International Journal of Physics
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International Journal of Physics. 2018, 6(5), 147-154
DOI: 10.12691/ijp-6-5-2
Open AccessArticle

Order Parameter Profiles near Surfaces of Super-weak Ferrimagnetic Materials Exhibiting a Paramagnetic-Ferrimagnetic Transition

A. Waddahou1, M. Chahid1, and A. Maârouf1

1Polymer Physics and Critical Phenomena Laboratory, Sciences Faculty Ben M’sik, Hassan II University of Casablanca, P. O. Box 7955, Casablanca, Morocco

Pub. Date: October 07, 2018

Cite this paper:
A. Waddahou, M. Chahid and A. Maârouf. Order Parameter Profiles near Surfaces of Super-weak Ferrimagnetic Materials Exhibiting a Paramagnetic-Ferrimagnetic Transition. International Journal of Physics. 2018; 6(5):147-154. doi: 10.12691/ijp-6-5-2


The aim of this paper is to determine the full order parameter profiles close to the surface for a system made of two strongly coupled paramagnetic sublattices of respective moments φ and ψ. The material exhibits a para-ferrimagnetic transition at some critical temperature Tc greater than the room temperature. The free energy describing the physics of the system is of Landau type, and involves, beside quadratic and quartic terms in both φ and ψ, a lowest-order coupling, -Coφψ where Co<0 is the coupling constant measuring the interaction between the two sublattices. We consider here below a film of thickness L, and the free energy is then a sum of bulk and surface contributions expanded in terms of the local order parameters. The magnetization at the surface are φs and ψs. As in a recent paper (whom the present work is an extension), we first reduce the model to an effective φ4 -theory written in terms of the overall magnetization =φ+ψ and the associated fraction of magnetization ηa=φ/(φ+ψ). For the ordinary transition where the extrapolation length is positive λs>0, we determine the order parameter profile near surfaces. We show, in particular, that the magnetization behavior is governed, in addition of λs and the bulk correlation length ξb-, by a new length La. We have interpreted this latter as the width of the ordered ferrimagnetic layer close to the surface. Finally, we examine the extraordinary (λs<0), special (λs=∞) and surface (at Tcs>Tc) transitions.

sublattices coupling paramagnetism ferrimagnetism transition order parameter profiles

