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Pressure dependence of the thermodynamic critical field $B_{\rm c}$ in elemental indium (In) and tin (Sn) superconductors was studied by means of the muon-spin rotation/relaxation. Pressure enhances the deviation of $B_{\rm c}(T)$ from the parabolic behavior, expected for a typical type-I superconductor, suggesting a weakening of the coupling strengths $\alpha=\langle\Delta\rangle /k_{\rm B}T_{\rm c}$ ($\langle\Delta\rangle$ is the average value of the superconducting energy gap, $T_{\rm c}$ is the transition temperature and $k_{\rm B}$ is the Boltzmann constant). As pressure increases from 0.0 to $\simeq 3.0$ GPa $\alpha$ decreases linearly, by approaching the limiting weak-coupling BCS value $\alpha_{\rm BCS}=1.764$. Analysis of the data within the framework of the Eliashberg theory reveals that only part of the pressure effect on $\alpha$ can be attributed to the effect of hardening of the phonon spectra, which is reflected by a decrease of the electron-phonon coupling constant. Nearly 40% of the effect is caused by increased anisotropy of the superconducting energy gap., Comment: 6 pages, 3 figures

$\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ is an itinerant ferromagnet with a Curie temperature, $T_C$, of $\sim$ 3.8 K and a remarkably small saturated moment of 0.1 $\mu_{B}/\text{Co}$. Here we present the growth and characterization of single crystals of the ${\text{La}_{5}\text{(Co}_{1-x}\text {Ni}_{x})_2\text {Ge}_{3}}$ series for 0.00 $\leq x \leq$ 0.186. We measured powder X-ray diffraction, composition as well as anisotropic temperature dependent resistivity, temperature and field dependent magnetization along with heat capacity on these single crystals. We also measured muon-spin rotation/relaxation ($\mu \text{SR}$) for some Ni substitutions ($x$ = 0.027, 0.036, 0.074) to study the evolution of internal field with Ni substitution. Using the measured data we infer a low temperature, transition temperature-composition phase diagram for ${\text{La}_{5}\text{(Co}_{1-x}\text {Ni}_{x})_2\text {Ge}_{3}}$. We find that $T_{C}$ is suppressed for low dopings, $x \leq 0.014 $; whereas for $0.036 \leq {x} \leq 0.186 $, the samples are antiferromagnetic with a Neel temperature, $T_{N}$, that goes through a weak and shallow maximum ($T_N \sim$ 3.4 K for $ x \sim$ 0.07) and then gradually decreases to 2.4 K by $x$ = 0.186. For intermediate Ni substitutions, $0.016 \leq {x} \leq 0.027 $, two transition temperatures are inferred with $T_N > T_C$. Whereas the $T-x$ phase diagram for ${\text{La}_{5}\text{(Co}_{1-x}\text {Ni}_{x})_2\text {Ge}_{3}}$ and the $T-p$ phase diagram determined for the parent $\text{La}_{5}\text{Co}_{2}\text{Ge}_{3}$ under hydrostatic pressure are grossly similar, changing from a low doping or low pressure ferromagnetic (FM) ground state to a high doped or pressure antiferromagnetic (AFM) state, perturbation by Ni substitution enabled us to identify an intermediate doping regime where both FM and AFM transitions occur., Comment: 24 pages, 40 figures

The unveiling of superconductivity in La$_3$Ni$_2$O$_{7-\delta}$ under pressure, following the suppression of a high-temperature density wave (DW) state, has attracted considerable attention. Notably, the nature of this competing DW order remains elusive, presenting a crucial question that demands further investigation. Here, we employ the muon-spin rotation/relaxation ($\mu$SR) technique combined with dipole-field numerical analysis to probe the magnetic response of La$_3$Ni$_2$O$_{7-\delta}$ as a function of hydrostatic pressure. At ambient pressure, $\mu$SR experiments reveal commensurate static magnetic order below $T_{\rm N} \simeq 151$K. The comparison of the observed internal magnetic fields with dipole-field calculations reveals the magnetic structure's compatibility with a stripe-type arrangement of Ni moments ($\simeq0.3-0.7$$\mu_{\rm B}$), characterized by alternating lines of magnetic moments and non-magnetic stripes. Experiments under pressure (up to $p\simeq2.3$~GPa) demonstrate an increase of the magnetic ordering temperature at a rate ${\rm d}T_{\rm N}/{\rm d}p\simeq 2.8$ K/GPa. This trend is opposite in sign and significantly smaller in magnitude compared to the changes observed in the DW order of unknown origin reported by Wang et al. [arXiv:2309.17378]. Our findings reveal that the ground state of the La$_3$Ni$_2$O$_{7-\delta}$ system is characterized by the coexistence of two distinct orders -- the spin density wave and, most likely, charge density wave -- with a notable pressure-induced separation between them., Comment: 8 pages, 4 figures

We report on measurements of the oxygen isotope ($^{16}$O/$^{18}$O) effect (OIE) on the transition temperature $T_{\rm c}$ and the zero-temperature in-plane magnetic penetration depth $\lambda_{\rm ab}(0)$ in the stoichiometric cuprate superconductor YBa$_2$Cu$_4$O$_8$ by means of muon-spin rotation/relaxation. An analysis of the temperature evolution of $\lambda^{-2}_{\rm ab}$ in terms of coexisting $s+d-$wave order parameters reveals that the OIE on the superfluid density $\rho_{\rm s}(0)\propto\lambda^{-2}_{\rm ab}(0)$ stems predominantly from the $d-$wave component while the contribution of the $s-$wave one is almost zero. The OIE on the transition temperature $T_{\rm c}$ is found to be rather small: $\delta T_{\rm c}/T_{\rm c}= -0.32(7)$%, compared to the total OIE on the superfluid density $\rho_{\rm s}(0)$: $\delta \rho_{\rm s}(0)/\rho_{\rm s}(0)= -2.8(1.0)$%. The partial OIE's on the corresponding $d-$wave and $s-$wave components of $\rho_{\rm s}(0)$ are $\delta \rho_{\rm s,d}(0)/\rho_{\rm s}(0)= -3.0(1.2)$%, and $\delta \rho_{\rm s,s}(0)/\rho_{\rm s}(0)= 0.2(1.2)$%, respectively. Our results demonstrate that polaron formation in the CuO$_2$ planes is the origin of the observed OIE in the $d-$wave channel. In the much weaker $s-$wave channel, fermionic quasiparticles are present, which do not contribute to the OIE on $\rho_{\rm s}(0)$. Our results support the original idea of K. Alex M\"{u}ller on the polaronic nature of the supercarries in high-temperature cuprate superconductors., Comment: Contribution to a special issue by Physica C: "Oxide superconductors and beyond - In memoriam of Professor Karl Alex M\"uller"

