Quasi Three-Dimensional Tetragonal SiC Polymorphs as Efficient Anodes for Sodium-Ion Batteries

: In the present work, we investigate, for the first time, quasi 3D porous tetragonal silicon − carbon polymorphs t (SiC) 12 and t (SiC) 20 on the basis of first-principles density functional theory calculations. The structural design of these q3-t (SiC) 12 and q3-t (SiC) 20 polymorphs follows an intuitive rational approach based on armchair nanotubes of a tetragonal SiC monolayer where C − C and Si − Si bonds are arranged in a paired configuration for retaining a 1:1 ratio of the two elements. Our calculations uncover that q3-t (SiC) 12 and q3-t (SiC) 20 polymorphs are thermally, dynamically, and mechanically stable with this lattice framework. The results demonstrate that the smaller polymorph q3-t (SiC) 12 shows a small band gap ( ∼ 0.59 eV), while the larger polymorph of q3-t (SiC) 20 displays a Dirac nodal line semimetal. Moreover, the 1D channels are favorable for accommodating Na ions with excellent (>300 mAh g − 1 ) reversible theoretical capacities. Thus confirming potential suitability of the two porous polymorphs with an appropriate average voltage and vanishingly small volume change (<6%) as anodes for Na-ion batteries.


INTRODUCTION
−5 Contrary to this, the large abundance of sodium and its environmentally friendly nature has garnered a lot of research efforts for realizing Na-ion batteries (NIBs), which can facilitate cost-effective large-scale energy storage.However, the considerably larger size of Na ions (i.e., r ion Na = 1.02Å vs r ion Li = 0.76 Å) necessitates the development of special anode materials that conform to sodium's electrochemical, kinetic, and thermodynamic requirements. 6−12 The lightest members of the tetrels (i.e., C and Si) have remained the go-to elements for the design of advanced and efficient anode materials. 6,10,13,14Their appeal for battery applications particularly stems from the availability of a large range of exotic electronic structures and mechanical features, which originate from a plethora of stable geometric configurations at varying degrees of dimensionality.For instance, the graphitic 15 and diamondoid modifications are among the best known to date for battery applications.The very recent study by He et al. 16 shows porous siliconedimondyne with excellent metal storability and diffusivity properties.In addition, low-dimensional forms such as fullerenes, 17 nanotubes, 18 and graphene sheets 19 have attracted a large number of research inquiries that have not only impacted the scientific and industrial sectors but have also encouraged further explorations of structure−property diversification.Consequently, numerous Si/C ratios have been investigated in recent years, which include graphene-like Si x C 1−x monolayers 13,20−23 penta-SiC 5 , 24 and SiC 2 silagraphene containing planar tetra-coordinate silicon (ptSi), 25 to name a few.These modifications of C and Si offer a wide array of electronic properties, which can yield trivial as well as nontrivial band topologies showing semiconducting, semimetallic, and metallic characters.
−28 The bulk binary SiC with a 1:1 ratio is the earliest known modification that has remained in production on an industrial scale since the accidental discovery of the Acheson process by E. G. Acheson in 1891. 29Since then, bulk SiC has been successfully synthesized in a large variety of polymorphs in laboratories with various stacking arrangements of SiC bilayers along the [111] or [0001] direction. 26Most forms of SiC are chemically bonded via a sp 3 -type hybridization and show a wide band gap semiconducting nature.Moreover, is the graphitic silicon carbide (1:1) predicted to be a semiconductor, 12,30−34 and even the graphene-like stable monolayer hSiC, with an alternating Si and C arrangement and sp 2 -type hybridization, shows an insulating band structure contrary to the Dirac cones found in pure graphene.Only the work by Qin et al. has revealed a semimetal siligraphene monolayer with C�C and Si�Si bonds in a paired configuration in a 1:1 ratio of Si and C, which facilitate the reinstatement of the Dirac cone along the G-Y k-path. 21The emergence of the Dirac cones in siligraphene originates from the coupling between Si−Si bonding states in the valence band (VB) and C−C antibonding states in the conduction band (CB).
The above discussion raises the important question of whether a 3D modification of 1:1 ratio SiC can retain a noninsulating band dispersion.To answer this, we note the recent upsurge in designing tetrel-based porous materials that offer unique opportunities for battery applications. 10,27,28,35aking the requirements of designing new anode materials suitable for NIBs into consideration, we follow, in this work, an intuitive approach to design a thermodynamically stable tetragonal SiC porous polymorph derived from siligraphene nanotubes.Our design strategy follows an atomic configuration that can be regarded as a three-dimensional interlocking fusion of armchair nanotubes constructed from a tetragonal siligraphene monolayer (tSiC) to form a quasi 3D (q3) tetragonal SiC system that contains large one-dimensional channels similar to those present in a nanotube. 42We consider two different polymorphs: a smaller q3-t(SiC) 12 system and a larger q3-t(SiC) 20 system.These are constructed by rolling up a monolayer tSiC into an (8, 8) or (12, 12) armchair nanotube, respectively.These structures, with their large porous open framework, are found not only to hold interesting mechanical and electronic properties, but we also demonstrate that the large 1D channels can be effectively employed for accommodating Na ions with low volume variations (ΔV), showing the potential for application as an anode material in NIBs.

