Kubo-Toyabe Relaxation Research Articles

Electronic structure of interstitial hydrogen in In-Ga-Zn-O semiconductor simulated by muon

We report on the local electronic structure of an interstitial muon (Mu) as pseudohydrogen in In-Ga-Zn oxide (IGZO) semiconductors studied by muon spin rotation/relaxation (μSR) experiment. In polycrystalline (c-) IGZO, it is inferred that Mu is in a diamagnetic state, where the μSR time spectra under zero external field are perfectly described by the Gaussian Kubo-Toyabe relaxation function with the linewidth Δ serving as a sensitive measure for the random local fields from In/Ga nuclear magnetic moments. The magnitude of Δ combined with the density functional theory calculations for H (to mimic Mu) suggests that Mu occupies Zn-O bond-center site (MuBC) similar to the case in crystalline ZnO. This implies that the diamagnetic state in c-IGZO corresponds to MuBC+, thus serving as an electron donor. In amorphous (a-) IGZO, the local Mu structure in the as-deposited films is nearly identical to that in c-IGZO, suggesting MuBC+ for the electronic state. In contrast, the diamagnetic signal in heavily hydrogenated a-IGZO films exhibits the Lorentzian Kubo-Toyabe relaxation, implying that Mu accompanies more inhomogeneous distribution of the neighboring nuclear spins that may involve a Mu− H−-complex state in an oxygen vacancy.

μSR study of stoichiometric NbFe2

The magnetic ground state of nominally stoichiometric single crystalline NbFe2 is investigated by bulk magnetisation and muon spin relaxation techniques. Magnetic order clearly emerges below the critical temperature TN=10.3K and is dominated by randomly orientated quasi-static moments. The local field distribution observed by muons can be explained by the phenomenological Gaussian-broadened-Gaussian Kubo Toyabe relaxation function. The observed short range order could be used to describe a new magnetic ground state, but a helical spin density wave with an incommensurate amplitude modulation cannot be ruled out. The sensitivity of μSR to the local magnetic field distribution in the vicinity of the quantum critical point (QCP) in NbFe2 is clearly demonstrated via comparison with already published work. This suggests detailed measurements of the muon relaxation as the QCP is approached will reveal further details of the field distribution and fluctuations in Nb1−yFe2+y.

A μSR study of the ruthenium perovskites ACu3Ru4O12 with A = Ca, Pr, Nd

The metallic ruthenium perovskites ACu3Ru4O12 with A = Ca, Pr,Nd were investigated by zero field (ZF) and weak transverse field (TF) muon spin rotation/relaxation (μSR) spectroscopy. The ZF spectra for CaCu3Ru4O12 show pure static Gaussian Kubo-Toyabe relaxation arising from the interaction between the muon spin and the nuclear moments of the Cu ions. This confirms that no atomic magnetic moment exists. A sudden increase of the lattice parameter a when heating above ~150 K had previously been detected by neutron diffraction. The root mean square field at the muon site Brms was found to be 0.15 mT independent of temperature, in particular around 150 K. Also, no change of spectral parameters is seen in the weak TF data in that temperature range. Those findings imply that the structural change around 150 K takes place without noticeably shifting atomic positions. The spectra for PrCu3Ru4O12 and NdCu3Ru4O12 are dominated by the interaction with the dynamic rare earth moments. The analysis requires the use of the electron-nuclear double relaxation formalism. The electronic part shows simple exponential relaxation typical for a paramagnet. It features, with reducing the temperature a steep increase of paramagnetic relaxation rate as is characteristic for an approach to a magnetic spin freezing transition from above. That result suggests that the magnetic ground states of PrCu3Ru4O12 and NdCu3Ru4O12 are spin frozen states, although bulk magnetic data give no direct evidence in that direction.

Low field magnetic response of the non-centrosymmetric superconductor YPtBi

The low-field magnetic response of the non-centrosymmetric superconductor YPtBi (Tc=0.77K) is investigated. Ac-susceptibility and dc-magnetization measurements provide solid evidence for bulk superconductivity with a volume fraction of ~70%. The lower critical field is surprisingly small: Bc1=0.008mT (T→0). Muon spin rotation experiments in a transverse magnetic field of 0.01T show a weak increase of the Gaussian damping rate σTF below Tc, which yields a London penetration depth λ=1.6±0.2μm. The zero-field Kubo–Toyabe relaxation rate σKT equals 0.129±0.004μs−1 and does not show a significant change below Tc. This puts an upper bound of 0.04mT on the spontaneous magnetic field associated with a possible odd-parity component in the superconducting order parameter.

