Strain-tuned properties of hybrid improper ferroelectric superlattices through first-principles calculations and machine learning

The hybrid improper ferroelectricity is due to the trilinear coupling between three symmetric phonon modes, and the free energy of the system can be described as ${F}_{\mathrm{tri}}\ensuremath{\sim} \ensuremath{\alpha}\phantom{\rule{4pt}{0ex}}{Q}₁{Q}₂{Q}_P$, where $\ensuremath{\alpha}$ is the coupling constant and ${Q}₁$ and ${Q}₂$ are the primary order parameters coupled to the polar mode ${Q}_P$. In the case of $Pnma$ symmetry, the ${Q}₁$ and ${Q}₂$ modes are rotation, ${Q}_{\mathrm{Rot}}({a}⁰{a}⁰{c}⁺$), and tilt, ${Q}_{\mathrm{Tilt}}({a}^{\ensuremath{-}}{\mathrm{a}}^{\ensuremath{-}}{\mathrm{c}}⁰$), of ${\mathrm{BO}}₆$ octahedra, respectively. Here, we perform density functional theory (DFT) calculations along with machine learning (ML) to investigate the effects of biaxial strain on the ferroelectric and magnetic properties for hybrid improper ferroelectric $({\mathrm{LaFeO}}₃{)}₁/({\mathrm{LnFeO}}₃{)}₁$ superlattices (SLs), where Ln represents Ce, Nd, Sm, Gd, Dy, Y, Tm, and Lu. We have investigated how the polarization, magnetization, and coupling coefficients are modified subject to the external strain. The shift in energy minima in the energy landscape, which is a function of both ${Q}_{\mathrm{Rot}}$ and ${Q}_{\mathrm{Tilt}}$, indicates that the ferroelectric switching can be modulated upon imposing strain. Strain can change the polarization up to $16%$ with respect to the zero-strain value. In the case of magnetization, we observe a change in the easy axis leading to a change in the magnetic configuration from ${G}_x{A}_y{F}_z$ to ${F}_x{C}_y{G}_z$ upon tensile strain. It is impractical to compute the strain response for all possible $({\mathrm{LnFeO}}₃{)}₁/({\mathrm{Ln}}^{\ensuremath{'}}{\mathrm{FeO}}₃{)}₁$ combinations. To overcome this limitation we perform ML with DFT calculated data within a subset of all the combinations. Using machine learning, we predict the change in polarization upon strain for tolerance $\ensuremath{\tau}$ ranging from 0.844 to 0.902, which covers all the possible $({\mathrm{LnFeO}}₃{)}₁/({\mathrm{Ln}}^{\ensuremath{'}}{\mathrm{FeO}}₃{)}₁$ SL combinations. The prediction by ML is in excellent agreement with the DFT calculations.