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High-Throughput Computational Approach to Li/Vacancy Configurations and Structural Evolution during Delithiation: The Case of LiMnO Surface 2

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Truong Vinh Truong Duy, Tsukuru Ohwaki, Tamio Ikeshoji, Yasushi Inoguchi, and Hideto Imai J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b12275 • Publication Date (Web): 16 Feb 2018 Downloaded from http://pubs.acs.org on February 18, 2018

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The Journal of Physical Chemistry

High-Throughput Computational Approach to Li/Vacancy Configurations and Structural Evolution during Delithiation: The Case of Li2MnO3 Surface Truong Vinh Truong Duy,∗,† Tsukuru Ohwaki,† Tamio Ikeshoji,† Yasushi Inoguchi,‡ and Hideto Imai† †Device Analysis Department, Nissan ARC, Ltd., 1 Natsushima-cho, Yokosuka, Kanagawa 237-0061, Japan ‡Research Center for Advanced Computing Infrastructure, Japan Advanced Institute of Science and Technology (JAIST), Asahidai 1-1, Nomi, Ishikawa 923-1211, Japan E-mail: [email protected] Phone: +81 46 867 5154. Fax: +81 46 866 5814

Abstract High-throughput prediction of chemical and physical properties of materials is a promising approach to the discovery and development of novel materials. We perform high-throughput first-principles calculations of more than 65,000 Li/vacancy configurations at different concentrations in the surface region of Li2 MnO3 to exhaustively investigate its structural evolution during the delithiation process. Through this computation, we establish a complete picture of all stable and unstable configurations of the material that is difficult to obtain with non-exhaustive calculations. Given the picture,

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we identify the free energy and the ground state energy together with their electrochemical potential by considering the energy levels of all configurations. Examination into the vacancy population and vacancy site map shows that the Li atoms are extracted from the surface region at first, and then from the bulk during the process. The Li extraction is found to destabilize the O atoms near the surface and help to generate surface oxygen loss from an early stage of charging that likely causes the material’s irreversible capacity loss. The formation of vacancies enables the migration of the Mn atoms in the surface that initiates the phase transformation to the spinel-like structure. These findings are also supported from the local structures made by vacancies. Oxygen atoms are less stable at highly coordinated state by vacancies either on the surface or in the bulk, and Mn atoms in the surface when surrounded by many vacancies will migrate more easily than those in the bulk. Furthermore, we analyze the Boltzmann distribution of the energy to unravel the contribution of each configuration, and suggest a simple, yet efficient approach based on energy sampling rate that is demonstrated to deliver satisfactory results with about 10% configurations for this class of structural search applications.

Introduction Surfaces and interfaces of a material define the interactions of the material with its surrounding environment, and consequently characterize the properties and functionalities of a wide range of materials from catalysis to semiconductor and biomedical devices. 1–4 Their crucial role is especially obvious in the field of electrochemistry and its applications in lithium-ion batteries (LIBs) with complex surface reactions among electrochemical components and constantly evolving surfaces due to charge-discharge cycling. 5,6 In fact, successful attempts to address long-standing problems with capacity fading and impedance buildup for improving the performance of LIBs including commercial ones were focused on re-engineering the surfaces. 7–9 Recent applications of LIBs in electric vehicles and large-scale energy storages have

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driven the demand for increasing the energy density of the batteries using materials with high capacity and high voltage but also inherent long-term instability that further require deep understanding of surfaces. One class of such materials is the layered lithium-rich oxide xLi2 MnO3 · (1 − x)LiMO2 (M = Mn, Ni, Co, etc., 0 < x < 1), 10–14 considered one of the most promising candidates for cathode materials. They exhibit very high capacities and operating voltages, but suffer from a large irreversible capacity loss that has been plaguing their commercialization and is still an open research problem. 15 Surface deficiency such as surface oxygen loss has been suggested as a leading culprit for the irreversible capacity loss of the materials. 16–21 For understanding the problem, the common core component Li2 MnO3 in the two-component structures is a spontaneous research target due to its essential role in the reversible capacity of the compounds. Although research on the bulk Li2 MnO3 is abundant, 22–27 there exist only a few works on the surface of Li2 MnO3 . Recently, a number of different crystal facets were constructed to investigate their surface stability against oxygen loss by first-principles calculations. 28 Kim et al. 21 assumed a sequence of Li extraction from the surface to the bulk of Li2 MnO3 during the delithiation process to examine the phase transformation from a layered structure to a spinel-like structure. 29 However, despite all these efforts, the underlying mechanism causing the irreversible capacity loss problem and the phase transformation to the spinel-like structure of the material is still unclear. Also, whether the electrochemical reaction of Li2 MnO3 is actually initiated from the surface or not is still unknown, as the mere assumption on the sequence of Li extraction 21 does not assure that the assumed structure correlated to each level of Li concentration under examination is in the ground state. In fact, this assumption is somehow understandable, as an exhaustive study of the sequence of Li extraction in the surface would require thousands of structures to be screened and investigated, making it virtually impossible without an appropriate approach. On this front, traditional approaches to constructing the phase diagram of a system include combining first-principles calculations and statistical-mechanics methods, for in-

