First-Principles Investigation on Electrochemical Performance of Na-Doped LiNi1/3Co1/3Mn1/3O2

The cathode material LiNi1/3Co1/3Mn1/3O2 for lithium-ion battery has a better electrochemical property than LiCoO2. In order to improve its electrochemical performance, Na-doped LiNi1/3Co1/3Mn1/3O2 is one of the effective modifications. In this article, based on the density functional theory of the first-principles, the conductivity and the potential energy of the Na-doped LiNi1/3Co1/3Mn1/3O2 are calculated with Materials Studio and Nanodcal, respectively. The calculation results of the band gap, partial density of states, formation energy of intercalation of Li+, electron density difference, and potential energy of electrons show that the new cathode material Li1−x Na x Ni1/3Co1/3Mn1/3O2 has a better conductivity when the Na-doping amount is x = 0.05 mol. The 3D and 2D potential maps of Li1−x Na x Ni1/3Co1/3Mn1/3O2 can be obtained from Nanodcal. The maps demonstrate that Na-doping can reduce the potential well and increase the removal rate of lithium-ion. The theoretical calculation results match well with experimental results. Our method and analysis can provide some theoretical proposals for the electrochemical performance study of doping. This method can also be applied to the performance study of new optoelectronic devices.


INTRODUCTION
Since Sony introduced commercial lithium-ion batteries (LIBs) that used LiCoO 2 as cathode, LIBs have been used in electric vehicles (EVs), hybrid electric vehicles (HEVs), and mobile electronic devices for their good cycle performance, long life, less self-discharge, high specific capacity, and working voltage [1]. In the current era of LIBs, there is an ever-growing demand for even higher energy densities to power mobile devices with increased power consumption.
Cathode materials are one of the key materials to determine the electrochemical performance of LIBs. Recently, the ternary cathode material LiNi x Co y Mn z O 2 (NCM, 0 < x, y, z < 1) has been paid much attention for its lower price and better electrochemical performance. LiNi 1/3 Co 1/3 Mn 1/3 O 2 , LiNi 0. 5 [2,3] are often studied by scholars and experts. Their precursors are made by hydroxide coprecipitation reaction combined with solid-phase sintering step [4]. LiNi 1/3 Co 1/3 Mn 1/3 O 2 is considered a promising cathode material for commercial LIBs due to its low price, high conductivity, excellent cycle performance, high reversibility, and stable structure [5]. However, the commercial LiNi 1/3 Co 1/3 Mn 1/3 O 2 has already reached its theoretical limit, with poor cycling performance at high rate and thermal stability at high cutoff voltage, which limit its further application in higher-power systems such as EVs and HEVs [6].
Samples and raw materials needed in the NCM ternary cathode material experiments are of high cost, and some experiments may be conducted several times; thus, the entire experiment will need a long time. However, based on firstprinciples [16][17][18], the physical and electrochemical properties can be calculated and analyzed by Materials Studio, Nanodcal, and MATLAB; the results can give some theoretical advice or investigation directions about the relevant experiments, and the theoretical calculation can effectively shorten the whole investigation period and reduce the cost [16]. In this article, the Na-doped layer-structure Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 is studied theoretically with density functional theory of firstprinciples; the results show that the electrochemical performance of Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 is affected by the proportion of Na-substitution and the best proportion of Nasubstitution in LiNi 1/3 Co 1/3 Mn 1/3 O 2 is in agreement with that of experiments [15].

PRINCIPLE The Theory Foundation of the First-Principles
Quantum mechanics is about the properties of molecules and atoms, which is closer to the nature of all things in the universe, so it is called the first principle. In fact, according to the nuclear charge number of atoms and some simulated environmental parameters (such as electron quantity and static mass), the first principle means to solve the Schrödinger equation as shown in Equation (1).
where H is the Hamiltonian and E is the system's energy. The solution of the Equation (1) is the eigenvalue (energy value) and eigenfunction (wave function), which are used to analyze the related properties of particles. However, for the multiparticle system, it is impossible to solve the Schrödinger equation exactly because of a complex electronic interaction in the system, so it is necessary to simplify and approximate it reasonably. The mass of the nucleus is much larger than that of the electron, and the velocity of the nucleus is much smaller than that of the electron. For the electron, the nucleus can be assumed to be stationary, and then the Schrödinger equation can be approximated adiabatically, as shown in Equation (2).
where H(R) is the adiabatical Hamiltonian and E(R) is the system's adiabatical energy.

