Ordering due to disorder in dipolar magnets on two-dimensional lattices

Sona Prakash; Christopher L. Henley

doi:10.1103/PhysRevB.42.6574

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Ordering due to disorder in dipolar magnets on two-dimensional lattices

Sona Prakash

Christopher L. Henley

Department of Physics, Boston University, Boston, Massachusetts 02215

Department of Physics, Boston University, Boston, Massachusetts 02215
Department of Physics, Cornell University, Ithaca, New York 14853

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Phys. Rev. B 42, 6574 – Published 1 October, 1990

DOI: https://doi.org/10.1103/PhysRevB.42.6574

Abstract

We calculate the effects of thermal fluctuations, dilution, and a uniform external field on the continuously degenerate ground state of dipolar magnets on the square and honeycomb lattices. In all cases, these perturbations make terms in the free energy that favor particular states, reducing the continuous degeneracy to a discrete symmetry. Dilution and thermal selection compete; i.e., they select different states. For the cases of thermal fluctuations and dilution, the discrete symmetry is fourfold on the square lattice and sixfold on the honeycomb lattice; for a uniform field it is twofold and threefold, respectively. In addition, dilution generates effective random axes, which disorder all of the selected phases in each system, except for the dilution-selected phase of the square lattice. We present a proposed phase diagram including speculations on the finite-temperature transition.

References (24)

C. L. Henley, J. Appl. Phys. 61, 3962 (1987).
C. L. Henley, Phys. Rev. Lett. 62, 2056 (1989).
J. Villain, R. Bidaux, J. P. Carton and R. Conte, J. Phys. (Paris) 41, 1263 (1980).
R. Bidaux, J. P. Carton, and J. Villain, in Disordered Systems and Localization, Vol. 149 of Lectures Notes in Physics, edited by C. Castellani et al. (Springer, Berlin, 1981); A. Danielian, Phys. Rev. 13, A1344 (1964).
Note that we have a three-dimensional dipole interaction even though the system is two dimensional. It can easily be verified that the ground states have spins lying in the plane: states where they tilt out of the plane are energetically unfavorable. Furthermore, when we consider fluctuations about these ground states, the δ $μ_{z}$ fluctuations are not coupled to the ( δ $μ_{x}$ , δ $μ_{y}$ ) fluctuations. The δ $μ_{z}$ spin waves also have an energy cost that is independent of the parameter phi, so they would not contribute to the selection terms computed in Secs. III and IV. Therefore, we do not lose any of the physics by confining the dipoles to the XY plane and shall do so henceforth.
G. O. Zimmerman, A. K. Ibrahim and F. Y. Wu, Phys. Rev. B 37, 2059 (1988).
H. Kawamura, J. Phys. Soc. Jpn. 53, 2452 (1984).
Of course, there will be higher-order responses of the spin deviations δ theta of O( $h^{2}$ ). One might worry that this produces two additional contributions when these are included in the O( δ $θ^{2}$ ) dipolar energy and the O( δ θ ) magnetization, but, in fact, these two terms cancel each other quite generally in any problem with this algebraic structure.
The degree of freedom phi in this case corresponds to a constrained rotation in the sense that the two groups of dipoles have to be rotated in opposite directions by the same angle.
J. F. Fernandez, Phys. Rev. B 38, 690 (1988).
J. F. Fernandez, Europhys. Lett. 5, 129 (1988).
J. V. José, L. P. Kadanoff, S. Kirkpatrick and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).
(a) Y. Imry and S. K. Ma, Phys. Rev. Lett. 35, 1399 (1975); for a review, see T. Natermann and J. Villain, Phase Transitions K. Binder11, 5 (1988). (b) , Z. Phys. B 50, 343 (1983).
Y. Y. Goldschmidt and B. Schaub, Nucl. Phys. B251, 77 (1985).
R. A. Pelcovits, E. Pytte and J. Rudnick, Phys. Rev. Lett. 40, 476 (1978); R. Fisch, Phys. Rev. B 39, 873 (1989), and references therein.
A. P. Young, in Ordering in Strongly Fluctuating Condensed Matter Systems, edited by T. Riste (Plenum, New York, 1980).
D. R. Nelson, in Critical Phenomena and Phase Transitions, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1983), Vol. 7.
B. J. Minchau and R. A. Pelcovits, Phys. Rev. B 32, 3081 (1985); M. Ma and P. A. Lee, ibid. 32, 5658 (1985).
E. F. Shender, Zh. Eksp. Teor. Fiz. 83, 326 (1982) [Sov. Phys.—JETP 56, 178 (1982)].
M. W. Long, J. Phys. Condens. Matter 1, 2857 (1989).
B. E. Larson and C. L. Henley, (unpublished).
S. Miyashita and H. Kawamura, J. Phys. Soc. Jpn. 54, 3385 (1985).
Q. Sheng and C. L. Henley (unpublished).
A. K. Ibrahim and G. O. Zimmermann, Phys. Rev. B35, 1860 (1987).

