Article pubs.acs.org/JPCC
Density Functional Theory Calculations of Alkali Metal (Li, Na, and K) Graphite Intercalation Compounds Yasuharu Okamoto* The Smart Energy Research Laboratories, NEC Corporation, 34 Miyukigaoka, Tsukuba, Ibaraki, 305-8501, Japan S Supporting Information *
ABSTRACT: First-principles calculations were done to examine the energetics of alkali metal intercalation into graphite. Calculations based on the exchange-correlation functionals that include a nonlocal correlation were found to give reasonable agreement with experiments concerning the crystal structure of graphite and LiC6, binding energy of graphene sheets, and Li intercalation potential. We found that K intercalation from KC8 to KC6 cannot be achieved through electrochemical reactions. We also found that the absence of the stage-I structures for Na graphite intercalation compounds such as NaC6 or NaC8 is linked to a relatively high redox potential of Na/Na+ compared to that of Li/Li+.
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INTRODUCTION Recent technological and industrial trends of expanding the applicability of lithium ion batteries (LIBs) from conventional power supply for mobile phones, portable music players, and notebook computers to new and emerging fields such as electric vehicles and stationary batteries reacknowledge their performance limitations in terms of capacity, safety, cyclability, and above all cost. This situation stimulates the study of various post Li-ion secondary batteries.1−5 Na-ion batteries (NIBs) are one of the post LIBs.4,6 The omnipresence of sodium compared to lithium is expected to be beneficial to reduce the material cost in addition to the advantage in a resource strategy toward a fabrication of minor metal-free secondary batteries. However, apart from the increasing instability of metallic Na in comparison with Li due to the decrease in the cohesive energy, it is known that Na graphite intercalation compounds (Na-GICs) do not form a stage-I structure where every graphene interlayer is occupied by guest Na ions,7 which is quite different from other alkali metal elements: both lithium and potassium form stage-I GICs (LiC6 and KC8).7 Thus, hard carbons are usually used as an anode active material for NIBs instead of graphite,8 which might have relevance to low capacity and poor cyclability of NIBs. Enoki et al. explained the reason for the absence of the stageI Na-GICs in terms of ionization potential (IP) and ionic radius.7 The ionization potential of alkali metal decreases from 5.390 (Li) to 3.893 eV (Cs) as its atomic number increases. The low IP is advantageous to transfer electrons from metal to graphite, which in turn facilitates the formation of GICs. Sodium is not much favorable on this point because of its relatively high IP (5.139 eV9). Moreover, its large ionic radius compared to Li is disadvantageous in accommodating Na ions in the graphitic galleries. Apart from somewhat large ionic radii © 2013 American Chemical Society
of alkali metal (2−4 Å) that are quoted in ref 7, their explanation seems to capture the essential chemistry underlying the alkali-metal GICs. Yet, it inevitably lacks the quantitative discussion to what extent the stage-I Na-GICs is hard to fabricate experimentally. In this context, first-principles calculations based on density functional theory (DFT) are helpful for shedding light on the fundamental features of the energetics of the alkali-metal intercalation into graphite. To address the energetics of GICs through the first-principles calculations, van der Waals (vdW) interaction must be handled properly. Including the vdW interaction within the framework of DFT has been a hot topic for a decade in this research field.10,11 There are two types of methods for including vdW interaction in DFT that are collectively referred to as vdW-DF12−14 and DFT-D.11,15,16 The former method includes a nonlocal correlation that can account for dispersion interaction. On the other hand, the latter method adds a pairwise force field that is inversely proportional to the sixth power of the interatomic distance with empirical parameter C6 to adjust the strength of the interaction. Historically, the prediction of the interlayer spacing of graphite had been a challenge for DFT because standard DFT methods such as generalized gradient approximation (GGA) cannot properly bind two graphene sheets whereas local density approximation (LDA) superficially binds the two sheets in spite of an improper treatment of dispersion force in the exchangecorrelation functional. In contrast, the vdW-DF and its family functionals that explicitly treat the vdW interaction succeeded Received: June 27, 2013 Revised: November 1, 2013 Published: December 19, 2013 16
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where μ(X) is chemical potential of species X. Thus, we obtain the intercalation potential measured from the redox potential of metallic M:
in reproducing the interlayer spacing as well as the binding energy of graphene sheets.14 Nonetheless, we note that there are still controversial issues with respect to the energetics of GICs based on DFT calculations. Lee et al. showed that the vdW-DF results in a wrong conclusion: Li-GIC (LiC6) is energetically less stable than the sum of isolated bulk graphite and metallic Li.17 Thus, instead of vdW-DF, they employed DFT-D2 correction for including vdW interaction in their calculation with adjusting the C6 parameter in DFT-D2. To adjust the parameter, they imposed a condition that the formation energy of LiC6 should be zero because they considered that LiC6 is the most lithiated state of graphite. However, as explained below, their condition is not appropriate for Li intercalation through electrochemical reactions. Moreover, more lithiated GICs than LiC6 are possible to fabricate.18,19 LiC2 is accessible by high-pressure synthesis and it decomposes slowly in several steps at ambient pressure to a stoichiometry of Li7C24 or Li9C24.19 Nalimova et al. showed that the Li highly saturated phase remains even after one year of keeping the samples under normal pressure and temperature.18 The redox potentials of LiC3, LiC3.4, and LiC4 are estimated to be 7.7, 11.0, and 14.9 mV (vs Li/Li+), respectively.18 These arguments also indicate the inadequacy of the condition to determine the C6 parameter in ref 17.
