Recently, nanoporous (NP) GaN has emerged as a promising photonic material in the III-N family. Due to its attractive properties, such as its large refractive index contrast and perfect lattice matching with GaN, as well as its good electrical conductivity, photonic components and devices involving NP GaN have been successfully demonstrated. However, further development of high-performance NP GaN based electrically injected devices, such as vertical-cavity surface-emitting lasers (VCSELs) and edge emitting lasers, requires efficient heat dissipation. Therefore, in this paper, we study thermal conductivity (TC) of NP GaN, especially when incorporated into a practical distributed Bragg reflector (DBR) in a VCSEL device. Through an effective medium model, we study the theoretical effect of NP GaN morphological properties over its TC. We then experimentally measure the TC of NP GaN, with different porosities and pore wall thicknesses, which shows a high agreement with the theoretical model. We also fabricate actual NP GaN DBRs and study the large tunability and interdependence among their TC (1–24 W/m K), refractive index (0.1–1.0), and electrical conductivity (100–2000 S/m) compared to other conventional DBRs. Finally, we perform a finite-element simulation of the heat dissipation within NP GaN-VCSELs, revealing their superior thermal dissipation compared to dielectric DBR based VCSELs. In this regard, this study lays the foundation for nanoscale thermal engineering of NP GaN optoelectronic and photonic devices and paves the way for their successful commercialization.

III-nitride photonic devices such as edge emitting lasers (EELs),1 vertical-cavity surface-emitting lasers (VCSELs),2,3 and other components for integrated photonic circuits are receiving much attention after the success in the conventional light emitting diode (LED) technology. A fundamental requirement for such photonic devices is the capability of engineering optical indices for the confinement and manipulation of photons. Such a requirement has been difficult to meet in the III-nitride material system. The ternary AlGaN system is commonly used in constructing photonic structures, yet it is severely constrained as the binary end compounds (GaN and AlN) have a large lattice mismatch (Δa/a ∼ 2.4%) with only a modest index contrast (Δn ∼ 0.4); surely enough, AlN/GaN distributed Bragg reflector (DBR) is still one of the most challenging topics in III-nitride laser diodes research after two decades of endeavors.4–9 It is worth noting that the AlInN ternary system latticed-matched to GaN presents an interesting solution10 for photonic designs, even though its eventual feasibility remains unclear due to the challenges in epitaxy.9 

Recently, we reported the growth and formation of nanoporous (NP) GaN by an electrochemical (EC) etching process,11,12 through which the creation of nanoscale voids (5–50 nm in diameter) in GaN dramatically reduces the index of refraction, making it a new form of epitaxial III-nitride material that is lattice-matched to GaN, highly conductive,13 and single crystalline. Photonic components and devices involving NP GaN, such as DBR mirrors,13–15 optically pumped VCSELs, and resonant-cavity LEDs, have been successfully demonstrated.16–18 Furthermore, by increasing the optical confinement with engineered NP GaN cladding layers, we have demonstrated through optical pumping a much reduced threshold in III-nitride EELs.19 

For electrically injected devices, especially those operating under high current densities such as EELs and VCSELs, efficient thermal management for rapid heat dissipation is essential for high device performance. Indeed, commercialization of GaN-VCSELs using two dielectric DBR mirrors have been impeded by their poor thermal conductivity (TC, ∼1 W/m K), in spite of their nearly ideal reflectance spectrum and reasonably low lasing threshold.20–22 In this regard, the purpose of this paper is to investigate the TC properties of NP GaN, especially when incorporated into a practical DBR structure in a VCSEL device. First, we study the theoretical dependence of TC of a porous medium on the volumetric porosity and the wall thickness using a modified effective medium theory. We then prepare, based on the model predictions, NP GaN with different porosities and wall thicknesses, by varying the doping concentration and the anodization voltage during EC etching. We first estimate the TC of the bulk GaN film on sapphire which depends on the thickness and dislocation density.23,24 Then, we measure the NP GaN samples’ TC experimentally and compare the measured values with the model predictions. Having established the TC of NP GaN, we fabricate NP GaN DBR mirrors and study their thermal properties. Finally, to benchmark our findings of NP GaN and to provide additional insight into heat transport, we carry out a finite-element modeling of heat dissipation in GaN-VCSELs with either dielectric DBRs or NP GaN DBRs. The result clearly indicates the advantage and promise of employing NP GaN in photonic devices to engineer the refractive index.

