The quantum paraelectric phase of SrTiO$_3$ from first principles

We demonstrate how the quantum paraelectric ground state of SrTiO$_3$ can be accessed via a microscopic $ab~initio$ approach based on density functional theory. At low temperature the quantum fluctuations are strong enough to stabilize the paraelectric phase even though a classical description would predict a ferroelectric phase. We find that accounting for quantum fluctuations of the lattice and for the strong coupling between the ferroelectric soft mode and lattice elongation is necessary to achieve quantitative agreement with experimental frequency of the ferroelectric soft mode. The temperature dependent properties in SrTiO$_3$ are also well captured by the present microscopic framework.

energy landscape along the coordinate of the so-called ferroelectric soft (FES) mode, such that at zero temperature the ground state is in a degenerate superposition of positively and negatively polarised states, which endows the material with an internal macroscopic polarization when the degeneracy is lifted by spontaneous symmetry breaking. While the FES mode of SrTiO 3 displays a similar characteristic softening, it stabilizes at low temperatures and no ferroelectricity is observed. This low temperature behaviour has been rationalized in terms of quantum fluctuations that prevent the formation of a macroscopic dipole [3].
This phenomenon of quantum paraelectricity and has been invoked to explain the behavior of other complex oxides [3,[23][24][25]. A number of models have been developed to describe the dielectric properties of this phase, in particular the Barrett and Vendik models have been widely used to understand quantum paraelectric behavior [3,23,[26][27][28][29]. The decisive role played by quantum fluctuations for the temperature dependent competition between the ferro-and paraelectric phases has been confirmed by quantum Monte Carlo calculations with an effective Hamiltonian for phenomenologically strained SrTiO 3 [30,31] as well as for other materials [24,25]. The quantum paraelectric phase is therefore now widely accepted as the explanation for the low temperature behaviour of SrTiO 3 [3,32,33], however the ground state of the quantum paraelectric phase and the frequency of the FES mode at low temperature have not yet been described by a microscopic theory.
The conventional first principles method to evaluate phonon frequencies of materials at zero temperature is density functional perturbation theory (DFPT) [34]. In DFPT calculations, which are based on a harmonic description of the lattice, an imaginary eigenvalue of the dynamical matrix indicates a phonon instability that can point towards a phase transition. However, DFPT does not include the quantum nuclear effects that are believed to stabilise the paraelectric phase in SrTiO 3 and therefore wrongly predicts an instability of the FES mode that suggests a phase transition to ferroelectrity [35][36][37]. This failure to describe quantum paraelectricity and its influence on the frequency of the FES mode, prevents the application of standard ab initio methods to give a microscopic explanation of recent experiments on quantum criticality and THz induced ferroelectricity in SrTiO 3 [6,7,[10][11][12].
In this letter, we unambiguously confirm the quantum nuclear nature of the ground state of SrTiO 3 based on DFT calculations and show that quantum fluctuations of the FES mode stabilise the paraelectric phase at low temperatures. To describe the quantum behavior of the lattice dynamics, we compute the potential energy surface obtained from DFT and construct a lattice-nuclear Schrödinger equation. We find that it is, indeed, not enough to only describe the FES mode as a quantum state in a 1D Schrödinger equation, but the nonlinear coupling to the lattice needs to be included in the quantum description. From this description we correctly reproduce frequency of the FES mode at zero temperature as well as the temperature dependence of the frequency and dielectric constant in the quantum pararelectric phase, which well agree with the experimental observations. We furthermore show that the crystal properties obtained with DFT strongly depend on the exchangecorrelation functional and that a correct description of both the lattice constants as well as the atomic positions are crucial to obtain the correct quantum paraelectric phonon energy.
