Spin-phonon coupling effect in $A$MnO$_3$ ($A$=Ca, Sr, Ba) and La$M$O$_3$ ($M$=Cr, Fe, Cr/Fe) from DFT+$U$ and hybrid functional methods

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 using the DFT+$U$ method with a $U$ that has been extracted by comparing with hybrid-functional (HSE) calculations. The phonon frequency shifts $\Delta \omega = \omega_{\rm AFM} - \omega_{\rm FM}$ obtained in this way 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,Ba), the $\Gamma$ ($R$) phonon frequency shift $\Delta \omega$ decreases (increases) as the $A^{2+}$ size increases. In La$M$O$_3$ ($M$=Cr, Fe, Cr/Fe), the phonon frequencies at $\Gamma$ decrease as spin order changes from AFM to FM for LaCrO$_3$ and LaFeO$_3$, but they increase for double perovskite La$_2$(CrFe)O$_6$. We discuss the prospects for bulk and superlattice forms of these materials to be useful as multiferroics.