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Magnetic phase stability of 3d-metals and plutonium
Journal of Magnetism and Magnetic Materials 140-144 (1995) 1355-1356
~i ~l~ ~1~
journalof magnetism and magnetic materials
ELSEVIER
Magnetic phase stability of 3d-metals and plutonium V.P. Antropov a,*, M. van Schilfgaarde b, B.N. Harmon a a
Ames Laboratory, Iowa State Uni~,ersity,Ames, IA 50011, USA b SRI International, Menlo Park, CA 94025, USA
Abstract We have performed self-consistent spin-polarized calculations for the magnetic structure and exchange parameters of pure 3d-metals and Pu. Local field effects and the strength of non-Heisenberg interactions are estimated. For a-Mn and a-Pu antiferromagnetic structures are dominant, and a complexe magnetic ordering was found for the low temperature structures. An analytical expression for the magnetic torque and a non-collinear low energy magnetic state of T-Fe are presented.
In spite of the fundamental importance of the pair exchange parameters between local magnetic moments, the number of first-principle calculations of such parameters is still very limited. Also the calculation schemes are very different, so that direct comparison is hard to perform. Some rigorous expressions for exchange parameters and spin-wave stiffness (SWS) were proposed [1,2], but applications of the formalism for real materials has been limited. We analyze the exchange parameters and the stability of magnetic phases in cubic 3d-metals and the a-phases of Mn and Pu using the LMTO-Green-function technique [3]. We start from expressions for the second order term in the Landau expansion introduced in Ref. [2] and rewritten in the form: A ~ = 2~r
de Im TrL{XijPi .pj},
(1)
and introduce the pairwise magnetic torque as
1 [EF , Vii= "~-~-j dE Im TrL{ XijPiXPj}
(2)
with the on-site perturbation having the form p = Tr(6-. /3)/2. The corresponding expression for the SWS D can be obtained using Aij. We corroborate the results of Wang et al. [4] about ferromagnetic (FM) type of interactions between nearest neighbors and antiferromagnetic (AFM) at longer range. Also we found stronger interactions in bcc Fe than in fcc Ni, where exchange weakly oscillates (we analyzed only fifteen shells of neighbors). The real elementary excitations connected with spin rotations are spin waves, and our parameter D can be
* Corresponding author.
directly compared with experiment. The agreement with experiment is satisfactory but not excellent. A possible explanation for that might be the local field effects which can be estimated if we go beyond the one-site approximation used in Ref. [2] and employ the effective vector on-site perturbation for a rotation of spin-density at one particular site [[p X T] × T] (with similar scalar expressions for the exchange). Using such corrections the expressions (both in real and reciprocal space) for the corresponding exchange parameters can be obtained [5]. For SWS the corresponding renormalized values are presented in Table 1. We note that including local field effects decreases the total value of D (about 10%) and leads to better agreement with experiment. To estimate the magnetic transition temperature we first use simple mean field expression for Tc and our values are listed in Table 1. Those numbers should be considered with some caution however, because the simple mean field expression assumes Heisenberg model parameters, whereas o u r Aij are just the second derivative of total energy and include non-Heisenberg types of interactions as well. This is appropriate for real excitations (as, for instance, spin waves), and for a valid estimation of Tc the general form of the model Hamiltonian should be specified. In this case the equation for Tc in mean field approximation is Tc=~
'
(3)
Table 1 Calculated spin wave stiffness (meV ,~2) and Curie temperature for 3d-systems Metal
D
D ioc
Dexp
Tc
Tcexp
Fe (bcc) Ni (fcc)
410 631
375 565
314,550 395,422
1220 380
1040 670
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSD10304-8853(94)00887-6
J - - - + - - - " ' 32
ILP. Antropol et al. /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1355-1356
1356
Table 2 Spin and orbital magnetic moments in c~-phase of Pu Site
1
2
M~ M1
3.68 - 0.75
3.37 - 0.41
3 2.67 0.64
4
5
6
7
8
- 0.4 0.42
0.63 - 0.25
- 2.83 0.85
4.13 - 0.70
5.25 - 0.81
where J , K and C are the sum of pair parameters of the pure Heisenberg model biquadratical Kij(SiS/) 2, and bicubic terms Ci/(SiSj) 3. Our estimation for bcc Fe (using only the corresponding expansion of one-electron contribution to the total energy) has shown that this expansion is converged quickly (whereas the zero-moment expansion of the pure Heisenberg model J = P X P PXPPXP-PXPPXPPXP-... converges much slower) and including the first 4 terms gives Tc ~ 850 K. The slow convergence suggests that using such truncated expansions to extract absolute numbers for biquadratical terms without also making an estimation of the corresponding terms from the double-counting contributions might be problematic (i.e. there could be important cancelation effects). We also found that the higher order terms in the Landau expansion have increasing importance for the fcc structure as one goes from Ni to Co to Fe, so that those terms are very important for the description of magnetism in frustrated fcc AFM (Fe, Mn and so on). We also note that the usual Landau expansion is more suitable for the discussion of stability of magnetic phases than the model spin-Hamiltonian [5]. We tried to optimize the magnetic structures by direct calculation of the magnetic torque V0 = EVoj (for simplicity we present an expression in the 3,-representation of TB-LMTO):
Jij(SiSj),
V=(CtR-C~/R)z×m°-(R-1/R)z×m
'
(4)
(where R = ( A ~ / A 1")1/2, A,,, C,~ are the standard potential parameters, m" the nth moment of the DOS, and z is
Table 3 Theoretical spin moments in tetragonal phases of c~-Mn and corresponding moments from experiment Site
1
2
3 (I)
3 (II)
4 (I)
4 (II)
Present Exp.
