Importance of anisotropic Coulomb interaction in ${\mathrm{LaMnO}}_3$

In low-temperature antiferromagnetic ${\mathrm{LaMnO}}_3$, strong and localized electronic interactions among Mn $3d$ electrons prevent a satisfactory description from standard local density and generalized gradient approximations in density functional theory calculations. Here, we show that the strong on-site electronic interactions are described well only by using direct and exchange corrections to the intraorbital Coulomb potential. Only DFT + U calculations with explicit exchange corrections produce a balanced picture of electronic, magnetic, and structural observables in agreement with experiment. To understand the reason, a rewriting of the functional form of the  + U corrections is presented that leads to a more physical and transparent understanding of the effect of these correction terms. The approach highlights the importance of Hund's coupling (intraorbital exchange) in providing anisotropy across the occupation and energy eigenvalues of the Mn $d$ states. This intraorbital exchange is the key to fully activating the Jahn-Teller distortion, reproducing the experimental band gap and stabilizing the correct magnetic ground state in ${\mathrm{LaMnO}}_3$. The best parameter values for ${\mathrm{LaMnO}}_3$ within the DFT (PBEsol)  + U framework are determined to be $U=8$ eV and $J=1.9$ eV.

Figure 1

$\left(001\right)$ face of A-type antiferromagnetic (A-AFM) ${\mathrm{LaMnO}}_3$: Mn in purple, La in green, and O in red. Arrows indicate direction of spin polarization on Mn ions.

Figure 2

$U_{\text{eff}}$ (left) and $U|J$(right) corrected density of states for fully relaxed A-AFM ${\mathrm{LaMnO}}_3$ as a function of energy, $E-E_{\text{F}}$. Black curves show the total density of states while red and yellow curves show Mn $d$ majority spin and minority spin densities of states.

Figure 3

The occupation of states is represented in the model ${\mathrm{Mn}}^{3+} d^4$ manifold (majority spin only). Orbital degeneracy is broken by octahedral crystal field (CF), Coulomb repulsion U${}_{\mathrm{eff}}$ (U in the figure) and exchange J following Eq. (4). Each vertical bar represents one unit of electron occupation. ${\pi }^{e_g\sigma }$ (denoted ${\pi }^{e_g}$ for the majority spin channel in the figure) is defined in Eq. (1), and three limits are examined: ${\pi }^{e_g\sigma }=0$ ($f_{x^2-y^2\sigma }=f_{3z^2-r^2\sigma }=0.5$), ${\pi }^{e_g\sigma }=+1$ ($f_{x^2-y^2\sigma }=f_{3z^2-r^2\sigma }+1=1$), and ${\pi }^{e_g\sigma }=-1 \left(f_{x^2-y^2\sigma }+1=f_{3z^2-r^2\sigma }=1\right)$.

Figure 4

Orbitals in the ${\mathrm{LaMnO}}_3$ unit cell from a $U|J=8|2$ eV calculation. (a) ${\mathrm{MnO}}_6$ octahedron with Jahn-Teller distorted plane and local basis vectors labeled. (b) Visualization of the occupation of the $3z^2-r^2$ and $x^2-y^2$ states in the rotated basis of the density matrix as well as their superposition for the total local $e_g$ occupancy (plotting the occupation times the orbital expressed in spherical harmonics). (c) The ordering of the occupied $e_g$ shell $\left(1.97z^2-0.58x^2-1.4y^2\right)$ in the $\left(010\right)$ Jahn-Teller distorted FM coupled plane. (d) The ordering of the occupied $e_g$ shell in the $\left(001\right)$ plane with AFM coupling along $b$. Note $x,y,z$ is the local octahedron basis, and $a,b,c$ lattice vectors correspond to $x^{\prime },y^{\prime },z^{\prime }$ global (prerotation) calculation basis.

Figure 5

${\mathrm{LaMnO}}_3$ densities of states (DOS) for $U|J=8|2$ and $8|3$ eV calculations. Majority spin corresponds to positive DOS and minority to negative DOS.

Figure 6

Local magnetic moment within the Bader volume for Mn cations and O anions [within the (010) basal plane], from DFT + $U_{\text{eff}}$ and DFT + $U|J$ methods. The notation $U_6|J$and $U_8|J$ indicates $U$ is fixed to 6 and 8 eV, respectively, while $J$ is varied. The experimental reference local magnetic moment is $3.7$${\mu }_{\text{B}}$ [51].

Figure 7

${\mathrm{LaMnO}}_3$ magnetic coupling constants $J_1$ and $J_2$ vs $U|J$ schema Hund's exchange parameter $J$, for $U=6$ (white squares), 7 (white circles), and 8 eV (black squares). $J_1$ and $J_2$ are defined in Eq. (11). The red-blue overlap (centered at $J=1.9$ eV) suggests a $J$ exchange value for the $U|J$ scheme that provides the correct sign for both coupling constants—see main text for discussion.

Figure 8

(Top left) Band gap of ${\mathrm{LaMnO}}_3$ vs $U-J$ within the $U_{\text{eff}}$ approach for fully relaxed structures where both electron-lattice and electron-electron interactions are active (dashed line, e-l, and e-e) and for structures with the Jahn-Teller distortion frozen out so only electron-electron interactions are active (solid line, $e\text{−}e$ only). (Bottom left and middle) Jahn-Teller normal modes vs $U-J$ within the $U_{\text{eff}}$ approach (white circles), and within the $U|J$ approach (black circles) for $U$ fixed to 8 and 6 eV with $J$ varied. (Right) Orbitally ordered and strongly Jahn-Teller active $ac$ plane, the local basis convention, and ${\mathbf{Q}}^{\text{Ortho}}$ and ${\mathbf{Q}}^{\text{Tetra}}$ modes.