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    Journals >Journal of Inorganic Materials >Volume 36 >Issue 8 >Page 865 > Article
    • Journal of Inorganic Materials
    • Vol. 36, Issue 8, 865 (2021)
    Mechanism of the Magnetic and Specific-heat Anomalies in Rare-earth Dodecaborides RB12 (R=Tb-Tm): an Effect of Crystal-field-splitting Order Parameter
    Jingsan HU1, Jianfei GU1, and Weiyi ZHANG1、2、*
    Author Affiliations
  • 11. School of Physics, Nanjing University, Nanjing 210093, China
  • 22. National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
    • show less
    DOI: 10.15541/jim20200623 Cite this Article
    Jingsan HU, Jianfei GU, Weiyi ZHANG. Mechanism of the Magnetic and Specific-heat Anomalies in Rare-earth Dodecaborides RB12 (R=Tb-Tm): an Effect of Crystal-field-splitting Order Parameter [J]. Journal of Inorganic Materials, 2021, 36(8): 865 Copy Citation Text show less

    Abstract

    Rare-earth dodecaborides RB12 (R=Tb, Dy, Ho, Er, Tm) are all characterized by a controversial and unresolved magnetic and specific-heat anomaly at low temperature (TC≈3-22 K) except LuB12 with closed f-shell. Although the bump feature in magnetic susceptibilities resembles that of antiferromagnetically ordered R3+ ions, no definite magnetic structure was identified by either neutron scattering or Mössbauer spectra. The anomalies in susceptibilities, entropy, and low temperature neutron diffraction pattern all pointed to the amplitude-modulated nature of complex magnetic structure. In view of the rather small crystal-field-splitting energies of the order of meV, we propose to treat crystal-field-splitting (CFS) energy as an order parameter rather than a quantum mechanical quantity. In this way, we found that not only the magnetic and specific-heat anomalies can be explained properly by a gradual quenching of R orbital moments, but also the low temperature satellite peaks of neutron scattering spectra can be understood in terms of the spontaneously (111) dimerised structure and inelastically absorbed rattling phonon modes.

    Keywords

    $\begin{matrix} {{H}_{0}}={{H}_{\text{SO}}}+{{H}_{\text{CF}}}+{{H}_{\text{ZS}}} \\ \end{matrix}$

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    $\begin{align} & {{Y}_{\text{l{m}'}}}{\sigma }'|{{H}_{\text{SO}}}|{{Y}_{\text{lm}}}\sigma =0.5\xi [\sqrt{(l-\sigma m)(l+\sigma m+1)} \\ & \ \ \ \ {{\delta }_{{m}'m+\sigma }}{{\delta }_{{\sigma }'\bar{\sigma }}}+\sigma m{{\delta }_{{m}'m}}{{\delta }_{{\sigma }'\sigma }}] \\ \end{align}$

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    $\begin{align} & \langle {{Y}_{\text{l{m}'}}}{\sigma }'|{{H}_{\text{ZS}}}|{{Y}_{\text{lm}}}\sigma \rangle =-{{\mu }_{\text{B}}}B\cdot \\ & \ \ \ \ [0.5({{n}_{x}}-i{{n}_{y}})\sqrt{(l-m)(l+m+1)}{{\delta }_{{m}'m+1}}{{\delta }_{{\sigma }'\sigma }}+ \\ & \ \ \ \ 0.5({{n}_{x}}+i{{n}_{y}})\sqrt{(l+m)(l-m+1)}{{\delta }_{{m}'m-1}}{{\delta }_{{\sigma }'\sigma }}+ \\ & \ \ \ \ ({{n}_{x}}+i\sigma {{n}_{y}}){{\delta }_{{m}'m}}{{\delta }_{{\sigma }'\bar{\sigma }}}+{{n}_{z}}\left( m+\sigma \right){{\delta }_{{m}'m}}{{\delta }_{{\sigma }'\sigma }}] \\ \end{align}$

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    $\begin{matrix} {{H}_{\text{CF}}}=\underset{i=1,\sigma }{\overset{7}{\mathop \sum }}\,{{J}_{i}}f_{i\sigma }^{\dagger }{{f}_{i\sigma }} \\ \end{matrix}$

