Li+ Defects in a Solid-State Li Ion Battery: Theoretical Insights with a

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Li+ defects in a solid-state Li ion battery: theoretical insights with a Li3OCl electrolyte Saskia Stegmaier, Johannes Voss, Karsten Reuter, and Alan C. Luntz Chem. Mater., Just Accepted Manuscript • Publication Date (Web): 26 Apr 2017 Downloaded from http://pubs.acs.org on April 27, 2017

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Chemistry of Materials

1

Li+ defects in a solid-state Li ion battery: theoretical insights with a Li3OCl electrolyte Saskia Stegmaier1, Johannes Voss2, Karsten Reuter3 and Alan C. Luntz2

1SUNCAT

Center for Interface Science and Catalysis, Department of Chemical Engineering, Stanford University, Stanford California 94305-5025, USA 2SUNCAT

Center for Interface Science and Catalysis, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, United States

3

Chair for Theoretical Chemistry and Catalysis Research Center, Technische Universität München, Lichtenbergstrasse 4, 85747 Garching, Germany

Abstract: In a solid-state Li ion battery, the solid-state electrolyte exits principally in regions of high externally applied potentials, and this varies rapidly at the interfaces with electrodes due to the formation of electrochemical double layers. We investigate the implications of these for a model solid-state Li ion battery Li|Li3OCl|C, where C is simply a metallic intercalation cathode. We use DFT to calculate the potential dependence of the formation energies of the Li+ charge carriers in superionic Li3OCl. We find that Li+ vacancies are the dominant species at the cathode while Li+ interstitials dominate at the anode. With typical Mg aliovalent doping of Li3OCl, Li+ vacancies dominate the bulk of the electrolyte as well, with freely mobile vacancies only ~ 10-4 of the Mg doping density at room temperature. We study the repulsive interaction between Li+ vacancies and find that this is extremely short range, typically only one lattice constant due to local structural relaxation around the vacancy and this is significantly shorter than pure electrostatic screening. We model a Li3OCl- cathode interface by treating the cathode as a nearly ideal metal using a polarizable continuum model with an εr = 1000. There is a large interface segregation free energy of ~ - 1 eV per Li+ vacancy. Combined with the short range for repulsive interactions of the vacancies, this means that very large vacancy concentrations will build up in a single layer of Li3OCl at the cathode interface to form a compact double layer. The calculated potential drop across the interface is ~ 3 V for a nearly full concentration of vacancies at the surface. This suggests that nearly all the cathode potential drop in Li3OCl occurs at the Helmholtz plane rather than in a diffuse space-charge region. We suggest that the conclusions found here will be general to other superionic conductors as well.

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1. Introduction Practical development of all-solid-state Li ion batteries using Li metal anodes could significantly enhance the applicability of electric vehicles by improving energy density and safety of the Li ion battery. There are now several excellent solid-state highly conducting Li+ electrolytes (SSEs), so-called superionic Li+ conductors, some of which exhibit conductivities that are even better than those of the liquid electrolytes used in conventional Li ion batteries.1 Unfortunately, current generations of all-solid-state Li ion batteries have limitations in power density and/or cycle life. It is generally believed that these limitations are principally due to buildup of interfacial impedances caused by electrochemical and mechanical instability issues at the electrode–SSE interfaces.2-4 However, even in the absence of such instability issues, it is possible that fundamental space-charge issues at ideal interfaces could limit either discharge or charge power density as well, especially when multi-electrolyte layers are utilized to avoid the electrochemical or mechanical instabilities at the electrode interfaces3, 5. Even if all-solid-state batteries do not become feasible in the near term, there is still considerable activity trying to incorporate Li metal anodes in conventional liquid electrolyte Li ion or Li-S batteries. At a minimum, these would still require a Li stable thin solid electrolyte separator with sufficient stiffness and density to block Li dendrite formation on cycling. Of the well-known highly conducting SSEs, cubic-Li7La3Zr2O12 (c-LLZO) and the antiperovskite Li3OCl (and Li3OBr, Li3OCl0.5Br0.5) seem to be electrochemically stable to reduction by Li metal and therefore directly compatible with a Li metal anode.2, 4, 6-7 There is now considerable experimental work trying to prevent dendrite formation using c-LLZO as a separator. However, even though the ceramic material is sufficiently stiff to block Li dendrites, they still seem to grow through grain boundaries at modest charging currents due to a complex set of reasons.8-9 Also, the native Li|LLZO interface has some microscopic gaps and consequently moderately high interfacial impedance because Li does not wet LLZO well. However, additional ultrathin conformal interfacial layers in between Li and LLZO may remedy this impedance issue by inducing good wetting of the electrolyte by Li.10 One significant advantage of Li3OCl as a solid separator is that it has a low melting point (282° C) and could in principle be made with minimal grain boundaries upon slow cooling of the melt, and with possible wetting by Li.11 In addition, heating and/or electrochemical cycling divalent-cation-doped Li3OCl samples (in the presence of minute water) can form a glass or amorphous phase with exceedingly high room temperature Li+ conductivity (> 10-2 S/cm).7 An amorphous phase SSE may significantly reduce dendrite formation and/or loss of capacity due to Li anode morphological changes by homogenizing the current distribution to the Li anode.8 Several authors have previously used density-functional theory (DFT) or molecular modeling (MM) to study Li3OCl as a SSE and its charge carrying defects in the absence of any applied potentials or interfaces.12-14 However, none have considered what actually happens to the charge carriers in Li3OCl as it occurs in an actual solid-


