The Role of Disorder in NaO2 and its Implications for Na-O2 Batteries

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The Role of Disorder in NaO and Its Implications for Na-O Batteries Oleg Sapunkov, Vikram Pande, Abhishek Khetan, and Venkatasubramanian Viswanathan J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b04753 • Publication Date (Web): 16 Jul 2018 Downloaded from http://pubs.acs.org on July 17, 2018

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The Journal of Physical Chemistry

The Role of Disorder in NaO2 and its Implications for Na-O2 Batteries Oleg Sapunkov, Vikram Pande, Abhishek Khetan, and Venkatasubramanian Viswanathan∗ Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 E-mail: [email protected]

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Abstract There is a need for more energy dense batteries and Na-O2 batteries have emerged as an attractive option. In this work, we present a DFT study utilizing the Hubbard U correction to probe structural and magnetic disorder in NaO2 , primary discharge product of Na-O2 batteries. We show that NaO2 exhibits a large degree of rotational and magnetic disorder. A 3-body Ising Model is necessary to capture the subtle interplay of this disorder. Our MC simulations demonstrate for the first time that energetically favorable, FM phases near room temperature consist of alternating bands of O2 dimers oriented along two of four cubic cell body diagonals. Using hybrid density functional theory calculations, we find that bulk structures are insulating, with a subset of FM structures showing a moderate gap (< 2 eV) in one spin channel. The insulating nature of NaO2 implies that growth of the discharge product is most likely occurring due to the solution mechanism pathway involving a chemical dissolution of NaO2 into Na+ and O− 2 , similar to what is seen in Li-O2 batteries.

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Introduction Electrification of road transportation, one of the most promising solutions to curbing worldwide greenhouse gas emissions, recently led to increased interest in high-energy-density rechargeable batteries, in particular, the metal-air batteries. 1 The Li-O2 battery, considered one of the most promising due to its high energy density, suffers from electrolyte 2,3 and electrode 4–6 instability, as well as limited discharge capacity. 7–9 Hartmann et al. demonstrated a rechargeable Na-O2 battery, with sodium superoxide (NaO2 ) as the predominant observed discharge product, which showed superior cycle life and rechargeability to state-of-the-art Li-O2 batteries. 10,11 Subsequently, there have been numerous theoretical and experimental efforts to understand the fundamental mechanisms that govern electrochemistry in Na-O2 batteries. 12–18 Thorough description of the Na-O2 battery reaction mechanisms 12,19 requires detailed understanding of electronic structure throughout the phase space of the sodium oxides. The roles of nucleation, 18,20,21 nanoscale stabilization, 12 and surface energetics of various sodium-oxygen compounds have been examined through a combination of density functional theory (DFT) calculations and electrochemical measurements. This has led to an improved understanding of Na-O2 battery reactions, which, as proposed, constitute a combination of surface and solution mechanisms of the NaO2 discharge product. 1,22 In certain cases, it should be noted, Na2 O2 has been reported as the discharge product, 23,24 and selectivity between NaO2 and Na2 O2 is not yet fully characterized. 25 Alkali superoxides are known to be highly disordered materials, both in bulk geometric structure and magnetic ordering. 26,27 Presently, there is limited understanding of geometric and magnetic disorder in NaO2 , due to the challenges in experimentally synthesizing NaO2 , as well as its long-term instability at room temperature conditions. It is crucial to map out disorder in room-temperature NaO2 , considering its importance in determining electronic structure, surface energetics, and growth properties relevant for Na-O2 batteries. In this work, we aim to build a comprehensive understanding of the magnetic and geometric ordering 3



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of bulk NaO2 . We perform DFT calculations incorporating the Hubbard U correction. We employ an Ising model accounting for magnetic and geometric degrees of freedom and find that both 2-body and 3-body nearest-neighbor interactions are critical to accurately describe bulk NaO2 . The Ising model is used within a Monte Carlo framework to characterize the complete phase space of bulk NaO2 as a function of temperature. Our simulations indicate that the low-temperature phase of NaO2 is characterized by oxygen dimers oriented along two of four cubic cell body diagonals, arranged in large planar structures. This finding is in good agreement with the experimental description of low-temperature bulk NaO2 . 28 In order to determine the effect of disorder on the electronic properties of NaO2 , we employ hybrid density functional theory calculations and find that the calculated bandgap is strongly affected by disorder. All investigated anti-ferromagnetic structures exhibit a direct bandgap around 4 eV and a small subset of ferromagnetic structures have direct bandgaps below 2 eV in one spin channel.

