⇦Chinese Physics Letters, 2021, Vol. 38, No. 3, Article code 036201 Novel Superconducting Electrides in Ca–S System under High Pressures Yun-Xian Liu (刘云仙), Chao Wang (王超)*, Shuai Han (韩帅), Xin Chen (陈欣), Hai-Rui Sun (孙海瑞), and Xiao-Bing Liu (刘晓兵)* Affiliations Laboratory of High Pressure Physics and Material Science (HPPMS), School of Physics and Physical Engineering, Qufu Normal University, Qufu 273165, China Received 22 December 2020; accepted 19 January 2021; published online 2 March 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 11704220, 11804184, 11974208 and 11804185), the Shandong Provincial Natural Science Foundation (Grant Nos. ZR2017BA020, ZR2018PA010, ZR2019MA054 and ZR2017BA012).
*Corresponding authors. Email: wangchao0531608@163.com; xiaobing.phy@qfnu.edu.cn Citation Text: Liu Y X, Wang C, Han S, Chen X, and Sun H R et al. 2021 Chin. Phys. Lett. 38 036201    Abstract Due to their unique structure properties, most of the electrides that possess extra electrons locating in interstitial regions as anions are insulators. Metallic and superconducting electrides are very rare under ambient conditions. We systematically search possible compounds in Ca–S systems stabilized under various pressures up to 200 GPa, and investigate their crystal structures and properties using first-principles calculations. We predict a series of novel stoichiometries in Ca–S systems as potential superconductors, including $P2_{1}/m$ Ca$_{3}$S, $P$4mbm Ca$_{3}$S, Pnma Ca$_{2}$S, Cmcm Ca$_{2}$S, Fddd CaS$_{2}$, Immm CaS$_{3}$ and $C2/c$ CaS$_{4}$. The $P4mbm$ Ca$_{3}$S phase exhibits a maximum $T_{\rm c}$ value of $\sim $20 K. It is interesting to notice that the $P2_{1}/m$ Ca$_{3}$S and Pnma Ca$_{2}$S stabilized at 60 and 50 GPa behave as superconducting electrides with critical temperatures $T_{\rm c}$ of 7.04 K and 0.26 K, respectively. More importantly, our results demonstrate that $P2_{1}/m$ Ca$_{3}$S and Pnma Ca$_{2}$S are dynamically stable at 5 GPa and 0 GPa, respectively, indicating a high possibility to be quenched to ambient condition or synthesized using the large volume press. DOI:10.1088/0256-307X/38/3/036201 © 2021 Chinese Physics Society Article Text Superconducting materials exhibit peculiar physical properties, which can be applied in energy, defense, transportation, computer technology and medicine. Since the discovery of superconductors, exploring new high temperature superconductors has been a hot topic in the fields of condensed matter physics and materials science. To date, cuprates,[1] iron pnictides,[2–4] hydrides[5–6] and some 2D compounds[7–11] have been reported. Although considerable research and understanding of superconductors have been greatly progressed, it is a long way to seek new high temperature superconductors which can be experimentally synthesized and available at room temperature. Electrides are extraordinary compounds with extra electrons locating in interstitial regions as anions.[12–13] Under ambient conditions, metallic and superconducting electrides are very rare. Among a few superconducting candidates, previous electrides are reported to exhibit low superconducting transition temperature $T_{\rm c}$, such as [Ca$_{24}$Al$_{28}$O$_{64}$]$^{4+}(e^{-})_{4}$ (C12A7:$e^{-})^{-}$,[14] Mn$_{5}$Si$_{3}$-type Nb$_{5}$Ir$_{3}$,[15] $Y_{2}$C and MgONa.[16] Further exploration of new superconducting electrides can be helpful for deep understanding their formation mechanisms and superconducting behavior. Since energies of both the interstitial and atomic orbitals are sensitive to pressure conditions, high-pressure techniques have been found to be a powerful way to synthesize novel electrides by effectively shortening the atomic spacing and increasing the overlap of adjacent electron orbitals to overcome high formation energy barriers.[13,17–18] Previous experimental and theoretical studies have reported that a series of electrides of elements (Li, Na, K and Mg) and related compounds [NaHe$_{2}$, Mg$_{3}$O$_{2}$, Mg–NG (NG = Xe, Kr, Ar), Li$_{n}$I, Al$_{2}$S and Ca$_{3}$Si] have already been stabilized under high pressures, exhibiting exotic properties as metallic-semiconductor-insulator transition, superconductivity.