Spin Ordering in Low-Dimensional Magnetic System Induced by Model Interaction


The magnetic susceptibility has been studied by using a new trial function for the model interaction between spins arranged in a low-dimensional square pyramidal system. We suggest that the exchange interaction depends on a power exponent which depends on the pyramidal structure in addition to the magnetic energy and thermal energy. We have calculated the magnetization as a function of the temperature and magnetic field keeping the exponent constant. Dependence of the susceptibility on a characteristic length related to the low-dimensional system exhibits novel features. Magnetic field dependence of the susceptibility has a maximum which shifts toward higher field as the temperature is increased. Even at higher fields, the susceptibility can be very small. The variations of the susceptibility with temperature exhibit maximum which depends on the external perturbing field.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    Yosida, K.: Paramagnetic susceptibility In superconductors. Phys. Rev. 110, 769 (1958)

    ADS  Article  Google Scholar 

  2. 2.

    Schnack, J., Luban, M.: Rotational modes in molecular magnets with antiferromagnetic Heisenberg exchange. Phys. Rev. B 63, 014418 (2000)

    ADS  Article  Google Scholar 

  3. 3.

    Schroder, C., Schmidt, H. J., Schnack, J., Luban, M.: Metamagnetic phase transition of the antiferromagnetic Heisenberg icosahedron. Phys. Rev. Lett. 94, 207203 (2005)

    ADS  Article  Google Scholar 

  4. 4.

    Strecka, J., Karlova, K., Madaras, T.: Giant magnetocaloric effect, magnetization plateaux and jumps of the regular Ising ployhedra. Physica B 466-467, 76–85 (2015)

    ADS  Article  Google Scholar 

  5. 5.

    Perenboom, J. A. A. J., Wyder, P., Meier, F.: Electronic properties of small metallic particles. Phys. Rep. 78, 173–292 (1981)

    ADS  Article  Google Scholar 

  6. 6.

    Lorenzo, A. D., Fazio, R., Hekking, F. W. J., Falci, G., Mastellone, A., Giaquinta, G.: Re-entrant spin susceptibility of a superconducting grain. Phys. Rev. Lett. 84, 550 (2000)

    ADS  Article  Google Scholar 

  7. 7.

    Thirion, C., Wernsdorfer, W., Jamet, M., Dupuis, V., Mélinon, P., Perez, A., Mailly, D.: Temperature dependence of switching fields of single 3 nm cobalt nanoparticles. J. Appl. Phys. 91, 7062 (2002)

    ADS  Article  Google Scholar 

  8. 8.

    Felner, I., Asaf, U., Levi, Y., Millo, O.: Coexistance of magnetism and superconductvity in R1.4Ce0.6RuSr2Cu2O10-δs (R = Eu and Gd). Phys. Rev. B 55, R3374 (1997)

    ADS  Article  Google Scholar 

  9. 9.

    Bernhard, C., Tallon, J. L., Niedermayer, C., Blasius, T., Golnik, A., Brucher, E., Kremer, R. K., Noakes, D. R., Stronach, C. E., Ansaldo, E. J.: Coexistence of ferromagnetism and superconductivity in the hybrid ruthenacuprate compound RuSr2GdCu2O8 studied by muon spin rotation and dc magnetization. Phys. Rev. B 59, 14099 (1999)

    ADS  Article  Google Scholar 

  10. 10.

    Fainstein, A., Winkler, E., Butera, A., Tallon, J.: Magnetic interactions and magnon gap in the ferromagnetic superconductor RuSr2GdCu2O8. Phys. Rev. B 60, R12597 (1999)

    ADS  Article  Google Scholar 

  11. 11.

    Bernhard, C., Tallon, J. L., Brucher, E., Kremer, R. K.: Evidence for a bulk Meissner state in the ferromagnetic superconductor RuSr2GdCu2O8 from dc magnetization. Phys. Rev. B 61, R14960 (2000)

    ADS  Article  Google Scholar 

  12. 12.

