Muon Stopping Sites Research Articles

Muon-spin relaxation investigation of magnetic bistability in a crystalline organic radical compound

We present the results of a muon-spin relaxation (μ+SR) investigation of the crystalline organic radical compound 4-(2-benzimidazolyl)-1,2,3,5-dithiadiazolyl (HbimDTDA), in which we demonstrate the hysteretic magnetic switching of the system that takes place at T=(274 ± 11)K caused by a structural phase transition. Muon-site analysis using electronic structure calculations suggests a range of candidate muon stopping sites. The sites are numerous and similar in energy but, significantly, differ between the two structural phases of the material. Despite the difference in the sites, the muon remains a faithful probe of the transition, revealing a dynamically-fluctuating magnetically disordered state in the low-temperature structural phase. In contrast, in the high temperature phase the relaxation is caused by static nuclear moments, with rapid electronic dynamics being motionally narrowed from the muon spectra.

Magnetic structure and crystal field states ofPr2Pd3Ge5:μSRand neutron scattering investigations

We present the results of muon spin relaxation ($\ensuremath{\mu}\mathrm{SR}$), neutron powder diffration (NPD), and inelastic neutron scattering (INS) investigations on polycrystalline ${\mathrm{Pr}}_{2}{\mathrm{Pd}}_{3}{\mathrm{Ge}}_{5}$. The polycrystalline ${\mathrm{Pr}}_{2}{\mathrm{Pd}}_{3}{\mathrm{Ge}}_{5}$ is known to exhibit two antiferromagnetic transitions near ${T}_{\mathrm{N}1}=8.3$ K and ${T}_{\mathrm{N}2}=7.5$ K which is confirmed by the heat capacity measurements on the present sample. In the magnetically ordered state the NPD data show clear evidence for magnetic Bragg peaks, which for $T<{T}_{\mathrm{N}2}$ could be indexed with a propagation vector k = (1, 0, 0), and with k = (1, 0, 0.07) for ${T}_{\mathrm{N}2}<T<{T}_{\mathrm{N}1}$. We have determined the magnetic structure and found a canted antiferromagnetic structure of ordered ${\mathrm{Pr}}^{3+}$ moments. At 7.6 K we find an amplitude modulated incommensurate antiferromagnetic arrangement of ordered moments with a maximum ordered moment of 1.64(3) ${\ensuremath{\mu}}_{\mathrm{B}}$/Pr lying in the $ab$ plane tilted at about ${26}^{\ensuremath{\circ}}$ from the $b$ axis. At 1.8 K the antiferromagnetic structure is found to be commensurate with an ordered moment of 2.80(6) ${\ensuremath{\mu}}_{\mathrm{B}}$/Pr lying in the $ab$ plane tilted at about ${28}^{\ensuremath{\circ}}$ from the $b$ axis. Our zero-field $\ensuremath{\mu}\mathrm{SR}$ spectra at 1.5 K and at shorter time scale (below 0.5 $\ensuremath{\mu}\mathrm{s}$) reveal two oscillatory frequencies further supporting a long-range-ordered magnetic ground state. We have calculated the muon sites using the density functional theory calculation and present the results of internal field calculations at the muon stopping sites, which are in good agreement with the experimental observations. In the paramagnetic state, the INS data show three well defined CEF excitations near 8.0, 12.3, and 18.1 meV arising from the crystal electric field (CEF) split $J=4$ ground state of ${\mathrm{Pr}}^{3+}$. The CEF level scheme is deduced by analyzing the INS data by a model based on the CEF model.

Identifying muon sites “by eye” in KPF6 and KBF4

Molecular magnets are one of the key research themes of µSR, but locating the muon stopping site in these compounds using density functional theory is often very challenging as their unit cells tend to contain a very large number of atoms. Nevertheless, many molecular magnets contain the [PF6]− and [BF4]− molecular ions, which, due to their fluorine nuclei, produce a distinctive µSR spectrum, which can give information about the muon stopping site. This paper details the calculation of the muon sites in the much simpler materials KPF6 and KBF4, providing insights which can be applied to situations where these molecular ions are found in complicated molecular magnets.

Muon spin rotation study of the layered oxyselenide Sr2CoO2Ag2Se2

The results of a muon spin rotation experiment on the layered oxyselenide ${\mathrm{Sr}}_{2}{\mathrm{CoO}}_{2}{\mathrm{Ag}}_{2}{\mathrm{Se}}_{2}$ are presented. The compound contains square-planar ${\mathrm{CoO}}_{2}$ layers and is found to exhibit a phase transition at 160.4(1) K to an antiferromagnetic configuration of the ${\mathrm{Co}}^{2+}$ ions. Density functional theory calculations were performed in order to determine the stopping site of the muon within the unit cell. The calculated magnetic dipole field at the muon stopping site was shown to be consistent with the proposed magnetic structure.

