Introduction

The controlled growth of metastable materials has been plagued by the competition of thermodynamically stable structures1. Single-crystal metal foil with high-index facets has emerged as a promising metastable platform for 2D epitaxy1,2,3,4,5,6,7,8,9,10,11,12, catalysis13, and electronics14. It is currently prepared by driving the abnormal grain growth (AGG) of an arbitrary facet under grain boundary energy or strain energy minimization, which is particularly attractive owing to the potential of obtaining diverse facet indices3,4,5,6,15. However, its deterministic growth is challenging because AGG needs to be selectively driven for as many kinds of metastable high-index facets as possible rather than the stable low-index ones3,16,17,18,19,20,21,22,23. Meanwhile, the net driving force (driving force minus retarding force) must be sufficiently large to kinetically initiate the AGG in a variety of metals. Due to the lack of high selectivity for diverse high-index facets, the prevalent strategies suffer from the trade-off between the diversity of facets and their deterministic growth. For example, surface pre-oxidation allows the random AGG of more than 30 kinds of facets driven by the grain boundary energy minimization, but the probability of high-index facets is less than 82% without combining the additional facet transfer method3. The probability is improved to >90% by applying strain to the metal foil as a whole, but the obtained high-index facets are significantly reduced to below 10 kinds4,6. This arises from the facet-dependent nature of the biaxial modulus, which permits only a limited number of high-index facets to be thermodynamically favorable under external strain. Moreover, a large net driving force has not been obtained for high-index facets by using current strategies, which arises from the small driving force and the ubiquitous thermal groove drag. The initiation of AGG is thus kinetically retarded or even suppressed. For the two high-index single-crystal Cu and Ni foils that are currently available3,4,6, it takes several hours to grow a decimeter-sized sample4, corresponding to a growth rate below 2 mm min−1. Many more materials have not been prepared such as the highly desirable single-crystal Au foil with high-index facets, which promises wide applications in epitaxy and catalysis10,11,12,24,25. Therefore, a deterministic, efficient and versatile growth strategy is in great demand.

Dislocations function as a universal source for tuning the free energy of diverse crystallographic facets, given their nature as inherent defects in polycrystalline metals that exhibit high tunability in energy and spatial distribution26. Here, we propose a dislocation-driving strategy for the controlled growth of single-crystal metal foils with high-index facets. This strategy selectively activates the AGG of diverse high-index facets and provides an enhanced net driving force, thus overcoming the determinacy/diversity trade-off and greatly improving the growth rate. As a result, we realized the fast (~20 mm min−1) deterministic growth of single-crystal Cu, Ni and Au foils with dozens of high-index facets.

Results and discussion

As shown in Fig. 1a, the fundamental principle is lifting the free energy of strong (100) texture (the only type of low-index facet in the foil) by using the random intragranular dislocations and the dislocations along its low-angle grain boundary. If the density of intragranular dislocation fluctuates significantly among grains, there is certain probability that one high-index facet is surrounded by several (100) facets with much higher energy associated with dislocations (referred to as dislocation energy Ed for simplicity). Thus-formed large dislocation energy difference (ΔEd) outweighs the adverse effect of surface energy difference (ΔEs, energy change per unit volume) to convert the growth of high-index facets to be thermodynamically favorable [ΔE = (Ed-(hkl) − Ed-(100)) + (Es-(hkl) − Es-(100)) = ΔEd + ΔEs < 0]. Since the texture suppresses the AGG of all (100) facets, a substantial driving force is established to direct the selective AGG of diverse high-index facets with improved kinetics. This principle is realized by annealing commercial polycrystalline metal foils with an appreciable ΔEd and a strong (100) texture (Fig. 1b). Note that ΔEd is preserved by using an ultrafast heating-up (>200 °C s−1) technique, which is crucial for activating the AGG of high-index facets (Supplementary Note 1). Meanwhile, the normal grain growth of strong (100) texture predominates in the entire foil, which dramatically reduces the number of AGG by consuming most high-index facets. The kinetic advantage of a certain high-index facet is further enhanced by using an in-plane temperature gradient for annealing. In this process, dislocations along the grain boundaries of (100) texture form a uniform dislocation network throughout the foil, thus maintaining a relatively constant ΔEd for the continuous AGG. Finally, one high-index facet grows overwhelmingly and transforms the polycrystalline foil into a single crystal in most cases (Supplementary Note 2 and Supplementary Fig. 1).

