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Effects of Off-Stoichiometry of LiC6 on the Lithium Diffusion Mechanism and Diffusivity by First Principles Calculations Kazuaki Toyoura,*,† Yukinori Koyama,† Akihide Kuwabara,‡ and Isao Tanaka*,†,‡ Department of Materials Science and Engineering, Kyoto UniVersity, Yoshida, Sakyo, Kyoto 606-8501, Japan, and Nanostructure Research Laboratory, Japan Fine Ceramics Center, Atsuta, Nagoya, 456-8587, Japan ReceiVed: October 23, 2009; ReVised Manuscript ReceiVed: December 29, 2009
Lithium diffusion in a stage-1 structure of LiC6 has been theoretically investigated from first principles on the basis of transition state theory. The calculated chemical diffusion coefficient of lithium atoms under the lithium excess conditions is much larger than that under the lithium deficient conditions, e.g., 2 × 10-7 cm2/s vs 1 × 10-10 cm2/s at room temperature. The calculated activation energies of the chemical diffusion coefficients under the lithium excess and deficient conditions are also different, e.g., 0.30 and 0.49 eV, respectively. The fast diffusion under the lithium excess conditions is attributed to the “interstitialcy” mechanism. These results mean that a small deviation of lithium composition from the exact composition of LiC6 causes a large difference in the lithium diffusion coefficient. The off-stoichiometry can be the major reason why the experimental diffusion coefficients and activation energies are widely scattered. I. Introduction Lithium-graphite intercalation compounds (Li-GICs) have been extensively investigated,1-6 because of their application to negative electrodes in lithium-ion rechargeable batteries. It is well-known that lithium atoms form a superstructure in interlayers of graphite, and that the graphene and lithium layers are stacked with periodicity. This periodic stacking, which is called “staging”, is characterized by a periodic sequence of lithium layers. The stage number n refers to the number of graphene layers separating two lithium layers. Lithium atoms are inserted into graphite with staging phase transitions repeated, finally to reach a stage-1 structure of LiC6. Lithium diffusion in each of the staging phases has been investigated experimentally using electrochemical methods, such as alternating-current (AC) impedance measurements and potentiostatic and galvanostatic intermittent titration techniques (PITT and GITT).4-6 However, the reported chemical diffusion coefficients, DLi, are widely scattered in the range of several orders of magnitude. For example, the reported DLi in the stage-1 structure of LiC6 are distributed between 10-12 and 10-7 cm2/s at room temperature.4-6 This implies the presence of unknown experimental parameters behind the wide scattering. The diffusion coefficient would be strongly dependent on the selection of experimental conditions. In the present study, we employ a theoretical approach to evaluate the lithium diffusion in Li-GICs, particularly focusing on the lithium in-plane diffusion in the stage-1 structure of LiC6. We have recently developed a technique to evaluate mean frequencies of atomic jumps in a crystal from first principles on the basis of transition state theory.7 The mean jump frequencies were quantitatively evaluated from the potential barriers and the phonon frequencies at both initial and saddlepoint states of the jumps under the harmonic approximation. The lattice vibrations were treated within quantum statistics, * To whom correspondence should be addressed. E-mail: k.toyoura0315@ gmail.com (K.T.);
[email protected] (I.T.). † Kyoto University. ‡ Japan Fine Ceramics Center.
