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Abstract
The effect of doping concentration on the temperature performance of the novel split-well resonant-phonon (SWRP) terahertz quantum-cascade laser (THz QCL) scheme supporting a clean 4-level system design was analyzed using non-equilibrium Green’s functions (NEGF) calculations. Experimental research showed that increasing the doping concentration in these designs led to better results compared to the split-well direct-phonon (SWDP) design, which has a larger overlap between its active laser states and the doping profile. However, further improvement in the temperature performance was expected, which led us to assume there was an increased gain and line broadening when increasing the doping concentration despite the reduced overlap between the doped region and the active laser states. Through simulations based on NEGF calculations we were able to study the contribution of the different scattering mechanisms on the performance of these devices. We concluded that the main mechanism affecting the lasers’ temperature performance is electron-electron (e-e) scattering, which largely contributes to gain and line broadening. Interestingly, this scattering mechanism is independent of the doping location, making efforts to reduce overlap between the doped region and the active laser states less effective. Optimization of the e-e scattering thus could be reached only by fine tuning of the doping density in the devices. By uncovering the subtle relationship between doping density and e-e scattering strength, our study not only provides a comprehensive understanding of the underlying physics but also offers a strategic pathway for overcoming current limitations. This work is significant not only for its implications on specific devices but also for its potential to drive advancements in the entire THz QCL field, demonstrating the crucial role of e-e scattering in limiting temperature performance and providing essential knowledge for pushing THz QCLs to new temperature heights.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The terahertz (THz) spectrum corresponds to wavelengths between 30 µm and 1000 µm (0.3 to 10 THz) and is situated between the microwave and infrared (IR) radiation domains. This portion of the electromagnetic field is called the “THz gap” due to the lack of available efficient sources in this specific range. THz radiation is not effectively accessible by either electronic or photonic widespread sources, although some advancements have been achieved in the last few years [1,2]. The most effective sources remain the THz QCLs, which have been extensively researched and developed since their demonstration in 2002 [3]. As opposed to devices reaching THz radiation by nonlinear frequency conversion [4–7], THz QCLs have much higher efficiency and power levels. Achieving access to the THz gap potentially enables, among others, non-invasive medical applications [8], spectroscopy [9,10], security [11], and astronomy [12].
Even though THz QCLs are very promising, the cooling requirements are still limiting the number of actual applications. Investigation is still ongoing and a maximum operating temperature (Tmax) of ∼250 K [13], and subsequently of ∼261 K were lately achieved [14]. However, these Tmax were reached only in pulsed operation and to date, other groups did not report similar Tmax values, indicating how big of a challenge this represents.
Our main goal is to study the physics and investigate the mechanisms that are keeping room temperature performance of THz QCLs unreachable. There are some mechanisms that have already been identified, such as the thermally activated longitudinal optical (LO) phonon scattering from the upper laser level (ULL) to the lower laser level (LLL) [15]. The strategy to overcome this mechanism is by using highly diagonal optical transitions in the THz QCL structures [16,17]. However, this could also cause detrimental effects on optical gain, especially in designs with high barriers. For this purpose, an optimal doping level for each scheme needs to be achieved, as typically gain tends to saturate with doping [18,19].
Limiting mechanisms that also needed to be addressed in diagonal structures, were leakage of carriers into the continuum [17], dominant in structures with barriers containing 15% Al, and leakage into the excited states, present in structures with higher barriers (30% Al) [20,21]. Optimization of these structures, by using high barriers and thin wells to push the excited states and continuum to higher energies, lead to the suppression of both these leakages [20,22].
Devices that pushed the excited states and continuum to higher energies were carefully engineered and a clean n-level system was obtained, n being the number of active levels [20,22,23]. In clean n-level systems, thermally activated leakage of electrons to excited and continuum states is suppressed, meaning that the electron transport occurs only within the laseŕs active levels, the lack of leakage was verified by the clear negative differential resistance (NDR) behaviour shown in the current voltage (I-V) curves all the way up to room temperature. It is worth noting that different periods may exhibit different temperatures and different biases leading to electric field domain formation [24,25], meaning that modelling THz QCL in continuous wave (CW) operation is highly unreliable [26]. Consequently, this may represent some challenges for clean n-level systems, as, if alignment of levels is not perfect, the structures may not perform as well as those that have parasitic levels to assist coupling between the periods [19,27].
