First-Principles Study of Sodium Intercalation in Crystalline Nax Si24 (0 ≤ x ≤ 4) as Anode Material for Na-ion Batteries

The search for Si-based anodes capable of undergoing low volume changes during electrochemical operation in rechargeable batteries is ample and active. Here we focus on crystalline Si24, a recently discovered open-cage allotrope of silicon, to thoroughly investigate its electrochemical performance using density functional theory calculations. In particular, we examine the phase stability of NaxSi24 along the whole composition range (0 ≤ x ≤ 4), volume and voltage changes during the (de)sodiation process, and sodium ion mobility. We show that NaxSi24 forms a solid solution with minimal volume changes. Yet sodium diffusion is predicted to be insufficiently fast for facile kinetics of Na-ion intake. Considering these advantages and limitations, we discuss the potential usefulness of Si24 as anode material for Na-ion batteries.

Forming a solid-state solution of crystalline silicon under electrochemical conditions is one of the best strategies to preserve the morphology of the anode during cycling. Indeed, lithium has been inserted electrochemically into crystalline silicon, with Li 3.75 Si as the most Li-rich phase known to date 31 . Crystalline Na-Si alloys have also been reported (e.g., NaSi 32 ), however, as previously noted, crystalline silicon is inactive towards electrochemical sodium insertion [21][22][23] . Finding novel crystalline silicon phases beyond cubic-diamond (silicon's most common form) that are capable of intercalating sodium is, therefore, of the outmost technological relevance. Fortunately, silicon shows a rich variety of bulk phases and allotropes [33][34][35][36] . And a thoroughly exploration of such complex free-energy landscape might yield very fruitful results. Open-cage structures are particularly compelling, since, in principle, they could easily accommodate alkali ions inside the cages. Actually, it has been found that several classes of silicon clathrates evidence negligible volume changes during electrochemical lithium insertion 37 . Yet little is known about the corresponding sodium counterparts.
Interestingly, the synthetic strategies that led to the discovery of many silicon clathrates and other open-cage structures rely on Na-based precursors (e.g., Na 4 Si 24 38 , Na x Si 136 39 , and Na 8 Si 46 39 ), which suggests that such compounds could show high sodium intercalation capacities and fast ion mobility 40 . This, together with the fact that often these compounds are metallic 41,42 , makes open framework structures of silicon attractive candidates for anode materials. Motivated by this, Marzouk et al. 43 have recently explored the insertion of sodium into different allotropic forms of crystalline Na x Si 24 (with x = 1, 2, 3, 4, 5, and 6): (i) the experimentally observed open-cage Si 24 33-38 and (ii) additional metastable allotropes found computationally using evolutionary metadynamics. They showed that sodiation in all these systems is energetically favourable up to four sodium atoms per Si 24 , with small volume expansions and low intercalation potentials. These results provide basic evidence for the potential benefits of using Si 24 as anode material. Follow-up studies to fully understand the electrochemical behaviour of such compounds are therefore timely and highly desirable.
Here, we focus on the most stable allotrope Si 24 and thoroughly assess its potential electrochemical performance as anode material for Na-ion batteries. To this end, we carried out a series of density functional theory (DFT) calculations to examine a range of key properties during battery operation. First, we computed the equilibrium phase diagram of Na x Si 24 (0 ≤ x ≤ 4) at 0 K in order to identify possible phase separation and solid solution regions. Second, we analysed volume and voltage changes along the whole composition range. Then, sodium mobility within the open-cage network of Si 24 was explored by computing Na-ion diffusion barriers. Finally, in light of our results, we discuss the advantages and drawbacks of Si 24 -based insertion anode for Na-ion cells and how future research directions might help to improve this class of anode materials.

Methods and computational details
We performed spin-polarized DFT calculations as implemented in the code VASP (version 5.4.1) 44,45 . Projector augmented wave potentials 46 were employed to treat valence-core interactions, whereas Na (2p, 3 s) and Si (2 s, 2p) valence electrons were expanded in plane-waves with a cut-off energy of 600 eV. The Perdew-Burke-Ernzerhof (PBE) 47 exchange-correlation functional was used together with a Monkhorst-Pack grid centred at the Γ point with at least 8 × 4 × 4 k-point sampling per 1 × 1 × 1 unit cell. The structures were fully optimized (cell parameters, volume cells, and atomic positions) with a residual force threshold of 0.02 eV/Å −1 . These computational settings guaranteed a tight convergence in total energies (ca. 3 meV per formula unit) and lattice parameters (differences are less than 2%). Additionally, we confirmed that the cut-off energy of 600 eV was sufficiently large to mitigate the problems of Pulay stress and incomplete basis set associated to changes of the volume in plane-waves calculations.
