Influence of Ti/V Cation-Exchange in Na2Ti3O7 on Na-Ion Negative ElectrodePerformance: an Insight from First-Principles Study

Chol-Jun Yu∗,a, Suk-Gyong Hwanga, Yong-Chol Paka, Song-Hyok Choea, Jin-Song Kima, Kum-Chol Ria

aChair of Computational Materials Design, Faculty of Materials Science, Kim Il Sung University, Ryongnam-Dong, Taesong District, Pyongyang, DemocraticPeople’s Republic of Korea

Abstract

Sodium-titanate Na2Ti3O7 (NTO) is regarded as a highly promising anode material with a very low voltage for Na-ion batteriesand capacitors, but suffered from relatively low specific capacity and poor electron conductivity. Here we report a first-principlesstudy of electrochemical properties of NTO and its vanadium-modified compounds, Na2Ti2VO7 and Na2TiV2O7 (NTVO), offeringan insight into their detailed working mechanism and an evidence of enhancing anode performance by Ti/V cation exchange. Ourcalculations reveal that the specific capacity can increase from 177 mAh g−1 in NTO to over 280 mAh g−1 in NTVO when usingNaTi3-xVxO7 (x = 1, 2) as a starting material for Na insertion due to higher oxidation state of V +5, together with lower voltages andsmall volume expansion rates below 3%. With Ti/V exchange, we obtain slightly higher activation energies for Na ion migrationsalong the two different pathways, but find an obvious improvement of electronic transport in NTVO.

Key words: Sodium titanium oxide, Anode material, Electrochemical property, Na-ion battery, First-principles

1. Introduction

Ever increasing demand for a better welfare in daily life hasdrawn an explosive production of portable electronic devicesand electric vehicles. Together with the urgent need for ex-ploiting renewable and clean energy sources, which are oftenintermittent, this requires to develop advanced electrochemicalenergy storage (EES) systems with high efficiency, stability andreliability. Primarily two types of devices exist for reversibleEES; secondary batteries, and electrochemical supercapacitors.The former delivers a high energy density, while the latter pro-vides a high power density with a longer life time. Up tillnow, Li-ion batteries (LIBs) have been dominating EES mar-kets, but recently, a great deal of renewed research attention hasbeen paid to Na-ion based technology, such as Na-ion batter-ies (NIBs) [1] and Na-ion capacitors (NICs) [2]. Due to thehuge abundance and low price of sodium resources, the use ofsodium instead of lithium is expected to overcome the probableshortage of lithium resources and to be suitable choice for thelarge-scale applications.

The key issue in realizing commercially viable NIBs andNICs is to search their proper electrode materials, especiallyanodic materials. Although graphite is a common anode ma-terial in LIBs, it does not properly intercalate Na+ ion dueto its larger ionic radius (1.02 Å) compared to Li+ ion (0.76Å). In this status, ternary graphite intercalation compoundswith organic molecules have been devised for using graphitein NIBs [3, 4, 5], and also non-graphitic carbon materials such

∗Corresponding authorEmail address: cj.yu@ryongnamsan.edu.kp (Chol-Jun Yu)

as soft and hard carbon have been developed, offering a signif-icantly high capacity of ∼300 mA h g−1 but with low Coulom-bic efficiency [6]. Titanium-based oxides are another promisingcandidate for the intercalation-type anodes because of their lowvoltage, low cost and non-toxicity [7]. Among them, sodium-titanate Na2Ti3O7 (NTO) with well-defined step-layered struc-ture was found to exhibit the lowest electrode potential (0.3 Vvs Na+/Na) and high theoretical capacity (311 mA h g−1 [8]),demonstrated for the first time by Tarascon’s group in 2011 [9].It was revealed that upon sodiation Na2Ti3O7 could transforminto Na4Ti3O7 with the sluggish kinetics of Na+ ion insertionand the large lattice expansion (6%), resulting in poor electro-chemical performance.

Thereafter, NTO-based compounds have been extensivelystudied, aiming at improving the performance. The synthe-sis was evolved from the simple solid-state reaction way [9,10, 11] to elaborate multistep reaction routes for specially de-signed phases [12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Ac-cordingly, the phase morphology was diversified from mi-crosize particles [9, 21] to various nanoscale phases, includ-ing nanotubes [16, 22, 23, 24, 8], nanosheets [13, 14, 25],nanowires [26], nanofibers [27] and so on. Such nanosizingwas proved to effectively enhance the Na+ ion storage kinet-ics, increase the surface area and suppress the volume expan-sion. Moreover, most nanoscale phases are in the form of com-plex with, for example, carbon [27, 28], N-doped carbon [12],graphene [29, 26] and reduced graphene oxide [30], from whichmade flexible electrodes offered excellent rate capability andcycling stability. Introducing other elements into NTO was re-ported to be another way for enhancing the electronic and ionicconductivity of electrode, such as lathanide-doping [31], Nb-doping [32] and Ti3+ self-doping [33, 34]. In spite of the ex-

