All-carbon-based porous topological semimetal forLi-ion battery anode materialJunyi Liua, Shuo Wanga, and Qiang Suna,b,1

aDepartment of Materials Science and Engineering, College of Engineering, Peking University, Beijing 100871, China; and bCenter for Applied Physics andTechnology, Peking University, Beijing 100871, China

Edited by George William Crabtree, Argonne National Laboratory, Argonne, IL, and approved December 16, 2016 (received for review October 31, 2016)

Topological state of matter and lithium batteries are currently twohot topics in science and technology. Here we combine these twoby exploring the possibility of using all-carbon-based poroustopological semimetal for lithium battery anode material. Basedon density-functional theory and the cluster-expansion method,we find that the recently identified topological semimetal bco-C16

is a promising anode material with higher specific capacity (Li-C4)than that of the commonly used graphite anode (Li-C6), and Li ionsin bco-C16 exhibit a remarkable one-dimensional (1D) migrationfeature, and the ion diffusion channels are robust against thecompressive and tensile strains during charging/discharging.Moreover, the energy barrier decreases with increasing Li inser-tion and can reach 0.019 eV at high Li ion concentration; theaverage voltage is as low as 0.23 V, and the volume changeduring the operation is comparable to that of graphite. Theseintriguing theoretical findings would stimulate experimentalwork on topological carbon materials.

carbon materials | topological semimetal | Li-ion battery | anode

With the growing demand for portable energy sources, it ismore and more urgent to improve the present lithium-ion

batteries (LIBs) (1, 2). Apart from the cathode and electrolyte,the anode as one of the most important parts in LIBs has beenextensively explored for better performance (3, 4). Although thegraphite anode has good stability and low cost, its theoreticalmaximum specific capacity is only 372 mAh/g, which cannot meetthe higher requirements of current and future technologies suchas advanced electrical vehicles (5). Therefore, efforts have beendevoted to finding new candidates with larger specific capacity.Among them, Si- and P-based materials are considered aspromising candidates because their specific capacities canreach as high as 4,200 and 2,596 mAh/g (6, 7), respectively,which are much higher than that of the graphite anode. How-ever, they both suffer from the poor reversibility caused byhuge volume expansion (Si > 300%; P > 300%) and slow ratecapability caused by low electronic conductivity (8, 9). Al-though the electrochemical performances can be improved viathe thin-film technique, porous structures, nanotube/nanowirearrays, carbon coating, etc. (10, 11), the complex syntheticprocedures and high fabrication costs prevent their practicalapplications. Thus, it remains a great challenge to develop ananode material with high capacity, good stability, and fast ki-netics as well as low cost.Considering the abundance in resources, flexibility in bonding,

and variety in morphology (12, 13), carbon materials have someunique advantages in anode applications. In fact, graphene,carbon nanotubes (CNTs), carbon nanofibers (CNFs), carbonnanorings, and porous carbon (14–16) have been extensivelyexplored for this purpose (17, 18). For instance, pristine gra-phene shows a weak adsorption of Li and has low capacitywhereas defective graphene can bind Li stably and has a highercapacity than graphite (19); the mesoporous graphene nano-sheets have achieved an ultrahigh initial discharge capacity of3,535 mAh/g (20). However, there are still some problems tobe solved for the porous carbon anodes: Firstly, owing to the

disorder of pores and structure defects, the electronic con-ductivity of amorphous porous carbon usually is relatively low,resulting in an unfavorable rate capability (18). Secondly, ahigh fraction of exposed edge planes in porous carbon willcause extremely irreversible side reactions with electrolytesolution which give rise to a low Coulombic efficiency (17). Forexample, the Coulombic efficiencies of the porous graphenenanosheets, CNFs, and CNTs are all smaller than 50% (15, 20).Considering that a full Li-ion cell has a limited Li inventory,this is a serious disadvantage in terms of achievable capacity. Asimple way to minimize the associated irreversible capacitywould be to decrease the direct exchange surface area. Thirdly,the ordered and small pores are highly desirable to improve therate performance and packing density. Recently, Sander et al.demonstrated that electrodes with aligned pore channels fab-ricated via magnetic template can deliver faster chargetransport kinetics with threefold higher area capacity thanthat of conventional Li-ion electrodes (21). However, thedistribution and size of the pores cannot be easily controlledin experiments by using the conventional porous carbon (17,18, 22). Motivated by these dissuasions, we wonder if we canfind a 3D carbon material with intrinsic ordered nanopores aswell as high electronic conductivity to avoid the problemsdiscussed above.Here we show that the predicted 3D topological semimetal