2.2 2.83
- 1.8 - 1.82
0.52 0.43
- 0.52 0.32
0.17 - 0.45
0.17 0.48
the unit vector along local magnetization axis). We found that the ground state of fcc Fe is non-collinear triple-k type of magnetic structure (with a local moment of about 0.8/XB), while for fcc Pu the double-k (or quadrupolar AFM) type of structure has lower energy. Both types of structure are stable with respect to the volume. Besides being useful for finding non-collinear magnetic ground states, the torque may prove useful for spin-dynamics simulations. Further investigating the magnetic ground state of Mn and Pu, we performed direct calculations of their a-phases. We found that if we restrict ourselves by consideration on only collinear structures, the magnetism of both metals is very similar (Tables 2 and 3). In both metals there are two layers of atoms (with 8 and 29 atoms for Pu and Mn respectively) per unit cell, which have large and opposite magnetic moments so that the ground state is a compensated antiferromagnet. For Mn we obtained reasonable agreement with experiment (and we confirmed that tetragonal distorted structure has lower energy than the cubic one) whereas for a-Pu our calculations are the first to consider magnetic ordering. Acknowledgements: This work was carried out at the Ames Laboratory, which is operated for the US Department of Energy by Iowa State University under Contract No. W-7405-82. The research was supported by the Director for Energy Research, Office of Basic Energy Sciences of the US Department of Energy. References
[1] S.H. Liu, Phys. Rev. B 15 (1977) 4281; T. Oguchi et al., J. Phys. F 13 (1983) 145. [2] A.l. Liechtenstein et al., J. Phys. F 14 (1984) L 125; V.P. Antropov and A.I. Liechtenstein, MRS Proc. V 253 (1992) 325. [3] O. Gunnarsson et al., Phys. Rev. B 27 (1983) 7144. [4] C.S. Wang et al., Phys. Rev. B 25 (1982) 5766. [5] V.P. Antropov et al. (unpublished).
~i ~l~ ~1~
journalof magnetism and magnetic materials
ELSEVIER
Magnetic phase stability of 3d-metals and plutonium V.P. Antropov a,*, M. van Schilfgaarde b, B.N. Harmon a a
Ames Laboratory, Iowa State Uni~,ersity,Ames, IA 50011, USA b SRI International, Menlo Park, CA 94025, USA
Abstract We have performed self-consistent spin-polarized calculations for the magnetic structure and exchange parameters of pure 3d-metals and Pu. Local field effects and the strength of non-Heisenberg interactions are estimated. For a-Mn and a-Pu antiferromagnetic structures are dominant, and a complexe magnetic ordering was found for the low temperature structures. An analytical expression for the magnetic torque and a non-collinear low energy magnetic state of T-Fe are presented.
In spite of the fundamental importance of the pair exchange parameters between local magnetic moments, the number of first-principle calculations of such parameters is still very limited. Also the calculation schemes are very different, so that direct comparison is hard to perform. Some rigorous expressions for exchange parameters and spin-wave stiffness (SWS) were proposed [1,2], but applications of the formalism for real materials has been limited. We analyze the exchange parameters and the stability of magnetic phases in cubic 3d-metals and the a-phases of Mn and Pu using the LMTO-Green-function technique [3]. We start from expressions for the second order term in the Landau expansion introduced in Ref. [2] and rewritten in the form: A ~ = 2~r
de Im TrL{XijPi .pj},
(1)
and introduce the pairwise magnetic torque as
1 [EF , Vii= "~-~-j dE Im TrL{ XijPiXPj}
(2)
with the on-site perturbation having the form p = Tr(6-. /3)/2. The corresponding expression for the SWS D can be obtained using Aij. We corroborate the results of Wang et al. [4] about ferromagnetic (FM) type of interactions between nearest neighbors and antiferromagnetic (AFM) at longer range. Also we found stronger interactions in bcc Fe than in fcc Ni, where exchange weakly oscillates (we analyzed only fifteen shells of neighbors). The real elementary excitations connected with spin rotations are spin waves, and our parameter D can be
* Corresponding author.
