Understanding the effect of doping on the charging performance of the Li-O2 battery: the role of hole polarons and lithium vacancies

In this work, we perform DFT calculations using the hybrid functional HSE to properly describe the insulating nature of lithium peroxide and study its more energetically favourable surfaces [0001], [11̄00] and [112̄0]. We then analyse how the insulating character and the correct description of the hole polarons at the Li2O2 surfaces affect the electrochemical steps of Li2O2 decomposition in the charging process of the Li-O2 battery. We then study the effect of doping and propose possible scenarios in which


Introduction
The Li-O 2 battery (LOB) is a promising energy storage system due to its exceptionally high theoretical energy density (∼ 3500 Whkg −1 ). 1,2 However, the LOBs technology is still facing important challenges and limitations that have to be sorted out. Among them, there are the high charging potential that leads to low discharge/charge cycle efficiency, the reduced capacity at high discharge rates and the poor cathode and electrolyte stability. 2,3 The main discharge product of the LOB is Li 2 O 2 whose insulating character deeply affects various aspect of the battery. 2,4,5 The dominant reversible reaction that takes place at the cathode of a non-aqueous Li-O 2 cell is During the discharge of the battery, the oxygen from the air is reduced at the cathode and combines with a Li ion to form solid Li 2 O 2 . During the recharge, the reverse decomposition of Li 2 O 2 occurs. The overpotential needed to recharge the battery determines its efficiency and depends on several factors that can have different origin. The source of the energy barriers originates in the intrinsic reversible reaction but also is affected by the morphology of the discharge product 2,6 or the presence of side-reaction secondary phases. 5,7 A profound knowledge of the electronic and transport properties of Li 2 O 2 both in bulk and at the surface is crucial to understand the reversible reaction of Eq. 1 and to propose new routes to circumvent the above mentioned difficulties.
The charge transport through the bulk of Li 2 O 2 has been studied extensively by ex-periments and theoretical calculations. [8][9][10][11][12][13] Particularly, Gerbig et al. observed the Li 2 O 2 insulating nature by impedance spectroscopy and DC conductivity measurements, showing that ionic lithium defects are the majority carriers while electronic conductivity is two order of magnitude smaller and of hole type. These results were confirmed later by Dunst et al. by means of nuclear magnetic resonance spectroscopy. 9 Independently, Ceder and coworkers showed, using Density Functional Theory (DFT) calculations with the Heyd-Scuseria-Ernzerhof (HSE) screened hybrid functional, that holes can be self-trapped at the peroxide sites (O 2 −2 ) forming small hole polarons (O −1 2 ). Furthermore, they noted that the nonpolaronic structure, which present metallic states, has significantly higher energy than the polaronic one. 11 Afterwards, Radin and Siegel, employing the same hybrid functional, concluded that the dominant charge carriers in bulk Li 2 O 2 are the lithium vacancies (V Li ) and hole polarons, and that the inclusion of exact exchange at some extent is essential for achieving their correct description. 13 Li 2 O 2 surfaces have also been object of study as possible conduction paths during the operation of the cell. Previous theoretical reports have shown that the more energetically favourable surface of Li 2 O 2 under ambient conditions is metallic. 14 These results were based on DFT calculations using functionals in the General Gradient Approximation (GGA In view of the well known self-interaction error of the GGA functional, the result of the surface metallicity was revisited 20 using the HSE correction and even the scGW method. 20 Interestingly, the metallic behaviour that was also obtained for these corrected functionals, arises as an artifact due to a finite size effect of a small 1x1 unit cell that does not allow for surface reconstruction. In fact, calculations done later for larger supercells and proper functionals confirm that all the Li 2 O 2 surfaces are insulating and that the only charge carrier conduction path available is the diffusion of defects 21 or through tunnelling in the case of thin films. 22 Considering the very high computational cost of HSE, specially for simulating large supercells, the DFT+U technique 23 was also used as an alternative to correct the underestimation of the electronic correlation characteristic of local or semilocal functionals as GGA. 12 After this winding road towards understanding the formation and decomposition of Li 2 O 2 in the LOB, there is still some confusion in the literature regarding these fundamental issues that hinders future efforts to make progress in this promising technology and a clarification is urging. With respect to the high charging overpotential, several strategies were proposed. Recently, Byon and coworkers showed that the nanostructuring of one-dimensional and amorphous Li 2 O 2 can get an improve of the round-trip efficiency of the LOB of ∼ 80%. DFT calculations reveal that the structural distortions of the amorphous structure lead to a weaker binding of LiO 2 , a key intermediate in the reaction, yielding smaller overpotentials in the delithiation process. 24 On the other hand, the Li 2 O 2 doping has been reported as another potential strategy to improve the efficiency of the LOB. [25][26][27][28][29][30] However, the mechanisms of how the incorporation of heteroatoms in the Li 2 O 2 can affect performance of the battery is still unclear. Experimentally, barium (Ba) was one of the heteroatoms considered, achieving a significant reduction in the charging overpotential. The authors ascribed this effect to an improvement in the charge transport as a consequence of Ba incorporation in Li 2 O 2 . 28 Afterward, Chen and coworkers showed that a LOB with Na-doped Li 2 O 2 as a discharge product also presents a lower charge overpotential as compared to the undoped system, since the Na + as the dopant induces lithium vacancies, which according their DFT calculations, lead to conducting states in Li 2 O 2 . 29 Nevertheless, in view of the above mentioned problems of self-interaction errors, this theoretical interpretation needs to be revised. In this work, we provide an insightful description of the electronic and structural prop-erties of the more energetically favourable Li 2 O 2 surfaces obtained with the HSE functional, modeling the supercell that allow for the polaron formation not only for in non-stoichiometric surfaces but also during all the electrochemical steps involved in the recharge of the battery.
We study the effect of the insulating nature of the surface in the lithium peroxide decomposition and how doping can induce lower energy barriers to decrease the charging overpotential.

