## Abstract

Highly nonlinear optical materials with fast third-order nonlinear optical response are crucial for the operation of all-optical photonic devices, such as switches for signal processing and computation, power limiters, and saturable absorbers. The nonlinear response of traditional optical materials is weak, thus requiring large light intensities to induce significant changes in their properties. Here we show that optical control of the coupling rate in subwavelength patch antennas coupled to intersubband transitions in multi-quantum-well semiconductor heterostructures can provide a giant third-order nonlinear response, on the order of ${{3.4 \times 10}}^{- 13}{{{m}}^2}/{{\rm{V}}^2}$, with a response time ${\lt}{{2}}\;{\rm{ps}}$. We utilize this effect to realize intersubband polaritonic metasurfaces and demonstrate their operation as highly nonlinear saturable and reverse saturable absorbers, enabling optical power limiters and other elements for all-optical modulation and control. Our approach enables a plethora of compact, low-power, highly nonlinear devices with spectral, temporal, and structured wavefront responses tailored by design.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Highly nonlinear metasurfaces are desirable for a wide range of applications, including wave mixing without phase-matching constraints, all-optical switching and beam steering, optical limiting, and phase conjugation. With these functionalities in mind, a number of approaches have been pursued to achieve large ${\chi ^{(2)}}$ and ${\chi ^{(3)}}$ in metasurfaces [1], using intrinsic metal [2,3] or dielectric [4–7] nonlinearities, epsilon-near-zero materials [8–13], and coupling to J-aggregates or other molecules [14–18]. It has recently been shown that intersubband polaritonic metasurfaces, based on coupling between subwavelength patch antenna resonators and intersubband transitions in multi-quantum wells (MQWs), can produce an effective ${\chi ^{(2)}}$ far exceeding other approaches [19,20]. Their unique nonlinear response stems from the combination of quantum engineering of the electronic states in the MQW heterostructure, which by itself enables one of the largest known nonlinear responses in bulk materials [21–24], with photonic engineering of the resonance field profiles. These engineered resonances support highly confined, strong electric fields ideally overlapped with the MQW transition dipole moment driving the nonlinearity. Using this approach, metasurfaces with a tailored nonlinear response have recently been demonstrated [25–28] across a wide range of frequencies, from the near to far infrared.

The nonlinear conversion efficiency of intersubband polaritonic metasurfaces is ultimately limited by intensity saturation [29]. Here, we show that this mechanism, driven by depletion of the ground state of the MQW, can be used to our advantage to demonstrate extreme third-order nonlinearity beyond the limits of perturbative responses [Figs. 1(a) and 1(b)]. Depletion of the ground state reduces the intersubband oscillator strength, which in turn reduces the effective coupling rate between the antenna resonance and the intersubband transition [30,31]:

where ${n_g}$ and ${n_e}$ are the ground and excited state occupancies in the MQW (assuming that it can be described by a two-level system), respectively, and ${\Omega _R}$ is the vacuum Rabi frequency [the splitting between peaks at low intensity is approximately $2{\Omega _R}$, see Fig. 1(c)].Nonlinear tunability of the coupling coefficient enables control over the system response: in the unsaturated case coherent energy transfer occurs between the antenna resonance and the intersubband transition [15,32], while in the saturated case only the resonator is effectively excited. In the frequency domain, this implies a transition from two resonant peaks in the unsaturated regime, due to polaritonic splitting, to a single peak in the spectrum when the transition is saturated [Figs. 1(c) and 1(d)]. This nonlinear transition from strong to weak coupling has been observed in, for example, semiconductor microcavities [31,33], plasmonic arrays coupled to molecules [14,15,18], and defect modes in metal/multi-quantum well photonic crystals [30,34]. Here, we demonstrate the nonlinear transition from strong to weak coupling in a polaritonic metasurface, where the constituent building blocks, or meta-atoms, are subwavelength metal patch antennas coupled to a multi-quantum well [see Fig. 1(a)]. The nonlinear optical response of these metasurfaces is principally determined by the local response of the meta-atom, which results in a largely angle-independent response (see Supplement 1), and it can be engaged without necessarily illuminating a very large area (i.e., with focused beams). Crucially, the approach demonstrated here opens up a pathway to structural and all-optical control of the wavefront of the reflected beam, analogous to previous demonstrations for metasurfaces [25,26].