The first-principles calculations and measurements of the magnetic penetration depths, the upper critical field, and the specific heat were performed for a family of Mo$_5$Si$_{3-x}$P$_x$ superconducotrs. First-principles calculations suggest the presence of a flat band dispersion, which gradually shifts to the Fermi level as a function of phosphorus doping $x$. The flat band approaches the Fermi level at $x\simeq 1.3$, thus separating Mo$_5$Si$_{3-x}$P$_x$ between the purely steep band and the steep band/flat band superconducting regimes. The emergence of flat bands lead to an abrupt change of nearly all the superconducting quantities. In particular, a strong reduction of the coherence length $\xi$ and enhancement of the penetration depth $\lambda$ result in nearly factor of three increase of the Ginzburg-Landau parameter $\kappa=\lambda/\xi$ (from $\kappa\simeq 25$ for $x\lesssim 1.2$ to $\kappa\simeq 70$ for $x\gtrsim 1.4$) thus initiating the transition of Mo$_5$Si$_{3-x}$P$_x$ from a moderate to an extreme type-II superconductivity., Comment: 6 pages, 4 figures

We report on measurements of the in-plane magnetic penetration depth ($\lambda_{\rm ab}$) in single crystals of Sr$_2$RuO$_4$ down to $\simeq 0.015$ K by means of muon-spin rotation/relaxation. The linear temperature dependence of $\lambda^{-2}_{\rm ab}$ for $T\lesssim 0.7$ K suggests the presence of nodes in the superconducting gap. This statement is further substantiated by observation of the Volovik effect, $i.e.$ the reduction of $\lambda_{ab}^{-2}$ as a function of the applied magnetic field. The experimental zero-field and zero-temperature value of $\lambda_{\rm ab}=124(3)$ nm agrees with $\lambda_{\rm ab}\simeq 130$ nm, calculated based on results of electronic structure measurements reported in [Phys. Rev X 9, 021048 (2019)]. Our analysis reveals that a simple nodal superconducting energy gap, described by the lowest possible harmonic of a gap function, does not capture the dependence of $\lambda_{\rm ab}^{-2}$ on $T$, so the higher angular harmonics of the energy gap function need to be introduced., Comment: 7 pages, 3 figures

We discuss the use of commercial high-power light emitting diodes (LEDs) as a light source for fluorescence pressure measurements. A relatively broad light emitting spectra of single color LEDs (in comparison with lasers) do not prevent producing narrow fluorescence lines at least for two widely used pressure indicator materials, namely ruby (Cr$^{3+}$:Al$_2$O$_3$) and strontium tetraborate (Sm$^{2+}$:SrB$_4$O$_7$). Strongest responses of both pressure indicators were detected for the green color LEDs with the average wavelength $\lambda_{\rm av}\sim 530$ nm. LEDs might be easily implemented for producing fiber coupled, as well as the parallel light sources. LEDs were found to be efficient to replace laser sources in piston-cylinder cell and diamond anvil cell fluorescence pressure measurement setups., Comment: 12 pages, 8 figures. Accepted for publication by "High Pressure Research"

The newly discovered kagome superconductors represent a promising platform for investigating the interplay between band topology, electronic order, and lattice geometry. Despite extensive research efforts on this system, the nature of the superconducting ground state remains elusive. In particular, consensus on the electron pairing symmetry has not been achieved so far, in part owing to the lack of a momentum-resolved measurement of the superconducting gap structure. Here we report the direct observation of a nodeless, nearly isotropic, and orbital-independent superconducting gap in the momentum space of two exemplary CsV$_3$Sb$_5$-derived kagome superconductors -- Cs(V$_{0.93}$Nb$_{0.07}$)$_3$Sb$_5$ and Cs(V$_{0.86}$Ta$_{0.14}$)$_3$Sb$_5$, using ultrahigh resolution and low-temperature angle-resolved photoemission spectroscopy (ARPES). Remarkably, such a gap structure is robust to the appearance or absence of charge order in the normal state, tuned by isovalent Nb/Ta substitutions of V. Moreover, we observe a signature of the time-reversal symmetry (TRS) breaking inside the superconducting state, which extends the previous observation of TRS-breaking CDW in the kagome lattice. Our comprehensive characterizations of the superconducting state provide indispensable information on the electron pairing of kagome superconductors, and advance our understanding of unconventional superconductivity and intertwined electronic orders.

A Muon Spin Rotation ($\mu$+SR) study was conducted to investigate the magnetic properties of SrCu2(BO3)2 (SCBO) as a function of temperature/pressure. Measurements in zero field and transverse field confirm the absence of long range magnetic order at high pressures and low temperatures. These measurements suggest changes in the Cu spin fluctuations characteristics above 21 kbar, consistent with the formation of a plaquette phase as previously suggested by inelastic neutron scattering measurements. SCBO is the only known realisation of the Shatry-Sutherland model, thus the ground state mediating the dimer and antiferromagnetic phase is likekly to be a plaquette state.

