Role of van der Waals Forces in Thermodynamics ... - ACS Publications

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Role of van der Waals Forces in Thermodynamics and Kinetics of Layered Transition Metal Oxide Electrodes: Alkali and Alkaline-Earth Ion Insertion into V2O5 Javier Carrasco* CIC Energigune, Albert Einstein 48, 01510 Miñano, Á lava, Spain S Supporting Information *

ABSTRACT: Layered transition metal oxides (LTMOs) have a long tradition of success as effective electrode materials for power storage applications. However, the growing demand for improved technologies has motivated a strong interest in developing new generations of this class of materials. First-principles calculations, in particular density functional theory (DFT), have become an important tool to gain atomic-level understanding and speed up the search of new materials in general. An important structural ingredient of LTMOs is the weak van der Waals (vdW) forces that hold layers together. Unfortunately, conventional DFT approaches have serious shortcomings to treat these dispersion interactions. This is an uneasy position for the role of DFT in describing such layered-type structural materials. Recent exciting developments in DFT allow us now to tackle this problem head on. Here we have employed newly developed vdW-inclusive methods based on improved nonlocal density functionals to thoroughly explore the role of vdW forces in key thermodynamic and kinetic properties of alkali (Li, Na, and K) and alkaline-earth (Mg, Ca, and Sr) ion insertion into α-V2O5. We find that vdW forces help to stabilize inserted ions and, therefore, increase average voltages compared to the values obtained with conventional non-vdW-inclusive DFT methods. Added to this, activation energies for ion diffusion significantly increase as a consequence of a proper account for vdW interactions. These results highlight the relevance of vdW forces to ion intercalation and dynamics in LTMOs in general.

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INTRODUCTION Layered transition metal oxides (LTMOs) are important electrode materials for chemical energy storage applications.1−3 Often these layered-type structural compounds meet the basic requirements of promising electrodes for solid-state rechargeable ion batteries; that is, they can easily intercalate and transport small alkali or alkaline-earth ions at suitable potentials.4 Prominent examples of LTMOs include mainly cathode materials such as AxMO2 (where A = Li, Na, and M = Co, Ni, Mn, Fe, or a combination of two or more of them).5−9 Some of these LTMOs are still used in most of today’s batteries; however, the power storage industry is undergoing a rapid expansion and interest in developing and discovering new LTMO-based cathodes continues to date. High-energy density, high-power capability, safety issues, structural stability, and low cost are pivotal requirements in the search for better materials. Meeting all of them is indeed a challenging task. From a theoretical viewpoint, first-principles computational methods, in particular density functional theory (DFT), are playing an increasingly important role in understanding and predicting the atomic and electronic structures of complex electroactive materials.10−16 DFT studies in this area are typically carried out using generalized gradient approximation (GGA) functionals in combination, when dealing with transition metal compounds, with electron self-interaction correction approaches. This theoretical framework has shown © 2014 American Chemical Society

a remarkable good performance to calculate, for example, ion intercalation voltages and the relative stabilities of different intermediate phases upon ion insertion (see, for example, ref 16 for a recent review). Semilocal GGA functionals, however, fail to account for ever-present van der Waals (vdW) forces.17 These forces are indeed crucial to satisfactorily describe sparse matter,18,19 such as LTMOs. The lack of binding forces between layers in DFT-GGA usually involves a poor agreement with experimental data, especially in the description of structural parameters, interlayer spacings, and energetic properties. These severe shortcomings cast a shadow over the descriptive and predictive capabilities of all conventional GGA functionals when applied to layered materials, where the interlayer spacing is, of course, a key parameter in the thermodynamic and kinetic properties of ion intercalation. Fixing the interlayer distance to an experimental value is a pragmatic approach that can be useful in the limit of low ion concentrations, but at moderate and high ion concentrations one needs to rely on the capabilities of the chosen functional.20 Therefore, it would be timely and very important to understand the role of vdW forces in ion insertion and diffusion from a first-principles perspective, without the need to consider ad hoc Received: June 11, 2014 Revised: August 1, 2014 Published: August 4, 2014 19599

