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Thermodynamic Properties of the Core/Shell Antiferromagnetic Ising Nanocube

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Abstract

Using the effective field theory with correlations, the effects of the exchange interaction on the thermal behaviors of the total magnetization, internal energy, specific heat, entropy, and free energy of a transverse antiferromagnetic Ising nanocube are investigated. The phase diagram is also calculated and discussed in detail.

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Acknowledgments

This work has been initiated with the support of URAC: 08, the project RS:02(CNRST) and the Swedish Research Links programme dnr-348-2011-7264 and completed during a visit of A. A. at the Max Planck Institut für Physik Komplexer Systeme Dresden, Germany. The authors would like to thank all the organizations.

Author information

Authors and Affiliations

  1. Laboratoire de Physique des Matériaux et Modélisation, des Systèmes, (LP2MS), Unité Associée au CNRST-URAC 08, Physics Department, Faculty of Sciences, University of Moulay, Ismail, B.P. 11201, Meknes, Morocco

    M. El Hamri, S. Bouhou, I. Essaoudi & A. Ainane

  2. Max-Planck-Institut für Physik Complexer Systeme, Nöthnitzer Str. 38, 01187, Dresden, Germany

    A. Ainane

  3. Condensed Matter Theory Group, Department of Physics and Astronomy, Uppsala University, 75120, Uppsala, Sweden

    A. Ainane & R. Ahuja

  4. Quantum Functional Semiconductor Research Center, Dongguk University, Chung gu, Seoul, 100-715, Korea

    R. Ahuja

  5. Laboratoire de Chimie et Physique des Milieux Complexes, (LCPMC) Institut de Chimie, Physique et Matériaux (ICPM), 1 Bd. Arago, 57070, Metz, France

    F. Dujardin

Authors
  1. M. El Hamri
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  2. S. Bouhou
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  3. I. Essaoudi
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  4. A. Ainane
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  5. R. Ahuja
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  6. F. Dujardin
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Corresponding author

Correspondence to A. Ainane.

Appendix: Equations of the Core and Surface Shell Magnetizations

Appendix: Equations of the Core and Surface Shell Magnetizations

Within the framework of the effective field theory with correlations, four longitudinal magnetizations on the core, namely \(m_{c_{1}}^{z}\), \( m_{c_{2}}^{z}\), \(m_{c_{3}}^{z}\), and \(m_{c_{4}}^{z}\), and six longitudinal magnetizations on the surface shell, namely \(m_{s_{1}}^{z}\), \(m_{s_{2}}^{z}\) , \(m_{s_{3}}^{z}\), \(m_{s_{4}}^{z}\), \(m_{s_{5}}^{z}\) and \(m_{s_{6}}^{z}\), can be obtained as:

Magnetization of the central spin c 1:

$$\begin{array}{@{}rcl@{}} m_{c_{1}}^{z} &=&\frac{1}{2^{(2N_{3})}}\sum\limits_{\mu_{1}=0}^{N_{3}}\sum\limits_{\mu_{2}=0}^{N_{3}}C_{\mu _{1}}^{N_{3}}C_{\mu_{2}}^{N_{3}}\left( 1-m_{c_{2}}^{z}\right)^{\mu_{1}}\left( 1+m_{c_{2}}^{z}\right)^{N_{3}-\mu_{1}} \\ &&\times \left( 1-m_{s_{4}}^{z}\right)^{\mu_{2}}\left( 1+m_{s_{4}}^{z}\right)^{N_{3}-\mu_{2}}\times f_{z}(J_{c}(N_{3}-2\mu_{1}) \\ &&+J_{cs}(N_{3}-2\mu_{2})+h,{\Omega}_{c}) \end{array} $$
(19)

Magnetization of the central spin c 2:

$$\begin{array}{@{}rcl@{}} m_{c_{2}}^{z} &=&\frac{1}{2^{(3N_{2})}}\sum\limits_{\mu_{1}=0}^{N_{2}}\sum\limits_{\mu_{2}=0}^{N_{2}}\sum\limits_{\mu _{3}=0}^{N_{2}}C_{\mu_{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{2}}\left( 1-m_{c_{1}}^{z}\right)^{\mu_{1}}\left( 1+m_{c_{1}}^{z}\right)^{N_{2}-\mu_{1}} \\ &&\times \left( 1-m_{c_{3}}^{z}\right)^{\mu_{2}}\left( 1+m_{c_{3}}^{z}\right)^{N_{2}-\mu_{2}}\left( 1-m_{s_{5}}^{z}\right)^{\mu_{3}}\left( 1+m_{s_{5}}^{z}\right)^{N_{2}-\mu_{3}} \\ &&\times f_{z}(J_{c}(2N_{2}-2(\mu_{1}+\mu_{2}))+J_{cs}(N_{2}-2\mu_{3})+h,{\Omega}_{c}) \end{array} $$
(20)