Fig. 1: Design and verification of the dislocation-driving strategy.
Fig. 1: Design and verification of the dislocation-driving strategy.
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a Schematic illustration of the thermodynamic mechanism, where Ed and Es refer to dislocation energy and surface energy per unit volume, respectively. b Schematic growth process of a single-crystal foil by driving the abnormal grain growth (AGG) of a high-index facet with the dislocation-driving strategy. The activated high-index facet is surrounded by a strong (100) texture, in which the adjacent (100) grains show higher dislocation density. The blue, green, yellow and red areas denote the density of geometrically necessary dislocation (GND) ranging from low to high. c–f Photographs of the pristine polycrystalline Cud/t foil (Cu with appreciable ΔEd and strong (100) texture) (c) and that after annealing for 20 s (d), 100 s (e) and 150 s (f), showing the abnormal growth of a high-index facet into a large single-crystal. The sample in (d–f) was oxidized at 180 °C for 2–5 min to show the growing high-index facet with a light color. g–j Electron backscatter diffraction (EBSD) inverse pole figure in the normal direction (IPZ) maps of the pristine polycrystalline Cud/t foil (g) and that after annealing for 20 s (h), 100 s (i) and 150 s (j), corresponding to the position marked by the white spot in (c–f). Low-angle and large-angle grain boundaries in h and i are highlighted in pink and white to correlate with the GND distribution in l and m, respectively. k–n EBSD GND density maps of the pristine polycrystalline Cud/t foil (k) and that after annealing for 20 s (l), 100 s (m) and 150 s (n). o–r EBSD inverse pole figure (IPF) contours of the pristine polycrystalline Cud/t foil (o) and that after annealing for 20 s (p), 100 s (q) and 150 s (r); color bar represents the signal intensity of different facets in IPF contour. The evolution of signal intensity indicated that the (100) texture first strengthened and then weakened due to its competitive growth with the high-index facets. Inset, IPF scatter plot shows the transformation from a polycrystal into a single crystal with the high-index facet.

We take the deterministic and efficient growth of diverse high-index single-crystal Cu foils as an example to demonstrate the effect of dislocation-driving strategy. We first investigated the crystalline structure evolution of a commercial polycrystalline Cu foil (Cud/t) to clarify the ΔEd-driven AGG of a high-index facet into a large single crystal (Fig. 1c-f). The as-received foil featured a strong (100)-oriented cube texture (50.8%) together with a fraction of high-index facets (Fig. 1g and Supplementary Figs. 2c, f and 3a). The proportion of (100) texture is significantly higher than the sum of (111) and (110) facets (~0.16%), which is common in other types of Cu foils (Supplementary Fig. 4). Meanwhile, it showed an average dislocation density of 0.29 × 1014 m−2 with the maximum reaching 1.44 × 1014 m−2, indicating the presence of appreciable ΔEd. If the conventional slow heating-up was used to minimize the ΔEd, only the normal grain growth of (100) texture (growth rate <0.2 mm min−1) was observed after annealing (Supplementary Figs. 5, 6 and 3b). In contrast, upon using the ultrafast heating-up to preserve the ΔEd, a high-index facet grew excessively into a ~ 10 mm grain after 20 seconds annealing, which then formed a centimeter-scale Cu(014) single crystal within 150 s (Fig. 1c-f, Supplementary Movie 1). Statistical analysis revealed a representative grain growth rate range of 16–23 mm min-1 (Supplementary Fig. 3c). This comparison also suggests the distinct roles of random intragranular dislocations and dislocations along low-angle grain boundaries in driving the AGG of high-index facets (Supplementary Note 3). Molecular dynamics (MD) simulations indicated that the AGG of Cu(014) was activated by consuming the adjacent (100) facet with higher dislocation energy (Supplementary Fig. 7). The corresponding inverse pole figure maps in the normal direction (IPZ maps) of electron backscatter diffraction (EBSD) measurements showed that Cu foil was dominated by a single type of (100) texture without (111) and (110) facets upon annealing (Fig. 1h, p and Supplementary Fig. 3a). GND density maps confirmed that dislocations were preferentially located along the low-angle grain boundaries of the (100) texture, forming a uniform network over the entire region (Fig. 1l). Note that a sustained difference in dislocation density was established between the predominant (100) texture and the growing (014) facet (Fig. 1m), corresponding to an absolute value of 2.99 × 104 J·m–3 for ΔEd (Supplementary Note 4). Such value is approximately five times that of ΔEs (~5.76 × 103 J·m−3), thus reversing the growth of (014) facet to be thermodynamically favorable. This unidirectional enhanced driving force ensured the fast growth of the (014) facet until the entire (100) texture was consumed (Fig. 1q, r). The above phenomenon is ubiquitous in other types of Cu foils as well as Au and Ni foils (Supplementary Fig. 8), laying the foundation for realizing the versatile growth.