not using the conventional treatment by Vineyard,8 corresponding to the classical limit. Lithium diffusion by the interstitial and vacancy mechanisms in LiC6 was examined as a model case. However, no information was given for realistic LiC6 in which different kinds of diffusion mechanisms play roles for the lithium diffusion depending upon experimental conditions, such as the temperature and the chemical potential of lithium. In the literature, the superstructure of lithium layers in LiC6 was reported to be stable up to 715 K.9 Therefore, the lithium diffusion under the temperature can be considered to be mediated by point defects, i.e., lithium vacancies and interstitials. However, dominant point defects that determine the diffusion coefficients should be different between lithium deficient and excess conditions, namely, in Li1-δC6 and Li1+δC6. The deviation from the stoichiometric composition, (δ, corresponds to the difference in concentration between lithium vacancies and interstitials in the present system. Therefore, the sign of the deviation is a crucial factor determining the dominant point defects, i.e., lithium vacancies in Li1-δC6 versus lithium interstitials in Li1+δC6, even though the deviation was reported to be small (e.g., below 0.01 up to 433 K).10 The present study aims at revealing the lithium diffusion mechanism and the diffusion coefficient under these two conditions from first principles. This would clarify the origin of the wide scattering in experimental diffusion coefficients. II. Computational Procedures On the basis of transition state theory, a mean frequency of a type of atomic jumps in a crystal is expressed as
ω)
(
kT ∆Emig + ∆Fvib exp h kT
)
(1)
where k is the Boltzmann constant, T is the temperature, h is Planck’s constant, ∆Emig is the potential barrier of the atomic jumps, and ∆Fvib is the change in vibrational free energy from the initial state to the saddle-point state for the jumps. In the present study, the potential barriers and the vibrational free
10.1021/jp910134u 2010 American Chemical Society Published on Web 01/15/2010
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Figure 1. (a) Crystal structure of LiC6 from the c-axis view. Schematic diagrams of the lithium migrations by (b) the vacancy mechanism, (c) the interstitial mechanism, and (d) the interstitialcy mechanism. The hexagons, open and solid circles, and squares denote the carbon network of graphene sheets, lithium atoms at regular sites and interstitial sites, and lithium vacancies, respectively. The open and solid arrows denote migrations of lithium atoms and defects, respectively.
energies at the initial and saddle-point states were evaluated from first principles, using the nudged elastic band (NEB) method11 and the frozen phonon method implemented in the “fropho” code,12,13 respectively. The first-principles calculations were performed using the projector augmented wave (PAW) method implemented in the VASP code.14 The local density approximation (LDA) or the generalized gradient approximation (GGA) is often used for the exchange-correlation term. It is well-known that the GGA does not reproduce weak interlayer interactions involving van der Waals interactions in graphite and the related compounds including lithium-graphite systems.15-17 Therefore, the LDA parametrized by Perdew and Zunger18 was chosen in the present study, though the binding energy between lithium atoms and graphene sheets tends to be overestimated by the LDA.15,16 The plane wave cutoff energy was 350 eV. The 2s and 2p orbitals were treated as valence states for both lithium and carbon. A supercell consisting of 3 × 3 × 2 unit cells of LiC6 was used with a k-point grid of 3 × 3 × 6. Atom positions were fully optimized until the residual forces became less than 0.02 eV/ Å. Since vibrational frequencies are sensitive to the residual forces of the structures before being displaced, the structures were precisely optimized with the convergence of 10-5 eV/Å of the residual forces. Each atom in the supercell was displaced by (0.01 Å in each of x, y, and z directions to obtain all of the interatomic force constants. A supercell consisting of 2 × 2 × 2 unit cells was used with a k-point grid of 5 × 5 × 6 for the phonon calculations in the interstitial and interstitialcy mechanisms. The reduction of the supercell size can be reasonable because the difference in ∆Fvib between 3 × 3 × 2 and 2 × 2 × 2 unit cells is small, e.g., 2 meV at 0 K and 8 meV at 1000 K in the case of the interstitial mechanism. III. Results and Discussion A. Classification of Diffusion Mechanisms in LiC6. Lithium atoms in the stage-1 structure of LiC6 form a (3 × 3)R30° superstructure in every interlayer of graphene sheets, and the graphene and lithium layers make an ARAR stacking (A, graphene layer; R, lithium layer). From the out-of-plane view (Figure 1a), lithium atoms regularly occupy one-third of the