To achieve strong spectral variations with intensity, as shown in Figs. 1(c) and 1(d), a large coupling rate with respect to the dephasing rates is required. In contrast, to achieve an ultrafast nonlinear response, the antenna and MQW linewidths have to be reasonably large. This implies that, for a strongly nonlinear response, the coupling constant has to approach the frequency of operation. This so-called ultrastrong coupling regime has recently been demonstrated in MQWs [35–38], with other systems rapidly following [39]. In this paper, we show that the synergy between MQW transitions and the antennas can dramatically enhance the coupling constant and nonlinear response, resulting in very low input powers required to trigger the transition at ultrafast speeds. The MQW transition is quantum engineered to have a large dipole moment and short upper state lifetimes (to achieve ultrafast speeds), while the antenna enhances the field in the MQW both through a large optical cross section and a moderate quality factor [40]. The resulting system is ideal to realize ultralow-power optical limiters based on reverse saturable absorption, which strongly reflect at low intensity but limit the amount of reflection at high intensity. At other frequencies, the system enables extremely sensitive saturable absorbers that work in reflection mode.

The paper is structured as follows: in the next two sections, we discuss the nonlinear optical properties of the polaritonic metasurface, followed by a demonstration of all-optical control in Section 4. Based on a theoretical model introduced and experimentally verified in Sections 2–4, we then evaluate the fundamental limits of resonant nonlinear optical limiters in Section 5, providing a pathway for further improvements to their performance. Finally, in Section 6, we numerically demonstrate all-optical nonlinear control of the wavefront of the reflected beam, highlighting the versatility of the platform.

## 2. POLARITONIC METASURFACES

An scanning electron microscope (SEM) image of the fabricated polaritonic metasurface is shown in Fig. 2(a). It is composed of an array of gold dipolar patch antennas [41–45], where the dielectric spacer layer between the gold substrate and a thin gold strip is an n-doped InGaAs/InAlAs MQW, as shown schematically in Figs. 1(a) and 1(b) (further details are provided in Supplement 1). The band diagram of a single period of the MQW layer is shown in Fig. 2(b). The MQW layer consisted of 26 period repetitions for a total thickness of approximately 400 nm. The transition frequency between the first and the second electron subbands in the MQW is ${\nu _{21}} \approx 40\;{\rm{THz}}$, as confirmed by absorption measurements (see Appendix A). The transition energy between the second and third subband is ${\nu _{32}} \approx 25\;{\rm{THz}}$, significantly detuned from ${\nu _{21}}$.

To find good overlap between the antenna resonance and the intersubband transition, we vary the patch antenna length ${l_y}$, as shown in Fig. 2(c). The measured linear absorption spectra of the metasurfaces [in the absorption units of $A = ({I_{{\rm{in}}}} - {I_{\rm{r}}})/{I_{{\rm{in}}}}$, where ${I_{{\rm{in}}}}$ is the incident and ${I_{\rm{r}}}$ is the reflected light intensity] shows a clear anticrossing when the patch resonance intersects the intersubband transition. The MQW transition dipole moment is out-of-plane, and hence, it cannot be excited by the normally incident radiation directly but couples very efficiently to the antenna [35,37,38,46]. The intersubband transition frequency, as expected from bulk absorption measurements, is shown by the white dashed line.

We model the interaction between the metasurface and the multi-quantum well by combining coupled-mode theory with a Maxwell–Bloch model [47–50], where we approximate the MQW by $N$ two-level quantum oscillators all coupled to the antenna with frequency $g$ in the rotating wave approximation (see Supplement 1 for details):

Here $a$ is the complex plasmon amplitude [normalized so that ${| a |^2}$ is the stored number of photons], $q$ tracks the evolution of the off-diagonal elements of the intersubband transition reduced density matrix [50], and $w = {n_e} - {n_g}$ is the inversion of the transition. The antenna and intersubband (ISB) transition have resonance angular frequencies ${\omega _c}$ and ${\omega _q}$, and the MQW is further described by the total decoherence rate ${\gamma _2} = 1/T{}_2$ and relaxation rate ${\gamma _1} = 1/{T_1}$. Only the antennas are excited by the incident field amplitude ${s_ +}$ (normalized so that $|{s_ +}{|^2}$ is the number of incident photons per second). The antenna decays through radiation (${\gamma _r}$) and absorption (${\gamma _a}$), as well as through a saturation-related decay rate ${\gamma _s}$. This additional decay rate has to be included in the model, because the MQW cannot truly be described by a two-level system, but instead, there are transitions between higher excited states that become relevant at high intensity. In fact, at very high intensity, all electrons are promoted into the continuum, and the electromagnetic response becomes similar to that of a hot electron gas. The Rabi frequency is given by ${\Omega _R} = \sqrt N g$ (see Supplement 1).