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3 state battery. In this paper, we use first principles electronic structure theory (DFT) to discuss the energetics of charge carriers in Li3OCl and how these manifest themselves in a typical solid-state Li ion battery. We calculate formation energies of the relevant Li+ defects and discuss how these change with the potential distribution within the battery. We find that Li+ interstitials are the stable ionic carriers at the anode, but that Li+ vacancies dominate in the cathode region of the SSE. The vacancies also dominate the bulk region of the electrolyte when formed by cation aliovalent doping. We discuss the interaction between Li+ vacancies and find that this is extremely short ranged, on the order of one lattice constant. We also discuss what happens in the region of the cathode where high concentrations of these vacancies must accumulate to screen the potential applied to the cathode. To model the interaction of the Li+ vacancies with an ideal metal as cathode, we treat the metal through a polarizable continuum model with a high relative dielectric constant ε r = 1000 . We find strong stabilization of the Li+ vacancies at the interface and that aggregation of the vacancies at this interface causes a considerable potential drop across the interface. Combining this latter with the short interaction lengths between defects, we suggest that nearly all the potential drop at the cathode occurs in the first (Helmholtz) layer and very little in the space charge region of the electrolyte. We believe that most of these conclusions are quite general for all ionic SSEs in solid-state batteries. 2. Basics of a solid-state Li ion battery Ionic conductivity of a SSE is commonly measured on samples placed between inert (ion-blocking) electrodes such as Au, e.g. Au|SSE|Au. The conductivity then depends only on the concentration and mobility of the charge carriers in the absence of any significant externally applied potential (typically only ~ 10 mV modulation of the electrodes is used in electrochemical impedance spectroscopy). In a Li ion battery, however, discharge and charge occur at a significant external potential that only differs by modest overpotentials from that at equilibrium. The battery equilibrium is given by two conditions; i.e. that the electron chemical potential µe is constant throughout the battery and secondly that the Li electrochemical potential is constant throughout the battery. The first simply aligns the bulk Fermi energy εF of all components, while the second defines the potentials applied to the various components of the battery that are necessary to achieve the equilibrium or open circuit condition. In a solid-state Li ion battery such as Li|SSE|C (where C is some intercalation cathode), the latter is achieved by applying potentials at C and the charge neutral bulk of the SSE relative to the Li anode. Therefore, it is important to understand what happens to the charge carrying defects in the bulk of the SSE at these potentials and in fact even at the interfaces where double layers are formed and potentials change rapidly. The open circuit voltage φOCV of a Li|SSE|C battery is defined by the Nernst equation that arises from equating

in C ( is defined as the zero of potential,

) with the bulk Li anode µ Li0 . When the anode


, with µ LiC the chemical


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4 potential of Li in C. For typical C, e.g. LiCoO2, φOCV ~ 4.2 V relative to the Li metal anode. To equate in the SSE with the Li anode, a potential φbulk must also be applied to the SSE as . Equating µ LiSSE − µ Li0 with the negative of the formation energy of a vacancy and a small entropic correction, gives  x  (1) eφbulk = E f VLi ′ + k BT ln  b   1− xb 

( )

( )

where E f VLi ′ is the formation energy of the Li+ vacancy, VLi′ , in the SSE in the absence of any external potential and xb is the mole fraction of the Li+ vacancy.3 The second term in equation (1) accounts for the mean-field configurational entropy contribution to µLiSSE . For typical doping densities, the entropy term is -0.1 to -0.2 eV,

( )

i.e. considerably smaller than E f VLi ′ . We shall show later that φbulk ~ 1.3 V for Li3OCl. A diagram of the typical variation of the potential φ relative to the Li anode in such a battery is given in Figure 1, based on a simple one dimensional model and assuming that VLi′ is the only Li+ charge carrier. 3 A potential drop occurs close to the electrodes in double layers, but φ is constant at potential φbulk in the charge neutral bulk of the SSE, where . In the double layer regions, the SSE deviates from charge neutrality to create the potential drops that occur near the electrodes. When VLi′ is the only charge carrier, this implies vacancy depletion at the anode and accumulation at the cathode as shown in Figure 1. During discharge of the battery, Li+ ions are created in the SSE at the Li anode and annihilated at the cathode C or equivalently VLi′ are created at C and destroyed at the anode. During charge, the reverse is true. How much of the deviation from charge neutrality and potential drop occurs in the first SSE layer next to the electrodes (Helmholtz plane) relative to that further away in diffuse space-charge regions is not yet clear for any SSE. At liquid electrolyte-electrode interfaces, measuring the capacitance with varying concentrations of ionic carriers can approximately separate the two regions. This has not yet been possible for SSE-electrode interfaces. 3. Basics of Li+ charge carriers in Li3OCl In contrast to most other solid Li ion superionic conductors, the pristine Li3OCl antiperovskite crystal structure features neither fractionally occupied Li sites nor large interstitial void space. Its crystal structure is outlined in Figure S1 and discussed in the Supplementary Information (SI). Computational studies indicate that ideal (defect-free) bulk Li3OCl is not a good Li ion conductor. 12, 14. However, its high temperature synthesis from LiOH + LiCl may induce charge-carrying defects that preserve overall charge neutrality. These include Schottky type LiCl defects with Li+ + Cl- vacancies ( VLi′ + VCl• ) to produce Li3-xOCl1-x, Schottky type Li2O defects