Preprint submitted to arXiv June 19, 2020

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cellent electrochemical performance, the nanophasing indeedgreatly eliminated the plateau feature of NTO electrode in volt-age profile. Instead, the slope-like voltage profiles were ob-served, indicating their usability as battery-type (pseudocapaci-tance) anodes for NICs [13, 15, 25, 35, 36, 37, 38]. For instance,Dong and coworkers fabricated the quasi-solid-state NICs using3D self-supported NTO nanoribbon array/graphene foam as theanode and graphene foam as the cathode, which delivered highspecific energy of 70 Wh kg−1, high specific power of 4000 Wkg−1, and superior cycling stability like capacitance retention73% over 5000 cycles [15].

In comparison with such extensive experimental studies,only a few number of theoretical works to reveal the sodia-tion/desodiation mechanism has been reported so far [8, 11, 39,40, 41]. Using first-principles simulations within the densityfunctional theory (DFT) framework, Jiao and coworkers calcu-lated formation energies of bulk NaxTi3O7 as increasing the Nacontent x, identifying the maximum value of x to be 5.5 thatgives the maximum theoretical capacity of 311 m Ah g−1 [8].Hu et al. calculated the band structure of NTO with a directbandgap of 2.75 eV, which was smaller than their experimen-tal value of 3.73 eV, and estimated the activation barriers forvacancy-mediated Na hopping along the layers and through thelayers, indicating the former path surely to occur [11]. Thetheoretical electrode voltage profiles in reasonable agreementwith experiment were calculated by Meng and coworkers [39].However, a comprehensive study has not yet been providedon NTO and furthermore its vanadium-modified compounds,which were proposed to offer a better performance [42].

Herein we presented the electrochemical properties of NTOand its two different Ti/V exchange compounds, Na2Ti2VO7(NTV1O) and Na2TiV2O7 (NTV2O), obtained by performingDFT simulations. We identified the possible sites for Na inser-tion into NTO and pathways for Na ion migrations based on thebond valence sum (BVS) analysis. As increasing the Na con-tent, we calculated the formation energies of compounds andderived the convex hull plots, which were used to plot the elec-trode voltage profiles. The activation energies for Na migra-tions along the paths identified through the BVS analysis werecalculated to reveal the ionic conductivity and Na storage mech-anism. We calculated the electronic density of states in thesecompounds, providing an insight into their electronic transport.Systematic comparison between NTO and NTVO compoundswas given, emphasizing the positive role of Ti/V exchange inimproving the electrode performance.

2. Methods

2.1. Structural models

The sodium titanate Na2Ti3O7 is known to crystallize in step-layered structure with a monoclinic space group P21/m, inwhich each step consists of three edge-sharing TiO6 octahedraconnected with the others by vertex-sharing [43]. The Na ionsare located in the interlayer space with two different crystallo-graphic coordinations (7 and 9). By doubling the unit cell (2formula units, 24 atoms) in (010) direction, we constructed a

supercell (4 formula units, 48 atoms) to investigate Na inser-tion and diffusion, as shown in Fig. 1(a). It should be noted thatthe size of the supercell is similar to those of other NIB cathodematerials in our previous works [44, 45], being expected to givereliable result for electrochemical properties of NTO.

In the present form of NTO, Ti is in +4 oxidation state, andupon insertion of additional Na+ ions, its oxidation state canchange to +3, allowing NTO to act as electrode active materialwith Ti 4+/Ti 3+ redox couple. Therefore, it is necessary to iden-tify probable positions for the additional Na+ ions, which canbe carried out by plotting the BVS isosurface of NTO [46, 47].We calculated BVS for the Na−O bond in NTO using the for-mula, B(r) =

∑i exp[(1.803 − Ri(r)/0.37] (R(r) = |r − Ri|, Ri

is the position vector of the i-th atom), and plotted the BVSisosurface at the value of 3, as shown in Fig. 1(a) and (b).

In the supercell containing 4 formula units, the eight host Naions are shown to occupy the most probable sites, and another 8equivalent positions are found to be unoccupied in the interlayerspace. Accordingly, 8 Na ions in total can be inserted one byone into the interlayer space of supercell, forming Na2+xTi3O7(x = 0 − 2), while some Ti ions are reduced from +4 to +3oxidation state. To reduce all the Ti 4+ ions to Ti 3+, 4 Na ionsshould be more inserted into the supercell, resulting in the for-mation of Na5Ti3O7. To search these positions, we again drewthe BVS isosurface plot at higher value of 14, identifying threesites more in the gap spaces between the neighboring up anddown steps of layer (see Fig. S1 in Supporting Information).However, we disregarded these sites because they were found atmuch higher value of BVS (compared with the ideal value of 1)and moreover the crystalline phase formed by filling these siteswith Na ions could be severely deviated from the original phaseof NTO, leading to large volume expansion and thus short cy-cling lifetime [8]. In fact, the Na4Ti3O7 compound rather thanNa5Ti3O7 was found to be the final production in the sodiumstorage process in the most experimental works [9].

