Unveiling hidden charge density waves in single-layer ${\mathrm{NbSe}}_2$ by impurities

We employ ab initio calculations to investigate the charge density waves in single-layer ${\mathrm{NbSe}}_2$, and we explore how they are affected by transition metal atoms. Our calculations reproduce the observed orthorhombic phase in single-layer ${\mathrm{NbSe}}_2$ in the clean limit, establishing the energy order between three different distorted structures, two consisting of triangular Nb-Nb clusters and a third, energetically unfavored, consisting of hexagonal Nb-Nb clusters. Such energy order, in agreement with known experimental work, is reversed by the adsorption of Co and Mn, which favor the formation of hexagonal Nb-Nb clusters; this CDW structure is indeed allowed from a symmetry point of view but hidden in pure single layers because it is at a higher energy. The other adsorbates, K and Ga, still favor one of the triangular Nb-Nb cluster, while suppressing the other. We report how the energy difference between such distorted structure varies with these adsorbates. Furthermore, transition metals induce magnetism and favor the reduction of the symmetry of the charge density distribution.

Figure 1

First row: relaxed CDW structures for pristine ${\mathrm{NbSe}}_2$ (top view): T-U (a), T-C (b), and HX (c) CDWs (see text for details). Second row: charge densities of the respective CDW structures. Volumetric data are represented as blue surfaces enclosing points whose electronic density is greater than or equal to 0.0075 electrons/${\mathrm{Bohr}}^3$. Atoms are represented by spheres, as illustrated in the legends (magnified); Nb-Nb bonds shorter than the equilibrium distance (3.45 Å) are represented by solid lines, in order to help visualizing the CDW structure pattern. Dashed lines mark the supercells borders.

Figure 2

Differences between the Fourier transform (FT) of the charge density distributions for the pristine CDW structures and the nonmodulated structure. The plots of T-U (a), T-C (b), and HX (c) CDWs charge density correspond to those in Fig. 1. The large dashed hexagon with vertices at $\left|\mathbf{q}\right|=\frac{2\pi }{a}$ (with $a=3.45\AA$) marks the characteristic Bragg peaks; the medium size dashed hexagon with vertices at $\left|\mathbf{q}\right|=\frac{2\pi }{3a}$ marks the (shorter) CDW peaks; the small green shaded hexagon ($\left|\mathbf{q}\right|=\frac{2\pi }{6a}$) maps points beyond the supercell borders.

Figure 3

First row: ground state $\mathrm{Co}|{\mathrm{NbSe}}_2$ HX-S (a), HX-A (b), and T-U (c) CDW structure. Second row: charge densities of the respective CDW structures. The isosurface value for the volumetric data is set in agreement with Fig. 1. Atoms are represented by spheres, as illustrated in the legends; Nb-Nb bonds shorter than the equilibrium distance (3.45 Å) are represented by solid lines, to help visualizing the CDW structure pattern. Dashed lines mark the supercells borders.

Figure 4

Magnetization densities of $\mathrm{Co}|{\mathrm{NbSe}}_2$ HX-S (a), $\mathrm{Co}|{\mathrm{NbSe}}_2$ HX-A (b), $\mathrm{Co}|{\mathrm{NbSe}}_2$ T-U at 1 (c), $\mathrm{Co}|{\mathrm{NbSe}}_2$ T-U at 3 (d), $\mathrm{Mn}|{\mathrm{NbSe}}_2$ HX-A (e), and $\mathrm{Mn}|{\mathrm{NbSe}}_2$ T-U at 1 (f). The magnetization densities in (c) and (d) are different although they have virtually the same structure and total energy; the magnetization in (c) is juxtaposed to (f) to compare structures with the same adsorption site. Volumetric data are represented as blue (red) surfaces enclosing points whose magnetization density is greater than or equal to 0.0075 (smaller than or equal to $-0.0075$) ${\mu }_B/{\mathrm{Bohr}}^3$. Atoms are represented by spheres, as illustrated in the legends; Nb-Nb bonds shorter than the equilibrium distance (3.45 Å) are represented by solid lines to help visualizing the CDW structure pattern. Dashed lines mark the supercells borders.

Figure 5

Differences between charge density distributions. $\mathrm{Co}|{\mathrm{NbSe}}_2$ T-U minus ${\mathrm{NbSe}}_2$ T-U (a), $\mathrm{Co}|{\mathrm{NbSe}}_2$ HX-A minus ${\mathrm{NbSe}}_2$ T-U (b), and $\mathrm{Mn}|{\mathrm{NbSe}}_2$ HX-A minus ${\mathrm{NbSe}}_2$ HX (c). Volumetric data are represented as blue (red) surfaces enclosing points whose electronic density is greater than or equal to 0.0025 (smaller than or equal to $-0.0025$) electrons/${\mathrm{Bohr}}^3$. Atoms are represented by spheres, as illustrated in the legends; Nb-Nb bonds shorter than the equilibrium distance (3.45 Å) are represented by solid lines to help visualizing the CDW structure pattern. Dashed lines mark the supercells borders.