Below 105 K, the crystal structure of SrTiO 3 forms a tetragonal unit cell with the oxygen octahedra rotated with respect to the cubic cell. This rotation counteracts the formation of ferroelectricity and is hence usually referred to as an anti-ferro-distortive (AFD) motion [35]. Therefore, the tetragonal geometry can be described as a √ 2 × √ 2 × 2 supercell of the primitive cubic perovskite ABO 3 unit cell with an additional AFD rotation, as depicted in Figs. 1(a) and 1(b). This AFD in-plane rotation is accompanied by an elongation and a contraction of the c and a lattice vectors relative to the cubic structure [35]. To investigate the optimized geometry and total energy, we perform DFT calculations using the Quantum Espresso package [41]. The projector augmented wave method is employed to describe core level atomic orbitals and a plane-wave basis set with 70 Ry energy cut-off is used. The Brillouin zone is sampled with 6 × 6 × 4 k-points. In Tab. 1, we summarize the lattice parameter a, c/a ratio and AFD rotation angle obtained with various DFT functionals. Comparing with experimental observation [38][39][40]42], local density approximation (LDA) [43] and Perdew-Berke-Ernzerhof revised for solid (PBEsol) [44] functionals provide a contracted lattice parameter a and a higher a/c ratio with an over-rotated AFD angle. Even though the Perdew-Berke-Ernzerhof (PBE) [45] functional and Heyd-Scuseria-Ernzerhof (HSE06) [46,47] hybrid functional describe elongated a and c lattices with overrotated AFD angle, these lattice parameters are closer to experimental observations than the former two functionals.  Fig. 1(f). The potential energy surface for the FES mode is evaluated by displacing the  [4,49].
Given the shallow double well found for the FES mode in SrTiO 3 , it is necessary to include quantum-nuclear effects, namely the zero-point motions of the atoms. We restrict our description of the lattice dynamics to the FES mode and the lattice motion in the cdirection, which we have established above to be intimately dependent. We sample the DFT total energy for 25 × 13 geometries along the FES mode parameterized by Q f and the lattice expansion parameterized by Q c . We then fit the potential energy surface; details on the f , whereP f and M f = 1.76 × 10 −25 kg are the momentum operator and the FES phonon mass, respectively and k f,i are the coefficients that parameterize the DFT potential energy surface. While for a ferroelectric one would expect a double degenerate ground state in the double well potential, the diagonalization of the 1DSE provides a non-degenerated ground (ψ 0 ) and 1st excited (ψ 1 ) state, which are depicted in To obtain the correct low temperature FES mode frequency, we find it is necessary to explicitly include its coupling to the lattice mode Q c in a two-dimensional (2D) latticenuclear Schrödinger equation (2DSE). The corresponding Hamiltonian given asĤ FES,c is built on the 2D potential energy surface, as shown in We then extend our microscopic approach to include the effect of finite temperatures.
Experimentally it has been shown that a flat temperature dependence of the FES mode is expected in the quantum paraelectric phase (T < 4K), [3,32] whereas for increasing temperatures a stiffening of the FES mode and a drop of the dielectric function are observed [6,32,51,52]. We first evaluate the temperature dependent FES frequency via ab initio molecular dynamics simulations with a thermostat [54]. As shown in Fig. 3(a) ). This indicates that PBE provides a realistic potential energy surface and that the effect of a slightly overestimated lattice is negligible [36]. To include temperature in our quantum lattice model we apply Kubo's formula for the linear response of a thermal state to a perturbationĤ (t) = −Z * Q f E(t), where Z * is the FES mode effective charge that we assume to be temperature independent. The resulting polarizability takes the form: where the dipole matrix and the thermal density matrix are defined as D ij = ψ i |Q f |ψ j and ρ i (T ) = e −( i − 0 )/k B T / j e −( j − 0 )/k B T , respectively. The temperature dependent frequency of the FES mode, is then evaluated by averaging over the polarizability as ω(T ) = ωIm[α(ω,T )]dώ Im[α(ω,T )]dω ; the results of this procedure with PBE are depicted in Fig. 3(a). Even though the 2D potential only includes two degrees of freedom (Q f and Q c ), the temperature dependence behavior is well reproduced and the typical flattening at low temperatures (< 10 K) is evident. We assigned the observed deviations of our model from the experiment to the effect of the phonon degrees of freedom that are not included in the model [6,51,52]. An attempt