We then used our strategy to grow a variety of high-quality high-index single-crystal Cu foils in a deterministic and efficient way. After annealing for merely 5 min, the Cud/t was converted into a decimeter-scale single crystal (Fig. 2a). The uniform and bright single-crystal region can be clearly observed after mild oxidation. Its crystallographic structure was analyzed by using multiple characterization techniques. The identical color of IPZ maps and consistent (001) PF maps at five regions suggested a large-scale homogeneous crystallographic orientation in the Cu(113) single crystal (Supplementary Fig. 9). The kernel average misorientation (KAM) map (Fig. 2b) revealed the small average misorientation between each measured point and its nearest neighboring points, indicating a homogeneous single-crystal structure. The characteristic peaks in X-ray diffraction (XRD) 2θ scan align with the Cu(113) single-crystal structure (Fig. 2c). The presence of a single prominent Cu(200) peak in the φ scan unambiguously indicates a homogeneous single-crystal structure without in-plane rotation. The (113) lattice structure was corroborated by using high-resolution transmission electron microscopy (HRTEM; Fig. 2d). Our strategy also enabled the preparation of a series of high-index single-crystal Cu foils, as confirmed by the distinct IPZ and XRD 2θ scan results (Fig. 2e, Supplementary Fig. 10, 11). Larger single-crystal foil was obtained by prolonging the annealing in a large-sized CVD furnace (Supplementary Fig. 12). Note that there is no trade-off between the diversity of facets and deterministic growth of high-index single crystals. Statistical analysis of 169 pieces of single-crystal Cu foils revealed that ~40 kinds of high-index facets with random orientations were obtained without low-index facets (Fig. 2f). The appearance frequency of high-index facets was primarily determined by their proportional distribution in the pristine Cu foils (Supplementary Note 5 and Supplementary Fig. 13). This result demonstrated the selective AGG of diverse high-index facets enabled by combining strong (100) texture with the appreciable ΔEd. This ΔEd is sufficiently large to drive the fast growth of high-index facets with an average rate of ~20 mm min−1. Comparison of single-crystal Cu foils prepared using different methods shows that our strategy enables one order of magnitude improvement in the growth rate over the reported results even at a lower annealing temperature (Fig. 2g)3,4,5,15,18,19,20,21,22,23.

