**1. Introduction**

It is a long time since Kohn and Sham pave the way to the self-consistent equation, based on the exchange and correlation effects in 1965, leading the Kohn–Sham (KS) Equation [1]. This has ignited the success of quantum physics and chemistry, specifically many-body problem, owing to the KS equation can be utilized for the ground state energy. Briefly stated, the KS equation formalism of

density functional theory (DFT) described the motion of electron nuclei, which separated to be two part: the energy of electron *Eelectron* and the Coulomb interactions between the nuclei *Enuclei* And what is more, the details of Ewald summations have been described extensively in Refs. [2, 3]. Apart from this, *Eelectron* and *Enuclei* were performed by using the pseudopotential approximation within the KS equation. It is because of the effects of Coulomb interactions between the nuclei *Enuclei*, as being in accordance with the the core electrons *Ecore*, that the terms *Eelectron* and *Ecore* used in the static crystal energy of materials relevant to the energies of valence electrons and pseudocores. Subsequently, the KS equation displayed the term *Eelectron* is from the summation of quasiparticle eigenvalues, corresponding the Kohn–Sham orbital, of occupied states.

Regarding thermodynamic properties, the Gibbs free energy is considered for the static crystal energy of materials; however, the KS equation formalism of DFT carried out at a temperature of 0 K. The Gibbs free energy therefore reduced to the Enthalpy. This, appearing at first glance to be high potential for high-pressure physics, is actually demonstrated the importance of superconductivity. According to the aforementioned theoretical findings by the KS equation formalism of DFT, resulting the exchange and correlation effects *Exc*. Following this, Perdew et al. [4] presented a simple derivation of a simple of generalized gradient approximations (GGA) with *Exc*. This methodology appropriated, it is well known to GGA with Perdew-Burke-Ernzerhof (PBE), for description of atoms, molecules, and solids. This is due to the fact that the GGA-PBE method give an accurate with the most energetically important. As a result of this, the role of the GGA-PBE method is key factor in achieving the ground-state energy of the static crystal materials. Herein, we preformed mainly the PBE formalism of GGA for calculations of lithium, strontium, scandium, and arsenic under high pressure.

The extensive studies of electronic structure were initiated chiefly by the KS equation formalism of DFT. In principle, one should note the quasiparticle eigenvalues of occupied states is useful for achieving the electronic band structure, density of sates, phonon dispersion. It is also interesting to note the DFT used mainly strong sides for prediction the metallicity, leading to the prediction of superconducting transition temperature. For considered the superconductivity, the PBE formalism of GGA for exchange-correlation energy is suitable for interpret the metallicity. This implied that the reliable theoretical study has quite a predictive potential, moreover, the GGA-PBE for the exchange-correlation energy give an accurate description of dynamical stability of crystal structure. One of the wellknown Bardeen-Cooper-Schrieffer (BCS) theory [5] were already discussed phonon mediated superconductivity, leading to the way to vast both experimental and theoretical studies on high-pressure research. At this stage, using the KS equation formalism of DFT with the GGA-PBE for the exchange-correlation energy were used to have unique features of phonon mediated superconductivity, showing towards the evidence of superconducting materials as well.

There is alternative way to use the KS equation formalism of DFT with the GGA-PBE. It is well known to *ab initio* random structure searching (AIRSS). The AIRSS method have been described extensively in Refs. [6, 7]. Especially, the AIRSS method is useful in achieving the high-pressure research owing to it can predict novel structure under compressed conditions. The reliable theory for ground-state structure can help to interpret experimental data. In fact, there is also quite some experimental observation cannot identify the atomic position and crystal symmetry. The AIRSS method is powerful tool and it can guide further experimental studies.

High pressure physics is important for structural phase transitions in materials [8–18]. Regarding a crystal structure of materials under high pressure, it can enhance electronic properties of materials [19–21]. Nowadays, superconductivity is *Superconductivity in Materials under Extreme Conditions: An* ab-initio*… DOI: http://dx.doi.org/10.5772/intechopen.99481*

one of the most charming in physical properties. Many materials were predicted to be a superconducting transition temperature (*Tc*), such as SH3 [22–25], LaH10 [26–28], YH10 [28–30], CeH10 [17, 31]. It is worth note that hydrogen (H) is a role important for promoting a *Tc*. For example, the case of LaH10 was shown that the *Tc* reached 250 K at 170 GPa [26]. The existence of H displayed that it can support pure element lanthanum (La), one can see that the *Tc* of pure element lanthanum is 5.88 K [32]. It is interesting to note that the physical property of pure elemental metal should be mentioned.

