Strain-Controlled Magnetic Phase Transitions through Anisotropic Exchange Interactions: A Combined DFT and Monte Carlo Study

TL;DR

This work investigates strain-controlled magnetic phase transitions in correlated materials by combining first-principles DFT and a semiclassical Monte Carlo treatment of an anisotropic Hubbard model. DFT on BiFeO$_3$ under epitaxial strain reveals direction-dependent exchange couplings that map to anisotropic hoppings, showing compressive strain reduces $J_z$ while increasing $J'_ot$, driving a G-type AF to C-type AF$(\pi,\pi,0)$. The s-MC analysis maps out ground-state phase diagrams as a function of $t_z$, $t'_ot$, and $t'_{xy}$, predicting that compressive strain stabilizes C-type AF and tensile strain stabilizes A-type AF or degenerate C-type AF states, with the in-plane NNN hopping mainly influencing $T_N$. Overall, strain emerges as a powerful parameter to engineer competing magnetic phases in correlated systems, offering design rules for strain-controlled magnetoelectric and spintronic materials.

Abstract

Epitaxial strain provides a powerful, non-chemical route to tune the properties of functional materials by manipulating the coupling between spin, charge, and lattice degrees of freedom. Using density functional theory (DFT) calculations and $\rm BiFeO_3$ as a model system, we first demonstrate how epitaxial strain exactly leads to anisotropic magnetic interactions where the exchange coupling along the $c$-axis differs from that in the $ab$-plane. We show that subtle structural modifications, specifically the distortion from a cubic to a tetragonal lattice, drive a magnetic phase transition from a G-type to a C-type antiferromagnetic (AF) phase. The anisotropy in magnetic interactions, which becomes prominent in the lower symmetry tetragonal phase, provides a direct link between the structural distortion and the potential change in magnetic ordering. For a more comprehensive study, we next investigate the role of strain in driving magnetic phase transitions within a half-filled one-band Hubbard model in three dimensions. In this framework, strain is introduced through anisotropic hopping processes between nearest- and next-nearest-neighbor sites, inspired by the DFT calculations. Using a semiclassical Monte Carlo (s-MC) approach, we construct ground state phase diagrams in the nonperturbative regime, which show how uniaxial strain stabilizes distinct magnetic ground states: Compressive strain drives a transition from a G-type to a C-type AF insulator, whereas tensile strain suppresses the C-type AF order, favoring an A-type AF phase. Overall, our combined DFT and s-MC calculations highlight that strain is a powerful tuning parameter for controlling competing magnetic phases by governing exchange coupling mechanisms in correlated systems, offering valuable insights for the design of strain-controlled materials.

Figures (12)

