Search

Funding: Deutsche Forschungsgemeinschaft (DFG) through the SFB 1143 and the Wurzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter–ct.qmat (EXC 2147, Project No. 390858490), as well as the support of the HLD at HZDR, a member of the European Magnetic Field Laboratory (EMFL). We gratefully acknowledge support from the European Research Council (through the QUESTDO project, 714193), the Leverhulme Trust, and the Royal Society. We thank the Elettra synchrotron for access to the APE-HE beamline under proposal number 20195300. The research leading to this result has been supported by the project CALIPSOplus under Grant Agreement 730872 from the EU Framework Programme for Research and Innovation HORIZON 2020. Part of this work has been performed in the framework of the Nanoscience Foundry and Fine Analysis (NFFA-MUR Italy Progetti Internazionali) project (www.trieste.NFFA.eu).

While the Heisenberg model for magnetic Mott insulators on planar lattice structures is comparatively well understood in the case of transition metal ions, the intrinsic spin-orbit entanglement of $4f$ magnetic ions on such lattices shows fascinating new physics largely due to corresponding strong anisotropies both in their single-ion and their exchange properties. We show here that the Yb delafossites, containing perfect magnetic ${\mathrm{Yb}}^{3+}$ triangular lattice planes with pseudospin $s=\frac{1}{2}$ at low temperatures, are an ideal platform to study these phenomena. Competing frustrated interactions may lead to an absence of magnetic order associated to a gapless spin liquid ground state with a huge linear specific heat exceeding that of many heavy fermions, whereas the application of a magnetic field induces anisotropic magnetic order with successive transitions into different long-range ordered structures. In this comparative study, we discuss our experimental findings in terms of a unified crystal-field and exchange model. We combine electron paramagnetic resonance (EPR) experiments and results from neutron scattering with measurements of the magnetic susceptibility, isothermal magnetization up to full polarization, and specific heat to determine the relevant model parameters. The impact of the crystal field is discussed as well as the symmetry-compatible form of the exchange tensor, and we give explicit expressions for the anisotropic $g$ factor, the temperature dependence of the susceptibility, the exchange-narrowed EPR linewidth, and the saturation field.

${\mathrm{TaSnS}}_{2}$ single crystal and polycrystalline samples are investigated in detail by magnetization, electrical resistivity, and specific heat as well as Raman spectroscopy and nuclear magnetic resonance (NMR). Studies are focused on the temperature and magnetic field dependence of the superconducting state. We determine the critical fields for both directions $B\ensuremath{\parallel}c$ and $B\ensuremath{\perp}\phantom{\rule{0.16em}{0ex}}c$. Additionally, we investigate the dependence of the resistivity, the critical temperature, and the structure through Raman spectroscopy under high pressure up to 10 GPa. At a pressure of $\ensuremath{\approx}3\phantom{\rule{0.28em}{0ex}}\mathrm{GPa}$ the superconductivity is suppressed below our minimum temperature. The Sn NMR powder spectrum shows a single line which is expected for the ${\mathrm{TaSnS}}_{2}$ phase and confirms the high sample quality. Pronounced de Haas-van Alphen oscillations in the ac susceptibility of polycrystalline sample reveal two pairs of frequencies indicating coexisting small and large Fermi surfaces. The effective mass of the smaller Fermi surface is $\ensuremath{\approx}0.5{m}_{\mathrm{e}}$. We compare these results with the band structures from DFT calculations. Our findings on ${\mathrm{TaSnS}}_{2}$ are discussed in terms of a quasi-two-dimensional BCS superconductivity.

