Finite-Temperature $\textit{ab initio}$ Structural Optimization of the Bilayer Nickelate Superconductor La$_3$Ni$_2$O$_7$

Abstract

We develop a first-principles framework for finite-temperature structural optimization that incorporates vibrational contributions to the free energy through anharmonic phonon theory. We extend and further improve the efficiency of the recent approach, enabling its application to systems in which the size of the primitive cell changes across structural phase transitions. Applying this framework to La$_3$Ni$_2$O$_7$, we establish its pressure-temperature phase diagram and find that the slope of the phase boundary between the high-symmetry and low-symmetry phases is negative, with a magnitude of approximately -60 K / GPa. The present results provide a theoretical foundation for discussing how changes in crystal symmetry influence the emergence of superconductivity.

Figures (5)

• Figure 1: Schematic figures representing two types of structural phase transitions: (a) BaTiO$_3$, whose primitive cell size remains unchanged before and after the structural phase transition, and (b) CaTiO$_3$, whose primitive cell size changes. The crystal structures were visualized using VESTAMomma:db5098.
• Figure 2: Iterative convergence behavior of two optimization schemes at 0GPa and 380K, showing the norm of atomic displacements versus iteration number.
• Figure 3: Phonon dispersion of the conventional cell of La$_3$Ni$_2$O$_7$ under different pressure conditions: (a) 0GPa and (b) 13GPa. The two lowest-energy modes at the M point are highlighted in orange. (c) Potential energy surface (PES) along the M-point modes at each pressure. As the pressure increases, the double-well potential becomes shallower and approaches a single-well shape. (d) Coefficients $a_2$ obtained by fitting the PES at each pressure with the quartic function $F(Q) = a_2 Q^2 + b_4 Q^4$. We performed linear fit of $a_2(p)$, and the intercept on the pressure axis (red triangle) was taken as the estimated critical pressure $p_s$.
• Figure 4: (a) Temperature dependence of atomic displacements obtained from finite-temperature structural optimization starting from the relaxed $I4/mmm$ structure. The onset of finite displacements indicates the symmetry change to $Amam$. (b) Temperature dependence of the SCP free energy. The free-energy difference $\Delta F_{\mathrm{SCP}}$ between the $Amam$ (blue) and $I4/mmm$ (red) phases indicates that the former becomes thermodynamically stable when $\Delta F_{\mathrm{SCP}}<0$. (c) Phonon band dispersions at $T=410$ and 420K, showing the splitting near the M point associated with the transition.
• Figure 5: Phase diagram of La$3$Ni$2$O$7$ obtained from self-consistent phonon (SCP) calculations. Blue points denote the transition temperatures $T\mathrm{s}$ calculated under fixed pressures from 0 to 8GPa. The blue dashed line represents a linear fit obtained using bootstrap resampling, and the shaded region indicates the corresponding confidence interval. The intercept of this fit with the $T=$ 0K axis provides the transition pressure $p\mathrm{s}$ estimated from SCP calculations. The red triangle indicates the transition pressure $p\mathrm{s}$ determined from potential energy surface. Green points show the $T_\mathrm{s}$ values corrected by the vibrational pressure $P_\mathrm{vib}(T_\mathrm{s})$, which accounts for the effect of finite-temperature lattice vibrations. The green dashed line corresponds to a bootstrap-based linear fit to the $p_\mathrm{vib}$-corrected data, and the shaded band gives its confidence range. The intercept of this line with the $p=$ 0GPa axis yields the corrected transition temperature $T_\mathrm{s}$ at 0GPa.