carbon hexagonal sites. These sites are called “regular sites”, and the rest of the unoccupied hexagonal sites are “interstitial sites” hereafter. In the present study, three mechanisms, i.e., the vacancy mechanism, the interstitial mechanism, and the interstitialcy mechanism, are taken as possible diffusion mechanisms in LiC6. The schematic diagrams of the three mechanisms are shown in Figure 1b-d. A lithium vacancy is surrounded by the six lithium atoms at regular sites, and one of them jumps into the vacancy. According to our previous work,7 the neighboring lithium atom does not migrate in a straight line trajectory but by way of a first-nearest-neighbor interstitial site. Therefore, the vacancy mechanism has 12 paths in total. A lithium interstitial is located at an unoccupied hexagonal site, i.e., an interstitial site, surrounded by three unoccupied interstitial sites and three regular sites. The interstitial mechanism corresponds to a direct jump of the lithium interstitial into one of the three neighboring interstitial sites. On the other hand, the interstitialcy mechanism proceeds in two stages: a lithium atom at one of the neighboring regular sites jumps into an unoccupied interstitial site, and then, the original interstitial subsequently jumps into the vacant regular site formed by the former jump. The interstitialcy mechanism has three types (I, II, and III) depending on the difference in symmetry of the jump direction, as shown in the figure. B. Mean Frequencies of Lithium Jumps in LiC6. The calculated energy profiles during lithium migration in types I and II of the interstitialcy mechanism are shown in Figure 2. Type III is not shown in the figure because there is no direct path of type-III jump as a result of the NEB method. The energy profiles of the vacancy mechanism and the interstitial mechanism in our previous report7 are also shown for comparison. The horizontal axis denotes the normalized migration path. Each type of the interstitialcy mechanism goes through a metastable state at the intermediate state, and consists of two elementary jumps: one is the initial jump of a neighboring lithium atom at a regular site, and the other is the subsequent jump of the original interstitial. The calculated potential barriers, ∆Emig, of the first and second jumps are 0.28 and 0.12 eV in type I and 0.30 and 0.13 eV in type II, respectively. There is a slight difference in
Effects of Off-Stoichiometry of LiC6
J. Phys. Chem. C, Vol. 114, No. 5, 2010 2377 is rate-determining. Compared with the vacancy and interstitial mechanisms, the mean frequencies of the rate-determining jumps in the interstitialcy mechanism are much higher than those in the other two mechanisms, e.g., by over 3 orders of magnitude at room temperature. C. Diffusion Coefficient under Lithium Deficient Conditions. The chemical diffusion coefficient of lithium atoms, DLi, is defined by Fick’s first law as
JLi ) -DLi∇CLi Figure 2. The calculated energy profiles during lithium migration by (a) type I and (b) type II of the interstitialcy mechanism (green lines). The calculated energy profiles in the vacancy and interstitial mechanisms are also shown by the blue and red broken lines, respectively. The horizontal axis denotes the normalized migration path. The migration distances in types I and II of the interstitialcy mechanism, the vacancy mechanism, and the interstitial mechanism are 5.0, 4.3, 4.3, and 2.5 Å, respectively.
(2)
where JLi and CLi are the flux and concentration of lithium atoms, respectively. Under the lithium deficient conditions, lithium vacancies mediate the lithium diffusion, and lithium interstitials are negligible. Therefore, JLi counterbalances the flux of vacancies, JV, i.e., JLi ) -JV. The summation of the two concentrations of lithium atoms and vacancies, CLi + CV, is equal to the constant concentration of regular sites, leading to ∇CLi + ∇CV ) 0. Hence, eq 2 can be rewritten as
JV ) -DLi∇CV
(3)
This equation can be interpreted as Fick’s first law of vacancies. DLi under the lithium deficient conditions is equal to the chemical diffusion coefficient of lithium vacancies, DV. According to the fluctuation dissipation theorem,19 the chemical diffusion coefficient of lithium vacancies, which can be considered to be independent in LiC6 due to their small concentrations,9,10 is equal to the self-diffusion coefficient, D*V. The chemical diffusion coefficient of lithium atoms, DLi, is then equal to the self-diffusion coefficient of lithium vacancies, D*V. The self-diffusion coefficient of independent particles is given by the following definitional equation:20
D* ) lim
nf∞
Figure 3. The calculated mean frequencies of the first and second jumps in types I (green lines) and II (black lines) of the interstitialcy mechanism. For reference, the mean jump frequencies of the first and second jumps in the vacancy mechanism (blue lines) and in the interstitial mechanism (red lines) are also shown in the figure. The solid and broken lines denote rate-determining and non-rate-determining jumps, respectively.