LiFePO$_4$ (LFPO) is an archetypical and well-known cathode material for rechargeable Li-ion batteries. However, its quasi-one-dimensional (Q1D) structure along with the Fe ions, LFPO also displays interesting low-temperature magnetic properties. Our team has previously utilized the muon spin rotation ($\mu^+$SR) technique to investigate both magnetic spin order as well as Li-ion diffusion in LFPO. In this initial study we extend our investigation and make use of high-pressure $\mu^+$SR to investigate effects on the low-$T$ magnetic order. Contrary to theoretical predictions we find that the magnetic ordering temperature as well as the ordered magnetic moment increase at high pressure (compressive strain)., Comment: 8 pages, 6 figures, submitted as a part of the International Conference on Muon Spin Rotation, Relaxation and Resonance 2022. Accepted Dec 2022

Pressure, together with temperature, electric and magnetic fields, alters the system and allows to investigate the fundamental properties of the matter. Under applied pressure the interatomic distances shrink, which modify interactions between atoms and may lead to appearance of a new (sometime exotic) physical properties as, e.g., pressure induced phase transition(s); quantum critical points(s), new structural, magnetic and/or superconducting states; changes of the temperature evolution and the symmetry of the order parameter(s) etc. The muon-spin rotation/relaxation ($\mu$SR) appears to be a commonly used powerful technique allowing to study the magnetic and superconducting responses of various materials under extreme conditions. At present, $\mu$SR experiments might be performed under the high magnetic field up to $\simeq 9$ T, temperatures down to $\simeq 10-15$ mK and hydrostatic pressure up to $\simeq 2.8$ GPa. In the following Perspective paper the requirements to the $\mu$SR under pressure experiments, the existing high-pressure muon facility at the Paul Scherrer Institute (Switzerland), and selected experimental results obtained by using the $\mu$SR under pressure technique are discussed., Comment: 14 pages, 14 figures

The recently discovered vanadium-based kagome metals $A$V$_{3}$Sb$_{5}$ ($A$~=~K,~Rb,~Cs) exhibit superconductivity at low-temperatures and charge density wave (CDW) order at high-temperatures. A prominent feature of the charge ordered state in this family is that it breaks time-reversal symmetry (TRSB), which is connected to the underlying topological nature of the band structure. In this work, a powerful combination of zero-field and high-field muon-spin rotation/relaxation is used to study the signatures of TRSB of the charge order in CsV$_3$Sb$_5$, as well as its anisotropic character. By tracking the temperature evolution of the in-plane and out-of-plane components of the muon-spin polarization, an enhancement of the internal field width sensed by the muon-spin ensemble was observed below $T_{\rm TRSB}=T_{\rm CDW}\simeq95$~K. Additional increase of the internal field width, accompanied by a change of the local field direction at the muon site from the $ab$-plane to the $c$-axis, was detected below $T^\ast\simeq30$~K. Remarkably, this two-step feature becomes well pronounced when a magnetic field of 8~T is applied along the crystallographic $c-$axis, thus indicating a field-induced enhancement of the electronic response at the CDW transition. These results point to a TRSB in CsV$_3$Sb$_5$ by charge order with an onset of ${\simeq}~95$~K, followed by an enhanced electronic response below ${\simeq}~30$~K. The observed two-step transition is discussed within the framework of different charge-order instabilities, which, in accordance with density functional theory calculations, are nearly degenerate in energy., Comment: 10 pages, 4 figures

The kagome metals of the family $A$V$_3$Sb$_5$, featuring a unique structural motif, harbor an array of intriguing phenomena such as chiral charge order and superconductivity. CsV$_3$Sb$_5$ is of particular interest because it displays a double superconducting dome in the region of the temperature-pressure phase diagram where charge order is still present. However, the microscopic origin of such an unusual behavior remains an unsolved issue. Here, to address it, we combine high-pressure, low-temperature muon spin relaxation with first-principles calculations. We observe a pressure-induced threefold enhancement of the superfluid density, which also displays a double peak feature, similar to the superconducting critical temperature. This leads to three distinct regions in the phase diagram, each of which features distinct slopes of the linear relation between superfluid density and the critical temperature. These results are attributed to a possible evolution of the charge order pattern from the superimposed tri-hexagonal Star-of-David phase at low pressures (within the first dome) to the staggered tri-hexagonal phase at intermediate pressures (between the first and second domes). Our findings suggest a change in the nature of the charge ordered state across the phase diagram of CsV$_3$Sb$_5$, with varying degrees of competition with superconductivity., Comment: 9 pages, 4 figures

We investigated the superconducting properties of the topological superconductor $\alpha$--PdBi$_2$ at ambient and external pressures up to 1.77~GPa using muon spin rotation ($\mu$SR) experiments. The ambient pressure measurements evince a fully gapped $s$-wave superconducting state in the bulk of the specimen. AC magnetic susceptibility and $\mu$SR measurements manifest a continuous suppression of $T_{\rm c}$ with increasing pressure. In parallel, we observed a significant decrease of superfluid density by $\sim$20\% upon application of external pressure. Remarkably, the superfluid density follows linear relation with $T_{\rm c}$ which was found before in some unconventional topological superconductors and hole doped cuprates. This finding signals a possible crossover from BEC to BCS in $\alpha$--PdBi$_2$., Comment: 6 pages, 4 figures

Measurement of the absolute value of the applied pressure in high-pressure muon and neutron experiments is a complicated task. It requires both the presence of a calibration material inside the sample volume as well as additional time for refining the response of the calibrant. Here we describe the use of optical calibrants for precise determination of the pressure value inside the piston-cylinder clamp cells. Utilizing the concept of separate volumes for the sample and the optical media, a new setup for conducting in-situ pressure measurements has been successfully tested. Pressures in both the `sample' and the `optical' volumes were proved to be the same within experimental accuracy. The use of SrB$_{4}$O$_7$:(0.01 Sm$^{2+}$, 0.03 Eu$^{2+}$) as a pressure calibrant allows for a high accuracy of pressure determination by considering up to eight fluorescence lines., Comment: 9 pages, 9 figures

