Ionic Conductivity of Lithium Phosphides

Abstract

We comprehensively study the ionic conductivity in lithium phosphides, promising materials for energy storage applications, by using a combination of first-principles computations and machine learning interatomic potentials. Using the quasiharminic approximation, we calculated convex hulls of the Li-P system at various temperatures and the temperature-composition phase diagram was obtained, delineating the stability regions of each phase. The ionic conductivity of stable (Li${}_3$P, LiP, Li${}_3$P${}_7$, Li${}_3$P${}_{11}$, LiP${}_7$) and metastable (Li${}_4$P${}_3$, Li${}_5$P${}_4$, LiP${}_5$) compounds was studied as a function of temperature. In some compounds we found have high ionic conductivity at room temperatures (10${}^{-3}$–10${}^{-2}$ S cm${}^{-1}$). Structures with the lowest ionic conductivity are LiP, Li${}_3$P${}_{11}$, and LiP${}_7,$ in which diffusion is negligible in the whole temperature range 300–500 K. In Li${}_3$P, Li${}_3$P${}_7$, and Li${}_4$P${}_3$, LiP, there is the 3D diffusion of Li atoms, while in Li${}_5$P${}_4$ the 2D mechanism prevails, and in LiP${}_5$ and LiP${}_7$ the 1D mechanism was observed. This study may provide insights for the development of Li-P materials in lithium ion and lithium metal battery applications.

1. Introduction

Lithium ion batteries (LIB) are widely used [1,2,3,4,5,6,7,8,9]: they are the most effective secondary energy sources for portable and mobile devices [10,11,12,13,14,15,16] and are also used in electric vehicles [17,18,19,20]. However, the maximum specific energy of modern commercial LIBs is about 250 Wh kg${}^{-1}$ (theoretical capacity of Li in griphite anode is 372 mAh g${}^{-1}$), which does not satisfy new needs [7,14,21,22,23]. To overcome the capacity limitations, the use of other anodes is suggested, such as Li-metal anode [2,3,4,5], or different lithium alloys [24], i.e., silicon anodes have the highest capacity of 3579 mAh g${}^{-1}$ [25]. The main problems with lithium ion, and especially Li-metal batteries, are dendrite formation and/or large structural changes under lithiation [24,25], leading to poor cyclability, a lower Coulomb efficiency, reduced service life, and short circuits [26]. There are several ways to solve these problems. Safety and durability can be improved by switching from liquid to solid electrolytes, more effectively preventing dendrite formation and short circuits, and thus increasing the safety and cyclability of batteries [21,27,28]. Moreover, the transition to all-solid-state batteries will allow for the use of lithium metal instead of a graphite anode, which will increase the specific capacity of batteries [14,21].

A solid electrolyte, just like a liquid one, must be a good ionic conductor and should not conduct electrons, be thermally stable and, when in contact with the anode, must form stable solid interphases, which must also be good ionic conductors and mechanically and thermally stable [21].

Li-P-S compounds are promising candidates for the role of a solid electrolyte in batteries with a metal lithium anode. For example, batteries with Li${}_2$S-P${}_2$S${}_5$ electrolyte were assembled and tested, their interfacial layers with lithium were studied, and the mechanisms of dissolution and precipitation of lithium in Li${}_2$S-P${}_2$S${}_5$ were determined using scanning electron microscopy methods [29,30,31]. Liu et al. [32] studied the effect of microstructure modification on the ionic conductivity of Li${}_3$PS${}_4$ and showed the possibility of its use at room temperature. Li-P-S compounds and the interfacial layers they form in contact with lithium were also studied by first-principles simulations. It was shown [33] that the Li${}_3$PS${}_4$-Li${}_3$PO${}_4$ electrolyte forms interfacial layers Li${}_3$PS${}_4$, Li${}_3$PO${}_4$, Li${}_2$O, Li${}_2$S, of which Li${}_3$PO${}_4$ is the least stable. In more complex systems, such as Li${}_7$P${}_3$S${}_{11}$, the ionic conductivity varies from 10${}^{-5}$ to 10${}^{-3}$ S cm${}^{-1}$ depending on the morphology [34], and substitution by iodine [35] or other halogens [36] can lead to an increase in ionic conductivity and improve the stability of these solid-state electrolytes.

