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# Local electronic structure of dilute hydrogen in $\beta$-Ga2O3 probed by
muons

M. Hiraishi  Muon Science Laboratory and Condensed Matter Research Center,
Institute of Materials Structure Science, High Energy Accelerator Research
Organization (KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan  Graduate School of
Science and Engineering, Ibaraki University 2-1-1 Bunkyo, Mito, Ibaraki,
310-8512, Japan.  H. Okabe  Muon Science Laboratory and Condensed Matter
Research Center, Institute of Materials Structure Science, High Energy
Accelerator Research Organization (KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan
Institute for Materials Research, Tohoku University (IMR), 2-1-1 Katahira,
Aoba-ku, Sendai 980-8577, Japan  A. Koda  Muon Science Laboratory and
Condensed Matter Research Center, Institute of Materials Structure Science,
High Energy Accelerator Research Organization (KEK-IMSS), Tsukuba, Ibaraki
305-0801, Japan  Department of Materials Structure Science, The Graduate
University for Advanced Studies (Sokendai), Tsukuba, Ibaraki 305-0801, Japan
R. Kadono  Muon Science Laboratory and Condensed Matter Research Center,
Institute of Materials Structure Science, High Energy Accelerator Research
Organization (KEK-IMSS), Tsukuba, Ibaraki 305-0801, Japan  Department of
Materials Structure Science, The Graduate University for Advanced Studies
(Sokendai), Tsukuba, Ibaraki 305-0801, Japan  T. Ohsawa  National Institute
for Materials Science (NIMS), Tsukuba, Ibaraki 305-0044, Japan  N. Ohashi
National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0044,
Japan  K. Ide  Materials and Structures Laboratory, Tokyo Institute of
Technology, Yokohama, Kanagawa 226-8503, Japan  T. Kamiya  Materials and
Structures Laboratory, Tokyo Institute of Technology, Yokohama, Kanagawa
226-8503, Japan  Materials Research Center for Element Strategy, Tokyo
Institute of Technology (MCES), Yokohama, Kanagawa 226-8503, Japan  H. Hosono
Materials Research Center for Element Strategy, Tokyo Institute of Technology
(MCES), Yokohama, Kanagawa 226-8503, Japan

###### Abstract

The local electronic structure of muons (Mu) as dilute pseudo-hydrogen in
single-crystalline $\beta$-Ga2O3 has been studied by the muon spin
rotation/relaxation ($\mu$SR). High-precision measurements over a long time
range of $\sim$25 $\mu$s have clearly identified two distinct Mu states: a
quasi-static Mu (Mu1) and fast-moving Mu (Mu2). By comparing this result with
predictions from the recently established ambipolarity model, these two states
are respectively attributed to the relaxed-excited states associated with the
donor ($E^{+/0}$) and acceptor ($E^{-/0}$) levels predicted by density
functional theory (DFT) calculations for the interstitial H. Furthermore, the
local electronic structure of Mu1 is found to be an OMu-bonded state with
three-coordinated oxygen. The structure is almost identical with the thermal
equilibrium state of H, and it is found to function as an electron donor. The
other Mu2 is considered to be in the hydride state (Mu-) from the ambipolarity
model, suggesting that it is in fast diffusion motion through the short-lived
neutral state due to the charge exchange reaction with conduction electrons
(Mu${}^{0}+e^{-}\rightleftarrows$Mu-).

##  I Introduction

Gallium trioxide ($\beta$-Ga2O3) is attracting attention as a material for
high-voltage power devices and other applications because of its large band
gap ($E_{\mathrm{g}}\sim$4.9 eV) and associated high critical electric field
Galazka (2018), where control of electrical activity by $p$/$n$-type carrier
doping is one of the critical issues. While Sn or Si doping is known to induce
$n$-type conductivity Ueda  _et al._ (1997); Orita  _et al._ (2000), it has
been pointed out that impurity hydrogen (H) is the possible cause of
unintentional carrier doping in as-grown samples Galazka  _et al._ (2020).

From infrared spectroscopic studies of $\beta$-Ga2O3 containing macroscopic
amounts of H, a variety of H states including O-H defects have been observed
Weiser  _et al._ (2018); Qin  _et al._ (2019). However, their relationship to
carrier doping is not always clear due to the lack of information on their
local electronic structures and H valence. Furthermore, since isolated H
exists only in trace amounts in solids, experimental means to study its
microscopic electronic state are limited. In this regard, theoretical studies
based on density functional theory (DFT) calculations have played an important
role, and for H in $\beta$-Ga2O3 DFT calculations have shown that interstitial
H may be indeed the origin of $n$-type doping Varley  _et al._ (2010, 2011);
Li and Robertson (2014). This prompted us to introduce muons as pseudo-H into
$\beta$-Ga2O3 for investigating the corresponding interstitial H states by
analyzing their electronic and dynamical properties in detail by muon spin
rotation ($\mu$SR).

The positive muon ($\mu^{+}$) is an elementary particle with a mass of
$0.1126m_{\mathrm{p}}$ (where $m_{\mathrm{p}}$ is the proton mass). The local
electronic structure of muon in matter is determined by the muon-electron
interaction which is practically equivalent to H, as can be seen from the fact
that the difference in reduced electron mass between a muon-electron bound
state (muonium) and a neutral H atom is only 0.43%. Therefore, Mu in matter
can be regarded as a light isotope of H ($=^{0.1126}$H; we introduce the
elemental symbol Mu below to denote muon as pseudo-H).

