Hydrogen-Ti$^{3+}$ Complex as a Possible Origin of Localized Electron Behavior in Hydrogen-Irradiated SrTiO$_3$

A recent muon spin rotation ($\mu^+$SR) study on a paramagnetic defect complex formed upon implantation of $\mu^+$ pseudo-proton into SrTiO$_3$ is reviewed with a specific focus on the relation with experimental signatures of coexisting delocalized and localized electrons in hydrogen-irradiated metallic SrTiO$_3$ films. The paramagnetic defect complex, composed of interstitial $\mu^+$ and Ti$^{3+}$ small polaron, is characterized by a small dissociation energy of about 30 meV. Density functional theory (DFT) calculations in the generalized gradient approximation (GGA) +$U$ scheme for a corresponding hydrogen defect complex reveal that a thermodynamic donor level associated with electron transfer from an H$^+$-Ti$^{3+}$ complex to the conduction band can form just below the conduction band minimum for realistic $U$ values. These findings suggest that the coexistence of delocalized and localized electrons can be realized in hydrogen-irradiated SrTiO$_3$ in electron-rich conditions.

conductivity [5,13], which proves that forcedly implanted hydrogen is mostly ionized and donates electrons into the conduction band. On the other hand, the existence of localized electrons has also been becoming clear in these films. For instance, a deep in-gap state, which can be associated with strongly localized electrons, was detected by photoemission spectroscopy in a heavily hydrogenated SrTiO 3 film prepared by hot H atom irradiation [13]. Thermal hysteresis in resistivity of an H 2 + ion-irradiated metallic SrTiO 3 film also implies the existence of trapped electrons together with delocalized electrons [5]. However, details of the localized state are still far from being fully understood.
In this review paper, a recent muon spin rotation (μ + SR) study on a paramagnetic defect complex formed upon implantation of μ + pseudo-proton into SrTiO 3 [14] is reviewed with a specific focus on the relation with the experimental signatures of coexisting delocalized and localized electrons in the hydrogen-irradiated metallic SrTiO 3 films. The thermodynamic stability of the paramagnetic defect complex, which is composed of interstitial μ + and Ti 3+ small polaron, is discussed on the basis of density functional theory (DFT) calculations in the generalized gradient approximation (GGA) +U scheme for a corresponding hydrogen defect complex.

II. MUON SPIN ROTATION EXPERIMENT
A. Modeling isolated hydrogen with muons Muonium (Mu) defects formed upon implantation of  + in insulators or semiconductors have been utilized as an experimentally accessible model of isolated hydrogen [15,16].
The electronic structures of isolated H and Mu defects are substantially identical, as can be seen when comparing atomic Mu 0 (= + e  ) and H 0 with respect to their reduced mass ratio ~1, where the superscript on the right of the (pseudo) element symbols indicates the total charge of corresponding atomic or defect states. Microscopic insight into Mu defects can be obtained with the  + SR spectroscopy, which is a -detected magnetic resonance technique using a spin-polarized  + beam.
Technical details of  + SR are given in Sec. II.B, which can also be found in literature, such as Ref. [17].
Characterizing a paramagnetic Mu 0 defect, or a paramagnetic muon-electron complex, is a relatively easy task in comparison with distinguishing two ionized Mu defect species, Mu + and Mu  . Hyperfine interactions between a muon spin I and an electron spin s in the paramagnetic Mu 0 defect under a magnetic field B can be modeled with the following spin Hamiltonian,  This measurement cycle is repeated typically 10 7 times to build up time histograms of the decay events. For the positron detector i(=U, D, …), the histogram has the following form, where is a factor that is proportional to the total dose, A is effective asymmetry, and n i is a unit vector pointing in the direction of the detector i from the sample position. In the configuration of Fig. 1, the scalar projection of P(t) onto n U ,

C. Mui + -bound Ti 3+ small polaron in SrTiO3
 + SR studies of SrTiO 3 have been performed by several groups [14,19,20]. Here, that by Ito et al. [14] is reviewed, focusing on the hyperfine structure of a paramagnetic Mu-  proximity to a Mu i + donor to form a polaronic center [22][23][24], as shown in Fig. 2(b).
The simple spin Hamiltonian in eq. (1) While it seems unusually small for the energy that separates the small polaron bound state and the unbound state, this situation can be understood by considering the energy balance for polaron formation in the presence of lattice distortion caused by the localized electron, as shown in Fig. 3(b). into the microsecond sensitivity range (Fig. 3(a)). A similar situation could arise in the hydrogenated SrTiO 3 film prepared by H 2 + ion irradiation at low temperature. Indeed, thermal hysteresis in resistivity of this film implies that the asimplanted mixture of H states excessively contains neutral species [5]. This neutral state is suggested to be an H i + -bound small polaron from the similarity to the μ + implantation experiment.

III. DFT CALCULATIONS A. Computational details
All calculations were performed within the DFT framework with a Hubbard U correction by using the QUANTUM ESPRESSO code [25,26]. The generalized gradient approximation (GGA) using Perdew-Burke-Ernzerhof (PBE) exchange correlation functional was adopted. PAW-type pseudopotentials with H(1s), Sr(4s, 4p, 5s), Ti(3s,3p,4s,3d) and O(2s, 2p) valence states were used. Wave functions were expanded in plane waves with a cutoff of 70 Ry for the kinetic energy and 600 Ry for the charge density.
The SrTiO 3 structure was described using a 3×3×3 supercell of the 5-atom cubic primitive cell. Brillouin zone sampling with a Monkhorst-Pack k-point mesh of 3×3×3 was applied.
The finer k-point mesh of 5×5×5 was used for plotting density of states (DOS). The Hubbard U potential on Ti 3d states, U Ti , was set to 4.74 eV according to Ref. [27]. Gaussian smearing with a broadening of 0.01 Ry was used for structure optimization calculations, where atomic positions were relaxed with keeping the lattice constant fixed at the optimized value for stoichiometric SrTiO 3 . In all cases, atomic forces were converged to within 7.7×10 -3 eV/Å. Then, the selfconsistent field calculation for the optimized structure was refined with the optimized tetrahedron method [28] for avoiding artifacts associated with the Gaussian smearing. The structure optimization calculations were started from spinunpolarized initial states unless otherwise specified.
The formation energy of an interstitial H defect in a charge state q, H , was calculated as described in Ref. [9,29]: where H is the total energy of a defective supercell in a charge state q, host is the total energy of a charge neutral host supercell, is the chemical potential of an H 2 molecule, is the chemical potential of the electron reservoir at the valence-band maximum (VBM), and E F is the Fermi energy with reference to . E corr is a small potential-alignment term for charged defective supercells, which was obtained in accordance with a standard procedure [30]. as shown in Fig. 4(b). On the other hand, this figure together with Fig. 4