Intercalation processes and diffusion paths of lithium ions in spinel-type structured Li1+xTi2O4: Density functional theory study M. Anicete-Santos,1,2 L. Gracia,1 A. Beltrán,1 J. Andrés,1,* J. A. Varela,3 and E. Longo3 1Departament de Química Física i Analítica, Universitat Jaume I, Campus de Riu Sec, Castelló E-12080, Spain 2LIEC, Departamento de Química, Universidade Federal de São Carlos-UFSCAR, P.O. Box 676, São Carlos, São Paulo 13565-905, Brazil 3LIEC, Instituto de Química, UNESP, P.O. Box 355, Araraquara, São Paulo 14801-907, Brazil �Received 20 May 2007; revised manuscript received 17 October 2007; published 14 February 2008� Intercalation processes and corresponding diffusion paths of Li ions into spinel-type structured Li1+xTi2O4 �0�x�0.375� are systematically studied by means of periodic density functional theory calculations for different compositions and arrangements. An analysis of the site preference for intercalation processes is carried out, while energy barriers for the diffusion paths have been computed in detail. Our results indicate that the Li insertion is thermodynamically favorable at octahedral sites 16c in the studied composition range, and Li migration from tetrahedral sites 8a to octahedral sites 16c stabilizes the structure and becomes favorable for compositions x�0.25. Diffusion paths from less stable arrangements involving Li migrations between tetra- hedral and octahedral sites exhibit the lowest energy barrier since the corresponding trajectories and energy profiles take place across a triangle made by three neighboring oxygen anions without structural modification. Theoretical and experimental diffusion coefficients are in reasonable agreement. DOI: 10.1103/PhysRevB.77.085112 PACS number�s�: 71.15.Mb I. INTRODUCTION In recent years, spinel-type structured lithium titanium ox- ides have been subject of a large number of publications because of their promising applications as anodes in solid- state high energy Li ion batteries.1–4 In particular, the Li- Ti-O spinel oxide studies are restricted in the composition range by two limiting compounds: LiTi2O4 and Li4Ti5O12, 5,6 originally synthesized and characterized as end members in the solid solution system Li1+yTi2−yO4 �0�y�1 /3�.7 These materials present good reversibility of the lithium in- sertion and/or extraction process without significant changes in cell volume �less than 2%� and show good capacity retention.5,8–10 The combination of high lithium mobility and zero strain insertion properties9,11,12 makes it an adequate anode for high rate battery applications. In addition, the fact that cells of Li-Ti-O spinel oxides produce no metallic lithium during charge makes them more attractive from a safety viewpoint compared to lithium cells that employ lithium-carbon anodes.8 The use of spinel LiTi2O4 as electrode material is based on its ability to host an additional amount of Li leading to composition Li2Ti2O4.13 The insertion capacity of the LiTi2O4 spinel toward Li addition was first reported by Mur- phy and co-workers in the early 1980s.13,14 Colbow et al. cycled this material between about 3 and 1 V, leading to compositions between x=−0.2 and 1 for Li1+xTi2O4; for val- ues of x between 0 and 1, the potential is constant around 1.34 V vs Li /Li+, and Li can be reversible cycled with little capacity loss.5 Furthermore, much attention has been paid to the composition of LiTi2O4, which was the first discovered superconducting oxide system having a transition tempera- ture Tc=11 K.6,15 These materials are attractive enough not only to be used for the practical purposes but also to be employed as a model material in the fundamental study on solid-state electro- chemical reactions. Experimental results offer extensive in- formation that is yet to be explored, but they cannot probe directly the atomistic features of crystal and electronic struc- ture along the intercalation processes and diffusion paths. Theory can complement these efforts and first-principles cal- culations are claimed to disclose a valuable complementary tool to help for understanding and providing insights into the relationship between the lithium composition and the nature of lithium intercalation and diffusion processes. Several computational studies on Li1+xTi2O4 �0�x�1� �Fd3̄m� spinel structure have been reported;16,17 however, to the best of our knowledge, no theoretical analysis of Li dif- fusion has been reported until now. In this paper, we analyze, by means of numerical simulations, the thermodynamic and electronic properties of Li1+xTi2O4 system �0�x�0.375�; in addition, since both fourfold and sixfold vacancies are