Negatively Charged Muonium and Related Centers in Solids

Computational studies on H impurities in semiconductors have revealed that correlations exist between their crystallographic sites and favorable charge states (Fig. 2).[26] However, it is difficult to identify the structures of H-related defects due to their random distribution as well as low concentrations. Therefore, the information on the structures of Mu centers obtained from $\mu^{+}$SR spectroscopy is vital with respect to identifying Mu/H charge states. The structures of paramagnetic Mu⁰ centers can be easily estimated from hyperfine anisotropies.[27, 28] This is also the case for shallow and polaronic Mu centers, where a Mu⁺ or Mu^- core weakly binds an unpaired electron or hole.[13, 14, 15, 16, 17, 18, 19, 20, 29, 21] Contrastingly, diamagnetic Mu^± centers are much more difficult to characterize because they do not have unpaired electrons, and, accordingly, they lack hyperfine interactions. In such cases, magnetic-dipolar and electric-quadrupolar interactions between a muon and surrounding nuclei can be used to identify the structures of diamagnetic Mu centers. These interactions can be efficiently studied using the muon level-crossing resonance ($\mu$LCR) technique,[1] as demonstrated in heavily doped GaAs.[30, 31]

Two paramagnetic and two diamagnetic Mu centers are primarily observed in GaAs.[1] In heavily doped samples, only the diamagnetic Mu centers are detectable. Theoretical studies on H impurities[26] suggest that they should be assigned to Mu^- at the tetrahedral site surrounded by Ga atoms (T_Ga) or Mu⁺ near the bond-center (BC) position (Fig. 2). The Mu${}^{-}_{\rm T}$ and Mu${}^{+}_{\rm BC}$ centers are theoretically stable in $n$-type and $p$-type carrier-rich environments, respectively.

Chow et al. investigated the diamagnetic Mu center formed in $n$-type GaAs (Si concentrations between $2.5\times 10^{18}$ and $5\times 10^{18}$ cm^-3) using $\mu$LCR and conventional transverse-field (TF) $\mu^{+}$SR techniques to establish the Mu${}^{-}_{\rm T}$ state from a structural point of view.[30] Herein, the time-integrated muon spin polarization $\bar{P}$ was obtained in the longitudinal field (LF) configuration[4] as a function of $B$. The spin Hamiltonian relevant to this experiment comprises the Zeeman interactions for the muon and surrounding nuclei, the magnetic-dipolar interactions between the muon and the nuclei, and the electric-quadrupolar interactions for the nuclei that have an electric field gradient (EFG) primarily caused by the diamagnetic Mu. Resonant cross relaxation between the muon and the nuclei, observed as dips in $\bar{P}(B)$, occurs when the muon Zeeman splitting matches the separation of the combined quadrupolar and Zeeman energy levels for the nuclei. The resonance position depends on the electric quadrupolar parameter $Q_{i}$ for the on-resonance nucleus $i$ and the angle $\theta_{i}$ between $\boldsymbol{B}$ and the muon-nucleus axis. For a given $Q_{i}$ and $\theta_{i}$, the intensity of the resonance line is a function of the muon-nuclear dipolar coupling $D_{i}$, which is dependent on the muon-nucleus distance $r_{i}$. Chow et al. observed a couple of doublets in the $\bar{P}$ spectrum under $\boldsymbol{B}$ $||~{}\langle 001\rangle$, which were assigned to the two spin-3/2 isotopes of ⁶⁹Ga and ⁷¹Ga. Another resonance was identified at the low-field end of the spectrum, assigned to ⁷⁵As with spin 3/2. The small number of lines and the structure of the Ga doublets indicate that the principal axis of the EFG tensor, or the muon-Ga axis, is parallel to $\langle 111\rangle$ at $\theta_{\rm Ga}\sim 54.7^{\circ}$, which is compatible with the BC, T_Ga, and AB_Ga (antibonding to a Ga atom) sites. The $r_{i}$ parameter was estimated as $r_{\rm Ga}=2.199(7)$ Å  and $r_{\rm As}=2.72(5)$ Å  from a combined analysis of the $\mu$LCR spectrum and the TF-$\mu^{+}$SR Gaussian width as functions of $\boldsymbol{B}$. The results are consistent with the Mu^- center at the T_Ga site, the $r_{\rm Ga}$ of which is roughly 10% shorter than expected for an undistorted GaAs lattice.