Weak quasistatic magnetism in the frustrated Kondo lattice Pr2Ir2O7

Abstract Muon spin relaxation experiments have been performed in the pyrochlore iridate Pr2Ir2O7 for temperatures in the range 0.025–250 K. Kubo–Toyabe relaxation functions are observed up to ≳ 200 K , indicating static magnetism over this temperature range. The T → 0 static muon spin relaxation rate Δ ( 0 ) ≈ 8 μ s - 1 implies a weak quasistatic moment ( ∼ 0.1 μ B ) . The temperature dependence of Δ is highly nonmean-field-like, decreasing smoothly by orders of magnitude but remaining nonzero for T T f . The data rule out ordering of the full Pr 3 + CEF ground-state moment ( 3.0 μ B ) down to 0.025 K. The weak static magnetism is most likely due to hyperfine-enhanced 141Pr nuclear magnetism. The dynamic relaxation rate λ increases markedly below ∼ 20 K , probably due to slowing down of spin fluctuations in the spin-liquid state. At low temperatures λ is strong and temperature-independent, indicative of a high density of low-lying spin excitations as is common in frustrated antiferromagnets.

U 0.94Y 0.06CoAl: a dilute-moment ferromagnet

Hexagonal UCoAl is considered to be a 5f-itinerant metamagnet, with no magnetic ordering in ZF and a rather low critical field of about 0.8 T at temperatures below ∼15 K . Low-field muon spin relaxation in it is slow and temperature independent. When small amounts of yttrium are alloyed on the uranium site, bulk magnetization finds weak ferromagnetism with T C ≃12 K . TF, ZF and LF-μSR data have been obtained for the composition of strongest ferromagnetism, U 0.94Y 0.06CoAl. The data show field-history effects, such as remanence (which is expected for a ferromagnet), at 12 K and below. In zero-field cooling, however, the data also show clear Lorentzian Kubo–Toyabe relaxation in ZF and LF at 12 K and below. The behavior is consistent with a dilute-moment system that manages to achieve ferromagnetic ordering. The Lorentzian relaxation-function shape is not consistent with a moment density as large as the yttrium concentration, so describing the mechanism by which the yttrium causes the moments becomes a challenging problem.

Motional properties of positive muonium in gallium III–V compounds

The motion of Mu + defect centers is investigated in heavily Zn-doped p-type GaAs, GaP, and GaSb using zero-field muon spin depolarization. Hop rates extracted from dynamic Kubo–Toyabe relaxation functions show activated motion above 200 K and changes in site or dynamics within 50 K of the onset. Barrier energies are extracted for the mobile centers. A second Mu + state at T V sites is suggested by these results. Additional diamagnetic states are present implying interactions with Zn acceptors involving mobile states, Mu T 0 at low T and Mu + at high T.

Toward Monte Carlo simulation of general cases of static muon spin relaxation in disordered magnetic materials: long-range magnetic order in alloys

Monte Carlo simulations of zero-field (ZF) muon spin relaxation (μSR) functions generated by long-range-ordered states with disorder are presented, for the completely static limit. Understanding of this is necessary before Monte Carlo simulation of the effect of short-range magnetic ordering on μSR in spin glasses can begin. Alloy disorder, controlled by the magnetic ion concentration parameter f m , and partial ordering of each moment, controlled by the order parameter f o, are considered. Qualitatively different behavior is seen depending on whether the dense moment, perfect-order limit ( f m=1 , f o=1) field at the muon site is non-zero, or cancels (as can happen in high-symmetry materials). Around the edges of the two-dimensional ( f m,f o ) parameter space, four limit cases with qualitatively different behavior are identified: (A) f o→0, the random frozen spin glass for arbitrary magnetic ion concentration; (B) f o→1, nearly perfect magnetic ordering in a alloy of arbitrary magnetic ion concentration; (C) f m →0, magnetic order developing (as f o increases) in a dilute magnetic alloy; (D) f m →1, magnetic order developing (as f o increases) in a dense magnetic material. Case A was discussed in a previous publication. The results for case D answer the question of how the Gaussian Kubo–Toyabe relaxation function for perfect disorder develops into an oscillating function as magnetic order develops in a material. Case C indicates that the effects of magnetic ordering in the dilute moment limit produce only subtle effects in ZF-μSR spectra that would be difficult to unambiguously identify as due to ordering in a real-world experiment. Case B generates complicated multi-frequency behavior.

