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Abstract
We apply the rigorous coupled-wave analysis (RCWA) to the design of a multi-layer plasmonic refractive index sensor based on metallic nanohole arrays integrated with a Ge-on-Si photodetector. RCWA simulations benefit from modularity, frequency-domain computation, and a relatively simple computational setup. These features make the application of RCWA particularly interesting in the case of the simulation and optimization of multi-layered devices in conjunction with plasmonic nanostructures, where other methods can be computationally too expensive for multi-parameter optimization. Our application example serves as a demonstration that RCWA can be utilized as a low-cost, efficient method for device engineering.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Plasmonic nanostructures can be used to control and manipulate light in the visible and infrared spectrum and have been utilized in combination with optoelectronic devices to selectively enhance light absorption [1–3] or to increase the efficiency of light-emission [4]. Biosensing is one particularly important application area: the properties of localized plasmons or propagating surface plasmon-polaritons (SPPs) are influenced by refractive index change in their vicinity, which enables the detection of ligand-analyte binding events [5]. This can be used e.g. for the detection of biomarkers to enable early diagnosis and treatment of diseases. Substrates with functionalized metallic layers have been commercialized for biosensing using surface plasmon resonances [6], but those approaches rely on bulky external instrumentation for readout. In recent years, there has been a development towards integrated nanoscale biosensors, in which plasmonic structures are combined with optoelectronic devices for miniaturized on-chip biosensing [7–10]. If such devices can be realized using industrial-scale CMOS (complementary metal-oxide-semiconductor) processes, this paves the way for low-cost, small footprint sensors, with the promise of affordable technology for a broad range of applications.
Due to the complex fabrication technology, those integrated devices often feature sophisticated stacks of several semiconductor and passivation layers with varying refractive indices and thicknesses. While those layers serve a practical purpose e.g. as etch-stop layers during processing or for the integration of Ge photodetectors on Si substrates, their geometric parameters can also serve as optimization parameters for device performance as an integrated refractive index sensor: Layer thicknesses have been shown e.g. to influence the shape of the Fano resonance in a device in which an Al nanohole-array is integrated into the metallization of Ge-on-Si photodetectors [7]. Solar cells are another example of devices that can benefit from plasmonic enhancement and that can feature complex multilayer stacks, especially in multi-junction devices [11].
In order to predict the performance of such devices, one has to solve Maxwell’s equations for stacks of multiple, periodically patterned layers. Moreover, the geometric parameters of the plasmonic structures as well as all layer thicknesses can, in principle, be used as optimization parameters. As a result, multi-parameter optimization can be computationally expensive for well-established methods such as finite-difference time domain (FDTD) approaches [12] or finite element methods (FEM) [13] that use a discretization mesh in real space. In contrast, the rigorous coupled-wave analysis (RCWA) [14] enables the computation of layers of different thicknesses within one device. In addition, layers can be treated independently: When only one layer is modified, one can skip the computation for the other layers, speeding up parameter sweeps and optimization.
Here, we use the example of an optoelectronic device in which metallic nanohole arrays (NHA) are combined with Ge-on-Si photodetectors (Fig. 1(a)) [7] in order to discuss the suitability of RCWA for the optimization of selected quantities that are particularly relevant for device operation as a refractive index sensor. The device exploits the wavelength-selective effect of extraordinary optical transmission (EOT) in a metallic nanohole array. The light passing through the array is then turned into a photocurrent in the integrated detector. RCWA has proven to be a valuable tool in the design and optimization of optoelectronic devices with dielectric nanohole arrays for broadband absorption. [15,16] Plasmonic devices featuring nanoholes in metal layers [17,18], however, are generally more challenging to simulate using RCWA, since they feature high spatial order evanescent waves, requiring a large number of spatial modes to be considered for convergence. In this work, we will discuss how RCWA can nevertheless facilitate rapid and simple simulation of integrated refractive index sensors, which is particularly relevant for device optimization in the presence of technological constrains imposed on device geometry.