Figure 6

Differences between FT of the charge density distribution of Co-adsorbed and Mn-adsorbed ${\mathrm{NbSe}}_2$ CDWs and the T-U CDW in pristine ${\mathrm{NbSe}}_2$. Ground state $\mathrm{Co}|{\mathrm{NbSe}}_2$ HX-A (a), $\mathrm{Co}|{\mathrm{NbSe}}_2$ HX-S (b), and $\mathrm{Mn}|{\mathrm{NbSe}}_2$ HX-A (c). The dashed hexagons mark the same regions as in Fig. 2 (see caption).

Figure 7

First row: ground state CDW structure, with K adsorbed on the hollow position (a), its charge density distribution (b), and the charge density for K adsorbed on the Nb site (c). Second row: Difference in the charge densities between the $\mathrm{K}|{\mathrm{NbSe}}_2$ ground state and ${\mathrm{NbSe}}_2$ T-U (d), ground state CDW structures for two $\mathrm{Ga}|{\mathrm{NbSe}}_2$, namely T-U (e) and HX-A (f). Atoms are represented by spheres, as illustrated in the legends; Nb-Nb bonds shorter than the equilibrium distance (3.45 Å) are represented by solid lines to help visualizing the CDW structure pattern. Dashed lines mark the supercells borders. The isosurface value for the volumetric data is set in agreement with Fig. 1.

Figure 8

Difference in the FT of the charge density distributions between the $\mathrm{K}|{\mathrm{NbSe}}_2$ on the hollow site (a), Nb site (b), and the T-U pristine CDW. The charge distribution has a $C_2$ symmetry for the adsorption on the hollow site and a $C_6$ symmetry for the adsorption on the Nb site. In (c), the FT difference between $\mathrm{K}|{\mathrm{NbSe}}_2$ on the hollow site and the pristine T-C CDW is shown.

Figure 9

Relaxed structures of the pristine CDWs as they appear in Fig. 1. 1 in the main text: T-U (a), T-C (b), and HX (c). Atoms are represented by spheres, as illustrated in the legends; Se-Se bonds shorter than the equilibrium distance (3.45 Å) are represented by solid lines, in order to help visualizing the CDW structure pattern. Dashed lines mark the supercells borders.

Figure 10

Total DOS of ${\mathrm{NbSe}}_2$ with the fully symmetric structure and with the HX, the T-C, and the T-U CDW structures, obtained for a $3\times 3\times 1$ supercell.

Figure 11

Fourier transform (FT) plots of the charge density distributions for the pristine T-U (a), T-C (b), and HX (c) CDW structures, as given in Fig. 1. The dashed hexagons mark the same regions as in Fig. 2 (see caption).

Figure 12

Relaxed structures of a variant of the HX CDW in a $9\times 9\times 1$ supercell, highlighting its Nb-Nb (a) and Se-Se (b) distance pattern. Atoms are represented by spheres, as illustrated in the legends; in (a) and (b), respectively, Nb-Nb or Se-Se bonds shorter than the equilibrium distance (3.45 Å) are represented by solid lines, in order to help visualizing the CDW structure pattern. Dashed lines mark the supercells borders.

Figure 13

Total DOS of $\mathrm{Co}|{\mathrm{NbSe}}_2$ and $\mathrm{K}|{\mathrm{NbSe}}_2$ with symmetric structures and with their CDW structures.

Figure 14

Differences of the FT of the charge density distributions, done with respect to pristine T-U, of Co-adsorbed (a), Mn-adsorbed (b), T-U, and Mn-adsorbed (c) HX-S.

Figure 15

Relaxed structures of $\mathrm{Co}|{\mathrm{NbSe}}_2$ in a $9\times 9\times 1$ supercell. Atoms are represented by spheres, as illustrated in the legends; Nb-Nb bonds shorter than the equilibrium distance (3.45 Å) are represented by solid lines, in order to help visualizing the CDW structure pattern. Dashed lines mark the supercells borders.

Figure 16

Direct space charge density distribution of $\mathrm{Co}|{\mathrm{NbSe}}_2$ in a $9\times 9\times 1$ supercell; (a) and (b) correspond to the respective structures in Fig. 15. The isosurface value for the volumetric data is set in agreement with the relative figures in the main text. Atoms are represented by spheres, as illustrated in the legends; Nb-Nb bonds shorter than the equilibrium distance (3.45 Å) are represented by solid lines, in order to help visualizing the CDW structure pattern. Dashed lines mark the supercells borders.

Figure 17

Magnetization densities of $\mathrm{Co}|{\mathrm{NbSe}}_2$ T-U (a) and $\mathrm{Mn}|{\mathrm{NbSe}}_2$ T-U (b) with details around Co and Mn, respectively. The sets of axes represent the unitary lattice vectors in red, green, and blue color, respectively. Volumetric data representation is set in agreement to the relative figures in the main text.