Ab Initio Molecular Dynamics Studies of Fast Ion Conductors

Chapter 7 Ab Initio Molecular Dynamics Studies of Fast Ion Conductors

Zhuoying Zhu, Zhi Deng, Iek-Heng Chu, Balachandran Radhakrishnan, and Shyue Ping Ong

7.1 Introduction

Fast ion conductors are a technologically important class of materials that have extensive applications in energy storage, ion-selective electrodes and sensors. For example, fast oxide conductors such as yttria-stabilized zirconia (YSZ) and CaO- doped zirconia have been used as solid electrolytes in solid oxide fuel cells and oxygen sensors in automotive applications, respectively [1–4]. In next-generation all- solid-state rechargeable alkali-ion (lithium-ion and sodium-ion) batteries (SSBs), the key enabling material is the solid electrolyte, which is a fast alkali-ion conductor. SSBs are safer and potentially more energy dense at the system level than traditional lithium-ion batteries based on ﬂammable organic liquid electrolytes, and the search for novel superionic alkali conductors is a topic of immense current interest [5–10]. As their name implies, fast ion conductors exhibit high ionic mobility in at least one species, such as oxide (O2) or alkali ions (LiC/NaC). There are in general two broad classes of ab initio techniques for the study of ionic diffusion and conduction in materials in the scientiﬁc literatures: transition state methods [11] (e.g., nudged elastic band [12] and kinetic Monte Carlo [13]) and molecular dynamics simulations [14–16]. In this chapter, we will focus on the latter, i.e., ab initio molecular dynamics (AIMD) simulation, and its application in the study and design of fast ion conductors. A more general discussion of all techniques can be found in several excellent reviews in the literature [5, 17, 18]. There are several reasons why AIMD techniques have become increasingly the method of choice in the study of fast ion conductors. First, fast ion conductors

Z. Zhu • Z. Deng • I.-H. Chu • B. Radhakrishnan • S. Ping Ong () Department of NanoEngineering, University of California San Diego, 9500 Gilman Dr, Mail Code 0448, 92093-0448, La Jolla, CA, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]

© Springer International Publishing AG 2018 147 D.Shin,J.Saal(eds.),Computational Materials System Design, https://doi.org/10.1007/978-3-319-68280-8_7 148 Z. Zhu et al. typically exhibit non-dilute mobile ion concentrations and disorder, which are more difficult for transition state methods to effectively model. Second, the high ionic conductivity means that converged diffusion statistics can usually be obtained in reasonable simulation time scales. Finally, advances in computing power have significantly enhanced our ability to simulate reasonable cell sizes and time scales. The remainder of this chapter is structured in the following manner. We will begin by outlining the broad theoretical underpinnings for AIMD simulations, including both the Car-Parrinello and Born-Oppenheimer variants, and the analysis of such simulations for diffusion properties. As defects, typically introduced via aliovalent doping, are a critical lever to tune ionic conductivity, we will also briefly discuss first principles techniques in which to assess the dopability of materials. We will then conclude with a review of application-driven examples wherein AIMD techniques have provided useful insights for materials design. Here, we will limit our discussion to crystalline fast ion conductors that can be represented effectively using relatively small cell sizes and, hence, are more amenable to the more computationally intensive ab initio techniques that are the subject of this chapter.

7.2 Ab Initio Molecular Dynamics

Molecular dynamics (MD) simulations model the motion of atoms at finite temperatures by integrating Newtonian equations of motion. A critical input that enables all MD simulations is the description of the interatomic interactions. In the more commonly used classical MD simulations, these interactions are predefined by empirical potentials, which are based on analytical formula and parameters fitted from experiments or first principles calculations [19–21]. Despite the significant lower computational costs, classical MD simulations do suffer from several serious drawbacks that arise from empirical potentials, such as the lack of transferability across chemistries and the difficulty in handling complex interatomic interactions using simple functional forms. In ab initio MD simulations, on the other hand, interatomic interactions are directly derived from solving the Schrödinger equation, albeit using various approximations. The minimal parameterization in ab initio methods means that AIMD can be generally applied across broad chemical spaces, and the accuracy of the interatomic interactions are limited only by the underlying approximations of ab initio method. There are two main variants of AIMD today: • In the Born-Oppenheimer (BO) variant, the motions of the ions and electrons are treated separately. The electronic structure part is reduced to solving the time-independent, stationary Schrödinger equation, while the ions are propagated according to classical mechanics with ionic forces obtained from electronic structure calculations. 7 Ab Initio Molecular Dynamics Studies of Fast Ion Conductors 149