carbon structure bco-C16 can be such a system (23), in which allcarbon atoms are sp2-bonded with ordered 1D channels, as seenfrom Fig. 1, providing good binding sites and efficient diffusionchannels for Li ions. Moreover, electronic band structure cal-culations prove that bco-C16 is a topological node-line semimetalwith a linear dispersion band structure near the Fermi level (23),

Significance

The 2016 Nobel Prize in Physics highlighted the importance oftopological state in science and technology. Here we explorethe possibility of using all-carbon-based topological semimetal(ACTS) for lithium-ion battery anode material based on themerits of intrinsic high electronic conductivity and orderedporosity. Using state-of-the-art theoretical calculations, we doshow, by taking topological semimetal bco-C16 as an example,that ACTS structures can be promising anode materials withhigh specific capacity, fast ion kinetics, and slight volumechange during operation. Our study would not only pave apathway for design of high-performance anode materials go-ing beyond the commercially used graphite but would alsoencourage further theoretical studies on topological phases ofcarbon materials.

Author contributions: Q.S. designed research; J.L. performed research; J.L., S.W., and Q.S.analyzed data; and J.L., S.W., and Q.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: sunqiang@pku.edu.cn.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1618051114/-/DCSupplemental.

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indicating a high electronic conductivity. Then, a question arisesnaturally: Can the bco-C16 be a promising anode material forLIBs? To answer this question, we perform a detailed study onthe Li intercalation and diffusion process in the bco-C16 based onthe first-principles calculations. The results show that Li ions areable to intercalate into the bco-C16 with a binding energy of−0.63 eV at a dilute concentration, and it remains a negativevalue of −0.23 eV when the bco-C16 are fully intercalated with Liions, indicating a stable Li adsorption instead of phase separa-tion. The maximum Li concentration is Li-C4, showing an im-proved specific capacity compared with the graphite anode.Because of the significant structural anisotropy, Li diffusion inbco-C16 exhibits a strong directional anisotropy. At a dilute Liconcentration, the migration barrier along the 1D channel (Apath) is 0.53 eV, whereas for the diffusion perpendicular to the1D channel (P path), it is a rather large value of 2.32 eV. At ahigh Li concentration, the migration barrier along 1D channelcan decrease to a rather small value of 0.019 eV, while it remainsa large value of 1.29 eV perpendicular to the 1D channel.Moreover, the 1D Li ion diffusion is robust against the com-pressive and tensile strains. The estimated average voltage is aslow as 0.23 V, and the volume change during the Li charge/discharge is comparable to that of graphite. Based on thesefindings, bco-C16 is expected to be promising as an anode ma-terial with a high specific capacity, a high rate capability, as wellas a low open-circuit voltage.

Results and DiscussionSingle Li Atom Insertion and Diffusion in bco-C16.As shown in Fig. 1,the crystal structure of bco-C16 can be regarded as 3D modifica-tion of graphite in AA stacking with benzene linear chainslinked by ethene-type planar π-conjugation. Considering that

the conventional exchange–correlation functionals of stan-dard density functional theory (DFT) cannot give a reason-able interlayer distance of graphite (24) because of the poordescription of the van der Waals (vdW) interactions, we firstrelaxed the crystal structure of bco-C16 with three differentmethods [Perdew–Burke–Ernzerhof (PBE) functional with-out vdW corrections; vdW-D2 with semiempirical corrections(25); and vdW-optPBE with vdW functional (26), respec-tively]. The results are shown in Table S1 and the graphite(27) is also included for comparison. We find that three dif-ferent methods give almost the same lattice parameters ofbco-C16, suggesting that vdW interactions in bco-C16 crystalare weak and negligible due to covalent bonding. Therefore,the subsequent computations of bco-C16 are based on thestandard PBE exchange–correlation functional without vdWcorrections.Then, we investigated the insertion of single Li atom into bco-