To investigate the influence of Ti exchange with V on sodiumstorage performance, we considered two different exchangemodels, NaxTi2VO7 and NaxTiV2O7. Since vanadium hashigher oxidation state of +5 than Ti, the Na content x can bebelow 2, indicating that the host Na ions can be removed fromNa2Ti3-xVxO7 (x = 1 and 2). In these Ti/V exchange com-pounds, the lowest values of Na content reduce to 1 and 0 forNTV1O and NTV2O, while the highest values are equal to 5.With these considerations, we can write the reduction processesoccurred on NTO, NTV1O and NTV2O upon insertion of Naions into their lowest Na content compounds as follows,

Na2Ti+43 O7 + 3Na→ Na5Ti+3

3 O7, (1a)

NaTi+42 V+5O7 + 4Na→ Na5Ti+3

2 V+3O7, (1b)

Ti+4V+52 O7 + 5Na→ Na5Ti+3V+3

2 O7. (1c)

Then, we can expect an enhancement of electrode specific ca-pacity by Ti/V exchange in NTO, as discussed below.

It is important to provide concrete evidence of formability asexperimental work on synthesis of the NTVO compounds couldnot yet found. As a first check for formability, we calculate

2

Figure 1: Optimized structure of supercell containing 4 formula units for bulk Na2Ti3O7 in (a) perspective and (b) top views. Yellow-colored isosurface plot ofbond valence sum at the value of 3 is shown to indicate the probable positions for additional inserting Na (ins Na) ions and their diffusion paths. Supercells for bulk(c) Na2Ti2VO7 and (d) Na2TiV2O7, selected as the lowest energy configurations. Black solid lines indicate the lattice of supercell.

cohesive energy Ec using the following equation,

Ec = ENa2TiaVbO7− (2Ei

Na + aEiTi + bEi

V + 7EiO) (2)

where ENa2TiaVbO7is the total energy per formula unit of

Na2TiaVbO7 bulk and EiX is the total energy of isolate atom

X. We also calculate the formation enthalpy of the compoundsfrom binary metal oxides including Na2O in cubic phase (spacegroup FM3M), anatase-type TiO2 and V2O5 in orthorhombicphase (PMMN). For the following chemical reactions for syn-thesis,

Na2O + 3TiO2 ⇒ Na2Ti3O7,

2Na2O + 4TiO2 + V2O5 ⇒ 2Na2Ti2VO7 + 12 O2,

Na2O + TiO2 + V2O5 ⇒ Na2TiV2O7 + 12 O2

(3)

the formation enthalpy Ebf can be calculated as follows,

Ebf = ENa2Ti3O7

− (ENa2O + 3ETiO2),

Ebf =

12

[2ENa2Ti2VO7

+ 12 EO2

− (2ENa2O + 4ETiO2+ EV2O5

)],

Ebf = ENa2TiV2O7

+ 12 EO2

− (ENa2O + ETiO2+ EV2O5

)(4)

where EO2is the total energy of isolated O2 molecule that can

be simulated by supercell with lattice constant of 15 Å. On theother hand, the synthesis was suggested to be performed by us-ing ternary sodium carbonate Na2CO3 instead of binary sodium

oxide Na2O [42] as,

Na2CO3 + 3TiO2 ⇒ Na2Ti3O7 + CO2,

2Na2CO3 + 4TiO2 + V2O5 ⇒ 2Na2Ti2VO7 + 2CO2 + 12 O2,

Na2O + TiO2 + V2O5 ⇒ Na2TiV2O7 + CO2 + 12 O2

(5)for which the formation enthalpy Et

f can be calculated as fol-lows,

Etf = ENa2Ti3O7

+ ECO2− (ENa2CO3

+ 3ETiO2),

Etf =

12

[2ENa2Ti2VO7

+ 2ECO2+ 1

2 EO2− (2ENa2CO3

+ 4ETiO2+ EV2O5

)],

Etf = ENa2TiV2O7

+ ECO2+ 1

2 EO2− (ENa2CO3

+ ETiO2+ EV2O5

)(6)

For the cases of bulk, we can assume that the enthalpic differ-ence term P∆V and the entropic term T∆S at room temperatureare negligible due to being 3 or 4 order smaller than the inter-nal energy difference. Meanwhile, for the cases of O2 and CO2gas we should consider both terms which can be taken fromexperimental data [48].