• In the Car-Parrinello (CP) [16, 22] variant, the ionic and electronic degrees of freedom are coupled via fictitious dynamics variables. Unlike BO-AIMD, minimization of the electronic energy is not required at each time step. However, a sufficiently small fictitious mass must be chosen for the electrons to maintain adiabaticity, and, correspondingly, the time step for integrating the equations of motions is typically smaller than that required for BO-AIMD [23–25]. The theoretical underpinnings and implementation of AIMD techniques are beyond the scope of this work but are extensively covered by many excellent reviews and books [22, 26]. The main disadvantage of AIMD methods is their significantly higher computational cost relative to classical MD, which places constraints on the accessible system sizes and simulation timeframes. Nevertheless, these disadvantages can be mitigated by continued computational power growth, as well as the features of the materials of interest, as discussed in the introduction. The main output from AIMD simulations is the trajectories of the ions, from which transport properties such as the diffusivity and conductivity can be derived. Currently, most AIMD studies of superionic conductors only simulate the self- diffusion of ions at equilibrium conditions (instead of under driving forces) under the assumption that the ionic displacements between ions are uncorrelated. From an AIMD simulation, the self-diffusion coefficient D in a 3D crystal structure with N mobile ions can be calculated from the velocity autocorrelation function through the Green-Kubo relation: Z 1 1 XN . / . / ; D D dt hvi t0 vi t C t0 it0 (7.1) 3N 0 iD1 where vi.t/ is the velocity of ion i at time t and angular bracket hit0 stands for ensemble average over time argument t0. More commonly, the diffusivity is computed from ionic displacements via the Einstein relation:

1 @ . /2 1 @ XN r t Œ 2 D D lim D lim h ri.t C t0/ ri.t0/ it0 ; (7.2) 6 t!1 @t 6N t!1 @t iD1 XN 2 1 2 r.t/ Á h Œr .t C t0/ r .t0/ i ; (7.3) N i i t0 iD1

2 where ri.t/ is the position of mobile ion i and r.t/ is the mean square displacement (MSD) of the mobile ions over time t as an ensemble average (over time argument t0). It should be noted that D from the Green-Kubo relation (Eq. 7.1) and the Einstein relation (Eq. 7.2) are strictly equivalent. However, there are practical considerations to prefer one method over the other, especially given the relatively short accessible time scale of AIMD simulations (up to a few hundreds of ps in 150 Z. Zhu et al. most cases). For example, the long-time tail of the integral in Eq. 7.1 may cause numerical inaccuracy in the estimate of diffusivity in short simulation time scales; as such, Eq. 7.2 is more reliable and commonly used for computing the diffusivity. The ionic conductivity can be calculated from the self-diffusion coefﬁcient D using the Nernst-Einstein equation:

z2F2 D D; (7.4) RT where and z are the molar density and the charge of the mobile ions in the unit cell, respectively. F is the Faraday constant and R is gas constant. It should be noted that the self-diffusion coefficient obtained from any computer simulations is essentially the tracer diffusivity in experiments, and the ionic conductivity from Eq. 7.4 assumes no ionic correlation. The ionic correlation can be reflected by the so-called Haven ratio, which is defined as the ratio of tracer diffusivity to charge diffusivity (D )[27]:

D HR Á ; (7.5) D " #2 1 @ XN D D lim ri.t/ : (7.6) 6N t!1 @t iD1

HR D 1 for uncorrelated ionic diffusion, whereas HR <1when concerted motions exist. Typical values of HR for highly correlated superionic conductors are between 0.3 and 0.6 [28–30]. Given that the convergence of D using Eq. 7.6 is often slow, only a few studies have attempted to estimate HR within the time scales accessible via AIMD simulations [31]. Under the assumption of no phase transitions and an abundance of defect carriers, the diffusivity D generally follows an Arrhenius relationship: Â Ã Ea D D D0 exp ; (7.7) kT where Ea is the activation energy of ionic diffusion and k and T are Boltzmann constant and temperature, respectively. D0 is the diffusivity at T !1.By calculating the diffusivities from AIMD simulations at multiple temperatures, Ea can be estimated from a linear ﬁtting of the log of D versus 1=T.AsAIMD simulations are usually performed at elevated temperatures to increase the number of diffusion events, Eq. 7.7 is also used to obtain diffusivities at room or lower temperatures. Care must be taken in the interpretation of these extrapolated values as the core assumption is that the fundamental mechanisms of diffusion remain unchanged between the lower and higher temperatures. In addition to diffusivity and conductivity estimates, the ionic trajectory data provide a treasure trove of information about the energy landscape in the material 7 Ab Initio Molecular Dynamics Studies of Fast Ion Conductors 151 and the mechanisms of ionic diffusion. These information can be extracted via several analyses: • The probability density function P.r/ plot can provide useful message on high- occupancy positions which are corresponding to low-energy sites in a superionic conductor, as well as theR migration pathways in the crystal structure [32]. P is normalized such that Pdr D 1 with being the volume of the unit cell. Site occupancies of geometrically distinct sites also can be acquired to further investigate the diffusion of dynamic process through AIMD simulations. • The van Hove correlation function can provide useful information about the correlation in the motion of ions. The van Hove correlation can be split into the self-part Gs and the distinct-part Gd, deﬁned as follows:

