Role of spin-orbit coupling in spin-spiral structures in Fe monolayer on W(110): A first-principles noncollinear magnetism study

Kohji Nakamura; Toru Akiyama; Tomonori Ito; A. J. Freeman

doi:10.1063/1.3070635

Role of spin-orbit coupling in spin-spiral structures in Fe monolayer on W(110): A first-principles noncollinear magnetism study

Kohji Nakamura, Toru Akiyama, Tomonori Ito, A. J. Freeman

Northwestern University

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8 Citations (Scopus)

Abstract

The stability of spin-spiral structures in an Fe monolayer on a W(110) substrate is investigated by means of the first-principles film full-potential linearized augmented plane-wave method, and the role of spin-orbit coupling (SOC) on the spin-spiral structures is determined. Our calculations demonstrate that without SOC, the spin-spiral structures are energetically favored over the ferromagnetic (FM) state, but that when the strong SOC at the Fe/W(110) interface is introduced, the formation of the spin-spiral structures is suppressed. Thus, the ground state of the system appears to be the FM state-as observed in experiments.

Original language	English
Article number	07C304
Journal	Journal of Applied Physics
Volume	105
Issue number	7
DOIs	https://doi.org/10.1063/1.3070635
Publication status	Published - 2009

ASJC Scopus subject areas

Physics and Astronomy(all)

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@article{e2592dd5f4fa42769254af813d43b6b5,

title = "Role of spin-orbit coupling in spin-spiral structures in Fe monolayer on W(110): A first-principles noncollinear magnetism study",

abstract = "The stability of spin-spiral structures in an Fe monolayer on a W(110) substrate is investigated by means of the first-principles film full-potential linearized augmented plane-wave method, and the role of spin-orbit coupling (SOC) on the spin-spiral structures is determined. Our calculations demonstrate that without SOC, the spin-spiral structures are energetically favored over the ferromagnetic (FM) state, but that when the strong SOC at the Fe/W(110) interface is introduced, the formation of the spin-spiral structures is suppressed. Thus, the ground state of the system appears to be the FM state-as observed in experiments.",

author = "Kohji Nakamura and Toru Akiyama and Tomonori Ito and Freeman, {A. J.}",