In some studies, phosphides or even phosporous materials are considered promising anode materials [37,38,39,40,41] because of their high theoretical capacity and low redox potentials, i.e., 2596 mAh g${}^{-1}$ for Na${}_3$P [42]. Amine et al. proposed using a composite material based on black phosphorous and showed that ball milling modification helps to achieve a high initial Coulomb efficiency, high reversible capacity and good electrochemical performance [43]. Fang et al. developed a composite phosphorous anode for Na-ion batteries with a high reversible capacity and great electrochemical performance [44]. A hierarchical red phosphorous composite anode for K-ion batteries was designed and showed a high reversible capacity of 566.7 mAh g${}^{-1}$ after 100 cycles at 0.1 A g${}^{-1}$ and long cycling stability [45]. Despite the rapid growth of research and recent progress, there are still many unresolved issues that need attention. From this point of view, a complex investigation of the stability and ionic conductivity of all possible compounds in Li-P is essential.

A search for the most stable and the best ionic conductor among Li-P compounds will help to improve existing lithium-based batteries and develop an efficient liquid and solid sodium and potassium ion batteries. This work reports an extensive computational search of all stable Li-P compounds and evaluation of their ionic conductivity by modern computational techniques.

2. Materials and Methods

All density functional theory (DFT) calculations were performed using Vienna Ab initio Simulation Package (VASP) [46,47] using the projector augmented-wave method [48] with a plane-wave basis set [49] and Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional [50]. Energy cutoff was set at 500 eV. Gamma-centered k-points mesh with 2$\pi \times$0.01 Å$-1$ spacing was used. Phonon were calculated using the supercell approach and the finite displacement method [51], as implemented in Phonopy package [52]. Ab Initio Molecular Dynamics (AIMD) [53], to collect datasets for the machine learning interatomic potential (MLIP), was run at 1000 K with NPT ensemble and Langevin thermostat [54,55].

Moment tensor potential (MTP) was used as an MLIP. The MTP potentials are non-parametric potentials that use linear regression and invariant polynomials [56]. They are based on tensors of inertia of atomic environments and can approximate any regular function with the necessary symmetries. MTP potentials are up to 150 times faster than GAP potentials with the same magnitude of force RMS error and up to 50 times faster, with an error that is 1.5 times smaller than GAP [56,57]. The MLIP code, which allows for active learning with MTP, has been successfully used in many applications, including a crystal structure search, phase transitions, phase diagram prediction, and thermodynamic and kinetic properties [58,59,60,61,62,63].

MLIP was constructed using MLIP-3 package [64,65] in several stages. At the first stage [56], MTP of the 20th level was pretrained using data collected from AIMD. A total of 24,000 configurations from AIMD calculations were collected, and each 100th configuration was taken for the initial MLIP. At the second stage, 240 unique configurations from the AIMD-dataset were added using select-add function of MLIP-3 package. At the final stage, eight consecutive active learning [58,65] procedures were performed for each structure on the convex hull (or low-lying metastable phases). The extrapolation threshold and threshold-brake grade parameters were set to 2.0 and 35.0, respectively. Molecular dynamics simulations for the active learning were performed in LAMMPS [66,67] code; structures were collected during heating from 0 K to 1000 K with a heating rate 2 × 10${}^6$ K s${}^{-1}$ and annealing at 1000 K for 10 ns.

The accuracy of the constructed MLIP is shown in Table 1 and Figure 1, Figure 2 and Figure 3. Root mean square (RMS) errors in energies per atom and forces predicted by MLIP are less than 0.019 eV/atom and 0.34 eV/Å, respectively, for the training set. RMS errors in energies and forces for the validation set are 0.012 eV/atom and 0.145 eV/Å.

Table 1. Errors of the MLIP. MEA, AAD and RMS denote maximal absolute difference, average absolute difference and RMS absolute difference, respectively, between DFT and MLIP data of the training set.

Number of configurations	1047
	MAE	AAD	RMS
Errors in energy per atom, eV/atom	0.063	0.0145	0.0185
Range of energies per atom in training set, eV/atom	−5.09: −3.20
Errors in forces, eV/ Å	1.97	0.27	0.335
Range of forces in training set, eV/ Å	−10.86: +18.18
Errors in stresses, kBar	37.76	5.576	8.0
Range of stresses in training set, kBar	−84.1: +98.2

Figure 1. Energy errors on the validation data set and their distribution over the 11,311 configurations. AAE and RMSE denote average absolute error and RMS error, respectively, between DFT and MLIP data of the validation set.

