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M. Hiraishi, H. Okabe, A. Koda, R. Kadono, [T. Ohsawa](https://orcid.org/0000-0001-7528-8940), [N. Ohashi](https://orcid.org/0000-0002-4011-0031), K. Ide, T. Kamiya, H. Hosono

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[Local electronic structure of dilute hydrogen in <math>  <mrow>    <mi>β</mi>    <mtext>−</mtext>    <msub>      <mi>Ga</mi>      <mn>2</mn>    </msub>    <msub>      <mi>O</mi>      <mn>3</mn>    </msub>  </mrow></math> probed by muons](https://mdr.nims.go.jp/datasets/b120944b-68a7-483a-8a23-f78f90767b01)

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arXiv:2401.02109v1  [cond-mat.mtrl-sci]  4 Jan 2024arXiv:2401.02109v1  [cond-mat.mtrl-sci]  4 Jan 2024Local electronic structure of dilute hydrogen in β-MnO2H. Okabe,1 M. Hiraishi,1 A. Koda,1, 2 S. Takeshita,1, 2 K. M. Kojima,3I. Yamauchi,4 T. Ohsawa,5 N. Ohashi,5 H. Sato,6 and R. Kadono1, 2, ∗1Muon Science Laboratory and Condensed Matter Research Center, Institute of Materials StructureScience, High Energy Accelerator Research Organization (KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan2Department of Materials Structure Science, The Graduate University for Advanced Studies (Sokendai), Tsukuba, Ibaraki 305-0801, Japan3Centre for Molecular and Materials Science TRIUMF, Vancouver, B.C. V6T 2A3, Canada4Graduate School of Science and Engineering, Saga University, Saga 840-8502, Japan5National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0044, Japan6Department of Physics, Chuo University, Tokyo 112-8551, JapanThe electronic and magnetic states of β-MnO2 in terms of hydrogen impurities have been investigated bymuon spin rotation (µSR) technique combined with density-functional theory (DFT) calculations for muon aspseudo-hydrogen. We found that 85 % of implanted muons are localized in the oxygen channels of the rutilestructure and behave as interstitial protons (Mu+) except those (7.6%) forming a charge-neutral state (Mu0) at2.3 K, which indicates that interstitial hydrogen acts as a shallow donor within less than 0.1 meV of ionizationenergy. The residual 15% of muons are attributed to those related to lattice imperfection as Mn vacancies.Detailed analyses combined with DFT approach suggested that the muon is localized at the center of the oxygenchannel due to its large zero-point vibration energy.I. INTRODUCTIONManganese dioxide (MnO2) is well known as a catalystfor a variety of chemical reactions such as the deoxidizationprocess of hydrogen peroxide, with which we are familiar inelementary school chemistry, or most recently, it is attract-ing a lot of attention as a highly efficient heterogeneous cata-lyst for the oxidation of various types of substrates includingbio-mass-derived compounds [1]. The most known applica-tions of MnO2 are in batteries; MnO2 synthesized by the elec-trolytic method (γ-MnO2) is used worldwide as the cathodematerial in lithium or zinc primary battery [2]. The artificiallysynthesized MnO2 generally contains hydrogen in the crystalstructure, which are supposed to influence various physical,chemical, and electrochemical properties.There are two types of hydrogens (protons) existing in γ-MnO2 proposed by Ruetschi and Giovanoli [4–6]. One is thatprotons associated with cation vacancies, termed ”Ruetschi”protons, which compensate for the charge instead of Mn4+cations. The other is that protons associated with Mn3+cations, termed ”Coleman” protons, which are located in thetunnels of the structure and more mobile than Ruetschi pro-tons. The existence of the two types of protons has been con-firmed experimentally by neutron diffraction and NMR stud-ies [7–11]. However, these protons are mainly distinguishedby differences in O-H bond lengths, therefore its specific po-sitions and electronic states remain unclear. The main reasonfor the problem is the difficulty of detecting dilute amounts ofhydrogen.Since hydrogen is the lightest and smallest of all the el-ements, and ubiquitous presence in nature, hydrogen easilypenetrates into materials to cause embrittlement of metals, orserves