[13,19–28] Recently, it is striking that a pressure-induced electride as Li$_{6}$P compound exhibits a highest $T_{\rm c}$ value of 39.3 K among all already known electrides.[29] Such a value is even higher than that of some conventional superconducting systems,[1–4] indicating a new way to search for high temperature superconductors. Nevertheless, the expense of trial-and-error synthesis under extremely high pressure condition (270 GPa) for the phase stabilization has severely hampered the deep investigation of detailed mechanism, much less to its potential commercial applications in the future. It is therefore of great importance to investigate novel superconducting electride systems under relatively low pressure conditions. On the other hand, most of the pressure-stabilized electrides are metastable materials and can be conserved after releasing the pressure, thus exploration for stable phases at ambient pressure is also highly deserved for practical applications. Previous studies have found that compounds consisting of elements with small electronegativity and large atomic radii are more likely to form electrides, including alkali metals, alkaline-earth metals and early transition metals in group IIIB.[30,31] In particular, the alkaline earth metal element, Ca, occurs more frequently than others. Additionally, considering that S can be comparable to P in electronegativity, we expect that Ca–S systems would have high possibility to realize electrides with superconductivity. Here, we systematically search the possible compounds in Ca–S systems under various pressure, and investigate their crystal structures and properties using first-principles calculations. A series of novel stoichiometries are found in Ca–S systems as potential superconductors, including $P2_{1}/m$ Ca$_{3}$S, $P4mbm$ Ca$_{3}$S, Pnma Ca$_{2}$S, Cmcm Ca$_{2}$S, Fddd CaS$_{2}$, Immm CaS$_{3}$ and $C2/c$ CaS$_{4}$. The $P4mbm$ Ca$_{3}$S phase exhibits a maximum $T_{\rm c}$ value of $\sim $20 K. Moreover, it is interesting to notice that the $P2_{1}/m$ Ca$_{3}$S and Pnma Ca$_{2}$S behave as superconducting electrides, which can be quenched to ambient conditions. Computational Details. We have explored the pressure-induced structural evolution in Ca–S systems with various Ca$_{m}$S$_{n}$ ($m = n = 1$–4) using the CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization)[32–33] and USPEX (Universal Structure Predictor: Evolutionary Xtallography) codes[34–44] at temperature of 0 K and at pressures of 0, 50, 100, 150 and 200 GPa. Such methods have been successfully employed in various systems.[37–55] We performed structural optimization and electronic structure calculations using the Vienna ab initio simulation package (VASP)[56] based on density functional theory (DFT), within the Perdew–Burke–Ernzerhof (PBE) version of generalized gradient approximation (GGA).[57] The projector-augmented-wave potentials (PAWs)[58] have been adopted, with 3$p^{6}4s^{2}$ and 3$s^{2}3p^{4}$ treated as valence electrons for Ca and S atoms, respectively. To ensure that total energy calculations were converged to less than 1 meV, we used the cutoff energy of 500 eV and fine Monkhorst–Pack $k$-meshes of 2$\pi \times 0.025$ Å$^{-1}$. We determined dynamic stability by calculating the phonon using a supercell approach with the finite displacement approach,[59] as implemented in the PHONOPY code.[60] We used the electron localization function (ELF)[61] and Bader's Quantum Theory of Atoms in Molecules (QTAIM) analysis[62–64] to analyze the degree of electron localization, and electronic charge transfer calculations are tested by the VASP code. The crystal orbital Hamilton populations (COHP)[65,66] was calculated using the LOBSTER code,[67] to analyze the interatomic interaction and chemical bonding. The electron–phonon coupling (EPC) calculations were performed using the Quantum ESPRESSO (QE) package.[68] The norm-conserving pseudopotentials and a kinetic energy cutoff of 90 Ry were employed.
