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), 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+/) 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/) predicted by DFT calculations for H. The known experimental results are then explained by the relative position of E+/0 and E0/ 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.

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 10151016/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 (μSR) is one of the techniques used to obtain the relevant information on H by implanting a positively charged muon (μ+, 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. For deliberately avoiding distinction between Mu+ and Mu, 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 2E0. 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 μSR 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.

TABLE I.

Comparison between the valence state of Mu in various oxides and DFT calculations for the interstitial H. E+/0, E0/−, and E+/− are the energy measured from valence band top (Ev) as the origin (so that E0/− = ɛ). MuX0 refers to the deep level Mu0 [corresponding to either donor (X = D) or acceptor (X = A) in DFT calculations], and MuS0 to the shallow donor level Mu0. For these marked by , the discrepancy can be attributed to E+/0 over indirect gaps around the Γ point (see text and Fig. 5) or to the ambiguity in the experimental background mimicking Mu+ signal, while those marked by ¶ exhibits a correlation with the polaronic (off-center) MuS0. For the case of α-Al2O3 (marked by ♦), see text. For the DFT calculations, the code packages, potentials, functionals, and supercell sizes are summarized (see text for Ref. 14).

Expt.DFT Calc.
EgCell sizeEgE0/−E+/−E+/0ɛ+
Mater.(eV)MuReferenceCodes, functionals(atoms)(eV)(eV)(eV)(eV)(eV)MuReference
BeO 10.6 MuX0 58, 59  VASP, GGA, HSE06 72 10.6 5.5 6.09 6.7 3.9 MuA/D0 59  
SiO2 9.0 MuX0, Mu+ 54, 55  CASTEP, GGA, HSE06 33–49 8.7 2.9 5.4 7.9 0.8 MuA0, MuD0 14  
α-Al2O3 8.8 MuX0, Mu+ 56, 75  CASTEP, GGA, HSE06 33–49 8.5 3.0 5.4 7.7 0.8 MuA0, MuD0 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−HfO2 6.0 MuX0, Mu+ 60  CASTEP, GGA, HSE06 33–49 5.8 1.6 4.0 6.3 −0.5 MuA0, MuD+ 14  
q−GeO2 6.0 MuX0, Mu+ 58, 55  CASTEP, GGA, HSE06 33–49 5.6 2.0 4.8 7.2 −1.6 MuA0, MuD+ 14  
Lu2O3 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  
ZrO2 5.5(3) MuX0, Mu+ 61  VASP, GGA, HSE06 96 5.4 2.1 3.5 4.8 0.6 MuA0, MuD0 76  
Y2O3 5.5 MuA0, Mu+ 62  VASP, GGA, HSE06 80 5.9 2.15 3.8 5.5 0.4 MuA0, MuD0 62  
La2O3 5.4(1) MuX0, MuS0 58  CASTEP, GGA, HSE06 33–49 5.2 0.3 3.0 6.2 −1 MuA0, MuD+¶ 14  
β-Ga2O3 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  
SnO2 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 MuD+¶ 81  
α-TeO2 3.4 MuA0, Mu+ 82  VASP, GGA, PBE 96 2.82 0.8 2.2 2.82 MuA0, MuD+ 82  
SrTiO3 3.2 MuS0, Mu+ 83, 70  VASP, GGA, HSE06 135 3.1 >4 3.8 3.5 −0.4 MuD+ 71, 84  
r−TiO2 3.2 MuS0, Mu+ 67, 68  CASTEP, GGA, HSE06 33–49 3.0 2.6 3.1 3.5 −0.5 MuD+ 14  
In2O3 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.
EgCell sizeEgE0/−E+/−E+/0ɛ+
Mater.(eV)MuReferenceCodes, functionals(atoms)(eV)(eV)(eV)(eV)(eV)MuReference
BeO 10.6 MuX0 58, 59  VASP, GGA, HSE06 72 10.6 5.5 6.09 6.7 3.9 MuA/D0 59  
SiO2 9.0 MuX0, Mu+ 54, 55  CASTEP, GGA, HSE06 33–49 8.7 2.9 5.4 7.9 0.8 MuA0, MuD0 14  
α-Al2O3 8.8 MuX0, Mu+ 56, 75  CASTEP, GGA, HSE06 33–49 8.5 3.0 5.4 7.7 0.8 MuA0, MuD0 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−HfO2 6.0 MuX0, Mu+ 60  CASTEP, GGA, HSE06 33–49 5.8 1.6 4.0 6.3 −0.5 MuA0, MuD+ 14  
q−GeO2 6.0 MuX0, Mu+ 58, 55  CASTEP, GGA, HSE06 33–49 5.6 2.0 4.8 7.2 −1.6 MuA0, MuD+ 14  
Lu2O3 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  
ZrO2 5.5(3) MuX0, Mu+ 61  VASP, GGA, HSE06 96 5.4 2.1 3.5 4.8 0.6 MuA0, MuD0 76  
Y2O3 5.5 MuA0, Mu+ 62  VASP, GGA, HSE06 80 5.9 2.15 3.8 5.5 0.4 MuA0, MuD0 62  
La2O3 5.4(1) MuX0, MuS0 58  CASTEP, GGA, HSE06 33–49 5.2 0.3 3.0 6.2 −1 MuA0, MuD+¶ 14  
β-Ga2O3 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  
SnO2 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 MuD+¶ 81  
α-TeO2 3.4 MuA0, Mu+ 82  VASP, GGA, PBE 96 2.82 0.8 2.2 2.82 MuA0, MuD+ 82  
SrTiO3 3.2 MuS0, Mu+ 83, 70  VASP, GGA, HSE06 135 3.1 >4 3.8 3.5 −0.4 MuD+ 71, 84  
r−TiO2 3.2 MuS0, Mu+ 67, 68  CASTEP, GGA, HSE06 33–49 3.0 2.6 3.1 3.5 −0.5 MuD+ 14  
In2O3 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  

In the actual μSR experiment, nearly 100% spin-polarized μ+ is implanted into a sample, and time-dependent spatial asymmetry (20%) of positrons emitted with high probability in the direction of spin polarization is observed upon beta decay. When μ+ 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 τμ=2.198μs), μ+ 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

H/=γμH(r)Sμ=12[2πA(r)]Sμ,
(1)

where γμ is the muon gyromagnetic ratio (=2π×135.54 MHz/T), Sμ 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 μSR frequency spectrum can easily distinguish the paramagnetic state (Mu0) from the diamagnetic Mu (Mu+ or Mu) due to the large difference in H(r) (see  Appendixes A and  B for more details). The distinction between Mu+ and Mu, on the other hand, necessitates high-precision chemical shift measurements (101 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 10τμ (20 μs), with t=0 defined by the arrival time of μ+.

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 104/cm2 per cross section. Muons with a typical incident energy of Tμ4 MeV have a stopping range of about 0.1–1 mm from the sample surface, so their volume concentration is less than 105 muons per cm3. Furthermore, they do not accumulate in the sample, as they disappear quickly (τμ). 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 Tμ along the muon track to the host lattice. This is also implied by experiments using a recently available low-energy muon beam (LEM, Tμ1–30 keV at Paul Scherrer Institute, Switzerland), in which muons are implanted into a region of 101102 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 Tμ,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.

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

Ξq(EF)=Et[Hq]Et[]+qEFnHμH,
(2)

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

Et[Muq]=Et[Hq],

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+/) is lower than the donor and acceptor levels (E+/0, E0/) 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 to H0).11,12 The electronic state of H in the thermal equilibrium is then determined by the relationship among Ξ+(EF), Ξ0(EF), and Ξ(EF). More specifically, only HD+ (EF<E+/) or HA (EF>E+/) will be realized, and, thus, E+/ will be the effective impurity level.

