Magnetic anisotropy energies of M–Fe wires (M = V–Co) on vicinal Cu(111)

Abstract

One-dimensional transition metal (TM)-Fe nanowires of single-atom width can be formed on a stepped Cu(111) surface. The electronic magneto-crystalline anisotropy and magnetic dipolar (shape) anisotropy energies are calculated for an isolated and deposited TM-Fe wire series that ranges from V to Co along the fourth period of elements. For linear atomic V chains, the shape anisotropy energy is comparable to the electronic energy. The shape anisotropy energy dominates for Cr/Mn chains while it has a minor influence for Co/Fe. The electronic contributions to the anisotropy energies are always dominant for TM-Fe wires in both isolated and deposited cases. All linear structures exhibit axial magnetization except for the Cr chain, which exhibits perpendicular magnetization. All isolated TM-Fe wires have large magnetic anisotropy energies and easy magnetization along the perpendicular axis. In deposited TM-Fe wires, the electronic anisotropy energies are higher for Mn–Fe and Fe–Fe and lower for V–Fe, Cr–Fe and Co–Fe wires. Deposited TM-Fe wires magnetize easily perpendicular to the Cu surface except for Co–Fe, which prefers in-plane magnetization. The large magnetic anisotropies of Fe–Fe and Mn–Fe wires points to potential application in ultra-high density data storage.

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I. Introduction

Magnetism at the nanometer scale has been an exciting research area over the past few decades due to fundamentally interesting physical properties such as magnetic anisotropy. In crystals, there are isotropic properties such as the Curie temperature, but the majority of properties tend to be anisotropic. This includes magnetic properties like the easy and hard directions of magnetization.

There are two possible reasons for magnetic anisotropy: the coupling of the electron orbits to the crystal electric field and the interaction of dipoles. The first is the magnetocrystalline anisotropy, or electron contribution, which originates from simultaneous occurrence of the electron spin–orbit interaction and spin-polarization in the magnetic system. The second is magneto-static in nature, and is stabilized by the shape anisotropy energy, that emerges from magnetic dipolar interactions in the solid.

Due to potential applications in memory devices, there has been recent interest in the magnetic anisotropy energy (MAE) of small magnetic nanostructures supported on nonmagnetic substrates. A minimum MAE of 1.2 eV per bit is required to inhibit magnetization reversal for a storage device, therefore, a reduction in the size of nanostructures carrying one bit of information requires an increase in MAE/atom.

Ideally, increasing the MAE would lead to efficient magnetization in two dimensions (surface), one dimensions (wire) or even zero dimensions (single atom). For example the magnetic and structural properties of finite free-standing gold atomic chains have been the focus of intensive experiments and theoretical studies since their discovery in 1998.^1,2 However, these free-standing atomic chains are unstable and can only exist at low temperatures on suitably chosen substrates.

Physically, stable magnetic nanowires deposited on metallic substrates are one of the most important nanostructures and a variety of techniques have been used to prepare and study them. For example, Gambardella et al.^3,4 prepared stepped, high density, parallel cobalt chains on a high-purity Pt(997) vicinal surface in a temperature range of 10–20 K. Using X-ray magnetic circular dichroism, it was shown that the Co wires exhibit a particularly high MAE of 9.3 ± 0.6 meV.⁴

As mentioned above, the extreme case of a system showing magnetic anisotropy is an atom on a nonmagnetic surface. A high magnetic anisotropy energy requires large spin and orbital magnetic moments as well as strong spin–orbit coupling. Cobalt has a large spin moment among the ferromagnetic 3d metals; platinum also exhibits strong spin–orbit coupling. In this case, Co chains supported on substrates of highly polarizable Pt are a viable route for tuning both the spin moments and the anisotropy energy. Therefore, the large MAE can be attributed to the large spin moment and reduced dimensionality of the Co atoms combined with the induced magnetic moments and strong spin–orbit coupling found in the Pt substrate.