cpl-38-3-036201-fig1.png
Fig. 1. Thermodynamic stabilities and stable pressure ranges of Ca$_{m}$S$_{n}$ compounds. (a) Formation enthalpies ($\Delta H$) per atom of the Ca$_{m}$S$_{n}$ phases with respect to their separated counterparts at 0 K and at different pressures. (b) Pressure ranges in which the corresponding phases of different Ca–S compounds are thermodynamically stable.
Results and Discussion. To investigate the phase stability of Ca–S systems, the thermodynamic stabilities were evaluated by calculating their formation enthalpies ($\Delta H$) with respect to constituent elements. As references, the energetically most favorable structures of face-centered-cubic, body-centered-cubic, and $\beta$-tin phases for Ca, $I4_{1}$/acd and $\beta$-Po phases for S were adopted in their corresponding stable pressure ranges. To account for the enthalpy relationships and all possible formation routes of each stoichiometry, we constructed the convex hulls for the Ca–S binary compounds at 0 and 50, 100, 150, and 200 GPa, as shown in Fig. 1(a) and Fig. S1 in the Supplementary Information. Formation enthalpies of compounds are located on the hull (solid lines), showing thermodynamically stability with relative to other binary stoichiometries or elements, while other phases located above the hull (dashes lines) are either unstable or metastable. At ambient conditions, CaS is the only stable stoichiometry, in agreement with the experimental results, and it remains stable to 200 GPa. With increasing pressure to 50 GPa, Ca$_{2}$S and CaS$_{3}$ are found to be stable. Upon further compression, Ca$_{3}$S gradually becomes stable with respect to decomposition into other compounds at 100 GPa. At 150 and 200 GPa, exception of the Ca$_{4}$S stoichiometry, all the other compounds are energetically favored over element. The present finding demonstrates that high pressure clearly favors the formation of Ca–S compounds. The predicted pressure ranges of thermodynamic stability for Ca–S are presented in Fig. 1(b). For CaS, the well-known structure $Fm\bar{3}m$ under ambient conditions was predicted to be stable till 39 GPa. Then it transforms into $Pm\bar{3}m$ and $I4_{1}$/amd at 39 and 161.5 GPa, respectively. We find that Ca$_{3}$S is stable in monoclinic $P2_{1}/m$ structure in pressure ranges of 56–71 GPa, then a tetragonal $P$4/mbm phase is energetically favorable. Turning to Ca$_{2}$S, Pnma is the most stable phase between 24 and 116.7 GPa, and then it is overtaken by an energetically favored Cmcm structure up to 200 GPa. For CaS$_{2}$ stoichiometry, the Fddd structure becomes favorable from 130 to 200 GPa. For CaS$_{3}$, $P\bar{4}2_{1}m$ is stable in a pressure range from 12 to 158 GPa, then Immm becomes more favored. For the CaS$_{4}$ compound, the predicted $C2/c$ is energetically stable above 178.5 GPa. Since the dynamical stability is one of the basic requirements for structures, we have calculated their phonon spectra for Ca–S compounds, as shown in Figs. S2 and S3 in the Supplementary Information. It is clear that the calculated phonon dispersion curves show no imaginary phonon frequency in the entire Brillouin zone for all the predicted structures, indicating that they are dynamically stable in the corresponding stable pressure ranges.
cpl-38-3-036201-fig2.png
Fig. 2. Predicted stable crystal structures of studied Ca-rich compounds under pressure: (a) $P2_{1}/m$ Ca$_{3}$S at 50 GPa, (b) $P$4/mbm Ca$_{3}$S at 200 GPa, (c) Pnma Ca$_{2}$S at 100 GPa, (d) Cmcm Ca$_{2}$S at 200 GPa, (e) $Fm\bar{3}m$ CaS at 0 GPa, (f) $Pm\bar{3}m$ CaS at 100 GPa, (g) $I4_{1}$/amd CaS at 200 GPa. Bice and yellow balls denote Ca and S atoms, respectively.
