We demonstrate our FEM model symmetry reduction with symmetry modes using two numerical examples, implemented with COMSOL Multiphysics
® 5.6 software with the Wave Optics Module. First, we consider models of a single hexagonal nanowire embedded in a semi-infinite superstrate on top of a a semi-infinite substrate for obtaining the nanowire absorption cross-section as a function of the plane wave incidence direction and polarization (as opposed to the common practice of modeling at just the normal incidence for convenience). Such models could be used e.g., in the analysis or design of a single-nanowire solar cell [
3]. Second, we consider models also with semi-infinite superstrate and substrate but with a periodic array of hexagonal nanowires arranged in an orthorhombic (rectangular) lattice. In this case, the primitive cell is rectangular, and we therefore get the full benefit of using the symmetry reduction (as opposed to a non-rectangular primitive cell).
Due to the restriction on the plane wave incidence angle (Equation (
13)), this second example with symmetry reduced unit cell for the array finds somewhat limited applicability. One application could be to use such an array as a grating and use similar models to compute the diffraction order efficiencies as a function of the nanowire dimensions (diameter and length) when the incidence is kept fixed at one of the
th Littrow configurations (for chosen wavelength and period). Alternatively, the array could act as a photodetector [
5] and the models be used to design the nanowire dimensions to maximize absorption at the desired wavelengths (again, with a fixed specific incidence that respects the restrictions). We lean towards the latter application in our example but also demonstrate an additional aspect of the symmetry reduction.
Therefore, in our periodic nanowire array example, we consider the incidence of two coherent antiphase plane waves at mirror incidence angles (
,
,
, and
) leading to polarization selective absorption in the nanowires (centered at the origin of the unit cell) with constructive interference for the p-polarization x-components and destructive interference for the s-polarization y-components (also destructive interference for the p-polarization z-components). We note that, with this incidence configuration and either p- or s-polarization, the background field is actually directly one of the symmetry modes:
for p- and
for s-polarization, respectively. Therefore, this is a simple example of such a case where the symmetry modes themselves could be of interest. We further note that exploiting such coherent absorption of multiple incident plane waves has wider interest [
21] and similar opportunities for symmetry reduction of models could arise in this context. With these models, we explore the array absorption as a function of the periods
and
, while maintaining the
and
Littrow configuration for the two incident plane waves, i.e.,
and
, meaning that
does not put restriction on the incidence angle.
3.1. Models
In the first single nanowire absorption cross-section model, we select the substrate material as Si, the nanowire material as GaAs, the nanowire diameter as
nm, and the nanowire length as
nm (viz. ref. [
3]). The wavelength (in free space) of the incident plane wave is chosen as
nm, at which we use the refractive index
for Si [
22] and
for GaAs [
23]. For simplicity, we consider the superstrate medium to be a polymer with a refractive index of
instead of a more realistic stack of air, transparent conductive oxide, and polymer (viz. ref [
3]). As already mentioned, with this simplification, the general background field expression of Equation (
12) reduces to a total of three plane wave components (incident and reflected plane wave in the superstrate and transmitted plane wave in the substrate). We select the simulation domain width (square cross-section), superstrate height, and substrate height such that they are sufficient to not result in numerical artifacts in the field solution (approximately one wavelength space between the nanowire and the simulation domain boundaries). We then cut the simulation domain with the
and
planes to one quarter with
and
. All the remaining original boundaries are terminated with PMLs, while for each SP, the background field is set as one of the symmetry modes of Equation (
2) and the cut plane boundaries are assigned either PMC or PEC boundary condition according to
Table 1. The simulation domain is meshed with a free tetrahedral mesh using the Fine preset settings in the software and adjusting the maximum mesh element size to check convergence of the results with mesh refinement. The PMLs are meshed with a swept mesh of six elements across. We selected the tightest maximum mesh element size as
, where
is the free space wavelength and
n is the refractive index (real part) in each domain, and found sufficient convergence in the obtained absorption cross-section values with maximum mesh element size of
. The electric field in the model is solved using a Wavelength Domain study at
nm and the direct MUMPS solver with the default settings, except for turning off the out-of-core functionality to avoid swapping data from RAM to hard drive storage if the system matrix becomes too large, which would lead to erroneous extracted values of maximum RAM use. We opted to use a direct solver instead of an iterative one since, although iterative solvers tend to use less RAM, direct solvers are more robust and reliable. For comparison, we also consider a second model without the symmetry reduction having the OP background field and all boundaries terminated with PMLs. In the following, we refer to these two models as “Sym. red. 1/4” and “No sym. red.”, respectively.
