Muon spin rotation has long been recognized as one of the few methods for experimentally accessing the electronic state of dilute hydrogen (H) in semiconductors and dielectrics, where muon behaves as a pseudo-H (designated by the elemental symbol Mu). Meanwhile, predictions on the electronic state of H in these materials by density functional theory (DFT) do not always agree with the observed states of Mu. Most notably, Mu frequently occurs in wide-gap oxides simultaneously in a neutral ($Mu0$) and a diamagnetic state ($Mu+$ or $Mu\u2212$), which DFT calculations do not explain; they predict that H is stable only in a diamagnetic state with the polarity determined by the equilibrium charge-transition level ($E+/\u2212$) vs the Fermi level. To address this issue, we developed a semi-quantitative model that allows a systematic understanding of the electronic states reported for Mu in the majority of oxides. Our model assumes that muons interact with self-induced excitons to produce relaxed-excited states corresponding to donor-like ($MuD$) and/or acceptor-like ($MuA$) states and that these states correspond to the non-equilibrium electronic level ($E+/0$ or $E0/\u2212$) predicted by DFT calculations for H. The known experimental results are then explained by the relative position of $E+/0$ and $E0/\u2212$ in the host’s energy band structure. In addition, the model sheds new light on the polaron-like nature of the electronic states associated with shallow donor Mu complexes.

## I. INTRODUCTION

Hydrogen (H) is traditionally classified as a group 1 element on the periodic table. This is due to the fact that H, like alkali metals, readily donates electrons in a variety of redox reactions. On the other hand, it is well known that H can exist as a relatively stable anion (hydride). In this case, H can be considered as a member of the same family as halogens (group 17). Since it was discovered in the 1980s that H can interact with both $n$-type and $p$-type impurities in silicon and significantly affect the electrical conductivity,^{1,2} H has attracted considerable attention in the field of semiconductors as a special impurity that exhibits ambipolarity.

Earlier studies have shown that the majority of the incorporated H forms complexes with other impurities and defect centers such as atomic vacancies, resulting in passivation (loss of electrical activity). Unlike the transfer of electrons between impurity levels (carrier compensation) that occurs when both $n$-type and $p$-type defect centers coexist, solid-state chemical reactions eliminate the impurity levels from within the bandgap in the passivation. Various experimental techniques have already been used to analyze such complex defects, and their local structures are being clarified.

Meanwhile, another critical issue is the electrical activity of H as a defect center. It is expected that a few ppm of H (equivalent to about $\u223c1015$–$1016/cm3$) will be unintentionally incorporated into the material during the manufacturing process. This is comparable to the carrier concentration caused by intentional impurity addition, which has a significant impact on conductivity on its own. Understanding the local electronic state of isolated H is, therefore, crucial in elucidating the entire mechanism of H’s contribution to electrical activity in semiconductors at the atomic level.

The muon spin rotation ($\mu $SR) is one of the techniques used to obtain the relevant information on H by implanting a positively charged muon ($\mu +$, hereafter simply called muon) into the target material and investigating its electronic state as pseudo-H. The muon is an unstable subatomic particle that can be obtained as a particle beam in specialized accelerator facilities. In terms of chemical properties, it behaves as a light radioactive isotope of the proton (with about 1/9 of the proton mass) when incorporated into matter. It is useful to have the elemental name in the discussion of muon as pseudo-H. In the following, the symbol Mu (corresponding to H for hydrogen) will be used, and the valence states of Mu will be denoted as $Mu+$, $Mu0$, and $Mu\u2212$. For deliberately avoiding distinction between $Mu+$ and $Mu\u2212$, we refer to them together as “diamagnetic Mu.” Because the muon mass is two orders of magnitude greater (about 206 times) than the electron mass, the adiabatic approximation is sufficient for understanding muon–electron interaction. In fact, the difference in the Bohr radius between a muon binding a single electron, known as muonium ($Mu0$), and the corresponding $H0$ atom is only 0.43%, implying that they have nearly the same electronic structure.

Nevertheless, the light mass of muon in comparison to H implies relatively large isotope effects on dynamical properties such as diffusion in solids. In a harmonic potential, for example, the zero-point energy $E0$ is proportional to the square root of the particle mass. Because muon/muonium has nearly three times the $E0$ of H, the activation energy for the former in the over-barrier hopping motion is reduced by $\u223c2E0$. The large zero-point motion also increases the probability of tunneling to neighboring sites, enhancing tunneling-mediated diffusion (quantum diffusion). As a result, when inferring H from dynamic properties of Mu, these factors must be taken into account.

Various electronic states and dynamics of Mu have been experimentally revealed in a wide variety of materials, including oxides, since the beginning of $\mu $SR research in the 1970s. Furthermore, recent advances in first-principles calculations using density functional theory (DFT) in conjunction with the advent of the computational environment have enabled us to discuss the local electronic structures of Mu/H in individual materials in great detail. Meanwhile, the development of a physical model that would allow us to understand the Mu states in a cross-material context is still in its early stages. In relation to this, relatively little attention has been paid to the ambipolarity of Mu/H, with the majority of research to date concerned with the donor-like behavior of Mu/H as a member of group 1 elements. It is noteworthy that the importance of ambipolarity for H in oxide semiconductors has become increasingly clear in recent years.^{3,4}

In this paper, we will demonstrate that the ambipolar property of Mu/H, including its acceptor-like behavior, is critical to a coherent understanding of the local electronic state. We also contend that the fact that such ambipolar Mu states have been experimentally observed is inextricably linked to another fact that the initial state of Mu is in a temporary non-thermal equilibrium state. The primary goal of this paper is to develop a semi-quantitative model for a unified understanding of electronic structure that takes these two factors into account.

To that end, we compared the experimentally observed electronic states of Mu in various oxides with those predicted by DFT calculations for H in the respective oxides published previously; it is not the scope of this paper to test the precision of previous calculations by performing new ones. In fact, the results of these existing calculations serve as a basis for our model, and it could be argued that the model’s success demonstrates that they are sufficiently reliable for our goal. While we provide a brief overview of the general aspects of the DFT calculations in Sec. III A, readers are encouraged to consult the individual references cited in Table I in Sec. IV B for more information.

. | Expt. . | DFT Calc. . | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

. | E_{g}
. | . | . | . | Cell size . | E_{g}
. | E^{0/−}
. | E^{+/−}
. | E^{+/0}
. | ɛ_{+}
. | . | . |

Mater. . | (eV) . | Mu . | Reference . | Codes, functionals . | (atoms) . | (eV) . | (eV) . | (eV) . | (eV) . | (eV) . | Mu . | Reference . |

BeO | 10.6 | $MuX0$ | 58, 59 | VASP, GGA, HSE06 | 72 | 10.6 | 5.5 | 6.09 | 6.7 | 3.9 | $MuA/D0$ | 59 |

SiO_{2} | 9.0 | $MuX0$, Mu^{+} | 54, 55 | CASTEP, GGA, HSE06 | 33–49 | 8.7 | 2.9 | 5.4 | 7.9 | 0.8 | $MuA0$, $MuD0\u22c6$ | 14 |

α-Al_{2}O_{3} | 8.8 | $MuX0$, Mu^{+} | 56, 75 | CASTEP, GGA, HSE06 | 33–49 | 8.5 | 3.0 | 5.4 | 7.7 | 0.8 | $MuA0$, $MuD0\u29eb$^{♦} | 14 |

MgO | 7.8 | $MuX0$, Mu^{+} | 55, 57 | CASTEP, GGA, HSE06 | 33–49 | 7.5 | 2.8 | 5.4 | 7.9 | −0.4 | $MuA0$, $MuD+$ | 14 |

m−HfO_{2} | 6.0 | $MuX0$, Mu^{+} | 60 | CASTEP, GGA, HSE06 | 33–49 | 5.8 | 1.6 | 4.0 | 6.3 | −0.5 | $MuA0$, $MuD+$ | 14 |

q−GeO_{2} | 6.0 | $MuX0$, Mu^{+} | 58, 55 | CASTEP, GGA, HSE06 | 33–49 | 5.6 | 2.0 | 4.8 | 7.2 | −1.6 | $MuA0$, $MuD+$ | 14 |

Lu_{2}O_{3} | 5.6(1) | $MuX0$, Mu^{+} | 63 | VASP, GGA, PBE/DFT+U | 80 | 4.0 | 1.1 | 2.46 | 4.2 | −0.2 | $MuA0$, $MuD+$ | 63 |

ZrO_{2} | 5.5(3) | $MuX0$, Mu^{+} | 61 | VASP, GGA, HSE06 | 96 | 5.4 | 2.1 | 3.5 | 4.8 | 0.6 | $MuA0$, $MuD0\u22c6$ | 76 |

Y_{2}O_{3} | 5.5 | $MuA0$, Mu^{+} | 62 | VASP, GGA, HSE06 | 80 | 5.9 | 2.15 | 3.8 | 5.5 | 0.4 | $MuA0$, $MuD0\u22c6$ | 62 |

