Anisotropic field-induced ordering in the triangular-lattice quantum spin liquid ${\mathrm{NaYbSe}}_2$

High-quality single crystals of ${\mathrm{NaYbSe}}_2$, which resembles a perfect triangular-lattice antiferromagnet without intrinsic disorder, are investigated by magnetization and specific heat, as well as the local probe techniques nuclear magnetic resonance (NMR) and electron spin resonance. The low-field measurements confirm the absence of any spin freezing or long-range magnetic order down to 50 mK, which suggests a quantum spin liquid ground (QSL) state with gapless excitations. Instability of the QSL state is observed upon applying magnetic fields. For the $H\perp c$ direction, a field-induced magnetic phase transition is observed above 2 T from the $C_{\mathrm{p}}\left(T\right)$ data, agreeing with a clear $\frac{M_s}{3}$ plateau of $M\left(H\right)$, which is associated with an up-up-down spin arrangement. For the $H\parallel c$ direction, a field-induced transition could be evidenced at a much higher field range (9–21 T). The ${}^{23}\mathrm{Na}$ NMR measurements provide microscopic evidence of field-induced ordering for both directions. A reentrant behavior of $T_{\mathrm{N}}$, originating from the thermal and quantum spin fluctuations, is observed for both directions. The anisotropic exchange interactions $J_{\perp }\simeq 4.7$ K and $J_z\simeq 2.33$ K are extracted from the modified bond-dependent XXZ model for the spin-$\frac{1}{2}$ triangular-lattice antiferromagnet. The absence of magnetic long-range order at zero fields is assigned to the effect of strong bond frustration, arising from the complex spin-orbit entangled $4f$ ground state. Finally, we derive the highly anisotropic magnetic phase diagram, which is discussed in comparison with the existing theoretical models for spin-$\frac{1}{2}$ triangular-lattice antiferromagnets.

Figure 1

Powder XRD pattern for a ${\mathrm{NaYbSe}}_2$ sample compared with the ICSD reference (409678). Inset: Single crystal of ${\mathrm{NaYbSe}}_2$.

Figure 2

(a) Schematic crystal structure of ${\mathrm{NaYbSe}}_2$. Yb triangular layers (green) are formed by edge-shared ${\mathrm{YbSe}}_6$ octahedra in the $ab$ plane. (b) ESR spectra measured at 20 K for both the $H\perp c$ and $H\parallel c$ directions. (c) Angular dependence of the magnetic susceptibility obtained at 3 T and 20 K for a ${\mathrm{NaYbSe}}_2$ single crystal.

Figure 3

(a) Magnetic susceptibilities $({\chi }_{\parallel }$ and ${\chi }_{\perp })$ of ${\mathrm{NaYbSe}}_2$ single crystals. Red dotted and solid lines correspond to the high-temperature Curie-Weiss (CW) fit and TLAF fitting (see text), respectively. The inset shows the low-temperature inverse magnetic susceptibility after subtracting ${\chi }_{VV}$ along with the CW fit for both directions. (b) Isothermal magnetization $M\left(H\right)$ measured at 470 mK for both directions. Dashed lines represent the Van Vleck contribution (see text).

Figure 4

(a) Zero-field specific heat of ${\mathrm{NaYbSe}}_2$ and ${\mathrm{NaLuO}}_2$. Inset: The calculated magnetic entropy as a function of temperature. The dotted line indicates the value of $R\ln 2$. (b) $C_{\mathrm{p}}\left(T\right)$ data in the ordered states for $H\perp c$ (5 T) and $H\parallel c$ (18 T), together with a $T^3$ power law (see text).

Figure 5

(a) Temperature dependence of zero-field heat capacity divided by temperature, $C_{\mathrm{mag}}/T$, for ${\mathrm{NaYbSe}}_2$. The sharp upturn towards low temperatures due to the nuclear contribution is fitted with $C_{\mathrm{nuc}}\propto \alpha /T^2$. (b) Temperature dependence of $C_{\mathrm{mag}}/R$. $T_l$ and $T_h$ represent the low-$T$ and high-$T$ broad maxima at $T_l\simeq 1.1$ K and $T_h\simeq 2.75$ K.

Figure 6

(a) Analysis for the high-field magnetization isotherm for $H_{\perp }c$. The field boundaries for the uud phase and fully saturated phase are marked as red dashed lines. (b) Magnetization isotherms $M\left(H\right)$ measured at different temperatures for $H_{\perp }c$. Note that the curves are shifted by a constant offset in the magnetization.

Figure 7

Temperature-dependent specific heat of ${\mathrm{NaYbSe}}_2$, measured at different applied fields with orientations as indicated.

Figure 8

${}^{23}\mathrm{Na}$ NMR spin-lattice relaxation rate in the magnetic ordered states for the field in plane at ${\mu }_0{H}_{\perp }c=5$ T and for the fields out of plane at ${\mu }_0{H}_{\parallel }c=16$ T.

Figure 9

Field-temperature phase diagram of ${\mathrm{NaYbSe}}_2$ for $H\perp c$ and $H\parallel c$.