note = "Funding Information: Nakamura Kohji 1 a) Akiyama Toru 1 Ito Tomonori 1 Freeman A. J. 2 1 Department of Physics Engineering, Mie University , Tsu, Mie 514-8507, Japan 2 Department of Physics and Astronomy, Northwestern University , Evanston, Illinois 60208, USA a) Email address: kohji@phen.mie-u.ac.jp . 01 04 2009 105 7 07C304 11 11 2008 08 09 2008 16 11 2008 19 02 2009 2009-02-19T12:28:36 2009 American Institute of Physics 0021-8979/2009/105(7)/07C304/3/ $25.00 The stability of spin-spiral structures in an Fe monolayer on a W(110) substrate is investigated by means of the first-principles film full-potential linearized augmented plane-wave method, and the role of spin-orbit coupling (SOC) on the spin-spiral structures is determined. Our calculations demonstrate that without SOC, the spin-spiral structures are energetically favored over the ferromagnetic (FM) state, but that when the strong SOC at the Fe/W(110) interface is introduced, the formation of the spin-spiral structures is suppressed. Thus, the ground state of the system appears to be the FM state—as observed in experiments. 20540334 DE-FG02-88ER45372 PROCEEDINGS OF THE 53RD ANNUAL CONFERENCE ON MAGNETISM AND MAGNETIC MATERIALS Austin, Texas (USA) 10-14 November 2008 Noncollinear magnetism in helical spin-density wave structures or spin-spiral structures, in which the magnetization is rotated along a certain direction in a crystal, has received much attention in fundamental and applied physics. Now, interest in the noncollinear magnetism at surfaces and interfaces has increased because of their potential technological importance, 1–3 in which a breaking of symmetry and an enhanced SOC arising from the reduced dimensionality would give rise to new and exotic features differing from those of bulk. In order to investigate such complex noncollinear magnetism, we have previously implemented the film full-potential linearized augmented plane-wave (FLAPW) method 4,5 including intra-atomic noncollinear magnetism, 6–8 which applies the generalized Bloch theorem 9,10 to treat spin-spiral structures. Moreover, we determined spin-spiral structures in an Fe monolayer on a W(110) substrate based on the local spin-density approximation (LSDA), and found that the spin-spiral structures are energetically favorable over the ferromagnetic (FM) state by only about 1 meV/atom. 8 It is therefore of interest to discuss the effect of spin-orbit coupling (SOC) on the spin-spiral structures since the strong SOC at the W substrate interface plays a key role in determining the magnetism in Fe/W(110) and so to determine the easy and hard magnetization axes. 8,11,12 However, since the spin-spiral order breaks the spatial translation symmetry in a crystalline solid through the SOC, the generalized Bloch theorem is no longer applicable for describing the spin-spiral states. Therefore, in the present work, we implement the FLAPW method in the second variation way 2,13,14 to treat large unit cells for the spin-spiral structures. The calculations demonstrate that when the SOC is introduced, the formation of the spin-spiral structures is suppressed due to the strong SOC at the Fe/W(110) interface, and so the system is found to be the FM state—as observed in experiments. 15 As a model of the Fe monolayer on the W(110) substrate, we employ a single slab consisting of five layers of W(110) and an Fe(110) monolayer on each side, and assume the experimental values of the lattice constant of bulk W ( a = 3.16 {\AA} ) and of the Fe–W interlayer distance ( d Fe – W = 1.94 {\AA} ) . 16 The magnetization in the spin-spiral structures is assumed to lie on a plane parallel to the surface since the magnetic easy axis is along the in-plane [ 1 1 ¯ 0 ] direction with a large magnetocrystalline anisotropy energy of 3 meV/atom and the hard axis along the out-of-plane [110] direction. Calculations are first carried out in the scalar relativistic approximation (SRA) for the valence electrons, i.e., without the SOC, and fully relativistically for the core electrons, based on the LSDA using the von Barth–Hedin exchange correlation. 17 In order to treat spin-spiral structures, here, we apply the generalized Bloch theorem 9,10 into the film-FLAPW method 4,5 with intra-atomic noncollinear magnetism, 6,7 in which the eigenstate at k in the first Brillouin zone (BZ) defined by a chemical unit cell is in the form; Ψ k SRA ( r ) = ∑ G ( C k + G − q / 2 , χ ↑ e − i q / 2 ψ k + G ( r ) C k + G + q / 2 , χ ↓ e + i q / 2 ψ k + G ( r ) ) , (1) where q is a wave vector of the spin-spiral structure, and e ∓ q / 2 ψ k + G ( r ) is described by the spin-independent LAPW basis. 7 The LAPW basis with a cutoff of | k + G ∓ q / 2 | ≤ 3.6 a .u . − 1 and muffin-tin (MT) sphere radii of 2.3 a.u. for the Fe and 2.43 a.u. for the W is used; lattice harmonics with angular momenta up to ℓ = 8 are employed to expand the charge and magnetization density, the vector potential, and eigenvectors. Next, the SOC is incorporated by the second variational method 2,14 employing the SRA eigenvectors. Since the spin-spiral order breaks the spatial translation symmetry in a crystalline solid through the SOC, the generalized Bloch theorem is no longer applicable for describing the spin-spiral states. Therefore, we newly implement the FLAPW method in the second variation way to treat large unit cells (supercells) with lattice constants corresponding to wavelengths of the spin-spiral structures, where commensurate spin-spiral orders are assumed. The eigenstate in the second variation at k s in the BZ redefined by the supercell is now specified by using the SRA eigenvectors as Ψ k s 2 nd ( r ) = ∑ n ∑ j d n , j Ψ k s + n q , j SRA ( r ) . (2) Here n is the number of the k points, those are connected by the q in the BZ of the chemical unit cell, and j represents states in the SRA eigenvectors. The Hamiltonian of the SOC, H soc , is approximated by using the nonmagnetic part of the spherical potential in the MT spheres of the chemical unit cell. The formation energy for the spin-spiral structures with and without SOC, calculated as the energy difference with respect to their FM states, as a function of the q 001 in the [001] direction, is shown in Fig. 1 , where q 001 = 0 indicates the FM state. The difference between the energies with and without SOC is shown in Fig. 2 . As presented previously, 8 without SOC, the spin-spiral structures are energetically favorable over the FM state by only about 1 meV/atom, where the energy minimum appears to be around q 001 = 0.05 ∼ 0.1 , corresponding to a wavelength of about 10 a ∼ 20 a . However, when the SOC is introduced, the formation energy of the spin-spiral states is no longer lower than that of the FM one. Moreover, we carried out the second variational calculations that artificially reduce the SOC contribution from the W substrate, and found that the formation energy does not change over that in the SRA (without SOC) results. This indicates that the magnetization rotation in the spin-spiral structures, which changes the orientation from the easy axis, increases the total energy due to the strong SOC at the W substrate interface. Thus, the formation of the spin-spiral structures is suppressed due to the SOC and the system appears to be the FM state—as observed in experiments. 15 Figure 3 shows the magnitude of the spin and orbital magnetic moments integrated in the MT spheres for the spin-spiral structure with q = 0.1 as a function of atomic position along the [001] direction. For the whole atomic position, the spin moments have almost the same value of 2.36 μ B . In contrast, and interestingly, the orbital moments periodically change with a large variation of the 0.03 μ B . It is noted that the orbital moments at site 1 in Fig. 3 , where the spins orient along the easy axis of the [ 1 1 ¯ 0 ] direction, have a maximum value ( 0.17 μ B ) , while those at site 2, where the spins orient along the [001] direction, have a minimum value ( 0.14 μ B ) . Thus, the orbital moments are strongly sensitive to the atomic position in the spin-spiral structures. In conclusion, the stability of the spin-spiral structures in the Fe monolayer on the W(110) substrate was investigated by means of the FLAPW method where we newly implemented the FLAPW method in the second variation way to treat large unit cells for the spin-spiral structures. The calculations demonstrated that the formation of the spin-spiral structures are energetically suppressed due to the strong SOC at the Fe/W(110) interface, and so the ground state of the system appears to be the FM state. Work at Mie University was supported by a Grant-in-Aid for Scientific Research (Grant No. 20540334) from the Japan Society for the Promotion of Science, and for computations performed at the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. Work at Northwestern University was supported by the U.S. DOE Grant No. DE-FG02-88ER45372. FIG. 1. Calculated formation energy of spin-spiral structures as a function of wave vector, q 001 , in the [001] direction for Fe/W(110). Open and closed circles represent results with and without SOC. FIG. 2. Difference between energies with and without SOC as a function of wave vector, q 001 , in the [001] direction for Fe/W(110). FIG. 3. Calculated spin and orbital magnetic moments integrated in Fe MT spheres as a function of atomic position along [001] direction for spin-spiral structure with q 001 = 0.1 of Fe/W(110). Open and closed circles represent the spin and orbital moments, respectively. ",

