ${L}_{o}/2$, are centered about $x=\pm ({L}_{x}-{L}_{o}/2)/2$. Note that ${L}_{o}+{L}_{s}<{L}_{x}$. 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 ${L}_{z}<z<{L}_{z}+{L}_{r}$. This back reflector is periodically corrugated in the $x$-direction with period ${L}_{x}$. The region ${L}_{z}<z<g(x)$ is filled with the window material, while the region $g(x)<z<{L}_{z}+{L}_{r}$ is filled with silver. The corrugation in the unit cell is specified by the function

## (1)

$$g(x)=\{\begin{array}{ll}{L}_{z}+{d}_{a}-{L}_{g}\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi x}{\zeta {L}_{x}}\right),& 2x/{L}_{x}\in (-\zeta ,\zeta )\\ {L}_{z}+{d}_{a},& 2x/{L}_{x}\notin (-\zeta ,\zeta )\end{array},$$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., ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$), 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 ${V}_{\mathrm{ext}}$.

By varying the proportion of indium relative to that of gallium, the bandgap of ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ can be engineered to take any value from ${E}_{g}^{\mathrm{InN}}=0.7\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$ (i.e., for InN when $\xi =1$) continuously through to ${E}_{g}^{\mathrm{GaN}}=3.42\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$ (i.e., for GaN when $\xi =0$). Thus, allowing for nonhomogeneity in the $z$-direction, the ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ alloy has bandgap given by

## (2)

$${E}_{g0}(z)=\xi (z){E}_{g}^{\mathrm{InN}}+[1-\xi (z)]{E}_{g}^{\mathrm{GaN}}-b\xi (z)[1-\xi (z)],\phantom{\rule[-0.0ex]{2em}{0.0ex}}z\in (0,{L}_{z}),$$^{43}To estimate $\xi $ for a material with a specific bandgap ${E}_{g0}$, Eq. (2) may be solved as

## (3)

$$\xi (z)=\frac{b+({E}_{g}^{\mathrm{GaN}}-{E}_{g}^{\mathrm{InN}})-\sqrt{4b[{E}_{g0}(z)-{E}_{g}^{\mathrm{GaN}}]+{(b+{E}_{g}^{\mathrm{GaN}}-{E}_{g}^{\mathrm{InN}})}^{2}}}{2b}.$$The electron affinity ${\chi}_{0}$ for ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ is modeled in an analogous manner to the bandgap. Hence

## (4)

$${\chi}_{0}(z)=\xi (z){\chi}^{\mathrm{InN}}+[1-\xi (z)]{\chi}^{\mathrm{GaN}}-b\xi (z)[1-\xi (z)],\phantom{\rule[-0.0ex]{2em}{0.0ex}}z\in (0,{L}_{z}),$$^{44}is assumed to apply, i.e.,

## (5)

$${\tau}^{{\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}}(z)=\xi (z){\tau}^{\mathrm{InN}}+[1-\xi (z)]{\tau}^{\mathrm{GaN}},\phantom{\rule[-0.0ex]{2em}{0.0ex}}z\in (0,{L}_{z}),$$## Table 1

Electronic data used for GaN and InN. The composition of InξGa1−ξN was estimated using Eq. (3), with linear interpolation used to estimate data for the semiconductor-filled region 0<z<Lz with bandgaps not presented here in all cases, except for the electron affinity χ0 which uses Eq. (4).

Symbol | Unit | GaN | InN | |
---|---|---|---|---|

Bandgap | ${E}_{g}^{*}$ | eV | 3.42 | 0.7 |

Electron affinity | ${\chi}_{0}$ | eV | 4.1 | 5.6 |

Density of states (conduction band) | ${N}_{C}$ | ${\mathrm{cm}}^{-3}$ | $2.3\times {10}^{18}$ | $9.1\times {10}^{17}$ |

Density of states (valence band) | ${N}_{V}$ | ${\mathrm{cm}}^{-3}$ | $4.6\times {10}^{19}$ | $5.3\times {10}^{19}$ |

Electron mobility 1 | ${\mu}_{n}^{(1)}$ | ${\mathrm{cm}}^{2}\text{\hspace{0.17em}}{\mathrm{V}}^{-1}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ | 295 | 1030 |

Electron mobility 2 | ${\mu}_{n}^{(2)}$ | ${\mathrm{cm}}^{2}\text{\hspace{0.17em}}{\mathrm{V}}^{-1}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ | 1460 | 14150 |

