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Chapter 7Ab Initio Molecular Dynamics Studies of FastIon 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 haveextensive applications in energy storage, ion-selective electrodes and sensors. Forexample, 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 traditionallithium-ion batteries based on flammable organic liquid electrolytes, and the searchfor 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 atleast one species, such as oxide (O2�) or alkali ions (LiC/NaC). There are ingeneral two broad classes of ab initio techniques for the study of ionic diffusionand conduction in materials in the scientific literatures: transition state methods[11] (e.g., nudged elastic band [12] and kinetic Monte Carlo [13]) and moleculardynamics simulations [14–16]. In this chapter, we will focus on the latter, i.e., abinitio molecular dynamics (AIMD) simulation, and its application in the study anddesign of fast ion conductors. A more general discussion of all techniques can befound in several excellent reviews in the literature [5, 17, 18].

There are several reasons why AIMD techniques have become increasingly themethod 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, MailCode 0448, 92093-0448, La Jolla, CA, USAe-mail: [email protected]; [email protected]; [email protected]; [email protected];[email protected]

© Springer International Publishing AG 2018D. Shin, J. Saal (eds.), Computational Materials System Design,https://doi.org/10.1007/978-3-319-68280-8_7

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148 Z. Zhu et al.

typically exhibit non-dilute mobile ion concentrations and disorder, which are moredifficult for transition state methods to effectively model. Second, the high ionicconductivity means that converged diffusion statistics can usually be obtained inreasonable simulation time scales. Finally, advances in computing power havesignificantly enhanced our ability to simulate reasonable cell sizes and time scales.

The remainder of this chapter is structured in the following manner. We willbegin by outlining the broad theoretical underpinnings for AIMD simulations,including both the Car-Parrinello and Born-Oppenheimer variants, and the analysisof such simulations for diffusion properties. As defects, typically introduced viaaliovalent doping, are a critical lever to tune ionic conductivity, we will also brieflydiscuss first principles techniques in which to assess the dopability of materials.We will then conclude with a review of application-driven examples whereinAIMD techniques have provided useful insights for materials design. Here, wewill limit our discussion to crystalline fast ion conductors that can be representedeffectively using relatively small cell sizes and, hence, are more amenable to themore computationally intensive ab initio techniques that are the subject of thischapter.

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 enablesall MD simulations is the description of the interatomic interactions. In the morecommonly used classical MD simulations, these interactions are predefined byempirical potentials, which are based on analytical formula and parameters fittedfrom experiments or first principles calculations [19–21]. Despite the significantlower computational costs, classical MD simulations do suffer from several seriousdrawbacks that arise from empirical potentials, such as the lack of transferabilityacross chemistries and the difficulty in handling complex interatomic interactionsusing simple functional forms.

In ab initio MD simulations, on the other hand, interatomic interactions aredirectly derived from solving the Schrödinger equation, albeit using various approx-imations. The minimal parameterization in ab initio methods means that AIMDcan be generally applied across broad chemical spaces, and the accuracy of theinteratomic interactions are limited only by the underlying approximations of abinitio method. There are two main variants of AIMD today:

• In the Born-Oppenheimer (BO) variant, the motions of the ions and electronsare treated separately. The electronic structure part is reduced to solving thetime-independent, stationary Schrödinger equation, while the ions are propagatedaccording to classical mechanics with ionic forces obtained from electronicstructure calculations.

7 Ab Initio Molecular Dynamics Studies of Fast Ion Conductors 149

• In the Car-Parrinello (CP) [16, 22] variant, the ionic and electronic degreesof 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 maintainadiabaticity, and, correspondingly, the time step for integrating the equations ofmotions is typically smaller than that required for BO-AIMD [23–25].

The theoretical underpinnings and implementation of AIMD techniques arebeyond the scope of this work but are extensively covered by many excellentreviews and books [22, 26]. The main disadvantage of AIMD methods is theirsignificantly higher computational cost relative to classical MD, which placesconstraints on the accessible system sizes and simulation timeframes. Nevertheless,these disadvantages can be mitigated by continued computational power growth, aswell 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, fromwhich 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) underthe assumption that the ionic displacements between ions are uncorrelated. From anAIMD simulation, the self-diffusion coefficient D� in a 3D crystal structure with Nmobile ions can be calculated from the velocity autocorrelation function through theGreen-Kubo relation:

D� D 1

3N

Z 1

0

dtNX

iD1

hvi.t0/ � vi.t C t0/it0 ; (7.1)

where vi.t/ is the velocity of ion i at time t and angular bracket h�it0 stands forensemble average over time argument t0.