The $\mu$LCR technique cannot be applied to systems that do not contain nuclei with a finite nuclear quadrupolar moment. Even in such cases, the local structures of diamagnetic Mu centers are occasionally determined via the magnetic-dipolar interactions between a muon and the surrounding nuclei. Characteristic oscillating features appear in zero field (ZF) $\mu^{+}$SR spectra when the muon is dipole coupled to a small number of nuclei, thus creating an entangled a-few spin system.[33, 34, 35, 36, 37, 38] Such a state often forms in materials containing fluorine[33, 34, 35] or hydrogen[36, 37, 38] with relatively strong electronegativity, which causes muons to preferentially localize in their vicinity. In addition, the ¹⁹F and ¹H nuclei with spin 1/2 have relatively large dipolar moments. Here we consider an entangled spin system consisting of a muon and a spin-1/2 nucleus. The powder-averaged muon spin relaxation function for the two-spin state (hereafter, referred to as 2S) in ZF can be expressed as

	$\displaystyle G_{\rm 2S}(t)$	$\displaystyle=\frac{1}{6}+\frac{1}{6}\cos(2\pi f_{d}t)$
		$\displaystyle+\frac{1}{3}\cos(\pi f_{d}t)+\frac{1}{3}\cos(3\pi f_{d}t),$		(1)
		$\displaystyle f_{d}=\frac{\mu_{0}\hbar\gamma_{\mu}\gamma_{I}}{8\pi^{2}d^{3}},$		(2)

where $d$ is the distance between $\mu^{+}$ and the spin-1/2 nucleus and $\gamma_{\mu}$ and $\gamma_{I}$ are the gyromagnetic ratios for $\mu^{+}$ ($\gamma_{\mu}/2\pi=135.53~{}{\rm MHz/T}$) and the nucleus ($\gamma_{I}/2\pi=42.58~{}{\rm MHz/T}$ for ¹H or $40.08~{}{\rm MHz/T}$ for ¹⁹F), respectively (Fig. 3).[34, 39] The muon-nucleus distance, $d$, obtained by fitting eq. (1) is useful with respect to estimating the local structure of a diamagnetic Mu center. This method was applied to identify an H^--Mu^- complex supposedly formed in the BaTiO_3-xH_x oxyhydride upon implantation of $\mu^{+}$ by Ito et al.[6], as reviewed in Section 3.3.

As the temperature increases, the Mu site changes and/or charge-state transitions are thermally activated. Characteristic energies relevant to these processes are obtained from the temperature dependences of diamagnetic and paramagnetic Mu amplitudes and muon spin relaxation rates.[1] The Mu species involved can be identified by mapping these energies to theoretically allowed transitions. The H/Mu defect-formation energy diagram, such as Fig. 1, obtained from first-principles calculations provides potential candidates for the transitions that are linked to Mu sites and charge states. In low-carrier systems, final Mu states can be different from promptly formed Mu states, which may be reached via a delayed process with a time constant comparable to the muon lifetime $\sim 2.2~{}\mu$s. The radio frequency (RF) $\mu^{+}$SR technique is suitable for investigating such final states, as demonstrated in the studies of Mu centers in Si, Ge, and GaAs.[40, 41, 42, 43, 5]

Longitudinal $T_{1}$ relaxation is often linked to cyclic charge-state transitions between Mu⁰ and Mu^- or Mu⁺ in thermal equilibrium. Chow et al. established that the Mu defect in heavily doped $n$-type GaAs:Si serves as a deep recombination center via the observation of a Mu${}^{-/0}_{\rm T}$ charge cycle.[44] The longitudinal relaxation rate 1/$T_{1}$ associated with the Mu${}^{-/0}_{\rm T}$ dynamics is expressed as a function of $\lambda_{-0}$, $\lambda_{-0}$, $A_{\mu}$, and $B$, where $\lambda_{-0}$ and $\lambda_{0-}$ are the rates of conversion for ${\rm Mu}_{\rm T}^{-}\rightarrow{\rm Mu}_{\rm T}^{0}$ and ${\rm Mu}_{\rm T}^{0}\rightarrow{\rm Mu}_{\rm T}^{-}$, respectively, and $A_{\mu}$ is the hyperfine coupling constant for Mu${}^{0}_{\rm T}$. The $\lambda_{-0}$ (hole capture rate) exhibits Arrhenius-like behavior with an activation energy of 1.66(7) eV. This value is approximal to that of the band gap in GaAs, suggesting that a band-gap excitation governs this process. Contrastingly, the $\lambda_{0-}$ (electron capture rate) is independent of temperature and much larger than the $\lambda_{-0}$. This is expected since the electron-carrier concentration is nearly temperature-independent in heavily doped $n$-type GaAs.