    Cardoso, C. A., Lanfredi, A. J. C., Chiquito, A. J., Araujo-Moreira, F. M., Awana, V. P. S., Kishan, H., de Almeida, R. L., de Lima, O.F.: Magnetic and superconducting properties of RuSr2Gd1.5Ce0.5Cu2O10-δ samples: Dependence on the oxygen content and aging effects. Phys. Rev. B 71, 134509 (2005)

    ADS  Article  Google Scholar 

  13. 13.

    Lv, Z. M., Wu, H. Y., Zhang, J. L., Li, M., Wei, Z., Li, H. O., Guo, X. F., Cao, L. Z., Ruan, K. Q.: Magnetic transition and Meissner shielding in ferromagnetic high-Tc superconductor RuSr2Gd1.4Ce0.6Cu2O10-δ. Eur. Phys. J. B 69, 161–165 (2009)

    ADS  Article  Google Scholar 

  14. 14.

    Canfield, P. C., Bud’ko, S. L., Cho, B. K.: Possible co-existence of superconductivity and weak ferromagnetism in ErNi2B2C. Phys. C 262, 249–254 (1996)

    ADS  Article  Google Scholar 

  15. 15.

    Kawano-Furukawa, H., Takeshita, H., Ochiai, M., Nagata, T., Yoshizawa, H., Furukawa, N., Takeya, H., Kadowaki, K.: Weak ferromagnetic order in the superconducting ErNi\(_{2}^{11}\)B2C. Phys. Rev. B 65, 180508 (2002)

    ADS  Article  Google Scholar 

  16. 16.

    For a review see Fischer, O. In: Buschow, K.H.J., Wohlfarth, E.P. (eds.) Ferromagnetic Materials, vol. 5. Amsterdam, Holland (1990)

  17. 17.

    Strecka, J., Richter, J., Derzhko, O., Verkholyak, T., Karl’ova, K.: Diversity of quantum ground states and quantum phase transitions of a spin-1/2 Heisenberg octadedral chain. Phys. Rev. B 95, 224415 (2017)

    ADS  Article  Google Scholar 

  18. 18.

    Deak, A., Szunyogh, L., Ujfalussy, B.: Thickness-dependent magnetic structure of ultrathin Fe/Ir(001) films: From spin-spiral states toward ferromagnetic order. Phys. Rev. B 84, 224413 (2011)

    ADS  Article  Google Scholar 

  19. 19.

    Antal, A., Udvardi, L., Ujfalussy, B., Lazarovits, B., Szunyogh, L., Weinberger, P.: Magnetic properties of a Cr trimer on Au(111) surface. J. Magn. Magn. Mater. 316, 118–121 (2007)

    ADS  Article  Google Scholar 

  20. 20.

    Karlova, K., Strecka, J., Verkholyak, T.: Cluster-based Haldane phases, bound magnon crystals and liquids of a mixed spin-1 and spin-1/2 Heisenberg octahedral chain. Phys. Rev. B 100, 094405 (2019)

    ADS  Article  Google Scholar 

  21. 21.

    Zheng, S. M.: Magnetic properties of disordered Ising systems with various probability distributions of the exchange integrals. Phys. Rev. B 52, 7260 (1995)

    ADS  MathSciNet  Article  Google Scholar 

  22. 22.

    Shanenko, A. A., Croitoru, M. D., Peeters, F. M.: Magnetic-field induced quantum-size cascades in superconducting nanowires. Phys. Rev. B 78, 024505 (2008)

    ADS  Article  Google Scholar 

  23. 23.

    Blount, E. I., Varma, C. M.: Electromagnetic effects near the superconductor-to-ferromagnet transition. Phys. Rev. Lett. 42, 1079 (1979)

    ADS  Article  Google Scholar 

  24. 24.

    Tachiki, M., Kotani, A., Matsumoto, H., Umezawa, H.: Spin-spiral ordering in magnetic superconductors. Solid State Commun. 31, 927–930 (1979)

    ADS  Article  Google Scholar 

  25. 25.

    Greenside, H. S., Blount, E. I., Varma, C. M.: Possible coexisting superconducting and magnetic states. Phys. Rev. Lett. 46, 49 (1981)

    ADS  Article  Google Scholar 

  26. 26.

