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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 oxy- gen 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 temper- atures 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 approx- imations. 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 displace- ment (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:  à 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.
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