Structural stability and electronic property in K 2 S under pressure †

In this work, the structures, phase sequence, and metallic properties of K 2 S have been systematically explored. We con ﬁ rm that the P 6 3 / mmc phase is the best possible candidate for the stable structure of K 2 S at low pressure range. Although the phases of P 6 3 / mmc and Cmcm K 2 S are semiconductors, two new structures of P 6/ mmm and P (cid:1) 3 m 1 emerge with metallic characters at high pressures. The analyses of electronic localization functions reveal that the conductivity mainly comes from the electrons surrounding S atom chains, which supplies a potential way to improve the conductivity of sulfur to enhance the electrode recharge ability and rate capability in alkali sul ﬁ de battery under pressure.


Introduction
The storage and conversion of energy continue to be important to society.Batteries, which interconvert chemical and electrical energy, are widely used in industry and for consumer applications, e.g.appliances and laptop computers.The battery energy storage system, e.g.lead-acid battery, solid oxide fuel battery, solar battery, lithium ion battery and so on, are focused and studied for a long time.The lead-acid battery is the most-used battery system with advantages including large current discharge, rechargeable, and low-cost.On the other hand, the disadvantage of lead-acid battery is obvious, for examples, the heavy weight and pollution. 1,2Solid oxide fuel battery is an energy conversion device that can convert chemical energy directly into electricity without pollution, nevertheless, the high operating temperature limits its widely application in our daily life. 3,4Solar battery is excepted to have large-scale applications if the problem of low efficiency is solved. 5,6Lithium ion battery is an attractive battery for the property of portable type, high energy density, and non-pollution, and is widely used for digital products, electric vehicles, household appliances, etc. [7][8][9] The principle of operation lithium ion battery is that Li is removed from an intercalation cathode, for example LiCoO 2 , on charging and is inserted between sheets of carbon atoms in the graphite negative electrode; discharge reverses this process. 10Though the compound of Li 2 S and Na 2 S with high specic energy (energy per unit weight) and energy density (energy per unit volume) are candidates for applications in solid state batteries, 11,12 it is confronted with major challenge, i.e., poor electrode recharge ability and limited rate capability owing to the insulating nature of sulfur and the solid reduction products. 135][16][17] The calculated band structures show the Li 2 S, K 2 S, and Rb 2 S are indirect bandgap materials, whereas Na 2 S is a direct gap material. 14At the same pressure, bulk modulus of the alkali metal suldes show a clear decrease with heavier mass of alkali atoms. 15The similar tendency of decrease with atomic mass of X in X 2 S has been also reported on the elastic constants under the same pressure.When the pressure is enhanced, except for the linear decrease of the elastic constant of C 44 , the bulk modulus and the elastic constant of C 11 , C 12 increase in the crystals of Rb 2 S, K 2 S and Na 2 S. Similar trends are also reported in the Li 2 S except for the elastic constants C 44 .It is reported that C 44 increases linearly with pressure which is related to the difference in the bonding nature of the bottom of the conduction band. 14 2 S has been extensively investigated by theoretical and experimental methods since it was rst synthesized by Zintl et al. 18 It is reported that K 2 S can crystallize in the antiuorite structure, i.e.Fm 3m 18 at ambient pressure, then transforms to an orthorhombic crystal with the space group of Pnma at about 2.7 GPa. 16 Theoretical prediction points out that Pnma crystal can transform to a distorted P6 3 /mmc at 4.4 GPa.16,19 Nonetheless, the Pnma phase is not observed by experiment to date.19 In this work, four stable phases of P6 3 /mmc, Cmcm, P6/mmm and P 3m1 have been found at different pressures through ab initio ELocR code. 20The low-pressure phase of Pmma proposed by an experimental analysis 19 is found unstable by our theoretical calculations.Based on the reported X-ray diffraction and our theoretical analysis, we update the low-pressure phase with a hexagonal crystal of P6 3 /mmc.Our calculations reveal the insulating character of the two phases of P6 3 /mmc and Cmcm, and the Raman spectra are subsequently analyzed for the experimental convenience.Electronic band structure and the partial density of states show the metallic properties of P6/mmm and P 3m1 at pressures.As shown in the chart of electronic localization functions (ELF), the delocalization of electrons around S atoms in the phases of P6/mmm and P 3m1 denotes the improvement of conductivity of sulfur in Li 2 S battery under pressure.