2.2. Computational detailsAll the DFT calculations have been carried out by applying

the pseudopotential plane wave method as implemented in theQUANTUM ESPRESSO package [49]. To describe the interactionbetween the valence electrons and ionic cores, we have usedthe ultrasoft pseudopotentials established in the package, wherethe valence electron configurations were set to Na:2s22p63s1,Ti:3s23p63d24s2, V:3s23p63d34s2, and O:2s22p4. The Perdew-Burke-Ernzerhof (PBE) formulation [50] within generalized

3

Table 1: Optimized lattice parameters and volume of supercell with 4 formula units, cohesive energy Ec (eV per atom), and formation enthalpies at T = 298.15 Kfrom binary oxides Eb

f and from ternary compound Etf (eV per formula unit) of the electrode materials

Compound a (Å) b (Å) c (Å) α (◦) β (◦) γ (◦) Vol (Å3) Ec Ebf Et

f

Na2Ti3O7 9.040 7.519 8.475 90.00 102.04 90.00 563.36 −5.57 −2.25 −0.10Expa 9.126 7.599 8.563 90.00 101.59 90.00

Na2(Ti2V)O7 8.948 7.483 8.433 89.99 102.07 90.00 552.15 −5.55 −2.21 −0.05Na2(TiV2)O7 8.885 7.465 8.393 90.00 101.65 90.00 545.27 −5.53 −2.19 −0.04

a X-ray diffraction data [11].

gradient approximation (GGA) was adopted for the exchange-correlation (XC) interaction between the valence electrons.The weak van der Waals (vdW) interaction between the layerswas considered through the semi-empirical Grimme’s function(DFT-D2) [51]. We have also applied the GGA + U approachin the simplified version of Cococcioni and de Gironcoli [52] totake into account the effect of localized 3d electrons of the tran-sition metal cations. The effective Hubbard parameter Ueff = 3eV was given to Ti and V atoms, consistent with the previouswork [39]. Structural optimizations of the supercells were car-ried out with cutoff energies of 50 Ry for wave function and 500Ry for electron density and a (2 × 2 × 2) k-point mesh until theforce on each atom and the pressure of crystalline lattice con-verged to 5 × 10−4 Ry Bohr−1 and 0.05 Kbar respectively. Tocalculate the density of states (DOS), a denser k-point mesh of(6 × 6 × 6) was adopted with consideration of spin-polarizationeffect, where ferromagnetic ordering was applied.

The activation energies for Na ion diffusion according tothe paths predicted by BVS in the supercells were calculatedby applying the climbing image nudged elastic band (NEB)method [53]. During the NEB simulation, the supercell sizeswere fixed at the optimized ones, and all the atoms were al-lowed to relax until the force converged to 0.02 eV Å−1. Thenumber of NEB image points was set to 7 or 13 according to theroute of Na ion migration. Visualization of crystalline lattice,and volumetric data of BVS and electronic density was carriedout by using the VESTA code [54]

3. Results and discussion

3.1. Structural variationFirstly, we determined the lattice parameters of NTO unit

cell with 2 formula units in monoclinic phase by using differ-ent XC functionals. With the PBE + DFT-D2 functional, theywere calculated to be a = 9.092 Å, b = 3.786 Å, c = 8.510Å, β = 101.88◦, which were slightly underestimated comparedwith the experimental values within a relative error of −0.5%.Such underestimation is due to the inclusion of vdW dispersionenergy, since the pure PBE functional gave the slight overesti-mation within the well-known trend of the GGA functional.

Then, the supercell with 4 formula units was constructed bydoubling the optimized unit cell in the (010) direction, whichwas again optimized. The polyhedral view of its crystal struc-ture is shown in Fig. 1 (a) and (b), together with the isosur-face plot of BVS. To construct the supercells for NTV1O andNTV2O crystalline solids, 4 and 8 Ti ions were exchanged withV ions in the supercell. By considering the crystallographic

equivalent positions, 15 different configurations for Ti/V ex-change were considered to be possible and the correspondingsupercells were also optimized. Among them, the lowest energyconfigurations were selected, and their optimized structures areshown in Fig. 1(c) and (d) for NTV1O and NTV2O, respec-tively. Upon such replacement of some Ti ions with V ions, thelattice parameters of supercells were found to decrease as in-creasing the content of V ions, as listed in Table 1. Accordingly,the volumes of the supercells were also observed to decrease by98.0% and 96.8% for NTV1O and NTV2O respectively, com-pared with NTO. These are possibly due to an enhancement ofinteraction between the transition metal cations and oxygen an-ions as reflected by the average M−O bond length within theMO6 octahedra (1.98 Å in TiO6 vs 1.93 Å in VO6). In Ta-ble 1 we also presented the cohesive energy, and binary andternary formation enthalpies calculated by applying Eq. 2, andEqs. 4 and 6 respectively. The cohesive energy and formationenthalpies were calculated to be negative for both NTV1O andNTV2O as well as NTO, indicating that they are thermodynam-ically stable and can be synthesized securely. As increasing theamount of V, the energy was found to slightly decrease in mag-nitude, implying the slackening of stability.

