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Anisotropy in random field systems: The phase diagram
Journal
of Magnetism
and Magnetic
ANISOTROPY
Materials
IN RANDOM
54-57
51
(1986) 51-53
FIELD SYSTEMS:
THE PHASE DIAGRAM
Gary S. GREST Exxon Research and Engineering
Co., Annandcrle. NJ 08801. USA
C.M. SOUKOULIS Ames Laboratory,
Iowa State Unruerslty. Ames, IA 50011, USA
and
K. LEVIN The James Franck Institute.
The Unioersrty
of Chrcago, Chkago. IL 60637, USA
We calculate the irreversibility and phase stability boundaries, using site dependent mean field theory. in the H-T plane for diluted uniaxially anisotropic Heisenberg antiferromagnets in magnetic fields H at temperatures T. Five distinct phases are found corresponding 10 various types of transverse and longitudinal order and their associated irreversibilities.
In the last few years considerable theoretical progress has been made in our understanding of random field systems. The work of Villain [l] and of Grinstein and Fernandez [2] has led to an understanding of the absence of long-range order (LRO) observed in some experimental situations [3] in three-dimensional (3d) systems. Furthermore these and related [4] approaches have predicted the time and field dependencies characterizing the relaxation towards the equilibrium state. The work of Imbrie [5] demonstrates conclusively that LRO is the zero temperature equilibrium state in 3d. Recent Monte Carlo simulations [6] have yielded additional insights into the nature of metastability in these systems. These previous theoretical studies have almost exclusively concentrated on Ising spin systems. It is the purpose of the present paper to discuss the random field problem in the presence of varying degrees of anisotropy. A particularly physical and simple way of understanding the random field behavior has been through the numerical solution of the site dependent mean field equations. Such an approach was originally proposed by the present authors as a means of explaining and predicting short time irreversible and metastable properties of spin glasses [7]. It has been applied to the random field ferromagnets and diluted antiferromagnets in finite field H by Yoshizawa and Belanger [8] and by Ro, Grest, Soukoulis and Levin [9]. Yoshizawa and Belanger were among the first to show that diluted antiferromagnets will form a domain state in field cooled (FC) processes whereas in zero field cooled (ZFC) measurements LRO will be obtained. In ref. [9] a detailed H-T diagram was obtained for the irreversibility and equilibrium phase boundaries for diluted Ising antiferromagnets and random field ferromagnets. Domain wall hysteresis and the
0304-8853/86/$03.50
0 Elsevier Science Publishers
various history dependent magnetizations and specific heats were also calculated [9]. Of particular interest was the observation that in equilibrium the transition from the domain to LRO state appears to be first order. The mean field equations, while correct at zero temperature, are clearly only approximate at finite T. Simple mean field theory is used primarily as a method of illustrating the general physical picture that the short time behavior of glassy systems reflects the evolution of the free energy surface. On these shift time scales, and because of large free energy barriers, irreversibility is associated with the disappearance of a free energy minimum as the temperature or fields are varied. This physical picture is not tied to the use of a particular mean field theory. Nevertheless in the absence of any satisfactory “ improved” field energy functionals. we adopt the simplest model here. The mean field approach has been extremely successful in understanding Ising random field systems. In addition to qualitatively explaining existing data there has been recent experimental confirmation [lo] on diluted antiferromagnets of predictions that on short time scales (i) history dependence sets in at a well defined phase boundary and (ii) upon heating the ZFC state, at large fields, there will be a weak first order-like jump near the irreversibility temperature corresponding to the transition from the LRO to domain state. This last observation is a direct consequence of the fact that the high temperature (paramagnetic) minimum of the free energy surface evolves into the domain state upon cooling, whereas the LRO minimum is a separate extremum which appears only at and below the ordering temperature (say TN) and is initially not as deep as that of the domain state. Only at temperatures somewhat below T, is the LRO state the deeper minimum. The existence of
B.V.