Fig. 2: Preparation of single-crystal Cu foils with high-index facets by the dislocation-driving strategy.
Fig. 2: Preparation of single-crystal Cu foils with high-index facets by the dislocation-driving strategy.
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a Photograph of a single-crystal Cu (113) foil after mild oxidation in air at 180 °C. The dark polycrystalline region on the right side was intentionally maintained as a reference for comparison. b Representative electron backscatter diffraction (EBSD) inverse pole figure in the normal direction (IPZ, top) and kernel average misorientation (KAM, bottom) maps of the single-crystal Cu(113) foil. c X-ray diffraction (XRD) 2θ scan spectrum of the single-crystal Cu foil with (113) orientation. Inset, azimuthal off-axis φ scan spectrum with a single prominent Cu(200) peak. d Representative atomically resolved high-resolution transmission electron microscopy (HRTEM) image of the single-crystal Cu(113) foil. Inset, fast Fourier transform pattern of the HRTEM image. e Several EBSD IPZ maps of single-crystal Cu foils with distinct high-index facets. f IPZ contour of statistical facet indices of single-crystal Cu foils obtained, demonstrating the deterministic growth of single crystals with diverse high-index facets. g Comparison of the grain growth rate and annealing temperature of our work (single-crystal Cu with high-index facets) with the reported results of single crystals prepared by means of abnormal grain growth.

Our strategy also applied to common Cu foils without combining the appreciable ΔEd and strong (100) texture (Supplementary Fig. 14). With only a strong (100) texture, the Cut foil was pre-stretched to increase the ΔEd, which outweighed the adverse effect of ΔEs on growing high-index facets. The original normal growth of (100) facets (Fig. 3a-left column and b, Supplementary Fig. 15 and Note 4) was then converted into the fast AGG of a (014) facet (Fig. 3a-right column and c, Supplementary Fig. 16). Statistical analysis of 64 pieces of thus-obtained single-crystal Cu foils verified the growth of ~30 kinds of high-index facets without low-index ones (Fig. 3d). For the foil without strong (100) texture and large ΔEd, (termed Cun), it was pre-annealed to strengthen the texture followed by the pre-stretching to obtain an appreciable ΔEd (Supplementary Figs. 14, 17). With this enhanced driving force, single-crystal Cu foil with high-index facet was obtained after several minutes of annealing, which would otherwise yield polycrystalline foil under the identical annealing condition.

Fig. 3: Selective abnormal grain growth of high-index facets in different types of Cu foils enabled by the dislocation-driving strategy.
Fig. 3: Selective abnormal grain growth of high-index facets in different types of Cu foils enabled by the dislocation-driving strategy.
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a Calculated driving force for grain growth in pristine and pre-stretched Cut foils, involving ΔEd and ΔEs as the driving forces that refer to dislocation energy difference and surface energy difference per unit volume, respectively (Supplementary Note 4). b Representative crystal orientation of pristine Cut foil after annealing for 1 min (top) and corresponding schematic mechanism of ΔEs-driven growth (bottom). c Representative crystal orientation of pre-stretched Cut foils after annealing for 1 min (top) and corresponding schematic mechanism of ΔEd-driven growth (bottom). d Inverse pole figure in the normal direction (IPZ) contour of statistical facet indices of single-crystal Cut foils obtained, demonstrating the deterministic growth of single crystals with diverse high-index facets. e Schematic mechanism of groove structure/drag variation of metal foil caused by grain growth under different driving force processes. The gray, pink and yellow lines denote the surface normal, the grain boundary with its extension along the thermal groove at an angle θ and that at the critical angle θ0, respectively. If θ > θ0 (blue line), the grain boundary is capable of migrating (Supplementary Note 6). f Optical microscope (OM) image of Cut foil after annealing for 2 h, which shows the intermittent migration of grain boundaries. The grain boundary is marked by the yellow dotted line and its migration direction is indicated by the white arrow. g OM image of the pre-stretched Cut foil after the 1 min annealing. The grain boundary is marked by the yellow dotted line. h–j Atomic force microscope (AFM) morphology of grooves at grain boundaries of Cut foil annealed for 1 min (h), 2 h (i), and for an additional 1 min after stretching the sample in i (2 h-s-1 min) (j). The angle and depth of grooves are marked in the image. k Statistical results of angle and depth of grooves in samples (h–j). Each error bar in k represents the standard deviation calculated from 51 experimental data.