FIG. 1.

(a) Schematic illustrations for the formation energy (Ξq) of Muq (q=0,±) vs Fermi energy (EF), (b) the local structure of a donor-like center (MuD at Site-D, bonded to an oxygen ion) and an acceptor-like center (MuA at Site-A, bonded to K cations) in binary compounds, and (c)–(f) the corresponding band diagrams derived from molecular orbital models (where arrows indicate spin-degrees of freedom for localized electrons); (+/) refers to the equilibrium charge-transition level, (+/0) to the donor level [(c) and (d)], and (0/) to the acceptor level [(e) and (f)].

FIG. 1.

(a) Schematic illustrations for the formation energy (Ξq) of Muq (q=0,±) vs Fermi energy (EF), (b) the local structure of a donor-like center (MuD at Site-D, bonded to an oxygen ion) and an acceptor-like center (MuA at Site-A, bonded to K cations) in binary compounds, and (c)–(f) the corresponding band diagrams derived from molecular orbital models (where arrows indicate spin-degrees of freedom for localized electrons); (+/) refers to the equilibrium charge-transition level, (+/0) to the donor level [(c) and (d)], and (0/) to the acceptor level [(e) and (f)].

Close modal

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 Ξq(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 α 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 μ=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 2×2×2. 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 1), 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 functional19 and Lee, Yang, and Parr correlation functional20 (B3LYP). These hybrid functions could be handled more efficiently if calculations could be performed using local basis set codes such as CRYSTAL21 or CP2K.22 As a couple of helpful recent examples, such calculations (CRYSTAL06 with B3LYP) have been performed for H in Ga2O323 and In2O3,24 although the calculated values of the bandgap are not given. CP2K calculations have been also performed for H in amorphous SiO225 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).

The attempt to interpret the electronic states of Mu by Ξq(EF) vs E+/ 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 μ+ implantation to rest corresponds to a relaxed-excited state upon rapid quenching from infinite temperature [i.e., β1/kBT0 in the partition function Z(β); 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 defects in the photodissociation reaction,

[OH]+hν[O](=[h+])+Hi0,
(3)

where [ ] refers to the anion substitutional site, hν 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+hν[X2](=[h+])+ei, where X represents the halogen atoms. (It has been established that holes in alkali halides consist of X2 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] is interpreted as the self-trapped h+, and Hi0 as the e captured by Hi+ (in place of h+). The Hi0 state in alkali halides is thought to be stabilized by the antibonding character with halogen atoms32 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 103eh+ 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 α-SiO2 when exposed to ionizing radiations at low temperatures.34–39 It is reasonable to assume that the excited electrons on Si 3p orbitals25,40 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 (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 bound to Mu0 upon the annihilation of μ+ by the beta decay, [X2]Mu0[X2]ei2X+hν, where the lattice relaxation for the [X2]ei 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, KO (with K denoting the divalent cations). The free exciton electrons and holes, denoted as e, h+, can interact with Mu to form the states corresponding to the MuA and MuD states, i.e.,

Mu++e[K2+]Mu0=MuA0,
(4)
MuA0+h+[O2]Mu+=MuD+.
(5)

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 α-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, FH+F, 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-Mu+F complexes have been reported in various alkali fluorides and alkaline earth fluorides. They are readily identified by the characteristic μSR signal due to a well-defined magnetic dipolar field exerted from the two 19F nuclei (spin I=1/2).48 The presumed formation process, Mu0 + h+ (=F2) FMu+F 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 Ξ+(EF)>Ξ0(EF) or Ξ(EF)>Ξ0(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/ towards E+/ within the observation time (<105 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+/ of Mu/H or other impurities, leaving the E+/0 level empty) with increasing temperature (kBTEcE+/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)].

FIG. 2.

(a) Adiabatic potential curves for Mu in binary compounds, where each curve represents Sites-D and -A with different valences. (NB: those for Mu+ and Mu depend on EF.) Since interstitial Mu is expected to induce lattice relaxation of the coordinating atoms, the position dependence of the potential is appropriately expressed in terms of the Mu-lattice configuration coordinate Q. If there is a potential barrier between Site-D and A (e.g., V and V between Mu0 states), both are observed simultaneously. (b) and (c) The electronic states of Mu relevant to the respective sites are shown by dashed circles in the Ξq vs EF diagram.

FIG. 2.

(a) Adiabatic potential curves for Mu in binary compounds, where each curve represents Sites-D and -A with different valences. (NB: those for Mu+ and Mu depend on EF.) Since interstitial Mu is expected to induce lattice relaxation of the coordinating atoms, the position dependence of the potential is appropriately expressed in terms of the Mu-lattice configuration coordinate Q. If there is a potential barrier between Site-D and A (e.g., V and V between Mu0 states), both are observed simultaneously. (b) and (c) The electronic states of Mu relevant to the respective sites are shown by dashed circles in the Ξq vs EF diagram.

Close modal

Furthermore, if the MuD and MuA states are separated on the configuration coordinate frame by an energy barrier V (and V) [Fig. 2(a)], then Mu can take two corresponding electronic states as initial states at low temperatures (V,VkBT). 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 (1012 s) to the completion of the lattice relaxation (1010 s).53 When V0 or V0, 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/ levels.

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) 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 (σD) is filled with two covalent electrons and sinks to a deep position in the valence band, while the antibonding orbital (σD) is pushed up to the conduction band.

FIG. 3.

Typical defect structures involving Mu and the associated electronic states; those in (a)–(c) indicate the states where OMu bonds are formed, (d) and (e) indicate the states where Mu forms multiple bonds with cations. Note that (c) is in a polaron state, where electrons are localized in cationic electron orbitals (e.g., d orbitals). (a)–(c) can have n-type activity, whereas (d) and (e) can be p-type active.

FIG. 3.

Typical defect structures involving Mu and the associated electronic states; those in (a)–(c) indicate the states where OMu bonds are formed, (d) and (e) indicate the states where Mu forms multiple bonds with cations. Note that (c) is in a polaron state, where electrons are localized in cationic electron orbitals (e.g., d orbitals). (a)–(c) can have n-type activity, whereas (d) and (e) can be p-type active.

Close modal

The hybridization of the σD 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+Mu0OMu(=MuD+)+e [Fig. 3(a)]. Meanwhile, when the bandgap (EgEcEv) 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 (EcE+/0kBT), the bound electron of MuD0 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 2p band, as discussed for Hi0 in alkali halides). The electron can enter the KMu molecular bond orbitals and act as an acceptor; when the associated bond level (σA) is close to the valence band (E0/EvkBT), Mu is promoted to a hydride (H)-like state by accommodating the second electron and supplying a hole to the valence band top [Fig. 3(e)], i.e., KMu0KMu (=MuA) +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 SiO2,54,55Al2O3,56 or MgO,55,57 where atomic Mu0 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 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) is observed to coexist with Mu0 in many of these materials (e.g., in SiO2, the yields of Mu0 and Mu+ are 65% and 35%, respectively55), 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/ shown in Fig. 3. In the case of SiO2, the position of E+/0 and E0/ 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<EcE+/0Eg) was discovered relatively recently in ZnO.64,65 This discovery was motivated by the prediction of a DFT calculation that laid E+/ 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 μSR study on powder ZnO samples, a single Mu0 state was observed with the hyperfine parameter described by A(θ,ϕ)=A+Dcos2θ with A=0.50(2) MHz and D=0.26(2) MHz, respectively.64 In this case, θ (ϕ) 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 and D are comparable, indicating that the hyperfine interaction is clearly anisotropic, the value of A is orders of magnitude smaller than that of atomic Mu0 in vacuum (A/Avac104), 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:

A(θ,ϕ)=Ac+Ad2(3cos2θ1),
(6)

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=AcAd/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 4s 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 Mu0 states in which the angular dependence of the hyperfine interaction was isotropic with respect to the rotation of the crystal (wurtzite type) around the 0001 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 0001 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 [AdAc, 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 3d-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 0.05μB, implying that the state is more extended (larger polaron-like, or with a greater distance between Mu+ and e). Furthermore, the emergence of the second Mu0 state with a greater Ad below 5 K coexisting with the diamagnetic Mu68 recalls the situation in ZnO. Recent μSR studies in SrTiO3 have reported a similar electronic state,70 where the localized electron moment at the Ti site is as large as 0.33μB, suggesting a more strongly localized state (small polaron-like) than TiO2.