different regions of the phase diagram separating metastable and stable LRO has also been deduced in recent analytical calculations [ll]. With one exception [12] mean field theory has so far led to qualitative although not necessarily quantitative agreement between theory and experiment in diluted antiferromagnets. In view of this success, we have solved the mean field equations for a 3d diluted Heisenberg antiferromagnet in the presence of uniaxial anisotropy. The system is described by the Hamiltonian H=
(1)
-xJ,,S/S,-D~(S,+~HS;. 1,
/
I
where
the nearest neighbor exchange constant J,, = and e, = 1 if site i is occupied and zero otherwise. Similarly the uniaxial anisotropy term involving D contributes only when the site i contains a spin. We studied systems with a range of concentrations of magnetic sites c (above the percolation threshold) and for simple cubic lattices as large as (40)‘. For large positive D. the thermodynamic properties of eq. (1) correspond to an lsing system. Within mean field theory, the free energy functional f[m,] where m, = (S,) may be obtained by a simple 3 x 3 matrix analysis for the case of -Jc,c,
S = 1. For simplicity the constants D and T are measured in units of J. The self-consistent equation for the [m,] are then obtained from the variational condition: SF/Sm, = 0. These are solved iteratively as in refs. [7-91 according to the experimental prescription for FC and ZFC states. In this paper we focus on the phase diagram which indicates the nature of the most stable phases and where irreversibility occurs. All states we studied were obtained either by FC or ZFC processes. Irreversibility in general is said to occur when the FC and ZFc states are different. In the most general case we found five regions of the phase diagram: (i) The high temperature paramagnetic phase which is totally reversible. (ii) The domain phase in which the lower free energy state consists of domains within which there is short range antiferromagnetic order. This state is irreversible. (iii) The “ Ising” state in which LRO is stable and largely confined to the z direction. In general this state is irreversible, except when the anisotropy constant D is small. (iv) The spin flop phase which occurs in high magnetic
6.0
5.0
4.0
3.0 H
2.0
1.0
0.0
(4 c=o.50
5.0
4.0
3.0 H
I
2.0
1.0
0.0 1
T
Fig. 1. Irreversibility and stability phase diagrams for diluted Heisenberg antifcrromagnets with a range of concentration of magnetic sites and dimensionless uniaxial anisotropy D = 2.0 on A 20’/lattice. Shaded regions correspond 10 stable domain states. Unless Indicated otherwise all labelled states are irreversible. The dashed Ime in la corresponds to a first-order transition.
G.S. Grest et al. / Anisotrqv in random field systems fields. Here long range antiferromagnetic order is stable and confined to the xy plane. There is also an appreciable induced magnetization in the z direction. For intermediate values of H the spin flop phase is irreversible. (v) The reversible spin flop phase. As in (iv) LRO is stable and in the xy plane. At sufficiently high fields both fc and zfc processes lead to the same spin flop phase. To illustrate these results in fig. 1, we indicate the various stable phases in the H-T plane. The shaded region corresponds to the stable domain phase. Figs. a-d correspond to varying the concentration c of magnetic sites for a fixed value of D = 2.0. Unless otherwise indicated all labelled states are irreversible, i.e., the FC and ZFC states are distinct. As the concentration is decreased the effect of the anisotropy is effectively increased. In the extreme Ising limit [9] the phase diagram consists of three regions corresponding to the shaded domain state, the “Ising” LRO phase and the paramagnetic state. As the concentration c decreases (or similarly the anisotropy D increases towards the Ising limit) more and more of the low T phase is occupied by the domain state. As in the Ising case, for low or moderate H, we always found that the LRO state was not the deeper minimum close to the irreversibility boundary. This boundary is indicated by the right-most line in each figure and as can be seen in all the figures there is a narrow shaded region which intercedes between the Ising phase the irreversibility boundary. This observation is important and reflects the fact that LRO is not the most stable state at temperatures near that at which this minimum first appears. Our phase diagram can be compared with that obtained by Aharony [13] for random field ferromagnets using a (site averaged) mean field theory as well as renormalization group arguments. The calculations of ref. [13] have not included the effects of history dependence so that the possibility of stable domain states and irreversible spin-flop phases are not considered. At a general level our phase diagrams are similar in that the low H phase is Ising-like while the high H phase is spin-flopped. For high c (or low D) we find, as in ref. [13], a first-order transition between these two types of LRO. We have looked for the “mixed phase” predicted by Aharony and find that it is difficult to confirm numerically whether it exists or not. This is because finite size effects generally lead to some transverse component of the spins even when the spins are predominantly ordered in the z direction and conversely to