Interestingly, the large driving force enabled by appreciable ΔEd reduced the retarding force of thermal groove and further increased the net driving force. Due to the insufficient driving force to overcome the groove drag (i.e. small net driving force), the grain boundary migration of common metal foil is usually intermittent (Fig. 3e, f) with a relatively small θ of groove (e.g. 82.39° for Cut in Fig. 3h), corresponding to a large retarding force (Supplementary Note 6). This situation was exacerbated if the conventional prolonged annealing was used, manifested as a further decrease in θ and increase in depth of grooves at grain boundaries (Fig. 3i, k), thus forming numerous remaining grooves (Fig. 3e, f, Supplementary Figs. 18 and 19, Note 6). In contrast, incorporating appreciable ΔEd to improve the driving force significantly reduced the retarding force of groove by accelerating the migration of grain boundary in the foil interior, as revealed by the increased θ (e.g. 85.63°) and decreased depth of grooves (Fig. 3j, k). For Cut, the retarding force was reduced by 43.9% on average (Supplementary Note 6). As a result, the increased net driving force initiated the AGG of high-index facets and dramatically improved the kinetics without forming the undesirable intermittent grooves (Fig. 3g, Supplementary Fig. 20). We then quantitatively analyzed the AGG kinetics of high-index facets by using the classic relation between grain growth rate and the net driving force (Supplementary Note 7). For both Cut and Cun, the increment in grain growth rate agrees with that in the net driving force (Supplementary Fig. 21a). Similarly, the superior constant grain growth rate of Cud/t arises from the large net driving force dominated by the sustained ΔEd (Supplementary Fig. 21b).

Single-crystal Au foil with high-index facets has not been grown using AGG as it has been restrained by the significant pinning effect of thermal groove drag. This issue was addressed by using our strategy to improve the net driving force (Supplementary Fig. 22). As shown in Fig. 4a and d, the grooves at the grain boundaries of pristine Au foil featured small θ (e.g. 80.87°) upon annealing for 1 min, corresponding to a large retarding force. As a result, the foil only showed the slow normal grain growth after annealing for 2 h with further decreased θ and deepened grooves (Fig. 4b, Supplementary Fig. 22). Incorporating appreciable ΔEd then initiated the AGG with the retarding force being reduced by 44.1% on average (Fig. 4c, d, Supplementary Fig. 22 and Note 6). Consequently, a 15 mm sized single-crystal Au foil was prepared in 2 min (Fig. 4e). Its high-index facet and uniform single-crystal structure were confirmed by the IPZ and KAM maps (Fig. 4f). Statistical analysis of 68 samples identified the growth of 27 kinds of high-index single-crystal Au foils without low-index ones (Fig. 4g, h). Similarly, we realized the deterministic and efficient growth of single-crystal Ni foils with high-index facets (Supplementary Fig. 23). Their single-crystal structures were consistently verified by using multiple characterization techniques (Supplementary Fig. 24). Furthermore, we demonstrated that the use of high-index single-crystal Cu as the epitaxial substrate enabled the growth of high-quality graphene with the excellent electrical performance (Supplementary Fig. 25). FET measurement indicated a carrier mobility as high as 288,000 cm2·V−1·s−1 (at 1.7 K), which rivals that of the mechanically exfoliated graphene27.

Fig. 4: Growth of single-crystal Au foils with high-index facets enabled by the dislocation-driving strategy.
Fig. 4: Growth of single-crystal Au foils with high-index facets enabled by the dislocation-driving strategy.
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a–c Atomic force microscope (AFM) morphology of Au foil annealed for 1 min (a), 2 h (b), and for an additional 1 min after stretching the sample in (b) (2 h-s-1 min) (c). The angle and depth of grooves are marked in the image. d Statistical results of angle and depth of grooves in samples (a–c). Each error bar in (d) represents the standard deviation calculated from 51 experimental data. e Photograph of a single-crystal Au(012) foil. f Representative electron backscatter diffraction (EBSD) inverse pole figure in the normal direction (IPZ, top) and kernel average misorientation (KAM, bottom) maps of the single-crystal Au(012) foil. g Representative IPZ maps of single-crystal Au foils with diverse high-index facets. h IPZ contour of statistical facet indices of single-crystal Au foils obtained, demonstrating the deterministic growth of single crystals with diverse high-index facets.