The $T_{1}$ relaxation can also occur when a Mu^+/0 charge cycle is activated. The values and behavior of charge-transition rates associated with this process in intrinsic Si are considerably different from those of $\lambda_{-0}$ and $\lambda_{0-}$ in heavily doped $n$-type GaAs.[45] This suggests that the charge state of diamagnetic Mu species involved in a Mu-charge cycle can be distinguished by carefully investigating the charge-transition rates.

The optical excitation of electrons trapped at point defects to the conduction band (CB) is a useful method of investigating the corresponding defect levels formed in the band gap. Such a technique, combined with $\mu^{+}$SR, is also helpful with respect to distinguishing between Mu^- and Mu⁺ in semiconductors. The photoexcitation of the second electron from Mu^- to the CB causes a paramagnetic Mu⁰ center, which can be described as

$${\rm Mu}^{-}+h\nu\rightarrow{\rm Mu}^{0}+e^{-}{\rm(CB)},$$

(3)

where $h\nu$ denotes a photon with energy corresponding to the optical excitation of the second electron to the CB minimum. The photogenerated Mu⁰ center is detectable using conventional $\mu^{+}$SR techniques due to the strong hyperfine interactions.

Shimomura et al. conducted optical $\mu^{+}$SR experiments to identify the Mu${}^{-}_{\rm T}$ acceptor in $n$-type GaAs using a high-power pulse laser.[46, 47] The experiments were performed at the port 2 of the RIKEN-RAL muon facility, UK, using a broadband OPO laser system pumped by a 355-nm beam from a Nd:YAG laser. The laser system was operated in pulse mode at a 25-Hz repetition rate and synchronized to the arrival of muon pulses to a single-crystalline GaAs wafer doped with $3\times 10^{16}$-cm^-3 Si. The decrease in $\mu^{+}$SR asymmetry associated with the Mu${}^{-}_{\rm T}\rightarrow{\rm Mu}^{0}_{\rm T}$ conversion was recorded as a function of photon energy, which was scanned from 0.8 to 1.5 eV. Shimomura et al. found a broad feature centered at around 1 eV after normalizing the asymmetry decrease by laser power. This is consistent with the results obtained from the temperature dependence of RF-$\mu^{+}$SR amplitude assigned for the Mu${}_{\rm T}^{-}\rightarrow{\rm Mu}_{\rm T}^{0}+e^{-}{\rm(CB)}$ excitation.[43] Moreover, a sharp feature was identified at around 1.5 eV, which was attributed to the spin or charge scattering between Mu${}_{\rm T}^{-}$ and photoexcited electrons. Subsequent experiments using circularly polarized laser light revealed that this effect depends on the polarization direction of conduction electrons with respect to the muon polarization direction; however, the exact mechanism is yet to be clarified.[48, 49]

Recent systematic ¹H-NMR studies in Ca and Sr-mayenites (C12A7 and S12A7) have revealed that ¹H chemical shifts are useful with respect to distinguishing nominally H⁺ and H^- species.[50] Mayenites comprise a class of cage-structured compounds that contain “extraframework” anions in their cages, such as O^2-, OH^-, H^-, and $e^{-}$.[51] Hayashi et al. prepared mayenite samples incorporating OH^- or H^- extraframework anions with concentrations approximal to theoretical maxima and measured the isotropic chemical shifts $\delta_{iso}$ for said H species using the magic-angle-spinning technique. They obtained a $\delta_{iso}$ of $+5.1$ ppm (+6.1 ppm) for H^- and $-0.8$ ppm ($-1.3$ ppm) for OH^- (formally H⁺) in C12A7 (S12A7) with respect to the tetramethylsilane (TMS) reference. Surprisingly, the observed values of $\delta_{iso}$ for H^- are larger than those for H⁺, falling within the typical range for the H⁺ state from +20 to 0 ppm.[52] This result suggests that the electron density around the ¹H nuclei, which is associated with the chemical-shielding effect, is larger for H⁺ than H^- species in mayenites. First-principles calculations using the periodic and embedded cluster approaches successfully reproduced such electronic states as well as the $\delta_{iso}$(H^-) $>$ $\delta_{iso}$(H⁺) relation. The experimental and theoretical results for $\delta_{iso}$s are consistent with each other within 1 ppm, demonstrating that the combined approach based on ¹H chemical shift measurements and first-principles calculations is useful with respect to identifying the charge state of H species. Systematic surveying of the relation between the ¹H chemical shift and the local structure around H species has also been conducted for many oxides, ionic hydrides, and mixed-anion hydrides. The research suggests that the distance between the H species and the coordinating atoms strongly affects the chemical shift and that $\delta_{iso}$ for both H⁺ and H^- is distributed in a similar range. This indicates that the H^± species can be separated by carefully analyzing the chemical shift; however, it does not serve as an easy fingerprint of the nominal charge states.

