Partial Long-range Order Research Articles

Spin-liquid signatures in the quantum critical regime of pressurized CePdAl

CePdAl is a prototypical frustrated Kondo lattice with partial long-range order (LRO) at ${T}_{\mathrm{N}}=2.7$ K. Previous bulk experiments under hydrostatic pressure found signatures for a quantum critical regime that extends from ${p}_{\mathrm{c}}\ensuremath{\approx}0.9$ GPa, where LRO disappears, up to $\ensuremath{\sim}1.7$ GPa. We employed extensive muon spin relaxation and rotation experiments under pressure. The continuous and complete suppression of LRO at ${p}_{\mathrm{c}}$ is confirmed. Above ${T}_{\mathrm{N}}(p)$ and beyond ${p}_{\mathrm{c}}$, an additional crossover scale ${T}^{*}(p)$ characterizes the change from pure to stretched exponential relaxation in zero field. Remarkably, ${T}^{*}(p)$ agrees with previously determined signatures of entropy accumulation above LRO. This coincidence microscopically evidences fluctuating frustrated spins at $T\ensuremath{\le}{T}^{*}$ with spin-liquid behavior. Power-law divergences of the temperature and longitudinal field dependences of the relaxation rate, with time-field scaling, at pressures between ${p}_{c}$ and 1.7 GPa characterize this regime as quantum critical.

Magnetic properties of Ni5Sn(O2BO3)2 ludwigite

The magnetic properties of site-disordered ${\mathrm{Ni}}_{5}\mathrm{Sn}{({\mathrm{O}}_{2}{\mathrm{BO}}_{3})}_{2}$ ludwigite have carefully been studied using local probes and bulk experimental methods in temperatures down to 1.6 K and magnetic fields up to 9 T. Our results have clearly shown the role of the effect of order/disorder site occupancy in these complex compounds with spin moments $S=1$. Different from most ludwigites, site-disordered ${\mathrm{Ni}}_{5}\mathrm{Sn}{({\mathrm{O}}_{2}{\mathrm{BO}}_{3})}_{2}$ ludwigite has shown three magnetic transitions at 80, 50, and 5 K. Local probes ($^{119}\mathrm{Sn}$ M\"ossbauer and muon spin spectroscopy) have indicated a partial long-range magnetic order with onsets at 80 and 50 K with predominant ferromagnetic interactions at 80 K and antiferromagnetic ones at 50 K. At lower temperatures, a partial spin-glass-like freezing of the moments has been developed concomitantly with fraction of the moments ordered at higher temperatures, a spin-glass-like and long-range magnetic order coexist and a spontaneous exchange bias effect has been observed. Applied magnetic fields have gradually reduced magnetic frustration, leading to a magnetically ordered state of the complete structure at 51 K under 9 T. The absence of double-exchange interactions in order/disorder ${\mathrm{Ni}}_{5}\mathrm{Sn}{({\mathrm{O}}_{2}{\mathrm{BO}}_{3})}_{2}$ ludwigites shows how the magnetic interactions that compete as direct exchange and double exchange are modified by the site occupancy of nonmagnetic Sn ions. Features as a third magnetic transition, the appearance of spontaneous exchange bias, and the enhancement of the coercive field at low temperatures were not found in the site-ordered isomer compound.

Fragmentation in Frustrated Magnets: A Review

Spin liquids are exotic phases of matter that often support emergent gauge fields and quasi-particle excitations. While spin liquids are commonly known for remaining disordered, their definition has been extended to include phases with broken symmetry corresponding to (partial) long-range order, such as chiral and nematic spin liquids for example. For Coulomb spin liquids, this ordering can be quantitatively understood via a Helmholtz decomposition between divergence-free and divergence-full terms. This phenomenon has been coined fragmentation, where spin degrees of freedom fragment into two components; the fluctuating disordered part and the ordered one. In this review, we will cover the theoretical and experimental aspects of this growing field, in particular its relation to magnetic monopoles in spin ice, its phase diagram and the possibility to observe it in solid-state crystal and artificial networks.

Ising antiferromagnet on the 2-uniform lattices.

The antiferromagnetic Ising model is investigated on the twenty 2-uniform lattices using the Monte Carlo method based on the Wang-Landau algorithm and the Metropolis algorithm to study the geometric frustration effect systematically. Based on the specific heat, the residual entropy, and the Edwards-Anderson freezing order parameter, the ground states of them were determined. In addition to the long-range-ordered phase and the spin ice phase found in the Archimedean lattices, two more phases were found. The partial long-range order is long-range order with exceptional disordered sites, which give extensive residual entropy. In the partial spin ice phase, the partial freezing phenomenon appears: A majority of sites are frozen without long-range order, but the other sites are fluctuating even at zero temperature. The spin liquid ground state was not found in the 2-uniform lattices.

