Determination of Activation Energy for Li Ion Diffusion in Electrodes

Feb 6, 2009 - Fast Li-Ion Insertion into Nanosized LiMn2O4 without Domain Boundaries. Masashi Okubo , Yoshifumi Mizuno , Hirotoshi Yamada , Jedeok ...
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J. Phys. Chem. B 2009, 113, 2840–2847

Determination of Activation Energy for Li Ion Diffusion in Electrodes Masashi Okubo,†,‡ Yoshinori Tanaka,‡ Haoshen Zhou,‡ Tetsuichi Kudo,‡ and Itaru Honma*,‡ Department of Applied Chemistry, Faculty of Science and Engineering, Chuo UniVersity, Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan, and National Institute of AdVanced Industrial Science and Technology (AIST), Umezono 1-1-1, Tsukuba, Ibaraki 305-0044, Japan ReceiVed: NoVember 11, 2008; ReVised Manuscript ReceiVed: December 26, 2008

Higher power Li ion rechargeable batteries are important in many practical applications. Higher power output requires faster charge transfer reactions in the charge/discharge process. Because lower activation energy directly correlates to faster Li ion diffusion, the activation energy for ionic diffusion throughout the electrode materials is of primary importance. In this study, we demonstrate a simple, versatile electrochemical method to determine the activation energy for ionic diffusion in electrode materials via temperature dependent capacitometry. A generalized form of the temperature dependence of the discharge capacity was derived from the diffusion equation. This method yielded activation energy values for Li ion diffusion in LiCoO2 comparable to those obtained from ab initio calculations. Introduction High power and high capacity Li ion rechargeable batteries have become an important technology for auxiliary power units in electric vehicles. Since a higher power output requires faster charge transfer reactions in the charge/discharge process, nanoelectrodes with a short Li ion diffusion path have been intensively investigated.1-5 The fundamental step for the charge transfer reactions is ionic diffusion throughout the electrode materials. Here, we report a simple, versatile electrochemical method to determine the chemical Li ion diffusion coefficient and activation energy for ionic diffusion in electrode materials via temperature dependent capacitometry. If the rate-limiting step is ionic diffusion in the host material, the temperature dependence of the capacity can be used to determine the activation energy of the diffusion process. From an application viewpoint, electrodes with lower activation energy (EA) have gathered increasing interest. One of the most promising approaches for finding superior electrode materials is computer simulation of Li ion diffusion dynamics in the electrodes. Kang et al.6 have recently predicted a lower EA for Li ion diffusion in LiMn0.5Ni0.5O2 using ab initio calculations. The low EA has resulted in fast Li ion diffusion and a high discharge rate capability. Consequently, determination of EA for Li ion diffusion in various electrodes is of primary importance, because a lower EA directly correlates to faster Li ion diffusion, thus resulting in higher power outputs from the electrode. The EA for Li ion diffusion is normally determined by analysis of the spin-lattice relaxation or motional narrowing of the 7Li NMR spectral line width. For example, EA for diffusion in LiCoO2 has been estimated to be 0.30 eV by analyzing the motional narrowing.7,8 Although 7Li NMR spectroscopy is an effective method for directly observing the motion of Li ions in the solid state, this method is known to frequently yield lower EA values and higher diffusion coefficients compared to those from diffusion/conductivity measurements. The differences arise because the excited state observed by 7Li NMR spectroscopy * Corresponding author. E-mail: [email protected]. † Chuo University. ‡ AIST.

