Ab Initio Study Of The Mechanisms For The Pressure -Induced Phase Transitions In Alkali Metals

The alkali metals have usually been considered as simple metals due to their monovalency, high conductivity and crystallization in high-symmetric structures. Under normal conditions the interaction between valence electrons and ionic cores is week and the simple nearly free electron (NFE) model has been considered a good approximation to analyze their properties, which can be easily checked by their almost spherical Fermi surface. However, under high pressure, the NFE description of alkali metals fails as evidenced by the complex sequence of phase transitions. Alkali metals having the body-centered cubic (bcc) structure at normal pressure and temperature transform to a face-centered cubic (fcc) structure for Li at 7.5 GPa, Na at 65 GPa, K at 11.5 GPa, Rb at 7 GPa, and Cs at 2.3 GPa, respectively. At higher pressure all alkali metals with fcc structure transform into more complex structures. Specifically, Li at 39 GPa, Na at 103 GPa, K at 19 GPa, Rb at 13 GPa, and Cs at 4.2 GPa transform to R-3m, I-43d, host-guest composite structure, C2221, and C2221 structures, respectively.There are many experimental and theoretical investigations related to phase transition of alkali metals, but the physical mechanism driving the phase transition is less studied. Electronic (Fermi surface nesting, FSN, charge transform), dynamical, and elastic constants instabilities are often responsible for phase transitions under pressure. In a previous study on Cs, the high-pressure phase transition of Cs from bcc to fcc and from fcc to C2221 are related to the phonon instabilities of the transverse acoustic modes (TA) near the first Brillouin zone (BZ) center. Recently, Rodriguez-Prieto and Bergara suggested that the bcc to fcc phase transition in lithium can be explained via a Hume-Rothery mechanism; the FSN at 30GPa and the TA phonon softening alongΓ? K direction could be the origin of the complex phase transition for fcc lithium. In this work, ab initio investigations of electronic properties, lattice dynamics, and elastic modulus are carried out to probe the nature of the pressure induced phase transformation of bcc and fcc alkali metals.The theoretical equilibrium lattice constants and equation of states (EOS) of the bcc alkali metals were determined by fitting the total energies as a function of volume to the 3rd-order Birch-Murnahan EOS. The calculated equilibrium lattice constants and EOS are in excellent agreement with experimental data. We investigate systemically the evolution of electronic properties, lattice dynamics, and elastic modulus with pressure in bcc alkali metals. We found that increasing pressure lead to a strong anisotropic deformation of Li, K, Rb, and Cs's Fermi surface (FS) and the FS touches the BZ boundary at 7 GPa, 10 GPa, 5 GPa, and 1 GPa , respectively, while the FS doesn't change in Na. From the analysis of the partial density of states (DOS) at Fermi level, it is shown that in Li, the p-DOS contribution increases steadily with increasing pressure at the expense of decreasing s contribution. The increased p contribution eventually led to the deformation of the FS. It is observed in Na, the p contribution dominates the total DOS at 0 GPa and is almost constant up to 120 GPa, while the d contribution increases slightly but remains small. This might be the reason for the persistence of the almost spherical Fs in Na at high pressure at 120 GPa even up to 240 GPa. For the heavier K, Rb, and Cs, we indeed observed the electronic s→d transition and at the pressure close to the FS touches the BZ boundary, the d contribution increases significantly and dominates the total DOS. This might be the reason for the deformation of FS in heavier alkali metals. From analysis the evolution of phonon dispersive curve and elastic constant with increasing pressure, we found that the TA phonons become unstable around the BZ center along theΓ? N direction at 4 GPa for Li, 80 GPa for Na, 15 GPa for K, 8 GPa for Rb, and 2.5 GPa for Cs, respectively. The elastic constant C′become negative for Li at 4GPa, Na at 100 GPa, K at 15 GPa, Rb at 8 GPa, and Cs at 2.3 GPa. It is quite clearly that at high pressure all five bcc structure alkali metals become dynamical unstable that may induce structural instability. The negative tetragonal shear elastic constant implies an instability towards the fcc structure through the tetragonal Bain's path. We studied the total energy E as a function of c /a along the tetragonal Bain's path at several pressures. At zero pressure, it is obviously that there are two minimum located at c / a = 1 and c / a = 1.4 which corresponding to the bcc and fcc structure, while the fcc structure is energetically favorable at T = 0, the large phonon entropy associated with the