The third-order nonlinearity in this model arises through the inversion $w$ [see Eq. (2)], which affects the coupling between the two-level systems and the patch nanoantenna. Particularly, when the excited and ground state populations are equal, the quantum oscillators can no longer be coupled to, and the coupled system reduces to the patch antenna alone. In contrast, at low intensities, $w \approx - 1$ and Eq. (2) simplifies to the well-known linear equations for coupled oscillators. We use the linearized equations to extract model parameters from the absorption measurements shown in Fig. 2(b) by fitting them simultaneously (see Supplement 1 for details). The linear response from the model is shown in Fig. 2(d), and it is in excellent agreement with the measurements shown in Fig. 2(c). In particular, for the metasurface with ${l_y} \approx {1.75}\;{\rm{\unicode{x00B5}{\rm m}}}$ shown in Fig. 2(a), we find ${\omega _c} = 251.1 \times {10^{12}}\;{\rm{rad/s}}$, ${\omega _q} = 248.2 \times {10^{12}}\;{\rm{rad/s}}$, ${\gamma _r} = 6.85\;{\rm{THz}}$, ${\gamma _a} = 1.34\;{\rm{THz}}$, ${\gamma _2} = 27.6\;{\rm{THz}}$, and ${\Omega _R} = 31.8 \times {10^{12}}\;{\rm{rad/s}}$. In the next section, we turn to the nonlinear response, and demonstrate multiple novel functional operating regimes for the nonlinear metasurface.

## 3. NONLINEAR INTENSITY DEPENDENCE

In Fig. 3(a), we show the metasurface absorption spectra at low (${\sim}{{70}}\;{\rm{kW/c}}{{\rm{m}}^2}$) and high peak intensity (${\sim}{{700}}\;{\rm{kW/c}}{{\rm{m}}^2}$), measured using a tunable ps laser (with an approximate 0.5 THz pulse bandwidth). At low peak intensity, we still observe the characteristic polaritonic splitting, but with reduced separation compared to the Fourier-transform infrared spectroscopy (FTIR) spectrum, while at higher peak intensity the polaritonic splitting vanishes, resulting in a single peak. This implies that the intersubband transition has been saturated sufficiently for it to be weakly coupled to the patch antenna. The experimental spectra are in good agreement with those predicted from the simple Maxwell–Bloch model, although the model only considers a single coupling strength to all $N$ two-level systems.

Next, we consider the power dependence for three separate fixed frequencies, shown by the gray bands in Fig. 3(a). In Fig. 3(b) we consider the response at 35.5 THz, where there is a peak in absorption due to the lower polariton branch. As we increase the intensity, the mode splitting decreases, resulting in a reduction in absorption. In this regime, the metasurface acts as a saturable absorber. At 38 THz [Fig. 3(c)], we are slightly detuned from the lower polariton branch, and the absorption increases with intensity as the mode splitting reduces. Finally, at 40 THz [Fig. 3(d)], we observe a strong increase in absorption with increasing intensity. This frequency is close to the bare plasmon and intersubband transition frequencies and, as a result, it lies in between both low-intensity polariton branches. As the intensity is increased, the mode splitting vanishes, and we ultimately obtain a single peak with strong absorption. Interestingly, as the intersubband transition saturates, the absorption rates decrease, yet the overall absorption increases. This is because the system is engineered to be overdamped at low intensities, and with increasing intensity it approaches critical coupling [51], where the absorption and radiation loss rates are equal and at resonance the metasurface fully absorbs. At 40 THz, we observe a change in absorption from 52% at low intensity up to 82% as the intensity increases.

