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Ab Initio Calculations of Exchange Interactions, Spin-Wave Stiffness Constants, and Curie Temperatures of Fe, Co, and Ni

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Ab Initio Calculations of Exchange Interactions, Spin-Wave Stiffness Constants, and Curie Temperatures of Fe, Co, and Ni PHYSICAL REVIEW B, VOLUME 64, 174402 Ab initio calculations of exchange interactions, spin-wave stiffness constants, and Curie temperatures of Fe, Co, and Ni M. Pajda,1 J. Kudrnovsky´,2,1 I. Turek,3,4 V. Drchal,2 and P. Bruno1 1Max-Planck-Institut fu¨r Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany 2Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-182 21 Prague 8, Czech Republic 3Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zˇizˇkova 22, CZ-616 62 Brno, Czech Republic 4Department of Electronic Structures, Charles University, Ke Karlovu 5, CZ-121 16 Prague 2, Czech Republic ͑Received 27 July 2000; revised manuscript received 20 December 2000; published 1 October 2001͒ We have calculated Heisenberg exchange parameters for bcc Fe, fcc Co, and fcc Ni using the nonrelativistic spin-polarized Green-function technique within the tight-binding linear muffin-tin orbital method and by em- ploying the magnetic force theorem to calculate total energy changes associated with a local rotation of magnetization directions. We have also determined spin-wave stiffness constants and found the dispersion curves for metals in question employing the Fourier transform of calculated Heisenberg exchange parameters. Detailed analysis of convergence properties of the underlying lattice sums was carried out and a regularization procedure for calculation of the spin-wave stiffness constant was suggested. Curie temperatures were calcu- lated both in the mean-field approximation and within the Green-function random-phase approximation. The latter results were found to be in a better agreement with available experimental data. DOI: 10.1103/PhysRevB.64.174402 PACS number͑s͒: 75.10.Ϫb, 71.15.Ϫm, 75.30.Ds I. INTRODUCTION ͑ii͒ The spin waves or magnons, which correspond to col- lective transverse fluctuations of the direction of the magne- The quantitative description of thermodynamic properties tization. Near the bottom of the excitation spectrum, the den- of magnetic metals is challenging solid-state theorists since sity of states of magnons is considerably larger than that of decades. Thanks to the development of density-functional corresponding Stoner excitations, so that the thermodynam- theory and its implementation to ab initio computational ics in the low-temperature regime is completely dominated schemes, an excellent understanding of their ground state by magnons and Stoner excitations can be neglected. There- ͑i.e., at Tϭ0 K) has been achieved. On the other hand, most fore it seems reasonable to extend this approximation up to of the progress towards a description of magnetic metals at the Curie temperature, and to estimate the latter by neglect- nonzero temperature has been based upon models in which ing Stoner excitations. This is a good approximation for fer- the electronic structure is oversimplified and described in romagnets with a large exchange splitting such as Fe and Co, terms of empirical parameters. Although this approach has but it is less justified for Ni that has a small exchange the great merit of emphasizing the relevant mechanisms and splitting. concepts, it cannot properly take into account the complex The purpose of the present paper is to describe the spin- details of the electronic structure and is therefore unable to wave properties of transition metal itinerant ferromagnets at yield quantitative predictions of the relevant physical quan- ab initio level. With thermodynamic properties in mind, we tities such as spin-wave stiffness, Curie temperature TC , etc., are primarily interested in the long-wavelength magnons for comparison with experimental data. with the lowest energy. We shall adopt the adiabatic approxi- It is therefore of a great importance to develop an ab mation in which the precession of the magnetization due to a initio, parameter-free, scheme for the description of ferro- spin wave is neglected when calculating the associated magnetic metals at TϾ0 K. Such an approach must be able change of electronic energy. Clearly, the condition of validity to go beyond the ground state and to take into account ex- of this approximation is that the precession time of the mag- cited states, in particular the magnetic excitations responsible netization should be large as compared to characteristic times for the decrease of the magnetization with temperature and of electronic motion, namely, the hopping time of an electron ϭ for the phase transition at T TC . Although density- from a given site to a neighboring one, and the precession functional theory can be formally extended to nonzero tem- time of the spin of an electron subject to the exchange field. perature, there exists at present no practical scheme