potential barriers between types I and II of the interstitialcy mechanism, while these potential barriers are much lower than those in the vacancy mechanism and the interstitial mechanism. The low potential barriers result in a remarkable difference in mean jump frequencies between the interstitialcy mechanism and the other two. Figure 3 shows the calculated mean frequencies for the first and second jumps in types I and II of the interstitialcy mechanism. For comparison, the calculated mean jump frequencies in the vacancy and interstitial mechanisms are shown together in the figure. The second jump of each type of the interstitialcy mechanism has a much higher frequency than that of the first, which means that the first jump
〈 〉 |Rn | 2 2dtn
(4)
where d is the dimension of the diffusion field (d ) 2 for inplane diffusion) and Rn and tn are the displacement vector and the time after n jumps, respectively. The self-diffusion coefficient of lithium vacancies in LiC6 can be analytically evaluated from the mean jump frequencies, though the kinetic Monte Carlo (KMC) simulation is often used for estimation of self-diffusion coefficients. The results of the analytical evaluation are reported in this paper, because we obtained the same results using both analytical evaluation and KMC. Under the lithium deficient conditions, eq 4 can be rewritten using the mean time of vacancy migration from one site to another, τv, as follows:
D*V )
RV2 2dτV
(5)
where Rv is the migration distance of a lithium vacancy, equal to 4.3 Å. τv is the summation of the two mean dwell times at the initial and metastable states in the vacancy mechanism, i.e., τv ) τv,initial + τv,metastable. τv,initial and τv,metastable are related to the mean frequencies of the first and second jumps, i.e., τv,initial ) 1/(6ωv,first) and τv,metastable ) 1/ωv,second. The two mean dwell times are shown in Figure 4. τv,metastable is much smaller than τv,initial at
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Toyoura et al. regular sites are essentially zero. Therefore, the flux and the concentration gradient of all of the lithium atoms are equal to those of the interstitials, i.e., JLi ) JInt and ∇CLi ) ∇CInt, respectively. Hence, eq 2 becomes equivalent to Fick’s first law of lithium interstitials:
JInt ) -DLi∇CInt
Figure 4. The mean dwell times of the initial (solid lines) and metastable (broken lines) states under the lithium excess (green lines) and deficient (blue lines) conditions.
(6)
DLi under the lithium excess conditions is definitely equal to the chemical diffusion coefficient of lithium interstitials, DInt. Furthermore, DInt is equal to the self-diffusion coefficient, D*Int, for the same reason as the lithium deficient conditions. The self-diffusion coefficient of lithium interstitials can also be analytically evaluated, though it is complicated due to the multiple mechanisms. A lithium interstitial has 12 migration paths of the interstitial and interstitialcy mechanisms: 3 paths of the interstitial mechanism and 3 in type I and 6 in type II of the interstitialcy mechanism. Note that each of the paths in the interstitialcy mechanism is additionally divided into two paths. There are two possible jumps as a second jump in the interstitialcy mechanism: one is the forward jump of the original interstitial into the vacant site formed by the first jump, and the other is the backward jump of the lithium atom having migrated by the first jump. The former jump means a successful process, and the latter means failure. The migration of the interstitial, therefore, has 21 paths in total. The self-diffusion coefficient of lithium interstitials is given by an equation similar to eq 5: 21
D*Int )
Figure 5. The calculated chemical diffusion coefficients of lithium atoms under the lithium excess conditions (green lines) and the lithium deficient conditions (blue lines) as a function of (a) temperature and (b) the inverse of temperature. For reference, the chemical diffusion coefficient by the interstitial mechanism only is also shown by a red line.