Muon diffusion and relaxation in YCo 2

Muons were expected to gain additional information about the spin fluctuations proposed for this enhanced paramagnetic system. At the lowest temperatures the μ +SR signal can be explained by a static Kubo–Toyabe relaxation function. On raising the temperature an interplay is suggested to occur between muon diffusion and electronic relaxation of “Co-clusters” surrounding a Co vacancy yielding a maximum in the fluctuation rate versus temperature curve. This behaviour is similar to that found in the case of paramagnetic impurities in nonmagnetic metals, the “Co-cluster” moment being the analogue to the impurity spin.

Magnetism in heavy-fermion U(Pt,Pd)3 studied by SR

We report µSR experiments carried out on a series of heavy-electron pseudobinary compounds U(Pt1-xPdx)3 (x0.05). For x0.005 the zero-field muon depolarization is described by the Kubo-Toyabe function. However the temperature variation of the Kubo-Toyabe relaxation rate KT(T) does not show any sign of the small-moment antiferromagnetic phase with TN6 K (signalled by neutron diffraction), in contrast to previous reports. The failure to detect the small ordered moment suggests it has a fluctuating (>10 MHz) nature, which is consistent with the interpretation of NMR data. For 0.01 x0.05 the muon depolarization in the ordered state is described by two terms of equal amplitude: an exponentially damped spontaneous oscillation and a Lorentzian Kubo-Toyabe function. These terms are associated with antiferromagnetic order with substantial moments. The Knight shift measured in a magnetic field of 0.6 T on single-crystalline U(Pt0.95Pd0.05)3 in the paramagnetic state shows two signals for Bc, while only one signal is observed for Bc. The observation of two signals for Bc, while there is only one muon localization site (0,0,0), points to the presence of two spatially distinct regions of different magnetic response.

Anomalous zero-field muon spin relaxation in highly disordered magnets

Low-temperature zero-field muon spin relaxation spectra from several disordered magnetic materials show static-field relaxation behavior that cannot be properly represented by the Gaussian or Lorentzian distributions commonly used in the analysis of such data (Kubo-Toyabe relaxation). In the case of ${\mathrm{CeCu}}_{0.2}{\mathrm{Ni}}_{0.8}\mathrm{Sn}$, a Gaussian distribution of second moments, convoluted with a Gaussian form of field distribution, produces a closed-form modified Kubo-Toyabe relaxation function that fits the data well. Its characteristic feature is a shallow polarization minimum before recovery to the 1/3 asymptote. Any microscopic model that reproduces such shallow static Kubo-Toyabe relaxation must do so by generating an excess of low-field sites.

Muonium localization in solid krypton.

Muonium spin relaxation in zero, longitudinal, and transverse magnetic fields has been studied in solid and liquid krypton in the temperature range of 2 to 120 K. In the solid at low temperatures, the spin dynamics exhibit features characteristic of a magnetically dilute crystal, permitting measurements of exceptionally low muonium diffusion rates. At the lowest temperatures, the static Kubo-Toyabe relaxation function is found to describe muon spin relaxation in the atomic muonium state, indicating strong interstitial localization in the Kr lattice; muonium is determined to be localized at the tetrahedral interstitial position. At high temperatures, muonium diffusion in solid Kr exhibits a nonclassical behavior.

A generalized Kubo-Toyabe formula for muon spin relaxation in crystals with uniaxial symmetry

The Kubo-Toyabe semiclassical formula, describing the time development of the polarization of a particle in zero external field at a lattice site with cubic local environment, is generalized for uniaxial site symmetry. The relaxation function and, in particular, its first moments and long time asymptotics obtained in a closed form depend on the angleθ between polarization and the crystalc-axis and are shown to vary sensitively with the asymmetrye of the field distribution at the particular muon site. Besides the exact “uniaxial” variant of the Kubo-Toyabe relaxation function, an approximate simple interpolation formula is also derived, which is correct for both short times and in its long time asymptotics. The two parameters (e, Δ1) in the “uniaxial” formulae can be determined by using the observed values of the second momentM 2 for two different crystal orientations.