(

)

with 2 Li+ + O2- vacancies 2VLi′ + VO•• to form Li3-2xO1-xCl, Frenkel defects with a Li+


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5 vacancy and a Li+ interstitial pair ( VLi′ + Lii• ) that maintains the Li3OCl stoichiometry, O substitution for Cl and a Li+ interstitial ( Lii• + OCl′ ) to produce Li3+xO1+xCl1-x and others. Here, the labels in parentheses are the conventional Kröger-Vink notations for the defects. The formation energies of all these defects are too high for them to be thermally populated at room temperature.15 They could, however, be formed during high temperature synthesis as metastable defects. It is difficult to control exactly how many defects are produced during synthesis, and it was suggested that unintentional doping during the synthesis may actually dominate the defects produced during synthesis.16 Previous DFT and MM studies have shown that both Li+ interstitials and vacancies are quite mobile in Li3OCl and other halide isomorphs.12, 14, 17 These are shown schematically in Figure S2, along with predicted migration paths. The Li+ interstitial exhibits a net +1 formal charge around a Li dumbbell, while the Li+ vacancy exhibits a net -1 formal charge because of the missing Li+ ion. Although calculated Bader charges for the remaining ions in the lattice are essentially unchanged from the bulk values, the local electron structure changes and geometric relaxations occur near the charged Li defects as described in Figure S4 and S5 and discussed in the SI. For the Li+ vacancy, these are dominated by motion of nearest neighbor O anions away from the vacancy by ~ 0.14 Å and the nearest neighbor Li ions toward the vacancy by ~0.18 Å. DFT predicts that the minimum calculated diffusion barriers for the Li+ interstitial and vacancy are ~175 meV and ~360 meV, respectively.12 This indicates that the Li+ interstitial is the most mobile Li+ charge carrier because of its two times lower diffusion barrier. However, the dominance of a Li+ charge carrier depends not only on its diffusion barrier, but also on its concentration. The latter in turn depends on the defect formation energy (and defect pair binding energy if it is formed as a defect pair). Since some of these aspects point in different directions, there has been some previous debate about which is the dominant charge carrier in Li3OCl.18 Abinitio molecular dynamics studies have also shown that the Li+ vacancy conductivity in the mixed halides Li3OCl1-xBrx can be enhanced relative to that in pure Li3OCl with a maximum at x = 0.25.19 Deliberate aliovalent doping with divalent species that creates Li+ vacancies, e. g. Li3+ 2xMgxOCl, causes significant increases in Li+ conductivity σ Li .13 In this case, it seems clear that VLi′ is the dominant charge carrier in the bulk of the doped samples. Extreme examples of aliovalent doping (and slight introductions of H2O and high temperature annealing) produce the glasses noted before with low glass temperatures Tg and very high room temperature σ Li+ > 10-2 S/cm. Neither the structure of these highly amorphous phases, nor the origin of the glass transition are known, but it is likely that VLi′ is still the dominant charge carrying defect since the stoichiometry of the glass is thought to be Li3-xHxOCl (with no aliovalent doping) or Li2.99Ba0.005OCl (or Li2.99Ba0.005OCl1-x(OH)x) with some Ba doping.7



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6

In addition to conductivity, the mechanical properties of a SSE are also critical if they are to be utilized with Li metal. Deng, et al.20 have used DFT to calculate elastic properties for a variety of solid-state electrolytes. They predict that Li3OCl has a shear modulus (G) that is almost as high as the ceramic oxides so that it should easily fulfill the MonroeNewman rules for blocking dendrite growth. However, they also predict that Li3OCl has a high Pugh’s ratio (G/B) so that the material is quite brittle. The latter is a concern for Li deposition in the SSE initiated at Griffith or other flaws that causes crack propagation with continued Li deposition in the crack. This mechanism can also can ultimately lead to Li shorts.21 4. Theoretical methods In order to investigate bulk Li3OCl and the properties of its charge carriers in a model battery, we use semi-local (generalized-gradient, GGA) and hybrid functional DFT. The DFT calculations were carried out with the Vienna Ab Initio Simulation Package (VASP) 22-23 using the projector-augmented wave (PAW) method24 and a plane wave basis set. PBEsol25-26 and HSE0627-28 functionals were employed for calculations at the GGA and hybrid level, respectively. Plane wave cutoff energies ≥ 500 eV were used for all calculations. The calculated Li3OCl lattice constants are a = 3.844 Å and a = 3.850 Å with the PBEsol and HSE06 functionals, respectively. The experimental lattice constant linearly extrapolated to 0 K is 3.86 Å.7 Given this agreement and the consistency in the defect formation energies presented later, the numerically most demanding interface calculations were only performed at the GGA level. For calculations of bulk Li metal used as a reference in the formation energies, Brillouin zone sampling was performed with Γ-point centered Monkhorst-Pack29 grids with spacings of at most ~0.03Å-1. For calculations of (insulating) Li3OCl, sparser grids with spacings of at most ~0.04Å-1 were used. Smearing of the Fermi surface was performed with a width of 0.2 eV for the Li reference to ensure converged sampling. The valence band edge of Li3OCl was smeared with a much lower width of 0.05 eV. Structural relaxation was performed until residual forces were less than 0.01 eV/Å for GGA and 0.02 eV/Å for hybrid functional calculations for a fixed unit cell volume of the bulk. For slab calculations, remaining forces were less than 0.01 eV/Å (except for the outermost layer z coordinate which was held fixed) or the change in total energy between two relaxation steps was less than 0.001 eV. Bulk formation energies of Li+ vacancies and interstitials were calculated in (3×3×3) supercells of Li3OCl (i.e. in supercells composed of 27 (5-atom) unit cells of Li3OCl). Both interstitial and vacancy calculations were performed with compensating background charges necessary to maintain charge neutrality of the extended systems. Vacancy calculations were also performed with Mg as an aliovalent dopant ensuring charge neutrality already at the formula unit level. Both approaches led to similar results in terms of Li+-defect-related electronic states and structural relaxation as will be discussed later.