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stance the well-known Monte Carlo methods or cluster variation method for obtaining finitetemperature thermodynamic properties, and employing the cluster expansion method for extrapolating the energy of an arbitrary configuration from a set of reference configurations. 30,31 In the past few years, high-throughput computational materials design has emerged as a powerful approach to solving this class of structural search problem by taking advantage of the computational power of supercomputers. 32 Examples for LIB research include, 33 where 515 stable lithiation reactions of some selected anode materials were enumerated based on a calculated energy data set of 291 compounds. Recently, the relationship between voltage and safety of LIBs was investigated using 1,400 cathode materials in an identical high-throughput fashion. 34 In this work, we perform an exhaustive investigation of more than 65,000 different configurations corresponding to the Li extraction in the surface region of Li2 MnO3 during the delithiation process through high-throughput first-principles calculations to yield the full picture of all stable and unstable Li vacancy configurations for elucidating the mechanism causing the irreversible capacity loss and the phase transformation of the material. Based on the picture, we identify the free energy and the ground state energy as well as their redox potential, examine the Li vacancy population and vacancy site map, establish the sequence of Li extraction, link the Li extraction with the stability of O atoms during the process, and clarify the contribution of each configuration to the energy. Local structures made by vacancies are also analyzed by the coordination number of vacancy around the atoms and vacancies themselves. The study is anticipated to provide comprehensive atomic-level insights into the surface configuration of the material to assist future design of high-energy batteries with better surface stability and ultimately, improved overall stability and quality.

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Computational methods Automation tool for high-throughput DFT calculations With the aim of enabling the exploration of large-scale structural search space, for instance to search for the most stable structures/configurations in thousands of candidate structures, or to find the structures corresponding with the experimental spectra, we have developed an automation software tool for high-throughput density functional theory (DFT) 35 calculations. The tool can effectively and reliably search for all configurations in a given search space, handle concurrent task executions, task and data management, optimize calculation parameters, and perform principle data analytics. Figure 1 illustrates the workflow and key features of our tool. Starting from a calculation model, we first define the specific structural search problem, and then determine the parameters for the search conforming to the problem, comprising the parameters for the search space and the parameters for the DFT calculations. The former involve determining the size of the search space that must be explored, while the latter are related to the accuracy of the calculations, including the choice of the basis functions, pseudopotentials, exchange-correlation functionals, initial spin states, energy cutoffs, k-mesh, etc. Next, we conduct the modeling with the structural search in the given space and mass-generation of the candidate structures found by the search. Once all the candidate structures have been created, we carry out the mass-execution of the calculations on parallel machines, which can be total energy or spectral calculations with respect to the problem, collection of data, and interpretation of the calculation results. Finally, based on the results, the structures can be clarified and output. The tool is equipped with three core features. The first one is the DFT engine; currently we employ the DFT code OpenMX 36 as the engine, though the tool can be extended to accommodate other codes with ease. Our automation tool is built on top of OpenMX as a wrapper of the code and can spawn many OpenMX instances/processes in parallel. In order to handle multiple concurrent OpenMX processes, our tool interacts directly with the

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？ Problem to be solved

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Modeling and mass-generation of candidate structures Determination of structural search parameters

Output

Mass-execution of calculations (Energy, spectral, etc.)

• The most stable structure in thousands of candidate structures? • The structure reflecting the experimental spectra?

