Dynamical stability of two-dimensional metals in the periodic table

We study the dynamical stability of two-dimensional (2D) metals from Li to Pb in the periodic table. Using a first-principles approach, we calculate the phonon band structure of 2D metals that have planar hexagonal (HX), buckled honeycomb (bHC), and buckled square (bSQ) lattice structures. The bHC and bSQ are dynamically stable structures for most transition metals, whereas the HX and bHC are dynamically stable structures for alkali earth and noble metals. Thin films thicker than bHC and bSQ are dynamically stable for group 5 elements and indium. We demonstrate that the trend in the dynamical stability of 2D metals is correlated with that of three-dimensional (3D) metals. This provides design principles of ordered alloys: 2D metals are building blocks for constructing 3D alloys, where the similarity regarding the dynamical stability of different 2D metals is important for creating dynamically stable alloys.

Figure 1

(a) The total energy variation as a function of $a$ and $\delta$ and (b) top and side views for the bHC structures of Ag. [(c) and (d)] Same as (a) and (b), respectively, but for the bSQ structure. The unit cell is enclosed by red lines. The layer thickness is equal to $2\delta$.

Figure 2

The $Z$ dependence of (a) $E_j$, (b) $a_j$, and (c) ${\delta }_j$ for $j=$ HX, bHC, and bSQ. The figure $X(=2,3,4,5,\mathrm{and}6)$ indicates the period in the periodic table.

Figure 3

Periodic table for elemental 2D materials. The present results are colored. The stable 2D structures are indicated (The structure in parenthesis indicates the metastable phase). If the HX structure is dynamically stable, the bHC structure is also stable. Stable crystal structures in 3D materials are from Ref. [1]. The previously investigated structures that have dynamical stability include semiconductors of C [9], Si, Ge, and Sn [11], P [19], and As and Sb [20], and metals of Cu [12], Ag [13], Au [14], B [15, 16], Ga [17], Bi [21], and Po [22].

Figure 4

The cohesive energy difference between $E_{\mathrm{bHC}}$ and $E_{\mathrm{HX}}$ as a function of $E_{\mathrm{HX}}$. The calculated data for the elements that are dynamically stable in the HX structure are plotted by filled symbols.

Figure 5

The phonon band structures of group 5 metals of V, Nb, and Ta: (left) 3HX and (right) 3SQ structures.

Figure 6

Same as Fig. 5 but for group 13 metals of Ga, In, and Tl.

Figure 7

The phonon band structures of (a) bHC Co and (b) bHC Ni. Spin-polarized calculations are performed.

Figure 8

The phonon band structure of Fe, Co, and Ni that have bSQ structure. Left for non-spin-polarized and right for spin-polarized calculations.

Figure 9

Relationship of the dynamical stability between 2D structures (HX, bHC, and bSQ) and 3D structures (HCP, FCC, and BCC) in elemental metals. Stable crystal structures in 3D metals are extracted from the periodic table depicted in Ref. [1]. Hg, Mn, and In and Sn have rhombohedral, cubic, and tetragonal structures as its ground state, respectively. Fe (nsp), Fe (sp), Co (nsp), and Co (sp) have HCP, BCC, FCC, and HCP structures, respectively, as predicted by DFT calculations [37]. The stability of multilayered structures (3SQ and 3HX) is investigated if the HX, bHC, and bSQ structures are unstable. Note that 2D Ga becomes dynamically stable when strains are imposed [17].

Figure 10

Schematic illustration of (a) ${\mathrm{B}}_h$ and (b) $\text{L1}{}_0$ structures. The phonon band structure of (c) ${\mathrm{B}}_h$ and (d) $\text{L1}{}_0$ AlCu alloys.

Figure 11

The phonon band structure of group 1 elements (Li, Na, K, Rb, and Cs) for (a) HX, (b) bHC, and (c) bSQ structures.

Figure 12

Same as Fig. 11 but for group 2 elements (Be, Mg, Ca, Sr, and Ba).

Figure 13

Same as Fig. 11 but for group 3 elements (Sc, Y, and Lu).

Figure 14

Same as Fig. 11 but for group 4 elements (Ti, Zr, and Hf).

Figure 15

Same as Fig. 11 but for group 5 elements (V, Nb, and Ta).

Figure 16

Same as Fig. 11 but for group 6 elements (Cr, Mo, and W).

Figure 17

Same as Fig. 11 but for group 7 elements (Mn, Tc, and Re).

Figure 18

Same as Fig. 11 but for group 8 elements (Fe, Ru, and Os).

Figure 19

Same as Fig. 11 but for group 9 elements (Co, Rh, and Ir).

Figure 20

Same as Fig. 11 but for group 10 elements (Ni, Pd, and Pt).

Figure 21

Same as Fig. 11 but for group 11 elements (Cu, Ag, and Au).

Figure 22

Same as Fig. 11 but for group 12 elements (Zn, Cd, and Hg).

Figure 23

Same as Fig. 11 but for group 13 elements (Al, Ga, In, and Tl).

Figure 24

Same as Fig. 11 but for group 14 elements (Sn and Pb).

Figure 25

The time evolution of $\mathit{u}$ for an Al atom in a unit cell of ${\mathrm{B}}_h$ AlCu.

Figure 26

The phonon band structure of ${\mathrm{B}}_h$ CuZn.