Two-dimensional magnetic correlations and partial long-range order in geometrically frustrated CaOFeS with triangle lattice of Fe ions

We report the results on the structure, transport, and magnetic properties of a layered oxysulfide CaOFeS with a stacked triangle lattice of Fe ions. The susceptibility data show a broad maximum near 120 K, indicating the existence of two-dimensional (2D) short-range ordering in this compound. Features associated with long-range antiferromagnetic (AFM) phase transition are seen below 40 K. Meanwhile, a very small heat-capacity anomaly is detected around 35 K, and most of the measured magnetic entropy is lost during the 2D ordering process. Both crystal and magnetic structures were studied by neutron powder diffraction at 300, 125, 40, and 6 K. The structure was refined based on space group $P{6}_{3}mc$ with $a=3.759\phantom{\rule{0.28em}{0ex}}98(4)$ and $c=11.383\phantom{\rule{0.28em}{0ex}}51(16)\phantom{\rule{0.16em}{0ex}}\AA{}$ at ambient temperature. Low-temperature diffraction reveals 2D magnetic correlations between Fe moments without showing significant structural distortion. Warren peak shape analysis of the neutron-diffraction data at $2\ensuremath{\theta}$ near 18\ifmmode^\circ\else\textdegree\fi{} is employed to characterize the correlation length in the 2D magnetic state with lowering temperature. The geometrically frustrated compound is found to gradually condense into a partial long-range ordered state with AFM coupled Fe layers between 40.6 and 26 K. The resulting partially ordered magnetic structure is a $G$-type Ising AFM with a propagation vector of $\mathbf{k}=(1/2,1/2,0)$ and an ordered magnetic moment of $2.59(3){\ensuremath{\mu}}_{\mathrm{B}}/\mathrm{Fe}$ along $c$ at 6 K.

Partial long-range order in antiferromagnetic Potts models

The Potts model plays an essential role in classical statistical mechanics, illustrating many fundamental phenomena. One example is the existence of partially long-range-ordered states, in which some degrees of freedom remain disordered. This situation may arise from frustration of the interactions, but also from an irregular but unfrustrated lattice structure. We study partial long-range order in a range of antiferromagnetic $q$-state Potts models on different two-dimensional lattices and for all relevant values of $q$. We exploit the power of tensor-based numerical methods to evaluate the partition function of these models and hence to extract the key thermodynamic properties -- entropy, specific heat, magnetization, and susceptibility -- giving deep insight into the phase transitions and ordered states of each system. Our calculations reveal a range of phenomena related to partial ordering, including different types of entropy-driven phase transition, the role of lattice irregularity, very large values of the critical $q_c$, and double phase transitions.

Critical properties of a spin-1 triangular lattice ising antiferromagnet

We employ Monte Carlo simulations in order to investigate critical behavior of a geometrically frustrated spin-1 Ising antiferromagnet on a triangular lattice in the presence of a single-ion anisotropy. It has been previously found that long-range order can exist in the isotropic system with a spin larger than some critical value estimated as 11/2. We show that the presence of the single-ion anisotropy can lead to a partial long-range order in the low-temperature region even below this critical value, namely for the spin 1, within a certain range of the anisotropy strength. At higher temperatures we identify another phase of the Berezinsky-Kosterlitz-Thouless type and using a finite-size scaling analysis evaluate the correlation decay exponent. We also study densities of various local spin patterns in the respective phases.

A μSR study of the magnetic properties of Tb2CuIn3

Tb2CuIn3 shows exceptional magnetic behavior within the series of R2CuIn3 intermetallics (R denoting one of the rare earth elements), where long-range magnetic order is absent in favor of spin liquid magnetic ground states as a consequence of frustration by competing exchange. In contrast, neutron studies suggested for the magnetic ground state of Tb2CuIn3 the coexistence of long-range antiferromagnetic and short-range order. The present μSR (muon spin rotation/relaxation) data show that at ∼45 K all Tb magnetic moments freeze into a spin-glass state. At TN = 33 K partial long-range order sets in, its volume fraction being ∼30%. On cooling to 10 K, the antiferromagnetic fraction reaches the saturation value of 70% and hence remains coexistent with the spin-glass state for T → 0. The antiferromagnetic state is characterized by considerable short-range spin disorder. At low temperatures the Tb moments in the two coexisting states approach the static limit. This absence of persistent moment fluctuations together with partial formation of long-range order indicates considerably lower frustration in Tb2CuIn3 compared to the rest of the R2CuIn3 compounds.