Clarification of structure

Collection of data and interpretation of results

Features Difficult-toconverge structures

Search space (SP)

SP1

OpenMX: Open source package for Material eXplorer

SP2

SP3

Automatic optimization Optimization of the number of SCF Optimization of Hubbard U

Machine A

Machine B

Machine C

Optimization of mixing methods

Collection of data

DFT engine: OpenMX

Parallel search of space

Automatic optimization of calculation parameters

Figure 1: Workflow and core features of our automation tool for high-throughput DFT calculations.

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job scheduler of the parallel computers via its commands to submit jobs, check job status, delete jobs, and so on. Another feature is the parallel space search on different computing platforms, where the original large space can be divided into several smaller subspaces on a couple of machines, which are then searched in parallel on each machine for fully utilizing their computational resources and accelerating the search operation. Lastly, we have also implemented the automatic optimization of the calculation parameters for dealing with the convergence problem in addition to the automatic execution. The automation of parameter optimization is undoubtedly needed, as manual optimization is impossible with thousands of structures and calculations. Moreover, this feature is particularly useful for surface and interface calculations that are known to be difficult to converge. It automatically conducts several optimizations step by step, from adjustment of the number of self-consistent field iterations (SCFs) and incremental setting of the Hubbard U value to mixed selection of mixing methods, to steadily make the calculations converge.

Computational details In this work, we employ the surface slab calculation model of Li2 MnO3 (010), with the equivalent cell of Li16 Mn8 O24 (Fig. 2a) and 16 Li vacancy sites (Fig. 2b) considering the exponential computational cost to the number of Li atoms as explained below. In accordance with the workflow in the previous section, we first cast the structural search problem as: “For x in Li2−x MnO3 , where 0 6 x 6 2 indicating that up to all of 16 Li atoms are subjected to removal, all possible placements of the remaining Li atoms in the lattice are considered, and the total energies of the resulting Li/vacancy structures are calculated to identify the full picture of stable and unstable Li/vacancy configurations for a thorough analysis on the free energy and the redox potential, Li vacancy population and vacancy site map, sequence of Li extraction, stability of O atoms during the delithiation process, and contribution of each configuration to the energy”. In other words, differently from previous works, we do not enforce any assumption on the sequence of Li extraction during the delithiation process, 7



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O(1)

16 15 14 12 11 10 8 7 6 4 3

13 9 5

1

(a) Surface slab calculation model. Li, Mn, and O atoms are shown in purple, green, and red spheres, respectively.

2

(b) Li vacancy site. Mn and O atoms are not shown for clarity.

O(1) O(2)

O(bulk)

O(2) O(1)

(c) Grouping of O atoms.

Figure 2: Calculation model and definition of vacancy and atomic sites. but exhaustively search for all possible structures to find out the stable and unstable ones. We then need to determine the parameters for the search space and the parameters for the DFT calculations. For the former, exploring all possible values of x needs 216 = 65, 536 structures to be calculated and examined, as depicted in Fig. 3. The calculation model has 16 Li atoms, with x increased from 0 to 2, accordingly from 0 to 16 Li atoms will be removed from the structure. When x is equal to 0, we have 0 atom removed, or 16 atoms in 1 structure; this is the starting structure, as a result of the mathematical combination C016 = 1. When x is increased to 0.125, 1 atom is eliminated from the structure; the eliminated atom can be any one in 16 atoms, and the number of structures in this case is the combination C116 = 16. Similarly, when x = 0.25, there are 2 atoms extracted with the extracted atoms being any two in 16 atoms, and the number of structures is C216 = 120. The number of structures keeps increasing to x and reaches a peak of C816 = 12, 870 at x = 1. Since then, it starts decreasing until all 16 Li atoms are extracted. In total, the size of the search space is P 16 16 the summation 16 = 65, 536 structures that are different from each other. We k=0 Ck = 2 8


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…

x=0

C016 = 1 structure

x = 0.125

C116 = 16 structures

……

x = 0.25

x = 1.0


Total search space:

16

C k 0

C816 = 12870 structures 16 k

…

……

x = 1.875


x=2

C1616 = 1 structure

 216  65,536 structures

Figure 3: Structural search space. Li atoms and vacancies are shown in purple and gray spheres, respectively. Mn and O atoms are not shown for clarity. note that the system does not exhibit any symmetrical property that can be exploited for reducing the number of configurations, as confirmed by the derivative structure enumeration library enumlib 37 and site-occupation disorder SOD code. 38 Regarding the latter, the calculations are performed with the spin-polarized GGA+U scheme for accounting for the strong correlation effect of transition metals, 39–41 norm-conserving pseudopotentials, 42 and pseudo-atomic localized basis functions. 43 The basis functions of Li8.0-s3p1, Mn8.0-s3p3d3, and O7.0-s2p2d1 are adopted for Li, Mn, and O, respectively, and the empty atom scheme is employed for the vacancy state. 43 Specifically, we use a cutoff radius of 8.0 a.u with three s-orbitals and one p-orbital for Li, a cutoff radius of 8.0 a.u with three s-orbitals, three p-orbitals, and three d-orbitals for Mn, and a cutoff radius of 7.0 a.u with two s-orbitals, two p-orbitals, and one d-orbital for O. 43 Among the three dorbitals used for Mn, the Hubbard U value of the first orbital, as defined by OpenMX in 9



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the context of pseudo-atomic localized basis functions, is set at 5 eV, as referred to previous studies. 22,25,44 Beside the total energy, the atomic energy of the oxygen atoms is also calculated by the orbitally decomposed total energy method. 36 Other important parameters include the charge mixing method RMM-DIIS for Hamiltonian matrix, 45 an energy cutoff for electron density of 4081.71 eV, and a k-point mesh of 3x3x1. During the automatic optimization of the calculation parameters, alternative charge mixing methods such as GR-Pulay 46 or Kerker 47 may be attempted, and the initial Hubbard U value may not be 5 eV but 1 eV, which is then gradually incremented to 5 eV for ease of convergence. It is worth noting that ultimately all calculation results are obtained under identical calculation conditions and accuracy. Also, the same cell parameters are applied to all configurations and not relaxed due to high computational cost. Once the parameters have been determined, the modeling and mass-generation of 65,536 candidate structures are performed, followed by their mass-execution on parallel machines. As mentioned previously, automatic optimization is applied to difficult-to-converge structures. In order to speed up the search and calculation processes, we concurrently make use of several machines, including the K computer at RIKEN AICS 48 and the Cray XC30 at JAIST. 49

Results and discussion Formation energy Figure 4 shows the calculation results in terms of the formation energy to the vacancy concentration x during the delithiation process. The formation energy Eif of configuration i with vacancy content x is defined from its total energy Ei and the energies of the two ending structures of the range at x = 0, ELi2 MnO3 , and x = 2, EMnO3 , as x x Eif = Ei − (1 − )ELi2 MnO3 − EMnO3 . 2 2 10


(1)

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All energies in capital letters are per Li + vacancy site, i.e. per a half of formula unit, here and hereafter. The formation energies from all different vacancy configurations are represented by the red circles. In total, there are 65,506 structures calculated successfully out of 65,536 ones with an SCF convergence rate of 99.95%, demonstrating the capability of our tool for realizing such a large-scale structural search. The ground state structures f for different Li compositions are identified by the smallest energies, Emin , located at the

bottom of the vertical lines, as highlighted by the purple line. The green squares are used to show the ground state energies that are on a convex hull and Af is the Helmholtz free energy which are explained in the next section. The structures at the ground state and those nearly at formation energy E f equal to 0, referred to as zero-formation-energy structures hereafter, will be analyzed and compared in subsequent sections for clarifying the sequence of Li extraction and the stability of O atoms.

Redox potentials from free energy and ground state in delithiation process Given the advantage of having the complete picture of all configurations, in this section we calculate two redox potentials, one from the free energy and the other from the ground state energy. We first identify the free energy, taking into account the effects of entropy and temperature. According to the Boltzmann distribution, 50–52 the probability of a state, or −k

structure/configuration i, with energy ei is proportional to e

ei BT

, where T is the tempera-

ture, kB is the Boltzmann constant, and the energy in small letters is the total energy of the system having n Li sites (n = 16 in this work). Degeneracy of each state is assumed to be 1. In order to calculate the delithiation potential, energy EiLi is defined against the Li metal energy ELi as EiLi = Ei − (2 − x)ELi − EMnO3 .

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(2)


0.6 0.5 0.4

Efi

Efmin Af Efmin on convex hull

0.3 Energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.2 0.1 0.0 -0.1 -0.2 0.00

0.25

0.50

0.75

1.00 1.25 x in Li2-xMnO3

1.50

1.75

2.00

Figure 4: Formation energy Eif for all vacancy configurations calculated from Eq. (1) and f Helmholtz free energy of formation Af . Emin is the ground state energy of formation at x. f The green squares are Emin assumed to be on a convex hull as explained in the text.