Interestingly, all of the charge-transition levels (E+/, E+/0, and E0/) 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.

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 Ξq(EF), and the relationship between E+/, E+/0, E0/, and the band structure. When E+/ 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+/ and Ec (measured from Ev). Figures 4(a) and 4(b) represent the deep E+/ level, Fig. 4(c) represents the shallow E+/ level, and Fig. 4(d) represents the case where there is no level in the gap and only H+ is stable.

FIG. 4.

Fermi level dependence of the formation energy Ξq of the charged state q (=0,±) obtained by DFT calculations for interstitial H in oxides. The symbols (0/), (+/), and (+/0) correspond to E0/, E+/, and E+/0 (the mutual intersection points among Ξq). The electrical activity of H is determined by the relationship between E+/ vs Ec and Ev, whereas that of Mu is governed by E+/0 and E0/ vs Ec and Ev. The latter is predicted by the sign and value of ε+ and ε in the four patterns, (a) ε+>0 and ε>0, (b) ε+0 and ε>0, (c) ε+<0 and ε>0, and (d) ε+<0 and ε>Eg.

FIG. 4.

Fermi level dependence of the formation energy Ξq of the charged state q (=0,±) obtained by DFT calculations for interstitial H in oxides. The symbols (0/), (+/), and (+/0) correspond to E0/, E+/, and E+/0 (the mutual intersection points among Ξq). The electrical activity of H is determined by the relationship between E+/ vs Ec and Ev, whereas that of Mu is governed by E+/0 and E0/ vs Ec and Ev. The latter is predicted by the sign and value of ε+ and ε in the four patterns, (a) ε+>0 and ε>0, (b) ε+0 and ε>0, (c) ε+<0 and ε>0, and (d) ε+<0 and ε>Eg.

Close modal

Early DFT studies suggested that in oxides, E+/ is aligned at a certain energy measured from the vacuum level (E+/Evac3 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/ instead of E+/ and that Mu is in the relaxed-excited states associated with E+/0 and E0/, 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/. The charge state changes of MuD0 and MuA0 were interpreted as related with E+/0 and E0/, respectively, and it was revealed that E+/ calculated by the interpolation of E+/0 and E0/ 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+/ is material independent, the results indicate that the change transition (ionization) of Mu0 is dependent on E+/0 and E0/ rather than E+/; note that the estimated E+/0 and E0/ 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/, and ε± [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 2p orbitals (represented by the LACO model in Fig. 1).

Figure 5 shows E+/0 and E0/ 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/ levels (e.g., Y2O362) we refer to MuD+ and MuA, 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/ is to MuA0 or MuA. 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/ levels in the relevant band structures (which are illustrated in Fig. 5). If the transition barrier between MuD and MuA (V, V in Fig. 2) is large enough, both MuD0 and MuA0 may be observed for the case of ε+>0 and ε>0 [Figs. 4(a) and 4(b)], MuA0 and MuD+ for ε+<0 and ε>0 [Fig. 4(c)], and only MuD+ for the case of Fig. 4(d). (When V or V0, only MuD or MuA is observed.)

FIG. 5.

The donor/acceptor levels (E+/0/E0/) in Table I plotted on the band structure aligned to the vacuum level. The parabolic curves just beneath the conduction band minimum in SiO2, ZrO2, and Y2O3 indicate the presence of dispersive bands with narrower indirect gaps around the Γ point. E0/ is unknown for c-IGZO, ZnO, and SrTiO3. (α-TeO2 is not included because the electron affinity is unknown.)

FIG. 5.

The donor/acceptor levels (E+/0/E0/) in Table I plotted on the band structure aligned to the vacuum level. The parabolic curves just beneath the conduction band minimum in SiO2, ZrO2, and Y2O3 indicate the presence of dispersive bands with narrower indirect gaps around the Γ point. E0/ is unknown for c-IGZO, ZnO, and SrTiO3. (α-TeO2 is not included because the electron affinity is unknown.)

Close modal

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 5 eV, which is in excellent agreement with the prediction of MuA0 based on the E0/ levels. The stable existence of neutral states in these oxides is consistent with DFT calculations, because they infer that E0/ is situated far above the valence band top (ε0.3 eV); the possibility for MuA0 to be promoted to MuA 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<0; see Fig. 2(a).] Among those in which the calculated E+/0 is in the bandgap to predict MuD0 states ( in Table I), SiO2, ZrO2, and Y2O3 have indirect gaps smaller than Eg around the Γ 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 Γ 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 α-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 Eg5 eV can be, respectively, attributed to MuA0 and the MuD+ states which are associated with E0/ 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 5 eV fall in the conduction band (ε+0) without exception. Moreover, the E0/ level is deep in the gap (large value of ε), merging to E+/0 with decreasing Eg (except for β-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/ to be above the midgap (corresponding to β-Ga2O3 and In2O3), it is necessary to know the position of ECNL before discussing whether E0/ can exist as an acceptor level within the bandgap. It has been argued that ECNL is almost equivalent to E+/ for H in binary compounds3 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 ECNLE+/E0/ 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/ is still expected to be present in these materials. In fact, our recent μSR study on β-Ga2O3 suggests the presence of two distinct Mu states, one of which seems to be linked with the E0/ level.92 

The electronic state of Mu0 that behaves like a shallow donor (abbreviated as MuS0, which is observed in oxides with Eg5 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.

TABLE II.

Fermi contact term (Ac) and magnetic dipole interaction (Ad) in Mu0 [see Eq. (6)]. For ZnO, two states (Mu1,2) have been observed from μSR experiments on single crystals, and the values for each are given.

Ac (MHz)Ad (MHz)Reference
ZnO 0.579(9) 0.177(5) 65  
 0.436(12) 0.286(7)  
TiO2 −0.06(5) 0.86(6)a 67  
SrTiO3 1.4(3) 15.5(2) 70  
GaN 0.079(22) 0.258(22) 86  
Ac (MHz)Ad (MHz)Reference
ZnO 0.579(9) 0.177(5) 65  
 0.436(12) 0.286(7)  
TiO2 −0.06(5) 0.86(6)a 67  
SrTiO3 1.4(3) 15.5(2) 70  
GaN 0.079(22) 0.258(22) 86  
a

0.05μB 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, also known as Vk centers) and the electron at the halogen vacancy (equivalent to the F center) located next to the X2 dimer.30,31,93,94 If we consider OMu 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. 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 (=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/OH 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,99TiO2,100 and SrTiO3101 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 complex. This invokes the precaution that the activation energy for the promotion from MuD0 to MuD+ cannot be attributed only to ε+; 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 α-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 (ε+0), 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 example86 (bottom row of Table I). Previous theoretical calculations suggested that ε+0, 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+/, which was positioned deep in the bandgap.) The reported hyperfine interaction, on the other hand, exhibits c-axis symmetry with A=0.337(10) MHz and A=0.243(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+/ 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 (3.6%),80 and it is unclear how it was distinguished from slow depolarization due to nuclear magnetic moments (Δ0.03 MHz;58 see  Appendix A for the definition of Δ) 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 μSR 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.