is not exactly the same as that of diffusion/conductivity.9,10 Furthermore, 7Li NMR spectroscopy is not always feasible. Here, we report a simple electrochemical method to determine EA for Li ion diffusion in electrodes, which should be applicable to the study of ionic diffusion in solid electrodes. Experimental Section LiCoO2 was purchased from Honjo Chemical Corp. Its powdered sample (50 mg) was ground into a paste with acetylene black (45 mg) and Teflon (5 mg). In order to suppress IR drops, we used 45 wt % carbon. A 1 M solution of LiClO4 in EC/DEC was used as the electrolyte. Lithium metal was used as the reference and counter electrodes. After the first charge-discharge cycle at a rate of 0.5C (4.2-2.6 V), the cell was recharged to 4.2 V at the same rate, but the discharge was performed at a rate of 10C. By charging to 4.2 V at the very slow rate, LiCoO2 was delithiated to almost the same stoichiometry (Li0.5CoO2) in all experiments, and we can analyze the current density dependent and temperature dependent discharge capacity. Concerning the temperature dependent capacitometry, the discharge temperature was varied from 273 to 303 K. The IR drop was corrected appropriately. Ab initio calculations were performed using the density functional theory (DFT) program CASTEP with the Perdew-Burke-Ernzerhof gradient corrected functional and ultrasoft pseudopotentials.11-13 The crystal structures of Li1-xCoO2 with x ) 0.5 and 0.17 were assumed to be a rock salt layered structure, and the unit cells for Li1-xCoO2 with x ) 0.5 and 0.17 were assumed to be a 2 × 1 × 1 hexagonal supercell. The electronic structures were determined from atomic positions obtained from geometric optimization. The wave functions were expanded in plane waves using a 600 eV cutoff. The Brillouin zone was sampled using a 5 × 10 × 2 mesh. The transition states were determined as stationary points that were energy maxima in the direction of the reaction coordinate and energy minima in all other directions. The transition state search was performed using synchronous transit methods.14,15 Along the reaction path, the total energy transit started from an energy minimum of a reactant, increased to a maximum at a transition state, and decreased to the energy of a product. The

10.1021/jp8099576 CCC: $40.75  2009 American Chemical Society Published on Web 02/06/2009

Li Ion Diffusion in Electrodes

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SCHEME 1: Equivalent Electric Circuit and Illustration of Li Ion Intercalation into the Host Cylindrical Material

energy difference between the energy at the transition state and that of the reactant or product was defined as EA for Li ion diffusion. Results and Discussion Let us consider an electrochemical intercalation of Li ion into host Li1-RCoO2 with a cylinder shape of radius d. Under equilibrium conditions, the initial open circuit voltage, E0, determines the Li ion concentration within the host material. The equivalent electric circuit and schematic illustration of the Li ion intercalation into the host material are shown in Scheme 1. When a potential step is applied to the system in equilibrium at the potential E0 [V vs Li/Li+] and stoichiometry R0, the diffusion equation

( dCdr )

D

)

r)d

I(t) , eS

I(t) )

Es(t) - Ef RT

gives the normalized transient current as16

I(t) ) I0

∞

(

∑ β 22L+ L2 exp -

n)1

n

βn2Dt d2

)

(1)

where D is the chemical Li ion diffusion coefficient, C is the Li ion concentration, r is the cylindrical coordinate, d is the diffusion length, Es(t) is the surface potential related to the Li ion concentration at the surface, Ef is the potential required for the equilibrium state after the potential step, e is the elementary electric charge, S is the surface area, RT is the sum of the resistance in the electrochemical cell, I0 is I(t)0), βn’s are the roots of βJ1(β) - LJ0(β) ) 0, Jn is the nth Bessel function, and

L)

I0d eDS(Cf - C0)

The surface area for the cylinder model of radius d can be calculated as

S)

2w Fd

where w [g] is the sample weight and F is the sample weight density (∼5.0 g/cm3 for LiCoO2). Thus, eq 1 can be transformed by defining the characteristic diffusion time, d2/D (≡τ), as

I(t) ) I0

∞

( )

βn2 2L exp - t 2 2 τ n)1 βn + L

∑

FI0 L ) τ 2ew(Cf - C0) Note that L/τ is a constant given by the model and the experimentally obtained I0. However, in the actual experiment, the above expression could not reproduce the experimental result. For example, the black solid line in Figure 1 shows the transient current for LiCoO2 after the potential step from 4.2 to 4.15 V. The triangles, circles, and squares are simulated results with the various τ values and a constant L/τ value of 8.52 × 10-4 derived from the model. As shown in Figure 1, no value of the characteristic diffusion time could reproduce the experimental result, especially in the long time range. In the long time approximation, eq 1 approaches

