Single-crystal two-dimensional material epitaxy on tailored non-single-crystal substrates

The use of single-crystal substrates as templates for the epitaxial growth of single-crystal overlayers has been a primary principle of materials epitaxy for more than 70 years. Here we report our finding that, though counterintuitive, single-crystal 2D materials can be epitaxially grown on twinned crystals. By establishing a geometric principle to describe 2D materials alignment on high-index surfaces, we show that 2D material islands grown on the two sides of a twin boundary can be well aligned. To validate this prediction, wafer-scale Cu foils with abundant twin boundaries were synthesized, and on the surfaces of these polycrystalline Cu foils, we have successfully grown wafer-scale single-crystal graphene and hexagonal boron nitride films. In addition, to greatly increasing the availability of large area high-quality 2D single crystals, our discovery also extends the fundamental understanding of materials epitaxy.


Contents
Supplementary Figure 10 The characterization of merging graphene islands.
Supplementary Figure 11 Characterization of the seamless coalescence of aligned graphene islands and following single-crystal graphene film.
Supplementary Figure 13 Graphene carrier mobility characterizations using FETs devices.
Supplementary Figure 14 The in-situ SEM observation of graphene growth on twinned copper substrate.
Supplementary Figure 15 Characterization of graphene grown near a twin boundary.
Supplementary Table 1 Experimentally obtained ( ∆ ) and theoretically predicted ( ∆ ) misalignment angles of graphene islands on various twinned Cu foils. Supplementary In order to calculate the surface index of a Cu grain in the Cu foil, we chose three arbitrary points of the Cu grain on the surface of foil, whose coordinates in the coordinate system of the single-crystal Cu grain are A (X1, Y1, Z1), B (X2, Y2, Z2), and C (X3, Y3, Z3). The normal direction <hkl> can be obtained by ⃑ × ⃑ .
Therefore, the surface index of a Cu grain is determined based on the three Euler angles.

Definition of θ and ψ of a twin in the Cu foil
A twin in a Cu foil has three degrees of freedom. The rotation along the axis normal to the surface of the foil is trivial and the other two degrees of freedom are described by θ and here. θ denotes the angle between the Cu foil surface and the twin boundary plane, and denotes the rotation angle of the twin crystal around the <1 1 1> co-axis of the two single crystals. In

Interfacial formation energies of various graphene edges attached to different types of Cu step edges
The orientation of a graphene island grown on a Cu substrate is determined by its interaction with the substrate, which includes the weak van der Waals interaction between the graphene bulk and substrate, and the strong graphene edge-metal surface interaction. During nucleation or when the graphene island is very small, graphene edge-metal surface interaction is the dominant interaction because of the strong chemical bonding between the edge carbon atoms and the metal surface, and the large edge to area ratio of the graphene island. On a metal surface with step edges, such as high index surfaces, graphene tends to nucleate near a step edge and hence, the graphene edge-metal step-edge interaction critically determines the alignment of graphene island on the metal surface. Here, we define the interfacial formation energy of a graphene edge attached to a Cu step edge as: where E is the formation energy of a freestanding graphene edge and E is the bonding energy of a graphene edge attached to a step edge of the Cu substrate, defined by: Here, E is the total energy of the system (graphene on the Cu substrate), and E and E are the energies of the freestanding Cu substrate and graphene, respectively.
In this work, we compared the interfacial formation energies of various graphene edges attached to different Cu step edges. Using the graphene zigzag edge (0˚ in the map) and Cu <110> (0˚ in the map), respectively, as reference directions to indicate the directions of graphene edges and Cu step edges, each configuration can be uniquely described by the angle of the graphene edge and that of the metal step edge. The longest <110> step edge segment is the direction that determines the zigzag edge orientation of the graphene islands. In order to calculate the direction of the longest (dominant) <110> step edge segment of an arbitrary Cu surface, we chose a single-crystal Cu foil with an arbitrary orientation that can be measured by the three Euler angles. By comparing the angles between the six <110> axes of the face-centered cubic (FCC) Cu and the Cu foil surface, the <110> axis with the smallest angle and its projected direction on the surface, or the direction of the longest <110> step edge segment of an arbitrary Cu surface, can be obtained.
The projected direction of a <110> axis of a Cu grain (denoted by ⃑ ) on the Cu foil can be obtained by calculating the projections of two points of the axis, represented by A (a1, b1, c1) and B (a2, b2, c2), on the Cu foil surface.
The plane equation of the Cu foil surface in the single-crystal coordinate system is defined as: hx + ky + lz + d = 0. Then the projected direction ⃑ can be calculated as: Therefore, the included angle α between a <110> axis and the Cu foil surface can be calculated as:

Calculating the misalignment angle between graphene islands grown on both sides of a twin boundary
Since a graphene zigzag edge prefers to attach to the longest <110> step edge segment of the Cu surface and therefore, the misalignment angle between two graphene islands grown on both sides of a twin boundary is the same as the angle between the two longest <110> step edge segments on both sides of the twin boundary. Here, the theoretical misalignment angles between graphene islands grown on both sides of the twin boundary, ∆ , are calculated. Comparing with experimentally measured values, ∆ , we conclude that our theoretical predictions are in very good agreement with experimental measurements, as listed in Supplementary Table 1  boundary. c-f, Optical images of graphene domains grown on Cu substrates with different twin densities (Scale bar is 100 μm). The nucleation density of graphene near twin boundaries is same as that on terrace.