dx.doi.org/10.1021/jp505821w | J. Phys. Chem. C 2014, 118, 19599−19607

The Journal of Physical Chemistry C

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Becke86b (optB86b) to improve the accuracy of vdW-DF32,33 and now has been successfully applied to a variety of systems, ranging from solids,33,34,48,49 adsorption of molecules on surfaces,50−55 and gas-phase clusters.32 These three optimized functionals are referred to as optPBE-vdW, optB88-vdW, and optB86b-vdW herein, respectively. In all calculations the core electrons were replaced by PBE-based projector augmented wave (PAW) potentials,56 whereas the wave functions of the valence electrons were expanded in plane-waves with a cutoff energy of 700 eV. The DFT+U scheme of Dudarev et al.57 was employed, in which the Hubbard U-like term (the difference between the Coulomb U and exchange J parameters, hereinafter referred to as simply U) was added to the corresponding XC functional. This pragmatic approach is necessary to describe the localized V 3d states upon V2O5 reduction,58−61 such as when alkali or alkaline-earth ions are intercalated in the material. Here, the chosen value of U is 4.0 eV, as the one proposed by Scanlon et al.62 to properly describe oxygen vacancies and intercalated Li ions in V2O5. Equilibrium lattice parameters of bulk V2O5 were performed, allowing the atomic positions, lattice constants, and cell shape to relax with a residual force threshold of 0.02 eV/Å. The cutoff of 700 eV was found to be sufficiently large to avoid the problems of Pulay stress and changes in the basis set that accompany volume changes in plane wave calculations. A Monkhorst−Pack63 grid with 2 × 4 × 4 k-point sampling was used. These computational settings guarantee a tight convergence (Table S1 of the Supporting Information). The optimized lattice parameters were then used for all subsequent calculations of alkali and alkaline-earth insertion and diffusion, which were performed at constant volume using the same calculation parameters, the same convergence criterion, a 1 × 3 × 3 supercell of V36O90, and a Monkhorst−Pack k-point grid of 2 × 2 × 2. The transition-state structures and activation barriers for ion diffusion were calculated using the climbing image nudged elastic band (CI-NEB) method64 with seven images along each pathway. The interlayer binding energy per unit cell of the bulk V2O5 (Eb) was computed by subtracting the total energy of an expanded bulk V2O5 (Eex) from the total energy of the optimized bulk V2O5 (Ebulk):

experimental inputs. This would be particularly relevant in the search for new electroactive layered materials. The ability to properly treat nonlocal vdW interactions in DFT is an ongoing quest. Fortunately, recent developments of vdW-inclusive methods are now showing great success for a range of different solids, extended surfaces, and nanoclusters (see, for example, ref 21). This has prompted a number of recent dispersion-corrected DFT studies on intercalation compounds, mainly focused on alkali metals intercalated in graphite.22−26 A common conclusion from these studies is that the inclusion of vdW forces into DFT-GGA often plays an important role in the structural and energetic stability of these compounds, resulting in better agreement with the experiment. In the last years, a number of vdW-inclusive schemes have been proposed to treat dispersion within DFT. Noteworthy examples include the DFT-D approach of Grimme,27 the vdW(TS) of Tkatchenko and Scheffler,28 and improved versions of the vdW(TS) scheme which includes many body dispersion and self-consistent screening.29,30 (For an overview of these and other methods the interested reader can consult the review of Klimeš and Michaelides21). Of the various methodologies available, the nonlocal vdW density functional (vdW-DF)31 and, in particular, modified versions of this32,33 have shown great performance when applied to soft-layered materials.18,22,34−39 However, to the best of our knowledge, the impact of vdW forces on ion insertion into LTMOs has not been thoroughly explored yet. Thanks to the recent computational developments outlined above, it is now possible to tackle this question head on. In this paper, we present results obtained with selected vdWDF offspring32 for investigating the role of vdW forces in the thermodynamic and kinetic properties of alkali (Li, Na, and K) and alkaline-earth (Mg, Ca, and Sr) ion insertion into bulk αV2O5 (hereinafter referred to as simply V2O5). The study only considers low ion concentrations (dilute regime). We have focused on V2O5 because it is a simple LTMO, where the impact of vdW forces on ion intercalation can conveniently be investigated as a model system for this class of materials. Actually, V2O5 is an interesting electrode material itself: widely studied as a cathode for Li-ion batteries (see, for example, ref 2 and, for more recent works, refs 40. and 41) and able to form intercalated compounds with Na42 and Mg43 as well. In the following, we first set out the computational details of the DFT calculations and approaches considered. Then the main results are reported and analyzed on the basis of calculated ion intercalation energies and diffusion energy barriers comparing the performance of vdW-inclusive and conventional GGA funtionals. Our findings show that the inclusion of vdW forces in the calculations has a significant impact on ion insertion energies, average voltages, and diffusion energy barriers.