COMPUTATIONAL METHODS
We have performed all the first-principles calculations using the Vienna ab initio simulation package (VASP) package 36 using the projector-augmented wave (PAW) method 37,38 with standard recommended pseudopotentials for the C, Si, and Na atoms (defualt energy cutoffs of 400, 245, and 260 eV respectively).The exchange− correlation potentials are modeled using the generalized gradient approximation with the parameterization proposed by Perdew et al. (PBE). 39In order to treat van der Waals (vdW) forces, we use the semiempirical corrections proposed by Grimme with a zero-damping function (DFT-D3) 40 together with the PBE functional.The plane wave cutoff energy is set to 500 eV, and all the structures have been fully relaxed until the total energy and residual forces converged to values less than 10 −5 eV and 0.01 eV Å −1 , respectively.The Brillouin zone has been sampled by 4 × 4 × 16 and 1 × 1 × 16 k-point grids 41 during structural optimizations, whereas denser 4 × 4 × 18 and 1 × 1 × 18 k-point grids have been employed for static energy calculations of the q3-t(SiC) 12 and q3-t(SiC) 20 polymorphs, respectively.For the electronic band structure, calculations are first performed using PBE-GGA.To improve the accuracy of our calculations, also, the Heyd− Scuseria−Ernzerhof (HSE06) 42,43 hybrid functional is used.Changes in the calculated total ground state energies are examined carefully for accurate estimation of the total energy dependent parameters for both pristine porous q3-t(SiC) 12 and q3-t(SiC) 20 and their sodiated (i.e., Na x t(SiC) 12 and Na x t(SiC) 20 ) modifications.
−46 The evaluation of the Si−Si and Si−C chemical bond interactions has been further conducted by chemical bonding analysis based on the crystal orbital Hamilton population (COHP) 47 as implemented in the LOBSTER 4.1.0package. 48,49To confirm dynamic stability, the phonon dispersion spectrum has been calculated using 1 × 2 × 3 supercells within the density functional perturbation theory as implemented in the Phonopy program. 50Ab initio molecular dynamics (AIMD) simulations have been performed to confirm thermal stability of the porous q3-t(SiC) 12 and q3-t(SiC) 20 polymorphs.In the AIMD simulations, we have considered a canonical ensemble (NVT) for 5 ps with a time step of 1 fs at a temperature of 500 K for the pristine porous q3-t(SiC) 12 and q3t(SiC) 20 polymorphs, and at a temperature of 300 K for the Na x t(SiC) 12 and Na x t(SiC) 20 within the Nose−Hoover heat bath method. 51Apart from this, the 2D in-plane Young's modulus of the two polymorphs has also been estimated to describe the in-plane stiffness and to confirm that the systems are mechanically stable.To obtain the migration energy for the minimum energy pathway of Na ions in the two polymorphs, the climbing image nudged elastic band (CI-NEB) method has been adopted. 52The migration pathway was constructed using seven linearly interpolated images between fully relaxed initial and final points.
The adsorption configurations of intermediate Na concentrations are investigated by first finding the maximum theoretical capacity.These are obtained by studying the formation energy for an increasing number of Na atoms.Different Na configurations are sampled randomly from a uniform distribution of the possible adsorption sites found for the stable fully sodiated system.The relative formation energies are evaluated using a convex-hull diagram.For both q3t(SiC) 12 and q3-t(SiC) 20 , we use a 1 × 1 × 2 supercell model for q3t(SiC) 12 to generate 100 and 132 different intermediate structures, respectively.