NP GaN was created through a conductivity selective EC etching process described in detail elsewhere.12,25 First, an epitaxial structure consisting of five n+-GaN layers with different doping concentrations, separated by UID-GaN layers, was grown on a c-plane sapphire. Afterward, standard test samples for TC measurement consist of an n+-type GaN of 500 nm in thickness to be porosified according to designed parameters. Underneath the highly doped layers, a moderately doped n-GaN layer (ND = 5 × 1018 cm−3) with a thickness of 1 μm was grown to ensure uniform distribution of the anodization bias across the entire sample during EC etching. Additionally, DBR samples were also prepared to examine and confirm the TC of optically engineered NP GaN DBRs. The DBR samples have an epitaxial structure grown on a c-plane sapphire consisting of 35 pairs of alternating n+-GaN (ND = 4 × 1019 cm−3) and UID-GaN layers.

After growth, the sample surface was covered with a silicon dioxide (SiO2) layer through plasma-enhanced chemical vapor deposition (PECVD), which was then lithographically patterned into 100 μm wide stripes, separated by 10 μm openings. The sample was then dry-etched by Cl-based reactive ion etching (RIE) to create trenches and to expose the sidewalls of the highly doped n+-GaN layers. EC etching was conducted in an acid electrolyte at room temperature. A positive bias was applied across the sample by a source meter (Keithley 2400), while a Pt wire was used as a cathode. After EC etching, the protective SiO2 was stripped off with a buffered oxide etch (BOE).

The morphology of the nanoporous samples was characterized by scanning electronic microscopy (SEM, Hitachi SU-70). SEM images were scanned digitally to an image processing software to measure the porosity, pore size, and wall thickness of NP GaN. The porosity was calculated by dividing the total area of the voids to the entire area of the porous medium. The reflectance spectrum of the GaN/NP GaN DBRs was measured using a microreflectance setup with a spot size of ∼20 μm, which was calibrated against a commercial single-crystal silver mirror. The reflectance of Si was also measured and compared with the published data, and the precision was determined to be within 0.1%.

The dislocation densities of unetched bulk GaN and NP GaN were estimated using X-ray rocking curve measurement. (0002) and (10–12) rocking curves were measured with Rigaku SmartLab X-ray Diffractometer. Full-width at half-maximum (FWHM) was extracted and used for estimation of the threading dislocation densities.

TC was measured using a microfabricated heater–sensor pattern. A spiral metal coil with a radius of 40 μm was used as both a microheater and a resistivity-based temperature sensor.26 Before metal deposition, we first deposited a 50 nm SiO2 layer by PECVD as an electrical insulation layer between the heating element (Ni coil pattern) and the NP GaN thin films. A spiral-coil pattern was obtained through E-Beam evaporation and a liftoff process of a 50 nm-thick nickel layer. A DC current was applied via the two contact pads at opposite ends of the spiral-coil microheater and the resulting resistance was sensed by a four-wire measurement after a stabilization time of 100 s.

A comparative study of the thermal conduction properties between NP DBR based VCSELs and dual-dielectric VCSELs was conducted through numerical modeling by the Photonic Integrated Circuit Simulator in 3D (PICS3D) simulation program.27 The thermal modeling used basic heat generation and thermal conduction theories that involved heat sources such as Joule/optical heating, recombination heating, radiative heating, and Thomson/Peltier heating.22,28–30 Temperature profiles were numerically modeled and recorded as heat maps in the cross section of the two different VCSEL structures (refer to the supplementary material for details).