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7 Slab calculations were used to investigate defects at model interfaces of the Li3OCl solid-state electrolyte, both at a vacuum interface and in contact with a model metal interface. Two possible terminations of the (001) surface of Li3OCl were considered, referred to as LiCl- and Li2O- terminations. We used equivalent terminations on both sides of symmetric slabs to avoid dipole corrections. A (3×3) surface unit area was chosen for the slabs. The optimized GGA bulk lattice constant leads to a slab thickness of 11.532 Å for 7 ideally spaced layers containing Li atoms with alternating density of one and two Li per unit area, respectively. The same thickness was employed as vacuum spacing between the periodic images of the slab. Within a supercell setup, use of compensating background charges leads to spurious doublelayers at the slab surfaces and spurious charging of vacuum. Therefore, Mg-dopants in central slab layers have been introduced instead to maintain charge neutrality of slab calculations with Li+ vacancies. To model the ideal metallic behavior of a field free electrode, the vacuum layer was replaced by an implicit solvent with a numerically large dielectric constant (εr = 1000) as implemented in VASPSOL.30 Setting εr = 1 in VASPSOL recovered the conventional vacuum interface calculation. As the interface between Li3OCl and the polarizable medium is supposed to mimic an interface with a rigid cathode, we do not relax the coordinate normal to the interface for the outermost layer of atoms, while all other coordinates are relaxed. Due to this structural constraint at the Li3OCl - implicit solvent interface, we did not need a surface tension to stabilize the interface. Although charge compensation with aliovalent Mg was preferred for the supercell, some calculations with εr = 1000 and background charge compensation were possible since the polarizable medium expels the field. The electronic energy levels for bulk, slab, neutral and charged supercell calculations do not share a common electrostatic potential reference. Therefore, to compare the electronic density of states and electrostatic potential plots for the different calculations, they are all aligned with respect to Li 1s states of bulk and bulk-like atoms in the center of slabs that are distant from the defects. Since this is nominally a core orbital, it rides on the average electrostatic potential in the bulk. The static ion-clamped dielectric constant of Li3OCl was calculated using density functional perturbation theory as implemented in VASP.31-32 The electronic contribution to the relative dielectric constant due to local fields in the Hartree approximation is 3.2. The ionic contribution is 11.6, giving a total εr ≈ 15. 5. Results 5.1 Bulk defects in Li3OCl Figure S3 shows the electronic density of states (DOS) for bulk Li3OCl calculated with the PBEsol GGA and the HSE06 hybrid functional. Except for the lower fundamental band gap (Δg = 4.76 eV vs. 6.4 eV for the GGA vs the HSE06 functionals respectively), the electronic structure as reflected by the DOS appears identical. Although the band gap Δg is not known experimentally, it is likely closer to the HSE06 value. In any event, it is clear that Li3OCl is a wide band gap insulator with



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8 limited electrical conductivity, in agreement with experiment.7 As noted earlier, the optimized lattice constant calculated for bulk with the two functionals is essentially identical. Both this bulk structure and the DOS are in excellent agreement with prior DFT calculations.7, 14 While the large band gap of Li3OCl suggests high electrochemical stability, the actual stability also depends on the alignment of the bands relative to the Li metal anode Fermi level and the cathode conduction band. Semi-local DFT studies of the phase stability of Li3OCl in contact with a Li source and sink suggests that Li3OCl is the thermodynamically stable phase over the range φ ≈ 0 − 3 V, but that phase decomposition to other neutral species containing different amounts of Li could occur beyond this potential range.2, 4, 12 Of course, there may always be additional kinetic barriers to the decomposition. Experiments based on cycling symmetric Li|Li3OCl|Li cells imply that Li3OCl is stable to Li metal.7, 33 However, in LiC6|Li3OCl|C batteries with a typical cathode such as LiCoO2, low coulombic efficiencies are obtained with cycling.33 This would imply that Li3OCl is not stable to typical C at potentials of ~ 4V and that it does not form an electrically insulating self-limiting solid-electrolyte interface (SEI). Therefore, practical solid-state batteries based on Li3OCl as the SSE will likely require a sandwich of Li3OCl with a second thin Li+ conducting electrolyte or “spacer” that is electrochemically stable at the cathode.3, 34 We have previously discussed possible space-charge issues that can arise when using spacer layers.3 Nevertheless, in this paper we neglect this issue and treat the Li3OCl as if it is electrochemically stable over the entire φ range in order to investigate generic SSE–cathode interface issues. The structure of Li+ vacancy and interstitial bulk defects are obtained by geometry optimization of a missing (added) Li+ with background charge compensation to keep a (3x3x3) supercell of Li3OCl charge neutral. Essentially identical relaxations were obtained at the PBEsol and the HSE06 level of theory. The local structural rearrangements around the defects are discussed in detail in the SI and summarized in Figures S4 and S5. Those around the Li+ vacancy are local and modest. In contrast, the rearangements are more pronounced for the interstitial where a Li dumbbelltype configuration is formed as noted earlier by Emly, et al.12 The electronic structure of the bulk Li3OCl and the Li defects as reflected by the DOS is given in Figure 2 for the PBEsol functional. The VLi′ gives a slightly split-off O state at the top of the bulk Li3OCl valence band, while the Lii• leads to a slightly split-off O state at the bottom of the bulk Li3OCl valence band. When using Mg aliovalent doping to generate free VLi′ (see later discussion), the DOS is nearly the same as that produced by charge compensation (except for the O-Mg interaction at the bottom of the valence band). Figure S6 shows that identical DOS structures are obtained for the bulk Li defects with the HSE06 functional except for the increased band-gap and somewhat lower resolution due to the lower number of k-points employed. At 0K, the formation energy Ef of a charged Li defect in the bulk of a material is defined as