year = "2009",

doi = "10.1063/1.3070635",

language = "English",

volume = "105",

journal = "Journal of Applied Physics",

issn = "0021-8979",

publisher = "American Institute of Physics Publising LLC",

number = "7",

}

TY - JOUR

T1 - Role of spin-orbit coupling in spin-spiral structures in Fe monolayer on W(110)

T2 - A first-principles noncollinear magnetism study

AU - Nakamura, Kohji

AU - Akiyama, Toru

AU - Ito, Tomonori

AU - Freeman, A. J.

N1 - Funding Information: Nakamura Kohji 1 a) Akiyama Toru 1 Ito Tomonori 1 Freeman A. J. 2 1 Department of Physics Engineering, Mie University , Tsu, Mie 514-8507, Japan 2 Department of Physics and Astronomy, Northwestern University , Evanston, Illinois 60208, USA a) Email address: kohji@phen.mie-u.ac.jp . 01 04 2009 105 7 07C304 11 11 2008 08 09 2008 16 11 2008 19 02 2009 2009-02-19T12:28:36 2009 American Institute of Physics 0021-8979/2009/105(7)/07C304/3/ $25.00 The stability of spin-spiral structures in an Fe monolayer on a W(110) substrate is investigated by means of the first-principles film full-potential linearized augmented plane-wave method, and the role of spin-orbit coupling (SOC) on the spin-spiral structures is determined. Our calculations demonstrate that without SOC, the spin-spiral structures are energetically favored over the ferromagnetic (FM) state, but that when the strong SOC at the Fe/W(110) interface is introduced, the formation of the spin-spiral structures is suppressed. Thus, the ground state of the system appears to be the FM state—as observed in experiments. 20540334 DE-FG02-88ER45372 PROCEEDINGS OF THE 53RD ANNUAL CONFERENCE ON MAGNETISM AND MAGNETIC MATERIALS Austin, Texas (USA) 10-14 November 2008 Noncollinear magnetism in helical spin-density wave structures or spin-spiral structures, in which the magnetization is rotated along a certain direction in a crystal, has received much attention in fundamental and applied physics. Now, interest in the noncollinear magnetism at surfaces and interfaces has increased because of their potential technological importance, 1–3 in which a breaking of symmetry and an enhanced SOC arising from the reduced dimensionality would give rise to new and exotic features differing from those of bulk. In order to investigate such complex noncollinear magnetism, we have previously implemented the film full-potential linearized augmented plane-wave (FLAPW) method 4,5 including intra-atomic noncollinear magnetism, 6–8 which applies the generalized Bloch theorem 9,10 to treat spin-spiral structures. Moreover, we determined spin-spiral structures in an Fe monolayer on a W(110) substrate based on the local spin-density approximation (LSDA), and found that the spin-spiral structures are energetically favorable over the ferromagnetic (FM) state by only about 1 meV/atom. 8 It is therefore of interest to discuss the effect of spin-orbit coupling (SOC) on the spin-spiral structures since the strong SOC at the W substrate interface plays a key role in determining the magnetism in Fe/W(110) and so to determine the easy and hard magnetization axes. 8,11,12 However, since the spin-spiral order breaks the spatial translation symmetry in a crystalline solid through the SOC, the generalized Bloch theorem is no longer applicable for describing the spin-spiral states. Therefore, in the present work, we implement the FLAPW method in the second variation way 2,13,14 to treat large unit cells for the spin-spiral structures. The calculations demonstrate that when the SOC is introduced, the formation of the spin-spiral structures is suppressed due to the strong SOC at the Fe/W(110) interface, and so the system is found to be the FM state—as observed in experiments. 15 As a model of the Fe monolayer on the W(110) substrate, we employ a single slab consisting of five layers of W(110) and an Fe(110) monolayer on each side, and assume the experimental values of the lattice constant of bulk W ( a = 3.16 Å ) and of the Fe–W interlayer distance ( d Fe – W = 1.94 Å ) . 16 The magnetization in the spin-spiral structures is assumed to lie on a plane parallel to the surface since the magnetic easy axis is along the in-plane [ 1 1 ¯ 0 ] direction with a large magnetocrystalline anisotropy energy of 3 meV/atom and the hard axis along the out-of-plane [110] direction. Calculations are first carried out in the scalar relativistic approximation (SRA) for the valence electrons, i.e., without the SOC, and fully relativistically for the core electrons, based on the LSDA using the von Barth–Hedin exchange correlation. 17 In order to treat spin-spiral structures, here, we apply the generalized Bloch theorem 9,10 into the film-FLAPW method 4,5 with intra-atomic noncollinear magnetism, 6,7 in which the eigenstate at k in the first Brillouin zone (BZ) defined by a chemical unit cell is in the form; Ψ k SRA ( r ) = ∑ G ( C k + G − q / 2 , χ ↑ e − i q / 2 ψ k + G ( r ) C k + G + q / 2 , χ ↓ e + i q / 2 ψ k + G ( r ) ) , (1) where q is a wave vector of the spin-spiral structure, and e ∓ q / 2 ψ k + G ( r ) is described by the spin-independent LAPW basis. 