allowing In other words, the spin-wave energies should be small as to implement it. One therefore has to rely on approximate compared to the bandwidth and to the exchange splitting. approaches. The approximations to be performed must be This approximation becomes exact in the limit of long- chosen on the basis of physical arguments. wavelength magnons, so that the spin-wave stiffness con- In itinerant ferromagnets, it is well known that magnetic stants calculated in this way are in principle exact. excitations are basically of two different types. This procedure corresponds to a mapping of the itinerant ͑i͒ Stoner excitations, in which an electron is excited from electron system onto an effective Heisenberg Hamiltonian an occupied state of the majority-spin band to an empty state with classical spins of the minority-spin band and creates an electron-hole pair of triplet spin. They are associated with longitudinal fluctua- ϭϪ ͑ ͒ Heff ͚ Jijei ej , 1 tions of the magnetization. iÞ j • 0163-1829/2001/64͑17͒/174402͑9͒/$20.0064 174402-1 ©2001 The American Physical Society PAJDA, KUDRNOVSKY´ , TUREK, DRCHAL, AND BRUNO PHYSICAL REVIEW B 64 174402 where Jij is the exchange interaction energy between two have been underestimated in a number of previous 2,5,13,14 particular sites (i, j), and ei ,ej are unit vectors pointing in studies, and the claimed agreement with experiment is the direction of local magnetic moments at sites (i, j), re- thus fortituous. We shall present a procedure allowing to spectively. The same point of view has been adopted in vari- overcome these difficulties. In addition, we shall demonstrate ous papers recently published on the same topic.1–14 that the evaluation of the spin-wave dispersion E(q) in the The procedure for performing the above mapping onto an real-space approach has to be also done carefully with re- effective Heisenberg Hamiltonian relies on the constrained spect to the convergency of results with the number of shells density-functional theory,15 which allows to obtain the included. ground-state energy for a system subject to certain con- Finally, to obtain thermodynamic quantities such as the straints. In the case of magnetic interactions, the constraint Curie temperature, we apply statistical mechanics to the ef- consists in imposing a given configuration of spin- fective Hamiltonian ͑1͒. In the present paper, we use two polarization directions, namely, along ei within the atomic different approaches to compute the Curie temperature. The cell i. Note that intracell noncollinearity of the spin polariza- first one is the commonly used mean-field approximation tion is neglected since we are primarily interested in low- ͑MFA͒. The limitations of this method are well known: it is ͑ energy excitations due to intercell noncollinearity. correct only in the limit of high temperatures above TC), Once the exchange parameters Jij are obtained, the spin and it fails to describe the low-temperature collective excita- dynamics4,16,17 can be determined from the effective Hamil- tions ͑spin waves͒. The second approach is the Green- tonian ͑1͒ and one obtains the result known from spin-wave function method within the random-phase approximation theories of localized ferromagnets: the spin-wave energy ͑RPA͒.21–26 The RPA is valid not only for high temperatures, E(q) is related to the exchange parameters Jij by a simple but also at low temperatures, and it describes correctly the Fourier transformation collective excitations ͑spin waves͒. In the intermediate re- ͑ gime around TC), it represents a rather good approximation 4␮ that may be viewed as an interpolation between the high- and ͑ ͒ϭ B ͑ Ϫ ͓ ͔͒ ͑ ͒ E q ͚ J0 j 1 exp iq R0 j , 2 low-temperature regimes. It usually yields a better estimate M jÞ0 • of the Curie temperature as compared to the MFA. It should ϭ Ϫ be noted, however, that both the MFA and RPA fail to de- where R0 j R0 Rj denote lattice vectors in the real space, q is a vector in the corresponding Brillouin zone, and M is scribe correctly the critical behavior and yield in particular ␮ ͒ incorrect critical exponents. the magnetic moment per atom ( B is the Bohr magneton . There are basically two approaches to calculate the ex- change parameters and spin-wave energies. The first one that II. FORMALISM we adopt in the present paper, referred to as the real-space The site-off diagonal exchange interactions J are calcu- approach, consists in calculating directly J by employing ij ij lated using the expression2 the change of energy associated with a constrained rotation 2 of the spin-polarization axes in cells i and j. In the frame- 1 work of the so-called magnetic force theorem2,18 the change ϭ ͵ ͓͕ "͑ ͒Ϫ #͑ ͖͒ " ͑ ͒ Jij ␲ Im trL Pi z Pi z gij z of the total energy of the system can be approximated by the 4 C " # # corresponding change of one-particle energies that signifi- ϫ͕P ͑z͒ϪP ͑z͖͒g ͑z͔͒dz, ͑3͒ cantly simplifies calculations. The spin-wave energies are j j ji then obtained from Eq.
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