Computational Methods The discharge process of the Na-O2 battery involves the Na+ ion coupled electron transfer reaction with O2 at the cathode, which produces NaO2 , the primary reported discharge product. 10,11 Na+ + O2 + e− NaO2(s)

(1)

A few studies do report the formation of Na2 O2 , 23 the thermodynamically stable structure in the Na-O phase diagram at room temperature. 12 The preferential formation of NaO2 as a discharge product in room-temperature Na-O2 batteries, over the thermodynamically stable Na2 O2 , still remains a puzzle. 25 The stable phases of bulk NaO2 is temperaturedependent. 12,28,29 Below 196 K, NaO2 takes the Pnnm space group, the marcasite structure, with a lattice parameter of a = 4.26 ˚ A. In this structure, superoxide ion dimers are observed to be oriented along two of four cubic cell body diagonals, arranged in a regular pattern. 28 4


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Between 196 and 223 K, NaO2 takes the Pa¯3 space group, the pyrite structure, with a = 5.46 ˚ A. Superoxide ion dimers in this structure are observed to be fixed, oriented along any of 4 cubic body diagonals, with short-range order within the structure. Above 223 K, NaO2 takes the Fm¯3m space group, the disordered pyrite structure, with a = 5.49 ˚ A. In this structure, superoxide ion dimers are under complete dynamic rotational disorder, switching between orientations parallel to any of the four cubic lattice body diagonals. This is the primary relevant structure for room-temperature Na-O2 batteries. 30 The oxygen dimer in NaO2 behaves similar to the superoxide anion, O− In the molec2. ∗ ular orbital picture, the highest occupied molecular orbital of O− 2 is π2p , occupied by 3

electrons, as illustrated in the SI. This means that the O− 2 dimer has 1 unpaired electron, and thus, is magnetic. The magnetic ordering of O− 2 dimers is one of the key contributions to the configurational disorder of bulk NaO2 . We considered bulk NaO2 in the Fm¯3m space group structure. In this structure, sodium atoms occupy the body centers and corners of the cubic cell, and oxygen dimers occupy face centers and edge centers of the cell, as illustrated in Fig. 1. We introduce the following naming scheme for these structures to describe both their geometric and magnetic arrangement. The first 4 letters, which can be A, B, C, or D, refer to the 4 possible geometric orientations of oxygen dimers. The next 4 letters, which can be P or N, refer to the net positive or negative magnetic moment of the corresponding dimers. Thus, a structure designated AAAA-PPPP (Fig. 1(a)) has all dimers mutually parallel to each other (AAAA), and all dimers of identical, positive magnetization (PPPP), providing a ferromagnetic structure. Alternatively, a structure designated ABCD (Fig. 1(b)) has all dimers mutually orthogonal to each other, while a structure designated PNPN or PPNN is antiferromagnetic. Formation enthalpies were calculated for 19 distinct possible structures to explore the phase space of bulk NaO2 . Self-consistent DFT calculations were performed using the Projector Augmented Wave Method as implemented in GPAW, 31 with the Revised PBE (RPBE) exchange correlation

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Figure 1: Unit cell of Fm¯3m NaO2 used in the study. Two configurations are presented. (a) In AAAA-PPPP, all of the oxygen dimers within the unit cell are mutually parallel, and have a net magnetization identical for all dimers. (b) In ABCD-PNPN, all of the oxygen dimers are mutually orthogonal, and are arranged in pairs with opposite magnetization. functional. 32 To correct electron localization in NaO2 , we incorporated the Hubbard U, applied on oxygen 2p states in NaO2 . Application of the Hubbard U on oxygen 2p states has been known to improve the accuracy of simulation results, with respect to experimental data, for a variety of materials. 33–36 Furthermore, it was found that calculations on GGAlevel DFT without the Hubbard U correction fail to converge any of the examined structures. Calculations were run with a real-space grid of 0.18 ˚ A, and 6×6×6 k-point sampling, following the Monkhorst-Pack scheme. 37 Fermi-Dirac smearing of 0.01 eV was used to facilitate convergence and broyden-type mixing of electron densities was used in the calculation. 38,39 Two schemes were used to calculate formation enthalpy of bulk NaO2 . The formation enthalpy of NaO2 is given by: Ref Ref ∆HFNaO2 = EDFT NaO2 − ENa − EO2 + ∆pV

