Phys. Status Solidi B 248, No. 9, 2027–2031 (2011) / DOI 10.1002/pssb.201147085 p s sb

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basic solid state physics

Rational designs ofcrystal solid-solution materialsfor lithium-ion batteries

Ting Luo1, Caixia Zhang2, Zhiyong Zhang3, Yingdan Zhu2, and Jun Li*,2

1College of Science and Technology, Ningbo University, Ningbo 315211, P.R. China2Ningbo Institute of Material Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, P.R. China3Stanford Nanofabrication Facility, Stanford University, 420 Via Palou Mall, Stanford, California 94305, USA

Received 21 February 2011, revised 23 June 2011, accepted 23 June 2011

Published online 22 July 2011

Keywords lithium-ion batteries, quantum modelling, rational design, solid solutions

*Corresponding author: e-mail lijun@nimte.ac.cn, Phone: þ86 057486688074, Fax: þ86 057486685043

We present a new strategy for computer-aided rational design

of quantum supercells of crystal solid-solution (SS) materials

for lithium-ion batteries. Layered SS materials Li(Nix-CoyMn1�x�y)O2 are used to demonstrate our design strategy.

Crystal structures obtained by rational design agree well with

those fully optimized by quantum total-energy calculations and

real materials measured from experiments. Compared with

conventional first-principles determination of structures, our

approach achieves a significant increase in efficiency without

loss of reliability and predictability for searching new

structures. We further discuss the necessary steps to develop a

generic strategy to rationally design new composite electrode

materials.

� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Advanced batteries substantiallyimpact the areas of energy storage, energy efficiency, hybridand plug-in electric vehicles, power tools, laptops, cellphones and many other mobile electronic and entertainmentdevices. Rechargeable lithium-ion batteries offer the highestenergy density of any battery technology and, therefore, arean attractive long-term technology that now sustainsa billion-dollar business. At the materials level, over the last30 years the major improvement in the performance oflithium batteries has been achieved through the discovery ofnew lithium cathode materials.

LiCoO2 is currently the most active cathode materialused in lithium-ion batteries since its discovery in the early1990s. However, the safety and high cost of cobaltsignificantly limits its application to the emerging high-capacity and high-power battery markets. Recent efforts inboth industrial and academic attempts to overcome theselimitations have been focused on compositional modifi-cation of LiCoO2, mainly by infusion with other transition-metal elements to form multicomponent layered cathodes[1–5], or new architectures for advanced compositematerials for cathodes [6, 7]. There has been a similarinterest in the development of an advanced anode using

alloyedmaterials since the commercialization of the graphiteanode accompanying the LiCoO2 cathode in the 1990s [8].Alloyed materials for an advanced anode and compositematerials for an advanced cathode are the mainstreamapproach for next-generation Li-ion battery technology.

Searching for new materials by empirical experimentalefforts is time consuming and expensive. Significant effortsare currently being made to use quantum modelling andsimulation on high-performance computers to accelerate thesearch for new and better materials for the battery industry.The goals of these efforts are: (1) to reduce the costs of theresearch and development of a product; (2) to accelerate thetime cycle for new material in a product from laboratory tomarket; and (3) to increase the scope of systematicimprovement in material designs [9–15]. Quantum simu-lations (QS), based on the first-principles density functionaltheory (DFT) or its equivalent, provide reliable computer-aided design to improve the application properties ofcurrently known battery materials, including cathode, anodeand electrolyte, on atomic-scale. The accuracy of quantumdesign of materials properties has been proven in a broadrange of applications from the semiconductor to thepharmaceutical industries.