pre- sented in this structure, we have also investigated all possible Li intercalation processes and corresponding diffusion paths at different Li compositions and arrangements. The remainder of the paper is organized as follows. Sec- tion II addresses the computing methods and model systems. In Sec. III A, we focus in the intercalation processes, giving values for their formation energies. Section III B is divided in three sections in which we describe and analyze the mechanisms of the diffusion paths at different compositions �Secs. III B 1 and III B 2�, while the corresponding jump dif- fusion coefficients are reported in Sec. III B 3. Finally, we summarize our main conclusions in Sec. IV. II. COMPUTING METHODS AND MODEL SYSTEMS First-principles total energy calculations were carried out within the periodic density functional theory framework us- ing the VASP program.18,19 The Kohn-Sham equations have been solved by means of the generalized gradient approxi- PHYSICAL REVIEW B 77, 085112 �2008� 1098-0121/2008/77�8�/085112�11� ©2008 The American Physical Society085112-1 http://dx.doi.org/10.1103/PhysRevB.77.085112 mation �GGA� proposed by Perdew and Wang,20,21 and the electron-ion interaction described by the projector aug- mented wave pseudopotentials.22 The plane-wave expansion was truncated at cut off energy of 400 eV, and the Brillouin zones have been sampled through a Monkhorst-Pack 4�4 �4 k-point mesh until the forces on all atoms were less than 0.03 eV /Å assuring geometrical and energetic convergence for the Li1+xTi2O4 structures considered in this work. All the crystal structures are optimized in both simultaneously the volume of the cell and the atom positions. A spin- unrestricted approach �ISPIN option� has been employed when Li were inserted and migrated into the lattice. This methodology has been used in other recent works involving Li mobility on layered lithium transition metal oxides,23 oli- vine LiFePO4,24,25 spinel LixMn2O4,26 lithium halides,27 TiO2,28,29 and WO3,30 as well as in diffusion studies of metal subsurface impurities.31,32 Therefore, we are confident with the data calculated at this computing level for this type of processes. LiTi2O4 spinel is characterized by a close cubic packing �space group Fd3̄m� with eight LiTi2O4 units per unit cell, in which lithium ions are located at 8a tetrahedral sites, tita- nium ions at 16d octahedral sites, and oxide ions at 32e sites. Since this unit cell has 64 tetrahedral and 32 octahedral holes, there are 56 empty tetrahedra and 16 empty octahedral sites. The unit cell can be partitioned into five types of poly- hedra that fill completely its volume: TiO6 octahedra, LiO4 tetrahedra, O6 octahedra, �O4�1, and �O4�2 tetrahedra, with multiplicities of 16, 8, 16, 8, and 48, respectively. Figure 1 outlines the unit cell of LiTi2O4 showing the occupied 8a, 16d, and 32e sites and the empty 16c octahedral sites. Inter- calation processes were considered by placing Li ions in these empty 16c octahedral sites during electrochemical re- actions in the Li1+xTi2O4 �0�x�1� systems, while alterna- tive diffusion paths of the Li ions from the occupied 8a tetrahedral sites to the empty 16c octahedral sites were ana- lyzed. The optimization procedure takes into account the atomic positions of the first and second nearest neighbors of the jumping Li ions during the migration processes. Similar diffusion channels have been considered previously for LixMn2O4 spinel systems, in which the lithium intersite hop- ping between 8a and 16c sites is presumably the primary lithium conduction process.33 The energy associated with intercalation processes Eint of Li ions in a LiTi2O4 bulk was calculated by the following equation: Eint = ELi+bulk − nELi − Ebulk, �1� where ELi+bulk is the binding energy of the Li1+xTi2O4 sys- tem, n is the number of intercalated Li, ELi is the thermody- namic chemical potential of a lithium, and Ebulk is the bind- ing energy of the bulk LiTi2O4. The reference chemical potential of lithium can be calculated in several ways,34 al- though is often referred to the solid Li. One can also relate it to the Li atom in the ground state, and in the present study, ELi is calculated by placing one Li in a large cell with the same lattice parameters of the host crystal �a=8.363 Å�. The diffusion coefficient of Li ion in electrode materials is a key parameter of the rate capability of rechargeable Li batteries and it depends on several factors. Li hopping re- quires at least an adjacent vacancy to hop into, and Li ion will have more possibilities to migrate if it is surrounded by many vacancies. This depends on the overall lithium