Computational methods
The structural searching of K 2 S has been carried out by ab initio ELocR code, 20 and the further conrmation with high accuracy of structural relaxations and electronic localization functions (ELF) are completed by the Vienna ab initio simulation package (VASP code) 21 with a plan wave cutoff energy of 1000 eV and Brillouin zone sampling grid with a spacing of 2p Â 0.04 ÅÀ1 .Total energy calculations are performed using the VASP codes with the Perdew-Burke-Ernzerhof (PBE) parameterization of generalized gradient approximation (GGA) 22 used to treat the exchange-correlation energy.The partial augmented wave (PAW) 23 method is adopted with the PAW potentials where 2s2p and 3s3p4s are treated as valence electrons.The electronic selfconsistent calculation is stopped until the energy convergence less than 1 Â 10 À5 eV.
The electronic projected density of states and the electronic band structure are calculated by CASTEP 24 code with the cutoff energy of 720 eV, GGA-PBE exchange-correlation functional, Brillouin zone sampling grid with a spacing of 2p Â 0.03 ÅÀ1 , and norm-conserving pseudopotentials.The Raman spectra are obtained within the density functional theory (DFT) formalism, and PBE exchange-correlation functional with norm-conserving pseudopotentials in the CASTEP code. 25The dynamical stability properties phonon calculations are carried out by a nite displacement approach 26 through the PHONOPY code. 27The supercell are: 2 Â 2 Â 2, 2 Â 2 Â 3, 2 Â 2 Â 3, 3 Â 3 Â 3 for P6 3 / mmc, Cmcm, P6/mmm, and P 3m1, respectively.The force constants are calculated from forces on atoms with atomic nite displacements, and the nite displacement size is 0.01 Å in this work.

Results and discussion
We have performed extensive structural searches on K 2 S compounds at selected pressures of 0, 50, 100, 150, and 200 GPa.Five phases with lowest enthalpy, i.e.Fm 3m, P6 3 /mmc, Cmcm, P6/mmm, and P 3m1, are predicted as presented in Fig. 1.According to our enthalpy curves, the K 2 S can crystallize in a face-centered cubic lattice with the space group of Fm 3m at ambient pressure, and keep the symmetry up to 3 GPa.The Fm 3m phase has been synthesized experimentally at ambient pressure and reported by Zintl et al. 18 as shown in Fig. 2(a).The unstable phase Pmma 19 is also rebuilt in Fig. 2(b) for the convenience of comparison.Elevating pressure above 3 GPa, a hexagonal phase with P6 3 /mmc symmetry emerges and is stable till 88 GPa, it has been reported by experiment. 28The layer stacking sequence in the c-axis can be denoted by the repeated ABAC stacking.K atoms occupy the crystallographic 2a position with 3m symmetry along a axis in A layers, the other nonequivalent K atoms occupy the crystallographic 2d position with 6m2 symmetry, locating in B and C layers.S atoms occupy the crystallographic 2c position with 6m2 symmetry and locate at C and B positions in B and C layers respectively, as shown in Fig. 2(c).The base-centered orthorhombic phase of Cmcm is found with the lowest enthalpy in the pressure range of 88 GPa to 136 GPa, which is about 0.03 eV lower than another bodycentered orthorhombic structure of Immm per K 2 S at 100 GPa.There are two nonequivalent atoms in the crystal lattice, and thus consists of one K 2 S unit as shown in Fig. 2(d).K atoms occupy the crystallographic 8g position (site symmetry is ..m) while S atoms occupy the crystallographic 4c position (site symmetry is m2m).Each K is surrounding by four S atoms to form a triangular pyramid geometry, and the distance between S and nearest K is 2.492 Å. Aer 136 GPa, another hexagonal crystal with P6/mmm symmetry takes over Cmcm phase and becomes most competitive on enthalpy up to 163 GPa.K atoms occupy the crystallographic 2d position with 6m2 symmetry, S atoms occupy the crystallographic 1a position with 6/mmm symmetry.There, K atoms form an interesting graphene-like layered structure in the ab plane, and the distance between S and nearest K is 2.590 Å as shown in Fig. 2(e).At high pressure range from 163 GPa to 200 GPa, a trigonal phase of P 3m1 obtains the minimum enthalpy, which includes two nonequivalent atoms of K and S consisting of layered K-S geometry.The layer stacking sequence in the c-axis can be denoted by the repeated ABA stacking.K atoms occupy the crystallographic 2d position with 3m.symmetry along b axis in A layers.S atoms occupy the crystallographic 1b position with 3m symmetry, locating in layers B, as shown in Fig. 2(f).The calculated structural parameters, volume, and Wyckoff positions for each phase are summarized in Table 1.
In order to describe the response of a material to an externally applied stress, the single-crystal elastic constants of proposal structures are calculated and analyzed, as shown in Table 2.The matrix of elastic constants C ij must be positive denite, 29 it is worth noting that negative values are not prohibited for C ij . 30For hexagonal structure P6 3 /mmc and P6/mmm, there are ve independent elastic constants (C 2 , indicating the mechanical stability of P 3m1.So, the proposal structures all satisfy Born-Huang criterion, 31 representing the stability of the mechanical property. The mechanically and thermodynamically stability of proposal phases have been discussed, and they are all stable at relevant pressure range.Furthermore, the phonon band structure and partial phonon density of states (PHDOS) for atoms with nonequivalent Wyckoff positions are calculated to judge its dynamical stability as shown in Fig. 3, and the absence of imaginary frequency modes in the entire Brillouin zone indicates dynamic stability of the structures.From the PHDOS on nonequivalent atoms in P6 3 /mmc crystal at 50 GPa as plotted in Fig. 3(a), we can see clearly that the low frequency modes mainly  dominated by the K atoms occupying the crystallographic 2a position, but the other nonequivalent K atoms occupying the crystallographic 2d position contribute more than K atoms with 2a position at high frequency modes.In the Cmcm phase, S atoms mainly contribute to the frequency upon 14 THz while the low frequency below 14 THz mainly comes from the vibrations of K atoms at 100 GPa, see the Fig. 3(b).At higher pressure range from 150 GPa to 200 GPa, there are no distinguishing characteristics in P6/mmm and P 3m1 K 2 S except K atoms contribute more than S atoms as a whole because the number of K atoms is twice the number of S atoms in K 2 S crystals as shown in Fig. 3