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[1]  Neumann K.U., Crangle J., Ziebeck K.R.A., “Magnetic order in Pd2TiIn: a new itinerant antiferromagnet?” J. Magn. Magn. Mater. 127 (1993) 47.
[2]  Chouteau G., Yazami R., Private Communication.
[3]  Thomas M.G.S.R., David W.I.F., Goodenough J.B., Groves P., “Synthesis and structural characterization of the Normal spinel Li[Ni2]O4” Mat. Res. Bull. 20 (1985) 1137.
[4]  Dahn J.R., Von Sacken V., Juskow M.W., Al Janaby H., J. Electrochem. “Rechargeable LiNiO2/Carbon Cells” Soc. 138 (1991) 2207.
[5]  Ohzuku T., Ueda A., Nagayama M., “Electrochemistry and structural chemistry of LiNiO2 (R3m) for 4 Volt Secondary Lithium Cells” J. Electrochem. Soc. 140 (1993) 1862.
[6]  Decker F., Passerini S., Pileggi R., Scrosati B., “The electrochromic process in non-stoichiometric nickel oxide thin film electrodes” Electrochim. Acta 37 (1992) 1033.
[7]  Neumann K.U., Lipinski S., Ziebeck K.R.A., “Superweak ferrimagnetism arising from strong coupling in paramagnetic systems” Solide State Commun. 91 (1994) 443.
[8]  Tolédano J.C., The Landau Theory of Phase Transitions, World Scientific, Singapore, 1987.
[9]  Stanley H.E., Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford, 1971.
[10]  Amit D., Field Theory, The Renormalization Group and critical phenomena, McGraw-Hill, New York, 1978.
[11]  Pauli W., Über Gasentartung und paramagnetismus Z. Physik 41 (1927) 81.
[12]  Kittel C., Physique de l’Etat Solide, Dunod and Bordas, Paris, 1983.
[13]  El Houari B., Benhamou M., El Hafidi M., Chouteau G., “Para-ferrimagnetic transition in strong coupling paramagnetic systems: Landau theory approach”, J. Magn. Magn. Mater. 166 (1997) 97.
[14]  El Houari B., Benhamou M., “Mean-field analysis of the superweak ferrimagnetism of strongly coupled paramagnetic systems: II”, J. Magn. Magn. Mater. 172 (1997) 259.
[15]  Chahid M., Benhamou M., “Spin time-relaxation within strongly coupled paramagnetic systems exhibiting paramagnetic-ferrimagnetic transitions”, J. Magn. Magn. Mater. 218 (2000) 287.
[16]  Chahid M., Benhamou M., “Field theoretical approach to the paramagnetic-ferrimagnetic transition in strongly coupled paramagnetic systems”, J. Magn. Magn. Mater. 213 (2000) 219.
[17]  Chahid M., Benhamou M., “Critical dynamics of strong coupling paramagnetic systems exhibiting a paramagnetic-ferrimagnetic transition”, Physica A 305 (2002) 521.
[18]  Collins J.C., Renormalization, Cambridge University Press, Cambridge, 1985.
[19]  Zinn-Justin J., Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989.
[20]  Itzykson C., Drouffe J.M, Statistical Field Theory: 1 and 2, Cambridge University Press, Cambridge, 1989.
[21]  Binder K., Phase Transitions, C. Domb and M.S. Green, eds, Academic Press, London, Vol. 8, 1983.
[22]  Brout R., Phase Transitions, Benjamin, New York, 1965.
[23]  Smart J.S. “Effective Field Theories of magnetism“, Saunders, London, 1966.
[24]  Burley D.M. “Phase Transitions and Critical Phenomena”, C. Domb and M.S. Green, eds, Academic Press, London, Vol. 2, P. 29, 1972.
[25]  Mills D. L., “Surface Effects in Magnetic Crystals near the Ordering Temperature” Phys. Rev. B, 3 (1971) 3887.
[26]  Binder K., Hoenberg P.C., “Phase Transitions and Static Spin Correlations in Ising Models with Free Surfaces” Phys. Rev. B, 6 (1972) 3461.
[27]  Guyon E., Mitescu C.D., “Comparative study of size effects in solid and liquid films” Thin solid films, 12 (1972) 355.
[28]  Kumar P., Phys. Rev., “Magnetic phase transition at a surface: mean-field theory” Phys. Rev. B, 10 (1974) 2928.
[29]  Lubensky J.C., Rubin M.H., ”Critical Phenomena in semi-infinite. II. Mean-field theory” Phys. Rev. B, 12 (1975) 3885.
[30]  Waddahou A., Chahid M., Maârouf A., Benzouine F., “Critical Behavior at Surfaces of Strong Coupling Paramagnetic Systems Exhibiting a Paramagnetic-Ferrimagnetic Transition”. International Journal of Physics, Vol. 6, no. 4 (2018) 116-121.
[31]  Weiss P., Forrer R., “Magnetization of Nickel and the magnetocaloric effect” Ann. Phys. Paris 5, (1926) 153.
[32]  Chahine C., Thermodynamique Statistique, Dunod and Bordas, Paris, 1986.
[33]  Rougier A., «Relation entre la structure et le comportement électrochimique des phases LixNi1-xMyO2(M=Al, Fe, Co) : matériaux d’électrodes positives pour batteries au lithium», Thèse d’Université, Bordeaux, France, 1995.
[34]  Ginzburg V.L., Landau L.D., “On the theory of superconductivity”, Zh. Eksp. Teor. Fiz. 20 (1950) 1064.
[35]  Bray A. J., Moore A. M., “Critical behaviour of semi-infinite systems” J. Phys. A, 10 (1977) 1927.