Taking as a reference the pristine Si 24 unit cell with three crystallographic non-equivalent Si positions (space group Cmcm) 38 , we simulated a range of Na x Si 24 (0 ≤ x ≤ 4) systems using supercells containing up to 16 formula units (Na x/4 Si 6 ) as shown in Fig. 1. We used the cluster assisted statistical mechanics (CASM) code 48 to generate all of the possible sodium and sodium vacancy () arrangements within each supercell. By so doing, we thoroughly took into account the configurational order/disorder associated with different Na- arrangements for a given sodium concentration. However, given the large number of possible Na- configurations for most of the Na concentrations, we selected only a fraction of them to be actually computed using DFT. The selection was driven by a fitting procedure using a cluster expansion scheme as implemented in CASM 48 . Overall, we computed with DFT a total of 235 different configurations and the specifically investigated Na x Si 24 compositions were x = 0.000, 0.250, 0.333, 0.500, 0.667, 1.000, 1.333, 1.500, 1.667, 2.000, 2.333, 2.500, 2.667, 3.000, 3.333, 3.500, 3.667, and 4.000. The DFT runs of the entire set of configurations (all input and output files) are hosted by NoMaD (Novel Materials Discovery) Repository in the following link http://dx.doi.org/10.17172/NOMAD/2017.03.17-2.
The transition-state structures and activation barriers for Na-ion diffusion were calculated using the climbing image nudged elastic band (CI-NEB) method 49 with seven images along the migration pathway. The calculations were performed at constant volume using the optimized lattice parameters of the lowest-energy Na 0.333 Si 24 and Na 3.667 Si 24 configurations. The inclusion of van der Waals (vdW) forces into plain DFT can play a role in the magnitude of the computed activation energies for ion diffusion in sparse matter (see, e.g., refs 50 and 51). Therefore, we used the vdW-inclusive DFT-D3 method with a Becke-Johnson damping function (DFT-D3BJ) 52 , which has shown good accuracy to describe vdW layered electroactive materials as compared with other vdW-inclusive approaches 53 , to test the effect of vdW forces on our computed activation energies. We found that the impact of vdW forces is negligible in Si 24 , with energy differences less than 2 meV between PBE and D3BJ.

Results
Phase stability as a function of sodium content. We examined the relative stability of Na x Si 24 with sodium concentrations ranging from x = 0 to x = 4. First, we computed the total energy of different Na- arrangements with composition x; Table S1 (Supplementary Material) summarises the ground state and second most Scientific RepoRts | 7: 5350 | DOI:10.1038/s41598-017-05629-x stable structures for each x, including their optimized cell parameters. Then, for each configuration, the corresponding formation energy per formula unit was calculated as where E a Si N x 24 is the total energy of a given Na x Si 24 configuration per Na x/4 Si 6 formula unit, E a Si N 4 24 is the total energy of the fully sodiated phase per NaSi 6 formula unit, and E i S 24 is the total energy of pristine Si 24 per Si 6 formula unit. Negative formation energies correspond to stable Na x Si 24 structures. Figure 2(a) shows Δ f E as a function of sodium content and includes the corresponding convex hull. The convex hull envelopes all the points in the plot by connecting the lowest-energy structures with straight lines. Given two inflexion points in the convex hull (stable structures) at sodium concentrations x i and x j , the straight line connecting them indicates a tendency of any intermediate Na , according to the application of the lever rule. The convex hull can therefore be interpreted as the zero-temperature mixing free energy of the system (i.e. neglecting configurational entropy).
