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We describe calculations that address the suitability at using silicon-germanium multiple quantum well (MQW) modulators in dense wavelength division multiplexed (DWDM) short reach optical interconnects that vary over a significant temperature range. Our calculations indicate that there is a tradeoff between the number of channels, the temperature range and laser power required. Twenty to forty DWDM channels at 100 GHz and 50 GHz channel spacing is possible in DWDM links with a ∼ 12° temperature range with less than a 1 dB laser power penalty compared to the optimum single channel, single temperature case. The same number of channels can be operated over a wider 37° temperature range with laser power penalties of 3 dB. It shows that, even for DWDM systems, silicon-germanium modulators might provide an alternative to ring and disk resonant modulators without the need for stringent (≪ 1°C) temperature control.
© 2013 Optical Society of America
1. Introduction
Interconnections between processors and between processors and memory for high performance computing and data center platforms seem to require an ever increasing bandwidth density at reduced powers per bit of information. Silicon photonics, with heterogeneous integration with state of the art CMOS electronics, is a promising approach to address both bandwidth density and power dissipation in optical links [1–9]. Dense wavelength division multiplexing (DWDM) techniques offer the highest bandwidth density solution, by virtue of the ability to send multiple signals in the same waveguides and fibers. It is likely that DWDM will be a necessity to achieve the costs required to compete adequately with electronic systems as well.
Silicon microring and microdisk modulators offer a path to DWDM by their resonant nature. In addition, the resonance allows one to use the relatively weak but broadband carrier induced change in the index of refraction to modulate light at very low energies [2–4]. A simple wavelength division multiplexed transmitter can be realized by making devices with different diameters on a common bus. The free-spectral range of the devices determines the maximum number of channels that can be achieved with a set of modulators. Modulators with a free spectral range exceeding 8 THz have been demonstrated [4]. The downside of the silicon photonics resonant modulators is that the resonant wavelength is very sensitive to both fabrication and temperature variations [5,6], and while a number of techniques have been applied both in fabrication and in circuitry to address this sensitivity [5–10], it has not yet been established that these approaches are practical on a large scale.
In contrast, optical waveguide electro-absorption modulators on silicon substrates have been made in both germanium and silicon-germanium strained materials [11, 12] and silicon-germanium quantum wells [13, 14] that make use in the change in optical absorption as a function of electric field via the Franz Keldysh effect in germanium and strained silicon-germanium and the quantum confined stark effect (QCSE) in the quantum well material. The electro-absorption is a much stronger effect than carrier induced refractive index, so these devices need no resonators, they can be quite small, and exhibit switching energies comparable to the smallest silicon photonics resonant modulators [14, 15]. In addition, it is well known that they have a relatively wide fabrication, wavelength, and temperature tolerance compared to resonant devices. While demonstration systems were built years ago using massively parallel free-space optics to interconnect single wavelength devices [16], to date there has been little work in trying to use these devices in DWDM systems. Recently, it was proposed to use VCSELS as a source in multimode guides near 850 nm to provide coarse wavelength division multiplexed signals to GaAs/AlGaAs MQW devices [17]. However, the multimode approach will have limitations in lowering receiver energies [18].
In this paper, we evaluate the use of MQW modulators in DWDM optical interconnects by evaluating the output figure of merit as a function of temperature and optical frequency (wavelength). We developed a model of silicon-germanium quantum wells and fit the results of the model to experimental results in [13]. The model includes the temperature dependence of the band edge and the phonon broadening of the absorption peak. We used the model to calculate the high and low state transmissions from the MQW modulator. The figure of merit is merely one half times the difference in the high-state and low state transmissions. In the next section, we will show that the figure of merit is directly equal to the insertion loss reduced by the extinction ratio power penalty. We describe the methodology to the calculations and present simulated results for the optical usable bandwidth as a function of temperature range for different figures of merit and different applied electric fields. Our results show the potential for up to 10s of DWDM channels, separated by 100 GHz, for a temperature variation of 37° with less than a 3 dB power penalty from the optimum value at a fixed temperature and wavelength. For the devices to be useful for DWDM optical interconnections, some degree of temperature control will be required; a 100 ° temperature swing will not allow uncompensated, uncontrolled quantum well modulators with an adequate figure of merit. We also acknowledge that a DWDM optical interconnect technology needs fiber interfaces, waveguides, wavelength multiplexers and demultiplexers, detectors, and perhaps polarization multiplexers and demultiplexers.