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However, due to the cubic scaling of the computing timeto the number of atoms in atomisticmodelling, computationsof supercell models on a first-principles basis remain aformidable task even for materials containing very simpleelements. For example, in recent highly accurate modellingof a carbon-based system of 180 atoms, the computing timeto determine the binding energy was 1500CPU h (or 63 CPUdays) using supercomputers [16]. Modelling composite orsolid-solution (SS) materials containing transition-metalelements for battery applications inevitably requires largesupercells that could contain from tens to hundreds of atomsin thousands of combinatorial mixtures. For example, inrecent studies, a supercell with 511 atoms has been used tomodel a dilute concentration of vacancies in LixCoO2

(x< 1%) [17].Thus, it is highly desirable to develop a fast and high-

throughput method to computationally design complexelectrode materials. The desired method should achievesignificant computing efficiency without loss of reliabilityand predictability for new structures. In thiswork,we discussthe main ideas and physical base of the Lego approachproposed by Li and Srivastava [18]. This discussion serves asour first rehearsal to make new materials by rational design.We present the efficiency and reliability if known charac-teristics of material structures are used in rational designs.Moreover, we elaborate the necessary developments toovercome limitations of the Lego approach in order toachieve rational design of generic composite materials.Materials used in our discussion are layered SS materialsLi(NixCoyMn1�x�y)O2 (LNCM) that are a promisingcathode material for next-generation lithium-ion batterytechnology.

2 Quantum total-energy computation methodThe first-principles DFT method used in this work is the all-electron, full potential linear augmented plane wave(FLAPW) software developed under license fromQuantum Materials Design, Inc. The exchange correlationenergy is treated within the generalized gradient approxi-mation (GGA). The muffin-tin radii chosen are 2.0, 1.9, 1.9,1.9 and 1.4 bohr for Li, Co, Ni, Mn and O, respectively. Theintegration over the Brillouin zone uses a 7� 7� 7Monkhorst–Pack mesh. Cutoffs of the plane wave basis(up to 8Ry) and potential representation (up to 50Ry), andspherical harmonics (up to 8) inside the muffin-tin sphereswere used for the FLAPW total-energy calculations. Theconvergence measured by the rms difference between inputand output charge density is better than 0.01me per (a.u.)3.

3 Materials design strategy It is always preferableto design properties of a big, complex system from theproperties of its constitutive components. The idea inthe Lego approach is to obtain reliable supercell structuresof a complex material from its constitutive components,which can often be efficiently determined by the state-of-the-art quantum total-energy calculations. Thus, as a generalprinciple, rational designs pursue not only a reliable design

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mechanism, but also an efficient implementation of compu-ter-aided design for complex materials. The substitutionalSSmaterial LNCM is an ideal, simple system to demonstratethe proof-of-concept of our rational design strategy. Theheavy computing cost of SS supercells can be substantiallyreduced if some characteristics of complex structures can bereadily computed from constitutive components by utilizingempirical rules or applicable principles.

The Hume–Rothery rule is a well-established empiricalrule that establishes a qualitative relationship between SSmaterials (mainly metal alloys) and constitutive alloyingelements. The Halpin–Tsai equations in general provide aquantitativemathematical model for the design of compositematerials from constitutive structural units. It is well knownthat substitutional SS materials bear crystalline character-istics of the constitutive parent crystals. Therefore, a fast andhigh-throughput approach is to design SS supercells frombuilding blocks, referred to as Legos, other than searchingstable structures on energy landscapes of big supercells. TheLegos can be thoroughly optimized by conventional trial-and-error highly accurate DFT calculations. The crystallinecharacteristics of solid solutions are then computed frommathematic models of the lattices of Legos. The accuracy ofSS structures can be systematically improved by adjustingmathematic models used to interpret the relation between SSand the Legos. Therefore, the heavy computing cost islargely avoided for lattice optimizations of supercells. Thisdesign scheme, coupled with the Halpin–Tsai equations, canachieve a systematic improvement.

4 Rational design of the ternary materialLi(NiCoMn)1/3O2 SS materials LNCM are promisingternary cathode materials for next-generation lithium-ionbattery technology because of their high-rate capability,relative safety and high flexibility to customize theirelectrochemical properties. Li(NiCoMn)1/3O2 (L333) is themost studied ternary SS material in this family. So far, twoatomistic models have been proposed for L333 in theliterature [19, 20]:Models A andB, shown in Fig. 1. Becauseof the complexity of cation distribution in the transition-metal layers, other models have not been explored inprevious works. As shown in the following, a systematicdesign scheme not only provides a modelling strategy, butalso offers a relatively complete picture of ternary L333materials at the atomic level.