Despite the challenges, the strategy of obtaining a clean n-level system was also used by the MIT group to reach the maximum operating temperatures of ∼250 K [13] and ∼261 K [14]. Taking this into account, creating clean n-level systems is the strategy we are following in this research as it showed the best performance so far, especially at elevated temperatures.
Some of the previously demonstrated designs supporting clean n-level systems are a resonant-phonon design [20], a split-well direct-phonon (SWDP) design [23], a two-well (TW) design [22], and more recently a two well injector direct phonon design [28]. In Ref. [13] published in 2021, a TW design, like the TW design in Ref. [22], was demonstrated and showed an improved Tmax of ∼250 K [13] and subsequently of ∼261 K [14].
Even though it was expected that the suppression of thermally activated leakages would improve the temperature performance, we did not see improvement in Tmax in the structures from Ref. [20,22,23]. Investigation into what still limits the temperature performance in these structures is ongoing. Some potential reasons for the lack of improvement may include MBE growth quality issues, fine tuning of design parameters, and negative effects of interface roughness scattering (IFR) [27,29–36] and ionized impurities scattering (IIS) [37–43].
2. Modelling
In this work, we study the effect that different doping density levels have on a novel highly diagonal split-well resonant-phonon (SWRP) design that was recently proposed [43,44], also supporting a clean n-level system. We combine data analysis and simulations that help us understand the effect that different scattering mechanisms have on the device through calculations based on non-equilibrium Green’s functions (NEGF) [36,45–48]. Various modelling approaches have been employed for the optimization of THz QCLs [49]. The density matrix method stands out due to its historical significance and success, achieving impressive results such as a prior Tmax of ∼200 K [50]. This highlights the importance of coherence and the utility of the density matrix model in understanding electron transport dynamics in THz QCLs [51–54]. While the density matrix approach has a strong foundation and has yielded remarkable insights in the field, in the current study, a different approach based on the NEGF method has been chosen for device optimization. Elastic and inelastic scattering processes, as well as other scattering mechanisms, are considered in the simulations. These mechanisms include IIS, IFR scattering, electron-electron (e-e) scattering, optical and acoustic phonons, and alloy disorders, as listed in prior studies [36,38,47,55]. A comprehensive quantum-kinetic approach, following linear response theory was employed to calculate the gain in a self-consistent manner [56]. In the NEGF approach, ee-scattering is handled within a static approximation, extending beyond the Hartree approximation. Similar to impurity scattering, the distribution of scatterers follows the electron distribution along the growth axis. This term is self-consistently updated during the NEGF loop as the electron distribution evolves. The self-consistent updating of the ee-scattering term during the NEGF loop enhances the precision of our simulations, contributing to the reliability and accuracy of our findings.
The SWRP design is based on the same principles as the SWDP scheme previously mentioned [23,27,37,47]. The excited states are pushed to higher energies by means of a thin intrawell barrier. The thickness of this barrier can be adjusted to make the energy separation between the LLL and the injector level match the LO-phonon scattering energy (36 meV for GaAs) [23], allowing this way a fast depopulation of the LLL [57–59]. Similar designs featuring four quantum wells have been also proposed before [60], however with lower aluminum content and without the distinct split well structure with a thin intrawell barrier which allows for the engineering of the energy separation between the lower laser level and the injector. This scheme supports a clean 4-level system, as shown by the NDR signature in the I-V curves [43]. Moreover, the overlap between the active laser states and the doped region was reduced in comparison to the SWDP design. Despite this, the maximum operating temperature achieved by this device was of only ∼131 K [43], which was still an improvement compared to the SWDP device from Ref. [27] that lased up to ∼120 K. Understanding of the effect of doping concentration on this design is crucial for comprehending the physics behind its temperature performance.
In an attempt to improve the temperature performance of this scheme, we looked back to experimental work done on SWDP, where the effect of doping concentration was also investigated [37,40]. In SWDP structures, the increase of the doping concentration had a significant negative effect. This contrasted with what was expected, as increasing the doping level was meant to compensate for the low oscillator strength in highly diagonal structures, and thus lead to higher gain [61]. The conclusion in Ref. [37] was that the temperature performance in direct phonon schemes did not improve due to the high overlap between the doped area and the active region states. The new SWRP was then proposed in order to decrease this overlap between the doped region and active laser states [43].