To place the nonlinear response of the metasurface into context of other nonlinear materials, we estimate an effective nonlinear coefficient $\chi _{{\rm{eff}}}^{(3)}$ from the change in absorption at low intensities. To do so, we consider the metasurface (comprising subwavelength patch resonators and multi-quantum well) as a single homogeneous material with an effective nonlinear absorption coefficient ${\beta _{{\rm{eff}}}} = \Delta A/2d$, where $\Delta A$ is the change in the absorption due to illumination with intensity $I$ after a double pass through the material with thickness $2d = 1\;{{\unicode{x00B5}{\rm m}}}$. From Fig. 3(b), we find ${\beta _{{\rm{eff}}}} = 1.6 \times {10^7}\;{\rm{cm/GW}}$, which can be converted into $\Im (\chi _{{\rm{eff}}}^{(3)}) = 3.4 \times {10^{- 13}}\;{{\rm{m}}^2}/{{\rm{V}}^2}$ (see Supplement 1 for details). These values exceed naturally occurring ultrafast ${\chi ^{(3)}}$’s by several orders of magnitude, and also appear stronger than photonic approaches to enhanced $\chi _{{\rm{eff}}}^{(3)}$’s based on plasmonics [1,3], ENZ [8,9], and two-dimensional [52,53] materials, or semiconductor saturable absorbers [54,55]. However, note that when comparing nonlinearity strengths across different frequency ranges, it must be considered that higher frequencies generally correspond to weaker material nonlinearities.

## 4. ALL-OPTICAL CONTROL

The change in absorption, both positive and negative, can be initiated by a separate pulse. To demonstrate this all-optical control in reflection, we show in Fig. 4 results from a degenerate pump-probe measurement. In these experiments, we excite the metasurface with a strong pump pulse at an angle of incidence around 30°. We simultaneously excite the metasurface under normal incidence with a weak probe pulse (${\lt}{{50}}\;{\rm{kW/c}}{{\rm{m}}^2}$) with variable delay. Depending on the center frequency, we obtain either an increase [Fig. 4(a), 35.5 THz] or a decrease in reflection [Fig. 4(b), 40 THz], as expected from the power dependence highlighted in Fig. 3. In both cases, we observe excellent agreement with the Maxwell–Bloch model (red solid line), where we have assumed that the pulse length is ${\sim}{1.6}\;{\rm{ps}}$ intensity full-width half-maximum (FWHM) and ${T_1} = 1.7{\rm{ps}}$, which is obtained from band structure calculations. Note that ${T_1}$ is dominated by nonradiative recombination, so that the optical environment has negligible effect on the decay rate. As evident from the pump-probe measurement, the metasurface response is ultrafast, despite the observed giant nonlinearity. The rise time of the system in our measurement is constrained by the pulse length, indicating that the system has nonlinear dynamics below the ps scale. The recovery time of the system is limited by the relaxation time of the excited state, ${T_1} = 1.7\;{\rm{ps}}$, enabling ultrafast all-optical low-power manipulation of optical signals. It should be noted that, with a broadband probe pulse, richer dynamics such as the frequency shifts of the polariton branches or Rabi oscillations can be observed [15,32]. We are currently expanding our experimental capabilities to investigate these phenomena.

## 5. FUNDAMENTAL LIMITS TO OPTICAL LIMITERS

Having verified the accuracy of our Maxwell–Bloch model, we are able to investigate the fundamental limits to nonlinear light manipulation of our polaritonic metasurface in terms of reflection contrast. An ideal optical limiter fully absorbs when saturated, and at the same time, it fully reflects at low intensities. In our metasurface, Ohmic losses in the antenna (which are not affected by the input intensity) and in the saturated multi-quantum well fundamentally limit the lowest possible reflection that we can achieve. By requiring that at high intensity the saturated metasurface is perfectly absorbing, i.e., critically coupled, we find from our model that the absorption contrast $|\Delta A|$ between high and low intensity is given by

An example of optical limiting is shown in the top of Fig. 5(a), where at high intensity the metasurface is critically coupled, while at low intensity, mode splitting results in low absorption at the same frequency. This example corresponds to $\eta = 0.031$ and ${f_s} = 0.08$, resulting in $|\Delta A| = 0.65$. The absorption contrast for other parameter combinations is shown in the bottom part of Fig. 5(a), where this particular example is highlighted with a circle. The contour plot highlights that optimal performance is simply governed by the two parameters $\eta$ and ${f_s}$, one corresponding to photonic engineering, the other to material engineering. The lower the two can be made, the larger the reflection contrast that can be achieved. Improved performance in our metasurfaces can be achieved using dielectric resonators rather than metallic, to minimize $\eta$ by avoiding metal absorption, and improving the MQW system to minimize the level of absorption in the saturation regime, for example by reducing the decoherence rate ${\gamma _2}$.