directly compared with experiment. The agreement with experiment is satisfactory but not excellent. A possible explanation for that might be the local field effects which can be estimated if we go beyond the one-site approximation used in Ref. [2] and employ the effective vector on-site perturbation for a rotation of spin-density at one particular site [[p X T] × T] (with similar scalar expressions for the exchange). Using such corrections the expressions (both in real and reciprocal space) for the corresponding exchange parameters can be obtained [5]. For SWS the corresponding renormalized values are presented in Table 1. We note that including local field effects decreases the total value of D (about 10%) and leads to better agreement with experiment. To estimate the magnetic transition temperature we first use simple mean field expression for Tc and our values are listed in Table 1. Those numbers should be considered with some caution however, because the simple mean field expression assumes Heisenberg model parameters, whereas o u r Aij are just the second derivative of total energy and include non-Heisenberg types of interactions as well. This is appropriate for real excitations (as, for instance, spin waves), and for a valid estimation of Tc the general form of the model Hamiltonian should be specified. In this case the equation for Tc in mean field approximation is Tc=~
'
(3)
Table 1 Calculated spin wave stiffness (meV ,~2) and Curie temperature for 3d-systems Metal
D
D ioc
Dexp
Tc
Tcexp
Fe (bcc) Ni (fcc)
410 631
375 565
314,550 395,422
1220 380
1040 670
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSD10304-8853(94)00887-6
J - - - + - - - " ' 32
ILP. Antropol et al. /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1355-1356
1356
Table 2 Spin and orbital magnetic moments in c~-phase of Pu Site
1
2
M~ M1
3.68 - 0.75
3.37 - 0.41
3 2.67 0.64
4
5
6
7
8
- 0.4 0.42
0.63 - 0.25
- 2.83 0.85
4.13 - 0.70
5.25 - 0.81
where J , K and C are the sum of pair parameters of the pure Heisenberg model biquadratical Kij(SiS/) 2, and bicubic terms Ci/(SiSj) 3. Our estimation for bcc Fe (using only the corresponding expansion of one-electron contribution to the total energy) has shown that this expansion is converged quickly (whereas the zero-moment expansion of the pure Heisenberg model J = P X P PXPPXP-PXPPXPPXP-... converges much slower) and including the first 4 terms gives Tc ~ 850 K. The slow convergence suggests that using such truncated expansions to extract absolute numbers for biquadratical terms without also making an estimation of the corresponding terms from the double-counting contributions might be problematic (i.e. there could be important cancelation effects). We also found that the higher order terms in the Landau expansion have increasing importance for the fcc structure as one goes from Ni to Co to Fe, so that those terms are very important for the description of magnetism in frustrated fcc AFM (Fe, Mn and so on). We also note that the usual Landau expansion is more suitable for the discussion of stability of magnetic phases than the model spin-Hamiltonian [5]. We tried to optimize the magnetic structures by direct calculation of the magnetic torque V0 = EVoj (for simplicity we present an expression in the 3,-representation of TB-LMTO):
Jij(SiSj),
V=(CtR-C~/R)z×m°-(R-1/R)z×m
'
(4)
(where R = ( A ~ / A 1")1/2, A,,, C,~ are the standard potential parameters, m" the nth moment of the DOS, and z is
Table 3 Theoretical spin moments in tetragonal phases of c~-Mn and corresponding moments from experiment Site
1
2
3 (I)
3 (II)
4 (I)
4 (II)
Present Exp.
2.2 2.83
- 1.8 - 1.82
0.52 0.43
- 0.52 0.32
0.17 - 0.45
0.17 0.48
the unit vector along local magnetization axis). We found that the ground state of fcc Fe is non-collinear triple-k type of magnetic structure (with a local moment of about 0.8/XB), while for fcc Pu the double-k (or quadrupolar AFM) type of structure has lower energy. Both types of structure are stable with respect to the volume. Besides being useful for finding non-collinear magnetic ground states, the torque may prove useful for spin-dynamics simulations. Further investigating the magnetic ground state of Mn and Pu, we performed direct calculations of their a-phases. We found that if we restrict ourselves by consideration on only collinear structures, the magnetism of both metals is very similar (Tables 2 and 3). In both metals there are two layers of atoms (with 8 and 29 atoms for Pu and Mn respectively) per unit cell, which have large and opposite magnetic moments so that the ground state is a compensated antiferromagnet. For Mn we obtained reasonable agreement with experiment (and we confirmed that tetragonal distorted structure has lower energy than the cubic one) whereas for a-Pu our calculations are the first to consider magnetic ordering. Acknowledgements: This work was carried out at the Ames Laboratory, which is operated for the US Department of Energy by Iowa State University under Contract No. W-7405-82. The research was supported by the Director for Energy Research, Office of Basic Energy Sciences of the US Department of Energy. References
[1] S.H. Liu, Phys. Rev. B 15 (1977) 4281; T. Oguchi et al., J. Phys. F 13 (1983) 145. [2] A.l. Liechtenstein et al., J. Phys. F 14 (1984) L 125; V.P. Antropov and A.I. Liechtenstein, MRS Proc. V 253 (1992) 325. [3] O. Gunnarsson et al., Phys. Rev. B 27 (1983) 7144. [4] C.S. Wang et al., Phys. Rev. B 25 (1982) 5766. [5] V.P. Antropov et al. (unpublished).