Spin excitations in the frustrated triangular lattice antiferromagnet ${\mathrm{NaYbO}}_2$

Here we present a neutron scattering-based study of magnetic excitations and magnetic order in ${\mathrm{NaYbO}}_2$ under the application of an external magnetic field. The crystalline electric field-split $J=7/2$ multiplet structure is determined, revealing a mixed $|m_z\rangle$ ground-state doublet, and is consistent with a recent report by Ding et al. [L. Ding, P. Manuel, S. Bachus, F. Grußler, P. Gegenwart, J. Singleton, R. D. Johnson, H. C. Walker, D. T. Adroja, A. D. Hillier, and A. A. Tsirlin, Phys. Rev. B 100, 144432 (2019)]. Our measurements further suggest signatures of exchange effects in the crystal-field spectrum, manifested by a small splitting in energy of the transition into the first excited doublet. The field dependence of the low-energy magnetic excitations across the transition from the quantum disordered ground state into the fluctuation-driven ordered regime is analyzed. Signs of a first-order phase transition into a noncollinear ordered state are revealed at the upper-field phase boundary of the ordered regime, and higher-order magnon scattering, suggestive of strong magnon-magnon interactions, is resolved within the previously reported up-up-down phase. Our results reveal a complex phase diagram of field-induced order and spin excitations within ${\mathrm{NaYbO}}_2$ and demonstrate the dominant role of quantum fluctuations across a broad range of fields within its interlayer frustrated triangular lattice.

Figure 1

(a) The ${\mathrm{NaYbO}}_2$ structure ($R\overline{3}m$, 1.6 K [54]) contains alternating layers of equilateral triangular ${\mathrm{YbO}}_6$ octahedra (Yb, purple; O, orange) and Na ions (black). (b) Top-down view of the local ${\mathrm{Yb}}^{3+}{D}_{3\text{D}}$ environment extended to two coordination shells. From the central Yb ion, six ${\mathrm{O}}^{2-}$, six ${\mathrm{Na}}^{+}$, and six ${\mathrm{Yb}}^{3+}$ ions reside at 2.24, 3.35, and 3.35 Å, respectively. (c) In the ${\mathrm{D}}_{3\text{D}}$ environment, Yb creates four Kramers doublets as observed from crystalline electric field (CEF) fits to inelastic neutron-scattering data. Listed errors correspond to instrumental resolutions at $E_i$ = 150 meV (solid lines) and $E_i$ = 60 meV (dashed lines). The ground state has anisotropic $g$ factors of $g_c$ = 1.73 and $g_{ab}$ = 3.30 [54]. Dashed lines next to the first excited CEF doublet represent experimentally observed splitting due to exchange-induced CEF dispersions observed in $E_i$ = 60 meV data. Intensity ratios of the second and third excitations ($I_2/I_1, I_3/I_1$) were obtained relative to the first CEF state. Uncertainties specified next to energy values are the calculated instrument energy resolution at those energies.

Figure 2

(a) INS spectrum $S(Q,\hslash \omega )$ at 5 K obtained on ARCS at the Spallation Neutron Source with $E_i$ = 150 meV. The three dashed black lines feature the three CEF excitations while the dashed white lines correspond to phonons seen in integrated cuts in panel (b). Uncertainties specified next to energy values are the calculated instrument energy resolutions at those energies. (b) Experimental fits to the $\left|Q\right|=[3.0,3.5]{\AA }^{-1}$ integrated cut at 5 K of the INS spectrum. Fit 1 has the lowest ${\chi }^2$ value to INS data (Table 1), while fit 2 closely resembles a point-charge model incorporating two coordination shells (1). (c) Inverse magnetic susceptibility of ${\mathrm{NaYbO}}_2$ at 20 Oe overlaid with calculated inverse magnetic susceptibilities from two comparable CEF models. The mean-field exchange interaction is introduced (solid lines) to account for deviations due to strong antiferromagnetic exchange in ${\mathrm{NaYbO}}_2$ [54]. Calculated susceptibilities that do not include the antiferromagnetic interaction (dashed lines) do not accurately reproduce the data.