Monoatomic wires have been investigated theoretically in a large number of studies.^5–15 The reduced coordination of magnetic atoms deposited on a metallic surface leads to enhancement of magnetic moments as compared to the bulk. Large magnetocrystalline anisotropy also stabilizes the direction of the enhanced magnetic moments. In ref. 16–19, the electronic and magnetic properties of TM chains at their equilibrium bond length are investigated. The presented study is focused on TM chains with the equilibrium Cu bond length.

Ab initio studies of magnetic anisotropy for 3d TM freestanding linear chains²⁰ revealed that freestanding Fe linear chains have a massive MAE. In ref. 20 the bond length of TM chains is not constrained to the equilibrium Cu bond length. Experimentally, copper and tungsten are good substrates for growth of Fe thin films.^21,22 TM-Fe wires can be formed on Cu(111) stepped surface.¹⁸

In this paper, ab initio DFT calculations including spin–orbit coupling (SOC) have been performed to calculate the magnetic anisotropy energies, including magnetocrystalline and shape anisotropies of TM-Fe wires in both isolated and deposited cases. The TM-Fe structures have been optimized for different orientations of the magnetization with respect to the crystallographic axes of the systems. The MAE and the anisotropies of spin and orbital moments have been determined. Particular attention has been paid to the correlation between the geometric and magnetic structures as discussed in detail in ref. 16–19. The Fe–Fe and Mn–Fe wires exhibit a large MAE, indicating that these nanowires would have applications in high density magnetic data storages. The substrate effect on the MAE of the wires is discussed in this paper.

The paper is organized as follows: in Section II., a brief description of the theory and computational details is given. In Section III., results for spin–orbit couplings and orbital magnetic moments of the linear 3d TM chains and the TM-Fe wires are presented. The calculated magnetic anisotropy energies and moments of linear TM chains and TM-Fe wires on Cu(111) stepped surface are presented in Section IV.

II. Computational methodology

The calculations were performed within the framework of spin-polarized density functional theory, using the Vienna ab initio simulation package (VASP).^23,24 The frozen-core full-potential projector augmented-wave method (PAW) was used,²⁵ applying the generalized gradient approximation of Perdew and Wang (PW91-GGA).²⁶

Computational details and convergence checks are the same as those in a previous study,¹⁸ although the computational details and convergence checks are modified as described below. A plane-wave cutoff energy of 340 eV is used for all 3d TM chains. The Methfessel–Paxton scheme²⁷ is employed for Brillouin zone integrations. The convergence of the calculated properties with respect to number of k-points and supercell size is also carefully checked. For linear chains, the nearest wire–wire distance between the neighboring chains is at least 13 Å. Because of its small magnitude, ab initio calculations of the MAE are computationally very demanding and must be carefully carried out. Here we use the total energy difference approach rather than the widely used force theorem to determine the MAE; the MAE is calculated as the difference in the full self-consistent total energies for the two different magnetization directions (e.g., parallel and perpendicular to the chain).

Because MAE is a delicate and controversial effect, approach-independence of the major conclusions is verified by performing calculations with two different software packages: VASP^23,24 and WIEN2K.²⁸ Using VASP, the total energy convergence criterion is 10⁻⁶ eV per atom. MAEs calculated with a dense 32 × 6 × 1 k-point mesh with σ = 0.001 eV shows little difference to MAEs calculated with a 20 × 5 × 1 k-point mesh (within 0.02 meV). The same k-point mesh is used for the band structures and density of states calculations.

To verify the MAE results of VASP, we used the all-electron full-potential linear augmented plane wave (FP-LAPW) method is also implemented in the WIEN2K code with the generalized gradient approximation (GGA).

III. Spin–orbit coupling and orbital magnetic moment

Spin–orbit coupling is necessary for orbital magnetization and magneto-crystalline anisotropy in solids, however it is weak in 3d transition metals. Thus, it was taken into account in the self-consistent calculations presented here.