1 XNd G .r; t/ D h ı.r jr .t0/ r .t C t0/j/i ; (7.8) s 4r2N i i t0 d iD1

1 XNd . ; / ı. . / . / / : Gd r t D 2 h r jri t0 rj t C t0 j it0 (7.9) 4r Nd i¤j

Here, ı./ is the one-dimensional Dirac delta function. Nd and r represent for the number of mobile ions in the unit cell and radial distance, respectively. is the average number density which serves as the “normalization factor” in Gd to ensure Gd ! 1 when r 1. The self-part Gs.r; t/ may be interpreted as the probability density that a particle diffuses away from its initial position by a distance of r after time t, while the distinct-part Gd.r; t/ describes the radial distribution of N1 particles at time t with respect to the initial reference particle. A peak near r D 0 is an indication of collective motions. Besides, Gd is reduced to the static pair distribution function when t D 0, which is often adopted to study the dynamics of structural changes.

7.2.1 Practical Considerations

For practical AIMD simulations of fast ion conductors, decisions need to be made about several key parameters as outlined below. In general, the trade-off is between more realistic models (whether in a spatial or temporal sense) and computational cost, and such decisions have to be made based on the system in question: 1. Simulation cell size. Simulation supercell sizes should be large enough to avoid introducing artiﬁcial correlated motion due to periodic boundary conditions. However, due to the high cost of AIMD methods, only a moderate-sized supercell of 10 Å in each lattice direction is usually used today. Depending on the number of mobile ions and the correlation between their motion, larger or smaller cells may be appropriate, and proper convergence tests should be performed 152 Z. Zhu et al.

for high-accuracy studies. For comparative/qualitative studies evaluating similar topologies and/or chemistries, strict convergence with respect to cell sizes may not be necessary. 2. Spin. Non-spin-polarized calculations can significantly speed up electronic convergence. The decision of whether to exclude spin has to be made based on the chemistry in question. For nonmagnetic s=p systems, excluding the effect of spin is usually a reasonable approximation. 3. Time step. For BO-AIMD simulations of fast ion conductors, a time step of 1–2 fs is typically sufficient, though smaller values may be required for fast proton conductors. For CP-AIMD, the simulation time step usually has to be considerably smaller ( 0.02–0.2 fs). 4. Total simulation time. The total simulation time necessary to achieve converged diffusion statistics depends on the diffusivity of the system and the simulation temperature. Fortunately, the generally high diffusivities of fast ion conductors mean a higher number of diffusion events per unit time, and good results for superionic conductors have been obtained with AIMD simulations of 100–200 ps at temperatures as low as 600–1200 K. 5. Simulation temperatures. Most AIMD simulations are performed at higher temperatures to increase the number of diffusion events for faster convergence of diffusion statistics. Typical temperatures range from 600–2000 K, depending on the chemistry. AIMD simulations of oxides, which usually have much higher melting points, can be performed at much higher temperatures. AIMD simulations of sulfides, on the other hand, are usually performed at lower temperatures. 6. Ensemble. Though the NpT (constant number of particles, pressure and temperature) ensemble is more representative of real-world environments, many AIMD studies of fast ion conductors are performed using the NVT ensemble (constant number of particles, volume and temperature) due to the smaller energy cutoff requirement [33, 34]. Furthermore, the use of the NVT ensemble allows simulations to be performed at higher temperatures without melting. In using the NVT ensemble, care should be taken to ensure that the initial volume is representative of the material. This initial volume is either obtained via experimental input or from ab initio structure optimization.

7.3 Dopability

Diffusion is defect-driven process. As such, tuning the concentration of defects, either intrinsically or via extrinsic doping, is a common strategy to enhancing ionic conductivity [35–38]. Indeed, AIMD simulations of fast ionic conductors based on reported stoichiometric compositions frequently result in diffusivity and conductivity estimates that are much lower than the reported experimental values [30, 39–41]. Doping can also be a means to introducing desired phase transitions 7 Ab Initio Molecular Dynamics Studies of Fast Ion Conductors 153

(e.g., to a higher conductivity phase) [42, 43] or to optimizing other properties such as cost or stability [28, 44, 45]. The feasibility of introducing a dopant into a crystal structure depends on many factors, including the ionic radius and oxidation state of the dopant relative to existing species in the crystal. One measure of dopability is the neutral dopant formation energy, which is deﬁned as follows [46]:

XN Ef Œdoped D EŒdoped EŒpristine nii: (7.10) iD1

Here, EŒdoped and EŒpristine are the total energies of the structure with and without the neutral dopant, respectively; i is the atomic chemical potential of species i that varies based on different experimental conditions; N is the total number of species in the doped structure; ni indicates the number of atoms of species i being added (ni >0) or removed (ni <0) from the pristine structure. i can further be decomposed as i D Ei C i, where i is the chemical potential of species i referenced to the elemental solid/gas with energy Ei. Equation 7.10 can be rearranged as

