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Hysteresis for anisotropic ±J Ising square lattices
Journal of Alloys and Compounds 369 (2004) 55–57
Hysteresis for anisotropic ±J Ising square lattices M.C. Salas-Solis a,∗ , F. Aguilera-Granja a , E.E. Vogel b , J. Cartes b a
b
Instituto de F´ısica, Universidad Autónoma de San Luis Potos´ı, Dr. Manuel Nava No. 6, Apartado Postal 629, Av. Alvaro Obregón, No. 64, San Luis Potosi 78290, Mexico Departamento de F´ısica, Universidad de la Frontera, Av. Francisco Salazar 01145, Casilla 54-D, Temuco, Chile
Abstract It is known that ±J square Ising lattices exhibit hysteresis in the isotropic case emulating some general behavior of spin glasses. On the other hand, some experimental curves of hysteresis in real spin glasses show additional features not explained by simulation on isotropic systems. In the present work, we explore the possibility of explaining some of these features by introducing anisotropic ±J exchange interactions so they are stronger along one direction by an anisotropy factor f (square symmetry is broken), which is varied in the interval 1–8. For each sample, hysteresis cycles are found varying temperature by means of Monte Carlo method. The analysis is extended over many samples for each size searching for average magnetization values under different realizations. Two interesting properties are established in this paper: (a) Loops break into sub loops; (b) Energy loss per cycle (area within the hysteresis loop) tends to increase with f, which is an indication for a more stable spin-glass phase in the presence of anisotropy. Comments comparing these results with experiment are also included. © 2003 Elsevier B.V. All rights reserved. Keywords: Ising square lattices; Anisotropy; Hysteresis; Spin glasses
The system in which ferromagnetic (F,−J) interactions coexist with antiferromagnetic interactions (AF, +J) known as ±J lattices was introduced by Edwards and Anderson with the idea of providing a basis for understanding spin-glass phenomena [1]. In such lattices, F and AF interactions are distributed at random in a concentration x and (1−x), respectively; periodic boundary conditions are imposed. For the sake of simplicity, we restrict here to the case of x = 0.5, with interactions to nearest neighbors, so exchange interactions coincide with bonds. It has been shown that simulations of hysteresis for isotropic ±J Ising lattices reproduce [2,3] some general aspects of hysteresis loops of real spin-glass systems such as the virgin curve laying outside of the main loop [4] and that the loop closes upon increasing temperature [4,5]. In the present paper, we want to study the implications on hysteretic behavior of ±J Ising square lattices due of anisotropy in the interactions. Namely, all interactions along one direction are stronger by an anisotropy factor f than interactions along the directions perpendicular to it. That is
∗
Corresponding author. Tel.: +52-444-826-2363; fax: +52-444-813-3874. E-mail address: [email protected] (M.C. Salas-Solis). 0925-8388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2003.09.055
to say, in addition to randomness and frustration present in the original isotropic ±J Ising lattices [6,7], we add a new disorder element which forces the loss of square symmetry. The main motivation for this work is the variety of behaviors exhibited by hysteresis loops in real spin glasses. We will show that some of these properties can be simulated by the inclusion of anisotropic interactions in the way described above. Then we will study the dependence of energy loss as function of f. We start with a square array with N spins, which defines the size of the system. Each particular distribution of F and AF bonds is called a sample. Many samples are randomly generated. Here, we report results based on average values for magnetization over 125 samples for size 144. The Ising Hamiltonian for system of N spins in the presence of a external magnetic field B and under an anisotropy condition described below, is given by: H(B) =
N ij
fij Jij Si Sj −
BSi ,
i
where Jij = ±J in random way for each sample, B is measured in units of J (which serves as a unit of energy as well); anisotropy is defined as fij = f for “horizontal” interactions, while fij = 1, otherwise.