C16. To avoid the interaction between two Li atoms, we adopteda 2 × 2 × 3 supercell of bco-C16. As seen from Fig. 1, threepossible initial insertion sites are considered: the bridge site M(above the midpoint of the C–C bond in a benzene linear chain),the hollow site H (above the center of the C hexagon), and theinterstitial site I (above the midpoint of the interstitial C–C bondbetween two adjacent benzene linear chains). We find that Liatoms on both M and I sites moved to the neighboring H sitesafter full relaxation, suggesting that H sites are the most stableadsorption sites for Li atoms. And, the adsorbed Li atoms on Hsites tend to get closer to the up C hexagon hollows for theupward bending of linking C–C bonds; the calculated distancebetween the Li atom and the up (d0)/down (d1) C hexagonhollows is 1.60/1.87 Å. The Bader charge population analysisshows that there is about 0.73 jej charge transfer from Li to bco-C16, namely, Li atoms donate almost all their s electrons andbecome Li ions, thus leading to a strong Coulomb repulsioninteraction between the adsorbed Li ions that prevents themfrom clustering.To further check the ionic binding with the substrate, we

calculate the Li binding energy (Eb), which is defined as

Eb =ELix−bco−C16 −Ebco−C16 − xμLi

x, [1]

where ELix-bco-C16and Ebco-C16

are the total energies of Li-insertedand pristine bco-C16 crystal structure, respectively. μLi is the chem-ical potential of Li which is taken as the cohesive energy per atom ofmetallic Li. The calculated Eb of Li adsorption on the H site of bco-C16 is −0.63 eV, which is comparable to that of graphite (−0.78 eVin our computations).The rate performance plays an important part in the electrode

material which is mainly determined by the Li-ion mobility. It isdesirable to estimate the diffusion of Li ion in the bco-C16.Different from the graphite where Li ion is located on the 2Disotropic graphene sheet, the bco-C16 shows obvious structuralanisotropy along the x- and y directions. Therefore, we need toinvestigate the diffusion barriers for Li ions along two differentpossible migration paths: one is path H0–H1, which is along thebenzene linear chains (A path); the other is the path H0−H2,which is perpendicular to the benzene linear chain (P path), asshown in Fig. 2 A and B.Based on climbing-image nudge elastic band (CI-NEB) cal-

culations, the energy profiles along the A path and P path arepresented in Fig. 2 C and D. For the case of Li diffusion alongthe benzene linear chain (A path), there is only one energypeak of 0.53 eV. The calculated diffusion barrier is slightlylarger than that of graphite (0.218–0.4 eV) (24, 28, 29) butcomparable to that of commercially used anode materialsbased on TiO2 with a barrier of 0.35–0.65 eV (30–33) (Table 1),indicating that Li ions can diffuse easily along the A path. In

Fig. 1. Structure of bco-C16 and the schematics of the possible Li-ion ab-sorption sites in top and side views.

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contrast, for the diffusion perpendicular to the benzene linearchain (P path), a rather large barrier of 2.32 eV is found, whichmeans an unfavorable Li diffusion along this path. To furthercompare the diffusion property between these two paths, thetemperature-dependent diffusion constant (D) can be evaluatedby the Arrhenius equation (29):

D∝ exp�−Ea

KBT

�, [2]

where Ea and KB are the diffusion barrier and Boltzmann’s con-stant, respectively, and T is the environmental temperature. Accord-ing to Eq. 2, the diffusion mobility of Li along the benzene linearchains is about 7.5 × 1029 times larger than that perpendicular to thebenzene linear chain at room temperature, suggesting that Li-iondiffusion along the P path at room temperature would not happendue to the large energy barrier (2.32 eV); this is similar to thatof 2D black phosphorus (34) and 3D titanium niobate (35).This remarkably 1D diffusion observed in bco-C16 is absent inother 3D carbon materials. According to the basic theory ofdiffusion, diffusion constant varies inversely with spatial dimen-sion, i.e. D∝ 1=ð2nÞ (36), where n is the dimensionality of space.When other conditions are the same, diffusion in 1D will be thefastest and will lead to the highest diffusion coefficient.