Ug − UM = [mμ(C) + μ(M) − μ(MCm)]/e
In the following, we designate Ug − UM as U[Cm→MCm]. Note that Vave(x1 ≤ x ≤ x2) in ref 21 is equivalent with U[Cm→ MCm] by changing Li → M, MnO3 → Cm, x1 → 0, and x2 → 1. It should be also noted that the intercalation potential U[C6→ LiC6] coincides with the formation energy of Li intercalation Ea(x) used in ref 17 with setting x = 1. For the discussion below, it is useful to consider the intercalation potential corresponding to the reaction of nMCm+n + mM → (m + n)MCn. We also designate this potential as U[MCm+n→MCn], which can be expressed by the following equation U[MCm + n→MCn]
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= 1/m{(m + n)U[Cn→MCn] − nU[Cm + n→MCm + n]}
RESULTS AND DISCUSSION A. Assessment of the Employed Methods. First, to check the validity of applying vdW-DF type exchangecorrelation functionals to graphite and Li-GICs, we calculated bulk graphite and LiC6 by using four different functionals (vdW-DF,12 vdW-DF2,13 vdW-DF-C09,14 and vdW-DF2C0914). Table 1 lists the calculated properties: lattice constants
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COMPUTATIONAL METHODS All our first-principles calculations were done by using the Quantum Espresso (version 5.0.2) program package.20 Calculations of electronic structure were made in accordance with the DFT framework under the periodic boundary conditions, where four exchange correlation functionals that can handle vdW interaction (vdW-DF,12 vdW-DF2,13 vdW-DFC09,14 and vdW-DF2-C0914) were used. Both hexagonal cell parameters (a and c) and all ionic positions in the cell were fully relaxed in all calculations by using the Broyden−Fletcher− Goldfarb−Shanno (BFGS) algorithm. Ultrasoft pseudopotential for C (C.pbe-rrkjus.UPF) and Troullier-Martins type soft pseudopotentials for alkali-metal elements (Li.pbe-mt_fhi.UPF, Na.pbe-mt_fhi.UPF, and K.pbe-n-mt.UPF) were employed in the calculation. Note that these pseudopotentials of alkali metals include the nonlinear core correction. Plane-wave basis sets with cutoff energies of 30 and 300 Ry were respectively used for the expansion of wave functions and charge density. Depending on the number of the graphene layers in the model, (9 × 9 × 11), (9 × 9 × 7), and (9 × 9 × 5) k-point samplings with Methfessel−Paxton smearing were used for Brillouin zone integration for 2-, 3-, and 4-layer models, respectively. To define the intercalation potential of metal M into graphite whose computational cell contains m carbon atoms (Cm), we considered the following two electrochemical reactions of graphite and metal M electrodes:
Table 1. Lattice Constants a and c, in Å, Binding Energy of Graphene Sheet Per C Atom, EB, in meV/atom, and Potential of Li Intercalation, U[C6→LiC6], in V (vs Li/Li+) vdWDF
vdWDF2
vdW-DFC09
vdW-DF2C09
graphite a graphite c LiC6 a LiC6 c EB
2.43 3.56 2.50 3.67 −57.1
2.43 3.52 2.50 3.72 −55.8
2.42 3.23 2.49 3.49 −79.6
2.42 3.28 2.49 3.51 −59.1
U[C6→ LiC6]
+0.04
+0.04
+0.29
+0.29
a
expt 2.456 (ref 7) 3.354 (ref 7) 3.7 (ref 7) −52 ± 5 (ref 21) +0.14a
Averaged value of the experimental profile in Figure 6 in ref 23.