E b = Ebulk − E ex

(1)

The expanded cell was calculated by increasing the bulk cell dimensions along the c axis over 15 Å. A larger increase of 30 Å has a minimal effect on Eb (less than 0.01 eV), indicating that 15 Å is sufficiently large. The energy of A (where A = Li, Na, K, Mg, Ca, or Sr) ion insertion into V2O5 was computed as

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E ins = E[A/V2O5] − E[V2O5] − E[A]

THEORETICAL METHODS Spin-polarized DFT+U and supercell periodic models were used within the Vienna ab initio simulation package (VASP) 5.3 code.44−46 Total energies and electron densities were computed with various different exchange-correlation (XC) functionals: three optimized nonlocal vdW-DF and, for comparison, the semilocal Perdew−Burke−Ernzerhof (PBE).47 The optimized vdW-DF functionals, as implemented in VASP, are modified versions of the vdW-DF of Dion et al.,31 where its original GGA exchange functional has been replaced by an optimized PBE (optPBE), optimized Becke88 (optB88), or optimized

(2)

where E[A/V2O5] is the total energy of the intercalated A ion in V2O5, E[V2O5] is the total energy of the V2O5 host, and E[A] is the total energy of metallic A as calculated in a bcc (A = Li, Na, and K), hcp (A = Mg), or fcc (A = Ca and Sr) optimized unit cell. In addition, the nonlocal correlation part, Enlc, of the total XC energy was considered to compute the corresponding nonlocal correlation contribution to the ion insertion energy, Enlc ins , as follows nlc E ins = E nlc[A /V2O5] − Enlc[V2O5] − Enlc[A]

19600

(3)



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bond), the interaction between bilayers is weak (vdW forces). This bilayer structure is ultimately responsible for making the material such an effective host for ion insertion. A suitable computational method should therefore be able to reproduce this structure accurately. To this regard, many groups have shown that conventional GGA calculations significantly overestimate the interlayer spacing in V2O5 when compared to experimental values.62,65−69 Adding vdW forces to the GGA calculations substantially improve the results.37,38,70 In particular, vdW-DF methods can provide fairly good agreement with experimental data when a proper choice of the underlying exchange functional is employed.37,38 Here we applied the vdW-DF method to the bulk V2O5, considering various optimized exchange functionals and PBE. The optimized lattice constants, selected equilibrium V−O bond distances (Figure 1), and Eb calculated with eq 1 are given in Table 1. As expected, PBE predicts a very small Eb (−0.22 eV) and overestimates the c lattice parameter by almost 7% with respect to the experimental value. In contrast, all the vdWDF methods significantly increase Eb and bring the c lattice constant closer to the experiment. In particular, the optPBEvdW shows a small relative deviation (+1.2%) from the experimental value; this deviation is indeed within the same range as the deviation in a and b lattice constants, +1.5% and +1.8%, respectively. For optB88-vdW and optB86b-vdW, the optimal values of c are smaller than the experimental values by −3.4% and −4.3%, respectively. This is likely due to the fact that these two functionals are less repulsive than optPBE-vdW

an analogous expression to eq 2, but where total energies have been substituted by their nonlocal correlation part.