Given the fact that the overlap between the doped region and active laser states is reduced in this new structure, an increase of the gain and consequently of the Tmax value is expected, similar to the positive effect of doping concentration observed in previous resonant-phonon designs [61]. Tmax in Ref. [61] was found to improve significantly when the doping concentration was increased from 3 × 1010 cm−2 to 6 × 1010 cm−2. The values reported were around ∼135 K and ∼177 K, respectively. We were expecting a similar improvement in our design, as it is also a resonant-phonon scheme.
Within this work, the performance of SWRP devices with a doping concentration of 3 × 1010 cm−2 was compared to a design with doubled doping concentration (6 × 1010 cm−2). Additionally, using NEGF calculations, we analyse the SWRP scheme for the THz QCLs, comparing the contributions of the different scattering mechanisms on these schemes.
The design that is studied here is a highly diagonal (oscillator strength ƒ∼0.22 of the radiative transition) GaAs/Al0.3Ga0.7As SWRP THz QCL. The composition of the barriers is 30% Al, the same composition as the one that showed good results in past works [13,20,22,23]. Two devices based on this scheme with different doping levels were fabricated, namely Device 1 (VB0846) with doping level of ∼3 × 1010 cm−2 and Device 2 (VB0845) with a doubled doping level of ∼6 × 1010 cm−2 (Table 1). The doped layers are the two wells by the sides of the thin intrawell barrier. The devices were fabricated and measured together from the same batch. More details regarding the design and more fabrication details and device parameters can be found in Tables 1 and 2. Note that all design parameters were similar for the two devices.
Table 1. Main nominal design parameters and device data.
Table 2. Device parameters and performance.
The structure of both devices can be seen in Fig. 1. There are four subbands in each module (all other levels are considered parasitic). The LLL is a doublet (levels 2 and 3 in the scheme) with an anticrossing of ∼4.7 meV, meaning the resonant tunnelling is very strong. The depopulation of the doublet into the injector level (level 1 in the scheme) involves a resonant phonon scattering scheme [62]. Resonant tunnelling occurs between the ULL (level 4 in the scheme) and the injector of the former module (level 5 in scheme), which are aligned with an anticrossing of ${\sim} $2 meV (Table 2). The radiative transition occurs between the ULL and the LLL doublet (levels 5, 4 and 3, 2 in the scheme). The relevant excited state is level 10 in the scheme, which is well above the ULL (∼120 meV) as can be seen in Fig. 1 and in Table 1. The first excited state (level 9 in scheme) is lower and closer to the ULL but there is no overlap between the two, as they are separated by two potential barriers. Negligible intermodule leakage can be assumed, as indicated by the ULL of the following module, which is at a lower energy than level 9. As mentioned before, the thin intrawell barrier allows an energy gap between the LLL and injector of ∼36 meV (Table 1).
Fig. 1. Band diagram of two sequential periods termed module i (left, marked by dashed-dotted box) and module i + 1 (right) of the SWRP THz-QCLs with Al0.3Ga0.7As barriers, corresponding to energy levels of Device 1 (VB0846) with doping level of ∼3 × 1010 cm−2 and Device 2 (VB0845) with a doubled doping level of ∼6 × 1010 cm−2. The doped region is marked in the structure (by the sides of the thin intrawell barrier). More details regarding the design and device parameters can be found in Table 1 and Table 2.
3. Data analysis
Pulsed light–current (L–I) measurements for both devices are shown in Fig. 2. Increasing the doping of Device 1 (Fig. 2(a)) by a factor of two (i.e., creating Device 2, Fig. 2(b)) causes the device performance to degrade, with Tmax being reduced from ∼131 K to ∼93 K. This drop in Tmax is mainly attributed to the significant additional gain broadening due to the higher doping concentration. Although the effect is negative, these results are more encouraging than the results obtained for the SWDP since the drop in Tmax when increasing the doping concentration is relatively low in the SWRP design as opposed to the large drop in Tmax in the SWDP design, where Tmax dropped from ∼170 K to ∼45 K [37]. The lasing frequency is of ∼ 4 THz for Device 1 and ∼3.3 THz for Device 2, unlike Ref. [60] dual lasing frequency is not observed in the THz QCL scheme.