On the other hand, when interpreting the results of the $\mu$SR experiment
based on the results of the DFT calculations, there is an important caveat
that should be pointed out at this stage. Generally, muons are implanted as an
ion beam of relatively high-energy (typically $\sim$4 MeV), and the associated
electronic excitations in the insulator crystals produce free carriers and
excitons ($\sim$103 per muon) Thompson (1974); Alig and Bloom (1975); Itoh
(1997). These often propagate rapidly in the crystal, and there is
experimental evidence that Mu acts as a capture center for them Prokscha  _et
al._ (2007). Consequently, the electronic state of Mu is a relaxed-excited
state generated by the interaction with free carriers and excitons, which can
be different from the electronic state under thermal equilibrium as expected
from the thermodynamic double charge conversion level ($E^{+/-}$) obtained by
DFT calculations.

Another well-known evidence that Mu is in a relaxed-excited state is that two
or more different electronic states of Mu are often observed simultaneously in
the same material Lichti  _et al._ (2008). In order to understand the origin
of this phenomenon, we have recently conducted an extensive investigation to
explore the regularities between the Mu valence states observed in various
oxides and the results of previous DFT calculations. As a result, assuming
that these relaxed-excited states are associated with acceptor ($E^{0/-}$) and
donor ($E^{+/0}$) levels that are predicted by DFT calculations as metastable
states, we found that the observed Mu states can be coherently explained by
the relationship between acceptor/donor levels and band structure Hiraishi
_et al._ (2022). We call it the “ambipolarity model” since such behavior of Mu
is a manifestation of the ambipolarity of H through the relaxed-excited
states.

In this study, we show that Mu in single-crystalline $\beta$-Ga2O3 exhibits
two different electronic structures corresponding to the relaxed-excited
states respectively associated with the donor and acceptor levels in the
ambipolarity model. One is in the OMu-bonded state (Mu${}_{1}^{+}$)
corresponding to H serving as a donor, and another is the hydride-like state
in rapid motion (Mu${}_{2}^{-}$). In particular, the Mu2 state is a component
overlooked in the previous $\mu$SR experiments King  _et al._ (2010); Celebi
_et al._ (2012), and it is speculated to be a transient state diffusing
rapidly along the $\langle 010\rangle$ axis while undergoing the charge
exchange reaction, Mu${}_{2}^{-}\rightleftarrows{\rm Mu}_{2}^{0}+e^{-}$. The
occurrence of charge exchange is supported by the observation that the
temperature dependence of the fractional yield of Mu2 exhibits a strong
correlation with that of the bulk carrier electron mobility and density. This
also implies that the interstitial H can take hydride state under electronic
excitation and may exhibit fast diffusion motion, depending on the bulk
electronic properties of the host.

##  II EXPERIMENTAL METHODS and DFT CALCULATIONS

The sample used in the $\mu$SR experiment was a slab of single crystal (sc-
Ga2O3, 10$\times$15$\times$0.6 mm3) with $\langle 001\rangle$ plane
synthesized by the edge-defined film-fed (EFG) method (provided by Novel
Crystal Technology, Inc.) Kuramata  _et al._ (2016). It is reported to have no
twin boundary and the lowest carrier density ($N_{\mathrm{e}}\sim
2\times$1017cm-3) commercially available Kuramata  _et al._ (2016). The
electrical conductivity and Hall effect measurements were performed using the
PPMS (Quantum Design Co.). Ohmic contacts were formed from vacuum-deposited Ti
at room temperature Víllora  _et al._ (2008); Irmscher  _et al._ (2011) and
gold paste (Seishin Trading Co. LTD., No. 8556). The impurity H content in the
sc-Ga2O3 sample was estimated to be 3.5$\times 10^{18}\mbox{cm}^{-3}$ by
thermal desorption spectrometry, which is sufficient to explain the above
described $N_{\mathrm{e}}$. Details are described in Supplemental Material
(SM) sm .

In this study, $\mu$SR measurements and data analysis were also performed on a
powder sample (99.99%, provided by Rare Metallics Co.) for comparison, and the
results were found to be significantly different from those for single
crystal. However, these results are excluded from the discussion in this
paper, because it is difficult to measure the bulk electronic properties of
powder samples which is necessary to consider the cause of the difference.
Instead, they are presented with a brief interpretation in SM sm .

Conventional $\mu$SR measurements were performed using the S1 instrument
(ARTEMIS) at the Materials and Life-science Experiment Facility, J-PARC Kojima
_et al._ (2014), where high-precision measurements over a long time range of
20–25 $\mu$s can be routinely performed using a high-flux pulsed muon beam
($\sim$$3\times 10^{4}$ $\mu^{+}$/cm2/s for the single-pulse mode at a proton
beam power of 0.8 MW). The $\mu$SR spectra [the time-dependent decay-positron
asymmetry, $A(t)$] which reflects the magnetic field distribution at the Mu
site, was measured from room temperature to 4 K under zero field (ZF), weak
longitudinal field (LF, parallel to the initial Mu polarization
${\bm{P}}_{\mu}$), and weak transverse field (TF, perpendicular to
${\bm{P}}_{\mu}$), and were analyzed by least-squares curve fitting Suter and
Wojek (2012). The background contribution from muons which missed the sample
was estimated from $\mu$SR measurements on a holmium plate of the same
geometry and subtracted from the asymmetry.