Muon spin rotation and relaxation study on Nb1−yFe2+y

We present a detailed study of the magnetic properties of weakly ferromagnetic/quantum critical Nb$_{1-y}$Fe$_{2+y}$ using muon spin rotation and relaxation ($\mu$SR). By means of an angular dependent study of the muon spin rotation signal in applied magnetic fields on a single crystal in the paramagnetic state we establish the muon stopping site in the crystallographic lattice of NbFe$_2$. With this knowledge we develop models to describe the muon spin rotation and relaxation signals in the weakly ferromagnetic, spin density wave and quantum critical phases of Nb$_{1-y}$Fe$_{2+y}$ and fit the corresponding experimental data. In particular, we quantify the $\mu$SR response for quantum critical behavior in Nb$_{1.0117}$Fe$_{1.9883}$ and extract the influence of residual weak structural disorder. From our analysis, Nb$_{1-y}$Fe$_{2+y}$ emerges to be uniquely suited to study quantum criticality close to weak itinerant ferromagnetic order.

MuFinder: A program to determine and analyse muon stopping sites

Significant progress has recently been made in calculating muon stopping sites using density functional theory. The technique aims to address two of the most common criticisms of the muon-spin spectroscopy (μ+SR) technique, namely, where in the sample does the muon stop, and what is its effect on its local environment. We have designed and developed a program called MuFinder that enables users to carry out these calculations through a simple graphical user interface (GUI). The procedure for calculating muon sites by generating initial muon positions, relaxing the structures, and then clustering and analysing the resulting candidate sites, can be done entirely within the GUI. The local magnetic field at the muon site can also be computed, allowing the connection between the muon sites obtained and experiment to be made. MuFinder will make these computations significantly more accessible to non-experts and help to establish muon site calculations as a routine part of μ+SR experiments. Program summaryProgram Title: MuFinderCPC Library link to program files:https://doi.org/10.17632/pwwt7p9hv8.1Developer's repository link:https://gitlab.com/BenHuddart/mufinderLicensing provisions: GPLv3Programming language: PythonNature of problem: To automate the process of calculating muon stopping sites using density functional theory, thereby making these calculations accessible to non-experts.Solution method: A Python-based graphical user interface (GUI) through which users can calculate muon stopping sites using the structural relaxation method. The program makes use of newly-developed algorithms for generating candidate initial muon positions and for clustering muon positions obtained from the structural relaxations. Analysis of the muon sites, including calculation of the local dipolar magnetic field, is also possible within the GUI.

Probing spin fluctuations of the quantum phase transition in Ce3Al by muon spin rotation

We report on the dynamics of a magnetic-field-driven antiferromagnetic-to-paramagnetic quantum phase transition in monocrystalline Ce3Al via transverse-field muon spin rotation (TF-µSR) experiments down to low temperature of sim 80 mK. The quantum phase transition is of a spin-flip type and takes place on the Ce–Al magnetic chains as a result of competition between the indirect exchange and the Zeeman interaction of the Ce moments with the external field, applied along the chain direction (also the direction of the antiferromagnetic axis). The Ce moments are not static at T to 0, but fluctuate in their direction due to the Heisenberg uncertainty principle. Upon applying the magnetic field sweep, the fluctuations exhibit the largest amplitude at the quantum critical point, manifested in a maximum of the muon transverse relaxation rate at the critical field. The quantum nature of fluctuations observed in the TF-µSR experiments is reflected in the temperature independence of the average local magnetic field component along the external magnetic field at the muon stopping site(s) and the muon transverse relaxation rate within the investigated temperature range 1.5 K–80 mK. Quantum fluctuations are fast on the muon Larmor frequency scale, tau_{0} < 10–10 s.

Spin order and fluctuations in the EuAl4 and EuGa4 topological antiferromagnets: A μSR study