as possible sources for unintentional carrier doping insemiconductors [12, 13]. Nevertheless, it is difficult to find∗ Corresponding author: ryosuke.kadono@kek.jpout the behavior of dilute hydrogen in matter due to its ownweak signals and/or false detections of background-originatedsignals from extraneous hydrogen in various diffraction anal-yses. It is therefore desirable to develop an effective methodto ascertain the behavior of hydrogen in host materials.To obtain microscopic information on the local electronicstructure of isolated hydrogen in matter, positive muon stud-ies have been making significant contributions. While positivemuon as pseudo-hydrogen exhibits a remarkable isotope ef-fect for kinetics of muon/proton in matter, its local electronicstructure as an atom determined by muon-electron interaction(resulting in various charged states designated as Mu+, Mu0,or Mu−) is almost identical with that of hydrogen. Thus, im-planted muon can be regarded as a simulator for the electronicstructure of interstitial hydrogen. We previously identified theelectronic structure of muons in oxide semiconductors usingmuon spin rotation (µSR) technique, and found evidence forhydrogen impurities as a possible origin of unintentional n-type conductivity [14–22].As a part of our continued effort to elucidate the behaviorof hydrogen in oxide semiconductors, we focused on β-MnO2as our next research target. MnO2 is known to exhibit a va-riety of crystalline polymorphs, which results in various tun-nel and layered structures [3], where β-MnO2 (pyrolusite) isthe simplest and thermodynamically stable phase. The crys-tal structure of β-MnO2 consists of MnO6 octahedra sharedby corners or edges with 1 × 1 tunnels (oxygen channels, seeFig. 1), which is also part of the basic structure of γ-MnO2.It would be interesting to note that the structure of β-MnO2 isidentical with rutile (TiO2), which we have intensively studiedby µSR [18, 19].β−MnO2 is regarded as a narrow gap semiconductor and itis known to exhibt a n-type conductivity [23, 24]. The n-typenature of β-MnO2 may stem from oxygen deficiency and/orlesser impurity interstitials like hydrogen. If oxygen vacancyconcentration is negligibly small, hydrogen could be a ma-jor source of carriers. However it is often difficult to separateall these different contributions in the bulk property measure-http://arxiv.org/abs/2401.02109v12FIG. 1. Crystal structure and electrostatic potential maps for β-MnO2calculated by using reported lattice parameters [25]. The Hartreepotential calculated using VASP are represented by the color contourfrom blue to red in the lattice planes (002) and (100). The solid linesindicate the unit cell.ments. On the other hand, µSR can provide useful informa-tion to extract intrinsic electronic properties out of those ex-trinsic contributions because muon monitors local magneticstates in the atomic scale. Even more conveniently, β-MnO2is known as a classical example of magnetic materials with awell-defined screw magnetic order below TN = 92 K [26, 27].Since muon is a direct probe of local magnetic fields, we candetermine the position and electronic state of hydrogen in β-MnO2 by µSR, using the magnetic structure as a guide.In this work, we used muon as a microscopic simulator forextrinsic hydrogen/proton to identify the local electronic andmagnetic structures stemmed from interstitial hydrogen in β-MnO2, by probing the local fields from surrounding Mn ionsthat exhibit the well-defined magnetic order. The paper pro-ceeds as follows. Section II describes the sample characteri-zation, µSR experimental, and computational details. SectionIII presents the results of transverse-field (TF) µSR, zero-field(ZF) µSR measurements, and density functional theory (DFT)calculations. Then, we discuss the positions of hydrogen andthe effect on the electronic and magnetic state of β-MnO2. Fi-nally, Sec. IV summarizes this paper.II. EXPERIMENTAL DETAILSPowder sample of β-MnO2 (3N, Soekawa Co., Ltd.) wasused in the present study. The crystal structure and phase pu-rity were checked using a Rigaku MultiFlex powder