7 The LAPW basis with a cutoff of | k + G ∓ q / 2 | ≤ 3.6 a .u . − 1 and muffin-tin (MT) sphere radii of 2.3 a.u. for the Fe and 2.43 a.u. for the W is used; lattice harmonics with angular momenta up to ℓ = 8 are employed to expand the charge and magnetization density, the vector potential, and eigenvectors. Next, the SOC is incorporated by the second variational method 2,14 employing the SRA eigenvectors. Since the spin-spiral order breaks the spatial translation symmetry in a crystalline solid through the SOC, the generalized Bloch theorem is no longer applicable for describing the spin-spiral states. Therefore, we newly implement the FLAPW method in the second variation way to treat large unit cells (supercells) with lattice constants corresponding to wavelengths of the spin-spiral structures, where commensurate spin-spiral orders are assumed. The eigenstate in the second variation at k s in the BZ redefined by the supercell is now specified by using the SRA eigenvectors as Ψ k s 2 nd ( r ) = ∑ n ∑ j d n , j Ψ k s + n q , j SRA ( r ) . (2) Here n is the number of the k points, those are connected by the q in the BZ of the chemical unit cell, and j represents states in the SRA eigenvectors. The Hamiltonian of the SOC, H soc , is approximated by using the nonmagnetic part of the spherical potential in the MT spheres of the chemical unit cell. The formation energy for the spin-spiral structures with and without SOC, calculated as the energy difference with respect to their FM states, as a function of the q 001 in the [001] direction, is shown in Fig. 1 , where q 001 = 0 indicates the FM state. The difference between the energies with and without SOC is shown in Fig. 2 . As presented previously, 8 without SOC, the spin-spiral structures are energetically favorable over the FM state by only about 1 meV/atom, where the energy minimum appears to be around q 001 = 0.05 ∼ 0.1 , corresponding to a wavelength of about 10 a ∼ 20 a . However, when the SOC is introduced, the formation energy of the spin-spiral states is no longer lower than that of the FM one. Moreover, we carried out the second variational calculations that artificially reduce the SOC contribution from the W substrate, and found that the formation energy does not change over that in the SRA (without SOC) results. This indicates that the magnetization rotation in the spin-spiral structures, which changes the orientation from the easy axis, increases the total energy due to the strong SOC at the W substrate interface. Thus, the formation of the spin-spiral structures is suppressed due to the SOC and the system appears to be the FM state—as observed in experiments. 15 Figure 3 shows the magnitude of the spin and orbital magnetic moments integrated in the MT spheres for the spin-spiral structure with q = 0.1 as a function of atomic position along the [001] direction. For the whole atomic position, the spin moments have almost the same value of 2.36 μ B . In contrast, and interestingly, the orbital moments periodically change with a large variation of the 0.03 μ B . It is noted that the orbital moments at site 1 in Fig. 3 , where the spins orient along the easy axis of the [ 1 1 ¯ 0 ] direction, have a maximum value ( 0.17 μ B ) , while those at site 2, where the spins orient along the [001] direction, have a minimum value ( 0.14 μ B ) . Thus, the orbital moments are strongly sensitive to the atomic position in the spin-spiral structures. In conclusion, the stability of the spin-spiral structures in the Fe monolayer on the W(110) substrate was investigated by means of the FLAPW method where we newly implemented the FLAPW method in the second variation way to treat large unit cells for the spin-spiral structures. The calculations demonstrated that the formation of the spin-spiral structures are energetically suppressed due to the strong SOC at the Fe/W(110) interface, and so the ground state of the system appears to be the FM state. Work at Mie University was supported by a Grant-in-Aid for Scientific Research (Grant No. 20540334) from the Japan Society for the Promotion of Science, and for computations performed at the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. Work at Northwestern University was supported by the U.S. DOE Grant No. DE-FG02-88ER45372. FIG. 1. Calculated formation energy of spin-spiral structures as a function of wave vector, q 001 , in the [001] direction for Fe/W(110). Open and closed circles represent results with and without SOC. FIG. 2. Difference between energies with and without SOC as a function of wave vector, q 001 , in the [001] direction for Fe/W(110). FIG. 3. Calculated spin and orbital magnetic moments integrated in Fe MT spheres as a function of atomic position along [001] direction for spin-spiral structure with q 001 = 0.1 of Fe/W(110). Open and closed circles represent the spin and orbital moments, respectively.

