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Insight into the Vibrational and Thermodynamic Properties of Layered Lithium Transition-Metal Oxides LiMO (M = Co, Ni, Mn): A First-Principles Study 2

Taoyuan Du, Bo Xu, Musheng Wu, Gang Liu, and Chuying Ouyang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b11891 • Publication Date (Web): 01 Mar 2016 Downloaded from http://pubs.acs.org on March 8, 2016

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The Journal of Physical Chemistry

Insight into the Vibrational and Thermodynamic Properties of Layered Lithium Transition-Metal Oxides LiMO2 (M = Co, Ni, Mn): A First-Principles Study Taoyuan Du, Bo Xu*, Musheng Wu, Gang Liu and Chuying Ouyang Department of Physics, Laboratory of Computational Materials Physics, Jiangxi Normal University, Nanchang 330022, China * [email protected] Abstract: Evaluation of the finite-temperature thermodynamic properties of the electrode materials generally helps to accurately describe the performance of Li-ion battery (LIBs). To know the characteristics of the layered lithium transition-metal oxides LiMO2 (M = Co, Ni, Mn) comprehensively, herein, the vibrational and related thermodynamic quantities of these electrode materials are investigated by using density functional perturbation theory (DFPT). Local density approximation (LDA) and generalized gradient approximation with the Hubbard model correction (GGA+U) yield similar results, either for the phonon dispersion or for the thermodynamic functions. Among the three layered lithium transition-metal oxides, the vibrational and thermodynamic properties of LiNiO2 is more close to that of LiMnO2, while relatively far away from that of LiCoO2, due to the same crystal structure of LiNiO2 and LiMnO2, which is different from that of LiCoO2. In addition, the corrections of average intercalation voltage as a function of temperature for Li0.75CoO2 and Li0.5CoO2 are evaluated when considering the contribution of vibrational entropy. Since our theoretical results for LiCoO2 agree well with those from experiments, we can provide the reliable thermodynamic data for the layered lithium transition-metal oxides.

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Ⅰ. INTRODUCTION Increasing demands for plug-in hybrid electric vehicle applications promote the development of higher energy and power Li-ion batteries (LIBs).1,2 However, serious thermal problems for these batteries at high power rates will occur due to the much more heat that is generated during the rapid charge and discharge cycles at high current level, especially when the battery size increases and many batteries are interconnected to form large packages.

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With the increase of battery size, therefore,

the temperature effect on the material properties should be considered during the development of large LIBs. Recently, first-principles calculations based on density functional theory (DFT) have been shown to be important in the ground state property analysis and prediction of materials in LIBs, and are now routinely applied in the investigations on material electronic and thermodynamic properties.4 The majority DFT investigations, however, only consider the thermodynamic properties at 0 K, not including the physical and chemical processed occurring at moderate or high temperatures.5,6 Owing to the effect of temperature on the battery material performance, there naturally exists certain differences between the predicted property with ground state calculations and the realistic performance measured at environment temperature. For example, the cathode material LiMn2O4 has a cubic structure at room temperature, while first-principles calculations only reproduce its low temperature orthorhombic structure.7,8 The phase transition of LIB electrode materials under finite temperature plays a crucial role in determining the battery performance, including the failure of electrode material at high

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or low temperature caused by intermediate phase. On the other hand, the accurate prediction of cell voltages is closely related to the temperature-dependent Gibbs energy functions of electrode materials. To exactly describe the performance of the cathode materials in LIBs, therefore, the thermodynamic quantities at finite temperature should be evaluated. An important contribution to the thermodynamic properties at finite temperature is the vibrational partition function, which can be evaluated by calculating the material's normal modes of lattice vibrations.9-11 Several authors have theoretically addressed the vibrational contribution to the material thermodynamic properties in LIBs. Liu et al. examined the lattice dynamics, finite-temperature thermodynamics of monoclinic Li2CO3, a critical component of the solid electrolyte interphase (SEI), with DFT method and various exchange-correlation functionals.12 Shang et al. employed first-principles phonon calculations with a mixed-space approach to probe the lattice dynamics and finite-temperature thermodynamic properties of LiMPO4 (M = Mo, Fe, Co, Ni).13 Chang et al. investigated the thermodynamic and electrochemical properties of Li-Co-O and Li-Ni-O systems at room temperature by means of first-principles calculations and empirical methods.14 In addition, our previous study has already included the vibrational contribution in calculating the intercalation potential of the complete delithiated state Li□CoO2.15 Layered LiMO2 (M = Co, Mn, Ni) and their mixture phases are widely applied as cathodes in LIBs.16 Knowledge of thermodynamic properties of layered LiMO2 (M = Co, Mn, Ni) is fundamental to study the stability and capacity of the LIBs with these