Since the $\mu$SR spectra were found to be dominated by signals from the
diamagnetic Mu (i.e., Mu+ or Mu-), the data under ZF/LF conditions were
analyzed using the dynamical Kubo-Toyabe (KT) function,
$G_{z}^{\mathrm{KT}}(t;\Delta,\nu,B_{\mathrm{LF}})$, where $\Delta$ is the
linewidth determined by random local fields from nuclear magnetic moments,
$B_{\mathrm{LF}}$ is the magnitude of LF, and $\nu$ is the fluctuation rate of
$\Delta$ Hayano  _et al._ (1979). The KT function is expressed analytically in
the case of static ($\nu=0$) and ZF ($B_{\mathrm{LF}}=0$) conditions,

$G_{z}^{\mathrm{KT}}(t;\Delta,0,0)=\frac{1}{3}+\frac{2}{3}\left(1-\Delta^{2}t^{2}\right)e^{-\frac{1}{2}\Delta^{2}t^{2}},$
(1)

The magnitude of $\Delta$ for a given Mu site is evaluated by calculating the
second moments of dipolar fields from nuclear magnetic moments by the
following equation,

$\displaystyle\Delta^{2}$
$\displaystyle\simeq\gamma_{\mu}^{2}\sum_{m}f_{m}\sum_{j}\sum_{\alpha=x,y}\sum_{\beta=x,y,z}\gamma_{m}^{2}({\bf\hat{A}}_{mj}{\bf
I}_{m})^{2}\sin^{2}\Theta_{j}$ (2) $\displaystyle{\bf\hat{A}}_{mj}$
$\displaystyle=A^{\alpha\beta}_{mj}=(3r_{mj}^{\alpha}r_{mj}^{\beta}-\delta_{\alpha\beta}r_{mj}^{2})/r_{mj}^{5}\quad(\alpha,\beta=x,y,z)$

where $\gamma_{\mu}/2\pi=135.539$ [MHz/T] is the muon gyromagnetic ratio,
${\bm{r}}_{mj}=(x_{mj},y_{mj},z_{mj})$ is the position vector from the muon
site to the $j$-th nucleus, ${\bm{\mu}}_{m}=\gamma_{m}{\bm{I}}_{m}$ is the
nuclear magnetic moment of the atom with the natural abundance of $f_{m}$,
$\Theta_{j}$ is the polar angle of ${\bm{r}}_{mj}$ Hayano  _et al._ (1979).
Since 69Ga with $f_{1}=0.604$ and 71Ga with $f_{2}=0.396$ have nuclear spins
of $I_{m}=3/2$, respectively, they have electric quadrupole moments. In this
case, ${\bm{\mu}}_{m}$ is subject to electric quadrupolar interactions with
the electric field gradient generated by the point charge of the diamagnetic
Mu, leading to the reduction of the effective ${\bm{\mu}}_{m}$ (by a factor
$\sin^{2}\Theta_{j}$) and to the modification of $\Delta$ that depends on the
initial muon spin direction relative to the crystal axis. (In the powder
sample, the spatial averaging leads to
$\langle\sin^{2}\Theta_{j}\rangle=2/3$.) The $\Delta$ of the candidate Mu
sites inferred from DFT calculations were evaluated using the Dipelec code
Kojima  with Eq. (2) implemented.

DFT calculations were performed to investigate in detail the local structure
of H-related defects using OpenMX Ozaki (2003) and VASP codes Kresse and
Hafner (1993). Structural relaxation calculations with H (to mimic Mu) using
the GGA-PBE exchange correlation function were performed on a $1\times 3\times
2$ superlattice with cutoff energies of 200 Ry for OpenMX (520 eV for VASP),
and $K$ points were set to $3\times 4\times 3$. Structures were relaxed until
the the maximum force on each atom was less than $3\times 10^{-4}$
(Hartree/Bohr) for OpenMX (0.01 eV/Å for VASP). Additional structural
relaxation calculations using the Heyd-Scuseria-Ernzerhof (HSE06) hybrid
functional implemented in VASP were performed with $K$ points of $2\times
3\times 2$ to confirm the structure of the minimum formation energy Krukau
_et al._ (2006). The Hartree-Fock mixing parameter was set to 0.35, which
reproduces the experimental value of the band gap Varley  _et al._ (2010,
2011).

##  III RESULTS

First, we investigated the initial asymmetry [$A(0)$] of the $\mu$SR spectrum
under TF = 2 mT and found that it is independent of temperature ($T$) and
nearly constant within experimental error over the entire measured temperature
range $4\leq T\leq 300$ K [see Fig. S5(a) in SM sm ]. Since the absence of a
paramagnetic Mu at 300 K is confirmed from the TF data, the $T$-independent
$A(0)$ values indicates that the muon is mostly in diamagnetic state(s) at all
temperatures. (For the possibility of Mu0 with extremely small hyperfine
parameters to exist, see Section IV.)

Figure 1:  (a) ZF and LF $\mu$SR time spectra observed at 300 K ($4.0\times
10^{8}$ positron events collected for ZF and $2.0\times 10^{8}$ for LF), which
consists of two components (Mu1 and Mu2). (b) ZF spectra at typical
temperatures ($3.0\times 10^{8}$ positron events collected). The solid curves
represent the least-square fit by Eq. (3). The horizontal dashed line shows
$f_{2}$. (c), (d) Temperature dependence of the linewidth $\Delta_{1}$ and
fluctuation rate $\nu_{1}$ for the Mu1 component.