Structural and magnetic properties of a quasi-one-dimensional spin-$\frac{1}{2}$ compound ${\mathrm{NaVOPO}}_{4}$ are explored by x-ray diffraction, magnetic susceptibility, high-field magnetization, specific heat, electron spin resonance, and $^{31}\mathrm{P}$ nuclear magnetic resonance measurements, as well as complementary ab initio calculations. Whereas magnetic susceptibility of ${\mathrm{NaVOPO}}_{4}$ may be compatible with the gapless uniform spin chain model, detailed examination of the crystal structure reveals a weak alternation of the exchange couplings with the alternation ratio $\ensuremath{\alpha}\ensuremath{\simeq}0.98$ and the ensuing zero-field spin gap ${\mathrm{\ensuremath{\Delta}}}_{0}/{k}_{\mathrm{B}}\ensuremath{\simeq}2.4\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ directly probed by field-dependent magnetization measurements. No long-range order is observed down to 50 mK in zero field. However, applied fields above the critical field ${H}_{c1}\ensuremath{\simeq}1.6\phantom{\rule{0.28em}{0ex}}\mathrm{T}$ give rise to a magnetic ordering transition with the phase boundary ${T}_{N}\ensuremath{\propto}{(H\ensuremath{-}{H}_{c1})}^{\frac{1}{\ensuremath{\phi}}}$, where $\ensuremath{\phi}\ensuremath{\simeq}1.8$ is close to the value expected for Bose-Einstein condensation of triplons. With its weak alternation of the exchange couplings and small spin gap, ${\mathrm{NaVOPO}}_{4}$ lies close to the quantum critical point.

We present the synthesis and a detailed investigation of structural and magnetic properties of polycrystalline ${\mathrm{NaVOAsO}}_{4}$ by means of x-ray diffraction, magnetization, electron spin resonance (ESR), and $^{75}\mathrm{As}$ nuclear magnetic resonance (NMR) measurements as well as density-functional band structure calculations. Temperature-dependent magnetic susceptibility, ESR intensity, and NMR line shift could be described well using an alternating spin-$1/2$ chain model with the exchange coupling $J/{k}_{\mathrm{B}}\ensuremath{\simeq}52$ K and an alternation parameter $\ensuremath{\alpha}\ensuremath{\simeq}0.65$. From the high-field magnetization measured at $T=1.5$ K, the critical field of the gap closing is found to be ${H}_{\mathrm{c}}\ensuremath{\simeq}16$ T, which corresponds to the zero-field spin gap of ${\mathrm{\ensuremath{\Delta}}}_{0}/{k}_{\mathrm{B}}\ensuremath{\simeq}21.4$ K. Both NMR shift and spin-lattice relaxation rate show an activated behavior at low temperatures, further confirming the singlet ground state. The spin chains do not coincide with the structural chains, whereas the couplings between the spin chains are frustrated. Because of a relatively small spin gap, ${\mathrm{NaVOAsO}}_{4}$ is a promising compound for further experimental studies under high magnetic fields.

Platelike high-quality ${\mathrm{NaYbS}}_{2}$ rhombohedral single crystals with lateral dimensions of a few mm have been grown and investigated in great detail by bulk methods such as magnetization and specific heat, but also by local probes such as nuclear magnetic resonance (NMR), electron-spin resonance (ESR), muon-spin relaxation ($\ensuremath{\mu}\mathrm{SR}$), and inelastic neutron scattering over a wide field and temperature range. Our single-crystal studies clearly evidence a strongly anisotropic quasi-two-dimensional magnetism and an emerging spin-orbit entangled ${J}_{\mathrm{eff}}=\frac{1}{2}$ state of Yb towards low temperatures together with an absence of long-range magnetic order down to 260 mK. In particular, the clear and narrow Yb ESR lines together with narrow $^{23}\mathrm{Na}$ NMR lines evidence an absence of inherent structural distortions in the system, which is in strong contrast to the related spin-liquid candidate ${\mathrm{YbMgGaO}}_{4}$ falling within the same space group $R\overline{3}m$. This identifies ${\mathrm{NaYbS}}_{2}$ as a rather pure spin-$\frac{1}{2}$ triangular-lattice magnet and a putative quantum spin liquid.

