Dynamical Stability of Face Centered Cubic Lithium at 25 GPa

Miguel Borinaga1,2, Unai Aseginolaza1,2, Ion Errea2,3 and Aitor Bergara1,2,4

1Centro de Física de Materiales CFM, CSIC-UPV/EHU, Paseo Manuel de Lardizabal 5,

20018 Donostia/San Sebastián,Basque Country, Spain 2Donostia International Physics Center (DIPC), Manuel Lardizabal pasealekua 4, 20018

Donostia/San Sebastián, Basque Country, Spain 3Fisika Aplikatua 1 Saila, Bilboko Ingeniaritza Eskola, University of the Basque Country

(UPV/EHU), Rafael Moreno “Pitxitxi” Pasealekua 3,

48013 Bilbao, Basque Country, Spain 4Departamento de Física de la Materia Condensada, University of the Basque Country

(UPV/EHU), 48080 Bilbao, Basque Country, Spain

(Received November 11, 2016)

We study the dynamical stability of face centered cubic lithium at 25 GPa within ab initio density

functional theory calculations. The system shows an extremely softened transverse acoustic mode

in the Γ-K high symmetry line, whose frequency is difficult to converge within a first-principles

harmonic approach. We estimate the anharmonic correction of this value within the frozen phonon

approximation assuming that the transverse acoustic mode only interacts with itself. By solving the

Schrödinger equation for the potential energy curve and displacements along the vibrational mode

of interest, we obtain the anharmonic eigenvalues and eigenfunctions. While the harmonic

approach yields a phonon frequency of -28.6 cm-1 for this mode, anharmonicity renormalizes

dramatically this value up to 115.3 cm-1. Thus, anharmonic effects seem to dynamically stabilize

this system.

1. Introduction

Lithium, the lightest metal on the periodic table, is a nearly-free-electron material which adopts a

bcc structure at ambient conditions [1]. Although it could be expected to evolve to an even more

free-electron like system with increasing pressure, it has been shown that pressure not only induces

several structural transformations [2,3,4], but also gives rise to a plethora of fascinating physical

properties [5] normally induced by nonlocality in the core pseudopotential with pressure[6,7]. For

instance, lithium becomes a semiconductor near 80 GPa [8], it melts below ambient temperature

(190 K) at around 50 GPa [2] and displays a periodic undamped plasmon according to theoretical

calculations [9]. Moreover, it presents one of the highest superconducting critical temperatures (Tc)

for an element, reaching values as high as 15 K at around 30 GPa [10-13]. Recently, the

measurement of an inverse isotope effect in the 20-26 GPa range [10] has put this element back

under the spotlight.

Experimental evidence [2,3,4,8] shows that Tc rapidly soars in the face centered cubic (fcc) phase,

which is adopted by lithium roughly between 7 and 40 GPa, pressure after which it transforms to

the rhombohedral hR1 phase. This phase is really similar to fcc as it consists of just a distortion

along the c axis if one switches to a hexagonal representation. The transformation to the cubic cI16

phase occurs shortly after, at around 43 GPa.

JJAP Conf. Proc. , 011103 (2017) https://doi.org/10.7567/JJAPCP.6.0111036Proc. 17th Int. Conf. on High Pressure in Semiconductor Physics & Workshop on High-pressure Study on Superconducting

© 2017 The Author(s). Content from this work may be used under the terms of the .Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

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Theoretical calculations within the harmonic approximation in fcc lithium show a highly softened

transverse acoustic branch in the Γ-K high-symmetry line [12,14,15,16,17]. At qinst=2π/a(2/3,2/3,0),

this anomalous mode presents a huge electron-phonon coupling, becoming a key factor to explain

the high Tc observed in lithium [12,14,16]. This softening is associated to a well-defined Fermi

surface nesting [14,17] and even yields imaginary phonon frequencies at pressures where fcc is

known to be stable; the instability emerges at pressures higher than 30 GPa in the local density

approximation, and at even lower pressures if one uses the generalized gradient approximation. As

seen in other systems, such as simple cubic calcium [18], anharmonicity is expected to have a

significant role stabilizing this structure and, due to phonon frequency renormalization, also

determining its superconducting properties.

In this work we analyze the mentioned anomalous vibrational mode in terms of ab-initio total

energy calculations. We study the convergence of the harmonic phonon frequency of the unstable

mode and we estimate the impact of anharmonicity assuming it does not interact with the rest of the

modes in the crystal, by solving the time independent Schrödinger equation for the potential

associated to the atomic displacements along the mode of interest.

2. Results and discussion

2.1 Computational details

Our density functional theory (DFT) calculations were done within the Perdew-Burke-Ernzerhof

parametrization of the generalized-gradient approximation [19]. The electron-ion interaction was

considered making use of an ultrasoft pseudopotential[20] as implemented in Quantum ESPRESSO

[21]. An energy cutoff of 885 eV was necessary for expanding the wave-functions in the

plane-wave basis. Converging the total electronic energy within 0.1 meV. required a 30x30x30

k-point mesh and a 0.136 eV (0.01 Ry) Hermite-Gaussian electronic smearing for electronic

integrations in the first Brillouin zone (BZ). Harmonic phonon frequencies were also calculated

within density functional perturbation theory (DFPT) [22,23].

2.2 Numerical convergence of harmonic phonons

In Fig. 1 we show the DFPT harmonic phonon spectrum of fcc lithium at 25 GPa. The dynamical

Fig. 1. DFPT harmonic phonon spectrum of fcc Li at 25 GPa.