Magnetization of the central spin c 3:

$$\begin{array}{@{}rcl@{}} m_{c_{3}}^{z} &=&\frac{1}{2^{(2N_{1}+N_{4})}}\sum\limits_{\mu_{1}=0}^{N_{4}}\sum\limits_{\mu _{2}=0}^{N_{1}}\sum\limits_{\mu_{3}=0}^{N_{1}}C_{\mu_{1}}^{N_{4}}C_{\mu_{2}}^{N_{1}}C_{\mu_{3}}^{N_{1}}\left( 1-m_{c_{2}}^{z}\right)^{\mu_{1}}\left( 1+m_{c_{2}}^{z}\right)^{N_{4}-\mu_{1}} \\ &&\times \left( 1-m_{c_{4}}^{z}\right)^{\mu_{2}}\left( 1+m_{c_{4}}^{z}\right)^{N_{1}-\mu_{2}}\left( 1-m_{s_{6}}^{z}\right)^{\mu_{3}}\left( 1+m_{s_{6}}^{z}\right)^{N_{1}-\mu_{3}} \\ &&\times f_{z}(J_{c}(N_{4}+N_{1}-2(\mu_{1}+\mu_{2}))+J_{cs}(N_{1}-2\mu_{3})+h,{\Omega}_{c}) \end{array} $$
(21)

Magnetization of the central spin c 4:

$$\begin{array}{@{}rcl@{}} m_{c_{4}}^{z} &=&\frac{1}{2^{(N_{6})}}\sum\limits_{\mu_{1}=0}^{N_{6}}C_{\mu_{1}}^{N_{6}}\left( 1-m_{c_{3}}^{z}\right) ^{\mu_{1}}\left( 1+m_{c_{3}}^{z}\right)^{N_{6}-\mu_{1}} \\ &&\times f_{z}(J_{c}(N_{6}-2\mu_{1})+h,{\Omega}_{c}) \end{array} $$
(22)

Magnetization of the surface spin s 1:

$$\begin{array}{@{}rcl@{}} m_{s_{1}}^{z}&=&\frac{1}{2^{N_{3}}}\sum\nolimits_{\mu_{1}=0}^{N_{3}}C_{\mu_{1}}^{N_{3}}\left( 1-m_{s_{2}}^{z}\right) ^{\mu_{1}}\!\left( 1+m_{s_{2}}^{z}\right)^{N_{3}-\mu_{1}} \\&&\times f_{z}(J_{s}(N_{3}-2\mu_{1})+h,{\Omega}_{s}) \end{array} $$
(23)

Magnetization of the surface spin s 2:

$$\begin{array}{@{}rcl@{}} m_{s_{2}}^{z} &=&\frac{1}{2^{(2N_{1}+N_{2})}}\sum\limits_{\mu_{1}=0}^{N_{1}}\sum\limits_{\mu _{2}=0}^{N_{1}}\sum\limits_{\mu_{3}=0}^{N_{2}}C_{\mu_{1}}^{N_{1}}C_{\mu_{2}}^{N_{1}}C_{\mu_{3}}^{N_{2}}\left( 1-m_{s_{1}}^{z}\right)^{\mu_{1}}\left( 1+m_{s_{1}}^{z}\right)^{N_{1}-\mu_{1}} \\ &&\times \left( 1-m_{s_{3}}^{z}\right)^{\mu_{2}}\left( 1+m_{s_{3}}^{z}\right)^{N_{1}-\mu_{2}}\left( 1-m_{s_{4}}^{z}\right)^{\mu_{3}}\left( 1+m_{s_{4}}^{z}\right)^{N_{2}-\mu_{3}} \\ &&\times f_{z}(J_{s}(2N_{1}+N_{2}-2(\mu_{1}+\mu_{2}+\mu_{3}))+h,{\Omega}_{s}) \end{array} $$
(24)

Magnetization of the surface spin s 3:

$$\begin{array}{@{}rcl@{}} m_{s_{3}}^{z} &=&\frac{1}{2^{2N_{2}}}\sum\nolimits_{\mu_{1}=0}^{N_{2}}\sum\nolimits_{\mu_{2}=0}^{N_{2}}C_{\mu _{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}}\left( 1-m_{s_{2}}^{z}\right)^{\mu_{1}} \\ &&\times \left( 1+m_{s_{2}}^{z}\right)^{N_{2}-\mu_{1}}\!\left( 1-m_{s_{5}}^{z}\right)^{\mu_{2}}\!\left( 1+m_{s_{5}}^{z}\right)^{N_{2}-\mu _{2}}\\&&\times f_{z}(J_{s}(2N_{2}-2(\mu_{1}+\mu_{2}))+h,{\Omega}_{s}) \end{array} $$
(25)