La_{2}O_{3} | 5.4(1) | $MuX0$, $MuS0$ | 58 | CASTEP, GGA, HSE06 | 33–49 | 5.2 | 0.3 | 3.0 | 6.2 | −1 | $MuA0$, $MuD+$¶ | 14 |

β-Ga_{2}O_{3} | 5.0 | $MuS0$, Mu^{+} | 77 | CASTEP, GGA, HSE06 | 33–49 | 4.8 | 3.2 | 4.9 | 6.4 | −1.6 | $MuD+$ | 14 |

c-IGZO | 3.68 | Mu^{+} | 78 | CASTEP, GGA, HSE03 | 116 | 3.1 | >3.1 | >3.1 | 4.8 | −1.7 | $MuD+$ | 79 |

SnO_{2} | 3.6 | $MuS0$, Mu^{+} | 58, 80 | CASTEP, GGA, HSE06 | 33–49 | 3.6 | 4.1 | 4.3 | 4.6 | −1 | $MuD+$ | 14 |

ZnO | 3.4 | $MuS0$, Mu^{+} | 64, 65 | VASP, GGA, HSE06 | 72–784 | 3.4 | ≥3.4 | ≥3.4 | 3.4 | 0 | $MuD+$¶ | 81 |

α-TeO_{2} | 3.4 | $MuA0$, Mu^{+} | 82 | VASP, GGA, PBE | 96 | 2.82 | 0.8 | 2.2 | 2.82 | 0 | $MuA0$, $MuD+$ | 82 |

SrTiO_{3} | 3.2 | $MuS0$, Mu^{+} | 83, 70 | VASP, GGA, HSE06 | 135 | 3.1 | >4 | 3.8 | 3.5 | −0.4 | $MuD+$¶ | 71, 84 |

r−TiO_{2} | 3.2 | $MuS0$, Mu^{+} | 67, 68 | CASTEP, GGA, HSE06 | 33–49 | 3.0 | 2.6 | 3.1 | 3.5 | −0.5 | $MuD+$¶ | 14 |

In_{2}O_{3} | 2.7(1) | Mu^{+} | 80 | VASP, LDA+U | 80 | 2.67 | 1.8 | 3.2 | 3.67 | −1 | $MuD+$ | 85 |

w-GaN | 3.48 | $MuS0$, Mu^{+} | 86 | (custom), LDA | 32–96 | 3.4 | 1.2 | 2.4 | 3.5 | −0.1 | $MuD+$¶ | 87 |

. | Expt. . | DFT Calc. . | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

. | E_{g}
. | . | . | . | Cell size . | E_{g}
. | E^{0/−}
. | E^{+/−}
. | E^{+/0}
. | ɛ_{+}
. | . | . |

Mater. . | (eV) . | Mu . | Reference . | Codes, functionals . | (atoms) . | (eV) . | (eV) . | (eV) . | (eV) . | (eV) . | Mu . | Reference . |

BeO | 10.6 | $MuX0$ | 58, 59 | VASP, GGA, HSE06 | 72 | 10.6 | 5.5 | 6.09 | 6.7 | 3.9 | $MuA/D0$ | 59 |

SiO_{2} | 9.0 | $MuX0$, Mu^{+} | 54, 55 | CASTEP, GGA, HSE06 | 33–49 | 8.7 | 2.9 | 5.4 | 7.9 | 0.8 | $MuA0$, $MuD0\u22c6$ | 14 |

α-Al_{2}O_{3} | 8.8 | $MuX0$, Mu^{+} | 56, 75 | CASTEP, GGA, HSE06 | 33–49 | 8.5 | 3.0 | 5.4 | 7.7 | 0.8 | $MuA0$, $MuD0\u29eb$^{♦} | 14 |

MgO | 7.8 | $MuX0$, Mu^{+} | 55, 57 | CASTEP, GGA, HSE06 | 33–49 | 7.5 | 2.8 | 5.4 | 7.9 | −0.4 | $MuA0$, $MuD+$ | 14 |

m−HfO_{2} | 6.0 | $MuX0$, Mu^{+} | 60 | CASTEP, GGA, HSE06 | 33–49 | 5.8 | 1.6 | 4.0 | 6.3 | −0.5 | $MuA0$, $MuD+$ | 14 |

q−GeO_{2} | 6.0 | $MuX0$, Mu^{+} | 58, 55 | CASTEP, GGA, HSE06 | 33–49 | 5.6 | 2.0 | 4.8 | 7.2 | −1.6 | $MuA0$, $MuD+$ | 14 |

Lu_{2}O_{3} | 5.6(1) | $MuX0$, Mu^{+} | 63 | VASP, GGA, PBE/DFT+U | 80 | 4.0 | 1.1 | 2.46 | 4.2 | −0.2 | $MuA0$, $MuD+$ | 63 |

ZrO_{2} | 5.5(3) | $MuX0$, Mu^{+} | 61 | VASP, GGA, HSE06 | 96 | 5.4 | 2.1 | 3.5 | 4.8 | 0.6 | $MuA0$, $MuD0\u22c6$ | 76 |

Y_{2}O_{3} | 5.5 | $MuA0$, Mu^{+} | 62 | VASP, GGA, HSE06 | 80 | 5.9 | 2.15 | 3.8 | 5.5 | 0.4 | $MuA0$, $MuD0\u22c6$ | 62 |

La_{2}O_{3} | 5.4(1) | $MuX0$, $MuS0$ | 58 | CASTEP, GGA, HSE06 | 33–49 | 5.2 | 0.3 | 3.0 | 6.2 | −1 | $MuA0$, $MuD+$¶ | 14 |

β-Ga_{2}O_{3} | 5.0 | $MuS0$, Mu^{+} | 77 | CASTEP, GGA, HSE06 | 33–49 | 4.8 | 3.2 | 4.9 | 6.4 | −1.6 | $MuD+$ | 14 |

c-IGZO | 3.68 | Mu^{+} | 78 | CASTEP, GGA, HSE03 | 116 | 3.1 | >3.1 | >3.1 | 4.8 | −1.7 | $MuD+$ | 79 |

SnO_{2} | 3.6 | $MuS0$, Mu^{+} | 58, 80 | CASTEP, GGA, HSE06 | 33–49 | 3.6 | 4.1 | 4.3 | 4.6 | −1 | $MuD+$ | 14 |

ZnO | 3.4 | $MuS0$, Mu^{+} | 64, 65 | VASP, GGA, HSE06 | 72–784 | 3.4 | ≥3.4 | ≥3.4 | 3.4 | 0 | $MuD+$¶ | 81 |

α-TeO_{2} | 3.4 | $MuA0$, Mu^{+} | 82 | VASP, GGA, PBE | 96 | 2.82 | 0.8 | 2.2 | 2.82 | 0 | $MuA0$, $MuD+$ | 82 |

SrTiO_{3} | 3.2 | $MuS0$, Mu^{+} | 83, 70 | VASP, GGA, HSE06 | 135 | 3.1 | >4 | 3.8 | 3.5 | −0.4 | $MuD+$¶ | 71, 84 |

r−TiO_{2} | 3.2 | $MuS0$, Mu^{+} | 67, 68 | CASTEP, GGA, HSE06 | 33–49 | 3.0 | 2.6 | 3.1 | 3.5 | −0.5 | $MuD+$¶ | 14 |

In_{2}O_{3} | 2.7(1) | Mu^{+} | 80 | VASP, LDA+U | 80 | 2.67 | 1.8 | 3.2 | 3.67 | −1 | $MuD+$ | 85 |

w-GaN | 3.48 | $MuS0$, Mu^{+} | 86 | (custom), LDA | 32–96 | 3.4 | 1.2 | 2.4 | 3.5 | −0.1 | $MuD+$¶ | 87 |

## II. NOTABLE FEATURES OF μSR AS A METHOD FOR STUDYING PSEUDO-HYDROGEN

In the actual $\mu $SR experiment, nearly 100% spin-polarized $\mu +$ is implanted into a sample, and time-dependent spatial asymmetry ($\u2243$20%) of positrons emitted with high probability in the direction of spin polarization is observed upon beta decay. When $\mu +$ is implanted into a solid material, it decelerates quickly (usually in less than 1 ns) and comes to rest in an interstitial position. From that point until the beta decay occurs (with a mean lifetime of $\tau \mu =2.198$ $\mu $s), $\mu +$ behaves as Mu, taking various valence states depending on the local environment. Precession of the muon spin occurs at a frequency proportional to the hyperfine interaction of Mu^{+} ($S=1/2$) with the surrounding electrons and/or nuclear spins. The hyperfine interaction is described by the Hamiltonian

where $\gamma \mu $ is the muon gyromagnetic ratio ($=2\pi \xd7135.54$ MHz/T), $S\mu $ is the muon spin operator, and $H(r)$ is the effective hyperfine field [$A(r)$ being the hyperfine parameter] at the Mu position $r$ in the crystalline lattice.^{5} The $\mu $SR frequency spectrum can easily distinguish the paramagnetic state ($Mu0$) from the diamagnetic Mu ($Mu+$ or $Mu\u2212$) due to the large difference in $H(r)$ (see Appendixes A and B for more details). The distinction between $Mu+$ and $Mu\u2212$, on the other hand, necessitates high-precision chemical shift measurements ($\u223c101$ ppm).^{6} The time evolution of muon polarization is observed as a statistical average of signals from a large number of Mu over a time period of about $10\tau \mu $ ($\u223c$20 $\mu $s), with $t=0$ defined by the arrival time of $\mu +$.