concen- tration and the equilibrium degree at short or long range. The jump diffusion coefficient DJ for lithium hopping to vacant neighboring sites has the form DJ=�l2; l and � are the hop- ping distance and rate, respectively. Here, � can be estimated by the activation barriers of the migration process as35 � = �0 exp�− �E/KBT� , �2� where �0 is the vibrational frequency of a migrating specie around its equilibrium position that is typically on the order 1013 s−1 �value used in this work� and �E is the mean jump activation energy, i.e., the difference in energy at the acti- vated state and the energy at the initial equilibrium state of the hop. The activated state is located at the maximum en- ergy point along the minimum energy path between the end points of the hop. III. RESULTS AND DISCUSSION A. Intercalation processes First, we have carried out a detailed study on the interca- lation processes of 1, 2, and 3 Li ions into the unit cell of the structure Li1+xTi2O4 �0�x�0.375�, resulting Li concentra- tions of x=0.125, 0.250, and 0.375, respectively. Different configurations were calculated corresponding to different 8a /16c ratios. The selected nomenclature was T-O, where T and O denote the numbers of Li ion located at tetrahedral �8a� and octahedral �16c� sites, respectively. Table I presents the values of both total and relative energies per unit cell of the spinel-type Li1+xTi2O4 �0�x�0.375� for the most stable arrangements of different configurations at each value of Li concentration x. It is important to remark that lithium can be inserted and extracted without significant changes of lattice dimensions. Then, the intercalation process provides a vol- ume change of less than 1% as the Li concentration is in- creased. The structure LiTi2O4, x=0, is labeled as configuration 8-0 and it presents all the Li located in 8a tetrahedral sites and has all 16c octahedral sites empty. In the configuration 7-1, seven Li ions remain in 8a sites and one Li ion is lo- cated in 16c site; in principle, there are 8 possibilities for 8a sites and 16 for 16c sites, but only 3 arrangements are non- equivalent. The most stable arrangement for this 7-1 configu- FIG. 1. Schematic figure of LiTi2O4 spinel structure. ANICETE-SANTOS et al. PHYSICAL REVIEW B 77, 085112 �2008� 085112-2 ration is depicted in Fig. 2�a�, being 1.55 eV less stable than the configuration 8-0, as presented in Table I. The configu- ration 6-2 presents four nonequivalent arrangements, and the most stable of them is shown in Fig. 2�b�. This arrangement is 2.01 eV less stable than configuration 8-0. At composition x=0.125, Li1.125Ti2O4, the most stable ar- rangement of the 8-1 configuration is depicted in Fig. 2�c�, while the most stable arrangements for the configurations 7-2 �see Fig. 2�d�� and 6-3 �see Fig. 2�e�� present values of rela- tive energies with respect 8-1 of 0.47 and 0.91 eV, respec- tively. In model Li1.250Ti2O4, x=0.250, the different arrange- ments for both configurations 8-2 and 7-3 as well as the diagram of relative energies are illustrated in Figs. 3 and 4, respectively. The three nonequivalent arrangements for the configuration 8-2 are depicted in Fig. 3�a� and the stability order is E18-2�E28-2�E38-2, where the type of configura- tion is referred as subscript. The most stable arrangement of the configuration 7-3 is shown as E1 �E17-3� in Fig. 3�b�, together with the 19 possible local Li arrangements of this configuration. An analysis of the results points out that E18-2 is less stable than E17-3 by 0.46 eV, while the most favorable arrangement of the configuration 6-4 �illustrated in Fig. 3�c�� is slightly less stable than the E18-2 by 0.02 eV �see Table I�. For composition x=0.375, the values of the relative ener- gies for the different arrangements of the configurations 8-3 and 7-4 are listed in Figs. 5�a� and 5�b�, respectively. In contrast with the results reported for x=0.250 �see Fig. 4�, four arrangements of configuration 7-4 are clearly more stable than the corresponding to 8-3 configuration. A global analysis of the results reported in Table I and Fig. 2–5 shows that for a given value of x, the lithium diffu- sion processes from 8a tetrahedral to 16c octahedral sites will be thermodynamically favorable only in the composition range of x=0.250 and 0.375, i.e., 7-3 and 7-4 are more stable than 8-2 and 8-3 configurations, respectively. Table II gives the values of intercalation energies per Li1+xTi2O4 unit cell of the most stable arrangements from different configura- tions, providing values in a range between −1.3 and −6.9 eV. When eight Li ions remain in 8a sites, each lithium interca- lated in