First-principles investigation of the impact of high pressure on the structural , electronic and elastic properties of the type-VIII barium-doped silicon clathrate Ba 8 Si

Nassim Ahmed MAHAMMEDI 1,2, Marhoun FERHAT 2, 3 1 Laboratoire de physique des matériaux LPM, Amar Télidji University of Laghouat, BP37G, Laghouat 03000, ALGERIA. 2 École Normale Supérieure de Laghouat (ENSL),RN01, Laghouat 03000, ALGERIA. 3 Department of physics, The University of the West Indies, Mona, Kingston 07, JAMAICA. Corresponding author Nassim Ahmed MAHAMMEDI n.mahammedi@lagh-univ.dz  Abstract


Introduction
The quest for a perfect semiconductor that combines several features is a challenging task and a revived competitive scientific topic for materials scientists and engineers. Silicon is one of the most investigated, handled, and utilized semiconductor in major high-tech devices including not only optics, electronics, microelectronics and photovoltaics, but also thermoelectrics, superconducting and future quantum computing devices. Additionally to the fact that silicon is the second abundant substance on the Earth's crust (about 25.8% of its global mass), it has several advantages that made its industry mature and very advanced. However, exploring crystalline phases beyond the cubic diamond structure will certainly overcome its several shortages and provide more technological alternatives for expensive and rare semiconductors.
Silicon clathrates are regarded as the most promising structures of silicon, they were first synthesized in 1965 by Kasper et al and Cros et al [1,2], and their discovery released prominent research for exploring and exploiting their many intriguing features including thermoelectrics [3,4], electronics, photovoltaics [5], superconductors [6,7], and even as anodes for Li-based batteries [8]. Si clathrates are cage-like framework of silicon that entraps guest atoms inside.
Relatively to their constituents and their chemical formula, a very large family of clathrates can be formed [9]. Etymologically, the term 'Clathrate' is derived from the Greek word 'Klethra' which originally means 'alder' [10] and more precisely from the Latin word 'Clathrus', meaning 'surrounded on all sides' [11], the term was used in the 1940s to describe clathrates of hydroquinone compounds by Powell, H.M. [12]. Theoretically, eight types of clathrates are described, and more clathrate-like forms can also be added [4], however, only three types are successfully synthesized so far, i.e. type-I, type-II and type-VIII. Type-I and type-VIII clathrates possess the same chemical formulas X8Si46. Guest free silicon clathrates are promising materials for semiconducting and optoelectronic applications, as well as for thermoelectrics. Synthesis of guest-free silicon clathrates is still a challenging task, where attempting to grow single crystals, powder, or thin films only succeeded for the type-II configuration [13,14]. Thin film Si136 was recently grown over a (111) Si substrate [15]. Norouzzadeh et al extensively investigated the type-VIII gust-free Si46 by first-principles calculations, their results showed very promising thermoelectric features of this material [16][17][18].
In this work, first-principles calculations in the frame of the DFT within the GGA approach are employed to investigate the main structural features and the impact of hydrostatic pressure on the structural, electronic and elastic properties of type-VIII Ba8Si46 silicon clathrates, thus, this paper is organized as follows: the first section outlines the theoretical approach including our computational details and structural aspects of the clathrate, the second section discusses the electronic properties under pressure, and then the elastic properties under pressure are discussed in detail.

II.
Theoretical aspects

II.1. Computational procedures
In this work, first-principles calculations within the density functional theory DFT [19,20] are performed using the Cambridge Serial Total Energy Package (CASTEP) code [21]. Plane-wave based norm-conserving pseudopotentials [22] [23] approach is selected to estimate the exchange and correlation energy XC. In order to ensure that the self-consistent convergence of the total energy reaches 10 -6 eV/atom, the kinetic plane-wave cut-off energy is set to 480 eV and the first irreducible Brillouin zone is sampled using a 6x6x6 Monkhorst-Pack scheme [24]. Structural optimization is primordial in order to get an equilibrium structure at a minimum total energy, for that purpose, the optimizations are being performed using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [25][26][27][28] and stopped until the Hellmann-Feynman interatomic forces reach 10 -5 eV/Å and the pressure reach a minimum value of 0.001 GPa for all calculations. All physical properties are calculated for equilibrium crystals optimized at minimum of the total energy.