Fig. 1. a) Schematic representation of the layer stack and contacts of the NHA device, as introduced by Augel et al. [7]. The lateral dimensions are not to scale. Only absorption in the 480 nm i-Ge layer (dark green) contributes to the photocurrent. b) Simulated optical responsivity spectra for the NHA geometry as in a), with water (H2O) and ethanol (CH3COOH) as superstrates, with example calculation for bulk sensitivity and FOM*;. The corresponding absorption spectra can found as Fig. S2 in the Supplement 1.
2. RCWA for device optimization
The device geometry as shown in Fig. 1(a) can be seen to be composed of several layers sandwiched between a semi-infinite homogeneous superstrate (liquid with a certain refractive index) and a semi-infinite homogeneous substrate (Si). Within each layer, the material parameters are constant along the z-direction and constant or 2D-periodic in the x-y-plane. For sample geometries that fulfill those two conditions the RCWA method is applicable. The RCWA method relies on expanding material parameters as well as the fields in a single layer in Fourier series [19]. For example, the component of the electric field in parallel with the $x$-axis (assuming a harmonic time dependence) in the nanohole array layer shown in Fig. 1(a) can be expanded as
Here, $\tilde n_x$ and $\tilde n_y$ are the indices of the reciprocal lattice vector in the plane wave expansion, and $\Lambda$ is the pitch of the square lattice. For practical computation, the sum over the indices has to be truncated so that only a certain number of Fourier modes are retained. Details on the method are derived in Supplement 1.
When using the RCWA approach, care has to be taken to avoid numerical instabilities that can originate from exponentially growing and decaying terms within the layers. Here, we focus on two different numerically stable solution algorithms, i.e. the enhanced transmittance matrix method (ETM) [14] and the S-matrix method [19,20]. While the enhanced transmittance matrix method is generally faster, the S-matrix method allows a larger degree of precomputation of the expansion coefficients for individual layers and can be beneficial in multi-layer structures such as ours.
While several RCWA implementations are publicly available, see e.g. [21–23], most of them [22,23] are written for pure optics simulations, and do not compute local absorption in multilayer devices, which is required for simulation of optoelectronic devices. Furthermore, most implementations [21,22] only feature the scattering-matrix (S-matrix) [19] algorithm. For ease of use and implementation, we, therefore, developed our own software based on Refs. [14,19,20,24]. Our implementation features both ETM [14] and S-matrix algorithms. It is derived fully within Supplement 1, and the source code is available online [25].
In NHA devices as shown in Fig. 1(a), optimization targets are related to device and sensor performance and include device responsivity as well as surface and bulk sensitivity and FOM*. Here, we briefly discuss how to extract those quantities from RCWA results. Diode responsivity, i.e. the photocurrent $I_{photo}$ generated in the i-Ge layer divided by the incident optical power $P_0$, is related to the absorbed power within the i-Ge layer and can be calculated from the difference of the relative radiant flux above and below that layer (see Supplement 1 for details). The power absorbed in the $l$-th layer $A_l$ is then:
where $P_{l+1}$ and $P_l$ are the relative radiant flux through the interfaces after and before the layer of interest, respectively. By multiplying this with the conversion yield $\eta$ and dividing by the photon energy $\frac {hc}{\lambda }$, one can then calculate the responsivity asThroughout this work, we neglect conversion losses, assuming unity conversion yield $\eta =1$. Figure 1(b) shows a simulated optical responsivity spectrum, which depends on the wavelength and the superstrate refractive index.
Another important figure is the bulk sensitivity $S_{\mathrm {bulk}}$, defined as the change in resonance wavelength per refractive index change [5]. For this work, we use the point where $R_{\mathrm {opt}}(\lambda )$ has maximum slope as the resonance wavelength (Fig. 1(b).
In plasmonic biosensing applications, detection of specific biochemical markers is generally accomplished via a surface functionalization [5], refractive index changes then only occur in a conformal film relatively close to the device surface [26,27]. This surface sensitivity $S_{\mathrm {surface}}$ can be computed through simulation with a varied-index zone close to the sample surface and a constant-index zone making up the rest of the superstrate.