the vacant 16c site contributes with a gain of stability of �1.4 eV. However, when seven or six Li ions are main- tained in 8a sites and one or two Li ions are located in empty 16c sites, respectively, each inserted lithium supplies an en- ergy gain of �2.4 eV in the resulting structure. These results indicate that lithium intercalation processes appear to be thermodynamically favorable in all studied range of compo- sition. This increase of stability in intercalation processes when 8a sites are not totally occupied points out that the diffusion processes from tetrahedral 8a sites to octahedral 16c sites are also energetically favorable from a thermody- namic point of view. B. Diffusion paths At a particular value of x, different Li diffusion paths in the structure Li1+xTi2O4 can be theoretically investigated in order to find energetically favorable channels from a thermo- dynamic and kinetic point of view. In agreement with previ- ous results obtained for intercalation processes, diffusion paths were analyzed at compositions x=0.250 and 0.375. Four sets for the diffusion channel have been considered, namely: �1� Li migrations from 8a to 16c sites involving the most stable Li arrangements for each configuration. Therefore, at x=0.250, a Li diffusion path from arrangements E18-2 and E17-3 has been considered, hereafter called path 1; for x TABLE I. Values of total and relative energies per Li1+xTi2O4 unit cell of the most stable arrangements from different configurations. Configuration Energy �eV� �E �eV� LiTi2O4 8-0 −458.350 0.00 7-1 −456.803 1.55 6-2 −456.343 2.01 Li1.125Ti2O4 8-1 −459.989 0.00 7-2 −459.515 0.47 6-3 −459.077 0.91 Li1.250Ti2O4 8-2 −461.630 0.00 7-3 −462.092 −0.46 6-4 −461.612 0.02 Li1.375Ti2O4 8-3 −463.335 0.00 7-4 −464.503 −1.17 FIG. 2. The most stable local Li arrangements of the configurations �a� 7-1, �b� 6-2, �c� 8-1, �d� 7-2, and �e� 6-3. The circles are 16c octahedral sites and the squares are 8a tetrahedral sites. The black circles and black squares are filled sites by Li, while the gray circles and gray squares are empty sites. INTERCALATION PROCESSES AND DIFFUSION PATHS… PHYSICAL REVIEW B 77, 085112 �2008� 085112-3 FIG. 3. �a� Local Li arrangements of the configuration 8-2, �b� local Li arrangements of the configuration 7-3, and �c� the most stable arrangement of the configuration 6-4 in the Li1.250Ti2O4 structure. The circles are 16c octahedral sites and the squares are 8a tetrahedral sites. The black circles and black squares are filled sites by Li, while the gray circles and gray squares are empty sites. Values of the total energies per Li1.250Ti2O4 unit cell are presented. ANICETE-SANTOS et al. PHYSICAL REVIEW B 77, 085112 �2008� 085112-4 =0.375, the analysis is carried out for the path that connects E18-3 to E17-4. Assuming that a wide range of different Li arrangements is formed during the Li insertion process due to internal ten- sions caused by voltage application, we can also consider Li migrations from 8a sites to 16c sites involving less stable Li arrangements. Therefore, the investigation of different Li dif- fusion paths involving less stable configurations can be clas- sified in the next sets: �2� Li diffusion path from the E18-3 to E37-4 arrange- ments, called path 2, at x=0.375. �3� Li diffusion paths from the arrangements E28-2 and E38-2 as starting points to reach E17-3, called paths 3.1 and 3.2, respectively, for x=0.250. For x=0.375, path 3.3 corre- sponds to Li migration from E88-3 to E17-4. �4� Li diffusion path from E48-3 to E27-4 arrangements, called path 4, at x=0.375. 1. At composition x=0.250 The results obtained for path 1 of set �1� are depicted in Fig. 6. The Li ion to be migrated can be located at both 8a and 16c sites. Therefore, three alternative diffusion channels can be delineated. �i� Path 1a where Li migration involves the shift of a Li ion from one 8a site to one 16c site to obtain E37-3; from this arrangement, one Li can be displaced from another 16c site to an adjacent 16c site. �ii� Path 1b where Li migrates between adjacent 16c sites to yield E38-2, and then other Li located at 8a site shifts to one 16c site. �iii� Path 1c which involves the concerted diffusion of the previous four Li migrations from E18-2 to E17-3 arrangements. The corre- FIG. 4. Diagram of relative energy values per Li1.250Ti2O4 unit cell containing nonequivalent Li arrangements of the configurations �a� 8-2 and �b� 7-3. Energy differences with regard to the arrange- ment E17-3 �−462.092 eV�. FIG. 5. Diagram of relative energy values per Li1.375Ti2O4 unit cell containing nonequivalent Li arrangements of the configurations �a� 8-3 and �b� 