Unlocking Static Polarization and Strain Density Waves in Perovskites by Softening a Hidden Antiferrodistortive Tilt Gradient Mode

Abstract

Spin density waves (SDWs) represent a fundamental paradigm of spatially modulated order in condensed matter systems, yet their electrical and mechanical analogues polarization and strain density waves (PDWs and StDWs) have remained elusive as equilibrium phases. Here, we introduce a general, symmetry-driven strategy to unlock static PDWs and StDWs in perovskites SrTiO3 and SrMnO3. Using first-principles calculations, we uncover a previously overlooked soft antiferrodistortive tilt gradient mode at small-q wavevector in the phonon dispersion of their presumed Ima2 ground state under moderate tensile strain. Group-theory analysis reveals that a hard polaracoustic phonon, which intrinsically carries PDWs and StDWs, is improperly destabilized by a trilinear coupling with this modulated tilt mode and an inherently uniform tilt mode. This interaction drives a structural transition from the Ima2 phase to a novel lower-energy Pmn21 phase that hosts long-range-ordered PDWs and StDWs. Strikingly, the engineered StDWs in SrMnO3 activate an electrically tunable SDW via the flexomagnetic effect. These discoveries fundamentally revise the strain-phase diagrams of prototypical perovskites and establish a unified phonon-engineering framework that links modulated phonon instabilities to targeted density-wave order, offering new pathways for designing advanced electromechanical and magnetoelectric functionalities.

Figures (5)

• Figure 1: Schematic of (a) transverse acoustic phonon with modulated atomic displacements, and (b) hybrid polar-acoustic phonon coupling shear strain to polarization propagating along the $x$ axis. The red circles, red balls, and olive arrows denote the $y$-axis displacement, strain, and polarization, respectively.
• Figure 2: Atomic theory of spontaneous PDW and StDW. The $x$, $y$, $z$ axes correspond to $[110]$, $[1\bar{1}0]$, and $[001]$ of the pseudocubic phase. (a) Phonon dispersion of $P4/mmm$ STO under 1.5% tensile strain; inset shows the corresponding atomic structure. Schematic representations of (b) in-plane polarization $P_x$, (c) out-of-phase tilt mode $R_{xy}$ (projected along $[1\bar{1}0]$, with rotation around $[110]$), and (d) $Ima2$ phase with $P_x$ and $R_{xy}$ distortions. (e) Phonon dispersion ($\Gamma$–X) of $Ima2$ STO under the same strain, revealing a hidden instability. (f) Modulated tilt mode $S_1$ and (g) hybrid polar-acoustic mode $\Sigma_2$ carrying PDW and StDW. (h) The $Pmn2_1$ ground state incorporating all four distortions ($P_x$, $R_{xy}$, $S_1$, $\Sigma_2$). (i) Schematic of biaxial tensile strain (brown arrows) inducing PDW and StDW in a freestanding membrane; the arrows in the circles represent the modulated total polarization vectors. The substrate-like layer is a conceptual reference to experimental setups using sacrificial layers, actual DFT calculations use periodic supercells without substrates. The green dashed line separates the conventional understanding (left) from our new findings (right) on the lowest-energy state of STO under moderate tensile strain.
• Figure 3: PESs of the (a) $S_{1}$ mode and the (b) $\Sigma_2$ mode without (orange curve) and with (blue curve) the fixing of $R_{xy}$ and $S_{1}$ modes. Here, 100% denotes the amplitude of distortions in STO under 1.5% tensile strain.
• Figure 4: Improper PDW and StDW in tensile-strained STO. The (a) out-of-plane Ti-O-Ti bond angle, (b) displacement field, (c) strain field, and (d) $P_y$ along the $x$ axis. These fields are extracted from the $10\sqrt{2} \times \sqrt{2} \times 2$ (blue curves) supercell and $20\sqrt{2} \times \sqrt{2} \times 2$ (red curves) supercell of $Pmn2_1$ STO under 1.5% tensile strain.
• Figure 5: Electrical and chemical control of density waves. (a) Atomic structure of the low-energy $Pmn2_1$ phase in $8\sqrt{2} \times \sqrt{2} \times 2$ SMO supercell under 3% tensile strain. (b) Corresponding global strain field and distribution of $P_y$ along the $x$ axis. (c) The spatial distribution of magnetization deviation along the $x$ axis under different in-plane electric field conditions. (d) Comparison of modulated strain field for Pb$_{0.84}$Sr$_{0.16}$TiO$_3$ grown on the DyScO$_3$ substrate without and with the involving of tilt distortions, highlighting the critical role of tilt distortions in suppressing higher-order harmonics and stabilizing single-$q$ dominant StDW.