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/) rather than the equilibrium charge-transition levels (E+/). The agreement between the electronic states of Mu predicted by the position of E+/0 and E0/ 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 (Eg5 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.

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).

The authors have no conflicts to disclose.

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).

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

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

H/=HZ/+Hd/,
(A1)
Hd/=γμγISμiA^diIi,
(A2)

where HZ represents the Zeeman interaction for muon and nuclear spins, Sμ is the muon spin, γμ=2π×135.53 MHz/T is the gyromagnetic ratio of muon spin, Ii is the nuclear spin at distance ri on the ith lattice point, γI is the gyromagnetic ratio of the nuclear spin, and A^di is the magnetic dipole tensor,

(A^di)αβ=1ri3(3αiβiri2δαβ)(α,β=x,y,z),
(A3)

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

H(r)=Hd=γIiA^diI¯i
(A4)

is used to obtain the effective Hamiltonian

H/=γμSμHd,
(A5)

and the time evolution of the muon spin polarization P(t)=Sμ(0)Sμ(t)/|Sμ(0)|2 can be obtained analytically using the density matrix of the muon-nucleus spin system for a small number of nucleons (where γII¯i is the effective magnetic moment considering the electric quadrupole interaction for Ii1).

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 (4), 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,

n(Hα)=δ(HαHα(r))r=γμ2πΔexp(γμ2Hα22Δ2)(α=x,y,z).
(A6)

Here, Δ is given by the second moment of Hd as

Δ2γμ2=iα,β[γI(A^di)αβI¯i]2,
(A7)

with β taking all x,y,z, and the α 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 θ,

Pz(t)=cos2θ+sin2θcos(γμHt),
(A8)

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

Gz(t)=Pz(t)Παnα(Hα)dHα=13+23(1Δ2t2)e12Δ2t2.
(A9)

The magnitude of Δ is sensitive to the size of the nearest-neighbor nuclear magnetic moment γII¯i 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 Δ 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.

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 1s orbit is given by

H/=[Hd+He+HMu]/,
(B1)
HMu/=γμγeSe[8π3δ(r)+A^d]Sμ
(B2)
=12(2πA)Sμ=12ωMuSμ,
(B3)

where Hd is the muon-nuclear spin system [Eq. (A2)], and He is the Hamiltonian of the electron system with γe being the gyromagnetic ratio of the electron (=2π×28.02421 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 |szμ,sze (szμ,sze=±1/2), 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,

G(t)=n<manmcosωnmt,
(B4)

are observed, where ωnm=ωnωm=(EnEm)/ are the spin rotation frequencies and anm are their amplitudes.

Now, taking z^ as the main axis of the hyperfine interaction ωMu with θ and ϕ being the polar and the azimuthal angles, Eq. (B3) is expressed as

HMu/=12ωMu(θ,ϕ)=122πA(θ,ϕ),A(θ,ϕ)=(Axcos2ϕ+Aysin2ϕ)sin2θ+Azcos2θ,
(B5)

with which we can sort out the qualitative relationship between the electronic structure of Mu0 with surrounding atoms and that of A(θ,ϕ).

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

Aωvac2π[mεme]3,
(B6)
aa0εmme,
(B7)
RRymε2me,
(B8)

where m is the effective mass of the electron in the conduction band, Ry is Rydberg’s constant, and ε is the relative permittivity at zero frequency [=ε(ω0)/ε]. 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), 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

A(θ,ϕ)=A+Dcos2θ,
(B9)

with Ax,yAAD/2 and AzAA+D. The first reported Mu0 with shallow electronic levels were found in II–VI compounds such as zinc oxide (ZnO),64,65 which also exhibits hyperfine interactions well described by Eq. (B9).

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). Provided that VV, the partition function for the two-level system is approximately given by

Z(β)=nD+nAeβV,
(C1)

where β1/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

fA=nAeβVnD+nAeβV,
(C2)
fD=nDnD+nAeβV.
(C3)

Note that fD<1 at finite temperatures (nAeβV>0) to reduce the free energy by gaining entropy.

Now, we presume that the initial site occupancy for Mu is that quenched from T= (β=0), so that fA=nA/(nA+nD)fA0, fD=nD/(nA+nD)fD0 (i.e., proportional to the number density of available sites). Then, the fractional yields observed by μSR 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 101μs. 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 eβV. The observed fraction of muons at Site-A can be approximately given by fA(T)fA0(1eβV), which is valid for low temperatures (VkBT) where the inverse hopping process is negligible. Thus, we have

fD(T)=1fA(T)fD0+fA0eβV,
(C4)

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.

FIG. 6.

(a) Schematic drawing of the two-site model based on the Boltzmann factor. (b) Temperature dependence of Eq. (C4) for various V with fD0=2/3, fA0=1/3.

FIG. 6.

(a) Schematic drawing of the two-site model based on the Boltzmann factor. (b) Temperature dependence of Eq. (C4) for various V with fD0=2/3, fA0=1/3.