RESULTS AND DISCUSSION
3.1.Structure and Stability.The atomic structures of the two porous polymorphs shown in Figure 1 can be regarded as the interlocking of (8, 8) armchair nanotubes and larger (12,  12) armchair nanotubes derived from the monolayer tSiC.Owing to this fact, we first carried out a detailed evaluation of the structural and electronic properties of the monolayer and the nanotube associated with the two porous polymorphs before confirming their stability and suitability for battery applications.For the monolayer tSiC, the optimization as well as the examination of its dynamic and mechanical stability is performed using DFT-D3.Compared to the earlier data reported by Qin et al., 21 we observe slight changes in the structural and energetic properties upon introducing dispersion corrections in our DFT calculations (Supporting Information Figure S1 and Table S1).Our calculations clearly show that both monolayer tSiC (Figure S2) and an armchair nanotube (Figure S3) derived from this monolayer exhibit robust Dirac cones in their electronic band structure.It is worth pointing out here that in addition to the PBE-based calculations, we have also computed the electronic band structure of monolayer tSiC using HSE06-based hybrid DFT calculations to confirm that the emergence of the Dirac cone in this system is not an artifact of the semilocal DFT functional (see Figure S2).This is important in view of the fact that noninsulating electronic properties in monolayer tSiC are beneficial for its application as an anode material.
For confirming the energetic stability of the monolayer tSiC and its armchair nanotube derivative, the cohesive energies have been computed using the relation where E T is the total ground state energy of the system containing N atoms and the summation is the over all isolated atomic species "i" with chemical potential μ i .For the case of graphene-like monolayer hSiC, 53 the cohesive energy calculated in the present work (E C = 6.74 eV/atom) is found to be intermediate between the pristine graphene and silicene monolayers. 21On the other hand, our results show that monolayer tSiC (E C = 6.43 eV/atom) has a slightly smaller cohesive energy compared to monolayer hSiC.This can be attributed to the intrinsic bonding strains in the monolayer tSiC where the Si−Si bond (2.23 Å) is smaller and the C−C bond (1.44 Å) is larger compared to the bond lengths in pristine silicene (2.28 Å) and graphene (1.42 Å), respectively.Nevertheless, our results clearly show that monolayer tSiC is dynamically and mechanically stable (supplementary material Figure S1).On constructing a (6, 6) armchair nanotube from monolayer tSiC, the atomic relaxation brought about by the curvature of the nanotube increases the Si�Si bond length to 2.25 Å. Surprisingly, the reduction in strain caused by rolling up a monolayer tSiC into a nanotube is found to be responsible for improving the cohesive energy of this system to E C = 6.47 eV/atom.This is in stark contrast to the case of rolling up a graphene monolayer where strain introduced by the curvature of the nanotube causes the cohesive energy to decrease. 54owever, it has already been shown in earlier works that porous 3D modifications derived from carbon nanotubes can have better energetic stability than C 60 fullerene and in some cases even better than the parent carbon nanotubes used for constructing these porous systems. 55In addition, the presence of sp 2 and sp 3 hybridization in these carbon allotropes makes them suitable for a variety of technological applications.Since these results indicate that a 3D porous system obtained from an armchair nanotube network of monolayer tSiC can be energetically more stable, it is worthwhile to examine the physical properties of this system.Encouraged by the above findings, the crystal structures of q3-t(SiC) 12 and q3-t(SiC) 20 are constructed based on interlocking fusion of (8, 8) and (12, 12) armchair nanotubes of tSiC, such that linking Si atoms results in sp 3 hybridization to form a fourfold chemical bond coordination as shown in the primitive interlocking chains (Figure 1a).This is close to the case of interpenetrating silicene networks (ISN). 56However, the unique atomic arrangement of q3-t(SiC) 12 and q3-t(SiC) 20 presented in Figure 1b,c demonstrates that only the larger q3t(SiC) 20 has a single sp 2 -hybridized SiC ring in between the sp 3 -hybridized Si−Si bonds. 57The optimized unit cells of the two q3-t(SiC) 12 and q3-t(SiC) 20 porous polymorphs shown in Figure 1b,c contain 12 and 20 atoms, respectively, of Si and C, and the space group adopted by these systems is 123, P4/mmm (D 4h -1).The fully relaxed lattice parameters of the two polymorphs q3-t(SiC  and C(sp 2 )−Si(sp 3 ) = 1.86 Å.In addition to these bonds, the q3-t(SiC) 20 also contains C(sp 2 )−Si(sp 2 ) and Si(sp 2 )−Si(sp 2 ) bonds with bond lengths of 1.83 and 2.22 Å.For the two polymorphs, bond angles within the primitive interlocking chains are centered at Si(sp 3 ) (i.e., Si(sp 3 )−Si(sp 3 )−Si(sp 3 ) and Si(sp 3 )−Si(sp 3 )−C(sp 2 )) and at C(sp 2 ) (i.e., Si(sp 3 )− C(sp 2 ) −C(sp 2 )).However, for the q3-t(SiC) 20 polymorphs, the aromatic ring between the two interlocking chains gives rise to additional bond angles centered at Si(sp 2 ) (i.e., C(sp 2 )− Si(sp 2 )−C(sp 2 ) and Si(sp 2 )−Si(sp 2 )−C(sp 2 )) and C(sp 2 ) (i.e., Si(sp 2 )−C(sp 2 )−Si(sp 2 ) and Si(sp 2 )−C(sp 2 )−C(sp 2 )).Despite these differences, the whole lattice arrangement forms two parallel octagonal 1D nanotube channels for the q3t(SiC) 