The magnetic phase diagram of Sr$_{1-x}$Ca$_x$Co$_2$P$_2$ as a function of hydrostatic pressure and temperature is investigated by means of high pressure muon spin rotation, relaxation and resonance ($\mu^+$SR). The weak pressure dependence for the $x\neq1$ compounds suggests that the rich phase diagram of Sr$_{1-x}$Ca$_x$Co$_2$P$_2$ as a function of $x$ at ambient pressure may not only be attributed to solely chemical pressure effects. The $x=1$ compound on the other hand reveals a high pressure dependence, where the long range magnetic order is fully suppressed at $p_{\rm c2}\approx9.8$~kbar, which seem to be a first order transition. In addition, an intermediate phase consisting of dilute ferromagnetic islands (FMI) is formed above $p_{\rm c1}\approx8$~kbar where they co-exist with a magnetically disordered state. Moreover, such FMI phase seems to consist of an high- (FMI-\textcircled{\small{1}}) and low-temperature (FMI-\textcircled{\small{2}}) region, respectively, separated by a phase boundary at $T_{\rm i}\approx20$~K., Comment: 10 pages, 8 figures

A three-wall piston-cylinder type high pressure cell for muon-spin rotation/relaxation experiments was designed, manufactured, tested and commissioned. The outer cylinder of the cell body is made from MP35N and the middle and the inner cylinders are made from NiCrAl nonmagnetic alloys. The mechanical design and performance of the pressure cell are evaluated and optimised using finite-element analysis. The outcomes of the experimental testing closely match the modelling results. The high-pressure cell is shown to reach pressures of up to 3.3 GPa at ambient temperature, corresponding to 3.0 GPa at low temperatures, without irreversible damage., Comment: 17 pages, 12 figures

The unconventional multi-gap superconductivity in elemental Pb were reported previously by surface sensitive tunneling experiments, as well as predicted by several theory works. To obtain bulk evidence for such multiple gap behavior, the thermodynamic critical field $B_{\rm c}$ was measured along three different crystallographic directions ([100], [110], and [111]) in a high-quality Pb single crystal by means of muon spin rotation/relaxation. No difference in temperature evolution of $B_{\rm c}$ for all three directions was detected. The average reduced gap $\alpha=\Delta/k_{\rm B}T_{\rm c}=2.312(3)$ ($\Delta$ is the zero-temperature gap value and $T_{\rm c}$ is the transition temperature) was further obtained by employing the phenomenological $\alpha-$model. Our results imply that the elemental Pb is an isotropic superconductor with a single energy gap., Comment: 6 pages, 3 figures. The version without `smiling' description to appear in Physical Review B as a Letter

The antiferromagnet and semimetal EuCd$_2$As$_2$ has recently attracted a lot of attention due to a wealth of topological phases arising from the interplay of topology and magnetism. In particular, the presence of a single pair of Weyl points is predicted for a ferromagnetic configuration of Eu spins along the $c$-axis in EuCd$_2$As$_2$. In the search for such phases, we investigate here the effects of hydrostatic pressure in EuCd$_2$As$_2$. For that, we present specific heat, transport and $\mu$SR measurements under hydrostatic pressure up to $\sim\,2.5\,$GPa, combined with {\it ab initio} density functional theory (DFT) calculations. Experimentally, we establish that the ground state of EuCd$_2$As$_2$ changes from in-plane antiferromagnetic (AFM$_{ab}$) to ferromagnetic at a critical pressure of $\,\approx\,$2\,GPa, which is likely characterized by the moments dominantly lying within the $ab$ plane (FM$_{ab}$). The AFM$_{ab}$-FM$_{ab}$ transition at such a relatively low pressure is supported by our DFT calculations. Furthermore, our experimental and theoretical results indicate that EuCd$_2$As$_2$ moves closer to the sought-for FM$_c$ state (moments $\parallel$ $c$) with increasing pressure further. We predict that a pressure of $\approx$\,23\,GPa will stabilize the FM$_c$ state, if Eu remains in a 2+ valence state. Thus, our work establishes hydrostatic pressure as a key tuning parameter that (i) allows for a continuous tuning between magnetic ground states in a single sample of EuCd$_2$As$_2$ and (ii) enables the exploration of the interplay between magnetism and topology and thereby motivates a series of future experiments on this magnetic Weyl semimetal., Comment: 8 figures in main text, 6 figures in Appendix; 14 pages total

The recently discovered kagome superconductor CsV$_3$Sb$_5$ ($T_c \simeq 2.5$ K) has been found to host charge order as well as a non-trivial band topology, encompassing multiple Dirac points and probable surface states. Such a complex and phenomenologically rich system is, therefore, an ideal playground for observing unusual electronic phases. Here, we report on microscopic studies of its anisotropic superconducting properties by means of transverse-field muon spin rotation ($\mu$SR) experiments. The temperature dependences of the in-plane and out-of-plane components of the magnetic penetration depth $\lambda_{ab}^{-2}(T)$ and $\lambda_{c}^{-2}(T)$ indicate that the superconducting order parameter exhibits a two-gap ($s+s$)-wave symmetry, reflecting the multiple Fermi surfaces of CsV3Sb5. The multiband nature of its superconductivity is further validated by the different temperature dependences of the anisotropic magnetic penetration depth $\gamma_\lambda(T)$ and upper critical field $\gamma_{\rm B_{c2}}(T)$, both in close analogy with the well known two-gap superconductor MgB$_2$. Remarkably, the high value of the $T_c/\lambda^{-2}(0)$ ratio in both field orientations strongly suggests the unconventional nature of superconductivity. The relaxation rates obtained from zero field $\mu$SR experiments do not show noticeable change across the superconducting transition, indicating that superconductivity does not break time reversal symmetry., Comment: 9 pages, 4 figures

There is considerable evidence that the superconducting state of Sr$_2$RuO$_4$ breaks time reversal symmetry. In the experiments showing time reversal symmetry breaking its onset temperature, $T_\text{TRSB}$, is generally found to match the critical temperature, $T_\text{c}$, within resolution. In combination with evidence for even parity, this result has led to consideration of a $d_{xz} \pm id_{yz}$ order parameter. The degeneracy of the two components of this order parameter is protected by symmetry, yielding $T_\text{TRSB} = T_\text{c}$, but it has a hard-to-explain horizontal line node at $k_z=0$. Therefore, $s \pm id$ and $d \pm ig$ order parameters are also under consideration. These avoid the horizontal line node, but require tuning to obtain $T_\text{TRSB} \approx T_\text{c}$. To obtain evidence distinguishing these two possible scenarios (of symmetry-protected versus accidental degeneracy), we employ zero-field muon spin rotation/relaxation to study pure Sr$_2$RuO$_4$ under hydrostatic pressure, and Sr$_{1.98}$La$_{0.02}$RuO$_4$ at zero pressure. Both hydrostatic pressure and La substitution alter $T_\text{c}$ without lifting the tetragonal lattice symmetry, so if the degeneracy is symmetry-protected $T_\text{TRSB}$ should track changes in $T_\text{c}$, while if it is accidental, these transition temperatures should generally separate. We observe $T_\text{TRSB}$ to track $T_\text{c}$, supporting the hypothesis of $d_{xz} \pm id_{yz}$ order., Comment: 14 pages, 8 Figures

