Synthesis, disorder and Ising anisotropy in a new spin liquid candidate PrMgAl₁₁O₁₉

Introduction

Quantum spin liquids (QSLs) represent an intriguing state where the spins remain disordered even at zero Kelvin due to quantum fluctuations, albeit with strong spin-spin couplings [1]. Achieving a QSL ground state is challenging because of the propensity for the spin sublattice to freeze as the temperature is lowered, especially around defects and/or disorder which act as pinning centers. For example, in the triangular lattice of YbMgGaO₄, the presence of site disorder between the nonmagnetic ions Mg and Ga induces a spin glass behavior [2]. However, disorder is not always detrimental to the QSL state. Studies on the pyrochlore oxides [3, 4] and 1T-TaS₂ [5] reveal that the quenched disorder does not compete, but rather cooperates with the frustration to induce strong quantum fluctuations, and may give rise to emergent spin disordered state responsible for the gapless excitations.

From an experimental point of view, a QSL state does not break any symmetry making it arduous to identify using a single technique [6]. Inelastic neutron scattering plays an important role in providing crucial evidence for a QSL state, as fractional excitations, such as spinons, manifest as a distinctive excitation continuum in the spectrum [7]. The availability of high-quality large single crystals is essential for neutron scattering due to the small scattering cross-section and low neutron flux. Similar requisites hold true for other measurements, such as spin thermal transport, to minimize any grain boundary effects as well as the anisotropic characterizations of magnetic behaviors [8, 9].

Here we present the single crystal growth of a hexaaluminate, PrMgAl₁₁O₁₉, using the optical floating zone technique. Crystal growth and basic magnetic property measurements on this series of lanthanide aluminates have been reported [10, 11], but a detailed characterization down to milikelvin is still lacking. Our previous results on the isostructural, polycrystalline PrZnAl₁₁O₁₉ already suggested the magnetic sublattice of Pr³⁺ ions, decorating a triangular lattice, has the potential to host a Dirac QSL [12, 13]. The triangular network of Pr ions reside within the ab plane, and are connected via intermediate O ions which are also within the triangular plane, forming a nearly linear Pr-O-Pr bond with an angle of about 176.7°. These layers are separated along the c-axis by ~11 Å while the nearest neighbor Pr-Pr bond is ~5.59 Å; see figure 1. As a comparison, the distance between triangular layers in YbMgGaO₄ are about 8 Å with a nearest neighbor bond length of ~3.40 Å [14]. Moreover, the large difference in the ionic radii between the magnetic and nonmagnetic ions inhibits any site mixing between the magnetic and nonmagnetic ions. These structural features indicate that this system is an ideal quasi-two-dimensional system free from chemical disorder, and great starting conditions for realizing the QSL state. Indeed, the signature of forming a QSL state has been observed in AC susceptibility measurements of PrZnAl₁₁O₁₉ revealing no spin freezing or ordering down to 50 mK, despite a considerable antiferromagnetic coupling strength of about -9 K. In addition, inelastic neutron scattering exhibits an abnormal broadening of low energy excitations at ~1.5 meV [13]. Unfortunately single crystals of PrZnAl₁₁O₁₉ are currently not available at the size needed for inelastic neutron scattering.

Figure  1.  The crystal structure of PrMgAl₁₁O₁₉ extracted from single-crystal XRD refinement. Green—Pr; grey—Al; red—O; orange—Mg. The mirror plane including the disordered Pr ions is depicted in the right panel.

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By substituting Zn with the less evaporative Mg element, we have succeeded in growing sizable single crystals of PrMgAl₁₁O₁₉ suitable for neutron scattering measurement. However, we have identified site disorder within the mirror plane, with about 7% of the Pr ions displaced from the ideal position. Magnetic susceptibility measurements show the moments lying perfectly along the c-axis (or perpendicular to the triangular plane) and do not freeze down to 50 mK. We argue that the system keeps fluctuating despite strong couplings and site disorder, which makes this system unique for a triangular system with Ising character.