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9

E f ( ε F , µ Li ) = E − E0 − nLi µ Li + q ( ε VBM + ε F ) + ∆ corr

(2)

where E is the total energy of the system with defect, E0 is the total energy of the system without defect, nLi is the number of Li added or taken away to form the defect, µ Li = µ LiSSE , the Li chemical potential in the SSE, q is the charge of the defect, ε VBM is the energy of the valence band maximum and ε F is the Fermi energy. The formation energies of the charged Li defects implicitly depend on ε F and µ Li since one adds/subtracts electrons to ε F and adds/subtracts Li atoms to bulk Li metal in their formation, with (nLi = −1, q = −1) for a Li+ vacancy and (nLi = +1, q = +1) for the Li+ interstitial. As discussed earlier, µ Li = µ Li0 − eφ , with µ Li0 the chemical potential of bulk Li metal. ε F is confined to 0 ≤ εF ≤ Δg and for bulk Li3OCl, Δg =6.4 eV from the HSE06 DFT calculations. ∆ corr is a correction term that aligns the average electrostatic potentials for E relative to E0 due to the use of background charge compensation in the calculation of the charged defect.35 This ensures a consistent comparison of formation energies for species with different charge states and also includes the correction for spurious interactions of the charged defects with their supercell-periodic images. ∆ corr ≈ 0.3 eV for VLi′ in the (3x3x3) Li3OCl supercell (and ~0.2 eV in a (4x4x4) supercell). ∆ corr ≈ 0.2 eV for Lii• in the (3x3x3) supercell using the sxdefectalign code by Freysoldt, et. al..36 The calculated formation energies for the Li+ vacancy and interstitial defects in intrinsic bulk Li3OCl are shown in Figure 3 a) as a function of εF in the range 0 ≤ εF ≤ Δg. This describes the Fermi-level dependence of Ef( VLi′ ) and Ef( Lii• ) in the absence of any external potential. In the bulk of the electrolyte, charge neutrality is required. Since these are the two dominant mobile charge carriers in Li3OCl in the battery configuration, i. e. with a source and sink of Li atoms and electrons, charge neutrality occurs where the concentrations of Lii• and VLi′ are equal, i. e. when Ef( VLi′ ) = Ef( Lii• ). From Figure 3 a) we thus obtain that εF is uniquely defined as εF = 0.62Δg. This is nearly midgap so that the Li3OCl remains an excellent electronic insulator. Note that even if there is aliovalent doping or charge neutral defect pairs that produce higher concentrations of VLi′ , the charge from these additional VLi′ are compensated by the dopant or defect pair and thus do not change εF from the value for intrinsic Li3OCl. Figure 3a) therefore uniquely determines φbulk in equation (1). Neglecting the small entropic term, φbulk = 1.3 V. Also, E f Lii• + E f (VLi′ ) ~ 2.6 eV is

( )

the formation energy of the Frenkel defect. This high formation energy is in qualitative agreement with another calculation of ~ 1.9 eV.12. Since the Li3OCl electrolyte in a battery exists in regions of different applied potentials as shown in Figure 1, Figure 3 b) shows the potential (φ) dependence of Ef( VLi′ ) and Ef( Lii• ) at the εF defined above. This arises from the potential dependence of µ Li in eq. (2). At φ = φbulk, Ef( VLi′ ) = Ef( Lii• ) so that there is no thermodynamic preference for either charge defect in the charge-neutral bulk



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10 region of the intrinsic electrolyte. However, with aliovalent cation doping Li+ vacancies will be the dominant charge carriers in the bulk Li3OCl electrolyte in a Li|Li3OCl|C battery (or symmetric Li|Li3OCl|Li cell). At the cathode where φ > 1.3 V, VLi′ accumulate to yield the potential drop at the cathode double layer (see Figure 1). Thus, as described earlier VLi′ are generated at the cathode during discharge and consumed during charging. However, near the Li metal anode where φ < 1.3 V, Lii• are more stable and are the species exchanging Li with the anode. This aspect was not included in Figure 1 where it was assumed that only VLi′ exchanged Li at the anode, i.e. that the electrolyte had only a single charge carrier. Equivalent results to Figure 3 using the PBEsol functional are shown in Figure S7. They are nearly identical to those in Figure 3 except for the smaller εF range due to the smaller Δg of the GGA functional. For the PBEsol functional, εF = 0.62Δεg, φbulk = 1.24 V (neglecting the entropy term) and a Frenkel formation energy of ~2.5 eV. Thus, except for the smaller Δg, all aspects of the bulk and defect electronic structure are equivalently obtained as in the hybrid HSE06 calculations. Again, the same conclusion is reached that VLi′ is the defect that exchanges Li at the cathode, while