• Figure 1: Representative magnetic spin configurations on a simple cubic lattice, characterized by distinct ordering vectors $\mathbf{q}$. (a) G-type AF: spins alternate along all three spatial directions, $\mathbf{q} = (\pi, \pi, \pi)$. (b) C-type AF$(\pi, \pi, 0)$: AF order in the $xy$-planes with ferromagnetic stacking along $z$, $\mathbf{q} = (\pi, \pi, 0)$. (c) C-type AF$(\pi, 0, \pi)$: AF alignment in $xz$-planes with ferromagnetic coupling along $y$, $\mathbf{q} = (\pi, 0, \pi)$. (d) C-type AF$(0, \pi, \pi)$: AF alignment in $yz$-planes with ferromagnetic stacking along $x$, $\mathbf{q} = (0, \pi, \pi)$. (e) A-type AF$(0, 0, \pi)$: spins align ferromagnetically within each $xy$-plane, with neighboring planes stacked antiferromagnetically along the $z$-axis; $\mathbf{q} = (0, 0, \pi)$. (f) A-type AF$(0, \pi, 0)$: ferromagnetic alignment in the $xz$-plane, with alternating spin orientation along the $y$-axis; $\mathbf{q} = (0, \pi, 0)$. (g) A-type AF$(\pi, 0, 0)$: ferromagnetic layers lie in the $yz$-plane, coupled antiferromagnetically along the $x$-direction; $\mathbf{q} = (\pi, 0, 0)$. (h) FM: uniform spin alignment across the lattice, $\mathbf{q} = (0, 0, 0)$.
• Figure 2: Crystal structure of tetragonal $\rm BiFeO_3$ shown in a $2 \times 2 \times 2$ supercell. Purple spheres represent Bi ions, brown spheres denote Fe ions located at the centers of corner-sharing $\rm FeO_6$ octahedra, and red spheres correspond to oxygen ions.
• Figure 3: Energies of G-type AF and C-type AF($\pi,\pi,0$) phases in $\rm BiFeO_3$ as a function of the out-of-plane lattice constant $c_s$, for two fixed in-plane lattice constants: (a) $a_s = 4.0$ Å and (b) $a_s = 3.8$ Å. As $a_s$ decreases from $4.0$ Å to $3.8$ Å, the magnetic ground state switches from G-type AF to C-type AF($\pi,\pi,0$). (c) Energy difference $\Delta E = E_G - E_C$ as a function of the in-plane lattice constant $a_s$, where the out-of-plane lattice parameter $c_o$ is fixed at its optimized value, which minimizes the total energy. As $a_s$ decreases (increasing compressive strain), $\Delta E$ gradually increases and eventually becomes positive, indicating a transition from the G-type AF state to the C-type AF($\pi,\pi,0$) state. (d) Exchange couplings extracted from Table \ref{['tab:Jvalues']} as a function of the in-plane lattice parameter $a_s$. For clarity, $J'_\perp$ and $J'_{xy}$ are scaled by a factor of $4$ and $2$, respectively, highlighting the point near $a_s \approx 3.9$ Å, where $E_G = E_C$. Results corresponding to the volume-relaxed tetragonal structure ($a_s = 3.73~$Å and $c_o = 4.88$ Å) are also shown in panels (c) and (d) using open symbols. The G-type AF and C-type AF phases are labeled G-AF and C-AF, respectively in all figures.
• Figure 4: (a) Temperature dependent of $S(\pi,\pi,\pi)$ for isotropic NN ($t_x = t_y = t_z = 1$) and NNN hoppings ($t'_{xy} = t'_{yz} = t'_{xz} = 0.25$). The system undergoes a G-type AF transition with $T_N \approx 0.18$, $0.13$, and $0.09$ for $U = 8, 12$, and $18$, respectively (see main text for details). (b) Temperature dependence of the specific heat $C_v$ (left axis, solid symbols) and resistivity along the $z$ direction $\rho_z$ (right axis, open symbols) for $U = 12$. The low-$T$ peak in $C_v$ marks the onset of long-range magnetic order, while the broad high-$T$ peak signals local-moment formation and coincides with the metal--insulator transition temperature $T_{\text{MIT}}$. (c) Temperature evolution of the local magnetic moment $M$ for $U = 12$; the magnitude of $M$ grows continuously upon cooling and saturates, reflecting the formation of localized magnetic moments at low temperatures. (d) DOS at $T = 0.01$ and $T = 0.2$, both showing a clear gap at the Fermi level ($\omega = 0$). While the gap is wider at $T = 0.01$, it remains distinctly visible even at $T = 0.2 > T_N$, confirming the robustness of the insulating state well above the magnetic ordering temperature. In all calculations, the DOS is shifted such that $\omega = 0$ corresponds to the Fermi level (chemical potential).
• Figure 5: Magnetic and transport properties for $t_z = 0.5$, with varying isotropic NNN hopping $t' = 0.0, 0.2,$ and $0.4$ ($t' = t'_{xy} = t'_{yz} = t'_{xz}$): (a) Magnetic structure factor $S(\mathbf{q})$ indicating G-type AF($\pi,\pi,\pi$) ordering for $t' = 0$ and $0.2$, and C-type AF($\pi,\pi,0$) order at $t' = 0.4$. (b) Specific heat $C_v$ displaying low-$T$ peaks that align with the magnetic transitions in (a), and high-$T$ peaks related to moment formation near $T_{\text{MIT}}$. (c) Resistivity $\rho_z$ showing insulating behavior for all $t'$. Notably, $T_{\text{MIT}} \gg T_N$, leading to a PM-I regime between the paramagnetic metal (PM-M) and magnetically ordered phases. Inset: Temperature dependence of the local magnetic moment $M$, which saturates at low $T$, reflecting moment formation well above $T_N$. (d) DOS at $T=0.01$ confirming insulating ground states with a finite gap at $\omega = 0$. Legends are consistent across all panels.