The chemical shift approach has also been applied to identify Mu^± species supposedly formed in C12A7:O^2- and C12A7:$e^{-}$ upon implantation of $\mu^{+}$ by Hiraishi et al.,[53] as reviewed in Section 3.1.

The extraframework H species in C12A7 mayenite have attracted much attention in association with persistent photoconductivity in C12A7:H^-[54] and highly efficient ammonia synthesis using Ru-loaded C12A7:$e^{-}$.[55, 56] Hiraishi et al. adopted the $\mu^{+}$SR technique to investigate the electronic structure and chemical activity of these centers using Mu analogs.[53] Here, we focus on their attempts to distinguish extraframework OMu^- (formally Mu⁺) and Mu^- species by measuring the Mu chemical shift $K_{\mu}$ according to the methodology outlined in Section 2.4. These species are supposedly formed in C12A7:O^2- and C12A7:$e^{-}$, respectively, via

$$\{\rm{O}^{2-}\}+\rm{Mu}^{0}\rightarrow\{\rm{OMu}^{-}\}+\{{\it e}^{-}\},$$

(4)

$$\{e^{-}\}+\rm{Mu}^{0}\rightarrow\{\rm{Mu}^{-}\},$$

(5)

where {X} indicates an extraframework anion $X$ in a cage.

Mu chemical shift measurements were conducted on single-crystalline samples of C12A7:O^2- (pristine insulator) and C12A7:$e^{-}$ (electride, $n\sim 10^{21}$ cm^-3) in the high-TF configuration[4] at TRIUMF, Canada, using a spin-polarized surface muon beam. A TF of 6 T was applied along the muon incident axis, which was nominally perpendicular to the (001) plane. For both samples, the value of $K_{\mu}$ was mostly independent of temperature below 300 K. The temperature-averaged $K_{\mu}$ was $+0.3(4)$ ppm for C12A7:O^2- and $+6.6(4)$ ppm for C12A7:$e^{-}$ using CaCO₃ as a zero-shift reference. These results are consistent with corresponding ¹H chemical shifts, specifically $\delta_{iso}=-0.8$ ppm for C12A7:OH^- and $\delta_{iso}=+5.1$ ppm for C12A7:H^-, using TMS as a reference.[50] On the basis of this similarity, Hiraishi et al. assigned the chemical state of diamagnetic muons in the C12A7:O^2- and C12A7:$e^{-}$ samples to {OMu^-} and {Mu^-}, respectively. However, there seems to be some ambiguity associated with the difference in the gyromagnetic ratios for $\mu^{+}$ and ¹H ($\gamma_{\mu}/\gamma_{\rm H}\sim 3.2$). Indeed, the value of $K_{\mu}$ for {Mu^-} in C12A7:$e^{-}$ is considerably smaller than expected from the ¹H chemical shift for C12A7:H^- ($5.1\times 3.2=16$ ppm). This discrepancy was tentatively attributed to the metallic environment of Mu^- in the C12A7:$e^{-}$ sample ($n\sim 10^{21}$ cm^-3).[53]

Oxygen vacancies can act as electron donors in SrTiO₃, causing a wide range of interesting phenomena, such as superconductivity[59] and ferromagnetism.[60] The oxygen-deficient SrTiO_3-x can be obtained by annealing the parent band insulator in highly reducing atmospheres. Hydrogen is often used as a reducing agent in this process. Systematic annealing studies have revealed that such a treatment does not only remove oxygen to obtain metallic SrTiO_3-x, but that it can also create other types of defects involving hydrogen.[61] The most common H-related defect in perovskite oxides is an interstitial H⁺ bound to an O^2- in the host lattice, which serves as a shallow donor in SrTiO₃[62]. This behavior has also been confirmed from Mu viewpoints[63, 29]. Contrastingly, subsequent hydrogen annealing of the metallic SrTiO_3-x causes a decrease in conductivity.[61] This contradictory behavior against the interstitial H⁺ implies that the site and charge state of hydrogen introduced by the subsequent annealing is considerably different from those of the interstitial H⁺. First-principles studies suggest that the most stable form of hydrogen in SrTiO_3-x is H^- located at the anion site, where doubly charged V${}^{2+}_{\rm O}$ changes into singly charged H${}_{\rm O}^{+}$.[11]