Li Storage Capacity and Vacancy Diffusion. Next, we explore thefully Li-intercalated bco-C16 which directly determines themaximum Li capacity. The binding energy Eb is calculated as−0.23 eV with all H sites occupied by the Li ions; the absolutevalue is larger than that of graphite (LiC6: −0.11 eV) and close tothat of VS2 monolayer (Li2VS2: −0.26 eV) (37), suggesting thatLi atom can be adsorbed stably in bco-C16 and the phase sepa-ration problem can be safely avoided at such a high Li concen-tration. The theoretical specific capacity of bco-C16 is 558 mAh/g,corresponding to the Li-C4, which is 1.5 times larger than that ofgraphite (Li-C6).In the previous sections, we only considered the single Li-ion

diffusion in bco-C16; to gain a more complete picture, it is nec-essary to estimate the diffusion in high Li concentration. Weremove one Li ion from the fully Li-intercalated bco-C16 andrelax the structure again. Accompanying the removal of the Liion, its nearest-neighboring Li ion will move to the M site be-cause of electrostatic repulsion, leading to a vacancy. Then weperform the calculations of migration energy barriers for anisolated vacancy in a 2 × 2 × 3 supercell along both the A pathand the P path; the results are presented in Fig. 3. The calculatedenergy barriers for the A path and the P path are 0.019 and 1.29 eV,respectively. Compared with that of single Li ion, the energybarriers for the isolated vacancy are both reduced dramatically;the reason can be ascribed to the enlarged size along the z di-rection and strong Coulomb repulsions between Li ions. Espe-cially for the A path, the value of the vacancy hopping energybarrier is about 15 times smaller than that of graphite (0.283 eV)(24), and close to the single Li-ion migration energy barrier of2D Ti3C2 (0.068 eV), Mo2C (0.043 eV), and black phosphorus(0.08 eV) (34, 38, 39) (Table 1). Therefore, it is reasonable toexpect that the mobility of Li ions becomes higher with the in-creasing Li concentration and achieves a high rate of perfor-mance at a high Li concentration. At the same time, we comparethe vacancy diffusion property between the A path and P pathby estimating the temperature-dependent diffusion constantaccording to Eq. 2. The mobility of Li along the A path is about2.57 × 1021 times larger than that along the P path at roomtemperature, suggesting that the significantly 1D diffusion canstill be observed with a high Li concentration at room temperature.

Effect of Li Concentration and Theoretical Voltage Profile. Open-circuit voltage is another key factor which is widely used forcharacterizing the performance of the Li battery. In theory, wecan obtain the open-circuit voltage curve by calculating the av-erage voltage over parts of the Li composition domain. Thecharge/discharge processes of bco-C16 comply with the followinghalf-cell reaction vs. Li/Li+:

Fig. 2. Schematics of the Li diffusion pathways along the A path (H0–H1)and P path (H0–H2) in the top (A) and side (B) views, respectively, where H0,H1, and H2 denote the initial and end points. The corresponding energyprofiles are presented in C and D.

Table 1. Comparison of specific capacity, diffusion barrier and open-circuit voltage of candidateanode materials for LIB

Materials

Specific capacity,mAh/g Diffusion barrier, eV

Open-circuitvoltage, eV

Theor. Expt. Theor. Expt. Theor.

Bco-C16 558 — 0.53 — 0.23Graphite 372 372 0.218–0.4(24, 28, 29) — 0.11(37)TiO2 200(41) 200(32) 0.3(33) 0.35–0.65(30–32) 1.8(41)2D black P 432(42) — 0.08(34) 2.9(34)2D MoS2 335(37) — 0.25(37) — 0.26(37)2D VS2 466(37) — 0.22(37) — 0.93(37)2D Ti3C2 447.8(39) — 0.068(39) — 0.43(39)2D Mo2C 526(38) — 0.043(38) — 0.68(38)

“––” means data unavailable.