Zero-field μSR study of CeRu2Si2 heavy fermion compound

Zero — field muon spin relaxation measurements have been carried out on a high-quality single-crystal sample of the paramagnetic heavy-fermion compound CeRu2Si2 from 5K to 300K. The observed relaxation at each temperature was well fitted by a function described by the product of a Kubo-Toyabe type and an exponential one. The temperature dependence of the Kubo-Toyabe relaxation rate σZF shows an almost constant value (0.03 μs−1) which is comparable to the calculated linewidth due to nuclear dipole fields from99Ru and101Ru atoms assuming the possible μ+ stopping sites for the ThCr2Si2 structure. On the other hand, we observed an increase of the exponential relaxation rate λZF to a value of ∼- 0.02 μs−1 with decreasing temperature below about 100K. This suggests a contribution of Ce 4f moments.

Muon spin relaxation in hydrogen-loaded amorphous Ni-Ti alloys

Muon spin relaxation in the amorphous alloy Ni33Ti65 containing interstitial hydrogen has been measured in longitudinal geometry at zero applied magnetic field, at temperatures in the range 18-340 K. The relaxation profiles at all temperatures were consistent with time-dependent magnetic fields at the muon site and were analysed by means of a dynamic form of the Kubo-Toyabe relaxation theory. The correlation time of the local field at the muon below 80 K was found to be about 6 mu s and to vary only slowly with temperature. Above approximately 250 K, where the diffusion hopping rate of the hydrogen atoms is about 107-108 s-1, as measured by nuclear magnetic relaxation methods, the activation energy of the correlation time is similar to the activation energy of the hydrogen diffusion. The relaxation of the muon spin at these temperatures is ascribed to the highly correlated motion of the muon and the hydrogen atoms. Some Monte Carlo calculations of the fluctuation rates of the local fields under such conditions are reported. According to the outcome of these calculations the experimental data are consistent with the muon having an intrinsically lower diffusion rate than the hydrogen atoms.

Muon spin rotation in the magnetic and superconducting ground states of (U,Th)Be13 and (U,Th)Pt3.

