Mermin-Wagner physics, $(H,T)$ phase diagram, and candidate quantum spin-liquid phase in the spin-$\frac{1}{2}$ triangular-lattice antiferromagnet ${\mathrm{Ba}}_{8}{\mathrm{CoNb}}_{6}{\mathrm{O}}_{24}$

Mermin-Wagner physics, $(H, T)$ phase diagram, and candidate quantum spin-liquid phase in the spin- $\frac{1}{2}$ triangular-lattice antiferromagnet ${Ba}_{8} {CoNb}_{6} O_{24}$

Y. Cui, J. Dai, P. Zhou, P. S. Wang, T. R. Li, W. H. Song, J. C. Wang, L. Ma, Z. Zhang, S. Y. Li, G. M. Luke, B. Normand, T. Xiang, and W. Yu

Phys. Rev. Materials 2, 044403 – Published 13 April 2018

Abstract

${Ba}_{8} {CoNb}_{6} O_{24}$ presents a system whose ${Co}^{2 +}$ ions have an effective spin 1/2 and construct a regular triangular-lattice antiferromagnet (TLAFM) with a very large interlayer spacing, ensuring purely two-dimensional character. We exploit this ideal realization to perform a detailed experimental analysis of the $S = 1 / 2$ TLAFM, which is one of the keystone models in frustrated quantum magnetism. We find strong low-energy spin fluctuations and no magnetic ordering, but a diverging correlation length down to 0.1 K, indicating a Mermin-Wagner trend toward zero-temperature order. Below 0.1 K, however, our low-field measurements show an unexpected magnetically disordered state, which is a candidate quantum spin liquid. We establish the $(H, T)$ phase diagram, mapping in detail the quantum fluctuation corrections to the available theoretical analysis. These include a strong upshift in field of the maximum ordering temperature, qualitative changes to both low- and high-field phase boundaries, and an ordered regime apparently dominated by the collinear “up-up-down” state. ${Ba}_{8} {CoNb}_{6} O_{24}$ , therefore, offers fresh input for the development of theoretical approaches to the field-induced quantum phase transitions of the $S = 1 / 2$ Heisenberg TLAFM.

Received 25 September 2017

DOI:https://doi.org/10.1103/PhysRevMaterials.2.044403

Physics Subject Headings (PhySH)

Frustrated magnetism

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Y. Cui¹, J. Dai¹, P. Zhou¹, P. S. Wang¹, T. R. Li¹, W. H. Song¹, J. C. Wang¹, L. Ma², Z. Zhang³, S. Y. Li^3,4, G. M. Luke^5,6, B. Normand⁷, T. Xiang^8,9, and W. Yu^1,*

¹Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials & Micro-nano Devices, Renmin University of China, Beijing 100872, China
²High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, China
³State Key Laboratory of Surface Physics, Department of Physics, and Laboratory of Advanced Materials, Fudan University, Shanghai 200433, China
⁴Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
⁵Department of Physics and Astronomy, McMaster University, Hamilton, Canada L8S 4M1
⁶Canadian Institute for Advanced Research, Toronto, Canada M5G 1Z8
⁷Neutrons and Muons Research Division, Paul Scherrer Institute, CH-5232 Villigen-PSI, Switzerland
⁸Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
⁹Collaborative Innovation Center of Quantum Matter, Beijing 100190, China