X-raydiffractometer under ambient conditions with Cu Kα radia-tion. The XRD data indicate that the sample crystallized intoβ-MnO2 single phase and the absence of secondary phase (seeSupplemental Material Fig. S1).The hydrogen content in the sample was observed by ther-mal desorption spectrometry (TDS) placed in the NationalInstitute for Material Science (NIMS). The sample was lin-early heated from room temperature to 1073 K in 2 hours andkept at 1073 K for 1 hour. During the heating, the amountof desorbed hydrogen was measured by a quadrupole massspectrometer (QMS). The QMS signal intensity of hydrogenmolecule from sample increase with increasing temperature,and 2.918 × 1019 cm−3 of hydrogen was detected (Fig. S2).The magnetic susceptibility χ was measured using a Quan-tum Design SQUID magnetometer from 5 to 400 K. The tem-perature dependence of χ shows a kink at 92 K, correspond-ing to the Néel temperature of β-MnO2 (Fig. S3). The peakanomaly around 40 K is probably due to hydrogen impurities,since it was suppressed after heat treatment at 723 K in air. χalso shows the Curie-Weiss behavior with the Curie constantC = 2.51 emu k/mol and the Weiss temperature Θ = −788 K,which are in good agreement with the earlier report [24] (Fig.S3, inset).Conventional µSR experiments were performed using theARTEMIS spectrometer installed in the S1 area at Muon Sci-ence Establishment (MUSE), Japan Proton Accelerator Re-search Complex (J-PARC). Time dependent µ-e decay asym-metry A(t) was measured to monitor the local field under zerofield (ZF), longitudinal field (LF), and weak transverse field(TF) conditions. Additional µSR experiments were conductedto resolve the detailed local field distribution using two in-struments, i.e., the LAMPF spectrometer on the M20 beam-line and NuTime spectrometer on the M15 beamline at TRI-UMF, Canada, for the magnetic and paramagnetic phases ofβ-MnO2, respectively. The µSR spectra have been analyzedusing the MUSRFIT software package [28].The DFT calculations were performed using the projectoraugmented wave approach [29] implemented in the Viennaab initio simulation package (VASP) [30] with the Perdew-Burke-Ernzerhof (PBE) exchange correlation potential [31].The cutoff energy for the plane-wave basis set was 500 eV.All atoms were relaxed until the Hellmann−Feynman forceson them were smaller than 0.01 eV/Å. The distribution of thelocal magnetic field at the muon sites was calculated usingDipelec program [32]. Crystal structures were visualized us-ing the VESTA program [33].III. RESULTS AND DISCUSSIONA. Local fields in the paramagnetic phaseAt first, we attempted to narrow down the position of hydro-gen/muon in β-MnO2 structure by examining the local field inthe paramagnetic phase. The local field provides useful infor-mation for identifying the muon stopping site, because it ismainly determined by the spatial arrangement of the nearestneighboring (nn) Mn d-electrons. As shown in Fig. 1, it is30.040.030.020.010.00real amplitude820815810805Frequency (MHz)���������������������������������������������FIG. 2. Fast Fourier transform of the µSR time spectra measuredunder a transverse field of 6 T.suggested from our preliminary calculation using VASP thatthe Hartree potential for the interstitial Mu+ exhibits minimaaround the center of the oxygen channels along the c-axis di-rection. When an external magnetic field is applied transverseto the initial muon polarization vector, the muon exhibits spinprecession at a frequency corresponding to the summation ofthe external and internal fields. Thus, by evaluating the shiftfrom the reference frequency of the external field, one can ex-tract the contribution of the internal field at a particular stop-ping site from the TF-µSR spectrum.Figure 2 shows the fast Fourier transform (FFT) of TF-µSR spectra under a transverse field of 6 T obtained at var-ious temperatures. One can see a single Gaussian-like line-shape at around 813 MHz, where the slight broadening oflinewidth with decreasing temperature is probably due to thegradual development of staggered magnetization related to theshort range magnetic