Figure 2. Force errors on the validation data set and their distribution over the 11,311 configurations (838,824 values of force components). AAE and RMSE denote average absolute error and RMS error, respectively, between DFT and MLIP data of the validation set.

Figure 3. Stress errors on the validation data set and their distribution over the 11,311 configurations (67,866 values of stress components). AAE and RMSE denote average absolute error and RMS error, respectively, between DFT and MLIP data of the validation set.

Ionic conductivity was simulated using LAMMPS at temperatures 300–500 K with the step of 50 K for supercells with 5% vacancies on lithium sites. The size of the supercells was chosen so the number of atoms was approximately equal to 1000 and lattice parameters was close to 3 nm (See Table 2). Each simulation in LAMMPS was performed with NPT ensempble and Nose–Hoover thermostat [68,69] with a 0.5 fs timestep for 0.5 ns.

Table 2. Details of supercells. N denotes number of atoms in the supercell.

	Super Cell	Lattice Parameters						N
		a, Å	b, Å	c, Å	$\mathit{\alpha }$, deg	$\mathit{\beta }$, deg	$\mathit{\gamma }$, deg
Li${}_3$P	6 × 6 × 3	25.36	25.36	22.66	90	90	120	864
Li${}_3$P${}_7$	4 × 3 × 3	30.52	29.63	31.71	90	90	90	1440
Li${}_3$P${}_{11}$	3 × 5 × 5	39.33	30.21	33.94	90	118.95	90	1680
Li${}_4$P${}_3$	2 × 7 × 3	22.33	27.17	30.52	90	90	90	1176
Li${}_5$P${}_4$	3 × 7 × 3	26.59	27.84	25.45	90	109.37	90	1134
LiP	5 × 5 × 3	27.975	24.82	30.77	90	118.16	90	1200
LiP${}_5$	3 × 5 × 4	31.45	33.39	26.7	90	90	90	1440
LiP${}_7$	2 × 2 × 2	27.29	27.29	29.74	90	90	90	1024

Diffusion coefficients were calculated using mean square displacements (MSD) of Li${}^{+}$ cations:

$$D=\frac{1}{6}\underset{t\to \infty }{lim}\frac{d}{dt}\frac{1}{N}\sum _{i=1}^N<{|r_i\left(t\right)-r_i\left(0\right)|}^2>$$

(1)

Activation energy E${}_a$ was calculated using Arrhenius formula:

$$D=D_{0}exp\left(\frac{-E_a}{k_{B}T}\right)$$

(2)

The ionic conductivity of lithium was calculated using Nernst–Einstein formula:

$$\sigma =\frac{n{q}^{2}D}{H_r{k}_{B}T}$$

(3)

where n is the Li${}^{+}$ ion density, q is the formal charge of the lithium (q = 1), k${}_B$ is Boltzmann constant, T is the temperature and H${}_r$ is Haven ratio (assumed to be 1).

3. Results and Discussion

Before calculations of ionic conductivity in Li-P systems, we collected information about known stable and metastable Li-P compounds. Computational databases have different information about the stability of various structures. For example, in the OQMD [70,71] database, Li${}_3$P${}_{11}$ appears to be stable, while in thr Materials Project [72] database, this structure is not presented at all. Mayo et al. [37] performed first-principles structure prediction and showed that Li${}_3$P${}_{11}$ is metastable. However, previously it was shown by Sen and Johari [73] that the methodology used by Mayo et al. [37,74] has led to incorrect results in similar systems. Later, Fei et al. [75] reported that Li${}_5$P${}_2$ and Li${}_4$P became stable under high pressures and could be observed at ambient conditions. In order to understand which compounds are stable, we recalculated all known Li-P structures and studied their thermodynamic stability. We also carried out USPEX [76,77,78] calculations in order to predict previously unknown Li-P structures. More than 70 generations of 200 structures each were calculated. We did not find new stable or low-lying metastable phases.