Density Functional Theory
The density functional theory originates from the Thomas-Fermi model proposed by Thomas [19] and Fermi [20]. This model ignores the exchange-correlation between electrons. On this assumption, the Schrödinger equation can be converted into a simpler wave equation, and then the function of the total energy about the electronic system is determined only by the electron density function r(r), which is called density functional theory (DFT). This model is too rough to be used directly. Based on the Thomas-Fermi model, under the Born-Oppenheimer approximation, Frontiers in Physics | www.frontiersin.org February 2021 | Volume 8 | Article 616066 nonrelativistic adiabatic approximation, and single-electron approximation, Hohenberg and Kohn proposed the more exact density functional method (HK theorems) [21,22]. The physical properties of matter can be analyzed by selfconsistent calculation. According to HK theorems, properties of particles' ground state can be expressed by particle number density ρ(r) instead of the wave function, the system degrees turning 3N into 3, so a lot of computing resources can be saved. However, HK theorems still have the following three problems: firstly, how to determine the particle number density function ρ(r); secondly, how to determine the kinetic energy functional T [ρ(r)]; lastly, how to determine the exchangecorrelation energy functional E xc [ρ(r)].

Kohn-Sham Equation and Exchange-Correlation Functional
The first and the second problems can be solved by the Kohn-Sham equation [23], as shown in Equation (3). where is the attraction potential between nuclei, V coul [ρ(r)] is the Coulomb potential between electrons, and V xc [ρ(r)] is the exchange-correlation potential. In Equation (3), the Hamiltonian with the interacting particles is replaced with that of the noninteracting particles, and the interacting particles are described by the exchange-correlation functional E xc [ρ(r)]; then, the single-electron equation can be obtained. The last problem is generally solved by the local density approximation (LDA) and the general gradient approximation (GGA). LDA can give more accurate results for the system with small change of electron density relative in the space. Considering the ununiformity of the electron density, GGA can give more exact results than LDA [24]. With the development of computer technology, computing power has been improved dramatically, and programs based on DFT have gradually become a universal means to study the properties of materials. Multielectron system is the research object of DFT, and HK theorems are the theoretical basis of this method.

METHOD AND MODEL
Based on PW91 (Perdew-Wang) [25] method with the PBE (Perdew-Burke-Ernzenhof) [26] exchange-correlation functional, DFT calculations have been carried out using GGA in the CAmbridge Serial Total Energy Package (CASTEP) program of Materials Studio 8.0. The CASTEP is the quantum mechanical procedure of the plane wave pseudopotential method. An ultrasoft pseudopotential is used to describe the Coulomb attraction potential between the inner layer electrons around the nucleus and that of the outer layer [27], A plane wave cutoff of 340 eV and a k-point mesh of 4 × 4 × 1 in the Monkhorst-Pack [28] sampling scheme are used. The parameter about the self-consistent calculation is 1 × 10 −6 eV/atom. To obtain the local stable structure of material, the structural geometry is optimized until the average force on every atom falls less than 0.3 eV/nm, the maximum displacement tolerance is less than 1 × 10 −4 nm, and the maximum stress tolerance less than 0.05 GPa.
For this theoretical calculation, the electronic conductivity of Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 is calculated by the CASTEP program with the virtual mixed atom method. The chemical formula assumes Li and Na as 1 mol, the doping amount of Na is x mol, and then the doping amount of Li is 1-x mol. The Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 cell model is shown in Figure 1. Li and Na occupy 3a, Ni 1/3 Co 1/3 Mn 1/3 occupies 3b, O atoms occupy 6c, and the space group is R3m. Based on the model (B) When x 0.01 mol, PDOS peak of Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 is higher than that of the pristine. (C) When x 0.05 mol, the PDOS peak rises to about 157 eV, its conductivity is best. (D) When x 0.10 mol, the PDOS peak declines dramatically to 5.5 eV, which is much lower than that of the pristine.

Band Gap and Partial Density of States
The band gap and the partial density of states (PDOS) of Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 were calculated when Na-doping amount x 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, and 0.10 mol. The electronic conductivity of the material depends on the number of electrons contained in the conduction band. The wider the band gap is, the less easily the valence band electrons can leap to the conduction band and the worse the conductivity of the material is. After Na-doping, the whole band structure of Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 is not affected, but the energy gap has changed significantly. The relevant data of the band gap is shown in Table 1, and the diagram of the relationship between x and the band gap is shown in Figure 2. Compared with the pristine, the change of the band gap is most obvious when x 0.01 mol. The results show that its band gap is decreasing when x increases, and Na-doping can effectively improve the conductivity of LiNi 1/3 Co 1/3 Mn 1/3 O 2 . The peak of the PDOS reflects the number of electrons in this energy. The bonding and the density of states near the Fermi level can be obtained by PDOS, and the influence of Na-doping on the conductivity of LiNi 1/3 Co 1/3 Mn 1/3 O 2 can be analyzed. Their PDOS are shown in Figure 3 when x 0, 0.01 and 0.05 mol. The colored lines represent the density of states of different orbitals. Figures 3A-D show that Na-doping has a great influence on the number of electrons near the Fermi level. When x 0.01 mol (shown in Figure 3B), the PDOS peak is about 148 eV that is higher than that of the pristine (shown in Figure 3A). When x 0.03-0.06 mol, the PDOS peak near Fermi level increases to 157 eV and its conductivity becomes better; when x 0.06-0.09 mol, the PDOS peak begins to decline, its conductivity becomes worse; when x 0.10 mol, the PDOS peak is lower than that of the pristine. Thus, the right amount of Na-doping should be x < 0.06 mol.