12 (q3-t(SiC) 20 ) where the larger channel is 13.17 Å (19.02 Å) wide, and the smaller channel is 6.97 Å (12.52 Å) wide as shown in Figure 1b,c.These octagonal channels allow the porous polymorphs to achieve a porous framework giving it a low density of ∼2.38 g/cm 3 for q3-t(SiC) 12 and ∼1.69 g/cm 3 for q3-t(SiC) 20 .The low density of q3-t(SiC) 12 is comparable to bulk phases of graphite (2.24 g/cm 3 ) and silicone (2.30 g/ cm 3 ) and significantly smaller than the bulk unit cells of SiC (6.40 g/cm 3 ) and diamond (3.53 g/cm 3 ).Moreover, the density of q3-t(SiC) 20 polymorphs is comparable to tC 24 57 (1.52 g/cm 3 ).The structure of porous polymorphs with a large volume to atom ratio is highly desirable for battery applications as it provides ideal conditions for storage of chemical species such as the Na ion. 16,28,57.2.Thermodynamic, Dynamic, Thermal, and Mechanical Stability.In order to establish the stability of freestanding porous polymorphs, we first compare the ground state energy per atom of q3-t(SiC) 12 and q3-t(SiC) 20 with the bulk cubic phases of Si, carbon, and SiC (Supporting Information Figure S4).It is interesting to note that both bulk SiC and porous polymorphs have energetic stability in between the bulk phases of Si and C. From eq 1, the cohesive energy estimated for q3-t(SiC) 12 is 6.56 eV/atom and for q3t(SiC) 20 is 6.49 eV/atom, which are clearly larger than the cohesive energy of the monolayer tSiC.However, since the ground state total energy of bulk SiC is ∼ −0.80 eV/atom (see Figure S4), which is more negative than for the two porous polymorphs, it is evident that this 3D framework of SiC derived from monolayer tSiC is a metastable phase compared to bulk cubic SiC.In order to examine the thermodynamic stability of the two porous polymorphs, we perform a crystal orbital Hamilton population analysis (COHP) 47 for all four types of chemical bonds present in this system.The COHP is an intuitive way of partitioning the electronic band energies into orbital pairs, which provide information regarding the degree of overlap between bonding and antibonding states in a chemical bond and therefore determine their stability.In Figure 2a,b, the total COHP and the orbital pairwise projected crystal orbital Hamilton population (pCOHP) for the two porous polymorphs are shown.The total COHP shows stable bonding types below the Fermi level (E f ), thus confirming a very stable chemical bond framework in q3-t(SiC) 12 and q3t(SiC) 20 .The pCOHP also shows that all bonds are stable below the Fermi level except for the C(sp 2 )−C(sp 2 ) bond, which shows a slightly occupied population in the antibonding states below the Fermi level.This behavior can be ascribed to the smaller electronegativity of Si that causes a small transfer of electronic charge from bonding Si−Si states to antibonding C−C states, weakening the bond compared to the other orbital pairwise bonding pairs.This feature is seen to be more pronounced in the porous q3-t(SiC) 20 .
This brings us to assess the dynamic stability of porous q3-tSiC polymorphs; the phonon band dispersion spectra are presented in Figure 2c,d.It is evident from Figure 2c,d that there are no imaginary phonon modes over the entire Brillouin zone for q3-tSiC, confirming the absence of any bonds prone to breakage and showing that porous q3-tSiC polymorphs are dynamically stable against atomic vibration and can be realized as a freestanding system.Since higher temperatures can easily dissociate weak bonds, it is equally necessary to examine the temperature stability of the two porous polymorphs.To this end, we performed AIMD simulations for a 1 × 1 × 3 supercell of q3-t(SiC) 12 and a 1 × 1 × 4 supercell of q3-t(SiC) 20 at a temperature of 500 K, which show complete structural integrity after a 5 ps run (see the inset in Figure 2e,f).Moreover, the time dependence of the potential energy in Figure 2d,e also makes it evident that the porous polymorphs, designed in this work, do not undergo an abrupt structural disruption or meltdown throughout the span of 5 ps.
Lastly, we carried out a comprehensive analysis of the mechanical properties of the two porous polymorphs by computing the nine independent elastic constants for the tetragonal lattice (Supporting Information Table S2).Our calculations show that both porous polymorphs fulfill all the required criteria for mechanical stability defined by Born− Huang 64 for a tetragonal structure.32 The mechanical properties such as the bulk modulus (B), Young's modulus (E), and shear modulus (G) for the two porous structures are also investigated based on Voigt−Reuss− Hill approximation (see Table S2). 58Although both porous polymorphs have lower B, E, and G values compared to the bulk unit cell of SiC, one can see that these systems are still harder than other SiC 4 polymorphs. 28The calculated Pugh's ratio values (G/B = 2.26 and 2.25) indicate their ductile nature under external strain, which is higher among the different polymorphs of SiC.We have also examined the anisotropy of the mechanical stability of the two porous polymorphs in terms of polar diagrams of Young's modulus (E (θ)) along ab-and ac-planes that are shown in Figure 3.It is evident that the Young's modulus of the two porous polymorphs along the caxis possesses larger values, while smaller values are achieved along the a and b directions.This shows that the framework of both porous polymorphs is significantly more rigid along the one-dimensional octagonal channels as compared to strain applied perpendicular to the c-axis.