Close modal
1.
J.
Pankove
and
N. M.
Johnson
,
Hydrogen in Semiconductors
(
Academic
,
New York
,
1991
).
2.
S. J.
Pearton
,
J. W.
Corbett
, and
M.
Stavola
,
Hydrogen in Crystalline Semiconductors
(
Springer
,
Berlin
,
1992
).
3.
C. G.
Van de Walle
and
J.
Neugebauer
, “
Universal alignment of hydrogen levels in semiconductors, insulators and solutions
,”
Nature
423
,
626
628
(
2003
).
4.
K.
Hayashi
,
P. V.
Sushko
,
Y.
Hashimoto
,
A. L.
Shluger
, and
H.
Hosono
, “
Hydride ions in oxide hosts hidden by hydroxide ions
,”
Nat. Commun.
5
,
3515
(
2014
).
5.
Muon Spectroscopy, edited by S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt (Oxford University Press, 2021).
6.
M.
Hiraishi
,
K. M.
Kojima
,
M.
Miyazaki
,
I.
Yamauchi
,
H.
Okabe
,
A.
Koda
,
R.
Kadono
,
S.
Matsuishi
, and
H.
Hosono
, “
Cage electron-hydroxyl complex state as electron donor in mayenite
,”
Phys. Rev. B
93
,
121201
(
2016
).
7.
T.
Prokscha
,
E.
Morenzoni
,
D. G.
Eshchenko
,
N.
Garifianov
,
H.
Glückler
,
R.
Khasanov
,
H.
Luetkens
, and
A.
Suter
, “
Formation of hydrogen impurity states in silicon and insulators at low implantation energies
,”
Phys. Rev. Lett.
98
,
227401
(
2007
).
8.
Time-Dependent Density Functional Theory, Lecture Notes in Physics Vol. 706, edited by M. A. L. Marques, C. A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E. K. U. Gross (Springer, Berlin, 2006).
9.
C. G.
Van de Walle
and
J.
Neugebauer
, “
First-principles calculations for defects and impurities: Applications to III-nitrides
,”
J. Appl. Phys.
95
,
3851
3879
(
2004
).
10.
P. W.
Anderson
, “
Model for the electronic structure of amorphous semiconductors
,”
Phys. Rev. Lett.
34
,
953
955
(
1975
).
11.
J.
Neugebauer
and
C. G.
Van de Walle
, “
Hydrogen in GaN: Novel aspects of a common impurity
,”
Phys. Rev. Lett.
75
,
4452
4455
(
1995
).
12.
A.
Yokozawa
and
Y.
Miyamoto
, “
First-principles calculations for charged states of hydrogen atoms in SiO2
,”
Phys. Rev. B
55
,
13783
13788
(
1997
).
13.
S.
Lany
and
A.
Zunger
, “
Assessment of correction methods for the band-gap problem and for finite-size effects in supercell defect calculations: Case studies for zno and gaas
,”
Phys. Rev. B
78
,
235104
(
2008
).
14.
H.
Li
and
J.
Robertson
, “
Behaviour of hydrogen in wide band gap oxides
,”
J. Appl. Phys.
115
,
203708
(
2014
).
15.
M. D.
Segall
,
P. J. D.
Lindan
,
M. J.
Probert
,
C. J.
Pickard
,
P. J.
Hasnip
,
S. J.
Clark
, and
M. C.
Payne
, “
First-principles simulation: Ideas, illustrations and the CASTEP code
,”
J. Phys.: Condens. Matter
14
,
2717
2744
(
2002
).
16.
J.
Heyd
,
G. E.
Scuseria
, and
M.
Ernzerhof
, “
Hybrid functionals based on a screened coulomb potential
,”
J. Chem. Phys.
118
,
8207
8215
(
2003
).
17.
J.
Heyd
,
G. E.
Scuseria
, and
M.
Ernzerhof
, “
Erratum: “Hybrid functionals based on a screened Coulomb potential” [J. Chem. Phys. 118, 8207 (2003)]
,”
J. Chem. Phys.
124
,
219906
(
2006
).
18.
Y.
Kumagai
and
F.
Oba
, “
Electrostatics-based finite-size corrections for first-principles point defect calculations
,”
Phys. Rev. B
89
,
195205
(
2014
).
19.
A. D.
Becke
, “
Density-functional exchange-energy approximation with correct asymptotic behavior
,”
Phys. Rev. A
38
,
3098
3100
(
1988
).
20.
C.
Lee
,
W.
Yang
, and
R. G.
Parr
, “
Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density
,”
Phys. Rev. B
37
,
785
789
(
1988
).
21.
R.
Dovesi
,
F.
Pascale
,
B.
Civalleri
,
K.
Doll
,
N. M.
Harrison
,
I.
Bush
,
P.
D’Arco
,
Y.
Noël
,
M.
Rérat
,
P.
Carbonnière
,
M.
Causà
,
S.
Salustro
,
V.
Lacivita
,
B.
Kirtman
,
A. M.
Ferrari
,
F. S.
Gentile
,
J.
Baima
,
M.
Ferrero
,
R.
Demichelis
, and
M.
De La Pierre
, “
The CRYSTAL code, 1976–2020 and beyond, a long story
,”
J. Chem. Phys.
152
,
204111
(
2020
).
22.
T. D.
Kühne
,
M.
Iannuzzi
,
M.
Del Ben
,
V. V.
Rybkin
,
P.
Seewald
,
F.
Stein
,
T.
Laino
,
R. Z.
Khaliullin
,
O.
Schütt
,
F.
Schiffmann
,
D.
Golze
,
J.
Wilhelm
,
S.
Chulkov
,
M. H.
Bani-Hashemian
,
V.
Weber
,
U.
Borštnik
,
M.
Taillefumier
,
A. S.
Jakobovits
,
A.
Lazzaro
,
H.
Pabst
,
T.
Müller
,
R.
Schade
,
M.
Guidon
,
S.
Andermatt
,
N.
Holmberg
,
G. K.
Schenter
,
A.
Hehn
,
A.
Bussy
,
F.
Belleflamme
,
G.
Tabacchi
,
A.
Glöss
,
M.
Lass
,
I.
Bethune
,
C. J.
Mundy
,
C.
Plessl
,
M.
Watkins
,
J.
VandeVondele
,
M.
Krack
, and
J.
Hutter
, “
CP2K: An electronic structure and molecular dynamics software package—Quickstep: Efficient and accurate electronic structure calculations
,”
J. Chem. Phys.
152
,
194103
(
2020
).
23.
Y.
Qin
,
M.
Stavola
,
W. B.
Fowler
,
P.
Weiser
, and
S. J.
Pearton
, “
Hydrogen centers in β-Ga2O3: Infrared spectroscopy and density functional theory
,”
ECS J. Solid State Sci. Technol.
8
,
Q3103
Q3110
(
2019
).
24.
W.
Yin
,
K.
Smithe
,
P.
Weiser
,
M.
Stavola
,
W. B.
Fowler
,
L.
Boatner
,
S. J.
Pearton
,
D. C.
Hays
, and
S. G.
Koch
, “
Hydrogen centers and the conductivity of In2O3 single crystals
,”
Phys. Rev. B
91
,
075208
(
2015
).
25.
A.-M.
El-Sayed
,
Y.
Wimmer
,
W.
Goes
,
T.
Grasser
,
V. V.
Afanas’ev
, and
A. L.
Shluger
, “
Theoretical models of hydrogen-induced defects in amorphous silicon dioxide
,”
Phys. Rev. B
92
,
014107
(
2015
).
26.
M.
Kaviani
,
V. V.
Afanas’ev
, and
A. L.
Shluger
, “
Interactions of hydrogen with amorphous hafnium oxide
,”
Phys. Rev. B
95
,
075117
(
2017
).
27.
F.
Kerkhoff
,
W.
Martienssen
, and
W.
Sander
, “
Elektronenspin-resonanz und photochemie des U2-zentrums in alkalihalogenid-kristallen