The TC of porous medium strongly depends on the porosity and size of nanopores, especially when its characteristic lengths are comparable to the phonon mean free path (MFP). For a more quantitative understanding, we used a modified effective medium model as a theoretical guideline31,32 to describe the heat transfer in porous medium. The model appears as κeff=κ0f(φ)1+l0D1, where φ is the porosity of NP GaN, κ0 is the TC of bulk GaN (130 W/m K), f(φ) = (1 − φ)/(1 + φ) is the effective medium modified by macroscopic porosity of a composite material,33 and the parenthetical term includes the effect of phonon scattering or phonon bottleneck34 in nanostructures. l0 is the MFP of phonons and D is the wall thickness (refer to the supplementary material). κ0 was determined to be 130 W/m K based on the thickness (2.5 μm) and dislocation density (5 × 108 cm−2).35 The dislocation density of the GaN film was a rough estimation based on the measurement of FWHM in the X-ray rocking curves, as shown in Fig. 1. As the wall thickness becomes close to the MFP of phonons, there will be a rapid decrease of TC due to phonon-nanopore scattering. On the other hand, the TC decreases as the porosity increases as a result of reduced effective medium. Thus, the two most important determining parameters for TC, as predicted by the model, are the porosity and wall thickness of the NP GaN. Experimentally, it is worth noting that the wall thickness D is often correlated with porosity φ, and there is a general trend of increase in D with decreasing φ.

FIG. 1.

Symmetric (0002) (a) and (10–12) (b) rocking curves of unetched bulk GaN and EC etched NP GaN (porosity ∼47%) films on sapphire.

FIG. 1.

Symmetric (0002) (a) and (10–12) (b) rocking curves of unetched bulk GaN and EC etched NP GaN (porosity ∼47%) films on sapphire.

Close modal

Figure 2 summarizes an EC etching phase diagram of GaN. The phase diagram consists of no-etching, nanoporous etching, and electropolishing regions, and provides guidelines to design the NP GaN nanostructures, which will be used to investigate the TC of NP GaN. In traversing from the “no-etching” (lower left corner) to “electropolishing” (upper right corner) regions, there is a continuous increase of the overall porosity. The effect of doping and applied bias on the pore morphology is somewhat intertwined. The general trend as depicted schematically by the white circles in Fig. 2 is summarized as follows: both a low doping and low bias lead to a small pore diameter. A low doping is effective in increasing the wall thickness (interpore separation) based on the depletion-region model.11 

FIG. 2.

Processing phase diagram for EC etching. Two sets of samples labeled as solid circles of three different colors were studied for the investigation of TC of NP GaN. The first set of samples (blue and green series) had the same doping levels. The second set of samples (red series) was designed to follow the isoporosity curve near the low-porosity side of the EC etching phase diagram.

FIG. 2.

Processing phase diagram for EC etching. Two sets of samples labeled as solid circles of three different colors were studied for the investigation of TC of NP GaN. The first set of samples (blue and green series) had the same doping levels. The second set of samples (red series) was designed to follow the isoporosity curve near the low-porosity side of the EC etching phase diagram.

Close modal

Based on the theoretical model, three sets of NP GaN samples were prepared with process parameters labeled as solid circles of three different colors in Fig. 2. The first two set of samples (B series and C series) had the same doping levels, and the third set of samples (A series) was designed to follow the isoporosity curve near the low-porosity side of the EC etching phase diagram.

The SEM images of samples A1–A3 and C1–C3 were shown in Figs. 3(a)3(c) and 3(d)3(f), respectively. For n+-GaN at higher doping levels such as 1 × 1020 cm−3 (samples B1 and B2), the pore size and porosity have been reported in our previous publications.13 Statistical results of porosity (φ) and average wall thickness (D) for all these NP GaN samples were summarized in Table I. As we could observe, the constant doping series (B and C) had similar D but different φ, while the isoporosity series (A) showed a major change in D and a slight change in φ. Based on φ and D, we used the modified effective model to calculate TC of the NP GaN samples, and the results were named as κNP GaN calculated and summarized in Table II.

FIG. 3.

Cross-sectional SEM images of samples etched at different doping levels and etching voltages. The sample numbers correspond to the annotations in Fig. 2.

FIG. 3.

Cross-sectional SEM images of samples etched at different doping levels and etching voltages. The sample numbers correspond to the annotations in Fig. 2.

Close modal
TABLE I.

The extracted porosity and wall thickness data from the SEM images.