XN Œ 0Œ ; Ef doped D Ef doped ni i (7.11) iD1 XN 0Œ Œ Œ : Ef doped D E doped E pristine niEi (7.12) iD1

The neutral dopant formation energy is thus a function of fig whose values depend on the synthesis conditions. The achievable values of fig should be constrained under equilibrium growth conditions. First, precipitation of all single elements should be avoided. Second, the doped structure should remain stable during synthesis. In other words, the decomposed products of the doped structure are not allowed to form. Third, fig should be selected such that the pristine structure remains stable.

7.4 Applications in Materials Design

In this section, we will provide an overview of how AIMD simulations have been applied in fast ion conductor design. Despite the fact that AIMD simulation as a technique has been around for decades, it is only in recent years that computational power has reached the levels necessary to enable its broader application. This section is organized by technological area, beginning with the classic AgI and similar superionic conductors, followed by fast oxide conductors for solid oxide 154 Z. Zhu et al. fuel cells and, ﬁnally, alkali superionic conductors for SSBs. It should be noted that the computational literature on fast ion conductors is extensive and encompasses a variety of techniques. Here, we have limited our focus to just studies in which AIMD techniques play a primary role.

7.4.1 Iodide Superionic Conductors

The ˛ phase of AgI, which becomes stable above 420 K [47], is one of the earliest known superionic conductors with an AgC conductivity in excess of 1.0 S/cm [48]. It is therefore often the subject of AIMD-based studies in an attempt to elucidate the factors for its extraordinary high conductivity and, perhaps, apply those insights to other technologically important fast ion conductors. Figure 7.1 shows the disordered structure of ˛-AgI, which comprises a body- centered cubic (bcc) sublattice occupied by the anions I (Wyckoff, 2a) with Ag occupying three energetically degenerate interstitial sites: the tetrahedral (Wyckoff, 12d), the octahedral (Wyckoff, 24h), and the trigonal sites (Wyckoff, 6b)[49]. While it had been established experimentally that the Ag ions occupy the 12d sites, the structure factor measured in experiments shows an uncharacteristic pre-peak [50], which has generally been correlated with the fast ionic motion of AgC [51]. The observed peak is nearly identical to those observed in cuprous iodides, wherein the Cu-Cu near-neighbor distances are very small and comparable to those of Cu-I near-neighbor distances. Similarly, AIMD simulations have conﬁrmed that the fast diffusion of Cu-ions facilitates such short-ranged interactions [52]. Numerous AIMD calculations have also been performed to resolve both the structural as well as pathway issues in ˛-AgI [53–55]. Shimojo et al. [53] performed AIMD simulations and computed the structure factor of ˛-AgI based on the radial distribution of Ag ions. Figure 7.2 shows the comparison of computed structure factor with experimental results [50]. Of particular interest is the emergence of a pre-peak at k D 1 Å1 which is not associated with a bcc framework. This pre- peak is a consequence of anomalously close Ag-Ag ions due to their fast diffusion resulting in ﬂuctuating spatial distribution of Ag ions. As we can see in Fig. 7.2,the

Fig. 7.1 Crystal structure of ˛-AgI with corresponding Wyckoff positions of Ag and I 7 Ab Initio Molecular Dynamics Studies of Fast Ion Conductors 155

Fig. 7.2 Comparison of the structure factor S.k/ of ˛-AgI between AIMD simulations and experiments [50]. The solid line is constructed from the AIMD simulations, while the circles are from experiments (Reproduced with permission from Ref. [53]. Copyright (2006) Physical Society of Japan.)

Fig. 7.3 (a) Isosurface of Ag C trajectories in ˛-AgI computed from AIMD simulations at 750 K; (b) cross section of the isosurface: darker region represents highly occupied interstitial sites (Reproduced from Ref. [54] with permission. Copyright (2006) American Physical Society.)

AIMD-simulated structure factor is in excellent agreement with the experimentally measured one. Sun et al. [55] and Wood et al. [54] have also studied the Ag partial occupancies of various interstitial sites using AIMD simulations. Both studies find that Ag pre- dominantly occupies the tetrahedral sites which is in agreement with experimental results [50]. In addition, Wood and Marzari [54] also computed the isosurface (Fig. 7.3) of the Ag trajectories to identify the diffusion pathway in ˛-AgI. It was found that Ag diffuses between the tetrahedral sites (darker regions in Fig. 7.3)via the octahedral sites (smeared patterns in Fig. 7.3). These AIMD predictions were confirmed by migration pathway energy calculations showing that the octahedral site-mediated pathway does indeed have a significantly lower barrier (193 meV) compared to direct hopping between tetrahedral sites [55]. 156 Z. Zhu et al.

