Atlas for the properties of elemental two-dimensional metals

Common two-dimensional (2D) materials have a layered three-dimensional (3D) structure with covalently bonded, atomically thin layers held together by weak van der Waals forces. However, in a recent transmission electron microscopy experiment, atomically thin 2D patches of iron were discovered inside a graphene nanopore. Motivated by this discovery, we perform a systematic density-functional study on atomically thin elemental 2D metal films, using 45 metals in three lattice structures. Cohesive energies, equilibrium distances, and bulk moduli in 2D are found to be linearly correlated to the corresponding 3D bulk properties, enabling the quick estimation of these values for a given 2D metal and lattice structure. In-plane elastic constants show that most 2D metals are stable in hexagonal and honeycomb, but unstable in square 2D structures. Many 2D metals are surprisingly stable against bending.

Figure 1

Part of the periodic table with the studied metals. Chemical symbol and proton number are given for each element.

Figure 2

Schematic representations of the 2D lattices. Solid line indicates the computational unit cells and dashed lines the Wigner-Seitz cells.

Figure 3

Calculated cohesive energies of 3D bulk, and hexagonal (hex), square (sq), and honeycomb (hc) 2D lattices as a function of proton number (a) and experimental 3D bulk cohesion (b). Lines show linear fits between experimental and calculated values. Experimental cohesive energies are from Ref. [48].

Figure 4

Calculated bond lengths for hexagonal (hex), square (sq), and honeycomb (hc) 2D structures as a function of proton number (a) and experimental 3D bulk bond length (b). Lines show linear fits between calculated and experimental values [48].

Figure 5

Calculated 2D bulk moduli for hexagonal (hex), square (sq), and honeycomb (hc) structures, as a function of the proton number (a) and experimental bulk moduli (b). Lines display the linear fits between calculated and experimental values [48].

Figure 6

Summary of calculated results for hexagonal 2D structures.

Figure 7

Elastic constants for hexagonal, square, and honeycomb 2D structures.

Figure 8

Determinant of the Hessian matrix of the energy density $U$. Triangles, squares, and hexagons correspond to hexagonal, square, and honeycomb structures, respectively. Logarithmic scale is used for clarity.

Figure 9

Schematic representation of the structures used in the calculation of bending moduli.

Figure 10

Calculated bending moduli for the hexagonal (hex) and square (sq) structures. Arrows indicate negative bending moduli and stars undetermined values for hexagonal (red) and square (blue) 2D structures.

Figure 11

The second moments of electron density in the direction perpendicular to the atomic plane of the 2D structures.

Figure 12

The electron density of Au in the atomic plane, for hexagonal (a), square (b), and honeycomb (c) latices, in atomic units. Black star shows the position of saddle point and white star shows the location of a minimum point.

Figure 13

The ratio between minimum electron density and the saddle-point density at the atomic plane, for hexagonal (hex), square (sq), and honeycomb (hc) lattices.

Figure 14

Density of states for different 2D Au structures.

Figure 15

Density of states at Fermi energy for all studied metals and 2D structures.