Exchange stiffness of ferromagnets
- PDF / 399,467 Bytes
- 8 Pages / 439.37 x 666.142 pts Page_size
- 85 Downloads / 200 Views
Exchange stiffness of ferromagnets M. D. Kuz’min1,a , K. P. Skokov2 , L. V. B. Diop3 , I. A. Radulov2 , O. Gutfleisch2 1 IM2NP, UMR CNRS 7334, Aix-Marseille Université, 13397 Marseille, France 2 Institut für Materialwissenschaft, Technische Universität Darmstadt, D-64287 Darmstadt, Germany 3 Institut Jean Lamour, CNRS, Université de Lorraine, 54011 Nancy, France
Received: 7 January 2020 / Accepted: 21 February 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Spin-wave and exchange stiffness constants of 22 ferromagnetic compounds have been deduced from their spontaneous magnetization, Ms , by using an improved technique. The improvement consists in utilizing the entire Ms (T ) curve, up to the Curie point, rather than just its low-temperature part, with T TC . For 17 of the 22 ferromagnets, literature data have been used, while 5 compounds have been studied anew, on single crystals grown for the purpose. Exchange stiffness A is the prefactor of the gradient term in the energy density of an inhomogeneously magnetized ferromagnet E = A |∇m x |2 + |∇m y |2 + |∇m z |2 , (1) where m is the unit vector in the direction of magnetization. A is an important characteristic of a ferromagnetic material. It is related to the Curie temperature, TC , and determines the width and energy of domain walls. It is a necessary ingredient of any micromagnetic calculation, the applications ranging from the coercivity of bulk materials, through composite (exchangespring) magnets, to nanoscopic devices such as the racetrack memory [1]. A way to determine A was proposed by Landau and put in writing by Lifshitz [2]. In this method, one first finds the spin-wave stiffness D (i.e., the prefactor in the magnon dispersion relation, h¯ ωq = Dq 2 ) by fitting a measured temperature dependence of spontaneous magnetization to Bloch’s law [3]: gμ B kT 3/2 Ms (T ) = M0 1 − 0.0586 . (2) M0 D Here, M0 is (volume) saturation magnetization. Knowing D, one then finds A from a simple proportionality relation that exists between both quantities [2]: A=
M0 D 2gμ B
(3)
The advantageous, model-independent character of this relation was pointed out by Herring and Kittel [4]. Later, in the light of the density functional theory (DFT), it was realized
In Memory of Dominique Givord. a corresponding author
0123456789().: V,-vol
123
301
Page 2 of 8
Eur. Phys. J. Plus
(2020) 135:301
that the decisive advantage of Eq. (3) is that A, D, and the proportionality factor—equal to one-half of the mean spin density—are all ground-state properties. (Our consideration is limited to materials where orbital contribution to M0 can be neglected and, therefore, g = 2.) One difficulty of the Landau–Lifshitz technique is that the Bloch’s law (2) is only valid at low temperature (strictly speaking, infinitesimally low), and it is from the infinitesimally small difference between Ms and M0 that D has to be extracted. Other methods have their own difficulties, and as a result, our knowledge of A and D rem
Data Loading...
