The chemical vapor deposition of CH4 on Ge(001) results in the anisotropic synthesis of graphene nanoribbons that are aligned to Ge⟨110⟩ and have faceted armchair edges, sub-10 nm widths, and lengths greater than 100 nm. The utilization of small graphene seeds to initiate nanoribbon synthesis provides control over the nanoribbon placement and orientation. However, in order to exclusively grow nanoribbons and suppress the concomitant growth of lower aspect ratio crystals, it is imperative to control the crystallographic orientation of the seeds with respect to the Ge lattice. Here, we demonstrate that when seeds are less than 18 nm in diameter, they are able to rotate upon annealing at 910 °C prior to nanoribbon synthesis. The effect of this rotation on the resulting nanoribbons’ orientation is characterized as a function of the diameter and initial crystallographic orientation of the seeds. The seeds preferentially rotate to an orientation in which an armchair direction of their lattice is parallel to Ge⟨110⟩—subsequently maximizing the anisotropy in growth kinetics. By exploiting this seed rotation phenomenon, we demonstrate the fabrication of seamless nanoribbon meshes and gain understanding that will affect future efforts to create arrays of unidirectionally aligned nanoribbons.

Graphene nanoribbons have promising exceptional electrical1,2 and thermal properties3 among other unique qualities4–6 of interest for next-generation electronics. Unlike graphene, which is a semimetal, graphene nanoribbons can be semiconducting when narrower than 10 nm. Theoretical studies indicate that their bandgap varies roughly inversely with their width and find that an armchair edge orientation provides the largest bandgaps.7 

However, synthesizing ribbons with the qualities required for device applications has been a challenge.8–10 Ribbons ideally should have sub-10 nm widths, lengths greater than 100 nm, controlled orientation and placement on substrates, and a smooth, armchair edge structure. Top-down techniques that lithographically pattern ribbons from monolayer graphene can address the issues of orientation, placement, and length; however, they suffer from poor edge structure, which degrades their electrical and thermal properties,3,11–17 and from challenges in achieving sub-10 nm widths. Alternative techniques such as the unzipping of graphite18 and nanotubes19,20 can result in narrow ribbons with smooth edges but do not provide control over placement, orientation, or edge chirality. Nanoribbons can also be fabricated via bottom-up polymerization followed by cyclodehydrogenation,21–26 which results in a smooth, armchair edge structure and sub-10 nm widths; however, this method lacks control over placement and thus far has been difficult to adapt to technologically useful semiconducting or insulating substrates.

Our group has demonstrated a bottom-up synthesis process in which graphene nanoribbons can be grown directly on Ge(001)27,28 and Ge(001) epilayers on Si(001)29 by the chemical vapor deposition (CVD) of CH4. The technique yields oriented ribbons (aligned to Ge⟨110⟩), faceted armchair edges, sub-10 nm widths, and lengths greater than 100 nm. These ribbons possess promising charge transport properties compared to others in the literature, demonstrating an on/off conductance ratio of up to 2 × 104 and an on-state conductance of 5 µS.30 

Nanoribbon placement and orientation can be controlled by decorating the Ge(001) surface with small graphene seeds in which each seed initiates the growth of a single nanoribbon.31 Lithographic control over seed placement thereby enables the synthesis of dense arrays of nanoribbons. Seeded growth significantly reduces polydispersity while still allowing for the creation of nanoribbons that are narrower than 10 nm and longer than 100 nm.31,32

A key aspect of seeding nanoribbons that have high-aspect ratios and are aligned is controlling the crystallographic orientation of the seeds with respect to the Ge surface. For example, we previously found that when an armchair direction of a seed is aligned with a Ge⟨110⟩ direction, the anisotropy of the resulting nanoribbon is maximized (aspect ratio >10), and the long axis of the nanoribbon is aligned along the same Ge⟨110⟩.31,32 On the other hand, if there is misalignment, the aspect ratio decreases (approaching 1 for the worst alignment), and the resulting graphene island is no longer aligned with Ge⟨110⟩.