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(2)

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The pressure-volume work term, ∆pV, can be neglected. 40 Internal energies of Na, O2 , and NaO2 were calculated using DFT simulations. It is well known that molecular oxygen is poorly described in DFT and water reference scheme is used, given by: 41

Exp DFT DFT ERef O2 = 2EH2 O − 2EH2 − ∆HH2 O

(3)

The Hubbard U correction was applied on the oxygen atoms in both H2 O and NaO2 . The second correction scheme was used to calculate reference energy of bulk Na. Formation enthalpies of alkali oxides, peroxides and superoxides are best described when the oxidation state of the metal in the reference compound matches the oxidation state of the metal in the compound under investigation. 42 The Na reference energy was corrected using the NaCl scheme, given by: 1 DFT Exp EN aRef = EDFT NaCl − ECl2 − ∆HNaCl 2

(4)

Currently, there is no universally accepted method for selecting the optimal Hubbard U to fit calculations to experimental data. In this study, we follow the scheme 43 that compares the calculated formation enthalpy of a substance under investigation to the experimentally measured formation enthalpy. The experimentally measured formation enthalpy of NaO2 is ∆f H0solid = −2.7eV. 44

Results & Discussions We find that the formation enthalpies of all considered bulk configurations of NaO2 lie within a band of 0.3-0.6 eV/NaO2 , at all investigated values of the Hubbard U, as shown in Fig. 2(a). The fully-organized, ferromagnetic (AAAA-PPPP) and fully disorganized, antiferromagnetic (ABCD-PPNN) structures were found to be least energetically stable,

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while the moderately disorganized structures with intermediate magnetic order, such as AABB-PPPN, were stabilized by 0.2 - 0.4 eV, relative to the fully-organized and fullydisorganized structures investigated. Many of these moderately disorganized structures were energetically degenerate at all examined values of the Hubbard U. We observe that U = 3 eV is matches the experimental formation energy and will be used for further analysis. It is worth highlighting that the conclusions regarding disorder in bulk NaO2 remain consistent for all examined values of the Hubbard U. Given the high degree of configurational degeneracy observed among examined structures of NaO2 , it becomes crucial to examine the role of thermal disorder and its role in determining electronic properties of NaO2 . The computation of disorder in larger supercells directly through DFT is computationally intractable. We employ the approach of determining a lattice Hamiltonian that can be used to describe the energetics of bulk NaO2 . In order to map out energetic interactions between the magnetic and geometric degrees of freedom, we utilize a modified Ising Model for the lattice Hamiltonian. As implemented, the model consists of a lattice of N sites i, whose filling is described by occupation terms, σi . System energy contributions due to particle interactions in neighboring sites are captured by the 2-body and 3-body interaction terms, j2i,k and j3i,k,l . The total energy of the N -site lattice is calculated as: E=−

X

j2i,k σi σk −

hiki

X

j3i,k,l σi σk σl

(5)

hikli

Formation enthalpies of NaO2 calculated by DFT simulations were used to fit Ising Model coefficients for our system. Formation enthalpy data at each value of the Hubbard U was used to derive a set of corresponding interaction coefficients, though a least-squares regression fit. The sole differences among the structures lay in the relative geometric and magnetic arrangement of the oxygen dimers, and these differences are reflected in the nearest-neighbor interaction terms j. A total of 4 different types of j2i,k terms are identified, along with 6 different types of j3i,k,l terms. Our naming scheme relies on comparison of dimers, e.g. j2AAP P , corresponds to the interaction between 2 dimers of parallel geometry and identical 8