II.2. Crystalline structure
Type-VIII silicon clathrates typically crystallize in the body centered cubic space group I4-3m N° 217 [29,30], their frameworks are initially composed from associations of polyhedral cages of silicon as a host material encapsulating alkali-metals or earth-like guest species.   We calculated the Gibb's free energy at 0 GPa using the formula G 0 =E 0 +PV-T S , where E0 is the total energy per unit cell and TS is the temperature. Since this DFT study is held at a ground state where TS= 0 K, the Gibb's free energy will be equal to the enthalpy H 0 =E 0 +pV 0 . The Enthalpy variation with pressure is illustrated in figure2. Not surprisingly, it is observed that the most stable structure is under zero pressure, and then the enthalpy increases linearly with pressure above 5 GPa. Compared with the type-I phase, it was reported that the type-VIII Ba8Ga16Sn30 clathrate has lower enthalpy and therefore it tend to be more stable under high-pressure [33]. We also estimated the formation energy of the VIII-Ba8Si46 from the following equation [16]: Where: H is the total formation energy, EM is the total energy of the intercalated atom calculated in its stable bulk phase, and ESi is the total energy of Si atom in the stable diamond phase. A negative value is supposed to indicate that the compound is energetically favorable to form, whereas, if it is positive, that means that the material could be synthesized but require an additional amount of energy. We report a value of H=-97.23 eV.
The evolution of the lattice volumes and the interatomic distances Si-Si and Ba-Si under highpressure are illustrated in figures 3 and 4 respectively. As can be noticed from figure 3, the cell volume tend to normally decrease when pressure is increased, however, the behavior is not linear, from 0 GPa to 5 GPa, the volume is drastically reduced by about 31%. Above 10 GPa, lattice volume will decrease in a quasi-linear manner with compressive pressure. At 30 GPa the overall volume is reduced by about 49%. According to Kim et al [34], a volume collapse above 20 GPa is expected for the type-VIII quaternary substituted silicon clathrate Sr8Al8Ga8Si30, therefore, in our case, no such change is observed. Leoni et al [35] has suggested an analytical model to analyze the phase transition from type-I ( phase) to type-VIII ( phase) clathrates.
High pressure behavior for type-I Ba8Si46 by Zhang et al [36] and K8Si46 by Tse et al [37] were studied by means of DFT codes. In both studies similar behaviors were observed with higher mechanical stability for the barium doped clathrate as a result of the larger Ba atoms and hence stronger interaction with the Si cage. A study by Li et al [33] on the high-pressure behavior of the type-VIII Ba8Ga16Sn30 has also predicted that the type-VIII clathrate is more stable than its type-I allotrope, and they found that the phase transition from type-I to type-VIII (→) cannot occur under hydrostatic pressure regardless its magnitude. This will be understood in the discussion of mechanical properties section. This monotonous behavior demonstrates the isotropic compression of the cages under pressure; this is certainly related to the enhanced integrity of the Si cages by the presence of Ba atoms.
Nevertheless, generally, the expansion of cages volume caused by the presence of large Ba atoms will loosen the Si-Si bonding, even if the Ba-Si interaction contributes to the integrity of the structure but the Si cage will be weakened.
We have also examined the evolution of angles in the silicon frameworks under high-pressure and their evolution is plotted in the figure 3(b). It can be seen that the Si(2-1-2) angle in the pentagonal rings are constant under pressure. Other angles such as the Si(4-2-4) in the six-fold rings and the Si(4-3-4) in the seven-fold rings are not affected by pressure. The Si(4-3-4) in the six-fold rings increases with pressure, whereas, the Si(2-4-3) and Si (4-4-3) in the six-and sevenfold rings respectively decrease rapidly with pressure. At 20 GPa, it can be noticed that the Si(1-2-4) and the Si (2-4-4) in the pentagonal rings became equal and cross at this pressure.
Anisotropic behavior of angles is a normal characteristic of cage-like structures and is a consequence of the isotropic behavior of the overall lattice volume. 3.8 Bond length (Å)  Site: Norouzzadeh et al [16,30], who has shown that Ba8Si46-VIII is a metallic-like compound with a fundamental bandgap of about 1eV (GGA-PBE) appearing between  and N high-symmetry points. We found that the fundamental bandgap between the two same points is about 1.07 eV.
The total and partial densities of states DOS are illustrated in the figure 4(b). Structural stability of a compound is directly associated with its bonding strength. It can be noticed that near the valence band, the Si atoms contribute with their p-states, while the contribution from Si-s states increases when going toward low energies. Ba atoms contribute through their s and p states mainly to the middle and lower parts of the valence band, and they are less present in the conduction band.

