Table of contents

To a particle physicist a muon is a member of the lepton family, a heavy electron possessing a mass of about 1/9 that of a proton and a spin of 1/2, which interacts with surrounding atoms and molecules electromagnetically. Since its discovery in 1937, the muon has been put to many uses, from tests of special relativity to deep inelastic scattering, from studies of nuclei to tests of weak interactions and quantum electrodynamics, and most recently, as a radiographic tool to see inside heavy objects and volcanoes. In 1957 Richard Garwin and collaborators, while conducting experiments at the Columbia University cyclotron to search for parity violation, discovered that spin-polarized muons injected into materials might be useful to probe internal magnetic fields. This eventually gave birth to the modern field of μSR, which stands for muon spin rotation, relaxation or resonance, and is the subject of this special issue of Journal of Physics: Condensed Matter.

Muons are produced in accelerators when high energy protons (generally >500 MeV) strike a target like graphite, producing pions which subsequently decay into muons. Most experiments carried out today use relatively low-energy (~4 MeV), positively-charged muons coming from pions decaying at rest in the skin of the production target. These muons have 100% spin polarization, a range in typical materials of about 180 mg cm^-2, and are ideal for experiments in condensed matter physics and chemistry. Negatively-charged muons are also occasionally used to study such things as muonic atoms and muon-catalysed fusion. The μSR technique provides a local probe of internal magnetic fields and is highly complementary to inelastic neutron scattering and nuclear magnetic resonance, for example. There are four primary μSR facilities in the world today: ISIS (Didcot, UK), KEK (Tsukuba, Japan), PSI (Villigen, Switzerland) and TRIUMF (Vancouver, Canada), serving about 500 researchers world-wide. A new facility, JPARC (Tokai, Japan), is currently being built to replace the current Japanese μSR capability at KEK. These μSR institutions provide scientists a variety of sample environments, including a range of temperatures, magnetic fields and applied pressure. In addition, very low-energy muon beams (< 1 keV) have been developed for studies of thin films and nano-materials. In 2002 this world-wide community founded the International Society of μSR Spectroscopy (http://musr.org/~isms/) in order to promote the health of this growing field of research.

The 20 papers presented in this volume are intended to highlight some of the current μSR research activities of interest to condensed matter physicists. It is not an exhaustive review. In particular, the active and exciting area of muonium chemistry is left to a future volume. The group of papers in section I addresses the physics of strongly correlated electrons in solids, one of the most active fields of condensed matter research today. Strong electron correlations arise from (Coulomb) interactions which render Landau's theory of electron transport for weakly interacting systems invalid. Included in this category are unconventional heavy-fermion superconductors, high-temperature copper-oxide superconductors, non-Fermi liquid (NFL) systems and systems with strong electron-lattice-spin coupling, such as the colossal magnetoresistance manganites. Two key properties often make the muon a unique probe of these materials: (1) the muon's large magnetic moment (~3 μ_p) renders it extremely sensitive to the tiny magnetic fields (~1 Gauss) found, for example, in many NFL systems and in superconductors possessing time-reversal-violating order parameters, and (2) the muon's spin 1/2 creates a simple μSR lineshape (no quadrupolar coupling), ideal for measuring spin-lattice-relaxation, local susceptibilities and magnetic-field distributions in ordered magnets and superconductors. Section II contains studies which exploit the unique sensitivities of μSR just noted to elucidate new and hidden properties of novel magnetic materials, including the use of very low energy muons to study thin films. Sections I and II are concerned with the bare positive muon as a probe of internal magnetic fields in metals. The papers in section III describe studies which exploit the fact that in semiconductors the muon appears as a light isotope of the hydrogen atom, called muonium. These studies provide important new information regarding the electronic structure and motion of light, dilute hydrogen-like impurities in semiconductors, which is useful for both semiconductor fundamentals and applications. Finally, in section IV experiments which probe electron transfer in large molecular systems are presented, including future prospects for investigating some materials of biological interest. These latter experiments exploit the sensitivity of muonium to the motion of electrons inside molecular systems, the so-called `labeled-electron' method.