During the charge/discharge process in NIBs, Na ions shouldbe inserted/deserted into/from the intercalation-type anode ma-terial, causing a volume change or even multi-step phase tran-sition. To simulate the charge process, we constructed the su-percells for NTO, NTV1O and NTV2O with increasing numberof Na ions from 1 to 8, which were placed on the sites iden-tified by BVS analysis as discussed above. It should be em-phasized that the final productions are Na4M3O7 based on theBVS analysis in this work, although Na5M3O7 is theoreticallypossible as proved in Eqs. 1a−1c. Meanwhile, we consideredthe supercells with remove of host Na ions up to 4 and 8 for thecases of NTV1O and NTV2O respectively. There were manyconfigurations for Na insertion and remove, which could be re-duced by considering the crystalline symmetry. After pickingout the lowest energy configuration of each intermediate phasewith structural optimization, we measured the lattice parame-ters and volume as increasing the number of Na ions.

In Fig. 2, we show the calculated lattice constants and vol-ume as a function of Na content x in NaxM3O7 compounds.For the case of M = Ti (NTO), the Na content varies from x = 2to 4, while for the cases of M = Ti2/3V1/3 and Ti1/3V2/3 it startsfrom x = 1. It should be noted that for the case of NTV2O,the supercell volume was found to abruptly decrease when re-moving the host Na ions by from 5 (x = 1) to 8 (x = 0) (seeFig. S2 in Supporting Information), implying that NaxTiV2O7

4

7.0

7.5

8.0

8.5

9.0

9.5

Latti

ce c

onst

ant (

Å)

NTONTV1O

NTV2O

a

b

c

(a)

1 2 3 4Na content

530

540

550

560

570

580

Vol

ume

(Å3 )

NTONTV1O

NTV2O

0

1

2

3

r vol (

%)

(b)

Figure 2: Variation of (a) lattice constants (a, b, c) and (b) volume of supercellwith 4 formula units as increasing the Na content x in NaxM3O7 (M = Ti,Ti2/3V1/3, Ti1/3V2/3), where the relative volume expansion rate rvol = (Vx −

Vx=2)/Vx=2 × 100(%) is shown by rectangular bar.

with x ∈ (0, 1) can not be expected to properly work as elec-trode active material. As increasing the Na content x, the lat-tice constants b are shown in Fig. 2(a) to gradually increaseand a to increase in more or less wavy way, whereas the inter-layer lattice constants c to decrease gradually first and rapidlyafter x = 3.25. This indicates that the insertion of Na ionsinto the interlayer space enhances the interaction between thelayers through the attractive Na−O bonding. In most cases ofNa content, the lattice constants became smaller as increasingV content, indicating that Ti/V exchange in NTO leads to en-hancement of M−O interaction. On the other hand, the super-cell volumes were found to gradually increase from x = 1 forNTVO and x = 2 for NTO to x = 3.25 and decrease afterx = 3.25, as can be seen in Fig. 2(b). The relative volume ex-pansion rates, rvol = (Vx − Vx=2)/Vx=2 × 100(%), were found tobe under 3%, which is quite small compared with other elec-trode materials, indicating their superior cycling stabilities. Forall the three cases of electrode materials, similar variation ten-dencies and moreover no phase transitions were observed. It isworth noting that from x = 1 to x = 2 for Ti/V exchange modelsthe volume changes are almost negligible.

3.2. Electrode potential

We estimated the relative stability of NTO-based compoundsupon insertion and extraction of Na ions by calculating the for-

0 0.5 1 1.5 2Inserted Na (x)

-0.2

-0.1

0.0

0.1

0.2

Ef (

eV/f.

u.)

NTONTV1O

NTV2O

Na2+xM3O7

(a)

0 0.5 1 1.5 2 2.5 3Inserted Na (x)

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

Ef (

eV/f.

u.)

NTV1O

NTV2O

Na1+xM3O7

(b)

Figure 3: Convex hull plot of formation energy per formula unit (f.u.) ofNa2Ti3O7 (NTO), Na2Ti2VO7 (NTV1O) and Na2TiV2O7 (NTV2O), calculatedwith PBE + vdW (DFT-D) functional. The curves are obtained by using thestarting compound as (a) Na2M3O7 for all the three cases of compounds and(b) NaM3O7 only for NTVO compounds.

mation energy of intermediate Nan+xM3O7 compound with re-spect to the two end compounds. As mentioned above, the start-ing compounds in this work were arranged to be Na2M3O7 forall the cases and NaM3O7 only for NTVO cases, correspond-ingly indicating n to be 2 and 1, while the final compoundswere consistently Na4M3O7. Then, the formation energies canbe written in two ways, as follows,

Ef = ENa2+xM3O7−

12

[xENa4M3O7

+ (2 − x)ENa2M3O7

], (7a)

Ef = ENa1+xM3O7−

13

[xENa4M3O7

+ (3 − x)ENaM3O7

], (7b)

where Ecomp is the DFT total energy of the compound.Figure 3 shows the formation energies as a function of