( )

β12 I(t) 2L exp ∼ 2 t I0 τ β1 + L2

It appears that the slope of the ln{I(t)/I0} vs t plot could give the value of τ, which is frequently employed in the potentiostatic intermittent titration technique (PITT). However, since β1 nearly equals (2L)1/2 for L , 1, the transient current could be further transformed as

I(t) 2 L ∼ exp - t I0 2+L τ

( )

Thus, the slope of the ln{I(t)/I0} vs t plot depends only on the constant L/τ value given by the model and, therefore, could not determine the diffusion coefficient in the host material. This result clearly suggests that the L/τ value derived from the model should include the experimental error. Therefore, in

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Figure 1. Transient current (solid line) for LiCoO2 after the potential step from 4.2 to 4.15 V. The triangles, circles, and squares are simulated results based on eq 1 with various characteristic diffusion times (see text).

order to analyze the actual experimental result, we should introduce the experimental error factor, f, into L/τ as

FI0 L ≡f τ 2ew(Cf - C0) Now we have a set of two parameters (τ, f) to simulate the experimental result. In an ideal case, f should be unity. Therefore, if f deviates significantly from unity as a result of the analysis, the analysis should be considered inappropriate. Figure 2a shows the resimulated result with (τ, f) ) (45, 1.6) for the transient current after the potential step from 4.2 to 4.15 V. The simulated result closely reproduces the experimental result, and the value of f near unity supports the appropriateness of the analysis. Figure 2a also shows the simulated results for various voltages, and the τ value was successfully obtained using an f value near unity. Figure 2b shows the stoichiometric dependence of the chemical Li ion diffusion coefficient assuming a Li ion diffusion length, d, of 100 nm. Although a slightly low D was obtained at the phase transition region from the monoclinic to the hexagonal phase (x ∼ 0.45),17 the value of D during the discharge process does not vary substantially, and is about 10-12 cm2/s. In ref 18, the value of D varied over 2 orders of magnitude (10-10-10-12 cm2/s) during the discharge process. However, it should be emphasized that the large deviation (10-12 cm2/s) from the constant diffusion coefficient (10-10 cm2/s) was recorded only at the two-phase region around 3.95 V, where the discharge process should be dominated not by the Li ion diffusion in the electrode, but by the other factors such as a moving boundary. Therefore, our result does not contradict the previous results. On the other hand, when the constant current, I, is forced to transverse the electrochemical cell, the diffusion equation

( ∂C∂r )

D

)

r)d

I , es

I ) constant

Cs ) C0 +

{

( )

∞ Rn2t 1 FτI 2t 1 exp + -2 2ew τ 4 τ R2 n)1 n

∑

of t/τ > 0.125, the solution approximates to

Cs ∼ C0 +

FτI 2t 1 + 2ew τ 4

(

}

where Rn’s are the positive roots of J1(Rn) ) 0. In the condition

)

which provides the current density (J, [A/g]) dependence of the discharge capacity:

1 Q(J) ) Q0 - τJ 8 where Q0 is the ideal open circuit voltage (OCV) discharge capacity. Note that, regardless of the diffusion dimensionality, the solution for the diffusion equation gives a similar current density dependence of the discharge capacity:16

1 Q(J) ) Q0 - τJ a

16

gives the Li ion concentration at the surface, Cs:

Figure 2. (a) Transient current (solid lines) and simulated results (triangles) for LiCoO2 after the potential step of 50 mV at various voltages. (b) Stoichiometric dependence of chemical Li ion diffusion coefficient for LiCoO2 obtained from PITT measurement. The diffusion length was assumed to be 100 nm.

(2)

where a is a constant related to the diffusion dimensionality (a ) 3, 8, and 15 for the thin film, cylinder, and sphere models, respectively). This equation is valid for Q > 0.5Q0. Note that D is assumed to be constant during the discharge process.