A. TM chains

Taking into account SOC, the spin moments for the isolated 3d transition chains are calculated as 3.02 μ_B for V, 3.99 μ_B for Cr, 4.01 μ_B for Mn, 3.24 μ_B for Fe, and 2.23 μ_B for Co. These spin moments are almost identical to the corresponding values calculated without taking into account SOC due to the weakness of SOC in 3d transition metals. While less important for the calculation of the spin moment, SOC has large contributions to orbital magnetic moments in some atomic chains; this allows us to determine the easy magnetization axis for these 3d atomic chains.

As summarized in Table 1, the orbital magnetic moments for parallel magnetization in the ferromagnetic (FM) state are 0.41 μ_B for Fe, 0.55 μ_B for Co, 0.02 μ_B for V, 0.01 μ_B for Cr, and 0.02 μ_B for Mn per atom. The orbital magnetic moments for V, Cr, and Mn chains are negligible, but in Fe and Co atomic chains there is considerable enhancement of the orbital magnetic moment, when compared with bulk samples and monolayers of the same material.^29,30

Table 1 Spin (m_s) and orbital (m_o) magnetic moments (in μ_B per atom) of the magnetic 3d TM linear chains at the Cu bond lengths with magnetization parallel (M∥ẑ) and perpendicular (M⊥ẑ) to the chain axis

		M∥ẑ		M⊥ẑ
		ms	mo	ms	mo
V	(FM)	3.02	0.02	3.01	0.01
Cr	(AF)	3.99	0.01	4.00	0.03
Mn	(AF)	4.01	0.02	4.07	0.07
Fe	(FM)	3.24	0.41	3.15	0.19
Co	(FM)	2.23	0.55	2.18	0.47

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The spin moment is found to be unaffected when SOC is taken into account even in 4d^31–34 and 5d¹⁴ TM linear atomic chains. Nonetheless, SOC still has a significant contribution to orbital moments in these chains. It was also shown in some studies^20,31 that the orbital moment has a strong dependence on bond length as well as the magnetization orientation.

As summarized in Table 1, the orbital moment for parallel magnetization is higher than that for perpendicular magnetization in the case of V, Fe, and Co chains, while the opposite is true for Cr and Mn chains. The orbital moments of the V, Fe, and Co chains with perpendicular magnetization are 0.01 μ_B, 0.19 μ_B, and 0.47 μ_B, while the orbital moments of Cr and Mn with perpendicular magnetization are only 0.03 μ_B and 0.07 μ_B, respectively.

It is well known that the magnetization direction with a larger orbital moment has a lower total energy. Therefore, the easy magnetization direction in the V, Fe, and Co chains is expected to be parallel to the chain, whereas in Cr and Mn chains, it is perpendicular to the chain. Note that the present results for the spin and orbital moments are in good agreement with previous calculations for the 3d atomic chains.^20,31

B. TM-Fe wires

As discussed in ref. 18, and using their notation, all the isolated TM-Fe wires have magnetic solutions, summarized in Table 2. The V–Fe wire is most stable in the F̲F̲ state, whilst the ground state of the Cr–Fe wire is the AF state and the Mn–Fe, Fe–Fe, and Co–Fe wires have the FF ground state.

Table 2 Spin (m_s) and orbital (m_o) magnetic moments (in μ_B per atom) of the TM-Fe wires with magnetization M in x̂, ŷ and ẑ directions

		x̂		ŷ		ẑ
		ms	mo	ms	mo	ms	mo
V–Fe	(F̲F̲)	2.59	0.00	2.59	0.01	2.61	0.02
Cr–Fe	(AF)	3.49	0.01	3.50	0.00	3.53	0.03
Mn–Fe	(FF)	4.06	0.01	4.07	0.01	4.07	0.02
Fe–Fe	(FF)	3.04	0.01	3.04	0.09	3.05	0.14
Co–Fe	(FF)	2.03	0.03	2.04	0.11	2.04	0.17

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The spin magnetic moments in isolated TM-Fe wires are generally smaller than in the corresponding TM single chains, due to an increase in the coordination number of the TM-Fe wires. When SOC is taken into account, there is a noticeable change in the spin magnetic moments of TM-Fe wires in a similar fashion to the linear chains. The orbital magnetic moments of the TM-Fe wires with the easy magnetization axis parallel to the wire are 0.02 μ_B for V, 0.03 μ_B for Cr, 0.02 μ_B for Mn, 0.14 μ_B for Fe and 0.17 μ_B for Co. It should be noted that these values are significantly lower than is seen in the corresponding linear atomic chains (Fig. 1).