Along with the evanescent electric field, the detectability of refractive index changes decays with the distance to the sample surface. Accordingly, $S_{\mathrm {surface}}$ is a function of the thickness of the surface film $t$
$\delta$ is the decay length of the exponential. It is most practical for 3D devices to measure $\delta$ reasonably far from the surface, so that the topography does not play a role.Methods based on a real-space grid like FDTD of FEM require numeric derivation or even fitting to obtain the decay length. In contrast, RCWA directly computes quantities analytically related to the decay length and $\delta$ can be employed as an optimization objective, without significant increase in computation time.
The quantity FOM* [28] is of particular relevance for single-wavelength operation of the device. It is defined as the ratio of the relative change of optical responsivity induced by a variation in superstrate refractive index divided by said variation at a fixed wavelength $\lambda$,
An example is shown in Fig. 1(b). Finally, we note that for all simulations, literature values were chosen for the permittivities of Si [29], Ge [30], SiO2 [31] and Al [32].
3. Device Simulation and Convergence
Computer simulations always have to discretize physical quantities in order to represent them in a digital format. The degree of this discretization is defined and adjusted by simulation hyperparameters. Generally, these impose a trade-off between computation accuracy and speed, and finding the right parameter values can be challenging. Since no PML, and no real-space mesh is required, the only hyperparameter to be adjusted in RCWA simulations is the number of plane waves considered. We use the maximum scattering order $N$ to quantify this. The simulation considers all indices $\tilde n_x$, $\tilde n_y$ of the reciprocal lattice that fulfill $-N\leq \tilde n_x,\tilde n_y\leq N$. For a given $N$, there are $(2N+1)^2$ reciprocal lattice vectors considered.
The numerical result converges to theory for $N\rightarrow \infty$. The value of $N$ required for reasonable accuracy depends on the device geometry and materials. Figure 2 shows spectra for the device submerged in DI water. We also compare with spectrum computed by FDTD obtained using the simulation package Lumerical [34]; this commercial simulation tool is widely used for the simulation of plasmonic nanostructures. For $N=0$, the NHA layer simplifies to an effective medium approximation, with permittivity $\varepsilon =\varepsilon _{Al}+\pi (d/2p)^2(\varepsilon _{H2O}-\varepsilon _{Al})$. As $N$ is increased, more and more modes are considered. The Fano resonance at 1280 nm is fully observable from $N=4$ onwards, higher values of $N$ continue to increase accuracy. The slow convergence of the Fano peak is due to its connection to high spatial order evanescent plasmonic modes [7] .
Fig. 2. Spectra for different $N$, stack by [7] in DI water (n=1.321) [33]. Hole diameter $d=480\mathrm {nm}$, pitch $\Lambda =950\mathrm {nm}$. FDTD result (Lumerical) for comparison,
In order to quantify convergence, we employ two metrics. Since the real spectrum for $N\rightarrow \infty$ is not known, we define iterative deviations, representing the change in the spectrum for an increase of $N$. The maximum incremental error $E_{max}(N)$ is the maximum relative deviation of all the wavelength points between the responsivity spectra for $N$ and $N+1$. The root mean square (RMS) error $E_{RMS}(N)$ is the square root of the mean of the squared relative deviations for all wavelengths.
Figure 3(a) illustrates the dependence of both error metrics on the maximum order $N$. Both show a trend to decrease exponentially with $N$, confirming convergence of the results. However, they do not decrease monotonically, but show an oscillating behavior. This can be observed more closely in the spectra for (Fig. 2). For example, the spectra for $N=2$ and $N=4$ agree more with each other than with the spectrum for $N=3$. The reason for this alternating behavior is that odd- and even-order modes cancel each other here. Care has to be taken when scaling the accuracy, as convergence can be deceiving. The incremental RMS error of an FDTD simulation of the same device, computed from absorption spectra for rectilinear grids with point distance of $5\mathrm {nm}$ and 4nm is shown for reference, the value is 0.078.
Fig. 3. a) Convergence of RCWA results for increasing $N$. RMS and maximum deviation between the spectra (Fig. 2) for $N$ and $N+1$. b) Computation time for 81 wavelength points for both RCWA algorithms as a function of $N$.