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cathode materials. The present work aims at providing a systematic investigation on the vibrational properties of LiMO2 (M = Co, Mn, Ni), and in turn, evaluating the corresponding finite-temperature thermodynamic properties. In our calculations, local density approximation (LDA) and generalized gradient approximation with the Hubbard model correction (GGA+U) yield similar phonon dispersions without soft modes and analogous thermodynamic functions for any of the layered lithium transition-metal oxides LiMO2 (M = Co, Mn, Ni). In addition, the phonon dispersion and thermodynamic functions of LiNiO2 are extremely close to that of LiMnO2, whereas the differences between LiNiO2 and LiCoO2 are relatively large. These results are the reflection of crystal structure. In addition, we study the intercalation voltage correction as a function of temperature by taking the vibrational contribution into account for the LiCoO2 compound at different delithiated concentration, which helps to accurately evaluate the cell voltage for the layered LiCoO2 systems.

Ⅱ. COMPUTATIONAL DETAILS The equilibrium structure calculations are performed using DFT method as implemented in Vienna Ab-initio Simulation Package (VASP) with a plane-wave basis set.17 The projector-augmented wave (PAW) method is used for the electron-ion interaction, and GGA in the form of the Perdew and Wang (PW91) functional is used to approximate exchange and correlation potential.18 The adopted pseudopotentials treat the Li-2s12p0, O-2s22p4, Co-3d84s1, Mn-3d64s1, Ni-3d94s1 shells as valence states. The cutoff energy of 550 eV is employed for the plane-wave basis expansion. The Brillouin zone is sampled with 7×7×1 Monkhorst-Pack k-point grid. The convergence

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criteria for the total energies and ionic forces are set to be 10-8 eV and 10−5 eV/Å in the formula unit. It is well known that GGA cannot adequately describe the strong exchange-correlation effects of

transition-metal 3d electrons,

thus

usually

underestimating the localization of transition-metal 3d electrons. In order to overcome this problem, GGA+U method is usually used to calculate the electronic structure of the system with transition-metal atoms.19, 20 Herein, the effective Coulomb repulsion parameter Ueff is set to be 3.9, 3.5 and 5.3 eV for the d orbital of Co, Mn and Ni, respectively, which are in agreement with those in the published literature.21 The phonon frequencies are calculated by using PHONOPY code, 22 which can directly use the force constants calculated by density functional perturbation theory (DFPT) 23 as supplemented in VASP. In addition, in order to investigate the effect of exchange-correlation approximations on the phonon frequencies, we also employ the LDA method in our calculations besides the GGA+U method. Knowledge of the entire phonon spectrum of a given system would further enable the calculation of its important thermodynamic quantities as a function of temperature (T). Considering the vibrational contribution, in order to obtain the temperature dependence of vibrational Helmholtz free energy (F), vibrational entropy (S) and constant-volume specific heat capacity (Cv), the integration over the phonon density of states is performed within the quai-harmonic approximation according to the following equations,24 F = 3 nNk B T ∫

ω max

0

 hω  ln  2 sinh   2 k BT 

    g (ω) d ω  

5


(1)


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S = 3nNk B ∫

ω max

0

C v = 3 nNk B ∫

 hω   hω   h ω       coth   − ln  2 sinh     g (ω ) d ω  2 k BT   2 k BT       2 k BT 

ω max

0

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(2)

2

 hω  2  hω    csch   g (ω ) d ω  2 k BT   2 k BT 

(3)

where kB is the Boltzmann constant, ωmax is the largest phonon frequency, n is the number of atoms per unit cell, N is the number of unit cells and g(ω) is the normalized vibrational density of states (VDOS) with ∫

ωmax

0

g (ω )d ω = 1 . In our calculations, the

thermodynamic quantities are also calculated by using PHONOPY code.

