Implications of the $\mathrm{DFT}+U$ method on polaron properties in energy materials

To model polaronic behavior in strongly correlated transition-metal oxides with ab initio methods, one typically requires a level of theory beyond that of local density or general gradient density functional theory (DFT) approximations to account for the strongly correlated $d$-shell interactions of transition-metal oxides. In the present work, we utilize density functional theory with additional on-site Hubbard corrections $\left(\mathrm{DFT}+U\right)$ to calculate polaronic properties in two lithium ion battery cathode materials, ${\mathrm{Li}}_x{\mathrm{FePO}}_4$ and ${\mathrm{Li}}_x{\mathrm{Mn}}_2{\mathrm{O}}_4$, and two photocatalytic materials, ${\mathrm{TiO}}_2$ and ${\mathrm{Fe}}_2{\mathrm{O}}_3$. We investigate the effects of the $+U$ on-site projection on polaronic properties. Through systematic comparison with hybrid functional calculations, it is shown that $+U$ projection in these model materials can impact upon the band gap, polaronic hopping barrier, and polaronic eigenstate offset from the band edges in a nontrivial manner. These properties are shown to have varying degrees of coupling and dependence on the $+U$ projection in each example material studied, which has important implications for arriving at systematic material predictions of polaronic properties in transition-metal oxides.

Figure 1

Schematic of the radial charge density distribution $\left(\rho \right)$ of a polaronic state taken from ${\mathrm{Fe}}_2{\mathrm{O}}_3$, centered at the TM site on which the polaron is localized. $r_O$ is the average metal-oxygen bond length, while in this example, $r_{\text{Fe8}}$ and $r_{\text{Fe16}}$ correspond to two different cutoff radii $\left(r_{\text{PAW}}\right)$ of a large core Fe potential $\left(r_{\text{PAW}}=2.3a_0\right)$ with eight valence electrons, and a small core Fe potential $\left(r_{\text{PAW}}=1.9a_0\right)$ with 16 valence electrons, respectively. These are also drawn approximately on the ${\mathrm{MO}}_6$ octahedral complex, which is cut out from the ${\mathrm{Fe}}_2{\mathrm{O}}_3$ solid. Other materials show similar behavior.

Figure 2

${\mathrm{TiO}}_2$ orbital projected DOS of the polaronic ground state. Inset: Real-space radial charge distribution of the polaronic state.

Figure 3

Calculated electronic properties of ${\mathrm{TiO}}_2$ with $\mathrm{DFT}+U$ and HSE06 as a function of different PAW potentials. (a) Band gap $E_g$; (b) polaronic gap state $E_p$; (c) formation energy $E_{\text{form}}$; (d) polaronic hopping barrier $E_a$.

Figure 4

Deviation from linearity as a function of fractional charge in ${\mathrm{TiO}}_2$. $r_{\text{PAW}}$ and the number of valence electrons are listed for each PAW potential. $U=4.2$ eV and $\alpha =25\%$ in all $\mathrm{DFT}+U$ (solid upright triangles) and HSE06 (hollow inverted triangles) cases, respectively.

Figure 5

${\mathrm{Fe}}_2{\mathrm{O}}_3$ orbital projected DOS of the polaronic ground state. Inset: Real-space radial charge distribution of the polaronic state.

Figure 6

Calculated electronic properties of ${\mathrm{Fe}}_2{\mathrm{O}}_3$ with $\mathrm{DFT}+U$ and HSE06 using different PAW potentials. (a) Band gap $E_g$; (b) polaronic gap state $E_p$; (c) formation energy $E_{\text{form}}$; (d) polaronic hopping barrier $E_a$.

Figure 7

Deviation from linearity as a function of fractional charge in ${\mathrm{Fe}}_2{\mathrm{O}}_3$. $r_{\text{PAW}}$ and the number of valence electrons are listed for each PAW potential. $U=4.3$ eV in all $\mathrm{DFT}+U$ calculations (solid upright triangles), and the two sets of HSE06 calculations are done with $\alpha =25\%$ (hollow inverted triangles) and $\alpha =12\%$ (hollow squares).

Figure 8

${\mathrm{FePO}}_4$ orbital projected DOS of the polaronic ground state. Inset: Real-space radial charge distribution of the polaronic state.

Figure 9

Calculated electronic properties of ${\mathrm{FePO}}_4$ with $\mathrm{DFT}+U$ and HSE06 using different PAW potentials. (a) Band gap $E_g$; (b) polaronic gap state $E_p$; (c) formation energy $E_{\text{form}}$; (d) polaronic hopping barrier $E_a$.

Figure 10

${\mathrm{LiFePO}}_4$ orbital projected DOS of the polaronic ground state. Inset: Real-space radial charge distribution of the polaronic state.

Figure 11

Calculated electronic properties of ${\mathrm{LiFePO}}_4$ with $\mathrm{DFT}+U$ and HSE06 using different PAW potentials. (a) Band gap $E_g$; (b) polaronic gap state $E_p$; (c) formation energy $E_{\text{form}}$; (d) polaronic hopping barrier $E_a$.

Figure 12

Deviation from linearity as a function of fractional charge in ${\mathrm{FePO}}_4$. $r_{\text{PAW}}$ and the number of valence electrons are listed for each PAW potential. $U=4.3$ eV and $\alpha =25\%$ in all $\mathrm{DFT}+U$ (solid upright triangles) and HSE06 (hollow inverted triangles) cases, respectively.

Figure 13

Deviation from linearity as a function of fractional charge in ${\mathrm{LiFePO}}_4$. $r_{\text{PAW}}$ and the number of valence electrons are listed for each PAW potential. $U=4.3$ eV and $\alpha =25\%$ in all $\mathrm{DFT}+U$ (solid upright triangles) and HSE06 (hollow inverted triangles) cases, respectively.

Figure 14

${\mathrm{MnO}}_2$ orbital projected DOS of the polaronic ground state. Inset: Real-space radial charge distribution of the polaronic state.

Figure 15

Calculated electronic properties of ${\mathrm{MnO}}_2$ with $\mathrm{DFT}+U$ and HSE06 using different PAW potentials. (a) Band gap $E_g$; (b) polaronic gap state $E_p$; (c) formation energy $E_{\text{form}}$; (d) polaronic hopping barrier $E_a$.

Figure 16

Deviation from linearity as a function of fractional charge in spinel ${\mathrm{MnO}}_2$. $r_{\text{PAW}}$ and the number of valence electrons are listed for each PAW potential. $U_{\text{Mn}}=4.5$ eV and $\alpha =25\%$ in all $\mathrm{DFT}+U$ (solid upright triangles) and HSE06 (hollow inverted triangles) calculations, respectively. We have included an additional set of calculations where we have set a $U_{\text{O}}=6.0$ eV on the O $2p$ states in addition to the existing $U_{\text{Mn}}=4.5$ eV (solid right facing triangles).

Figure 17

Real-space radial charge distribution around a polaronic Fe site in ${\mathrm{Fe}}_2{\mathrm{O}}_3$. The blue curve and area corresponds to the polaronic ground state (POL). The red curve and area corresponds to the transition state (TS), where half of the electron density is centered on this site and the other half is centered on a neighboring site. The PAW radii shown correspond to the large $(r_{\text{PAW}}=2.3a_0$, eight valence electrons) and small core Fe $(r_{\text{PAW}}=1.9a_0$, 16 valence electrons) potentials.