It is our hope that even this limited perspective shows the extraordinary degree to which the μSR technique is contributing deeply to our understanding of condensed matter physics. Richard Garwin's early experiments have indeed borne unexpected fruit!

The editor is very grateful to all the invited authors for their timely contributions to this special section of Journal of Physics: Condensed Matter.

The behaviour of the so-called weak-moment antiferromagnetic states, observed in the heavy-fermion superconductors UPt₃ and URu₂Si₂, is discussed in view of recent μSR results obtained as functions of control parameters such as chemical substitution and external pressure. In UPt₃, the Pd substitution for Pt reveals the dynamical character of the weak-moment order. On the other hand, μSR measurements performed on samples in which Th substitutes U suggest that crystallographic disorder on the magnetic sites deeply affects the fluctuation timescale. In URu₂Si₂, a phase separation between the so-called hidden order state, present at ambient pressure, and an antiferromagnetic state, occurring under pressure, is observed. In view of the pressure–temperature phase diagram obtained by μSR, it is deduced that the respective order parameters have different symmetries.

We show that the field dependence of the magnetic penetration depth (λ), for which muon spin rotation (μSR) is an excellent microscopic probe, provides useful information on the degree of anisotropy of the superconducting order parameter. In type II superconductors associated with anisotropic order parameters, λ is sensitive to the quasiparticle excitation induced by the Doppler shift due to a supercurrent around magnetic vortices. The presence of such low energy excitations manifests itself in the non-zero slope of λ against an external magnetic field. We review recent results on the field dependence of λ obtained from the application of μSR to novel superconductors that exhibit unconventional characters associated with the anisotropic order parameter.

Muon spin rotation (μSR) studies of the oxygen isotope (¹⁶O/¹⁸O) effect (OIE) on the in-plane magnetic field penetration depth λ_ab in cuprate high-temperature superconductors (HTS) are presented. First, the doping dependence of the OIE on the transition temperature T_c in various HTS is briefly discussed. It is observed that different cuprate families show similar doping dependences of the OIE on T_c. Then, bulk μSR, low-energy μSR, and magnetization studies of the total and site-selective OIE on λ_ab are described in some detail. A substantial OIE on λ_ab was observed in various cuprate families at all doping levels, suggesting that cuprate HTS are non-adiabatic superconductors. The experiments clearly demonstrate that the total OIE on T_c and λ_ab arise from the oxygen sites within the superconducting CuO₂ planes, demonstrating that the phonon modes involving the movement of planar oxygen are dominantly coupled to the supercarriers. Finally, it is shown that the OIE on T_c and λ_ab exhibit a relation that appears to be generic for different families of cuprate HTS. The observation of these unusual isotope effects implies that lattice effects play an essential role in cuprate HTS and have to be considered in any realistic model of high-temperature superconductivity.

In this work we present µSR experiments to examine spin stripe order in layered nickelate and cuprate systems. We discuss the signature of static spin stripe order in a zero field (ZF) µSR experiment. In the nickelate La_1.67Sr_0.33NiO₄ well separated signals from the antiferromagnetic (AF) domains and the domain walls are identified, whereas in Nd and Eu doped La_2−xSr_xCuO₄ one asymmetrically broadened line is found. In both systems we observe slow fluctuations of the charge stripes. By changing the rare-earth and Sr doping levels in La_2−x−yRE_ySr_xCuO₄ we prove the strong correlation of static spin stripe order with the structural distortion in the low temperature tetragonal (LTT) phase, and demonstrate the competition of static spin stripe order with bulk superconductivity. Finally, evidence for a similar magnetic inhomogeneity in the electron doped high-T_C cuprate Pr_2−xCe_xCuO₄ is presented.

Muon spin rotation and relaxation (μSR) experiments have yielded evidence that structural disorder is an important factor in many f-electron-based non-Fermi-liquid (NFL) systems. Disorder-driven mechanisms for NFL behaviour are suggested by the observed broad and strongly temperature-dependent μSR (and NMR) linewidths in several NFL compounds and alloys. Local disorder-driven theories (Kondo disorder, Griffiths–McCoy singularity) are, however, not capable of describing the time-field scaling seen in muon spin relaxation experiments, which suggest cooperative and critical spin fluctuations rather than a distribution of local fluctuation rates. A strong empirical correlation is established between electronic disorder and slow spin fluctuations in NFL materials.