As mentioned above, a structural prediction is a key factor for achieving a *Tc*. We referred the original predictions regarding superconductivity in strontium (Sr), it is beginning to show that the *Tc* of the predicted phase increased with a increasing pressure [33]. The case of strontium is interested. At high pressure, Sr. displayed structural phase transition from a simple structure to a complex structure [34–37]. We can see that Sr. is a normal metal at ambient pressure, with a increasing pressure, Sr. is a metallicity, indicating that it is a superconductivity [33]. Moreover, Sr. is not only superconducting phase at high pressure, but also calcium (Ca) indicating possible increasing of the *Tc* also at high pressure [38].

A curious aspect of a *Tc* increased with increasing pressure. We found that Ca is one of a periodic table, indicating the highest *Tc* among the periodic table [38–40]. Moreover, it is not always clear whether or not that the increased pressure and *Tc* are increased. We note that scandium (Sc) [14] and arsenic (As) [12], shown a possible decreasing of *Tc* with a increasing pressure [12, 14]. Hence, the focus is on pure elemental metals are interesting. This is because that the prediction discovered to novel structure [12, 14], leading to the superconductivity at high pressure.

According to the aforementioned superconductor findings, the characteristic of electronic structure is often attributed to the *Tc* [14, 17, 41]. It is interesting that Lithium (Li) has the second highest *Tc* among the elemental metals [42–45]. The electron localization function (ELF) is one of the tools can determine the *Tc* [14, 17]. The nature of chemical bonding is directly shown in the ELF, it is considered to be consistent with the highest *Tc*; this implies that a strong bonding supports the *Tc* of the metals [46].

Regarding superconductivity in the metals [12, 14, 17, 41], a lattice dynamic is a key factor for consideration a stable structure. In practice, we can achieve the superconducting structure through electron–phonon coupling (EPC) [12, 14, 17, 41]. For example, recent work on LaH10 has shown that the quantum effect is important for the stabilization and destabilization [27]. In fact, both thermodynamically and dynamically structures have to consistent. Generally, the solution of dynamically structure is a harmonic phonon but the case of LaH10 shown that it displayed an anharmonic phonon. This because the EPC exhibited the destabilized structure. Hence, it is worth to note that Sr. is possible to be an anharmonic phonon in the Sr-III structure (the *β*-tin structure) At this point, we found that there is a discrepancy between a experimental observation and a theoretical study [33– 37, 47]. Herein we review the superconductivity in the elemental metals both the experimental observation and the theoretical study under high pressure. In this review, we provide the success of the metals [12, 14] is BCS-type superconductor [12, 14, 17, 19, 41, 48–53]. Also, we hope that this review is useful for those interested readers in superconductivity in elemental metal under high pressure.

### **2. Methodology**

In considered in the present work, we performed the first-principles calculations, based on the density functional theory, to examine the thermodynamic

stability as a function of pressure. The static crystal energy of materials was considered at a temperature of 0 K. The calculation details of stable structure were determined by neglecting the entropy contributions. This is because the calculations were carried out at 0 K, indicating that the ground-state energy can confirm phase stability. Here, the KS equation formalism of DFT with the GGA-PBE for the exchange-correlation energy were used for Li, Sr., Sc, and As. For further details of the energy cutoff for plane waves and the Monkhorst–Pack k-point mesh as well as the DFT software have been described extensively in Refs. [10, 12–14]. Our works used the AIRSS technique, based on the density functional theory, to predict the novel structure. Following the AIRSS method, we calculated the enthalpies of the phases at any pressure using the simple linear approximation [7]. For each relaxed structure, the structures were simulated to be a non-symmetry and randomly placed in atomic position. During the calculations of the structures, it started to relax from bias until it reaches unbias. The shape is generating by shaking within a reasonable pressure range. It led to higher-symmetry space groups obtained in a search. The AIRSS technique is the approach in the local minima by giving the lowest enthalpy. We have studied the phonon mediated superconductivity by using isotropic Eliashberg theory, as implemented in the quantum espresso (QE) [54, 55]. Following the result of isotropic Eliashberg theory, the Allen-Dynes modify McMillan Equation [56] was used to estimated the superconducting transition temperature.