Figure 2 illustrates the crystal structures for the predicted stable Ca-rich compounds. For Ca$_{3}$S, it is observed that the $P2_{1}/m$ structure consists of Ca-sharing SCa$_{10}$ polygons [Fig. 2(a)], while in the $P$4/mbm Ca$_{3}$S phase, each S atom is surrounded by 12 Ca [Fig. 2(b)]. It is a common phenomenon that the coordination number increases under high pressure. Ca$_{2}$S exhibits two orthorhombic structures Pnma [Fig. 2(c)] and Cmcm [Fig. 2(d)], in which each S is surrounded by ten Ca atoms and the nearest Ca–S distance varies from 2.42 to 2.31 Å with the increased pressure. For CaS, the well-known semiconducting $Fm\bar{3}m$ structure and new cubic $Pm\bar{3}m$ phases were predicted, which could be viewed as NaCl-type and CsCl-type [Figs. 2(e) and 2(f)], respectively. The high-pressure stable $I4_{1}$/amd CaS phase is found to be composed of Ca-sharing SCa$_{8}$ polygons with a closest Ca–S distance of 2.34 Å, as depicted in Fig. 2(g). It should be pointed out that, compared to the well-known semiconducting $Fm\bar{3}m$ CaS structure, the Ca-rich phases possess higher coordination number and denser polyhedral packing, which may result in distinct electronic properties.
cpl-38-3-036201-fig3.png
Fig. 3. Predicted stable crystal structures of studied S-rich compounds under pressure: (a) Fddd CaS$_{2}$ at 200 GPa, (b) $P\bar{4}2_{1}m$ CaS$_{3}$ at 100 GPa, (c) Immm CaS$_{3}$ at 200 GPa, (d) $C2/c$ CaS$_{4}$ at 200 GPa.
Then it turns to the predicted S-Rich compounds. In the Fddd CaS$_{2}$ structure, we observe folded layers formed by sulfur atoms, between which S–S bonds exist. They form the three-dimensional network [Fig. 3(a)]. There are two distances between closest S–S of 2.18 and 2.05 Å. With the increase of S, the CaS$_{3}$ with $P\bar{4}2_{1}m$ symmetry is constituted by S$_{3}$ units and isolated Ca atoms [Fig. 3(b)]. Moreover, it is interesting to notice that the Immm CaS$_{3}$ phase is formed by alternating four-grid chains (composed of S) and Ca atoms [Fig. 3(c)]. For the S-richest compound, sulfur atoms in the predicted $C2/c$ CaS$_{4}$ structure form fence with the S–S distances of 2.08 and 2.04 Å [Fig. 3(d)]. From the calculated electron localization function (ELF), it can be seen in Figs. S4 and S5 that high ELF values for S-rich compounds are distributed between S atoms, indicating the presence of S–S covalent bonds. Moreover, we calculated their integrated-crystal orbital Hamilton populations (ICOHPs) to scale the bond strength in compounds by counting the energy-weighted population of wave function between two atomic orbitals. Our results demonstrate that the forming S–S interaction strength is much larger than that of Ca–S bonds. For example, the value of ICOHP of S–S and Ca–S are 3.19 and 0.45 eV per pair in the $C2/c$ CaS$_{4}$, respectively. Therefore, we conclude that the S–S interactions are mainly responsible for the structural stabilities in the S-rich compositions.