Figure 3(b)) shows how computation time increases exponentially with $N$ for $N\geq 5$. All computation times are measured on a single computing node of a workstation cluster, featuring 24 virtual cores. For lower $N$, computation time is higher than expected, since only larger matrices benefit from parallelized algebra. The ETM algorithm requires consistently $\sim 2.5$ times less computation time, and should thus be the preferred algorithm for single-device spectra. For reference, a FDTD [34] simulation with comparable accuracy (see Fig. 3(a)) requires $1.5\cdot 10^3$ seconds on the same computing platform.
With only one hyperparameter in the system, scaling between low-$N$ spectra for rapid initial design and high-$N$ single-wavelength/few-wavelength optimizations at higher simulation time for performance maximization is simple.
4. Parameter sweeps
RCWA treats and computes layers independently to concatenate them later. Because of this, it can benefit from parameter sweeps where only single layers are modified. The most common example of a parameter sweep for a refractive index sensor is the modification of the refractive index of the analyte. This affects the properties of the superstrate and the nanohole array layer. In a multilayer stack, all other layers can be precomputed, saving computation time. We also evaluated the computational effort of sweeping the thickness of any of the intermediate dielectric layers.
Fig. 4 shows this performance improvement for both algorithms. One can see how the computational cost per element decreases with the number of computed elements for all curves. The performance increase for a refractive index sweep with the ETM method is minuscule. This is because one can only precompute the modes of the unpatterned, dielectric layers, which do not involve any matrix inversions and eigenvalue computations. When sweeping the thickness of any of the dielectric layers, a greater performance increase is achievable since the electromagnetic field eigenstates of the NHA can be precomputed.
Fig. 4. Performance increase through modularization in sweeps. Computation times per device for sweeps with $N=11$. 81 wavelength points were simulated for each device.
The scattering matrix methods benefits much more from parallelization in sweeps. This is because one can precompute the scattering matrixes of all constant layers, as well as the concatenations of adjacent layers. Per sweep element, only the scattering matrices of the variable layers and the final concatenation need to be computed. While it is generally slower than the ETM algorithm, the scattering matrix algorithm achieves lower per-element computation times for large parameter sweeps. This is because computation-heavy eigenvalue computation of the NHA is precomputed, while the modified homogenous layer only requires matrix-vector multiplications (S72).
With more complex layer stacks, especially such that feature several patterned layers, this improves further, since the fraction of modified layers by unmodified layers decreases. In multiscale sweeps, and in iterative optimizations, a large number of similar devices differing in only few layers is evaluated. Here, the computational effort per device can be decreased even further.
5. Selected results in device simulation and optimization
While $\mathrm {FOM}^*$ as defined in Eq. (7) is an ideal target metric for a single-wavelength refractive index sensor, setups including the photodiode as discussed here also require a significant optical responsivity. This is because the electrical signal requires a reasonable signal-to-noise ratio to be accurately detectable. Thus, device geometry has to be optimized for both objectives. We employed a cost function
to be minimized via the Nelder-Mead algorithm. Here, $r$ is a weighting hyperparameter used to prioritize either $\mathrm {FOM}^*$ or $R_{\mathrm {opt}}$. The parameter $\lambda$ is the fixed target wavelength at which the metric is evaluated, motivated by the fact that an external single-wavelength laser light source would be expected in a compact refractive index sensing setup. Since $R_{\mathrm {opt}}$ enters into the denominator of $\mathrm {FOM}^*$, achieving high values for both metrics simultaneously cannot be done. Table 1 shows how the selection of $r$ balances both metrics against each other. We used $\lambda =1300 \mathrm {nm}$, $n=1.35$ and $\Delta n=0.002$ for the evaluation of all metrics. While not optimizing for them, we also show the bulk and surface sensitivities $S_{\mathrm {bulk}}$ and $S_{\mathrm {surface}}$ as well as the field decay length $\delta$ for a $t=20\mathrm {nm}$ surface film.
Table 1. Outcome of optimization for $\mathrm {FOM}^*$ and $R_{\mathrm {opt}}$ for different weighting factors $r$, and comparison with the device by Augel et al. [7]. The other metrics discussed in this work are given for reference.