Na content, calculated by adopting the starting compoundas Na2M3O7 for the three different kinds of models, and asNaM3O7 only for Ti/V exchange models. In Fig. 3(a) forNa2+xM3O7 models, the formation energies of the intermedi-ated phases with x ≤ 1.25 are shown to be negative, indicat-ing their safe formation, whereas the phases with x = 1.5 andx = 1.75 can not be said to be naturally formed due to theirpositive formation energies. This is consistent with the abruptdecrease of the interlayer lattice constant c and volume decrease

5

with these Na contents, shown in Fig. 2. In these cases, the low-est energy configurations were found at x = 0.75 for NTO andNTV2O, while at x = 1 for NTV1O model. When comparedNTVO models with NTO, the formation energies are in overallhigher, indicating that Ti/V exchange enhances the interactionsbetween the cations and anions.

On the other hand, the formation energies, calculated by ap-plying Eq. 7b and using the two end compounds of NaM3O7and Na4M3O7 for NTVO models, were found to be negativefor the whole range of inserted Na content x ∈ (0, 3), as shownin Fig. 3(b). In these cases, the intermediate phases with thelowest formation energy were identically found at x = 1, i.e.Na2M3O7, which is agreed well with the fact that it is the ac-tual starting compound. When compared between two differ-ent Ti/V exchange models of NTV1O and NTV2O, the formercompounds were always shown to be more readily formed dueto their slight lower formation energies. We also calculated theformation energies of the intermediate phases with respect tothe same starting compounds and Na metal in body-centeredcubic (bcc) phase, and Na binding energies by using Na reser-voir in gas state (see Fig. S3 in Supporting Information). As theNa and V contents increase, the formation and Na binding en-ergies were found to decrease in agreement with the above dis-cussion. The optimized structures for Na2+xTi3O7 (x ∈ (0, 2)),Na1+xTi2VO7 (x ∈ (0, 3)) and NaxTi2VO7 (x ∈ (0, 4)) wereshown in Fig. S4-S6 in Supporting Information.

In order to shed light on the enhancement of electrode perfor-mance by Ti/V exchange, we estimated the electrode potentialV as a function of specific capacity. The step discharge volt-age between the adjacent Na contents, identified as placed onthe convex hull curves in Fig. 3, with respect to the Na/Na+

counter electrode can be calculated as follows,

V = −ENax j M3O7

− ENaxi M3O7+ (x j − xi)ENabcc

(x j − xi)e, (8)

where e is the elementary charge.

Due to the two different working ways of electrodes distin-guished by selecting the base compound, which is Na2M3O7 orNaM3O7, we plotted also two different step voltage curves, asshown in Fig. 4. With Na2M3O7 as the starting compound, themaximal specific capacities were estimated to be 177.6, 175.8and 174.1 mAh g−1 in Na4Ti3-xVxO7 for x =0, 1 and 2 respec-tively. As shown in Fig. 4(a), such lowering in capacity by Ti/Vcation exchange is negligible but voltage lowering is clearly dis-tinctive. Together with the calculated step voltages below 1 V,this indicates that these compounds can work as anodic materi-als for NIB and Ti/V exchange can increase the full-cell voltageand energy density. More interestingly, due to the higher oxida-tion state of V +5, NaTi3-xVxO7 (x =1 and 2 in this work) can bethought as the base compound, resulting in a great increase ofspecific capacity. As shown in Fig. 4(b), the maximal capacitiesin these cases were determined to be 285.3 and 282.2 mAh g−1

in Na4Ti3-xVxO7 for x = 1 and 2, being much higher than thoseobtained by using Na2M3O7 as the base compound.

0 50 100 150Capacity (mAh/g)

0.0

0.2

0.4

0.6

0.8

1.0

Vol

tage

(V

)

NTONTV1O

NTV2O

NTO Exp

(a)

Na2+xM3O7

0 50 100 150 200 250Capacity (mAh/g)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Vol

tage

(V

)

NTV1O

NTV2O

(b)

Na1+xM3O7

Figure 4: Electrode potential as a function of specific capacity of of NTO,NTV1O and NTV2O, calculated by using the DFT total energies of the com-pounds identified as placed on the convex hull line (Fig. 3). (a) Na2+xM3O7(x ∈ (0, 2)) for all the models, where the experimental curve for NTO is fromRef. [19], and (b) Na1+xM3O7 (x ∈ (0, 3)) for NTVO models.

3.3. Sodium-ion conductivity

Ionic conductivity plays a decisive role in electrochemicalperformance of rechargeable batteries and capacitors. To getan insight into ionic conductance, we determined the activa-tion energies for vacancy-mediated Na ion migrations along themost plausible pathways, which were preliminary predicted bythe BVS analysis. As shown in Fig. 5, the pathways can beformed in the interlayer space along the crystallographic (010)and (100) directions. The (010) direction pathway can be char-acterized by a slightly wavy line without step, whereas alongthe (100) direction a step-line-step pathway can be formed. Inaccordance with the exclusion of possible Na sites in the gapspace between up and down steps, we did not consider the mi-gration through the layer, for which the activation energies weredetermined to be much higher (∼1.75 eV) than those for migra-tions along the layer with first-principles NEB calculation [11].

