Role of relativity in high-pressure phase transitions of thallium

We demonstrate the relativistic effects in high-pressure phase transitions of heavy element thallium. The known first phase transition from h.c.p. to f.c.c. is initially investigated by various relativistic levels and exchange-correlation functionals as implemented in FPLO method, as well as scalar relativistic scheme within PAW formalism. The electronic structure calculations are interpreted from the perspective of energetic stability and electronic density of states. The full relativistic scheme (FR) within L(S)DA performs to be the scheme that resembles mostly with experimental results with a transition pressure of 3 GPa. The s-p hybridization and the valence-core overlapping of 6s and 5d states are the primary reasons behind the f.c.c. phase occurrence. A recent proposed phase, i.e., a body-centered tetragonal (b.c.t.) phase, is confirmed with a small distortion from the f.c.c. phase. We have also predicted a reversible b.c.t. → f.c.c. phase transition at 800 GPa. This finding has been suggested that almost all the III-A elements (Ga, In and Tl) exhibit the b.c.t. → f.c.c. phase transition at extremely high pressure.

Scientific RepoRts | 7:42983 | DOI: 10.1038/srep42983 upon the structural phase transition from the h.c.p to f.c.c. But this dependence can not be seen in case of NR calculation within both L(S)DA and GGA schemes. This is because of the fact that the enthalpy of the f.c.c. phase is lower than the h.c.p. phase in that range of pressures. In other cases that consider the SR and FR effects, the h.c.p. → f.c.c. phase transition can be revealed. The result using FPLO with L(S)DA and FR shows reasonable agreement with the experimental outcomes 1-4 with a transition pressure of 3 GPa, whereas, the other methods yield the overestimating transition pressures. Interestingly, the transition pressure using GGA and FR within FPLO method is higher than that from the L(S)DA and the PAW-GGA reveals the transition pressure as three times higher than the experimental value. This issue will be discussed later. The fitting parameters and transition pressures of both the phases calculated with other methods are shown in Table 1. For fitting parameters, the results agree following the conventional trend that GGA overestimates the volume and underestimates the bulk modulus, but they are different while using L(S)DA scheme. It is found that the fitting parameters using FPLO with L(S)DA and FR also mostly agree with experiments as well. Therefore, this scheme will be reliable and appropriate option to perform the structural and mechanical properties of thallium.
The electronic band structures and density of states (DOSs) of the h.c.p. and f.c.c. phase of Tl at selected pressures as depicted in Fig. 2 show the influence of relativistic effects in particular L(S)DA method. For the h.c.p. phase at 0 GPa (top row), the 6p states are the majority across the Fermi level. The DOS shapes for the FR and SR consideration are different in term of smoothness as it is less smooth between 1.0 and 2.0 eV energy range in the former case. This difference also reflects in the energy dispersion of zone-boundary at M-and K-points, where crossing of dispersion lines in the SR is observed, while a small splitting right at H-point in the FR is seen. The DOS and energy dispersions in case of the SR and FR are similar below the Fermi level between − 4.0 and − 10.0 eV, dominantly occupied by 6s states. There is a separation between 6p and 6s states that is observed around 3 eV below the Fermi level. It leads to the conclusion that there is no s-p hybridization, but only bonding of the p states exist that forms the h.c.p. phase at ambient condition. In contrast to this separation in case of the SR and FR, both 6p and 6s states play the important role of bonding electrons in the NR case. This leads to the disagreement with experiment regarding the phase transformation of the h.c.p. → f.c.c. in cases of NR calculations. In  the lower energy range between − 15 and − 10 eV, an apparent difference in relativistic levels is contributed by 5d states, which originates together at around − 11.8 eV and − 14.0 eV in the SR and NR, respectively. But in the FR case, there is a splitting of 5d states into 5d 5/2 and 5d 3/2 states as allocated at − 10.24 eV and − 12.55 eV. This splitting is resulted due to the spin-orbit (SO) coupling in this case, and is in good agreement with the previous experiments 12, 13 . Furthermore, an overlap of 6s and 5d 5/2 at Γ -point is observed in case of FR. The evidences of the h.c.p. → f.c.c. transformation are revealed at 3 GPa, as shown in second and third row of Fig. 2 for h.c.p. and f.c.c., respectively. For the first one, a mixing state between 6s and 6p appears in the h.c.p. and f.c.c. phase at this pressure for both SR and FR. In contrast, the second evidence of a valence-core overlap of 6s and 5d 5/2 state appears in the FR case only. The transition pressure for the h.c.p. → f.c.c. phase transition of the FR calculation has been determined as 3 GPa. For the SR calculation, it has been found that the valence-core overlap arises at 3 GPa for the h.c.p. phase and increases with increasing pressure. The bottom row of Fig. 2 displays the band structures and DOS of the f.c.c. phase at 8 GPa (the transition pressure in the SR case) and the overlap of 6s and 5d states is also appearing in the SR case. In the NR case, the valence-core overlap is unobserved up to 8 GPa as the energy tail of 6s states being still far from the 5d states.
The calculation results using both FPLO and PAW approach with GGA functional reveal the similar tendency in accord with the previous discussion. But, due to the overestimation of volume and as a consequence to the change in the electronic band structure, the transition pressure is also overestimated using GGA. Moreover, it is found that the use of GGA and FR within FPLO obtains the compatible transition pressure with previous reports using GGA and including SO coupling within PAW method 3 .
The bonding in case of the h.c.p. phase originates from 6p states and the valence-core overlapping of 6s and 5d 5/2 states is appeared. The external pressure induces s-p mixing of valence state and partial-valence state of 5d electrons leading to the h.c.p. → f.c.c. phase transformation. This finding corresponds to forming the f.c.c. phase in III-A metal group, consisting of Ga and In as reported in previous investigations 11,14,15 . It should be emphasized here that the electronic structure of Tl strongly depends on the level of relativistic application.
We have done further calculations to confirm the existence of the b.c.t phase. at high pressure reported by Kotmool et al. Firstly, in order to search the other possible distorted f.c.c. phases, primarily a body-centered orthorhombic (b.c.o.) phase, which has been found with more structural distortion than the b.c.t. phase, is considered. The calculations while varying b/a and c/a in range of 1.0 to 1.8 at 100 GPa and 200 GPa are reported as depicted in Fig. 3. The both contours have a diagonal symmetry. The lowest enthalpy zone is represented by the black area of contours as lay on the middle of the axes. The shape of the black zones along both the axes confirm the probability of existence of the b.c.t. phase, rather than the lower symmetry b.c.o. phase. Secondly, we also predict the high-pressure phases of Tl using USPEX code from 300 GPa to 1 TPa. At 300-600 GPa, the b.c.t. phase still prevails as the most stable phase with lowest enthalpy comparing to the f.c.c., b.c.o. and monoclinic (i.e., C2/m) phases. But the f.c.c. phase emerges to be the most stable once again at 800 and 1000 GPa. The relationships between relative energy and c/a at fixed volumes are calculated using FPLO with L(s)DA and FR calculation as shown in Fig. 3. The results confirm the previous reports 1-4 that the f.c.c. phase (20 GPa, Fig. 4(a)) is induced by increasing pressure to be the b.c.t. phase (Fig. 4(b-f)) with a small distortion of |( 2*a-c)/c| < 1%. Intriguingly, at 800 GPa (Fig. 4(g)), the lowest energy is corresponding to the f.c.c. phase with c/a = 2 which is in agreement with the prediction. These results indicate that the perfect f.c.c. phase can be recovered by pressure of 800 GPa. This finding is quite important because it is a fulfillment suggesting that almost the III-A elements (Ga, In and Tl) exhibit the b.c.t. → f.c.c. phase transition under high pressure.