We found the formation of a stable intermediate phase at x = 0.5 that governs the convex hull ( Figure S1 in Supplementary Material). Accordingly, sodiation of Si 24 is predicted to occur through two successive phase transitions: (i) between Si 24 and Na 0.5 Si 24 , and (ii) between Na 0.5 Si 24 and Na 4 Si 24 . Similarly, desodiation of Na 4 Si 24 should follow the inverse process. However, notice that the formation energies are very low along the whole composition range, and they are easily accessible at room temperature (kT = 25.6 meV at 25 °C), especially below x = 2. In total 189 formation energies are below kT with respect to the hull line (grey shaded area in Fig. 2(a)). This provides accessible low-energy Na- configurations through intermediate Na x Si 24 structures and, therefore, it suggests the formation of a solid solution regime along the whole range of sodium compositions. These results are in agreement with experimental observations 38 showing that the thermal release of sodium from Na 4 Si 24 occurs spontaneously at temperatures as low as 47 °C, with complete sodium removal after eight days under vacuum at 127 °C.
Average voltage versus Na/Na + . DFT calculations can also be used to predict average equilibrium voltages (sodiation potential) as a function of capacity, V(x) 54 . The average voltage for sodium compositions between x i and x j (with x i < x j ) is given by the negative change in the corresponding Gibbs free energy (ΔG ij ) per charge carrier. For zero-temperature calculations, ΔG ij (≡ΔE ij + PΔV ij + TΔS ij ) can be approximated by simply considering the change in total energy (ΔE ij ) 54 , neglecting zero-point energy contributions and the term PΔV ij :  24 24 where F is Faraday's constant, e is the electron charge, E Na Si is the total energy of Na Si x 24 j ground state structure, and E Na(s) is the total energy (per atom) of bulk bcc sodium. The approximation is indeed good at room temperature as well, since finite-temperature entropic effects are only expected to smoothen the voltage curve. Figure 2(b) shows the computed voltage curve along the minimum energy path of formation energies ( Fig. 2(a)). Our computed average voltage between the fully sodiated (x = 4) and desodiated (x = 0) phases is 0.30 V. In addition, we predict a small voltage jump of 0.11 V at x = 0.5, which corresponds to the formation of the intermediate Na 0.5 Si 24 structure. These results are consistent with the intercalation potentials reported by Marzouk et al. 43 for Na 1 Si 24 , Na 2 Si 24 , Na 3 Si 24 , and Na 4 Si 24 . In particular, their average voltage considering these four structures is also ~0.30 V.
Volume expansion as a function of sodium content. In Fig. 2(a) we used a colour gradient scale to follow the volume change (%v) associated to each Na x Si 24 structure. Taking as a reference the desodiated pristine Si 24 phase, the volume change was calculated as x 0 0 where v x and v 0 are the volumes of the sodiated Na x Si 24 and pristine Si 24 phases, respectively.
Our results indicate that the (de)sodiation process along the whole composition range is accompanied by small volume changes (less than 2.3%). Further analysis shows that, considering only the lowest-energy Na x Si 24 structure for each x value, there is a systematic volume expansion when increasing the sodium content, as shown in Fig. 2(c). This is in good agreement with previous DFT calculations and experimental evidence for a limited number of sodium concentrations 38 . Interestingly, the volume changes associated to different Na- arrangements within a given sodium content are generally very small (less than 0.2%). This suggests that the geometrical flexibility of the silicon framework is independent of how sodium and sodium vacancies get distributed. This property is, in principle, beneficial to avoid possible mechanical stresses which could be associated to the eventual visit of high-energy Na- configurations during electrochemical operation.
The observed small volume changes are a direct consequence of the low-density and cage-like structure of allotrope Si 24 , which facilitates that Na ions enter and leave the cages with minimal distortion of the open-framework arrangement of Si. Essentially, Si 24 presents cages large enough to easily accommodate guest Na ions.

Sodium diffusion kinetics.
We examine now the migration of sodium in Na x Si 24 at dilute (x = 0.333) and concentrated (x = 3.667) regimes. These two concentration limits allow us to easily assess the effect of Na-Na interactions on diffusion barriers. In the dilute limit, the diffusing sodium ion is completely surrounded by sodium vacancies and the impact of Na-Na interactions on the diffusion barrier is minimal. In contrast, by imposing a high sodium concentration Na-Na interactions are maximized. In principle, diffusion barriers along intermediate sodium concentrations are expected to range between these two limits.