Muon spin rotation (µSR) has emerged as the leading experimental probe of the effective size of magnetic vortices in type-II superconductors. µSR data on several different classes of type-II superconductors show that the inner structure of a vortex can depend quite strongly on temperature and the strength of the external magnetic field. In this paper it is shown that these behaviours are related to the quasiparticle excitation spectrum both inside and outside of the vortex cores. Here we establish that the vortex core size determined by means of µSR is particularly sensitive to the nature of the superconducting energy gap(s) and the symmetry of the Fermi surface. A survey of results for different superconductors arrives at the conclusion that the large vortex core size observed in YBa₂Cu₃O_7−δ is due to CuO chain superconductivity.

To find a primary factor determining T_c and a pairing mechanism in high-T_c cuprates, we combine the muon spin relaxation results on n_s/m^* (superconducting carrier density/effective mass), accumulated over the last 17 years, with the results from neutron and Raman scattering, scanning tunnelling microscopy, specific heat, Nernst effect, and angle-resolved photoemission spectroscopy measurements. We identify the neutron magnetic resonance mode as an analogue of the roton minimum in the superfluid ⁴He, and argue that n_s/m^* and the resonance mode energy play a primary role in determining T_c in the underdoped region. We propose a picture wherein roton-like excitations in the cuprates appear as a coupled mode, which has resonance modes for spin and charge responses at different momentum transfers but the same energy transfer, as detected respectively by means of the neutron S = 1 mode and the Raman S = 0 A1_g mode. We shall call this the 'hybrid spin/charge roton'. After discussing the role of dimensionality in condensation, we propose a generic phase diagram for the cuprates with spatial phase separation in the overdoped region as a special case of the Bose–Einstein to Bardeen–Cooper–Schrieffer crossover conjecture where the superconducting coupling is lost rapidly in the overdoped region. Using a microscopic model of charge motion resonating with antiferromagnetic spin fluctuations, we propose the possibility that the hybrid spin/charge roton and higher-energy spin fluctuations mediate the superconducting pairing. In this model, the resonance modes can be viewed as a meson analogue and the 'dome' shape of the phase diagram can be understood as a natural consequence of departure from the competing Mott insulator ground state via carrier doping.

We report a series of measurements in (La, Ca, Pr)MnO₃ compounds which show that ferromagnetic (FM) CMR compounds develop two Mn-ion spin–lattice relaxation channels as the materials are cooled below their insulator-to-metal transition temperature T_MI. This result is in contrast to conventional FMs and is attributed to the presence of both insulating and conducting FM regions below T_MI which coexist on a microscopic scale. The coexistence of different phases is found in both polycrystalline and single-crystalline materials, though the single crystal exhibits a narrower temperature region of phase coexistence below its FM critical temperature T_C. Possible differences between crystalline and granular materials which could give rise to these findings are discussed. These results could have important implications for the use of CMR materials in spintronics devices, which rely on conducting surfaces in thin film multilayers below T_C.

We present the results of muon spin relaxation (μ⁺SR) studies on low dimensional molecular magnet systems. μ⁺SR measurements have been carried out on the Cu-based chain compounds CuX₂(pyz) (where X = Br, Cl, NCS and pyz = pyrazine) as a function of temperature and applied longitudinal magnetic field. Oscillations in the time dependence of the muon polarization, characteristic of magnetic order at two distinct muon sites, are detected in both CuBr₂(pyz) (below T_N = 3.6(1) K) and CuCl₂(pyz) (below T_N = 3.2(2) K). No evidence of magnetic order is observed in Cu(NCS)₂(pyz) down to 0.35 K. The results are discussed in terms of the estimated Cu–X–Cu and Cu–(pyz)–Cu exchange constants. The theory of μ⁺SR in high spin molecule (HSM) systems, which are effectively zero-dimensional magnets, is discussed and results are presented on [Ni₁₂(chp)₁₂(O₂CMe)₁₂(H₂O)₆(THF)₆] (S = 12), [Mn₉O₇(OAc)₁₁(thme)(py)₃(H₂O)₂] (S = 17/2) and [Fe₁₄(bta)₆(O)₆(OMe)₁₈ Cl₆] (). Measurements made in applied longitudinal magnetic fields on HSM materials at dilution refrigerator temperatures strongly suggest that dynamic local magnetic field fluctuations are responsible for the relaxation of the muon spin ensemble. Trends in temperature and field dependent behaviour in these systems, as probed by the muon, are discussed.

