- Email: [email protected]
Modeling of magnetic and crystal disorder in magnetic oxides
F_J[ ELSEVIER
Physica B 241 243 (1998)442 444
Modeling of magnetic and crystal disorder in magnetic oxides A. Mellerg~rd a'*, R.L. McGreevy b a Material Physics. Royal Institute o[" Technology, S-IO0 44 Stockholm, Sweden b Studsvik Neutron Research Laboratory, S-611 82 Nykb)~ing. Sweden
Abstract
Many crystalline materials have interesting properties, both scientifically and technologically, which are related to the detailed magnetic and crystalline structures on a local scale. Recent advances in e.g., GMR (Giant Magnetoresistance) materials, indicate an intimate coupling of the local spin structure to the conduction properties. However, almost all diffraction work on these and related materials concentrate on the long-range crystal and magnetic structures. Only a few studies of the local order have been performed and these have not simultaneously considered magnetic and atomic scattering. We have therefore developed a new reverse Monte Carlo (RMC) method for modeling both lattice and magnetic disorder in powder crystalline materials by direct calculation of the structure factor. The method, and results from modeling test data on the lattice and magnetic structure of MnO around the N6el temperature are presented. The prospects for modeling GMR materials are good. ,~, 1998 Elsevier Science B.V. All rights reserved. K e v w o r d s : Computer modeling; Diffuse scattering
When studying atomic and magnetic structure with powder neutron diffraction techniques the data is usually analyzed in terms of the time-averaged long-range order corresponding to the observed Bragg peaks. The diffuse scattering, being the direct result of static and dynamic displacements, is rarely considered and so information is lost. A general, self-consistent method of including the effect of static and dynamic disorder is provided by the reverse Monte Carlo (RMC) technique [1]. The basic approach of this method is to fit the calculated scattering cross-section of an atomic model to the experimental data. The scattering *Corresponding author. Fax: + 46 08 24 91 31; e-mail: [email protected].
cross-section is obtained as the Fourier transform of the pair distribution function as calculated from the model. Atoms are moved randomly and after each move the resulting pattern is checked for improvement of the fit. The method has been applied to various systems, including magnetic structure [1]. For materials with long-range order the RMC transform method can lead to serious truncation errors due to the finite model size. It is also not possible to include the effect of the experimental resolution. These problems can be avoided by using a direct calculation of the total structure factor [-2,3]. This is done by making the model a supercell of the crystal unit cell, with periodic boundary conditions. Thus, the model scattering is described by
0921-4526/98/$19.00 ~ 1998 Elsevier Science B.V. All rights reserved PII S092 1-4526(97)006 14-5
A. Mellerghrd, R.L. McGreevy / Physica B 241 243 (1998) 442-444
the allowed Bragg peaks of the supercell. For a perfectly ordered model, scattering will be non-zero only for the subset of Bragg peaks of the crystal cell. If atoms are displaced from their equilibrium crystal sites then intensity will also appear in additional peaks, these effectively being the diffuse scattering of the disordered model crystal. In the limit of large models this will approach the smooth diffuse scattering of experimental data. We have modified this RMC method to include also the magnetic scattering [3]. The total cross-section is then simply the sum of the squared lattice and magnetic structure factors summed over all reciprocal vectors. This cross-section is calculated on an absolute scale so suitable data must be measured, i.e. with proper background corrections and normalization. Full details of the algorithm are given elsewhere [3]. To evaluate this method we have measured total neutron scattering cross-sections for MnO at six temperatures between 15 and 200 K using the SLAD diffractometer at N F L Studsvik. This antiferromagnetic system, first studied with neutron diffraction by Shull and Smart [4], has a large effective moment. Thus, it is easy to study the development of magnetic diffuse scattering around the Nbel temperature, TN = 120 K. The crystal structure is similar to rock-salt but has a small rhombohedral distortion close to and below TN. Initially, a model was set up for the 15 K data using lattice parameters and Debye-Waller factors obtained from Rietveld refinement. Both the refinement and RMC modeling were performed using the corresponding hexagonal unit cell to take care of the enlarged magnetic cell. The initial model, consisting of 6 × 6 × 4 hexagonal unit cells (6912 atoms and 3456 spins), was produced by applying Gaussian distributed random displacements (with widths given by the refined thermal factors) to atoms at the equilibrium lattice sites. The refinement also gave the spin orientation as the [1 - 1 0] direction in the pseudo-cubic cell, which was used for the initial spin configuration. The magnitude of the effective moment was found to be 5.9~tB in agreement with the expected spin ~. All temperatures were modeled using some 50000 accepted moves and 30000 spin rotations, after which no further improvement in the fit was found. We
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rlA Fig. I. Top: Results of powder RMC modeling for MnO at 200 K. Experimental F(Q) (solid line), RMC total F(Q) (dashed line) with magnetic diffuse part (dot) and lattice diffuse part (short dash), the two latter offset by -0.1. Bottom: The spin-spin correlation function (#(O).tt(r)) for 15 K (dash), 100 K (dot) and 200 K (solid curve).