7-4. Energy differences with regard to the arrange- ment E17-4 �−464.503 eV�. TABLE II. Values of intercalation energies per Li1+xTi2O4 unit cell of the most stable arrangements of different configurations, calculated by Eq. �1�. The value of energy for one Li is −0.274 eV. LiTi2O4 Li1.125Ti2O4 Li1.250Ti2O4 Li1.375Ti2O4 Configuration 8-0 8-1 8-2 8-3 Eint �eV� 0.00 −1.37 −2.73 −4.16 Configuration 7-1 7-2 7-3 7-4 Eint �eV� 0.00 −2.44 −4.74 −6.88 Configuration 6-2 6-3 6-4 Eint �eV� 0.00 −2.46 −4.72 INTERCALATION PROCESSES AND DIFFUSION PATHS… PHYSICAL REVIEW B 77, 085112 �2008� 085112-5 sponding energy profiles, energy vs hoping distance, for both paths 1a and 1b are presented in Figs. 6�a�–6�c�, respectively, while a tridimensional surface of path 1c is depicted in Fig. 6�c�. An analysis of the results shows that the first stage of path 1a is associated with a continuous increase of the energy from E18-2 to E37-3, 0.50 eV; while the second step presents a high barrier of 3.92 eV at 1.5 Å. These energetic values corresponding to the different migrations can be associated with the Li overlapping with neighboring oxygen anions along its motion toward the vacant site. Therefore, the key role of oxide ions along the electronic exchange during the migration processes has been analyzed. In the first stage of path 1a, Li hopes from the 8a to the 16c site and this channel takes place across the center of a triangle made by three neighboring oxygen anions without significant structural modifications; as a result, this path presents a lower barrier height. However, in the second step of path 1a, Li moves FIG. 6. Energy profiles for Li diffusion paths belonging to set �1� for x=0.250: �a� path 1a involving two Li migrations from E18-2 to E17-3; �b� path 1b implying two Li migrations from E18-2 to E17-3; �c� simultaneous path 1c. ANICETE-SANTOS et al. PHYSICAL REVIEW B 77, 085112 �2008� 085112-6 between adjacent 16c sites implying the direct pass through two oxygen anions which undergo an increase of their nega- tive charge to �6.5%; thus, at the maximum in the energy profile, a large value of electronic charge from the migrating Li is transferred to them. In addition, from a structural point of view, a noticeable enlargement of the distance between these two oxygen anions takes place from 3.07 to 3.28 Å. Then, the value of energy barrier of the diffusion path in- creases on going from the first to the second stage of path 1a. These remarks are similar to those employed in the Li mo- bility in LixCoO2 by Van der Ven et al.36 and lithium halides by Kishida et al.27 Path 1b presents an opposite trend with regards to path 1a: the first stage presents a barrier height of 4.40 eV and the jumping Li ion overlaps significantly with neighboring oxygen counterions while it passes through, re- sulting in a significant energy barrier. This structural hin- drance is avoided along the second stage since Li ion moves through the center of the triangle made by the three neigh- boring oxygen anions. Path 1c takes place via a maximum, located at 1.2 and 1.5 Å, with a relative energy with respect E18-2 of 3.84 eV. As can be seen in Fig. 4, the Li1.250Ti2O4 structure has only one arrangement of the configuration 7-3 �E17-3� more stable than the arrangements of the configuration 8-2. The investigation of Li diffusion processes of set �3� involving less stable arrangements as starting points, E28-2 and E38-2, to reach the final most stable arrangement, E17-3, is carried out by the study of paths 3.1 and 3.2, respectively. The mi- gration paths from E28-2 to E17-3 were also analyzed in three different forms. �i� Path 3.1a, where one Li migrates from one 8a site to one 16c site to reach E37-3, and then a different Li ion shifts between 16c sites, achieving the arrangement E17-3. �ii� Path 3.1b, where Li migrates between 16c sites, and then another Li moves from one 8a site to one 16c site also reaching the arrangement E17-3. �iii� Path 3.1c, which implies the simultaneous diffusion of the previous four Li migrations from E28-2 to E17-3. An analysis of the corre- sponding energy profiles for both paths 3.1a and 3.1b pre- sented in Figs. 7�a� and 7�b�, respectively, shows an increase of the energy from E28-2 to E37-3 of 0.46 eV in the first stage of path 3.1a and a high barrier of 3.92 eV associated with the second stage; however, for path 3.1b, the first stage implies the overcoming of an energy barrier of 3.98 eV and in the second one, the achievement of the arrangement E17-3 along a barrierless process. Finally, in Fig. 7�c�, a tridimensional representation for path 3.1c is depicted, where concerted Li migration