Low-energy structures of K atoms in expanded ${\mathrm{K}}_3{\mathrm{C}}_{60}$ monolayers: Ab initio pseudopotential density-functional calculations

We have performed ab initio pseudopotential density-functional calculations to study spatial distributions of potassium atoms in expanded ${\mathrm{K}}_3{\mathrm{C}}_{60}$ monolayers and their effects on the conduction bands from the ${\mathrm{C}}_{60}$’s lowest unoccupied molecular orbitals (LUMOs). Our results show that, as the lattice constant $a$ of the monolayer increases, the lowest-energy configuration for the potassium atoms changes from a honeycomb $(a<10.8\mathrm{\AA })$ to a kagome lattice $(a>11.4\mathrm{\AA })$, with an intermediate phase appearing in between. The calculated electronic structures show that the ${\mathrm{C}}_{60}$-LUMO-derived conduction bands are deformed sensitively by the presence and location of the dopants in the intermediate and the kagome-lattice phases, deviating greatly from the rigid-band picture. The sensitivity to the dopants may provide a method for tailoring the ${\mathrm{C}}_{60}$-derived electronic structures.

Figure 1

Top views of the lowest-energy locations of potassium atoms in ${\mathrm{K}}_3{\mathrm{C}}_{60}$ monolayers for various lattice constants $a$: (a) a honeycomb-lattice structure of K atoms at $a=10\mathrm{\AA }$, (b) an intermediate structure at $a=11\mathrm{\AA }$, (c) a kagome-lattice structure at $a=12\mathrm{\AA }$, and (d) at $a=13\mathrm{\AA }$. (e) and (f) are schematic diagrams of honeycomb and kagome lattices, respectively. The three K atoms in a unit cell are labeled with K1 (on the top side), K2, and K3 (the other two on the bottom side). At $a=10\mathrm{\AA }$, the lateral positions of K1 and K3 are the same.

Figure 2

The lowest-energy positions of potassium atoms and the corresponding total energy of the ${\mathrm{K}}_3{\mathrm{C}}_{60}$ monolayers. (a) The $x$, (b) $y$, and (c) $z$ coordinates of the three K atoms in a unit cell for lattice constants from $10.0\text{to}12.0\mathrm{\AA }$. The $x$ and $y$ coordinates are their in-plane positions from a center of a fullerene, and the $z$ coordinates are their out-of-plane positions from the central plane. (d) The total energy of the ${\mathrm{K}}_3{\mathrm{C}}_{60}$ monolayers. The filled circles are the lowest total energy of the system when the K atoms are fully relaxed with no constraints. The empty circles represent the minimal total energy when the K atoms are relaxed only along the $z$ direction, while their in-plane positions are fixed in a perfect honeycomb lattice. Arrows indicate structural phase boundaries. K1, K2, and K3 are defined as in Fig. 1.

Figure 3

${\mathrm{C}}_{60}$-LUMO-derived conduction-band dispersions and the density of states (DOS) in the ${\mathrm{K}}_3{\mathrm{C}}_{60}$ monolayers and the undoped ${\mathrm{C}}_{60}$ monolayers for $a=10$, 11, and $12\mathrm{\AA }$, respectively. The unit of DOS is states/eV ${\mathrm{C}}_{60}$. For easy comparison, half-filling is assumed for the Fermi level of the undoped ${\mathrm{C}}_{60}$ monolayers.

Figure 4

(Color online) ${\mathrm{C}}_{60}$-LUMO-derived conduction-band properties of ${\mathrm{K}}_3{\mathrm{C}}_{60}$ monolayers. (a) Full and (b) occupied bandwidths of the conduction bands as a function of the lattice constant $a$. Filled and empty circles and empty boxes are bandwidths in the ${\mathrm{K}}_3{\mathrm{C}}_{60}$ monolayers, the constrained ${\mathrm{K}}_3{\mathrm{C}}_{60}$ monolayers [which are obtained as in Fig. 2d], and the undoped ${\mathrm{C}}_{60}$ monolayers, respectively. Half-filling of the conducton bands is assumed for the undoped ${\mathrm{C}}_{60}$ monolayers. (c) The density of states at the Fermi level $\left[\mathrm{DOS}\left(E_F\right)\right]$. Symbols are defined as in (a). (d) The electron density of the occupied conduction bands for the ${\mathrm{K}}_3{\mathrm{C}}_{60}$ monolayer of $a=12\mathrm{\AA }$ plotted on the central plane. In the plot, solid (red), dashed (green), thin dashed (blue), and thin dotted (magenta) lines represent ${10}^{-2.0}$, ${10}^{-2.5}$, ${10}^{-3.0}$, and ${10}^{-3.5}\text{electrons}∕{\mathrm{\AA }}^3$, respectively.