started with the data at 15 K and the fitted configurations were then used as initial models for successively higher temperatures. The agreement of calculated and experimental data is generally good at all temperatures (Fig. 1, top). Both Bragg and diffuse intensity are well reproduced. The 200 K show excellent agreement at all Q. For the lower temperatures the Bragg scattering is properly calculated but there are small deviations in the diffuse scattering at Q --~ 2 ~,-1.
444
A. Mellerghrd. R.L. McGreeey / Physica B 241 243 (1998) 442 444
This can be understood by examining the width aL of the diffuse line at 1.2 A - 1 ag ~ 0.1 A - 1 implies a mean spin correlation length of 2 ~ / a g 60 A which is larger than the smallest dimension of the model ( ~ 40 A) and therefore cannot be fitted perfectly. For the 200 K data we instead have az ~ 0.5 A - 1, i.e. a correlation length of the order of 13 A, well below the model size. The experimental data show a distinct difference from a pure paramagnetic structure factor even at 200 K, so there has to be some local magnetic order in the models at all temperatures. To see this clearly we plot the spin-spin correlation as a function of r in Fig. 1, bottom. The spin spin correlation function is the integral of la(0)" ~(r) for atoms at a distance between r and r + dr from a central magnetic atom averaged over all magnetic atoms as centers, per unit volume. Due to the rhombohedral distortion spin-up and spin-down sublattices are clearly resolved at 15 K for nearest neighbors, at r ~ 3.1 A. At 200 K nearest neighbors are all at about the same distance, so any ferro- and antiferromagnetic components cancel. Looking at next-nearest distances, however, r ~ 4.4 A, short-range order is still evident at 200 K; for larger distances the correlations are rapidly damped. More detailed discussion
and interpretation of these models for the magnetic structure of M n O will be presented elsewhere. In conclusion, we have shown that a direct calculation R M C technique can be used to model both spin and lattice disorder in a simple transition-metal oxide. We emphasize how well the model scattering accounts for the total experimental pattern without having to invoke any arbitrarily fitted backgrounds or scale factors. We believe that this also makes the technique applicable to more complicated oxides, possibly involving mixed spin states. Thus we are planning to use the method for studies of local disorder in G M R materials and high-To superconductors. We gratefully acknowledge the support of the Swedish Natural Sciences Research Council.
References [1] R.L. McGreevy,Nucl. Instr. and Meth. A 354 (1995) 1. [2] W. Montfrooij et al., J. Appl. Crystallogr. 29 (1996) 285. [3] A. Mellerg~rd, R.L. McGreevy,Physica Scripta; submitted see also http://www.studsvik.uu.se/rmc/conference/abs.htm. [4] C.G. Shull, J.S. Smart, Phys. Rev. 76 {1949) 1256.