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SCHEME 2: Potential Energy Diagram of Li Ion Diffusion from a Regular Site to an Interstitial Site

SCHEME 3: Schematic Representation of Li Ion Diffusion through an Interstitial Tetrahedral Site in LiCoO2

In general, Li ion diffusion in electrodes is governed by EA, i.e., the energy difference between the regular Li ion sites and the transient Li ion sites (Scheme 2). Thus, D depends on the temperature in relation to the Arrhenius equation:

D ) Θ(T, R)

( )

Rl2ν0 EA exp 2n kT

where n is the diffusion dimensionality, l is the jumping distance, and ν0 is the temperature-independent jumping frequency.19 Here, Θ(T,R) is a thermodynamic factor expressed as

Θ(T, R) )

F dE (1 - R) RT dR

Therefore, D is a function of temperature and stoichiometry. However, assuming the Nernst equation

E ) E0 +

R RT ln F 1-R

(

)

is valid, D is expressed as

D)

( )

l2ν0 EA exp 2n kT

(3)

Since this expression indicates that D is constant during the discharge process, we can combine eqs 2 and 3 to obtain the discharge current density (J) and temperature (T) dependence of the discharge capacity, Q(J,T), as

( )

ln{Q0 - Q(J, T)} ) ln

EA 2n d2 J + 2 alν kT

(4)

0

As is well-known, the chemical and self Li ion diffusion coefficients depend on the stoichiometry, and eq 4 derived from the constant chemical Li ion diffusion coefficient seems to be oversimplified. However, as discussed above, the PITT measurement shown in Figure 2 suggests that the chemical Li ion diffusion coefficient does not vary remarkably, even if the structural phase transition and the two-phase region occur during the discharge process of LiCoO2. Thus, the use of the constant Li ion diffusion coefficient and the Nernst equation is verified as a first approximation to establish a simple method for the

determination of the diffusion coefficient and the activation energy. In fact, eq 4 could provide the Li ion diffusion coefficient and the activation energy comparable to the PITT measurement and the ab initio calculation, as will be discussed in the following section. Figure 3 shows the simulated Q(J,T) vs EA relationship for the cylinder model at discharge rates of 1C and 10C. The parameters used were Q0 ) 137 mAh/g, ν0 ) 10-13 s-1, l ) 2.82 Å, and d ) 100 nm. As shown in Figure 3, the temperature dependence of the discharge capacity at a discharge rate of 1C could hardly be used to determine EA for Li ion diffusion in this assumed model if the value of EA is below 0.35 eV. However, the temperature dependence can provide EA by increasing the discharge rate until the Li ion diffusion in the host material becomes the rate-limiting step. Figure 3 demonstrates that the temperature dependence of the discharge capacity at a 10C discharge rate could be used to determine EA between 0.3 and 0.35 eV. In order to confirm the validity of the dependence of the discharge capacity on current density (eq 2) and temperature (eq 4), we determined the characteristic diffusion time, τ, and the activation energy, EA, for Li ion diffusion in LiCoO2. In general, the Li ions in the regular tetrahedral sites in LiCoO2 diffuse through interstitial tetrahedral sites (Scheme 3). Thus, EA can be defined as the energy difference between the two sites. EA for Li ion diffusion in LiCoO2 has been determined by using 7Li NMR spectroscopy and ab initio calculations,20 as shown in Table 1. Figure 4a shows the current density dependence (T ) 298 K) of the discharge curve. As expected from eq 2, the discharge capacity decreased with an increase in the current density. The decrease in the discharge capacity apparently results from slow Li ion diffusion in LiCoO2. Figure 4b shows the discharge

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Figure 3. Calculated Q(J,T) vs EA relationship for the cylinder model at discharge rates of 1C and 10C. The parameters used were Q0 ) 137 mAh/g, ν0 ) 10-13 s-1, r ) 2.82 Å, and d ) 100 nm.