In the first periodic nanowire array absorption model, we select the substrate as Si, the nanowire material as In
Ga
As, the nanowire diameter as
nm, and the nanowire length as
nm (viz. ref. [
5], but we choose a thicker nanowire for increased absorption, although not optimized [
8]). The wavelength (in free space) of the incident plane wave is chosen as the popular telecom wavelength
nm, at which we use the refractive index
for Si [
24] and
for In
Ga
As [
23]. Again, for simplicity, we consider the superstrate medium to be a polymer with a refractive index of
instead of a more realistic stack of air, transparent conductive oxide, and polymer (viz. ref [
5]). We select the superstrate and substrate heights such that they are sufficient to not result in numerical artifacts in the field solution (approximately one wavelength space between the nanowire and the simulation domain boundaries above and below). We then cut the simulation domain with the
and
planes to one quarter with
and
. The top and bottom boundaries are terminated with PMLs, and the background field with the two incident plane waves is, as mentioned, the symmetry mode
for p- and
for s-polarization (Equation
2 and the cut plane boundaries are assigned either PMC or PEC boundary condition according to
Table 1). The simulation domain is meshed with a free tetrahedral mesh using the Fine preset settings and adjusting the maximum mesh element size to check convergence of the results with mesh refinement. The PMLs are meshed with a swept mesh of six elements across. We selected the tightest maximum mesh element size as
, where
is the free space wavelength and
n is the refractive index (real part) in each domain, and found sufficient convergence in the obtained absorption values with maximum mesh element size of
, except for small
. The electric fields in the model are solved using a Wavelength Domain study at
nm and the direct MUMPS solver with the default settings, except for turning off the out-of-core functionality. For the periodic models, iterative solvers are not even a practical option since they, in general, tend to not work well with Floquet boundary conditions. Therefore, we use the selected direct MUMPS solver throughout. In the following, this first periodic model is referred to as “APMI sym. red. 1/4”.
For comparison, we also consider four additional models: one with the same background field but without the symmetry reduction and three single plane wave background field (
and
) models: without symmetry reduction, with symmetry reduction to one half, and with symmetry reduction to one quarter, respectively. In the following, we refer to these four models as “APMI no sym. red.”, “SI no sym. red.”, “SI sym. red. 1/2”, and “SI sym. red. 1/4”, respectively. The model “APMI no sym. red.” has the original simulation domain with Floquet boundaries at the sides. Note that, since the wave vector
x-components of the two incident plane waves differ only in sign and the
y-components are zero,
and the Floquet condition on the boundaries
reduces to field continuity. The only difference between the models “SI no sym. red.” and “APMI no sym. red.” is the background field. These two models without symmetry reduction serve as the reference point for the symmetry reduced models. The “SI sym. red. 1/2” model uses the symmetry modes of Equation (
4) which simply reduce to the p- and s-polarized component of the background field. Boundary conditions for the
and
planes are determined from
Table 2 while the
and
planes have the Floquet boundary condition. This model actually corresponds to the aforementioned conventional mirror plane incidence symmetry reduction case where the incident light is along the mirror plane. Finally, background field in the “SI sym. red. 1/4” corresponds to the aforementioned special case of the symmetry modes of Equation (
2), where
for p-polarization and
for s-polarization, reducing the SPs to two for each polarization. Otherwise, the four models are set up the same as the “APMI sym. red. 1/4” model. For clarity, we have summarized the SPs solved with each model in
Table 4.
With the field solutions from the solved models, we compute the single nanowire absorption cross-section and nanowire array absorption results. The nanowire absorption cross-section is computed as
, where
is the power absorbed in the nanowire (
integrated over the nanowire volume, where
is the induced current density) and
is the incident intensity (
, where
is the free space impedance). The nanowire array absorption is computed as
, where
is the incident power into the unit cell and
is the power absorbed in the nanowire in the unit cell. Note that there is no power flow between the unit cells, so the incident power in one unit cell is given by integrating the incident field Poynting vector
z-component over the cross-section area (
). In the “APMI” models, the incident power is doubled with two plane waves incident instead of one (the interference does not alter the total power, only its distribution). We perform the computation of
in the symmetry reduced domain with Equation (
7) or Equation (
8) and sum the results to obtain the value in the full domain.
