Spin-phonon coupling effects in transition-metal perovskites: A DFT + $U$ and hybrid-functional study

Spin-phonon coupling effects, as reflected in phonon frequency shifts between ferromagnetic (FM) and $G$-type antiferromagnetic (AFM) configurations in cubic CaMnO${}_3$, SrMnO${}_3$, BaMnO${}_3$, LaCrO${}_3$, LaFeO${}_3$, and La${}_2$(CrFe)O${}_6$, are investigated using density-functional methods. The calculations are carried out both with a hybrid-functional Heyd-Scuseria-Ernzerhof (HSE) approach and with a DFT + $U$ approach using a $U$ that has been fitted to HSE calculations. The phonon frequency shifts obtained in going from the FM to the AFM spin configuration agree well with those computed directly from the more accurate HSE approach, but are obtained with much less computational effort. We find that in the $A$MnO${}_3$ materials class with $A=$ Ca, Sr, and Ba, this frequency shift decreases as the $A$ cation radius increases for the $\Gamma$ phonons, while it increases for $R$-point phonons. In La$M$O${}_3$ with $M=$ Cr, Fe, and Cr/Fe, the phonon frequencies at $\Gamma$ decrease as the spin order changes from AFM to FM for LaCrO${}_3$ and LaFeO${}_3$, but they increase for the double perovskite La${}_2$(CrFe)O${}_6$. We discuss these results and the prospects for bulk and superlattice forms of these materials to be useful as multiferroics.

Figure 1

$AB$O${}_3$ perovskite structure doubled along the [111] direction. $A$ atoms (largest) shown in green; $B$ atoms (medium sized) shown in violet; O atoms (smallest) shown in red.

Figure 2

Three idealized polar modes and $R_5^{-}$ mode in simple perovskites. (a) Slater mode; (b) Last mode; (c) Axe mode; (d) $R_5^{-}$ mode.

Figure 3

Variation of PBEsol + $U$ total energy difference $\Delta E=E_{\mathrm{AFM}}-E_{\mathrm{FM}}$ with $U$ for LaCrO${}_3$, $\Delta E^{+U}$ is the energy difference with volume fixed to the optimized volume from HSE and $\Delta E_x^{+U}$ relaxed volume for each $U$. (The fitted value of $U$ occurs at the crossing with the $\Delta E^{\mathrm{HSE}}$ value.)

Figure 4

Effect of $U$ on spin-phonon coupling in SrMnO${}_3$. (a) ${\Gamma }_4^{-}$ and $R_5^{-}$ phonon frequency shifts ($\Delta \omega ={\omega }_{\mathrm{AFM}}-{\omega }_{\mathrm{FM}}$) versus $U$. (b) S-${\Gamma }_4^{-}$ and $R_5^{-}$ phonon frequencies versus $U$ for AFM and FM states.

Figure 5

Frequency shifts $\Delta \omega ={\omega }_{\mathrm{AFM}}-{\omega }_{\mathrm{FM}}$ in $AM$O${}_3$ from HSE calculations.

Figure 6

Changes to metal-O-metal bond angle (solid blue line) resulting from Slater (a) and ${\Gamma }_5^{-}$ (b) phonon modes. Open and solid dots indicate ideal and displaced positions, respectively. Small red dots are oxygen; larger purple dots are metal atoms.

Figure 7

Frequency shifts for La$M$O${}_3$ (HSE for LaCrO${}_3$ and LaFeO${}_3$; PBEsol + $U$ for La${}_2$CrFeO${}_6$).