More commonly, the diffusivity is computed from ionic displacements via theEinstein relation:

D� D 1

6lim

t!1@�r.t/2

@tD 1

6Nlim

t!1@

@th

NXiD1

Œri.t C t0/ � ri.t0/�2it0 ; (7.2)

�r.t/2 � 1

Nh

NXiD1

Œri.t C t0/ � ri.t0/�2it0 ; (7.3)

where ri.t/ is the position of mobile ion i and �r.t/2 is the mean square displace-ment (MSD) of the mobile ions over time t as an ensemble average (over timeargument t0).

It should be noted that D� from the Green-Kubo relation (Eq. 7.1) and theEinstein relation (Eq. 7.2) are strictly equivalent. However, there are practicalconsiderations to prefer one method over the other, especially given the relativelyshort 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 causenumerical 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:

� D �z2F2

RTD�; (7.4)

where � and z are the molar density and the charge of the mobile ions in the unitcell, respectively. F is the Faraday constant and R is gas constant. It should benoted that the self-diffusion coefficient obtained from any computer simulationsis essentially the tracer diffusivity in experiments, and the ionic conductivity fromEq. 7.4 assumes no ionic correlation. The ionic correlation can be reflected by theso-called Haven ratio, which is defined as the ratio of tracer diffusivity to chargediffusivity (D� ) [27]:

HR � D�

D�

; (7.5)

D� D 1

6Nlim

t!1@

@t

"NX

iD1

�ri.t/

#2

: (7.6)

HR D 1 for uncorrelated ionic diffusion, whereas HR < 1 when concertedmotions exist. Typical values of HR for highly correlated superionic conductors arebetween 0.3 and 0.6 [28–30]. Given that the convergence of D� using Eq. 7.6 isoften slow, only a few studies have attempted to estimate HR within the time scalesaccessible 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:

D D D0 exp

�� Ea

kT

�; (7.7)

where Ea is the activation energy of ionic diffusion and k and T are Boltzmannconstant and temperature, respectively. D0 is the diffusivity at T ! 1. Bycalculating the diffusivities from AIMD simulations at multiple temperatures, Ea

can be estimated from a linear fitting of the log of D versus 1=T . As AIMDsimulations are usually performed at elevated temperatures to increase the numberof diffusion events, Eq. 7.7 is also used to obtain diffusivities at room or lowertemperatures. Care must be taken in the interpretation of these extrapolated valuesas the core assumption is that the fundamental mechanisms of diffusion remainunchanged between the lower and higher temperatures.

In addition to diffusivity and conductivity estimates, the ionic trajectory dataprovide a treasure trove of information about the energy landscape in the material


and the mechanisms of ionic diffusion. These information can be extracted viaseveral 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 superionicconductor, as well as the migration pathways in the crystal structure [32]. P isnormalized such that

R�

Pdr D 1 with � being the volume of the unit cell.Site occupancies of geometrically distinct sites also can be acquired to furtherinvestigate the diffusion of dynamic process through AIMD simulations.

• The van Hove correlation function can provide useful information about thecorrelation in the motion of ions. The van Hove correlation can be split into theself-part Gs and the distinct-part Gd, defined as follows:

Gs.r; t/ D 1

4�r2Ndh

NdXiD1

ı.r � jri.t0/ � ri.t C t0/j/it0 ; (7.8)

Gd.r; t/ D 1

4�r2�Ndh

NdXi¤j

ı.r � jri.t0/ � rj.t C t0/j/it0 : (7.9)

Here, ı.�/ is the one-dimensional Dirac delta function. Nd and r representfor the number of mobile ions in the unit cell and radial distance, respectively.� is the average number density which serves as the “normalization factor” inGd to ensure Gd �! 1 when r � 1. The self-part Gs.r; t/ may be interpretedas the probability density that a particle diffuses away from its initial positionby a distance of r after time t, while the distinct-part Gd.r; t/ describes the radialdistribution of N�1 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 reducedto the static pair distribution function when t D 0, which is often adopted tostudy the dynamics of structural changes.