Microscopic aspects of magnetism and superconductivity have been studied in the heavy-fermion superconducting alloy systems (U,Th)${\mathrm{Be}}_{13}$ and (U,Th)${\mathrm{Pt}}_{3}$ using the technique of positive-muon (${\ensuremath{\mu}}^{+}$) spin rotation and relaxation (\ensuremath{\mu}SR). In ${\mathrm{U}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Th}}_{\mathrm{x}}$${\mathrm{Be}}_{13}$, x=0.033, a striking increase of zero-field Kubo-Toyabe relaxation rate ${\ensuremath{\sigma}}_{\mathrm{KT}}$ is observed below the temperature ${T}_{c2}$\ensuremath{\simeq}0.4 K at which a second phase transition occurs in the superconducting state. This jump is firm evidence for the onset of weak static magnetism below ${T}_{c2}$. The observed increase of \ensuremath{\sim}1.5 Oe in the ${\ensuremath{\mu}}^{+}$ local field corresponds to an effective moment of ${10}^{\mathrm{\ensuremath{-}}3}$--${10}^{\mathrm{\ensuremath{-}}2}$ ${\ensuremath{\mu}}_{B}$/U atom.The ${\ensuremath{\mu}}^{+}$ Knight shift ${K}_{\ensuremath{\mu}}$ in the normal states of ${\mathrm{U}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Th}}_{\mathrm{x}}$${\mathrm{Be}}_{13}$, x=0 and 0.033, is proportional to the bulk susceptibility and yields a transferred f-spin-${\ensuremath{\mu}}^{+}$ hyperfine field of -1.99\ifmmode\pm\else\textpm\fi{}0.12 kOe/${\ensuremath{\mu}}_{B}$. In the superconducting states of ${\mathrm{U}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Th}}_{\mathrm{x}}$${\mathrm{Be}}_{13}$, x=0, 0.01, and 0.033, a large reduction of the magnitude of ${K}_{\ensuremath{\mu}}$ is observed for x=0 but not for x=0.033, and an intermediate reduction is found for x=0.01. This behavior suggests (1) spin-singlet pairing in undoped ${\mathrm{UBe}}_{13}$, and (2) conventional reduction of the spin susceptibility suppression in (U,Th)${\mathrm{Be}}_{13}$ by spin-orbit scattering from Th impurities. In undoped ${\mathrm{UPt}}_{3}$ small increases of ${\ensuremath{\sigma}}_{\mathrm{KT}}$ and the transverse-field Gaussian relaxation rate ${\ensuremath{\sigma}}_{G}$ below \ensuremath{\sim}6 K are again evidence for the onset of weak static magnetism similar to that found in (U,Th)${\mathrm{Be}}_{13}$. Recent spin-polarized neutron scattering studies of normal-state ${\mathrm{UPt}}_{3}$ have observed a commensurate antiferromagnetic (AF) structure with a sublattice magnetization of \ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}2}$ ${\ensuremath{\mu}}_{B}$/U atom, in rough agreement with the \ensuremath{\mu}SR results. For ${\mathrm{U}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Th}}_{\mathrm{x}}$${\mathrm{Pt}}_{3}$, x=0.05, zero-field ${\ensuremath{\mu}}^{+}$ precession frequencies below 6.5 K indicate an AF structure with a considerably larger moment (\ensuremath{\sim}0.6 ${\ensuremath{\mu}}_{B}$/U atom). The average ${\ensuremath{\mu}}^{+}$ Knight shift in the normal state of undoped ${\mathrm{UPt}}_{3}$ is much smaller than in ${\mathrm{UBe}}_{13}$, which suggests multiple ${\ensuremath{\mu}}^{+}$ stopping sites with hyperfine couplings of opposite signs.

Long-Time Spin Relaxation in Zero-Field

The long-time tail of the Kubo-Toyabe relaxation function of spin polarization in zero-field is examined by a variational method and a numerical simulation. The asymptotic exponential decay was found to be of the form, 0.2362 exp (-1.562 v t ), where v is the decay constant of the perturbing Gaussian random field. This decay constant is shown to be determined by the lowest eigenvalue of a confluent hypergeometric differential equation.

Observation of an oscillatory behavior of the zero-field μ + spin relaxation function in pure copper

High precision measurements were carried out on the zero field (ZF) spin relaxation function of μ + in copper at 14.8 K, where μ + is nearly localized. The time spectrum obtained shows an oscillatory deviation from the dynamical Kubo-Toyabe relaxation function. The deviation seems to come mainly from the distribution of local field different from a gaussian distribution.

Skewed-field technique in muon-spin rotation: Kubo-Toyabe longitudinal and transverse relaxation functions.

For any arbitrary external-field direction, three distinct muon-spin relaxation functions can be defined; that is, there is a longitudinal relaxation function in the direction of the external field and two transverse relaxation functions in the plane perpendicular to the field direction. One of these latter functions, termed the coplanar transverse relaxation function, lies in the plane defined by the field and the incoming muon-spin polarization, while the other, termed the perpendicular transverse relaxation function, is perpendicular to both the field direction and the incoming spin polarization. All three relaxation functions can be measured simultaneously if the applied external field is not in either of the standard geometries, namely longitudinal or transverse. This suggests that a skewed-field arrangement provides an experimental technique in which the traditional relaxation functions for a given sample may be determined in a single experiment with the use of one apparatus. Thus, at the very least, such an experimental alignment eliminates the down time required when changing from one geometry to the other in the determination of the relaxation functions for a given sample. With the use of the static Kubo-Toyabe theory as an illustrative example, explicit expressions for these relaxation functions are obtained in terms of a pair of one-dimensional integrals. In particular, an alternate, but equal, expression for the standard longitudinal Kubo-Toyabe relaxation function is obtained. An analytic expression for the coplanar transverse Kubo-Toyabe relaxation function is presented which agrees with the numerical results derived by Kubo, while an analytic expression for the new perpendicular transverse Kubo-Toyabe relaxation function is given.