Physica B 241 243 (1998)442 444
Modeling of magnetic and crystal disorder in magnetic oxides A. Mellerg~rd a'*, R.L. McGreevy b a Material Physics. Royal Institute o[" Technology, S-IO0 44 Stockholm, Sweden b Studsvik Neutron Research Laboratory, S-611 82 Nykb)~ing. Sweden
Abstract
Many crystalline materials have interesting properties, both scientifically and technologically, which are related to the detailed magnetic and crystalline structures on a local scale. Recent advances in e.g., GMR (Giant Magnetoresistance) materials, indicate an intimate coupling of the local spin structure to the conduction properties. However, almost all diffraction work on these and related materials concentrate on the long-range crystal and magnetic structures. Only a few studies of the local order have been performed and these have not simultaneously considered magnetic and atomic scattering. We have therefore developed a new reverse Monte Carlo (RMC) method for modeling both lattice and magnetic disorder in powder crystalline materials by direct calculation of the structure factor. The method, and results from modeling test data on the lattice and magnetic structure of MnO around the N6el temperature are presented. The prospects for modeling GMR materials are good. ,~, 1998 Elsevier Science B.V. All rights reserved. K e v w o r d s : Computer modeling; Diffuse scattering
When studying atomic and magnetic structure with powder neutron diffraction techniques the data is usually analyzed in terms of the time-averaged long-range order corresponding to the observed Bragg peaks. The diffuse scattering, being the direct result of static and dynamic displacements, is rarely considered and so information is lost. A general, self-consistent method of including the effect of static and dynamic disorder is provided by the reverse Monte Carlo (RMC) technique [1]. The basic approach of this method is to fit the calculated scattering cross-section of an atomic model to the experimental data. The scattering *Corresponding author. Fax: + 46 08 24 91 31; e-mail: [email protected].
cross-section is obtained as the Fourier transform of the pair distribution function as calculated from the model. Atoms are moved randomly and after each move the resulting pattern is checked for improvement of the fit. The method has been applied to various systems, including magnetic structure [1]. For materials with long-range order the RMC transform method can lead to serious truncation errors due to the finite model size. It is also not possible to include the effect of the experimental resolution. These problems can be avoided by using a direct calculation of the total structure factor [-2,3]. This is done by making the model a supercell of the crystal unit cell, with periodic boundary conditions. Thus, the model scattering is described by
0921-4526/98/$19.00 ~ 1998 Elsevier Science B.V. All rights reserved PII S092 1-4526(97)006 14-5
A. Mellerghrd, R.L. McGreevy / Physica B 241 243 (1998) 442-444
the allowed Bragg peaks of the supercell. For a perfectly ordered model, scattering will be non-zero only for the subset of Bragg peaks of the crystal cell. If atoms are displaced from their equilibrium crystal sites then intensity will also appear in additional peaks, these effectively being the diffuse scattering of the disordered model crystal. In the limit of large models this will approach the smooth diffuse scattering of experimental data. We have modified this RMC method to include also the magnetic scattering [3]. The total cross-section is then simply the sum of the squared lattice and magnetic structure factors summed over all reciprocal vectors. This cross-section is calculated on an absolute scale so suitable data must be measured, i.e. with proper background corrections and normalization. Full details of the algorithm are given elsewhere [3]. To evaluate this method we have measured total neutron scattering cross-sections for MnO at six temperatures between 15 and 200 K using the SLAD diffractometer at N F L Studsvik. This antiferromagnetic system, first studied with neutron diffraction by Shull and Smart [4], has a large effective moment. Thus, it is easy to study the development of magnetic diffuse scattering around the Nbel temperature, TN = 120 K. The crystal structure is similar to rock-salt but has a small rhombohedral distortion close to and below TN. Initially, a model was set up for the 15 K data using lattice parameters and Debye-Waller factors obtained from Rietveld refinement. Both the refinement and RMC modeling were performed using the corresponding hexagonal unit cell to take care of the enlarged magnetic cell. The initial model, consisting of 6 × 6 × 4 hexagonal unit cells (6912 atoms and 3456 spins), was produced by applying Gaussian distributed random displacements (with widths given by the refined thermal factors) to atoms at the equilibrium lattice sites. The refinement also gave the spin orientation as the [1 - 1 0] direction in the pseudo-cubic cell, which was used for the initial spin configuration. The magnitude of the effective moment was found to be 5.9~tB in agreement with the expected spin ~. All temperatures were modeled using some 50000 accepted moves and 30000 spin rotations, after which no further improvement in the fit was found. We
1,0
u
443
!
