Ferromagnetic Anisotropy and the Itinerant Electron Model

Brooks, Harvey

doi:10.1103/physrev.58.909

Cited by 233 publications

(79 citation statements)

References 25 publications

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“…This is reasonable because the contribution due to the slightly different Fermi surface in the two approaches, which to order 2 can be explicitly expressed as (E F /⍀ BZ )͐ ␦V F dk, is zero because the number of electrons is fixed. In surface-interface systems, it is well known that the dominant contribution to MAE is of the order of 2 , and so as a practical method the state tracking approach is a very good approximation in determining the leading order contribution to the MAE for these systems. Indeed, when the MCA force theorem is to be applied with a limited number of k points, the state tracking method is necessary to ensure that the change of charge and spin densities be of order 2 and thus to suppress the Fermi surface fluctuations.…”

Section: ⑀͓ ͑ ͒; ͔ϩEј͓ ͑ ͔͒ ͑4͒mentioning

confidence: 99%

“…In surface-interface systems, it is well known that the dominant contribution to MAE is of the order of 2 , and so as a practical method the state tracking approach is a very good approximation in determining the leading order contribution to the MAE for these systems. Indeed, when the MCA force theorem is to be applied with a limited number of k points, the state tracking method is necessary to ensure that the change of charge and spin densities be of order 2 and thus to suppress the Fermi surface fluctuations. 9 Since for transition metals, contributions to the MAE from terms of higher order than 2 was estimated to be ϳ0.1 meV/atom, 9 different methods of determining the Fermi surface should agree within this range.…”

Section: ⑀͓ ͑ ͒; ͔ϩEј͓ ͑ ͔͒ ͑4͒mentioning

confidence: 99%

“…[2][3][4][5][6][7][8][9][10][11][12] Unfortunately, despite the great advances in local-spin-density electronic structure theory and computational power in the past decades, the accurate determination of MAE still remains difficult and computationally demanding. In the traditional approach, the value of MAE is calculated through comparing the total energies of a given system for two different magnetization orientations ͑in-plane and perpendicular directions for surface-interface systems͒.…”

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Torque method for the theoretical determination of magnetocrystalline anisotropy

et al. 1996

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We propose a torque method for the theoretical determination of the magnetocrystalline anisotropy ͑MCA͒ energy for systems with uniaxial symmetry. While the dependence of the total energy on the angle between the magnetization and the normal axis ( ) can be expressed as E( )ϭE 0 ϩK 2 sin 2 ( )ϩK 4 sin 4 ( ), we show that the MCA energy ͓defined as E MCA ϭE( ϭ90°)ϪE( ϭ0°)ϭK 2 ϩK 4 ͔ can be easily evaluated through the expectation value of the angular derivative of the spin-orbit coupling Hamiltonian ͑torque͒ at an angle of ϭ45 o . Unlike other procedures, the proposed method is independent of the validity of the MCA force theorem, or of the absolute accuracy of two total energy calculations. Calculated MCA energies for the free Fe monolayer with different lattice constants are analyzed and compared with results of other ab initio calculations, especially those obtained with our previously reported state tracking method. ͓S0163-1829͑96͒02926-8͔Recent developments in magnetic thin films and overlayers that show perpendicular magnetic anisotropy 1 with their tremendous practical implications for high-density magnetooptical storage media, have stimulated ab initio theoretical determinations of the magnetocrystalline anisotropy energy ͑MAE͒. 2-12 Unfortunately, despite the great advances in local-spin-density electronic structure theory and computational power in the past decades, the accurate determination of MAE still remains difficult and computationally demanding. In the traditional approach, the value of MAE is calculated through comparing the total energies of a given system for two different magnetization orientations ͑in-plane and perpendicular directions for surface-interface systems͒. Since the spin-orbit coupling ͑SOC͒ is very weak in 3d transition metals, the so-called magnetocrystalline anisotropy ͑MCA͒ force theorem 6 is usually adopted and the MAE is calculated by merely comparing the band energies between the two magnetic orientations. ͑We call them direct approaches.͒ The main difficulty associated with the direct approaches concerns the numerical stability of calculating a very small difference of two large numbers. In fact, in order to eliminate numerical fluctuations, [3][4][5][6][7]9 extremely fine sampling meshes are required for the k-space integrations.In this paper, we propose a torque method for the determination of MCA. The main advantage that this method offers is that one only needs to calculate MAE at one particular magnetic orientation and to do the k-space integration with the single Fermi surface at this orientation. In the following, we first present the method, then give results for free standing Fe monolayers as a test case, and compare them with the results of other methods. [3][4][5]9 To demonstrate the idea, let us recall that the total energy of an uniaxial system can be well approximated in the formwhere is the angle between the magnetization and the normal axis. If we define the torque, T( ), as the angular derivative of the total energy, i.e.,it is easy to show that M AEϵE͑ ϭ...