To increase the reliability of ionic conductivity test, we con-sidered two different Na concentrations, one Na atom insertedinto Na2M3O7 supercells (Na2.25M3O7, denoted as Na1) andthree inserted Na atoms (Na2.75M3O7; Na3). Figure 5 presentsthe ball-and-stick view of Na ion migrations in the case ofNa2.25TiV2O7 (similar pathways were constructed for NTO and

6

NTV1O with two different Na concentrations). Here, one cycleof Na ion migration consists of two processes; firstly the hostNa ion at Nh position moves toward vacancy at N1

i position (rep-resented by green-colored balls) and then the inserted Na ion atN2

i position moves toward the generated vacancy at Nh position(represented by orange-colored balls). This is called sequentialmove way. In the case of (100) direction, interestingly, althoughwe let two Na ions move along the step-line-step pathway in thesequential way, they were observed in the NEB simulation tomove simultaneously for all the cases of compounds. This canbe called concurrent move way, which was already found in thecathodic material NaFe(SO4)2 [44].

Figure 6 shows the energy profiles computed with the NEBmethod for Na ion migrations along these pathways. Based onthe obtained activation energy Ea, the Na ion diffusion coeffi-cient can be estimated by D = a2ν exp(−Ea/kBT ), where a ≈ 4Å, ν ≈ 1011 Hz and kBT = 0.026 eV [11]. Table 2 summarizethe calculated activation energies and Na ion diffusion coeffi-

Figure 5: Pathways of Na ion migration along (a) (010) direction with a wavy-line character and (b) (100) direction with a step-line-step character, formed inthe interlayer space and indicated by yellow-colored isosurface plot of BVS atthe value of 3. Top and bottom panels show top and perspective views. Fixedand migrating Na atoms are denoted as yellow and green or orange balls, andTi−O and V−O bonds as blue-red and purple-red sticks, respectively.

Table 2: Activation energy values (Ea) for Na ion migrations along the (010)sequential and (100) concurrent pathways, and corresponding Na ion diffusioncoefficient D at room temperature.

(010) (100)Compound Ea (eV) D (cm2/s) Ea (eV) D (cm2/s)NTO Na1 0.325 5.96×10−10 0.362 1.44×10−10

Na3 0.418 1.67×10−11 0.365 1.28×10−10

NTV1O Na1 0.411 2.18×10−11 0.441 6.88×10−12

Na3 0.393 4.36×10−11 0.579 3.41×10−14

NTV2O Na1 0.408 2.45×10−11 0.451 4.69×10−12

Na3 0.574 4.13×10−14 0.464 2.84×10−12

cients. For the case of NTO, they were determined to be 0.325and 0.362 eV in (010) sequential and (100) concurrent ways,and slightly increased at higher Na concentration, being agreedwell with the experimental value of 0.39 eV [55]. It turned outthat the Ti/V exchange causes a slight increase of activation en-ergy. This might be due to the aforementioned enhancement ofcation−anion interactions by Ti/V exchange in NTO. Neverthe-less, the calculated activation energies were below 0.579 eV inall the cases, which are not so high compared with other ex-perimental data for NTO (0.67 eV) [55]. Therefore, such slightworsening of ionic conductivity would not be said to spoil theenhancement of electrode performance by Ti/V exchange.

3.4. Electronic transport properties

It was known that NTO is electrically insulating materialwith a direct bandgap of 3.73 eV [11], which is quite disadvan-tageous to electrode of batteries or capacitors. What is the effectof Ti/V exchange on the electronic transport? To answer thisquestion, we investigated the atomic resolved partial densityof states (PDOS) in the compounds under study, as shown inFig. 7. The bandgap of Na2Ti3O7 was calculated to be 3.45 eV,which is very close to the experimental value of 3.73 eV [11],indicating that the GGA + U method with U = 3 eV for Ti

0

0.1

0.2

0.3

0.4

0.5

0.6

∆E

(eV

)

NTO Na1

NTO Na3

NTV1O Na1

NTV1O Na3

NTV2O Na1

NTV2O Na3

(a)

0 1 2 3 4Path (Å)

0

0.1

0.2

0.3

0.4

0.5

0.6

∆E

(eV

)

(b)

Figure 6: Energy profile calculated with the NEB method for the migrationsof Na ions along (a) (010) direction wavy-line pathway in sequential move wayand (b) (100) direction step-line-step pathway in concurrent move way, depictedin Fig. 5. Two different Na concentrations, one Na atom inserted into Na2M3O7supercells (denoted as Na1 with solid line) and three inserted Na atoms (Na3with dashed line) are considered for NTO, NTV1O and NTV2O.