7.2.1 Practical Considerations

For practical AIMD simulations of fast ion conductors, decisions need to be madeabout several key parameters as outlined below. In general, the trade-off is betweenmore realistic models (whether in a spatial or temporal sense) and computationalcost, 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 avoidintroducing artificial correlated motion due to periodic boundary conditions.However, due to the high cost of AIMD methods, only a moderate-sized supercellof � 10 Å in each lattice direction is usually used today. Depending on thenumber of mobile ions and the correlation between their motion, larger or smallercells may be appropriate, and proper convergence tests should be performed

152 Z. Zhu et al.

for high-accuracy studies. For comparative/qualitative studies evaluating similartopologies and/or chemistries, strict convergence with respect to cell sizes maynot be necessary.

2. Spin. Non-spin-polarized calculations can significantly speed up electronicconvergence. The decision of whether to exclude spin has to be made based onthe chemistry in question. For nonmagnetic s=p systems, excluding the effect ofspin is usually a reasonable approximation.

3. Time step. For BO-AIMD simulations of fast ion conductors, a time step of1–2 fs is typically sufficient, though smaller values may be required for fastproton conductors. For CP-AIMD, the simulation time step usually has to beconsiderably smaller (� 0.02–0.2 fs).

4. Total simulation time. The total simulation time necessary to achieve convergeddiffusion statistics depends on the diffusivity of the system and the simulationtemperature. Fortunately, the generally high diffusivities of fast ion conductorsmean a higher number of diffusion events per unit time, and good results forsuperionic conductors have been obtained with AIMD simulations of 100–200 psat temperatures as low as 600–1200 K.

5. Simulation temperatures. Most AIMD simulations are performed at highertemperatures to increase the number of diffusion events for faster convergenceof diffusion statistics. Typical temperatures range from 600–2000 K, dependingon the chemistry. AIMD simulations of oxides, which usually have muchhigher melting points, can be performed at much higher temperatures. AIMDsimulations of sulfides, on the other hand, are usually performed at lowertemperatures.

6. Ensemble. Though the NpT (constant number of particles, pressure and tem-perature) ensemble is more representative of real-world environments, manyAIMD studies of fast ion conductors are performed using the NVT ensemble(constant number of particles, volume and temperature) due to the smallerenergy cutoff requirement [33, 34]. Furthermore, the use of the NVT ensembleallows simulations to be performed at higher temperatures without melting.In using the NVT ensemble, care should be taken to ensure that the initialvolume is representative of the material. This initial volume is either obtainedvia 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 enhancingionic conductivity [35–38]. Indeed, AIMD simulations of fast ionic conductorsbased on reported stoichiometric compositions frequently result in diffusivity andconductivity estimates that are much lower than the reported experimental values[30, 39–41]. Doping can also be a means to introducing desired phase transitions


(e.g., to a higher conductivity phase) [42, 43] or to optimizing other properties suchas cost or stability [28, 44, 45].

The feasibility of introducing a dopant into a crystal structure depends on manyfactors, including the ionic radius and oxidation state of the dopant relative toexisting species in the crystal. One measure of dopability is the neutral dopantformation energy, which is defined as follows [46]:

Ef Œdoped� D EŒdoped� � EŒpristine� �NX

iD1

ni�i: (7.10)

Here, EŒdoped� and EŒpristine� are the total energies of the structure with andwithout the neutral dopant, respectively; �i is the atomic chemical potential ofspecies i that varies based on different experimental conditions; N is the totalnumber of species in the doped structure; ni indicates the number of atoms of speciesi being added (ni > 0) or removed (ni < 0) from the pristine structure. �i canfurther be decomposed as �i D Ei C ��i, where ��i is the chemical potential ofspecies i referenced to the elemental solid/gas with energy Ei. Equation 7.10 can berearranged as

Ef Œdoped� D E0f Œdoped� �

NXiD1

ni��i; (7.11)

E0f Œdoped� D EŒdoped� � EŒpristine� �

NXiD1

niEi: (7.12)

The neutral dopant formation energy is thus a function of f��ig whose valuesdepend on the synthesis conditions. The achievable values of f��ig should beconstrained under equilibrium growth conditions. First, precipitation of all singleelements should be avoided. Second, the doped structure should remain stableduring synthesis. In other words, the decomposed products of the doped structure arenot allowed to form. Third, f��ig should be selected such that the pristine structureremains stable.