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Section: ⑀͓ ͑ ͒; ͔ϩEј͓ ͑ ͔͒ ͑4͒mentioning

confidence: 99%

Section: ⑀͓ ͑ ͒; ͔ϩEј͓ ͑ ͔͒ ͑4͒mentioning

confidence: 99%

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confidence: 99%

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Torque method for the theoretical determination of magnetocrystalline anisotropy

et al. 1996

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“…One of the oldest problems which has been addressed in the past is the calculation of the magneto-crystalline anisotropy energy (MAE) 13,14,15,16,17 of magnetic materials. The MAE is defined as the difference of total energies with the orientations of magnetization pointing in different, e.g., (001) and (111), crystalline axis.…”

Section: Magnetic Anisotropy Of Ferromagnetsmentioning

confidence: 99%

Electronic Structure and Magnetic Properties of Solids

et al. 2005

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“…The quenching of the orbital moment in transition metals is the result of the competition between the mixing of the states by the SOC and the splitting of the states by the band structure: 26 The conduction electron states are split into a set of bands. The energy separation is of the order of the bandwidth W. The expectation value of the orbital moment would vanish for all eigenstates in the absence of the SOC.…”

Section: A Competition Between Band Structure and Socmentioning

confidence: 99%

Spin-orbit induced noncubic charge distribution in cubic ferromagnets. II. Tight-binding analysis

2002

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The orbital moment and the noncubic charge distribution in ferromagnetic transition metals with cubic lattice symmetry are investigated within the tight-binding model. By combining the tight-binding approximation, perturbation theory, and the Green's function formalism for impurity scattering, approximate expressions for both effects are derived that depend only on the spin-orbit coupling strength and the density of states of the system without spin-orbit coupling. The basic relations between the orbital moment, the noncubic charge distribution, and the band structure are derived from the form of these expressions and from their application to various model band structures: We explain in this way the scaling with the spin-orbit coupling strength and bandwidth, the typical order of magnitude, the variation as a function of the band filling, the sensitivity to band structure details, and the role of the splitting between spin-up and spin-down states. For the noncubic charge distribution we derive the form of the dependence on the direction of the magnetization and show how the sign and magnitude of this anisotropy are related to the different energy distributions of e g and t 2g states. This tight-binding analysis is finally applied to the 5d impurities in Fe. The local densities of states without spin-orbit coupling are obtained by self-consistent augmented plane-wave calculations using a supercell method. The special features of the 5d impurities in Fe with respect to the band structure, the orbital moment, and the noncubic charge distribution are discussed. The general trend of the systematics is interpreted as a band filling effect. The prevailing sign of the anisotropy is ascribed to the concentration of the e g states near the Fermi energy. The results of the tight-binding analysis are compared with the experiment and a more rigorous calculation.

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Ferromagnetic Anisotropy and the Itinerant Electron Model

Cited by 233 publications

References 25 publications

Torque method for the theoretical determination of magnetocrystalline anisotropy

Torque method for the theoretical determination of magnetocrystalline anisotropy

Electronic Structure and Magnetic Properties of Solids

Spin-orbit induced noncubic charge distribution in cubic ferromagnets. II. Tight-binding analysis

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