4.1 Quantum Lego database The ternary SSLNCM contain three transition-metal elements: Ni, Co andMn.Over awide range of (x, y) ratios, LNCMshows the samecrystalline characteristics as the layered LiCoO2 structure(space groupR-3m). Therefore, in the design process, LNCMare viewed as a composition from three constitutive Legos:LiCoO2 (LCO), LiNiO2 (LNO) and LiMnO2 (LMO). TheLego structure is defined by two lattice parameters, a and c,and contains 4 atoms per primary cell. All structuralparameters are determined by the DFT method with the

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Figure 1 L333 models.

FLAPW program and are collected in the quantum Legodatabase.

Since the crystal parameters of LiCoO2 and LiNiO2 havebeen measured, experimental data are used as starting pointsin the DFT structural optimization. Table 1 compares DFToptimized structures of LiCoO2 and LiNiO2 to experimentalstructures.

The structural parameters of both models agree withmeasurements to within about 1%. The layered LiMnO2 istreated as a hypothetical crystal, because it has not beenreported in the same layered phase as the layered LCO.Therefore, its lattice parameters and atomic positions aretotally determined by the DFT trial-and-error strategy. Thestructural parameters of optimized LiCoO2 were used asthe initial parameterswhenCo is replacedwithMn. TheDFToptimization of LMO generates over 197 structural points,including both lattice and atomic position relaxations, toconverge on a stable conformation. The total computing timeis about 15CPU h, which is easily handled on moderatecluster computers.

4.2 Rational designs of SS structures Once theLego database is determined by DFT calculations, LNCMcan be designed for any mixing ratio of (x, y) according totwomathematic relations between the SS and the Legos. Thefirst is to express the lattices of SS, or LNCM, as a function of

Table 1 Structural determination of Legos by DFT trial-and-error stonly the oxygen position is shown).

LiCoO2

FLAPW exp. [21]

a (A) 2.8473 (þ1.1%) 2.815c (A) 13.9214 (�0.9%) 14.05oxygen 0.2603 (�0.1%) 0.2606

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the lattices of Legos, or LCO, LNO and LMO. A genericfunction form is the power series, and the linear weightedfunction of mixing ratio is the simplest form: F(A,C)¼SaLegofLego(a, c), where F(A, C) and f(a, c) stands forlattice constants of the SS and the Legos, respectively, andtheweighting factors {aLego} aremixing ratios for each Legoand are normalized to one.Our experience indicates that sucha linear superposition represents good solubility amongLegos, complying with the Hume–Rothery rule. Deviationfrom a linear function is often associated with localdistortions of lattices due to alloying effects of transition-metal ions in active layers.

The cation distribution in transition-metal layers is theother relation that needs to be handled carefully. Thedifficulty in characterizing a SS structure is largely attributedto the disordered nature of cation distribution patterns, calledsuperstructures. A recent experiment indicates a tendencytowards random distribution[23]. The inherent disorder canbe systematically addressed at the level of supercells byrational design. AnR30 superlattice (shown in Fig. 1), whichis a supercell of LCO, was used as a template to presentpatterns of superstructures.

TheR30 structure has 36 atoms per unit cell and providesa template to model configurations of cation distribution.Each configuration has a different distribution of Ni, Coand Mn in transition-metal layers. The total number ofconfigurations of L333 is 1680, each corresponds to aspecific atomic superstructure if repeating due to latticeperiodicity is ignored. Previously proposed L333models arespecific cases of rational design: Model A (space groupP3112) and Model B (space group P3m1).

Due to the large number of supercell models, it is notpractical to rationally design materials of lithium-ionbatteries. Computing the material properties of a supercellof 36 atoms may be just a moderate cost for accurate DFTcalculation. If computing one structure requires 1 day, a fullscan of the 1680 models by DFT would take 4.6 years.Characterization of cation distributions is associated withidentification of superstructures of SS, which originate fromthe clustering tendency of transition-metal ions and arefundamentally determined by the d-orbital interactions.