Caughey–Thomas doping power (electrons) | ${\delta}_{n}$ | — | 0.71 | 0.6959 |

Caughey–Thomas critical doping density (electrons) | ${N}_{\mathrm{n}}^{\mathrm{crit}}$ | ${\mathrm{cm}}^{-3}$ | $7.7\times {10}^{16}$ | $2.07\times {10}^{16}$ |

Hole mobility 1 | ${\mu}_{p}^{(1)}$ | ${\mathrm{cm}}^{2}\text{\hspace{0.17em}}{\mathrm{V}}^{-1}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ | 3 | 3 |

Hole mobility 2 | ${\mu}_{p}^{(2)}$ | ${\mathrm{cm}}^{2}\text{\hspace{0.17em}}{\mathrm{V}}^{-1}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ | 170 | 340 |

Caughey–Thomas doping power (holes) | ${\delta}_{p}$ | — | 2 | 2 |

Caughey–Thomas critical doping density (holes) | ${N}_{\mathrm{p}}^{\mathrm{crit}}$ | ${\mathrm{cm}}^{-3}$ | $1\times {10}^{18}$ | $8\times {10}^{17}$ |

Auger recombination factor (electrons) | ${C}_{n}$ | ${\mathrm{cm}}^{6}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ | $1.5\times {10}^{-30}$ | $1.5\times {10}^{-30}$ |

Auger recombination factor (holes) | ${C}_{p}$ | ${\mathrm{cm}}^{6}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ | $1.5\times {10}^{-30}$ | $1.5\times {10}^{-30}$ |

Direct recombination factor | ${C}_{\mathrm{rad}}$ | ${\mathrm{cm}}^{3}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ | $1.1\times {10}^{-8}$ | $2\times {10}^{-10}$ |

Slotboom reference energy | ${E}_{\mathrm{ref}}$ | eV | $9\times {10}^{-3}$ | $9\times {10}^{-3}$ |

Slotboom reference concentration | ${N}_{\mathrm{ref}}$ | ${\mathrm{cm}}^{-3}$ | $1\times {10}^{17}$ | $1\times {10}^{17}$ |

Conduction-band fraction | ${\alpha}_{oc}$ | — | 0.9 | 0.9 |

Adachi refractive-index parameter ${A}_{A}$ | ${A}_{A}$ | — | 9.31 | 13.55 |

Adachi refractive-index parameter ${B}_{A}$ | ${B}_{A}$ | — | 3.03 | 2.05 |

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 ${E}_{g}={E}_{g0}-\mathrm{\Delta}{E}_{g}$, while the electron affinity reduces to $\chi ={\chi}_{0}-{\alpha}_{oc}\mathrm{\Delta}{E}_{g}$, with the conduction-band fraction ${\alpha}_{oc}$ being a material-specific parameter. Here, the bandgap narrowing is estimated as

## (6)

$$\mathrm{\Delta}{E}_{g}={E}_{\mathrm{ref}}\left\{\mathrm{ln}\right(\frac{{N}_{d}+{N}_{a}}{{N}_{\mathrm{ref}}})+\sqrt{{\left[\mathrm{ln}\right(\frac{{N}_{d}+{N}_{a}}{{N}_{\mathrm{ref}}}\left)\right]}^{2}+\frac{1}{2}}\},$$The real part of the optical refractive index of ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ is

## (7)

$$\mathrm{Re}\{n\}=\mathrm{Re}\left(\sqrt{{A}_{A}(\xi )\left\{{\left[\frac{{E}_{g}(\xi )}{{E}_{\gamma}}\right]}^{2}\right[2-\sqrt{1+\frac{{E}_{\gamma}}{{E}_{g}(\xi )}}-\sqrt{1-\frac{{E}_{\gamma}}{{E}_{g}(\xi )}}]+{B}_{A}(\xi )}\right\}),$$^{29}where ${A}_{A}$ and ${B}_{A}$ are parameters given in Table 1. The photon energy is ${E}_{\gamma}=h{c}_{0}/{\lambda}_{0}$, where $h=6.62607004\times {10}^{-34}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}\text{\hspace{0.17em}}\mathrm{kg}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ is the Planck constant, ${c}_{0}=299792485\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ is the speed of light in free space, and ${\lambda}_{0}$ is the free-space wavelength. The imaginary part of the refractive index is The absorption coefficient ${\alpha}_{\mathrm{opt}}$ is modeled in Ref. 29