(c) and (d).
There is a discussion needed with the proposal phase in low pressure range.Experiment proposes a crystal with the space group of Pmma 19 at 4.4 GPa, which is not consistent with the structure of P6 3 /mmc proposed in this work.We rebuild Pmma phase according to the space group and atomic coordinates given by the experiment report, and calculate the phonon dispersion and elastic constants.The matrix of elastic constants C ij are as shown in ESI, † which contains C 11 < 0, C 55 < 0, C 66 < 0, C 11 + C 22 À 2C 12 < 0, C 11 + C 33 À 2C 13 < 0. For the orthorhombic structure, elastic constants do not satisfy Born-Huang criterion, 31 representing the instability of the mechanical property.Phonon band structure is added in ESI.† There are imaginary frequency modes in the entire Brillouin zone indicates dynamic instability of the structures.Comparing the experimental X-ray powder diffractograms as shown in Fig. 4(a) and (c), the peak position and intensity of P6 3 /mmc K 2 S coincide well with reported experiment observation.So, the reported Pmma structure is unstable theoretically, and the P6 3 /mmc phase is the best possible candidate for the stable structure of K 2 S at low pressure range.
Moreover, Raman spectroscopies of P6 3 /mmc and Cmcm are simulated and display in Fig. 4(b) and (d).The Raman spectroscopy of P6 3 /mmc can be classied by the irreducible representation of the point group D 6h , a weak characteristic peak is observed at 96 cm À1 correspond to E 2g , the another peak appear at 196 cm À1 also belong to E 2g irreducible representation.Cmcm vibrational modes belong to the D 3h point group, the peaks at 82 cm À1 , 442 cm À1 and 519 cm À1 are related to A g irreducible representation, the peaks at 253 cm À1 , 341 cm À1 and 484 cm À1 are correlated with B 1g .The following peaks at 255 cm À1 and 551 cm À1 are connected with B 3g , the positions of the peak at 337 cm À1 is corresponding to B 2g .The two close peaks at 253 cm À1 and 255 cm À1 produce an intense peak, and the other two    close peaks at 337 cm À1 and 341 cm À1 combine the most intense peak.
In order to analyze the electronic properties of phases P6 3 / mmc, Cmcm, P6/mmm, and P 3m1, the electronic band structure and the partial density of states are further explored in Fig. 5. Band gap is discovered in the P6 3 /mmc phase at 50 GPa, revealing the nonmetallic character which coincides with previously report. 14The electronic band structure of Cmcm phase shows the semiconductor character, as displayed in Fig. 5(c).The band gap decreases with pressure from P6 3 /mmc to Cmcm by contrast to the values of gaps at 50 GPa and 100 GPa in Fig. 5, respectively.Elevating the pressure to 150 GPa, three bands marked with magenta, green, and blue crossing over the Fermi level contribute large total electronic density distribution, and revel the strong metallic character of P6/mmm, as displayed in Fig. 5(b).Three bands intersecting the Fermi level, the P 3m1 phase also behaves as a metal, and the bands colored by blue and magenta closing to Fermi level are found to contribute more to the density of states (DOS) of Fermi level, as displayed in Fig. 5(d).From the partial density of states (PDOS) in Fig. 5(b) and (d), the metallic character of P6/mmm and P 3m1 is conrmed by the high level of total electronic density distribution at Fermi level.The majority of occupied states come from K(p) state, whereas the contribution from K(s), S(s) and S(p) to the states around the Fermi level is quite small.The electronic localization functions (ELF) is derived from an earlier idea of Lennard-Jones 32 and rst reported by A. D. Becke and K. E. Edgecombe. 33It has been calculated to characterize the degree of electron localization in the proposal K 2 S crystals.ELF as dened runs from 0 to 1, the values of ELF equaling to 1.0 and 0.5 reect the extremely strong localization and homogeneous electrons distribution, respectively. 33,34hree-dimensional electron location function (3D ELF) of P6/ mmm and P 3m1 show free-electron channels with isosurface value of 0.5 (ELF ¼ 0.5) around S atoms, which reveals that the conductivity comes from the contributions of the electrons around the S atoms as presented in Fig. 6(a) and (c).Furthermore, the two-dimensional electron location function (2D ELF) conrm the character of conductivity in two crystals, where connected regions with the ELF equaling to 0.5 are surrounding S atoms as shown in Fig. 6(b) and (d).Potassium is an alkali element and always described as a typical free-electron-like metal at ambient pressure; on the contrary, sulfur is oen described as a good insulator at the same condition.Nevertheless, free electrons assemble around only S atom chains demonstrating the conductivity, and not K atoms under high pressure.At ambient pressure, K 2 S is crystallized in facecentered cubic antiuorite structure with Fm 3m symmetry which is an ionic crystal 14 and performed insulator characters.Nevertheless, the two phases of P6/mmm and P 3m1 exhibit metallic features as discussed above, which implies the available improvement of the conductivity of sulfur to enhance the electrode recharge ability and rate capability in alkali sulde battery under pressure.