Fig. 1 Schematic structure diagram for (a) the isolated TM chain and (b) TM-Fe wire.

IV. Magnetic anisotropy energy (MAE)

The total energy of the orientation dependent magnetization M(ϕ,θ) of TM wires (see Fig. 2) in the lowest non-vanishing terms can be written²⁰ as

E_t = E₀ + sin² θ(E₁ − E₂ cos² ϕ) (1)

where θ is the polar angle of the magnetization M from the wire axis (z-axis) and ϕ is the azimuthal angle in the x–y plane perpendicular to the wire, measured from the x axis. For the isolated chain, the azimuthal anisotropy energy constant E₂ is zero, due to rotational invariance. The axial anisotropy energy, E₁, is calculated as the total energy difference between magnetization along the y(x) and z axes, i.e., E₁ = E^y − E^z (E^x = E^y). If E₁ is positive, the easy magnetization axis is along the chain (z) axis. For TM-Fe wires which are in the x–z plane, E₂ is not zero and can be calculated as the total energy difference between magnetization along the x and y axes, i.e., E₂ = E^y − E^x.

Fig. 2 Schematic representation of the magnetization orientation (θ,ϕ) of a TM-Fe wire on the vicinal surface.

As mentioned earlier, there are two major components to MAE for a magnetic solid, the magnetocrystalline energy and the shape anisotropy energy. It is known that shape anisotropy energy is zero for cubic systems such as bcc Fe and fcc Ni, and very small for solids such as hcp Co. However, the shape anisotropy energy is significant for anisotropic structures such as magnetic Fe and Co monolayers.^30,35 The shape anisotropy is calculated by summing over the classical dipole–dipole energies in the system. Individual dipole–dipole interactions do not have a significant contribution to the MAE, but, because dipole–dipole interactions are long-ranged, when summed over the whole material, the dipolar interactions are non-negligible.

Furthermore, the shape anisotropy energy of the 3d TM chains and TM-Fe wires are also non-negligible as will be discussed in detail later. Therefore, their magnetic dipole–dipole interaction energies are calculated.

To check for potential collinearity between the Fe and TM magnetic moments (M_TM∥M_Fe), fully relaxed magnetic structures have been calculated using the all-electron full-potential linear augmented plane wave (FP-LAPW). The ground state calculations estimated a collinear magnetic structure for TM-Fe wires on a Cu(111) stepped surface. However, there is a negligible angle between the Fe and TM magnetic moments. This angle is less than 2° for all of the TM-Fe wires. Therefore, the magnetic anisotropy energy calculations are restricted to the collinear cases in the present paper.

For collinear magnetic systems, this magnetic dipolar energy (E^d) is calculated as (in atomic Rydberg units)

andwhere M_rr′ is the magnetic dipolar Madelung constant, c = 274.072 is the speed of light, R are the lattice vectors, r are the atomic position vectors in the unit cell and m_r is the atomic magnetic moment on site r. Note that in atomic Rydberg units, one Bohr magneton (μ_B) is .

The dipolar energy convergence criterion of 10⁻⁶ eV per atom was reached by taking 300 terms (300 neighboring magnetic moments) of eqn (3).

A. Free-standing TM chains

Electronic (E^e₁) and shape anisotropy (E^d) energies for free-standing TM chains, free-standing TM-Fe wires and TM-Fe wires on a Cu(111) stepped surface are listed in Tables 3–5, respectively. Table 3 shows MAEs for single 3d TM chains in their ground states.

