Magnetic moment and magnetic anisotropy of linear and zigzag $4d$ and $5d$ transition metal nanowires: First-principles calculations

An extensive ab initio study of the physical properties of both linear and zigzag atomic chains of all $4d$ and $5d$ transition metals (TMs) within the generalized gradient approximation by using the accurate projector-augmented wave method, has been carried out. The atomic structures of equilibrium and metastable states were theoretically determined. All the TM linear chains are found to be unstable against the corresponding zigzag structures. All the TM chains, except Nb, Ag, and La, have a stable (or metastable) magnetic state in either the linear or zigzag or both structures. Magnetic states appear also in the sufficiently stretched Nb and La linear chains and in the largely compressed Y and La chains. The spin magnetic moments in the Mo, Tc, Ru, Rh, W, Re chains could be large $\left(\ge 1.0{\mu }_B/\text{atom}\right)$. Structural transformation from the linear to zigzag chains could suppress the magnetism already in the linear chain, induce the magnetism in the zigzag structure, and also cause a change in the magnetic state (ferromagnetic to antiferromagnetic or vice verse). The calculations including the spin-orbit coupling reveal that the orbital moments in the Zr, Tc, Ru, Rh, Pd, Hf, Ta, W, Re, Os, Ir, and Pt chains could be rather large $\left(\ge 0.1{\mu }_B/\text{atom}\right)$. Importantly, large magnetic anisotropy energy $\left(\ge 1.0\text{meV}/\text{atom}\right)$ is found in most of the magnetic TM chains, suggesting that these nanowires could have fascinating applications in ultrahigh-density magnetic memories and hard disks. In particular, giant magnetic anisotropy energy $\left(\ge 10.0\text{meV}/\text{atom}\right)$ could appear in the Ru, Re, Rh, and Ir chains. Furthermore, the magnetic anisotropy energy in several elongated linear chains could be as large as 40.0 meV/atom. A spin-reorientation transition occurs in the Ru, Ir, Ta, Zr, La, Ta, and Ir linear chains when they are elongated. Remarkably, all the $5d$ as well as Tc and Pd chains show the colossal magnetic anisotropy (i.e., it is impossible to rotate magnetization into certain directions). Finally, the electronic band structure and density of states of the nanowires have also been calculated in order to understand the electronic origin of the large magnetic anisotropy and orbital magnetic moment as well as to estimate the conduction electron spin polarization.

Figure 1

(Color online) Schematic structure for (a) the linear and (b) zigzag atomic chains.

Figure 2

(Color online) (a) Equilibrium bond length (angstrom), (b) magnetization energy $\left(\Delta E\right)$ (i.e., the total energy of a magnetic state relative to that of nonmagnetic state) $\left(\Delta E=E^{\text{FM}\left(\text{AF}\right)}-E^{\text{NM}}\right)$, and (c) spin magnetic moments $\left({\mu }_B\right)$ of all the $4d$ TM linear atomic chains in the NM, FM, and AF states.

Figure 3

(Color online) (a) Equilibrium bond length (angstrom), (b) magnetization energy $\left(\Delta E\right)$ (i.e., the total energy of a magnetic state relative to that of nonmagnetic state) $\left(\Delta E=E^{\text{FM}\left(\text{AF}\right)}-E^{\text{NM}}\right)$, and (c) spin magnetic moments $\left({\mu }_B\right)$ of all the $5d$ TM linear atomic chains in the NM, FM, and AF states.

Figure 4

(Color online) Spin (left panels) and orbital (right panels) magnetic moments as a function of interatomic distance of the ferromagnetic Y, La, Zr, Hf, Nb, Ta, Os, Ir Pd, and Pt linear chains. In the left panels, “no-SOC” means the results from the scalar-relativistic calculations. “Para” (perp) denotes the magnetization being parallel (perpendicular) to the chain axis. The spin magnetic moment for the Pd chain goes to zero at $\sim 3.6\text{Å}$.

Figure 5

Band structures of the Zr (upper panels) and Ir (lower panels) linear chains at $2.6\text{Å}$. Left panels: the scalar-relativistic band structures; the middle and right panels: the fully relativistic band structures with the magnetization parallel to and perpendicular to the chain axis, respectively. In the left panels, the solid and dashed lines represent (spin-up) and (spin-down) bands, respectively. The Fermi level (the dotted horizontal line) is at the zero energy.

Figure 6

(Color online) Density of states of the FM $4d$ TM linear atomic chains at the equilibrium bond length. The Fermi level (dotted vertical lines) is at the zero energy.

Figure 7

(Color online) Density of states of the FM $5d$ TM linear atomic chains at the equilibrium bond length. The Fermi level (dotted vertical lines) is at the zero energy.

Figure 8

[(a)–(c)] Scalar-relativistic and [(d)–(f)] fully relativistic band structures of the Re, W, and Ir zigzag atomic chains in the FM state. In (d)–(f), the spin magnetization is along the chain direction (i.e., the $z$ axis). The Fermi level (the dotted horizontal line) is at the zero energy.

Figure 9

(Color online) The cohesive energy of the $4d$ TM zigzag chains in the NM, FM, and AF states. For comparison, the ground-state cohesive energy of the corresponding linear chains is also plotted (solid circles). The ground-state magnetic configuration of the linear chains is labeled as NM or FM or AF near each solid circle.

Figure 10

(Color online) The cohesive energy of the $5d$ TM zigzag chains in the NM, FM, and AF states. For comparison, the ground-state cohesive energy of the corresponding linear chains is also plotted (solid circles). The ground-state magnetic configuration of the linear chains is labeled as NM or FM or AF near each solid circle.

Figure 11

(Color online) Magnetocrystalline anisotropy energy $\left(E_1^e\right)$ of the selected $4d$ and $5d$ transition metal linear atomic chain as a function of interatomic distance. The upper panels contains the TM linear chain with larger MAE. A positive value of $E_1^e$ means that the magnetization would be parallel to the chain axis while a negative value would means that the easy magnetization axis would be perpendicular to the chain.