In summary, taking Cu, Ni and Au as examples, we demonstrate the deterministic and highly efficient growth of single-crystal metal foils with high-index facets by using the dislocation-driving strategy. The distinctive combination of appreciable ΔEd with a strong (100) texture selectively drives the AGG of diverse high-index facets, thus overcoming the determinacy/diversity trade-off. The enhanced driving force reduces the groove drag to further improve the net driving force, which enables not only the preparation of high-index single-crystal Au foils by AGG but also the fast growth of the three typical single crystals. Furthermore, we demonstrate the high-index single-crystal Cu foil to be a robust epitaxial substrate for growing high-quality graphene. These findings provide thermodynamic and kinetic insights into the precise growth strategies of metastable structures, and promote the application of diverse high-index single-crystal metal foils in 2D epitaxy, catalysis and electronics.

Methods

Preparation of single-crystal high-index copper foil

In our experiments, Cu foils are divided into three types, namely Cud/t, Cut and Cun. The subscripts d and t represent ΔEd and (100) texture, respectively. The term Cud/t refers to the Cu foil combining the appreciable ΔEd and strong (100) texture. Similarly, the term Cut refers to the Cu foil with only strong (100) texture yet without large ΔEd, while the term Cun refers to the Cu foil without large ΔEd and strong (100) texture. For Cud/t foil (Cu-46365, 99.8%, 25 µm, Alfa Aesar), the treatment was performed by directly annealing the foil using a home-made tube CVD furnace equipped with push-pull rods to push the sample into high-temperature zone for ultrafast heating-up (Hefei Kejing Materials Technology Co., Ltd.). Specifically, the foil was loaded into the quartz tube outside the high-temperature zone of the furnace. The quartz tube was first purged by using 500 sccm Ar and 70 sccm H2 for more than 10 min prior to annealing. The Cud/t foil was then rapidly pushed into the high-temperature zone (1020 °C) and annealed for several minutes, followed by a rapid push-out to terminate the annealing.

Commercially available Cu foils with low average dislocation density (Cu-1031, 99.8%, 25 μm, Sichuan Oriental Stars Trading Co. Ltd; Cu-KS, 99.8%, 25 µm, Kunshan Luzhifa Electronic Technology Co., Ltd) were treated using two different processes: (1) For Cut foil (Cu-1031), the treatment was performed by pre-stretching the foil (1.5 cm × 3 cm) to obtain a 2%-4% plastic deformation (see Supplementary Fig. 26), followed by a short annealing for 1–2 min to obtain centimeter-sized high-index single-crystal grains. (2) For Cun foil (Cu-KS), the treatment was performed by short annealing the foil (1.5 cm × 3 cm) for 1 min to strengthen the (100) texture and then stretching to incorporate a large ΔEd, followed by the short annealing (1020 °C, 1–2 min).

Preparation of single-crystal high-index Au foil

Commercially available Au foils (99.99%, 50 µm, ZhongNuo Advanced Material Technology Co., Ltd, Au-YJ10J22) were first cut into rectangular foils of 2 cm × 4 cm and annealed at 1000 °C for 1–10 min to strengthen the (100) texture and then stretched, followed by the short annealing (1–2 min, 1020 °C) to obtain 1–2 cm high-index single-crystal foils.

Preparation of single-crystal high-index Ni foil

Commercially available Ni foils (99.994%, 100 µm, Alfa Aesar) show relatively strong (100) texture and large ΔEd, and thus pre-stretching is not required. The foils were first cut into rectangle foils of 2 cm × 4 cm and then directly annealed at 1020 °C for 1–2 min to obtain 1–2 cm single-crystal high-index Ni foils. Short annealing at 1000 °C for 20–30 s was also used to observe the annexation of (100) grains by high-index grains.