    Pickett, W. E., Weht, R., Shick, A. B.: Superconductivity in ferromagnetic RuSr2GdCu2O8. Phys. Rev. Lett. 83, 3713 (1999)

    ADS  Article  Google Scholar 

  27. 27.

    Nesterov, K. N., Alhassid, Y.: Thermodynamics of ultrasmall metallic grains in the presence of pairing and exchange correlations: Mesoscopic fluctuations. Phys. Rev. B 87, 014515 (2013)

    ADS  Article  Google Scholar 

  28. 28.

    Jamet, M., Wernsdorfer, W., Thirion, C., Mailly, D., Dupuis, V., Melinon, P., Perez, A.: Magnetic anisotropy of a single cobalt nanocluster. Phys. Rev. Lett. 86, 4676 (2001)

    ADS  Article  Google Scholar 

  29. 29.

    Volokitin, Y., Sinzig, J., de Jongh, L. J., Schimd, G., Vargaftik, M. N., Moiseevi, I. I.: Quantum-size effects in the thermodynamic properties of metallic nanoparticles. Nature 384, 621–623 (1996)

    ADS  Article  Google Scholar 

  30. 30.

    Gladilin, V. N., Fomin, V. M., Devreese, J. T.: Magnetic susceptibility of ultrasmall superconductor grains. Phys. Rev. B 70, 144506 (2004)

    ADS  Article  Google Scholar 

  31. 31.

    Chubukov, A. V., Pines, D., Schmalian, J.: . In: Bennemann, K.H., Ketterson, J.B. (eds.) The Physics of Conventional and Unconventional Superconductors, vol. 1. Springer, Berlin (2003)

  32. 32.

    Zhang, L. -F., Covaci, L., Milosevic, M. V., Berdiyorov, G. R., Peeters, F. M.: Vortex states in nanoscale superconducting squares: The influence of quantum confinement. Phys. Rev. B 88, 144501 (2013)

    ADS  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Ajay Kumar Ghosh.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix 1

Appendix 1

The Hamiltonian of the system can be written as follows:

$$ \mathcal{H}=-\frac{1}{2}\left[\sum\limits_{i=1}^{4}{J_{ij}\left( \textbf{S}_{i}.\textbf{S}_{j}\right)}+\sum\limits_{j_{i}=1}^{4}{J_{j_{i}j_{j}}.\left( \textbf{S}_{j_{i}}.\textbf{S}_{j_{j}}\right)}\right]-\textbf{H}.\left( \textbf{S}_{i}+{\sum}_{j=1}^{4}{\textbf{S}_{j}}\right) $$

Here, Si is the apex spin. Sj’s are the spins on the plane.

$$ J_{ij}=J_{0}\left[1+{\Gamma}\left( \frac{a}{r_{ij}}\right)^{n}\coth{\left( \frac{k_{B}T}{\mu_{B}H}\right)}\right] $$

For \(J_{j_{i}j_{j}}\), d = 0 that is rij = a. From the geometry of the model system, the expression of rij is:

$$ r_{ij}=\frac{1}{\sqrt{2}}\sqrt{a^{2}+2d^{2}} $$

Magnetization of the model system can be written as:

$$ M=\left<S_{i}\right>+4\left<S_{j}\right> $$

Applying mean field theory, we can write:

$$ \left<S_{i}\right>=\frac{tr S_{i}e^{\upbeta S_{i}\mathcal{H}_{eff}^{i}}}{tr e^{\upbeta S_{i}\mathcal{H}_{eff}^{i}}}=\frac{1}{2}\tanh{\left( \frac{1}{2k_{B}T}\mathcal{H}_{eff}^{i}\right)} $$

Where, \({\mathscr{H}}_{eff}^{i}=2J_{ij}\left <S_{j}\right >+H\). Putting the value of \({\mathscr{H}}_{eff}^{i}\) into the expression, we can obtain the expression of \(\left <S_{i}\right >\), which is given as follows:

$$ \left<S_{i}\right>=\frac{1}{2}\tanh{\left[\frac{1}{2k_{B}T}\left( 2J_{ij}\left<S_{j}\right>+H\right)\right]} $$