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RESULTS a. Bulk V2O5. The α-phase of V2O5 has orthorhombic symmetry (space group Pmmn). The primitive cell contains two formula units where each V atom is connected to five O atoms forming distorted VO5 square-based pyramids (Figure 1). The

Figure 1. Unit cell of V2O5 bulk structure. Some selected atoms are indicated for further reference. The labels a, b, and c denote the lattice vectors. Color code: V, blue; O, red.

O1 is a terminally bound oxygen, forming a double bond to V (vanadyl group), which is oriented nearly perpendicular to the (001) plane. These pyramidal units are then joined by edge and corner sharing to form two-dimensional periodic stacked layers separated by a distance that equals the c lattice constant (Figure 2a). Each layer contains two V atoms at two different heights along c, and, therefore, we hereinafter refer to them as bilayers. While the V−O bonds within the bilayers are strong (covalent

Figure 2. Atomic structure and localization of V4+ ions upon alkali and alkaline-earth ion insertion into V2O5. (a) Supercell (V36O90) containing three stacked bilayers and one generic inserted ion in its most favorable site. (b) Intercalation energy of Li, Na, and Mg as a function of the position of V4+ ions that are formed upon ion insertion. The labels used in (b) are defined in (c−e) for each bilayer in (a). Color code: alkali (alkaline-earth) ion, yellow; V, blue; and O, red. 19601



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on top of these four vanadyl groups and close to another four O atoms (O2 and O3) from the bilayer above. The interatomic distances between each considered ion and the nearest neighbor O atoms are given in Table 2. The smallest ions (Li and Mg) are less symmetrically placed in their site than larger ions, as indicated by the observed range of dA‑O1. For example, in the case of Li, the shortest A−O1 distance is 2.154 Å, whereas the other three A−O1 distances are larger than 2.403 Å. Similarly, one of the two A−O3 distances, 2.158 Å, is much shorter than the other, 2.847 Å. Overall, the values shown in Table 2 suggest that the ion oxygen coordination number of large ions (Na, K, Ca, and Sr) would be essentially 8, whereas for Li and Mg this value is reduced to 6−7. One key observation is that the insertion of A+ (A2+) ions into V2O5 results in the electron transfer of one (two) s electrons to the d band of V2O5, forming one (two) V4+ ions, as revealed, for example, from a simple d-projected local magnetization analysis or the presence of localized V 3d states near the Fermi level in the density of states (Figures S1 and S2 of the Supporting Information). In the case of Li intercalation, this is consistent with previous DFT works,62 and the observed localized state in the band gap using ultraviolet and X-ray photoemission spectroscopy (XPS).75 The reduction of V5+ to V4+ ions has also been observed using XPS for Na and Mg insertion in V2O5 thin films and vanadium oxide nanotubes, respectively.42,43 In principle, many possible locations of these reduced V centers in the V2O5 lattice are possible. Here we have systematically explored a large range of configurations for Li, Na, and Mg insertion (Figure 2b), where the V4+ ions were located at different distances from the inserted ion (Figure 2, panels c−e). We considered up to 17 (28) configurations for Li and Na (Mg). In the cases of K, Ca, and Sr, a reduced number of configurations were computed because we found very similar energy profiles to those shown in Figure 2b. This search showed that the insertion energy is significantly affected by the position of V4+ ions. Energy differences between global minima and other configurations can be as large as 0.22 eV for monovalent ions and up to 0.85 eV in the case of Mg2+ ion (Figure 2b). These large energy differences are associated with the atomic relaxation around the reduced V centers, whose size is larger than that of V5+ ions. We found that V4+ ions prefer to be located close to the inserted ion site. In the case of monovalent ions, the preferential location is the V5 position in Figure 2c. In the case of bivalent ions, the configurational space is more complex due to the presence of two V4+ ions per supercell, but again we found that configurations with both V4+ ions close to the inserted ion are preferred. Interestingly, configurations where each V4+ ion is located in different bilayers (V5−V18, V5−V22, and V5−V17 in Figure 2b) are substantially more stable than others with two V4+ ions in the same bilayer (e.g., V5−V9 or V5−V10 in Figure 2b). This result is not surprising given the covalent nature of the bonding within a bilayer; the presence of V4+ ions imposes a geometric strain to the V 2 O 5 host, which is more effectively accommodated if each bilayer minimizes its content of reduced V centers. In addition, configurations with V4+ ions close to each other and far from the inserted ion are particularly unfavorable (e.g., V4−V8 or V28−V32 in Figure 2b). Consequently, in the following, we reported calculations only considering the lowest-energy configuration found (i.e., the V5 configuration for alkali ions and the V5−V18 configuration for alkaline-earth ions).