1.   Methods

Polycrystalline samples were prepared using a standard solid-state reaction technique. Raw materials of Pr₆O₁₁ (99.99%), MgO (99.99%), and Al₂O₃ (99.99%) were dried at 900 °C over night prior to reaction to avoid moisture contamination. Then, stoichiometric amounts of the raw materials were mixed and ground thoroughly, pressed into pellets and sintered at 1400 °C ~ 1600 °C with several intermediate grindings. The powder sample was mixed with about 1% ~ 2% excess of MgO and pressed into a cylindrical rod of ~6 mm in diameter and ~140 mm (~35 mm) in length as a feed (seed) rod using a hydrostatic pressure of 70 MPa. The obtained feed and seed rods were sintered at 1500 °C for 2 h. Subsequent single crystal growth was conducted in an optical floating zone furnace (HKZ300) in pure argon atmosphere at 10 bar. After floating zone growth, the several-centimeter-sized as-grown crystal was annealed at 1000 °C in flowing O₂ for 24 h and then slowly cooled down to room temperature in order to avoid any possible oxygen vacancy.

The single crystal x-ray diffraction (XRD) measurement was performed on a XtaLAB Synergy diffractometer (Rigaku) at room temperature using the Mo-K_α radiation. The experimental conditions are tabulated in table 1. Part of the single crystal was crushed into powders for powder XRD measurement performed on a MiniFlex diffractometer (Rigaku) with the Cu-K_α radiation. JANA [15] and FULLPROF [16] packages were used for crystal structure refinement.

Table  1.  Experimental conditions for the single crystal XRD measurements, and agreement factors for the refinement.

Formula	PrMgAl11O19
Space group	$P{6}_3/mmc$ (No. 194)
a, b (Å)	5.587 00(10)
c (Å)	21.8732(6)
V (Å3)	591.29(2)
Z	2
2Θ (°)	3.72 - 82.16
No. of reflections, $R_{\mathrm{int}}$	125 34, 4.11%
No. of independent reflections	614
No. of parameters	48
Index ranges	-9 $\text{⩽}$H$\text{⩽}$ 10,
	-10 $\text{⩽}$K$\text{⩽}$ 10,
	-39 $\text{⩽}$L$\text{⩽}$ 39
R, wR2a	1.69%, 4.60%
Goodness of fit on F2	1.42
Largest difference peak/hole (e/Å3)	0.34/-0.38

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Heat capacity measurements were carried out on the Physical Property Measurement System (PPMS, Quantum Design) equipped with a dilution insert using the relaxation method. The DC magnetic susceptibility between 2 and 350 K was measured using the vibrating sample magnetometer (VSM) option of the PPMS. The AC magnetic susceptibility between 0.05 and 15 K was measured using the ACMS-II and ACDR options of the PPMS. For the AC susceptibility measurement, a driven field of 1-3 Oe in amplitude was used.

2.   Results

The single crystal growth was initially attempted in a pure oxygen atmosphere, which turns out to be unstable and results in cavities along the as-grown crystals. This occurrence indicates the dissolution of oxygen in the melt, forming stable bubbles during the growth process. Therefore, subsequent growths were performed in a pure Ar atmosphere at 10 bar pressure in order to minimize the evaporation of the Mg element. The upper and lower rods were counter-rotated at a rate of 20 rpm, and moved downwards simultaneously at a rate of 2 mm h^-1. These conditions led to the successful growth of centimeter-sized single crystals, as depicted in the inset of figure 2(a). The single-domain nature was confirmed by Laue diffraction measurements along the rod. A typical Laue pattern is shown in figure 2(a) with the x-ray beam nearly parallel to the c-axis. Laue measurements also confirm that the crystal was grown along the a direction.

Figure  2.  X-ray studies of PrMgAl₁₁O₁₉. (a) A Laue pattern with the x-ray beam approximately along the c axis. The inset shows a photo of the as-grown single crystal. (b)-(d) Precession images within the HK0, 0KL and H0L planes constructed from the single-crystal XRD measurements.