We report on systematic muon-spin rotation and relaxation ($\mu$SR) studies of the magnetic properties of EuAl$_4$ and EuGa$_4$ single crystals at a microscopic level. Transverse-field $\mu$SR measurements, spanning a wide temperature range (from 1.5 to 50 K), show clear bulk AFM transitions, with an almost 100% magnetic volume fraction in both cases. Zero-field $\mu$SR measurements, covering both the AFM and the paramagnetic (PM) states, reveal internal magnetic fields $B_\mathrm{int}(0) = 0.33$ T and 0.89 T in EuAl$_4$ and EuGa$_4$, respectively. The transverse muon-spin relaxation rate $\lambda_\mathrm{T}$, a measure of the internal field distribution at the muon-stopping site, shows a contrasting behavior. In EuGa$_4$, it decreases with lowering the temperature, reaching its minimum at zero temperature, $\lambda_\mathrm{T}(0) = 0.71$ $\mu$s$^{-1}$. In EuAl$_4$, it increases significantly below $T_\mathrm{N}$, to reach 58 $\mu$s$^{-1}$ at 1.5 K, most likely reflecting the complex magnetic structure and the competing interactions in the AFM state of EuAl$_4$. In both compounds, the temperature-dependent longitudinal muon-spin relaxation $\lambda_\mathrm{L}(T)$, an indication of the rate of spin fluctuations, diverges near the onset of AFM order, followed by a significant drop at $T < T_\mathrm{N}$. In the AFM state, spin fluctuations are much stronger in EuAl$_4$ than in EuGa$_4$, while being comparable in the PM state. The evidence of robust spin fluctuations against the external magnetic fields provided by $\mu$SR may offer new insights into the origin of the topological Hall effect and the possible magnetic skyrmions in the EuAl$_4$ and EuGa$_4$ compounds.

Megahertz dynamics in skyrmion systems probed with muon-spin relaxation

We present longitudinal-field muon-spin relaxation (LF $\mu$SR) measurements on two systems that stabilize a skyrmion lattice (SkL): Cu$_2$OSeO$_3$, and Co$_x$Zn$_y$Mn$_{20-x-y}$ for $(x,y)~=~(10,10)$, $(8,9)$ and $(8,8)$. We find that the SkL phase of Cu$_2$OSeO$_3$ exhibits emergent dynamic behavior at megahertz frequencies, likely due to collective excitations, allowing the SkL to be identified from the $\mu$SR response. From measurements following different cooling protocols and calculations of the muon stopping site, we suggest that the metastable SkL is not the majority phase throughout the bulk of this material at the fields and temperatures where it is often observed. The dynamics of bulk Co$_8$Zn$_9$Mn$_3$ are well described by $\simeq~2$ GHz excitations that reduce in frequency near the critical temperature, while in Co$_8$Zn$_8$Mn$_4$ we observe similar behavior over a wide range of temperatures, implying that dynamics of this kind persist beyond the SkL phase.

From magnetic order to quantum disorder in the Zn-barlowite series of S\u2009=\u20091/2 kagom\xe9 antiferromagnets

We report a comprehensive muon spectroscopy study of the Zn-barlowite series of S=frac{1}{2} kagomé antiferromagnets, ZnxCu4−x(OH)6FBr, for x = 0.00 to 0.99(1). By combining muon spin relaxation and rotation measurements with state-of-the-art density-functional theory muon-site calculations, we observe the formation of both μ–F and μ–OH complexes in Zn-barlowite. From these stopping sites, implanted muon spins reveal the suppression of long-range magnetic order into a possible quantum spin liquid state upon the increasing concentration of Zn-substitution. In the parent compound (x = 0), static long-range magnetic order below TN = 15 K manifests itself in the form of spontaneous oscillations in the time-dependent muon asymmetry signal consistent with the dipolar fields expected from the calculated muon stopping sites and the previously determined magnetic structure of barlowite. Meanwhile, in the x = 1.0 end-member of the series—in which antiferromagnetic kagomé layers of Cu2+S=frac{1}{2} moments are decoupled by diamagnetic Zn2+ ions—we observe that dynamic magnetic moment fluctuations persist down to at least 50 mK, indicative of a quantum disordered ground state. We demonstrate that this crossover from a static to dynamic magnetic ground state occurs for compositions of Zn-barlowite with x > 0.5, which bears resemblance to the dynamical behaviour of the widely studied Zn-paratacamite series that contains the quantum spin liquid candidate herbertsmithite.

Magnetism and its coexistence with superconductivity in CaK(Fe0.949Ni0.051)4As4 : Muon spin rotation/relaxation studies