Sodium migration in the open framework of Si 24 corresponds to a hop path between two low-energy sites along the a lattice axis, within the linear channels formed by eight-member silicon rings (see insets in Fig. 3). Sodium diffusion along the alternative b and c lattice axes is blocked by the presence of too narrow six-and five-member silicon rings, respectively. The computed migration energy profiles at x = 0.333 and x = 3.667 using CI-NEB calculations are shown in Fig. 3. The diffusion barrier for the dilute limit is 683 meV, and it increases to 811 meV in the case of high concentration (Table 1). Further analysis shows that, for the transition state structure at x = 0.333, the eight-member ring along the diffusion pathway expands up to 0.46 Å with respect to the initial positions of the silicon atoms. For x = 3.667, however, the presence of sodium atoms in nearby cages hinders the outward relaxation of silicon atoms, with an expansion of the eight-member ring at the transition state structure of ≤0.43 Å. This geometrical constrain in turn explains the observed trend in the diffusion barriers when moving from the dilute to concentrated regime.

Discussion
Our first-principles computations show that crystalline Na x Si 24 exhibits useful electrochemical properties and is, therefore, an interesting anode material for Na-ion batteries. First, Na x Si 24 forms a solid solution along the whole range of sodium compositions. Secondly, it shows negligible volume expansion upon sodium insertion. Thirdly, the operating voltage versus Na/Na + is conveniently low.
The formation of a solid solution implies that Na x Si 24 will evolve through a single-phase regime, rather than by nucleation and growth of other phases, which has significant consequences for the (de)sodiation process. Essentially, Na x Si 24 grains should insert or remove sodium as rapidly as their bulk and interfacial diffusion kinetics allow. Therefore, during electrochemical operation the potential required to drive (de)sodiation will be nearly constant and low. In addition, the potential negative effects of phase separation (e.g., mechanical stresses, sluggish kinetics, or problems of irreversibility) are effectively avoided. Notice, however, that the equilibrium state of the Na x Si 24 system is actually governed by a ground state at x = 0.5 ( Fig. 2(a)). Hence, at sufficiently large relaxation times (thermodynamic conditions), a Na x Si 24 anode should evolve to its equilibrium two-phase state of Si 24 / Na 0.5 Si 24 or Na 0.5 Si 24 /Na 4 Si 24 depending on the initial sodium content.
The main drawback of current Si-based anode materials is their massive volume change during ion insertion and extraction. Hence, even though the formation of a solid solution is beneficial, it should be accompanied by minimal volume changes along the whole sodium composition range. Our results indicate that the structural retaining ability of Na x Si 24 is indeed excellent, with a maximum volume expansion of 2.3% along the whole sodium composition range. This is in contrast to the massive volume expansion of other binary sodium compounds, such as a-Na 0.76 Si (%v = 114%) 55 , a-NaSi (%v = 230%) 30 , a-Na 3 Sb (%v = 293−390%) 56, 57 , or a-Na 3.75 Sn (%v = 480%) 30 . Interestingly, the volume changes of Na x Si 24 are even smaller than those reported for some   . Na x Si 24 is therefore expected to offer a very competitive structural function as it cycles sodium in and out. The average voltage of a suitable anode material should be low (typically around 0-1 V versus Na/Na + ) to maximize the overall cell potential and in turn its energy density. For instance, promising sodium anodes such as NiP 3 60 , disodium terephthalate (Na 2 C 8 H 4 O 4 ) 61, 62 , and some oligomeric-Schiff bases 63, 64 present operating voltages of 0.3-0.8 V, 0.4-0.6 V, and 0.6-1.2 V versus Na/Na + , respectively. Electrodes with voltages too close to 0 V are indeed highly prone to sodium plating on charge. This issue makes them unsafe and, in general, operating voltages slightly above 0 V are desirable. The predicted average voltage of Na x Si 24 (0.30 V) and its flat profile are therefore beneficial to improve safety as anode material while offering a good energy density.
All of these predicted properties are certainly encouraging, specially taking into account that the search of open framework allotropes of silicon as electrode materials for batteries is at its early stage. Yet, from a critical viewpoint, further developments are required in order to advance such compounds towards actual practical applications. For instance, our calculations pointed out a need for enhancing sodium mobility in bulk Na x Si 24 . The computed diffusion barriers (683-811 meV) are lower than that of bulk silicon (~1000 meV) 65 , but about twice larger than the values reported for a-Na 0.76 Si (380 meV) 55 or a-NaSi (310 meV) 30 . In general, good ionic conductors should show bulk diffusion barriers lower than 400 meV 66 .