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The quality of the single crystal was further characterized by x-ray diffraction of single crystals and crushed crystals. All peaks in the powder diffraction pattern can be indexed by the space group $P{6}_3/mmc$, indicating an impurity-free phase; see the supplementary materials (SM) [17]. The precession images within the HK0, 0KL, and H0L planes from the single crystal XRD measurements are shown in figures 2(b)-(d). The sharp peaks confirm the high quality of the crystal. The structure refinement is consistent with the findings reported by Kahn et al for LaMgAl₁₁O₁₉ [11]. The final structure parameters extracted from the single crystal refinement are listed in table 2. It was found that about 7% of the Pr ions at the 2d (Pr1) site were shifted about 0.6 Å away from the ideal position towards the 6h (Pr2) site. Moreover, within the mirror plane of the structure, the Al5 ion displayed an anomalous atomic displacement parameter (ADP) along the c-axis, U₃₃, which amounts to 0.018 Ų, compared to the others of around 0.007 Ų. Therefore, it is set to be slightly off the mirror plane with half occupancy at the 4e site. Note, that with the resolution of our lab-based powder x-ray diffractometer, this anomalous ADP would be within errors and not highlighted. The last unsolved issue is the occupied position of the Mg ions. It is difficult to distinguish Mg²⁺ from Al³⁺ by means of x-ray diffraction because of the same number of electrons for both ions. However, neutron powder diffraction measurement on its sister compound CeMgAl₁₁O₁₉ [18] indicated that Mg is likely positioned at the Al3 site with an AlO₄ tetrahedron coordination, which also shows an anomalous ADP if solely occupied by Al ions. Therefore, the Mg ions are mixed with the Al3 ion at the 4f site with 0.5 occupancy. The refined structure is shown in figure 1 and those particular ions are highlighted.

Table  2.  Refined crystal structure parameters.

Atom	occ.	x	y	z
Pr1 (2d)	0.928(10)	1/3	2/3	0.75
Pr2 (6 h)	0.024(3)	0.271(5)	0.729(5)	0.75
Al1 (2a)	1	0	0	0
Al2 (4f)	1	1/3	2/3	0.189 98(4)
Al3 (4f)	0.5	1/3	2/3	0.472 75(4)
Mg (4f)	0.5	1/3	2/3	0.472 75(4)
Al4 (12k)	1	0.167 38(4)	0.334 77(9)	0.608 39(2)
Al5 (4e)	0.5	0	0	0.2428(3)
O1 (6 h)	1	0.181 22(15)	0.3624(3)	0.25
O2 (12k)	1	0.152 14(11)	0.3043(2)	0.446 09(6)
O3 (12k)	1	-0.0112(2)	0.494 42(11)	0.348 08(5)
O4 (4f)	1	1/3	2/3	0.558 37(10)
O5 (4e)	1	0	0	0.348 18(9)

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Moving to the magnetic property measurements. The x-ray photoelectron spectroscopy (XPS) measurements indicate a pure Pr³⁺ state [17]. Figure 3(a) shows the temperature dependence of the magnetic susceptibility measured along different crystallographic directions. Note, that the c- and c-directions are identical. The data shows pronounced anisotropy behavior with the moment predominantly aligned along the c-axis. No difference between the zero-field-cooled (ZFC) and field-cooled (FC) curves can be observed, consistent with a dynamical (paramagnetic) state. The temperature dependence of ${\chi }_c^{-1}$ is shown in figure 3(b). A Curie-Weiss (CW) fit, ${\chi }_c^{-1}=(T-{\theta }_{\text{CW}})/C$, to the data above 250 K yields a positive Curie temperature θ_CW of 67 K. It should be noted that at these elevated temperatures we expect a large contribution from the crystal electric field (CEF) effect since two CEF levels were observed at ~12 and 35 meV [13] in the isostructural PrZnAl₁₁O₁₉. At low temperatures where only the low-lying CEF ground state is occupied, the influence from the higher excitation levels is negligible, a CW-like fit then provides a measure of the intersite interactions. A fit below 10 K yields θ_CW of -8 K, indicating strong antiferromagnetic couplings; see figure 3(b). The effective moment is 5.18 µ_B/Pr, which is close to the expected value of $g_c^{\text{ESR}}\sqrt{(S(S+1)}=4.42{\mu }_B$/Pr, where $g_c^{\text{ESR}}$ = 5.1 is the g factor determined from ESR measurement [17], and S = 1/2, implying that the low temperature magnetic properties are governed by an effective S = 1/2 state. AC susceptibility measurements along the c-axis show no sign of long-range ordering, nor any spin freezing down to 50 mK; see figure 3(c). The field dependence of isothermal magnetization (figure 3(d)) reveals negligible magnetization along the a direction compared to that of the c direction. It is about 20 times larger along the c axis at 2 K and 14 T, which can be easily explained by a slight misalignment (2 degrees) of the crystal during the measurement. Thus, the system can be considered as a well-defined Ising system.