The structural and magnetic properties of the square-cupola antiferromagnet Sr(TiO)Cu$_{4}$(PO$_{4}$)$_{4}$ are investigated via x-ray diffraction, magnetization, heat capacity, and $^{31}$P nuclear magnetic resonance experiments on polycrystalline samples, as well as density-functional band-structure calculations. The temperature-dependent unit cell volume could be described well using the Debye approximation with the Debye temperature of $\theta_{\rm D} \simeq $ 550~K. Magnetic response reveals a pronounced two-dimensionality with a magnetic long-range-order below $T_{\rm N} \simeq 6.2$~K. High-field magnetization exhibits a kink at $1/3$ of the saturation magnetization. Asymmetric $^{31}$P NMR spectra clearly suggest strong in-plane anisotropy in the magnetic susceptibility, as anticipated from the crystal structure. From the $^{31}$P NMR shift vs bulk susceptibility plot, the isotropic and axial parts of the hyperfine coupling between $^{31}$P nuclei and the Cu$^{2+}$ spins are calculated to be $A_{\rm hf}^{\rm iso} \simeq 6539$ and $A_{\rm hf}^{\rm ax} \simeq 952$~Oe/$\mu_{\rm B}$, respectively. The low-temperature and low-field $^{31}$P NMR spectra indicate a commensurate antiferromagnetic ordering. Frustrated nature of the compound is inferred from the temperature-dependent $^{31}$P NMR spin-lattice relaxation rate and confirmed by our microscopic analysis that reveals strong frustration of the square cupola by next-nearest-neighbor exchange couplings.

$^{75}$As NMR measurements were performed on a polycrystalline sample of spin-1/2 alternating-spin-chain Heisenberg antiferromagnet AgVOAsO$_4$. Temperature-dependent NMR shift $K(T)$, which is a direct measure of the intrinsic spin susceptibility, agrees very well with the spin-1/2 alternating-chain model, justifying the assignment of the spin lattice. From the analysis of $K(T)$, magnetic exchange parameters were estimated as follows: the leading exchange $J/k_{\rm B} \simeq 38.4$ K, alternation ratio $\alpha = J'/J \simeq 0.68$, and spin gap $\Delta/k_{\rm B} \simeq 15$ K. The transferred hyperfine coupling between the $^{75}$As nucleus and V$^{4+}$ spins obtained by comparing the NMR shift with bulk susceptibility amounts to $A_{\rm hf} \simeq 3.3$ T/$\mu_{\rm B}$. Our temperature-dependent spin-lattice relaxation rate $1/T_1(T)$ also shows an activated behaviour at low temperatures, thus confirming the presence of a spin gap in AgVOAsO$_4$.

We report the thermodynamic properties, magnetic ground state, and microscopic magnetic model of the spin-1 frustrated antiferromagnet ${\mathrm{Li}}_{2}{\mathrm{NiW}}_{2}{\mathrm{O}}_{8}$, showing successive transitions at ${T}_{\mathrm{N}1}\ensuremath{\simeq}18$ K and ${T}_{\mathrm{N}2}\ensuremath{\simeq}12.5$ K in zero field. Nuclear magnetic resonance and neutron diffraction reveal collinear and commensurate magnetic order with the propagation vector $\mathbf{k}=(\frac{1}{2},0,\frac{1}{2})$ below ${T}_{\mathrm{N}2}$. The ordered moment of $1.8{\ensuremath{\mu}}_{B}$ at 1.5 K is directed along $[0.89(9),\ensuremath{-}0.10(5),\ensuremath{-}0.49(6)]$ and matches the magnetic easy axis of spin-$1\phantom{\rule{4pt}{0ex}}{\mathrm{Ni}}^{2+}$ ions, which is determined by the scissor-like distortion of the ${\mathrm{NiO}}_{6}$ octahedra. Incommensurate magnetic order, presumably of spin-density-wave type, is observed in the region between ${T}_{\mathrm{N}2}$ and ${T}_{\mathrm{N}1}$. Density-functional band-structure calculations put forward a three-dimensional spin lattice with spin-1 chains running along the $[01\overline{1}]$ direction and stacked on a spatially anisotropic triangular lattice in the $ab$ plane. We show that the collinear magnetic order in ${\mathrm{Li}}_{2}{\mathrm{NiW}}_{2}{\mathrm{O}}_{8}$ is incompatible with the triangular lattice geometry and thus driven by a pronounced easy-axis single-ion anisotropy of ${\mathrm{Ni}}^{2+}$.

