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Phase transition in randomly diluted spin system with competing interactions
Journal of Magnetism and Magnetic Materials 90 & 91 (1990) 326-328 North- Holland
326
Phase transition in randomly diluted spin system with competing interactions Kazuko Kawasaki, K. Tanaka, M. Hara and R.A. Tahir-Kheli Physics Department, Nara Women's University. Nara 630, Japan Physics Department, Temple University Philadelphia. PA 19122. USA
The wave vector dependent spin correlation function S(k) is evaluated for the diluted Heisenberg system with competing interactions on the basis of high temperature series expansion. including terms up to f36. By analyzing S(k). features of the evolving local order in the critical regime are studied for triangular and fcc lattices.
1. Introduction
The diffuse magnetic scattering of slow neutrons. which is related directly to the wave vector dependent spin correlation function S(k) is well known to be central to the understanding of the relevant critical phenomena [1). In addition, it often provides insight into the structure of the long range order that appears below the critical point, i.e. at T < 1;;. In order to clarify how the lattice structure, magnetic dilution, as well as the magunitude and sign of exchange interactions playa role in determing some of the features of the long range order, we evaluate the correlation function S(k) by means of the high temperature series expansion including terms up to p6 for a randomly diluted Heisenberg system. Detailed studies are made on triangular and fcc magnetic systems with both first and second neighbor interactions. J I , J2 •
Fig. 1. S( k) on k x - k y plane for a triangular anti ferromagnet with J I = -1, J2 = 0 at k BT/ I J I I = 50.
2. The triangular lattice Despite the fact that isotropically coupled Heisenberg spins on a triangular lattice with J I and J 2 do not exhibit a phase transition at finite temperature (2), characteristic critical behavior is, neverthless, observed over a range of decreasing temperatures. First, let us examine S(k) for the case, J I < 0, J2 = 0, and S = 3/2 which is clearly a frustrated system. In fig. 1, S(k) is plotted on the kx-k y plane for the case J I = -1, J2 = 0 and kBT/1 J I I = 50. Here the maximum of S(k) occurs at positions ±(k a + 2k b )/ 3, ±(2k a + k b )/ 3, ±(k a k b)/3, where the primitive vectors k a and k b for the reciprocal lattice are given respectively by (2'l1, - 2':1/ If) and (0,4':1/ If) for a lattice constant of unit length.
Fig. 2. The first Brillouin zone of a triangular lattice is given by bold (dashed) lines for the lattice with only the J) (J2 ) interaction. The position at which S(k) becomes a maximum is denoted by F,. if I n > 0 and An if I n < O.
0304-8853/90/$03.50
327
K. Kawasaki et al. / Phase transition in randomly diluted spin system
F--
-1.0
P=l 5=7/2 Fig. 4. The phase diagram of fcc magnetic lattice. AF 1 and AF2 mean type I and II anti ferromagnetism. Fig. 3. The: contour map for lattice with J) = -I'and J2 = explained for fig. 2) is marked The "ridge" is seen
an anti ferromagnetic triangular -0.7. The position Al (notation by the notation X and A 2 by o. along the dashed line.
The peaks in S(k) herald the structure of the short range ordering. Thus in fig. 2, we show the location of such peaks for the cases where either J) or 12 . is zero. The first Brillouin zone is marked by a bold (or a broken) line for the case where J2 (or J) is zero. Clearly for the cases with positive interaction, the ground state is ferromagnetic and thus the maximum occurs at the zone center (i.e., the T point) marked by a cross (X). For negative J's, the location is marked by the notation A) (when J2 = 0) and A 2 (when J 1 = 0). For the general case when both J) and J2 are present, the results are summarized as follows: (1) J 1 > 0, J 2 > 0; the peak position stays at the T point (2) J 1 > 0, J2 < 0; the peak position moves along the line 1: connecting points rand M. Therefore, screw type spin configurations are expected in the direction of 1:. Their pitchs change from 0 to 2'lT/3 depending on the magnitude of J) and J 2 • (3) J 1 < 0, J2 > 0; The position A) coincides with the r-point of the second Brillouin zone of the triangular lattice with J t = 0 and finite J2 • So the antiferromagnetic tendency is enhanced. (4) J 1 < 0, J 2 < 0; This is the most complicated case. Here, a certain combination of J 1 and J2 , S(k) yields a .. ridge" like behavior instead of simple peaks, as shown in the contour map (sec fig. 3).
done [4] on this system. For example, it is well known that apart from the ferromagnetic ordering, antiferromagnetic ordering of various types can also occur depending on the couplings J 1 , J2 • Noting that there are four high-symmetry points rro, 0, 0). X(O, 0, I) , L(1/2, 1/2, 1/2) and W(l, 1/2,0) in the first Brillouin zone of fcc lattice, these antiferromagnetic phases are characterized as follows : If S(k) is maximum 'at the X-point we should have an antiferromagnetic structure of type I, the L-point is associated with type II . and the W-point should give type III anti ferromagnetism. Thus, we can estimate the phase
5(k)
1.0 --------3. The fcc lattice Since the pioneering work of Anderson (3). who used the molecular field approximation. much work has been
L
--fC/= X
W
r
Fig. 5. The wave number dependence of S(k) near the phase boundary between ferro and type II antiferromagnetic phases.