Here, we report a more complex result. We show that the above behaviors are true only when the graphene seeds are relatively large, with diameters greater than 20 nm. In contrast, when the diameter is reduced below 10 nm, nanoribbons that have high-aspect ratios and are aligned along Ge⟨110⟩ evolve from seeds in nearly every instance, regardless of the original crystallographic orientation of the seed and its alignment or misalignment with respect to Ge⟨110⟩. This observation indicates that small graphene seeds are able to rotate on the Ge(001) surface prior to nanoribbon synthesis and that the most energetically favorable seed orientation occurs when one of its armchair directions is parallel to Ge⟨110⟩.

A theoretical study by Zhang et al. has predicted that small graphene islands can rotate on Cu(111), Pt(111), Ni(111), Ru(0001), and Co(111) surfaces when below a critical diameter (5 nm–19 nm, depending on the metal).33 However, the rotation of graphene islands or seeds on Ge(001) has not been studied. Here, we elucidate the effect of the seed diameter (0 nm–50 nm) and crystallographic orientation on this rotation and take advantage of this rotation phenomenon to fabricate meshes of interconnected nanoribbons.

To experimentally create seeds, we employ a fabrication process used previously.31,32 Briefly, hexagonal islands of graphene are grown on Cu foil. Then, the hexagons are wet-transferred to Ge(001) via etching of the Cu foil with the assistance of a sacrificial polymer layer. The crystallographic orientation of each transferred graphene hexagon is determined by mapping the orientation of its zigzag edges with respect to the Ge substrate. Next, electron-beam lithography and metal deposition are used to define small (diameter of 20 nm–50 nm and thickness of 10 nm) discs of Ni on the graphene, which are used as etch masks in conjunction with directional reactive ion etching to produce roughly circular seeds of graphene (also ∼20 nm to 50 nm in diameter) on Ge. The angle between the armchair direction of each seed and Ge[110] after lithographic patterning is known from the orientation of the graphene hexagon from which each seed is patterned, as described above.

The graphene seeds on Ge(001) are then annealed at 910 °C for 45 min–67.5 min in 33% H2 in Ar at 1 atm to remove carbon-based adsorbates and impurities from the Ge surface, which otherwise cause nucleation of unseeded nanoribbons. During this anneal, the seed diameter decreases at a rate of 0.6 nm min−1, as previously measured.32 The final seed diameter, ds, is controlled by varying the anneal duration or the initial seed diameter, as set by lithography. Immediately after annealing, without cooling the substrate, nanoribbon growth is initiated from the seeds at 910 °C in an environment of 66% Ar, 33% H2, and 0.66% CH4 at 1 atm for 6 h. After growth, the resulting graphene islands (e.g., high-aspect ratio graphene nanoribbons or low-aspect ratio graphene islands) are imaged via scanning electron microscopy (SEM).

The schematic diagram in Fig. 1 illustrates the convention used in this study to describe the relative orientation of the seed lattice with respect to the Ge lattice. Each graphene seed features three armchair orientations, labeled AC1, AC2, and AC3. The AC most closely aligned with Ge[110] is defined as AC1, and the angle between AC1 and Ge[110], which can vary from −30° to 30°, is denoted as θseed. Note that θseed does not exceed ±30° because of the sixfold symmetry of the graphene lattice. Moreover, graphene ribbons and islands that evolve from seeds with θseed are mirror images (with respect to Ge[110]) of those that evolve from seeds in which AC1 has the same magnitude, but opposite sign (i.e., −θseed), due to the fourfold symmetry of the Ge lattice. Therefore, the discussion in this paper is limited to positive θseed (0°–30°).

FIG. 1.

Schematic of the relative orientation of the graphene seed lattice (yellow) with respect to the Ge lattice (blue). The three armchair directions of the graphene seed are shown as AC1 (red), AC2 (purple), and AC3 (green), where the angle between AC1 and Ge[110] is defined as θseed. Note that the exact edge structure of the graphene seed is unknown.

FIG. 1.

Schematic of the relative orientation of the graphene seed lattice (yellow) with respect to the Ge lattice (blue). The three armchair directions of the graphene seed are shown as AC1 (red), AC2 (purple), and AC3 (green), where the angle between AC1 and Ge[110] is defined as θseed. Note that the exact edge structure of the graphene seed is unknown.