Ⅲ. RESULTS AND DISCUSSION A. Structure We first examine the structures of the layered lithium transition-metal oxides, LiMO2 (M = Co, Mn, Ni). LiCoO2 adopts the α-NaFeO2 structure with consecutive alternating CoO2 and Li layers, and having rhombohedral structure with R 3 m space group, as shown in Fig. 1(a). LiNiO2 also has the α-NaFeO2 structure. However, DFT calculations suggested the transformation from rhombohedral lattice to monoclinic lattice (C2/m space group), which is induced by the cooperative Jahn-Teller (JT) distortion25. The local Jahn−Teller distortion was previously observed by extended X-ray absorption fine structure (EXAFS) measurement.26 As for LiMnO2, layered structure is metastable and synthesized by ion-exchange techniques.27-29 Layered LiMnO2 also exhibits a small monoclinic (C2/m space group) deformation of the ideal rhombohedral structure due to Jahn-Teller distortion of Mn3+. Therefore, the monoclinic structure with C2/m space group, as shown in Fig. 1(b), is adopted for the LiNiO2 and LiMnO2 in this work. The corresponding structural parameters are listed

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in Table 1. It is worth mentioning that the optimized structures here are based on the GGA+U method. Actually, all of the LiMO2 (M = Co, Mn, Ni) structures are also relaxed at the LDA level so that these structures can be used in the following phonon frequency calculations. B . Phonon dispersion Calculations of vibrational frequencies may reveal structural instabilities through low or imaginary phonon frequencies. Therefore, phonon dispersion could give a criterion for the crystal stability and the possible phase transitions by the prediction of soft modes, such as in SrTiO3.30 If all phonon frequencies are positive, the crystal is stable. If some frequencies are imaginary (soft modes), then the system is unstable. The calculated phonon dispersion curves of LiMO2 (M = Co, Ni, Mn) along several high symmetry lines in the first Brillouin zone are plotted in Fig. 2. As there are four atoms in the primitive cell of LiMO2 (M = Co, Ni, Mn), three acoustic modes and nine optical modes are presented in the dispersion curves. From Fig. 2, it is found that the phonon dispersion curves of LiMO2 (M = Co, Ni, Mn) with GGA+U method are basically similar to those with LDA method. No soft modes are observed in the phonon dispersion curves. For the LiNiO2 and LiMnO2 compounds, the phonon frequencies from LDA method are higher to some extent than those obtained from GGA+U method around the high symmetry point M. This mainly results from the general underestimation of bond length according to LDA calculations when compared with GGA+U method, thus corresponding to the stronger bond and higher vibrational frequencies. Despite this, the similar results of phonon dispersion for LDA and GGA+U methods are helpful conclusion. As we know, the computational cost of 7