K. Kawasaki et al. / Phase transition in randomly diluted spin system
328
L
r
x
w
r
Fig. 6. The wave number dependence of S(k) for an fcc magnet with J 1 = - J 2 = 1 and p = 0.6.
transition temperatures by applying the power CAM theory [5] to S(k) at these high symmetry points. A calculated phase diagram is given in fig. 4. In contradiction to some of the previous results, there are regions of the phase diagram over' which we can not specify the critical point. For instance, let us examine S( k) in some detail near the phase boundary between ferromagnetic and type II anti ferromagnetic orderings. As shown in fig. 5, S(k) peaks at the f-point for J) = 1 and J 2 = -0.9 and L-point for J) = 1 and J 2 = -1.1. These tendencies suggest that ferromagnetic phase is plausible for the former case and type II antiferrornagnet for the latter ordering case. However, it is difficult to assess the stability of these orderings because of the diffuseness of the maxima and the existence of another
peak situated at the X-point. More striking is the fact that for J) = -J2 = 1, which is the phase boundary of the moleculer field theory, S(k) is constant along the A-line that links the f- and the L-points. This means that in the vicinity of the J) = -J2 = 1 point, disordered magnetic structure coexists along the body diagonal direction of the fcc lattice with these other orderings. To show the effect of dilution, S(k) is also drawn in fig. 6 for the case of magnetic concentration p = 0.6. Notice that the peak at the X-point has disappeared and instead a weak k-dependence is observed along the A-line. In conclusion, it might be stated that the magnetic orderings in the neighborhood of the usually accepted boundary point J) = 1, J2 = -1 appear to lack stability with dilution. References (1) G.L. Squires, Introduction to the Theory of Thermal Neu-
tron Scattering (Cambridge Univ. Press. Cambridge, 1978). (2) N.D. Merrnin and Wagner, Phys. Rev. Lell. 17 (1966) 1133. (3) P.W. Anderson, Phys. Rev. 79 (1950) 705. (4) J.S. Smart, Phys. Rev. 86 (1952) 968. J. Villan, Phys. Chern. Solids 11 (1959) 303. R.A. Tahir-Kheli, H.B. Callen and H.S: Jarrell, Phys. Chern. Solids 27 (1966) 23. (5) M. Suzuki, J. Phys. Soc. Jpn. 56 (1987) 4221.
326
Phase transition in randomly diluted spin system with competing interactions Kazuko Kawasaki, K. Tanaka, M. Hara and R.A. Tahir-Kheli Physics Department, Nara Women's University. Nara 630, Japan Physics Department, Temple University Philadelphia. PA 19122. USA
The wave vector dependent spin correlation function S(k) is evaluated for the diluted Heisenberg system with competing interactions on the basis of high temperature series expansion. including terms up to f36. By analyzing S(k). features of the evolving local order in the critical regime are studied for triangular and fcc lattices.
1. Introduction
The diffuse magnetic scattering of slow neutrons. which is related directly to the wave vector dependent spin correlation function S(k) is well known to be central to the understanding of the relevant critical phenomena [1). In addition, it often provides insight into the structure of the long range order that appears below the critical point, i.e. at T < 1;;. In order to clarify how the lattice structure, magnetic dilution, as well as the magunitude and sign of exchange interactions playa role in determing some of the features of the long range order, we evaluate the correlation function S(k) by means of the high temperature series expansion including terms up to p6 for a randomly diluted Heisenberg system. Detailed studies are made on triangular and fcc magnetic systems with both first and second neighbor interactions. J I , J2 •
Fig. 1. S( k) on k x - k y plane for a triangular anti ferromagnet with J I = -1, J2 = 0 at k BT/ I J I I = 50.
2. The triangular lattice Despite the fact that isotropically coupled Heisenberg spins on a triangular lattice with J I and J 2 do not exhibit a phase transition at finite temperature (2), characteristic critical behavior is, neverthless, observed over a range of decreasing temperatures. First, let us examine S(k) for the case, J I < 0, J2 = 0, and S = 3/2 which is clearly a frustrated system. In fig. 1, S(k) is plotted on the kx-k y plane for the case J I = -1, J2 = 0 and kBT/1 J I I = 50. Here the maximum of S(k) occurs at positions ±(k a + 2k b )/ 3, ±(2k a + k b )/ 3, ±(k a k b)/3, where the primitive vectors k a and k b for the reciprocal lattice are given respectively by (2'l1, - 2':1/ If) and (0,4':1/ If) for a lattice constant of unit length.
Fig. 2. The first Brillouin zone of a triangular lattice is given by bold (dashed) lines for the lattice with only the J) (J2 ) interaction. The position at which S(k) becomes a maximum is denoted by F,. if I n > 0 and An if I n < O.