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The average energy, i.e. internal energy, of the delithiation, U Li (x), at a vacancy content x is given by eLi

U Li (x) =

P

i Li − kB T x ei e

1 n P

x

eLi

,

(3)

−k iT

e

B

Li where eLi i = nEi . The summation is over all configurations whose vacancy content is equal to

x. The denominator of this equation is the partition function, Z(x) at x, and the Helmholtz free energy A(x) is calculated as − n1 kB T log Z(x), 50,51,53,54 i.e. Li

X − ei 1 A(x) = − kB T log e kB T . n x

(4)

Entropy S Li (x) is calculated from the thermodynamic relation of A = U − T S 52 or temperature integration of U as 1 S (x) = U Li (x) − ALi (x) = T Li

Z

T

dU Li (x)/T.

(5)

0

The formal delithiation potential, V (x), at x correlated with the redox potential from x to x + dx can be consequently evaluated as the derivatives of the free energy as

V (x) = −

dA(x) . dx

(6)

Contribution from the entropy is VST =

T dS(x) . dx

(7)

On the other hand, we can also derive the delithiation potential correlated with the redox potential from an amount x1 to x2 as 21,25

VGS (x1 ≤ x ≤ x2 ) = −

ELi2−x1 MnO3 − ELi2−x2 MnO3 − (x2 − x1 )ELi , x2 − x1

(8)

where E refers to the calculated energies of the ground states on a convex hull connecting

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6 V VGS 5 VST

4 Potential (V)

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3

2

1

0

-1 0.00

0.25

0.50

0.75

1.00 1.25 x in Li2-xMnO3

1.50

1.75

2.00

Figure 5: Delithiation potentials V (Eq. (6)) from the free energy and VGS (Eq. (8)) from the ground states on the convex hull, and the contribution from the entropy VST (Eq. (7)).

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the selected minimum energies smoothly. The delithiation potentials derived from the thermodynamic free energy and ground states on the convex hull are presented in Fig. 5, in conjunction with the entropy contribution. We expected that the potential V derived from the free energy to be as smooth as that from the ground state VGS , since the free energy requires all configurations and is influenced by the energies of many states. However, this is not the case because the difference between the free energies at different x is much scattered than kB T (approximately 26 mV at 300 K). A larger cell may be needed to calculate the vacancy energy, particularly in the x- and y-directions. The number of the Li sites, n = 16, in this work seems to be large enough and the potential obtained with this size is believed to closely resemble that with n = ∞ in the ideal solid solution, as the entropy difference between an infinite cell and a finite one is anticipated to be relatively small. 50,55 Nevertheless, the initial gradual potential growth of VGS at low charging state and the successive long plateau above 4.5 V vs. Li/Li+ at charging state are in agreement with the experimental measurements. 21

Vacancy population and vacancy site map Next, we analyze the Li vacancy population along the z-direction and the Li vacancy site map of all configurations to find out stable vacancy site configurations. Figure 6 presents the vacancy population of different vacancy content x as viewed from the z-direction of the calculation model. It is apparent that the population of the first layer at the surface is very high, whereas the population of the bulk is almost identical for all the numbers of vacancies. This observation perhaps points out that Li tends to extract from the surface and makes the surface an active reaction area. It should be noted that, however, the observation may be affected by the cell’s horizontal size, which is quite small in this calculation. The vacancy sites of 1, 2, and 16 (Fig. 2b) are preferable sites for the lower energy states. Because the vacancy sites of 1, 2, and 16 are the outermost sites from the surface, the configurations with vacant surface sites appear to be electrochemically preferable. 15



1.0

0.8 Number of vacancies Vacancy population

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.6 0.250 0.875 1.250 1.500

0.4

0.2

0.0 0

2

4

6

8 z (Å)

10

12

14

16

Figure 6: Vacancy population viewed from the z-direction for various vacancy content x. The dashed lines approximate the data with a smoothing spline for guiding the eyes.