Magnetic frustration in three dimensions (3D) manifests itself in the spin-$\frac12$ insulator Li$_2$CuW$_2$O$_8$. Density-functional band-structure calculations reveal a peculiar spin lattice built of triangular planes with frustrated interplane couplings. The saturation field of 29 T contrasts with the susceptibility maximum at 8.5 K and a relatively low Neel temperature $T_N\simeq 3.9$ K. Magnetic order below $T_N$ is collinear with the propagation vector $(0,\frac12,0)$ and an ordered moment of 0.65(4) $\mu_B$ according to neutron diffraction data. This reduced ordered moment together with the low maximum of the magnetic specific heat ($C^{\max}/R\simeq 0.35$) pinpoint strong magnetic frustration in 3D. Collinear magnetic order suggests that quantum fluctuations play crucial role in this system, where a non-collinear spiral state would be stabilized classically.

We report a $^{7}\mathrm{Li}$ nuclear magnetic resonance (NMR) study of a frustrated three-dimensional spin-$\frac{1}{2}$ antiferromagnet ${\text{Li}}_{2}{\text{CuW}}_{2}{\text{O}}_{8}$ and also explore the effect of nonmagnetic dilution. The magnetic long-range ordering in the parent compound at ${T}_{\mathrm{N}}\ensuremath{\simeq}3.9\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ was detected from the drastic line broadening and a peak in the spin-lattice relaxation rate ($1/{T}_{1}$). The NMR spectrum above ${T}_{\mathrm{N}}$ broadens systematically, and its full width at half maximum (FWHM) tracks the static spin susceptibility. From the analysis of FWHM vs static susceptibility, the coupling between the Li nuclei and ${\text{Cu}}^{2+}$ ions was found to be purely dipolar in nature. The magnitude of the maximum exchange coupling constant is ${J}_{\mathrm{max}}/{k}_{\mathrm{B}}\ensuremath{\simeq}13\phantom{\rule{0.28em}{0ex}}\mathrm{K}$. NMR spectra below ${T}_{\mathrm{N}}$ broaden abruptly and transform into a double-horn pattern reflecting the commensurate nature of the spin structure in the ordered state. Below ${T}_{\mathrm{N}}$, $1/{T}_{1}$ follows a ${T}^{5}$ behavior. The frustrated nature of the compound is confirmed by persistent magnetic correlations at high temperatures well above ${T}_{\mathrm{N}}$. The dilution of the spin lattice with nonmagnetic Zn atoms has dramatic influence on ${T}_{N}$ that decreases exponentially similar to quasi-one-dimensional antiferromagnets, even though ${\text{Li}}_{2}{\text{CuW}}_{2}{\text{O}}_{8}$ has only a weak one-dimensional anisotropy. Heat capacity of doped samples follows power law (${C}_{\mathrm{p}}\ensuremath{\propto}{T}^{\ensuremath{\alpha}}$) below ${T}_{\mathrm{N}}$, and the exponent ($\ensuremath{\alpha}$) decreases from 3 in the parent compound to 1 in the 25% doped sample.

