Calculations of Spin-Wave Energy Gap in Transition Metals
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CALCULATIONS OF SPIN-WAVE ENERGY GAP IN TRANSITION METALS 2
1
V.P.ANTROPOV and A.I.LIECHTENSTEIN Max-Planck-Institut fiir Festkorperforschung Heisenbergstr. 1,D-7000 Stuttgart 80, FRG Abstract Analytical expressions for the total energy derivative with respect to the magnetic moment rotations have been derived in the relativistic local-spin density functional approach and multiple scattering theory. The spin-wave stiffness constant as well as the gap in the spin-wave spectrum due to relativistic anisotropy effects is calculated in KKR-ASA approximation for bcc Fe. This is compared with other theoretical calculations and available experimental data. KEYWORDS: Total energy, Energy gap in spin-wave spectrum, Local-force theorem, Magnetic anisotropy, Exchange interactions
Introduction Recently significant progress has been achieved in the first-principle calculations of groundstate magnetic properties for transition metals (spin and orbital magnetic moments, exchange interactions, spin-wave stiffness etc.) [1]. Using a spin-spiral band calculation [2] it is possible to investigate a spin-wave energy spectrum for different q-vectors in the Brillouin zone. In order to complete the description of the elementary excitations in a ferromagnetic system we discuss in the present paper the role of spin-orbit coupling. This relativistic effect determines the energy gap in the spin-wave spectrum for 41=0 [3]. In order to analyze such an energy gap, one has to calculate the second derivative of total energy with respect to the deviation of the magnetization axis from the easy direction. We generalize the method proposed for the calculation of exchange interaction parameters in ferromagnetic metals [4] to the relativistic spin-polarized case, and obtain an analytical expression for such a kind of perturbation. Numerical calculations have been done for bcc Fe.
Method Our purpose is to find a rigorous expression for the variation of the total energy with respect to small magnetic excitations. In the local-spin density functional (LSDF) approximation, the total energy can be written in the following form [5]:
E
=
EP
=
Edc
=
Esp-Edc
nr)n_
+ j[,(cc
6n[n)+
)d__
;
m""--2:) 6m
-nc.,cldi"(1
It can be shown that the first variations bE/ln and E/l6m are equal to the variation of singlepartical terms in (1) with the ground-state spin-density being kept fixed ("local-force" theorem) [6,4]. Therefore we can express b E in terms of the density of states:
6E =
61
cn(c)dE = CF6Z - I"
N(c)de
Mat. Res. Soc. Symp. Proc. Vol. 253. @1992 Materials Research Society
(2)
326
In the case of magnetic excitations the total number of electrons is constant and 6Z = 0. For solving the one-electron problem, we used a spin-polarized, fully-relativistic multiple scattering method in the atomic sphere approximation [7]. In this case the single site t-matrix (or potential function P(e)) has a non-diagonal matrix element in 1, mi, a -representation. According to Lloyd's formula [8] one can write the following expression for the number of st
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