Motivated by these theoretical predictions, Shimomura et al. conducted $\mu^{+}$SR experiments in oxygen-deficient SrTiO_3-x at the D1 area of J-PARC MUSE, Japan, with the expectation that implanted muons are preferentially trapped in V_O where they form a substitutional Mu^- center.[57] The SrTiO_3-x sample was prepared by reducing single-crystalline wafers of SrTiO₃ with the (110) surface according to the procedure outlined in Ref. [\citenjalan08]. The $\mu^{+}$SR measurements were performed using a double-pulsed surface muon beam and the D$\Omega$1 spectrometer (maximum asymmetry $\sim$ 0.20) in the LF geometry[4] under a weak TF of 2 mT applied perpendicular to the [110] direction.

$\mu^{+}$SR asymmetry spectra were fitted to a function composed of two exponentially-damped cosines with a shared diamagnetic frequency. Following [\citenshimomura14], the fits were refined by taking into account double-pulse correction.[58] Figure 4 shows the partial asymmetries $A_{1}$ and $A_{2}$ and the exponential relaxation rates $\lambda_{1}$ and $\lambda_{2}$ for the two (fast and slow) components as functions of temperature. The $A_{2}$ for the fast component was set to zero for fits below 160 K. We can find a decrease in the total diamagnetic asymmetry, $A_{1}+A_{2}$, below $\sim$ 50 K from the maximum value of $\sim$ 0.20. Similar behavior was also observed in nominally undoped SrTiO₃, where muons should stay at interstitial sites, and the asymmetry drop was attributed to the formation of shallow hydrogen-like Mu[63] or Mu⁺-bound small polaron[29]. This similarity implies that an interstitial Mu⁺ state is partially involved in the observed diamagnetic signal. However, the diamagnetic amplitude below 50 K is significantly larger than that in undoped SrTiO₃. Shimomura et al. ascribed the increased diamagnetic fraction in oxygen-deficient SrTiO_3-x to the hydridic Mu^- center formed in V_O.

The $\lambda_{2}$ for the fast component markedly increases with increasing temperature above 160 K. Shimomura et al. tentatively attributed this anomalous behavior to the activation of cyclic Mu^-/0 charge-state transitions on the basis of similarity with the temperature dependence of $1/T_{1}$ in $n$-type GaAs[44] (see Section 2.2). However, there seem to be some difficulties in this interpretation. In particular, the Mu^-/0 charge cycle model for $n$-type materials assumes that the Mu^- defect creates a deep level in the band gap and serves as a recombination center.[44] However, according to computational studies, the defect level for the substitutional H^- falls in the valence band.[11] Moreover, a characteristic energy of $\sim$ 0.1 eV estimated from the increase in $\lambda_{2}(T)$ is much smaller than the band-gap energy of 3.2 eV. Accordingly, a different model may be necessary to explain the unusual exponential relaxation possibly associated with Mu-state dynamics in SrTiO_3-x.

Oxyhydrides of perovskite titanates $A$TiO_3-xH_x ($A={\rm Ba}$, Sr, and Ca) comprise a new class of hydrogen ion conductors, which can be obtained from $A$TiO₃ parent insulators by CaH₂ reduction.[25, 64, 65] Conventional diffraction techniques can be applied to identify the H site in $A$TiO_3-xH_x since a large amount of H up to $x\sim 0.6$ can be incorporated. A combined analysis of x-ray and neutron diffraction data revealed that O^2- ions in the perovskite lattice are randomly substituted by H^- ions without creating any detectable amount of V_O. The substitutional H^- in $A$TiO_3-xH_x is in sharp contrast with the interstitial protonic hydrogen (formally H⁺) bound to an O^2- ion, which is a common impurity often found in $A$TiO₃. Macroscopic gas analysis revealed that the hydrogen in the solid phase is mobile and exchangeable in hydrogen gas environments above $\sim$ 400^∘C. Moreover, the reduction treatment changes the parent band insulators into paramagnetic metals. These transport characteristics suggest that $A$TiO_3-xH_x compounds are suitable for application in mixed electron/hydrogen ion conductors and hydrogen membranes.