The crystallographic, magnetic, and thermal properties of polycrystalline $\mathrm{Bi}{\mathrm{Mn}}_{2}{\mathrm{PO}}_{6}$ and its nonmagnetic analog $\mathrm{Bi}{\mathrm{Zn}}_{2}{\mathrm{PO}}_{6}$ are investigated by x-ray diffraction, magnetization $M$, magnetic susceptibility $\ensuremath{\chi}$, heat capacity ${C}_{p}$, and $^{31}\mathrm{P}$ nuclear magnetic resonance (NMR) measurements versus applied magnetic field $H$ and temperature $T$ as well as by density-functional band theory and molecular-field calculations. Both compounds show a strong monotonic lattice softening on cooling, where the Debye temperature decreases by a factor of two from ${\ensuremath{\Theta}}_{\mathrm{D}}\ensuremath{\sim}650$ K at $T=300$ K to ${\ensuremath{\Theta}}_{\mathrm{D}}\ensuremath{\sim}300$ K at $T=2$ K. The $\ensuremath{\chi}(T)$ data for $\mathrm{Bi}{\mathrm{Mn}}_{2}{\mathrm{PO}}_{6}$ above 150 K follow a Curie-Weiss law with a Curie constant consistent with a Mn${}^{+2}$ spin $S=5/2$ with $g$ factor $g=2$ and an antiferromagnetic (AFM) Weiss temperature ${\ensuremath{\theta}}_{\mathrm{CW}}\ensuremath{\simeq}\ensuremath{-}78$ K. The $\ensuremath{\chi}$ data indicate long-range AFM ordering below ${T}_{\mathrm{N}}\ensuremath{\simeq}30$ K, confirmed by a sharp $\ensuremath{\lambda}$-shaped peak in ${C}_{\mathrm{p}}(T)$ at 28.8 K. The magnetic entropy at 100 K extracted from the ${C}_{\mathrm{p}}(T)$ data is consistent with spin $S=5/2$ for the Mn${}^{+2}$ cations. The band-theory calculations indicate that $\mathrm{Bi}{\mathrm{Mn}}_{2}{\mathrm{PO}}_{6}$ is an AFM compound with dominant interactions ${J}_{1}/{k}_{\mathrm{B}}\ensuremath{\simeq}6.7$ K and ${J}_{3}/{k}_{\mathrm{B}}\ensuremath{\simeq}5.6$ K along the legs and rungs of a Mn two-leg spin-ladder, respectively. However, sizable and partially frustrating interladder couplings lead to an anisotropic three-dimensional magnetic behavior with long-range AFM ordering at ${T}_{\mathrm{N}}\ensuremath{\simeq}30$ K observed in the $\ensuremath{\chi}$, ${C}_{\mathrm{p}}$, and NMR measurements. A second magnetic transition at $\ensuremath{\approx}$10 K is observed from the $\ensuremath{\chi}$ and NMR measurements but is not evident in the ${C}_{\mathrm{p}}$ data. The ${C}_{\mathrm{p}}$ data at low $T$ suggest a significant contribution from AFM spin waves moving in three dimensions and the absence of a spin-wave gap. A detailed analysis of the NMR spectra indicates commensurate magnetic order between 10 and 30 K, while below 10 K additional features appear that may arise from an incommensurate modulation and/or spin canting. The commensurate order is consistent with microscopic density functional calculations that yield a collinear N\'eel-type AFM spin arrangement both within and between the ladders, despite the presence of multiple weak interactions frustrating this magnetic structure of the Mn spins. Frustration for AFM ordering and the one-dimensional spatial anisotropy of the three-dimensional spin interactions are manifested in the frustration ratio $f=|{\ensuremath{\theta}}_{\mathrm{CW}}|/{T}_{\mathrm{N}}\ensuremath{\simeq}2.6$, indicating a suppression of ${T}_{\mathrm{N}}$ from 68 K in the absence of these effects to the observed value of about 30 K in $\mathrm{Bi}{\mathrm{Mn}}_{2}{\mathrm{PO}}_{6}$.

We present a combined experimental and theoretical study of the spin-$\frac{1}{2}$ compound CuP${}_{2}$O${}_{6}$ that features a network of two-dimensional (2D) antiferromagnetic (AFM) square planes, interconnected via one-dimensional (1D) AFM spin chains. Magnetic susceptibility, high-field magnetization, and electron spin resonance (ESR) data, as well as microscopic density-functional band-structure calculations and subsequent quantum Monte Carlo simulations, show that the coupling ${J}_{2\mathrm{D}}\ensuremath{\simeq}40$ K in the layers is an order of magnitude larger than ${J}_{1\mathrm{D}}\ensuremath{\simeq}3$ K in the chains. Below ${T}_{N}\ensuremath{\simeq}8$ K, CuP${}_{2}$O${}_{6}$ develops long-range order, as evidenced by a weak net moment on the 2D planes induced by anisotropic magnetic interactions of Dzyaloshinsky-Moriya type. A striking feature of this 3D ordering transition is that the 1D moments grow significantly slower than the ones on the 2D units, which is evidenced by the persistent paramagnetic ESR signal below ${T}_{N}$. Compared to typical quasi-2D magnets, the ordering temperature of CuP${}_{2}$O${}_{6}$ ${T}_{N}/{J}_{2\mathrm{D}}\ensuremath{\simeq}0.2$ is unusually low, showing that weakly coupled spins sandwiched between 2D magnetic units effectively decouple these units and impede the long-range ordering.