Fig. 2. Pulsed light–current measurements of (a) Device 1 (VB0846), (b) Device 2 (VB0845), with their lasing spectra as insets. Maximum operating temperatures are indicated. More details on these structures can be found in Table 1, Table 2 and Fig. 1. Current-voltage curves of: (c) Device 1 (VB0846) at low, around maximum operating temperature, and room temperatures, (d) Device 2 (VB0845) at low, around maximum operating temperature, and room temperatures.
The I–V curves for both devices in Fig. 2(c) and 2(d) show clear NDR behavior up to room temperature. The curves become flatter as the temperature increases and the maximum current (Jmax) decreases at higher temperatures. Even though Jmax did increase with the doping concentration, its value did not double as expected.
The threshold current (Jth) is plotted in Fig. 3(a) as a function of temperature. It can be observed that Jth is almost doubled when the doping is doubled. Although Jth increases by a factor of two when the doping is doubled (Table 2), Jmax does not. Hence, the dynamic range ΔJd = (Jmax – Jth) scales up by a much lower factor (from 360 A/cm2 in Device 1 to 218 A/cm2 in Device 2, Table 2). Consequently, the Tmax value drops. This is like what was observed for SWDP structures [37], but the drop there was much more significant.
Fig. 3. (a)Threshold current versus temperature of the SWRP THz-QCL devices. (b) Maximum current versus temperature of the SWRP THz-QCLs. Device 1 (VB0846) in blue circles and Device 2 (VB0845) in red circles. The three regions defined in the text and maximum operating temperatures –Tmax, (black arrows) are marked on the graph for each of the devices.
We can also learn from Fig. 3(a) that T0 rises from ∼ 237 K in Device 1 to ∼301 K in Device 2. This can be explained by the fact that in Device 2 the lasing frequency is lower, this indicates a higher activation energy required for the non-radiative ULL to LLL transition.
In Fig. 3(b), ${{\boldsymbol J}_{{\boldsymbol {max}}}}$ as a function of temperature is plotted all the way up to room temperature for both devices. The analysis of ${{\boldsymbol J}_{{\boldsymbol {max}}}}$ in clean n-level systems is a powerful tool that might explain how different mechanisms affect the temperature performance [23,37]. Assuming the resonant tunneling of the LLL doublet is very strong and does not limit ${{\boldsymbol J}_{{\boldsymbol {max}}}}$, we can relate to our system as an effective three-level system and describe it by the Kazarinov-Suris formula [23,63]:
As we can see from the plot (Fig. 3(b)), the graph is not monotonic, and we can identify three different regions that correspond to: the strong coupling regime under lasing conditions (Region 1 in Fig. 3(b)), the strong coupling regime under non-lasing conditions (Region 2 in Fig. 3(b)) and the weak coupling regime (Region 3 in Fig. 3(b)).
Region 1 in the graph is the strong coupling regime under lasing condition. Here the transport is strongly limited by the ULL lifetime, ${{\boldsymbol J}_{{\boldsymbol {max}}}} \sim \frac{1}{{\boldsymbol \tau }} \approx \frac{1}{{{{\boldsymbol \tau }_{{\boldsymbol st}}}}}$ thus, the maximum current density decreases while the temperature increases [64]. The sudden drop observed at the end of Region 1 indicates the termination of lasing immediately, as explained in [43] and was also seen in the SWDP design with 30% aluminium in Ref. [27]. This is attributed to the instability of the laser output power.
Region 2 in the graph, is the strong coupling regime under non-lasing condition. This region starts at around ${\sim} $115 K for Device 1 and at around ${\sim} $65 K for Device 2. If lasing is achieved above this temperature, it is strongly affected by output power instability. In this region, the dominant process is the non-radiative scattering, ${{\boldsymbol J}_{{\boldsymbol {max}}}} \sim \frac{1}{{\boldsymbol \tau }} \approx \frac{1}{{{{\boldsymbol \tau }_{{\boldsymbol nr}}}}}$, and ${{\boldsymbol J}_{{\boldsymbol {max}}}}$ will start to increase. The nonradiative lifetime ${{\boldsymbol \tau }_{{\boldsymbol nr}}}$, which gets shorter when the temperature rises, will now affect the ULL lifetime. This behavior will continue until the current is limited by the resonant tunneling between the injector and the ULL, at around ${\sim} $190 K for Device 1 and ${\sim} $160 K for Device 2.