The normalized $\mu$SR time spectra [$G_{z}(t)\equiv A(t)/A(0)$] at 300 K
under ZF and LF ($\parallel\langle 001\rangle$) are shown in Fig. 1(a). The
ZF-$\mu$SR spectrum shows that it consists of two components, one we call Mu1,
which partially recovers at $t\gtrsim 12$ $\mu\mathrm{s}$ following slow
Gaussian relaxation, and Mu2, which exhibits almost no relaxation. Note that
the recovery of $G_{z}(t)$ characteristic to the 1/3-tail of the Kubo-Toyabe
function in Eq. (1) is clearly visible only for $t\gtrsim$15 $\mu$s. This
could not have been detectable by previous $\mu$SR measurements under TF alone
King  _et al._ (2010) or those using a continuous muon beam Celebi  _et al._
(2012), and has been first revealed by precise ZF-$\mu$SR measurements using
the high-flux pulsed muon beam at J-PARC. The suppression of the Gaussian
relaxation by the weak LF indicates that the relaxation for the Mu1 component
is induced by the quasi-static random local fields from Ga nuclear magnetic
moments.

It is clear from Fig. 1(b) that the relative yield of Mu1 vs Mu2 apparently
depends on $T$ at lower temperatures. Furthermore, the lineshape exhibits
change from a Gaussian to exponential-like behavior at low temperatures [see
$G_{z}(t)$ for $t\lesssim 6$ $\mu$s at 30 K]. Considering these features, we
analyzed the ZF and LF spectra by global curve-fits using the following
function,

$G_{z}(t)=f_{1}G_{z}^{\mathrm{KT}}(t;\Delta_{1},\nu_{1},B_{\mathrm{LF}})e^{-\lambda
t}+f_{2},$ (3)

where $f_{i}$ ($i=1,2$) are the relative yields of the Mui components. The
exponential damping with the relaxation rate $\lambda$ is introduced to
describe the possible influence of the fluctuating magnetic fields from
unpaired electrons (including excited carriers and excitons localized nearby
Mu). As shown by the solid lines in Figs. 1(a) (b), curve fit using Eq. (3)
provides reasonable agreement with the data, and $\Delta_{1}=0.140(1)$ MHz is
obtained from the fit at 300 K. The $T$ dependencies of $\Delta_{1}$ and
$\nu_{1}$ obtained from the fit are shown in Figs. 1(c) and (d), and those of
$f_{2}$ and $\lambda$ in Figs. 2(a) and (c), respectively.

Figure 2: Temperature dependence of (a) fractional yield for the Mu2 component
[$f_{2}\equiv A_{2}/A(0)$] which is characterized by the changes at
$T_{\alpha}\simeq 30$ K, $T_{\beta}\simeq 55$ K and $T_{\gamma}\simeq 80$ K,
(b) bulk carrier density ($N_{\mathrm{e}}$) multiplied by mobility [$\mu_{\rm
e}$, shown in (d)], (c) exponential relaxation rate ($\lambda$) at ZF and LF =
3 mT. Note that the vertical axis of (b) is on a logarithmic scale.

Above $T_{\gamma}\simeq 80$ K where the $f_{1}$ ($=1-f_{2}$) is nealy
independent of $T$, $\Delta_{1}$, $\nu_{1}$ and $\lambda$ also show similar
trends; $\Delta_{1}$ increases slightly to $\sim$0.15 MHz as $T$ decreases,
while $\nu_{1}$ and $\lambda$ shows a constant value close to zero
($\ll\Delta_{1}$). This indicates that the Mu1 component is quasistatic for
$T\gtrsim T_{\gamma}$. Meanwhile, the rapid motion of Mu2 is inferred from the
fact that there is no interstitial sites free of local magnetic fields from Ga
nuclear magnetic moments (with 100% natural abundance); $\Delta$ calculated at
any site is always larger than $\sim$0.15 MHz for the unrelaxed lattice, which
is only compatible with the situation where the relaxation due to the local
field (with a linewidth $\Delta_{2}\gtrsim 0.15$ MHz) is suppressed by the
motional averaging, namely,

$G_{z}(t)=f_{1}G_{z}^{\mathrm{KT}}(t;\Delta_{1},\nu_{1},B_{\mathrm{LF}})e^{-\lambda
t}+f_{2}G^{\rm KT}_{z}(t;\Delta_{2},\nu_{2}),$ (4)

with $G^{\rm KT}_{z}(t;\Delta_{2},\nu_{2})\simeq 1$ for
$\nu_{2}\gg\Delta_{2}$.

Below $T_{\gamma}$, $\nu_{1}$ increases with decreasing $T$ in correlation
with $f_{2}$, and tends to approach a constant value or decrease for
$T\lesssim T_{\alpha}\simeq 30$ K. A similar correlation is observed between
$f_{2}$ and $\lambda$; $\lambda$ gradually increases with decreasing $T$ to
exhibit a small peak around $T_{\gamma}$, then decreases towards $T_{\beta}$,
followed by an increase to reach another maximum near $T_{\alpha}$. Here, we
point out that, as shown in Fig. 2(b), the complicated $T$ dependence of
$f_{2}$ and the increase of $\lambda$ occur in correlation with the decrease
of $N_{\mathrm{e}}\mu_{\rm e}$ (with $\mu_{\rm e}$ being the mobility) of the
sample used for $\mu$SR measurements. More specifically, $N_{\mathrm{e}}$
exhibits gradual decrease around $T_{\gamma}$ with decreasing $T$, followed by
a sharp decrease below $T_{\beta}$ and an increase below $T_{\alpha}$ (see
Fig. S2 in SM sm ). As shown in Fig. 2(d), $\mu_{\rm e}$ decreases with
decreasing $T$ below $\sim$120 K, reaching a minimum around $T_{\beta}$, and
then increases to a maximum at $T_{\alpha}$. The behavior of $\mu_{\rm e}$ for
$T>T_{\gamma}$ is qualitatively in line with previous reports Parisini and
Fornari (2016); Kabilova  _et al._ (2019). This implies that the electronic
state of Mu is highly sensitive to the quality of samples, which is also
supported by the fact that the $T$ dependence of $f_{2}$ (and $f_{1}$) in
powder sample differs significantly from that in single crystal (see Fig.
S5(b) in SM sm ).