Computational methods
First-principle calculations are performed with the generalized gradient corrected approxi- The surface energy, for T=300 K and P =1 atm, can be calculated as: where G slab is the free energy of the surface supercell, A is the area of the exposed surface, The free energy change between reaction steps is given by: where n represents an intermediate reaction step, E n is the total energy of the configuration at the step n, ∆N Li and ∆N O 2 are the number of Li and O 2 atoms that are removed from the surface in the step n respect to the step (n − 1), and eU is electron energy under the applied charging potential U .

Results and discussion
Surface energy and electronic structure  Table 1 presents the calculated surface energies corresponding to the cases shown in Figure 1, obtained for T=300 K and P=1 atm, using both GGA and HSE. The γ energies are in good agreement with previous reports. 14,21    . The corresponding projected DOS shows that the polaron states lye in the band-gap of the 1st layer (solid red in Figure 2).
The polaron formation at the O-rich surface emerges as a consequence of a surface reconstruction, which in Ref. 14 was blocked by using a 1x1 supercell. Breaking the symmetry using a 2x1 unit cell, the reconstruction is now possible and the polaron is stabilized being 30.3 meV/Å 2 more stable than the delocalized configuration. The same effect occurs for Li 2 O 2 bulk, where the extra charge also prefers to be localized at the oxygen dimers. 11,13,27 On the other hand, Figure 3