Above these temperatures is the weak coupling regime (Region 3 in the graph). The dephasing time between the injector and ULL (i.e., ${{\boldsymbol \tau }_\parallel }$) declines as the temperature increases, and ${{\boldsymbol J}_{{\boldsymbol {max}}}}$ will start to decrease again with the temperature since ${{\boldsymbol J}_{{\boldsymbol {max}}}} \sim {{\boldsymbol \tau }_\parallel }$. For both devices, transport enters this region at a relatively early stage, meaning these structures are governed by dephasing at temperatures above ${\sim} $190 K and ${\sim} $160 K respectively. This region is of particular interest, as it is an experimental indication of line broadening and subsequent gain broadening.
The impact of doping concentration is clear from the comparison of the maximum current Jmax versus temperature data for Device 1 and Device 2. Device 2 demonstrates an earlier transition from the strong coupling regime (Regions 1 and 2) to the weak coupling regime (Region 3), which indicates the negative effect of the doping density on the electron transport and the resonant tunneling.
We can also observe a faster drop in the maximum current for Device 2 in Region 3. The faster drop of Jmax indicates a faster dephasing rate as the temperature increases, causing additional line broadening. This also indicates the negative effect of the doping density on the dephasing time and consequently the resonant tunneling. This result also supports our interpretation that the main reason for the reduced Tmax is an additional gain broadening due to the increased doping concentration. While Device 2 exhibits a more noticeable slope, it's worth highlighting that this slope, even though pronounced, does not reach the same level of sharp decline as the one observed in SWDP when the doping is increased from 3 × 1010 cm−2 to 6 × 1010 cm−2 [37].
4. Simulation results
As previously stated, we recognized greater line and gain broadening in our scheme as the cause of its performance constraints. (More information regarding our simulations can be found in Supplement 1.) In order to better understand the mechanisms contributing to an increased gain and line broadening in the SWRP scheme we conducted a systematic analysis using NEGF simulations. We employed NEGF calculations as a crucial tool to investigate the impact of various scattering mechanisms on the optical gain and spectral functions. Our analysis encompassed both devices, and the insights gained from these calculations carry significant implications for our study. The results for the gain calculations, as depicted in Fig. 4, shed light on the dominance of the IFR mechanism in restricting the performance of the SWRP design, particularly in Device 1 (Fig. 4(a)). This finding aligns with our comparative analysis to the SWDP design [47], revealing a similar predominance of the IFR mechanism. Notably, our investigation also underscores the substantial influence of e-e scattering on the optical gain of the SWRP design, underscoring its pivotal role in shaping the device's performance characteristics.
Fig. 4. (a) Contribution of the different scattering mechanisms to the optical gain of the SWRP design at temperature of 10 K with doping density of ∼3 × 1010 cm−2 and (b) with doping density of ∼6 × 1010 cm−2.
Our results indicate that in Device 1 (Fig. 4(a)), by mitigating the IFR scattering mechanism, we can achieve a remarkable enhancement in optical gain, approximately 50%. Likewise, the elimination of e-e scattering results in an impressive gain increase of approximately 40%. For Device 2 (Fig. 4(b)) the results show even further improvement. Here, the gain without the consideration of the IFR scattering rises by almost 80% and by eliminating the e-e scattering the gain impressively rises by approximately 170%. We identify a noteworthy increase in the contribution of e-e scattering, which emerges as the most dominant limiting mechanism in the SWRP design, particularly when coupled with a high doping level. This observation aligns with prior studies that have underscored the challenges posed by e-e scattering in RP designs [36].
Additionally, our analysis reveals the influence of IIS in the SWRP designs. As depicted in Fig. 4(a), suppressing IIS results in an enhancement of about 22%. However, in the case of Device 2 (Fig. 4(b)), with a higher doping density, the influence of IIS becomes more pronounced, approaching the significance of IFR scattering. These outcomes emphasize the intricate interplay between diverse scattering mechanisms and their collective contributions to the overall gain performance of the SWRP THz QCL design.