Elastic anomalies and long/short-range ordering effects: A first-principles investigation of the AgcPd1−csolid solution

We investigate the elastic properties of the binary alloy Ag-Pd. The lattice constant of this system shows significant deviations from the linear behavior anticipated by the semi-empirical Vegard's rule. This effect was formerly studied by assuming total substitutional disorder, and described by the coherent potential approximation (CPA). Theoretical phase diagram investigations have however suggested three ordered phases at low temperatures, and we extend our first-principles investigation to include such scenarios through the adoption of an extended unit cell representation and a recently developed multisublattice generalization of the original CPA. This allows us to explore equilibrium lattice constant and bulk modulus within a unified approach even in the presence of partial long-range order. We obtain significant variations of the bulk modulus in comparison to the totally disordered picture, and again very rich deviations from more intuitive predictions of a simple linear behavior. We follow former suggestions to analyze the different regimes in connection with topological transitions of the Fermi surface, examined through Bloch spectral function calculations.

The success story of crystallography

Diffractionists usually place the birth of crystallography in 1912 with the first X-ray diffraction experiment of Friedrich, Knipping and Laue. This discovery propelled the mathematical branch of mineralogy to global importance and enabled crystal structure determination. Knowledge of the geometrical structure of matter at atomic resolution had revolutionary consequences for all branches of the natural sciences: physics, chemistry, biology, earth sciences and material science. It is scarcely possible for a single person in a single article to trace and appropriately value all of these developments. This article presents the limited, subjective view of its author and a limited selection of references. The bulk of the article covers the history of X-ray structure determination from the NaCl structure to aperiodic structures and macromolecular structures. The theoretical foundations were available by 1920. The subsequent success of crystallography was then due to the development of diffraction equipment, the theory of the solution of the phase problem, symmetry theory and computers. The many structures becoming known called for the development of crystal chemistry and of data banks. Diffuse scattering from disordered structures without and with partial long-range order allows determination of short-range order. Neutron and electron scattering and diffraction are also mentioned.

The success story of crystallographyThis Laue centennial article has also been published inZeitschrift für Kristallographie[Schwarzenbach (2012).Z. Kristallogr.227, 52–62

Diffractionists usually place the birth of crystallography in 1912 with the first X-ray diffraction experiment of Friedrich, Knipping and Laue. This discovery propelled the mathematical branch of mineralogy to global importance and enabled crystal structure determination. Knowledge of the geometrical structure of matter at atomic resolution had revolutionary consequences for all branches of the natural sciences: physics, chemistry, biology, earth sciences and material science. It is scarcely possible for a single person in a single article to trace and appropriately value all of these developments. This article presents the limited, subjective view of its author and a limited selection of references. The bulk of the article covers the history of X-ray structure determination from the NaCl structure to aperiodic structures and macromolecular structures. The theoretical foundations were available by 1920. The subsequent success of crystallography was then due to the development of diffraction equipment, the theory of the solution of the phase problem, symmetry theory and computers. The many structures becoming known called for the development of crystal chemistry and of data banks. Diffuse scattering from disordered structures without and with partial long-range order allows determination of short-range order. Neutron and electron scattering and diffraction are also mentioned.

Coupled Al/Si and O/N order/disorder in BaYb[Si4–x Al x O x N7–x ]sialon: neutron powder diffraction and Monte Carlo simulations