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Sequence of Li extraction Assisted by the analysis on the vacancy population and vacancy site map, here we establish the sequence of Li extraction during the delithiation process utilizing the ground state structures with respect to the vacancy content in Fig. 4. Figure 7a displays the ground state sequence of Li extraction with increasing vacancy content x, where the Li atoms are shown to be extracted one after another from left to right. It can be seen that from the original structure where x = 0, first one atom at the surface is removed where x = 0.125, then another atom at the surface is extracted where x = 0.25, followed by the elimination of another atom in the surface region where x = 0.375, too. The whole delithiation process confirms that the atoms are apparently extracted in the surface-to-bulk order. For comparison, the sequence of Li extraction with the structures at zero-formation-energy is also given in Fig. 7b. In contrast to the ground state case where the removal of Li atoms is initiated from the surface, the atoms in the surface region seem to resist to the elimination, and there is no clear difference in the sequence of extraction between the atoms in the surface region and those in the bulk region. The consequence again affirms the origination of the material’s reaction from the surface and that this reaction is electrochemically favorable. Therefore, such states do not contribute to the electrochemical potential as shown later.

Stability of O The aforementioned surface oxygen loss has been proposed as a main cause of the poor initial performance of the material. 16–21 In this section, we look into the stability of O during the delithiation process by way of calculating the atomic energy of individual O atoms with the orbitally decomposed total energy method 36 for several states near/at the ground state and zero-formation-energy structures. We then divide the O atoms into three different groups in line with their positions from the surface and average the atomic energies of the atoms in the same group. Figure 2c depicts three different groups of O atoms: O(1), O(2), and O(bulk) as the outermost O atoms in the first layer from the surface, the second outermost 17



0

0.25

0.5

0.75

1

1.25

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1.5

1.75

2

1.5

1.75

2

x

(a) Ground state.

0

0.25

0.5

0.75

1

1.25

x

(b) Zero-formation-energy state.

Figure 7: Sequence of Li extraction to x in Li2−x MnO3 . Li atoms and vacancies are shown in purple and gray spheres, respectively. Mn and O atoms are not shown for clarity.

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O atoms in the second layer from the surface, and the remaining O atoms inside the bulk region, respectively. The average atomic energy of the O atoms for each group with regard to the vacancy content is given in Fig. 8a for the ground state, and Fig. 8b for the zero-formation-energy state. It is obvious that in both cases, the atomic energies of the O atoms keep increasing according to the Li extraction, indicating that they become more and more electrochemically unstable during the delithiation process. Also, the O atoms in the surface region are generally more unstable than those in the bulk owing to their higher energies. Furthermore, in the ground state case, the atomic energies of the O atoms near the surface surge sharply even at low charging state when 0.0 ≤ x ≤ 0.5. Actually, it is during this low charging period that the Li atoms near the surface are also being extracted. The observation agrees with the experimentally known generation of oxygen molecules above a certain potential. 56,57 In contrast, this behavior is not clearly observed in the zero-formation-energy case, implying that there is almost no difference between the surface region and the bulk region in this electrochemically unfavorable state. As a consequence, the electrochemical instability of the O atoms near the surface is highly likely caused by the vacancies localized in the surface region due to the removal of Li atoms and degrades the performance of the material.

Local structure made by vacancies From the Boltzmann distribution, it is possible to find the equilibrium local structure at 300 K in the same way as the average energy in Eq. (3). Figure 9 shows the coordination number by nearest neighbor vacancies, Ncv (V) and Ncv (O), around vacancies themselves and oxygen atoms, respectively, at vacancy content x. In the bulk region, Ncv (V) at 300 K is lower than that with all probabilities equal to 1, i.e. at T = ∞, as shown in Fig. 9a. Vacancies tend to avoid making vacancy-vacancy pairs in the bulk, even though they make higher density at the surface, as already revealed by Fig. 6. In the surface, Ncv (V) at 300 K is higher than that at T = ∞. On the other hand, although Ncv (O) in Fig. 9b is slightly larger than 19



-442

Average atomic energy (eV)

-444

-446

-448

-450

-452 0.00

O(1) O(2) O(bulk) 0.25

0.50

0.75

1.00 1.25 x in Li2-xMnO3

1.50

1.75

2.00

(a) Ground state. -442

-444 Average atomic energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 35

-446

-448

-450

-452 0.00

O(1) O(2) O(bulk) 0.25

0.50

0.75

1.00 1.25 x in Li2-xMnO3

1.50

1.75

(b) Zero-formation-energy state.

Figure 8: Average atomic energies of oxygen atoms.