TABLE 1: Activation Energy for Li Ion Diffusion in LiCoO2 Estimated Using Various Methods EA (eV) stoichiometry 7

7

Li NMR (spin-lattice relaxation) Li NMR (motional narrowing)8 ab initio20 ab initio (this work) this work 7

a

LiCoO2

Li0.8CoO2

Li0.6CoO2

Li0.5CoO2

0.076 0.30 0.31 0.32a

0.30 0.26

0.08 -

0.21

Average between LiCoO2 and Li0.5CoO2.

capacity as a function of the current density with a cutoff voltage of 3.85 V, which corresponds to the discharge process from Li0.5CoO2 to LiCoO2. The solid line in Figure 4b denotes the results fitted using eq 1. Here, τ was determined to be 242 s. Figure 5a shows the temperature dependence of the discharge curve with 1370 mA/g (10C). The discharge curve at T ) 298 K with 68.5 mA/g (0.5C) is also plotted (the broken line) as Q0. As shown in Figure 5a, the discharge capacity drastically decreased with a decrease in the temperature, which suggests that Li ion diffusion in LiCoO2 is the rate-limiting step under these conditions. Figure 5b shows the discharge capacity as a function of temperature at a cutoff voltage of 3.85 V (the discharge process from Li0.5CoO2 to LiCoO2): the solid line was fitted using eq 4. Here, EA was determined to be 0.32 eV. As shown in Figure 5b, the linear decrease in the discharge capacity agrees with eq 4. Compared with the previously reported values (Table 1), our value of EA at the cutoff voltage of 3.85 V is consistent with that obtained from the ab initio calculations reported by Kang et al. (0.31 eV for LiCoO2). However, note that our value is an average value during the discharge process. Therefore, in order to make our discussion more explicit, we determined EA using ab initio calculations with stoichiometries of Li0.83CoO2 and Li0.5CoO2.

The insets in Figure 6a show the optimized crystal structures of LiCoO2, Li0.83CoO2, and Li0.5CoO2, which were used for the calculation of the electronic structure. From the structural optimization, the c-axis became elongated when Li ions were extracted from the host. This result is consistent with the experimental results reported by Amatucci et al.17 Figure 6a shows the total density of states for LiCoO2, Li0.83CoO2, and Li0.5CoO2. Regardless of the stoichiometry, the density of states near the Fermi level mainly consists of the 2p orbitals of the oxygen atoms and t2g orbitals of the cobalt ions, which indicates strong orbital hybridization of the oxygen and cobalt orbitals, as has been reported previously.21 These various consistencies with the experimental results support the appropriateness of the optimized structure. Using the optimized structure, EA for the Li ion diffusion was determined by performing the transition state search with synchronous transit methods. Table 1 summarizes the obtained values of EA. As more Li ions were extracted, EA decreased from 0.32 to 0.21 eV. Kang et al. have pointed out that cobalt ions are oxidized as Li ions are extracted, increasing the Coulombic repulsion between the cobalt and Li ions in the interstitial tetrahedral sites, which results in a higher EA.20 However, as mentioned above, the c-axis is elongated through the extraction of Li ions. Kang et al. have also reported that the

Li Ion Diffusion in Electrodes

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Figure 4. (a) Discharge curves at 298 K and various discharge rates up to 100C (13 700 mA/g). (b) Discharge capacity as a function of current density at a cutoff voltage of 3.85 V.

Figure 5. (a) Discharge curves at a discharge rate of 10C and various temperatures. (b) Temperature dependence of discharge capacity at a cutoff voltage of 3.85 V.

distance between the Li and cobalt ions is crucial.20 The elongated distance between the ions should reduce the Coulombic repulsion, resulting in lower EA. In other words, EA decreased upon extraction of the Li ions, although the change in EA was only about 0.1 eV (Figure 6b). In comparison with the calculated results (EA ) 0.31-0.21 eV), the temperature dependence of the discharge capacity seemed to give a slightly higher EA. However, note that cation mixing of the Co3+ and Li+ ions was ignored in the ab initio calculations, although mixing is expected to exist, to a certain degree, in real LiCoO2. Since cation mixing in LiCoO2 causes a decrease in the unit cell volume and Li ion slab distance,22 the experimentally obtained EA should be higher than that obtained from the ab initio calculations on the optimized crystal structures. It should also be emphasized that the theoretical deficiencies in our model could also be responsible for the higher EA. Figure 7 shows the cutoff voltage dependence of D and EA assuming a Li ion diffusion length of 100 nm. Note that the value of x in Figure 7 indicates that D or EA is an average value between those of Li0.5CoO2 and Li1-xCoO2. As the cutoff voltage became higher, the Li ion diffusion became faster, whereas EA became