For the opposite process, the design of optimal metasurfaces with saturable absorption functionality, there are two approaches available. We can still work at the bare resonance frequencies but start with a critically coupled system at low intensity that becomes undercoupled at high intensity. In this case, the absorption contrast is exactly the same as derived for optical limiting in Eq. (3). The other approach to metasurface saturable absorption is shown in Fig. 5(b), where the frequency is aligned with a critically coupled normal mode at low intensity. In this case, the absorption contrast depends on three variables (${\gamma _s}/{\Omega _R}$, ${\gamma _a}/{\Omega _R}$, and ${\gamma _2}/{\Omega _R}$), and the final expression for the absorption contrast is more complicated (see Supplement 1 for details). We will therefore just highlight different regimes: one with very small ${\gamma _2}/{\Omega _R} = 0.1$ [Fig. 5(b)] and one with modest ${\gamma _2}/{\Omega _R} = 0.75$ [Fig. 5(c)] MQW dephasing rates, the latter example being more representative of the present experiment. We again find the largest contrast for small ${\gamma _a}$ and ${\gamma _s}$, but we also see that a low ${\gamma _2}/{\Omega _R}$ clearly makes the system more forgiving, again making the case for further engineering of the MQW materials to reduce the decoherence rate. Note that these results are derived assuming that the rotating wave approximation is valid, which implies that when ${\Omega _R}/{\omega _q}\gtrapprox 0.1$, the applicability of these limits should be verified using numerical simulations (see Supplement 1). Deep in the ultrastrong coupling regime, these limits may differ significantly.

In the experiment presented in this manuscript, we find that our designs yield ${f_s} \approx 0.32$ and $\eta \approx 0.09$, respectively, predicting a maximum optical limiting contrast $|\Delta A| = 0.2$. Interestingly, this is below the change in absorption that we actually observe ($|\Delta A| \approx 0.3$), which is a consequence of the finite reflection at high intensity: the polaritonic metasurface is underdamped (${\gamma _2}$ is too small) to be critically coupled at high intensity. Increasing ${\gamma _2}$ would also result in higher absorption at low intensity, reducing the overall absorption contrast. Furthermore, our theory demonstrates that in our metasurface residual absorption in the MQW is the limiting factor to achieve larger contrast: by reducing ${\gamma _s}$ in the MQW, we may expect better optical limiting, especially when combined with lower loss metals or dielectric designs and narrower MQW linewidths. We are currently exploring these venues.

## 6. NONLINEAR PHASE-GRADIENT METASURFACE

Highly nonlinear optical response in deeply subwavelength nanoresonators composing our metasurface can enable all-optical manipulation of the reflected beam wave front using relatively low pumping intensities. This capability is demonstrated in Fig. 6, where we show the simulation results of power dependent beam deflection. An imprinted phase profile on the reflected beam is used to deflect the incident beam at arbitrary angles [56]. Due to the nonlinear response, the incident beam at low powers is reflected specularly [Fig. 6(a)], while at high power it is deflected into a single diffracted order at 30° [Fig. 6(b)].

The nonlinear metasurface consists of the same antennas studied in the previous sections. We now form a supercell consisting of six antennas, the length of each adjusted to have the correct phase in reflection following the design procedure outlined in Ref. [57]. The unit cell is now 2.5 by 15 µm, and the six antennas have lengths 740, 1195, 1242, 1270, 1302, and 1379 nm, respectively, so that the local reflection phase of the antennas themselves is 120, 60, 0, ${-}{{60}}$, ${-}{{120}}$, and ${-}{{180}}$, respectively. In the low power regime, due to polaritonic splitting, at 42 THz the antenna is barely excited, and we obtain specular reflection (because no phase profile is imprinted). This can be seen in Fig. 6(c), where the reflection into each diffracted order is shown. At high intensity, however, the polaritonic splitting vanishes, and the beam is largely reflected into the first diffracted order, as shown in Fig. 6(d). These results demonstrate that, besides optical limiting (Fig. 4) and all-optical reflectivity control (Fig. 5) discussed in detail earlier, the highly nonlinear meta-atoms demonstrated in this work enable other, more complicated functionalities. These more advanced functionalities will be the subject of further theoretical and experimental analysis in a future study.

