Partial Up-Up-Down Order with the Continuously Distributed Order Parameter in the Triangular Antiferromagnet ${\mathrm{TmMgGaO}}_4$

We show that frustrated quasidoublets without time-reversal symmetry can host highly unconventional magnetic structures with continuously distributed order parameters even in a single-phase crystal. Our study comprises a comprehensive thermodynamic and neutron diffraction investigation on the single crystal of ${\mathrm{TmMgGaO}}_4$, which entails non-Kramers ${\mathrm{Tm}}^{3+}$ ions arranged on a geometrically perfect triangular lattice. The crystal electric field randomness caused by the site-mixing disorder of the nonmagnetic ${\mathrm{Mg}}^{2+}$ and ${\mathrm{Ga}}^{3+}$ ions merges two lowest-lying crystal electric field singlets of ${\mathrm{Tm}}^{3+}$ into a ground-state quasidoublet. Well below $T_c\sim 0.7\mathrm{K}$, a small fraction of the antiferromagnetically coupled ${\mathrm{Tm}}^{3+}$ Ising quasidoublets with small inner gaps condense into two-dimensional up-up-down magnetic structures with continuously distributed order parameters, and give rise to the columnar magnetic neutron reflections below ${\mu }_0{H}_c\sim 2.6\mathrm{T}$, with highly anisotropic correlation lengths, ${\xi }_{ab}\ge 250a$ in the triangular plane and ${\xi }_c<c/12$ between the planes. The remaining fraction of the ${\mathrm{Tm}}^{3+}$ ions remain nonmagnetic at 0 T and become uniformly polarized by the applied longitudinal field at low temperatures. We argue that the similar model can be generally applied to other compounds of non-Kramers rare-earth ions with correlated ground-state quasidoublets.

Popular Summary

Geometrical frustration describes a collection of spins on some symmetric lattice where not all of the interactions can simultaneously minimize energy. This kind of system cannot achieve conventional low-temperature ordering, where neighboring spins are oriented parallel to one another (ferromagnets) or in opposing directions (antiferromagnets). Specifically, an ideal “triangular-lattice Ising antiferromagnet” (where spins can be oriented only along a special direction, up or down) remains completely disordered even down to absolute zero, retaining a large residual entropy that violates the third law of thermodynamics. However, to date, the relevant real materials are still rare. Recently, researchers synthesized the triangular-lattice Ising antiferromagnet ${\mathrm{TmMgGaO}}_4$. Here, we report on a thorough single-crystal investigation of the low-temperature magnetism of ${\mathrm{TmMgGaO}}_4$.

Its structural sibling ${\mathrm{YbMgGaO}}_4$ was proposed as a quantum spin-liquid candidate. Here, the ${\mathrm{Yb}}^{3+}$ ion has an odd number of the $4f$ electrons, and thus the crystal electric field (CEF) always produces the lowest-lying doublet (the effective spin-$1/2$ moment), according to Kramers theorem. However, in ${\mathrm{TmMgGaO}}_4$ the situation is different: The ${\mathrm{T}\mathrm{m}}^{\mathrm{3}\mathrm{+}}$ ion has an even number of the $4f$ electrons, and CEF singlets are expected. Our present work indicates that the inner energy gap between the two lowest-lying singlets is small, and the site-mixing disorder of the nonmagnetic ${\mathrm{M}\mathrm{g}}^{2+}$ and ${\mathrm{G}\mathrm{a}}^{\mathrm{3}\mathrm{+}}$ ions with different valences distributes that inner gap throughout the crystal. These two lowest-lying CEF singlets are characterized as a “ground-state quasidoublet.”

We observe a very unconventional and complex magnetic order in ${\mathrm{TmMgGaO}}_4$ at low temperatures. A similar geometrically frustrated model can be generally applied to other non-Kramers rare-earth magnets with similar disorder-induced ground-state quasidoublets.