order [24, 27]. To interpret the line-shape of these spectra, we calculated frequency distribution inpolycrystalline sample (powder pattern) for several candidatemuon sites [34].The calculated powder patterns of putative muon sites areshown in Fig. 3 with additional Gaussian broadening due tothe local field inhomogeneity and the truncation of Fouriertransforms by the finite time window (t ≤ 10 µs). These muonsites have been selected by considering the electrostatic po-tential maps (see Fig. 1) and the distance from nn O ions. Thelineshape has two edges determined by the anisotropy of thehyperfine coupling (A‖ and A⊥), which is related with the lo-cal χ induced by the nn Mn spins. One can see that the muonsite near a certain Mn ion, e.g. 2b site (0, 0, 1/2), shows amarked asymmetric feature with broad distribution. Since theexperimental spectra show no signs of these shoulder-like fea-tures, implanted muons tend to be located away from Mn ions,at a high symmetry position in the Mn sublattice. Compari-son of the spectra with that simulated for the O defect position(vacancy) also infers that muons have little chance to find Ovacancies within their lifetime.Considering previous neutron and NMR studies [7–11], itis reasonable to assume that muons are localized near the cen-ter of the oxygen channel in the rutile structure. Our resultfurther suggests that muons do not occupy low symmetry po-sitions including oxygen vacancies. On the other hand, thereFIG. 3. Simulated powder patterns for the putative muon stoppingsites in β-MnO2 under a transverse field of 6 T. Each interstitial crys-tallographic position is represented in a Wycoff letter: 4c (1/2, 0, 0),8i (0.42667, 0.11185, 0), 16k (0.48954,0.26299, 0.18754), 2b (0, 0,1/2). The black lines represent the calculated powder patterns. Thered lines represent the additional Gaussian broadening (see the maintext) superimposed on the powder patterns.remains a possibility that muons are localized in Mn vacan-cies due to the lack of frequency resolution (see the lineshapeat the Mn defect position in Fig. 3). Therefore, it requiresadditional information to resolve this issue (see the next sec-tion).B. Local fields in the magnetically ordered phaseβ-MnO2 undergoes a helical magnetic transition at TN =92 K with incommensurate magnetic modulation [propaga-tion vector qm = (0, 0, ∼2/7)] without much temperature de-pendence [26, 27]. Given that the magnetic structure is al-ready known, a spontaneous long-range magnetic order is of-ten helpful to identify the muon sites by comparing the ob-served field distribution with that calculated for the candidatesites using the known magnetic structure.Figure 4(a) shows a ZF-µSR time spectrum at 2.5 K for thesample from the same batch used in the TF measurements.One can see a clear oscillation in the early time domain (<0.25µs), implying the long-range magnetic order. The FFT spec-trum [Fig. 4 (b)] indicates that the oscillating signal is dom-inated by two frequency components ( f1 ≃ 97 and f2 ≃ 124MHz). While this feature is usually attributed to the presenceof two distinct muon sites, it turns out to be not necessarilythe case in β-MnO2.For an incommensurate magnetic structure under zero field,the local field distribution D(Bloc) is given by40.200.150.100.050.00-0.05Asymmetry0.250.200.150.100.050.00Time (µs)0.0150.0100.0050.000Fourier amplitude250200150100500Frequency (MHz)(a)f2f1T = 2.3 K, ZFT = 2.3 K, ZF(b)FIG. 4. (a) µSR time spectrum at 2.5 K under a zero external field.The solid line represents the result of least-squares fitting (see themain text). (b) Fast Fourier transform of the time spectrum. f1 ( f2)indicates a peak frequency.-80-60-40-20020406080Phase (deg)100806040200T (K)140120100806040200Frequency (MHz)0.200.150.100.050.00Asymmetry(a) (c)(d)ATALγµBaveγµBdifφcφb403020100Relaxation rate  (µs-1)100806040200T (K)(b)λTλLFIG. 