2. Figure of Merit
In Fig. 1 we show a typical modulator based optical interconnect. The photocurrent generated in the receiver photodiode is given by:
where Plaser is the power in the laser in Watts, Tmod is the transmission of the modulator, Toptics is the transmission of the optical path, both before the modulator and between the modulator and the detector, and RPD is the responsivity of the photodiode in A/W.
Fig. 1 One directional photonic link showing laser, modulator, detector, and optical loss. The graphic shows the photocurrents for the two states and the ideal receiver threshold between the states. The driving function for the output of the TIA is proportional to the difference between the photocurrent and the threshold for the two states, the exact formulation (e. g. is it a driving current, driving voltage, in what stage, etc.) depending on the details of the receiver design.
The modulator has two states given by TH and TL. Typically a threshold is set in the receiver given by the average of the two states of (TH + TL)/2 (with PlaserTopticsRPD = 1). The useful photocurrent that drives the output of a receiver is given by the difference in the signal levels and a threshold. Assuming the threshold is set to the midpoint (average) of the two signal levels, the useful photocurrent for these two cases are given by TH − (TH + TL)/2 and TL − (TH + TL)/2. In either case, the useful photocurrent is (TH − TL)/2.
The figure of merit can also be derived from the insertion loss minus the power penalty as is typically done for telecommunications systems [19]. The insertion loss of the device is 10log[(TH + TL)/2]; it is the average power after the modulator as measured with a power meter, not the high-state loss. The power penalty from the finite extinction ratio is 10log[(E + 1)/(E − 1)] where E is the extinction ratio [19]. The figure of merit (in dB) is merely the insertion loss minus the power penalty. Substituting TH/TL for E, gives a result of 10log[(TH − TL)/2] as the figure of merit, and this is the same as the result derived from photocurrent arguments.
Of course, the figure of merit will depend on the receiver choice; a review of many of the possible receivers is described in [20]. Some of these will have different figures of merit, but a figure of merit for all can be derived similarly to that above.
Including non-uniform signal levels, either from optical path or modulator state variations, to derate the figure of merit by assuming the lowest of high-state signals are concurrently used with the highest of low state signals, similar to the analysis for optical logic gates in [21], can also be done by substituting TH(1 − f) and TL(1 + f) for the high and low modulator states respectively where f is the fractional variation in the high and low state transmissivities.
Importantly, the figure of merit is directly proportional to the useful photocurrent available to change the receiver state, at a given laser power as can be seen from Fig. 1 and the analysis above. Hence we can consider that the figure of merit is inversely proportional to laser power; a doubling of the figure of merit for a given photocurrent will lead to a halving of the laser power. Because laser power is currently one of the highest (if not the highest) contributor to the energy consumption of the optical link, the figure of merit as defined here is an important metric for device performance.
The function of the optical modulator is of course to convert an electrical digital signal into on optical one. We would like to maximize the overall conversion efficiency, as this would help minimize the power dissipation in the system. The conversion efficiency can be broken into two parts; the first part is the dissipation related to electrical dissipation of the modulator (which in this case is dominated by the charging of the modulator capacitance) and the second part is the optical transfer function (losses as a function of wavelength, voltage, and temperature) of the modulator that impacts the amount of power available at the receiver for conversion back to an electrical signal. For a given receiver sensitivity, the modulator loss directly impacts the required laser power and thus the dissipation of the optical link. That is, even an ideal modulator with zero capacitance and zero current draw has poor conversion efficiency if it has high loss and a poor extinction ratio as indicated by the figure of merit that we define here. The on-chip dissipation is independent of wavelength and has been covered in detail in reference [15]; as such, the electrical dissipation component of the conversion efficiency is less relevant in determining its suitability for DWDM short reach interconnects than the optical characteristics of the modulator as a function of wavelength and temperature. This is why we chose a figure of merit only related to the optical characteristics of the modulator and not the electrical characteristics.
By optimizing the figure of merit as we have defined it, we simultaneously optimize the insertion loss and extinction ratio of the device in a way that minimizes the optical power required in the system. Stated differently, the FOM provides one way to combine the insertion and extinction ratio metrics into one number which makes the optimization task relatively easier compared to simultaneous optimization of two metrics.
