Frustrated Heisenberg $J_1-J_2$ model within the stretched diamond lattice of $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$

We investigate the magnetic properties of $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$, containing a three-dimensionally frustrated, diamondlike lattice via neutron scattering, magnetization, and heat capacity measurements. The stretched diamond network of ${\mathrm{Yb}}^{3+}$ ions in $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ enters a long-range incommensurate, helical state with an ordering wave vector $\mathbf{k}=(0.384,\pm 0.384,0)$ that “locks-in” to a commensurate $\mathbf{k}=(1/3,\pm 1/3,0)$ phase under the application of a magnetic field. The spiral magnetic ground state of $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ can be understood in the framework of a Heisenberg $J_1-J_2$ Hamiltonian on a stretched diamond lattice, where the propagation vector of the spiral is uniquely determined by the ratio of $J_2/\left|J_1\right|$. The pure Heisenberg model, however, fails to account for the relative phasing between the Yb moments on the two sites of the bipartite lattice, and this detail as well as the presence of an intermediate, partially disordered, magnetic state below 1 K suggests interactions beyond the classical Heisenberg description of this material.

Figure 1

(a) Crystal structure of $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ with $\mathrm{Yb}{\mathrm{O}}_6$ octahedra shaded in green and black spheres noting the positions of Li ions. (b) The frustrated $J_1-J_2$ model on the diamond lattice consists of two interpenetrating FCC sublattices, A and B, with a $J_1$ (black) magnetic interaction connecting the two sublattices and a $J_2$ (orange) spanning interactions within an FCC sublattice. When this structure is stretched along one of the cubic axes, the $I{4}_1/amd$ lattice of $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ is reproduced where the dashed green line represents the unit cell origin of $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ shown in panel (c). In $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$, the stretched bond (5.909 Å, dashed orange) is 1.527 Å longer than the in-plane $J_2$ (4.382 Å, solid orange). In the present model for $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$, the stretched bond is assumed negligible in strength relative to the shorter $J_2$. (c) NN ($J_1$) and NNN ($J_2$) exchange pathways between Yb ions in $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ with Yb ions in the $A$ and $B$ sublattices shaded differently for clarity.

Figure 2

The series of $A\mathrm{Ln}{X}_2$ ($A$ = alkali, Ln = lanthanide, $X$ = chalcogenide) compounds crystallize in several structures governed by the ratio of lanthanide radius divided by the sum of the alkali and chalcogenide radii derived from tabulated ionic radii and reported crystal structures [45, 46, 47, 48, 49, 50, 51, 52, 53]. Crossover between differing phases occurs at the dashed lines, and materials on these lines can crystallize in both neighboring space groups (e.g., $\mathrm{Na}\mathrm{Er}{\mathrm{O}}_{2}R\overline{3}m$ and $C2/c$ [48]).

Figure 3

(a) INS spectrum $S(\mathbf{Q},\hslash \omega )$ collected at 5 K and $E_i=300$ meV on the ARCS spectrometer with full width half max resolution in the elastic line of 12.8 meV. Three CEF levels indicated by dashed black lines were observed. (b) $\mathbf{Q}$-integrated cut from panel (a) overplotted with model line shapes derived from the CEF fits in Table 1 and convolved with the instrumental resolution in MantidPlot [42]. (c) The $D_{2d}J=7/2{\mathrm{Yb}}^{3+}$ ion generates four Kramers doublets centered at 45, 63, and 128 meV determined via CEF fits to the INS data. Errors shown correspond to instrumental resolution, and the well-separated ground-state doublet has an average $g$ factor $g_{\mathrm{avg}}=3.0$. The intensity ratios $I_2/I_1$ and $I_3/I_1$ were determined relative to the 45-meV CEF excitation. (d) INS spectrum $S(\mathbf{Q},\hslash \omega )$ collected at 5 K and $E_i=150$ meV from the ARCS spectrometer. The first two CEF excitations observed with $E_i=300$ meV show splitting of $\approx 7$ meV, highlighted by the dashed black lines. At $E_i=300$ meV, this splitting is not resolvable due to the poorer energy resolution. [(e) and (f)] Integrating the split CEF excitations shows that the ratio of integrated intensities of CEF mode 2 (${\mathrm{I}}_{2a}+I_{2b}$) to CEF mode 1 (${\mathrm{I}}_{1a}+I_{1b}$) correlates to the ratios of the integrated intensities in (b). The splitting of CEF excitations may represent two distinct chemical environments within $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ that are not resolvable within structural neutron powder diffraction experiments.