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magnetization, while j2ABP N , corresponds to the interaction between 2 dimers of orthogonal geometry and opposite magnetization. Likewise, j3AAAP P P , corresponds to the interaction among 3 dimers of mutually parallel orientation and mutually identical magnetic moment, while j3ABCP P N corresponds to the interaction among 3 oxygen dimers of mutually orthogonal orientation, where 2 dimers share an identical magnetization that differs from that of the third dimer. The plot of fitted interaction coefficients is shown in Fig. 2(b). The plot demonstrates that among the 2-body interaction terms, j2AAP N was most energetically stabilizing, while among the 3-body interaction terms, j3AABP P P was most energetically stabilizing. Our DFT calculations suggest that the most thermodynamically stable configurations correspond to bulk structures of intermediate disorder, both in geometric and magnetic arrangement (e.g. AABB-PPPN). Subsequent Ising Model calculations supported this observation, with j2AAP P the most-stabilizing 2-body interaction term, and j3AABP P P the most-stabilizing 3-body interaction term. It should be noted that using a purely 2-body Ising Model of NaO2 , we found that the most-stabilizing nearest-neighbor interaction term was j2ABP N , implying that the fully-disorganized antiferromagnetic structure (e.g. ABCD-PNPN) would be the predominant energetically stable structure. This was not consistent with our DFT calculations, and therefore showed the need to incorporate 3-body interaction terms into the model. Interaction coefficients j derived from the Ising Model were used in Metropolis-Hastings Monte Carlo (MHMC) simulations, to characterize larger bulk cells with higher degree of disorder, at finite temperature. Supercell structures were constructed as N×N×N arrays of cubic unit cells of Na4 O8 , the structures studied in DFT simulations. Simulations were run for supercells of size 4 to 7, to study convergence of bulk properties as the overall simulated system size increases. Periodic boundary conditions in all directions were implemented and the systems were studied using the Markov Chain Monte Carlo Method, modified with the Metropolis-Hastings Algorithm. 45 Interaction coefficients corresponding to a Hubbard U correction of 3 eV were used with the MHMC simulations, to most closely match the

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Figure 2: (a) Formation enthalpies of NaO2 bulk structures were calculated for 19 4-formulaunit unit cells, using the H2 O-NaCl correction scheme, at all examined values of Hubbard U. (b) Interaction terms j as function of U, which were used to calculate formation enthalpies, scaled as per interaction within the unit cell. The 3-body interactions play an important contribution to predicting behavior of NaO2 at intermediate and low temperatures. experimental formation enthalpy of NaO2 . The results would remain qualitatively consistent for other values of U, and there would be minor changes in the observed phase transition temperatures. At the initialization of each MHMC simulation, a fully-organized supercell bulk structure was set up, with all dimers initialized in identical geometric orientation and magnetization (corresponding to a whole supercell consisting of AAAA-PPPP unit cells). Each individual trial step consisted of switching a randomly selected O2 dimer in the full bulk superstructure to one of the other available configurations; an orientational or a magnetic change. The effect of this switch on total system energy was calculated. The standard Metropolis-Hastings Algorithm was used to decide whether to accept the trial step or not: every configuration change which lowered the net system energy was accepted, and configuration changes which

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raised the system energy were accepted or rejected dependent on the temperature of the simulation at that step. At the end of every step, system formation enthalpy and entropy were recorded for further analysis. Configurational entropy was calculated as the logarithm of the number of configurations with the same proportions of oxygen dimers of different types. If there are M distinct dimer types available, the total number of possible configurations, Ω, of the full structure can be calculated using the complete multinomial coefficient: M PM Y i=j Ni Ω= Nj j=1