low temperatures, and the difference between the two decreases with increasing temperature. τv can be approximated by τv,initial in this temperature range. The calculated chemical diffusion coefficients of lithium atoms, DLi, under the lithium deficient conditions, which is equal to D*V, are shown in Figure 5. DLi at room temperature is 1 × 10-10 cm2/s, and the calculated activation energy from 100 to 1000 K is 0.49 eV. D. Diffusion Coefficient under Lithium Excess Conditions. Under the lithium excess conditions, the lithium diffusion is mediated by lithium interstitials, and lithium vacancies are negligible. All of the regular sites can be considered to be occupied by lithium atoms, as long as the initial and final states in the migration of lithium interstitials are focused. This means the flux and the concentration gradient of lithium atoms at
∑ pk|rk|2
k)1
2dτInt
(7)
where τInt is the mean time per interstitial migration, pk is the probability that path k is selected, and rk is the jump vector of 21 ωk,1st, using the mean path k. pk is expressed as pk ) ωk,1st/∑k)1 frequencies of the first jumps. τInt is the summation of the two mean dwell times at the initial state and the metastable states of the interstitialcy mechanism, τInt,initial and τInt,metastable. The two mean dwell times are shown in Figure 4. The relation between τInt,initial and τInt,metastable shows the same tendency as that under the lithium deficient conditions, and the two mean dwell times become almost the same at 1000 K. The calculated chemical diffusion coefficients of lithium atoms under the lithium excess conditions ()D*Int) are shown in Figure 5. The reason why DLi in the Arrhenius plot slightly bends around 1000 K is that τInt,metastable is not negligible in the high-temperature range. DLi under the lithium excess conditions is 2 × 10-7 cm2/s at room temperature, and the calculated activation energy from 100 to 1000 K is 0.30 eV. DLi by the interstitial mechanism only is also shown in the figure, which is much smaller than DLi by both interstitial and interstitialcy mechanisms, e.g., by 4 orders of magnitude at room temperature. This indicates that the “interstitialcy” mechanism is dominant under the lithium excess conditions. E. Comparison of Theoretical Diffusion Coefficients with Experimental Data. DLi under the lithium excess conditions is much larger than that under the lithium deficient conditions, e.g., 2 × 10-7 cm2/s vs 1 × 10-10 cm2/s at room temperature. This suggests that the off-stoichiometry of the lithium composition in LiC6 can be the major reason why the reported DLi are widely scattered. The calculated apparent activation energies
Effects of Off-Stoichiometry of LiC6 are 0.30 and 0.49 eV under the lithium excess and deficient conditions, respectively. There are some reports on the activation energy of lithium jumps in LiC6 using quasi-elastic neutron scattering (QENS) and nuclear magnetic resonance (NMR) measurements. The reported activation energies are 1.0 eV21 by QENS and 0.222 and 0.6 eV23 by NMR. The calculated activation energies are in good agreement with the two by NMR. The high activation energy by QENS would result from the high temperature range of the measurement (630-675 K). They pointed out in ref 23 that the high activation energy probably contains a significant contribution from the enthalpy of vacancy formation. IV. Conclusion Lithium diffusion in a stage-1 structure of LiC6 has