7.4 Applications in Materials Design

In this section, we will provide an overview of how AIMD simulations have beenapplied in fast ion conductor design. Despite the fact that AIMD simulation as atechnique has been around for decades, it is only in recent years that computationalpower has reached the levels necessary to enable its broader application. Thissection is organized by technological area, beginning with the classic AgI andsimilar 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 thatthe computational literature on fast ion conductors is extensive and encompassesa variety of techniques. Here, we have limited our focus to just studies in whichAIMD 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 earliestknown 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 thefactors for its extraordinary high conductivity and, perhaps, apply those insights toother 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 Agoccupying three energetically degenerate interstitial sites: the tetrahedral (Wyckoff,12d), the octahedral (Wyckoff, 24h), and the trigonal sites (Wyckoff, 6b) [49]. Whileit had been established experimentally that the Ag ions occupy the 12d sites, thestructure factor measured in experiments shows an uncharacteristic pre-peak [50],which has generally been correlated with the fast ionic motion of AgC [51]. Theobserved peak is nearly identical to those observed in cuprous iodides, whereinthe Cu-Cu near-neighbor distances are very small and comparable to those of Cu-Inear-neighbor distances. Similarly, AIMD simulations have confirmed that the fastdiffusion of Cu-ions facilitates such short-ranged interactions [52].

Numerous AIMD calculations have also been performed to resolve both thestructural as well as pathway issues in ˛-AgI [53–55]. Shimojo et al. [53] performedAIMD simulations and computed the structure factor of ˛-AgI based on the radialdistribution of Ag ions. Figure 7.2 shows the comparison of computed structurefactor with experimental results [50]. Of particular interest is the emergence of apre-peak at k D 1 Å�1 which is not associated with a bcc framework. This pre-peak is a consequence of anomalously close Ag-Ag ions due to their fast diffusionresulting in fluctuating spatial distribution of Ag ions. As we can see in Fig. 7.2, the

Fig. 7.1 Crystal structure of˛-AgI with correspondingWyckoff positions of Agand I


Fig. 7.2 Comparison of thestructure factor S.k/ of ˛-AgIbetween AIMD simulationsand experiments [50]. Thesolid line is constructed fromthe AIMD simulations, whilethe circles are fromexperiments (Reproducedwith permission fromRef. [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 experimentallymeasured one.

Sun et al. [55] and Wood et al. [54] have also studied the Ag partial occupanciesof various interstitial sites using AIMD simulations. Both studies find that Ag pre-dominantly occupies the tetrahedral sites which is in agreement with experimentalresults [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 wasfound that Ag diffuses between the tetrahedral sites (darker regions in Fig. 7.3) viathe octahedral sites (smeared patterns in Fig. 7.3). These AIMD predictions wereconfirmed by migration pathway energy calculations showing that the octahedralsite-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 madeup of two porous electrodes and a good oxide conductor as electrolyte. Fuels likehydrogen, hydrocarbon, or even carbon can react with the anode, while the cathodeis active to oxygen [38, 56]. The solid electrolyte should transport oxide ions andelectronically 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 mostcommercialized electrolyte, YSZ still encounters problems like relatively low oxideion conductivity under 1000 ıC [59, 60]. New alternatives like doped ceria [61]and perovskite-type conductor La1�xSrxGa1�yMgyO3�0:5.xCy/ (LSGM) [38] weresuggested as promising candidates for their high oxide conductivity at intermediatetemperature (600–800 ıC). However, doped ceria becomes a mixed electronic/oxideion conductor in the reducing fuel-rich atmosphere [59] since Ce(IV) can bepartially reduced to Ce(III), while chemical compatible problem [62, 63] betweenLSGM and NiO at high temperature and relatively high cost of Ga are both practicalissues [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 arelatively large supercell to achieve realistic dopant concentrations. This workshows the existence of many locally unstable vacancy configurations, and also givesthe assumption that multiple-vacancy concerted jumps may help to stabilize thoselocally unstable arrangements. As concluded in the end of this study, the vacancy-vacancy interactions on oxygen diffusion may not be fully understood and theauthors pointed out the need of improving existing lattice models.