(6)

where i, j refer to the available configurations of dimers and Ni , Nj refer to the number of dimers of a particular configuration present in the supercell. The configurational entropy is calculated using SConf = kB log Ω. To explore high-temperature, high-energy phases of bulk NaO2 , each system simulation was initially raised to a high temperature of 1252 K. Once the structure reached a stable series of configurations at the initialization temperature, the system was annealed 46 in temperature steps of 0.25 K, down to 2 K. At each 50 K increment, the system was held until stabilized, and formation enthalpy and configurational entropy data was collected for analysis from the stabilized structures. This scheme worked robustly and the evaluated thermodynamic quantities converged to the same values for all supercell sizes investigated. MHMC simulations were used to determine effect of temperature on the structural disorder, and in turn, the free energy of bulk NaO2 , as shown in Fig. 3(a) and Fig. 3(b). When the temperature is high, most of the structures could become accessible, as the entropic term dominates. Thus, the structure at high temperature was predominantly fully-disordered, both in geometry and in magnetic moment as shown in Fig. 4(a). Around 650 K, the structure underwent a phase change, wherein the bulk NaO2 formed large, parallel planes of alternating geometry of O2 dimers, as depicted in Fig. 4(b). Throughout the bulk, only

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2 possible orientations for the oxygen dimers, out of the original four orientations, were retained. The specific pair of retained orientations depended on the particular structure simulated at higher temperatures. This behavior agrees with the experimental behavior observed by Carter et al., who described the low-temperature structure of NaO2 as organized along only 2 of the 4 body diagonals. 28 In this phase, the magnetic moment was still largely disordered, within and across the monolithic-geometry planes. Around 350 K, the bulk structure underwent a second phase change, to a fully ferromagnetic structure, and the geometry maintained the alternating planar structure attained at intermediate temperatures. The degree of disorder in the system could strongly affect the electronic properties of NaO2 , so we explored this further using hybrid density functional theory calculations.

Figure 3: Results of Monte Carlo simulations using the Ising model for (a) the formation enthalpy and (b) configurational entropy of NaO2 throughout a wide range of temperatures. Three main temperature-dependent phases are identified: above 700 K, the bulk structure of NaO2 is fully disorganized, both in geometric and magnetic arrangement; between 400 and 550 K, the bulk is geometrically organized, but maintains general magnetic disorder; below 250 K, the bulk structure also gains full magnetic uniformity. Our simulations demonstrate that at room temperature, bulk NaO2 is expected to be 12


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Figure 4: Characteristic high-temperature (A) and low-temperature (B) structures of NaO2 as calculated through Monte Carlo simulations. The geometric orientation of each oxygen dimer in the bulk material is represented by the orientation of the corresponding element, as well as the color. The magnetic state of each oxygen dimer is represented by the shape of the corresponding element - cylinders represent net-spin-up dimers, rectangular prisms represent net-spin-down dimers. At high temperature, the bulk structure is fully disordered, both in the geometric and the magnetic arrangement. At low temperature, long-distance geometric ordering and complete magnetic ordering are observed, if both the two-body and three-body Ising Model coefficients are taken into account. magnetically disorganized, with long-range geometric ordering. As shown further in the paper, these phases are associated with higher bandgaps. However, if NaO2 growth on the cathode is externally controlled, for instance, by a substrate with specific magnetic ordering, it could be possible to induce an alternative magnetic phase in a thin-film deposit of NaO2 . 47,48 Through the choice of an appropriate electrode, it could be possible to induce growth of different metastable magnetic phases. HSE06 can accurately capture bandgaps for various semiconducting and insulating materials with mean absolute error (MAE) varying from 0.26−0.4 eV for different classes. 49–53 HSE+U calculations can provide an improved description of the electronic structure as well as crystal structure and energetics ithan hybrid functional calculations employing HSE functional without an additional Hubbard term. 54 Here, we perform non-self-consistent hybrid density functional theory calculations employing the HSE06 55 functional on the ground state 13



density obtained using RPBE+U of NaO2 . The aim of this work is to highlight the diverse covariance between structural/magnetic ordering and the electronic structure, at a very nominal loss of accuracy. Thus, HSE06 was applied non-self-consistently, due to the high computational cost of self-consistent application of the hybrid functional for a limited improvement in bandgap calculation accuracy. 50,53 This is also necessitated by the fact that without applying Hubbard correction, there are convergence issues for certain disorder phases of NaO2 . Bandgap values were calculated across all k-points and the minimum was reported for each phase of NaO2 for Hubbard U = 3-7 eV. We find that there is an increase in bandgap with increasing U as shown in Table 2 in the supplementary information. The widening of ∗ oxygen the bandgap with increasing U is due to enhanced localization of electrons in the π2p

bands in NaO2 . We also observe that the bandgap in both spin channels is identical for antiferromagnetic structures, and markedly different for ferromagnetic structures, as shown in Table 1. This behavior is observed as the additional electrons in O− 2 ions only occupy one spin channel. In the AFM structures, electrons are equally distributed in the spin chan∗ nels, and the gap is between pairs of π2p states. In the FM structures, the bandgap in the ∗ ∗ state and is thus higher than AFM; while and σ2p occupied spin channel is between π2p ∗ states and is thus lower than that in AFM. for the other channel, it is between deeper π2p

