COMPUTATIONAL MATERIALS SCIENCE Computational Materials Science 10 (1998) 217-220
Improved angular convergence in noncollinear magnetic orders calculations Daniel C.A. Stoeffler *, Clara C. Cornea IPCMS - Gemme, 23, rue du Loess, F-67037 Strasbourg, France
Abstract The method we use for determining the noncollinear orders in metallic superlattices is presented. The electronic structure is obtained within a semi-empirical tight binding framework and the real space recursion technique which allows to study nonsymmetric systems. We focus our attention on the iterative procedure for reaching the angular self-consistency which is extremely slow to converge. We show that the usual mixing scheme is unsuited (because the output angle is directly equal to the next input) and we give an example of possible speed increase by extrapolating the angles resulting from mixing over a few iterations. Copyright 0 1998 Elsevier Science B.V. Keywords: Noncollinear magnetism; Convergence acceleration
1. Introduction Noncollinear magnetism in the electronic structure framework is a relatively new research area. However, most of the work has been done fixing the direction
of each local quantization axis and determining the length of the magnetic moments for these angular frozen noncollinear states. Spin spiral [l] or nearly helicoidal  magnetic configurations are illustrations of such studies. It is only during these last years that the angular degree of freedom has been included in the self-consistency resulting from the increase of computer capabilities. The major computing time increase comes not from a higher complexity of the electronic structure description (larger matrix, complex Hamiltonian, three components of the magnetic * Corresponding author. Tel.: +33 3 88 10 70 65; fax: +33 3 88 10 72 49; e-mail: [email protected]
cal moment at the end of the calculation for magnetic configurations nearly converged. In this paper, we illustrate this behaviour in the case of metallic superlattices and show that the convergence becomes extremely slow when a mixing scheme is used. In the second part, we give a very simple method allowing a speed up of the convergence of more than 3 by extrapolation of the angular variations.
2. Method of calculation for noncollinear magnetism We use a tight binding Hamiltonian which can be written as the sum of a band H&d and an exchange
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relaxation during the iterative calculation. This is mainly due to the small torque acting on each lo-
D.C.A. Sloefleer; C.C. Cornea/Computational
Ii; 1; m)[(~yl ,. m
10 (1998) 217-220
given by the direction
of the “d” local magnetic
More details about the self-consistency
given in another paper of these proceedings
Finally, the tight-binding
are either ob-
tained from the literature  or determined reproduce
. in order to
the ab initio results of a pre-
study in the collinear case .
Hexc=~(-jl,lMi~)li:i:m)(i:i;mi 3. Angular self-consistency 1
cos 0 eivpi sin
where EP, .I5m is the energy level of the site i for the symmetry (1, m); Ui,l A Ni,l is the intrasite Coulomb contri-
In a simple mixing scheme, the next input variable is chosen between the input and the output ones with a mixing factor x corresponding to the weight of the
bution from the local charge transfer A Ni.l, Ui.’lbeing 50
the intrasite effective Coulomb parameter; r(:‘;,mm, 1 1 is the so-called hopping integral between orbitals of site i for the symmetry (1, m) and of site i’ for the symmetry (1’, m’); Zil is the intrasite effective exchange parameter and Mil the local magnetic moment whose
direction is given by the two angles (@i, vi) in the usual spherical representation; the [2 x 21 matrix represents the spin part. Such a Hamiltonian allows the determination of the electronic structure for a set of
~-- Simple Mixing
vectorial magnetic moments. In our case, we determine the electronic structure with the real space recursion technique which allows
(averagedover 3 iterations) . - An=7
of the local density of states pro-
jected on an arbitrary local quantization
axis by rotat-
ing the initial element of the recursion basis. We use a cluster containing approximately 5000 atoms for determining 24 levels in the continuous fraction for each “ s,” “p,” “d” symmetry, each spin and each quantization axis. The resulting magnetic moment vector Mi = M,.i&+ Mei& + M,i& is obtained by the successive calculation of three components Mr ,MO, Mq in the local spherical basis (li,, i&, li,)i We use an iterative method to obtain the angular self-consistent solution meaning that the output vector must be the same as the input one. In our case, the angular self-consistency is achieved when the “d” part of the components Me and Mq perpendicular to the local quantization axis are equal to zero (smaller than 5 x 10P5). This is equivalent to the assumption that the local quantization axis
Iterations Fig. I. Variations of the total energy during the iterations relative to the converged value obtained with a simple complete mixing scheme (thick solid line) and with the mixing-extrapolation method for An = 6 (open circles) for CosMn7 superlattices. The insert shows the comparison between the two previous converging cases and the nonconverging variations for An = 7; the variations for An = 6 have been averaged over three iterations. The scale of the horizontal axis has been renormalized by the speed up: the smallest (largest) axis labels correspond (respectively) to the accelerated (normal) calculation.