(a and c) of graphite and LiC6, binding energy of graphene sheet per C atom (EB), and potential of Li intercalation (U[C6→LiC6]). It was found that the in-plane lattice constant of graphite, a, does not much depend on the employed exchange-correlation functionals and the predicted values agree well with the experimental one.7 On the other hand, the interlayer spacing, c, calculated by using vdW-DF and vdWDF2 is significantly longer than that by vdW-DF-C09 and vdWDF2-C09 in both cases with and without Li between the graphene sheets. As a result, vdW-DF-C09 and vdW-DF2-C09 are suitable for predicting the interlayer spacing of graphite whereas vdW-DF and vdW-DF2 are suitable for predicting the interlayer spacing of LiC6. Moreover, to check the reliability of the employed Li pseudopotential (Li.pbe-mt_fhi.UPF) which does not explicitly include 1s electrons, we calculated U[C6→LiC6] with an ultrasoft pseudopotential (Li.pbe-s-van_ak.UPF) that treats 1s as valence electrons: calculated U[C6→LiC6] are +0.27 and
Graphite: Cm + M+solv + e− ⇌ MCm Metal M: M ⇌ M+solv + e−
By introducing the equilibrium potential of graphite (Ug) and metal M (UM) electrodes, we obtain the following equations: Graphite: mμ(C) + μ(M+solv) − eUg = μ(MCm) Metal M: μ(M) = μ(M+solv) − eUM 17
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+0.05 V for vdW-DF2-C09 and vdW-DF2, respectively. These values agree with the ones obtained by using Li.pbemt_fhi.UPF (Table 1). Concerning the binding energy of graphene sheet per carbon atom, we found that these functionals other than vdW-DF-C09 successfully provide a value that can be comparable with a recently estimated value (−52 ± 5 meV/atom) as extrapolated from the measurement of the adsorption energy of polyaromatic hydrocarbon molecules on graphite.22 The intercalation potential U[C6→LiC6] obtained by using vdW-DF and vdW-DF2 is lower than that by the vdW-DF-C09 and vdWDF2-C09 functionals. It seems that vdW-DF and vdw-DF2 somewhat underestimate the potential whereas vdW-DF-C09 and vdW-DF2-C09 overestimate the potential. It is noteworthy that the functional dependence of the intercalation potential is similar to that of the interlayer spacing in Table 1. This suggests that there is a significant correlation between the intercalation potential and the interlayer spacing of LiC6. Contrary to the result reported by Lee et al.17 the positive intercalation potential of Li in Table 1 means that LiC6 is energetically more stable than the sum of bulk Li and graphite. While the origin of the discrepancy between the present calculation and Lee’s work is unclear, this might be caused by the sensitivity of the employed pseudopotentials in predicting the interlayer spacing of LiC6 in vdW-DF type calculations. The longer interlayer spacing means the longer Li−C distance, which in turn, decreases the energy gained by Li intercalation through the electrostatic interaction between Li+ and C6−. As stated, the formation energy, Ea(x=1), in ref 17 can be regarded as the Li intercalation potential of U[C6→LiC6]. The setting of Ea(x=1) being zero means that the potential is also zero. It should be noted that unless the potential is identically zero, this implies that there is a negative potential region where the precipitation of metallic Li is more favorable than Li intercalation into graphite in terms of thermodynamics. This was explained in detail in the Supporting Information, S1. Thus, it is not appropriate to adjust the C6 parameter in DFT-D2 by imposing the Ea(x=1) = 0 condition when the Li intercalation occurs electrochemically. We consider that Table 1 confirms the validity in applying vdW-DF type functionals to the calculations of Li-GICs, yet further studies for explaining the discrepancy between the present study and ref 17 might be needed. B. Energy Profile of Li Intercalation. We then examined the stage formation and voltage profile of Li intercalation into graphite. In the following discussion, we used vdW-DF2 and vdW-DF2-C09 functionals that seem to be preferable in predicting structural and energetic properties of Li-GICs from Table 1. In addition to the above examined Li-GIC in stage-I, we calculated the stage-II (LiC12) and -III (LiC18) Li-GICs by assuming √3 × √3 Li ordering and the AABB (LiC12) and AAB (LiC18) stacking sequence. Figure 1 shows a voltage profile of Li intercalation that consist of the intercalation potentials of Li-undoped graphite to stage-III U[C6→LiC18], stage-III to stage-II U[LiC18→LiC12], and stage-II to stage-I U[LiC12→LiC6]. The voltage profile has a desirable decreasing step function character as the Li intercalation progresses without introducing an ad hoc empirical parameter for dispersion energy between Li empty graphene layers as was done in ref 23. Moreover, it was found that U[LiC12→LiC8] is 0.176 and 0.004 V (vs Li/Li+) by using vdW-DF2-C09 and vdW-DF2, respectively. These values are lower than U[LiC12→ LiC6] at x = 0.75 in Figure 1. This means that the two-phase
Figure 1. Voltage profile of Li intercalation into bulk graphite. The solid (dashed) line represents the result by using vdW-DF2-C09 (vdW-DF2) functional.