DFT+U corrections applied to Li 2 O 2
The DFT+U technique has been also used to study Li 2 O 2 as a less computationally demanding method to correct the well known problems of the GGA functionals.
Previously, it has been reported that DFT+U calculations as implemented in the GPAW code, 32 are able to describe the stabilization of hole polarons in Li 2 O 2 for physically relevant values of the Hubbard U parameter. 12,33 However, as mentioned later in Ref. 34, care should be taken when applying DFT+U techniques to extended orbitals as the p states. For instance, in the VASP code, the implemented DFT+U formalism is sensitive to the PAW augmentation radius that it might be smaller than the spatial extension of the orbital that one intends to correct. On the other hand, the GPAW implementation of the DFT+U reduces the effect of the augmentation radius dependence by normalizing the integral of the projected atomic orbitals within the augmentation sphere, scaling the corresponding overlaps accordingly.
In this work, we aim to asses the effect of the DFT+U correction as implemented in the widely used VASP code when applied to extended p-states as the ones present in Li 2 O 2 . 23 For this, we vary the U parameter between 0 and 16 eV in order to observe the effect of the Hubbard correction on the crystalline and electronic structure for Li 2 O 2 bulk and its more stable surfaces.
Since the final conclusion of this assessment is that applying the U correction to the   Figure S5 in the SI file. On the contrary, a hole gets trapped at a surface O 2 dimer within HSE, forming a polaron. This step has a higher energy barrier within HSE than within GGA, since the former functional leads to an extra elastic cost. The polarons are again schematized as purple bonded O 2 dimers in the insets of Figure 4. The second step is the formation of a second V Li that gives rise to another polaron within HSE whereas within GGA more metallic states are generated (see Figure S5 in the in SI file). In general, the second V Li has a lower formation energy than the first one because the surface structure was already distorted by the first vacancy; the system is less bound. The subsequent two reaction steps are the liberation of two LiO 2 . It is important to remark that these steps present a higher energy barrier for the functional GGA. The reason for this is that the energy cost is comparatively larger within GGA due to the overbinding energy of the delocalized charge. On the other hand, within HSE the system gets rid of the elastic cost of one polaron in each of the last two steps so that the final reaction free energy turns out to be smaller than within GGA. In particular, the last step is the limiting one and will determine the value of the overpotential. In the present work, we do not aim to obtain an absolute value of the overpotential because we are not considering other complex details such as the role of the solvent, the Li and O 2 diffusion and charge transport through the Li 2 O 2 , among others. The main message of this section is that some care should be taken when describing the recharge process within GGA because the energy cost of the limiting step is overestimated.

Effect of doping in the Li 2 O 2 decomposition
In the previous section we have clarified the electronic structure and the energetics of the Li 2 O 2 surface and its mechanism for the delithiation in the recharge process of the LOB. We used a proper functional for the exchange and correlation potential as HSE that corrects the self-interaction error introduced by the standard GGA. In the present section, we proceed to study the effect of Na doping in the OER. As mentioned in the Introduction, Chen and coworkers showed a decrease in the charging overpotencial when Na + ions were dissolved in the electrolyte, doping the formed Li 2 O 2 in the LOB. 29 The explanation of this desired effect was ascribed to the generation of more lithium vacancies in the presence of Na + that supposedly gives rise to metallic states as described by their DFT calculations. These calculations were performed using HSE with α=0.207 for which the obtained insulating band gap of ∼ 4 eV is still lower than the one calculated with α=0.48 and the more accurate formalism GW (∼ 6 eV), indicating that the theoretical metallic states obtained in the presence of lithium vacancies with α=0.207 are an artifact because the self-interaction correction is not completely accomplished.
In this work, we study the effect of Na + doping in the electronic structure and the OER of surface. We explore different Li + substitutional sites for Na doping (see Figure S6 and Table   S2 in the SI material). For the more energetically stable Na-doping site, we calculate the lithium vacancy energy formation for different remaining Li + sites (see Figure S7 and Table   S3 in the SI file). In line with the experimental observation, the formation energy in the presence of Na + for the more favorable vacancy site is 150 meV smaller than for the pristine surface (see Table S3 in the SI file).
There is another scenario in which dissolved Na + can promoted lithium vacancies. If an electrolyte with high donor number is used in the experiment, as was the case of Ref. 29, the solution mechanism of Li 2 O 2 formation in the LOB is more likely. 35 Considering the solution mechanism, two LiO 2 dissolved in the electrolyte disproportionate to form Li 2 O 2 releasing one O 2 molecule. As mentioned before, the doping with K + has also been probed to promote the tendency to O -2 and the generation of lithium vacancies. 30 In general, when a cation X is dissolved in the electrolyte, the following disproportionation We first calculate the energetics of the disproportionation when X is equal to Li, Na and K for comparison. The corresponding disproportionation energy reads: In Figure 5, we show the three cases studied and their calculated E dis (X). We obtain  Considering as a fact that Na + or K + doping promotes V Li generation, we next focus on the Na + case to study the OER for the Na-doped [1100] ST-3 surface in the presence of one V Li as initial state. In this Section, we will only show the results obtained with HSE α=0.48. The initial state of the simulated OER process for the Na-doped surface is depicted in the inset a) of Figure 6. It can be observed that there are a Na surface ion (in yellow), a hole polaron at the surface dimer O 2 (in purple) and the initial lithium vacancy is indicated with a cross. As mentioned before, we remark that both the Na substitutional site and the initial lithium vacancy are the ones that resulted more energetically stable. Figure 6 shows the calculated reaction free energy of the OER process at U=0 V for the In this line, our results suggest that K-doping will be even more effective in reducing the overpotential since more lithium vacancies and more structural distortions are expected for this heavier heteroatom.
It is worth mentioning that when these calculations of the OER are performed with GGA, we also obtain an improvement of the OER barrier (731 meV for the simulated supercell) for the Na-doped surface as compared to the pristine one. However, this energy gain is strongly overestimated as a consequence of the GGA overbinding of the pristine Li 2 O 2 surface.