The stability of Li-P compounds was analyzed by the calculation and construction of the convex hull, as shown in Figure 4a. Zero-point energy (ZPE) contribution was taken into account during the calculation of the convex hull. We found seven phases on the convex hull, namely Li −$Im\overline{3}m$, Li${}_3$P −$P6{}_3/mmc$, [79] LiP −$P2{}_1/c$, [80] Li${}_3$P${}_7$−$P2{}_{1}2{}_{1}2{}_1$, [81] Li${}_3$P${}_{11}$−$Cm$, LiP${}_7$−$I4{}_1/acd$, [82] P −$Cmca$ (black phosphorous).

Figure 4. (a) Calculated convex hull of the Li-P system at T = 0 K with ZPE. Blue circles represent stable compounds; red diamonds show metastable structures. (b) Composition–temperature phase diagram; solid line shows temperature region where the structure is stable, while the dashed line shows regions where the structure is metastable. (c) Crystal structures of considered Li-P compounds; big green atoms and small purple atoms are Li and P, respectively.

Experimentally known LP${}_5$−$Pnma2{}_1$ [82] is 16 meV/atom above the convex hull according to our DFT calculations. Mayo et al. [37] predicted that Li${}_4$P${}_3$−$P2{}_{1}2{}_{1}2{}_1$ was close to the convex hull and was dynamically stable, i.e. had no imaginary phonon frequencies. In our calculation, the Li${}_4$P${}_3$−$P2{}_{1}2{}_{1}2{}_1$ phase is 4 meV/atom above the convex hul; see Figure 4a. Li${}_5$P${}_4$−$C2/m$ was constructed using Na${}_5$As${}_4$ [83] as a prototype and is also very close to the convex hull (9 meV/atom). Fei et al. [75] predicted two stable compounds, Li${}_5$P${}_2$−$P3m1$ and Li${}_4$P −$R\overline{3}m$, claiming that they become stable at high pressure and can be observed at ambient conditions. According to our calculations, Li${}_5$P${}_2$−$P3m1$ and Li${}_4$P −$R\overline{3}m$ are 44 and 60 meV/atom above the convex hull, respectively.

We calculated phonon densities of states (PhDOS) for 20 stable and low-energy metastable structures shown in (Figure 4a). Integration over the phonon DOS allowed us to calculate the zero-point energy and vibrational contribution to the Helmholtz free energy for all studied compounds and construct the convex hulls at finite temperatures. Stability of the predicted compounds at different temperatures is summarized in the composition–temperature phase diagram (Figure 4b). Our calculations show that at temperatures higher than 550 K, Li${}_3$P${}_{11}$−$Cm$ becomes metastable; Li${}_5$P${}_4$−$C2/m$, on the other hand, becomes stable at T = 700 K and appears on the convex hull.

All Li${}_x$P structures can be classified into four categories according to the arrangement of P atoms [37]: (1) 0 ≤x≤ 0.5: tubes, cages, 3-D network—LiP${}_7$, LiP${}_6$, LiP${}_5$, Li${}_3$P${}_{11}$, Li${}_3$P${}_7$; (2) 0.5 ≤x≤ 1.33: infinite or finite chains—LiP${}_2$, Li${}_3$P${}_4$, LiP, Li${}_5$P${}_4$, Li${}_4$P${}_3$; (3) 1.33 ≤x≤ 2: P dumbbells—Li${}_3$P${}_2$, Li${}_2$P, Li${}_5$P${}_2$; (4) 1.33 ≤x≤ 2: isolated P ions—Li${}_3$P, Li${}_4$P, Li${}_5$P, Li${}_6$P. We discuss below how structural topology affects the ionic conductivity.

Figure 5 shows the ionic conductivity of stable systems (Li${}_3$P, LiP, Li${}_3$P${}_7$, Li${}_3$P${}_{11}$, LiP${}_7$) and low-lying metastable phases (Li${}_4$P${}_3$, Li${}_5$P${}_4$, LiP${}_5$) with 5% vacancy concentration as a function of temperature. Most structures have high ionic conductivity at room temperature about 10${}^{-3}$–10${}^{-2}$ S cm${}^{-1}$; however, for LiP, such values are obtained only at temperatures higher than 400 K, while in LiP${}_7$ diffusion is negligible it the whole temperature range 300–500 K. Curiously, experimental data are controversial: Wegner et al. [84] found the ionic conductivity of Li${}_3$P to be 10${}^{-6}$–10${}^{-4}$ S cm${}^{-1}$, but in article [85] 10${}^{-4}$–10${}^{-2}$ S cm${}^{-1}$ at temperatures 300–500 K. We explain such divergence by differences in the microstructure of materials: Wegner et al. [84] studied cold-pressured material, while Nazri [85] studied polycrystalline material, which might have more defects and vacacies. Our results are in good agreement with the earlier work [85]. In a polycrystalline material, diffusion is enhanced at grain boundaries, which explains the difference. In our simulations, we see almost negligible ionic conductivity for most systems without vacancies; however, once vacancies are introduced, ionic conductivity can become very high.