LaCrGe$_3$ has attracted attention as a paradigm example of the avoidance of ferromagnetic (FM) quantum criticality in an itinerant magnet. By combining thermodynamic, transport, x-ray and neutron scattering as well as $\mu$SR measurements, we refined the temperature-pressure phase diagram of LaCrGe$_3$. We provide thermodynamic evidence (i) for the first-order character of the FM transition when it is suppressed to low temperatures and (ii) for the formation of new phases at high pressures. From our microscopic data, we infer that short-range FM ordered clusters exist in these high-pressure phases. These results suggest that LaCrGe$_3$ is a rare example, which fills the gap between the two extreme limits of avoided FM quantum criticality in clean and strongly disordered metals., Comment: 7 pages, 4 figures plus Supplemental Information

Pressure dependence of the thermodynamic critical field $B_{\mathrm{c}}$ in elemental aluminum was studied by means of the muon-spin rotation/relaxation technique. Pressure enhances the deviation of $B_{\mathrm{c}}(T)$ from the parabolic behavior, expected for a typical type-I superconductor, thus suggesting the weakening of the gap ratio $\langle\alpha\rangle=\langle \Delta\rangle/k_{\mathrm{B} }T_{\mathrm{c}}$ ($\langle\Delta\rangle$ is the average value of the superconducting energy gap, $T_{\mathrm{c}}$ is the transition temperature and $k_{\mathrm{B}}$ is the Boltzmann constant). With the pressure increase from 0.0 to $\simeq1.6$ GPa, $\langle\alpha\rangle$ decreases almost linearly from 1.73 to 1.67. Our results imply, therefore, that in elemental aluminum the gap ratio $\langle\alpha\rangle$ is smaller than the weak-coupled BCS prediction $\alpha_{\mathrm{BCS}}\simeq1.764$ and it is even further reduced under pressure., Comment: 6 pages, 3 figures

The magnetic properties of iron-based superconductors $A$Fe$_2$As$_2$ ($A=$K, Cs, and Rb), which are characterized by the V-shaped dependence of the critical temperature ($T_{\rm c}$) on pressure ($P$) were studied by means of the muon spin rotation/relaxation technique. In all three systems studied the magnetism was found to appear for pressures slightly below the critical one ($P_{\rm c}$), i.e. at pressure where $T_{\rm c}(P)$ changes the slope. Rather than competing, magnetism and superconductivity in $A$Fe$_2$As$_2$ are coexisting at $P\gtrsim P_{\rm c}$ pressure region. Our results support the scenario of a transition from one pairing state to another, with different symmetries on either side of $P_{\rm c}$., Comment: 7 pages, 3 figures

It is a well-established fact that the physical properties of compounds follow their crystal symmetries. This has especially pronounced implications on emergent collective quantum states in materials. Specifically, the effect of crystal symmetries on the properties of superconductors is widely appreciated, although the clarification of this relationship is a core effort of on-going research. Emergent phenomena on honeycomb lattices are of special interest, as they can give rise to spectacular phenomenology, as manifested by the recent discovery of correlated states in magic-angle graphene, or by the high-temperature superconductivity in MgB$_2$. Here, we report on the structural and microscopic superconducting properties of a class of ternary superconductors with Al/Si honeycomb layers, i.e. Ca$_{1-x}$Sr$_{x}$AlSi. We show that this solid solution is a remarkable model system with a highly tunable two-gap to single-gap superconducting system on a honeycomb lattice, where the superconductivity is enhanced by a subtle structural instability, i.e. the buckling of the Al/Si layers.

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)]., Comment: 15 pages, 14 figures

The puzzling situation where some thermodynamic quantities require a single-gap description, while others need a more complex gap scenario, is discussed. Our approach reveals that in some cases, the conclusions based on measurements of only one thermodynamic quantity may lead to conflicting results. As an example, temperature evolutions of the electronic specific heat [$C_{\rm e}(T)$] and the thermodynamic critical field [$B_{\rm c}(T)$] of BeAu superconductor were reconsidered within the framework of the single-gap and the two-gap scenarios. The analysis shows that the single-gap approach describes the temperature dependencies of $C_{\rm e}(T)$ and the superfluid density $\rho_{\rm s}(T)$ satisfactorily. However it fails in the case of $B_{\rm c}(T)$. The self-consistent two-gap model, in contrast, is able to describe all of the mentioned thermodynamic quantities by using a similar set of parameters. Our results reveal that a proper description of the thermodynamic quantities, related to the superconducting pairing mechanism, requires the use of similar model containing the same set of parameters., Comment: 8 pages, 5 figures

Precise measurements of the thermodynamic critical field ($B_{\rm c}$) in type-I noncentrosymmetric superconductor BeAu were performed by means of the muon-spin rotation/relaxation technique. The temperature evolution of $B_{\rm c}$ can not be described within the single gap scenario and it requires the presence of at least two different types of the superconducting order parameters. The self-consistent two-gap approach, adapted for analysis of $B_{\rm c}(T)$ behavior, suggests the presence of two superconducing energy gaps with the gap to $T_{\rm c}$ ratios $2\Delta/k_{\rm B}T_{\rm c}\simeq4.52$ and $\simeq2.37$ for the big and the small gap, respectively. This implies that the superconductivity in BeAu is unconventional and that the supercarrier pairing occurs at various energy bands., Comment: 6 pages, 2 figures