Table 3 Total (E^t₁), electronic (E^e₁) and dipolar (E^d₁) magnetic anisotropy energies (in meV per atom) of the 3d TM linear chains for the ground state magnetic structure (MS). If E^t₁ is positive, the easy magnetization axis is along the chain; otherwise, the easy magnetization axis is perpendicular to the chain

	MS	E^t1	E^e1	E^d1
V	(FM)	1.62	0.88	0.74
Cr	(AF)	−1.29	−0.32	−0.97
Mn	(FM)	1.04	−0.27	1.31
Fe	(FM)	5.69	4.83	0.86
Co	(FM)	2.72	2.31	0.41

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Table 4 The total (E^t₁, E^t₂), electronic (E^e₁, E^e₂) and dipolar (E^d₁, E^d₂) magnetic anisotropy energies (in meV per atom) as well as the easy magnetization axis (M) of the free-standing TM-Fe wires for the ground state magnetic structure (MS). E₁ = E^y − E^z; E₂ = E^y − E^x

	MS	E^e1	E^e2	E^d1	E^d2	E^t1	E^t2	M
V–Fe	(F̲F̲)	4.42	−0.08	0.27	−0.11	4.69	−0.19	x
Cr–Fe	(AF)	3.33	−0.77	−0.04	−0.14	3.29	−0.91	x
Mn–Fe	(FF)	2.37	−0.74	0.65	0.18	3.02	−0.56	x
Fe–Fe	(FF)	1.39	0.07	0.51	0.14	1.90	0.21	x
Co–Fe	(FF)	1.92	0.55	0.38	0.09	2.30	0.64	x

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Table 5 The total (E^t₁, E^t₂), electronic (E^e₁, E^e₂) and dipolar (E^d₁, E^d₂) magnetic anisotropy energy constants (in meV per atom) as well as the easy magnetization axis (M) of the TM-Fe wires on Cu(111) stepped surface for the ground state magnetic structure (MS). E₁ = E^y − E^z; E₂ = E^y − E^x, see eqn (2)

	MS	E^e1	E^e2	E^d1	E^d2	E^t1	E^t2	M
V–Fe	(F̲F̲)	1.03	0.90	0.16	−0.19	1.19	0.71	x
Cr–Fe	(AF)	1.09	0.44	−0.08	−0.11	1.01	0.33	x
Mn–Fe	(FF)	5.15	0.31	0.57	0.17	5.72	0.48	x
Fe–Fe	(FF)	4.32	0.89	0.44	0.15	4.76	1.04	x
Co–Fe	(FF)	0.88	−1.61	0.33	0.11	1.21	−1.50	y

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It is immediately clear that shape anisotropy energies are comparable to the electronic contributions. Furthermore, V, Mn, Fe and Co (3d elements with FM ground state) have easy magnetization along the z-axis while Cr chains have easy magnetization perpendicular to the z-axis. This is expected behavior since the shape anisotropy energy always favors the direction of the longest dimension; in the case of FM chains it is the z-axis and in AF chains it is perpendicular to the z-axis. Therefore, magnetic anisotropy in the hard direction should originate from electronic anisotropy when it is sufficiently large to overcome the shape anisotropy.

Table 3 shows that in FM linear chains, the electronic anisotropy energy would favor a perpendicular anisotropy in the Cr and Mn chains but prefer the chain axis in the V, Fe and Co chains. Nevertheless, the easy magnetization direction is predicted to be the chain axis in all the 3d FM chains except Mn. But the perpendicular electronic anisotropy in the Mn chain is not sufficiently large to overcome the axial shape anisotropy.

In the AF state, Cr chains have the easy axis perpendicular to the chain. Remarkably, Fe chains with MAE of 5.69 meV have a large axial anisotropy energy, being in the same order of magnitude of that in the 4d TM linear chains.^31,36 In the 4d transition metals, the SOC splitting is large – about ten times larger than the 3d transition metals – and thus the large MAE in the 4d TM linear chains may be expected. The axial anisotropy energy for the V, Cr, Fe, and Co chains are also generally larger than the corresponding monolayers.^30,35

Our results show that major contributions from the occupied and empty d states in the vicinity of the Fermi level coupled by SOC, can explain the magnetocrystalline anisotropy. This finding is consistent with other reports in the literature.^20,37