Close modal

Likewise, each graphene ribbon or island that evolves from a seed has three AC orientations. The angle between AC1 of a ribbon or island and Ge[110] is denoted as θfinal. Here, θfinal is quantified by mapping the orientation of ribbon and island edges with respect to the Ge substrate; electron diffraction data show that the edges of ribbons and islands that grow from seeds are roughly parallel to the armchair crystallographic directions of graphene—both when seed rotation does (Fig. S1) and does not occur.31 

Representative examples of graphene ribbons and islands that evolve from seeds that do not self-rotate on the Ge surface (typically because the seeds are large) are shown in Fig. 2 as a function of θseed. When θseed = 0°, 7°, 15°, and 30°, ribbons and islands evolve from the seeds with a θfinal of ∼0°, 7°, 15°, and 30°, respectively. Thus, the crystallographic orientation of each nanoribbon or island matches that of its parent seed. The shape and aspect ratio of each ribbon or island are dictated by the relative orientation of each armchair edge with respect to Ge[110] or Ge1¯10. Armchair edges that are roughly parallel to Ge[110] or Ge1¯10 grow slowly, whereas edges misaligned from these axes grow quickly. It therefore follows that seeds with θseed = 0° [Fig. 2(a)] evolve into high-aspect ratio nanoribbons pointed along Ge[110] [Fig. 2(e)] because AC1 is parallel to Ge[110] and seeds with θseed = 30° [Fig. 2(d)] evolve into high-aspect ratio nanoribbons pointed along Ge1¯10 [Fig. 2(h)] because AC2 is parallel to Ge1¯10. When θseed is 7° [Fig. 2(b)] or 15° [Fig. 2(c)], the growth rate in the width direction accelerates because of the misalignment, and low-aspect ratio islands result [Figs. 2(f) and 2(g)]. These data match previously reported observations.31 

FIG. 2.

[(a)–(c)] Schematic of different orientations of graphene seeds (yellow) on the Ge(001) surface (blue) in which the angle between the armchair direction (AC1, red atoms) of the graphene seed and Ge[110] after lithographic patterning, θseed, is 0° (a), 7° (b), 15° (c), and 30° (d). The two other armchair directions (AC2, pink, and AC3, green) are depicted as well. [(e)–(h)] SEM images of resulting graphene ribbons and islands grown from seeds with θseed of 0° (e), 7° (f), 15° (g), and 30° (h) in which the angle between the armchair direction of the resulting ribbon or island and Ge[110], θfinal, is ∼0° (e), 7° (f), 15° (g), and 30° (h). Scale bar is 200 nm.

FIG. 2.

[(a)–(c)] Schematic of different orientations of graphene seeds (yellow) on the Ge(001) surface (blue) in which the angle between the armchair direction (AC1, red atoms) of the graphene seed and Ge[110] after lithographic patterning, θseed, is 0° (a), 7° (b), 15° (c), and 30° (d). The two other armchair directions (AC2, pink, and AC3, green) are depicted as well. [(e)–(h)] SEM images of resulting graphene ribbons and islands grown from seeds with θseed of 0° (e), 7° (f), 15° (g), and 30° (h) in which the angle between the armchair direction of the resulting ribbon or island and Ge[110], θfinal, is ∼0° (e), 7° (f), 15° (g), and 30° (h). Scale bar is 200 nm.

Close modal

In contrast, significant deviations from these behaviors are observed when the seed diameter, ds, is reduced. Every graphene crystal in Fig. 3 is grown from misaligned seeds with θseed in the range of 10°–15°. When ds is larger than 20 nm, the majority of the seeds evolve as low-aspect ratio parallelograms, and θfinal matches θseed [Fig. 3(g)], similar to Fig. 2. However, when ds is reduced below 20 nm, some seeds begin to evolve as high-aspect ratio ribbons in which θfinal does not match θseed [Fig. 3(f)]. When ds is further reduced to less than 10 nm, nearly all (over 90%) of the seeds evolve as high-aspect ratio nanoribbons in which θfinal does not match θseed [Fig. 3(e)]. Several important conclusions can be drawn from these data.

FIG. 3.