coexistence of stage-I LiC6 and stage-II LiC12 is thermodynamically more stable than the formation stage-I LiC8 from stage-II LiC12. These results indicate that the vdW-DF2 and vdW-DF2-C09 functionals are able to describe a qualitative behavior of the energetics of Li intercalation into graphite. However, a detailed comparison of Figure 1 and the experimental profile in Figure 6 in ref 23 shows that the profile by vdW-DF2 (vdW-DF2-C09) somewhat underestimates (overestimates) the experimental profile at each potential step, which suggests a need for improvement of the functionals. C. Na and K Graphite Intercalation Compounds. Finally, we examined the energetics of stage-I GICs of Na and K. The crystal structure of the GICs was assumed to be (√3 × √3) metal cation ordering with AA stacking for MC6 and to be p(2 × 2) having four graphene-layer periodicity along c-axis for MC8 based on the model of KC8 studied by Ziambaras et al.24 The calculated results are listed in Table 2. Table 2. Intercalation Potential of M (Na or K) into Graphite, in V (vs M/M+) U[C6→NaC6] U[C8→NaC8] U[C6→KC6] U[C8→KC8] U[KC8→KC6]
vdW-DF2
vdW-DF2-C09
−0.237 −0.237 −0.011 +0.106 −0.362
−0.140 −0.157 +0.079 +0.172 −0.200
We found that the positive intercalation potential U[C8→KC8] is consistent with the fact that KC8 formation is experimentally possible. It may be noted that U[C6→KC6] is also positive in vdW-DF2-C09 whereas the potential is negative in vdW-DF2. Considering that KC6 cannot be achieved even through a vapor-phase intercalation reaction, vdW-DF2-C09 seems to overestimate the intercalation potential approximately 0.1 V as observed in Li intercalation. However, it should be noted that both functionals predict negative U[KC8→KC6], which means that metallic K precipitates from a solution containing K+ ions instead of KC6 formation. Thus, both functionals succeed in explaining that KC8 does not lead to KC6 through an electrochemical reaction. In contrast to K intercalation, both U[C6→NaC6] and U[C8→NaC8] are negative irrespective of the employed 18
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(4) Komaba, S.; Murata, W.; Ishikawa, T.; Yabuuchi, N.; Ozeki, T.; Nakayama, T.; Ogata, A.; Gotoh, K.; Fujiwara, K. Electrochemical Na Insertion and Solid Electrolyte Interphase for Hard-Carbon Electrodes and Application to Na-Ion Batteries. Adv. Funct. Mater. 2011, 21, 3859−3867. (5) Levi, E.; Levi, M. D.; Chasid, O.; Aurbach, D. Review on the Problems of the Solid State Ions Diffusion in Cathodes for Rechargeable Mg Batteries. J. Electroceram. 2009, 22, 13−19. (6) Ellis, B. L.; Nazar, L. F. Sodium and Sodium-Ion Energy Storage Batteries. Curr. Opin. Solid State Mater. Sci. 2012, 16, 168−177. (7) Enoki, T.; Suzuki, M.; Endo, M. Graphite Intercalation Compounds and Applications; Oxford University Press: Oxford, UK, 2003. (8) Stevens, D. A.; Dahn, J. R. The Mechanisms of Lithium and Sodium Insertion in Carbon Materials. J. Electrochem. Soc. 2001, 148, A803−A811. (9) David, R. E., Ed. Handbook of Chemistry and Physics, 75th ed.; CRC Press, Inc.: Boca Raton, FL, 1994. (10) Björkman, T.; Gulans, A.; Krasheninnikov, A. V.; Nieminen, R. M. Are We Van der Waals Ready? J. Phys.: Condens. Matter 2012, 24, 424218−11. (11) Grimme, S. Density Functional Theory with London Dispersion Corrections. Comput. Mol. Sci. 2011, 1, 211−228. (12) Dion, M.; Rydberg, H.; Schröder, E.; Langreth, D. C.; Lundqvist, B. I. Van der Waals Density Functional for General Geometries. Phys. Rev. Lett. 2004, 92, 246401−4. (13) Lee, K.; Murray, É. D.; Kong, L.; Lundqvist, B. I.; Langreth, D. C. Higher-Accuracy Van der Waals Density Functional. Phys. Rev. B 2010, 82, 0811001(R)−4. (14) Cooper, V. R. Van der Waals Density Functional: An Appropriate Exchange Functional. Phys. Rev. B 2010, 81, 161104(R)−4. (15) Grimme, S. Semiempirical GGA-Type Density Functional Constructed with a Long-Range Dispersion Correction. J. Comput. Chem. 2006, 27, 1787−1799. (16) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate Ab Initio