Figure 4

(a) Temperature dependence of the inverse magnetic susceptibility of $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$. Solid line shows the a Curie-Weiss fit to the data between $20<T<100$ K. (b) Field dependence of the magnetization collected at a variety of temperatures. (c) 2 K isothermal magnetization curve with a linear fit in the saturated state above 10 T. The 0-T intercept ($g_{\mathrm{avg}}{\mu }_B/2$) provides a powder-averaged $g_{\mathrm{avg},VV}$ and the slope provides ${\chi }_{VV}$. (d) ac magnetic susceptibility $\chi `\left(T\right)$ data collected for $330\mathrm{mK}<T<3.5$ K at zero-field. The two dashes lines at 1.13 K and 0.45 K mark the onset of peaks observed in zero-field heat capacity data.

Figure 5

[(a)–(d)] Specific heat $C\left(T\right)$ of $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ collected as a function of temperature under ${\mu }_{0}H=0$, 3, 4, and 9 T. The integrated magnetic entropy $\delta S_M$ is overplotted with the data as a black line. Results from a Debye model of lattice contributions to $C\left(T\right)$ are shown as orange lines. The horizontal dashed lines represent $Rln\left(2\right)$.

Figure 6

Neutron powder diffraction data collected for $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ at HB-2A at the High Flux Isotope Reactor. (a) Fits to the elastic scattering data at 1.5 K reveal only one structural phase. (b) Temperature-subtracted diffraction data ($T-1.5$ K) revealing a series of new magnetic peaks on cooling. Additionally, at 270 mK and 3 T, another set of magnetic peaks arise. Intensity near 1.5 ${\AA }^{-1}$ results from slight under/over subtraction of the structural peak at that position in (a) and is not a magnetic Bragg reflection. (c) Helical magnetic structure fit below the ordering transition $T_{N2}$. (d) The 270 mK data collected under zero field with the 1.5 K structural data subtracted. Green line shows the resulting fit using the magnetic structure described in the text. (e) The 830 mK data collected under zero field with the 1.5 K structural data subtracted. The orange line shows the partially disordered, intermediate helical state described in the text and the green line shows a fit using the fully ordered helical structure for comparison. (f) The 270 mK data collected under $\mu H=3$ T with the 1.5 K structural data subtracted. The red line shows the fit to the commensurate magnetic structure describe in the text.

Figure 7

Low-energy INS spectra $S\left(\right|Q|,\hslash \omega )$ collected on the DCS spectrometer at (a) ${\mu }_{0}H=0$ T and 36 mK, (b) ${\mu }_{0}H=0$ T and 800 mK, and (c) ${\mu }_{0}H=3$ T and 36 mK. All data have data collected at 36 mK and 10 T subtracted, where $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ enters a field-polarized state, indicated by isothermal magnetization data from Fig. 4.

Figure 8

Phase diagram of magnetic order in the $J_1-J_2$ Heisenberg model, assuming $J_2>0$, where ferromagnetic (FM), incommensurate (IC) spiral, and antiferromagnetic (AFM) Néel order exist.

Figure 9

The classical ground-state condition ${\mathit{S}}_{▵,1}+{\mathit{S}}_{▵,2}+\frac{J_1}{2J_2}{\mathit{S}}_{▵,3}=0$.

Figure 10

(a) Spin wave spectrum (red lines) and the structure factor simulation for $J_1=1.42565J_2>0$. Both along the (110) direction. (b) Angular averaged structure factor for $J_1=1.42565J_2>0$.

Figure 11

Proposed powder-averaged, low-temperature ($H, T$) diagram of $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ extracted from a combination of specific heat ($C_p$) measurements and elastic neutron powder diffraction data. At high temperature, $\mathrm{Li}\mathrm{Yb}{\mathrm{O}}_2$ is in the paramagnetic ($PM$) phase. Below approximately 10 K, specific heat shows a broad feature where roughly half of the magnetic entropy of $Rln\left(2\right)$ is released and signifies the onset of short-range magnetic correlations. A sharp anomaly at 1.13 K at 0, 3, and 5 T and 1.40 K at 9 T in specific heat measurements shows where long-range magnetic order sets in. Combining specific heat data with neutron powder diffraction data suggests that the temperature regime between 0.45 K and 1.13 K consists of a helical magnetic structure with disordered phasing between the two interpenetrating Yb sublattices. The system undergoes a lock-in phase transition from an incommensurate helical structure at zero field to a commensurate structure at 3 T.