van der Waals Interactions in Layered Lithium ... - ACS Publications

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The Journal of Physical Chemistry

van der Waals Interactions in Layered Lithium Cobalt Oxides Muratahan Aykol, Soo Kim, and C. Wolverton∗ Department of Materials Science and Engineering, Northwestern University, Evanston IL, 60208, USA E-mail: [email protected]

∗

To whom correspondence should be addressed

1 ACS Paragon Plus Environment


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Abstract The role of van der Waals (vdW) interactions in density functional theory (DFT) + U calculations of the layered lithium-ion battery cathode Lix CoO2 (x = 0 − 1) is investigated using (i) dispersion corrections in the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation functional, (ii) vdW density functionals and (iii) the Bayesian error estimation functional with vdW correlation. We find that combining vdW corrections or functionals with DFT+U can yield lithiation voltages, relative stabilities and structural properties that are in much better agreement with experiments for the phases O1-CoO2 , O3-CoO2 , layered-Li0.5 CoO2 , spinel-Li0.5 CoO2 and LiCoO2 than using DFT+U or vdW-inclusive methods alone, or using the hybrid Heyd-Scuseria-Ernzerhof functional. Contributions of vdW-interactions to the lithiation voltages are estimated to have a similar magnitude with that of applying a typical U in the range 2–4 eV for cobalt, each accounting for 5 to 10% of calculated voltages relative to PBE. Relative stabilities of O1 and O3-CoO2 , as well as layered- and spinelLi0.5 CoO2 are correctly predicted with vdW-inclusive methods combined with the +U correction.

Introduction Density functional theory (DFT) calculations with the common exchange-correlation (XC) functionals based on local density approximation (LDA) and generalized gradient approximation (GGA) can accurately predict the properties of a wide spectrum of materials. 1 Such (semi)-local XC functionals, however, cannot describe the non-local correlation effects that lead to London dispersion forces and are often insufficient to account for binding and stabilities in, for example, biological systems, molecular and two-dimensional solids, adsorption on surfaces, and even certain textbook examples of ionic crystals like CsCl. 2–7 Quantitative investigations of the weak dispersion forces that bind the layers in layered transition metal compounds are now possible using new van der Waals (vdW) inclusive DFT methods. 7,8


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A layered oxide material particularly important for energy storage is the R¯3m LiCoO2 –the most widely used Li-ion battery cathode with a wide interlayer spacing that allows efficient (de)insertion of Li ions between the CoO2 layers comprised of edge-sharing CoO6 octahedra. 9 Removal of Li ions from this vdW gap leads to several phase transformations 9–11 yielding the experimentally characterized Lix CoO2 phases shown in Fig.1. Delithiation of LiCoO2 leads to formation of a layered monoclinic (P 2/m) phase around x = 0.5. 9,10 Extensive cycling, on the other hand, results in partial transformation of this layered phase into a spinel phase of similar composition and degrades the electrochemical performance. 12–14 In addition, excessive removal of Li (x → 0) leads to a shift from the O3-type stacking of CoO6 polyhedra to an O1-type stacking as illustrated in Fig.1. Structural and thermodynamic properties of these phases in the Lix CoO2 system have been studied extensively with DFT, 15–24 and the discrepancies with experiments are often attributed to the self-interaction error and the lack of van der Waals (vdW) forces in LDA or GGA. While the “+U ” correction to DFT 25 has been widely employed to mitigate the self-interaction errors, 24,26 there has been no systematic analysis of the extent of vdW interactions in determining properties of Lix CoO2 .

Co O Li

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Figure 1: Schematics of the Lix CoO2 structures investigated: (a) O3-LiCoO2 , (b) O3-CoO2 , (c) O1-CoO2 , (d) layered-Li0.5 CoO2 and (e) spinel-Li0.5 CoO2 . All layered structures are viewed perpendicular to the Co-O layers. Longer axes correspond to the c-axes in O1 and O3 structures. Here we investigate the inclusion of vdW interactions in DFT+U calculations of the