Similarly, the expression of \(\left <S_{j}\right >\) is:

$$ \left<S_{j}\right>=\tanh{\left[\frac{1}{k_{B}T}\left( 2J_{ij}\left<S_{i}\right>+J_{j_{i}j_{j}}\left<S_{j}\right>+4H\right)\right]} $$

Adding \(\left <S_{i}\right >\) and \(4\left <S_{j}\right >\), we have the expression of magnetization M:

$$ \begin{array}{@{}rcl@{}} M&=&\frac{1}{2}\tanh{\left[\frac{1}{2k_{B}T}\left( 2J_{ij}\left<S_{j}\right>+H\right)\right]}\\ &&+4\tanh{\left[\frac{1}{k_{B}T}\left( 2J_{ij}\left<S_{i}\right>+J_{j_{i}j_{j}}\left<S_{j}\right>+4H\right)\right]} \end{array} $$

The susceptibility is defined as follows:

$$ \chi=\left( \frac{\partial M}{\partial H}\right)_{T} $$

Now putting the expression of M following equation, we have:

$$ \chi=\frac{\partial\left<S_{i}\right>}{\partial H}+4\frac{\partial\left<S_{j}\right>}{\partial H} $$

Using derivative \(\frac {\partial \left <S_{i}\right >}{\partial H}\), we have:

$$ \frac{\partial\left<S_{i}\right>}{\partial H}=\frac{1}{4T}\textup{sech}^{2}{\left[\frac{2J_{ij}\left<S_{j}\right>+H}{2T}\right]}\left( 1+2J_{ij}^{\prime}\left<S_{j}\right>+2J_{ij}\frac{\partial\left<S_{j}\right>}{\partial H}\right) $$


$$ \frac{\partial\left<S_{j}\right>}{\partial H}=\frac{4+J_{j_{i}j_{j}}^{\prime}\left<S_{j}\right>+2J_{ij}^{\prime}\left<S_{i}\right>+2J_{ij}\frac{\partial\left<S_{i}\right>}{\partial H}}{T\cosh^{2}{\left( \frac{4H+J_{j_{i}j_{j}}\left<S_{j}\right>+2J_{ij}\left<S_{i}\right>}{T}\right)-J_{j_{i}j_{j}}}} $$


$$ J_{j_{i}j_{j}}=J_{0}\left[1+{\Gamma}\coth{\left( \frac{k_{B}T}{\mu_{B}H}\right)}\right] $$
$$ J_{ij}^{\prime}=\left( \frac{a}{r_{ij}}\right)^{n}\frac{k_{B}T}{\mu_{B}H^{2}}\textup{cosech}^{2}{\left[\frac{k_{B}T}{\mu_{B}H}\right]} $$
$$ J_{j_{i}j_{j}}^{\prime}=\frac{k_{B}T}{\mu_{B}H^{2}}\textup{cosech}^{2}{\left[\frac{k_{B}T}{\mu_{B}H}\right]} $$

The final expression of the susceptibility \(\left (\chi \right )\) is given as follows:

$$ \begin{array}{@{}rcl@{}} \chi=\frac{1}{4T}\textup{sech}^{2}{\left[\!\frac{2J_{ij}\left<S_{j}\right>+H}{2T}\!\right]}\!\left( \!1+2J_{ij}^{\prime}\left<S_{j}\right>+2J_{ij}\frac{\partial\left<S_{j}\right>}{\partial H}\!\right)\\ +4\frac{4+J_{j_{i}j_{j}}^{\prime}\left<S_{j}\right>+2J_{ij}^{\prime}\left<S_{i}\right>+2J_{ij}\frac{\partial\left<S_{i}\right>}{\partial H}}{T\cosh^{2}{\left( \frac{4H+J_{j_{i}j_{j}}\left<S_{j}\right>+2J_{ij}\left<S_{i}\right>}{T}\right)-J_{j_{i}j_{j}}}} \end{array} $$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Haldar, S., Ghosh, A.K. Spin Ordering in Low-Dimensional Magnetic System Induced by Model Interaction. J Supercond Nov Magn (2020). https://doi.org/10.1007/s10948-020-05566-3

Download citation


  • Spin ordering in nanosystems
  • Molecular magnet
  • Magnetic properties of superconductor