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GGA+U method is largely higher than that of LDA method. Though the electronic structures of the transition-metal oxides are affected significantly by the exchange-correction approximation, the difference of the vibrational property between LDA and GGA+U methods is acceptable. Based on this consideration, LDA method could be used to replace GGA+U method in calculating the phonon frequencies for the lithium mixed transition-metal oxides system (Li-Ni-Mn-O or Li-Ni-Co-Mn-O), which would save the computational cost and be helpful to treat larger systems. To further analyze the contribution of each atom to the lattice vibration, the VDOS figures are also plotted, as shown in the right panels of Fig. 2. For the three compounds LiMO2 (M = Co, Ni Mn), the high frequency modes, basically above 14 THz, are mainly from the motion of oxygen and transition-metal atoms, while the medium and low frequency modes are contributed from the motion of lithium and transition-metal atoms. In contrast to the case of LiCoO2, oxygen atoms also essentially contribute to the medium frequency modes for LiNiO2 and LiMnO2, which means that some vibrational frequencies of oxygen atoms are lower to some extent. From the structure point of view, the symmetry of LiNiO2 and LiMnO2 (C2/m space group) is lower than that of LiCoO2 (R 3 m space group). Therefore, it seems that the decrease of the structural symmetry make the vibrational frequencies of oxygen atoms lower. In fact, however, the variation of the oxygen vibrational frequencies results from the Jahn-Teller effect of Mn3+ and Ni3+, which leads to the reduction of crystalline symmetry from R 3 m to C2/m. It is known that the Jahn-Teller distortion of Mn3+ and Ni3+ is able to result in the elongation of some Mn-O and Ni-O bonds when

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compared with other Mn-O and Ni-O bonds. Elongation of bonds, suggesting the weaker interaction between metal atom and oxygen atom, subsequently lowers the vibrational frequencies of oxygen. C. Zone center phonon modes Next, the vibrational frequencies and the symmetry at zone center (Γ point) are studied. According to group theoretical analysis, the acoustical and optical modes at zone center can be classified into the following irreducible representations, LiCoO2: Γaco = A 2u + 2Eu and Γ opt = 2A 2u + 2Eu + A1g + E g LiMO2 (M = Ni, Mn): Γaco = A u + 2Bu and Γ opt = 2A u +4Bu + 2A g + Bg The corresponding phonon frequencies and activities are listed in Table 2, where the notations of IR and Raman indicate infrared active and Raman active modes, respectively. Besides the calculated results, some experimental data are also given in Table 2. On one hand, the experimental phonon frequencies agree well with our theoretical values (LDA or GGA+U), which further confirms the reliability of our calculations. On the other hand, the frequencies that have not been observed will provide the important information for the future experiments. In addition, the optical phonon modes of LiMO2 (M = Co, Ni, Mn) with hexagonal phase and monoclinic phase are plotted in supporting information (Figure S1).

D. Thermodynamic quantities Figure 3 shows the thermodynamic quantities, vibrational Helmholtz free energy (F), vibrational entropy (S) and constant-volume specific heat capacity (Cv), as a function of temperature (T) for LiMO2 (M = Co, Ni, Mn) compounds, which are calculated according to the Eqs. (1), (2) and (3). In this figure, the results from GGA+U and LDA methods are both given. It is obviously found that the curves of the

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thermodynamic quantities based on GGA+U method almost coincide with those based on LDA method. This also indicates that the different exchange-correlation functional has little effect on the thermodynamic properties, which is in agreement with the phonon dispersion results above. In addition, the constant-volume specific heat capacities (Cv) of the three compounds all tend to the limit value of 99.77 J mol-1 K-1 (equal to 12R, where R is the gas constant measured in joule per kelvin per mole) with the increase of temperature. According to the Eq. (3), the Cv value of 12R can be obtained when the temperature T tends to positive infinity. On the other hand, the Dulong-petit law indicates that the specific heat capacity per mole of element is 3R. As four atoms are included in the unit cell of LiMO2 (M = Co, Ni, Mn), the specific heat capacity for per primitive cell is 12R according to Dulong-petit law. Obviously, the Dulong–petit law is naturally embedded in our fomula. As is well known, the Dulong-Petit law offers fairly good prediction for the specific heat capacity of many solids at high temperature. Therefore, our results of LiMO2 (M = Co, Ni, Mn) specific heat capacities are meaningful. In order to better compare the thermodynamic quantities among LiMO2 (M = Co, Ni, Mn), we put the results together in Fig. 4, as well as the experimental data available in the literature. All of the dada in Fig. 4 are based on the calculations with GGA+U method. For LiCoO2, the constant-volume specific heat capacities (Cv) increases quickly from 0 to about 80 J mol-1 K-1 when the temperature increases from 0 to 400 K, as shown in Fig. 4(a). After that, the heat capacity increase slowly, eventually approaching the classical Dulong-Petit asymptotic limit that is discussed above. Compared with the result of LiCoO2, the Cv of LiNiO2 and LiMnO2 are larger to some extent, especially for the temperature region from 200 to 400 K. More interestingly, it is noted that the Cv curve of LiNiO2 is extremely close to that of