AIMD simulations also have been used in the optimization of new oxide ionconductors in recent years. For example, He et al. [67] conducted an AIMDstudy to optimize the conductivity of the sodium bismuth titanate, Na0:5Bi0:5TiO3(NBT), a family of oxide conductors using different dopants. The NBT familywas 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 siteand Bi deficiency, oxygen vacancies are introduced into the structure to formNa0:5Bi0:49Ti0:98Mg0:02O2:965, which has a conductivity comparable to well-known oxide conductors such as La0:9Sr0:1Ga0:9Mg0:1O2:9 (LSGM) [69] andCe0:9Gd0:1O1:95 (GDC) [70]. In He et al.’s work, the cubic phase of NBT was chosenfor AIMD simulations because it can be stabilized at high temperatures relativeto the rhombohedral and tetragonal phases. Using a supercell with compositionNa0:5Bi0:5Ti0:96Mg0:04O2:96 (with similar oxide vacancy concentration as theexperimental composition), AIMD simulations were carried out from 1200 K to2800 K. The calculated activation energy is 610 meV, and the extrapolated oxygendiffusivity and conductivity at 900 K are 2.1�10�8 cm2/s and 8 mS/cm, respectively[67]. These AIMD results are in excellent agreement with experimental valuesof 1.17�10�8 cm2/s for diffusivity at 905 K and 8 mS/cm for conductivity at873 K [68].


Fig. 7.4 Decomposition energies �G (measure of dopability) at 1500 K for doped NBT com-pounds. The lower the decomposition energy, the more stable the doped structure is relative to theundoped structure (Reproduced from Ref. [67] with permission. Copyright (2015) Royal Societyof 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 oxidemigration activation energy, He et al. also explored alternative doping strategies inboth A and B sites. The calculated decomposition energies (�G) at 1500 K (seeFig. 7.4) identify Cd and Zn as potential dopants on the B site and Na and Kas 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= Zn 2C and Cd 2C) 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 higherconductivity compared to the B-site doped structures (see Fig. 7.5). The calculatedactivation 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 ispredicted 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 inrecent years are alkali superionic conductors, which are of immense interest as solidelectrolytes in SSBs. It should be acknowledged that most of the major discoveriesin these areas, for example, Li10GeP2S12 [9], Li7P3S11 glass-ceramic [10], Na3PS4[6], cubic Li7La3Zr2O12 (LLZO) [71], etc., were made experimentally withoutcomputational guidance or input. Nevertheless, AIMD techniques have provideduseful insights into the performance of these materials and are playing an increasingrole in guiding design [30, 72–75]. Given the large number of works in this area inrecent years, we have chosen to highlight only a few particularly significant worksin various chemistries that have led to concrete strategies in materials optimizationand design.

7.4.3.1 Thiophosphates

Among the known alkali superionic conductors, the thiophosphates generallyhave the highest conductivity, with some approaching the alkali conductivity oftraditional organic liquid electrolytes at room temperature.

The Li10GeP2S12 (LGPS) superionic conductor (Fig. 7.6) was first discoveredby 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 fromc direction (Reproduced from Ref. [72] with permission. Copyright (2012) Royal Society ofChemistry.)


Fig. 7.7 Arrhenius plots of (left) isovalent (Si4C

and Sn4C

) and (right) aliovalent (P5C

and Al3C

)substituted Li10GeP2S12. The Si and Sn analogues have relatively similar diffusivities as the Gecompound at room temperature, while the Al analogue has a slightly higher diffusivity. Theseresults 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 thereasons for its extraordinarily high conductivity, as well as potential improvements.Using AIMD simulations, Mo et al. [76] first demonstrated that contrary to theinitial belief that LGPS was a 1D conductor along the c direction, the a and bdirections were found to exhibit reasonable diffusivity as well, a critical insightgiven that true 1D conductors are not expected to perform well on a macroscopicscale. To mitigate the high cost of Ge, Ong et al. [72] subsequently investigatedthe potential for replacing Ge with the much cheaper Si and Sn, as well as withaliovalent dopants such as Al and P. Using first principles calculations and AIMDsimulations, it was shown that the Si and Sn analogues would have similar stabilitiesand ionic conductivities as LGPS, while the Al-doped analogue (with an increasein Li concentration) would have a slightly higher ionic conductivity (see Fig. 7.7).These predictions were subsequently confirmed experimentally via the synthesis ofthese compounds [28, 44, 45]. More recently, the Na analogue of Li10SnP2S12 hasalso been predicted via AIMD and synthesized by Richards et al. [31] with NaCconductivity of 0.4 mS/cm.