Muons with 100% spin polarization and whose energy can be continuously varied from 0.5 to 30 keV provide a novel extension of the μSR technique allowing depth dependent studies of thin films and multilayered structures in the range from to  nm. For example it is possible to study magnetic field profiles near the surface of superconductors and directly determine fundamental quantities such as the magnetic penetration depth. Low energy muons (LE-μ⁺) also allow mapping the spin polarization in multilayered structures, for instance that of the conduction electrons of a non-magnetic buried layer in between thin ferromagnetic layers.

This paper describes the impact of dynamic spin fluctuations on the muon spin relaxation signal in the longitudinal field set-up, namely, when a field is applied along the initial muon spin direction. Our main objective is to show that the μSR technique can do more than determine the correlation time of the spins in the system under investigation. It can, in fact, determine if the concept of correlation time is valid to begin with, and if not suggest alternatives. Consequently, the paper shows what to expect from the muon signal over a range of situations starting from a simple antiferromagnetic hopping model to more complicated models involving power laws and other types of correlation functions. The application of all models to experimental data is demonstrated. The possibility that the muon, by itself, generates dynamic fluctuations is critically examined by comparing muon and neutron scattering data.

The exquisite sensitivity of zero-field muon spin relaxation to both static magnetic order and dynamic fluctuations has been used to study the ordering processes in a variety of exchange frustrated magnetic materials. Two distinct ordering events are identified, and they each clearly show both dynamic and static signatures. Phase diagrams for two classes of frustration are presented. We also provide an extensive discussion of both approximate and complete solutions to dynamic fitting functions, and show that the correct functions used in our analysis yield more accurate fits and quantitative agreement with independent measurements on the same samples.

A review is presented of µSR measurements on CeB₆, Ce_1−xLa_xB₆, CeAg, PrCu₂, HoB₂C₂, DyPd₃S₄ and UPd₃ which reveal multipolar, in particular quadrupolar, effects. μ⁺ Knight shift data imply that the spin polarization of conduction electrons at the μ⁺ may acquire an unusual temperature dependence and anisotropy in the presence of a non-spherical charge distribution of the f-electrons. This feature is also discussed theoretically. The interplay of dipolar magnetic and electric quadrupolar order is shown to possibly explain properties of the spontaneous fields at the μ⁺ in the magnetically ordered state, as well as aspects of the μ⁺ spin lattice relaxation. The possibility of magnetic-octupolar effects is briefly considered.

We first discuss magnetic excitation and relaxation processes which have already been recognized experimentally by muon spin relaxation experiments. Our examples are taken from experiments performed on strongly correlated electronic systems, magnetic metals and geometrically frustrated magnetic compounds. In another part, focusing on the flux-line lattice in the Bragg-glass phase of superconductors, we show that a muon spin experiment can provide information on the in-plane correlation length of the lattice.

The discovery and significance of weakly bound muonium states with low hyperfine constants in a number of compound semiconductors of the II–VI and III–V (nitride) families are briefly reviewed. With ionization energies of several tens of meV, these imply that their hydrogen counterparts would act as shallow donors and that hydrogen could, either as an impurity or a deliberate dopant, be a source of electronic conductivity in the relevant materials. We examine whether, in their neutral undissociated states, the electron orbitals can be described in the effective-mass approximation and are correspondingly dilated, made up of conduction-band states. The best evidence that this is so comes from novel double-resonance measurements of the electron g-factors, devised for the ISIS pulsed muon source, and so far undertaken for ZnO, CdS, CdSe and CdTe. The respective values are |g| = 1.97, 1.86, 0.51 and 1.68; these results discount orbitally quenched compact states and are fully consistent with literature values for known shallow dopants in these compounds. They also illustrate the potential for µSR detection and characterization of such states in new electronic materials where hydrogen-induced conductivity is suspected or predicted.