To further understand the impact of the different scattering mechanisms we present the spectral functions in Fig. 5, these calculations were done using NEGF simulations and include all elastic and inelastic scattering mechanisms, as explained before. The different scattering mechanisms contribute differently to the broadening of the laser levels. We present the spectral functions of only one of the ground states in Fig. 5, as the widths of all the ground states are similar under same conditions. We can observe in Fig.5a that when doubling the doping, the levels are broader (by ∼ 21%), regardless of the scattering mechanisms. Moreover, the full widths at half maximum (FWHM) of the spectral functions vary when subtracting different scattering mechanisms; this can be observed in Fig. 5(b) where the spectral functions of Device 2 were plotted for different scattering mechanisms. It is clear the e-e scattering is the most dominant mechanism. When eliminating the e-e scattering mechanism the width of the spectral function is reduced by ∼33%. When eliminating IIS the width is reduced by only ∼24%. Both mechanisms have crucial contributions to the detriment of the line broadening, although the effect of IIS is milder, assumingly due to the reduced overlap between doped region and active laser states.
Fig. 5. (a) Spectral functions of the injector level (as a representative level of the ground states) of Device1 and Device 2 including all the scattering mechanisms for comparison. Device 1 is plotted with a dash-dotted line in blue and Device 2 plotted with a continuous line in red. (b) Spectral functions of the injector level (as a representative level of the ground states) of Device 2, showing the contribution of the different scattering mechanisms by subtraction of one at a time . Spectral functions including all scattering mechanisms are marked with a continuous curve in light blue, without the e-e scattering with a dash-dotted curve in orange and without IIS with a dashed curve in green.
5. Discussion
From our calculations, we conclude that the e-e scattering, IFR scattering, and IIS all have significant contributions to the deterioration of the scheme’s temperature performance and the increase of the gain broadening. Additionally, we found that at higher doping density levels, the e-e scattering raised considerably. This was also observed in previous works regarding TW structures [58], where e-e scattering was also highlighted as one of the factors affecting lasing at high temperatures. On the other hand, the contributions of the IIS, and IFR scattering did not rise as much. We attribute the fact that the IIS did not notably rise with the doping to the reduced overlap between the doping profile and the active laser states. These findings highlight the critical role of e-e scattering in constraining the performance of SWRP THz QCLs and underscore the necessity for further investigations and optimization of design parameters to overcome these limitations.
The detrimental effect that the IIS have on the structure could be potentially diminished by further optimizing the doping profile. Potential further optimizations could include different doping densities as well as different positions and profiles of the doped region within the scheme. Additionally, optimizing the quality of the interfaces and MBE growth could lead to better temperature performance and further optimization of the barriers such as reducing the Al composition or utilizing designs with mixed potential barriers (barriers with different Al content) could also reduce effects related to the IFR. Regarding the e-e scattering, this scattering mechanism is independent of the doping location, making efforts to reduce overlap less effective. Optimization of the e-e scattering could be reached by fine tuning of the doping density in the devices.
6. Conclusions
In conclusion, we analysed the effect of doping concentration in a SWRP scheme supporting a clean 4-level system using NEGF calculations. This design has a reduced overlap of the doped region with the active laser states in comparison to the SWDP design. Experimental research showed that increasing the doping concentration in these designs led to better results compared to the SWDP design, which has a larger overlap between its active laser states and the doping profile. However, further improvement in the temperature performance was expected, which led us to assume there was an increased gain and line broadening when increasing the doping concentration despite the reduced overlap between the doped region and the active laser states. Through simulations based on NEGF calculations we were able to study the contribution of the different scattering mechanisms on the performance of these devices. We concluded that the main mechanism affecting the lasers’ temperature performance is electron-electron (e-e) scattering, which largely contributes to gain and line broadening was observed. Interestingly, this scattering mechanism is independent of the doping location, making efforts to reduce overlap between the doped region and the active laser states less effective. Optimization of the e-e scattering thus could be reached only by fine tuning of the doping density in the devices. By uncovering the subtle relationship between doping density and e-e scattering strength, our study not only provides a comprehensive understanding of the underlying physics but also offers a strategic pathway for overcoming current limitations. This work is significant not only for its implications on specific devices but also for its potential to drive advancements in the entire THz QCL field, demonstrating the crucial role of e-e scattering in limiting temperature performance and providing essential knowledge for pushing THz QCLs to new temperature heights.
Funding
Israel Science Foundation (1755/23).
Acknowledgments
The authors would like to acknowledge the Israel Science Foundation (ISF 1755/23) for its grant. This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. DOE’s National Nuclear Security Administration under contract DE-NA-0003525. The views expressed in the article do not necessarily represent the views of the U.S. DOE or the United States Government.
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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