Lii• is the dominant defect at the anode. 5.2 Doping induced defects As discussed previously, aliovalent cation substitution for Li has been used as a strategy to introduce Li+ vacancies into Li3OCl and to increase its Li+ conduction. Mg, Ca and Ba are typical dopants. The heavier (and bigger) elements cause more lattice distortion and can result in glass formation at low temperatures.7 However, Mg2+ was found to cause only small structural distortions with no significant change in unit cell volume anticipated. We therefore study bulk vacancies in crystalline Li3 2xMgxOCl supercells. Different configurations for VLi′ + Mg Li defect pairs have been considered in a (3×3×3) supercell (Li79MgO27Cl27). The relative energies of the

(

)

different configurations are simply given by Erel = E VLi ′ + Mg•Li − Ecis where Ecis corresponds to shortest possible separation between the vacancy and the Mg ion pair (see below). In Figure 4 a) these relative energies are plotted as a function of the distance between the Li+ vacancy and the Mg2+ dopant pair, d(V’Li-MgLi). The configurations with V’Li and MgLi as close as possible are energetically most favorable due to the attractive interaction between the positive Mg2+ and the negatively charged VLi′ . They are labeled cis and trans and their geometries are illustrated in Figure S8. However, except for these two closest configurations the defect pair formation energy is nearly independent of d(V’Li-MgLi), i.e. the attractive interaction between the Mg2+ and the VLi′ is heavily screened. In the cis and trans configurations, the ( VLi ′ + MgLi• ) pair share a next-neighbor O atom. We interpret Figure 4a) as a binding energy Eb ≈ 0.24 eV for the V’Li-MgLi pair. A qualitatively similar interpretation has previously been suggested for other defect pairs, e. g. ( VLi′ + VCl• ) 15. Figure 2 shows that once the pair is separated, the electronic structure


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11 of the Li+ vacancy as defined by the DOS is equivalent to that of the free defect obtained by background charge compensation. An Eb > 0 implies that the unbound or free carrier mole fraction x (VLi′ ) is obtained as

x (VLi′ ) ≈

x ( Mg•Li )

(

)

• exp ( −Eb k BT ) , where x MgLi is the doping density per Li3OCl unit

3 cell. The Boltzmann factor is ~10-4 at T = 300K. For M = Mg (Ca or Ba), experimentally doped samples Li3-2xMxOCl3, typically have x = 0.005 and we obtain x (VLi′ ) ~ 10 −7 for the concentration of free Li+ ion carriers in the bulk of the

electrolyte. This x (VLi′ ) corresponds to a concentration of ~10-5 mol/L of defects in the bulk of the electrolyte. Note that this concentration is much higher than those predicted for the Schottky and Frenkel defects in the stoichiometric material at 300 K of < 10-15 due to their high formation energies.15, 17-18 Of course nonstoichiometry induced in synthesis could significantly enhance these defects and induce a depth distribution to them. 17-18 However, as stated previously, aliovalent doping studies indicate that the doping generates VLi′ as the dominant charge

carrying species in actual batteries. At x (VLi′ ) ~ 10 −7 the Debye screening length λD is quite long, λD ~ 100 nm. However, as shown in Figure 4 this is not a particularly relevant screening length for the SSEs. Within the employed finite supercells, our calculations correspond in practice to x MgLi• = 1/27. However, because of the very

(

)

short length scale for interaction between defects that is discussed below, the obtained defect formation energies are also consistent with both higher and lower doping as well. 5.3 Interaction between bulk defects Both in the highly doped “glassy phase” of Li3OCl and in the space charge region of the cathode, higher x (VLi′ ) likely exist. The latter occurs in the double layer region due to the breakdown of charge neutrality (see Figure 1). Therefore, it is important to consider the interaction between VLi′ defects as well. Calculations were carried out for bulk structures with two charged Li+ vacancy defects in a (3×3×3) supercell using background charge compensation (Li79O27Cl27 + 2e−). The relative energy of two such defects E2V ' vs the average energy of two well separated defects E2,∞ is Li

given as Erel = E

2VLi ′

− E2,∞ and shown in Figure 4 b) as a function of distance

between the two vacancies, d(V’Li-V’Li). As expected for two negatively charged vacancies, repulsive interactions exist. However, except for the cis and trans configurations with vacancies sharing one next neighbor oxygen atom, there is again minimal repulsive interaction. This implies that when another nearby defect does not frustrate the electronic structure changes or relaxation around an isolated VLi′ , then the interaction between defects is negligible. In fact, the dotted line shows the purely repulsive interaction of two unit point charges with a dielectric constant εr = 15. The much faster decay observed in Li3OCl emphasizes that structural