Sites and dynamics obtained for muonium (Mu) defect centres provide a good experimental model for the behaviour of the equivalent hydrogen impurities. We discuss the dynamic properties of Mu centres in the III–V nitrides focusing on features common to the three materials. Muon spin depolarization data in zero magnetic field provide motional dynamics for the Mu⁺ and Mu⁻ charge states and field dependent longitudinal relaxation rates probe motion of Mu⁰ centres. The data also show dynamics associated with metastable locations, either intrinsic to the wurtzite structure or defect related, including trapping and release transitions. A general picture of the behaviour of H in the III–V nitrides is developed from these measurements for comparison to theoretical results.

At temperatures above 600 K in silicon, unlike at lower temperatures, the partitioning of muonium between its neutral paramagnetic states and its charged or electronically diamagnetic states corresponds closely to thermodynamic equilibrium. The individual charge states are short lived, with many cycles of carrier capture and release occurring within the muon lifetime. The resultant intermittent hyperfine interaction depolarizes the muons strongly, with longitudinal and transverse relaxation rates remaining distinct up to about 700 K but becoming equal at still higher temperatures. Data up to 900 K are presented and interpreted. The muon spin rotation spectrum in transverse magnetic fields, although collapsed to a single broad line in this charge exchange regime, is shifted substantially from the muon Larmor frequency, the shift being non-linear in field and only in small part due to electron polarization. A new density matrix treatment shows how all three observables can be accounted for with a consistent set of transition rates. These in turn may be interpreted in terms of effective donor and acceptor energy levels appropriate to this high-temperature regime, confirming negative-U behaviour and providing the first estimate, for muonium, of this elusive parameter. At temperatures where passivation complexes are dissociated, these findings provide a guide to, and microscopic models for, the electrical activity of hydrogen.

Various processes of muonium atom formation in semiconductors via electron capture by a positive muon have been studied using μSR techniques, including those with applied electric field. Experiments in GaAs, GaP and CdS suggest that the electron is initially captured into a highly excited state, from which the cascade down to the muonium ground state goes through an intermediate weakly bound state determined by the electron effective mass and the dielectric constant of the host. The electronic structure of this weakly bound state is shown to be hydrogenic. The nature of the final (on the μSR timescale) muonium state depends on the energy releasing mechanisms in the cascade process. We suggest that muonium dynamics in semiconductors (including the effects of electric and magnetic fields and temperature) reflect the electron dynamics in weakly bound muonium state(s) in which the electron is delocalized over distances of about 100 Å.

The use of implanted muons to probe the dynamics of electronic excitations in conducting polymers is reviewed. Early work on polyacetylene showed evidence for mobile solitons performing one-dimensional diffusion in the trans isomer and localized spins in the cis isomer. Subsequent muon studies on a range of conducting polymers have shown evidence for mobile polaronic excitations and microscopic transport properties for these polarons have been derived from the measurements. A theoretical framework was developed by Risch and Kehr to describe the intermittent hyperfine coupling between a static muon and an electron diffusing randomly through a chain of sites. This theory predicts a specific form for both the muon spin relaxation function and the field dependence of the relaxation rate. The experimental data are found to be described well by this model. Intrachain diffusion rates can be extracted from the data; in several cases an interchain diffusion rate can also be measured. The anisotropy of diffusion rates can be as high as 10⁴ at low temperatures, reducing typically to 10² or less at room temperature. The importance of molecular vibrational modes in controlling the electronic motion in the polymer has been shown.

Electron-transfer phenomena in biological macromolecules are among the most important processes of life science. So far, a limited number of microscopic studies exist, and most knowledge is based on macroscopic studies. In order to overcome this situation, a labelled electron method with positive muons was recently developed and successfully applied to directly explore microscopic electron-transfer phenomena in representative proteins, such as cytochrome c, myoglobin and cytochrome c oxidase, and DNA. The principle, some details of each experiment and future perspectives are described.