Table 1. Calculated Lattice Constants (in Å), V−O Bond Distances (in Å), and Interlayer Binding Energies per Unit Cell (in eV) Using PBE, optB86b-vdW, optB88-vdW, and optPBE-vdW Compared to Available Experimental Values

a b c dV−O1 dV−O2 dV−O3 dV−O4 dV′−O1 Eb

PBE

optB86bvdW

optB88vdW

optPBEvdW

11.571 3.632 4.682 1.610 1.808 1.915 2.031 3.073 −0.22

11.715 3.621 4.188 1.621 1.801 1.900 2.035 2.568 −1.43

11.741 3.622 4.224 1.623 1.804 1.903 2.041 2.602 −1.44

11.690 3.628 4.423 1.620 1.810 1.912 2.046 2.804 −1.19

exptl 11.51271 11.50872 3.56471 3.55972 4.36871 4.36772 1.58171 1.78071 1.88171 2.02271 2.79371

at short interatomic separations, as observed before for many other solids.33 The equilibrium V−O bond distances follow a similar trend to the lattice constants: V−O bonds within the bilayers (dV−O1, dV−O2, dV−O3, and dV−O4) are almost equally well-described with any of the considered methods, whereas between bilayers (dV′−O1) optPBE-vdW is the closest to the experiment. Given that optPBE-vdW recovers a structure of V2O5 close to that found by the experiments, in the following, we have only focused on the optPBE-vdW to investigate the role of vdW forces in ion intercalation and diffusion. For comparison, we have also included calculations using the PBE functional. b. Ion Intercalation. The insertion of alkali (Li, Na, and K) and alkaline-earth (Mg, Ca, and Mg) ions into V2O5 was studied using a 1 × 3 × 3 supercell of V36O90, which corresponds to a dilute ion concentration (A0.056V2O5). We first looked for the most favorable ion insertion site within the V2O5 lattice by exploring a range of possible configurations using optPBE-vdW. It was found that the lowest energy site (see Figure 2) is indeed very similar for all the considered ions. As observed before for Li62,73,74 and Mg,74 the inserted ion causes a small perturbation of the stoichiometric V2O5 structure. The four vanadyl oxygen atoms close to the inserted ion are pulled inward toward the ion (except in the case of the largest K ion, where the atomic relaxation is slightly outward), with small changes of V−O bond distances (0.008−0.062 Å) in relation to the empty V2O5 host (dV−O1 in Table 2). The inserted ion sits Table 2. Calculated V−O1, A−O1, A−O2, and A−O3 Interatomic Distances (in Å) for A0.056V2O5 (A = Li, Na, K, Mg, Ca, or Sr) Using optPBE-vdW dV−O1