3. Quantum well modeling
3.1. Fitting experimental room temperature electroabsorption
The quantum well structure used for discussion in the present paper is the one experimentally measured by Kuo et al [13]. The structure consists of a p-i-n diode having 10 periods of 10 nm Ge quantum wells between 16 nm Si0.15Ge0.85 barriers. The entire structure is capped by 100 nm of Si0.1Ge0.9 spacer layers on top and bottom. Reference [13] contains spectra for this structure measured at bias voltages of 0, 1, 2, 3, and 4 V.
It is possible to perform an ab initio calculation to obtain quantum well absorption spectra at various bias voltages. Such a calculation requires precise knowledge of several parameters like transition matrix elements, band offsets, subband structure at desired electric fields, band dispersion (E – K relationship), and exciton binding energies and linewidths presence and absence of electric fields. Because of relative novelty of Ge/SiGe quantum wells, many of these parameters are either unknown or have to estimated with considerable uncertainty. Further, even ab initio models of quantum well absorption require adjustable parameters to get numerical values for the absorption coefficient [22]. We have therefore chosen to fit directly the experimental bias-dependent absorption spectra reported in reference [13].
The phenomenological model (fit function), described in reference [23] for GaAs/AlGaAs MQWs, consists of two Gaussian distributions to model the electron-heavy hole and the electron-light hole exciton transition and an exponential factor to model the continuum. In this model, the absorption coefficient is given by:
In Eq. (2), E is the energy at which the absorption coefficient α is desired. The rest of the parameters in Eq. (2) are fitting parameters. The subscripts ‘hh’, ‘lh’ and ‘c’ stand, respectively for heavy hole, light hole, and the continuum. The parameters αi, Ei and Wi are the relative strength, the energy onset and the width of the of the transition i (i = hh, lh or c). Ry is the Rydberg constant of the materials and depends on its relative permittivity and the electron and hole effective masses.A non-linear least-squares fitting with the Levenberg-Marquardt algorithm [24] was performed on the experimental data to obtain the fitting parameters. The obtained values are reported in Table 1 and Fig. 2 pictorially shows the obtained fit quality.
Table 1:. Values of the fitting parameters in Eq. (2) and the fitting uncertainty.
Fig. 2 Experimental (light blue) and fitted (black) electroabsorption curves at bias values of V = 0, 1, 2 and 3 V. The fit was performed usingEq. (2) with the fitting parameters listed in Table 1.
3.2. Electroabsorption spectra at elevated temperatures
Because modern high-performance computers have widely varying temperatures at different locations, one of the figures of merit for the optical link performance is the number of DWDM channels supported at temperatures higher than the room temperature (∼ 300K). We therefore need values of bias-dependent quantum well absorption as a function of temperature. Since electroabsorption modulators rely on band-edge absorption values, the main mechanisms that we need to consider in modeling high temperature behavior of modulators are the shift of the absorption edge and the phonon broadening of the exciton peaks.
We have modeled the band-edge shift of the quantum wells using the widely used, but phenomenological temperature-bandgap relationship [25–27]:
where Eg is the bandgap at temperature T and α and β are the Varshney parameters with values of α = 4.77 × 10−4 eV/K and β = 235 K for Ge [26]. Note that Eq. (3) predicts a red shift of the band-edge wavelength (i.e., decreasing bandgap) with increasing temperature.The second temperature-dependent effect in excitonic electroabsorption is the phonon broadening of the absorption peak. At higher temperatures, there is an increased abundance of phonons which can ionize the quantum well exciton. The width of the exciton resonance, W, is proportional to the phonon population whose distribution is controlled by the Bose-Einstein function. The broadening of the exciton resonance peak is therefore expressed as [23]:
where W is the width of the exciton resonance, T is the temperature, k is the Boltzmann constant, and ELO = 36 meV is the LO phonon energy in Ge. The parameter γ quantifies the strength of the exciton-phonon coupling. The subscript i indicates that phonons can impact each of the three transitions in Eq. (2)(i = hh, lh, and c) in Eq. (2) with potentially different strengths γi for each transition. Our literature search, however, did not yield any experimentally measured values of γi for these transitions in bulk Ge or SiGe quantum wells. We have therefore used the value of γi for AlGaAs quantum wells reported in [23]. We can justify the use of Al-GaAs γ in our calculations using the following reasons. First, quantum well absorption spectra are relatively insensitive to precise values of γi as seen from our variation study in