## 7. CONCLUSIONS

In summary, we have experimentally demonstrated ultra-fast mid-infrared nonlinear polaritonic metasurfaces that display strong optical limiting and power saturation in reflection. The ultrafast switching of the metasurface reflectivity is based on the optical control of the coupling constant between a MQW intersubband transition and an optical mode in a patch antenna. To the best of our knowledge, the presented metasurface has the largest ultrafast third-order nonlinear absorption coefficient reported in the mid- or near-infrared range, enabling dramatic spectral changes at low input powers with large speed. We discussed what limits the absorption contrast in these structures, and determined the relevant design parameters, in terms of material and photonic engineering, required to achieve optimal performance for power limiting and saturable absorption. We foresee applications of these ultrathin metasurfaces for optical limiters, all-optical low-power switches, and saturable absorbers. By tailoring the meta-atoms across different unit cells, incorporating, e.g., a geometric phase, these metasurfaces may provide advanced linear and intensity-dependent nonlinear responses that may be used for optical control of the intensity, polarization state, and wavefront of the reflected beam.

## APPENDIX A: METHODS

**Sample Fabrication:** Fabrication of the intersubband polaritonic metasurface described in this work began with growth of the MQW structure by molecular beam epitaxy on an InP substrate. The wafer was first metallized via e-beam evaporation with Ti, Pt, and Au layers of 10, 20, and 200 nm, respectively, and thermo-compressively bonded to a foreign InP substrate with identical metal layers to form a metal backplane. The original substrate was then removed via mechanical polishing and selective wet chemical etching. 10 nm of Ti and 100 nm of Au were evaporated to form the top metal, followed by a 300-nm-thick SiNx hard mask layer. The dipole resonators were then defined via electron-beam lithography and successive dry etching steps, and finally, the hard mask was removed. To account for the uncertainty of material parameters such as metal loss and experimental radiative loss of the dipole array, we fabricated dipole antennas with varying lengths and unit cell sizes.

**Intensity-Dependent Response:** The mid-IR output of a picosecond optical parametric oscillator (OPO)/difference frequency generation (DFG) system (Carmina, APE) with ${\sim}{1.6}\;{\rm{ps}}$ intensity FWHM pulses at a 40 MHz repetition rate was focused onto the metasurface under study through a mid-IR transmissive BaF_{2} lens. We control the intensity of the pump source by using a wire-grid linear polarizer pair, where the first polarizer is rotated to attenuate the polarized output of the OPG/OPA throughout the intensity-dependent measurements, and the second polarizer is rotated and held constant to ensure light is polarized along the IPM dipole axis. The incident and reflected power are measured simultaneously by two Mercury–Cadmium–Telluride (MCT) photodetectors placed on either side of a wedge ZnSe beam-splitter in front of the BaF_{2} lens. We simultaneously measure a fraction of the incident and reflected beams using two separate detectors to account for any variations in laser power, and normalize results from our IPMs to an Au mirror reference, which does not produce an intensity-dependent optical response.

**Degenerate Pump-Probe Experiment:** The mid-IR output of a ps-OPO/DFG system is split into strong pump and weak probe beam paths. The pump beam is focused onto the metasurface under an angle, while the probe is directed toward a retroreflector mounted on a delay stage, and then focused at normal incidence onto the metasurface at the same spatial location. The incident and reflected powers of the probe path are simultaneously monitored by way of two MCT photodetectors placed on either side of a wedge ZnSe beam-splitter in front of the sample, and time-dependent reflectivity is retrieved by monitoring the reflectivity recovered from the probe beam as a function of the pump-probe delay.

## Funding

Defense Advanced Research Projects Agency; Air Force Office of Scientific Research; Office of Naval Research; U.S. Department of Defense; Nederlandse Organisatie voor Wetenschappelijk Onderzoek (019.172EN.019); U.S. Department of Energy; National Nuclear Security Administration (DE-NA0003525).

## Acknowledgment

IB and JK acknowledge support by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. Sample processing was done at the University of Texas at Austin Microelectronics Research Center, which is a member of the National Science Foundation National Nanotechnology Coordinated Infrastructure (NSF NNCI).

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## Supplemental document

See Supplement 1 for supporting content.

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