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ðx2 − x1Þ  Li++ðx2 − x1Þ  e−+Lix1−bco−C16 ↔Lix2−bco−C16.[3]

Thus, neglecting the volume, pressure, and entropy effects, theaverage voltage of Lix-bco-C16 in the concentration range of x1 <x < x2 can be estimated as (40)

V ≈ELix1−bco−C16 −ELix2−bco−C16 + ðx2 − x1ÞELi

ðx2 − x1Þ , [4]

where ELix1-bco-C16, ELix2-bco-C16 and ELi are the total energy ofLix1-bco-C16, Lix2-bco-C16, and metallic Li, respectively.Before calculating V, we first search the most stable Li occu-

pying configurations at different intermediate Li concentrations.The formation energy for a given Li-vacancy arrangement withcomposition x in Lix-bco-C16 is defined as

Ef ðσÞ=ELix−bco−C16 − ð1− xÞEbco−C16 − x  ELi, [5]

which reflects the relative stability of the specific Lix-bco-C16structure with respect to phase separation into a fraction x ofLi and a fraction (1 − x) of bco-C16. The energies of 150 config-urations with different Li concentrations are calculated with ac-curate first-principles methods, then the cluster-expansion (CE)Hamiltonian of Lix-bco-C16 is constructed by fitting to the first-principles energies. The cross-validation (CV) score of this CEfitting is 15 meV, indicating that the obtained CE Hamiltonian isaccurate enough to predict the energies of any Li-vacancy con-figurations of Lix-bco-C16. Based on the CE Hamiltonian, theformation energies of all of the symmetry-inequivalent Lix-bco-C16 structures within 60 atoms per cell are calculated. As shownin Fig. 4, five stable configurations at intermediate concentra-tions are found; the corresponding concentration x (LixC4) is0.167, 0.25, 0.5, 0.667, and 0.75, respectively. The correspondingconfigurations are presented in Fig. S1 A–E. Then we check thebinding energies of these stable structures and find that they areall negative, indicating that the Li ions can be adsorbed stablyinstead of clustering. On the other hand, the absolute value ofbinding energy decreases gradually with the increasing Li con-centration, due to the enhanced repulsive interaction betweenthe Li ions and the reduced interaction between the bco-C16host and Li ions, as the charge transfer from the metal atom to

bco-C16 at high concentrations is also reduced. Based on theseresults, we calculate the average voltage using Eq. 4. As it canbe seen in Fig. 4, the voltage profile displays three prominentvoltage platforms, and there is a dramatic drop from 0.63 Vwhen x < 0.25; then, the voltage profile decreases slowly to0.05 V with the Li concentration increasing to 1. The averagevoltage by numerically averaging the voltage profile is calcu-lated to be 0.23 V, which is rather low and is only slightlylarger than that of graphite (0.11 V) (37), but much smallerthan that of 2D VS2 (0.93 V), Mo2C (0.68 V), TiO2 (1.5–1.8 V),and black P (1.8–2.9 V) (38, 40–43) (Table 1). The low averagevoltage of bco-C16 anode suggests that once connected to cathodethe LIB can supply a higher operating voltage with largerenergy capacity.

Assessments of the Cycling Stability and Strain Effect on Li IonsDiffusion. Cycling stability is another factor that needs to beconsidered, which is mainly determined by the volume changesduring the Li charging /discharging. We compare the fullylithiated bco-C16 with the pristine one and find that no bondbreaking occurs and the total volume expansion is 13.4%,which is comparable to that of graphite (10%). Moreover,being similar to that of graphite, the volume changes aremainly in the z direction (10.3%). This anisotropic volumeexpansion can be understood from the changes of chemicalbonds. Along the x and y directions, the elastic deformationinvolves the stretch of in-plane C–C bond length, which needs alarger energy, whereas for the z axis it involves the rotation ofan out-of-plane C–C bond, which corresponds to a smallerenergy. Thus, when Li ions are inserted into bco-C16, it tends toexpand first along the z axis; moreover, the rotation of the C–Cbond can tolerate a large volume change without bond breaking.To further assess the cyclic stability, the stress−strain relations of

Fig. 4. (A) Formation energies predicted by CE method for the 150 differentLi configurations with 5 stable intermediate phases. (B) Correspondingvoltage profile (marked in red) and binding energy profile (marked in blue)calculated along the minimum energy path.