7

and V is reasonable to give confidential results for electronicstructure. In this compound, the valence bands (VBs) closeto the Fermi level (EF) were found to be dominated by O 2pstates, while the conduction bands (CBs) to be characterized byTi 3d states in partial hybridization with O 2p states. To in-quire into how the bandgap changes upon Ti/V exchange, wecalculated the PDOS of NaTi2VO7 and TiV2O7, in which Tiand V have the highest oxidation state of +4 and +5 (cf. Eqs 1band 1c). As indicated in Fig. 7(b) and (c), the bandgaps werefound to decrease as increasing the V content, such as 2.51 eVin NaTi2VO7 and 2.26 eV in TiV2O7, revealing that Ti/V ex-change can improve the electronic transport of electrode ma-terials. In these compounds, the VBs are governed by O 2pelectrons as well, but the CBs are dominated by V 3d stateswith a minor contribution from O 2p states.

When inserting a Na atom into these base compounds, theFermi level was found to move upward and be placed on theCBs consistently in the three kinds of models, resulting inchange of electric state of material from insulating to metallic.As already revealed in our previous work [44], this might be as-sociated with the ionization of the inserted Na atom. The elec-tronic bands around the Fermi level were found to be formedmainly from Ti 3d spin-up and -down electrons for NTO andV 3d spin-up electrons for NTVO models with O 2p electrons.Both the occupied and empty states of Na were observed farfrom the Fermi level, indicating that the inserted Na atom isfully ionized and the released electron moves to the Ti or V 3dand O 2p states around the conduction band minimum. As aresult, the Fermi level was pushed ahead, and the impurity-typebands were formed around it. When increasing the amount ofinserted Na atom, the occupied bands below EF were found togradually increase and the gap between the valence band maxi-mum and conduction band minimum to get closer.

To clarify the electron transfer upon insertion of Na atom, weinvestigated the electronic charge density difference defined as,

∆n = nNa2.25M3O7− (nNa2M3O7

+ 0.25nNa), (9)

where ncomp is the electron density of the compound. Here, allthe atomic positions were fixed in the same supercell dimen-sion for the three compounds. As shown in Fig. 8, remarkableredistribution of electronic charge density was observed aroundthe inserted Na atom and nearby Ti, V and O atoms. One cansee charge depletion around the Na atom, charge accumula-tion around the O atoms, and both accumulation and depletionaround Ti and V atoms. It can be thought that at the formationof Na2M3O7 the Na, Ti and V atoms offer their 3s and 3d elec-trons to the O atoms, becoming Na+, Ti 4+, V 5+ and O 2 – ions,and upon insertion of Na atom the released electrons transferto metal ions, becoming Na+, Ti 3+ and V 3+ ions. This indi-cates that Ti and V atoms act as redox couples of Ti 4+/Ti 3+ andV 5+/V 3+ upon insertion/desertion of Na atom. Such electrontransfer was found to occur more remarkably when performingTi/V exchange, indicating the stronger ionic bonding betweenthe cations and anions in NTVO models.

4. Conclusions

Using first-principles calculations, we have investigated theelectrochemical properties of Na2Ti3O7 and its two differentTi/V exchange compounds Na2Ti2VO7 and Na2TiV2O7, whichare thought to be promising anodic materials for Na-ion bat-teries and capacitors. The lowest energy configurations for thetwo different exchange models have been determined with thestructural optimizations combined with the crystalline symme-try consideration. Through the bond valence sum analysis, wehave identified the plausible positions for Na atoms insertinginto Na2Ti3O7, from which the final sodiation compounds havebeen fixed to be Na4M3O7. We have found that as increasingthe Na content x in NaxM3O7, the supercell volumes increasefirst and decrease after x = 3.25 with the maximum relativevolume expansion rates below 3%, revealing the shrinkage ofcell volume by Ti/V exchange because of their enhanced M−Ointeraction. Based on the convex hull plot of formation energiesof Na2+xM3O7, lower electrode voltages have been obtained inTi/V exchange models with the maximal specific capacity ofabout 175 mAh g−1, and by considering the higher oxidationstate of V +5, the specific capacity could increase over 280 mAhg−1. We have computed the activation energies for Na ion mi-grations along the (010) direction wavy pathway in sequentialmove and (100) direction step-line-step pathway in concurrentmove, finding that Ti/V exchange leads to slightly higher acti-vation barriers. Through the analysis of electronic density ofstates, we have found an improvement of electronic transportproperties by Ti/V exchange in Na2Ti3O7.

Acknowledgments

This work is supported as part of the basic research project“Design of Innovative Functional Materials for Energy and En-vironmental Application” (No. 2016-20) by the State Com-mission of Science and Technology, DPR Korea. Computationwas done on the HP Blade System C7000 (HP BL460c) that isowned by Faculty of Materials Science, Kim Il Sung University.

Notes

The authors declare no competing financial interest.

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