PY - 2009

Y1 - 2009

N2 - The stability of spin-spiral structures in an Fe monolayer on a W(110) substrate is investigated by means of the first-principles film full-potential linearized augmented plane-wave method, and the role of spin-orbit coupling (SOC) on the spin-spiral structures is determined. Our calculations demonstrate that without SOC, the spin-spiral structures are energetically favored over the ferromagnetic (FM) state, but that when the strong SOC at the Fe/W(110) interface is introduced, the formation of the spin-spiral structures is suppressed. Thus, the ground state of the system appears to be the FM state-as observed in experiments.

AB - The stability of spin-spiral structures in an Fe monolayer on a W(110) substrate is investigated by means of the first-principles film full-potential linearized augmented plane-wave method, and the role of spin-orbit coupling (SOC) on the spin-spiral structures is determined. Our calculations demonstrate that without SOC, the spin-spiral structures are energetically favored over the ferromagnetic (FM) state, but that when the strong SOC at the Fe/W(110) interface is introduced, the formation of the spin-spiral structures is suppressed. Thus, the ground state of the system appears to be the FM state-as observed in experiments.

UR - http://www.scopus.com/inward/record.url?scp=65249182013&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=65249182013&partnerID=8YFLogxK

U2 - 10.1063/1.3070635

DO - 10.1063/1.3070635

M3 - Article

AN - SCOPUS:65249182013

VL - 105

JO - Journal of Applied Physics

JF - Journal of Applied Physics

SN - 0021-8979

IS - 7

M1 - 07C304

ER -

Role of spin-orbit coupling in spin-spiral structures in Fe monolayer on W(110): A first-principles noncollinear magnetism study

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