,”
Z. Phys.
173
,
184
202
(
1963
).
28.
J. M.
Spaeth
and
M.
Sturm
, “
ESR and ENDOR investigation of interstitial hydrogen atoms in alkali halides
,”
Phys. Status Solidi B
42
,
739
748
(
1970
).
29.
S. P.
Morato
and
F.
Lüty
, “
Hydrogen defects from UV photodissociation of OH centers in alkali halides
,”
Phys. Rev. B
22
,
4980
4991
(
1980
).
30.
K. S. Song and R. T. Williams, Self-Trapped Excitons (Springer, Berlin, 1996).
31.
N.
Itoh
, “
Bond scission induced by electronic excitation in solids: A tool for nanomanipulation
,”
Nucl. Instrum. Methods Phys. Res. Sect. B
122
,
405
409
(
1997
).
32.
K.
Cho
,
H.
Kamimura
, and
Y.
Uemura
, “
Electronic structures of the U2-center in KCl and KBr:II—The effect of the configuration mixing
,”
J. Phys. Soc. Jpn.
21
,
2244
2252
(
1966
).
33.
H.
Baumeler
,
R. F.
Kiefl
,
H.
Keller
,
W.
Kündig
,
W.
Odermatt
,
B. D.
Patterson
,
J. W.
Schneider
,
T. L.
Estle
,
S. P.
Rudaz
,
D. P.
Spencer
,
K. W.
Blazey
, and
I. M.
Savic
, “
Muonium centers in the alkali halides
,”
Hyperfine Interact.
32
,
659
665
(
1986
).
34.
T. E.
Tsai
,
D. L.
Griscom
, and
E. J.
Friebele
, “
Medium-range structural order and fractal annealing kinetics of radiolytic atomic hydrogen in high-purity silica
,”
Phys. Rev. B
40
,
6374
6380
(
1989
).
35.
L.
Verdi
and
A.
Miotello
, “
Hydrogen dimerization process: A probe for investigation of the α-SiO2 structure
,”
Phys. Rev. B
47
,
14187
14192
(
1993
).
36.
T.
Ichikawa
and
V.
Kurshev
, “
Spin–lattice relaxation of the hydrogen atom in a fused quartz
,”
J. Chem. Phys.
99
,
5728
5735
(
1993
).
37.
I. A.
Shkrob
and
A. D.
Trifunac
, “
Time-resolved EPR of spin-polarized mobile H atoms in amorphous silica: The involvement of small polarons
,”
Phys. Rev. B
54
,
15073
15078
(
1996
).
38.
I. A.
Shkrob
and
A. D.
Trifunac
, “
Spin-polarized H/D atoms and radiation chemistry in amorphous silica
,”
J. Chem. Phys.
107
,
2374
2385
(
1997
).
39.
I. A.
Shkrob
,
B. M.
Tadjikov
,
S. D.
Chemerisov
, and
A. D.
Trifunac
, “
Electron trapping and hydrogen atoms in oxide glasses
,”
J. Chem. Phys.
111
,
5124
5140
(
1999
).
40.
D.
Muñoz Ramo
,
P. V.
Sushko
, and
A. L.
Shluger
, “
Models of triplet self-trapped excitons in SiO2, HfO2, and HfSiO4
,”
Phys. Rev. B
85
,
024120
(
2012
).
41.
P. Y. Yu and M. Cardona, Fundamentals of Semiconductors—Physics and Materials Properties (Springer, Berlin, 2005).
42.
R.
Kadono
,
A.
Matsushita
, and
K.
Nagamine
, “
Muonium fluorescence: Anomalous muonium center and relaxed excited state in KBr
,”
Phys. Rev. Lett.
67
,
3689
3691
(
1991
).
43.
R.
Kadono
,
A.
Matsushita
,
K.
Nishiyama
, and
K.
Nagamine
, “
Muon-induced luminescence in KBr
,”
Phys. Rev. B
46
,
8586
8588
(
1992
).
44.
V.
Storchak
,
S. F. J.
Cox
,
S. P.
Cottrell
,
J. H.
Brewer
,
G. D.
Morris
,
D. J.
Arseneau
, and
B.
Hitti
, “
Muonium formation via electron transport in silicon
,”
Phys. Rev. Lett.
78
,
2835
2838
(
1997
).
45.
D.
Eshchenko
,
V.
Storchak
, and
G.
Morris
, “
Electron transport to positive centers in GaAs
,”
Phys. Lett. A
264
,
226
231
(
1999
).
46.
D. G.
Eshchenko
,
V. G.
Storchak
,
S. P.
Cottrell
, and
S. F. J.
Cox
, “
Excited muonium state in CdS
,”
Phys. Rev. B
68
,
073201
(
2003
).
47.
G. A.
Jeffery
,
An Introduction to Hydrogen Bonding
(
Oxford University Press
,
New York
,
1997
).
48.
J. H.
Brewer
,
S. R.
Kreitzman
,
D. R.
Noakes
,
E. J.
Ansaldo
,
D. R.
Harshman
, and
R.
Keitel
, “
Observation of muon-fluorine ‘hydrogen bonding’ in ionic crystals
,”
Phys. Rev. B
33
,
7813
7816
(
1986
).
49.
B. D.
Patterson
, “
Muonium states in semiconductors
,”
Rev. Mod. Phys.
60
,
69
159
(
1988
).
50.
R. C.
Vilão
,
R. B. L.
Vieira
,
H. V.
Alberto
,
J. M.
Gil
,
A.
Weidinger
,
R. L.
Lichti
,
P. W.
Mengyan
,
B. B.
Baker
, and
J. S.
Lord
, “
Barrier model in muon implantation and application to Lu2O3
,”
Phys. Rev. B
98
,
115201
(
2018
).
51.
R. C.
Vilão
,
H. V.
Alberto
,
J. M.
Gil
, and
A.
Weidinger
, “
Thermal spike in muon implantation
,”
Phys. Rev. B
99
,
195206
(
2019
).
52.
M.
Toulemonde
,
J. M.
Costantini
,
C.
Dufour
,
A.
Meftah
,
E.
Paumier
, and
F.
Studer
, “
Track creation in SiO2 and BaFe12O19 by swift heavy ions: A thermal spike description
,”
Nucl. Instrum. Methods Phys. Res. Sect. B
116
,
37
42
(
1996
).
53.
R. C.
Vilão
,
R. B. L.
Vieira
,
H. V.
Alberto
,
J. M.
Gil
, and
A.
Weidinger
, “
Role of the transition state in muon implantation
,”
Phys. Rev. B
96
,
195205
(
2017
).
54.
G. G.
Myasishcheva
,
Y. V.
Obukhov
,
V. S.
Roganov
, and
V. G.
Firsov
, “
A search for atomic muonium in chemically inert substances
,”
Sov. Phys. JETP
26
,
298
301
(
1968
).
55.
D. P.
Spencer
,
D. G.
Fleming
, and
J. H.
Brewer
, “
Muonium formation in diamond and oxide insulators
,”
Hyperfine Interact.
18
,
567
573
(
1984
).
56.
E. V.
Minaichev
,
G. G.
Myasishcheva
,
Y. V.
Obukhov
,
V. S.
Roganov
,
G. I.
Savel’ev
, and
V. G.
Firsov
, “
Paschen-Back effect for the muonium atom
,”
Sov. Phys. JETP
31
,
849
852
(
1970
).
57.
R. F.
Kiefl
,
W.
Odermatt
,
H.
Baumeler
,
J.
Felber
,
H.
Keller
,
W.
Kündig
,
P. F.
Meier
,
B. D.
Patterson
,
J. W.
Schneider
,
K. W.
Blazey
,
T. L.
Estle
, and
C.
Schwab
, “
Muonium centers in the cuprous halides
,”
Phys. Rev. B
34
,
1474
1484
(
1986
).
58.
S. F. J.
Cox
,
J. L.
Gavartin
,
J. S.
Lord
,
S. P.
Cottrell
,
J. M.
Gil
,
H. V.
Alberto
,
J.
Piroto Duarte
,
R. C.
Vilão
,
N.
Ayres de Campos
,
D. J.
Keeble
,
E. A.
Davis
,
M.
Charlton
, and
D. P.