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    $\begin{align} & V(\overset{\scriptscriptstyle\rightharpoonup}{r})=\frac{\alpha {{e}^{2}}}{4\text{ }\!\!\pi\!\!\text{ }{{\epsilon }_{0}}a}+\frac{280Z{{e}^{2}}}{4\text{ }\!\!\pi\!\!\text{ }{{\epsilon }_{0}}{{a}^{5}}}\left[ ({{x}^{4}}+{{y}^{4}}+{{z}^{4}})-\frac{3}{5}{{r}^{4}} \right]- \\ & \ \ \ \ \frac{1344Z{{e}^{2}}}{4\text{ }\!\!\pi\!\!\text{ }{{\epsilon }_{0}}{{a}^{7}}}\left[ ({{x}^{6}}+{{y}^{6}}+{{z}^{6}})+\ \frac{15}{4}({{x}^{2}}{{y}^{4}}+{{y}^{2}}{{z}^{4}}+ \right. \\ & \ \ \ \ \left. {{z}^{2}}{{x}^{4}}+{{x}^{2}}{{z}^{4}}+{{y}^{2}}{{x}^{4}}+{{z}^{2}}{{y}^{4}})-\frac{15}{14}{{r}^{6}} \right]~ \\ \end{align}$

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    $\begin{matrix} \text{ }\!\!\Delta\!\!\text{ }_{1}^{0}=\frac{2Z{{\text{e}}^{2}}}{4\text{ }\!\!\pi\!\!\text{ }{{\epsilon }_{0}}a}\left[ \frac{560}{33}\frac{\langle {{r}^{4}}\rangle }{{{a}^{4}}}-\frac{6720}{143}\frac{\langle {{r}^{6}}\rangle }{{{a}^{6}}} \right]~ \\ \end{matrix}$

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    $\begin{matrix} \text{ }\!\!\Delta\!\!\text{ }_{2}^{0}=\frac{2Z{{\text{e}}^{2}}}{4\text{ }\!\!\pi\!\!\text{ }{{\epsilon }_{0}}a}\left[ \frac{336}{11}\frac{\langle {{r}^{4}}\rangle }{{{a}^{4}}}-\frac{2240}{143}\frac{\langle {{r}^{6}}\rangle }{{{a}^{6}}} \right]~ \\ \end{matrix}$

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    $\begin{matrix} \Delta V=\frac{Z}{N}\frac{{\alpha }'{{\text{e}}^{2}}}{4\text{ }\!\!\pi\!\!\text{ }{{\epsilon }_{0}}}\frac{2}{\sqrt{3}}\frac{Q}{{{a}^{2}}} \\ \end{matrix}$

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    $\begin{matrix} {{T}_{\text{C}}}\approx N{{\theta }_{\text{D}}}\sqrt{\frac{2{{k}_{\text{B}}}{{\theta }_{\text{D}}}{{M}_{\text{R}{{\text{B}}_{12}}}}}{{{\hbar }^{2}}{{(2\text{ }\!\!\pi\!\!\text{ }/a)}^{2}}}}\frac{4\text{ }\!\!\pi\!\!\text{ }{{\epsilon }_{0}}a}{Z{\alpha }'{{e}^{2}}}\text{ }\!\!\Delta\!\!\text{ }_{2}^{0} \\ \end{matrix}$

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    $\begin{matrix} \langle r_{i}^{n}\rangle \to {{\left[ 1+\beta \frac{{{a}_{i}}-{{a}_{\text{Tm}}}}{\sqrt[n]{\langle r_{i}^{n}\rangle }}+\gamma \right]}^{n}}\langle r_{i}^{n}\rangle \\ \end{matrix}$

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    $\begin{matrix} \text{ }\!\!\chi\!\!\text{ }_{n}^{N}=\frac{{{\mu }_{\text{B}}}}{B}\frac{\sum\limits_{I}^{C_{14}^{N}}{\mu _{I}^{N}\text{exp(}-E_{I}^{N}/{{k}_{\text{B}}}T\text{)}}}{\sum\limits_{I}^{C_{14}^{N}}{\text{exp}(-E_{I}^{N}/{{k}_{\text{B}}}T)}} \\ \end{matrix}$

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    $C_{V}^{N}=\frac{1}{{{k}_{\text{B}}}{{T}^{2}}}\left\{ \frac{\sum\limits_{I}^{C_{14}^{N}}{{{(E_{I}^{N})}^{2}}\exp (-E_{I}^{N}/{{k}_{\text{B}}}T)}}{\sum\limits_{I}^{C_{14}^{N}}{\exp (-E_{I}^{N}/{{k}_{\text{B}}}T)}}- \right.\ \left. {{\left[ \frac{\sum\limits_{I}^{C_{14}^{N}}{E_{I}^{N}\exp (-E_{I}^{N}/{{k}_{\text{B}}}T)}}{\sum\limits_{I}^{C_{14}^{N}}{\exp (-E_{I}^{N}/{{k}_{\text{B}}}T)}} \right]}^{2}} \right\}$

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    Jingsan HU, Jianfei GU, Weiyi ZHANG. Mechanism of the Magnetic and Specific-heat Anomalies in Rare-earth Dodecaborides RB12 (R=Tb-Tm): an Effect of Crystal-field-splitting Order Parameter [J]. Journal of Inorganic Materials, 2021, 36(8): 865
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