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layered Lix CoO2 system for the well-characterized phases relevant to the Li-ion batteries; namely, O1-CoO2 , O3-CoO2 , O3-LiCoO2 , layered-Li0.5 CoO2 and its competing spinelLi0.5 CoO2 polymorph (structures shown in Fig.1). We use two main approaches to consider the vdW interactions: (i) correcting the DFT total energy and (ii) modifying the XC functional to include non-local correlations. A widely employed vdW-correction scheme is that proposed by Grimme 27 where the vdW interactions are represented by a pair-wise force field (DFT+D2), followed by improved versions where the dispersion coefficients depend on the local geometry (DFT+D3) 28 and where the zero-damping function in dispersion correction is replaced with Becke-Johnson damping (DFT+D3/BJ). 29 For these vdW-correction approaches our calculations are based on the Perdew-Burke-Ernzerhof (PBE) XC functional 30,31 in modeling Lix CoO2 compounds. The second approach is based primarily on the vdW-density functional (vdW-DF) 32 which modifies the XC functional to add the nonlocal correlations. Based on the vdW-DF, Klimeˇs et al. 4,5 introduced the more accurate “opt” type functionals, where the GGA exchange functionals are optimized for the correlation functional used. These functionals include optB86b, optB88 and optPBE which refer to the optimization of B86, B88 and PBE exchange functionals, respectively. 4,5 In addition, we also use the Bayesian error estimation functional (BEEF)–vdW, where the XC was trained using machine learning, and includes the vdW effects with a non-local correlation term. 33 We show that while almost all vdW-inclusive methods are capable of improving the agreement between the DFT calculations and experiments for thermodynamic and structural properties in the Lix CoO2 system, including voltages, phase stabilities, crystal volumes and c/a ratios in the layered compounds, the “opt” functionals by Klimeˇs et al. 4,5 give the most accurate results consistently.


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Figure 2: Average voltages in regions x = 0 − 1, x = 0 − 0.5 and x = 0.5 − 1 calculated using PBE, PBE+D2, PBE+D3, PBE+D3/BJ, optB86b, optB88, optPBE and BEEF as a function of U in the Lix CoO2 system. Values calculated with HSE here, and previous LDGGA+U and constant-U (U = 3.3 eV) reports from Ref. 24 are also shown. Experimental data is extracted from Ref. 34 and compared to that from Ref. 9, where we estimated the difference to be less than about 0.1 V.

Methods All first-principles calculations were carried out with the Vienna Ab-initio Simulation Package (VASP) 35–38 with the projector augmented wave (PAW) potentials. 39 A plane-wave basis set cut-off energy of 520 eV and Γ-centered k-meshes with approximate density of 8000 kpoints per reciprocal atom were used in all calculations. The rotationally invariant DFT+U method was used to treat Co-3d states. 40 We searched for lower energy DFT+U solutions using the U ramping method. 41 For the vdW-DF by Dion et al. 32 we use the scheme of Román-Pérez and Soler, 42 as implemented in VASP by Klimeˇs et al. 3,5 Hybrid density functional calculations were carried out using the Heyd-Scuseria-Ernzerhof (HSE) functional. 43,44 The Co3+ and Co4+ ions were found to adopt low-spin magnetic moments of 0 and 1 µB , respectively. High-spin Co-3d electron configurations were found to be always higher in energy than the lower spin configurations. VESTA 45 was used to generate the visuals in Fig.1.



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Results and discussion The voltage corresponding to a lithiation reaction Lix1 CoO2 + (x2 − x1 )Li → Lix2 CoO2 can be calculated as, 15,20

V =−

E[Lix2 CoO2 ] − E[Lix1 CoO2 ] − (x2 − x1 )E[Li] (x2 − x1 )e

where E denotes the calculated total energy. We report the voltages calculated for the composition ranges x = 0 − 1, x = 0 − 0.5 and x = 0.5 − 1 using PBE with/without vdWcorrection methods and various vdW-DFs as a function of U together with HSE in Fig.2. When vdW interactions are ignored, the average voltages are always underestimated with PBE or PBE+U and cannot be reproduced in any of the composition ranges. A detailed comparison of constant-U PBE+U voltages in this work and that in Ref. 24 can be found in Supplemental Information (SI). By providing a better description of the local electron correlation effects, local environment dependent (LD) GGA+U 24 provides a clear improvement in predicted voltages over regular PBE+U that could not be achieved by simply varying the constant U . However, LD-GGA+U also still underestimates the experimental voltages. The HSE functional which includes a fraction of the Hartree-Fock exchange does not improve the voltage prediction compared to PBE+U in agreement with previous findings, 23 and simply overestimates the voltage for around the same amount that PBE+U underestimates it. All vdW-inclusive methods, on the other hand, improve the voltage estimations to a certain extent. Therefore, the reason underlying the under-prediction of voltage by PBE or PBE+U is clearly associated with the lack of non-local electron correlation effects. vdW-inclusion methods mostly provide a nearly constant increase in voltage (ranging between 0.2 and 0.5 V except D2) at all U over PBE+U. We find that very accurate prediction of all three voltages is possible when D3-type corrections or opt-type vdW density functionals are combined with a typical U value in the range of 2 to 4 eV applied to cobalt 3d states. Besides voltages, relative stabilities of polymorphs is also important for the accurate


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Figure 3: Stability of (a) O1-CoO2 relative to O3 − CoO2 and (b) spinel (S) Li0.5 CoO2 relative to layered (L) form as a function of U using PBE, vdW correction methods and density functionals as well as HSE.