Several theoretical models have been proposed regarding the stability of H species in $A$TiO_3-xH_x and their kinetics.[22, 10, 11, 23] These studies commonly conclude that the substitutional H^- configuration is most stable in $n$-type carrier-rich environments. With respect to the dynamical aspect of H in the solid phase, two types of scenarios have been proposed. One is based on the concept of correlated migration of H^-, O^2-, and V_O in the network of the anion site [type-I H migration, Fig. 5(a)].[25, 22, 66] The other model [type-II H migration, Fig. 5(b)] involves two metastable H configurations: the interstitial H⁺ bound to O^2-, and two H atoms trapped at the same anion site in place of O^2-. The former can support rapid proton diffusion via the Grotthuss mechanism[23], and the latter can act as a hydrogen exchange center, which is expected to temporarily form due to the interaction between a substitutional H^- and an incoming interstitial H⁺. Two charge configurations, hydridic 2H^-[11] and molecular H₂[23], have been proposed for this center.

The $\mu^{+}$SR technique is useful for creating and investigating experimentally accessible models of such metastable H-related centers. This is based on the fact that the as-implanted mixture of Mu states is far from equilibrium, often involving metastable excited states.[5] Ito et al. conducted $\mu^{+}$SR investigations of prototypical BaTiO_3-xH_x to obtain microscopic insight into the metastable H configurations associated with type-II H migration using Mu as a pseudoisotope of H.[6] The $\mu^{+}$SR measurements of powder samples of BaTiO_3-xH_x ($x=0.1$, 0.2, 0.3 and 0.5) were carried out in the D1 area of J-PARC MUSE, Japan, and in the port 2 of RIKEN-RAL, U.K., using a spin-polarized surface muon beam. Muons implanted in the samples lose their kinetic energy via electromagnetic interactions with host atoms and are then trapped in local potential minima (not necessarily in the global minimum). In BaTiO_3-xH_x, implanted muons are expected to serve as incoming excess hydrogen, creating Mu analogs of the metastable configurations involved in the type-II H migration model together with H^- in the host lattice. Information on their charge states was obtained via the local structure approach outlined in Section 2.1.

Figure 6(a) shows the time evolution of muon spin polarization $P(t)$ at 15 K in ZF for the $x=0.3$ and 0.5 samples. The $P(t)$ curves have a damped cosine-like feature superposed on a Gaussian relaxation curve. The oscillating feature is a signature of the formation of the entangled 2S state composed of diamagnetic Mu and H. The Gaussian relaxation part is usually ascribed to muons that interact with a large number of surrounding nuclei without creating such a special magnetic coupling[67]. Therefore, the two components were tentatively assigned to the metastable configurations involved in the type-II model: the 2S component for a Mu analog of the hydrogen exchange center [Fig. 6(e)] and the Gaussian component for the interstitial Mu⁺ [Fig. 6(f)].

Accordingly, the ZF spectra at 15 K were fitted to the following function,

$$P(t)=p_{\rm 2S}e^{-\lambda t}G_{\rm 2S}(t;f_{d})+(1-p_{\rm 2S})e^{-\Delta^{2}t^{2}},$$

(6)

where $\lambda$, $p_{\rm 2S}$, and $G_{\rm 2S}(t;f_{d})$ denote the relaxation rate, the fraction, and the ZF relaxation function in eq. (1) for the 2S state, respectively. The distance $d$ between a muon and a nearby H can be obtained from $f_{d}$ through eq. (2). $d$ is independent of $x$ [Fig. 6(b)], which is expected since the Mu-H structure should be robust against changes in $x$. The average value of $d=1.64(1)$ Å  is consistent with the theoretical H^--H^- distances of 1.64 and 1.67 Å  for symmetric and asymmetric 2H^- centers in SrTiO_3-x[11]; however, this does not match $d=0.74$ Å  for a molecular MuH configuration. On the basis of these results, Ito et al. assigned the 2S component to a hydridic (Mu^-, H^-) complex formed at V_O. Meanwhile, the $x$ dependence of the Gaussian relaxation rate $\Delta$ associated with the interstitial Mu⁺ configuration can be reproduced by calculating the rms width of the nuclear dipolar field at the interstitial Mu⁺ site, as shown in Fig. 6(d) with a solid curve.

The temperature dependence of $\Delta$ for the $x=0.5$ sample reveals that interstitial Mu⁺ diffusion (activation energy $\sim$ 0.1 eV) and subsequent trapping in an unknown deep potential well occur above 100 K within the $\mu^{+}$SR time window. This re-trapping process may function as a rate-limiting step of macroscopic H transport in the BaTiO_3-xH_x lattice. It is not taken into account in the type-II H migration model.