For mixed cases (e.g. PPPN), we expect the bandgap to be bounded by the AFM and FM ∗ cases, due to electron occupation of the π2p bands. With the exception of single-spin-channel

bandgaps around 1.3 eV, the bandgaps found in the conducted simulations were higher than those reported by Hartmann et al., 21 also carried out with the HSE functional, and reported bandgaps in the vicinity of 2 eV. One reason for the disparity between reported bandgaps is that Hartmann et al. reported a bandgap calculated for a Density of States averaged over all of the Densities of States calculated for the examined phases of NaO2 , whereas we report bandgaps calculated for individual phases independently. Our obtained results were closer to the values reported by Yang et al, who conducted their simulations with both the HSE functional and the G0 W0 functional. 13 For these functionals, the reported bandgaps were

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around 2 eV and 5.3 eV, respectively. We also note that an earlier HSE calculation, by Lee et al., reported a bandgap of 1.06 eV for a similar structure of NaO2 . 56 Table 1: Calculaed bandgap values (eV) for various ferromagnetic and antiferromagnetic configurations of NaO2 , for U=3 eV using HSE06 functional. Structure AAAA(PPPP) AAAB(PPPP) AABB(PPPP) AABC(PPPP) ABCD(PPPP) AAAA(PPNN) AAAB(PPNN) AABB(PPNN) AABB(PNPN) AABC(PPNN) ABCD(PPNN)

Spin Up 6.65 7.43 7.03 7.3 5.91 3.82 3.91 4.13 4.21 4.39 3.94

Spin Down 1.26 3.69 3.96 3.92 1.32 3.82 3.91 4.13 4.21 4.13 3.94

Conclusions In conclusion, we demonstrate the need and effectiveness of a 2-body and 3-body nearestneighbor Ising Model in describing energetics of bulk NaO2 . This model recovers the experimentally known long-range structural order of low-temperature NaO2 . Our study demonstrates that at low temperatures, close to room temperature and below, the predominant phase of NaO2 is ferromagnetic, with alternating planes of oxygen dimers in consistent geometric orientations, along 2 of 4 cubic cell body diagonals. The system is expected to exhibit some degree of magnetic disorder even at room temperature. Our study on the electronic properties shows that NaO2 is a wide-bandgap insulator, with a bandgap around 4 eV and we expect it to have poor electrical conductivity at room-temperature. In the context of Na-O2 batteries, this implies that growth of the discharge product is most likely occurring due to the solution mechanism pathway involving a chemical dissolution of NaO2 into Na+ and O− 2 , similar to what is seen in Li-O2 batteries. However, our analysis shows that prefer-

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ential nucleation of certain magnetic phases could be possible through appropriate electrode choice.

Supporting Information Available • A. Computational Parameters • B. Density of States • C. Formation Enthalpy Calculations • D. Ising Model • E. Metropolis Monte Carlo • F. Tables of Calculated Values

Acknowledgement The authors acknowledgment the partial support of this research work by the National Science Foundation award CBET-1604898.

References (1) Sapunkov, O.; Pande, V.; Khetan, A.; Choomwattana, C.; Viswanathan, V. Quantifying the promise of ‘beyond’ Li-ion batteries. Translational Materials Research 2015, 2, 045002. (2) Burke, C. M.; Pande, V.; Khetan, A.; Viswanathan, V.; McCloskey, B. D. Enhancing electrochemical intermediate solvation through electrolyte anion selection to increase nonaqueous Li-O2 battery capacity. Proc. Natl. Acad. Sci. USA 2015, 112, 9293–9298.

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The Role of Disorder in NaO2 and its Implications for Na-O2 Batteries

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