III.2.b Under pressure
In order to study the impact of hydrostatic pressure on the electronic properties, we compare the band structures and densities of states (DOS) of type-VIII Ba8Si46 under 0, 15, and 30 GPa. In and a smaller effective mass at this point is estimated which can lead to higher electronic mobility [16], and therefore to better semiconducting and thermoelectric properties. The fundamental band gap slightly increases under pressure; it goes from 1.07 eV at 0 GPa to 1.12 eV at 15 GPa, and to 1.17 eV under 30 GPa. This can be explained in terms of partial density of states. In Figure 6, contributions from Ba and Si states become stronger under high pressure, Due to the strong hybridization of the Si-s and Ba-p states at Fermi level at higher pressures; at 30 GPa we observe an additional peak of the Ba-p states at 15 eV energy level. Also, the contribution from silicon states near and beyond the Fermi level increases with increasing pressure.

III.3.a. At zero pressure
The type-VIII Ba8Si46 clathrate crystallize into a body centered cubic structure; therefore, only three independent elastic constants (C11, C12 and C44) are sufficient to describe the elastic behavior of the material. Defining these elastic constants and investigating their behavior under high-pressure is primordial to predict the mechanical stability of the investigated material under such conditions during synthesis. For sake of comparison, elastic constants and modulus at zero pressure were also computed for the type-VIII guest-free Si46, and for the type-I Si46 and Ba8Si46 by means of the same GGA-PBE approach and settings. Results are compared with the available literature. The total energy of a given symmetrical crystal at zero pressure and at its equilibrium volume state V0 is given by: The bulk modulus B is obtained by applying a hydrostatic strain tensor and then by fitting the Munaghan equation of state. The shear constant C' is obtained through applying a volumeconserving tetragonal stress tensor: Application of such tensor will change the total energy as follows: The third equation gives the C44 constant through applying an orthorhombic stress tensor This total energy is obtained as: Then the shear modulus G is given for a cubic system as: 11 12 44 The Young's modulus E is given as: Additionally, the Young's modulus E along <100> and <111> directions are respectively described as: 11 12 11 12 (100) 11 12 ( 2 )( ) The average speed of sound v in a material is given as [39]: Where vlong and vtran are the longitudinal and transversal sound velocities, which are expressed as: Where:  represents the mass density (Kg/m 3 ).
First, we examine the elastic properties of the Ba8Si46-VIII at zero pressure. The numerical values of the above-described elastic constants are estimated using the CASTEP code by applying a small set of deformations to the crystal and taking the second derivatives of the total energy as expressed in equation (1). Our results of elastic constants Cij, the bulk modulus B, the shear modulus G, the Young's modulus E, and the average velocity of sound v, calculated by GGA-PBE for Ba8Si46-VIII at 0 GPa are listed in table 2, and compared with reported data for type-I and type-VIII Si clathrates. Our results are in a fair agreement with the literature. We notice from table 2 that VIII-Ba8Si46 having a bulk modulus of B=75.804 GPa, lower than that of the type-I Ba8Si46 (B=76 GPa), and that of the guest-free Si46 type-I (B=76.92 GPa) and type-VIII (B=78.87 GPa). The bulk modulus B, which presents a measure of resistance of the material against external deformation, provides much information about both the hardness and the bonding strength of the studied material. Our results confirm that the type-VIII Ba8Si46 is less resistant to external deformation than its counterparts. However, resistance to plastic deformation is also proportional to the elastic shear modulus G. Additionally, the shear modulus G, the Young's modulus E, and the average velocity of sound v are all larger for the type-VIII Ba8Si46 than those for the type-I Ba8Si46 but reduced when compared with guest-free counterparts Si46.
Ductility and fragility play a critical role during the synthesis of materials. In general, a ductile material is able to deform a lot before it is broken unlike a fragile material which breaks without or with less deformation, we can therefore predict if the material is fragile or ductile upon the well-known Pugh ratio G/B [40], if this ratio is equal or lower than 1.74 the material is least malleable (fragile), otherwise it is more malleable (ductile). In our case we predict that B/G = 1.681, slightly lower than the given value, this indicates that the material should be fragile at zero pressure. Elastic constants C11, C12 and C44 are slightly reduced for the type-VIII Ba8Si46.
The first elastic constant C11, which quantitatively represents the uniaxial deformation along the <001> direction has the highest value, indicating the incompressible character of the compound under the <001> uniaxial stress and is relatively smaller in type-I than in the type-VIII Si46. One can approximately define the melting temperature Tm of a cubic crystal from C11 through the equation suggested by Fine et al [41] where Tm=553+(5.91 C11) ± 300 K (Tm in K, C11 in GPa), we therefore predict for the type-VIII Ba8Si46 that Tm=1274.634 K ± 300 K. Considering the Born's stability criteria for cubic crystals [42] C11>0, C11-C12>0, C11+2C12>0 and C44 >0, the two types are mechanically stable. The Si46-VIII seems to be harder than its type-I counterpart. Our results are in good agreement with those reported by Norouzzadeh et al [17], who predicted that Si46-VIII is a strongly-isotropic material, and stable against shear and high-pressure, similarly to the type-I Si46. However, empty clathrates, in general, tend to have lower bulk modulus than the guest-containing clathrates, the guest-host hybridization usually plays an important role that can lead to hard 'diamond-like' materials [10]. But in contrary, our result shows that intercalation of Si-Si bonding on one hand [43], and in the other hand, the electron exchange enforces the integrity of the clathrates frameworks [37]. The main challenge for obtaining a guest-free clathrate Si46, whether in type-I or VIII phases, is the fact that removing the guest atoms from the cages will induce a collapse of the system and hence a phase transformation towards more stable phases such as diamond structure Si2 [10,37]. Nevertheless, guest-free type-II Si136 clathrate, was the only guest-free silicon clathrate to be successfully synthesized as crystalline [13], or thin film forms [15].