dA‑O1

dA‑O2 dA‑O3

a

Li

Na

K

Mg

Ca

Sr

1.626 1.628 1.635 1.637 2.154 2.403 2.684 2.937 2.101 2.291 2.158 2.847

1.630 1.632 1.632 1.632 2.444 2.469 2.493 2.514 2.398 2.419 2.585 2.641

1.626 1.627 1.628 1.628 2.637 2.646 2.663 2.671 2.781 2.809 2.670 2.672

1.650 1.661 1.668 1.682a 2.145 2.149 2.252 2.471 2.147 2.230 2.474 2.717

1.652 1.653 1.658 1.680a 2.369 2.399 2.460 2.479 2.395 2.434 2.578 2.619

1.648 1.650 1.655 1.666a 2.493 2.519 2.567 2.579 2.528 2.562 2.673 2.705

In this case, the corresponding vanadium atom is reduced (V4+). 19602



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Further analysis shows that the interlayer spacing between V2O5 bilayers has a minor impact on the computed Eins when using PBE. In all cases, PBE insertion energies are reduced when decreasing the interlayer distance along c. In particular, fixing the unit cell lattice constants to those optimized with optPBE-vdW (PBE@optPBE-vdW bars in Figure 3), results in similar insertion energies to those obtained with PBE, especially in the case of small ions; differences are less than 0.02 eV for Li, Na, and Mg, whereas for the rest of ions insertion energies differ up to 0.22 eV. This suggests that nonlocal correlation plays an important role in the stabilization of inserted ions. While optPBE-vdW is able to capture these attractive interactions, PBE lacks them, resulting in smaller Eins than optPBE-vdW. Finally, the V5+/V4+ voltage associated with the insertion of a given ion into V2O5 at 0 K can be estimated as Eins/z, since the Gibbs free energies can typically be approximated by the total energies.78 Here z is the number of electrons transferred: one (two) in the case of alkali (alkaline-earth) ions. In principle, one should also account for the configurational entropy in a proper manner (see, for example, ref 79), but, in the limit of low ion concentrations and assuming that ion−ion interactions are repulsive, a good estimate of the upper limit voltage can easily be obtained, using the lowest energy configuration, as discussed by Scanlon et al.62 For example, assuming the optPBE-vdW Eins value for a Li ion (Figure 3), our computed upper limit voltage in the limit of dilute Li would be 3.50 eV (the calculated voltages for the rest of ions are given in Table 3), which compares reasonably well with the 3.2−3.4 V range

Moving now to the calculated ion insertion energies (eq 2), Figure 3 summarizes our results using both optPBE-vdW and

Figure 3. Calculated energies of alkali and alkaline-earth ion insertion into V2O5 using optPBE-vdW and PBE. The nonlocal correlation contribution to the optPBE-vdW intercalation energy (Enlc ins ) is also included. PBE calculations using the unit cell lattice constants optimized with optPBE-vdW (PBE@optPBE-vdW) are also shown.

Table 3. Calculated Voltages (in V) for A0.056V2O5 (A = Li, Na, K, Mg, Ca, or Sr) Using optPBE-vdW and PBE

PBE functionals.76 The first observation is that optPBE-vdW yields larger insertion energies than PBE. In general, the energy differences between the two functionals range from 0.22 to 0.30 eV for alkali ions, whereas in the case of alkaline-earth ions the discrepancies are much larger: 0.40, 0.65, and 1.71 eV for Mg, Ca, and Sr, respectively. Interestingly, PBE insertion energies are nearly constant for the alkali and the alkaline-earth series: ca. −3.3 eV and ca. −5.0 eV, respectively. These results suggest that, according to PBE, the increase of the ionic radius when moving along each series (Li < Na < K and Mg < Ca < Sr) does not play a major role in the overall stability of the inserted ion. In contrast, the inclusion of vdW forces leads to a less regular behavior, especially in the case of alkaline-earth ions where a progressive increase of Eins is observed when increasing the ion size. In order to gain insight into the effect of vdW forces on Eins, we calculated the nonlocal correlation part, Enlc ins , of Eins (gray bars in Figure 3), according to eq 3. This analysis revealed that Enlc ins is a small fraction of the total Eins, but it helps to rationalize the observed trends. In particular, Enlc ins increases when going from Li to K and from Mg to Sr; this is consistent with the fact that the ionic polarizabilities follow similar progressions (Li+ < Na+ < K+ and Mg2+ < Ca2+ < Sr2+).77 Indeed, larger polarizabilities imply larger attractive nonlocal correlation energies, resulting, in general, in larger Eins. The lack of such attractive energy contributions in PBE could ultimately explain why the insertion energies calculated with PBE are lower than those from optPBE-vdW and rather insensitive to the nature of the ion. Overall, these results highlight the fact that vdW forces are an important ingredient to properly describe the interaction of inserted ions in V2O5, especially for large and highly polarizable species.