Fig. 3. Schematics of the vacancy diffusion pathways along the A path(H0–H1) and P path (H0–H2) are presented in the top (A) and side (B) views,respectively, where H0, H1, and H2 denote the initial and end points of va-cancies. The corresponding energy profiles are presented in C and D.

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bco-C16 are calculated within 15% uniaxial compressive strain andtensile strain. Three kinds of uniaxial load conditions along the x, y,and z axes are considered, respectively; the corresponding resultsare presented in Fig. 5 A–C. For all three directions, the stressesincrease continuously with the growing strain and exhibit an ap-proximate linear dependence within a large range of compressiveand tensile strain; no abrupt decrease or nonlinear platformoccurs, indicating that the bco-C16 is able to sustain a largestrain. According to the continuum mechanics, the Young’smodulus E can be derived from the slope of stress−straincurves with strain up to 4%. The calculated Young’s modulialong the x and y directions are 795 and 824 GPa, respectively,close to that of graphene (1,000 GPa) (44), whereas they aremuch smaller in the z direction (only 150 GPa). These an-isotropic Young’s moduli are consistent with the volume ex-pansion. Considering the modest volume expansion of Liinsertion and the large fracture strains, it can be concludedthat the bco-C16 anode is able to accommodate the volumechanges during lithiation/delithiation and shows a goodcycling stability.Next we investigate the strain effect on Li-ion diffusion for

further understanding of the underlying mechanism and possibleimprovements. The diffusion barriers for Li along two differentmigration paths are calculated under the 5% uniaxial compres-sive and tensile strains, respectively. The corresponding resultsare summarized in Table 2, the detailed energy profiles areshown in Fig. S2 A–F. It can be found that uniaxial compressivestrains along the x (z) axis and tensile strains along the z (x) axisare able to reduce (increase) the energy barrier of both A pathand P path, respectively, whereas the strains along the y axis havelittle influence on energy barriers. By comparing the strainedbco-C16 structures, we find that both uniaxial compressive strainsalong the x (z) axis and tensile strains along the z (x) axis canincrease (reduce) the channel size along the z direction and re-duce (increase) the interaction between Li ions and bco-C16 host.However, the strains along the y axis have little influence on thechannel size along the z direction. Therefore, the channel sizealong the z direction plays a decisive role in the migration energybarrier, which is similar to that of graphite where interlayerdistance has a significant effect on the in-plane Li diffusion (24).

Meanwhile, the calculated ratio of diffusion constant along the Apath (DA) and P path (DP) at 300 K according to Eq. 2 remains alarge value under different strains, suggesting that the Li diffu-sion along the P path is prohibited and the 1D Li-ion diffusion isrobust against the strains.

SummaryIn this study we have explored the possibility of using all-carbon-based topological semimetal for LIB anode material which hasthe merits of intrinsic high electronic conductivity and orderedporosity for Li-ion transport; moreover, carbon is abundant inresources, flexible in bonding, and ever-changing in morphology.Taking bco-C16 as an example, we systematically examined theenergetics and kinetics of Li-ion insertion and diffusion, and candraw the following conclusions: (i) Li ions can be inserted intobco-C16 stably without clustering; (ii) The maximum capacity ofbco-C16 is 558 mAh/g, corresponding to the Li-C4, which is muchhigher than that of graphite (Li-C6); (iii) The average voltage is0.23 V, which is rather low as an anode and can supply largeroperating voltage and capacity once connected with the cathode;(iv) The Li ions in bco-C16 exhibit a remarkable 1D Li-ion dif-fusion; moreover, the migration energy barrier can reach a quitelow value at high Li-ion concentration due to the enlarged size ofthe channel, strong Coulomb repulsions from the neighboring Liions; (v) The 1D Li-ion diffusion is robust against the compres-sive and tensile strains; and (vi) The volume changes during theLi insertion/de-insertion are comparable to that of graphite,suggesting a good reversibility. With all of these extraordinarycharacteristics, bco-C16 should have a great potential to be ap-plied as anode material for LIBs. It is encouraging to note thepossible presence of boc-C16 phase in detonation and chimneysoot as suggested by an excellent match between the simulatedand measured X-ray diffraction patterns (23). We hope that thepresent study can stimulate further experimental effort on thissubject to develop all-carbon-based porous structures with con-ducting topological properties for novel LIB anode materials.