Another issue is the moderate capacity of Si 24 . Considering the volume of desodiated Si 24 as a reference, the theoretical volumetric capacity is 337 mAh/cm 3 , which is far from the values of bulk sodium (~1100 mAh/cm 3 ) or NaSi (~950 mAh/cm 3 ), but similar to that of, for example, hard carbon (Na 0.81 C 6 with ~375 mAh/cm 3 ) 67 . The theoretical gravimetric capacity of 159 mAh/g (assuming 4 sodium ions are reversibly inserted) is similar to that delivered by some sodium titanates and titanium phosphates that have also been proposed as anodes for Na-ion batteries 7 . Still, these values are quite behind the ~300 mAh/g of reversible capacity offered by hard carbon, which is currently a very attractive candidate material due to its low cost. Inserting additional sodium ions into Si 24 structures would be a good strategy to mitigate this issue. Indeed, Marzouk et al. 43 computationally explored such possibility by considering alternative stable forms of Si 24 . By breaking different Si-Si bonds, they were able to generate additional open cages within each of those allotropes and in turn rearrange the unit cell space to accommodate more alkali ions. This virtual experiment showed that those new allotropes could indeed allocate up to 7 lithium ions instead of only 4, leading therefore to a substantial capacity increase. In contrast, the results were less successful for the sodium counterparts due to the larger size of sodium as compared to lithium; insertion beyond 4 sodium ions was shown to be unfavourable. However, given the rich free-energy landscape of silicon, these studies open an interesting molecular-level strategy for the future design of improved crystalline allotropes.
Finally, it is important to notice that synthesizing low-cost Si 24 is of paramount relevance for their actual success as anode materials. Although Si 24 is stable at ambient conditions, its synthesis involves high pressures 38 . Likewise, current production of other open-framework silicon compounds is also expensive. Therefore, the development of new cost effective synthesis methods are mandatory. Fortunately, silicon is the focus of a number of large semiconductor industries, which makes such research attractive to them. Another reason to be optimistic is that other Si-based anode materials, firstly synthesized at high temperatures and high pressures, have then been prepared by more simple and affordable techniques. In particular, Duveau et al. 68 reported that SiP 2 (first prepared at >1000 °C and 1-4 GPa) can be obtained by ball milling of silicon and phosphorous powders at room temperature.

Conclusions
In this work we focused on the recently discovered open-cage allotrope Si 24 and, using DFT calculations, we assessed several of its thermodynamic and kinetic properties of relevance for the potential use as anode material for emerging Na-ion technologies. Our results shed light into the phase stability of Na x Si 24 along the whole composition range (0 ≤ x ≤ 4), predicting the formation of a beneficial solid solution at room temperature. In addition, we found that the (de)sodiation process is accompanied by low volume changes (less than 2.3%) and low average voltage (0.30 V). These characteristics make Si 24 an interesting anode material for Na-ion batteries.
However, we also identified a series of current limitations and concerns for its practical use. First, the computed sodium diffusion barriers in bulk Na x Si 24 indicate that sodium ions are expected to be only moderately mobile, likely leading to too sluggish kinetics for competitive electrochemical performance. Secondly, the specific capacity of Si 24 is acceptable, but clearly lower than that of other competing materials such as hard carbon. And finally, current Si 24 synthesis techniques need to improve in terms of cost effectiveness.
Overall, while Si 24 shows interesting opportunities as anode for Na-ion batteries, further improvements are required to unveil the full potential of such class of materials. Aware of the predicted advantages and current limitations of Si 24 , we put forward that the superior volume change behaviour of Si 24 makes this material suitable for applications where mechanical reliability and duration are more important than cost or high capacities. For instance, Si 24 -based electrodes could be interesting for all-solid state microbatteries for aerospace, medical implants, stand-alone microelectronic devices, nanometer-sized autonomous sensors, and so on. Furthermore, the physical qualities predicted for Na x Si 24 make interesting to explore the insertion of other ions into open framework silicon compounds for their potential use, for instance, in K-and Mg-ion batteries.