Page 12 of 32

12 reorganizations in the solid allow for significant screening additional to that from continuum electrostatics. We have also considered even higher bulk concentrations of defects in order to see if there are stronger synergistic effects than those simply given by a sum of nearestneighbor interactions. In bulk calculations using background charge compensation, we would have to add different numbers of background electrons for the different numbers of defects. In this case, we anticipate Δcorr would vary considerably with defect concentration. We therefore calculate different numbers of defects in the (3x3x3) supercell for the reaction Li81−2nMgnO27Cl27 + 1 Mg → Li81−2(n+1)Mg(n+1)O27Cl27 + 2 Li with n the number of ( VLi ′ + MgLi• ) pairs in the unit cell. These pairs are all spaced to avoid cis and trans configurations. The energy to add the n + 1 defect to the nth defect is ∆E ( n + 1) = En+1 − En + 2E Li − EMg , with Ei the electronic energy of the various defects and Li and Mg reservoirs. If` the defects can be regarded as separated and non-interacting, then ΔE should be independent of n. For ∆n = 1 , ∆E ( n + 1) ≈ En+1 − En + c , where c = 2ELi − EMg = E1 − E0 is a constant since ELi and EMg ∆n are independent of n. Figure 5 shows that up to a defect density of n = 6 in a (3x3x3) supercell, corresponding to x = 0.22 in Li3−2xMgxOCl that in fact ΔE/Δn is nearly constant. Including configurational entropy to describe the chemical potential of the ( VLi ′ + Mg•Li ) pairs only stabilizes the n = 1 by ~ 0.05 eV relative to n = 6. Therefore, except at very low or high bulk defect concentrations, the chemical potential of the

VLi ′ defect is nearly constant with concentration in the bulk. This means that even in the concentrations likely present in the space-charge region of the cathode, dilute electrolyte theory is an appropriate approximation. 5.4 Li3OCl vacancies at a model cathode interface At the SSE–C interface, a description of VLi ′ in terms of its properties as a bulk defect is not adequate. In the Helmholtz plane, the defect is both at the surface of the SSE and in contact with the (nearly) metallic C. To include both of these effects in a simple model, that is not overly dependent on the detailed structure of the interface, we consider the interface of the Li3OCl electrolyte with an implicit solvent representing an ideal metal electrode. In this model, we utilize (3x3) Li3OCl slabs with a dielectric continuum between the slabs in one direction, with εr = 1 describing vacuum and εr = 1000 approximating a metal cathode. Details of the calculations are as described in section 4. In the bulk of Li3OCl, all Li atoms are equivalent. However, since the surface breaks symmetry, Li atoms in the surface region are no longer equivalent. In this work, the (001) surface of Li3OCl is considered with two different nonequivalent planes of Li atoms, referred to as the LiCl- and the Li2O- planes. This leads to two different possible surface terminations of the (001) surface that we denote with corresponding labels. This surface was


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13 chosen for the interface since the average surface energy of the two possible terminations of (001) is ~ 3-4 times smaller than that of the (111) and (110) surfaces (obtained from DFT calculations of stoichiometric slabs normalized to the energy of equivalent number of bulk Li3OCl). For calculations of defects at the interface, we used symmetric slabs that have equal terminations on each side so that they are non-stoichiometric but charge balanced and without Dipoles. A diagram of the symmetric slabs for both terminations is shown in Figure S9. The DOS for the LiCl- termination of Li3OCl with one VLi′ at an interface with εr = 1 and εr =1000 polarizable continuum is given in Figure S10 and compared to a bulk defect. The valence band electron structure of either interface is nearly identical to that of the bulk. However, strong lowering or broadening of the Li 2s conduction band occurs at the SSE|electrode interface. Layer resolved DOS show that this lowering/broadening is limited to the single interfacial layer. Equivalent results are obtained for the Li2O termination as well. The small lowering of the conduction band with εr = 1 is consistent with many other studies of oxides that show such shifts at the vacuum interface. However, there is a much more dramatic lowering/broadening at the εr =1000 interface. A similar dramatic lowering of the interfacial conduction band is also predicted in DFT calculations of a Li|Li3PO4 interface.37 If general, this lowering could have profound implications for electrochemical stability of the SSE at the electrode interfaces. However, there is also likely band bending due to a Schottky barrier at the cathode interface due to the small workfunction of Li3OCl that may in part compensate this effect. We calculate VLi ′ defects at or near the interface by using a Mg2+ dopant deep in the slab for charge compensation. We define l as the number of complete lattice constants measured perpendicularly from the interface. Since VLi ′ can occupy two different layer configurations near the interface (LiCl- and Li2O-), half integral values of l correspond to the other layer relative to the interfacial layer. For example, with a LiCl- interface, l = 0,1, etc correspond to LiCl- layers and l = 0.5, 1.5, etc correspond to Li2O layers (see Figure S9). As shown in Figure S10, the valenceband DOS with VLi ′ located at a depth l = 1.5 away from the LiCl- interface is already nearly indistinguishable from the bulk situation. The energy of VLi ′ in layer l with

MgLi• in the center of the slab [l = 1.5] relative to having both VLi ′ and MgLi• in the center of the slab, but avoiding interacting cis and trans configurations, defines approximately the segregation energy of VLi ′ to the interface. This is given as

(

)

(

Eseg ( l ) = Eslab VLi ′ [ l ] + MgLi• [ l = 1.5 ] − Eslab VLi ′ [ l = 1.5 ] + MgLi• [ l = 1.5 ]

)

(3)

Results for Eseg ( l ) are shown in Figure 6 for the two terminations. There is strong stabilization of VLi ′ at the surface/interface for both terminations with εr = 1000.