the appendix. Second, γi seems to have a very weak dependence on the well width and the degree of confinement (bulk vs quantum well) [28]. Third, many references place the value of this parameter in the 5–10 meV range for most semiconductors and quantum wells [23,28]. Finally, experimental measurements of electroabsorption in Ge quantum wells in the 25–90° C shows little observable extra broadening at higher temperatures [29]. In spite of the comparatively weaker impact of broadening on electroabsorption spectra, we have included it in our model for completeness and for extensibility when the appropriate experimental values become available in the future.The predictions of the present study are sensitive to the thermal modeling of quantum wells. To gauge the accuracy of the thermal models, we have plotted in Fig. 3 the predicted absorption spectra at higher temperatures of the quantum wells measured by Kuo et al [13]. The accuracy of the thermal modeling can be gauged by comparing the predicted shift of the exciton peak and its broadening with published experimental data for similar MQW structures. Our modeling predicts an exciton resonance red shift of 11.9 meV and 12.5 meV respectively going from 300 K to 331 K and from 331 K to 363 K. We were not able to find temperature-dependent absorption studies of the MQW structure reported in [13]. The authors have, however, reported measurements of a slightly different MQW structure at the above mentioned temperatures in reference [29]. The measurements in [29] are performed on an MQW structure consisting of 12.5 nm Ge quantum wells between 5 nm Si0.175Ge0.825 barriers at a reverse bias of 0.5 V at temperatures of 300 K, 331 K, and 363 K. The measured exciton resonance shifts for this structure are 15.8 meV and 13.9 meV for 300 K–331 K and 331 K–363 K temperature shifts. These shifts are comparable to the model predictions. Further, the significant barrier width difference (16 nm in reference [13] vs 5 nm in reference [29]) causes the exciton the be less well-defined in reference [29] and may also lead to greater shifts for the same temperature difference.
Fig. 3 Calculated absorption spectrum of quantum well stack in reference [13] at temperatures of 300 K, 331 K and 363 K. The MQW structure consists of 10 periods of 10 nm Ge quantum wells between 16 nm Si0.15Ge0.85 barriers. Note that the spectrum at 300 K is fitted to experimental data at same temperature. The spectra at higher temperatures are calculated using the modeling described in section 3.2.
4. Modulator performance at elevated temperatures
With the models for temperature dependent electroabsorption at various bias values between 0V to 4 V, we can compute the modulator performance as a function of wavelength, temperature, and length. As mentioned in the Section 2, the modulator performance is quantified by the single parameter termed as the figure of merit (FOM). Our procedure to calculate the FOM is to insert the desired temperature into equations (3) and (4) and then to compute the total absorption at the desired energy (wavelength) using Eq. (2) and Table 1. Finally, by fixing the modulator length and the on and off bias voltages, we can calculate the FOM as:
Note that the sign of the FOM depends on the modulation wavelength: At the modulation wavelength λ, if α(λ, T, VON) > α(λ, T, VOFF) then FOM is negative. Otherwise, it is positive. The sign of the FOM is not of any particular concern since it can always be inverted electronically. Often this variation in the sign is described as “normally on” or “normally off” depending on whether the modulator has its highest output at 0V or under positive bias. The operating points and the maximum values discussed below have therefore been derived on the basis of the absolute values of the FOM.Each panel in Fig. 4 shows the modulator FOM as a function of wavelength and temperature between 300K to 400K. Different panels show the variation of length (4μm and 8μm) and the high bias voltage (1V to 4V). The low voltage is always assumed to equal 0V. From these maps of the FOM, it is possible to obtain a number of quantities that help us evaluate the suitability of quantum well modulators for interconnects.
Fig. 4 Colormaps of the figure of merit (FOM = (TH − TL)/2) as a function of temperature and wavelength for on voltage values of 1V ≤ V ≤ 4V and modulator lengths of L = 4μm and 8μm.
As explained in section 2, FOM combines the insertion loss and the extinction ratio metrics into one metric proportional to the useful photocurrent generation at the receiver. It is relatively straightforward to extract the insertion loss and the extinction ratio from the FOM. We calculate the values of TH and TL and plug them into the definitions of insertion loss and extinction ratio. We can generate the maps of these quantities similar to that of the FOM shown in Fig. 4. Figure 5 shows as an example the maps of insertion loss and extinction ratio generated for a 5 μm long modulator for on and off voltages of 4V and 0V respectively.