56
M.C. Salas-Solis et al. / Journal of Alloys and Compounds 369 (2004) 55–57
For a given sample, the magnetization per site can be evaluated at different anisotropy factors f and temperature T:
B, dB m |dB|, f, T
N
1 = Si , N i=1
where dB/|dB| = −1(+1) for decreasing (increasing) the external field B, and temperature will be measured in units of J (kB = 1.0 in these units). The hysteresis calculations are done using a Monte Carlo (MC) method with the Metropolis algorithm [8] using 10,000 (MC) steps to allow thermalization: We do not go into details due to the space limitations; however, interested readers may consult [3] were a description of the procedure is done. In Fig. 1, we present hysteresis curves, at T = 0.1, for four different anisotropy factors beginning by the isotropic case f = 1.0. Magnetic field B∗ corresponds to a rescaled field according to B∗ = 4B/2(f + 1), aiming to map all hysteresis curves within the same energy framework. Fig. 1 represents just part of calculations done at T = 0.1 for this size, since several other f values were included up to f = 8. Additionally, for each of the f factors we varied temperature between 0.001 and 1.6. Several comments are in order. First, for f = 1.0, we recover the behavior of the isotropic case presenting four sectors, with tendency to close the loop at intermediate values of the magnetic field where the most significant Barkhausen jumps occur (−2, 0, +2).
Fig. 1. Hysteresis loops obtained as an average over 125 randomly prepared samples of size N = 144 at T = 0.1, for four anisotropy factors.
The tendency to close the loop at the center is shown experimentally by some spin glasses like it is the case for NiMn [4]. Second, upon the slightest anisotropy (as already evident in Fig. 1b) additional Barkhausen jumps occur tending to open the loop. Third, critical values of normalized magnetic field for which Barkhausen jumps can occur in the case of square lattices (coordination 4) are B∗ = 4f/(f + 1), 4(f − 1)/(f + 1), 4/(f + 1), 0, −4/(f + 1), −4(f − 1)/(f + 1), and −4f/(f + 1). Fourth, hysteresis cycle presents itself in the form of two ladders with irregular steps. In general, the number of steps are nine when going from one saturation field (−4 say) to the other saturation field (+4), due to the seven expressions for critical fields B∗ , plus the two values at the extremes of the cycle. However, for some particular f values (like f = 2.0 for example, not shown here), some steps can accidentally disappear. So the number or width of steps can be very irregular, depending on coordination number and relative strength of the anisotropies present in a system. This is actually the case in some experimental hysteresis curves of real spin glasses or diluted antiferromagnets [9] Fifth, for the case f = 3.0 (Fig. 1c) ladders formed by steps of same width are obtained with jumps at B∗ = −3, −2, −1, 0, 1, 2, and 3; for any other anisotropy factor, ladders are formed by steps of irregular widths (height are generally irregular). Sixth, for large anisotropy factors, large sectors of constant magnetization arise (as shown for f = 6.0 in Fig. 1d); It is interesting to point out that such behavior in hysteresis curves is presented in the case of some systems with mixed magnetism [10]. Seventh, low-temperature loops (T < 0.1) show sharp steps; additionally, in the extremely anisotropic case (f → ∞) loops tend to show just two sectors, namely m(B∗ = 0) ∼ = 0. Eighth, when temperature is increased rounding off effects are quite notorious; For even higher T, the ladder shape is lost and finally the loop tends to close as it might be expected. Let us now get deeper into temperature effects, for a fixed anisotropy factor. Let us do the analysis for f = 3.0, varying temperature between 0.001 and 1.6. This is shown in Fig. 2, where upon increasing temperature we notice rounding off effects at the steps; then, steps are beginning to disappear;
Fig. 2. Average hysteresis curves for 125 samples of size 144 for f = 3.0, as temperature increases discretely from 0.001 to 1.6.
M.C. Salas-Solis et al. / Journal of Alloys and Compounds 369 (2004) 55–57
57
On one hand, anisotropy produces irregularities in the low-temperature hysteresis curve due to Barkhausen spin avalanches of different sizes and occurring at different values of magnetic field. On the other hand, energy loss per cycle tends to increase with anisotropy, meaning that more energy is needed to overturn all spins after completion of the hysteresis cycle; this can be interpreted as a larger stability for an anisotropic spin-glass as compared to a similar isotropic spin glass.