Here, we would like to highlight some practical aspects for Mu in mimicking H. First, the penetrating power of the muon beam is sufficiently high that it should not be affected by the surface condition of the sample (bulk-sensitive). Even in a pulsed beam experiment where a large number of muons are injected at once, the number of muons present in the sample at the same time is at most $\u223c104/cm2$ per cross section. Muons with a typical incident energy of $T\mu \u22434$ MeV have a stopping range of about 0.1–1 mm from the sample surface, so their volume concentration is less than $\u223c105$ muons per $cm3$. Furthermore, they do not accumulate in the sample, as they disappear quickly ($\u223c\tau \mu $). As a result, muons offer an excellent opportunity to observe the electronic state of pseudo-H in its true dilute limit.

Meanwhile, as detailed in Sec. III, the implanted Mu is mostly in a relaxed-excited (metastable) state in non-metallic materials due to the interaction with the electron–hole pairs (or excitons) generated by the transfer of $T\mu $ along the muon track to the host lattice. This is also implied by experiments using a recently available low-energy muon beam (LEM, $T\mu \u22431$–30 keV at Paul Scherrer Institute, Switzerland), in which muons are implanted into a region of $101$–$102$ nm from the sample surface. Although the Mu density is still in the dilute limit in this case, it has been shown that the fractional yields of Mu in different valence states are strongly dependent on $T\mu $,^{7} indicating the definitive influence of Mu-exciton interaction in determining the final Mu states.

The fact that the electronic state of Mu does not always correspond to the thermal equilibrium state of H may appear to imply that it is insufficient as a source of information for H. However, as will be discussed further below, it is this non-equilibrium nature that allows us to use Mu to experimentally evaluate the ambipolarity of H. Moreover, many electronic materials, including oxides, are used in devices under a variety of electronic excitation such as electric fields and optical irradiation. In this regard, the data obtained from Mu will provide microscopic clues for clarifying the effect of H in those materials on their performance under such electronic excitations (for example, see Sec. IV C). Thus, Mu works in conjunction with H to reveal the complete picture of H’s behavior in matter.

From the theoretical viewpoint, how to model non-equilibrium states using DFT that assumes the system to be at equilibrium is a challenging problem. This has been addressed in the DFT community for many years, and time-dependent DFT, for example, is still being studied for that purpose.^{8} Our model suggests that the electronic states of muons can serve as a testing ground for such theoretical studies.

## III. Mu STUDY—AN APPROACH FROM NON-THERMAL EQUILIBRIUM STATES

### A. DFT calculations for H defect centers

In general, DFT calculations estimate the formation energy $\Xi q$ of H defect centers as a function of the Fermi level $EF$ using the following equation:

where $Et[Hq]$ and $Et[\u2212]$ denote the total energy of a supercell involving H$q$ and a perfect cell of the host material, respectively, calculated for charge $q$ ($=\xb1,0$), $nH$ is the number of H atoms, and $\mu H$ is the reference chemical potential for H.^{9} Provided that

which is valid within the adiabatic approximation, Eq. (2) gives the formation energy for $Muq$ as schematically shown in Fig. 1(a). Although Mu/H can play the role of either a cation or an anion with respect to the host, their local structures can be different from each other [Fig. 1(b)]. Therefore, we refer to them as Site-$D$ (donor-like, associated with anions) and Site-$A$ (acceptor-like, associated with cations). In addition, among the three charge-transition energies, the equilibrium charge-transition level ($E+/\u2212$) is lower than the donor and acceptor levels ($E+/0$, $E0/\u2212$) in most cases [see Fig. 1(a)]. This behavior is characteristic for systems with strong electron–phonon coupling,^{10} indicating that the effective onsite Coulomb repulsion energy ($U$) is negative. The negative-$U$ character combined with the ambipolarity leads to a tendency of charge disproportionation for H (i.e., preferring $H+$ or $H\u2212$ to $H0$).^{11,12} The electronic state of H in the thermal equilibrium is then determined by the relationship among $\Xi +(EF)$, $\Xi 0(EF)$, and $\Xi \u2212(EF)$. More specifically, only $HD+$ ($EF<E+/\u2212$) or $HA\u2212$ ($EF>E+/\u2212$) will be realized, and, thus, $E+/\u2212$ will be the effective impurity level.

To predict the electrical activity of H defects using DFT calculations, the band structure of the defect-free host must be predicted with the accuracy comparable with that for $\Xi q(EF)$. However, early DFT calculations had a tendency to significantly underestimate the bandgap. Furthermore, the charged defect levels are sensitive to supercell size corrections, and various correction methods have been developed to mitigate these issues.^{13} The majority of the quoted DFT calculations, as summarized in Table I, are based on the Generalized Gradient Approximation (GGA) combined with the Heyd–Scuseria–Ernzerhof hybrid functional (HSE06) to improve the gap energy. Table I displays the calculated bandgap energies for comparison with experimental values.

Meanwhile, it is expected that differences in the details of other conditions will skew the results of these calculations. In this regard, it is worth noting that 9 of the 18 oxides in Table I on which our model is based rely on the same set of calculations performed by Li and Robertson.^{14} This could be useful in determining the DFT calculation’s systematic errors. The plane wave pseudo-potential code CASTEP was used to perform their calculations.^{15} The atomic potentials were represented using norm-conserving pseudo-potentials. They also used the HSE06 functional to correct the bandgap errors of the pure GGA approach, in which they varied a fraction $\alpha $ of the short-range separated part of the Hartree–Fock (HF) exchange to fit the bandgap for systems with a larger gap. The screening length was set to $\mu =0.106$/bohrs.^{16,17} The lattice parameters were set to experimental values for the defect calculations, and only the internal atomic coordinates for the interstitial H were relaxed. The H atom was placed in an arbitrary location near the center of the open interstitial site. The cutoff energy for plane waves was 800 eV, and the $k$ point mesh was $2\xd72\xd72$. The defect formation energies were calculated using Eq. (2), where the reference chemical potential was defined using the method described in Ref. 13. They were limited to small supercells for each oxide as a tradeoff for using the HSE06 hybrid functional; they adopted relatively small supercells of 33–49 atoms depending on the degree of localization for the electronic states. However, as shown in Table I, the calculated bandgap is very close to the experimental value. Moreover, because the hydrogen charge is not large ($+1$ or $\u22121$), the supercell-size dependence is not regarded as a major issue in terms of electrostatic interactions.^{18}

It is known that these plane-wave calculations have limitations for supercell size correction and that hybrid functions are costly to perform with this type of code. A way around this problem would be to incorporate, for example, Becke’s three-parameter hybrid exchange functional^{19} and Lee, Yang, and Parr correlation functional^{20} (B3LYP). These hybrid functions could be handled more efficiently if calculations could be performed using local basis set codes such as CRYSTAL^{21} or CP2K.^{22} As a couple of helpful recent examples, such calculations (CRYSTAL06 with B3LYP) have been performed for H in $Ga2O3$^{23} and $In2O3$,^{24} although the calculated values of the bandgap are not given. CP2K calculations have been also performed for H in amorphous $SiO2$^{25} and $HfO2$.^{26} The bandgaps were calculated to be 8.1 and 5.9 eV, respectively, which are in good agreement with the measured values (9 and 6 eV).