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Figure 4: Energy as a function of interlayer spacing for O1-CoO2 calculated using PBE and optB88 at U = 0 eV and U = 3 eV. Energy is given with respect to infinite separation. 7 ACS Paragon Plus Environment


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description of the thermodynamics in the Lix CoO2 system. Even though the O1-type CoO2 is more stable than the O3-type in experiments, 9,11 both structures are predicted to be nearly degenerate with PBE+U at all U values as shown in Fig.3(a). In fact, O1 structure can be formed with a simple glide of the CoO2 layers separated by the vdW gap in the O3 structure as shown in Fig.1. Therefore their relative stabilities are likely to be controlled by vdW interactions, and as expected, all vdW methods predict the O1 structure to be more stable in Fig.3(a). Unlike PBE or HSE, 46 the previous LDA calculations also predicted the O1 phase to be more stable than the O3 phase, 18 but this was presumably related with the known error cancellation between the over-binding in LDA and the lack of non-local correlations. 47 The relative stability of O1 with respect to O3 is found to be larger at lower U values in PBE+U for all vdW methods, and vanishes beyond U ≈ 2 − 3 eV. The results above imply that (i) vdW interactions are essential to correctly predict the relative stabilities of O1 and O3-CoO2 , and (ii) the appropriate Co U values that ensure this stability are lower than ∼ 3 eV. In Fig.4 we show the binding energy curves for O1-CoO2 calculated with PBE and optB88, at both U = 0 and U = 3 eV. PBE and PBE+U calculations predict there is practically no binding (< 1 meV/atom) between the layers of CoO2 . This ‘zero-binding’ of course is not likely to be true and is a consequence of the lack of non-local correlations in (semi)-local XC functionals. 47 Otherwise the CoO2 phase could not endure any mechanical strain associated with Li insertion/extraction and lose the integrity of its crystalline form, opposite the experimental evidence that it can be accessed with complete deintercalation. 9 Once we account for the vdW interactions, a significant binding is substantiated between layers of CoO2 , which slightly weakens with the application of U . The localization of electrons with U and the subsequent reduction in mixing of O-2p and Co-3d orbitals can lead to a reduced polarizability of the charge density, 48 and in turn to a weaker vdW binding observed in Fig.4. The spinel to layered transformation was first predicted by LDA calculations, 20 and later on was also observed in experiments and suggested as a degradation mechanism for Lix CoO2


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PBE+D2 PBE+D3 PBE+D3/BJ optB86b optB88 optPBE BEEF PBE HSE Exp.

Figure 5: Crystal volumes of (a) O1-CoO2 , (b) O3-LiCoO2 , (c) layered- and (d) spinelLi0.5 CoO2 as a function of U with vdW correction methods and functionals. Shaded regions cover ±2% of the experimental volumes. 9,12 cathodes. 13,14,49 For Li0.5 CoO2 , HSE, standard PBE(+U) and vdW-inclusive methods all predict the spinel structure to be more stable than the layered polymorph at all U as shown in Fig.3(b). The relative stability of the spinel phase decreases with increasing U for all methods with +U correction, and the energy difference shows some variation among the methods with D2 and D3 as outliers. But still, applying U and/or vdW-interactions do not induce a change in the relative stability over standard PBE, and the stability of the spinel Li0.5 CoO2 over the layered polymorph is ubiquitous. In addition, stability of the layered Li0.5 CoO2 with respect to the end-members CoO2 and LiCoO2 is proportional to the difference in the average voltages above and below x = 0.5 in Fig.2, 24 for which all methods can reproduce the experimental value in the range U = 2 − 3 eV (See SI for details of differences between current constant-U results and that in Ref. 24). In Fig.5 we observe that the experimental crystal volumes for all compounds are overestimated by PBE at all U , and captured reasonably well with most of the vdW-inclusive methods within ∼2% of the experimental volumes. HSE is found to have a slight tendency to underestimate the volume except at x = 0, where, like PBE, it significantly overestimates it. The vdW effects on crystal volumes become more pronounced at x = 0 as expected. When all Li is taken out, the electron density within the gap is small, and the semi-local XC functionals cannot induce any considerable binding between layers as electrons are mostly shared between Co and O within the layers. The interlayer spacing with HSE, PBE or 9 ACS Paragon Plus Environment


PBE+U is therefore inaccurate, and actually even very hard to locate precisely since the binding energy well is very shallow and binding forces are negligible as shown in Fig.4. We see in Fig.5 that calculated volumes are less sensitive to the U values than they are to the inclusion of different vdW approaches. As we have also concluded for the voltages, D3-type corrections and opt-type vdW-DF methods within a U value range of 2–4 eV consistently provide accurate volumes.