The fractions of aluminium, [Al]/[Al + Si], and oxygen, [O]/[O + N], in crystallographically distinct sites of BaYb[Si4–x Al x O x N7–x ] oxonitridoaluminosilicate (space group P63 mc, No. 186) were refined based on the results of neutron powder diffraction for a synthetic sample with the composition of x = 2.2(2) and simulated as functions of temperature for the compositions x = 2 and x = 2.3 using a combination of static lattice energy calculations (SLEC) and Monte Carlo simulations. The SLEC calcu lations have been performed on a set of 800 structures differing in the distribution of Al/Si and O/N within the 2 × 2 × 2 supercell containing 36 formula units of BaYb[Si4–x Al x O x N7–x ]. The SLEC were based on a transferable set of empirical interatomic potentials developed within the present study. The static lattice energies of these structures have been expanded in the basis set of pair-wise ordering energies and on-site chemical potentials. The ordering energies and the chemical potentials have been used to calculate the configuration energies of the oxonitridoaluminosilicates (so-called sialons) using a Monte Carlo algorithm. The simulations suggest that Al and O are distributed unevenly over two non-equivalent T(Si/Al) and three L(N/O) sites, respectively, and the distribution shows strong dependence both on the temperature and the composition. Both simulated samples exhibit order/disorder transitions in the temperature range 500–1000 K to phases with partial long-range order below these temperatures. Above the transition temperatures the Si/Al and N/O distributions are affected by short-range ordering. The predicted site occupancies are in a qualitative agreement with the neutron diffraction results.

Coupled Al/Si and O/N order/disorder in BaYb[Si4–xAlxOxN7–x]sialon

The fractions of aluminium, [Al]/[Al + Si], and oxygen, [O]/[O + N], in crystallographically distinct sites of BaYb[Si4–xAlxOxN7–x] oxonitridoaluminosilicate (space group P63mc, No. 186) were refined based on the results of neutron powder diffraction for a synthetic sample with the composition of x = 2.2(2) and simulated as functions of temperature for the compositions x = 2 and x = 2.3 using a combination of static lattice energy calculations (SLEC) and Monte Carlo simulations. The SLEC calcu lations have been performed on a set of 800 structures differing in the distribution of Al/Si and O/N within the 2 × 2 × 2 supercell containing 36 formula units of BaYb[Si4–xAlxOxN7–x]. The SLEC were based on a transferable set of empirical interatomic potentials developed within the present study. The static lattice energies of these structures have been expanded in the basis set of pair-wise ordering energies and on-site chemical potentials. The ordering energies and the chemical potentials have been used to calculate the configuration energies of the oxonitridoaluminosilicates (so-called sialons) using a Monte Carlo algorithm. The simulations suggest that Al and O are distributed unevenly over two non-equivalent T(Si/Al) and three L(N/O) sites, respectively, and the distribution shows strong dependence both on the temperature and the composition. Both simulated samples exhibit order/disorder transitions in the temperature range 500–1000 K to phases with partial long-range order below these temperatures. Above the transition temperatures the Si/Al and N/O distributions are affected by short-range ordering. The predicted site occupancies are in a qualitative agreement with the neutron diffraction results.

Geometric frustration in a new material system M 2X(OH) 3

Muon spin relaxation/rotation (μSR) are carried out on a new magnetic system of M 2X(OH) 3 (M=transition metal, X=halogen), which we show to be a new geometric frustration system for d-electron spins. Contrasting behaviors were observed for clinoatacamite Cu 2Cl(OH) 3 and Co 2Cl(OH) 3. Clinoatacamite Cu 2Cl(OH) 3 undergoes antiferromagnetic transition at T N=18 K with distinct muon spin rotations, and then it transforms abruptly into a metastable state with nearly complete depolarization of muon spins which suggests strong fluctuation. The system then enters a state in which partial long-range order and spin fluctuation coexist down to the lowest experimentally attainable temperature of 20 mK. For Co 2Cl(OH) 3, although dynamic interactions exist at temperatures higher than 400 K, static order only forms at a much lower temperature of T C=10 K with coexistence of spin fluctuation. This work demonstrates the coexistence of long-range order and spin fluctuation in a novel system of d-electron transition metal oxides.

Structural and magnetic properties of CrPt3(111) films grown on WSe2(0001)

We report on the growth-induced properties of 30 Å thick CrPt3(111) deposits on surfaces of WSe2(0001) with a van der Waals interaction. Assemblies of (111)-oriented fcc grains are obtained with average lateral sizes ranging from 5 to 15 nm for growth temperatures increasing from 50 °C to 550 °C. The grains develop a partial L12-type chemical long-range order starting at a deposition temperature as low as 50 °C, much lower than reported for films grown on other suitable substrate surfaces. An ordering parameter of about 0.3 was found over a wide temperature range between 200 °C and 550 °C, for a deposition rate of 0.1 Å/s. Lowering the deposition rate by a factor of 5 results in a higher ordering parameter of about 0.4. Above 400 °C, ordering is accompanied by a pronounced segregation of Se atoms toward the free surface, forming a (4×4) superstructure. All samples exhibit superparamagnetic behavior with blocking temperatures that scale with the activation volume of the ferrimagnetic grains. This work shows the importance of substrate choice in reducing the ordering temperature of thin CrPt3 films, an effect that might also occur in other alloy systems.