Formation Energy of Intercalation of Li + and Cell Volume
The greater the formation energy of metal oxide is, the more stable the matter is, the more difficult it is for atoms to separate from molecules. The formation energy of the intercalation of Li + (E f ) is shown in Equation (4).
where E t is the total energy of the supercell, E dl is the deintercalation energy of Li + , and E dil is the energy of the supercell after the deintercalation of Li + . The 4 × 3 × 2 supercell models are built before Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 is intercalated/deintercalated of Li + and when it is intercalated/ deintercalated of Li + and deintercalated of Li + . The calculation of the cell's volume shows that the volume of Na-doped LiNi 1/ 3 Co 1/3 Mn 1/3 O 2 is not extremely getting bigger though Na + is bigger than Li + and the structure keeps stable very well. The results of the energies in Equation (4) show that E f has changed with the Na-doping amount (shown in Figure 4). It can be seen  that Na-doping can effectively enhance the cycling performance of LiNi 1/3 Co 1/3 Mn 1/3 O 2 . E f decreases when x < 0.05 mol, and it rises gradually when x > 0.06 mol, so the right amount x is 0.05-0.06 mol.

Electron Density Difference
The change of electrons' distribution near the local atoms with Na-doping amount x is analyzed by the electron density difference of Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 . The results are shown in Figure 5. Compared with the pristine (Figure 5A), there is no change in the color of the electron cloud around atoms when x 0.01 mol ( Figure 5B), but the electron cloud has more coverage, which exhibits that its conductivity is becoming better; when x 0.05 mol, their color starts to get darker, meaning that the free electrons increase, but the electron cloud's coverage shrinks a little ( Figure 5C). The electron cloud's color gets darker than before when x > 0.05 mol, and the electron cloud's coverage expands again ( Figure 5D). So, Na-doping can improve the conductivity of LiNi 1/3 Co 1/3 Mn 1/3 O 2 , and the best amount of Na-doping is x 0.05-0.10 mol.

Potential Energy of Electrons
For a system, its internal energy cannot change when there is no external force. In a potential well, the particles have the greatest kinetic energy and smallest potential, and particles can move freely in this potential well. If there is no external energy, it is difficult for them to cross over the potential well to other positions in the molecule. Therefore, the potential energy of Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 can be analyzed to judge how difficult it is for electrons to escape from the potential well after doping. The 3D potential energy map of Li 0.95 Na 0.05 Ni 1/3 Co 1/3 Mn 1/3 O 2 is shown in Figure 6. The blue represents the smaller negative potential energy and the red represents the larger positive potential energy. To analyze the influence of Na-doping on the potential energy well, the 2D potential energy diagram of Li 0.95 Na 0.05 Ni 1/3 Co 1/3 Mn 1/3 O 2 is also obtained (shown in Figure 7). If electrons in a potential well (marked blue "*") get enough energy to escape from the potential well, they can move to other places in the molecule. At the same time, the barriers will prevent electrons from moving along the x-axis, so electrons must move along the minimum potential energy path marked blue "*" (shown in Figure 7). Finally, the electrons are deintercalated from the cathode to the anode.
In Figure 7, the blue mark "*" corresponds to the position and the potential energy. All the potential energy decreases from 17 to 1, and the change in each set is not exactly the same. As the amount of Na-doping increases, the average of the minimum potential energy decreases, which indicates that Na-doping will make the Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 potential well lower and it is easier for the electrons to escape from the potential well, and so it is for the charging reaction. Thus, its rate performance of Li 1−x Na x Ni 1/3 Co 1/ difference of Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/3 O 2 , it has the best conductivity when x 0.05-0.10 mol. In conclusion, the Na-doped LiNi 1/3 Co 1/3 Mn 1/3 O 2 has better electrochemical performance when x 0.05 mol, which is coherent with experimental results. The sample of Li 1−x Na x Ni 1/3 Co 1/3 Mn 1/ 3 O 2 can be synthesized by coprecipitation reaction combined with solid-phase sintering steps. Though the electrochemical performance of Li 0.95 Na 0.05 Ni 1/3 Co 1/3 Mn 1/3 O 2 is proved theoretically and experimentally to be better than that of LiNi 1/3 Co 1/3 Mn 1/3 O 2 , energy densities and specific capacity should be improved further with other modification methods. Our method and analysis based on the first-principles can provide some theoretical proposals for the electrochemical performance study of doping, and this method can be applied to the performance study of new optoelectronic devices.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

AUTHOR CONTRIBUTIONS
YG designed models, analyzed some data, and wrote the manuscript. KS, PL, and WY carried out calculations. LL and FC gave some proposals. XH analyzed some results.