Growth and transfer of graphene

Monolayer graphene was grown on the single-crystal Cu foil by using the atmospheric pressure CVD. First, Cu foil was annealed at 1060 °C under a 500 sccm Ar flow for 1 min. Then, 500 sccm Ar, 20 sccm H2 and 0.7 sccm CH4 were introduced and held for 1–10 min. Finally, the growth was terminated by rapidly pushing the sample out of the heating zone. The as-grown graphene film was transferred onto the target substrate (such as SiO2/Si) using a PMMA-based process for characterization and device fabrication28. Typically, PMMA solution (PMMA, 950 K A4, MicroChem Corp.) was spin-coated onto one side of the graphene/Cu sample at 1000 rpm for 60 s, followed by drying on a hot plate at 80 °C for 3 min. Oxygen plasma was then used to etch the graphene on the backside of the sample. Subsequently, sodium persulfate (1 mol/L, Sigma-Aldrich) was employed to etch the Cu foil, and the PMMA/graphene film was floated on the solution surface. The PMMA/graphene film was rinsed with distilled water several times to remove residual etchant. After rinsing, the PMMA/graphene film was scooped out by a SiO2/Si substrate at room temperature and subsequently dried overnight to minimize trapped water between the graphene and the substrate. Finally, the PMMA was removed by soaking the sample in an acetone bath, and the sample was then dried with compressed nitrogen gas.

Device fabrication and electrical measurement

The transferred monolayer graphene was encapsulated with the relatively thick (~50 nm) hexagonal boron nitride (hBN) crystals by using a hot pick-up technique28,29. Specifically, an hBN flake was picked up at 52 °C using a PPC/polydimethylsiloxane (PDMS) stack on a glass slide, which was attached to a micromanipulator. The as-formed hBN/PPC/PDMS stack was then used to pick up the graphene from the SiO₂/Si substrate at 52 °C. This pick-up process is feasible because the van der Waals forces between hBN and graphene are relatively stronger than that between SiO₂/Si and graphene. Subsequently, the graphene/hBN/PPC/PDMS stack was brought into contact with another hBN flake by releasing the graphene/hBN from the PPC surface at 70 °C, resulting in the formation of the final hBN/graphene/hBN heterostructure. Hall bar geometry was patterned by Electron-beam lithography and reactive ion etching (RIE). Cr/Au (3/50 nm) electrodes were deposited by electron-beam evaporation for forming one-dimensional contacts. The electrical properties of the fabricated devices were characterized with the conventional lock-in technique. An AC current Ids with a root mean square amplitude of 100 nA at 12.666 Hz (SR 830) was applied between the source and drain terminals. Besides, the four-point longitude voltage drop Vxx and transverse voltage drop Vxy were measured with lock-in amplifiers. The charge density tuning in the graphene channel was achieved by applying different back gate voltages Vg. To eliminate the negative effect of oxygen and water in air on the device performance, the devices were tested in an inert helium environment (1.7 K, Attocube AttoDRY 2100 system). The longitude resistance Rxx was calculated from Rxx = Vxx/Ids, and Rxy (Hall resistance) was calculated from Rxy = Vxy/Ie. The ρxx (longitude resistivity) was calculated from ρxx = Rxx × W/L, where W is the width of the conducting channel, L is the length of the channel between the probed contacts. The longitude conductivity σxx was obtained via σxx = 1/ ρxx. The Hall carrier density n was determined by n = B/(eRxy), where B is the magnetic induction, e is the elementary charge. According to the Drude model, mobility (µ = σxx/(ne)) was estimated from the linear regions in the transfer curve.