Magnetization of the surface spin s 4:

$$\begin{array}{@{}rcl@{}} m_{s_{4}}^{z} &=&\frac{1}{2^{(N_{1}+2N_{2})}}\sum\limits_{\mu_{1}=0}^{N_{1}}\sum\limits_{\mu _{2}=0}^{N_{2}}\sum\limits_{\mu_{3}=0}^{N_{2}}C_{\mu_{1}}^{N_{1}}C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{2}}\left( 1-m_{c_{1}}^{z}\right)^{\mu_{1}}\left( 1+m_{c_{1}}^{z}\right)^{N_{1}-\mu_{1}} \\ &&\times \left( 1-m_{s_{2}}^{z}\right)^{\mu_{2}}\left( 1+m_{s_{2}}^{z}\right)^{N_{2}-\mu_{2}}\left( 1-m_{s_{5}}^{z}\right)^{\mu_{3}}\left( 1+m_{s_{5}}^{z}\right)^{N_{2}-\mu_{3}} \\ &&\times f_{z}(J_{s}(2N_{2}-2(\mu_{2}+\mu_{3}))+J_{cs}(N_{1}-2\mu_{1})+h,{\Omega}_{s}) \end{array} $$
(26)

Magnetization of the surface spin s 5:

$$\begin{array}{@{}rcl@{}} m_{s_{5}}^{z} &=&\frac{1}{2^{(3N_{1}+N_{2})}}\sum\limits_{\mu_{1}=0}^{N_{1}}\sum\limits_{\mu _{2}=0}^{N_{1}}\sum\limits_{\mu_{3}=0}^{N_{2}}\sum\limits_{\mu_{4}=0}^{N_{1}}C_{\mu_{1}}^{N_{1}}C_{\mu _{2}}^{N_{1}}C_{\mu_{3}}^{N_{2}}C_{\mu_{4}}^{N_{1}} \\ &&\times\left( 1-m_{c_{2}}^{z}\right)^{\mu_{1}}\left( 1+m_{c_{2}}^{z}\right)^{N_{1}-\mu_{1}} \left( 1-m_{s_{3}}^{z}\right)^{\mu_{2}} \\ &&\times\left( 1+m_{s_{3}}^{z}\right)^{N_{1}-\mu_{2}}\left( 1-m_{s_{4}}^{z}\right)^{\mu_{3}}\left( 1+m_{s_{4}}^{z}\right)^{N_{2}-\mu_{3}} \\ &&\times \left( 1-m_{s_{6}}^{z}\right)^{\mu_{4}}\left( 1+m_{s_{6}}^{z}\right)^{N_{1}-\mu_{4}}f_{z}(J_{s}(2N_{1}+N_{2}\\&&\qquad-2(\mu_{2}+\mu_{3}+\mu_{4})) \\ &&+J_{cs}(N_{1}-2\mu_{1})+h,{\Omega}_{s}) \end{array} $$
(27)

Magnetization of the surface spin s 6:

$$\begin{array}{@{}rcl@{}} m_{s_{6}}^{z} &=&\frac{1}{2^{(N_{1}+N_{4})}}\sum\limits_{\mu_{1}=0}^{N_{1}}\sum\limits_{\mu_{2}=0}^{N_{4}}C_{\mu _{1}}^{N_{1}}C_{\mu_{2}}^{N_{4}}\left( 1-m_{c_{3}}^{z}\right)^{\mu_{1}}\\&&\times\left( 1+m_{c_{3}}^{z}\right)^{N_{1}-\mu _{1}} \left( 1-m_{s_{5}}^{z}\right)^{\mu_{2}} \\ &&\times \left( 1+m_{s_{5}}^{z}\right)^{N_{4}-\mu_{2}}f_{z}(J_{s}(N_{4}-2\mu_{2})\\&&+J_{cs}(N_{1}-2\mu_{1})+h,{\Omega}_{s}) \end{array} $$
(28)

With N 1=1, N 2=2, N 3=3, N 4=4 and N 6=6 denote respectively the coordination number, and \({C_{k}^{l}}\) are the binomial coefficients \({C_{k}^{l}}=\frac {l!}{k!(l-k)!}\).

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Hamri, M.E., Bouhou, S., Essaoudi, I. et al. Thermodynamic Properties of the Core/Shell Antiferromagnetic Ising Nanocube. J Supercond Nov Magn 28, 3127–3133 (2015). https://doi.org/10.1007/s10948-015-3143-1

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