Electronic Structure.
Having established the stability of these porous polymorphs, we calculate the distribution of the electron charges and electronic band structure in order to examine their potential for battery applications.The ELF is widely used for electron distribution analysis in materials where low (0) and high (1) values of this function help identify the bonding nature as well as regions where electrons are paired or not. 59Already, from the results presented in Figure 1, the ELF distributions suggest covalent localization between bonding pairs throughout the porous frameworks.However, the electron distribution is also illuminating with partial localization (∼0.5) in the green regions (Figure 1), which seems to be an artifact of metallic electronic states.To further investigate this assumption, we have computed the electronic band structure within the first Brillouin zone of the two porous polymorphs using the HSE06 (for the PBE-GGA level electronic band structure, see Supporting Information Figure S5) levels as shown in Figure 4b,c.The two porous polymorphs show distinct electronic band dispersion along the first Brillouin zone (Figure 3b).The q3-t(SiC) 12 polymorph shows a narrow band gap opening, with a small indirect band gap of 0.59 eV between the conduction band minima at point A (0.5 0.5 0.5) and the valence band maxima at point Γ (0.0 0.0 0.0).On the contrary, the q3-t(SiC) 20 clearly shows semimetal states along the A�M, Γ�Z, and R�X symmetry lines.The semimetallic nature of this system originates from the Dirac points laying slightly above the Fermi level in the case of aromatic rings.In fact, the electronic band structure of q3-t(SiC) 20 exhibits Dirac nodal lines appearing across the high symmetry lines passing along the 1D channels present in this crystal lattice (Figure S5).
For the sake of completeness, we draw a comparison by plotting the atomic projected density of states (PDOS) of the two porous polymorphs in Figure 4d,e.This specifies that distinct metallic states in q3-t(SiC) 20 with nearly linear dispersion originate from the aromatic Si and C atomic orbitals.On the other hand, the orbital PDOS confirms that these states are explicit projections of the p x and p y atomic orbitals (Figure 4e).In comparison to the PBE band structure (see Figure S5), the band gaps open up for HSE06 calculations.However, considering the nature of electronic bands in the Brillouin zone together with the presence of onedimensional channels, it is clear that porous polymorphs can be a potential anode material for NIBs.