Page 14 of 32

14 These strong stabilizations of the defect at a metallic interface (by ~ 1 eV) mean VLi ′ will preferentially segregate to a cathode interface. For εr = 1, there is also strong stabilization for the LiCl- termination, but none for the Li2O- termination. The slight positive energy for interface segregation to the Li2O- termination likely reflects the fixed z constraint at the interface. The energy of defects at a surface/interface relative to those in the bulk is generally a mix of bonding, mechanical and electrostatic/image effects. At these interfaces, we anticipate that the bonding effect is largest and is dominated by the interaction of the vacancy with the surrounding O anions. In the bulk, a Li+ vacancy generates two missing O-Li bonds. For the Li2O- termination, two O-Li bonds are also missing so that minimal stabilization should occur for VLi ′ at the εr = 1 surface relative to that of the bulk. On the other hand, for the LiCl- termination, only one O-Li bond is missing at the surface instead of two in the bulk. This suggests that there should be some bonding stabilization at the surface relative to the bulk. We also believe that the polarizable medium for the εr = 1000 interface provides a local compensation for the missing cation by means of the bound charge in the medium. Thus, the O bonding compensation by the medium for the Li2O- termination (where two O anions modified by the missing Li+ are at the interface) should be large and produce significant stabilization at the εr = 1000 interface relative to that of the εr = 1 interface. For the LiCl- termination (where one modified O anion is a half lattice constant back from the interface), this effect should be much smaller. The structural constraint of a fixed z coordinate in the interfacial layer seems to add an uncertainty of < 0.2 eV to the contribution of mechanical energy to the interfacial energy (by comparing this constraint to relaxation of all coordinates for εr = 1). We will suggest later that the electrostatic image stabilization term for this model is also estimated as only ~0.2 eV for both terminations. Thus, we believe that bonding effects dominate the interfacial stabilization of the vacancy and these are reasonably well represented by the simple interfacial model of a polarizable continuum. In a battery configuration, the interfacial segregation of VLi ′ must induce an interfacial potential drop between the cathode and the bulk of the Li3OCl electrolyte, φc − φb ~ 3 V . Conventional double layer theory suggests this is split between a drop over the Helmholtz plane ∆φ H and a diffuse space-charge or Stern layer ∆φS , with φc − φb = ∆φH + ∆φS . Our DFT model of the interface ignores the space-charge regions entirely and treats the interface simply as a Helmholtz plane. To obtain ∆φS would require some kind of self-consistent solution of the distribution of defects and the potential as in the Poisson-Boltzmann equation for a continuum model. This is far beyond the capabilities of the present DFT model so that we simply assume ∆φS = 0.

∆φ H has two contributions; dipole changes at the interface arising from the segregation of charged defects to the interface and the screening charge that occurs in the metallic C in response to those defects near the interface. In the polarizable ACS Paragon Plus Environment

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15 continuum model for C, the potential across the Li3OCl|C interface is defined by taking an electron from ε F in the bulk of Li3OCl deep into the polarizable continuum. Because the polarizable continuum must remain charge neutral, it does not allow a true screening charge to build up at the interface in response to the negative VLi ′ adjacent to the interface. However, it does polarize the medium to introduce a local screening charge as evidenced in the bound charge shown in Figure S11. Therefore, it is unclear if the potential across the interface in this model represents the full

∆φ H induced by VLi ′ at the interface. Figure 7 shows the planar averaged electrostatic potential φel through the slab for the LiCl- termination and polarizable medium as a function of the number of VLi ′ at the interface relative to ε F . In the finite supercell geometry, there are a total of 9 available vacancy sites at the interface. Thus, placing n vacancies at the interface, corresponds to an interfacial vacancy concentration of n/9. Without any VLi ′ at the interface, the workfunction defined as Φ = φel ( PC ) − ε F = ∆φH ~ - 1 V, where φel ( PC ) is the constant electrostatic potential in the polarizable continuum. The change in potential due to the segregation of VLi′ at the interface is given by the change in this workfunction. Figure 7 shows that Φ increases by ~ +0.3 V per VLi ′ (corresponding to a concentration increase of 1/9) at the interface relative to the polarizable medium. Approximately 3 defects are necessary to reach the potential of zero charge (PZC). The potential changes at the interface are well described by the traditional capacitor model of an electrochemical double layer, enVLi′ ( l = 0 ) q , with nVLi′ ( l = 0 ) the number of defects at the interface, ∆φ H = Φ = = CH Acell C H Acell the area of the supercell at the interface and CH the Helmholtz layer capacitance. Application of this equation to the inset in Figure 7 gives CH ~ 35 μF/cm2. Corresponding numbers for the Li2O- interface are also ~ 0.3V per VLi ′ for εr = 1000 and the same CH. Thus, both terminations support roughly the same potential drop at the interface.

At the Li3OCl- vacuum interface, the workfunction is quite small, ~ 0.4 eV and increases slightly ~ +0.35 V/ VLi ′ with VLi ′ at the interface. Corresponding results for the Li2O- termination are a workfunction of ~0.2 eV and an increase of ~0.55 V/

VLi ′ . The somewhat smaller slope of workfunction with VLi ′ for εr = 1000 relative to εr = 1 is likely due to the bound charge in the polarizable medium that opposes the dipolar induced potential change of the defect at the interface.



Page 16 of 32

16 Actual cathodes have an εr >> 1 and εr

Li+ Defects in a Solid-State Li Ion Battery: Theoretical Insights with a

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