53
some longitudinal component in the spin flopped phase. Finally, we have studied the magnetic field hysteresis of the position of the domain walls for low field domain states which are initially obtained by field cooling. Here the effects of anisotropy are quite pronounced. In the Ising case, we found [9] that the domain walls were essentially pinned by impurities so that they were frozen in position even when the field was then turned off. By contrast, in less anisotropic systems the domain grow considerably when H is reduced to zero and are not substantially affected when the field is then increased back to its initial value. This ability to respond to a reduction in the magnetic field is a consequence of the extra spin degrees of freedom which allow the system to find “paths” to larger domain states for finite values of the anisotropy constant D. These features, which have been observed, experimentally [3] are quite important and suggest that the domain wall dynamics in weakly anisotropic Heisenberg systems may be more like that of (Ising) random field ferromagnets. since impurity pinning plays a less significant role. This will be discussed in detail in a future publication. This work was partially supported by the National Science Foundation under Grant No. DMR 81-15618, the Materials Research Laboratory (Grant No. NSFODMR 16892) and the Department of Energy (Grant No. W-7405-Eng-82). HI J. Villain, Phys. Rev. Lett. 52 (1984) 1543. and J. Fernandez. Phys. Rev. B 29 (1984) 6389. I31 For a review of experiments see for example, D.P. Belanger. A.R. King and V. Jaccarino. J. App. Phys. 55 (1984) 2383. On weakly anisotropic systems see also R.A. Cowley, H. Yoshizawa, G. Shirane. M. Hagan and R.J. Brigineau, Phys. Rev. B 30 (1984) 6650. (41 R. Bruinsma and G. Aeppli. Phys. Rev. Lett. 52 (1984) 1547. ]51 J.Z. Imbrie, Phys. Rev. Lett. 53 (1984) 1747. Phys. Rev. Lett. 54 (1985) ]61 A.P. Young and M. Nauenberg. 2429. ]71 See, C.M. Soukoulis. K. Levin and G.S. Crest. Phys. Rev. B 29 (1983) 1495 and the companion paper. ]81 H. Yoshizawa and D.P. Belanger. Phys. Rev. B 30 (1984) 5220. ]91 C. Ro. G.S. Crest, C.M. Soukoulis and K. Lrvin, Phys. Rev. B 31 (1985) 1682. 1101 R.J. Birgineau, R.A. Cowley, G. Shirane and H. Yoshizawa, Phys. Rev. Lett. 54 (1985) 2147. [ll] D. Andelman and J. Joanny, unpublished. [12] In ref. [lo] it is found that the FC state is not reversible close to TN in contrast to the results of ref. [9]. [13] A. Aharony. Phys. Rev. B 18 (1978) 3328.
PI G. Grinstein
of Magnetism
and Magnetic
ANISOTROPY
Materials
IN RANDOM
54-57
51
(1986) 51-53
FIELD SYSTEMS:
THE PHASE DIAGRAM
Gary S. GREST Exxon Research and Engineering
Co., Annandcrle. NJ 08801. USA
C.M. SOUKOULIS Ames Laboratory,
Iowa State Unruerslty. Ames, IA 50011, USA
and
K. LEVIN The James Franck Institute.
The Unioersrty
of Chrcago, Chkago. IL 60637, USA
We calculate the irreversibility and phase stability boundaries, using site dependent mean field theory. in the H-T plane for diluted uniaxially anisotropic Heisenberg antiferromagnets in magnetic fields H at temperatures T. Five distinct phases are found corresponding 10 various types of transverse and longitudinal order and their associated irreversibilities.
In the last few years considerable theoretical progress has been made in our understanding of random field systems. The work of Villain [l] and of Grinstein and Fernandez [2] has led to an understanding of the absence of long-range order (LRO) observed in some experimental situations [3] in three-dimensional (3d) systems. Furthermore these and related [4] approaches have predicted the time and field dependencies characterizing the relaxation towards the equilibrium state. The work of Imbrie [5] demonstrates conclusively that LRO is the zero temperature equilibrium state in 3d. Recent Monte Carlo simulations [6] have yielded additional insights into the nature of metastability in these systems. These previous theoretical studies have almost exclusively concentrated on Ising spin systems. It is the purpose of the present paper to discuss the random field problem in the presence of varying degrees of anisotropy. A particularly physical and simple way of understanding the random field behavior has been through the numerical solution of the site dependent mean field equations. Such an approach was originally proposed by the present authors as a means of explaining and predicting short time irreversible and metastable properties of spin glasses [7]. It has been applied to the random field ferromagnets and diluted antiferromagnets in finite field H by Yoshizawa and Belanger [8] and by Ro, Grest, Soukoulis and Levin [9]. Yoshizawa and Belanger were among the first to show that diluted antiferromagnets will form a domain state in field cooled (FC) processes whereas in zero field cooled (ZFC) measurements LRO will be obtained. In ref. [9] a detailed H-T diagram was obtained for the irreversibility and equilibrium phase boundaries for diluted Ising antiferromagnets and random field ferromagnets. Domain wall hysteresis and the