optPBE-vdW PBE

Li

Na

K

Mg

Ca

Sr

3.50 3.24

3.62 3.32

3.33 3.11

2.70 2.50

2.79 2.46

3.34 2.48

reported for a charged Li−LixV2O5 cell.80 This comparison suggests that optPBE-vdW can provide quantitative energetics for ion insertion, but, in general, more experimental measurements of cell potentials on well-defined systems remains highly desirable. c. Ion Diffusion. The migration of Li, Na, K, Mg, Ca, and Sr ions in V2O5 at the dilute limit was investigated using the CINEB method and the same 1 × 3 × 3 supercell employed to study the energetics of ion insertion. Essentially, the migration process corresponds to a hop path between two low-energy ion inserted positions. We assumed that the most probable migration path involves diffusion within the interlayer spacing, in the parallel direction to the b lattice axis, and along the channels formed by vanadyl groups (Figure 4, panels a and b). We notice that diffusion along the alternative a lattice axis is blocked by the presence of a vanadyl group. A similar route has been proposed before for Li73,74 and Mg,74 whereas other pathways, including the diffusion through a bilayer along the c direction, were found to be much less favorable.74 It is important to stress that this ion hop process should also take into account the localization of V4+ ions in both the initial and the final states. As pointed out by Braithwaite et al.,73 the diffusion barriers might significantly depend on this issue. Here, we ensured that our initial and final states were indeed identical with respect to V4+ localization (both at their lowest Eins); 19603



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Figure 4. (a) Top and (b) side views of the initial (left), transition (middle), and final (right) atomic structures of an ion diffusing along the optPBEvdW minimum energy migration paths (c and d). The atomic structures in (a and b) reflect the general atomic relaxation along the migration process for all considered ions, but they actually correspond to the optimized structures of a diffusing Na ion; similar qualitative atomic displacements are found for other ions. In the case of the inserted alkaline-earth ions, two V4+ species are present in the supercell at positions V5 and V18 (Figure 2), as indicated with arrows; in the case of alkali ions, only one V4+ species is present at position V5 (Figure 2). Notice that the location of V4+ ions does not change during the whole migration process. For simplicity, only two V2O5 bilayers in (a and b) are shown. The lines connecting the points in (c and d) correspond to a spline fitted to the calculated CI-NEB energies. Color code: alkali (alkaline-earth) ion, yellow; V5+, blue; V4+, black; O, red.

added to this, the V4+ ions remain at the same positions during the whole migration process (Figure 4, panels a and b). The calculated minimum energy migration paths for alkali and alkaline-earth ions using optPBE-vdW are shown in Figure 4, panels c and d, respectively. Clearly, the activation energy of alkali or alkaline-earth migration increases with the size of the diffusing ion along each series (see Figure S3 of the Supporting Information). This is a consequence of the transition state atomic structure, where the diffusing ion forms a trigonal planar geometry with three nearest-neighbor O atoms within the a−c plane (Figure 4b): two vanadyl O1 atoms and one bridging O2 atom (Figure 1). The three O atoms are pulled outward away from the ion (Table S2 of the Supporting Information), resulting in a relatively large local distortion in relation to the empty V2O5 host structure. This distortion increases with the

size of the ion and, therefore, the activation energy of the process also increases. We discuss now the effect of vdW forces on the migration process. To this end, we first compare the results obtained with optPBE-vdW and PBE (Figure 5). The diffusion barriers computed with PBE are systematically lower than the corresponding optPBE-vdW values. For alkali ion diffusion, the energy differences range from 0.16 to 0.49 eV, whereas in the case of alkaline-earth ions, the range is 0.20−0.35 eV. For example, the diffusion barrier of Li using PBE (0.15 eV) is 50% lower than that obtained with optPBE-vdW (0.31 eV). These are large discrepancies between the two methods, which can lead to even larger differences when considering ion diffusion coefficients (see, for example, refs 81 and 82). These results indicate that vdW forces play an important role in the ion migration process. Essentially, the inclusion of vdW 19604