[(a)–(c)] Schematics of graphene seeds of varying seed diameter, ds, in which θseed = 15°. (d) Plot of the fraction of rotated seeds (θseed = 10°–15°) vs ds showing that more seeds rotate as ds decreases. The red line is the fit of the data to a normal cumulative distribution, and the intersection of the red line with the gray dashed line indicates that the critical seed diameter at which 50% of seeds rotate, ds,50, is 18 nm. [(e)–(g)] SEM images of seeds that fully rotated, yielding nanoribbons with θfinal ≈ 0° or 30° (e); partially rotated, yielding both nanoribbons with θfinal ≈ 0° or 30° and low-aspect ratio islands with θfinal ≈ 15° (f); and did not rotate, yielding low-aspect ratio islands with θfinal ≈ 15° (g). Scale bars are 200 nm.

FIG. 3.

[(a)–(c)] Schematics of graphene seeds of varying seed diameter, ds, in which θseed = 15°. (d) Plot of the fraction of rotated seeds (θseed = 10°–15°) vs ds showing that more seeds rotate as ds decreases. The red line is the fit of the data to a normal cumulative distribution, and the intersection of the red line with the gray dashed line indicates that the critical seed diameter at which 50% of seeds rotate, ds,50, is 18 nm. [(e)–(g)] SEM images of seeds that fully rotated, yielding nanoribbons with θfinal ≈ 0° or 30° (e); partially rotated, yielding both nanoribbons with θfinal ≈ 0° or 30° and low-aspect ratio islands with θfinal ≈ 15° (f); and did not rotate, yielding low-aspect ratio islands with θfinal ≈ 15° (g). Scale bars are 200 nm.

Close modal

First, the fact that there are nanoribbons in which θfinal does not match θseed indicates that the graphene seeds rotate during annealing and/or synthesis processes—both of which occur at 910 °C. The data further indicate that the percentage of seeds that rotate increases with decreasing ds. Fitting the data in Fig. 3(d) to a normal cumulative distribution (red curve) indicates a critical seed diameter at which 50% of the seeds rotate, ds,50, of 18 nm, which is similar to the values found by Zhang et al. for graphene rotation on metal surfaces.33 

Second, the fact the graphene seeds rotate so that θfinal ≈ 0° (horizontal ribbons in Fig. 3 in which AC1 is roughly parallel to Ge[110]) or 30° (vertical ribbons in Fig. 3 in which AC2 is roughly parallel to Ge1¯10) indicates that the alignment of AC with Ge⟨110⟩ is energetically favored. This is important because θfinal ≈ 0° or 30° is the alignment at which the growth anisotropy is maximized, thereby driving nanoribbon formation rather than the growth of low-aspect ratio islands.31 Similarly, in previous syntheses on Ge(001) without seeds, nearly all (>90%) graphene crystals that nucleate evolve as nanoribbons with high-aspect ratios.27 One explanation for this observation is that ∼90% of graphene crystals that nucleate without seeds have θnuclei ≈ 0° or 30° (where θnuclei is the angle between AC1 of a naturally occurring nucleus and Ge[110]). Alternatively, the data above indicate that it is also possible that the graphene nuclei without seeds have θnuclei ≠ 0° or 30°; however, shortly after the graphene islands nucleate (i.e., when they are small), they have the freedom to rotate and lock in to an energetically preferred orientation such that θfinal ≈ 0° or 30°. In either case (i.e., θfinal ≈ 0° or 30°), the anisotropy of the growth kinetics is then maximized, driving the evolution of nanoribbons with high-aspect ratios from the nuclei.

The above data indicate that seeds can self-rotate on Ge(001). Next, we measure the tendency for this self-rotation to occur toward θfinal ≈ 0° in which AC1 is parallel to Ge[110] vs θfinal ≈ 30° in which AC2 is parallel to Ge1¯10. This analysis is conducted as a function of θseed and ds in Fig. 4. Note that here, only seeds that evolve as high-aspect ratio nanoribbons are analyzed (θfinal ≈ 0° or 30°). For example, when examining growth from seeds with θseed = 10°–15° and ds > 20 nm, only 0%–5% of the seeds rotate to result in nanoribbons (consistent with Fig. 3).

FIG. 4.

(a) Plot of the fraction of graphene nanoribbons with their long axis aligned to Ge[110] vs ds for seeds with θseed of 0°–5° (red squares), 5°–10° (blue circles), and 10°–15° (green, triangles). [(b)–(g)] SEM images of nanoribbon arrays that grow from seeds with θseed = 0°–5° (b), θseed = 5°–10° (c), and θseed = 10°–15° (d) with a ds of 15 nm. SEM images of nanoribbon arrays that grow from seeds with θseed = 0°–5° with decreasing ds of 11 nm (e), 8 nm (f), and 4 nm (g). Pitch between ribbons in the x- and y-directions are 300 × 500 nm [(c), (f), and (g)] and 400 × 500 nm [(b), (d), and (e)]. Scale bars are 200 nm. Error bars denote one standard deviation.