### B. Mu as relaxed-excited states and acceptor/donor levels

The attempt to interpret the electronic states of Mu by $\Xi q(EF)$ vs $E+/\u2212$ fails to account for the existence of the paramagnetic $Mu0$ state reported in many wide-gap oxides (see Table I). This necessitates the introduction of the hypothesis that the initial Mu state immediately after $\mu +$ implantation to rest corresponds to a relaxed-excited state upon rapid quenching from infinite temperature [i.e., $\beta \u22611/kBT\u21920$ in the partition function $Z(\beta )$; see Appendix C]. An intriguing fact to remember when considering the origin of $Mu0$ is that interstitial paramagnetic H centers ($Hi0$) are produced by irradiating H-containing ionic crystals with ultraviolet (uv) light at low temperatures. For example, $Hi0$ (also known as $U2$-centers) is produced in alkali halides containing $OH\u2212$ defects in the photodissociation reaction,

where [ ] refers to the anion substitutional site, $h\nu $ to the photons, and $h+$ to the hole.^{27–29} This is thought to be a process similar to that of self-trapped-exciton (STE) formation via electronic excitation of the halogen sublattice, $2X\u2212+h\nu \u2192[X2\u2212](=[h+])+ei\u2212$, where $X$ represents the halogen atoms. (It has been established that holes in alkali halides consist of $X2\u2212$ dimers.^{30}) Because excited electrons are not self-trapped,^{31} the STE formation is viewed as the capture (localization) of excited electrons via the Coulomb interaction with self-trapped holes. Thus, [$O\u2212$] is interpreted as the self-trapped $h+$, and $Hi0$ as the $e\u2212$ captured by $Hi+$ (in place of $h+$). The $Hi0$ state in alkali halides is thought to be stabilized by the antibonding character with halogen atoms^{32} and the bonding character with alkali metals: it should be noted that the excited electron is a dangling bond for the cation. It is well established that the atomic $Mu0$ state observed in alkali halides can be regarded as the counterparts of the $Hi0$ center,^{33} where the electronic excitation is induced by the kinetic energy of incident muon; it is estimated that $\u223c103$$e\u2212$–$h+$ pairs (excitons) are produced from a 4 MeV muon.^{31} In the case of oxides, a process similar to Eq. (3) has been observed in OH-containing $\alpha $-$SiO2$ when exposed to ionizing radiations at low temperatures.^{34–39} It is reasonable to assume that the excited electrons on Si $3p$ orbitals^{25,40} are eventually captured by $Hi+$ (similar to how $h+$ is localized on O).

These findings strongly suggest that $Hi0$ centers (and corresponding $MuA0$ states) exist as the relaxed-excited state, with the electron in the acceptor level. In other words, the $Hi+$ ($MuA+$) state formed immediately after the electronic excitation serves as a center of complex formation analogous to the “acceptor-bound exciton.”^{41} Evidence for the interaction between $Mu+$ and excitons (rather than just electrons) can be found in the blueshift ($\u223c$0.5 eV) of the luminescence from muon-induced STE’s in KBr.^{42,43} This blueshift can be now attributed to the formation of the ionized $Mu0$-bound exciton, where the luminescence occurs between $h+$ and $e\u2212$ bound to $Mu0$ upon the annihilation of $\mu +$ by the beta decay, $[X2\u2212]Mu0\u2192[X2\u2212]ei\u2212\u21922X\u2212+h\nu $, where the lattice relaxation for the $[X2\u2212]ei\u2212$ pair is presumed to be smaller than that for the native STE.

Given that Mu in oxides acts as a trapping center for self-induced free excitons, the initial electronic state of the ambipolar Mu is not limited to the acceptor-like state. Consider the case of mono-oxides, $K$O (with $K$ denoting the divalent cations). The free exciton electrons and holes, denoted as $e\u2212\u2217$, $h+\u2217$, can interact with Mu to form the states corresponding to the $MuA$ and $MuD$ states, i.e.,

These reactions are in line with the fact that the yield of $MuA0$ is actually bottlenecked by the electron supply from ionization trails.^{7,44–46} While both electrons and holes are not self-trapped in many oxides including $Al2O3$, MgO, ZnO, and crystalline $SiO2$, holes are self-trapped in $\alpha $-$SiO2$ and in alkali halides.^{31} In the latter case, the yield of $MuD+$ may depend on the mobility of $MuA0$.

$MuD+$ has a good chance of capturing another electron and converting into $MuD0$ (which may be equivalent with “donor-bound excitons”). In fact, donor-like H/Mu defect centers have been observed in ionic compounds, including oxides. The hydrogen bifluoride complex, $F\u2212$–$H+$–$F\u2212$, is a classical example in the alkali halides. This is also known as a prototype of x-ray-induced strong hydrogen bonding.^{47} The analogous $F\u2212$-$Mu+$–$F\u2212$ complexes have been reported in various alkali fluorides and alkaline earth fluorides. They are readily identified by the characteristic $\mu $SR signal due to a well-defined magnetic dipolar field exerted from the two $19$F nuclei (spin $I=1/2$).^{48} The presumed formation process, $Mu0$ + $h+\u2217$ ($=F2\u2212$) $\u2192$ $F\u2212Mu+F\u2212$ demonstrates the donor-like character. It is worth noting that these donor-like states frequently coexist with $MuA0$.^{33} The presence of two distinct paramagnetic centers corresponding to $MuD0$ and $MuA0$ is well established in elemental (group 4) and group 13–15 compound semiconductors.^{1,49}

This updated “radiolysis model” with an emphasis on the Mu-*exciton* interaction not only provides a microscopic model of $Mu0$ formation but also explains the finite yield of $Mu0$ by electron-supply-limited processes. However, the model fails to account for the increase in the initial $Mu0$ yield with increasing temperature in place of the diamagnetic Mu, for example, in $Lu2O3$.^{50} This resulted in the development of the “thermal spike” model.^{51} In this model, the effective local lattice temperature is presumed to be temporally elevated by phonon excitation (within the time scale of sub-picoseconds) around the Mu stopping position. The thermal spike model has been used to understand the localized damage (e.g., atomic displacements) around ion tracks during irradiation of materials with heavy ions and fission products.^{52} The model for the initial Mu states, on the other hand, is concerned with the kinetic energy range lower than that leading to atomic displacements by the knock-on processes.

Given the preceding discussions, we assume that the initial state of Mu is determined in roughly two steps. First, Mu forms ambipolar relaxed-excited states, which are represented by an adiabatic potential as shown in Fig. 2(a).^{31} The key hypothesis here is that these states correspond to $MuA0$ and/or $MuD0$ in Fig. 1(a), which were originally predicted for H by the *ab initio* DFT calculations. The situation can be characterized by a temporary shift of $EF$ from thermal equilibrium to the region $\Xi +(EF)>\Xi 0(EF)$ or $\Xi \u2212(EF)>\Xi 0(EF)$. The variation of these states with temperature is then interpreted to reflect the degree of relaxation for $EF$ from around $E+/0$ or $E0/\u2212$ towards $E+/\u2212$ within the observation time ($<10\u22125$ s) at each temperature. For instance, regarding $MuD$ in Fig. 2(b), the transition from $MuD0$ to $MuD+$ occurs as the temporal $EF$ decreases from the middle between $E+/0$ and $Ec$ to the equilibrium level (either $E+/\u2212$ of Mu/H or other impurities, leaving the $E+/0$ level empty) with increasing temperature ($kBT\u226bEc\u2212E+/0$). This is interpreted as an electron being promoted from the $E+/0$ level to the conduction band [see Fig. 1(d)]. Therefore, if $E+/0$ is located within the bandgap, $MuD0$ can be realized as the initial state. Meanwhile, if $E+/0$ is in the conduction band ($E+/0>Ec$) and there is no barrier associated with charge conversion, $MuD0$ will immediately ionize and take the $MuD+$ state [Fig. 1(c)], meaning that it behaves as an $n$-type impurity regardless of temperature. The same is true for the $p$-type activity of Site-$A$ [Figs. 2(c), 1(e), and 1(f)].

Furthermore, if the $MuD$ and $MuA$ states are separated on the configuration coordinate frame by an energy barrier $V$ (and $V\u2032$) [Fig. 2(a)], then Mu can take two corresponding electronic states as initial states at low temperatures ($V,V\u2032\u226bkBT$). The yield of each state is proportional to its relative density, which is also temperature dependent (see Appendix C). Recently, an attempt was made to evaluate this potential from experimentally observed yields of $Mu0$ and $Mu+$ in $Lu2O3$, assuming that the relative yields of these states are determined by the potential similar to that shown in Fig. 2(a), within a short time from muon stopping ($\u223c10\u221212$ s) to the completion of the lattice relaxation ($\u223c10\u221210$ s).^{53} When $V\u22640$ or $V\u2032\u22640$, only one of these will be realized as the initial state, and its ionization is observed with increasing temperature. Since such initial states cannot be readily realized in an experiment for H under normal conditions, it is a major advantage of Mu study to allow direct access to $E+/0$ and $E0/\u2212$ levels.

## IV. ELECTRONIC STRUCTURE OF Mu DEFECTS AS RELAXED EXCITED STATES

### A. A brief overview

In conventional oxides with strong ionic bonding, the bottom of the conduction band is dominated by the cation $s$ band, while the top of the valence band is dominated by the O $2p$ band. As a result, the electronic states associated with H/Mu-related Site-$D$ and -$A$ in Fig. 1 can be understood qualitatively as those governed by interactions with the ligand oxygen ($O2\u2212$) and cation ($Kn+$), respectively. Figure 3 illustrates the schematic local structure of Mu as assumed here. The typical state at Site-$D$ is that associated with interstitial Mu forming OH bonds with oxygen [Figs. 3(a)–3(c)]. In Fig. 1, the bonding orbital between O $2p$ and Mu $1s$ ($\sigma D$) is filled with two covalent electrons and sinks to a deep position in the valence band, while the antibonding orbital ($\sigma D\u2217$) is pushed up to the conduction band.