## (9)

$${\alpha}_{\mathrm{opt}}(\xi )={10}^{5}\sqrt{C(\xi )({E}_{\gamma}-{E}_{g})+D(\xi ){({E}_{\gamma}-{E}_{g})}^{2}}\text{\hspace{0.17em}}{\mathrm{nm}}^{-1},$$## (10)

$$\begin{array}{l}C(\xi )=(3.525-18.28\xi +40.22{\xi}^{2}-37.52{\xi}^{3}+12.77{\xi}^{4})\text{\hspace{0.17em}}{\mathrm{eV}}^{-1}\\ D(\xi )=(-0.6651+3.616\xi -2.460{\xi}^{2})\text{\hspace{0.17em}}{\mathrm{eV}}^{-2}\end{array}\},$$^{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 ${\mu}_{n}$ and the hole mobility ${\mu}_{p}$ with temperature and doping. Thus

## (11)

$${\mu}_{\ell}={\mu}_{\ell}^{(1)}{\left(\frac{T}{{T}_{\mathrm{ref}}}\right)}^{{\alpha}_{ct,\ell}}+\frac{{\mu}_{\ell}^{(2)}{\left(\frac{T}{{T}_{\mathrm{ref}}}\right)}^{{\beta}_{\ell}}-{\mu}_{\ell}^{(1)}{\left(\frac{T}{{T}_{\mathrm{ref}}}\right)}^{{\alpha}_{ct,\ell}}}{1+{\left[\right(\frac{T}{{T}_{\mathrm{ref}}}{)}^{-{\gamma}_{\ell}}\left(\frac{{N}_{d}+{N}_{a}}{{N}_{\ell}^{\mathrm{crit}}}\right)]}^{{\delta}_{\ell}}},\phantom{\rule[-0.0ex]{2em}{0.0ex}}\ell \in \{n,p\},$$^{29}in lieu of experimental data, we assumed that ${\alpha}_{ct,\ell}={\beta}_{\ell}={\gamma}_{\ell}=1$ for all simulations reported here. With these assumptions, the carrier mobilities simplify to

## (12)

$${\mu}_{\ell}={\mu}_{\ell}^{(1)}+\frac{{\mu}_{\ell}^{(2)}-{\mu}_{\ell}^{(1)}}{1+{\tilde{N}}_{\ell}^{{\delta}_{\ell}}},\phantom{\rule[-0.0ex]{2em}{0.0ex}}\ell \in \{n,p\},$$^{29}who also demonstrated that a relatively large metal work-function $\mathrm{\Phi}$ for the Schottky-barrier contact improves efficiency. Accordingly, the relatively large value of $\mathrm{\Phi}=6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$ was chosen here.

The bandgap profile of ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ for the solar cells simulated in this study is given by

## (13)

$${E}_{g0}(z)={E}_{g}^{*}-A(1-{\left\{\frac{1}{2}\right[\mathrm{sin}(\frac{2\pi z}{{L}_{p}}-2\pi \phi )+1\left]\right\}}^{\alpha}),$$## 2.2.

### 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 $x\in (-{L}_{x}/2,{L}_{x}/2]$ and $z\in (0,{L}_{z})$. Normally incident monochromatic light sampled at 5-nm intervals on the ${\lambda}_{0}$-scale across the AM1.5G spectrum, 50% $s$ polarized and 50% $p$ polarized, was activated by the Periodic ports option at $z=-{L}_{c}-{L}_{w}-{L}_{\text{Air}}$. Diffraction Order ports for diffraction orders $\{-3,-2,-\mathrm{1,1},\mathrm{2,3}\}$ 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 ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$, 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 $z>{L}_{z}+L+f$ 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 ${J}_{\mathrm{SC}}^{\mathrm{Opt}}$, assuming that every absorbed photon creates an EHP in the ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer and that no recombination takes place. Neglect of recombination implies that ${J}_{\mathrm{SC}}^{\mathrm{Opt}}$ is necessarily larger than the short-circuit current density ${J}_{\mathrm{SC}}$, which is the electronically simulated current density that flows when the solar cell is illuminated and no external bias is applied (i.e., when ${V}_{\mathrm{ext}}=0$).