7.4.2 Solid Oxide Fuel Cells

Oxide conductors have significant applications in solid oxide fuel cells (SOFCs), oxygen separation membranes, and sensors [38, 56–58]. A typical SOFC is made up of two porous electrodes and a good oxide conductor as electrolyte. Fuels like hydrogen, hydrocarbon, or even carbon can react with the anode, while the cathode is active to oxygen [38, 56]. The solid electrolyte should transport oxide ions and electronically insulating. The most commonly used solid electrolyte in SOFC is yttria-stabilized zirconia (YSZ) because of its superior stability [58]. Though applied widely as the most commercialized electrolyte, YSZ still encounters problems like relatively low oxide ion conductivity under 1000 ıC[59, 60]. New alternatives like doped ceria [61] and perovskite-type conductor La1xSrxGa1yMgyO30:5.xCy/ (LSGM) [38]were suggested as promising candidates for their high oxide conductivity at intermediate temperature (600–800 ıC). However, doped ceria becomes a mixed electronic/oxide ion conductor in the reducing fuel-rich atmosphere [59] since Ce(IV) can be partially reduced to Ce(III), while chemical compatible problem [62, 63] between LSGM and NiO at high temperature and relatively high cost of Ga are both practical issues [59]. Computational works have been applied to traditional electrolyte such as YSZ [64–66]. Previous AIMD study on YSZ by Pietrucci et al. [66] has applied a relatively large supercell to achieve realistic dopant concentrations. This work shows the existence of many locally unstable vacancy configurations, and also gives the assumption that multiple-vacancy concerted jumps may help to stabilize those locally unstable arrangements. As concluded in the end of this study, the vacancy- vacancy interactions on oxygen diffusion may not be fully understood and the authors pointed out the need of improving existing lattice models. AIMD simulations also have been used in the optimization of new oxide ion conductors in recent years. For example, He et al. [67] conducted an AIMD study to optimize the conductivity of the sodium bismuth titanate, Na0:5Bi0:5TiO3 (NBT), a family of oxide conductors using different dopants. The NBT family was first reported to show significant promise as an electrolyte in intermediate- temperature SOFCs by Li et al. in 2013 [68]. With Mg doping on the Ti site and Bi deficiency, oxygen vacancies are introduced into the structure to form

Na0:5Bi0:49Ti0:98Mg0:02O2:965, which has a conductivity comparable to well- known oxide conductors such as La0:9Sr0:1Ga0:9Mg0:1O2:9 (LSGM) [69] and Ce0:9Gd0:1O1:95 (GDC) [70]. In He et al.’s work, the cubic phase of NBT was chosen for AIMD simulations because it can be stabilized at high temperatures relative to the rhombohedral and tetragonal phases. Using a supercell with composition

Na0:5Bi0:5Ti0:96Mg0:04O2:96 (with similar oxide vacancy concentration as the experimental composition), AIMD simulations were carried out from 1200 K to 2800 K. The calculated activation energy is 610 meV, and the extrapolated oxygen diffusivity and conductivity at 900 K are 2.1108 cm2/s and 8 mS/cm, respectively [67]. These AIMD results are in excellent agreement with experimental values of 1.17108 cm2/s for diffusivity at 905 K and 8 mS/cm for conductivity at 873 K [68]. 7 Ab Initio Molecular Dynamics Studies of Fast Ion Conductors 157

Fig. 7.4 Decomposition energies G (measure of dopability) at 1500 K for doped NBT compounds. The lower the decomposition energy, the more stable the doped structure is relative to the undoped structure (Reproduced from Ref. [67] with permission. Copyright (2015) Royal Society of Chemistry.)

Fig. 7.5 Arrhenius plots for (a)Na0:5Bi0:5Ti0:96B0:04O2:96 (B = Mg, Zn, Cd); (b) Na0:54Bi0:46TiO2:96 and Na0:5K0:04Bi0:46TiO2:96 (Reproduced from Ref. [67] with permission. Copyright (2015) Royal Society of Chemistry.)

As nudged elastic band calculations show that the Mg dopant increases the oxide migration activation energy, He et al. also explored alternative doping strategies in both A and B sites. The calculated decomposition energies (G) at 1500 K (see Fig. 7.4) identify Cd and Zn as potential dopants on the B site and Na and K as potential dopants on the A site according to their low decomposition energies.

AIMD simulations were carried out on B-site doped Na0:5Bi0:5Ti0:96B0:04O2:96 (B 2C 2C =Zn and Cd ) and two A-site doped compounds Na0:5A0:04Bi0:46TiO2:96 (A = NaC and KC). The A-site doped structures were found to exhibit signiﬁcantly higher conductivity compared to the B-site doped structures (see Fig. 7.5). The calculated activation energies of Na0:54Bi0:46TiO2:96 and Na0:5K0:04Bi0:46TiO2:96 are 380 meV 158 Z. Zhu et al. and 320 meV, respectively, which are greatly reduced compared to Mg-doped NBT

(610 meV) [67]. Moreover, the conductivity at 900 K for Na0:5K0:04Bi0:46TiO2:96 is predicted to be 96 mS/cm [67], one order of magnitude higher than Mg-doped NBT (8 mS/cm) [68].