The magnetic response of CaK(Fe$_{0.949}$Ni$_{0.051}$)$_4$As$_4$ was investigated by means of the muon-spin rotation/relaxation. The long-range commensurate magnetic order sets in below the N\'{e}el temperature $T_{\rm N}= 50.0(5)$~K. The density-functional theory calculations have identified three possible muon stopping sites. The experimental data were found to be consistent with only one type of magnetic structure, namely, the long-range magnetic spin-vortex-crystal order with the hedgehog motif within the $ab-$plane and the antiferromagnetic stacking along the $c-$direction. The value of the ordered magnetic moment at $T\approx3$ K was estimated to be $m_{\rm Fe}=0.38(11)$ $\mu_{\rm B}$ ($\mu_{\rm B}$ is the Bohr magneton). A microscopic coexistence of magnetic and superconducting phases accompanied by a reduction of the magnetic order parameter below the superconducting transition temperature $T_{\rm c}\simeq 9$ K is observed. Comparison with 11, 122, and 1144 families of Fe-based pnictides points to existence of correlation between the reduction of the magnetic order parameter at $T\rightarrow 0$ and the ratio of the transition temperatures $T_{\rm c}/T_{\rm N}$. Such correlations were found to be described by Machida's model for coexistence of itinerant spin-density wave magnetism and superconductivity [Machida, J. Phys. Soc. Jpn. 50, 2195 (1981) and Bud'ko et al., Phys. Rev. B 98, 144520 (2018)].

Estimation of Muon Stopping Site in CoCr2O4 Using Density Functional Theory

Spinel material Cobalt Chromite (CoCr2O4) is a ferrimagnetic material with TC of about 93 K. This compound has attracted much attention due to the presence of electrical polarization which related to spiral magnetic transition at TS = 26 K. [Physical Review B, 70, 214434, 2004]. This spiral magnetic transition phenomena could be explored by experiment using Muon Spin Rotation, Relaxation, and Resonance (liSR) . Positive muon in liSR plays a role as a local magnetic probethus, we need to determine the muon stopping site information. In this study, we investigate the muon stopping site of CoCr2O4 with ferrimagnetic ground state using spin polarized Density Functional Theory implemented in Full-potential Linearized Augmented Plane Wave (FLAPW) method. The exchange-correlation effect was considered in the scheme of GGA+Ueff approximation. Based on the minimum energy of electrostatic potential, we obtain the muon stopping site around the three Oxygen ions in the tetrahedral site of Cobalt.

Computational prediction of muon stopping sites: A novel take on the unperturbed electrostatic potential method.

Finding the stopping site of the muon in a muon-spin relaxation experiment is one of the main problems of muon spectroscopy, and computational techniques that make use of quantum chemistry simulations can be of great help when looking for this stopping site. The most thorough approach would require the use of simulations, such as Density Functional Theory (DFT), to test and optimize multiple possible sites, accounting for the effect that the added muon has on its surroundings. However, this can be computationally expensive and sometimes unnecessary. Hence, in this work, we present a software implementation of the Unperturbed Electrostatic Potential (UEP) Method: an approach used for finding the muon stopping site in crystalline materials. The UEP method requires only one DFT calculation, necessary to compute the electronic density. This, in turn, is used to calculate the minima of the crystalline material's electrostatic potential and the estimates of the muon stopping site, relying on the approximation that the muon's presence does not significantly affect its surroundings. One of the main UEP's assumptions is that the muon stopping site will be one of the crystalline material's electrostatic potential minima. In this regard, we also propose some symmetry-based considerations about the properties of this crystalline material's electrostatic potential, in particular, which sites are more likely to be its minima and why the unperturbed approximation may be sufficiently robust for them. We introduce the Python software package pymuon-suite and the various utilities it provides to facilitate these calculations, and finally, we demonstrate the effectiveness of the method with some chosen example systems.

Benchmarking Full-Potential Linearized Augmented Plane Wave (FLAPW) Method for Determination of Muon Stopping Sites in LiF

Muon stopping sites in Lithium Fluoride have been determined based on the minimum electrostatic potential calculation using density functional theory implemented in the full-potential linearized augmented planewave method. The isosurface of the electrostatic potential obtained in our calculation is similar to the calculation obtained by using pseudopotential-based plane wave (PPPW) method reported by Bernadini et al. [Physical Review B, 87 (2013) 115148]. This yields to the two possible muon sites inside the cage structure of Li-F forming tetrahedral coordination. One of the muon sites is located at the center of the tetrahedral while the other is at the equivalent site of the tetrahedral. In spite of the similar isosurface results, our global minimum is found at the center of the tetrahedral in contrast to the previous result obtained at the tetrahedral sites. The strategy to determine the muon possible sites based on the minimum of the total energy of the system will also be considered including the muon sites position between the two fluorine ions (F-).