##  IV DISCUSSION

In the ambipolarity model, the electronic state of the interstitial Mu is
determined by where the donor and acceptor levels associated with H, predicted
from the Fermi energy ($E_{F}$) dependence of defect formation energy
($\Xi^{q}$) for Hq ($q=\pm,0$), is located in the energy band structure of the
host material. Figure 3(a) shows $\Xi^{q}(E_{F})$ for each valence state of H
obtained by previous DFT calculations Li and Robertson (2014), where H is
substituted with Mu. The relationship between the donor/acceptor levels and
band structure predicts that Mu can take two different diamagnetic states: a
donor-like state in which it releases an electron into the conduction band to
become Mu+, and a hydride (Mu-) state, which is qualitatively consistent with
the present observations. Interestingly, the $E^{+/-}$ level, which determines
the charge state of H at thermal equilibrium, is in the conduction band as is
the donor level, so the behavior of H is expected to be the same as that of
the donor-like Mu.

Figure 3: a) The formation energy ($\Xi^{q}$) of interstitial Mu (Mu${}^{q}_{\rm i}$, $q=0,\pm$) vs the Fermi level ($E_{F}$) in $\beta$-Ga2O3 inferred from DFT calculations for Hq Li and Robertson (2014). The donor/acceptor levels are determined as cross points between $\Xi^{\pm}(E_{F})$ and $\Xi^{0}(E_{F})$. The dashed line shows the intrinsic charge neutral level ($E_{F}^{\rm int}$) obtained from the DFT calculation. b) Schematic band diagrams for the electron energy associated with the donor/acceptor levels in a).  Table 1: Comparison of the linewidth ($\Delta$) between those at the H sites obtained from structural relaxation calculations and the experimental value. $\Delta_{\rm sc}$, and $\Delta_{\mathrm{pwdr}}$ for single crystalline (sc) and powder (pwdr) samples, respectively. $d_{\mathrm{OH}}$ is the distance from H to the nearest oxygen (see also Fig. 4). Site |  $d_{\mathrm{OH}}$ [nm] |  $\Delta_{\rm sc}$ (MHz) |  $\Delta_{\mathrm{pwdr}}$ (MHz)
---|---|---|---
HI | 0.0966 | 0.136 | 0.137
HII | 0.1036 | 0.150 | 0.153
HIII | 0.1003 | 0.161 | 0.156
H- | – | 0.345 | 0.299
$\Delta_{1}$ (300 K) |  | 0.140(1) | 0.136(1)

To determine the local defect structure of the Mu1 state, $\Delta_{1}$ was
compared with those calculated for the candidate sites, and the results are
summarized in Table 1. The value for the HI site bonded to the three-
coordinated OI [see Fig. 4(a)] is in good agreement with $\Delta_{1}$ at 300
K, indicating that H${}_{\rm I}^{+}\approx{\rm Mu}_{1}^{+}$ in the dilute
limit ($\sim$105 cm-3). This confirms the earlier point that, aside from not
knowing the exact location of $E^{+/-}$, HI can serve as donor Varley  _et
al._ (2011). The existence of H in the HI structure hss been also inferred
from the infrared spectroscopy of hydrogenated $\beta$-Ga2O3 Qin  _et al._
(2019). The possibility that Mu1 corresponds to H trapped in Ga and O
vacancies is excluded by the comparison of $\Delta_{1}$ with those for Mu in
these vacancies (shown in Table S1 in SM sm ).

As seen in Fig. 1(c), $\Delta_{1}$ exhibits a slight increase with decreasing
$T$, approaching the value at the HII/HIII sites [see Fig. 4(b), (c)]. This
can be interpreted as reflecting the fact that the initial population of
relaxed-excited states immediately after muon implantation is a random
sampling of available metastable sites. Considering that the difference in the
formation energy estimated by DFT calculations is small among these sites
(e.g., $\sim$30 meV between HI and HII), the decrease in $\Delta_{1}$ with
increasing $T$ above $T_{\gamma}$ suggests that the Mu site distribution
approaches the lowest energy state by the annealing process. A similar $T$
dependence of $\Delta$ exhibited by donor-like Mu has been reported for
InGaZnO4 in which multiple H sites are available Kojima  _et al._ (2019).

Figure 4: Local structures H-related defects corresponding to (a) HI, (b) HII,
(c) HIII and (d) H- state. HI-III are bonded to O, while H- forms bonding with
two Ga3+ ions Li and Robertson (2014). The crystal structures were displayed
using VESTA Momma and Izumi (2011).