20


2.00

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the bulk because of the accumulation of vacancies in the surface, there is no large difference between the surface and bulk regions. We also analyze these features from the atomic energy point of view. Oxygen and manganese atomic energies obtained for several structures near/at the ground state energy and zero-formation-energy are plotted as a function of the vacancy coordination number in Fig. 10. Oxygen atoms surrounded by more vacancies are less stable, and oxygen does not like to be with the vacancies. Lithium has the opposite tendency, making Li-vacancy pair more stable. Mn atomic energy does not change by the coordination number so significantly as can be seen in Fig. 10b. Nevertheless, the atomic energy of the Mn atoms in the surface is notably higher than that of those in the bulk, indicating the thermodynamic instability of the Mn atoms in the surface when surrounded by many vacancies. Further investigation into the surface Mn atoms shows that the Mn atoms in the outermost layer from the surface have a higher energy ((−104.77, −104.74) eV in Fig. 10b) than those in the second- and third-outermost layers, and will migrate more easily as a consequence. This fact therefore strongly suggests that the phase transformation from the layered structure to the spinel-like structure starts from the surface and is initiated by the migration of the unstable Mn atoms in the surface.

Contribution of each configuration to energy As discussed above, the free energy at a specific vacancy concentration is determined based on the probability of each energy state/configuration following the Boltzmann distribution. As some states may have higher probability and contribute to the free energy more than others, the contribution of each state should be clarified to give some statistically useful information that may help scale down the scope of the structural search problem, and thus, accelerating the search process. Figure 11a summarizes the energy and the corresponding contribution of each configuration of all 65,506 configurations (left), arranged in the order for increasing x and increasing 21



10

In the bulk at T = ∞ In the bulk at T = 300 K In the surface at T = ∞ In the surface at T = 300 K

8

Ncv(V)

6

4

2

0 0.00

0.25

0.50

0.75

1.00 1.25 x in Li2-xMnO3

1.50

1.75

2.00

(a) Vacancy coordination number around vacancies, Ncv (V). 10

In the bulk at T = ∞ In the bulk at T = 300 K In the surface at T = ∞ In the surface at T = 300 K

8

6 Ncv(O)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 35

4

2

0 0.00

0.25

0.50

0.75

1.00 1.25 x in Li2-xMnO3

1.50

1.75

2.00

(b) Vacancy coordination number around oxygen atoms, Ncv (O).

Figure 9: Average vacancy coordination numbers around vacancies and oxygen atoms as a function of vacancy content x. 22


Page 23 of 35

-16.15 -16.20

O atomic energy (eV)

-16.25 -16.30 -16.35 -16.40 -16.45 -16.50 -16.55 in the bulk in the surface

-16.60 0

1

2 Ncv(O)

3

4

(a) Oxygen atomic energies in the bulk and surface. -104.65

in the bulk in the surface

-104.70

Mn atomic energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-104.75

-104.80

-104.85

-104.90

-104.95 0

2

4

6

8

10

Ncv(Mn)

(b) Manganese atomic energies in the bulk and surface.

Figure 10: Average atomic energies as a function of vacancy coordination numbers, Ncv .

23



energy with the same x for consolidating meaningful contributions to the left side of x, and shows configurations with higher probability for various vacancy content x (right). The figure reveals some certain configurations give much more meaningful contribution to the free energy, and the resultant output delithiation potential, than others. This suggests that virtually similar results may also be obtained by carefully choosing and calculating a much smaller set of configurations including the minimum energy states out of the exhaustive 65,536 ones. The problem is how to choose stable configurations for calculation and unstable configurations for approximation. One obvious approach is to create a small number of configurations for performing calculations, analyze the results to relate the energetic change to the structural change, and then shift the calculation to the configurations that appear to be stable. For example in the surface region of this material, the configurations with vacant surface sites should be stable and selected for calculation, while other configurations can be approximated from a few representative calculations, because previous sections have unraveled that Li is extracted from the surface to bulk and Li vacancy sites at the surface are preferable. However, in practice finding out the relationship between the energetic and structural changes is not a trivial task without a full picture of all configurations. We therefore suggest another much simpler approach based on random selection of configurations. In our approach, the configurations for each vacancy content are randomly chosen for calculation with respect to an energy sampling rate. Figure 11b demonstrates the effect of the energy sampling rate on the delithiation potential. As can be seen from the figure, the higher the sampling rate is, the finer the potential is. Apparently, the potential with a sampling rate of 100%, where all 65,506 configurations are taken into account, is the potential derived from the free energy in Fig. 5. A sampling rate of 10%, with 6,550 configurations randomly selected, can reproduce the potential much better than a sampling rate of 1% with 655 configurations. We also notice that around the region of x = 1 with the number of configurations exceeding 4,000, a sampling rate of 10% proves to be enough, whereas around the regions of