higher. These results contradict each other, because a higher EA should give slower diffusion. Therefore, the fast D and high EA obtained at the high cutoff voltage should be ascribed to the theoretical deficiencies in our model, in which the structural phase transition during the discharge process is completely ignored. As shown in Figure 2b, the discharge curve at 0.5C clearly shows abnormal behavior caused by a structural phase transition around 4.15 V.23 Nevertheless, by lowering the cutoff voltage and averaging over the wide stoichiometric range, our method could provide the appropriate values of D and EA. On the other hand, Figure 7 also shows a fast D and a high EA with a cutoff voltage below 3.8 V. This was also ascribed to the theoretical deficiencies in our model involving the constant diffusion coefficient and the Nernst equation. Since the latter ignores the interaction between the Li ions, it is only valid for dilute concentrations of Li ions. However, as a result of the Li ion intercalation during the discharge process, the Li ion concentration below 3.85 V is too high to be neglected. Thus, the Nernst equation is not valid below 3.85 V. Furthermore, as shown in the discharge curve, the chemical Li ion diffusion is accelerated remarkably by the thermodynamic factor below 3.85 V. Therefore, the assumption of the constant Li ion diffusion

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Figure 7. Cutoff voltage dependence of characteristic diffusion time (τ ) d2/D) and activation energy (EA). The value of x indicates that τ or EA is an average value between those of Li0.5CoO2 and Li1-xCoO2.

Conclusion In conclusion, we have developed a simple electrochemical method to determine the diffusion coefficient and activation energy for Li ion diffusion in electrodes. It should be emphasized that the temperature dependence of the discharge capacity can be applied to any electrode material, regardless of the diffusion dimensionality. Thus, our method will be a powerful technique for developing higher power batteries as well as electrode materials with faster ionic diffusion.

Figure 6. (a) Total density of states for LiCoO2, Li0.83CoO2, and Li0.5CoO2. Insets show the optimized crystal structures used for the calculation of the density of states. (b) Stoichiometric dependence of the activation energy (EA).

coefficient is no longer valid below 3.85 V, which should result in the fast D. Concerning the Nernst equation, there are the two-phase region and phase transition during the discharge process of LiCoO2, in which the Nernst equation is not necessarily valid. However, judging from the obtained values, the approximation using the Nernst equation above 3.85 V is not far from the real system, as long as it is averaging over the wide stoichiometric range. Taking these results into account, the cutoff voltage should be chosen carefully in order to estimate accurately EA for diffusion in the electrode materials by using the temperature dependence of the discharge capacity. However, this method can be used to easily obtain the D and EA values that are comparable to those obtained from ab initio calculations.

Acknowledgment. M.O. and Y.T. made equal contributions to this work. The fruitful discussions with Prof. Sohrab Rabii (Department of Electrical and Systems Engineering, University of Pennsylvania) are gratefully acknowledged. This work was supported by the New Energy and Industrial Technology Development Organization Japan, under a grant for Research and Development of Nanodevices for Practical Utilization of Nanotechnology (Nanotech Challenge Project). Glossary List of Symbols EA d E0 R0 D C r t Es(t) Ef e

activation energy for Li ion diffusion [eV] Li ion diffusion length [cm] initial open circuit voltage [V vs Li/Li+] stoichiometry of Li1-RCoO2 corresponding to E0 chemical Li ion diffusion coefficient [cm2/s] Li ion concentration [cm-3] cylindrical coordinate [cm] time after the potential step [s] surface voltage related to the Li ion concentration at the surface [V vs Li/Li+] voltage required for the equilibrium state after the potential step [V vs Li/Li+] elemental electric charge [C]