been theoretically investigated from first principles on the basis of transition state theory. The calculated chemical diffusion coefficients are much different between the lithium excess and deficient conditions, e.g., 2 × 10-7 cm2/s vs 1 × 10-10 cm2/s at room temperature. The fast lithium diffusion under the lithium excess conditions is attributed to the interstitialcy mechanism. The calculated activation energies of the chemical diffusion coefficients under the lithium excess and deficient conditions are also different, 0.30 and 0.49 eV, respectively, which are in good agreement with the reported values by the NMR measurements. These calculated results suggest that a small deviation of lithium composition from LiC6 causes a large difference in the lithium diffusion coefficient. The off-stoichiometry can be the major reason why the experimental diffusion coefficients and activation energies are widely scattered. Acknowledgment. This study was supported by Grant-inAid for Scientific Research on Priority Areas “Nano Materials Science for Atomic Scale Modification 474” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT)
J. Phys. Chem. C, Vol. 114, No. 5, 2010 2379 of Japan and the global COE program. K.T. also thanks the Japan Society for the Promotion of Science. References and Notes (1) Guerard, D.; Herold, A. Carbon 1975, 13, 337. (2) Fischer, J. E.; Fuerst, C. D.; Woo, K. C. Synth. Met. 1983, 7, 1. (3) Dahn, J. R. Phys. ReV. B 1991, 44, 9170. (4) Ogumi, Z.; Inaba, M. Bull. Chem. Soc. Jpn. 1998, 71, 521. (5) Levi, M. D.; Markevich, E.; Aurbach, D. Electrochim. Acta 2005, 51, 98. (6) NuLi, Y.; Yang, J.; Jiang, Z. J. Phys. Chem. Solids 2006, 67, 882. (7) Toyoura, K.; Koyama, Y.; Kuwabara, A.; Oba, F.; Tanaka, I. Phys. ReV. B 2008, 78, 214303. (8) Vineyard, G. H. J. Phys. Chem. Solids 1957, 3, 121. (9) Robinson, D. S.; Salamon, M. B. Phys. ReV. Lett. 1982, 48, 156. (10) Woo, K. C.; Mertwoy, H.; Fischer, J. E.; Kamitakahara, W. A.; Robinson, D. S. Phys. ReV. B 1983, 27, 7831. (11) Henkelman, G.; Uberuaga, B. P.; Jonsson, H. J. Chem. Phys. 2000, 113, 9901. (12) Togo, A.; Oba, F.; Tanaka, I. Phys. ReV. B 2008, 77, 184101. (13) Togo, A.; Oba, F.; Tanaka, I. Phys. ReV. B 2008, 78, 134106. (14) Blo¨chl, E. P. Phys. ReV. B 1994, 50, 17953. Kresse, G.; Hafner, J. Phys. ReV. B 1993, 47, RC558. Kresse, G.; Hafner, J. Phys. ReV. B 1993, 48, 13115. Kresse, G.; Hafner, J. Phys. ReV. B 1994, 49, 14251. (15) Valencia, F.; Romero, A. H.; Ancilotto, F.; Silvestrelli, P. L. J. Phys. Chem. B 2006, 110, 14832. (16) Martı´nez, J. I.; Cabria, I.; Lopez, M. J.; Alonso, J. A. J. Phys. Chem. C 2009, 113, 939. (17) Liu, W.; Zhao, Y. H.; Li, Y.; Jiang, Q.; Lavernia, E. J. J. Phys. Chem. C 2009, 113, 2028. (18) Perdew, J. P.; Zunger, A. Phys. ReV. B 1981, 23, 5048. (19) Gomer, R. Rep. Prog. Phys. 1990, 53, 917. Van der Ven, A.; Ceder, G.; Asta, M.; Tepesc, P. D. Phys. ReV. B 2001, 64, 184307. (20) Murch, G. E. In Phase Transformations in Materials; Kostorz, G., Ed.; Vch Verlagsgesellschaft Mbh: Weinheim, 2001; p 171. (21) Magerl, A.; Zavel, H.; Anderson, I. S. Phys. ReV. Lett. 1985, 55, 222. (22) Estrade, H.; Conard, J.; Lauginie, P.; Heitjans, P.; Fujara, F.; Buttler, W.; Kiese, G.; Ackermann, H.; Guerard, D. Physica 1980, 99B, 531. (23) Freila¨nder, P.; Heitjans, P.; Ackermann, H.; Bader, B.; Kiese, G.; Schirmer, A.; Stockmann, H. J.; Van der Marel, C.; Magerl, A.; Zabel, H. Z. Phys. Chem. 1987, 151, 93.
JP910134U