Parameterization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu. J. Chem. Phys. 2010, 132, 154104. (17) Lee, E.; Persson, P. A. Li Absorption and Intercalation in Single Layer Graphene and Few Layer Graphene by First Principles. Nano Lett. 2012, 12, 4624−4628. (18) Nalimova, V. A.; Guérard, D.; Lelaurain, M.; Fateev, O. V. X-ray Investigation of Highly Saturated Li-Graphite Intercalation Compound. Carbon 1995, 33, 177−181. (19) Bindra, C.; Nalimova, V. A.; Sklovsky, D. E.; Kamitakahara, W. A.; Fischer, J. E. Statics and Dynamics of Interlayer Interactions in the Dense High-Pressure Graphite Compound LiC2. Phys. Rev. B 1998, 57, 5182−5190. (20) Giannozzi, P.; et al. QUANTUM ESPRESSO: A Modular and Open-source Software Project for Quantum Simulations of Materials. J. Phys.: Condens. Matter 2009, 21, 395502−19. (21) Koyama, Y.; Tanaka, I.; Nagao, M.; Kanno, R. First-Principles Study on Lithium Removal from Li2MnO3. J. Power Sources 2009, 189, 798−801. (22) Zacharia, R.; Ulbricht, H.; Hertel, T. Interlayer Cohesive Energy of Graphite from Thermal Desorption of Polyaromatic Hydrocarbons. Phys. Rev. B 2004, 69, 155406−7. (23) Persson, K.; Himuma, Y.; Meng, Y. S.; Van der Ven, A.; Ceder, G. Thermodynamic and Kinetic Properties of the Li-Graphite System from First-Principles Calculations. Phys. Rev. B 2010, 82, 125416−9. (24) Ziambaras, E.; Kleis, J.; Schröder, E.; Hyldgaard, P. Potassium Intercalation in Graphite: A Van der Waals Density-Functional Study. Phys. Rev. B 2007, 76, 155425−10.
functionals. This shows that the stage-I structures of NaC6 and NaC 8 cannot be formed even through a vapor-phase intercalation reaction. The magnitude of U[C6→NaC6] and U[C8→NaC8] (approximately 0.14−0.24 V from Table 2) is suggestive in considering the reason why sodium does not form stage-I GICs. The redox potential of Na/Na+ is higher than that of Li/Li+ by 0.33 V according to ref 9. Namely, due to the high potential of Na/Na+, metallic Na precipitates before it is intercalated into graphite. Thus, if the redox potential of metallic sodium were as low as lithium, sodium would form stage-I GICs as lithium does. Although, as explained, it is hard to intercalate Na in graphite, it may be necessary to consider Na adsorption on graphite edges as a possible Na storage configuration in sodium ion batteries. However, since we focused on the graphite intercalation compounds, this is beyond the scope of the present study. We show our preliminary results of Na adsorption on zigzag and armchair graphene edges in Supporting Information, S2 and S3. Both edge types bind Na atoms. In particular, the zigzag edge strongly binds the Na atoms, which means a high anode potential and leads to low available voltage.
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CONCLUSIONS First-principles calculations were done to elucidate the energetics of alkali metal graphite intercalation compounds (M-GICs) based on the method that explicitly includes a nonlocal correlation to account for dispersion interaction. Our calculations based on vdW-DF type exchange correlation functional reproduced the lattice constant of graphite and LiC6, binding energy of graphene sheets. In addition, unlike the previous study,17 we obtained the Li intercalation potential from graphite to LiC6 which is comparable with experimental values. Also, we successfully obtained the voltage profile of Li intercalation into graphite that has a desirable decreasing step function behavior as the stage structure of Li intercalation progresses. Moreover, we showed that K intercalation does not progress from KC8 to KC6 through electrochemical reactions. Our calculations also showed that a relatively high redox potential of Na/Na+ compared to Li/Li+ is the reason why Na does not form stage-I GICs.
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ASSOCIATED CONTENT
S Supporting Information *
Supporting method and figures. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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