Characterizations and mechanical treatment

The as-annealed Cu foils were heated on a hot plate at 180 °C (Chemat KW-4AH) for 2–5 min to preliminarily determine the single-crystal area based on the color contrast. The morphology of as-annealed Cu foils was characterized by OM (Nikon LV100D) and SEM (Verios G4 UC). EBSD (Oxford Instruments C-NANO,), XRD 2θ scan measurement (Bruker D8 Advance system with a silver target, Panalytical Empyrean system with a silver target) and XRD φ scan measurement (PANalytical Empyrean system with a copper target), HRTEM (FEI Talos F200X, under 200 kV) were used to characterize the crystal orientation. The angle θ and depth of thermal groove were measured by AFM (Bruker Multimode 8). Confocal Raman spectrometer (JY HR800, 532 nm laser) was used to characterize the structure and quality of graphene. The pre-stretching treatment of metal foils was performed using a universal testing machine (Tinius Olsen Testing Machine Co., Inc., 5ST Series). To enhance the credibility of sample dimension, the ruler-inclusive versions of Figs. 1c–f, 2a, 4e are presented in Supplementary Fig. 27.

Calculation of surface energy

To make a valid evaluation and comparison of the effect of surface energy, the crystallographic orientation dependence of surface energy was calculated in this work, which was performed through first-principle density functional theory (DFT) calculations using the Vienna Ab-initio Simulation Package (VASP)30,31. Electron exchange and correlation were treated at the general-gradient approximation (GGA) level using the Perdew-Burke-Ernzerhof (PBE) functional32,33. The electron-core interaction was described by the projector augmented-wave (PAW) potential34,35. The plane wave basis set kinetic cutoff energy was set to be 520 eV. All calculations were performed with the spin polarization. Each surface model was constructed by incorporating >10 atomic layers with a certain Cu facet index. And a vacuum layer of 15 Å was introduced in the unit cell to avoid the interaction between adjacent periodic structures. The convergence tolerances for geometrical optimization were set to be 10−8 eV for the total energy and 0.01 eV Å−1 for the forces acting on the ions. The Monkhorst–Pack schem36 was used to sample the Brillouin zones: k-points mesh of 9 × 9 × 9 for the bulk and k-points mesh of 6 × 6 × 1 for the surfaces, respectively.

Surface energy \({\gamma }_{{{{\rm{s}}}}}\) is defined as

$${\gamma }_{{{{\rm{s}}}}}=\frac{1}{2A}({E}_{{{{\rm{s}}}}}-N\times {E}_{{{{\rm{b}}}}})$$

where \({E}_{{{{\rm{b}}}}}\) represents the bulk energy of per copper atom, N the number of atoms in the surface model, \({E}_{{{{\rm{s}}}}}\) the corresponding total energy of the copper surface and A the total area of the surface considered.

Molecular dynamics simulations

Molecular dynamics (MD) simulations of the grain boundary migration of the Cu(014)/(100) bicrystal were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)37. Two models were established to elucidate the influence of dislocations on grain boundary migration: a ΔEs-driven model with negligible ΔEd between (014) and (100) grains, and a ΔEd -driven model where the (100) grain contained a higher density of dislocations. To ensure structural stability of the defective crystal at high simulation temperatures, the following strategies were employed: first, periodic boundary conditions were applied along the x and y axis, while a 40 Å vacuum layer was introduced along the z axis; second, dislocations were constructed by removing close-packed half atomic planes, with dislocation lines oriented perpendicular to the interface to facilitate the motion of dislocation; finally, the modified embedded-atom method (MEAM) potential38 was adopted, which better captured the high-temperature kinetic behavior. The obtained atomic structures were analyzed in the open visualization tool OVITO39, including the Dislocation Extraction Algorithm (DXA) and Common Neighbor Analysis (CNA) algorithms. CNA was used to identify the structural arrangement of each atom, while DXA converted the atomic model of the dislocated crystal into a line-based model of the dislocation network. Before the annealing process, the energy minimization simulation was performed on the initial bicrystal structure. Subsequently, the NVT ensemble (constant-temperature, constant-volume) was employed at 1300 K to relax the structure with a time step of 1 fs for a total of 10,000,000 steps, corresponding to a total duration of 10 ns.