Figure  3.  Temperature and field dependence of the dynamic susceptibility of PrMgAl₁₁O₁₉. (a) Temperature dependence of the magnetic susceptibility measured with the magnetic fields applied along the reciprocal a and c directions. (b) Temperature dependence of the inverse susceptibility, ${\chi }_c^{-1}(T)$. The red and blue curves are according to the CEF fit and Curie-Weiss fit, respectively. The inset highlights the low-temperature region. (c) Temperature dependence of the real component of the ac susceptibility, ${\chi }_c^{\prime }$(T), measured at various frequencies. (d) Isothermal magnetization measured at various temperatures. The solid lines represent the magnetization calculated from the CEF parameters.

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In order to have more insights into the magnetic ground state, we performed CEF analysis and fit the CEF Hamiltonian to ${\chi }_c^{-1}(T)$. Although there are two inequivalent Pr sites, most of the Pr ions ($>$90%) are still located at the 2d site with a $D_{3h}$ symmetry. Thus, we start with a model disregarding the disorder effect at the Pr site. In such a case, the CEF Hamiltonian can be expressed as

where $O_n^m$ are the Stevens operators, and only the CEF parameters $B_2^0$, $B_4^0$, $B_6^0$ and $B_6^6$ are nonzero. The presence of the $O_6^6$ term admixes those states $|J,m_J⟩$ differ in m_J by 6. For Pr³⁺, the total angular momentum number J = 4. Thus, there are three doublets: two are formed by a linear combination of $|\pm 4⟩$ and $|\mp 2⟩$, and the other is $|\pm 1⟩$. The three singlets are consisting of $|3^a⟩=1/\sqrt{2}(|3⟩+|-3⟩)$, $|3^b⟩=1/\sqrt{2}(|3⟩-|-3⟩)$, and $|0⟩$. Therefore, if the ground state is a doublet and well separated from the first excited state, it will be highly anisotropic since $⟨J^{\pm }⟩$ = 0. To estimate the energy separation, we use the PyCrystalField package [19] to fit ${\mathcal{H}}_{\text{CEF}}$ to the magnetic susceptibility data. To do so, we calculate the magnetic susceptibility ${\chi }_{\mathrm{CEF}}^c=M_{\text{CEF}}^c/H$ based on the CEF Hamiltonian, and treat the influence of the surrounding ions to a mean field level, so that ${\chi }_{\mathrm{cal}}^c={\chi }_{\mathrm{CEF}}^c/(1-\lambda {\chi }_{\mathrm{CEF}}^c)$. By minimizing ${\chi }^2=\sum _i(1/{\chi }_{\mathrm{cal},i}^c-1/{\chi }_{\mathrm{obs},i}^c{)}^2$, the best fit yields $B_2^0$ = -0.841 81 meV, $B_4^0$ = -0.005 16 meV, $B_6^0$ = -0.000 12 meV, $B_6^6$ = -0.006 02 meV, and λ = -1.19 T/µ_B. The eigenvectors and eigenenergies can be found in the SM [17]. The negative λ indicates an antiferromagnetic interaction among the spins. From $\lambda =z{J}_c/({\mu }_B{g}_J{)}^2$, where z is the number of nearest neighbors, the exchange constant J_c can be estimated as ~ -0.73×10^-2 meV (-0.08 K). The small exchange constant is partly due to the large effective moment of Pr³⁺ (J = 4) [20]. In the effective spin S = 1/2 picture, it will be enhanced by a factor of $\frac{J(J+1)}{S(S+1)}$ = 26.7 (~ -2.1 K). Correspondingly, the CW temperature at low temperature can be estimated as ${\theta }_{\text{CW}}^{\mathrm{cal}.}=S(S+1)z{J}_c/3k_B\sim$ -3 K, which is roughly consistent with the experimental value. The calculated $({\chi }_{\mathrm{cal}}^c{)}^{-1}$ is shown in figure 3(b), which reproduces the overall data very well. Using the CEF parameters and λ, the calculated isothermal magnetization is shown in figure 3(d). Again, the calculated curves agree well with the experimental values. The small discrepancy below ~10 K may suggest the onset of spin-spin interactions. Moreover, the g factors corresponding to the ground state doublets amount to $g_{ab}=g_J⟨\mp |J^{\mp }|\pm ⟩$ = 0 and $g_c=2g_J⟨\pm |J_z|\pm ⟩$ = 5.6, consistent with the Ising character as inferred from the susceptibility measurements. The calculated g_c is also close to the ESR result [17]. These good agreements indicate that the magnetic ground state is well captured by our CEF analysis, although the excited state may be overestimated.