For sodium-ion chemistry, one of the most exciting chemistries to emerge inrecent 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 Nainterstitial defects are critical to achieving reasonable conductivity in this material(see Fig. 7.8). Such interstitial defects can be introduced via aliovalent doping, forexample, via Si 4C substitution for P 5C, and indeed, the AIMD predicted conductiv-ities for these doped materials are much closer to the 0.4–0.8 mS/cm that have beenachieved experimentally. Sn 4C doping was proposed as an alternative dopant thatholds the potential to further enhance the conductivity beyond 10 mS/cm, albeit atthe cost of higher dopant formation energies. The probability density plot of theAIMD 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.

Fig. 7.8 (left) Arrhenius plots for undoped c-Na3PS4 and Na3CxMxP1�xS4 (M D Si, Ge, Sn;x D 6:25% or 12.5%). (right) Isosurfaces of Na ion probability density distribution P forNa3CxSixP1�xS4 (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 ofair and moisture sensitivity [77]. They also tend to be intrinsically less stableelectrochemically, though this may be mitigated via the formation of passivatingphases [72, 78–80]. Oxides, on the other hand, are generally more stable in termsof air and moisture sensitivity as well as electrochemically. Despite the fact thatseveral oxide superionic conductors have been known for decades, e.g., the NAtriumSuperIonic CONductor family and perovskite (Li, La)TiO3 [81, 82], only the garnetfamily 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 10�3 mS/cmat room temperature but is stable against Li metal. The Zr modification withformula Li7La3Zr2O12 (LLZO) exhibits conductivity as high as 1 mS/cm in thecubic phase [42, 83], though the thermodynamically stable tetragonal form has aconductivity that is two orders of magnitude lower. [84] As shown in Fig. 7.9, thecrystal structures of cubic and tetragonal LLZO differ mainly in the disorder andoccupancies in the Li sublattice.

Jalem et al. [75] studied the diffusion mechanism in cubic LLZO using AIMDsimulations. The overall activation energy calculated from multiple temperatureAIMD simulations is 331 meV, and the extrapolated ionic conductivity at 300 K is1:06�10�1 mS/cm, both in line with results from experiments [42]. Li ion migrationoccurs between two neighbor 24d tetrahedral sites along the path consisting twonearest 96h octahedral sites. The migration process is triggered by the motion ofLi ions sitting on tetrahedral sites, leading to reconfiguration of Li sublattice toaccommodate the initial movement. Such asynchronous mechanism, however, doesnot happen in tetragonal LLZO due to the lack of vacant Li sites [85].


Fig. 7.9 Li sublattice of cubic IaN3d LLZO (left) and tetragonal I41=acd LLZO (right). Tetrahedraland octahedral sites are displayed in yellow and green, respectively. Partially colored spheresrepresent partially occupied sites. In cubic LLZO, Li ions and vacancies are distributed in a ratioof 7:9 over the 24d tetrahedral and 96h octahedral sites. In tetragonal LLZO, 8a tetrahedral and16f C 32g octahedral sites are fully occupied

The role of Li vacancies in the phase transformation of LLZO was investigatedvia first principles calculations and variable cell-shape AIMD simulations byBernstein et al. [74] Standard first principles structural relaxation calculations showthat the introduction of vacancies results in the cubic phase becoming increasinglyfavored over the tetragonal phase, with the crossover occurring at � 0.4 per formulaunit, in excellent agreement with experimental findings [86]. From the variablecell-shape AIMD simulations, Bernstein et al. [74] identified the tetragonal-cubicphase 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 octahedraland tetrahedral sites are plotted along with cell-shape variations in the middle andbottom panels of Fig. 7.10, respectively. It can be seen that the occupancy of the8a tetrahedral sites in the tetragonal phase drops sharply upon transformation tothe cubic phase, as other vacant tetrahedral sites (16e) start to be occupied. Theseresults confirm experimental findings that the stabilization of cubic LLZO is largelydue to unintentional doping of Al into the lattice [87], with the accompanyingintroduction of Li vacancies. This is an important insight, given the vastly differentionic conductivities between the two phases.

In addition, Miara et al. [88] studied the effect of Ta and Rb dopants on theionic conductivity of cubic LLZO using AIMD simulations. Generally Ta dopedstructures 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 concen-tration, while further doping leads to drastic decrease in conductivity. Topologicalanalysis shows that the enhancement in ionic conductivity is attributed to thechanges 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 systemwith 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: Tetrahedral24dc (black), 8at (red) and 16et (blue) lattice site occupations (Reproduced from Ref. [74] withpermission. Copyright (2012) American Physical Society.)