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., $0\le x\le {L}_{x}/2$) 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 ${V}_{\mathrm{ext}}$ was applied between the ohmic and Ideal Schottky contacts. Thermionic currents, with standard Richardson coefficients of ${A}_{n}^{*}=110\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{A}\text{\hspace{0.17em}}{\mathrm{K}}^{-2}\text{\hspace{0.17em}}{\mathrm{cm}}^{-2}$, and ${A}_{p}^{*}=90\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{A}\text{\hspace{0.17em}}{\mathrm{K}}^{-2}\text{\hspace{0.17em}}{\mathrm{cm}}^{-2}$ 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 ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ Schottky-Barrier Solar Cells

## 3.1.

### 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 ${J}_{\mathrm{SC}}^{\mathrm{Opt}}$ as a function of the back-reflector period ${L}_{x}\in (\mathrm{300,1000})\text{\hspace{0.17em}}\mathrm{nm}$ for a Schottky-barrier solar cell with a homogeneous ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer of thickness ${L}_{z}\in \{\mathrm{200,400,600,860,1000}\}\text{\hspace{0.17em}}\mathrm{nm}$. For this set of simulations, ${L}_{g}=130\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, ${d}_{a}=135\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, and $\zeta =0.5$ were determined as optimal for a solar cell with a homogeneous ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer characterized by ${E}_{g0}=1.6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$, ${L}_{x}=600\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, and ${L}_{z}=860\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$. While this optimization is far from exhaustive, significant optical variation was seen to occur for thinner ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ solar cells: a 6% variation in ${J}_{\mathrm{SC}}^{\mathrm{Opt}}$ was calculated for a solar cell with ${L}_{z}=200\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, with a maximum ${J}_{\mathrm{SC}}^{\mathrm{Opt}}$ of $24.9\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mA}\text{\hspace{0.17em}}{\mathrm{cm}}^{-2}$ when ${L}_{x}=480\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$. Importantly, for solar cells with ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layers thicker than about 600 nm, ${J}_{\mathrm{SC}}^{\mathrm{Opt}}$ saturates as ${L}_{x}$ increases beyond a threshold value. Therefore, ${L}_{x}=600\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ was chosen to be optimal. While the peak value of ${J}_{\mathrm{SC}}$ of a Schottky-barrier solar cell with a 600-nm-thick ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer is obtained for larger periods than this (${L}_{x}\approx 680\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$), ${J}_{\mathrm{SC}}$ falls more rapidly for devices with back reflectors of overly large periods than for those with periods that are too small.

## Table 2

Summary of parameters used for the simulations.

Symbol | Value | |
---|---|---|

Device width | ${L}_{x}$ | 600 nm |

$n$-doped ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ thickness | ${L}_{z}$ | 600 nm |

Back reflector insulating layer thickness | ${d}_{a}$ | 135 nm |

Corrugation height | ${L}_{g}$ | 130 nm |

Corrugation duty cycle | $\zeta $ | 0.5 |

Metal thickness | ${L}_{m}$ | 100 nm |

Contact region thickness | ${L}_{c}$ | 5 nm |

Ohmic contact width | ${L}_{o}$ | 140 nm |

Schottky contact width | ${L}_{s}$ | 440 nm |

AR window thickness | ${L}_{w}$ | 100 nm |

Air thickness | ${L}_{\text{Air}}$ | 500 nm |

Predoping bandgap | ${E}_{g0}$ | [0.7,3.42] eV |

Bandgap-nonhomogeneity amplitude | $A$ | $\ge 0\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$ |

Bandgap-nonhomogeneity phase | $\phi $ | [0,1] |

Bandgap-nonhomogeneity shaping parameter | $\alpha $ | (0,32) |

Bandgap-nonhomogeneity period | ${L}_{p}$ | $>0\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ |

Bandgap-nonhomogeneity ratio | $\kappa $ | $>0$ |

## 3.2.

### Effect of Periodic Nonhomogeneity of ${\mathsf{In}}_{\mathsf{\xi}}{\mathsf{Ga}}_{\mathsf{1}-\mathsf{\xi}}\mathsf{N}$ Layer

The essential effects of including periodic nonhomogeneity in the thickness direction in the ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer may be inferred from Fig. 4. For a fixed bandgap-nonhomogeneity amplitude $A\in \{0,0.2,0.4,0.6,0.8,1\}$, the baseline bandgap ${E}_{g}^{*}$ is shown to dramatically affect the simulated efficiency

## (14)

$$\eta =\underset{{V}_{\mathrm{ext}}}{\mathrm{max}}\text{\hspace{0.17em}}\frac{{V}_{\mathrm{ext}}J({V}_{\mathrm{ext}})}{1000\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{W}/{\mathrm{m}}^{2}},$$a. the bandgap-nonhomogeneity amplitude $A$ is the greatest and

b. when the maximum bandgap ${E}_{g}^{*}\approx 1.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$.