B. TM-Fe wires

The axial MAEs E₁ of free-standing TM-Fe wires with easy magnetization along the z-axis are 4.42 meV for V, 3.33 meV for Cr, 2.37 meV for Mn, 1.39 meV for Fe and 1.92 meV for Co. E₁ in the Fe and Co chains diminishes as the structures change from chains to wires; the MAE for Fe–Fe wires is almost three times smaller than the Fe linear chain. The MAEs of V–Fe, Cr–Fe and Mn–Fe wires, on the other hand, are considerably enhanced when compared to their linear chain analogues. The largest enhancement is in the V–Fe wire where the MAE E₁ in the V–Fe wire is almost three times larger than the V chain value. The shape anisotropy energies (E^d₁) generally decrease as the structure reduces in dimensionality from linear chain to TM-Fe wire.

A large MAE (E₂) is also present in the x–y plane perpendicular to the z-axis. E^d₂, albeit smaller than E^d₁. The x-axis is the easy magnetization axis for all TM-Fe wires (see Table 4).

Isolated TM atoms have large spin and orbital magnetic moments according to Hunds rules. However, electron delocalization and crystal field effects compete with intra-atomic Coulomb interactions leading to overall reduction of spin moments as well as quenching of orbital moments in TM impurities dissolved in nonmagnetic metal hosts. Theoretical calculations predict such effects to be strongly reduced at surfaces owing to the decreased coordination of TM impurities.³⁸ Therefore, it is important to also look at the effect the substrate has on the MAE of TM-Fe wires.

For deposited V–Fe, Cr–Fe, Mn–Fe and Fe–Fe wires, easy magnetization is along the x-axis (perpendicular to the Cu(111) surface); only the Co–Fe wire has easy magnetization along the y-axis (see Table 5). Experimentally, Shen et al. also demonstrated that Fe nanostripes on Cu(111) vicinal surface were characterized by a perpendicular anisotropy.^39,40 Similarly, Tung et al. found that the Fe wire presents a perpendicular magnetic anisotropy while the Co wire shows an in-plane anisotropy on Cu(001) surface.⁴¹

V. Conclusions

Non-collinear spin-polarized DFT is employed to calculate magnetocrystalline and shape anisotropy energies of TM-Fe wires in both isolated and deposited cases. The TM-Fe structures are optimized for different orientations of the magnetization with respect to the crystallographic axes of the systems. The magnetocrystalline contributions to the anisotropy energies are always dominant for TM-Fe wires in both isolated and deposited cases. In the linear structures, V, Mn, Fe, and Co chains exhibit axial magnetization while Cr chains display perpendicular magnetization. All isolated TM-Fe wires have large MAEs as well as easy magnetization along the perpendicular axis. Electronic anisotropy energy is enhanced for deposited Mn–Fe and Fe–Fe wires, but diminished for deposited V–Fe, Cr–Fe and Co–Fe wires. Most deposited TM-Fe wires displayed easy magnetization perpendicular to the Cu surface except for Co–Fe which preferred axial magnetization. Deposited Fe–Fe and Mn–Fe wires exhibit large MAE, a strong indication that these nanowires have potential applications in high-density magnetic data storages.

Systematic investigation of MAEs, can be used as a starting point for studying the finite temperature magnetic properties of TM-Fe wires. The calculated MAEs, the intrawire¹⁶ magnetic coupling, and the interwire⁴² magnetic coupling can be used to set up a classical Heisenberg model to study finite temperature properties of TM-Fe wires embedded on Cu(111) surfaces. Subsequent Monte Carlo calculations can be used to determine the magnetic properties of deposited TM-Fe wires at finite temperature.

Acknowledgements

We thank W. Hergert and V. Stepanyuk for fruitful discussions. The work was supported by the cluster of excellence Nanostructured Materials of the state Saxony-Anhalt and the International Max Planck Research School for Science and Technology of Nanostructures. We acknowledge the resources provided by the University of Michigans Advanced Research Computing.

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