van der Werf
, “
Oxide muonics: II. Modelling the electrical activity of hydrogen in wide-gap and high-permittivity dielectrics
,”
J. Phys.: Condens. Matter
18
,
1079
1119
(
2006
).
59.
A. G.
Marinopoulos
,
R. C.
Vilão
,
R. B. L.
Vieira
,
H. V.
Alberto
,
J. M.
Gil
,
M. V.
Yakushev
,
R.
Scheuermann
, and
T.
Goko
, “
Defect levels and hyperfine constants of hydrogen in beryllium oxide from hybrid-functional calculations and muonium spectroscopy
,”
Philos. Mag.
97
,
2108
2128
(
2017
).
60.
R. B. L.
Vieira
,
R. C.
Vilão
,
H. V.
Alberto
,
J. M.
Gil
,
A.
Weidinger
,
B. B.
Baker
,
P. W.
Mengyan
, and
R. L.
Lichti
, “
High-field study of muonium states in HfO2 and ZrO2
,”
J. Phys.: Conf. Ser.
551
,
012048
(
2014
).
61.
R. B. L.
Vieira
,
R. C.
Vilão
,
A. G.
Marinopoulos
,
P. M.
Gordo
,
J. A.
Paixão
,
H. V.
Alberto
,
J. M.
Gil
,
A.
Weidinger
,
R. L.
Lichti
,
B.
Baker
,
P. W.
Mengyan
, and
J. S.
Lord
, “
Isolated hydrogen configurations in zirconia as seen by muon spin spectroscopy and ab initio calculations
,”
Phys. Rev. B
94
,
115207
(
2016
).
62.
E. L.
Silva
,
A. G.
Marinopoulos
,
R. C.
Vilão
,
R. B. L.
Vieira
,
H. V.
Alberto
,
J.
Piroto Duarte
, and
J. M.
Gil
, “
Hydrogen impurity in yttria: Ab initio and μSR perspectives
,”
Phys. Rev. B
85
,
165211
(
2012
).
63.
E. L.
da Silva
,
A. G.
Marinopoulos
,
R. B. L.
Vieira
,
R. C.
Vilão
,
H. V.
Alberto
,
J. M.
Gil
,
R. L.
Lichti
,
P. W.
Mengyan
, and
B. B.
Baker
, “
Electronic structure of interstitial hydrogen in lutetium oxide from DFT+U calculations and comparison study with μSR spectroscopy
,”
Phys. Rev. B
94
,
014104
(
2016
).
64.
S. F. J.
Cox
,
E. A.
Davis
,
S. P.
Cottrell
,
P. J. C.
King
,
J. S.
Lord
,
J. M.
Gil
,
H. V.
Alberto
,
R. C.
Vilão
,
J.
Piroto Duarte
,
N.
Ayres de Campos
,
A.
Weidinger
,
R. L.
Lichti
, and
S. J. C.
Irvine
, “
Experimental confirmation of the predicted shallow donor hydrogen state in zinc oxide
,”
Phys. Rev. Lett.
86
,
2601
2604
(
2001
).
65.
K.
Shimomura
,
K.
Nishiyama
, and
R.
Kadono
, “
Electronic structure of the muonium center as a shallow donor in ZnO
,”
Phys. Rev. Lett.
89
,
255505
(
2002
).
66.
C. G.
Van de Walle
, “
Hydrogen as a cause of doping in zinc oxide
,”
Phys. Rev. Lett.
85
,
1012
1015
(
2000
).
67.
K.
Shimomura
,
R.
Kadono
,
A.
Koda
,
K.
Nishiyama
, and
M.
Mihara
, “
Electronic structure of mu-complex donor state in rutile TiO2
,”
Phys. Rev. B
92
,
075203
(
2015
).
68.
R. C.
Vilão
,
R. B. L.
Vieira
,
H. V.
Alberto
,
J. M.
Gil
,
A.
Weidinger
,
R. L.
Lichti
,
B. B.
Baker
,
P. W.
Mengyan
, and
J. S.
Lord
, “
Muonium donor in rutile TiO2 and comparison with hydrogen
,”
Phys. Rev. B
92
,
081202
(
2015
).
69.
A. T.
Brant
,
S.
Yang
,
N. C.
Giles
, and
L. E.
Halliburton
, “
Hydrogen donors and Ti3+ ions in reduced TiO2 crystals
,”
J. Appl. Phys.
110
,
053714
(
2011
).
70.
T. U.
Ito
,
W.
Higemoto
,
A.
Koda
, and
K.
Shimomura
, “
Polaronic nature of a muonium-related paramagnetic center in SrTiO3
,”
Appl. Phys. Lett.
115
,
192103
(
2019
).
71.
Y.
Iwazaki
,
T.
Suzuki
, and
S.
Tsuneyuki
, “
Negatively charged hydrogen at oxygen-vacancy sites in BaTiO3: Density-functional calculation
,”
J. Appl. Phys.
108
,
083705
(
2010
).
72.
Ç.
Kılıç
and
A.
Zunger
, “
n-type doping of oxides by hydrogen
,”
Appl. Phys. Lett.
81
,
73
75
(
2002
).
73.
S. F. J.
Cox
,
J. S.
Lord
,
S. P.
Cottrell
,
J. M.
Gil
,
H. V.
Alberto
,
A.
Keren
,
D.
Prabhakaran
,
R.
Scheuermann
, and
A.
Stoykov
, “
Oxide muonics: I. Modelling the electrical activity of hydrogen in semiconducting oxides
,”
J. Phys.: Condens. Matter
18
,
1061
1078
(
2006
).
74.
R. L.
Lichti
,
K. H.
Chow
, and
S. F. J.
Cox
, “
Hydrogen defect-level pinning in semiconductors: The muonium equivalent
,”
Phys. Rev. Lett.
101
,
136403
(
2008
).
75.
R. F.
Kiefl
,
E.
Holzschuh
,
H.
Keller
,
W.
Kündig
,
P. F.
Meier
,
B. D.
Patterson
,
J. W.
Schneider
,
K. W.
Blazey
,
S. L.
Rudaz
, and
A. B.
Denison
, “
Decoupling of muonium in high transverse magnetic fields
,”
Phys. Rev. Lett.
53
,
90
93
(
1984
).
76.
A. G.
Marinopoulos
, “
Incorporation and migration of hydrogen in yttria-stabilized cubic zirconia: Insights from semilocal and hybrid-functional calculations
,”
Phys. Rev. B
86
,
155144
(
2012
).
77.
P. D. C.
King
,
I.
McKenzie
, and
T. D.
Veal
, “
Observation of shallow-donor muonium in Ga2O3: Evidence for hydrogen-induced conductivity
,”
Appl. Phys. Lett.
96
,
062110
(
2010
).
78.
K. M.
Kojima
,
M.
Hiraishi
,
H.
Okabe
,
A.
Koda
,
R.
Kadono
,
K.
Ide
,
S.
Matsuishi
,
H.
Kumomi
,
T.
Kamiya
, and
H.
Hosono
, “
Electronic structure of interstitial hydrogen in In-Ga-Zn-O semiconductor simulated by muon
,”
Appl. Phys. Lett.
115
,
122104
(
2019
).
79.
H.
Li
,
Y.
Guo
, and
J.
Robertson
, “
Oxygen vacancies and hydrogen in amorphous In-Ga-Zn-O and ZnO
,”
Phys. Rev. Mater.
2
,
074601
(
2018
).
80.
P. D. C.
King
,
R. L.
Lichti
,
Y. G.
Celebi
,
J. M.
Gil
,
R. C.
Vilão
,
H. V.
Alberto
,
J.
Piroto Duarte
,
D. J.
Payne
,
R. G.
Egdell
,
I.
McKenzie
,
C. F.
McConville
,
S. F. J.
Cox
, and
T. D.
Veal
, “
Shallow donor state of hydrogen in In2O3 and SnO2: Implications for conductivity in transparent conducting oxides
,”
Phys. Rev. B
80
,
081201
(
2009
).
81.
F.
Oba
,
A.
Togo
,
I.
Tanaka
,
J.
Paier
, and
G.
Kresse
, “
Defect energetics in ZnO: A hybrid Hartree-Fock density functional study
,”
Phys. Rev. B
77
,
245202
(
2008
).
82.
R. C.
Vilão
,
A. G.
Marinopoulos
,
R. B. L.
Vieira
,
A.
Weidinger
,
H. V.