With Mu1 identified as being in the donor-like state, the paired Mu2 component
is presumed to be in the acceptor-like diamagnetic state (Mu${}_{2}^{-}$) from
the ambipolarity model. According to DFT calculations, the corresponding H-
state is bonded to the two nn Ga ions, as shown in Fig. 4(d) Li and Robertson
(2014). The $E^{0/-}$ level is near the conduction band, and it can still
accommodate two electrons as it is located below the charge neutral level
($E_{F}^{\rm int}$, see Fig. 3). This means that it is possible for Mu2 to
exchange electrons with the conduction band. Moreover, the conduction band has
a relatively large dispersion (effective mass $\sim$0.28$m_{e}$) around the
$\Gamma$ point Peelaers and Van de Walle (2015), allowing fast migration of
electrons.

While Mu${}_{2}^{-}$ is unlikely to exhibit fast diffusion by itself due to
the bonding to cations, there are many reported examples of fast diffusive
motion of acceptor-like Mu0 in semiconductors and alkali halides Patterson
(1988); Kadono  _et al._ (1994); Gxawu  _et al._ (2005); Chow  _et al._
(2000). Since $N_{\rm e}\mu_{\rm e}$ and $f_{2}$ show almost the same $T$
dependence for $T>T_{\beta}$, it is inferred that the Mu2 state diffuses
rapidly for $T>T_{\gamma}$ through the neutral state temporally attained by
the charge exchange reaction Mu${}_{2}^{-}\rightleftarrows{\rm
Mu}_{2}^{0}+e^{-}$. The reaction rate is given by

$r_{\mathrm{ex}}=\frac{1}{\tau_{-0}+\tau_{0-}}=\frac{r_{-0}r_{0-}}{r_{-0}+r_{0-}},$
(5)

where $r_{-0}$ ($=1/\tau_{-0}$) is the ionization probability ($\propto
e^{-\varDelta E/k_{B}T}$, with $\varDelta E\sim E_{c}-E^{0/-}$ and $E_{c}$ the
CBM energy) and $r_{0-}$ ($=1/\tau_{0-}$) is the capture rate of the free
carriers ($\propto\sigma_{\rm c}N_{\mathrm{e}}\mu_{\rm e}V_{r}$, with
$\sigma_{\rm c}$, $\mu_{\rm e}$, and $V_{r}$ being the cross section, carrier
mobility, and electric field exerted from the Mu-Ga complex state,
respectively). Note that $\lambda$ due to the charge exchange is also quenched
by the motional averaging when $r_{\mathrm{ex}}$ is much greater than the
hyperfine parameter. Such motional effects have also been observed in SiO2 and
are thought to be responsible for the disappearance of anisotropy in the
hyperfine interaction of Mu0 with increasing $T$ Chow  _et al._ (2000). The
existence of the non-relaxing Mu component has also been reported for variety
of materials including NaAlH4, LaScSiHx, and FeS2 Kadono  _et al._ (2008);
Okabe  _et al._ (2018); Hiraishi  _et al._ (2021), implying the ubiquitous
nature of acceptor-like Mu states.

It is natural to assume that Mu2 diffuses along the $\langle 010\rangle$
direction because $\beta$-Ga2O3 has an open channel structure along the
$\langle 010\rangle$ direction, In fact, secondary ion mass spectrometry
analysis of 2H implanted $\beta$-Ga2O3 Reinertsen  _et al._ (2020) revealed
that 2H diffuses more easily in the $\langle 010\rangle$ direction than
perpendicular to the $\langle-201\rangle$ surface. It has also been reported
for rutile TiO2 that hydrogen diffusion is more likely to occur in open
channels along the $c$-axis than in the $a$-$b$ plane Bates  _et al._ (1979).

There are three different open channels in the $\langle 010\rangle$ direction
in $\beta$-Ga2O3. The potentials associated with diffusion in each channel
were investigated by total energy calculations (see Sect. III in SM sm  for
details), and it was found that hydrogen tends to diffuse in the same channel
as the Mu1 site. Since the two charged states Mu${}_{1}^{+}$ and
Mu${}_{2}^{-}$ accompany mutually different lattice distortions, it is assumed
that the potential barriers caused by these distortions allow Mu2 to diffuse
while maintaining its state. However, because the adopted calculations do not
take structural relaxation into account, it is a future task to evaluate using
the nudged-elastic band (NEB) method.

The details of what causes the decrease in $f_{2}$ (increase in $f_{1}$) and
recovery below $T_{\gamma}$, which is apparently correlated with the onset of
slow dynamics for Mu1 (as inferred from the $T$ dependence of $\Delta_{1}$,
$\nu_{1}$, and $\lambda$), are currently unknown. Nevertheless, when analyzing
the time spectra in this $T$ range using Eq. (4) with $\Delta_{1}$ and
$\Delta_{2}$ as free parameters, $\Delta_{i}$ converge to mutually close
values ($\sim$0.13–0.14 MHz). A curve fit performed with $\Delta_{1}$ fixed at
0.136 MHz and $\Delta_{2}$ at 0.345 MHz (corresponding to the stationary
hydride state) and the other parameters free fails to reproduce the data, as
indicated by a large reduced chi-square (see Fig. S6 in SM). Thus, the
decrease in $f_{2}$ suggests that a transition from Mu2 to Mu1 occurs in the
relevant $T$ range. One possible scenario is that Mu${}_{2}^{-}$ can be
ionized by thermal excitation and transitions to Mu${}_{1}^{+}$ for
$T<T_{\beta}$, but the ionization is prevented by increasing $N_{\rm
e}\mu_{\rm e}$ for $T>T_{\beta}$ [i.e., $r_{0-}\gg r_{-0}$ in Eq. (5)],
allowing it to continue to exist as a metastable state again above
$T_{\gamma}$.