7.4.3.3 High-Throughput Screening

Given the inherently high cost of AIMD simulations, it may appear counterintuitiveto utilize it in high-throughput (HT) efforts. Nevertheless, by careful exploitationof the desired properties for superionic conductors and statistical techniques, HTAIMD simulations can still be fruitfully applied to materials design and screeningfor novel materials. The key here is to identify observables from AIMD simulationsas exclusionary criteria. Because room temperature superionic conductors bydefinition exhibit high alkali diffusivity at 300 K, they tend to exhibit low activationbarriers for alkali diffusion and a certain minimum level of diffusivity of alkaliions, which is related to the mean square displacement via Eq. 7.2. Unconvergedestimates for these can be obtained via relatively short AIMD simulations at onlytwo temperatures and used to exclude low conductivity materials efficiently.

The authors of this chapter recently applied such an approach to the discovery ofnovel lithium superionic conductors [89]. Figure 7.11 shows the plot of the meansquare displacement at 1200 K (MSD1200K) versus that at 800 K (MSD800K) for a


Li3OClxBr1-x

doped-Na3PS4

Li10GeP2S12

Li10Si1.5P1.5S11.5Cl0.5

Li10SnP2S12

Li10SiP2S12

pristine-Na3PS4

Li4P2S6

Li15P4S16Cl3 LiZnPS4

LiAl(PS3)2

MSD

800K = 5

Li7P3S11

Li5PS4Cl2

Li3Y(PS4)2

MSD 1200K = 7 MSD 800K

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 ordersof magnitude differences in diffusivities. Square markers are for known materials, while circlemarkers indicate novel predicted materials in this work. Well-known superionic conductors likeLi7P3S11 and LGPS family and doped Na3PS4 all fall into the predicted superionic conductorregion (white trapezoid zone). Poor conductors like Li3OClxBr1 � x and pristine Na3PS4 are locatedin the shadow area (Reproduced from Ref. [89] with permission. Copyright (2017) AmericanChemical Society.)

wide range of alkali conductors for 50 ps of AIMD simulations. It may be observedthat all known alkali superionic conductors fall within the white region bounded

by MSD1200K

MSD800K< 7 (providing an estimated activation energy < 400 meV), and

MSD800K > 5 Å2 (a minimum level of alkali diffusivity at 800 K). The authorsapplied 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 fromsubstituting 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 excellentcombination of good phase stability, very high ionic conductivity (> 1 mS/cm), andgood electrochemical compatibility with the electrodes [89]. The authors furtherdemonstrated that the ionic conductivity of the Li3Y.PS4/2 material can potentiallybe further enhanced multifold via aliovalent doping with Ca or Zr. It should benoted that a key enabler for such HT efforts is the development of a sophisticatedsoftware framework for the automation of AIMD workflow and data collection bythe authors [90].

164 Z. Zhu et al.

7.5 Conclusion

To conclude, AIMD simulations have emerged as a powerful tool in the studyof ionic diffusivity and conductivity (among other properties). Though the roleof methodological advancements over the past few decades cannot be overlooked[16, 25, 33, 91–94], a major contribution to its increasing accessibility to moreresearch groups and organizations stems from advances in computing power thathave continued to grow in accordance to Moore’s law. We expect this coupledmethodological-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 aspecialized tool with associated strengths and weaknesses. Clearly, the ability toaddress diverse chemical spaces is a critical advantage, especially in materialsdesign. Also, because it is an ab initio technique, the predictive power and accuracyof AIMD predictions have been nothing short of extraordinary, as demonstrated inthe examples highlighted. Its main weakness lies in its significant computationalcost relative to classical MD or even other ab initio methods such as nudgedelastic band calculations. The limitations that this weakness places on computationalmaterials 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 mate-rials, such as an electrode and the electrolyte). Even today, AIMD simulationsof such interfaces are rare due to the much larger cell sizes needed for effectivemodels. Also, the term “fast” ion conductor is not a binary classification and spans acontinuum. For moderately fast conductors, obtaining converged diffusion statisticsis still a challenge with modern computing resources.

Ultimately, like all computational techniques, AIMD simulations should beapplied judiciously based on the scientific problem at hand, and the answer toaccessing 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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