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

Fig. 1

Schematic illustration of a Schottky-barrier solar cell. Only one back-reflector period is shown [i.e., $-(L_x/2)<x<(L_x/2)$].

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

Absorption of the normally incident solar flux with AM1.5G spectrum⁴⁰ 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 $E_g^{\mathrm{InN}}=0.7\mathrm{eV}$ (i.e., for InN when $\xi =1$) continuously through to $E_g^{\mathrm{GaN}}=3.42\mathrm{eV}$ (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

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

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).

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

The real part of the optical refractive index of ${\mathrm{In}}_{\xi }{\mathrm{Ga}}_{1-\xi }\mathrm{N}$ is

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

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

Fig. 2

Bandgap profiles given by Eq. (13) for $\alpha \in \{0.2,\mathrm{1,10}\}$, when $\kappa =3$ and $\phi =0.75$.

2.2.

Computational Model

A 2-D finite-element optoelectronic model was implemented in the COMSOL Multiphysics (V5.1) software package⁴⁸ in two major steps, as described now. Let it be noted that terms in block capitals are COMSOL Multiphysics⁴⁸ 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 $A_n^{*}=110\mathrm{A}{\mathrm{K}}^{-2}{\mathrm{cm}}^{-2}$, and $A_p^{*}=90\mathrm{A}{\mathrm{K}}^{-2}{\mathrm{cm}}^{-2}$ 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.

3.1.

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 $L_x\in (\mathrm{300,1000})\mathrm{nm}$ for a Schottky-barrier solar cell with a homogeneous ${\mathrm{In}}_{\xi }{\mathrm{Ga}}_{1-\xi }\mathrm{N}$ layer of thickness $L_z\in \{\mathrm{200,400,600,860,1000}\}\mathrm{nm}$. For this set of simulations, $L_g=130\mathrm{nm}$, $d_a=135\mathrm{nm}$, 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 $E_{g0}=1.6\mathrm{eV}$, $L_x=600\mathrm{nm}$, and $L_z=860\mathrm{nm}$. 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 $L_z=200\mathrm{nm}$, with a maximum $J_{\mathrm{SC}}^{\mathrm{Opt}}$ of $24.9\mathrm{mA}{\mathrm{cm}}^{-2}$ when $L_x=480\mathrm{nm}$. 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, $L_x=600\mathrm{nm}$ 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 ($L_x\approx 680\mathrm{nm}$), $J_{\mathrm{SC}}$ 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.

Fig. 3

Optical short-circuit current density $J_{\mathrm{SC}}^{\mathrm{Opt}}$ as a function of the back-reflector period $L_x$ for a Schottky-barrier solar cell with ${\mathrm{In}}_{\xi }{\mathrm{Ga}}_{1-\xi }\mathrm{N}$ layer of thickness $L_z\in \{\mathrm{200,400,600,860,1000}\}\mathrm{nm}$. The absorbing layer of ${\mathrm{In}}_{\xi }{\mathrm{Ga}}_{1-\xi }\mathrm{N}$ is homogeneous with $\xi =0.6$, i.e., the bandgap $E_{g0}=1.45\mathrm{eV}$. Thin solid lines show $J_{\mathrm{SC}}^{\mathrm{Opt}}$ for the equivalent solar cells with a flat back-reflector.

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

Fig. 4

Efficiency $\eta$ as a function of baseline bandgap $E_g^{*}$ for bandgap-nonhomogeneity amplitude $A\in \{0,0.2,0.4,0.6,0.8,1\}\mathrm{eV}$, when $\kappa =3$, $\alpha =2$, and $\phi =0.75$.

Indeed, at $E_g^{*}=1.5\mathrm{eV}$, the efficiency is 13.26% for $A=0$ and 16.81% for $A=0.8\mathrm{eV}$, 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 $A=1\mathrm{eV}$, $E_g^{*}=1.7\mathrm{eV}$ 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 $L_z=600\mathrm{nm}$ 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.

Fig. 5

Efficiency $\eta$ as a function of the ratio $\kappa =L_z/L_p$ for the ${\mathrm{In}}_{\xi }{\mathrm{Ga}}_{1-\xi }\mathrm{N}$ layer when $L_z=600\mathrm{nm}$, $E_g^{*}=1.5\mathrm{eV}$, $A=0.8\mathrm{eV}$, $\alpha =2$, and $\phi =0.75$. The horizontal red line indicates the efficiency when the ${\mathrm{In}}_{\xi }{\mathrm{Ga}}_{1-\xi }\mathrm{N}$ layer is homogeneous.

Fig. 6

Efficiency $\eta$ as a function of thickness $L_z$ of the ${\mathrm{In}}_{\xi }{\mathrm{Ga}}_{1-\xi }\mathrm{N}$ layer, which is either homogeneous (solid black curve) or periodically nonhomogeneous with $L_p=200\mathrm{nm}$ (dashed red curve) or $L_p=400\mathrm{nm}$ (dotted blue curve), when $E_g^{*}=1.54\mathrm{eV}$, $A=0.8\mathrm{eV}$, $\alpha =5$, and $\phi =0.75$. Integer values of the ratio $\kappa =L_z/L_p$ are identified.

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 ($L_p=200\mathrm{nm}$) is found to perform more efficiently than the one with larger period ($L_p=400\mathrm{nm}$). Distinct maximums can be seen at $\kappa =1.5$ and $\kappa =3$, which correspond to $L_z=300\mathrm{nm}$ and $L_z=600\mathrm{nm}$, 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%.

Fig. 7

Efficiency $\eta$ plotted against bandgap-nonhomogeneity shaping parameter $\alpha$, when $E_g^{*}=1.54\mathrm{eV}$, $A=0.8\mathrm{eV}$, $L_z=600\mathrm{nm}$, $\kappa =3$, and $\phi =0.75$.

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 $A$.