0304-8853/86/$03.50
0 Elsevier Science Publishers
various history dependent magnetizations and specific heats were also calculated [9]. Of particular interest was the observation that in equilibrium the transition from the domain to LRO state appears to be first order. The mean field equations, while correct at zero temperature, are clearly only approximate at finite T. Simple mean field theory is used primarily as a method of illustrating the general physical picture that the short time behavior of glassy systems reflects the evolution of the free energy surface. On these shift time scales, and because of large free energy barriers, irreversibility is associated with the disappearance of a free energy minimum as the temperature or fields are varied. This physical picture is not tied to the use of a particular mean field theory. Nevertheless in the absence of any satisfactory “ improved” field energy functionals. we adopt the simplest model here. The mean field approach has been extremely successful in understanding Ising random field systems. In addition to qualitatively explaining existing data there has been recent experimental confirmation [lo] on diluted antiferromagnets of predictions that on short time scales (i) history dependence sets in at a well defined phase boundary and (ii) upon heating the ZFC state, at large fields, there will be a weak first order-like jump near the irreversibility temperature corresponding to the transition from the LRO to domain state. This last observation is a direct consequence of the fact that the high temperature (paramagnetic) minimum of the free energy surface evolves into the domain state upon cooling, whereas the LRO minimum is a separate extremum which appears only at and below the ordering temperature (say TN) and is initially not as deep as that of the domain state. Only at temperatures somewhat below T, is the LRO state the deeper minimum. The existence of
B.V.
different regions of the phase diagram separating metastable and stable LRO has also been deduced in recent analytical calculations [ll]. With one exception [12] mean field theory has so far led to qualitative although not necessarily quantitative agreement between theory and experiment in diluted antiferromagnets. In view of this success, we have solved the mean field equations for a 3d diluted Heisenberg antiferromagnet in the presence of uniaxial anisotropy. The system is described by the Hamiltonian H=
(1)
-xJ,,S/S,-D~(S,+~HS;. 1,
/
I
where
the nearest neighbor exchange constant J,, = and e, = 1 if site i is occupied and zero otherwise. Similarly the uniaxial anisotropy term involving D contributes only when the site i contains a spin. We studied systems with a range of concentrations of magnetic sites c (above the percolation threshold) and for simple cubic lattices as large as (40)‘. For large positive D. the thermodynamic properties of eq. (1) correspond to an lsing system. Within mean field theory, the free energy functional f[m,] where m, = (S,) may be obtained by a simple 3 x 3 matrix analysis for the case of -Jc,c,
S = 1. For simplicity the constants D and T are measured in units of J. The self-consistent equation for the [m,] are then obtained from the variational condition: SF/Sm, = 0. These are solved iteratively as in refs. [7-91 according to the experimental prescription for FC and ZFC states. In this paper we focus on the phase diagram which indicates the nature of the most stable phases and where irreversibility occurs. All states we studied were obtained either by FC or ZFC processes. Irreversibility in general is said to occur when the FC and ZFc states are different. In the most general case we found five regions of the phase diagram: (i) The high temperature paramagnetic phase which is totally reversible. (ii) The domain phase in which the lower free energy state consists of domains within which there is short range antiferromagnetic order. This state is irreversible. (iii) The “ Ising” state in which LRO is stable and largely confined to the z direction. In general this state is irreversible, except when the anisotropy constant D is small. (iv) The spin flop phase which occurs in high magnetic
6.0
5.0
4.0
3.0 H
2.0
1.0
0.0
(4 c=o.50
5.0
4.0
3.0 H
I
2.0
1.0
0.0 1
T
Fig. 1. Irreversibility and stability phase diagrams for diluted Heisenberg antifcrromagnets with a range of concentration of magnetic sites and dimensionless uniaxial anisotropy D = 2.0 on A 20’/lattice. Shaded regions correspond 10 stable domain states. Unless Indicated otherwise all labelled states are irreversible. The dashed Ime in la corresponds to a first-order transition.