Density Functional Theory Studies of Muon Stopping Sites and Hyperfine Interaction in [Au25(SR)18]0 Nanocluster

The Density Functional Theory method was applied to determine muon stopping sites in [Au25(SR)18]0 nanocluster and the associated hyperfine interaction. The molecular electrostatic potential map indicates that the inner region of the Au13 icosahedron is more favorable for muon stopping sites. On the other hand, the potential in the region surrounding the ligands is positive and it is unlikely for muon to stop near any of the carbon atoms. Based on the DFT calculated energy, four muon sites near different Au atoms were found to be inside the icosahedron, while eight other Au sites are positioned slightly outside. The sites near Au atoms in the staple motifs were found to be energetically unstable. Local minima corresponding to muon sites near the S atoms in the staple motif have higher energies as compared to the Au sites. Three of the four sites inside the icosahedron, MAu10, MAu6, and MAu5, are the most stable with similar energies. The calculated Fermi contact coupling constant for these three Au sites are −2.85, 62.63, and 3.56 MHz, respectively.

FeTi2O5 : A spin Jahn-Teller transition enhanced by cation substitution

We have used muon-spin rotation, heat capacity and x-ray diffraction measurements in combination with density functional theory and dipole field calculations to investigate the crystal and magnetic structure of FeTi$_2$O$_5$. We observe a long range ordered state below $T_{\rm N}$=41.8(5) K with indications of significant correlations existing above this temperature. We determine candidate muon stopping sites in this compound, and find that our data are consistent with the spin Jahn-Teller driven antiferromagnetic ground state with $\boldsymbol{k}$=(1/2,1/2,0) reported for CoTi$_2$O$_5$ ($T_{\rm N}$=26 K). By comparing our data with calculated dipolar fields we can restrict the possible moment size and directions of the Fe$^{2+}$ ions.

Local magnetism, magnetic order and spin freezing in the ‘nonmetallic metal’ FeCrAs

We present the results of x-ray scattering and muon-spin relaxation (SR) measurements on the iron-pnictide compound FeCrAs. Polarized non-resonant magnetic x-ray scattering results reveal the 120° periodicity expected from the suggested three-fold symmetric, non-collinear antiferromagnetic structure. SR measurements indicate a magnetically ordered phase throughout the bulk of the material below K. There are signs of fluctuating magnetism in a narrow range of temperatures above involving low-energy excitations, while at temperatures well below behaviour characteristic of freezing of dynamics is observed, likely reflecting the effect of disorder in our polycrystalline sample. Using density functional theory we propose a distinct muon stopping site in this compound and assess the degree of distortion induced by the implanted muon.

Comparison between density functional theory and density functional tight binding approaches for finding the muon stopping site in organic molecular crystals.

Finding the possible stopping sites for muons inside a crystalline sample is a key problem of muon spectroscopy. In a previous study, we suggested a computational approach to this problem when dealing with muonium, the pseudoatom formed by a positive muon that has captured an electron, using density functional theory software in combination with a random structure searching approach that relies on a Poisson sphere distribution. In this work, we test this methodology further by applying it to muonium in three organic molecular crystal model systems: durene, bithiophene, and tetracyanoquinodimethane. Using the same sets of random structures, we compare the performance of density functional theory software CASTEP and the much faster lower level approximation of Density Functional Tight Binding provided by DFTB+ combined with the use of the 3ob-3-1 parameter set. We show the benefits and limitations of such an approach, and we propose the use of DFTB+ as a viable alternative to more cumbersome simulations for routine site-finding in organic materials. Finally, we introduce the Muon Spectroscopy Computational Project software suite, a library of Python tools meant to make these methods standardized and easy to use.

Computational prediction of muon stopping sites using ab initio random structure searching (AIRSS).

The stopping site of the muon in a muon-spin relaxation experiment is in general unknown. There are some techniques that can be used to guess the muon stopping site, but they often rely on approximations and are not generally applicable to all cases. In this work, we propose a purely theoretical method to predict muon stopping sites in crystalline materials from first principles. The method is based on a combination of ab initio calculations, random structure searching, and machine learning, and it has successfully predicted the MuT and MuBC stopping sites of muonium in Si, diamond, and Ge, as well as the muonium stopping site in LiF, without any recourse to experimental results. The method makes use of Soprano, a Python library developed to aid ab initio computational crystallography, that was publicly released and contains all the software tools necessary to reproduce our analysis.

Unconventional magnetism on a honeycomb lattice inα−RuCl3studied by muon spin rotation

Muon spin rotation measurements have been performed on a powder sample of a-RuCl3, a layered material which previously has been proposed to be a quantum magnet on a honeycomb lattice close to a quantum spin liquid ground state. Our data reveal two distinct phase transitions at 11 K and 14 K which we interpret as originating from the onset of three-dimensional order and in-plane magnetic order, respectively. We identify, with the help of density functional theory calculations, likely muon stopping sites and combine these with dipolar field calculations to show that the two measured muon rotation frequencies are consistent with two inequivalent muon sites within a zig-zag antiferromagnetic structure proposed previously.