Figure 1

(a) Magnetization of ${\mathrm{TmMgGaO}}_4$ measured at 1.9 K in the fields parallel and perpendicular to the $c$ axis. The colored lines show the linear fits to the data above 8 T. Inset: Curie-Weiss fits to the susceptibilities of ${\mathrm{TmMgGaO}}_4$ and ${\mathrm{Tm}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$ measured at 0.1 T along the $c$ axis. (b) Magnetic heat capacities ($C_m$) of ${\mathrm{TmMgGaO}}_4$, ${\mathrm{Tm}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$, ${\mathrm{YbMgGaO}}_4$, and ${\mathrm{Yb}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$ at 0 T. The red and blue lines show, respectively, the fits to the data for ${\mathrm{Tm}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$ and ${\mathrm{Yb}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$ with the Lorentzian distributions of $E_2-E_1$. (c) Temperature dependence of susceptibilities measured in the field of 0.05 T applied both parallel and perpendicular to the $c$ axis. The inset shows the CEF levels from the combined CEF fit, with the black and violet lines for the CEF doublets and singlets, respectively. The GS quasidoublet is shown by the dark violet line. (d) Temperature dependence of the magnetic heat capacity measured at 0 T. The inset presents a sketch of the 2D three-sublattice magnetic dipole structure with the green and red lines showing the triangular lattice and the magnetic unit cell, respectively. The lines show the combined CEF fit to the magnetic data of ${\mathrm{TmMgGaO}}_4$ above 90 K in both (c) and (d). (e) Magnetic neutron diffraction of ${\mathrm{TmMgGaO}}_4$ measured on D23 at 60 mK and 0 T in the $ab$ plane ($L=0$). (f) $L$ dependence of selected static structure factors measured on POLI at 60 mK and 0 T. The magnetic structure factors are normalized by the magnetic form factor of ${\mathrm{Tm}}^{3+}$.

Figure 2

Combined fits to the temperature dependence of (a) susceptibility and (b) magnetic heat capacity measured at $\sim 0\mathrm{T}$, as well as (c) the field dependence of magnetization measured at 40 mK, on the single crystal of ${\mathrm{TmMgGaO}}_4$, using models 0, 1, 2, 3, and 4, respectively (see main text). (d) Magnetic field dependence of structure factors measured on the selected magnetic reflections at 60 mK on D23. The colored lines present the calculated values per Tm (multiplied by 1800 Tm, see Appendix pp4) at 60 mK using the above models 1, 2, 3, and 4. Models 1 and 2 give zero static structure factors on the magnetic reflections. (e) Magnetic entropy of ${\mathrm{TmMgGaO}}_4$ measured at 0 T. The colored lines show the calculated values using models 3 and 4.

Figure 3

2D continuous uud order in ${\mathrm{TmMgGaO}}_4$ calculated by the least-$R_p$ optimized model no. 4 under an external longitudinal field, ${\mu }_0{H}_{\parallel }=1.5\mathrm{T}$. The 9-site cluster with PBC is used in the ED calculation. The calculated (Ising) magnetic dipole structures at selected values of $\mathrm{\Delta }=E_2-E_1$ are shown in (a)–(d), respectively, with the numbers standing for the order parameter of the uud phase, as described in the text. (e) Lorentzian probability distribution of $\mathrm{\Delta }$, $P(\mathrm{\Delta })$. A fraction ($\sim 11\%$) of ${\mathrm{Tm}}^{3+}$ ions within the range of $2\delta$ (marked in red) give rise to the uud phase with the continuous distribution of the order parameter [see (a)–(c), for example], while the remaining large fraction marked by gray results in the uniform polarization of the spin system [see (d), for example]. (f) Field dependence of $\delta$, $\delta (H_{\parallel })$, calculated using the same Hamiltonian parameters of the fitted model no. 4 (9-site ED) on the 9-site (black line) and 12-site (olive line) clusters with PBC.

Figure 4

Calculated ground state in the presence of the inhomogeneous randomness using the (a) 9-site and (b) 12-site clusters with different PBC, under ${\mu }_0{H}_{\parallel }=1.5\mathrm{T}$. The fitted parameters of model no. 2 ($\overline{\mathrm{\Delta }}=5.71\mathrm{K}$, $J_1^{zz}=10.9\mathrm{K}$, $J_2^{zz}=1.11\mathrm{K}$, and $g_{\parallel }=13.6$) are used. The black number in the circle is the inner gap of each pseudospin (the multiple of $\overline{\mathrm{\Delta }}$), and the red and blue numbers are the spin-up and spin-down static dipole moments,$⟨S_i^z⟩$. The numbers in the triangles display the local order parameter calculated for the three spins at the corners of the respective triangle. The green lines depict the triangular lattice.

Figure 5

(a) Temperature and (b) magnetic field dependence of the intrinsic reflection width (blue) ${\omega }_L$, as well as the in-plane correlation length (red) ${\xi }_{ab}$, extracted from the magnetic reflection $(\frac{2}{3},-\frac{1}{3},0)$ measured on ${\mathrm{TmMgGaO}}_4$ at 0 T and 60 mK, respectively. (c) Phase diagram of ${\mathrm{TmMgGaO}}_4$ extracted from the neutron diffraction, heat capacity, magnetization, susceptibility, and magnetocaloric effect measurements. Long-range order (LRO, ${\xi }_{ab}\ge 2000\AA$), quasi-long-range order (QLRO, ${\xi }_{ab}\sim 1000\AA$), and short-range order (SRO, ${\xi }_{ab}\le 600\AA$) regions are marked roughly according to the magnetic neutron diffraction and heat capacity data.