The hybridization of the $\sigma D\u2217$ orbital and the conduction band determines the state of the remaining one electron, and if the hybridization with $s$-$p$-like orbitals is strong, the electron enters the bottom of the conduction band ($Ec$), which contributes to conductivity regardless of temperature: $O2\u2212+Mu0$ $\u2192OMu\u2212(=MuD+)+e\u2212$ [Fig. 3(a)]. Meanwhile, when the bandgap ($Eg\u2261Ec\u2212Ev$) is large and hybridization is weak, Mu strengthens the character of the isolated center and $E+/0$ falls within the gap [Figs. 3(b) and 3(c)]. Even in this case, when $E+/0$ is close to the bottom of the conduction band ($Ec\u2212E+/0\u2272kBT$), the bound electron of Mu$D0$ is thermally excited to the conduction band at a finite temperature, and the valence state change is observed at elevated temperatures.

In the case of Site-$A$ in Fig. 3(d) [see also Fig. 1(b)], this state is stabilized by the formation of multiple bonding with cations (and/or by the antibonding character of hybridization with oxygen 2$p$ band, as discussed for $Hi0$ in alkali halides). The electron can enter the $K$Mu molecular bond orbitals and act as an acceptor; when the associated bond level ($\sigma A$) is close to the valence band ($E0/\u2212\u2212Ev\u2272kBT$), Mu is promoted to a hydride ($H\u2212$)-like state by accommodating the second electron and supplying a hole to the valence band top [Fig. 3(e)], i.e., $KMu0\u2192KMu\u2212$ (=$MuA\u2212$) $+h+$. Mu is observed as an atomic $MuA0$ when the corresponding electron level is near the center of the gap. As discussed below, the $Mu0$ states observed in wide-gap oxides with large hyperfine parameters are interpreted as this state, where the multipolar interaction with cations is weakest.

The investigation of Mu in oxides began with insulators with large $Eg$, such as $SiO2$,^{54,55} $Al2O3$,^{56} or MgO,^{55,57} where atomic Mu$0$ states were observed with hyperfine interactions as large as those found in vacuum [$|A|=Avac=4463.30$ MHz; see Eq. (B5) in Appendix B]. Following studies have found atomic $Mu0$ almost without exception in insulators with $Eg$ values generally above 6–7 eV.^{58–63} However, there are two issues with comprehending these experimental results. One is that, as previously stated, $H+$ or $H\u2212$ is supposed to be always more stable than $H0$ regardless of $EF$ due to its negative-$U$ character, which contradicts the fact that $Mu0$ is observed (given that Mu were also in thermal equilibrium). Another point is that the diamagnetic Mu ($Mu+$ or $Mu\u2212$) is observed to coexist with $Mu0$ in many of these materials (e.g., in $SiO2$, the yields of $Mu0$ and $Mu+$ are $\u223c$65% and $\u223c$35%, respectively^{55}), but the origin of these diamagnetic components remains unknown.

These two issues can be solved by assuming that the observed electronic states of Mu correspond to the non-equilibrium states and that Mu can simultaneously adopt donor- and acceptor-like states near $E+/0$ and $E0/\u2212$ shown in Fig. 3. In the case of $SiO2$, the position of $E+/0$ and $E0/\u2212$ inferred from DFT calculations for H suggests that the diamagnetic Mu and $Mu0$ state correspond to $MuD+$ [Fig. 3(a)] and $MuA0$ [Fig. 3(d)], respectively (see Sec. IV B for more details).

In contrast to the preceding examples, the first example of $Mu0$ with a “shallow donor level” ($0<Ec\u2212E+/0\u226aEg$) was discovered relatively recently in ZnO.^{64,65} This discovery was motivated by the prediction of a DFT calculation that laid $E+/\u2212$ around 0.4 eV below $Ec$,^{66} resulting in an extensive search for shallow donor H/Mu in oxides. According to an earlier report of the $\mu $SR study on powder ZnO samples, a single $Mu0$ state was observed with the hyperfine parameter described by $A(\theta ,\varphi )=A\u2217+Dcos2\u2061\theta $ with $A\u2217=0.50(2)$ MHz and $D=0.26(2)$ MHz, respectively.^{64} In this case, $\theta $ ($\varphi $) is the polar (azimuthal) angle with respect to the symmetry axis of $A$ [see Eq. (B9) in Appendix B for more details].

Aside from the fact that the values of $A\u2217$ and $D$ are comparable, indicating that the hyperfine interaction is clearly anisotropic, the value of $A\u2217$ is orders of magnitude smaller than that of atomic $Mu0$ in vacuum ($A\u2217/Avac\u224310\u22124$), which led to the consensus that the electronic state is qualitatively understood by the effective mass model with a large Bohr radius [corresponding to Fig. 3(b); see Eq. (B6) in Appendix B]. To quantify the origin of the hyperfine interaction, we consider the Fermi contact term $Ac$ and the dipole field $Ad$ from the localized moment on the symmetry axis, resulting in the hyperfine parameters are expressed in the following form:

where the second term corresponds to the case in which the principal axis of the tensor $A^d$ representing the electronic dipole field is taken in the $z$ direction [see Eq. (B2) in Appendix B]. Then, using the relationships $A\u2217=Ac\u2212Ad/2$ and $D=3Ad/2$, we can calculate $Ac=0.579(9)$ MHz and $Ad=0.177(5)$ MHz. These values imply that the electrons associated with $Mu0$ are rather close to the intermediate situation between Figs. 3(b) and 3(c). The electron responsible for $Ad$ is assumed to be in the 4$s$ orbital of Zn with a high degree of delocalization. The fact that $Ad$ takes a value comparable to $Ac$ suggests the formation of an off-center polaron state in which the centers of positive and negative charges do not match.

Subsequent measurements on single crystals revealed two distinct Mu$0$ states in which the angular dependence of the hyperfine interaction was isotropic with respect to the rotation of the crystal (wurtzite type) around the $\u27e80001\u27e9$ axis. This implies that the local structure of these two $Mu0$ states corresponds to the bond-center and antibonding positions along the Zn–O bond parallel to the $\u27e80001\u27e9$ axis,^{65} which is consistent with theoretical predictions.^{66} From the standpoint of ambipolarity, the two observed states may correspond to donor/acceptor-like states, with the bond-center $Mu0$ tentatively assigned to $MuD0$ and another at the antibonding position surrounded by Zn to $MuA0$. This raises an additional question about the origin of the diamagnetic Mu that coexists with these two $Mu0$ states at low temperatures, which will be addressed later (see Sec. IV C).

A typical example corresponding to the limit of the off-center polaron state [$Ad\u226bAc$, Fig. 3(c)] is the shallow donor-like $Mu0$ state observed in $TiO2$ (rutile). This is a complex state involving Mu, O, and Ti, with the accompanying electron loosely localized in the 3$d$-orbitals of neighboring Ti atoms.^{67,68} In a previous ENDOR study, an H-related paramagnetic center with a similar electronic structure was discovered in a chemically reduced sample.^{69} In such an electronic state, the hyperfine interaction is dominated by the magnetic dipole interaction [the second term of Eq. (6)], so that the hyperfine parameters may satisfy the relation $TrA^d=0$. In fact, this relationship is nearly satisfied for $TiO2$, although there are minor differences in the literature.^{67–69} Notably, the size of the localized moment estimated from $Ad$ is only about $\u223c0.05\mu B$, implying that the state is more extended (larger polaron-like, or with a greater distance between $Mu+$ and $e\u2212$). Furthermore, the emergence of the second $Mu0$ state with a greater $Ad$ below $\u223c$5 K coexisting with the diamagnetic Mu^{68} recalls the situation in ZnO. Recent $\mu $SR studies in $SrTiO3$ have reported a similar electronic state,^{70} where the localized electron moment at the Ti site is as large as $\u223c0.33\mu B$, suggesting a more strongly localized state (small polaron-like) than $TiO2$.

Interestingly, all of the charge-transition levels ($E+/\u2212$, $E+/0$, and $E0/\u2212$) inferred from previous first-principles DFT calculations for H in ZnO, $TiO2$, and $SrTiO3$ lie within the conduction band.^{14,66,71} In contrast to the experimental observations, a naïve application of our model to those oxides would predict only $Mu+$. These disparities were previously regarded as individual anomalies, but we will show in Sec. IV C that considering the polaronic state leads to an alternative model for the origin of these shallow states.

### B. Electronic state of implanted Mu determined by E^{+/0} and E^{0/−}

As shown in Fig. 4, the quoted results of DFT calculations on the interstitial H in various oxides can be qualitatively classified into four patterns based on the $EF$ dependence of $\Xi q(EF)$, and the relationship between $E+/\u2212$, $E+/0$, $E0/\u2212$, and the band structure. When $E+/\u2212$ acts as the pinning level for $EF$ (i.e., there is a significant amount of H present), the electric activity of H is determined by the relationship between $E+/\u2212$ and $Ec$ (measured from $Ev$). Figures 4(a) and 4(b) represent the deep $E+/\u2212$ level, Fig. 4(c) represents the shallow $E+/\u2212$ level, and Fig. 4(d) represents the case where there is no level in the gap and only $H+$ is stable.