Single crystals of monoclinic La$_{5}$Co$_{2}$Ge$_{3}$ were grown using a self-flux method and were characterized by room-temperature powder X-ray diffraction, anisotropic temperature and field dependent magnetization, temperature dependent resistivity, specific heat, and muon spin rotation. La$_{5}$Co$_{2}$Ge$_{3}$ has a Curie temperature ($T_\mathrm{C}$) of 3.8~K and clear signatures of ferromagnetism in magnetization and $\mu SR$ data, as well as a clear loss of spin disorder scattering in resistivity data and a sharp specific heat anomaly. The magnetism associated with La$_{5}$Co$_{2}$Ge$_{3}$ is itinerant, has a change in the entropy at $T_\mathrm {C}$ of $\simeq$0.05 R ln2 per mol-Co, and has a low-field saturated moment of $\sim 0.1 \mu_\mathrm B$/Co, making it a rare, itinerant, small moment, low $T_\mathrm C$ compound.

High-quality single crystals of NdFeAsO$_{0.65}$F$_{0.35}$ (the transition temperature $T_{\rm c} \simeq 30.6$~K) were studied in zero-field (ZF) and transverse-field (TF) muon-spin rotation/relaxation ($\mu$SR) experiments. An upturn in muon-spin depolarization rate at $T\lesssim 3$~K was observed in ZF-$\mu$SR measurements and it was associated with the onset of ordering of Nd electronic moments. Measurements of the magnetic field penetration depth ($\lambda$) were performed in the TF geometry. By applying the external magnetic field $B_{\rm ex}$ parallel to the crystallographic $c$-axis ($B_{\rm ex}\| c$) and parallel to the $ab$-plane ($B_{\rm ex}\| ab$), the temperature dependencies of the in-plane component ($\lambda_{ab}^{-2}$) and the combination of the in-plane and the out of plane components ($\lambda_{ab,c}^{-2}$) of the superfluid density were determined, respectively. The out-of-plane superfluid density component ($\lambda_{c}^{-2}$) was further obtained by combining the results of $B_{\rm ex} \| c$ and $B_{\rm ex} \| {ab}$ set of experiments. The temperature dependencies of $\lambda_{ab}^{-2}$, $\lambda_{ab,c}^{-2}$, and $\lambda_{c}^{-2}$ were analyzed within the framework of a self-consistent two-gap model despite of using the traditional $\alpha$-model. Interband coupling was taken into account, instead of assuming it to be zero as it stated in the $\alpha$-model. A relatively small value of the interband coupling constant $\Lambda_{12} \simeq 0.01$ was obtained, thus indicating that the energy bands in NdFeAsO$_{0.65}$F$_{0.35}$ are only weakly coupled. In spite of their small magnitude, the coupling between the bands leads to the single value of the superconducting transition temperature $T_{\rm c}$., Comment: 10 pages, 8 figures

The majority of the iron-based superconductors (FeSCs) exhibit a two-dimensional square lattice structure. Recent reports of pressure-induced superconductivity in the spin-ladder system, BaFe$_2$X$_3$ (X =S,Se), introduce a quasi-one-dimensional prototype and an insulating parent compound to the FeSCs. Here we report X-ray, neutron diffraction and muon spin relaxation experiments on BaFe$_2$Se$_3$ under hydrostatic pressure to investigate its magnetic and structural properties across the pressure-temperature phase diagram. A structural phase transition was identified at a pressure of 3.7(3) GPa. Neutron diffraction measurements at 6.8(3) GPa and 120 K show that the block magnetism persists even at these high pressures. A steady increase and then fast drop of the magnetic transition temperature $T\rm_N$ and greatly reduced moment above the pressure $P_s$ indicate potentially rich and competing phases close to the superconducting phase in this ladder system.

The temperature dependent measurements of the thermodynamic critical field and the specific heat for the pressure stabilized Ga-II phase of elemental Gallium are presented. The discussion of these and other Ga phases data in the context of elemental and binary phonon-mediated type-I superconductors allowed to establish simple scaling relations between BCS quantities such as $B_{\rm c}(0)/T_{\rm c}\sqrt{\gamma_{\rm e}}$ and the specific heat jump at $T_{\rm c}$ versus the coupling strength $2\Delta/k_{\rm B} T_{\rm c}$ [$\Delta$ and $B_{\rm c}(0)$ are the zero-temperature values of the superconducting energy gap and the thermodynamic critical field, respectively, $T_{\rm c}$ is the transition temperature, and $\gamma_{\rm e}$ is the electronic specific heat]. The scaling relations can be analytically expressed by taking into account strong-coupling corrections to BCS theory. Such correlations can naturally explain the linear relation between $B_{\rm c}(0)$ and $T_{\rm c}$, which holds for type-I superconducting materials., Comment: 6 pages, 3 figures

The application of the muon-spin rotation/relaxation ($\mu$SR) technique for studying type-I superconductivity is discussed. In the intermediate state, i.e. when a type-I superconducting sample with non-zero demagnetization factor $N$ is separated into normal state and Meissner state (superconducting) domains, the $\mu$SR technique allows to determine with very high precision the value of the thermodynamic critical field $B_{\rm c}$, as well as the volume of the sample in the normal and the superconducting state. Due to the microscopic nature of $\mu$SR technique, the $B_{\rm c}$ values are determined directly via measurements of the internal field inside the normal state domains. No assumptions or introduction of any type of measurement criteria are needed. Experiments performed on a 'classical' type-I superconductor, a cylindrically shaped $\beta-$Sn sample, allowed to reconstruct the full $B-T$ phase diagram. The zero-temperature value of the thermodynamic critical field $B_{\rm c}(0)=30.578(6)$ mT and the transition temperature $T_{\rm c}=3.717(3)$ K were determined and found to be in a good agreement with the literature data. An experimentally obtained demagnetization factor is in very good agreement with theoretical calculations of the demagnetization factor of a finite cylinder. The analysis of $B_{\rm c}(T)$ dependence within the framework of the phenomenological $\alpha-$model allow to obtain the value of the superconducting energy gap $\Delta=0.59(1)$ meV, of the electronic specific heat $\gamma_e=1.781(3)$ ${\rm mJ}/{\rm mol}\; {\rm K}^2$ and of the jump in the heat capacity ${\Delta C(T_c)}/{\gamma T_{\rm c}}=1.55(2)$., Comment: 14 pages, 15 pictures