Figure 4(a) shows the specific heat measurements for PrMgAl₁₁O₁₉ under various magnetic fields. A nonmagnetic sample LaMgAl₁₁O₁₉ was also measured as a comparison. A broad hump can be observed at ~4 K in the zero field data. With an increase in magnetic field, the peak is shifted towards high temperatures, reminiscent of the Schottky anomaly. However, upon subtracting the phonon contributions using LaMgAl₁₁O₁₉ as a reference sample, the magnetic component, C_m, can be obtained and shown in figure 4(b). It is apparent that C_m of PrMgAl₁₁O₁₉ does not exhibit a typical (multilevel, broadened) Schottky behavior [21, 22], especially at the high temperature side where it displays a continuous increase up to 40 K. As a comparison, figure 4(d) shows the C_m of NdMgAl₁₁O₁₉ which displays the prototypical Schottky signature of a 2-level system. Along with the experimental data, a two-level Schottky fit, $C_m=p\cdot R(\frac{\mathrm{\Delta }}{k_{B}T}{)}^2\frac{\exp (\mathrm{\Delta }/k_{B}T)}{[1+\exp (\mathrm{\Delta }/k_{B}T){]}^2}$, where R is the ideal gas constant, Δ is the gap between the two levels, k_B is the Boltzmann’s constant, and p represents the fraction of free ions [23] is depicted. In polycrystalline PrZnAl₁₁O₁₉, the C_m follows a power law in the temperature range between 0.2 and 2 K [13]. However in our PrMgAl₁₁O₁₉ data (figure 4(b)), C_m decreases faster than anticipated by a power law, suggesting the presence of a gap not seen in the Zn-analogue. On the other hand, the released entropy increases monotonically with increasing temperature, as shown in figure 4(c), in a similar manner to that seen in the polycrystalline PrZnAl₁₁O₁₉ sample [13].

Figure  4.  Heat capacity from hexaaluminate single crystals. (a) $C_p/T$ versus T measured under various magnetic fields. The dashed line represents the phonon contribution obtained from the nonmagnetic counterpart LaMgAl₁₁O₁₉. (b) The magnetic specific heat is obtained by subtracting the phonon contributions (LaMgAl₁₁O₁₉) from the data of PrMgAl₁₁O₁₉. The red line represents a fit, $C_m=A{T}^{\alpha }$, to the zero field data in the temperature range between 0.2 and 2 K, with α = 1.91(1). (c) Representative change of the magnetic entropy in zero field. (d) Temperature dependence of the magnetic specific heat for NdMgAl₁₁O₁₉ together with a two-level Schottky fit. The inset shows the evolution of the gap as a function of the applied magnetic fields. A fit of $\mathrm{\Delta }=g_c{\mu }_{B}H$ to the gap yields g_c of 3.92.