Alberto
,
J. P.
Duarte
,
J. M.
Gil
,
J. S.
Lord
, and
S. F. J.
Cox
, “
Hydrogen impurity in paratellurite α-TeO2: Muon-spin rotation and ab initio studies
,”
Phys. Rev. B
84
,
045201
(
2011
).
83.
Z.
Salman
,
T.
Prokscha
,
A.
Amato
,
E.
Morenzoni
,
R.
Scheuermann
,
K.
Sedlak
, and
A.
Suter
, “
Direct spectroscopic observation of a shallow hydrogen-like donor state in insulating SrTiO3
,”
Phys. Rev. Lett.
113
,
156801
(
2014
).
84.
J. B.
Varley
,
A.
Janotti
, and
C. G.
Van de Walle
, “
Hydrogenated vacancies and hidden hydrogen in SrTiO3
,”
Phys. Rev. B
89
,
075202
(
2014
).
85.
S.
Limpijumnong
,
P.
Reunchan
,
A.
Janotti
, and
C. G.
Van de Walle
, “
Hydrogen doping in indium oxide: An ab initio study
,”
Phys. Rev. B
80
,
193202
(
2009
).
86.
K.
Shimomura
,
R.
Kadono
,
K.
Ohishi
,
M.
Mizuta
,
M.
Saito
,
K. H.
Chow
,
B.
Hitti
, and
R. L.
Lichti
, “
Muonium as a shallow center in GaN
,”
Phys. Rev. Lett.
92
,
135505
(
2004
).
87.
S.
Limpijumnong
and
C. G.
Van de Walle
, “
Stability, diffusivity, and vibrational properties of monatomic and molecular hydrogen in wurtzite GaN
,”
Phys. Rev. B
68
,
235203
(
2003
).
88.
S. S.
Nekrashevich
and
V. A.
Gritsenko
, “
Electronic structure of silicon dioxide (a review)
,”
Phys. Solid State
56
,
207
222
(
2014
).
89.
T. V.
Perevalov
,
A. V.
Shaposhnikov
,
K. A.
Nasyrov
,
D. V.
Gritsenko
,
V. A.
Gritsenko
, and
V. M.
Tapilin
, “Electronic structure of ZrO2 and HfO2,” in Defects in High-k Gate Dielectric Stacks, edited by E. Gusev (Springer, Dordrecht, 2006), pp. 423–434.
90.
R. C.
Vilão
,
A. G.
Marinopoulos
,
H. V.
Alberto
,
J. M.
Gil
,
J. S.
Lord
, and
A.
Weidinger
, “
Sapphire α-Al2O3 puzzle: Joint μSR and density functional theory study
,”
Phys. Rev. B
103
,
125202
(
2021
).
91.
P. D. C.
King
,
T. D.
Veal
,
P. H.
Jefferson
,
J.
Zúñiga Pérez
,
V.
Muñoz Sanjosé
, and
C. F.
McConville
, “
Unification of the electrical behavior of defects, impurities, and surface states in semiconductors: Virtual gap states in CdO
,”
Phys. Rev. B
79
,
035203
(
2009
).
92.
M.
Hiraishi
,
H.
Okabe
,
A.
Koda
,
R.
Kadono
,
K.
Ide
,
T.
Kamiya
, and
H.
Hosono
, “Local electronic structure of hydrogen in wide-gap semiconductor β-Ga2O3,” in MRM2021 Materials Research Meeting, Yokohama, Japan, 13–17 December 2021 (2021).
93.
R. T.
Williams
,
K. S.
Song
,
W. L.
Faust
, and
C. H.
Leung
, “
Off-center self-trapped excitons and creation of lattice defects in alkali halide crystals
,”
Phys. Rev. B
33
,
7232
7240
(
1986
).
94.
K.
Kan’no
,
T.
Matumoto
, and
Y.
Kayanuma
, “
Parity-broken and -unbroken self-trapped excitons in alkali halides
,”
Pure Appl. Chem.
69
,
1227
1235
(
1997
).
95.
B. K.
Meyer
,
H.
Alves
,
D. M.
Hofmann
,
W.
Kriegseis
,
D.
Forster
,
F.
Bertram
,
J.
Christen
,
A.
Hoffmann
,
M.
Strassburg
,
M.
Dworzak
,
U.
Haboeck
, and
A. V.
Rodina
, “
Bound exciton and donor-acceptor pair recombinations in ZnO
,”
Phys. Status Solidi B
241
,
231
260
(
2004
).
96.
B. K.
Meyer
,
J.
Sann
,
S.
Lautenschläger
,
M. R.
Wagner
, and
A.
Hoffmann
, “
Ionized and neutral donor-bound excitons in ZnO
,”
Phys. Rev. B
76
,
184120
(
2007
).
97.
M.
Gallart
,
T.
Cottineau
,
B.
Hönerlage
,
V.
Keller
,
N.
Keller
, and
P.
Gilliot
, “
Temperature dependent photoluminescence of anatase and rutile TiO2 single crystals: Polaron and self-trapped exciton formation
,”
J. Appl. Phys.
124
,
133104
(
2018
).
98.
R.
Heinhold
,
A.
Neiman
,
J. V.
Kennedy
,
A.
Markwitz
,
R. J.
Reeves
, and
M. W.
Allen
, “
Hydrogen-related excitons and their excited-state transitions in ZnO
,”
Phys. Rev. B
95
,
054120
(
2017
).
99.
X.
Yang
and
N. C.
Giles
, “
Hall effect analysis of bulk ZnO comparing different crystal growth techniques
,”
J. Appl. Phys.
105
,
063709
(
2009
).
100.
E.
Yagi
,
R. R.
Hasiguti
, and
M.
Aono
, “
Electronic conduction above 4k of slightly reduced oxygen-deficient rutile TiO2x
,”
Phys. Rev. B
54
,
7945
7956
(
1996
).
101.
A.
Spinelli
,
M. A.
Torija
,
C.
Liu
,
C.
Jan
, and
C.
Leighton
, “
Electronic transport in doped SrTiO3: Conduction mechanisms and potential applications
,”
Phys. Rev. B
81
,
155110
(
2010
).
102.
K.
Ide
,
K.
Nomura
,
H.
Hosono
, and
T.
Kamiya
, “
Electronic defects in amorphous oxide semiconductors: A review
,”
Phys. Status Solidi A
216
,
1800372
(
2019
).
103.
A.
Janotti
,
C.
Franchini
,
J. B.
Varley
,
G.
Kresse
, and
C. G.
Van de Walle
, “
Dual behavior of excess electrons in rutile TiO2
,”
Phys. Status Solidi RRL
7
,
199
203
(
2013
).
104.
S.
Nakamura
,
T.
Mukai
, and
M.
Senoh
, “
In situ monitoring and Hall measurements of GaN grown with GaN buffer layers
,”
J. Appl. Phys.
71
,
5543
5549
(
1992
).
105.
D. G.
Chtchekine
,
Z. C.
Feng
,
G. D.
Gilliland
,
S. J.
Chua
, and
D.
Wolford
, “
Donor-hydrogen bound exciton in epitaxial GaN
,”
Phys. Rev. B
60
,
15980
15984
(
1999
).
106.
M. H.
Dehn
,
J. K.
Shenton
,
S.
Holenstein
,
Q. N.
Meier
,
D. J.
Arseneau
,
D. L.
Cortie
,
B.
Hitti
,
A. C. Y.
Fang
,
W. A.
MacFarlane
,
R. M. L.
McFadden
,
G. D.
Morris
,
Z.
Salman
,
H.
Luetkens
,
N. A.
Spaldin
,
M.
Fechner
, and
R. F.
Kiefl
, “
Observation of a charge-neutral muon-polaron complex in antiferromagnetic Cr2O3
,”
Phys. Rev. X
10
,
011036
(
2020
).
107.
M. H.
Dehn
,
J. K.
Shenton
,
D. J.
Arseneau
,
W. A.
MacFarlane
,
G. D.
Morris
,
A.
Maigné
,
N. A.
Spaldin
, and
R. F.
Kiefl
, “
Local electronic structure and dynamics of muon-polaron complexes in Fe2O3
,”
Phys. Rev. Lett.
126
,
037202
(
2021
).