Anode Material for Na-Ion Batteries.
For the screening of high-performance anode materials, theoretical descriptors such as binding energy, diffusion barrier, specific theoretical capacity, and ion adsorption concentration should be evaluated. 60For this reason, we have performed a thorough assessment of these descriptors for the two porous polymorphs considered in this work to identify their potential for NIBs.As a first step, it is imperative to determine the most suitable  position for binding of a single Na ion with porous polymorphs.In the present study, we have recognized different nonequivalent sites (designated A x ) that are shown in Figure 5a,c.Using a 1 × 1 × 2 supercell of q3-tSiC, the binding energy of a Na atom has been computed using where E Na[q3] and E [q3] are the total energies of the two porous polymorph supercells with a Na adsorbed and the pristine system.μ Na corresponds to the chemical potential of Na with reference to the ground state energy of Na in its most stable bulk phase.After structural relaxation, it is found that for both q3-t(SiC) 12 and q3-t(SiC) 20 , the Na inside the smaller channel at the hollow sites on top of the sp 3 -linked Si atoms (i.e., A 1 , see in Figure 5a,c  (i.e., E B = −0.171eV and E B = − 0.29 eV) show relatively small negative binding energies for q3-t(SiC) 12 and q3-t(SiC) 20 .For the sake of comparison, we have also computed the binding energies of the Na ion in the monolayer tSiC and a (6, 6)  armchair nanotube derived from this monolayer (Supporting Information Figure S6).For the case of monolayer tSiC, the binding energies of Na-ion adsorption at the two distinct hexagonal hollow sites are found to be smaller (i.e., E B = − 0.14 eV for M 1 and E B = − 0.13 eV for M 2 ) than all the binding energy values obtained for q3-t(SiC) 12 .The most negative binding energy of Na in the (6, 6) armchair nanotube of tSiC is found to be E B = − 0.41 eV (see Figure S6).Thus, it is clear that both porous polymorphs with their 1D channels facilitate a much more promising binding for Na as compared to the 1D and 2D modifications of tSiC.In order to get a deeper insight into the reaction involved in the Na adsorption, we visualize the charge transfer between the q3-tSiC and Na through differential charge analysis, which is defined as In eq 4, ρ Na[q3] and ρ [q3] represent the charge densities of the porous polymorph with and without Na adsorption, respectively, while ρ Na is the charge density of the isolated Na atom. Figure 5b,d shows the charge depletion and accumulation of the Na and the porous polymorphs for the most stable adsorption site (i.e., A 1 ) .In both cases, it is evident that sufficient charge transfer takes place from Na to q3t(SiC) 12 and q3-t(SiC) 20 upon adsorption, which reduces at sites with smaller binding energies (see Supporting Information Table S3).For more quantitative examination, we have also computed the Bader charges of Na.The Bader charge analysis for Na adsorption reveals that 0.87 e − charge is transferred from Na to the respective porous polymorph at site A 2 and 0.77 e − charge is transferred from Na to the porous polymorph at site A 1 .This is in accord with the highest value of binding energy obtained for Na at the A 1 site of the porous polymorph.
From an anodic performance point of view, in the charging process of the anode, sodium would be continuously absorbed in the q3-t(SiC) 12 and q3-t(SiC) 20 structure until the chemical potential of Na on the anode (μ Na[q3] ) equals that of the Na metal (μ Na ).This requires that [ ] Na q3 Na (5) The chemical potential of Na on the porous polymorphs equals to where G = E + PV − TS is the Gibbs free energy, T is the temperature, P is the pressure, and N is the total number of atoms.If we ignore the PV and TS terms, eq 6 can be written as Hence, combing eqs 5 and 7, we find that the sodium charging process requires If we further define the formation energy (E f ) as the following it is possible to combine eqs 8 and 9 to give Therefore, the sodium loading process requires the slope of the formation energy curve to be negative.The maximum amount of adsorbed Na corresponds to the concentration at which the slope of the formation energy curve becomes positive, and we can determine the Na storage capacity, as shown in Figure 5e,f.This corresponds to the stoichiometric formula of maximum Na insertion for the porous polymorphs to be Na 5.5 (SiC) 12 and Na 11 (SiC) 20 , respectively.From these, the theoretical specific capacity can be calculated using the relation where F is the Faraday constant, n is the number of electrons, and M r is the molecular weight of the material.Using eq 11, the porous q3-t(SiC) 12 and q3-t(SiC) 20 allow impressive theoretical specific capacities of Q = 306.35mAh g −1 and Q = 334.22mAh g −1 , which are comparable to the values provided in Table 1 and clearly larger than the specific capacities of ISN, 56 Si 24 , 61 SiC 4 , 28 and tC 24 . 57Overall, the negative formation energy slope from the initial to the final concentration of Na indicates a stable and spontaneous storage of Na in both porous polymorphs.However, with the gradual increase of the Na concentration, the electrons transferred to the porous polymorphs' lattice make further Na storage inside the porous crystal lattice lesser negative.The large channel sizes of the two porous polymorphs allow sufficient storage space for Na to bind at least ∼4.0 Å apart from each other inside the porous polymorphs.This also ensures the possibility of the Na to bind with the porous polymorphs spontaneously and much less likely to form Na clusters, for the onedimensional octagonal channels to attain fully states.The migration behavior of Na ions in the two porous polymorphs is explored using the CI-NEB method (Figure 6a,b).Based on the structural symmetry and the most stable Na binding sites for smaller and larger octagonal channels, we select two paths along the channels (or c-axis) of the two porous polymorphs for sites A 1 and A 3 such that the nearest neighbors Na are ∼6.21Å apart.The migration barrier energy profiles along the first path (corresponding to binding site A 1 ) and the second path (corresponding to binding site A 3 ) are shown in Figure 6a,b, which reveal small energy barriers of E a = 0.143 eV and E a = 0.217 eV, for the q3-t(SiC) 12 , and E a = 0.101 eV and E a = 0.143 eV for q3-t(SiC) 20 , respectively.The difference in the migration barrier heights for the two paths shown in Figure 6a,b is mainly due to the different types of chemical bonds that Na need to cross in the two channels.In the case of the smaller channel (i.e., the A 1 site), Na has to cross a localized σ bond, a Si−Si bridge with a localized electronic cloud.On the other hand, in the larger channel (i.e., the A 3 site), Na needs to cross over a more delocalized π bond, a Si�Si bridge.However, it is evident that the energy barriers estimated for Na ions in porous polymorphs are lower than the barriers calculated for black phosphorus (E a = 0.18 eV), 62 open cage Si 24 (E a = 0.68 eV), 61 monolayer C 2 N 63 (E a = 3.3 eV), and HYG 64 (E a = 0.32 eV) but higher than the isostructural tC 24 (E a = 0.05 eV and E a = 0.131 eV) 57 (see Table 1).Overall, the low migration energy barriers are comparable to LIBs, 65 which indicates excellent rate capabilities during the charge/ discharge cycles when the two porous polymorphs are used as an anode material for NIBs.
For smooth battery performance, evaluation of the opencircuit voltage (V OC ) is another important characterization parameter.This can be given as the numerical average voltage of the whole variation during the charging/discharging process. 69For that, we first define the half-cell reaction of the charging/discharging process of the two porous polymorphs as )Na ( )e Na q3 Na q3 where x 1 and x 2 are the initial and final Na-ion reaction concentration.By considering that the change in entropy, volume, and pressure is rather negligible, it is possible to plot V OC as the function of Na-ion concentration x, in a defined range of x 1 < x < x 2 as where where x′ is the fractional concentration relative to the maximum theoretical capacity, with a total energy E′ Na[q3] (i.e., Na  The average open-circuit voltage is calculated to be V OC = 0.454 V, i.e., greater than Si 24 (V OC = 0.30 V) 61 but small in comparison to tC 24 (V OC = 0.54 V) 57 and ISN (V OC = 1.35 V). 56 Similarly, the relative formation energy convex-hull plot, generated from 132 different intermediate structures for a 1 × 1 × 2 supercell model of q3-t(SiC) 20 (Figure 7b), shows five stable intermediate concentrations to be Na 0.5 (SiC) 20 , Na 1.5 (SiC) 20 , Na 2 (SiC) 20 , Na 8.5 (SiC) 20 , and Na 9 (SiC) 20 , respectively.The stable intermediate phases have been used to plot the potential profile using eq 14, which is shown in Figure 7b.It is found that q3-t(SiC) 20 starts at relatively high V OC = 1.36 V, but it gradually drops to a minimum of V OC = 0.46 V passing through 5 prominent plateaus (Figure 7b).This analysis shows both q3-t(SiC) 12 and q3-t(SiC) 20 porous polymorphs provide evidence of safe battery operation and a diminishing possibility of the formation of sodium dendrites with the two porous polymorphs as an anode material.
For the maximum specific theoretical capacity of the porous polymorphs, our calculated DOS plots show a noninsulating nature (Supporting Information Figure S7), which would help to sustain good electron conductance ensuring and excellent columbic efficiency for anodes during the battery operation.This brings us finally to assess the cycling stability of the porous polymorphs as an anode material in NIBs.For this, we have calculated the percentage of the volume change after full Na adsorption, which is found to be around ∼4% in the case of q3-t(SiC) 12 and less than ∼6% for the q3-t(SiC) 20 with larger pores.Since this volume change is much smaller than the reported values for other systems (27.5% and comparable to 2.8% for ISN 56 and 2.3% for Si 24 61 ), one can infer that the two porous polymorphs can possess excellent cycling stability.This is also confirmed by the AIMD simulation of Na 5.5 (SiC) 12 and Na 10 (SiC) 20 carried out at 300 K for 5 ps (Figure 7c,d) where the potential energy variation and final structure show strong structural integrity of the anode material after being fully loaded with Na ions.