SampleDoping concentration (cm−3)Applied bias (V)Porosity (%) (±2)Wall thickness (nm) (±2)
A1 8 × 1019 1.25 30 
A2 2 × 1019 2.0 21 14 
A3 1 × 1019 3.0 10 31 
B1 1 × 1020 1.8 62 
B2 1 × 1020 2.5 74 
C1 4 × 1019 1.3 24 10 
C2 4 × 1019 1.5 36 
C3 4 × 1019 2.0 51 
DBR (1.3 V) 4 × 1019 1.3 24 10 
DBR (1.5 V) 4 × 1019 1.5 36 
SampleDoping concentration (cm−3)Applied bias (V)Porosity (%) (±2)Wall thickness (nm) (±2)
A1 8 × 1019 1.25 30 
A2 2 × 1019 2.0 21 14 
A3 1 × 1019 3.0 10 31 
B1 1 × 1020 1.8 62 
B2 1 × 1020 2.5 74 
C1 4 × 1019 1.3 24 10 
C2 4 × 1019 1.5 36 
C3 4 × 1019 2.0 51 
DBR (1.3 V) 4 × 1019 1.3 24 10 
DBR (1.5 V) 4 × 1019 1.5 36 
TABLE II.

The measured and calculated TC (κNP GaN) data obtained at the input power of 100 mW.37 

SampleCalculated TC from effective medium model κNP GaN calculated (W/m K)Measured TC from microheater–sensor κNP GaN measured (W/m K)
A1 4.0 3.37 
A2 10.1 9.45 
A3 25.2 24 
B1 1.7 1.27 
B2 1.1 0.83 
C1 7.9 7.23 
C2 4.0 3.18 
C3 2.6 2.24 
DBR (1.3 V)  6.96 
DBR (1.5 V)  3.29 
SampleCalculated TC from effective medium model κNP GaN calculated (W/m K)Measured TC from microheater–sensor κNP GaN measured (W/m K)
A1 4.0 3.37 
A2 10.1 9.45 
A3 25.2 24 
B1 1.7 1.27 
B2 1.1 0.83 
C1 7.9 7.23 
C2 4.0 3.18 
C3 2.6 2.24 
DBR (1.3 V)  6.96 
DBR (1.5 V)  3.29 

To directly measure the TC of NP GaN and test the model predictions, a differential temperature technique was used, where an 80-μm diameter, spiral Ni coil was patterned onto the described 8 samples (Fig. 2) as well as a reference sample containing no NP GaN (Fig. 4). In such a technique, the thermal conductivity of the test samples can be determined with the knowledge of generated heat flux (Q, controlled by the microheater) and corresponding temperature increase ΔT (measured by the Ni-thermometer).26,36 The ΔT changed almost linearly with heat flux within the input power range (Fig. 5). To reduce the error from the convection and radiation heat loss, a very small ΔT and heat flux in Fig. 5 were extracted from the linear regions of the curves to calculate the TC. Here, we have a simple evaluation of the convection and radiation heat losses. Under the heat flux of 100 mW, the maximum temperature increase in our reference sample and NP GaN sample is about 5 °C and 15 °C, respectively. In static air, the heat transfer coefficient from air convection is below 20 W/m2K. As a result, the maximum convection heat loss in our samples is about 1.5 μW. The power radiated through black-body radiation can also be estimated to be about 2.6 μW. Clearly, the convection and radiation heat loss are negligible compared to the heat flux of 100 mW.

FIG. 4.

Schematics of the structures for TC measurement. (a) Cross-sectional schematics of the reference sample. (b) Cross-sectional schematics of the sample structure with NP GaN. (c) A top-view optical micrograph image of a sample with the spiral-coil patterned Ni microheater.

FIG. 4.

Schematics of the structures for TC measurement. (a) Cross-sectional schematics of the reference sample. (b) Cross-sectional schematics of the sample structure with NP GaN. (c) A top-view optical micrograph image of a sample with the spiral-coil patterned Ni microheater.

Close modal
FIG. 5.

Temperature increases of the Ni microheater (ΔT) as a function of Joule heating power at thermal equilibrium in NP GaN samples and reference structure for (a) A series together with C2 in C series, (b) C series, and (c) B series.

FIG. 5.

Temperature increases of the Ni microheater (ΔT) as a function of Joule heating power at thermal equilibrium in NP GaN samples and reference structure for (a) A series together with C2 in C series, (b) C series, and (c) B series.

Close modal

By calculating the difference in ΔT between the reference and test samples (ΔT1 and ΔT2), the thermal conductivity from the different NP GaN layers (κNP GaN measured) was successfully extracted by κNPGaN=QΔT2ΔT1ΔlA, where l was the thickness of NP GaN and A was the area of the Ni pattern (see the supplementary material). The results were summarized in Table II. As presented, based on the porosity and wall thickness, the TC of NP GaN can be varied from below 1 to more than 20 W/m K. Such a strong dependence is expected when the feature size of the porous medium is comparable or less than the phonon MFP in GaN crystal.