At this stage, we focus on providing a concise picture ofpossible superstructure characterization. The clusteringtendency can be quantified by a local order matrix. Weintroduce a local order matrix that counts the number oftransition-metal ions in the nearest-neighbour around each

rategy (percentage is referring to corresponding experimental data;

LiNiO2 LiMnO2

FLAPW exp. [22] FLAPW

2.9108 (þ1.1%) 2.880 2.761414.1099 (�0.6%) 14.19 14.77400.2602 (þ0.5%) 0.259 0.2553

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Table 2 Characterization of superstructures of L333 by traces oflocal order matrix.

group 1 group 2 group 3 group 4 group 5

trace 0 4 6 10 18num. 216 972 324 162 6liter. Model A Model B

active site in transition-metal layers. As shown in Fig. 1,different models have very different matrices, whichmeasure transition-metal clustering patterns. This matrixalso automatically removes the repeating due to latticeperiodicity. From the order matrix, the 1680 models areclassified into only five different groups, as shown inTable 2.

We note that our local order matrix calculation does nottake into account the actual interactions and distancesbetween transition-metal ions, but indeed offers a simplemeasure of the topology of superstructures. In this sense,models in the same group have the same patterns in thetransition-metal-ion arrangement. Therefore, only a limitednumber of representatives are necessary to model super-structures of L333 without loss of generality. Thus, thedesign scheme provides a relatively complete picture ofstructures of L333 at the atomic level. Meanwhile, the localorder matrix offers a convenient mapping of the structuralentropy of synthesis of pure LCO, LNO and LMO. FromTable 2, it is obvious that neitherModel A norModel B is thestatistically favoured model for L333. Our FLAPW total-energy calculations indicate that neithermodel is the ground-state structure. Detailed analysis will be discussed in futurepublications.

From the above description, crystal supercells of SSmaterials can be designed rationally by three key steps: (1)choose an appropriate superlattice, which is a supercell ofconstitutive Lego’s primary cell. For example, R30 structureis the supercell of LCO’s R-3m unit cell; (2) calculate thecrystal lattice constants of SS structures from the linearsuperposition of Lego’s lattice constants; (3) scattertransition-metal ions among all allowed active sites ofsupercells to form all possible superstructure patterns, whichcan be further classified into groups by an order matrixwithout detailed energy calculation at this stage of designs.

We have found that supercells determined by such arational design agree well with the measured lattice data andare very close to stable points on the multidimensional

Table 3 Validation of designed structures comparing to exper-iment and fully optimized DFT structures (percentage is referringto computer designed lattice constants).

lattice constant a(A)

lattice constant c(A)

experimental data 2.863 (�0.8%) 14.247 (þ0.2%)rational design 2.8398 14.2684DFT Model A 2.8827 (�1.5%) 14.1067 (þ0.3%)DFT Model B 2.8401 (<0.1%) 14.0579 (þ1.5%)

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energy landscape of SS structures. As presented in Table 3,rationally designed structures are within 1–2% of fully DFToptimized structures. This accuracy is within the generalerror bar of the DFT methods, referring to the comparisonshown in Table 1. We attribute this surprising agreement tothe SS nature of LNCM, which is an excellent substitutionalSS material. This at least proves the reliability of rationaldesign for a specific family of materials.

4.3 Analysis of efficiency The efficiency of rationaldesigns can be addressed at two levels. First, the latticerelaxation is largely avoided. As indicated in Table 3, thedesigned lattices are within 1%of experimental lattices. Thiserror is good enough formost practical evaluation ofmaterialproperties. Therefore, for a fast and high-throughput designof new structures, one can use the computer-aided design asit is. The heavy computing cost of O(N3) for large N-atomsupercells is reduced to a linear summation of a few limitedcomputing cost of O(n3) for small n-atom Legos. In theexample of R30 L333 model, the computational scaling ofthe order of 363 was reduced to three calculations withscaling in the order of 43.The reduction is 243 times,assuming other factors are largely unchanged. Since theLego database is one time cost, the efficiency is obvious fordesigning even larger models. For a 48-atom SS model, thecomputing reduction increases to 576 times when using thesame quantum Lego database. This validates rational designas a fast and high-throughput method for the research anddevelopment of new lithium-ion battery materials.