low-energy mode in alkali metals stabilizes the bcc structure upon heating. With increasing pressure, the minimum at c / a = 1 shifts to smaller values around c / a = 0.9 and local maximum appears at c / a = 1 for Li at 4 GPa, Na at 110 GPa, K at 14 GPa, Rb at 8GPa, and Cs at 2.5 GPa, while the c / a = 1.4 become the global minimum of the E( c /a ) curves. The tetragonal distortions at that pressure result in a lower energy phase, the fcc structure. The dynamical and elastic instabilities might be the reason of phase transition for heavier bcc alkali metals. The softening TA [110] branch with polarization along [110] together with an elastic strain associated with C′, gives a possible path for the bcc to fcc transformations.The calculated equilibrium lattice constants and EOS of fcc alkali metals are also in good agreement with experimental data. We found that the fcc alkali metals have the similar pressure-induced charge transformation with the bcc alkali metals. This behaviors lead to the FS change of Li, K, Rb, and Cs and the Fermi surface nesting (FSN) feature alongΓ? K direction have been identified. At 30 GPa the (110) plane shows a FSN feature along theΓ? K direction with a nesting vector of 0.7 (110) for Li. The FSN suggests the possibility of a structural instability which might be related to structural phase transition of Li. There are two different FSN features in K, Rb, and Cs. The first FSN feature appears at 8 GPa, 3 GPa, and 0 GPa in (110) plane along theΓ? K direction with a nesting vector of 0.7 (110) for K, Rb, and Cs, while disappear at high pressure. The electronic instability due to this FSN might be the reason of the reversible phase transition of fcc to bcc under uncompressing for the heavier fcc alkali metals. The second FSN feature appears at (100) plane along theΓ? K direction with a nesting vector of 0.58 (110) for K at 18 GPa, Rb at 13 GPa, and Cs at 4 GPa, respectively. The pressures of the appearance of the second FSN feature coincide with the observed structural phase transition pressures for heavier alkali metals. It is observed that the TA phonon become unstable under high pressure but the elastic modulus remain stable for all five fcc alkali metals. Imaginary phonon modes for Li, Na, K, Rb, and Cs starts to appear at 33 GPa, 103 GPa, 28 GPa, 17 GPa, and 4 GPa, respectively. The predicted pressures correlated nicely with the experimental measured structural transition pressures of 39 GPa, 103 GPa, 19 GPa, 13 GPa, and 4.2 GPa in fcc Li, Na, K, Rb, and Cs, respectively. It is found that FSN and soft phonons can coexist at pressure close to the observed structural transition pressure. In principle, both electronic (FSN) and phonon softening can induce a structural instability, thus resulting in a phase transition. It is nontrivial to distinguish the dominating factor driving the phase transition. But several important aspects of the transition must be considered. For fcc Li, Na, and K, although the TA phonon frequencies along [110] soften and become imaginary frequency at similar incommensurate softening vectors of ~ (0.7, 0.7, 0) or (0.6, 0.6, 0) near the K symmetry point with increasing pressure but the evolution of the FS and the appearance of the FSN behavior were very different. Specifically, a FSN in the (110) plane alongΓ? K direction with a nesting vector of 0.7 (110) was observed in Li. No FSN feature that might lead to structural instability was found in Na up to 240 GPa. In K the FSN feature appears in the (100) plane alongΓ? K direction with an incommensurate nesting vector of 0.58 (110). These differences were found resulting in different high-pressure structures. The nature of the pressure induced phonon softening predicted for Rb and Cs is quite different from that for Li, Na, and K. For Rb and Cs, the soft phonon was found to occur near the BZ center with an incommensurate softening vector of (0.3, 0.3, 0). They have the similar phonon softening and FSN behavior and also exhibit a similar phase transition sequence and closely related high-pressure structures. For both metals, the succeeding high-pressure phases have a complex and modulated structure. Analysis of the calculated results suggested that it is crucial to understand the pressure-induced phase transitions in alkali metals from fcc to lowly coordinated structures by combining the FSN and the phonon softening.The pressure-induced phase transition mechanism has been studied using ab initio method within the density functional theory. We found that the electronic, dynamical, and elastic instabilities are crucial to understand the pressure-induced phase transition in alkali metals. Our results are helpful to understand the pressure-induced phase transition in elements and the structural transition from high-symmetric structure to complex and low-coordinate structure.