Characteristic fast H− ion conduction in oxygen-substituted lanthanum hydride

Fast ionic conductors have considerable potential to enable technological development for energy storage and conversion. Hydride (H−) ions are a unique species because of their natural abundance, light mass, and large polarizability. Herein, we investigate characteristic H− conduction, i.e., fast ionic conduction controlled by a pre-exponential factor. Oxygen-doped LaH3 (LaH3−2xOx) has an optimum ionic conductivity of 2.6 × 10−2 S cm−1, which to the best of our knowledge is the highest H− conductivity reported to date at intermediate temperatures. With increasing oxygen content, the relatively high activation energy remains unchanged, whereas the pre-exponential factor decreases dramatically. This extraordinarily large pre-exponential factor is explained by introducing temperature-dependent enthalpy, derived from H− trapped by lanthanum ions bonded to oxygen ions. Consequently, light mass and large polarizability of H−, and the framework comprising densely packed H− in LaH3−2xOx are crucial factors that impose significant temperature dependence on the potential energy and implement characteristic fast H− conduction.


Supplementary Note 1
For the NPD, first we refined the crystal structure of LaD2O0.5 by using a space group of Fm-3m (fcc structure) in which the oxygen is disordered at T-site and the off-centering of D at O-site is considered. However, the reliable factor for the fitting, Rwp = 17.5% and χ 2 = 8.24, were not good, and the net charge neutrality between La 3+ , O 2− and D − was not preserved in the refined structure. Then, next we considered an oxygen-ordering that frequently observed in rare earth oxyfluorides. The crystal structure of oxyfluorides, so called Vernier-phase, crystallizes in a pseudo-fluorite structure with a tetragonal or orthorhombic distortion. 1 Among those complex structures, we adopted a tetragonal structure with a space group of P4/nmm as the simplest model with the smallest lattice constants. 2 This model successfully reduced the reliable factors to Rwp = 5.93% and χ 2 = 7.31 and some of small peaks which were not indexed when using Fm-3m can be indexed (Supplementary Figure 1a, b), and the refined chemical composition agreed well with that measured by TDS. We also performed the NPD for LaD1.5O0.75 and analyzed using same structure model as used at xnom. = 0.5. The profile is shown in Supplementary Figures 1c and d. In Supplementary Table 1, we summarized the structural parameters refined using the tetragonal P4/nmm at xnom. = 0.5 and 0.75.
In order to know an accurate space group, we carried out the electron diffraction measurement on LaH2O0.5, and pursued an additional spots derived from the incommensurate structure. However, we only observed spots derived from the fcc structure as shown in Supplementary Figure 2 .m. Then, we synthesized the sample at the same temperature and pressure as described in main text, and took the SEM images and conductivity data. Supplementary  Table S2). On the other hand, the conductivity values from R2 corresponding to the lower frequency region were 6.5×10 −6 Scm −1 and 1.

Supplementary Note 5
A simplest way to add anharmonicity into harmonic potential is to take into account effects of thermal expansion of bond length on warming and a resulting frequency reduction of elongated bonds. 3 This effect can be treated by assuming that the temperature slope, b, corresponds αγ (quasi-harmonic approximation), where α is thermal expansion coefficient and γ is Grüneisen parameter: where ω and V are vibration frequency of atom and volume, respectively. In this case, the mean square displacement of vibrating atom, <u 2 >, increases more compared to the harmonic case, as expressed to <u 2 > = kBT/k (1 + 2αγT), where the k is force constant. 3 The Grüneisen parameter γ is a measure of anharmonicity. For the simplest example, we consider an one-dimensional system, i. e., a linear monatomic chain, with only onetype force constant, k (second derivatives of the potential energy) and a length of L = Na, where N is the number of atom and a is the equilibrium length of spring. In this case the γ is expressed as k'a/2k in which k' is a third derivatives of the potential energy. 4−6 harmonic case, as expressed to <u 2 > = kBT/k (1 + 2αγT), where the k is force constant. 3 In the following, we describe how the anharmonicity affects the prefactor. First, we explain the expression of prefactor based on the random walk theory, and then add the anharmonic effect into the prefactor.
The conductivity of charged species is given by 18 where c is the number of carrier in unit cell, V is the volume of unit cell per chemical formula, e is the ionic charge, and μ is the mobility of the charge carrier. The mobility is related to the corresponding diffusion coefficient by the Nernst-Einstein relation: The diffusion coefficient, D, is related to its mean jump frequency f by where d is the jump distance. Eq.(4) is based on the random walk theory. Then, f is given by: where f0 is the jump frequency in one specific direction, z is the number of directions in which the jump may occur, Hm is the jump activation enthalpy, and f0 is given by Sm being the migration entropy and υ0 an appropriate lattice vibration frequency.
Combining eq.s(2)−(6) gives the Arrhenius relation: where A is the prefactor in case of random walk theory.
In Table S3 we summarized values of each physical parameter of LaH3−2xOx described above. Here we considered H − hopping from tetrahedral site to octahedral site 19 which is observed in the molecular dynamics simulations. Then, the d corresponds to √3/4a where a is lattice constant of tetragonal LaH3−2xOx. z is 4 because the tetrahedral site is surrounded by four octahedral sites. For the value of υ0 we used the vibrational frequency of hydrogen that occupies tetrahedral site of LaH3 (125meV). 7 It is difficult to estimate the migration entropy, so that we referred the Sm/k of alkali halides which are in range from 10 to 60. 8 You can see that the calculated prefactor A seriously underestimates values observed experimentally which are in range from ~10 8 to ~10 12 . To add the anharmonic effect into the calculated prefactor, we adopt a quasiharmonic approximation. The quasi harmonic approximation proposed by Wills B. T. M.

Supplementary
is to replace the coefficients of second order derivative of potential energy (force constant), k, with the temperature-dependent term, k0(1−2αγT), where k0 is force constant at T = 0, α is volumetric thermal expansion coefficient, and γ is Grüneisen parameter. 3 Here we consider the temperature dependence of the quasi-harmonic potential energy: where R is the equilibrium bond length of oscillator, and U0 is the potential at r = R. We set U0 = 0, and consider the potential energy at r = r' in which activation energy barrier is formed. In the following, we assume that the temperature dependency of the quasiharmonic potential energy can be regarded to be similar with that of activation energy.
In main text, we suggest that Hassoc can be equated to a − bT, and that the prefactor becomes greater than A by a factor of exp(b/kB). If we put r' into r of the eq.(10), and adopt that a corresponds to k0(r' − R) 2 and b does 2αγk0(r' − R) 2 , the equation (10) is where b corresponds to 2αγa. Now we can quantitatively calculate the enlarged prefactor by using α, γ, and k0(r' − R) 2 . In main test, we attributed the large difference in experimental (~1.20eV) and calculated activation energies (~0.1eV) to be a = Hassoc0 (see equation (2)