Finally, we comment on the relationship between our experimental results and
the prior $\mu$SR studies on $\beta$-Ga2O3. In Ref. [King  _et al._ , 2010],
TF-$\mu$SR measurements are reported for a powder sample from a commercial
vendor, and it is observed that about 10% of the signal exhibits slight
increase of relaxation rate from 0.08 MHz to 0.12 MHz below 50–100 K. Ref.
[Celebi  _et al._ , 2012] reports results for a single-crystal sample
(orientation unknown) and observes a 4–6% decrease in the initial asymmetry
below 50–100 K in both ZF and TF-$\mu$SR measurements and an increase in
$\Delta$ of the component described by $G_{z}^{\rm KT}(t)$ from 0.10–0.11 MHz
to 0.15 MHz in the ZF-$\mu$SR spectra. Although detailed comparisons with our
results are not possible because no time spectra are available for either
case, these changes appear to correspond to the increase in $f_{1}$ at
$T<T_{\gamma}$ that we observed in both the powder and single-crystalline
samples [see Fig. S5(b) in SM sm ]. Meanwhile, these reports are silent about
the component corresponding to Mu2. We stress that the existence and origin of
Mu2 are clearly identified for the first time in this study with the help of
the ambipolarity model.

###### Acknowledgements.

This work was supported by the MEXT Elements Strategy Initiative to Form Core
Research Center for Electron Materials (Grant No. JPMXP0112101001) and JSPS
KAKENHI (Grant No. 19K15033). The $\mu$SR experiments were conducted under
user programs (Proposal No. 2019MS02) at the Materials and Life Science
Experimental Facility of the J-PARC. We also acknowledge the Neutron Science
and Technology Center, CROSS for the use of PPMS in their user laboratories.

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Supplemental Material: Local electronic structure of dilute hydrogen in
$\beta$-Ga2O3 probed by muons

M. Hiraishi et al.

##  I Properties of Single Crystalline $\beta$-${\bf Ga_{2}O_{3}}$

###  I.1 Thermal Desorption Spectrometry (TDS)

Figure S1 shows the TDS results for the $\langle 001\rangle$ sample, which was
linearly heated from room temperature to 800∘C and held at 800∘C for 1 hour.
Temperature is shown on the right axis and the signal intensities obtained by
the quadrupole mass spectrometer (QMS) are shown on the left vertical axis.
The analysis revealed that the $\langle 001\rangle$ sample contains 3.5$\times
10^{18}\mathrm{cm}^{-3}$ of hydrogen ($m/z=2$, with $m$ being the mass number,
$z$ the ion valence). Considering that the desorption process is diffusion-
limited for crystalline specimens, this value may correspond to the lower
limit. The signal intensity for $m/z=18$ (1H${}_{2}^{16}$O, 18O) is greater
than that for $m/z=16$ (16O) multiplied by 0.002 (natural abundance of 18O),
indicating that $m/z=18$ is derived from water (H2O). The similar temperature
dependence of $m/z=1$ and 17 to that of 18 strongly suggests that $m/z=1,17$
are also water-related signals. The steep increase below 100∘C ($\sim$20 min.)
is probably due to residual moisture on the sample surface, chamber, and so
on.

Figure S1: TDS spectra of the $\beta$-Ga2O3 sample used for $\mu$SR
measurements. Purple, green, yellow, red, and gray are signals for $m/z=$1, 2,
16, 17, and 18, respectively ($m$: mass number, $z$: ion valence). Temperature
history is shown on the right vertical axis.

###  I.2 Carrier concentration

The temperature dependence of carrier concentration obtained from Hall
coefficient measurements for the $\beta$-Ga2O3 sample (with $\langle
001\rangle$ orientation) used for $\mu$SR measurements is shown in Fig. S2.
The sign of the Hall coefficient indicates that majority carriers are
electrons.

Figure S2: Temperature dependence of the electron carrier concentration in the
single crystalline sample of $\beta$-Ga2O3.

##  II Result of DFT calculations

The relaxed structures obtained by DFT calculations are shown in Fig. S3, and
the simulated muon spin relaxation rates $\Delta$ in the Kubo-Toyabe function
for each structure are shown in Table S1.

Figure S3: Left: Crystal structure of $\beta$-Ga2O3. Isolated pink balls indicate the initial positions of hydrogen in the structural relaxation calculations shown in (a)-(f), respectively corresponding to HI, HII, HIII, HIV, HV, and HVI in the Table S1. The crystal structures were drawn using VESTA Momma and Izumi (2011). The nuclear dipolar linewidth $\Delta$ calculated for each structure is shown in Table S1. Table S1: The nuclear dipolar linewidth $\Delta$ for the various local structures involving interstitial H obtained from structural relaxation calculations. $\bigstar$ indicate the results of VASP:HSE06 calculations Kresse and Hafner (1993); Krukau  _et al._ (2006), while others were those by OpenMX:GGA-PBE Ozaki (2003). $d_{\mathrm{OH}}$ denotes the distance between H and the nearest O. $\Delta_{\langle 001\rangle}$ and $\Delta_{\mathrm{pwdr}}$ are the linewidths for single crystalline and powder samples, respectively.  Type | Config. | nn oxygen | $d_{\mathrm{OH}}$ | $\Delta_{\langle 001\rangle}$ | $\Delta_{\mathrm{pwdr}}$
---|---|---|---|---|---
(nm) | (MHz) | (MHz)
Interstitial★ | HI | OI | 0.0966 | 0.136 | 0.137
Interstitial★ | HII | OII | 0.1036 | 0.150 | 0.153
Interstitial★ | HIII | OII | 0.1003 | 0.161 | 0.158
Interstitial | HIV | OIII | 0.1024 | 0.157 | 0.165
Interstitial | HV | OIII | 0.0996 | 0.158 | 0.170
Interstitial | HVI | OI | 0.0989 | 0.175 | 0.162
Vacancy | V${}_{\mathrm{Ga_{I}}}$-H | OI | 0.0983 | 0.106 | 0.117
Vacancy | V${}_{\mathrm{Ga_{II}}}$-H | OI | 0.0988 | 0.108 | 0.115
Vacancy | V${}_{\mathrm{O_{I}}}$-H | — | — | 0.286 | 0.250
Vacancy | V${}_{\mathrm{O_{II}}}$-H | — | — | 0.168 | 0.235
Vacancy | V${}_{\mathrm{O_{III}}}$-H | — | — | 0.229 | 0.238