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LiMnO2, indicating that the vibrational contribution has the same effect on the specific heat capacity for LiNiO2 and LiMnO2. Similar results are also found for the thermodynamic quantities of F and S, as shown in Fig. 4(b). However, the results of LiCoO2 are relatively far away from that of LiNiO2 and LiMnO2. For LiCoO2, specifically, the free energy decreases from about 25 to -25 KJ mol-1, and the entropy increases from 0 to 137 KJ mol-1, when the temperature increases from 0 to 800 K. From the mass point of view, the difference between Mn and Ni atoms is larger than that between Mn and Co, or Ni and Co. Then, why is the thermodynamic results of LiNiO2 more close to that of LiMnO2? Obviously, the factor of mass is not the principle reason. Considering the factor of structure, it can be understood by the crystalline symmetry. The space groups of LiNiO2 and LiMnO2 are both C2/m, while

R 3 m for LiCoO2. The same crystalline symmetry brings the similar thermodynamic properties. According to the calculated results of the thermodynamic quantities, therefore, it is concluded that for the LiMO2 (M = Co, Ni, Mn) compounds, the crystalline symmetry has more significant effect on the thermodynamic properties than mass. This is in consistent with the VDOS results discussed above. It is worth noting that the agreement between our calculations and the experimental data is excellent for the specific heat capacity of LiCoO2 below room temperature, as shown in Fig. 4(a). Actually, the experimental results, which is from the supporting information of Ref 33, corresponds to the constant-pressure heat capacity Cp. Considering the thermal expansion, the relationship between the heat capacity at constant-pressure Cp and at constant-volume Cv can be expressed as the following well-known thermodynamic formula C p = Cv + α 2 BVT ,34 where α is the coefficient of volumetric thermal expansion, V is the unit cell volume, and B is the bulk modulus. Because the second term of the equation is negligible below room 11



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temperature for the solid materials, the Cp and Cv is comparable at the temperature below 300 K. Hence, the experimental results fully illustrate the accuracy of our calculations.

E. Voltage correction Generally, accurate estimation of intercalation/ extraction voltage for the electrode materials is important to evaluate the energy density of LIBs. The intercalation voltage is originated from the change of the Gibbs free energy in the battery system during the process of lithium intercalation. Taking Li metal as anode and LiMO2 (M = Co, Ni, Mn) as cathode, the intercalation voltage can be calculated by using the following equation 35, 36

Vave = -∆G / nF

(4)

where ∆G is the change of the Gibbs free energy for the intercalation reaction, F stands for the Faraday constant, and n represents the transferred electrons (Li ions) in the intercalation process.37 According to the knowledge of thermodynamics, the PV and -TS terms are included in the Gibbs free energy besides the inner energy E. Because the P∆V term is very small for solid state materials and the contribution of entropy is zero when the temperature is 0 K, the Gibbs free energy is approximately chosen as equal to the inner energy when the intercalation voltage is calculated, which could be seen in the published literature.35,

36

With the temperature increasing,

however, the contribution of entropy to the Gibbs free energy becomes more and more important. Despite that the change of volume also exists with the increase of temperature, the estimated P∆V term for the solid state material is of the order of 10-5

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eV.36 During the charging/discharging process, the potentials of common cathode materials in LIBs are about in the range of 3–4 V, which corresponds to the change of the total Gibbs free energy of 3–4 eV for per formula lithium atom. Obviously, the P-

∆V term is negligible when compared to the change of the total Gibbs free energy during the intercalation process. As a result, the contribution from the vibrational entropy to the Gibbs free energy, which in turn results in the correction of the intercalation/extraction voltage, should be considered at the environment temperature. Taking the contribution of the zero point energies into account, we calculated the voltage correction resulting from the lattice vibration for the LiCoO2/Li half battery system at different temperatures, as shown in Fig. 5(b). The voltage correction is given by the following formula,