1. Agreement between the model and measurement

Comparing the measured TC using the microheater method with the calculated κeff using the model (blue and green circles in Fig. 6, respectively), we observe a good agreement between the experimental data and calculation results, which lends credibility to the estimated TC values for NP GaN from the model. The grey solid curves represent calculated TC at different wall thicknesses as a function of porosity. It should be noted that pore size Rp, wall thickness D, and porosity φ have a coupling relationship, i.e., any one of the three parameters can be determined by the other two assuming a certain void geometry. In our case, the NP GaN structure is composed of parallel aligned nanopores thus Rp ∼ D × φ1/2. Hence, we simplified our analysis of the TC of NP GaN only using the porosity and wall thickness. A somewhat abrupt, upward transition of experimental TC values from A1 to A3 in Fig. 6 as porosity reduces can be attributed to the coupled effect of an increase in wall thickness.

FIG. 6.

TC as a function of porosity. Blue and green circles represent the experimental results and calculation results using the experimental porosity and wall thickness D, respectively. Solid grey lines correspond to the calculations for different wall thicknesses D, i.e., 2, 6, 10, 14, 18, 22, 26, and 30 nm, respectively.

FIG. 6.

TC as a function of porosity. Blue and green circles represent the experimental results and calculation results using the experimental porosity and wall thickness D, respectively. Solid grey lines correspond to the calculations for different wall thicknesses D, i.e., 2, 6, 10, 14, 18, 22, 26, and 30 nm, respectively.

Close modal

It is worth noting that the large tunability of TC by changing nanoporous physical parameters, especially the nanoporous wall thickness. On the one hand, a reduction in porosity leads to a moderate improvement in TC, as indicated as the black curve in Fig. 6, as a result of an increased effective medium. On the other hand, by widening the pore wall thickness, the TC can be sharply improved, owing to much reduced phonon interaction at the nanopores. For NP GaN layers with large wall thicknesses and small porosities, the TC is capable of reaching more than 20 W/m K, a result very encouraging for the practical usage of NP GaN in photonic devices requiring fast heat dissipation.

2. Design of real photonic layers, using a DBR as a case study

As previously described, a potential interesting application of NP GaN is the development of DBR mirrors for VCSELs. Given the stringent requirement for the DBRs, different physical properties need to be considered and engineered. Based on our present and previous studies, we can summarize the optical (index contrast),13 electrical,13,38 and thermal properties of NP GaN, juxtaposed with data from other DBRs (AlInN/GaN, AlN/GaN, and SiO2/Ta2O5) reported in the literature (Fig. 7).22,39,40 As observed, the tunability of NP GaN offers the most flexible range of design. Particularly, in the absence of DBR mirrors with high TC for the state-of-the-art GaN-VCSELs, NP GaN/GaN DBR offers an engineering choice with a comparable TC with AlInN/GaN DBRs, and 10 times increased TC compared with that of dielectric DBRs.

FIG. 7.

A parameter plot for NP GaN in comparison with other materials.

FIG. 7.

A parameter plot for NP GaN in comparison with other materials.

Close modal

To validate our study and design choices, we prepared two NP GaN DBR structures consisting of 35 alternating n+-GaN (ND = 4 × 1019 cm−3) and UID-GaN layers, etched at 1.3 and 1.5 V. Figures 8(a)8(c) show the schematic structure and cross-sectional SEM images of GaN/NP GaN DBR mirrors etched at 1.3 and 1.5 V, which have porosities of 24% and 36%, respectively. Reflectance of the two samples [Fig. 8(d)] shows peak reflectance exceeding 99.8%, at a designed central wavelength of 460 nm, with a stop band of more than 50 nm. Furthermore, the measured TC of the NP GaN layers in the DBRs etched at 1.3 and 1.5 V were 6.96 and 3.29 W/m K, respectively, resulting in an average TC of 12.3 and 6.0 W/m K in the NP DBR, respectively.

FIG. 8.

(a) Schematic structure of DBRs mirror with NP GaN layers. Cross-sectional SEM images of DBRs with NP GaN layers etched at (b) 1.3 V and (c) 1.5 V, respectively. (d) Experimentally measured reflectance spectra from a DBRs region. (e) The temperature increases of the Ni microheater ΔT as a function of Joule heating power at thermal equilibrium in DBRs samples and reference sample.