Early DFT studies suggested that in oxides, $E+/\u2212$ is aligned at a certain energy measured from the vacuum level ($E+/\u2212\u2212Evac\u223c\u22123$ eV).^{72} This model has also been applied to Mu cases in the literature,^{58,73} predicting the presence of shallow donor states in $Bi2O3$, HgO, $Sb2O3$, and other oxides, but no such state has been actually found. However, by considering $EF$ vs $E+/0$ and $E0/\u2212$ instead of $E+/\u2212$ and that Mu is in the relaxed-excited states associated with $E+/0$ and $E0/\u2212$, a coherent interpretation for Mu becomes conceivable.

Early investigations of Mu in highly covalent semiconductors, where both donor- and acceptor-like $Mu0$ states (equivalent to $MuD0$ and $MuA0$, respectively) were found to coexist,^{1,49} are a seminal example of the necessity of considering $E+/0$ and $E0/\u2212$. The charge state changes of $MuD0$ and $MuA0$ were interpreted as related with $E+/0$ and $E0/\u2212$, respectively, and it was revealed that $E+/\u2212$ calculated by the interpolation of $E+/0$ and $E0/\u2212$ using Eq. (2) was located around the charge neutral level ($ECNL$) common to materials in question.^{74} While the focus of this study was on whether or not the position of $E+/\u2212$ is material independent, the results indicate that the change transition (ionization) of $Mu0$ is dependent on $E+/0$ and $E0/\u2212$ rather than $E+/\u2212$; note that the estimated $E+/0$ and $E0/\u2212$ were also in semi-quantitative agreement with the predictions of DFT calculations.^{3} Let us now examine if this assumption leads to a consistent understanding of the Mu valence state in oxides.

In the left columns of Table I, the experimental findings of Mu in different oxides for which DFT calculations have been made are given in decreasing order of the size of $Eg$, and the observed electronic states of Mu are shown. The energies $Eg$, $E+/0$, $E0/\u2212$, and $\epsilon \xb1$ [see Fig. 4(a) for the definition] calculated by DFT are displayed in the right columns. These compounds are a subgroup of oxides in which the conduction and valence bands are made up of empty cation $ns0$ and O 2$p$ orbitals (represented by the LACO model in Fig. 1).

Figure 5 shows $E+/0$ and $E0/\u2212$ vs band structure, where all energy levels are aligned to the vacuum level by considering the electron affinity.^{14} Because the local structures of Mu might differ between $E+/0$ and $E0/\u2212$ levels (e.g., $Y2O3$^{62}) we refer to $MuD+$ and $MuA\u2212$, respectively, as illustrated in Figs. 3(a)–3(e). The electronic state of Mu observed near $E+/0$ is then assigned to $MuD0$ or $MuD+$, and that near $E0/\u2212$ is to $MuA0$ or $MuA\u2212$. The second column from the right predicts the initial state of Mu at low temperatures based on the location of the $E+/0$ and $E0/\u2212$ levels in the relevant band structures (which are illustrated in Fig. 5). If the transition barrier between $MuD$ and $MuA$ ($V$, $V\u2032$ in Fig. 2) is large enough, both $MuD0$ and $MuA0$ may be observed for the case of $\epsilon +>0$ and $\epsilon \u2212>0$ [Figs. 4(a) and 4(b)], $MuA0$ and $MuD+$ for $\epsilon +<0$ and $\epsilon \u2212>0$ [Fig. 4(c)], and only $MuD+$ for the case of Fig. 4(d). (When $V$ or $V\u2032\u22640$, only $MuD$ or $MuA$ is observed.)

It is clear from the left columns in Table I that $Mu0$ with large hyperfine parameters (denoted as $MuX0$) are observed in oxides with $Eg$ greater than $\u223c$5 eV, which is in excellent agreement with the prediction of $MuA0$ based on the $E0/\u2212$ levels. The stable existence of neutral states in these oxides is consistent with DFT calculations, because they infer that $E0/\u2212$ is situated far above the valence band top ($\epsilon \u2212\u22730.3$ eV); the possibility for $MuA0$ to be promoted to $MuA\u2212$ by capturing holes would be negligible even at ambient temperatures. (This also supports the attribution of the diamagnetic Mu to $MuD+$ below.)

Except for BeO, the diamagnetic Mu states are found in all of these oxides, and they are reasonably ascribed to $MuD$. [According to the DFT calculation, BeO corresponds to Fig. 4(a) where the fact that only a single $Mu0$ is experimentally observed suggests $V$ or $V\u2032<0$; see Fig. 2(a).] Among those in which the calculated $E+/0$ is in the bandgap to predict $MuD0$ states ($\u22c6$ in Table I), $SiO2$, $ZrO2$, and $Y2O3$ have indirect gaps smaller than $Eg$ around the $\Gamma $ point in the energy band structure, where the bottoms of the dispersive bands extend below (or near) the $E+/0$ level.^{62,88,89} The dashed parabolic arcs in Fig. 5 reflect the energy extent of the band dispersion around the $\Gamma $ point figuratively. It is worth noting that electrons at the $E+/0$ level can be promoted over the indirect gap in thermal activations via the “umklapp” scattering. As a result, the diamagnetic states in these oxides are also interpreted as $MuD+$. Another exceptional case is $\alpha $-$Al2O3$ (♦ in the Table I), for which a recent study reveals the presence of intricate interactions between Mu, phonons, and excitons.^{90}

Thus, we conclude that the $Mu0$ and diamagnetic Mu in oxides with $Eg\u22735$ eV can be, respectively, attributed to $MuA0$ and the $MuD+$ states which are associated with $E0/\u2212$ and $E+/0$ [corresponding to Figs. 4(b) or 4(c)].

In contrast to the case of wide-gap oxides, the predicted $E+/0$ levels in those with a bandgap of less than $\u223c$5 eV fall in the conduction band ($\epsilon +\u22640$) without exception. Moreover, the $E0/\u2212$ level is deep in the gap (large value of $\epsilon \u2212$), merging to $E+/0$ with decreasing $Eg$ (except for $\beta $-$Ga2O3$ and $In2O3$). Therefore, only the $Mu+$ state is considered to be stable in these materials. However, there are a number of cases where the $Mu0$ state, which is regarded as a shallow donor, is observed experimentally. We will argue in Sec. IV C that these can be understood consistently by considering a common feature that the $Mu0$ state exhibits polaron-like electronic structures and by taking into account the strong electron–phonon coupling exhibited by the host.

Regarding the situation shown in Fig. 4(c), where DFT calculation predicts $E0/\u2212$ to be above the midgap (corresponding to $\beta $-$Ga2O3$ and $In2O3$), it is necessary to know the position of $ECNL$ before discussing whether $E0/\u2212$ can exist as an acceptor level within the bandgap. It has been argued that $ECNL$ is almost equivalent to $E+/\u2212$ for H in binary compounds^{3} and in oxides,^{14,91} which is consistent with experimental evidence for Mu in semiconductors with a relatively narrow gap.^{80} It is reasonable to infer at this stage that the condition $ECNL\u2243E+/\u2212\u2265E0/\u2212$ holds true for H/Mu in general. (This may alternatively be seen as H, with its ambipolarity, probing the charge neutral level via its own charge state.) Thus, $E0/\u2212$ is still expected to be present in these materials. In fact, our recent $\mu $SR study on $\beta $-$Ga2O3$ suggests the presence of two distinct Mu states, one of which seems to be linked with the $E0/\u2212$ level.^{92}

### C. Polaron states mimicking donor-like Mu

The electronic state of $Mu0$ that behaves like a shallow donor (abbreviated as $MuS0$, which is observed in oxides with $Eg\u22725$ eV) is spatially extended, and anisotropy in hyperfine interactions provides insights for the origin of these states. As previously stated in Sec. IV, those in ZnO, $TiO2$, and $SrTiO3$ studied, thus, far all have an off-center polaronic character. The values of the hyperfine parameters in these materials are summarized in Table II, where they are represented using Eq. (6). In each case, $Ad$ is comparable to or greater than $Ac$, indicating that the simple effective mass model of atomic H with a large Bohr radius is insufficient for explaining such electronic structures.