cpl-38-3-036201-fig4.png
Fig. 4. Electronic and dynamic properties of $P2_{1}/m$ Ca$_{3}$S and Pnma Ca$_{2}$S: (a) the calculated 2D ELF in (010) plane, (b) 3D ELF plot with an isosurface value of 0.65 for $P2_{1}/m$ Ca$_{3}$S at 60 GPa, (c) 2D ELF for Pnma Ca$_{2}$S at 50 GPa, (d) the projected density of states (PDOS) for $P2_{1}/m$ Ca$_{3}$S at 60 GPa, (e) the calculated phonon dispersion curves of $P2_{1}/m$ Ca$_{3}$S at 5 GPa, (f) Pnma Ca$_{2}$S at 0 GPa.
Among all our predicted stable phases, it is interesting to notice that the Pnma Ca$_{2}$S and $P2_{1}/m$ Ca$_{3}$S compounds behave as electrides. Therefore, we mainly focus on the analysis and discussion about the two phases of the Pnma Ca$_{2}$S and $P2_{1}/m$ Ca$_{3}$S in the following. Figures 4(a) and 4(c) show the calculated two-dimensional (2D) ELF in (010) plane for Pnma Ca$_{2}$S at 50 GPa and for $P2_{1}/m$ Ca$_{3}$S at 60 GPa, respectively. To clearly observe the distribution of electron, the three-dimensional (3D) ELF plot with an isosurface value of 0.65 for $P2_{1}/m$ Ca$_{3}$S is also shown in Fig. 4(b). It can be clearly seen that extra electrons locate at the positions of interstitial regions, which can be regarded as interstitial quasi-atoms (ISQs). In addition, we calculated the Bader charges for $P2_{1}/m$ Ca$_{3}$S, and found that nearly two electrons reside in the interstitials. Such behavior provides strong evidence for Pnma Ca$_{2}$S and $P2_{1}/m$ Ca$_{3}$S as electrides.$^{12,13}$ More importantly, our results demonstrate that both the Pnma Ca$_{2}$S and $P2_{1}/m$ Ca$_{3}$S electrides exhibit metallic behavior. It is well known that the interstitial electrons can occupy shallow bands and largely determine the physical properties. Figure 4(d) gives the calculated projected density of states (PDOS) of the $P2_{1}/m$ Ca$_{3}$S. We found that, although the main contribution near the Fermi level ($E_{\rm F}$) is from Ca, the contribution of interstitial electrons to the metallic state is also important. Moreover, it is striking to note that the electrides of $P2_{1}/m$ Ca$_{3}$S and Pnma Ca$_{2}$S were also predicted to be dynamically stable at 5 GPa and 0 GPa [Figs. 4(e) and 4(f)] from the calculated phonon spectra. Such results provide high possibility for the high pressure phases to be quenched to ambient condition or synthesized using the large volume press at industrial scale production for practical applications.
cpl-38-3-036201-fig5.png
Fig. 5. The calculated values of $T_{\rm c}$ for different electrides in the pressure range from 0 to 300 GPa. Insert: 3D ELF plot for $P2_{1}/m$ Ca$_{3}$S at 60 GPa. Blue balls and red rhombus denote electrides and superconductors, respectively.