5. Temperature dependence of (a) the transverse (longitudinal)asymmetry AT (AL), (b) the transverse (longitudinal) relaxation rateλT (λL), (c) the precession frequencies γµBave and γµBdif, and (d) theinitial phase ϕb and ϕc. The open circle in (b) indicates λL obtainedat J-PARC MUSE. The solid line in (c) indicates the result of least-squares curve-fit by the power-law function (see the main text).D (Bloc) =2πBloc√B2loc − B2min√B2max − B2loc,where it has two characteristic cutoff field (Bmin andBmax) [37]. Unfortunately, the muon polarization function intime domain associated with D(Bloc) cannot be obtained ana-lytically, we adopt an approximate distribution which is calledshifted-Overhauser distributionD′(Bloc),D′ (Bloc) =1π1√B2dif − (Bloc − Bave)2,where Bdif = (Bmax − Bmin)/2 and Bave = (Bmax + Bmin)/2, isused for the incommensurate case [38]. According to thisapproximation, the oscillatory part of the muon polarizationfunction is written by the product of the zeroth order Besselfunction and the cosine function.Based on the above approximation, we analyzed the spec-trum by following curve-fit function,A (t) = AT J0(γµBdift + φb)cos(γµBavet + φc)exp (−λT t)+ ALexp (−λLt)where the amplitude AT (AL) is proportional to the fractionperpendicular (parallel) to initial polarization direction, J0 isa Bessel function of the first kind, γµ (= 2π × 135.53 MHz/T)is the muon gyromagnetic ratio, ϕb (ϕc) is the initial phaseof each function, respectively, and λT (λL) is the transverse(longitudinal) relaxation rate. The solid curve in Fig. 4(a)shows the result of least-squares curve-fit. The temperaturedependence of respective parameters below TN are shown inFigs. 5.The slight decrease of AT at lower temperatures is attributedto the partial loss of initial muon polarization (which is dis-cussed below). The development of the magnetic order pa-rameter with decreasing temperature manifests in the evolu-tion of the frequency γµBave as shown in Fig. 5(c). The tem-perature dependence of γµBave shows a critical behavior de-scribed by the power law, γµBave(T )∝ (TN − T )β with TN =92.7(2) K and β = 0.182(9). The value of critical exponentβ is in good agreement with those obtained from the syn-chrotron x-ray magnetic scattering [26] and the neutron scat-tering [27]. Although the phase parameter ϕc is nearly zeroexcept around TN , ϕb tends to be shift by ∼ π/4 compared withϕc. It may be related to the π/4 phase shift between the Besselfunction and the cosine function [37].The temperature dependence of the observed signal frac-tion, [AT + AL]/A0 (red circle) and the longitudinal fraction,AL/A0 (blue circle) are shown in Fig. 6, where A0 (= 0.23) isestimated from the full asymmetry above TN . One can clearlysee that the signal fraction below TN is less than 90 % of thetotal volume fraction. In contrast, the longitudinal fraction(z component) is almost one third of the total volume fraction,indicating that all the implanted muons are subject to isotropicstatic magnetic order. These facts indicate that there is an un-resolved signal component (missing fraction) which is depo-larized within the dead time (t <0.01 µs) of the experiment.The missing fraction at 2.3 K is estimated to be about 15 %of the total volume fraction. Such a fast depolarization is at-tributed to the strong disorder of D(Bloc). Considering that thepossibility of muons occupying O vacancy is unlikely as in-ferred from D(Bloc) (at least above TN , see Fig.3), it is reason-able to attribute the origin of the missing fraction to the muonsoccuping Mn vacancies that corresponds to naturally includedRuetschi protons. This also leads to the hypothesis that theresidual 85 % of muons corresponds to interstitial hydrogens(Coleman protons) which occupies the oxygen channels.In close examination of the temperature dependence ofAL/A0 in Fig. 6, one can find that it slightly declines fromthe one third line as temperature decreases. This is attributed5to the formation of paramagnetic Mu0 state which is subject tospin/charge exchange process with diamagnetic state (Mu++e− → Mu0). The reduction of AL strongly suggests that theelectronic state bound to the interstitial muon is not polaronic(i.e., the electron spin not being locked along the Mn spins),but rather atomic to allow spin-triplet (F = 1) or spin-singlet(F = 0) state relative to the muon spin