processes involve a barrier height of 3.77 eV with respect E28-2, and this maximum is located at 1.3 and 1.5 Å. Path 3.2 from E38-2 to E17-3 classified inside set �3� only involves one migration process between 8a and 16c sites without energy barrier. This path has been previously de- scribed in Fig. 6�b� and can be envisioned as the second segment of path 1b. 2. At composition x=0.375 For x=0.375, four arrangements belonging to configura- tion 7-4 are more stable than eight of the configuration 8-3 �see Fig. 5�, and consequently, there are several possibilities of Li diffusion paths between both configurations. Therefore, we have selected for each set of diffusion processes, �1�–�4�, the corresponding path that involves a minor number of Li migrations between the different sites of the Li1.375Ti2O4 structure. The diffusion path belonging to set �1� implies the Li migration from the arrangement E18-3 to the E17-4. This channel involves at least three migration processes, namely, two migrations between 16c sites and one migration between 8a and 16c sites. The energy profile depicted in Fig. 8 is associated with a Li migration process between 16c sites, which presents a large value for the energy barrier, 4.22 eV. Therefore, the subsequent steps have not been considered. The diffusion path of set �2� at x=0.375, related to the Li mobility from E18-3 to E37-4 arrangements, involves at least three migration processes between 8a and 16c sites. This path, called path 2, was investigated by two different ways and is depicted in Fig. 9�a�. The first one, path 2a, is a step- wise involving three consecutive processes: �i� the Li hoping between Tet1 �black square� and Oct4 �white circle�, �ii� the Li migration between Tet2 �black square� and Oct5 �white circle�, and �iii� the Li migration between Oct1 �black circle� and Tet2 �black square�. The second one corresponds to a concerted movement of the Li ions along the three different migrations above mentioned, path 2b. The energy profile for both path 2a �black square points� and path 2b �white square points� is presented in Fig. 9�b�. In the first step of path 2a, one Li migrates from the tetrahedral site Tet1 to the octahe- dral site Oct4 through the points 1, 2, 3, 4, and 5. In the second step, one Li migrates from the tetrahedral site Tet2 to the octahedral site Oct5 through the points 1�, 2�, 3�, 4�, and 5�. In the third step, one Li migrates from the octahedral site Oct1 to the tetrahedral site Tet2 through the points 1�, 2�, 3�, 4�, and 5�. The continuous path 2b of set �2� can be de- scribed in the following way. First, Li from sites Tet1, Tet2, and Oct1 migrate simultaneously to the points 1, 1�, and 1�, respectively. Second, one Li from points 1, 1�, and 1� mi- grate simultaneously to the points 2, 2�, and 2� and thus successively until to reach the arrangement E37-4. This con- tinuous path 2b presents an overall barrier height of 0.48 eV, lower than the barrier of path 2a �0.54 eV� and also is more realistic because the repulsive forces act on the three Li at the same time. These paths resulted in a lower energy barrier when compared with the barrier of 4.22 eV from one path segment of set �1� involving Li migration between octahedral sites �Fig. 8�. In Fig. 10�a�, one diffusion path belonging to set �3� for x=0.375 is illustrated, where one Li migrates from the ar- rangement E88-3 to the arrangement E17-4. This path 3.3 only involves one migration process between 8a and 16c sites, and its energy profile graph shows the Li migration to the more stable �1.44 eV� arrangement E17-4 without energy bar- rier. The diffusion path of set �4� at x=0.375 is presented in Fig. 10�b�. It has been studied the Li mobility along path 4 from E48-3 to E27-4, which only involves one migration be- tween 8a and 16c sites. As depicted in Fig. 10�b�, Li in 8a site �square� migrates to a 16c site �circle� reaching a more stable �0.48 eV� arrangement E27-4 without potential barrier. This path can also be classified as a Li migration path seg- ment of set �1� because it is a continuity of the path from Fig. 8. INTERCALATION PROCESSES AND DIFFUSION PATHS… PHYSICAL REVIEW B 77, 085112 �2008� 085112-7 3. Jump diffusion coefficient It is interesting to elucidate the variation of the hopping barrier and the jump diffusion coefficient with composition, along the different diffusion paths found between each con- figuration. Since the hop frequency � is proportional to the exponential of the activation barrier, the jump diffusion co- efficient DJ can be easily obtained taking into account the hopping distance. For composition x=0.250, the simulta- neous path 1c from