Vcor = -∆Fvib / nF

(5)

and

∆Fvib = Fvib ( LiCoO2 ) − Fvib ( Lix CoO2 ) − (1 − x) Fvib ( Li) Where

Fvib ( LiCoO2 ) ,

Fvib ( LixCoO2 ) and

Fvib ( Li )

(6)

represent the vibrational

Helmholtz free energy (zero point energy plus -TS) of LiCoO2, delithiated LixCoO2 and Li in bulk phase, respectively. In our calculations, the corrections of the average intercalation voltage with two different lithium concentrations, namely Li0.75CoO2 and Li0.5CoO2, are considered. To illustrate the calculated results better, the vibrational Helmholtz free energies as a function of temperature for LiCoO2, Li0.75CoO2 , Li0.5CoO2 and Li metal are plotted in Fig. 5(a). Our results show that the voltage correction decreases with the increase of temperature. When 25% Li are extracted

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from the perfect LiCoO2 (forming Li0.75CoO2), the correction of the average intercalation voltage decreases from -46 mV to -218 mV with the temperature increasing from 0 to 700 K. Especially, the voltage correction reaches -100 mV at the temperature of 300 K, while -114 mV at 350 K. When the operation temperature increases from 300 to 350 K, the voltage of the battery system will decrease 14 mV. Generally, the operation voltage of cathode for LIBs ranges between 3 V and 4 V. Considering the non-linear relationship between the discharge capacity and the voltage, the voltage decrease of 14 mV would result in a decrease of the discharge capacity by about 3%-4%. Consequently, the performance change of the battery system could be noticeable although the voltage correction induced by the temperature effect is small. In addition, the voltage correction curve as a function of temperature for the 50% delithiation state (Li0.5CoO2) is close to that of 25% delithiated state (Li0.75CoO2), indicating that the voltage correction is insensitive to the concentration of lithium.

F. Discussion on temperature effect Although the thermodynamic quantities of LiMO2 (M = Co, Ni, Mn) compounds as a function of temperature are discussed above, the related phonon frequencies are calculated by DFT method at 0 K. A problem is naturally raised. Are these calculated frequencies affected at the finite temperature discussed? Our answer is yes, but the effect is very limited. According to the knowledge of solid state physics, with the increase of temperature the number of phonon will increase, while phonon frequencies not. Phonon frequencies are determined by the interatomic interaction that

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is generally influenced by the bond lengths or the volume. If not considering the volume change, the phonon frequencies of the compounds are kept when temperature increases. Actually, the increase of temperature usually results in the increase of volume for solid materials. However, the thermal expansion coefficients of solid cathode materials in LIBs are of the order of 10-5 K-1 or less.38,39 Obviously, the volume expansion in the range of the temperature discussed is negligible. Therefore, the phonon frequencies calculated at 0 K are reliable for the discussion at the finite temperature.

Ⅳ. CONCLUSIONS In terms of first-principles phonon calculations, the phonon dispersion and thermodynamic quantities of the layered lithium transition-meal oxides LiMO2 (M = Co, Ni, Mn) are investigated. The phonon spectra without soft modes are predicted by our calculations for the three layered lithium transition-metal oxides, indicating the structural stability for the considered phases. According to the calculated phonon dispersion, the finite-temperature thermodynamic quantities, namely, specific heat capacity, vibrational Helmholtz free energy and vibrational entropy, are obtained. Combining with the vibrational and thermodynamic properties, it is found that the phonon dispersion and thermodynamic quantities of LiCoO2, LiNiO2 and LiMnO2 from LDA and GGA+U methods are extremely similar, showing that the phonon calculations are basically insensitive to the selection of the exchange-correction functional. In addition, the vibrational and thermodynamic properties of LiNiO2 is very close to that of LiMnO2, but relatively far away from that of LiCoO2, due to the

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same crystalline symmetry of LiNiO2 and LiMnO2, which is different from that of LiCoO2. The calculated phonon frequencies at zone center and the heat capacities below room temperature for LiCoO2 compound are in good agreement with the results from experiments. Our results of the thermodynamic properties are helpful to better understand the performance of LIBs at finite temperature. As an example, the intercalation voltage correction of LiCoO2 cathode material is discussed in this study.