Li Ion Diffusion in Electrodes S RT I(t) I0 Jn L βn w F τ f Rn Q J Q0 a Θ l ν0 n R F T k

surface area of the cylinder [cm2] resistance in the electrochemical cell [Ω] transient current [C/s] I(t)0) [C/s] nth Bessel function I0d/[eDS(Cf - C0)] positive roots of βJ1(β) - LJ0(β) ) 0 sample weight [g] sample weight density [g/cm3] characteristic diffusion time ()d2/D) [s] experimental error factor positive roots of J1(R) ) 0 discharge capacity [mAh/g] current density [mA/g] ideal discharge capacity [mAh/g] constant determined by the diffusion dimensionality thermodynamic factor [K] Li ion jump distance [cm] temperature-independent jumping frequency [s-1] diffusion dimensionality gas constant Faraday constant temperature Boltzmann constant

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J. Phys. Chem. B, Vol. 113, No. 9, 2009 2847 (4) (a) Okubo, M.; Hosono, E.; Kim, J.-D.; Enomoto, M.; Kojima, N.; Kudo, T.; Zhou, H. S.; Honma, I. J. Am. Chem. Soc. 2007, 129, 7444. (b) Okubo, M.; Hosono, E.; Kudo, T.; Zhou, H. S.; Honma, I. J. Phys. Chem. Solids 2008, 69, 2911. (5) Arico`, A. S.; Bruce, P. G.; Scrosati, B.; Tarascon, J.-M.; Schalkwijk, W. V. Nat. Mater. 2005, 4, 366. (6) Kang, K.; Meng, Y. S.; Bre´ger, J.; Grey, C. P.; Ceder, G. Science 2006, 311, 977. (7) Nakamura, K.; Ohno, H.; Okamura, K.; Yoshitake, M.; Moriga, T.; Nakabayashi, I.; Kanashiro, T. Solid State Ionics 2006, 177, 821. (8) Nakamura, K.; Yamamoto, M.; Okamura, K.; Yoshitake, M.; Nakabayashi, I.; Kanashiro, T. Solid State Ionics 1999, 121, 301. (9) Viefhaus, T.; Bolse, T.; Mu¨ller, K. Solid State Ionics 2006, 177, 3063. (10) Heitjans, P.; Ka¨rger, J. Diffusion in Condensed Matter; Springer: Berlin, 2005. (11) Segall, M. D.; Lindan, P. J. D.; Probert, M. J.; Pickard, C. J.; Hasnip, P. J.; Clark, S. J.; Payne, M. C. J. Phys.: Condens. Matter 2002, 14, 2717. (12) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77, 3865. (13) Vanderbilt, D. Phys. ReV. B 1990, 41, 7892. (14) Halgren, T. A.; Lipscomb, W. N. Chem. Phys. Lett. 1977, 49, 225. (15) Govind, N.; Petersen, M.; Fitzgerald, G.; King-Smith, D.; Andzelm, J. Comput. Mater. Sci. 2003, 28, 250. (16) Crank, J. C. Mathematics of Diffusion, 2nd ed.; Oxford University Press: Oxford, 1975. (17) Amatucci, G. G.; Tarascon, J. M.; Klein, L. C. J. Electrochem. Soc. 1996, 143, 1114. (18) Levi, M. D.; Salitra, G.; Markovsky, B.; Teller, H.; Aurbach, D.; Heider, U.; Heider, L. J. Electrochem. Soc. 1999, 146, 1279. (19) Kudo, T.; Fueki, K. Solid State Ionics; Kodansha: Tokyo, 1990. (20) Kang, K.; Ceder, G. Phys. ReV. B 2006, 74, 094105. (21) Yoon, W.-S.; Kim, K.-B.; Kim, M.-G.; Lee, M.-K.; Shin, H.-J.; Lee, J.-M.; Lee, J.-S.; Yo, C.-H. J. Phys. Chem. B 2002, 106, 2526. (22) Gummow, R. J.; Thackeray, M. M.; David, W. I. F.; Hull, S. Mater. Res. Bull. 1992, 27, 327. (23) Ven, V.; Ceder, G. Electrochem. Solid State Lett. 2000, 3, 301.

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