##  III $\langle 010\rangle$ Axial diffusion: Evaluation based on total
energy.

To investigate the Mu2 diffusion path, $\langle 010\rangle$ axis position $y$
dependence of the total energy $E_{\mathrm{tot}}$ was calculated in a $1\times
2\times 1$ superlattice. Figure S4 shows the variation of the total energy,

$\displaystyle\Delta E_{\mathrm{tot}}\equiv
E_{\mathrm{tot}}({\bm{r}})-E_{\mathrm{min}},$

where $\bm{r}$ is the position vector of channel $c_{i}$ $(i=1,2,3)$ and
$E_{\mathrm{min}}$ is the global minimum of $E_{\mathrm{tot}}(\bm{r})$. Here,
${\bm{r}}$ for $c_{1}=(0.25,y,0)$, $c_{2}=(0.5,y,0.5)$ and $c_{3}=(0.5,y,0)$,
respectively. The $\Delta E_{\mathrm{tot}}$ of $c_{2}$ and $c_{3}$ show large
changes of more than 0.5 eV, whereas the $\Delta E_{\mathrm{tot}}$ of $c_{1}$
is almost independent of $y$ and the difference is only 0.04 eV, suggesting
that $c_{1}$ is a the possible diffusion path for Mu2.

Figure S4: Hydrogen position on the $\langle 010\rangle$ axis dependence of
the total energy calculated at the center $c_{i}$ $(i=1,2,3)$ of each channel,
shown in left panel, using the $1\times 2\times 1$ structure.

##  IV Supplemental $\mu$SR Results

###  IV.1 $\mu$SR results on powder Sample

$\mu$SR measurements and data analysis for the powder sample were performed as
for the single crystal. The temperature dependence of each parameter obtained
by curve fitting of the $\mu$SR time spectra is shown in Fig. S5. For
comparison, the results for the single-crystal sample presented in the main
text are also shown.

Figure S5: Temperature dependence of (a) $G_{z}(0)$ determined by TF-$\mu$SR,
(b) the fractional yield $f_{1}$, (c) the Kubo-Toyabe linewidth $\Delta_{1}$,
(d) the fluctuation rate $\nu_{1}$, and (e) the exponential relaxation rate
$\lambda$ for the Mu1 component. The result for the single crystalline sample
is quoted from the main text for comparison.

As can be seen in Fig. S5(a), the most significant difference is that the
yield $f_{1}$ of the Mu1 component in the powder sample is greatly increased
in place of the Mu2 component. In addition, it is observed for the powder
sanple that $\nu_{1}$ (the fluctuation rate of $\Delta_{1}$) is almost zero
(static) for $T\lesssim T_{\gamma}\simeq 80$ K, whereas it takes on finite
values for $T\gtrsim T_{\gamma}$, suggesting slow diffusive motion of the Mu1
component. These behaviors are in remarkable contrast to the single-crystal
sample. On the other hand, the increase in $f_{1}$ (decrease in $f_{2}$) from
$T_{\beta}$ ($\simeq 55$ K) to $T_{\gamma}$ is common. From these
observations, it may be interpreted that for $T<T_{\beta}$, the component
corresponding to Mu2 in the powder sample is bounded by some defects, etc.,
and that a part of it is detrapped and begins to exhibit fast diffusion for
$T_{\beta}<T<T_{\gamma}$, which is gradually suppressed above $T_{\gamma}$.

###  IV.2 Comparison of curve fits for the data around $T_{\beta}$

To test the possibility that quasi-static Mu1 and Mu2 coexist in the $\mu$SR
spectrum near the temperature at which $f_{2}$ decreases significantly, we
performed global curve fits for the ZF and LF-$\mu$SR spectra with the
linewidth $\Delta_{1}$ and $\Delta_{2}$ as free parameters in each state, as
well as for those fixed to the values expected for the corresponding sites
($\Delta_{1}=0.136$ MHz and $\Delta_{2}=0.345$ MHz). The results of the
respective analyses for the spectra at 57 K are shown below in Fig. S6(a) and
(b). The fits with $\Delta$ fixed to the expected values at the two sites show
large residual chi-squares, clearly indicating that the data are not well
reproduced.

Figure S6: Analyzed $\mu$SR spectra at 57 K (global fit of the entire ZF/LF
data). (a) with $\Delta_{1}$ and $\Delta_{2}$ as free parameters, (b) fixed to
$\Delta_{1}=0.136$ MHz and $\Delta_{2}=0.345$ MHz.