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(optPBE-vdW). Our calculations reveal that the inclusion of vdW interactions quantitatively changes ion insertion energies, voltages, and activation energies for ion diffusion with respect to the widely used PBE functional. Analysis of our results shows that a proper account for nonlocal interactions enhances ion insertion energies and voltages, especially in the case of more polarizable ions. Thus, conventional GGA functionals are expected to systematically underestimate these two key thermodynamic properties. In addition, we find that a lack of vdW interactions results in the underestimation of activation energies for ion diffusion too. The attractive vdW forces reduce the interlayer spacing, resulting in a more compact layered structure which imposes a larger strain to the transition state geometry and, therefore, increase its energy. This geometric effect is consistent with the fact that larger inserted ions present larger activation energies than small ions. Overall, our findings indicate that vdW-inclusive DFT calculations are essential for accurately describing key electrochemical properties of ion insertion into V2O5. Since a suitable description of interlayer interactions is a key element for describing layered compounds, it is likely that vdW forces will be of importance to ion insertion into other LTMOs in general and not just the exemplar V2O5 system considered here. Although the search for more sophisticated vdW-inclusive DFT methods is an ongoing quest, we put forward that vdW density functionals, in particular optPBE-vdW, are useful candidates to account efficiently and accurately for vdW interactions in these types of electroactive materials. To this regard, the recent study by Eames et al.49 of Li insertion in layered LiFeSO4OH using optPBE-vdW is quite encouraging, showing that this functional can provide accurate cell voltages and structures for this material as well.

Figure 5. Calculated diffusion barriers of alkali and alkaline-earth ions in V2O5 using optPBE-vdW and PBE. PBE calculations using the unit cell lattice constants optimized with optPBE-vdW (PBE@optPBEvdW) are also shown.

forces brings the V2O5 bilayers closer to each other; this imposes an additional geometrical strain to the transition state structure and, consequently, the activation energy for ion diffusion is enlarged in relation to larger interlayer spacings. An evidence for the imposed geometrical strain by vdW forces is given by the interatomic distances between the diffusing ion at its transition structure and the nearest neighbor O and V atoms (Table S2 of the Supporting Information). PBE distances are, in general, larger than those computed using optPBE-vdW; differences between the two functionals can be as large as 0.55 Å, depending on the inserted ion. Interestingly, if one artificially compresses the interlayer spacing, it is possible to obtain suitable diffusion barriers using PBE. For example, we carried out PBE calculations using the unit cell lattice constants optimized with optPBE-vdW (PBE@optPBE-vdW bars in Figure 5). The diffusion barriers computed in this manner are systematically larger than those obtained using a PBE-based unit cell and very close (0.04−0.12 eV) to optPBE-vdW values in all cases. This is in contrast to the situation of ion insertion energies discussed before, for which PBE and PBE@optPBEvdW essentially yielded similar results. Thus, these results suggest that the transition state structure is dominated by shortranged interactions, which are well-described by both PBE and optPBE-vdW functionals. Nevertheless, vdW forces govern the overall strain imposed by the interlayer spacing to the transition state and, hence, they impact significantly on the magnitude of the activation energy for the diffusion process.

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ASSOCIATED CONTENT

S Supporting Information *

Detailed results (density of states, activation energy for diffusion as a function of ion size, convergence tests, selected interatomic distances, and optimized coordinates of selected atomic structures). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +34 94 529-7108. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Computer time provided by the Barcelona Supercomputer Center (BSC), i2BASQUE, and the Supercomputing Center of Galicia (CESGA) is acknowledged. The author is a Ramón y Cajal fellow and Newton Alumnus supported by the Spanish Government and The Royal Society.

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REFERENCES

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CONCLUSIONS We have examined the role of vdW forces in alkali (Li, Na, and K) and alkaline-earth (Mg, Ca, and Sr) ion insertion into V2O5 by considering an optimized nonlocal vdW density functional 19605



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