Fig. 5 (a) The insertion loss (IL) and (b) the extinction ratio (ER) as a function of wavelength and temperature for a 5μm long modulator device. For this figure the IL and the ER are defined as IL = 10log10 [(TH + TL)/2] and ER = 10log10 (TH/TL). The IL and ER provide an alternate view of modulator performance by breaking down the FOM into two metrics corresponding roughly to the average loss (IL) and on/off ratio (ER).
An important metric for evaluation of DWDM systems is the number of channels that can be supported as a function of temperature for a given threshold value of the FOM. In other words, we pick the operating temperature range (say room temperature + 40° C) and the minimum value of the FOM tolerable for our system. We then find the two wavelengths (frequencies) at which the threshold FOM value is attained. These wavelength (frequency) values together define the operating bandwidth which, with our assumed channel spacing, gives the number of supported channels at any temperature. Alternatively, we could invert the process and start with the number of desired DWDM channels and the threshold FOM values. The model then gives us the operating temperature range of the system based on this modulator.
The operating bandwidth calculated using the above procedure is plotted in Fig. 6 for different values of on bias (1 ≤ V ≤ 4), modulator length (L = 5μm), and threshold FOM (Th. FOM = 0.125, 0.15, 0.175, and 0.20).
Fig. 6 Optical bandwidth as a function of temperature variation at four values (0.125, 0.15, 0.175, 0.2) of the FOM and for a modulator length of 5μm. The number of channels can be approximated by dividing the optical bandwidth by the channel spacing and adding one.
A table of resonant modulators is presented in [4] that gives the insertion loss and extinction ratio of several silicon photonics resonant modulators. From the table, we can calculate the figure of merit for the different modulators. Ignoring one modulator that had very high loss, the figure of merit ranged from 0.17 to 0.35 with an average of 0.25, assuming the less than 1 dB loss was accurate and assumed to be 1 dB. Note the values in the reference for loss are the on-state loss, not the average insertion loss. However, all the silicon photonic resonant modulators will need accurate temperature stabilization. If we assume a 1 dB additional power penalty for variation because of imperfect temperature stabilization, the figure of merit range becomes 0.13 to 0.28 with a mean of 0.20. In a silicon photonics resonant modulator, a 0.2 C temperature change will give about a 1 dB power penalty in a sample case, but it potentially varies from device to device.
In Fig. 6 we show the optical bandwidth as a function of maximum operating temperature (assuming the minimum is room temperature) with four figures of merit of 0.125, 0.15, 0.175 and 0.2 for a device with an optical path length of 5μm, which is near the optimum. The maximum figure of merit at a single wavelength and temperature was ∼0.25 at 4V. By choosing an allowed figure of merit, operating voltage, and temperature, we can directly read the optical bandwidth from the graph. By further choosing the channel spacing, we can easily determine the number of channels from the channel bandwidth. Fairly obvious tradeoffs are evident. If we constrain the temperature variation to a greater extent, we can have more optical bandwidth and hence more channels. If we reduce the voltage, reducing the energy of the modulator itself, we reduce the number of channels or the allowed temperature variation. Several data points are summarized below:
- The MQW modulator cannot effectively be used in DWDM systems without temperature control over very large (> 60° C) temperature ranges.
- If we constrain the temperature variation to a rather large 37° C, we can potentially have ∼2 THz of optical bandwidth, meaning 20 to 40 channels at 100 or 50 GHz channels spacing respectively with a power penalty of 3 dB compared to the optimum single wavelength and temperature case.
- If we constrain the temperature further to about ∼ 12° C, we can then have the 20–40 channels for a figure of merit of 0.2; that figure of merit is about equal to what can be done with silicon photonics resonant ring or disk modulators with active temperature control to ±0.2° C.
So far, we have ignored the charging and discharging energy of the modulator as an optimization parameter, and optimized the length for maximum figure of merit of the optical path from laser to receiver. On-chip electro-absorption modulator power consumption is described in detail in [15]. It would not be difficult to extend the analysis presented here to include this capacitive charging term if we choose values for receiver sensitivity, laser efficiency, and optical path length losses. However, it is our belief based on recent results for both resonant and quantum well modulators that the modulator charging energy is likely to be low compared to the laser energy, because the device areas are so small.