ASSOCIATED CONTENT Supporting Information The zone center phonon modes of LiMO2 (M = Co, Ni, Mn) with hexagonal phase and monoclinic phase (Figure S1). This information is available free of charge via the Internet at http://pubs.acs.org/.

AUTHOR INFORMATION Corresponding Author * [email protected] Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work is supported by the Natural Science Foundation of China (Grand Nos. 11264014 and 11564016) and Natural Science foundation of Jiangxi Province (Grand No. 20133ACB21010, 20142BAB212002 and 20152ACB21014). B. Xu is also supported by the oversea returned project from the Ministry of Education. The computations are performed on TianHe-1(A) at the National Supercomputer Center in

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Tianjin.

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Table 1. Optimized structural parameters of LiMO2(M= Co, Ni, Mn) layered oxides. dM-O is the distance between transition metal atom and nearest neighboring O atom. Jahn-Teller (JT) distortion rate is defined as ( d M -O − d M -O ) / d M -O . long

short

System

Lattice parameter

LiCoO2

a=b=2.836 Å, c=14.150 Å

long

dM-O (Å)

JT distortion rate (%)

2.107

0

2.144/1.899

11.4

2.348/1.951

16.7

α= β =90°, γ=120° LiNiO2

a=5.515 Å, b=2.790 Å, c= 5.141 Å α= γ=90°, β=112°

LiMnO2

a=5.515 Å, b=2.848 Å, c= 5.463 Å α= γ=90°, β=116°

Table 2. Experimental and calculated zone center optical phonon modes for LiMO2 (M = Co, Ni, Mn).

Symmetry

Activity

Calculated ωm(cm-1) LDA

Experimental ωm (cm-1)

GGA

LiCoO2 1 g

A Eg 1 Eu 2 Eu 1 A2u 2 A2u

Raman Raman IR IR IR IR

604.3 483.4 570.2 242.1 538.0 380.9

576.2 468.8 524.9 232.0 578.4 386.5

59631 48631

LiNiO2 1 g 2 g

A A Bg 1 Bu 2 Bu 3 Bu 4 Bu 1 Au 2 Au

Raman Raman Raman IR IR IR IR IR IR

584.0 235.2 472.3 399.5 300.2 245.0 232.9 593.0 584.8

546.2 310.6 358.1 386.0 301.1 230.9 227.6 582.7 516.9

LiMnO2 1 g 2 g

A A Bg 1 Bu

Raman Raman Raman IR

624.6 228.7 436.4 424.8

578.2 244.4 386.1 381.2

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60332 42232

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2

Bu 3 Bu 4 Bu 1 Au 2 Au

IR IR IR IR IR

271.2 210.0 150.6 602.1 488.9

218.8 211.7 93.6 581.8 456.1

Figure 1. Structures of layered oxides for (a) LiCoO2 with R 3 m space group, and (b) LiNiO2 (LiMnO2) with C2/m space group. The medium red balls denote oxygen ions, while the small green and big blue ones are Li ions and transition-metal ions, respectively.

Figure 2. Phonon dispersion and VDOS of LiMO2 (M=Co, Ni, Mn).

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Figure 3. Calculated thermodynamic quantities of (a) LiCoO2, (b) LiNiO2 and (c)LiMnO2.

Figure 4. The comparison of the thermodynamic quantities of (a) heat capacity and (b) vibrational Helmholtz free energy and vibrational entropy for LiMO2(M = Co, Ni, Mn).

Figure 5. Vibrational free energies (a), and the intercalation voltage correction (b) as a function of temperature.

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Table of Contents (TOC) Graphic

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