Ab-initio calculation of spin-waves stiffness constant and Curie 
temperature of Fe, Co, and Ni

M. Pajda and P. Bruno
in cooperation with
J. Kudrnovský
Institute of Physics, Academy of Sciences of the Czech Republic
Na Slovance 2, CZ-18040 Praha 8, Czech Republic


Although magnetic interactions in transition metals are rather well understood, nevertheless, some discrepancies between theory and experiment remain for the Curie temperature and stiffness constant. On the theoretical side a traditional approach towards the Curie temperature is based on so-called mean-field approximation which neglects some important factors such as, for example, correlated fluctuations of the spins around their equilibrium values. As a result calculated values are either overestimated (Fe, Co) or underestimated (Ni) [1]. Moreover, hitherto reported calculations of the spin-wave stiffness constants [2, 3] are based upon non-convergent expression.

Those facts have thus motivated us to look into the problem of calculations of  the quantities, mentioned above,  for transition metals from first principles. The calculations are performed using scalar-relativistic Green function TB-LMTO method [4]. Employing the force theorem [5] to the local density approximation we calculate the change in the total energy of the system associated with infinitesimal rotation of the local moment within the adiabatic approximation. The energy is mapped onto a Heisenberg hamiltonian. Following the formalism of Liechtenstein et al. [6] we calculate the Heisenberg exchange parameters $J$_ij and hence the spin-wave stiffness constant $D$ which is given by where $M$ is a value of magnetic moment per atom (in _B). As it stands, eq.(1) cannot be used directly, because the long-range oscillatory character of $J$_ij versus $R$_ij in a metal makes it non convergent. Therefore we use instead the following expression which is formally equivalent but numerically convergent:



Next, we use the statistical mechanics of the effective Heisenberg hamiltonian to calculate the Curie temperature. Within the mean-field approximation (MFA),one gets

where  is the spin-wave energy. The MFA is notoriously a poor approximation because it disregards the collective transverse excitations (magnons) which are the most important ones due to their low energy. A much better approximation is provided by the Green's function method within the random phase approximation (RPA) [7] which yields:



where  is a numerical factor ( =1.240). As appears clearly from eqs. (3) and (4), the MFA and the RPA differ essentially in the way they weight the various $J$_ij: in the RPA, far neighbours play a more important role than in the MFA.

The results of calculation along with corresponding experimental data are presented in Table 1. As far as spin-waves stiffness constant is concerned we obtained very good agreement with experiment for Fe, worse but still reasonable for Co and overestimated result in case of Ni. This deterioration of results coincides with a reduction in both the magnetic moments and the range of the exchange interactions. Regarding Curie temperature results, our calculations based on mean-field approximation yield overestimated values for Fe and Co and considerably underestimated value for Ni. Qualitatively similar results were obtained in the past by a number of authors employing this approach. On the other hand using the random phase approximation we improve distinctively theoretical estimation of the Curie temperatures. In contrast to the case of the spin-wave stiffness constants the best result for Curie temperature was obtained for Ni and the worst for Fe.

References

1

 M. van Schilfgaarde and V.P. Antropov, J. Appl. Phys. 85, 4827-4829 (1999).

2

 V.P. Antropov, M.J. Katsnelson,and B.N. Harmon, Physica B 237, 336-340 (1997).

3

 D. Spisak, J. Hafner, J. Magn. Magn. Matter 168, 257-268 (1996).

4

 I. Turek, V. Drchal, J. Kudrnovsky, M. Sob, P. Weinberger, Electronic Structure of Disordered Alloys, Surfaces and Interfaces, (Kluwer Academic Publishers Boston/London/Dordrecht 1997).

5

 A.R. Machintosh and O.K. Andersen, Electrons at the Fermi Surface, ed. M. Springford (Cambridge Univ. Press, London, 1980) p. 149.

6

 A.I. Liechtenstein, M.I. Katsnelson, V.P. Antropov, and V.A. Gubanov, J. Magn. Magn. Matter 67, 65-74 (1987).

7

 R.A. Tahir-Kheli and H.S. Jarrett, Phys. Rev. 135, 4A, A1096 (1964).

8

 H.A. Mook, J.W. Lynn, and M.R. Nicklow, Phys. Rev. Lett. 30, 556 (1973).

9

 R. Pauthenet, J. Appl. Phys. 53, 2029 and 8187 (1982).

10

 G. Shirane, V.J. Minkiewicz, and R. Nathans, J. Appl. Phys. 39, 383 (1968).

Tab. 1
Calculated spin-waves stiffness constants and Curie temperatures with comparison to
experimental data.
 

Metal	$D$Th[meV2]	$D$Ex[meV2]	$T$CMFA[K]	$T$CRPA[K]	$T$Ex[K]
Ni(fcc)	$727$	$555A,422B$	$382$	$641$	$624-631$
Co(fcc)	$667$	$580*B$	$1645$	$1613$	$1388-1398*$
Fe(bcc)	$263$	$280B,330C$	$1427$	$859$	$1044-1045$

*[0.5cm]
-------------------
'*'- data refer to hcp Co
'A'- neutron scattering measurement at 4.2 K [8]
'B'- magnetization measurement [9]
'C'- neutron scattering measurement extrapolated to 0 K [10]