FIG. 4.

(a) Plot of the fraction of graphene nanoribbons with their long axis aligned to Ge[110] vs ds for seeds with θseed of 0°–5° (red squares), 5°–10° (blue circles), and 10°–15° (green, triangles). [(b)–(g)] SEM images of nanoribbon arrays that grow from seeds with θseed = 0°–5° (b), θseed = 5°–10° (c), and θseed = 10°–15° (d) with a ds of 15 nm. SEM images of nanoribbon arrays that grow from seeds with θseed = 0°–5° with decreasing ds of 11 nm (e), 8 nm (f), and 4 nm (g). Pitch between ribbons in the x- and y-directions are 300 × 500 nm [(c), (f), and (g)] and 400 × 500 nm [(b), (d), and (e)]. Scale bars are 200 nm. Error bars denote one standard deviation.

Close modal

The red squares in Fig. 4(a) show the fraction of nanoribbons that evolve with θfinal ≈ 0° (with their long axis along Ge[110]) when seeds start from an already well-aligned orientation with θseed in the range of 0°–5°. The fraction is nearly 100% for ds > 15 nm [Fig. 4(b)], showing again that when seeds are large, their rotation is suppressed. However, this fraction decreases to 80% as ds decreases below 5 nm [Figs. 4(e)–4(g)]. These data indicate that the energetic potential well holding the seeds near θfinal ≈ 0° is large enough to mostly (but not completely) suppress rotation at 910 °C when ds is small.

For comparison, the green triangles in Fig. 4(a) show the fraction of nanoribbons that evolve with θfinal ≈ 0° (with their long axis along Ge[110]) when seeds start from a misaligned orientation with θseed in the range of 10°–15°. The fraction varies between 50% and 60% for all ds [Fig. 4(d)]. These data indicate that rotation of the graphene seed to θfinal ≈ 0° is only slightly favored over rotation to θfinal ≈ 30°. The blue circles in Fig. 4(a) show similar phenomena for nanoribbons that evolve from seeds with intermediate θseed in the range of 5°–10°. In this case, the fraction of nanoribbons that grow with θfinal ≈ 0° (with their long axis along Ge[110]) varies between 70% and 80% for all ds [Fig. 4(c)], falling between the outcomes of seeds with θseed = 0°–5° and θseed = 10°–15°.

An alternative presentation of the data in Fig. 4(a) is shown in Fig. 5, which plots the fraction of nanoribbons with θfinal ≈ 30° (their long axis aligned along Ge1¯10), as a function of θseed from 0°–30° and binned by ds. It is clear that regardless of ds, when θseed ≈ 0°, the long axis of the nanoribbons is preferentially aligned along Ge[110], while when θseed ≈ 30°, the long axis of the nanoribbons is preferentially aligned along Ge1¯10. The data show that the probability of rotation toward Ge1¯10 vs Ge[110] is low for seeds with θseed ≈ 0° and increases with increasing θseed.

FIG. 5.

Plot of the fraction of graphene nanoribbons aligned to Ge1¯10 (left axis) and Ge[110] (right axis) vs θseed, where the data points are binned by ds. As θseed increases, the fraction aligned along Ge1¯10 increases for all ds.

FIG. 5.

Plot of the fraction of graphene nanoribbons aligned to Ge1¯10 (left axis) and Ge[110] (right axis) vs θseed, where the data points are binned by ds. As θseed increases, the fraction aligned along Ge1¯10 increases for all ds.

Close modal

Next, this rotation phenomenon is leveraged to synthesize meshes of interconnected nanoribbons with controlled placement, pitch, and width (Fig. 6). Square arrays of seeds spaced by 100 nm are fabricated with ds = 4 nm and 17 nm and θseed of 10°–15° [Figs. 6(a) and 6(c)]. These seeds rotate so that their armchair direction is aligned along Ge[110]/Ge1¯10 with a 60/40 split. The nanoribbons that evolve from the square arrays of seeds merge together to form a continuous network [Figs. 6(b) and 6(d)]. By controlling ds and the growth conditions, meshes with a desired width can be fabricated. The widths of the nanoribbons in Figs. 6(b) and 6(d) are 9.1 ± 2.6 nm and 20.8 ± 2.9 nm, respectively. While this approach is promising, more work is required to improve the nanoribbon width uniformity and to explore the electrical properties of these mesh networks.