. | A_{c} (MHz)
. | A_{d} (MHz)
. | Reference . |
---|---|---|---|

ZnO | 0.579(9) | 0.177(5) | 65 |

0.436(12) | 0.286(7) | ||

TiO_{2} | −0.06(5) | 0.86(6)^{a} | 67 |

SrTiO_{3} | 1.4(3) | 15.5(2) | 70 |

GaN | 0.079(22) | 0.258(22) | 86 |

. | A_{c} (MHz)
. | A_{d} (MHz)
. | Reference . |
---|---|---|---|

ZnO | 0.579(9) | 0.177(5) | 65 |

0.436(12) | 0.286(7) | ||

TiO_{2} | −0.06(5) | 0.86(6)^{a} | 67 |

SrTiO_{3} | 1.4(3) | 15.5(2) | 70 |

GaN | 0.079(22) | 0.258(22) | 86 |

^{a}

0.05*μ*_{B} on the nearest neighbor Ti.

Interestingly, these electronic states show distinct similarities to the off-centered STE in alkali halides, which consists of the hole localized on a halogen dimer ($X2\u2212$, also known as $Vk$ centers) and the electron at the halogen vacancy (equivalent to the F center) located next to the $X2\u2212$ dimer.^{30,31,93,94} If we consider $OMu\u2212$ as an analog of the hole-localized halogen dimer, the electron attracted to it is presumed to avoid the Coulomb repulsion from neighboring anions by being isolated at the cation. Thus, the electronic structure of $MuS0$ may be interpreted as a compromise among the strong electron–phonon coupling that favors the electron localization, the Coulomb attraction from $Mu+$, and the Coulomb repulsion from the neighboring $O2\u2212$. In other words, $MuS0$ is a STE-like state involving Mu, mimicking the shallow donor $MuD0$ state.

This similarity suggests that the Mu-exciton interaction discussed earlier contributes to the formation of $MuS0$, despite that the local charge polarity of $OMu\u2212$ (=$MuD+$) is opposite to the interstitial $MuA+$. Indeed, excitons bound to various donor/acceptor impurities have been found in ZnO and $TiO2$, and their local electronic structure has been investigated using photoluminescence spectroscopy.^{95–97} A very recent report provides a variety of H-related bound excitons in ZnO,^{98} although the one corresponding to the bond-center $H0$ appears to be lacking (probably due to the small yield). Therefore, the polaron-like bound states observed in these materials can be qualitatively understood, including the reason for the off-centered electronic structure, by adopting a reversed viewpoint that $OMu\u2212/OH\u2212$ serves as an electron trap to form a complex state analogous to the donor-bound exciton.^{41}

The electron–phonon interaction promotes electron localization, which has been inferred to be strong in ZnO,^{99} $TiO2$,^{100} and $SrTiO3$^{101} from the decrease of electron mobility at higher temperatures. In contrast to these cases (including ZnO), pristine a-IGZO exhibits a monotonous increase in carrier mobility with increasing temperature,^{102} in which no shallow state is observed for Mu localized around the Zn–O bond center.^{78} Given that the local atomic arrangement of Mu/H in ZnO and IGZO is nearly identical,^{78} the strong electron–phonon interaction is another essential factor in enhancing electron localization around the $OMu\u2212$ complex. This invokes the precaution that the activation energy for the promotion from $MuD0$ to $MuD+$ cannot be attributed only to $\epsilon +$; it also includes the contribution of the adiabatic potential barrier for the electron between the localized and continuous states.^{70} (A similar argument might be made for $\alpha $-$Al2O3$ in Sec. IV B.) The effect of the electron–phonon coupling has also been studied theoretically in terms of the duality of the conduction band carriers in $TiO2$ and $SrTiO3$, which can be in both the continuous and the deep level (self-trapped) state in the gap.^{103}

More generally, when the donor-like $Mu0$ ($MuD0$) is observed experimentally despite that the DFT calculation predicts $E+/0$ is in the conduction band ($\epsilon +\u22720$), the electron localization at $MuD+$ enhanced by the electron–phonon coupling might be the primary cause. This is not confined to oxides; it is also likely for the “shallow donor $Mu0$” observed in GaN, for example^{86} (bottom row of Table I). Previous theoretical calculations suggested that $\epsilon +\u223c0$, and interstitial H in GaN does not form shallow donor levels.^{11} (It was also a difficulty that the electrical state of Mu was described in respect to $E+/\u2212$, which was positioned deep in the bandgap.) The reported hyperfine interaction, on the other hand, exhibits $c$-axis symmetry with $A\u2225=0.337(10)$ MHz and $A\u22a5=\u22120.243(30)$ MHz. Deriving the parameters of Eq. (6) from these values yields the values indicated in Table II, suggesting that the magnetic dipole interaction is dominant, as in $TiO2$. The suppression of hole mobility in GaN at high temperatures indicated the presence of strong electron–phonon interaction.^{104} Photoluminescence spectroscopy has also proven the existence of H-exciton interaction.^{105} These circumstances imply that $Mu0$ in GaN is akin to a donor-bound exciton, in which an electron is localized near $MuD+$ via electron–phonon coupling to simulate the shallow donor state.

Now, assuming that the bound-exciton model correctly describes the $MuS0$ state, it is envisaged that electron localization around Mu will be dictated by stochastic processes that may be influenced by the presence of other impurities and defects nearby. As a result, the diamagnetic Mu in these oxides can be attributed to the isolated $MuD+$ predicted by DFT calculations, with the relative yield between $MuS0$ and $MuD+$ presumably governed by the local carrier (exciton) density.

The $E+/\u2212$ levels derived by DFT calculations for oxides not only contradict the finding of neutral $Mu0$ states but also do not correlate to “shallow” donor levels. Our proposed model departs from such a naïve interpretation in that the shallow donor-like $Mu0$ corresponds to a relaxed-excited state, where the bound state is formed by the capture of the exciton electron to the $E+/0$ level through strong electron–phonon interactions.

At this point, it may be worth commenting on the assertion of $Mu0$ with shallow levels in $SnO2$.^{58,80} The reported hyperfine parameter is as small as $A/2=0.045(1)$ MHz with a small relative yield ($\u223c$3.6%),^{80} and it is unclear how it was distinguished from slow depolarization due to nuclear magnetic moments ($\Delta \u22430.03$ MHz;^{58} see Appendix A for the definition of $\Delta $) and its apparent reduction due to muon diffusion (which mimics the promotion of $Mu0$ to $Mu+$). The essential fact is that the bulk of the implanted Mu is in a diamagnetic, which may be attributed the donor-like $Mu+$ state.

Finally, we discuss the implications of these findings for the fields of magnetism for local spin systems where $\mu $SR is commonly employed. The electrons associated with Mu are localized in the $3d$-orbitals of the Cr/Fe ions, as revealed by a detailed comparison between the magnitude of the internal magnetic field at the Mu position observed in their magnetically ordered phase and the local electronic state predicted by DFT calculations for in $Cr2O3$ and $Fe2O3$.^{106,107} Because the electron–phonon coupling is known to be strong in these materials, these off-centered electronic states suggest that a similar mechanism is at work in the formation of polarons. In any case, this means that muons can change the valence of the nearest neighboring magnetic ions, and, thus, the magnitude of the magnetic moment may not always be evaluated from the magnitude of the internal magnetic field.

## V. CONCLUSION

In this study, we propose a model that coherently explains the behavior of Mu and H in wide-gap oxides. In order to build a realistic model that is compatible with the experimental observation of neutral $Mu0$ states, non-equilibrium factors related with Mu-exciton interactions must be included. This leads to the conclusion that the information gained from Mu is about the donor/acceptor levels ($E+/0/E0/\u2212$) rather than the equilibrium charge-transition levels ($E+/\u2212$). The agreement between the electronic states of Mu predicted by the position of $E+/0$ and $E0/\u2212$ levels in the band structure estimated by first-principles DFT calculations for H and those experimentally seen for Mu in oxides supports the conclusion. Furthermore, by developing a model that allows for such a systematic understanding, it has been shown that $MuD+$-bound excitons are the origin of “shallow donor”-like Mu states in conventional transparent oxides ($Eg\u22725$ eV), as inferred from their STE-like electronic structure. This will result in a significant shift in our knowledge on the electronic structure of Mu as well as H in oxides.

## ACKNOWLEDGMENTS

The works of the authors quoted in this paper were conducted in collaboration with many colleagues. We would like to appreciate helpful discussions with K. Asakura, K. Fukutani, K. Ide, S. Iimura, T. U. Ito, W. Higemoto, Y. Kamiya, R. F. Kiefl, K. M. Kojima, R. L. Lichti, W. A. MacFarlane, S. Matsuishi, H. Miwa, F. Oba, N. Ohashi, T. Ohsawa, T. Prokscha, J. Robertson, M. Saito, K. Shimomura, A. L. Shluger, A. Suter, S. Tsuneyuki, and R. Vilão. This work was supported by the MEXT Elements Strategy Initiative to Form Core Research Center for Electron Materials (Grant No. JPMXP0112101001), the JSPS KAKENHI (Grant Nos. 19K15033 and 17H06153), the Core-to-Core Program (No. JPJSCCA20180006), and the JST MIRAI Program (No. JPMJMI21E9).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**M. Hiraishi:** Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (supporting). **H. Okabe:** Data curation (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (equal); Validation (supporting). **A. Koda:** Data curation (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (equal); Validation (equal). **R. Kadono:** Conceptualization (lead); Data curation (equal); Investigation (lead); Methodology (supporting); Project administration (lead); Resources (lead); Supervision (lead); Validation (equal); Writing – original draft (lead). **H. Hosono:** Conceptualization (equal); Funding acquisition (lead); Project administration (supporting); Resources (equal); Supervision (supporting); Validation (equal); Writing – original draft (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

### APPENDIX A: *μ*SR SPECTRUM WITHOUT UNPAIRED ELECTRONS

In the following, we summarize the typical cases of the hyperfine interactions between muon and nuclear/electron spins that is represented by the effective local magnetic field $H(r)$ and the corresponding time variation of the muon spin polarization (time spectrum) to be observed.