D.C.A. Stoefler; C.C. Cornea/Computational Materials Science 10 (1998) 217-220
I!_.__ 1 -_
Iterations Fig. 2. Variations of the local 0 angle on selected sites obtained with a simple complete mixing scheme (solid line) and with the mixing-extrapolationmethod for An = 7 (open symbols)for CogMn7 superlattices. The scale of the horizontal axis has been renormalized by the speed up: the smallest (largest) axis labels correspond (respectively) to the accelerated (normal) calculation.
output values, 1 - x being the weight of the input values. This is the case for the charge transfers (with x = 0.1-0.2) and the magnitude of the magnetic moments (with x = 0.5-l) and can be the case for the local quantization axis (the input and the output magnetic moments vectors defining a plane and the new input is in the same plane between these two vectors). However, the angular relaxation is so slow that a complete mixing can be used: the next input direction of the quantization axis is directly the output one. This corresponds to the fastest variations for a mixing scheme. Unfortunately, even with a complete mixing the convergence needs to much iterations. When the number of atoms in the cell is large, like in superlattices, it is possible that different noncollinear solutions are obtained. We have then to compare their energies which gives the most stable solution .
Fig. 3. Variations of the local 19 angle on the same selected sites as in Fig. 2 obtained with a simple complete mixing scheme (solid line) and with the mixing-extrapolationmethod for An = 7 (open symbols) for CosMn7 superlattices. The inserts correspond to a magnification for the last iterations. The scale of the horizontal axis has been renormalized similar to Figs. 1 and 2.
The figures of this paper are illustrations of the calculations of the magnetic properties for CosMn7 superlattices with ppi = 0 [3,5] and show the total energy (Fig. 1) and the angles on selected sites (Figs. 2-4) during the iterations n. With the simple mixing, the convergence is obtained after ii = 454 iterations. Fig. 2 shows clearly that the angles vary very smoothly during the calculations. This is why, it is reasonable to assume that they follow regular curves which can be interpolated. This has been done using a mixing-extrapolation method which consists in (i) calculating three points (6$(n), 8j(n + l), 8i(n + 2)) with the usual complete mixing, (ii) interpolating these three angle values as a function of the iteration index n with a quadratic polynomial &(n), and (iii) extrapolating up to a large iteration index jumping a few iterations An. The next input is then given by
D.C.A. StoefJleer; C.C. Cornea/Computational
Materials Science 10 (1998) 217-220
and the angle (Fig. 3) oscillate
around an incorrect
value. For An = 8, the noise level increases An=8
the iterations and the calculation
diverges as shown by
the variations of the angles in Fig. 4.
In this paper, we have shown that the angular convergence is very slow and needs a large number of iterations to be achieved when a simple complete mixing scheme is used. However, the smooth variations of the angle values can be interpolated in order to increase the convergence speed. This first step of our convergence improvement has been illustrated, for metallic superlattices, by a mixing-extrapolation method allowing a speed up nearly equal to 3. Unfortunately, this -0.25 L
is limited by the noise coming from the extrapolation.
? -LooOCOOCaCL 0L 5,L -A-Y
641200 96/300 Iterations
Fig. 4. Variations of the local 0 angle on selected sites obtained with a simple complete mixing scheme (solid line) and with the mixing-extrapolation method for (open symbols) for CosMn7 superlattices. The scale of the horizontal axis has been renormalized similar to Figs. 1 and 2.
until the convergence is achieved. In principle, if the interpolating function follows the regular curve, the convergence speed up is equal to 1 + An/3. For the situation presented in Figs. 1 and 2 the convergence is obtained after 161 iterations with An = 6: the speed up is then equal to 2.8 representing 93% of the best one. This shows the validity of the mixingextrapolation technique. Of course, the speed up increases when An is augmented. However, the extrapolation introduces a little noise which increases with An and the calculation will become unstable for large An values. This is illustrated by Figs. 3 and 4 for An = 8 and An = 7. For, An = 7 the noise level is nearly constant so that the calculation does not converge but the energies (Fig. 1)
In future work, we will try to enhance the stability of the acceleration by controlling the noise level.
Acknowledgements The calculations have been mainly realized on the T3E parallel computer at the Institut du DCveloppement et des Ressources en Informatique Scientifique (IDRIS) of the CNRS and on the SP2 parallel computer of the Centre National Universitaire
References [l] L.M. Sandratskii and J. Kubler, Phys. Rev. B 47 (1993) 4854.  D. Stoeffler and F. Gautier, J. Magn. Magn. Mater. 121 ( 1993) 259-265.  C.C. Cornea and D. Stoeffler, these proceedings.  O.K. Andersen, 0. Jepsen and D. Gloetzel, in: Highlights of Condensed Mater Theory, eds. F. Bassani, F. Fumi and M.P. Tosi (North-Holland, Amsterdam, 1985) p. 59.  CC. Cornea-Borodi, D.C.A. Stoeffler and F. Gautier, J. Magn. Magn. Mater. 165 (1997) 450-453; D.C.A. Stoeffler, to be published.