3.2. Simulation Results
With the two single nanowire absorption cross-section models “Sym. red. 1/4” and “No sym. red.”, we considered both p- and s-polarization and swept the polar incidence angle
from
to
with
step and the azimuth incidence angle
from
to
with
step (due to our definition of normal incidence,
when
in the sweep). These ranges for the incidence angles covered all unique incidences from the upper half-space, due to the hexagonal nanowire cross-section (
symmetry).
Figure 4 shows the nanowire absorption cross-section for
obtained from the simulations and the maximum DOF, total solver time and maximum RAM used for the whole sweep. These reported results were obtained with a local desktop computer Dell Precision 3640 MT with Intel Core i7-10700 8-core CPU and 128 GB RAM. As seen in
Figure 4a, the good agreement in the absorption cross-section values between the two models indicates that the used symmetry reduction was set up correctly.
The results shown in
Figure 4b were in line with our expectations. Since the symmetry reduced domain in the “Sym. red. 1/4” model was a factor of 4 smaller than the full domain in the “No sym. red.” model, the DOF was also approximately a factor of 4 smaller and so was the maximum RAM use, accordingly. Interestingly, a superlinear dependence of the solver time on the system matrix size showed up in the results yielding also a much smaller total solver time for the “Sym. red. 1/4” model. We did take into account the special incidence angle cases in the sweep where only part of the SPs needed to be solved, but this concerned only the normal incidence and
(which constituted only approximately 1/30 of the steps in the sweep) and was hence not the reason for the observed reduced total solver time with symmetry reduction. We noticed that the overhead from setting up the solver was increased for the “Sym. red. 1/4” model when several SPs were solved instead of the one OP. However, as long as the overhead time was small compared to the actual solver time, as is likely to be the case with tight mesh for good convergence, this was not an issue for obtaining performance increase with the symmetry reduction.
With the periodic nanowire array absorption models “APMI sym. red. 1/4”, “APMI no sym. red.”, “SI no sym. red.”, “SI sym. red. 1/2”, and “SI sym. red. 1/4”, we considered both p- and s-polarization and swept the period
such that, with the (2,0) Littrow configuration, the corresponding polar incidence angle
was swept from
to
with
step. We first also considered different
periods from 300 nm (less than one diameter
D space between the nanowires) to 1107 nm, corresponding to
, with coarser steps and meshing and then selected
nm for this example due to good convergence in the absorption results with the maximum mesh element size of
.
Figure 5 shows the periodic nanowire array absorption obtained from the simulations with the models “APMI sym. red. 1/4” and “APMI no sym. red.” and the maximum DOF, total solver time, and maximum RAM used for the sweep.
Figure 6 shows the periodic nanowire array absorption obtained from the simulations with the models “SI no sym. red.”, “SI sym. red. 1/2”, and “SI sym. red. 1/4” and the maximum DOF, total solver time, and maximum RAM used for the
(or equivalently
) sweep. These reported results were also obtained with the local desktop computer Dell Precision 3640 MT with Intel Core i7-10700 8-core CPU and 128 GB RAM. The good agreement in the absorption values between the models in both sets (as seen in
Figure 5a and
Figure 6a) indicated that the used symmetry reductions were set up correctly. Furthermore, the expected difference in the polarization dependence of the absorption between the case of two antiphase plane waves at mirror incidence and the case of a single plane wave incidence was also demonstrated.
Interestingly, the results shown in
Figure 5b and
Figure 6b were much better than what could be expected based on the DOF reduction alone: although the DOF scaled as expected, the maximum RAM use and total solver time with all the symmetry reduced models were much smaller than that would entail. With the “APMI sym. red. 1/4” model the maximum RAM use was approximately a factor of six smaller and the total solver time was approximately a factor of 10 smaller than with the “APMI no sym. red.” model. With the “SI sym. red. 1/2” model the maximum RAM use was approximately a factor of three smaller and the total solver time was approximately a factor of five smaller than with the “SI no sym. red.” model. Finally, with the “SI sym. red. 1/4” model the maximum RAM use was approximately a factor of nine smaller and the total solver time was approximately a factor of seven smaller than with the “SI no sym. red.” model. We attribute this result to the aforementioned effect of Floquet (or field continuity) boundary conditions making the system matrix more dense than the simpler PMC or PEC boundary conditions. Indeed, the symmetry reduced periodic array models having the additional benefit of not only making the system matrix smaller but also less dense, clearly could exhibit strongly reduced computational cost.