Indeed, at ${E}_{g}^{*}=1.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$, the efficiency is 13.26% for $A=0$ and 16.81% for $A=0.8\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$, i.e., the relative increase in efficiency attributable to the periodic nonhomogeneity of the ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer is 26.8%, which is this paper’s most significant result. For $A=1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$, ${E}_{g}^{*}=1.7\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$ 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 ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer is fixed at ${L}_{z}=600\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ while the ratio $\kappa $ varies from 0.5 to 4. The efficiency of the solar cell is seen to reach a maximum value of about 17% at $\kappa =1.4$ and $\kappa =3$. Between these two values of $\kappa $, only a slight decrease in $\eta $ was found. Significantly, the efficiency for all values of $\kappa >0$ is greater than it is when the ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer is homogeneous.

Figure 6 illustrates the effect of varying the thickness ${L}_{z}$ of the ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer when the nonhomogeneity period ${L}_{p}$ is fixed. Irrespective of the period, the solar cell with the nonhomogeneous ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer performs better than the one with a homogeneous ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer. Furthermore, the solar cell with the smaller period (${L}_{p}=200\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$) is found to perform more efficiently than the one with larger period (${L}_{p}=400\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$). Distinct maximums can be seen at $\kappa =1.5$ and $\kappa =3$, which correspond to ${L}_{z}=300\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ and ${L}_{z}=600\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, 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 ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer.

By varying the bandgap-nonhomogeneity shaping parameter $\alpha $, the spatial profile of the ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ nonhomogeneity can be varied. Thus, $\alpha =1$ holds for a sinusoidal bandgap profile, while smaller and larger values of $\alpha $ yield bandgap profiles with steeper gradients, as illustrated in Fig. 2. For all three values of $\alpha $ in Fig. 7, the efficiency of the solar cell is substantially greater than it is for the corresponding solar cell with a homogeneous ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ 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 ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer to 16.88% for its analog with a nonhomogeneous ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer with $\alpha =2$, which is a relative increase in efficiency of 28.5%.

Let us note here that as $\alpha $ 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 $\alpha $, 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 ${E}_{g0}(z)$ and tunneling through those peaks, especially for large amplitudes $A$.

## 4.

## Closing Remarks

A 2-D finite-element model was devised to simulate the combined optical and electrical performances of ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ Schottky-barrier solar cells. First, it was found that a periodically corrugated back reflector of a period of 600 nm is optimal for photon absorption in a solar cell containing a homogeneous layer of ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$. Second, the effects of periodic nonhomogeneity of the ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ 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 ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ was found to increase by 26.8% when suitable periodic nonhomogeneity was incorporated. Thus, the incorporation of a periodically nonhomogeneous ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ layer in a Schottky-barrier solar cell can substantially increase the efficiency as compared to the analogous solar cell with a homogeneous ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ 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 herein—represents 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 light-to-electricity conversion efficiency that arises following the introduction of periodic nonhomogeneity, but this too is matter for future study.

The feasibility of producing ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ alloys with the prescribed spatial variation in $\xi $ 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 ${\mathrm{In}}_{\xi}{\mathrm{Ga}}_{1-\xi}\mathrm{N}$ 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.

## Acknowledgments

T.H.A. thanks the Charles Godfrey Binder Endowment for partial financial support during a six-month stay at Pennsylvania State University. T.G.M. acknowledges the support of EPSRC Grant No. EP/M018075/1. A.L. thanks the National Science Foundation for partial financial support under Grant No. DMS-1619901, and he is grateful to the Charles Godfrey Binder Endowment for ongoing support of his research.

## References

## Biography

**Tom H. Anderson** received his BSc degree in mathematics and physics from the University of Edinburgh in 2011. He received his MSc degree in fusion energy from the University of York in 2012 and his PhD in applied mathematics from the University of Edinburgh in 2016. His research interests include the optical and electrical modeling of thin-film solar cells.

**Tom G. Mackay** is a reader at the School of Mathematics, the University of Edinburgh, and an adjunct professor in the Department of Engineering Science and Mechanics at Pennsylvania State University. He is a graduate of the Universities of Edinburgh, Glasgow, and Strathclyde and a fellow of the Institute of Physics (UK) and SPIE. His research interests include the electromagnetic theory of complex materials, including homogenized composite materials.

**Akhlesh Lakhtakia** received degrees from the Banaras Hindu University and the University of Utah. He is the Charles Godfrey Binder professor of engineering at the Pennsylvania State University. His research interests include surface multiplasmonics, bioreplication, forensic science, solar cells, sculptured thin films, and mimumes. He is a fellow of Optical Society of America, SPIE, Institute of Physics, American Association for the Advancement of Science, American Physical Society, Institute of Electrical and Electronics Engineers, and Royal Society of Chemistry. He received the 2010 SPIE Technical Achievement Award and the 2016 Walston Chubb Award for Innovation.