E18-2 to E17-3 �set �1�� involves a barrier height of 3.84 eV, whereas a minimum value of 3.77 eV is obtained for the simultaneous path 3c starting from E28-2 �set �3��. The DJ for both is in order of �10−19 cm2 s−1 �900 °C�. For composition x=0.375, one segment of the Li migration corresponding to set �1� involves a potential barrier of 4.22 eV, which is a DJ similar to previous composition. Nev- ertheless, the concerted path 2b corresponding to set �2� de- scribed in Fig. 9 has a lower potential barrier of 0.48 eV, which results in a DJ=2.8�10−5 cm2 s−1 for a hopping dis- tance l=1.81 Å �900 °C�. To our knowledge, there are only two diffusion experiments of lithium ion in Li-Ti-O spinel oxides. One of them, using neutron radiography, obtained values of the jump diffusion coefficient in the range of �1.3–2.2��10−6 cm2 s−1 at 860, 880, and 900 °C, respec- tively, for the Li1.33Ti1.67O4 structure.37 Recently, a second FIG. 7. Energy profiles for Li diffusion paths belonging to set �3� for x=0.250: �a� path 3.1a involving two Li migrations from E28-2 to E17-3; �b� path 3.1b implying two Li migrations from E28-2 to E17-3; �c� simultaneous path 3.1c. ANICETE-SANTOS et al. PHYSICAL REVIEW B 77, 085112 �2008� 085112-8 study for the same structure reported a jump rate value of 4.9�108 s−1 above 420 K by means of NMR and impedance spectroscopy.38 These reported studies can be interpreted as a resulting measure of the total migration possibilities com- pared to our theoretical results, which can be understood as upper bounds. A comparison to the similar LixMn2O4 spinel can be performed since reported data using one-dimensional NMR spectral intensities33 have determined activation ener- gies for jumping Li from 8a to 16c sites and vice versa; their value of 0.5 eV for the activation energy is in good agree- ment with the value found in this study, and they reported DJ values of 10�10−16 cm2 s−1 for a hopping distance l =1.78 Å, although at a range of temperature between 300 and 400 K. IV. CONCLUSIONS In this study, the ability of Li1+xTi2O4 spinel structure to be used as electrode material was investigated by means of first-principles calculations based on the density functional theory under the GGA approximation. The results presented have strong implications toward improved intercalation and diffusion processes in spinel-type structured Li1+xTi2O4 �0 �x�0.375�, which forms the basis of a number of solid- state electrochemical applications and needs to a deep theo- retical comprehension in order to identify the more favorable stages of the diffusion process during the Li insertion and/or extraction. The different arrangements for each studied com- position have been characterized in depth, and energy pro- files for Li migration paths as well as the corresponding jump diffusion coefficients have been reported. The main results can be summarized as follows. �i� Calculations predict that the Li1+xTi2O4 spinel be- comes more stable after the Li intercalation process in 16c sites for all range studied of x, whereas Li diffusion pro- FIG. 8. Energy profile for a Li diffusion path segment of set �1� for x=0.375 involving a Li migration process from E18-3 to E48-3. FIG. 9. �a� Li diffusion paths belonging to set �2� for x=0.375, in which Li moves from E18-3 to E37-4 involving three Li migration processes between 8a and 16c sites. �b� Energy profiles for path 2a �black squares�; path 2b continuous �white squares�. INTERCALATION PROCESSES AND DIFFUSION PATHS… PHYSICAL REVIEW B 77, 085112 �2008� 085112-9 cesses from tetrahedral 8a sites to octahedral 16c sites are thermodynamically favorable only in the compositions x �0.250. �ii� The diffusion process from the most stable Li arrange- ment of a configuration to the most stable arrangement of another configuration involves at least one Li migration be- tween 16c sites. The Li migrations between adjacent 16c sites present higher energy barriers ��4 eV� implying the direct pass through two oxygen anions which undergo an increase of their negative charge to �6.5% as well as a no- ticeable enlargement of the distance between them at the maximum in the energy profile. �iii� Lithium ions located in less stable Li local arrange- ments become free toward the 8a-16c diffusion with minor energetic cost than the lithium ions positioned in more stable arrangements, being the Li bindings stronger in more stable arrangements than in less stable arrangements. Therefore, the diffusion paths from less stable arrangements involving Li migrations between 8a and 16c sites are more favorable be- cause the Li migrates without or with lower potential barriers ��0.5 eV�, and