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Figure 6: Lattice parameters a and c as a function of x in Lix CoO2 calculated using PBE, PBE+U, optB88 and optB88+U and compared to the experimental values reported by Amatucci et al. 9 For the monoclinic Li0.5 CoO2 , a and c are converted to the hexagonal counterparts as described in Ref. 50. We further show the variation of the characteristic a and c lattice parameters in the hexagonal setting as a function of x for the layered Lix CoO2 system in Fig.6 for PBE and optB88 calculations. At x = 1, similar c parameters are obtained by both methods, all in good agreement with experiment. Applying U slightly lowers the a parameter for both PBE and optB88 towards the experimental value. The same trends at x = 1 are also observed at x = 0.5, with some expansion in c and contraction in a with U . Up to here, while both U and vdW effects result in a shift towards experimental values for both a and c, all calculated lattice parameters are within 2% of the experimental values. More significant changes occur at x = 0, where experiment shows that while a increases only slightly compared to x = 0.5, 10 ACS Paragon Plus Environment

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the interlayer spacing “collapses”; that is, the c parameter drops by more than 10%. Thus our results indicate that vdW-interactions are responsible for the well-known drop in the c lattice parameter (or the c/a ratio) at small x in Lix CoO2 . Having tested a series of vdW-inclusive methods combined with DFT+U, we can further discuss their overall performance in predicting properties of the Lix CoO2 system. The discrepancy between D2 and experimental thermodynamic and structural data is mostly because the dispersion coefficients (C6 ) are fixed in D2 whereas in actuality they strongly vary based on the environment of the atom. 3 However, even for the alternative methods D3 and Tkatchenko-Scheffler (TS) 51 which take the variations in the environment into account, there still exists the particular challenge of finding a consistent dispersion correction that works for both metallic and non-metallic systems. In the former case the bonding environment is less well-defined and the vdW interactions are strongly screened by the delocalized electrons, which may not be described well with the ‘atoms-in-molecules’ concept. 3,51–53 Inclusion of such screening effects for metals or semi-conductors in dispersion-corrected DFT methods is an ongoing pursuit. 53,54 In the current analysis the body-centered cubic (bcc) Li metal is clearly the phase most prone to such issues. In fact, our analysis excludes the the self-consistent TS method, 52,55 because the effective C6 coefficient for Li in the elemental bcc-phase or compounds remain too large after volumetric rescaling of atomic C6 , which results in an overestimation of vdW energies especially for bcc-Li and hampers the accuracy of the voltage predictions. The D3 method, which shares similarities with the TS method but instead uses a coordination number dependent rescaling of C6 , seems to yield an acceptable agreement with experimental voltages. However, this agreement is realized by using a much smaller reference C6 (compared to the atomic Li) derived from a Li-H molecule which has a somewhat questionable relevance for the current system. In addition, we observe that while D3 and D3/BJ yield mostly consistent results and one might expect some minor improvements using D3/BJ, 29 they have resulted in significantly different relative energies for spinel and layered Li0.5 CoO2 as shown in Fig.3. Such a strong dependence on the form of



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the damping function; i.e. ‘zero-damping’ in D3 vs. Becke-Johnson damping in D3/BJ, is rather concerning, because it implies a major change in the physical description of the current system as one switches from zero to BJ damping. Based on the discussions above, we conclude that the C6 type dispersion-correction methods are currently not as well-suited as the vdW-DF methods for the lithium-cobalt-oxide system. The BEEF-vdW functional consistently provides more accurate structural and thermodynamic properties than PBE at all U , but it still is not as accurate as the opt-type functionals. Therefore, among all the vdW-methods evaluated here, the opt-type vdW-DFs, which also provide the most accurate predictions, are currently the most relevant methods to account for vdW interactions in DFT calculations for Li-Co-oxides.