i
0,6
5" o41 0,2,
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0,0. ! '""
2
: .....
i
,
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6
8
10
Q/A-1 1,5-
1,o] 0r5.
A •~
J .:
~
,
.:
0,0.
O " =L - " -0,5V -1,0. -1,5.
rlA Fig. I. Top: Results of powder RMC modeling for MnO at 200 K. Experimental F(Q) (solid line), RMC total F(Q) (dashed line) with magnetic diffuse part (dot) and lattice diffuse part (short dash), the two latter offset by -0.1. Bottom: The spin-spin correlation function (#(O).tt(r)) for 15 K (dash), 100 K (dot) and 200 K (solid curve).
started with the data at 15 K and the fitted configurations were then used as initial models for successively higher temperatures. The agreement of calculated and experimental data is generally good at all temperatures (Fig. 1, top). Both Bragg and diffuse intensity are well reproduced. The 200 K show excellent agreement at all Q. For the lower temperatures the Bragg scattering is properly calculated but there are small deviations in the diffuse scattering at Q --~ 2 ~,-1.
444
A. Mellerghrd. R.L. McGreeey / Physica B 241 243 (1998) 442 444
This can be understood by examining the width aL of the diffuse line at 1.2 A - 1 ag ~ 0.1 A - 1 implies a mean spin correlation length of 2 ~ / a g 60 A which is larger than the smallest dimension of the model ( ~ 40 A) and therefore cannot be fitted perfectly. For the 200 K data we instead have az ~ 0.5 A - 1, i.e. a correlation length of the order of 13 A, well below the model size. The experimental data show a distinct difference from a pure paramagnetic structure factor even at 200 K, so there has to be some local magnetic order in the models at all temperatures. To see this clearly we plot the spin-spin correlation as a function of r in Fig. 1, bottom. The spin spin correlation function is the integral of la(0)" ~(r) for atoms at a distance between r and r + dr from a central magnetic atom averaged over all magnetic atoms as centers, per unit volume. Due to the rhombohedral distortion spin-up and spin-down sublattices are clearly resolved at 15 K for nearest neighbors, at r ~ 3.1 A. At 200 K nearest neighbors are all at about the same distance, so any ferro- and antiferromagnetic components cancel. Looking at next-nearest distances, however, r ~ 4.4 A, short-range order is still evident at 200 K; for larger distances the correlations are rapidly damped. More detailed discussion
and interpretation of these models for the magnetic structure of M n O will be presented elsewhere. In conclusion, we have shown that a direct calculation R M C technique can be used to model both spin and lattice disorder in a simple transition-metal oxide. We emphasize how well the model scattering accounts for the total experimental pattern without having to invoke any arbitrarily fitted backgrounds or scale factors. We believe that this also makes the technique applicable to more complicated oxides, possibly involving mixed spin states. Thus we are planning to use the method for studies of local disorder in G M R materials and high-To superconductors. We gratefully acknowledge the support of the Swedish Natural Sciences Research Council.
References [1] R.L. McGreevy,Nucl. Instr. and Meth. A 354 (1995) 1. [2] W. Montfrooij et al., J. Appl. Crystallogr. 29 (1996) 285. [3] A. Mellerg~rd, R.L. McGreevy,Physica Scripta; submitted see also http://www.studsvik.uu.se/rmc/conference/abs.htm. [4] C.G. Shull, J.S. Smart, Phys. Rev. 76 {1949) 1256.