Structural, electronic, and magnetic properties of $3d$ transition metal monatomic chains: First-principles calculations

In this paper we investigated structural, electronic, and magnetic properties of $3d$ (light) transition metal atomic chains using first-principles pseudopotential plane-wave calculations. Infinite periodic linear, dimerized linear, and planar zigzag chain structures, as well as their short segments consisting of finite number of atoms have been considered. Like Cu, the periodic, linear chains of Mn, Co, and Ni correspond to a local shallow minimum. However, for most of the infinite periodic chains, neither linear nor dimerized linear structures are favored; to lower their energy the chains undergo a structural transformation to form planar zigzag and dimerized zigzag geometries. Dimerization in both infinite and finite chains is much stronger than the usual Peierls distortion and appears to depend on the number of $3d$ electrons. As a result of dimerization, a significant energy lowering occurs which, in turn, influences the stability and physical properties. Metallic linear chain of vanadium becomes half-metallic upon dimerization. Infinite linear chain of scandium also becomes half-metallic upon transformation to the zigzag structure. An interplay between the magnetic ground state and the atomic as well as the electronic structure of the chain has been revealed. The end effects influence the geometry, the energetics, and the magnetic ground state of the finite chains. Structure optimization performed using noncollinear approximation indicates significant differences from the collinear approximation. Variation of the cohesive energy of infinite- and finite-size chains with respect to the number of $3d$ electrons is found to mimic the well-known bulk behavior. The spin-orbit coupling of finite chains is found to be negligibly small.

Figure 1

(Color online) Various structures of $3d$ TM atomic chains. (a) Infinite and periodic structures; L—the infinite linear monatomic chain of TM atom with lattice constant $c$. LD—the dimerized linear monatomic chain with two TM atoms in the cell. $ϵ$ is the displacement of the second atom from the middle of the unit cell. ZZ—the planar zigzag monatomic chain with lattice parameter $c$ and unit cell having two TM atoms. $c_1\sim c_2$ and $59°<\alpha <62°$. ZZD—the dimerized zigzag structure $c_1\ne c_2$. WZ—the wide angle zigzag structure $c_1\sim c_2$, but with $\alpha >100°$. (b) Various chain structures of small segments consisting of finite number $\left(n\right)$ of TM atoms, denoted by ${\left(\text{TM}\right)}_n$.

Figure 2

(Color online) The energy versus lattice constant $c$ of various chain structures in different magnetic states. FM—ferromagnetic, AFM—antiferromagnetic, NM—nonmagnetic, FMD—ferromagnetic state in the linear or zigzag dimerized structure, and AFMD—antiferromagnetic state in the dimerized linear or zigzag structure. The energy is taken as the energy per unit cell relative to the free constituent atom energies in their ground state (see text for definition). In order to compare the energy of the L structure with that of the LD, the unit cell (and also lattice constant) of the former is doubled in the plot. Types of structures identified as L, LD, ZZ, ZZD, and WZ are described in Fig. 1.

Figure 3

(Color online) The plot of charge accumulation, namely, the positive part of the difference between the charge density of the interacting system and that of the noninteracting system for the linear (L) and the dimerized linear structure (LD) of Cr monatomic chains. The contour spacings are equal to $\Delta \rho =0.0827e/{\text{Å}}^3$. The outermost contour corresponds to $\Delta \rho =0.0827e/{\text{Å}}^3$. The dark balls indicate the Cr atom.

Figure 4

(Color online) Variation of the nearest neighbor distance of $3d$ TM atomic chains and the bulk structures. For the linear and zigzag structures the lowest energy configuration (i.e., symmetric or dimerized one) has been taken into account. The experimental values of the bulk nearest neighbor distances have been taken from Ref. 47.

Figure 5

(Color online) Variation of the cohesive energy $E_c$ (per atom) of $3d$ TM monatomic chains in their lowest energy linear, zigzag, and bulk structures. For the linear and zigzag structures the highest cohesive energy configuration (i.e., symmetric or dimerized one) has been taken into account. The experimental values of the bulk cohesive energies have been taken from Ref. 47.

Figure 6

(Color online) Variation of the binding energy difference $\Delta E$ (per atom) between the lowest antiferromagnetic and ferromagnetic states of $3d$ TM monatomic chains. The open squares and filled circles are for the symmetric zigzag ZZ and dimerized zigzag ZZD chains, respectively.

Figure 7

(Color online) Energy band structures of $3d$ TM atomic chains in their L and LD structures. The zero of energy is set at the Fermi level. The gray and black lines are the minority and majority bands, respectively. In the antiferromagnetic state majority and minority bands coincide. The energy gaps between the valence and the conduction bands are shaded.

Figure 8

(Color online) Energy band structures of $3d$ TM atomic chains in their ZZ and ZZD structures. The zero of energy is set at the Fermi level. The gray and black lines are minority and majority spin bands, respectively. The gray and dark lines coincide in the antiferromagnetic state. Only the dark lines describe the bands of nonmagnetic state. The energy gap between the valence and the conduction bands is shaded.

Figure 9

(Color online) Variation of the average cohesive energy of small segments of chains consisting of $n$ atoms. (a) The linear chains; (b) the zigzag chains. In the plot, the lowest energy configurations for each case obtained by optimization from different initial conditions.

Figure 10

(Color online) The atomic magnetic moments of some finite chains of $3d$ transition metal atoms. Numerals on the atomic sites stand for the value of the atomic magnetic moments. Positive and negative numerals are for spin-up and spin-down polarizations, respectively. Because of finite-size of the zigzag chains, the end effects are usually appear by different values of magnetic moments on atoms at the end of the chain.

Figure 11

(Color online) Atomic magnetic moments of ${\text{Ni}}_6$ and ${\text{Mn}}_6$ planar zigzag chains calculated by noncollinear approximation including spin-orbit interaction. The magnitudes and directions of magnetic moments are described by the length and direction of arrows at each atom.