Conclusions
In this work, we have studied the electronic structure of different Li 2 O 2 surfaces using the hybrid functional HSE that is capable of adequately describing the electronic structure and modeling the localization of hole polarons. The insulating nature of the all studied surfaces has been confirmed. In the non-stoichiometric terminations with low-coordinated surface oxygen atoms, we have also confirmed a similar charge self-trapping behaviour as found in Li 2 O 2 bulk. Then, we have examined the Li 2 O 2 decomposition that occurs during the recharge process using HSE to take into account the presence of the surface polarons in the electrochemical intermediate steps. These results have been compared with the ones obtained using the GGA functional, which tend to delocalize the excess of charge carriers.
We have shown that the free energy variation of the limiting step is overestimated within GGA when there are spurious metallic states instead of hole polarons.
Finally, we have investigated the Na-doping effect on the lithium vacancy generation and on the Li 2 O 2 surface decomposition. On the one hand, it is found that Na dopant reduces the lithium vacancy formation energy compared to the value for pristine surface, promoting the vacancy generation in agreement with experimental results. On the other hand, we have presented a plausible scenario to enhance lithium vacancies when the formation of the lithium peroxide occurs via a solution mechanism and Na + or K + ions are dissolved in the electrolyte.
Based on the analysis of the energetics for different superoxide disproportionations, we have proposed that NaO 2 formed in the electrolyte can get trapped on the discharge product, concomitantly with the formation of a lithium vacancy. We have also predicted that Kdoping should be even more efficient in reducing the overpotential than Na-doping.
At last, the Na-doped Li 2 O 2 decomposition has been studied. The removal of the second LiO 2 intermediate is found to be the limiting step during charge, in both Na-doped and pristine Li 2 O 2 surfaces. We have found that Na-doping decreases the energy barrier of the limiting step, contributing to a reduction of the charging overpotential, in line with the experimental results. Our calculations indicate that the origin of this decrease are the lattice distortions associated with doping that weaken the LiO 2 binding, and not the emergence of surface metallic states as previously reported.

Supporting Information Available
In the Supplementary information file we include: • Detailed results of the DFT+U calculations performed with the VASP code.
• DOS plot of each electrochemical step of the Li 2 O 2 decomposition reaction calculated with GGA and HSE (α=0.48).
• The crystal structure of with the different sites evaluated for the dopants and lithium vacancies and their corresponding energy of formation.