Conclusion
In this work, we have thoroughly investigated the influence of the relativistic effects on the high-pressure phase transition of Tl. The transformation of the h.c.p. to f.c.c. phase is carried out by using various relativistic levels and exchange correlation functionals. It has been found that the relativistic effect strongly controls the transition pressure and structural parameters under high pressure. The transition pressure has been found in range of 3 to 16 GPa with the scalar and fully relativistic (SR and FR) schemes, which is in reasonable agreement with experimental result, specially with L(S)DA coupled with FR. Moreover, the SO coupling has been found to have inevitable role behind the h.c.p. → f.c.c. phase transformation. The previously reported b.c.t. phase has also been confirmed in this work. The search for high-pressure phase up to 1 TPa envisages the re-occurrence of the f.c.c. phase, that has been recovered at 800 GPa.

Methods
We have performed all the electronic structure calculations based on density functional theory (DFT) formalism. In addition to the local spin-density approximation (L(S)DA-PW92) 16 and generalized gradient approximation (GGA-PBE) 6 , the non-relativistic (NR), scalar relativistic (SR) and full relativistic (FR) schemes have also been considered. We have employed full-potential with local-orbital minimum-basis set implemented FPLO package 7,8 and projector augmented wave (PAW) 5 implemented Vienna Ab initio Simulation Package (VASP) 17,18 for performing all the calculations. By using the FPLO method, the converged k-points were set to be 19 × 19 × 12, 16 × 16 × 16 and 19 × 19 × 16 for the h.c.p., f.c.c. and b.c.t. phase, respectively. For using VASP calculation, energy cutoff of 500 eV is used and the automatic k-point mesh with setting spacing between k-points of 0.2 Å −1 is considered (corresponding 10 × 10 × 6 and 11 × 11 × 11 for the h.c.p. and f.c.c. phase, respectively). All the setup values of VASP and FPLO methods are verified for all the calculations to ensure the energy convergence of 1 meV/ atom. The third-order Birch-Murnaghan equation of state has been used to fit the energy change with volume 19 .
The structural searching method based on the evolutionary algorithm, which has been implemented in USPEX code 20,21 interfacing with VASP code, has been used to extend the previous prediction up to 1 TPa (i.e., 300, 400, 600, 800 and 1000 GPa). The searches are performed by varying cell size up to 8 atoms per cell. During the prediction, number of initial randomized structures in the 1st generation is set as 40 structures. The next is carried out 30 population structures by heredity and mutations of the lowest enthalpy structure of the previous generation by 50% and 30% respectively, and remaining 20% of populations are obtained by randomization. The prediction would be done when a lowest enthalpy structure would be continuously survival within 30 generations.