Coexistence of Long-Range Order and Spin Fluctuation in Geometrically Frustrated ClinoatacamiteCu2Cl(OH)3

Muon spin rotation experiments are carried out on clinoatacamite, Cu2Cl(OH)3, which is a new geometrically frustrated system featuring a three-dimensional network of corner-sharing tetrahedral 3d Cu2+ spins. A long-range antiferromagnetic order occurs below 18.1 K with a surprisingly small entropy release of about 0.05Rln2/Cu. Below 6.5 K, the static long-range order transforms abruptly into a metastable state with nearly complete depolarization of muon spins which suggests strong fluctuation. The system then enters a state in which partial long-range order and spin fluctuation coexist down to the lowest experimentally attainable temperature of 20 mK. This work presents a novel system for studying geometric frustration.

Importance of thermal disorder on the properties of alloys: Origin of paramagnetism and structural anomalies in bcc-basedFe1−xAlx

The bcc-based Fe_{1-x}Al_{x} exhibit interesting magnetic and anomalous structural properties as a function of composition and sample processing conditions arising from thermal or off-stoichiometric chemical disorder, and, although well studied, these properties are not understood. In stoichiometric B2 FeAl, including the effects of partial long-range order (i.e., thermal antisites), we find the observed paramagnetic response (with non-zero local moments), in contrast to past investigations which find ferromagnetism based on local density approximation (LDA) to density functional theory or which find a non-magnetic state from LDA+U, both of which are inconsistent with experiment. Moreover, from this magneto-chemical coupling, we are able to determine the origins of the lattice constant anomalies found in Fe_{1-x}Al_{x} for x=25-50, as observed from various processing conditions.

Partial order in the non-Fermi-liquid phase of MnSi.

Only a few metallic phases have been identified in pure crystalline materials. These include normal, ferromagnetic and antiferromagnetic metals, systems with spin and charge density wave order, and superconductors. Fermi-liquid theory provides a basis for the description of all of these phases. It has been suggested that non-Fermi-liquid phases of metals may exist in some heavy-fermion compounds and oxide materials, but the discovery of a characteristic microscopic signature of such phases presents a major challenge. The transition-metal compound MnSi above a certain pressure (p(c) = 14.6 kbar) provides what may be the cleanest example of an extended non-Fermi-liquid phase in a three-dimensional metal. The bulk properties of MnSi suggest that long-range magnetic order is suppressed at p(c) (refs 7-12). Here we report neutron diffraction measurements of MnSi, revealing that sizeable quasi-static magnetic moments survive far into the non-Fermi-liquid phase. These moments are organized in an unusual pattern with partial long-range order. Our observation supports the existence of novel metallic phases with partial ordering of the conduction electrons (reminiscent of liquid crystals), as proposed for the high-temperature superconductors and heavy-fermion compounds.

Site-specific long-range order in 57Fe3Al measured by Mössbauer diffractometry

Mössbauer powder diffractometry was used to study partially ordered 57Fe3Al. The intensities of fundamental and superlattice Bragg diffractions were measured at 89 Doppler velocities through all nuclear resonances in the sample. The measurements were analysed to provide data on the long-range order of Fe atoms having different numbers of Al neighbours. Energy spectra of the Bragg diffractions of Mössbauer radiations were calculated with both kinematic theory and dynamic theory. Comparing experimental data with calculations showed that Fe atoms having three and five Al atoms as first-nearest neighbours (1NNs) have partial sc long-range order, similar to that of Fe atoms with four Al 1NNs. The Fe atoms with two Al 1NNs had partial fcc order similar to that of Fe atoms with zero Al 1NN. No evidence was found for B32 order for any of the Fe environments.

What neutrons do tell us about the nature of (spin) glasses?

A common feature of all glassy systems is the decisive role of dynamics in the evolution of macroscopic properties, while the short and eventually partial long-range order shows little difference between the different phases. Our recent neutron scattering results on a Cu–Mn spin glass dramatically illustrate this point, and provide unambiguous evidence against recent suggestions that behaviour in Cu–Mn is driven by spin density waves rather than glassy dynamics. Recent findings in Au–Fe provide the first circumstantial evidence for the theoretically predicted extremely long correlation lengths at the spin-glass transition. Recent experimental advances opened up the way to investigate dynamics in glasses and liquids on extended length scales, where only spin correlations could be studied by now.