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3.   Discussions

One essential discovery in the present study is the quenched disorder observed from single crystal x-ray diffraction refinements. The role of disorder in QSL is still elusive, and varies significantly from sample to sample. While it may drive a system into a glassy state, it may also enhance quantum fluctuations and potentially facilitate the formation of a QSL state, as demonstrated both theoretically and experimentally [3-5, 24]. The AC susceptibility data in figure 3(c) and the lack of frequency-dependent in the data down to 50 mK, indicate that the system is still dynamic even though there is measurable site disorder. One possible scenario is the strong fluctuations resulting from the geometric frustration of antiferromagnetically coupled, easy-axis spins on the triangular lattice cannot be relieved by the site disorder (~7%), which would intuitively result in the destruction of the spin liquid state. Alternatively, the site disorder may cooperate with the frustration and stabilize a spin-liquid-like state as proposed for YbMgGaO₄ [2, 25-29]. The present study does not have sufficient information to distinguish between these possibilities. Systematic studies on samples with different degrees of disorder will be instructive to clarify this point. Similar questions have been addressed in the pyrochlore oxide Yb₂Ti₂O₇, where a sharp peak in the specific heat for polycrystalline samples, due to a magnetic transition at ~265 mK [30], was not observed in many single crystals, where only a broad peak at lower temperature was observed [31, 32]. To investigate the small degree of site disorder found here within powder samples will need higher spatial resolution measurements, such as neutron/x-ray pair distribution function (PDF) method and extended x-ray absorption fine structure (EXAFS) to illustrate the local structure differences.

Another prominent feature of PrMgAl₁₁O₁₉ is the distinct Ising anisotropy revealed by the magnetization measurements. While the Heisenberg model usually predicts a magnetically ordered state for a triangular antiferromagnet, the Ising model can result in a macroscopically degenerated spin liquid state [33]. Recent examples illustrating this concept is the TmMgGaO₄ [34-36] and neodymium heptatantalate, NdTa₇O₁₉ [37]. However, TmMgGaO₄ shows a partial order state below 0.7 K, and a lack of single crystals for NdTa₇O₁₉ hinders further explorations such as inelastic neutron scattering, as well as the exotic magnetic behaviors related to the crystalline directions. In this sense, the availability of large single crystals for PrMgAl₁₁O₁₉ provides a promising opportunity to investigate the triangular Ising model in depth.

4.   Conclusions

In summary, we have successfully synthesized centimeter-sized single crystals of a spin liquid candidate PrMgAl₁₁O₁₉ by means of the optical floating zone technique. Single crystal structure refinement unveiled the presence of about 7% quenched disorder at the Pr site in our sample. Directional magnetization measurements show a well-defined out-of-plane Ising anisotropy, which can induce strong fluctuations at a triangular lattice, as confirmed by the AC susceptibility measurements down to 50 mK. The availability of single crystals for this compound paves the way to explore the exotic magnetic properties by means of neutron scattering in the future.

While under review we became aware of a parallel work by Ma et al, who also conclude this sample is an Ising spin on the triangular lattice [38]. Here, by combining the CEF calculation, magnetization data and estimated g-factors by ESR results, we present more robust evidence on the Ising anisotropy. Moreover, without single crystal structure refinement, the disorder at the Pr site was not reported in that work.

5.   Future perspectives

While disorder or defect is inevitable in real materials, many researchers endeavour to produce crystals as perfect as possible to obtain their intrinsic properties, allowing one to validate theoretical models. On the other hand, in recent years, it has been realized that disorder may result in exotic phases such spin-liquid-like random-singlet state. Unveiling the role of disorder systematically turns out to be as challenging as producing ideal crystals. In the title compound, the site disorder is within the triangular magnetic sublattice, which is unique compared to, e.g. YbMgGaO₄ where the site disorder occurs completely at the nonmagnetic site. By substituting Pr with another rare earth element, we expect to find evidence of different degrees of site disorder at the magnetic site (presumably this is also true when the nonmagnetic ions are substituted too), which will be helpful in future studies to manipulate the disorder in a controllable way. Substituting the rare-earth ion will also change the local spin character decorating the triangular lattice due to a change in the crystalline electric field scheme. This provides researchers with another tuning parameter for the magnetism in these hexaaluminates and should result in the discovery of systems with the spins confined to the triangular plane or with more exotic spin textures. As with the magnetic pyrochlore oxides [39], the availability of large single crystals and the large number of chemical substitutions that present themselves, we envisage this avenue of research to be very fruitful, resulting in materials with a variety of properties including superconductivity, emergent quantum phenomena, exotic spin texture and other QSLs.