We identified $r=100$ as a reasonable value to achieve high $\mathrm {FOM}^*$ while keeping $R_{\mathrm {opt}}$ reasonably high at around $200\frac {\mathrm {mA}}{\mathrm {W}}$. The optimization result is shown in Fig. 5. The starting values for the geometric dimensions were taken from Augel et al. [7]. We varied the array pitch $\Lambda$, the nanohole diameter $d$, the aluminum layer thickness $h_{Al}$ and the silicon dioxide thickness $h_{SiO2}$ (Fig. 1(a)). The original and optimized ($r=100$) spectra are shown in Fig. 5(a).
Fig. 5. a) Spectra for the optimized device for two different refractive indices. The spectra for the original geometry are shown with dashed lines. b) Optical responsivity $R_{\mathrm {opt}}$ (mean of the values for the different refractive indices) for varied wavelength and SiO2 thickness. c) Difference in optical responsivity $\Delta R_{\mathrm {opt}}$ for the two refractive indices. All geometry parameters except for $h_{SiO2}$ are taken from Tab. 1, for $r=100$.
The influence of the SiO2 thickness on the optical responsivity is visualized in Fig. 5(b)) and c), all other parameters are taken from the optimized geometry. Figure 5(b) shows that peaks with the highest possible optical responsivity can be found for a large thickness of the SiO2 layer. This can be attributed to the similarity in refractive index values of the superstrate and the SiO2 layer, which generally leads to an improved EOT through the NHA and thus an increased photocurrent in the detector. By contrast, the largest (absolute value of) refractive-index related optical responsivity change $\Delta R_{\mathrm {opt}}$ is found for thin SiO2 layers (Fig. 5(c)). This optimization, thus, allows fine tuning of device parameters depending on whether large values for $R_{\mathrm {opt}}$ or a large index-dependence of $R_{\mathrm {opt}}$ is prioritized
As a frequency-domain method, RCWA has an advantage over time-domain methods, since the absorption only needs to be computed twice for each parameter set. With $N=10$, the single-wavelength optimization took approximately five hours to compute. With the computation speeds of (single-wavelength) RCWA and (multi-wavelength) FDTD as shown in Fig. 3, using FDTD would lead to an optimization time of about 650 hours, beyond practicality. The five-hour optimization time is due to the relatively large number of free design parameters. Restricting the optimization to NHA pitch and diameter only would lead to an optimization time of just 40 minutes (or 90 hours via FDTD).
6. Conclusion
The simulation-based optimization of optoelectronic devices containing both plasmonic nanostructures and a multi-layered device stack poses particular challenges, since a large number of geometry and material parameters are potentially relevant. Here, we show that RCWA as a frequency-domain method can provide a fast and computationally efficient means of tackling these problems. Using the combination of metallic nanohole arrays with Ge-on-Si photodetectors as a device example for applications in refractive index sensing, we give estimates of the benefits of modularity and the trade-off between computation speed and accuracy. In particular, we compare the performance of the scattering matrix method and the enhanced transmission matrix method, illustrating how the former can outperform the latter in extensive multi-parameter optimization due to its larger degree of modularity. Furthermore, we showcase the application of the methodology in a single-wavelength optimization problem that is of particular relevance for the improvement of sensor performance. This example serves to show that RCWA can make multi-parameter optimization problems computationally accessible that could not be addressed by e.g. FDTD approaches. Thus, RCWA can enable simple and rapid engineering of integrated multilayer optoelectronic devices. It is an interesting objective for future work to provide a comparative overview of several numeric methods for application to this or other multilayer device types. This work is accompanied by a full and consistent derivation of the RCWA formulation and algorithms in a Supplement 1, and a practical, modular open-source implementation of the algorithms.
Funding
Bundesministerium für Bildung und Forschung (16ES1128K).
Acknowledgements
The work was supported by the North-German Supercomputing Alliance (HLRN). We thank Andreas Krebs of BTU Cottbus-Senftenberg for valuable support with the computing cluster implementation.
Disclosures
The authors declare no conflicts of interest.
Data Availability
The RCWA source code is available online [25]. All RCWA simulations are performed with literature values for the relative permittivities. [29–32] Lumerical FDTD [34] simulations use material parameters included in the proprietary software package.
Supplemental document
See Supplement 1 for supporting content.
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