24


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Page 25 of 35

0.0

1.0

ELi i

x=0.125 x=0.625 x=1.000 x=1.125 x=1.375 x=1.750 x=1.875

ELi min

-0.5 -1.0

0.8

-2.0

Probability

Energy (eV)

-1.5

-2.5 -3.0

0.6

0.4

-3.5 -4.0

0.2

-4.5 -5.0

0.0 0

10000

20000

30000 40000 Configuration

50000

60000

70000

0

5

10 15 20 Configuration with higher probability

25

30

(a) Contribution of each configuration energy, EiLi , to delithiation potential V in Eq. (6) (left) and configuration with higher probability for various vacancy content x (right). 6

25000 V (100%; 65,506 configurations) V (10%; 6,550 configurations) V (1%; 655 configurations) Number of configurations 20000

4

15000

3

10000

2

5000

V (V)

5

1 0.00

0.25

0.50

0.75

1.00 1.25 x in Li2-xMnO3

1.50

1.75

Number of configurations

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 2.00

(b) Delithiation potential obtained with different energy sampling rate and the sampling number at x for 100% sampling.

Figure 11: Contribution of each configuration to delithiation potential and effect of sampling rate.

25



x = 0 and x = 2 with a smaller number of configurations, a sampling rate of nearly 100% is required for producing satisfactory potentials. The number of sampling configurations would become only around 12% of the total number of configurations. The outcome demonstrates that our approach is efficient, yet simple for narrowing the search scope to speed up the search process. It is worth mentioning that our approach is not limited to this material but general and can be applied to this class of structural search applications. In this regard, a sample size of 10,000 configurations is anticipated to be sufficient to deliver gratifying results, even for applications having more than 65,536 configurations, especially when the higher probability states, as demonstrated by Fig. 11a, are included in the samples.

Conclusion We have reported a high-throughput DFT investigation of Li vacancy configurations in the surface region of Li2 MnO3 during the delithiation process. In doing so, we have established a full picture of all stable and unstable structures of the material, which is not easy to be established with non-exhaustive calculations, to unravel the mechanism causing its irreversible capacity loss and phase transformation. We have determined not only the minimum energy structures but also the free energy along with their electrochemical potential. We have also examined the vacancy population, vacancy site map, and coordination numbers to clarify the sequence of Li extraction, where the Li atoms are extracted from the surface region to the bulk region. The vacancies caused by the Li removal have been identified to destabilize the O atoms near the surface and help to generate surface oxygen loss during the process. The formation of vacancies has shown to trigger the phase transformation to the spinel-like structure by enabling the migration of the Mn atoms in the surface. The oxygen atomic energy increases with increasing coordination by vacancies both in the surface and in the bulk, and the Mn atoms in the surface when surrounded by many vacancies are more unstable than those in the bulk.

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Finally, we have proposed an approach for accelerating the structural search process that is able to deliver satisfactory results with 10,000 configurations by analyzing the contribution of each configuration in the Boltzmann distribution of the energy. The study may provide hints for rational designs of future high-energy batteries with improved stability and quality by efficient surface controlling of oxygen retention. In addition, this exhaustive examination offers a substantial database of more than 65,000 different structures that can be utilized in the emerging field of materials informatics for data mining and big data analytics.

Acknowledgement The calculations in this work were partly performed using the computational resources of the K computer provided by the RIKEN Advanced Institute for Computational Science through the HPCI System Research project (Project ID: hp150187) and CDMSI (Post-K-Computer Priority issues 7 - Creation of new functional Devices and high-performance Materials to Support next-generation Industries). We are also grateful to Dr. Itoh of Nissan ARC for fruitful discussions.

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Graphical TOC Entry

Li

Formation energy (eV)

Mn

High-throughput DFT

High-throughput DFT

Li2MnO3 surface

1.0


15000

0.8

12500

0.6

10000

0.4

7500

0.2

5000

0.0

2500

-0.2 0.0

0.5

1.0 1.5 x in Li2-xMnO3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 2.0

Complete picture of 65,506 configurations

35


Vacancy

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