Fig. 3. Energy loss per hysteresis cycle (area within hysteresis loop) at different temperature values as functions of the anisotropic factor f. Temperature is varied as in Fig. 2.
for higher temperatures (T = 0.8 say) resemblance to the original ladder structure is lost with the tendency of showing the typical S shape of hysteresis loops that are near closing. For even higher values of T, hysteresis loop tends to close, as the hysteresis loop of any system would do. This means that main manifestations of anisotropy are to be found at low temperatures. Finally, let us turn our attention to the area within the loop, which represents the energy dissipated in each hysteresis cycle. Results are given in Fig. 3, for N = 144, for the same temperature values used in Fig. 2 (represented by the same symbols) and varying anisotropy factor between 1.0 and 8.0. As it can be seen at each temperature, the energy loss per cycle increases for higher anisotropy, which means that anisotropy tends to favor spin-glass phase since it becomes more difficult to overturn clusters of spins in the extremely anisotropic case. On the other hand, temperature needed to close the loop (no energy lost in the cycle) increases with increasing the value of the anisotropy factor f, which is another way of visualizing the increase of stability of the spin glass phase with anisotropy. Let us close by summarizing our two main conclusions (a few others were implied in previous discussion).
Acknowledgements EEV thanks Fondecyt (Chile) under projects 1020993. MCSS and EEV thank Millennium Scientific Initiative (Chile) under contract P-99-135-F. FAG also acknowledges Fondecyt under project 7010511, and also to CONACyT G25851,25083E grants. We also benefited from discussions with Prof. P. Vargas.
References [1] S.F. Edwards, P.W. Anderson, J. Phys. F5 (1975) 965. [2] E.E. Vogel, J. Cartes, P. Vargas, D. Altbir, S. Kobe, T. Klotz, M. Nogala, Phys. Rev. B 59 (1999) 3325. [3] E.E. Vogel, J. Cartes, P. Vargas, D. Altbir, Physica B 284–288 (2000) 1211. [4] S. Senoussi, J. Phys. 45 (1984) 315. [5] I.A. Campbell, S. Senoussi, F. Varret, J. Teillet, A. Hamzi´c, Phys. Rev. Lett. 50 (1983) 1615. [6] K. Binder, A.P. Young, Rev. Mod. Phys. 58 (1986) 801. [7] M. Mezard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, 1987. [8] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [9] A. Paduan-Filho, C.C. Becerra, V.B. Barbeta, Y. Shapira, J. Campo, F. Palacio, J. Magn. Magn. Mater. 140–144 (1995) 1925; P. Monod, J.J. Prejean, B. Tissier, J. Appl. Phys. 50 (1979) 7324. [10] J. Kushauer, R. van Bentum, W. Kleemann, D. Bertrand, Phys. Rev. B 53 (1996) 11647.
Hysteresis for anisotropic ±J Ising square lattices M.C. Salas-Solis a,∗ , F. Aguilera-Granja a , E.E. Vogel b , J. Cartes b a
b
Instituto de F´ısica, Universidad Autónoma de San Luis Potos´ı, Dr. Manuel Nava No. 6, Apartado Postal 629, Av. Alvaro Obregón, No. 64, San Luis Potosi 78290, Mexico Departamento de F´ısica, Universidad de la Frontera, Av. Francisco Salazar 01145, Casilla 54-D, Temuco, Chile
Abstract It is known that ±J square Ising lattices exhibit hysteresis in the isotropic case emulating some general behavior of spin glasses. On the other hand, some experimental curves of hysteresis in real spin glasses show additional features not explained by simulation on isotropic systems. In the present work, we explore the possibility of explaining some of these features by introducing anisotropic ±J exchange interactions so they are stronger along one direction by an anisotropy factor f (square symmetry is broken), which is varied in the interval 1–8. For each sample, hysteresis cycles are found varying temperature by means of Monte Carlo method. The analysis is extended over many samples for each size searching for average magnetization values under different realizations. Two interesting properties are established in this paper: (a) Loops break into sub loops; (b) Energy loss per cycle (area within the hysteresis loop) tends to increase with f, which is an indication for a more stable spin-glass phase in the presence of anisotropy. Comments comparing these results with experiment are also included. © 2003 Elsevier B.V. All rights reserved. Keywords: Ising square lattices; Anisotropy; Hysteresis; Spin glasses
The system in which ferromagnetic (F,−J) interactions coexist with antiferromagnetic interactions (AF, +J) known as ±J lattices was introduced by Edwards and Anderson with the idea of providing a basis for understanding spin-glass phenomena [1]. In such lattices, F and AF interactions are distributed at random in a concentration x and (1−x), respectively; periodic boundary conditions are imposed. For the sake of simplicity, we restrict here to the case of x = 0.5, with interactions to nearest neighbors, so exchange interactions coincide with bonds. It has been shown that simulations of hysteresis for isotropic ±J Ising lattices reproduce [2,3] some general aspects of hysteresis loops of real spin-glass systems such as the virgin curve laying outside of the main loop [4] and that the loop closes upon increasing temperature [4,5]. In the present paper, we want to study the implications on hysteretic behavior of ±J Ising square lattices due of anisotropy in the interactions. Namely, all interactions along one direction are stronger by an anisotropy factor f than interactions along the directions perpendicular to it. That is
∗