Let us first consider non-magnetic materials where there are no unpaired electrons. In these cases, the origin of $H(r)$ is none other than the nuclear magnetic moments of the host. In general, the term “hyperfine interaction” includes both magnetic dipole interaction and the Fermi contact interaction. However, since both nuclei and muons are well localized in the ground state, the interaction between them is mainly magnetic dipole interaction. The Hamiltonian is then given as

where $HZ$ represents the Zeeman interaction for muon and nuclear spins, $S\mu $ is the muon spin, $\gamma \mu =2\pi \xd7135.53$ MHz/T is the gyromagnetic ratio of muon spin, $Ii$ is the nuclear spin at distance $ri$ on the $i$th lattice point, $\gamma I$ is the gyromagnetic ratio of the nuclear spin, and $A^di$ is the magnetic dipole tensor,

representing the hyperfine interaction between muon-nuclear magnetic moments. In the case of zero-external field ($HZ=0$), the effective magnetic field expressed as

is used to obtain the effective Hamiltonian

and the time evolution of the muon spin polarization $P(t)=\u27e8S\mu (0)\u22c5S\mu (t)\u27e9/|S\mu (0)|2$ can be obtained analytically using the density matrix of the muon-nucleus spin system for a small number of nucleons (where $\gamma II\xafi$ is the effective magnetic moment considering the electric quadrupole interaction for $Ii\u22651$).

On the other hand, if the coordination of the nuclear magnetic moment viewed from the muon is isotropic and the number of coordination is sufficiently large ($\u22734$), the classical spin treatment is easier, and the density distribution $n(H)$ of $H(r)$ is approximated by a Gaussian distribution with zero mean value,

Here, $\Delta $ is given by the second moment of $Hd$ as

with $\beta $ taking all $x,y,z$, and the $\alpha $ over the $x,y$ components that are effective for longitudinal relaxation when $z^$ is the longitudinal direction; the $z$ component does not contribute to the relaxation because it gives a magnetic field parallel to the muon spin. In this case, the spin polarization $G(t)$ is given by the motion of one muon spin projected onto $H$ with the angle between the magnetic field $H$ and the $z^$ axis as $\theta $,

which is averaged by $n(H)$ in Eq. (A6) to yield the Kubo–Toyabe function

The magnitude of $\Delta $ is sensitive to the size of the nearest-neighbor nuclear magnetic moment $\gamma II\xafi$ and the distance $ri$ from the muon, and the position occupied by the muon as pseudo-hydrogen can be estimated by comparing the experimentally obtained $\Delta $ with the calculated value at the candidate sites. In particular, in recent years, the reliability of the first-principles calculations based on density functional theory (DFT) have been improved, and by using this method to narrow down the candidate sites, the muon sites can be estimated with higher credibility.

### APPENDIX B: *μ*SR SPECTRUM IN THE PRESENCE OF UNPAIRED ELECTRONS

As in Sec. A, the host is assumed to be a nonmagnetic material. In this case, the unpaired electron originates from that bound to the muon ($Mu0$). In general, the Hamiltonian for the magnetic interaction between muon and unpaired electron on the Mu 1$s$ orbit is given by

where $Hd$ is the muon-nuclear spin system [Eq. (A2)], and $He$ is the Hamiltonian of the electron system with $\gamma e$ being the gyromagnetic ratio of the electron ($=2\pi \xd728.02421$ GHz/T). The first term in $HMu$ is for the Fermi contact interaction, and the second term is for the magnetic dipolar interaction.

Provided that the magnitudes of interactions between nuclear spins and muons/electrons are negligible, Eq. (B1) is a two-spin Hamiltonian whose eigenstates are given by the linear combination of the muon electron spin eigenfunctions $|sz\mu ,sze\u27e9$ ($sz\mu ,sze=\xb11/2$), with four corresponding eigenenergies ($Em$, $m=1$–4). When an external magnetic field $H0$ is applied, the spin rotation signals corresponding to the allowed transitions between these eigenstates,

are observed, where $\omega nm=\omega n\u2212\omega m=(En\u2212Em)/\u210f$ are the spin rotation frequencies and $anm$ are their amplitudes.

Now, taking $z^$ as the main axis of the hyperfine interaction $\omega Mu$ with $\theta $ and $\varphi $ being the polar and the azimuthal angles, Eq. (B3) is expressed as

with which we can sort out the qualitative relationship between the electronic structure of Mu$0$ with surrounding atoms and that of $A(\theta ,\varphi )$.

The reason for the formation of bound states is the relatively weak local dielectric shielding (determined by the permittivity $\epsilon $) that leads to the long-range Coulomb interaction between muons and electrons. If the bound electron is in a 1$s$ orbital-like state, the hyperfine interaction is dominated by the Fermi contact term and is isotropic with positive sign as a whole ($Ax\u2243Ay\u2243Az>0$). In this case, the absolute value of $A$, the effective Bohr radius, and the depth of the bound level are estimated to be

where $m\u2217$ is the effective mass of the electron in the conduction band, $Ry$ is Rydberg’s constant, and $\epsilon \u2032$ is the relative permittivity at zero frequency [$=\epsilon (\omega \u21920)/\epsilon $]. This is thought to be one of the mechanisms by which shallow donor levels are induced by interstitial hydrogen in semiconductors with high permittivity.

However, such a Jellium model is not sufficient for actual materials, and the electronic states of H/Mu are anisotropically distributed due to interactions with surrounding atoms. One such example is $Mu0$ located near the bonding center between host atoms, which has been known for a long time in elemental semiconductors with diamond structure and in group 13–15 compound semiconductors such as GaAs with zinc blende structure.^{49} In these examples, the hosts have a four-coordinate configuration with $sp3$ hybrid orbitals that are strongly covalent, and Mu/H breaks this bond to make a new bond with the anion ($I\u2212$), and the excess electrons become a dangling bond on the cation ($K+$). In this case, the hyperfine interaction has an anisotropy symmetric around the axis connecting the muon and the electron, and by taking the symmetry axis to $z^$, Eq. (B5) is reduced to

### APPENDIX C: A MODEL FOR THE H/Mu SITE OCCUPANCY OF THE ASYMMETRIC DOUBLE-WELL POTENTIAL

We consider a simple model for the temperature dependence of the H/Mu site occupancy in the presence of two sites [Site-$A$ and -$D$; see Fig. 2(a)] with asymmetric double-well potential separated by a potential barrier ($V$ and $V\u2032$). Provided that $V\u2032\u226bV$, the partition function for the two-level system is approximately given by

where $\beta \u22611/kBT$, $nA$ and $nD$ are the degeneracies of the each site in the unit cell. The fractional occupancy of Mu/H for the respective sites in the equilibrium state is then described by

Note that $fD<1$ at finite temperatures ($nAe\u2212\beta V>0$) to reduce the free energy by gaining entropy.

Now, we presume that the initial site occupancy for Mu is that quenched from $T=\u221e$ ($\beta =0$), so that $fA=nA/(nA+nD)\u2261fA0$, $fD=nD/(nA+nD)\u2261fD0$ (i.e., proportional to the number density of available sites). Then, the fractional yields observed by $\mu $SR at the finite temperature correspond to the average fraction of muons over the annealing process from this initial distribution to the thermal equilibrium distribution [Eqs. (C2) and (C3)] in the time scale of $\u223c101$ $\mu $s. Such a relaxation process is generally described by the fluctuation–dissipation theorem within the linear response theory for the macroscopic systems.

However, since the implanted Mu as microscopic entity probes the local fluctuations only, we assume that the observed temperature dependence of $fD$ is determined by the migration from Site-$A$ to Site-$D$ via thermally activated hopping over a potential barrier $V$ [see Fig. 6(a)], where the migration probability is proportional to $e\u2212\beta V$. The observed fraction of muons at Site-$A$ can be approximately given by $fA(T)\u2243fA0(1\u2212e\u2212\beta V)$, which is valid for low temperatures ($V\u2032\u226bkBT$) where the inverse hopping process is negligible. Thus, we have

for the temperature dependence of the Mu occupancy at Site-$D$. Figure 6(b) shows examples of $fD(T)$ given by Eq. (C4) for various $V$.

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