In calculating the FOM and quantities derived from it, we used the absorption coefficient for surface-normal incidence. We anticipate, however, that the devices ultimately used would be fabricated in a waveguide-geometry. Depending on the size and the material used, the optical confinement factor in waveguides is less than 100%, effectively reducing the strength of interaction of light with quantum wells. This reduced interaction is accounted for by multiplying the quantum well absorption coefficient α with the confinement factor Γ (Γ < 100%). The confinement factor can be tailored by choice of waveguide dimensions, and sub-micron single mode silicon waveguides with Γ approaching 80% are routinely fabricated [30–32]. While there are clearly some differences in the electro-absorption characteristics of waveguide versus surface normal devices, it is likely that the conclusions drawn here are still relevant, independent of device geometry. Also, in a waveguide geometry, one can tradeoff the length, applied field, and applied voltage across the device. Indeed [14] presents and example of SiGe MQW modulator capable of operating at 1 V, although with a reduced figure of merit compared to our results here at the higher voltages. Applying the analysis here would allow the designer to optimize the number of wells and the length of the device, under the constraint of operating at CMOS compatible voltages, for best performance in an optical interconnect system environment.
Importantly, all the MQW devices in our DWDM link analysis were the same; we did not assume that we could make different devices, for example by epitaxial re-growth, for the different wavelengths, which would obviously be a substantial benefit. If we did choose to design different quantum well structures for different wavelengths, we could use the analysis presented here to perform the same analysis, and the results would show a multiplicative increase in the number of channels or the temperature range compared to the case of a single modulator design that we analyzed.
5. Conclusion
We have done preliminary calculations of the suitability of silicon-germanium MQW modulators for use in DWDM optical interconnects, by analyzing a figure of merit as a function of temperature and optical bandwidth. We used a simple model of the quantum wells that was fit to previously published data for silicon-germanium quantum wells. The results of our calculations show the potential for a common multiple quantum well design to be used in a DWDM system with a modest (20–40) number of channels provided a modest (∼ 12° C) temperature control is provided, with the same number of channels operating over 37°C with 3dB laser power penalty compared to the optimum single wavelength and temperature case. Using MQW modulators may be advantageous over silicon photonics micro-resonant modulators if the temperature control circuitry to achieve sub-degree tolerances in those devices proves too complex. Nonetheless even in that scenario, silicon photonics for optical multiplexing and routing must accompany MQW modulators for high-density low energy interconnects.
Appendix 1: Sensitivity of thermal model to exciton-phonon coupling parameter γ
Here we explore the impact of the exciton-phonon coupling parameter γ in Eq. (4) on the predicted high-temperature absorption of quantum wells. As noted in section 3.2, because of the relative difficulty in finding reported measured values of γ, we have used γi = 5.5 meV for all transitions (i = e-hh, e-lh, and continuum). Our choice was motivated by the observation that 5 meV ≤ γ ≤ 10 meV for most common semiconductors. It is important, however, to check the relative sensitivity of our thermal model to this parameter. To that end, we have calculated the predicted zero-bias absorption spectrum of SiGe MQW studied in this paper by varying γ from 2 meV to 10 meV. The spectra are plotted in Fig. 7.
Fig. 7 Predicted zero-applied-bias absorption spectrum of SiGe MQW’s at 360 K calculated using values of exciton-phonon coupling parameter of γ = 2.0, 5.0 and 10.0 meV. The thick light-blue line is the experimental zero-bias absorption at 300K for comparison.
Note the increased broadening of the exciton resonance for the same temperature increase when higher values of γ are used. However, even for a 5× variation in γ, the predicted high temperature absorption spectra are relatively insensitive to its precise value. Therefore, while we recognize the potential limitation to our model due to the lack of precise knowledge of γ, we do not anticipate any significant change to our findings when the precise values of gamma are input to the model as long as they are in the 1–10 meV range.
Appendix 2: Implementation note
The non-linear least squares fitting using the Levenberg-Marquardt algorithm was carried out using the Igor-Pro™ data analysis software [33]. The calculation of absorption spectrum, its temperature dependance, and search for the threshold FOM points were implemented in C++ code steered by a Python script and the graphing was performed using the Python [34] programming language.
Acknowledgments
This work was supported by Sandia National Laboratories’ Laboratory Directed Research and DevelopmentSandia National Laboratories’ Laboratory Directed Research and Development. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lock-heed Martin Company, for the United States Department of Energys National Nuclear Security Administration under contract DE-AC04-94AL85000DE-AC04-94AL85000.
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