Journal of Magnetism and Magnetic Materials 54-57 (1986) 965-966 965 LSDF-APPROACH TO THE THEORY OF EXCHANGE INTERACTIONS IN MAGNETIC METALS A.I. LIECHTENSTEIN *, M.I. KATSNELSON **, V.P. ANTROPOV ** and V.A. GUBANOV * • Institute of Chemisto' and ** Institute of Metal Physics, Sverdlovsk, GSP-145, USSR In the framework of the KKR-Green's function formalism and the local spin-density functi, onal (LSDF) approach the rigorous expressions applicable for band-structure calculations of effective exchange interactions parameters and spin-wave stiffness constant are presented. The Curie temperature for ferromagnetic metals are also calculated making use of the mean-field approximation. Recently a significant progress has been achieved in the quantitative theory of band magnetism. In the works of Oguchi et al. [1,2] the calculation of effective ex- change parameters in the transition metals and their compounds have been carried out on the basis of a KKR- CPA treatment of the paramagnetic electronic structure. The indirect spin interactions between pairs of magnetic impurities in Cu and Ag were investigated in ref. [3] by means of the KKR-Green's function method. Unfortunately, there are some difficulties in such theories arising from (i) ignoring the double counted terms when calculating the ferro-antiferromag- netic total energy (TE) differences [1] and (ii) a strong dependence of exchange parameters on magnetic con- figurations [3]. We generalised the approach by Oguchi et al. [1,2] and derived a rigorous expression for ex- change interaction parameters in the LSDF-scheme ap- plicable for arbitrary magnetic configuration's in crystals. The crucial point of our approach is the analy- sis of TE variations due to the small deviation of local magnetic moments connected with several muffin-tin (MT) spheres. As follows from the "local force" theo- rem derived by Andersen [4] in the framework of the LSDF approach, the first-order variation 8E with re- spect to fluctuations of charge density 8n(r) or spin density 8m(r) equals the sum of the variations of one-electron energies calculated with the fixed ground state potential. In terms of multiple-scattering theory one has [1]: 1 f~ 8E=~J ddmTrln(1-St-'T), (1) where T=(t -1- G) -1 is scattering path operator in the ground state, G is the structural Green's function matrix in the KKR method, t is the single site scatter- ing matrix. Variation of the t l-matrix due to deviation of the local magnetic moment at site i from its initial direction 8e~ can be expressed as follows: 8t7' = ~(tT,' - ',-?) ~," °- (2) Here a are Pauli matrices. If we consider a ferromagnetic crystal where local magnetic moments are directed along the (0, 0, 1) axis and change the direction of the magnetic moment of a given atom by a small angle 0, 8e i = ( - sin 0, 0, 1 - cos 0)8 m the variation of spin density appears to be of the second order in 0. Deriving the TE variation along eq. (1) and comparing the results with those of the classical Heisenberg model: ~iEex = ~y2Jo/(1 - cos 0) = Jo 02 one can obtain the following formulae for the effective exchange interaction parameters [5,6]: 1 f',.ddmTr{~ao(T~O_ T~O)+,aoT~OaoTOO} Jo=-~ = E J0j, (3) 1~-[) where 1 t'c~ ij # Jil= ~ J dclmTrA,T, AIT+, here A, = tl-r I - ti-+ ~, and Tr means the trace over orbital indices of scattering matrices. As follows from the spin-fluctuation theories [7], only effective classical Hei- senberg Hamiltonian can be introduced for metals. This very concept allow us to determine the exchange param- eters (3) for a given magnetic configuration for which the T-matrices have been calculated. In order to investigate the spin-wave spectrum we calculate the energy E(q) of classical spin spirals for which 0, = q- R, (% = 0, q is the wave vector, R, is the lattice vector). For small values of q, the variation of spin densities has the order of qZ, and eq. (1) leads to the exact expression for the spin-wave stiffness constant [5]: 2 O2E q=O 1 J'"d¢ImTr~] { 2~-M r aT{ ~T~ ] - k ~'~-~'~-3~~ )' (4) where M is the magnitude of local magnetic moment, T k is the scattering path operator in k-representation, the k summation is carried out over the Brillouin zone. 0304-8853/86/$03.50 © Elsevier Science Publishers B.V.