The superconductivity in the Bi-II phase of elemental Bismuth (transition temperature $T_{\rm c}\simeq3.92$ K at pressure $p\simeq 2.80$ GPa) was studied experimentally by means of the muon-spin rotation as well as theoretically by using the Eliashberg theory in combination with Density Functional Theory calculations. Experiments reveal that Bi-II is a type-I superconductor with a zero temperature value of the thermodynamic critical field $B_{\rm c}(0)\simeq31.97$~mT. The Eliashberg theory approach provides a good agreement with the experimental $T_{\rm c}$ and the temperature evolution of $B_{\rm c}$. The estimated value for the retardation (coupling) parameter $k_{\rm B}T_{\rm c}/\omega_{\rm ln} \approx 0.07$ ($\omega_{\rm ln}$ is the logarithmically averaged phonon frequency) suggests that Bi-II is an intermediately-coupled superconductor., Comment: 6 pages, 2 figures

The magnetic penetration depth anisotropy $\gamma_\lambda=\lambda_{c}/\lambda_{ab}$ ($\lambda_{ab}$ and $\lambda_{c}$ are the in-plane and the out-of-plane components of the magnetic penetration depth) in a CaKFe$_4$As$_4$ single crystal sample (the critical temperature $T_{\rm c}\simeq 35$ K) was studied by means of muon-spin rotation ($\mu$SR). $\gamma_\lambda$ is almost temperature independent for $T\lesssim 20$ K ($\gamma_\lambda\simeq 1.9$) and it reaches $\simeq 3.0$ by approaching $T_{\rm c}$. The change of $\gamma_\lambda$ induces the corresponding rearrangement of the flux line lattice (FLL), which is clearly detected via enhanced distortions of the FLL $\mu$SR response. Comparison of $\gamma_\lambda$ with the anisotropy of the upper critical field ($\gamma_{H_{\rm c2}}$) studied in Phys. Rev B {\bf 94}, 064501 (2016), reveals that $\gamma_\lambda$ is systematically higher than $\gamma_{H_{\rm c2}}$ at low-temperatures and approaches $\gamma_{H_{\rm c2}}$ for $T \rightarrow T_{\rm c}$. The anisotropic properties of $\lambda$ are explained by the multi-gap nature of superconductivity in CaKFe$_4$As$_4$ and are caused by anisotropic contributions of various bands to the in-plane and the out-of-plane components of the superfluid density., Comment: 6 pages, 4 figures

Identifying the symmetry of the wave function describing the Cooper pairs is pivotal in understanding the origin of high-temperature superconductivity in iron-based superconductors. Despite nearly a decade of intense investigation, the answer to this question remains elusive. Here we use the muon spin rotation/relaxation (muSR) technique to investigate the underlying symmetry of the pairing state of the FeSe superconductor, the basic building block of all iron-chalcogenide superconductors. Contrary to earlier muSR studies on powders and crystals, we show that while the superconducting gap is most probably anisotropic but nodeless along the crystallographic c-axis, it is nodal in the ab-plane, as indicated by the linear increase of the superfluid density at low temperature. We further show that the superconducting properties of FeSe display a less pronounced anisotropy than expected., Comment: 13 Pages, 8 figures (Accepted in PRB Rapid)

Magnetism induced by external pressure ($p$) was studied in a FeSe crystal sample by means of muon-spin rotation. The magnetic transition changes from second-order to first-order for pressures exceeding the critical value $p_{{\rm c}}\simeq2.4-2.5$ GPa. The magnetic ordering temperature ($T_{{\rm N}}$) and the value of the magnetic moment per Fe site ($m_{{\rm Fe}}$) increase continuously with increasing pressure, reaching $T_{{\rm N}}\simeq50$~K and $m_{{\rm Fe}}\simeq0.25$ $\mu_{{\rm B}}$ at $p\simeq2.6$ GPa, respectively. No pronounced features at both $T_{{\rm N}}(p)$ and $m_{{\rm Fe}}(p)$ are detected at $p\simeq p_{{\rm c}}$, thus suggesting that the stripe-type magnetic order in FeSe remains unchanged above and below the critical pressure $p_{{\rm c}}$. A phenomenological model for the $(p,T)$ phase diagram of FeSe reveals that these observations are consistent with a scenario where the nematic transitions of FeSe at low and high pressures are driven by different mechanisms., Comment: 10 pages, 7 figyres

The temperature dependence of the in-plane magnetic penetration depth ($\lambda_{ab}$) in an extensively characterized sample of superconducting CaKFe$_4$As$_4$ ($T_{\rm c}\simeq35$ K) was investigated using muon-spin rotation ($\mu$SR). A comparison of $\lambda_{ab}^{-2}(T)$ measured by $\mu$SR with the one inferred from ARPES data confirms the presence of multiple gaps at the Fermi level. An agreement between $\mu$SR and ARPES requires the presence of additional bands, which are not resolved by ARPES experiments. These bands are characterized by small supercondcting gaps with an average zero-temperature value of $\Delta_{0} =$ 2.4(2) meV. Our data suggest that in CaKFe$_4$As$_4$ the $s^\pm$ order parameter symmetry acquires a more sophisticated form by allowing a sign change not only between electron and hole pockets, but also within pockets of similar type., Comment: 6 pages, 4 figures. Accpeted for publication in Phys.Rev.B (Rapid)