Figure 6

(a) X-ray diffraction for the ${\mathrm{TmMgGaO}}_4$ single crystal on the $ab$ plane. The inset presents an enlargement of the strongest Bragg peak (009), where the angle ($2\mathrm{\Theta }$) difference between the nearest-neighbor data points is 0.01°. (b) Single crystals of ${\mathrm{TmMgGaO}}_4$ cut along the $ab$ plane. (c) Laue x-ray diffraction pattern of ${\mathrm{TmMgGaO}}_4$ on the $ab$ plane. (d) X-ray diffraction for the ${\mathrm{Tm}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$ single crystal on the $ab$ plane. The inset presents an enlarged plot of the strongest Bragg peak (009). (e) Single crystals of ${\mathrm{Tm}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$ cut along the $ab$ plane. (f) Laue x-ray diffraction pattern of ${\mathrm{Tm}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$ on the $ab$ plane. (g) X-ray diffraction for the ${\mathrm{Yb}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$ single crystal on the $ab$ plane. The inset presents an enlargement of the strongest Bragg peak (009). (h) Single crystals of ${\mathrm{Yb}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$ cut along the $ab$ plane. (i) Laue x-ray diffraction pattern of ${\mathrm{Yb}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$ on the $ab$ plane.

Figure 7

(a) Powder x-ray diffraction measured on the polycrystalline sample of ${\mathrm{TmMgGaO}}_4$ at 300 K. The black bars show the reflections calculated with the crystal structure data reported by Cevallos et al. [32]. The inset shows the strongest Bragg peaks (0,0,9) measured on the single crystals of ${\mathrm{TmMgGaO}}_4$, ${\mathrm{Tm}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$, and ${\mathrm{Yb}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$, respectively. (b) Temperature dependence of the ac susceptibility (the real part) measured on the single crystal of TmMgGaO4 down to 1.8 K. Thermal relaxation data of ${\mathrm{TmMgGaO}}_4$ single crystal measured (c) at 0 T at $\sim 0.1\mathrm{K}$, (d) at 0.5 T at $\sim 0.2\mathrm{K}$, with the lines representing the least-squares fits using the two-$\tau$ model [39]. For the definition of adj. $R^2$, see Ref. [50].

Figure 8

(a) Heat capacity of the ${\mathrm{TmMgGaO}}_4$ and ${\mathrm{LuMgGaO}}_4$ single crystals measured at 0 T. The inset presents an enlarged plot of the low-$T$ data with the black line showing the Debye heat-capacity fit (${\mathrm{\Theta }}_D=158\mathrm{K}$). (b) Magnetic heat capacity of ${\mathrm{TmMgGaO}}_4$ measured at selected fields. The phonon or lattice contribution was subtracted by the heat capacity of the nonmagnetic ${\mathrm{LuMgGaO}}_4$. The observed upturns below $\sim 0.3\mathrm{K}$ are fitted by considering the nuclear spin contributions. (c) Field dependence of the susceptibility ($d{M}_{\parallel }/d{H}_{\parallel }$) with the red line showing the three-peak Lorentzian fit. (d) Field dependence of the magnetic Grüneisen ratio measured at 0.09, 0.2, 0.3, and 2 K. (e) Field dependence of the heat capacity measured at 0.3 K.

Figure 9

Field dependence of (a) magnetization ($M$) and (b) susceptibility ($dM/dH$) of ${\mathrm{TmMgGaO}}_4$ measured in fields both parallel and perpendicular to the $c$ axis. The data were measured in the same vibrating sample magnetometer PPMS using the same single crystal of ${\mathrm{TmMgGaO}}_4$ (96.90 mg). The colored lines in (a) represent the linear fits above 8 T. We stopped the $M_{\perp }$ measurement at 12 T, as the force acting on the crystal may become too large, thus breaking the crystal itself or the sample holder [39]. (c) Temperature dependence of the susceptibility measured in the field of 0.1 T applied perpendicular to the $c$ axis. The straight red line is a guide to the eye demonstrating that no Curie-Weiss behavior is observed between 30 and 60 K [the temperature range where the magnetic entropy in Fig. 2 shows a plateau that indicates a paramagnetic regime not affected by spin-spin correlations and CEF excitations to higher levels]. (d) Temperature dependence of the dc susceptibility measured on the single crystal of ${\mathrm{Tm}}_{0.04}{\mathrm{Lu}}_{0.96}{\mathrm{MgGaO}}_4$ along the $c$ axis. The red line shows the Curie-Weiss fit to the data between 30 and 60 K.