the pass through the center of a triangle made by three neighboring oxygen anions takes place nearly without structural modification. �iv� Path 3.2 from E38-2 to E17-3 for x=0.250 and path 3.3 from E88-3 to E17-4 and path 4 from E48-3 to E27-4 for x =0.375 only involve one migration process between 8a and 16c sites, being the most favorable diffusion pathways with- out overcoming potential barriers. In addition, the concerted path 2b has a lower potential barrier �0.48 eV� which results in a diffusion coefficient of 2.8�10−5 cm2 s−1. ACKNOWLEDGMENTS This work was partially supported by DGI �Project No. CT2006-15447-CO2-O1/BQU�, by Generalitat Valenciana �Project No. ACOMP06/122�, and Fundación Bancaixa-UJI �Projects Nos. P1 1B2005-20 and P1 1B2005-13�. L. Gracia acknowledges support by the Postdoctoral grant �APOSTD07� from Generalitat Valenciana. M. Anicete- Santos also acknowledges the Brazilian agency CNPq. *andres@qfa.uji.es 1 J. M. Tarascon and M. Armand, Nature �London� 414, 359 �2001�. 2 J. Thomas, Nat. Mater. 2, 705 �2003�. 3 A. Hammami, N. Raymond, and M. Armand, Nature �London� 424, 635 �2003�. 4 B. Scrosati, Nature �London� 373, 557 �1995�. 5 K. M. Colbow, J. R. Dahn, and R. R. Haering, J. Power Sources 26, 397 �1989�. 6 D. C. Johnston, H. Prakash, W. H. Zachariasen, and R. Viswanathan, Mater. Res. Bull. 8, 777 �1973�. 7 A. Deschanvres, B. Raveau, and Z. Sekkal, Mater. Res. Bull. 6, 699 �1971�. 8 E. Ferg, R. J. Gummow, A. Kock, and M. M. Thackeray, J. Elec- trochem. Soc. 141, L147 �1994�. 9 T. Ohzuku, A. Ueda, and N. Yamamoto, J. Electrochem. Soc. 142, 1431 �1995�. 10 D. Peramunage and K. M. Abraham, J. Electrochem. Soc. 145, 2609 �1998�. 11 K. Ariyoshi, R. Yamato, and T. Ohzuku, Electrochim. Acta 51, 1125 �2005�. 12 F. Ronci, P. Reale, B. Scrosati, S. Panero, V. R. Albertini, P. Perfetti, M. di Michiel, and J. M. Merino, J. Phys. Chem. B 106, 3082 �2002�. 13 D. W. Murphy, M. Greenblatt, S. M. Zahurak, R. J. Cava, J. V. Waszczak, G. W. Hull, and R. S. Hutton, Rev. Chim. Miner. 19, 441 �1982�. 14 R. J. Cava, D. W. Murphy, S. Zahurak, A. Santoro, and R. S. Roth, J. Solid State Chem. 53, 64 �1984�. 15 D. C. Johnston, J. Low Temp. Phys. 25, 145 �1976�. 16 Y. Takahashi, Y. Gotoh, and J. Akimoto, J. Phys. Chem. Solids 63, 987 �2002�. 17 W. Y. Ra, M. Nakayama, Y. Uchimoto, and M. Wakihara, J. Phys. Chem. B 109, 1130 �2005�. 18 G. Kresse and J. Hafner, Phys. Rev. B 49, 14251 �1994�. 19 G. Kresse and J. Furthmuller, Comput. Mater. Sci. 6, 15 �1996�. FIG. 10. Energy profiles for Li diffusion paths belonging to sets �3� and �4� for x=0.375: �a� path 3.3 belonging to set �3� involving one Li migration from E88-3 to E17-4; �b� path 4 belonging to set �4�, which involves a Li migration from E48-3 to E27-4. ANICETE-SANTOS et al. PHYSICAL REVIEW B 77, 085112 �2008� 085112-10 20 J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 �1992�. 21 J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 �1992�. 22 G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 �1999�. 23 K. Kang and G. Ceder, Phys. Rev. B 74, 094105 �2006�. 24 Chuying Ou-Yang, Siqi Shi, Z. Wang, X. Huang, and L. Chen, Phys. Rev. B 69, 104303 �2004�. 25 S. Miao, M. Kocher, P. Rez, B. Fultz, R. Yazami, and C. C. Ahn, J. Phys. Chem. A 111, 4242 �2007�. 26 C. Y. Ouyang, S. Q. Shi, Z. X. Wang, H. Li, X. J. Huang, and L. Q. Chen, Europhys. Lett. 67, 28 �2004�. 27 I. Kishida, Y. Koyama, A. Kuwabara, T. Yamamoto, F. Oba, and I. Tanaka, J. Phys. Chem. B 110, 8258 �2006�. 28 F. Tielens, M. Calatayud, A. Beltrán, C. Minot, and J. Andrés, J. Electroanal. Chem. 581, 216 �2005�. 29 H. Iddir, S. Ögüt, P. Zapol, and N. D. Browning, Phys. Rev. B 75, 073203 �2007�. 30 L. Gracia, J. Garcia-Cañadas, G. Garcia-Belmonte, A. Beltrán, J. Andrés, and J. Bisquert, Electrochem. Solid-State Lett. 8, J21 �2005�. 31 L. Gracia, M. Calatayud, J. Andrés, C. Minot, and M. Salmeron, Phys. Rev. B 71, 033407 �2005�. 32 G. W. Peng, A. C. H. Huan, E. S. Tok, and Y. P. Feng, Phys. Rev. B 74, 195335 �2006�. 33 V. W. J. Verhoeven, I. M. de Schepper, G. Nachtegaal, A. P. M. Kentgens, E. M. Kelder, J. Schoonman, and F. M. Mulder, Phys. Rev. Lett. 86, 4314 �2001�. 34 Z. Lodziana and J. Piechota, Phys. Rev. B 74, 184117 �2006�. 35 G. H. Vineyard, J. Phys. Chem. Solids 3, 121 �1957�. 36 A. Van der Ven, G. Ceder, M. Asta, and P. D. Tepesch, Phys. Rev. B 64, 184307 �2001�. 37 S. Takai, M. Kamata, S. Fujine, K. Yoneda, K. Kanda, and T. Esaka, Solid State Ionics 123, 165 �1999�. 38 M. Wilkening, R. Amade, W. Iwaniak, and P. Heitjans, Phys. Chem. Chem. Phys. 9, 1239 �2007�. INTERCALATION PROCESSES AND DIFFUSION PATHS… PHYSICAL REVIEW B 77, 085112 �2008� 085112-11