direction. The preces-sion frequency for the F = 0 state usually exceeds the limitdetermined by experimental time resolution [35]. Therefore,only residual polarization for the F = 1 state can be observed,resulting in a reduction of the longitudinal fraction by half. IfMu0 is present, the longitudinal asymmetry AL is reproducedby following linear combinationAL =13(1 −12fMu)A0,where fMu is the fractional yield of Mu0.The temperature dependence of longitudinal signal frac-tion fL (see Fig.6 inset) is adequately described by the ther-mal activation model fL = fdexp(−Ea/2kBT ), where fd is thediamagnetic longitudinal fraction ( fL = fd, when fMu = 0),Ea is the characteristic energy, and kB is Boltzmann’s con-stant. A fit of fL (blue line in the inset of Fig.6) below 50K yields fd = 0.3420(8) and Ea = 0.0292(8) meV/K. Consid-ering the value of fL = 0.316(1) at 2.3 K, fMu is estimatedto be 0.0760(4) (which may be a lower bound for the Mu0yield). This means that the electrons from the hydrogen im-purity levels are mostly promoted to the conduction band inmost temperature ranges. In other words, the donor level as-sociated with hydrogen is estimated to be extremely shallow,probably less than 0.1 meV. Since the diamagnetic muons inthe oxygen channels (corresponding to the Coleman proton)can be attributed to Mu0 in its ionized state, they are presumedto serve as electron donors. For these reasons, the interstitialhydrogen is expected to be a source of n-type conductivity inβ-MnO2. We note that the hydrogen concentration (2.918 ×1019 cm−3) obtained from our TDS measurement is compara-ble with the carrier density (n = 4.8 × 1019 cm−3) reported inthe previous work [24], which is perfectly in line with thisscenario.C. First-principles calculationsIn order to gain more understanding of hydrogen states in β-MnO2, the DFT calculations were performed using VASP forcomparisons with the µSR experiments. As the first step, wecalculated the local field Bloc at the most energetically stableposition 4c (1/2, 0, 0) with the screw periodicity q = 2c*/7 andMn4+ moment size of 2.34µB [27]. Under zero-external field,Bloc is determined by a vector sum of the magnetic dipolarfield of the Mn ions and the Fermi contact interaction withunpaired electron in the muon 1s orbital [36],Bloc =∑iÂiµi +83πS eδ (r)1.00.80.60.40.20.0Signal fraction100806040200T (K)0.360.350.340.330.320.310.30longitudinal signal fraction50403020100T (K)FIG. 6. Temperature dependence of the observed signal fraction (redcircle) and the longitudinal signal fraction fL (blue circle). The insetshows the enlargement below 50 K with the fitting curve using thethermal activation model (see main text). The broken line representsthe 1/3 of the total fraction.where µi is the magnetic moment of the i-th Mn ion located atdistance ri = (xi, yi, zi) from the muon site, S e is the electronspin magnetic moment, and δ(r) is the delta function, andÂi = Aαβi=1r3i3αiβir2i− δαβ, (α, β = x, y, z)is the dipolar tensor. The Fermi contact term is basically neg-ligible except for the Mu0 state.Figure 7(b) shows a simulated distribution of Bloc for the 4csite, which is in remarkable agreement with the experimentalresult [Fig. 7(a)]. A comparative study with DFT calculationprovides useful information for identifying the muon’s loca-tion and state. For example, if all the nn Mn4+ (S = 3/2) ionsbear the full moment (3µB), the calculated distribution ofBlocbecomes broader, leading to the shift of central frequency by∼ 30 MHz (not shown), which is in disagreement with the ex-perimental result. This also supports the moment reductionof Mn ions (2.34µB) from the viewpoint of local magnetismprobed by µSR.We further extended the simulation of Bloc to investigatethe influence of structural relaxation by a muon/hydrogen in-sertion. The Brillouin zone for a 2 × 2 × 3 supercell wassampled using a cutoff energy of 500 eV and a 4 × 4 × 4 k-point mesh for calculations. The structure optimization wasperformed using lattice parameters and all of the atomic po-sitions were allowed to relax until the forces acting on eachion became less than 0.01 eV/Å. Note