Motivated by their metallic behavior and high stability, we further explored their possible superconductivity of $P2_{1}/m$ Ca$_{3}$S and Pnma Ca$_{2}$S. Their superconducting critical temperatures ($T_{\rm c}$) are evaluated through electron–phonon coupling (EPC) calculations, which is based on the Allen–Dynes-modified McMillan equation:[69] $$ T_{\rm c}=\frac{\omega _{\log}}{1.2}\exp\Big[\frac{1.04(1+\lambda)}{\lambda-\mu^* (1+0.62\lambda)}\Big]. $$ Other stable phases with metallic properties in the Ca–S systems are also included for comparison (Table 1). With the Coulomb parameter set $\mu^* = 0.1$, seven stoichiometries were predicted as potential superconductors, including $P2_{1}/m$ Ca$_{3}$S, $P4mbm$ Ca$_{3}$S, Pnma Ca$_{2}$S, Cmcm Ca$_{2}$S, Fddd CaS$_{2}$, Immm CaS$_{3}$ and $C2/c$ CaS$_{4}$. The $P4mbm$ Ca$_{3}$S phase exhibits a maximum $T_{\rm c}$ value of $\sim $20 K. More importantly, we note that the two stable electrides of $P2_{1}/m$ Ca$_{3}$S and Pnma Ca$_{2}$S phases exhibit $T_{\rm c}$ values of 7.04 and 0.26 K, respectively. The details are summarized in Fig. 5. Notably, it is clear that the relatively low pressure conditions (50–60 GPa) are needed for the stabilization of the $P2_{1}/m$ Ca$_{3}$S and Pnma Ca$_{2}$S phases among the known pressure-induced electrides in previous theoretical and experimental results.$^{12,19,21,28,29}$ Further calculations found that the compounds of Ca$_{3}$S and Ca$_{2}$S can be formed not only by elements Ca and S, but also by CaS and Ca (Ca + CaS $\to$ Ca$_{2}$S/Ca$_{3}$S). Taking into account of their quenchable behavior, we suggest that such new superconducting electrides are highly valuable for further research and possess a high possibility to be synthesized.
Table 1. The calculated value of $\lambda$, the logarithmic average phonon frequency $\omega_{{\log}}$, electronic density of states at the Fermi level $N(E_{\rm F}$) (states$\cdot$spin$^{-1}$$\cdot$Ry$^{-1}$$\cdot$f.u.$^{-1}$), and superconducting critical temperatures $T_{\rm c}$ for Ca–S compounds.
Structures Pressure (GPa) $\lambda$ $\omega_{\log}$ (K) N($E_{\rm F}$) $T_{\rm c}$ (K) for $\mu^*=0.1$ Type
$P2_{1}/m$ Ca$_{3}$S 60 0.71 193.24 47.54 7.04 Electride
$P$4/mbm Ca$_{3}$S 200 1.11 246.53 16.03 19.93 Superconductor
Pnma Ca$_{2}$S 50 0.32 289.03 36.82 0.26 Electride
Cmcm Ca$_{2}$S 150 0.35 412.24 13.16 0.78 Superconductor
$I4_{1}$/amd CaS 200 0.25 283.67 2.19 0.016 Superconductor
Fddd CaS$_{2}$ 150 1.03 114.37 11.87 8.33 Superconductor
Immm CaS$_{3}$ 200 0.40 532.46 6.03 2.13 Superconductor
$C2/c$ CaS$_{4}$ 200 0.63 460.16 18.11 11.88 Superconductor
In summary, the variable stoichiometries and properties of Ca–S system have been systematically explored under pressure up to 200 GPa. Six pressure-induced Ca–S compounds are found. The results show that $P2_{1}/m$ Ca$_{3}$S, $P4mbm$ Ca$_{3}$S, Pnma Ca$_{2}$S, Cmcm Ca$_{2}$S, Fddd CaS$_{2}$, Immm CaS$_{3}$ and $C2/c$ CaS$_{4}$ are superconductors with the $T_{\rm c}$ of 7.04, 19.93, 0.26, 0.78, 8.33, 2.13, 11.88 K at different pressures, respectively. Furthermore, we find that the pressure-induced superconducting compounds of $P2_{1}/m$ Ca$_{3}$S and Pnma Ca$_{2}$S become electrides at 60 and 50 GPa. These two electrides are dynamically stable at 5 and 0 GPa, showing quenchable under ambient conditions. Our findings provide a feasible direction for future experimental study on Ca–S systems under pressure. Some of the calculations were performed in the High Performance Computing Center (HPCC) of Qufu Normal University. 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