G.S. Grest et al. / Anisotrqv in random field systems fields. Here long range antiferromagnetic order is stable and confined to the xy plane. There is also an appreciable induced magnetization in the z direction. For intermediate values of H the spin flop phase is irreversible. (v) The reversible spin flop phase. As in (iv) LRO is stable and in the xy plane. At sufficiently high fields both fc and zfc processes lead to the same spin flop phase. To illustrate these results in fig. 1, we indicate the various stable phases in the H-T plane. The shaded region corresponds to the stable domain phase. Figs. a-d correspond to varying the concentration c of magnetic sites for a fixed value of D = 2.0. Unless otherwise indicated all labelled states are irreversible, i.e., the FC and ZFC states are distinct. As the concentration is decreased the effect of the anisotropy is effectively increased. In the extreme Ising limit [9] the phase diagram consists of three regions corresponding to the shaded domain state, the “Ising” LRO phase and the paramagnetic state. As the concentration c decreases (or similarly the anisotropy D increases towards the Ising limit) more and more of the low T phase is occupied by the domain state. As in the Ising case, for low or moderate H, we always found that the LRO state was not the deeper minimum close to the irreversibility boundary. This boundary is indicated by the right-most line in each figure and as can be seen in all the figures there is a narrow shaded region which intercedes between the Ising phase the irreversibility boundary. This observation is important and reflects the fact that LRO is not the most stable state at temperatures near that at which this minimum first appears. Our phase diagram can be compared with that obtained by Aharony [13] for random field ferromagnets using a (site averaged) mean field theory as well as renormalization group arguments. The calculations of ref. [13] have not included the effects of history dependence so that the possibility of stable domain states and irreversible spin-flop phases are not considered. At a general level our phase diagrams are similar in that the low H phase is Ising-like while the high H phase is spin-flopped. For high c (or low D) we find, as in ref. [13], a first-order transition between these two types of LRO. We have looked for the “mixed phase” predicted by Aharony and find that it is difficult to confirm numerically whether it exists or not. This is because finite size effects generally lead to some transverse component of the spins even when the spins are predominantly ordered in the z direction and conversely to
53
some longitudinal component in the spin flopped phase. Finally, we have studied the magnetic field hysteresis of the position of the domain walls for low field domain states which are initially obtained by field cooling. Here the effects of anisotropy are quite pronounced. In the Ising case, we found [9] that the domain walls were essentially pinned by impurities so that they were frozen in position even when the field was then turned off. By contrast, in less anisotropic systems the domain grow considerably when H is reduced to zero and are not substantially affected when the field is then increased back to its initial value. This ability to respond to a reduction in the magnetic field is a consequence of the extra spin degrees of freedom which allow the system to find “paths” to larger domain states for finite values of the anisotropy constant D. These features, which have been observed, experimentally [3] are quite important and suggest that the domain wall dynamics in weakly anisotropic Heisenberg systems may be more like that of (Ising) random field ferromagnets. since impurity pinning plays a less significant role. This will be discussed in detail in a future publication. This work was partially supported by the National Science Foundation under Grant No. DMR 81-15618, the Materials Research Laboratory (Grant No. NSFODMR 16892) and the Department of Energy (Grant No. W-7405-Eng-82). HI J. Villain, Phys. Rev. Lett. 52 (1984) 1543. and J. Fernandez. Phys. Rev. B 29 (1984) 6389. I31 For a review of experiments see for example, D.P. Belanger. A.R. King and V. Jaccarino. J. App. Phys. 55 (1984) 2383. On weakly anisotropic systems see also R.A. Cowley, H. Yoshizawa, G. Shirane. M. Hagan and R.J. Brigineau, Phys. Rev. B 30 (1984) 6650. (41 R. Bruinsma and G. Aeppli. Phys. Rev. Lett. 52 (1984) 1547. ]51 J.Z. Imbrie, Phys. Rev. Lett. 53 (1984) 1747. Phys. Rev. Lett. 54 (1985) ]61 A.P. Young and M. Nauenberg. 2429. ]71 See, C.M. Soukoulis. K. Levin and G.S. Crest. Phys. Rev. B 29 (1983) 1495 and the companion paper. ]81 H. Yoshizawa and D.P. Belanger. Phys. Rev. B 30 (1984) 5220. ]91 C. Ro. G.S. Crest, C.M. Soukoulis and K. Lrvin, Phys. Rev. B 31 (1985) 1682. 1101 R.J. Birgineau, R.A. Cowley, G. Shirane and H. Yoshizawa, Phys. Rev. Lett. 54 (1985) 2147. [ll] D. Andelman and J. Joanny, unpublished. [12] In ref. [lo] it is found that the FC state is not reversible close to TN in contrast to the results of ref. [9]. [13] A. Aharony. Phys. Rev. B 18 (1978) 3328.
PI G. Grinstein