7.4.3 Alkali Superionic Conductors

A class of materials where AIMD simulations have found major applications in recent years are alkali superionic conductors, which are of immense interest as solid electrolytes in SSBs. It should be acknowledged that most of the major discoveries in these areas, for example, Li10GeP2S12 [9], Li7P3S11 glass-ceramic [10], Na3PS4 [6], cubic Li7La3Zr2O12 (LLZO) [71], etc., were made experimentally without computational guidance or input. Nevertheless, AIMD techniques have provided useful insights into the performance of these materials and are playing an increasing role in guiding design [30, 72–75]. Given the large number of works in this area in recent years, we have chosen to highlight only a few particularly signiﬁcant works in various chemistries that have led to concrete strategies in materials optimization and design.

7.4.3.1 Thiophosphates

Among the known alkali superionic conductors, the thiophosphates generally have the highest conductivity, with some approaching the alkali conductivity of traditional organic liquid electrolytes at room temperature.

The Li10GeP2S12 (LGPS) superionic conductor (Fig. 7.6) was ﬁrst discovered by Kamaya et al. [9] in 2011 with a reported LiC conductivity of 12 mS/cm.

Fig. 7.6 Crystal structure of Li10GeP2S12 (left) viewed from a direction and (right) viewed from c direction (Reproduced from Ref. [72] with permission. Copyright (2012) Royal Society of Chemistry.) 7 Ab Initio Molecular Dynamics Studies of Fast Ion Conductors 159

C C C C Fig. 7.7 Arrhenius plots of (left) isovalent (Si4 and Sn4 ) and (right) aliovalent (P5 and Al3 ) substituted Li10GeP2S12. The Si and Sn analogues have relatively similar diffusivities as the Ge compound at room temperature, while the Al analogue has a slightly higher diffusivity. These results demonstrate the potential for developing a signiﬁcantly cheaper analogue for Li10GeP2S12 (Reproduced from Ref. [72] with permission. Copyright (2012) Royal Society of Chemistry.)

Since its discovery, AIMD simulations have played a major role in clarifying the reasons for its extraordinarily high conductivity, as well as potential improvements. Using AIMD simulations, Mo et al. [76] first demonstrated that contrary to the initial belief that LGPS was a 1D conductor along the c direction, the a and b directions were found to exhibit reasonable diffusivity as well, a critical insight given that true 1D conductors are not expected to perform well on a macroscopic scale. To mitigate the high cost of Ge, Ong et al. [72] subsequently investigated the potential for replacing Ge with the much cheaper Si and Sn, as well as with aliovalent dopants such as Al and P. Using first principles calculations and AIMD simulations, it was shown that the Si and Sn analogues would have similar stabilities and ionic conductivities as LGPS, while the Al-doped analogue (with an increase in Li concentration) would have a slightly higher ionic conductivity (see Fig. 7.7). These predictions were subsequently confirmed experimentally via the synthesis of these compounds [28, 44, 45]. More recently, the Na analogue of Li10SnP2S12 has also been predicted via AIMD and synthesized by Richards et al. [31] with NaC conductivity of 0.4 mS/cm. For sodium-ion chemistry, one of the most exciting chemistries to emerge in recent years is the cubic Na3PS4 superionic conductor discovered by Hayashi et al. [6, 35] Using AIMD simulations, Zhu et al. [30] showed that the introduction of Na interstitial defects are critical to achieving reasonable conductivity in this material (see Fig. 7.8). Such interstitial defects can be introduced via aliovalent doping, for example, via Si 4C substitution for P 5C, and indeed, the AIMD predicted conductivities for these doped materials are much closer to the 0.4–0.8 mS/cm that have been achieved experimentally. Sn 4C doping was proposed as an alternative dopant that holds the potential to further enhance the conductivity beyond 10 mS/cm, albeit at the cost of higher dopant formation energies. The probability density plot of the AIMD trajectories correctly identifies the high-occupancy Na1 sites as the low- energy sites, with diffusion being mediated via the Na2 sites. 160 Z. Zhu et al.

D Fig. 7.8 (left) Arrhenius plots for undoped c-Na3PS4 and Na3CxMxP1xS4 (M Si, Ge, Sn; x D 6:25% or 12.5%). (right) Isosurfaces of Na ion probability density distribution P for Na3CxSixP1xS4 (x D 6:25%) at 800 K (Reproduced from Ref. [30] with permission. Copyright (2015) American Chemical Society.)