Corresponding author. Tel.: +52-444-826-2363; fax: +52-444-813-3874. E-mail address: [email protected] (M.C. Salas-Solis). 0925-8388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2003.09.055
to say, in addition to randomness and frustration present in the original isotropic ±J Ising lattices [6,7], we add a new disorder element which forces the loss of square symmetry. The main motivation for this work is the variety of behaviors exhibited by hysteresis loops in real spin glasses. We will show that some of these properties can be simulated by the inclusion of anisotropic interactions in the way described above. Then we will study the dependence of energy loss as function of f. We start with a square array with N spins, which defines the size of the system. Each particular distribution of F and AF bonds is called a sample. Many samples are randomly generated. Here, we report results based on average values for magnetization over 125 samples for size 144. The Ising Hamiltonian for system of N spins in the presence of a external magnetic field B and under an anisotropy condition described below, is given by: H(B) =
N ij
fij Jij Si Sj −
BSi ,
i
where Jij = ±J in random way for each sample, B is measured in units of J (which serves as a unit of energy as well); anisotropy is defined as fij = f for “horizontal” interactions, while fij = 1, otherwise.
56
M.C. Salas-Solis et al. / Journal of Alloys and Compounds 369 (2004) 55–57
For a given sample, the magnetization per site can be evaluated at different anisotropy factors f and temperature T:
B, dB m |dB|, f, T
N
1 = Si , N i=1
where dB/|dB| = −1(+1) for decreasing (increasing) the external field B, and temperature will be measured in units of J (kB = 1.0 in these units). The hysteresis calculations are done using a Monte Carlo (MC) method with the Metropolis algorithm [8] using 10,000 (MC) steps to allow thermalization: We do not go into details due to the space limitations; however, interested readers may consult [3] were a description of the procedure is done. In Fig. 1, we present hysteresis curves, at T = 0.1, for four different anisotropy factors beginning by the isotropic case f = 1.0. Magnetic field B∗ corresponds to a rescaled field according to B∗ = 4B/2(f + 1), aiming to map all hysteresis curves within the same energy framework. Fig. 1 represents just part of calculations done at T = 0.1 for this size, since several other f values were included up to f = 8. Additionally, for each of the f factors we varied temperature between 0.001 and 1.6. Several comments are in order. First, for f = 1.0, we recover the behavior of the isotropic case presenting four sectors, with tendency to close the loop at intermediate values of the magnetic field where the most significant Barkhausen jumps occur (−2, 0, +2).
Fig. 1. Hysteresis loops obtained as an average over 125 randomly prepared samples of size N = 144 at T = 0.1, for four anisotropy factors.
The tendency to close the loop at the center is shown experimentally by some spin glasses like it is the case for NiMn [4]. Second, upon the slightest anisotropy (as already evident in Fig. 1b) additional Barkhausen jumps occur tending to open the loop. Third, critical values of normalized magnetic field for which Barkhausen jumps can occur in the case of square lattices (coordination 4) are B∗ = 4f/(f + 1), 4(f − 1)/(f + 1), 4/(f + 1), 0, −4/(f + 1), −4(f − 1)/(f + 1), and −4f/(f + 1). Fourth, hysteresis cycle presents itself in the form of two ladders with irregular steps. In general, the number of steps are nine when going from one saturation field (−4 say) to the other saturation field (+4), due to the seven expressions for critical fields B∗ , plus the two values at the extremes of the cycle. However, for some particular f values (like f = 2.0 for example, not shown here), some steps can accidentally disappear. So the number or width of steps can be very irregular, depending on coordination number and relative strength of the anisotropies present in a system. This is actually the case in some experimental hysteresis curves of real spin glasses or diluted antiferromagnets [9] Fifth, for the case f = 3.0 (Fig. 1c) ladders formed by steps of same width are obtained with jumps at B∗ = −3, −2, −1, 0, 1, 2, and 3; for any other anisotropy factor, ladders are formed by steps of irregular widths (height are generally irregular). Sixth, for large anisotropy factors, large sectors of constant magnetization arise (as shown for f = 6.0 in Fig. 1d); It is interesting to point out that such behavior in hysteresis curves is presented in the case of some systems with mixed magnetism [10]. Seventh, low-temperature loops (T < 0.1) show sharp steps; additionally, in the extremely anisotropic case (f → ∞) loops tend to show just two sectors, namely m(B∗ = 0) ∼ = 0. Eighth, when temperature is increased rounding off effects are quite notorious; For even higher T, the ladder shape is lost and finally the loop tends to close as it might be expected. Let us now get deeper into temperature effects, for a fixed anisotropy factor. Let us do the analysis for f = 3.0, varying temperature between 0.001 and 1.6. This is shown in Fig. 2, where upon increasing temperature we notice rounding off effects at the steps; then, steps are beginning to disappear;
Fig. 2. Average hysteresis curves for 125 samples of size 144 for f = 3.0, as temperature increases discretely from 0.001 to 1.6.