that the spin degree offreedom and the electron correlation parameter U are not in-corporated, because the difference between optimized atomicpositions of Mn around the hydrogen is small irrespective of6the presence or absence of U(= 4 eV). Therefore, it is not anissue in this study.The result of calculation is shown in Fig. 8. The hydro-gen occupies slightly off-center of the oxygen channel whereit forms O-H bond with a bond length of 0.99 Å and donat-ing electron to the host lattice. The surrounding oxygen showlarge displacement from their initial crystallographic positionsdue to hydrogen bond. On the other hand, the surroundingcationic sub-lattice of Mn ions shows a slight expansion dueto the electrostatic repulsion between H+ and Mn4+ ion; re-sulting in 1.4∼7.2% Mn-Mn bonds elongation compared tothose without H insertion. Taking into account the displace-ment of neighboring six Mn ions from H, we have simulatedBloc using a 686-octahedron 7 × 7 × 7 supercell to reproducethe partially distorted β-MnO2 lattice considering the periodof the helical magnetic structure.When muon (or hydrogen) is bound to an oxygen, the do-nated electron can localize on a neighboring transition metald orbital to form a charge-neutral complex (i.e., polaron)[18, 19, 39]. In light of this possibility, we simulated Bloc withtwo different charge distributions: a polaronic electron dis-tributed on the nn Mn ion only [Fig. 7(c)] or the nearby threeMn ions [Fig. 7(d)], that are not obtained from the DFT cal-culation. Although subtle differences in shape are observed incomparison with the non-distorted case [Fig. 7(b)], they stillhave a characteristic two peak structure. Note that when thestructure was optimized by fixing hydrogen at the 4c site, theMn-H distance was uniformly extended for the all Mn ions.Therefore, no significant change in the shape of the local mag-netic field was obtained.Hydrogen generally takes an asymmetric O-H···O configu-ration with a short O-H covalent bond and a long H···O hy-drogen bond. However, it resides equidistantly between twonearest oxygen atoms under high pressures. This phenomenonis called H-bond symmetrization and often observed in high-pressure phase of ice [40]. The H-bond symmetrization isclosely linked to zero-point vibration, thus the isotope effectis expected to be more pronounced [41, 42]. For example, δ-AlOOH, which has a distorted-rutile-type framework, showsthe H-bond symmetrization under high pressures with a sig-nificant isotope effect; the deuteration induces a shift of thesymmetrization to a higher pressure [43]. It is commonlyknown that a lighter isotope possesses a larger zero-point en-ergy. The presumption that muon locates in the middle of twonearest oxygen atoms to form a symmetric O-µ-O bond dueto the isotope effect would be a possible scenario to explainthe muon localization at the 4c site. We have evaluated thezero-point motion energy of muon and hydrogen from the po-tential energy distribution along O-4c-O direction by the DFTcalculation using VASP (see Supplemental Material Fig. S4).Given the energy difference between the ground state (E0) andthe first excited state (E1) of muon or hydrogen, muon can tun-nel between the two wells at a very fast timescale as far as themuon spin precession frequency is much lower than the tun-neling frequency. This picture can be approximated as a kindof resonant state where muon locates at the center of the tun-nel. In such a case where the quantum nature is pronounced,muon may not be able to simulate the behavior of dilute hy-43210P(f) (arb. units)0.4 0.6 0.8 1.0 1.2Field (T)0.0150.0100.0050.000Fourier amplitude86420P(f) (arb. units)6420P(f) (arb. units)0.4 0.6 0.8 1.0 1.2Field (T)(b)(a) (c)(d)FIG. 7. (a) Local field distribution (Bloc) at 2.3 K under a zero exter-nal field. (b) Simulated distribution of Bloc at the 4c site. Simulateddistribution of Bloc at the structurally-optimized site with a polaronicelectron (not obtained from the DFT calculation) distributed on thenearest Mn ion only (c) or the nearby three Mn ions (d).H0.99 ÅO1.82 ÅMnbaFIG. 