[c1] C. L. Henley, J. Appl. Phys. 61, 3962 (1987).

[c2] C. L. Henley, Phys. Rev. Lett. 62, 2056 (1989).

[c3] J. Villain, R. Bidaux, J. P. Carton and R. Conte, J. Phys. (Paris) 41, 1263 (1980).

[c4] R. Bidaux, J. P. Carton, and J. Villain, in Disordered Systems and Localization, Vol. 149 of Lectures Notes in Physics, edited by C. Castellani et al. (Springer, Berlin, 1981); A. Danielian, Phys. Rev. 13, A1344 (1964).

[c5] Note that we have a three-dimensional dipole interaction even though the system is two dimensional. It can easily be verified that the ground states have spins lying in the plane: states where they tilt out of the plane are energetically unfavorable. Furthermore, when we consider fluctuations about these ground states, the δ $μ_{z}$ fluctuations are not coupled to the ( δ $μ_{x}$ , δ $μ_{y}$ ) fluctuations. The δ $μ_{z}$ spin waves also have an energy cost that is independent of the parameter phi, so they would not contribute to the selection terms computed in Secs. III and IV. Therefore, we do not lose any of the physics by confining the dipoles to the XY plane and shall do so henceforth.

[c6] G. O. Zimmerman, A. K. Ibrahim and F. Y. Wu, Phys. Rev. B 37, 2059 (1988).

[c7] H. Kawamura, J. Phys. Soc. Jpn. 53, 2452 (1984).

[c8] Of course, there will be higher-order responses of the spin deviations δ theta of O( $h^{2}$ ). One might worry that this produces two additional contributions when these are included in the O( δ $θ^{2}$ ) dipolar energy and the O( δ θ ) magnetization, but, in fact, these two terms cancel each other quite generally in any problem with this algebraic structure.

[c9] The degree of freedom phi in this case corresponds to a constrained rotation in the sense that the two groups of dipoles have to be rotated in opposite directions by the same angle.

[c10] J. F. Fernandez, Phys. Rev. B 38, 690 (1988).

[c11] J. F. Fernandez, Europhys. Lett. 5, 129 (1988).

[c12] J. V. José, L. P. Kadanoff, S. Kirkpatrick and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).

[c13] (a) Y. Imry and S. K. Ma, Phys. Rev. Lett. 35, 1399 (1975); for a review, see T. Natermann and J. Villain, Phase Transitions K. Binder11, 5 (1988). (b) , Z. Phys. B 50, 343 (1983).

[c14] Y. Y. Goldschmidt and B. Schaub, Nucl. Phys. B251, 77 (1985).

[c15] R. A. Pelcovits, E. Pytte and J. Rudnick, Phys. Rev. Lett. 40, 476 (1978); R. Fisch, Phys. Rev. B 39, 873 (1989), and references therein.

[c16] A. P. Young, in Ordering in Strongly Fluctuating Condensed Matter Systems, edited by T. Riste (Plenum, New York, 1980).

[c17] D. R. Nelson, in Critical Phenomena and Phase Transitions, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1983), Vol. 7.

[c18] B. J. Minchau and R. A. Pelcovits, Phys. Rev. B 32, 3081 (1985); M. Ma and P. A. Lee, ibid. 32, 5658 (1985).

[c19] E. F. Shender, Zh. Eksp. Teor. Fiz. 83, 326 (1982) [Sov. Phys.—JETP 56, 178 (1982)].

[c20] M. W. Long, J. Phys. Condens. Matter 1, 2857 (1989).

[c21] B. E. Larson and C. L. Henley, (unpublished).

[c22] S. Miyashita and H. Kawamura, J. Phys. Soc. Jpn. 54, 3385 (1985).

[c23] Q. Sheng and C. L. Henley (unpublished).

[c24] A. K. Ibrahim and G. O. Zimmermann, Phys. Rev. B35, 1860 (1987).

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Ordering due to disorder in dipolar magnets on two-dimensional lattices

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