Using muon-spin rotation the pressure-induced superconductivity in the Bi-III phase of elemental Bismuth (transition temperature $T_{\rm c}\simeq7.05$ K) was investigated. The Ginzburg-Landau parameter $\kappa=\lambda/\xi=30(6)$ ($\lambda$ is the magnetic penetration depth, $\xi$ is the coherence length) was estimated which is the highest among single element superconductors. The temperature dependence of the superconducting energy gap [$\Delta(T)$] reconstructed from $\lambda^{-2}(T)$ deviates from the weak-coupled BCS prediction. The coupling strength $2\Delta/k_{\rm B}T_{\rm c}\simeq 4.34$ was estimated thus implying that Bi-III stays within the strong coupling regime. The Density Functional Theory calculations suggest that superconductivity in Bi-III could be described within the Eliashberg approach with the characteristic phonon frequency $\omega_{\rm ln}\simeq 5.5$ meV. An alternative pairing mechanism to the electron-phonon coupling involves the possibility of Cooper pairing induced by the Fermi surface nesting., Comment: 5 pages, 4 figures

Zero-field and transverse-field muon spin rotation/relaxation ($\mu$SR) experiments were undertaken in order to elucidate microscopic properties of a strongly-coupled superconductor Mo$_{8}$Ga$_{41}$ with $T_{\text{c}}=9.8$ K. The upper critical field extracted from the transverse-field $\mu$SR data exhibits significant reduction with respect to the data from thermodynamic measurements indicating the coexistence of two independent length scales in the superconducting state. Accordingly, the temperature-dependent magnetic penetration depth of Mo$_{8}$Ga$_{41}$ is described using the model, in which two s-wave superconducting gaps are assumed. The V for Mo substitution in the parent compound leads to the complete suppression of one superconducting gap, and Mo$_{7}$VGa$_{41}$ is well described within the single s-wave gap scenario. The reduction in the superfluid density and the evolution of the low-temperature resistivity upon the V substitution indicate the emergence of a competing state in Mo$_{7}$VGa$_{41}$ that may be responsible for the closure of one of the superconducting gaps.

The results of the Fe isotope effect (Fe-IE) on the transition temperature $T_c$ obtained up to date in various Fe-based high temperature superconductors are summarized and reanalyzed by following the approach developed in [Phys. Rev. B 82, 212505 (2010)]. It is demonstrated that the very controversial results for Fe-IE on $T_c$ are caused by small structural changes occurring simultaneously with the Fe isotope exchange. The Fe-IE exponent on $T_c$ [$\alpha_{\rm Fe}=-(\Delta T_c/T_c)/(\Delta M/M)$, $M$ is the isotope mass] needs to be decomposed into two components with the one related to the structural changes ($\alpha_{\rm Fe}^{\rm str}$) and the genuine (intrinsic) one ($\alpha_{\rm Fe}^{\rm int}$). The validity of such decomposition is further confirmed by the fact that $\alpha_{\rm Fe}^{\rm int}$ coincides with the Fe-IE exponent on the characteristic phonon frequencies $\alpha_{\rm Fe}^{\rm ph}$ as is reported in recent EXAFS and Raman experiments., Comment: 7 pages, 4 figures. The paper is partially based on the results published in [New J. Phys. 12, 073024 (2010) = arXiv:1002.2510] and [Phys. Rev. B 82, 212505 (2010) = arXiv:1008.4540]

The magnetic order induced by the pressure was studied in single crystalline FeSe by means of muon-spin rotation ($\mu$SR) technique. By following the evolution of the oscillatory part of the $\mu$SR signal as a function of angle between the initial muon-spin polarization and 101 axis of studied crystal it was found that the pressure induced magnetic order in FeSe corresponds either to the collinear (single-stripe) antiferromagnetic order as observed in parent compounds of various FeAs-based superconductors or to the Bi-Collinear order as obtained in FeTe system, but with the Fe spins turned by 45$^{\rm o}$. The value of the magnetic moment per Fe atom was estimated to be $\simeq 0.13-0.14$~$\mu_{\rm B}$ at $p\simeq 1.9$~GPa., Comment: 4 pages, 4 figures. The Supplemental Information is available upon request

The temperature-pressure phase diagram of the ferromagnet LaCrGe$_3$ is determined for the first time from a combination of magnetization, muon-spin-rotation and electrical resistivity measurements. The ferromagnetic phase is suppressed near $2.1$~GPa, but quantum criticality is avoided by the appearance of a magnetic phase, likely modulated, AFM$_Q$. Our density functional theory total energy calculations suggest a near degeneracy of antiferromagnetic states with small magnetic wave vectors $Q$ allowing for the potential of an ordering wave vector evolving from $Q=0$ to finite $Q$, as expected from the most recent theories on ferromagnetic quantum criticality. Our findings show that LaCrGe$_3$ is a very simple example to study this scenario of avoided ferromagnetic quantum criticality and will inspire further study on this material and other itinerant ferromagnets.

Metal-to-insulator transitions (MITs) are a dramatic manifestation of strong electron correlations in solids1. The insulating phase can often be suppressed by quantum tuning, i.e. varying a nonthermal parameter such as chemical composi- tion or pressure, resulting in a zero-temperature quantum phase transition (QPT) to a metallic state driven by quantum fluctuations, in contrast to conventional phase transitions driven by thermal fluctuations. Theories of exotic phenomena known to occur near the Mott QPT such as quantum criticality and high-temperature superconductivity often assume a second-order QPT, but direct experimental evidence for either first- or second-order behavior at the magnetic QPT associated with the Mott transition has been scarce and further masked by the superconducting phase in unconventional superconductors. Most measurements of QPTs have been performed by volume-integrated probes, such as neutron scattering, magnetization, and transport, in which discontinuous behavior, phase separation, and spatially inhomogeneous responses are averaged and smeared out, leading at times to misidentification as continuous second-order transitions. Here, we demonstrate through muon spin relaxation/rotation (MuSR) experiments on two archetypal Mott insulating systems, composition-tuned RENiO3 (RE=rare earth element) and pressured-tuned V2O3, that the QPT from antiferromagnetic insulator to paramagnetic metal is first-order: the magnetically ordered volume fraction decreases to zero at the QPT, resulting in a broad region of intrinsic phase separation, while the ordered magnetic moment retains its full value across the phase diagram until it is suddenly destroyed at the QPT. These findings call for further investigation into the role of inelastic soft modes and the nature of dynamic spin and charge fluctuations underlying the transition., Comment: Main text: 9 pages, 3 figures; Supplementary Information: 2 pages, 2 figures