8. Deformation of the host lattice on hydrogen intercalation in-ferred from the DFT (VASP) calculation. The broken line representsthe H···O hydrogen bond.drogen defects correctly.D. Electronic and magnetic statesFinally, we discuss the electronic and magnetic states of β-MnO2 based on the results obtained so far. The formal mag-netic moment of Mn4+ ions should be 3µB as the localized spinsystem, because β-MnO2 shows an insulating behavior in thelow-temperature region (T <50 K). However, the Mn orderedmoment is markedly reduced to 2.34µB [27], and this fact hasbeen supported by the present µSR result as described above.The discrepancy in the size of the Mn moment may be at-tributed to the localized eg electrons originating from unin-tended impurity defects. If the Hund coupling is fairly large,eg electrons can become localized at a part of the Mn sites andperfectly polarized along the localized t2g moments at the ex-pense of their itinerancy [24]. However, massive amounts of7impurities or oxygen defects would be required to induce suchsignificant Mn moment reduction (3µB → 2.34µB). Moreover,Hund coupling usually enhances the magnetic moment, e.g.,double-exchange interaction, thus it would lead to a contradic-tory result. Although there is a possibility that the delocaliza-tion effect of t2g electrons remains at low temperatures [24],we can rule it out because our µSR results clearly indicatethat the local field distribution of β-MnO2 is well reproducedby the summation of the dipolar interaction from localized Mnmoments.Here, we propose an alternative scenario to explain the Mnmoment reduction in β-MnO2. Charge redistribution from aformal valence between a metal and its ligand is a fundamen-tal process in transition metal oxides [45]. For example, thereare slightly-trivalent Cu ions (Cu2+L, L: a hole on the oxygenligand sites) in hole-doped high-TC superconductors [46] oroxygen holes in Co3+/Co4+ mixed valence state in LixCoO2[47]. The degree of the charge transfer from O2p orbital totransition-metal 3d orbital depends on the energy differencebetween both orbitals, and also relates to their electronegativ-ity. In the case of Mn3+/Mn4+ mixed-valent oxides, significantcharge transfer between Mn and O ions has been observed inthe x-ray Compton scattering [48]. It is also in good agree-ment with the theoretical prediction that Mn–O bond is notpurely ionic but partially covalent [44]. Such an intermediatecharacter between ionic and covalent Mn-O bond may play animportant role in the catalytic activity and the performance forelectrochemical storage of manganese oxides.Based on these facts, we assumed the charge-transfer pro-cess from the O2p orbital to the Mn 3d-t2g orbital in β-MnO2.If the covalent character of Mn-O bond suppresses the in-crease of Mn3+ ionic radius and stabilizes a low-spin elec-tronic configuration (t42g) absolutely, the Mn3+ moment wouldbe significantly reduced (2µB) compared to that (4µB) of ahigh-spin configuration (t32ge1g). As a reference, a low spinstate of Mn3+ ion is often found in some tunnel-structure man-ganese oxides [49, 50]. The experimental value of the re-duced Mn moment (2.34µB) in β-MnO2 can be reproduced bythe charge transfer of about 0.1 electron from each of the sur-rounding six oxygen ions (or even less than this, if the oxygenspin polarization is antiparallel to Mn spins [51, 52]). Fur-ther studies of each wave-function character, e.g., Comptonscattering, will be required to support our argument.IV. SUMMARYWe have studied the electronic and magnetic state of β-MnO2 in terms of hydrogen impurities by µSR technique com-bined with the first-principles calculations. We have foundthe following experimental results: (1) 85 % of muons local-ize in the oxygen channels of the rutile structure as intersti-tial protons, (2) residual 15% of muons are associated withlattice imperfection as Mn vacancies, and (3) 7.6% of thosemuons (in the oxygen channels) form charge-neutral Mu stateat 2.3 K, which suggests that hydrogen interstitials act as shal-low donors whose ionization energy is less than 1 meV. 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