FIG. 6.

SEM images of Ni etch masks patterned on graphene [(a) and (c)] and nanoribbon meshes grown from the graphene seeds [(b) and (d)] in which ds = 4 nm [(a) and (b)] and 17 nm [(c) and (d)] and the seeds have θseed of 10°–15°. The pre-growth anneal time is 30 min. Nanoribbon growth is then initiated from the seeds at 910 °C in an environment of 66% Ar, 33% H2, and 0.66% CH4 at 1 atm for 3 h. Scale bar is 200 nm. Note that Ni etch masks patterned on graphene are shown in [(a) and (c)] instead of the as-patterned graphene seeds because the graphene seeds have much lower contrast, and ds is smaller than the Ni etch mask diameter because the seed size is reduced via annealing prior to graphene growth, as described above.

FIG. 6.

SEM images of Ni etch masks patterned on graphene [(a) and (c)] and nanoribbon meshes grown from the graphene seeds [(b) and (d)] in which ds = 4 nm [(a) and (b)] and 17 nm [(c) and (d)] and the seeds have θseed of 10°–15°. The pre-growth anneal time is 30 min. Nanoribbon growth is then initiated from the seeds at 910 °C in an environment of 66% Ar, 33% H2, and 0.66% CH4 at 1 atm for 3 h. Scale bar is 200 nm. Note that Ni etch masks patterned on graphene are shown in [(a) and (c)] instead of the as-patterned graphene seeds because the graphene seeds have much lower contrast, and ds is smaller than the Ni etch mask diameter because the seed size is reduced via annealing prior to graphene growth, as described above.

Close modal

In this work, we have determined that if the diameter of graphene seeds is less than 18 nm, the seeds can rotate so that an armchair direction becomes aligned with Ge⟨110⟩. If graphene seeds with θseed = 10°–15° are sufficiently small to rotate, they rotate so that an armchair direction is along Ge[110]/Ge1¯10 with a 60/40 statistical split. Even graphene seeds with θseed = 0°–5° can rotate if the seed diameter is small enough; however, these seeds have a strong preference to resist rotation, resulting in only 20% of the seeds rotating from Ge[110] to Ge1¯10.

The data in Figs. 3–5 show that seed rotation is viable at 910 °C in a 67% Ar/33% H2 environment but does not explore other conditions. Additional work will be needed to elucidate the temperature dependence for seed rotation, the activation energy for this rotation, and its dependence on growth environment. Moreover, simulations specifically analyzing the rotation of graphene islands on Ge(001) will be needed to understand the driving forces for seed rotation. The driving forces on Ge(001) are likely similar to those on metal surfaces, whereby graphene islands have energetically favorable orientations on substrates with barriers to rotation restricting the rotation from one orientation to the other. The orientations that are preferential and the size of the energy barriers are dictated by graphene–substrate interactions at both the interior and edges of the islands.33 

Future work utilizing the rotation of graphene seeds could allow for large-area meshes/networks of nanoribbons to be fabricated for applications in thin-film electronics, sensors, and biosensors. Additionally, creating arrays of unidirectionally aligned sub-10 nm nanoribbons will require additional strategies to prevent unwanted seed rotation, such as the use of miscut Ge surfaces, which are already known to preferentially align nanoribbons along a single direction even without seeds.34 

See the supplementary material for transmission electron microscopy (TEM) and selective-area electron diffraction (SAED) of rotated graphene nanoribbons.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This research was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0016007 (A.J.W., V.S., R.M.J., and M.S.A.) for graphene seed and nanoribbon synthesis experiments and characterization. A.J.W. also acknowledges support from the NSF Graduate Research Fellowship Program (Award No. DGE-1256259). The authors acknowledge the use of facilities and instrumentation supported by the NSF through the University of Wisconsin Materials Research Science and Engineering Center (Grant No. DMR-1720415).

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Supplementary Material