Figure 3

(a) INS spectrum $S(Q,\hslash \omega )$ at 5 K obtained on ARCS at the Spallation Neutron Source with $E_i$ = 60 meV. The two dashed black lines indicate the split peak observable at $E_i$ = 60 meV, and the dashed white lines correspond to nearby phonons. Uncertainties specified next to energy values are the calculated instrument energy resolutions at those energies. (b) $\left|Q\right|$ integrated cut of the 5-K INS spectrum from $[2.5,3]{\AA }^{-1}$ illustrating that the first CEF excitation in ${\mathrm{NaYbO}}_2$ is broadened by exchange-induced CEF dispersions. Fits to two resolution-limited Gaussian peaks separated by 2.1 meV accurately capture the splitting. The instrumental resolution is denoted by error bars beneath each peak.

Figure 4

Low-energy INS spectrum $S(Q,\hslash \omega )$ of ${\mathrm{NaYbO}}_2$ powder at varying fields collected on DCS. With increasing field, ${\mathrm{NaYbO}}_2$ evolves from a gapless quantum disordered ground state (0–2 T) into an up-up-down equal moment magnetic structure (3–8T) and a field-polarized state at high field (9–10 T). Data were collected with longer scans at 0, 5, and 10 T to increase resolution. Detector spurions occur at $[0.5{\AA }^{-1}$, 1.8 $\mathrm{meV}]$ and $[1.75{\AA }^{-1}$, 0.4 $\mathrm{meV}]$. Data were collected between 67 and 100 mK.

Figure 5

(a) Low-energy INS of ${\mathrm{NaYbO}}_2$ from CNCS at 5 T plotted on a logarithmic scale showing a weak third magnetic peak ($E^{*}$) centered at 1.5 meV above the flat, powder-averaged bands from the up-up-down phase. The third magnetic mode arises from a multimagnon process convolving the two lower-energy bands of the up-up-down phase. At 5 T, this multimagnon band is readily visible as it is well separated from the powder-averaged up-up-down structure. (b) Low $\left|Q\right|$-integrated energy cut of INS data of ${\mathrm{NaYbO}}_2$ from DCS data (4) showing the evolution of the flat up-up-down mode as a function of field.

Figure 6

(a) Neutron powder-diffraction data collected from an elastic line cut on the DCS instrument at the NIST Center for Neutron Research with 5-Å incident neutrons between $E=[-1,1]$ meV. In zero field, no new magnetic reflections arise. Data at 0 T were collected at 67 mK and 5-T data were collected at 74 mK. (b, c) Integrated intensity of the $\mathit{Q}=(1/3,1/3,0)$ and $(1/3,1/3,2)$ peaks, respectively, upon sweeping the field from 0 $\to$ 10 T and 10 $\to$ 0 T between 67 mK (0 T) and 92 mK (10 T). Data were analyzed via Gaussian fits to elastic line DCS data. At 0 and 10 T, there was no observable intensity at the $\mathit{Q}=(1/3,1/3,0)$ position, indicating ${\mathrm{NaYbO}}_2$ is in its quantum disordered ground state and field-polarized state, respectively. No quantifiable intensity is ascribable to the $\mathit{Q}=(1/3,1/3,2)$ peak on field ramping from 0 $\to$ 10 T. However, upon ramping downward, 10 $\to$ 0 T, the $\mathit{Q}=(1/3,1/3,2)$ peak appears between 9 and 3 T.

Figure 7

Classical phase diagram of the 2D XXZ model on a triangular lattice in the presence of magnetic field. The 3D phase space is parametrized by $(h_{xy},A^{-1},h_z)$, where $h_{xy}=\sqrt{h_x^2+h_y^2}$. Only the first octant ($h_{xy}\ge 0,A^{-1}\ge 0,h_z\ge 0$) is considered. The blue surface separates the Y and the V phases; the red plane separates the phases between the $A<1$ and the $A>1$ regions; the green surface separates the canted-I and the paramagnetic phases in the region $A<1$, and the orange surface separates the V and the paramagnetic phases in the region $A>1$.

Figure 8

Linear spin-wave theory (LSWT) calculations showing $S(Q,\hslash \omega )$ as a function of field for powder-averaged ${\mathrm{Yb}}^{3+}$ ions on a two-dimensional triangular lattice assuming three-sublattice ordering derived from the proposed spin model for ${\mathrm{NaYbO}}_2$ [54]. At 0 T, ${\mathrm{NaYbO}}_2$ does not show magnetic ordering, and therefore LSWT fails to capture the continuum of excitations from the quantum disordered ground state.