7.4.3.2 Garnet Superionic Conductors

Though sulﬁdes have excellent ionic conductivities, they suffer from issues of air and moisture sensitivity [77]. They also tend to be intrinsically less stable electrochemically, though this may be mitigated via the formation of passivating phases [72, 78–80]. Oxides, on the other hand, are generally more stable in terms of air and moisture sensitivity as well as electrochemically. Despite the fact that several oxide superionic conductors have been known for decades, e.g., the NAtrium

SuperIonic CONductor family and perovskite (Li, La)TiO3 [81, 82], only the garnet family has been extensively studied via AIMD simulations.

The garnet family of superionic conductors are based on cubic Li5La3Ta2O12 [71], which has only a moderate ionic conductivity on the order of 103 mS/cm at room temperature but is stable against Li metal. The Zr modiﬁcation with formula Li7La3Zr2O12 (LLZO) exhibits conductivity as high as 1 mS/cm in the cubic phase [42, 83], though the thermodynamically stable tetragonal form has a conductivity that is two orders of magnitude lower. [84] As shown in Fig. 7.9,the crystal structures of cubic and tetragonal LLZO differ mainly in the disorder and occupancies in the Li sublattice. Jalem et al. [75] studied the diffusion mechanism in cubic LLZO using AIMD simulations. The overall activation energy calculated from multiple temperature AIMD simulations is 331 meV, and the extrapolated ionic conductivity at 300 K is 1:06101 mS/cm, both in line with results from experiments [42]. Li ion migration occurs between two neighbor 24d tetrahedral sites along the path consisting two nearest 96h octahedral sites. The migration process is triggered by the motion of Li ions sitting on tetrahedral sites, leading to reconﬁguration of Li sublattice to accommodate the initial movement. Such asynchronous mechanism, however, does not happen in tetragonal LLZO due to the lack of vacant Li sites [85]. 7 Ab Initio Molecular Dynamics Studies of Fast Ion Conductors 161

N Fig. 7.9 Li sublattice of cubic Ia3d LLZO (left) and tetragonal I41=acd LLZO (right). Tetrahedral and octahedral sites are displayed in yellow and green, respectively. Partially colored spheres represent partially occupied sites. In cubic LLZO, Li ions and vacancies are distributed in a ratio of 7:9 over the 24d tetrahedral and 96h octahedral sites. In tetragonal LLZO, 8a tetrahedral and 16f C 32g octahedral sites are fully occupied

The role of Li vacancies in the phase transformation of LLZO was investigated via first principles calculations and variable cell-shape AIMD simulations by Bernstein et al. [74] Standard first principles structural relaxation calculations show that the introduction of vacancies results in the cubic phase becoming increasingly favored over the tetragonal phase, with the crossover occurring at 0.4 per formula unit, in excellent agreement with experimental findings [86]. From the variable cell-shape AIMD simulations, Bernstein et al. [74] identified the tetragonal-cubic phase transformation from the change in the ratio of the cell dimensions ax;y=az from 1.04 to 1 (top panel Fig. 7.10). The time-dependent occupancies of octahedral and tetrahedral sites are plotted along with cell-shape variations in the middle and bottom panels of Fig. 7.10, respectively. It can be seen that the occupancy of the 8a tetrahedral sites in the tetragonal phase drops sharply upon transformation to the cubic phase, as other vacant tetrahedral sites (16e) start to be occupied. These results confirm experimental findings that the stabilization of cubic LLZO is largely due to unintentional doping of Al into the lattice [87], with the accompanying introduction of Li vacancies. This is an important insight, given the vastly different ionic conductivities between the two phases. In addition, Miara et al. [88] studied the effect of Ta and Rb dopants on the ionic conductivity of cubic LLZO using AIMD simulations. Generally Ta doped structures are predicted to have higher ionic conductivity than undoped cubic LLZO, and optimized ionic conductivity is found at composition Li6:75La3Zr1:75Ta0:25O12. However, for Rb, improved ionic conductivity is observed at low doping concentration, while further doping leads to drastic decrease in conductivity. Topological analysis shows that the enhancement in ionic conductivity is attributed to the changes in the Li concentration rather than that in the size of diffusion pathways. 162 Z. Zhu et al.

Fig. 7.10 Evolution over time of structural and site occupation quantities for a sample system with nvac D 2 at T D 600 K. Top: unit cell shape (ax=az blue, ay=az red) and volume (black). Middle: Octahedral 96hc (black) and 16f t C 32gt (red) lattice site occupation. Bottom: Tetrahedral 24dc (black), 8at (red) and 16et (blue) lattice site occupations (Reproduced from Ref. [74] with permission. Copyright (2012) American Physical Society.)