Conclusion We have performed a systematic analysis of the contribution of vdW interactions on structural and thermodynamic properties of the Lix CoO2 system in DFT calculations. Our results show that different vdW-inclusive methods provide voltages much closer to the experiments than using regular PBE+U alone for typical Co U values within 2–4 eV, or the hybrid HSE functional. We have found that PBE(+U) yields practically no binding between the Co-O layers in CoO2 , and the relative stabilities of O1- and O3-CoO2 can be correctly predicted using vdW-inclusive methods with Co U values smaller than ∼ 3 eV. Despite the small variations in their relative energies among different methods, we have found that the spinel Li0.5 CoO2 is stable over its monoclinic layered polymorph at all U values tested (from U = 0 to 5 eV) in PBE+U and with all vdW-inclusive methods as well as HSE. Predictions of crystal volumes of phases at x = 1 and x = 0.5 in the Lix CoO2 system are only slightly improved by including a +U term and/or vdW-interactions, whereas a reasonably accurate interlayer spacing (vdW-gap) at x = 0 can only be achieved using vdW methods. Finally, drawing on the fact that including the vdW interactions has an effect on the order of 5 to 10 % on


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voltages in the current system, it is a fair question to ask whether +U corrections applied in other layered transition-metal oxide systems are over-correcting the standard GGA-PBE to reproduce voltages or reaction energies of experiments. Therefore we recommend revisiting PBE+U calculations with vdW-inclusive methods in such transition metal oxide systems.

Acknowledgement M. A. and C. W. were supported by The Dow Chemical Company. S. K. was supported by Northwestern-Argonne Institute of Science and Engineering (NAISE). This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Authors are grateful for fruitful discussions with Dr. Shahab Naghavi. M.A. is grateful for the enlightening discussions with Dr. Alexandre Tkatchenko.

Supporting Information Available Details of locating a lower energy DFT+U solution for Li0.5 CoO2 . This material is available free of charge via the Internet at http://pubs.acs.org/.

Notes and References (1) Hafner, J.; Wolverton, C.; Ceder, G. Toward Computational Materials Design : The Impact of Density Functional. MRS Bull. 2006, 31, 659–668. (2) Langreth, D. C.; Lundqvist, B. I.; Chakarova-Käck, S. D.; Cooper, V. R.; Dion, M.; Hyldgaard, P.; Kelkkanen, A.; Kleis, J.; Kong, L.; Li, S. et al. A Density Functional for Sparse Matter. J. Phys.: Condens. Matter 2009, 21, 084203. (3) Klimeˇs, J.; Michaelides, A. Perspective: Advances and Challenges in Treating Van der



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Waals Dispersion Forces in Density Functional Theory. J. Chem. Phys. 2012, 137, 120901. (4) Klimeˇs, J.; Bowler, D. R.; Michaelides, A. Chemical Accuracy for the Van der Waals Density Functional. J. Phys. Condens. Matter 2010, 22, 022201. (5) Klimeˇs, J.; Bowler, D. R.; Michaelides, A. Van der Waals Density Functionals Applied to Solids. Phys. Rev. B 2011, 83, 195131. (6) Zhang, F.; Gale, J. D.; Uberuaga, B. P.; Stanek, C. R.; Marks, N. A. Importance of Dispersion in Density Functional Calculations of Cesium Chloride and its Related Halides. Phys. Rev. B 2013, 88, 054112. (7) Carrasco, J. Role of Van der Waals Forces in Thermodynamics and Kinetics of Layered Transition Metal Oxide Electrodes: Alkali and Alkaline-Earth Ion Insertion into V2 O5 . J. Phys. Chem. C 2014, 118, 19599–19607. (8) Björkman, T.; Gulans, A.; Krasheninnikov, A. V.; Nieminen, R. M. Van der Waals Bonding in Layered Compounds From Advanced Density-Functional First-Principles Calculations. Phys. Rev. Lett. 2012, 108, 235502. (9) Amatucci, G.; Tarascon, J.; Klein, L. CoO2, The End Member of the Lix CoO2 Solid Solution. J. Electrochem. Soc. 1996, 143, 1114–1123. (10) Reimers, J. N.; Dahn, J. Electrochemical and In Situ X-Ray Diffraction Studies of Lithium Intercalation in Lix CoO2 . J. Electrochem. Soc. 1992, 139, 2091–2097. (11) Chang, K.; Hallstedt, B.; Music, D.; Fischer, J.; Ziebert, C.; Ulrich, S.; Seifert, H. J. Thermodynamic Description of the Layered O3 and O2 Structural LiCoO2 -CoO2 Pseudo-Binary Systems. Calphad 2013, 41, 6–15. (12) Gummow, R.; Liles, D.; Thackeray, M. Spinel Versus Layered Structures for Lithium Cobalt Oxide Synthesised at 400 C. Mater. Res. Bull. 1993, 28, 235–246. 14 ACS Paragon Plus Environment

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Li+

vdW gap


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van der Waals Interactions in Layered Lithium ... - ACS Publications

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