Figure 5. Calculated ionic conductivity of Li-P compounds as a function of temperature. Experimental ionic conductivities were taken from article [85] and refitted by us.

The trajectories of Li${}^{+}$ diffusion are shown in Figure 6. Li${}_3$P, Li${}_3$P${}_7$, Li${}_4$P${}_3$, LiP are 3D conductors, but in Li${}_3$P${}_7$, the diffusion of Li ions prevails along the z-direction: on the Li trajectory, there are screw chains, which are connected though x- and y- directions. In LiP, there are a few Li jumps and small amount of trajectories, indicating a low diffusion rate with high activation energy. Li${}_5$P${}_4$ is a 2D ionic conductor; trajectory lines are located along the x–y plane and correspond to the diffusion of Li atoms located between P-chain layers. Lithium atoms, which have strong bonds with the four-atom P-chains, do not diffuse, forming a non-conductive layer, and there are only a few, rare Li jumps from non-conductive to conductive layers. One-dimensional diffusion appears in structures with a 3D network of P atoms, so in 3D conductor Li${}_3$P${}_7$, 1D conductivity along the y-axis prevails and LiP${}_5$ and LiP${}_7$ are 1D conductors. In LiP${}_7$, diffusion proceeds through two channels along the z-direction, which are connected to each other and form an extended cross; however, there are only a few trajectory lines, which appear only at elevated temperatures (500–600 K), indicating low diffusion and high activation energy; the ionic conductivity of LiP${}_7$ at 500 K is about 10${}^{-3}$ S cm${}^{-1}$ and negligible at lower temperatures. Li${}_3$P${}_{11}$ has dumbbell chains, which are not connected to each other; thus, no spatial diffusion is obtained. Our results indicate that not only does Li${}_3$P have high ionic conductivity, but almost all other stable and low-energy metastable phases (except Li${}_3$P${}_{11}$ and LiP${}_7$).

Figure 6. Calculated trajectories of Li-ion diffusion and their topology (in parentheses). For a better representation, ions are not shown. Trajectories were collected at 600 K for 0.2 ns after 0.1 ns heating.

4. Conclusions

We performed an extensive study of lithium–phosphorous compounds as promising candidates for solid-state electrolytes, anodes or components of electrolyte interphase in Li-ion or Li-metal batteries. Our first-principles calculations allowed us to determine the stability regions of all considered structures in the temperature range from 0 to 700 K. We found that Li${}_3$P −$P6{}_3/mmc$, LiP −$P2{}_1/c$, Li${}_3$P${}_7$−$P2{}_{1}2{}_{1}2{}_1$, Li${}_3$P${}_{11}$−$Cm$, LiP${}_7$−$I4{}_1/acd$ are stable at room temperature, while at temperatures higher than 550 K Li${}_3$P${}_{11}$ becomes metastable, and at temperatures higher than 700 K Li${}_5$P${}_4$ becomes stable. Our results indicate a variety of metastable compounds, some of which might exist at solid electrolyte interphase during battery operation. The diffusion of Li atoms was studied using molecular dynamics simulations with machine learning interatomic potentials. High lithium diffusion was observed for Li${}_3$P, LiP${}_5$, Li${}_5$P${}_4$, Li${}_4$P${}_3$, and Li${}_3$P${}_7$, and their ionic conductivity was in the range 10${}^{-4}$–10${}^{-2}$ S cm${}^{-1}$ at room temperature. We also determined the qualitative dependence of ionic conductivity on the arrangement of phosphorous atoms; namely, the compounds with a 3D network of phosphorus possess 1D conductivity (LiP${}_5$ and LiP${}_7$), compounds with phosphorus chains or broken chains display a 2D type of conductivity (i.e., Li${}_5$P${}_4$), while other arrangements lead to 3D conductivity. Lithium phosphides are a promising class of electrolytes for lithium metal batteries.