III.3.b. Under high-pressure
In order to probe the impact of hydrostatic pressure on the mechanical behavior of VIII-Ba8Si46,  , , P is the pressure in GPa.

C C 
that is equal to: 11 12 2 C C P   and which presents the precedent stability criterion under highpressure. When pressure increases, our calculations show that the values of 11 12 2 C C P   will not remain positive under all the pressure range, and the conditions are not fulfilled though, therefore, we predict that type-VIII Ba8Si46 will not be stable for pressures above 24 GPa.
Henceforward, the compound VIII-Ba8Si46 is expected to remain mechanically stable during and after synthesis below 24 GPa, however, we have to clarify that this criterion is not sufficient alone to predict any phase transition that can occur below 24 GPa, which is needed to be proven through experiments. Young's modulus E which represents the resistance of the material to uniaxial strains and provides the strength degree is found to rapidly increase from 0 to 10 GPa and then starts to slowly increase beyond that pressure, this indicated the incremented stiffness of the material towards pressure. The shear modulus G which describes the resistance towards shearing forces is less affected by pressure, but it tends to slightly increase and stabilizes after 10 GPa at about 56.50 GPa. The values of E and G both have similar behaviors towards pressure, even slowly, this means that, the capacity of the resist stiffness and shear deformation of type-VIII Ba8Si46 are increased with the increase of pressure respectively. However, the type-VIII Ba8Si46 is globally more resistive to volume compressions than to shear deformation since in all cases B > G for the pressure range.

IV. Conclusion
In this paper, first-principles calculations in the frame of the Density Functional theory DFT were solicited to investigate the high-pressure behavior of the structural, electronic and mechanical properties of the type-VIII Ba8Si46 silicon clathrate. The structural evolution shows a stable behavior with pressure with no significant indices of possible phase transition or volume collapse. Electronic properties are highly affected by the presence of pressure, the metallic character of the compound is not changed throughout the pressure range, however, the interaction between barium atoms and their surrounding Si environment becomes stronger with pressure, and as a consequence, the Fermi level decreases. Elastic properties are also affected by pressure, the major mechanical properties were computed for relaxed and stressed materials, and our results showed that the type-VIII Ba8Si46 is relatively hard and resistant against deformation  [30] e: data from ref [46] LDA (SIESTA) f: data from ref [47] Hartree-Fock HF (CRYSTAL 95) g: data from ref [36] h: data from ref [34] i: data from ref [48] and shear forces and might collapse if pressure exceeds 24 GPa, however, our predictions are to be confirmed through experiments.