^*wqyu_phy@ruc.edu.cn

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Vol. 2, Iss. 4 — April 2018

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Figure 1
Lattice structure and magnetic properties of ${Ba}_{8} {CoNb}_{6} O_{24}$ . (a) The triangular-lattice planes formed from the ${CoO}_{6}$ units are separated by many nonmagnetic units, specifically eight ${Ba}^{2 +}$ and six ${NbO}_{6}$ layers. (b) A plane of ${CoO}_{6}$ octahedra, showing their triangular-lattice configuration and the neighboring, corner-sharing ${NbO}_{6}$ octahedra that mediate the antiferromagnetic interactions. (c) Magnetization, $M$ ; blue and green straight lines represent linear fits to the low- and high-field data, whose origins lie, respectively, in the effective-spin and van Vleck paramagnetic contributions. Inset: $g$ -factor as a function of temperature. (d) dc susceptibility measured under a field of 0.1 T. The solid green line is a fit to a Curie-Weiss form, $χ_{s} (T) = a / (T + θ)$ , combined with an offset of $χ_{VV}$ , the constant van Vleck contribution determined from $M$ . The fitting parameters for the low-temperature regime are $a = 1.7$ emu K/mol and $θ = 3.5 \pm 0.5$ K. Inset: low-temperature spin susceptibility; the blue line shows HTSE results for the Heisenberg TLAFM, adapted from Ref. [25]; the green line shows Monte Carlo results for the Heisenberg SLAFM, adapted from Ref. [34]. Both curves use the parameters $J = 1.66$ K and $g = 4.13$ . (e) Specific heat at zero field. The solid line is a fit to $C_{p} = b T^{3}$ . Inset: low-temperature magnetic specific heat, $C_{m}$ , after subtracting the phonon contribution; the green line is the HTSE result for the Heisenberg TLAFM, adapted from Ref. [25] with $J = 1.66$ K. (f) Magnetic entropy, $S_{m} (T)$ , obtained by integrating the specific-heat data above 0.08 K. The solid line shows the high-temperature limit, $S_{m} (\infty) = R ln (2 S + 1)$ in a spin- $S$ system, for $S = 1 / 2$ . Inset: low-temperature magnetic entropy, showing for comparison RSBMF results (see the text) adapted from Ref. [23] with $J = 1.66$ K. Blue shading represents the temperature region excluded from our analysis.
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Figure 2
NQR measurements on ${Ba}_{8} {CoNb}_{6} O_{24}$ . (a) Spin-lattice relaxation rate, $1 /^{93} T_{1}$ , measured at zero field as a function of temperature. The upturn below 0.2 K indicates a progressive increase of the correlation length upon cooling, which is cut off abruptly below 0.1 K. Inset: zero-field ${}^{93}{Nb}$ NQR spectra for temperatures from 0.8 to 0.028 K, with the measured intensity multiplied by $T$ . Red lines represent the Gaussian fit to the spectrum at 0.028 K overlaid on all four datasets. (b) $1 /^{93} T_{1}$ data on logarithmic axes. The straight-line fit (blue) indicates an approximate power-law form, $1 /^{93} T_{1} \propto T^{α}$ with $α = 1.51 \pm 0.06$ . (c) $1 /^{93} T_{1}$ data on semilog axes as a function of inverse temperature. The straight-line fit (blue) indicates an approximate activated form with an energy gap $Δ = 0.070 \pm 0.003$ K.
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Figure 3
NMR response of ${Ba}_{8} {CoNb}_{6} O_{24}$ over a range of applied fields. (a) ${}^{93}{Nb}$ field-sweep spectra displayed as a function of frequency, $f - f_{0}$ , at a fixed temperature $T = 30$ mK. The reference frequency $f_{0} =^{93} γ H$ is defined for each spectrum from the measurement field, $H$ , given in the legend. The spectra are offset vertically for clarity. Arrows mark the spectral peak where the spin-lattice relaxation rate, $1 /^{93} T_{1}$ , is measured. (b) $1 /^{93} T_{1}$ shown as a function of temperature for selected low fields. (c) $1 /^{93} T_{1} (T)$ for selected high fields. Blue arrows mark the positions of the peaks we take to indicate the magnetic transition. Red arrows for certain field values in panel (b) mark a lower characteristic feature (discussed in the text).
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Figure 4
Field-temperature phase diagram of ${Ba}_{8} {CoNb}_{6} O_{24}$ . $(H, T)$ phase diagram constructed from the spin-lattice relaxation rate. Solid blue squares represent the temperature of the peak in $1 /^{93} T_{1}$ , taken from Figs. 3 and 3, and open gray circles mark the lower temperature scale found in $1 /^{93} T_{1}$ at low fields [Fig. 3] (discussed in the text). The shaded region demarcates the candidate quantum spin liquid phase. The solid line is a fit of the upper transition to the functional form $T_{c} (H) = c {(H_{s} - H)}^{0.5}$ . Green arrow symbols and dashed lines represent the phases and phase boundaries established by Monte Carlo simulations of the classical TLAFM, adapted from Refs. [40] and [41] and scaled to $J = 1.66$ K. Red ticks on the field axis at $T = 0$ mark the boundaries of the up-up-down phase for the $S = 1 / 2$ Heisenberg TLAFM, established by many authors and taken in this case from Ref. [42]. Red dashed lines mark schematic finite-temperature phase boundaries anticipated for the up-up-down phase in the $S = 1 / 2$ system, where it is favored by quantum fluctuations. Inset: field dependence of $1 /^{93} T_{1}$ shown at fixed temperatures of 50, 100, and 150 mK. The presence of only two peaks suggests that the entire intermediate regime accessible in experiment may have only one type of field-induced order.
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Physical Review Materials

Mermin-Wagner physics, (H,T) phase diagram, and candidate quantum spin-liquid phase in the spin-12 triangular-lattice antiferromagnet Ba8CoNb6O24

Y. Cui, J. Dai, P. Zhou, P. S. Wang, T. R. Li, W. H. Song, J. C. Wang, L. Ma, Z. Zhang, S. Y. Li, G. M. Luke, B. Normand, T. Xiang, and W. Yu

Phys. Rev. Materials 2, 044403 – Published 13 April 2018

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Mermin-Wagner physics, $(H, T)$ phase diagram, and candidate quantum spin-liquid phase in the spin- $\frac{1}{2}$ triangular-lattice antiferromagnet ${Ba}_{8} {CoNb}_{6} O_{24}$