0304-8853/90/$03.50
327
K. Kawasaki et al. / Phase transition in randomly diluted spin system
F--
-1.0
P=l 5=7/2 Fig. 4. The phase diagram of fcc magnetic lattice. AF 1 and AF2 mean type I and II anti ferromagnetism. Fig. 3. The: contour map for lattice with J) = -I'and J2 = explained for fig. 2) is marked The "ridge" is seen
an anti ferromagnetic triangular -0.7. The position Al (notation by the notation X and A 2 by o. along the dashed line.
The peaks in S(k) herald the structure of the short range ordering. Thus in fig. 2, we show the location of such peaks for the cases where either J) or 12 . is zero. The first Brillouin zone is marked by a bold (or a broken) line for the case where J2 (or J) is zero. Clearly for the cases with positive interaction, the ground state is ferromagnetic and thus the maximum occurs at the zone center (i.e., the T point) marked by a cross (X). For negative J's, the location is marked by the notation A) (when J2 = 0) and A 2 (when J 1 = 0). For the general case when both J) and J2 are present, the results are summarized as follows: (1) J 1 > 0, J 2 > 0; the peak position stays at the T point (2) J 1 > 0, J2 < 0; the peak position moves along the line 1: connecting points rand M. Therefore, screw type spin configurations are expected in the direction of 1:. Their pitchs change from 0 to 2'lT/3 depending on the magnitude of J) and J 2 • (3) J 1 < 0, J2 > 0; The position A) coincides with the r-point of the second Brillouin zone of the triangular lattice with J t = 0 and finite J2 • So the antiferromagnetic tendency is enhanced. (4) J 1 < 0, J 2 < 0; This is the most complicated case. Here, a certain combination of J 1 and J2 , S(k) yields a .. ridge" like behavior instead of simple peaks, as shown in the contour map (sec fig. 3).
done [4] on this system. For example, it is well known that apart from the ferromagnetic ordering, antiferromagnetic ordering of various types can also occur depending on the couplings J 1 , J2 • Noting that there are four high-symmetry points rro, 0, 0). X(O, 0, I) , L(1/2, 1/2, 1/2) and W(l, 1/2,0) in the first Brillouin zone of fcc lattice, these antiferromagnetic phases are characterized as follows : If S(k) is maximum 'at the X-point we should have an antiferromagnetic structure of type I, the L-point is associated with type II . and the W-point should give type III anti ferromagnetism. Thus, we can estimate the phase
5(k)
1.0 --------3. The fcc lattice Since the pioneering work of Anderson (3). who used the molecular field approximation. much work has been
L
--fC/= X
W
r
Fig. 5. The wave number dependence of S(k) near the phase boundary between ferro and type II antiferromagnetic phases.
K. Kawasaki et al. / Phase transition in randomly diluted spin system
328
L
r
x
w
r
Fig. 6. The wave number dependence of S(k) for an fcc magnet with J 1 = - J 2 = 1 and p = 0.6.
transition temperatures by applying the power CAM theory [5] to S(k) at these high symmetry points. A calculated phase diagram is given in fig. 4. In contradiction to some of the previous results, there are regions of the phase diagram over' which we can not specify the critical point. For instance, let us examine S( k) in some detail near the phase boundary between ferromagnetic and type II anti ferromagnetic orderings. As shown in fig. 5, S(k) peaks at the f-point for J) = 1 and J 2 = -0.9 and L-point for J) = 1 and J 2 = -1.1. These tendencies suggest that ferromagnetic phase is plausible for the former case and type II antiferrornagnet for the latter ordering case. However, it is difficult to assess the stability of these orderings because of the diffuseness of the maxima and the existence of another
peak situated at the X-point. More striking is the fact that for J) = -J2 = 1, which is the phase boundary of the moleculer field theory, S(k) is constant along the A-line that links the f- and the L-points. This means that in the vicinity of the J) = -J2 = 1 point, disordered magnetic structure coexists along the body diagonal direction of the fcc lattice with these other orderings. To show the effect of dilution, S(k) is also drawn in fig. 6 for the case of magnetic concentration p = 0.6. Notice that the peak at the X-point has disappeared and instead a weak k-dependence is observed along the A-line. In conclusion, it might be stated that the magnetic orderings in the neighborhood of the usually accepted boundary point J) = 1, J2 = -1 appear to lack stability with dilution. References (1) G.L. Squires, Introduction to the Theory of Thermal Neu-
tron Scattering (Cambridge Univ. Press. Cambridge, 1978). (2) N.D. Merrnin and Wagner, Phys. Rev. Lell. 17 (1966) 1133. (3) P.W. Anderson, Phys. Rev. 79 (1950) 705. (4) J.S. Smart, Phys. Rev. 86 (1952) 968. J. Villan, Phys. Chern. Solids 11 (1959) 303. R.A. Tahir-Kheli, H.B. Callen and H.S: Jarrell, Phys. Chern. Solids 27 (1966) 23. (5) M. Suzuki, J. Phys. Soc. Jpn. 56 (1987) 4221.