Conclusion
In summary, we have systematically explored phase sequence of K 2 S at pressures from ambient pressure to 200 GPa using the ab initio ELocR, and stable structures of Fm 3m, P6 3 /mmc, Cmcm, P6/mmm, and P 3m1 have been explored thoroughly.Based on enthalpies, phonon band structure, elastic constants, and X-ray power diffraction, we conrm that the reported Pmma structure is unstable and can distort into P6 3 /mmc by our calculations.Moreover, we also simulate the Raman spectroscopies of P6 3 / mmc and Cmcm phases for the convenience of the further studies of experiments.The two phases of P6 3 /mmc and Cmcm K 2 S are semiconductors under pressure, P6/mmm and P 3m1 emerge with the nature of metallic characters at higher pressures.The analyses of ELF show that the conductivity comes from the electrons surrounding S atom chains, which implies the potential improvement of the conductivity of sulfur to enhance the electrode recharge ability and rate capability in alkali sulde battery under pressure.

Fig. 1
Fig.1The calculated enthalpy values per K 2 S for various structures relative to previously reported Pmma 19 structure as a function of pressure.

Fig. 2
Fig. 2 The structures of K 2 S. (a) The Fm3m phase at ambient pressure, (b) the unstable Pmma 19 phase at 4.4 GPa, (c) the P6 3 /mmc phase at 10 GPa, (d) the Cmcm phase at 100 GPa, (e) the P6/mmm phase at 150 GPa, (f) the P3m1 phase at 200 GPa.Purple and yellow atoms are K and S, respectively.

Fig. 3
Fig. 3 The phonon band structure and partial phonon density of states for atoms with nonequivalent Wyckoff positions are plotted for the proposal phases.(a) P6 3 /mmc phase at 50 GPa, (b) Cmcm phase at 100 GPa, (c) P6/mmm phase at 150 GPa, (d) P3m1 phase at 200 GPa, respectively.

Fig. 4
Fig. 4 (a) Calculated difference X-ray power diffraction for P6 3 /mmc and Pmma which is rebuilt though data from experiment at 4.4 GPa, (b) Raman active modes of P6 3 /mmc phase, (c) experiment X-ray power diffraction for Pmma at 4.4 GPa, 19 (d) Raman active modes of Cmcm phase.

Fig. 5
Fig. 5 The electronic band structure and PDOS of K 2 S. (a) The electronic band structure of P6 3 /mmc, (b) the electronic band structure and PDOS of P6/mmm, (c) the electronic band structure of Cmcm, (d) the electronic band structure and PDOS of P3m1.

Table 1
Structural information of K 2 S phases