7.4.3.3 High-Throughput Screening

Given the inherently high cost of AIMD simulations, it may appear counterintuitive to utilize it in high-throughput (HT) efforts. Nevertheless, by careful exploitation of the desired properties for superionic conductors and statistical techniques, HT AIMD simulations can still be fruitfully applied to materials design and screening for novel materials. The key here is to identify observables from AIMD simulations as exclusionary criteria. Because room temperature superionic conductors by deﬁnition exhibit high alkali diffusivity at 300 K, they tend to exhibit low activation barriers for alkali diffusion and a certain minimum level of diffusivity of alkali ions, which is related to the mean square displacement via Eq. 7.2. Unconverged estimates for these can be obtained via relatively short AIMD simulations at only two temperatures and used to exclude low conductivity materials efﬁciently. The authors of this chapter recently applied such an approach to the discovery of novel lithium superionic conductors [89]. Figure 7.11 shows the plot of the mean square displacement at 1200 K (MSD1200K) versus that at 800 K (MSD800K)fora 7 Ab Initio Molecular Dynamics Studies of Fast Ion Conductors 163

Li3Y(PS4)2 LiAl(PS3)2 Li P S Cl Li P S 15 4 16 3 LiZnPS4 7 3 11 Li10SnP2S12

Li10GeP2S12 Li5PS4Cl2

pristine-Na3PS4 doped-Na3PS4 MSD Li10SiP2S12 800K Li OCl Br 800K 3 x 1-x Li10Si1.5P1.5S11.5Cl0.5 = 7 MSD = 5

1200K MSD Li4P2S6 Superionic conductor region

Fig. 7.11 Plot of mean square displacement at 1200 K (MSD1200K ) versus that at 800 K (MSD800K) for a wide range of alkali conductors. A log-log scale is used for better resolution across orders of magnitude differences in diffusivities. Square markers are for known materials, while circle markers indicate novel predicted materials in this work. Well-known superionic conductors like Li7P3S11 and LGPS family and doped Na3PS4 all fall into the predicted superionic conductor region (white trapezoid zone). Poor conductors like Li3OClxBr1 x and pristine Na3PS4 are located in the shadow area (Reproduced from Ref. [89] with permission. Copyright (2017) American Chemical Society.) wide range of alkali conductors for 50 ps of AIMD simulations. It may be observed that all known alkali superionic conductors fall within the white region bounded by MSD1200K <7(providing an estimated activation energy < 400 meV), and MSD800K 2 MSD800K > 5Å (a minimum level of alkali diffusivity at 800 K). The authors applied this screening criteria (in addition to other stability and topological criteria) to all materials in the Li-M-P-S system, including novel structures predicted from substituting Ag with Li in the Ag-M-P-S system. Two new superionic conductors, . / Li3Y PS4 2 and Li5PS4Cl2, were identiﬁed that are predicted to exhibit an excellent combination of good phase stability, very high ionic conductivity (>1mS/cm), and good electrochemical compatibility with the electrodes [89]. The authors further . / demonstrated that the ionic conductivity of the Li3Y PS4 2 material can potentially be further enhanced multifold via aliovalent doping with Ca or Zr. It should be noted that a key enabler for such HT efforts is the development of a sophisticated software framework for the automation of AIMD workﬂow and data collection by the authors [90]. 164 Z. Zhu et al.

7.5 Conclusion

To conclude, AIMD simulations have emerged as a powerful tool in the study of ionic diffusivity and conductivity (among other properties). Though the role of methodological advancements over the past few decades cannot be overlooked [16, 25, 33, 91–94], a major contribution to its increasing accessibility to more research groups and organizations stems from advances in computing power that have continued to grow in accordance to Moore’s law. We expect this coupled methodological-computing advancement to continue for at least the near future, enabling larger cell size simulations of longer time scales. Despite these advances, we would caution that AIMD simulations remain a specialized tool with associated strengths and weaknesses. Clearly, the ability to address diverse chemical spaces is a critical advantage, especially in materials design. Also, because it is an ab initio technique, the predictive power and accuracy of AIMD predictions have been nothing short of extraordinary, as demonstrated in the examples highlighted. Its main weakness lies in its significant computational cost relative to classical MD or even other ab initio methods such as nudged elastic band calculations. The limitations that this weakness places on computational materials system design are on both a spatial and temporal scale. For example, macroscopic conductivity in many fast ion conductors may be limited by internal (e.g., grain boundaries) and external interfaces (e.g., between two different materials, such as an electrode and the electrolyte). Even today, AIMD simulations of such interfaces are rare due to the much larger cell sizes needed for effective models. Also, the term “fast” ion conductor is not a binary classification and spans a continuum. For moderately fast conductors, obtaining converged diffusion statistics is still a challenge with modern computing resources. Ultimately, like all computational techniques, AIMD simulations should be applied judiciously based on the scientific problem at hand, and the answer to accessing larger system sizes or time scales may lie in its integration with force- field and/or continuum methods in a multi-scale model.

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