MethodsOur first-principles calculations are based on density-functional theory (DFT)implemented in Vienna Ab initio Simulation Package (VASP) (45) with theexchange–correlation functional in the PBE form (46). The projector-augmented

Fig. 5. Compressive and tensile stress σ as a function of uniaxial strain « along the (A) x, (B) y, and (C) z direction, respectively.

Table 2. Calculated diffusion barriers along A path and P path, and the corresponding ratio ofdiffusion constant at 300 K under different compressive and tensile strains, respectively

Strains

−5% 0% 5%

x y z —– x y z

A path 0.31 eV 0.53 eV 0.81 eV 0.53 eV 0.77 eV 0.56 eV 0.39 eVP path 2.15 eV 2.28 eV 2.59 eV 2.32 eV 2.33 eV 2.40 eV 2.07 eVDA/DP 9.4 × 1030 2.9 × 1029 9.2 × 1029 1.4 × 1030 1.8 × 1026 9.4 × 1030 1.9 × 1028

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wave (47) method is used to describe the electrons, and the cutoff energy ofthe plane-wave basis set is set to be 550 eV. Based on the convergence test, a(7 × 11 × 16)/(3 × 4 × 6) Monkhorst–Pack k-point mesh is adopted to representthe reciprocal space of the unit cell/(2 × 2 × 3 supercell), respectively, duringthe whole computational process. The conjugated gradient method is appliedto optimize the structure, and convergence criteria of total energy and forcecomponents are set to be 1 × 10−4 eV and 0.01 eV/Å, respectively. The diffusionbarriers for Li ions are calculated on the basis of the CI-NEB method asimplemented in the VASP transition state tools (48, 49). The convergence cri-terion of force is also set to be 0.01 eV/Å.

As there are a large number of possible Li-vacancy configurations at anintermediate Li concentration, it is impossible to calculate all of their ener-gies. Here, we use the CE method to describe the configurational energy andto search for the most energy-favorable Li configurations, which has beenwidely used in the previous studies of electrode materials such as Lix-graphite(24), LixCoO2 (50, 51), etc. Based on the CE method, the Lix-bco-C16 can betreated as alloy systems. For each possible Li site i, we can represent it withan occupation variable σi, which takes the value 1 if Li resides at that site and−1 if a vacancy is at that site; the configuration-dependent Hamiltonian canbe mapped onto a generalized Ising Hamiltonian:

EðσÞ= J0 +Xi

Jiσi +Xj<i

Jijσiσj +Xk<j<i

Jijkσijk + ...= J0 +Xα

Jαφα, [6]

where the indices i, j, and k range over all occupation sites, φα is the productof occupation variables σi, σj . . . σk that form a cluster configuration α, whichcan be a single point, a pair cluster, a triplet cluster, etc. Jα is correspondingeffective cluster interactions (ECI), which means the contribution of thespecific cluster configuration α to the total energy. Although Eq. 6 hasendless sum terms, in practice it is able to represent any Li-vacancy config-urations energy only by an appropriate selection of limited number ofcluster configurations because the ECI (Jα) will converge to zero at largedistance. The parameter Jα can be determined by fitting first-principles en-ergies of selected configurations, the fitted CE quality is measured by the CVscore, and the fitting process can be performed with Alloy Theoretic Auto-mated Toolkit (ATAT) code (52). When the CE fittings reach a reasonably lowCV score, we can easily get the lowest-energy configurations.

ACKNOWLEDGMENTS. This work is partially supported by the National KeyResearch and Development Program of China (Grant 2016YFB0100200), andthe National Natural Science Foundation of China (Grants NSFC-11274023and 21573008).

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