CONCLUSIONS
In summary, we have carried out first-principles density functional theory calculations for designing 3D porous materials based on 1:1 compositions of carbon and silicon in the form of tetragonal silicon−carbon polymorphs q3-t(SiC) 12 and q3-t(SiC) 20 and have examined their suitability for employment as an anode material in Na-ion batteries.We adopt an intuitive rational approach for designing the two porous polymorphs from armchair nanotubes derived from monolayer tSiC where C�C and Si�Si bonds are arranged in a paired configuration.For establishing the energetic stability, we have computed cohesive energies of the tSiC nanotubes derived from monolayer tSiC where reduction in strain caused by rolling up the monolayer into a nanotube is found to be responsible for improving the energetic stability of this system.For the two q3-t(SiC) 12 and q3-t(SiC) 20 polymorphs, the phonon band structures demonstrate that both these systems are dynamically stable.Moreover, the calculated bulk modulus, Young's modulus, and shear modulus also confirm mechanical stability of these q3 porous structures.The thermal stability is also examined by performing AIMD simulations at a temperature of 500 K, which show complete structural integrity after a 5 ps run.This clearly suggests that the two porous polymorphs designed in this work do not undergo an abrupt structural disruption or meltdown.The electronic and bonding properties are examined in terms of electronic band structures, electron localization function, Bader charges, and COHP analysis.Our results demonstrate that the smaller polymorph q3-t(SiC) 12 has a small band gap, while the larger polymorph of q3-t(SiC) 20 displays a Dirac nodal line semimetal behavior.Although the application of the HSE06 functional slightly increases the band gap for q3-t(SiC) 12 , the metallic states in q3-t(SiC) 20 with nearly linear dispersion are found to persist.The orbital projected DOS shows that these states originated from p x and p y atomic orbitals.Despite the differences in their electronic properties, the large porous open frameworks of q3-t(SiC) 12 and q3-t(SiC) 20 contain 1D channels that are found to be favorable for accommodating Na ions.We also examine various descriptors for assessing the performance of the q3-t(SiC) 12 and q3-t(SiC) 20 in NIBs, which include binding of Na ions at various sites, differential charge analysis, formation energies, specific capacity, migration barrier, and open-circuit voltage.Our results provide crucial information regarding the application of these systems as anode materials for next-generation Na-ion batteries.
Crystal structures, mechanical properties, electronic properties, binding energy comparisons, and summary of Bader population analysis (PDF) ■