Figure 10

Temperature dependence of the calculated CEF thermodynamic properties. The least-$R_P$ fitted average CEF parameters of ${\mathrm{TmMgGaO}}_4$ are used, without any CEF randomness (broadening) and without any intersite magnetic couplings. The dashed blue lines show the Curie fits to the calculated susceptibility along the $c$ axis (${\chi }_{\parallel }\sim 1/T$), at $30\le T\le 60\mathrm{K}$ and $T>4000\mathrm{K}$, respectively. The entropy (black) and dc susceptibilities (red) are calculated at 0 and 0.05 T (measuring field), respectively.

Figure 11

(a) High-temperature neutron diffraction background measured at 2 K and 0 T on D23. The much weaker peaks are magnetic field independent at 60 mK, and should not originate from the intrinsic magnetic signal of ${\mathrm{TmMgGaO}}_4$. The ring-shaped signals originate from the copper sample holder. (b) $\mathrm{\Omega }$ scans measured on the broad magnetic reflection $(\frac{2}{3},-\frac{1}{3},0)$ in 0 and 1.5 T applied along the $c$ axis, as well as on the nuclear reflection $(\overline{1},\overline{1},0)$ in 0 T at 60 mK on D23. The blue line shows the Gaussian fit with the resolution of ${\sigma }_{\mathrm{\Omega }}=0.65(2)°$, and the red lines are the fits to the data with a combination of the Gaussian and Lorentzian functions [see Eq. (d1)]. $L$ dependence of selected magnetic structure factors measured on POLI in (c) 0 T and (d) 1.5 T at 60 mK. (e) Longitudinal magnetic field dependence of the integral intensities of five nuclear Bragg reflections measured at 60 mK on D23. The colored lines are the combinations of the field-independent nuclear contribution and the calculated magnetic part using different models. The scale and magnetic form factors are included in the calculated magnetic contribution [see Eq. (e4)].

Figure 12

Magnetic neutron diffraction of TmMgGaO4 measured on D23 at 60 mK under the magnetic field of (a) 1.5 T and (b) 3 T, applied along the $c$ axis. Selected $\mathrm{\Omega }$ scans measured on the $(\frac{2}{3},-\frac{1}{3},0)$ magnetic reflection (c) at 0 T and (d) at 60 mK. The colored lines show the corresponding Lorentzian fits. (e) Temperature and (f) magnetic field dependence of the magnetic reflection intensities measured at 0 T and at 60 mK, respectively. The colored lines show the combined critical fits. (g) Temperature and (h) magnetic field dependence of the magnetic reflection FWHM measured at 0 T and at 60 mK, respectively.

Figure 13

Magnetic neutron diffraction of ${\mathrm{TmMgGaO}}_4$ measured at 60 mK in (a) 0 T and (c) 1.5 T. The calculated spectra are shown at (b) 0 T and (d) 1.5 T using model no. 4 by Eq. (e3) with the magnetic form factor. The isotropic resolution is used in the momentum ($\mathbf{Q}$) space, $\sim 0.015$ in both (b) and (d). (e) Static structure factor per Tm calculated by model no. 4 [see Eq. (e3)] along $[\frac{1}{3},-\frac{2}{3},L]$ at 0 and 1.5 T at 60 mK. (f) Thermodynamic properties calculated on the 9-site (solid lines) and 12-site (dashed lines) clusters with different PBC. The same Hamiltonian parameters of the fitted model no. 2 are used. The red, blue, and black (in the inset) lines show the calculated dc susceptibility at 0.1 T (measuring field), heat capacity at 0 T, and magnetization at 60 mK, respectively.

Figure 14

Calculated spin-wave excitations by Spinw at the longitudinal fields of 0 T and 1.5 T, using (a) model 1, (b) model 2, (c) model 3, and (d) model 4. Calculated INS spectra by the ED using the 9-site cluster with PBC at the longitudinal fields of 0 T and 1.5 T, at 50 mK, using (e) model 1, (f) model 2, (g) model 3, and (h) model 4 [see Eq. (e5)]. The instrumental resolution of 0.114 meV is used at $E_i=4.8\mathrm{meV}$, and the linear $\mathbf{Q}$-scan ranges are kept the same as Ref. [43].