M.C. Salas-Solis et al. / Journal of Alloys and Compounds 369 (2004) 55–57
57
On one hand, anisotropy produces irregularities in the low-temperature hysteresis curve due to Barkhausen spin avalanches of different sizes and occurring at different values of magnetic field. On the other hand, energy loss per cycle tends to increase with anisotropy, meaning that more energy is needed to overturn all spins after completion of the hysteresis cycle; this can be interpreted as a larger stability for an anisotropic spin-glass as compared to a similar isotropic spin glass.
Fig. 3. Energy loss per hysteresis cycle (area within hysteresis loop) at different temperature values as functions of the anisotropic factor f. Temperature is varied as in Fig. 2.
for higher temperatures (T = 0.8 say) resemblance to the original ladder structure is lost with the tendency of showing the typical S shape of hysteresis loops that are near closing. For even higher values of T, hysteresis loop tends to close, as the hysteresis loop of any system would do. This means that main manifestations of anisotropy are to be found at low temperatures. Finally, let us turn our attention to the area within the loop, which represents the energy dissipated in each hysteresis cycle. Results are given in Fig. 3, for N = 144, for the same temperature values used in Fig. 2 (represented by the same symbols) and varying anisotropy factor between 1.0 and 8.0. As it can be seen at each temperature, the energy loss per cycle increases for higher anisotropy, which means that anisotropy tends to favor spin-glass phase since it becomes more difficult to overturn clusters of spins in the extremely anisotropic case. On the other hand, temperature needed to close the loop (no energy lost in the cycle) increases with increasing the value of the anisotropy factor f, which is another way of visualizing the increase of stability of the spin glass phase with anisotropy. Let us close by summarizing our two main conclusions (a few others were implied in previous discussion).
Acknowledgements EEV thanks Fondecyt (Chile) under projects 1020993. MCSS and EEV thank Millennium Scientific Initiative (Chile) under contract P-99-135-F. FAG also acknowledges Fondecyt under project 7010511, and also to CONACyT G25851,25083E grants. We also benefited from discussions with Prof. P. Vargas.
References [1] S.F. Edwards, P.W. Anderson, J. Phys. F5 (1975) 965. [2] E.E. Vogel, J. Cartes, P. Vargas, D. Altbir, S. Kobe, T. Klotz, M. Nogala, Phys. Rev. B 59 (1999) 3325. [3] E.E. Vogel, J. Cartes, P. Vargas, D. Altbir, Physica B 284–288 (2000) 1211. [4] S. Senoussi, J. Phys. 45 (1984) 315. [5] I.A. Campbell, S. Senoussi, F. Varret, J. Teillet, A. Hamzi´c, Phys. Rev. Lett. 50 (1983) 1615. [6] K. Binder, A.P. Young, Rev. Mod. Phys. 58 (1986) 801. [7] M. Mezard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, 1987. [8] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [9] A. Paduan-Filho, C.C. Becerra, V.B. Barbeta, Y. Shapira, J. Campo, F. Palacio, J. Magn. Magn. Mater. 140–144 (1995) 1925; P. Monod, J.J. Prejean, B. Tissier, J. Appl. Phys. 50 (1979) 7324. [10] J. Kushauer, R. van Bentum, W. Kleemann, D. Bertrand, Phys. Rev. B 53 (1996) 11647.