FIG. 8.

(a) Schematic structure of DBRs mirror with NP GaN layers. Cross-sectional SEM images of DBRs with NP GaN layers etched at (b) 1.3 V and (c) 1.5 V, respectively. (d) Experimentally measured reflectance spectra from a DBRs region. (e) The temperature increases of the Ni microheater ΔT as a function of Joule heating power at thermal equilibrium in DBRs samples and reference sample.

Close modal

It is worth pointing out that the measured TC for DBRs in Fig. 8 and Table II are based entirely on vertical heat flow, where the TC is dominated by the NP GaN layers in series with GaN layers. In actual VCSEL devices, the lateral heat flow involves summing up the thermal “conductance” of layers in parallel, which will be dominated by the GaN layers. So, the TC of the DBR layers behaves as porouslike in the vertical direction and GaN-like in the lateral direction. Such an anisotropy in the lateral and vertical directions is advantageous for fast heat dissipation in the lateral direction. Here, it should be noted that since the nanopores are parallel aligned in the lateral direction and thus have a geometry anisotropy, there might also be an anisotropy in TC along the lateral and vertical directions. The lateral TC is not studied but is expected to be enhanced by the continuum of pore walls along the pore alignment direction, and thus probably exceed the TC in the vertical direction. For simplicity, we assume an isotropic TC of NP GaN in Sec. III D 3.

3. Numerical simulation of heat dissipation in VCSELs

Numerical modeling was conducted on two different VCSEL designs, a NP DBR based VCSEL and a double-dielectric VCSEL, to compare the effect of the DBR materials on the heat dissipation property. The two VCSEL structures were simulated under a continuous wave (CW) operation of 10 mW (4 V and 2.5 mA). Current crowding effect was also included in this study. The heat generated from the active region primarily dissipates through the metal contacts and a heat sink. Figure 9 showed the temperature profiles in the two VCSEL structures, respectively. As shown in Fig. 9, the double-dielectric VCSEL shows a heat built-up of ∼100 °C (372.6 K) in the active region, indicating the difficulty in heat transport through the dielectric bottom DBR. In contrast, the NP DBR based VCSEL shows much improved hear transfer, thanks to the high lateral TC; the highest temperature in the active region was 50 °C lower than in the double-dielectric VCSEL case. With the advantage of improved thermal conduction properties, the NP DBR based VCSELs will be able to work at much lower operation temperatures, which should benefit the threshold, power, and efficiency of devices.

FIG. 9.

Comparison in temperature profiles between a double-dielectric VCSEL (a) and a hybrid NP dielectric VCSEL (b) working at CW operation with an injection power of 10 mW.

FIG. 9.

Comparison in temperature profiles between a double-dielectric VCSEL (a) and a hybrid NP dielectric VCSEL (b) working at CW operation with an injection power of 10 mW.

Close modal

To summarize, in this study, we investigated the TC of NP GaN, especially when incorporated into a DBR structure for VCSELs. To clarify the respective influence on TC, key parameters were varied systematically associated with NP GaN, including volumetric porosity, average pore size, and average wall thickness. A theoretical effective medium thermal conduction model has been put forward and agrees well with the direct measurement by a spiral-coil microheater/sensor method. Considering the trade-offs in employing NP GaN as a low-index layer, parameter space and pathways were identified where the need of strong confinement with a low-index layer using a high porosity and the need of good TC with a low porosity can be balanced. To benchmark our findings of NP GaN and to provide operational insight into heat transport, we carried out a finite-element modeling of heat dissipation in GaN-VCSELs with either dielectric DBRs or NP GaN DBRs. The result clearly indicated the advantage and promise of employing NP GaN as a low-index layer for injected lasers including EELs and VCSELs.

See the supplementary material for the details of (a) parameter considerations in the modified effective medium model, (b) derivation of the equations for thermal conductivity measurement, and (c) the VCSEL structure and parameter settings in the simulation of heat dissipation.

This research was supported by the National Science Foundation (NSF) (Grant No. ECCS-1709149), and the facilities used were supported by YINQE and NSF MRSEC DMR (No. 1119826).

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Supplementary Material