011103-2JJAP Conf. Proc. , 011103 (2017) 6

matrices have been explicitly calculated in a 9x9x9 q-point grid and Fourier interpolated afterwards

to the chosen path in the BZ. The transverse acoustic branch T1 is clearly softened in the Γ-K path,

displaying even imaginary (negative) frequencies at qinst. This point is part of the 9x9x9 grid in

which an explicit DFPT dynamical matrix calculation was performed, which discards it being an

artifact of the Fourier interpolation.

In order to better understand the nature of this anomaly, we obtained its associated potential energy

profile by calculating the total electronic energy for different displacements following the

polarization vector of the T1 mode at qinst. In Fig. 2 we plot the total energy along the path

described by the anomalous mode for different choices of k-point grids and smearing widths for

electronic integrations. While for displacements larger than 0.05 Å the curves do not vary much

among the different calculations, for small displacements the chosen convergence criteria of 0.1

meV is clearly not enough for describing the profile properly. As one increases the BZ sampling

and decreases the smearing width, the double-well shape of the energy surface sharpens. As the

harmonic phonon frequency is obtained by calculating the second derivative at the equilibrium

position, we expect the convergence of this value to be harsh. This is confirmed by DFPT

calculations. In Fig.3 we can see that, while the frequency of the L longitudinal and T2 transverse

modes converge within 1 cm⁻¹, the frequency of the T1 transverse anomalous mode decreases

monotonically as smearing values are reduced for the densest samplings of the BZ. A proper

convergence would require an even denser grid and a smaller smearing width. However, such small

energies and displacements are physically meaningless considering the zero point fluctuations.

Thus, we hold to the choice of a 30x30x30 k-point mesh and a 0.01 Ry smearing width.

We estimated the harmonic frequency of the T1 mode at qinst by calculating the second derivative at

the equilibrium position numerically. By fitting a parabola to the three data-points closest to the

Fig. 2. Electronic energy vs. displacement along the T1 mode at qinst for different choices of

k-point grids and smearing width values. In red (30x30x30) and blue (39x39x39) curves the

same smearing width values have been chosen as in the black ones (21x21x21). The inset

shows a zoomed view of the small displacements region. The origin of energy is set at the zero

displacement energy for a 30x30x30 grid and 0.01 Ry smearing width. b) Convergence analysis

of the DFPT harmonic frequency of the three normal modes at qinst.

011103-3JJAP Conf. Proc. , 011103 (2017) 6

equilibrium position (see Fig.4), we have estimated a phonon frequency of -28.6 cm-1. The value

obtained within DFPT for the same calculation parameters is -22.1 cm-1, in quite good agreement

with the previous frozen phonon result.

2.3 Solving the Schrödinger equation

Due to the highly anharmonic shape of the energy profile of the mode under analysis, we solved the

one-dimensional time-independent Schrödinger equation for the potential associated to the

vibrational mode. This way, we are neglecting the interaction of the analyzed mode with the rest of

the normal modes of the lattice. We fitted a 10th degree polynomial to the energy vs. displacement

data points and solved the eigenvalue problem using ParametricNDSolve in Mathematica[24]. We

assumed that for displacements larger than 0.6 Å the eigenfunctions vanish, discarding tunneling to

adjacent positions.

In Fig. 4 we show the data points, the fitted polynomial and the obtained three lowest energy

eigenfunctions and eigenvalues, with values 6.3, 20.6 and 36.3 meV respectively. The possible

transitions between states yield phonon frequencies of 14.3, 15.7 and 30.0 meV (115.3, 121.8 and

241 cm-1) respectively. If we compare the value of the most probable transition, 115.3 cm-1, which

is the transition between the ground state and the first excited state, with the harmonic value of

-28.6 cm-1, we see a huge difference. First, the magnitude of the anharmonic correction is even

larger than the harmonic value itself. Second, the shape and width of the wavefunctions and the

magnitude of the eigenvalues confirms not only the lack of physical relevance of the convergence

problems observed when computing the second derivative, but also the absolute failure of the

harmonic approach for describing this vibrational mode. Finally, anharmonicity changes the sign of

the frequency, and, thus, guarantees the dynamical stability of the system within this frozen-phonon

approach.

3. Conclusions

We estimated the anharmonic correction of the anomalous transverse phonon at 2/3 ΓK in fcc

lithium at 25 GPa within the frozen-phonon approximation by solving the Schrödinger equation.

Fig. 3. Convergence analysis of the DFPT harmonic frequency of the three modes at qinst.

011103-4JJAP Conf. Proc. , 011103 (2017) 6

While within the harmonic approach one obtains a frequency of -28.6 cm-1, anharmonicity

renormalizes dramatically this value up to 115.3 cm⁻¹. Thus, it seems that anharmonicity is

dynamically stabilizing this system. Plus, this strong phonon renormalization could potentially

cause the inverse isotope effect [10] by making the electron-phonon coupling constant differ for the

two stable lithium isotopes considering that in PdH anharmonicity explains the inverse isotope

effect [25]. Anyway, our frozen-phonon calculations neglect the interaction of the analyzed

vibrational mode with the rest of the modes in the crystal. Therefore, a proper quantitative analysis

would require of a method taking into account the phonon-phonon interaction also among different

modes.

Acknowledgment

The authors acknowledge financial support from the Spanish Ministry of Economy and

Competitiveness (FIS2013-48286-C2-2-P) and the Department of Education, Universities and

Research of the Basque Government and the University of the Basque Country (IT756-13). MB is

also thankful to the Department of Education, Language Policy and Culture of the Basque

Government for a predoctoral fellowship (Grant No. PRE-2015-2-0269). Computer facilities were

provided by the DIPC.

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