Secondly, we give an estimate of the savings of time byadopting computationally designed structures as startingpoints in a fullDFToptimization. Table 4 gives a comparisonof DFT optimization between LiMnO2 and L333 Model A.

Even though these two models are quite different, thepath passed from starting points to stable points on the energysurface gives a clear perspective on the savings of time forlarge structures. As shown in Table 4, comparedwith the 197total points investigated on the searching path to a stableLiMnO2 phase, the trial points of L333Model A are reducedby a factor of 10 if the searching started with computerdesigned supercell. A broad range of energy points isavoided for the determination of large supercell. The ratherextensive and expensive DFT total-energy minimization isonly used to fine tune the rationally designed structures. Thisoffers a good starting point for more strict and systematicimprovement.

5 Generic rational designs It is much more pre-ferred to design new electrode materials based on deepunderstanding of the basic principles governing the relationbetween the structures and the target properties. This is oftencalled the bottom–up approach. The idea of rational design isto use applicable principles to accelerate materials inno-vation with controllable design errors. Designing newstructures is always a necessary step before designing newmaterial properties. Such designed structures are preferablywithin 1–2% of the real structures if applicable, because this

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Table 4 DFT computing cost comparison between LMO and SSL333 Model A (the same lattice mesh used, atomic positions fullyrelaxed on each lattice point).

LiMnO2 Model A

starting structure optimized LiCoO2 rational designunit cell size 4 atoms 36 atomstotal points 197 20total time forfull optimization

15CPU h 2026CPU h

design error is in the acceptable range of current DFTaccuracy for structural determination. However, we believethat the high accuracy and efficiency of our L333 designs islargely due to the SS characteristics of LNCM. The simpleLego approach is unlikely to achieve similar accuracy andefficiency easily for other more complicated materials, suchas lithium-rich layer-layer Li2MnO3–LNCM.

Therefore, we believe that significant development isneeded to overcome the limitations of the simple scheme ofthe Lego approach so that an advanced rational designscheme can be applicable to much more generic compositematerials than substitutional SS only. The first barrier lies inthe determination of Lego structures. A general compositestructure often has different crystalline characteristics fromthe Legos. Therefore, the simple scheme of the Legoapproach needs to be modified for a general composite.Secondly, the lattice relation between composites and Legosis often beyond a simple linear function, or a well-definedmathematic expression. It is not clear so far how to achievethe efficiency of lattice relaxation during global minimiz-ation of total energy. However, it is worth exploring the ideaof utilizing empirical rules or applicable principles to pursuerational designs of new battery materials.

6 Conclusion In this work, we discuss a new methodto design structures of SS materials by quantum modellingwith significant reduction of computation cost. The structuredesigned by the method is within 1–2% of otherwise fullyDFT optimized structure. The designed structure is nearby atotal-energy minimum without any ad hoc input fromexperimental measurements. This will narrow significantlythe range of the search landscape needed for the complex SSmaterials when one prefers highly accurate structuredetermination or further systematic improvement. And it isthe first proof-of-concept that rational design can signifi-cantly advance the process of materials innovation.

The computing efficiency mainly comes from savings oflattice relaxations by rational design, which reduces thecubic dependence of large SS models into linear summationof rather small Lego units and offers systematic applicationto a broad range of SS. The total computing costs are ratherlow. Computer-aided design provides amore concise pictureof SS materials at the atomic scale and can be used as a fast

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and high-throughput design platform for new electrodematerials research.

Acknowledgements Ting Luo acknowledges support fromthe Scientific Research Fund of Zhejiang Provincial EducationDepartment, grant no. Y201016652. Jun Li acknowledges supportfrom the New Faculty Startup Fund of Ningbo Institute of MaterialTechnology and Engineering, Chinese Academy of Sciences andprograms supported by the Ningbo Natural Science Foundation,grant no. 2011A610207. The authors thank the NationalNanotechnology Infrastructure Network (NNIN) for providingpartial computational time for this project. Correspondenceand requests for materials should be addressed to Jun Li(lijun@nimte.ac.cn).

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