Figure 1 .
Figure 1.Perspective view of the (a) primitive interlocking SiC chain along with the extended porous cross-sectional and side profiles of crystal structures of (b) q3-t(SiC) 12 and (c) q3-t(SiC) 20 , with the unit cell indicated by red dashed lines.ELF distribution on the two slices for the (d) primitive interlocking SiC chain along with the extended 2 × 2 × 1 supercells of (e) q3-t(SiC) 12 and (f) q3-t(SiC) 20 .The inset labels indicate ELF along the slice through Si and C cross sections, where the 0 (blue) to 1 (red) color temperature scale is assigned from a complete absence to highly localized electronic density distributions, respectively.

Figure 2 .
Figure 2. (a,b) HSE06-calculated crystal orbital Hamilton population (COHP) plots of total average and orbital-wise Si−Si, Si−C, and Si−C.(c,d) Phonon band dispersion along the whole Brillouin zone path (inset) of q3-t(SiC) 12 and q3-t(SiC) 20 .(e,f) AIMD simulation for supercells of q3t(SiC) 12 and q3-t(SiC) 20 , where the insets show a snapshot of the final geometric configuration after a simulation time of 5 ps.

Figure 3 .
Figure 3. Polar diagrams of the in-plane Young's modulus along the (a) ab-plane and (b) ac-plane, where the purple and pink plots represent the q3-t(SiC) 12 and q3-t(SiC) 20 , respectively.

Figure 4 .
Figure 4. (a) HSE06-calculated electronic band structures for porous (a) q3-t(SiC) 12 and (b) q3-t(SiC) 20 along (c) the first Brillouin zone.(d) Total density of states (TDOS) in gray and atomic projected density of states (PDOS) for Si and C for q3-t(SiC) 12 and (e) TDOS in gray and atomic PDOS for inequivalent Si and C along with the orbital PDOS for sp 2 -hybridized C and Si of q3-t(SiC) 20 .
) results in the most negative value (i.e., E B = − 0.68 eV and E B = − 0.93 eV, respectively).These strong binding energies can be attributed to the fact that the Na-ion site A 1 transfers around ∼0.87 e − and induces alternating sp 2 and sp3 Si bridges along the length of the channel.For the second site inside the smaller channel (i.e., A 2 ), the Na ion ends up at the A 1 site after structural relaxation in the case of the q3-t(SiC) 12 , whereas for the q3-t(SiC) 20 , it also shows a relatively strong binding (E B = − 0.60 eV) at the aromatic ring site inside the smaller channel.Inside the larger octagonal channel, the Si atoms connecting the hexagonal hollow are labeled site A 3 in Figure5a,cand are found to have the most negative value of E B = − 0.31 eV and E B = − 0.46 eV, while the A 4 site (i.e., E B = − 0.178 eV and E B = − 0.31 eV) and site A 5

Figure 5 .
Figure 5. (a,b) q3-t(SiC) 12 and (c,d) q3-t(SiC) 20 cross-sectional side views of the inequivalent adsorption sites (colored separately) along the smaller channel (blue) and larger channel (orange) planes considered in the present work for a single Na atom along with the differential charge density diagrams for the most stable Na adsorption site (i.e., A 1 ).The red and blue isosurfaces indicate charge depletion and charge accumulation, respectively.(e,f) Formation energy per atom change for gradual sodiation of porous q3-t(SiC) 12 and q3-t(SiC) 20 .

Figure 6 .
Figure 6.(a) Perspective view of the Na diffusion paths for the A 1 site of the q3-t(SiC) 12 and q3-t(SiC) 20 along with the comparison of diffusion energy barriers as red and green, respectively.(b) Perspective view of the Na diffusion paths for the A 3 site of the q3-t(SiC) 12 and q3-t(SiC) 20 along with the comparison of diffusion energy barriers as red and green, respectively.
evident that only one prominent voltage plateau is formed in the range of ∼0.42 V.This starts at the sharp voltage drop created at the Na 0.5 (SiC) 12 concentration and ends up at the Na 4 (SiC) 12 concentration to finally decrease to 0.12 V to reach the final concentration of Na 5.5 (SiC) 12 .It can be noted that the voltage remains positive throughout the half-cell reaction, indicating that the reaction can proceed spontaneously until it reaches the final reversible specific capacity of 306.35 mAh g −1 .

Figure 7 .
Figure 7. Relative formation energy (E rel convex hull along with the open-circuit voltage (V OC )) profiles of (a) q3-t(SiC) 12 and (b) q3-t(SiC) 20 for the gradually increasing concentration of Na. (c,d) Variation of potential energy obtained from AIMD simulation of a 1 × 1 × 4 supercell of Na 5.5 (SiC) 12 and Na 10 (SiC) 20 after a simulation time of 5 ps at a constant temperature of 300 K.The inset in (c) and (d) is a snapshot of the last geometric configuration after completion of the AIMD simulation.

Table 1 .
Comparison of Specific Capacity (Q), Migration Energy Barrier (E a ), Open-Circuit Voltage (V oc ), Volume Change (ΔV), and Electronic Properties of q3-t(SiC) 12 and q3-t(SiC) 20 with Other Anode Materials for Na-Ion Batteries a a "−" means data unavailable.
and μ Na are the total energies of Na xd 1 [q3], Na xd 2 [q3], and the bulk Na, respectively.To calculate V OC for intermediate stable phases, we first plot the relative formation energy E rel , for intermediate concentrations to determine the stable intermediate phase.E rel is defined as follows: 5.5(SiC)12or Na 10 (SiC) 20 ).The negative relative energy (E rel ) indicates stability of the possible formation of an intermediate state during the charging/discharging process as shown in Figure5a,b.Our formation energy convex-hull plot between, constructed of 100 different intermediate structures for a 1 × 1 × 2 supercell model of q3-t(SiC)12, shows two stable intermediate concentrations to be Na 0.5 (SiC)12and Na 4 (SiC) 12 , respectively.Subsequently, these stable intermediate phases have been used to plot the potential profile using eq 14, which is shown in Figure7a.It is