Measuring Density Functional Parameters from Electron Diffraction Patterns

We integrate density functional theory (DFT) into quantitative convergent-beam electron diffraction (QCBED) to create a synergy between experiment and theory called QCBED-DFT. This synergy resides entirely in the electron density which, in real materials, gives rise to the experimental CBED patterns used by QCBED-DFT to refine DFT model parameters. We use it to measure the Hubbard energy $U$ for two strongly correlated electron systems, NiO and ${\mathrm{CeB}}_6$ ($U=7.4\pm 0.6\text{eV}$ for $d$ orbitals in NiO and $U=3.0\pm 0.6\mathrm{eV}$ for $f$ orbitals in ${\text{CeB}}_6$), and the boron position parameter $x$ for ${\mathrm{CeB}}_6$ ($x=0.1992\pm 0.0003$). In verifying our measurements, we demonstrate an accuracy test for any modeled electron density.

Figure 1

Conventional QCBED (CQCBED). A few parameters are adjusted to fit a calculated CBED pattern to an experimental one, including the structure factors $V_h$, to which the intensities are most sensitive. These form a very small subset of all structure factors $V_g$, required for a full dynamical electron scattering calculation. The unrefined remainder $V_{\mathbf{m}}$ are obtained from an IAM, i.e., $V_{\mathbf{m}}^{\mathrm{IAM}}$. As the refinement progresses, the modified structure factors that start with IAM values $V_h^{\mathrm{IAM}}$ take on new values $V_h^{\mathrm{CQCBED}}$. The bonding potential is the difference between the CQCBED-refined and IAM potentials and is equivalent to computing the Fourier sum using the difference structure factors $\mathrm{\Delta }{V}_h$. Here, the bonding potential in aluminum from [43] has been used as an example of CQCBED.

Figure 2

QCBED-DFT. The DFT model parameters are refined which, when adjusted, change the simulated electron density in real space, thereby changing all structure factors $V_g^{\mathrm{DFT}}$ of the crystal potential returned to the electron scattering calculations. By optimizing the fit between the calculated and experimental CBED patterns, the DFT model parameters are refined (see Supplemental Material [62] for details). The present example involves antiferromagnetic NiO—a subject of this Letter.

Figure 3

The electronic structure of NiO determined by QCBED-DFT with $U=7.4\mathrm{eV}$ and $J=0.95\mathrm{eV}$ for $d$ orbitals [24, 91, 98] at $T=0\mathrm{K}$. The valence electron density ${\rho }_{\mathrm{val}}(\mathbf{r})$ is mapped in the (020) plane with a contour interval of $0.1e^{-}{\AA }^{-3}$ (a). Nearest-neighbor nickel atoms (gray) have opposite spins indicated by the black and green arrows parallel and antiparallel to [112]. The bonding and electron spin densities, (b) $\mathrm{\Delta }\rho (\mathbf{r})$ and (c) $s(\mathbf{r})$, respectively, are plotted for the same region as (a) ${\rho }_{\mathrm{val}}(\mathbf{r})$, but with a contour interval of $0.05e^{-}{\AA }^{-3}$ and the zero contour thickened.

Figure 4

Determination of the atomic structure parameter $x$ and the Hubbard energy $U$ for $f$ orbitals in ${\mathrm{CeB}}_6$ using QCBED-DFT. The boron atom positions are defined by $x$ (a),(b). Sets of seven CBED patterns from (a) near $⟨001⟩$ and (b) $⟨011⟩$ were matched by fixing $U$ and refining $x$. The mean values of $x$ from each zone axis, ${\overline{x}}_{⟨uvw⟩}$, are graphed with their uncertainties (c) for each value of $U$ (see Tables S6 and S7 in [62]). Fixing $x=\overline{{\overline{x}}_{⟨001⟩}}=0.19922$ (the mean of ${\overline{x}}_{⟨001⟩}$ over all values of $U$), $U$ was then refined and results for the $⟨001⟩$ CBED data ($U_{⟨001⟩}$), the $⟨011⟩$ data ($U_{⟨011⟩}$), and all data combined ($U_{\text{combined}}$) are reported (d). This was repeated for $x={\overline{x}}_{⟨001⟩(U=3\mathrm{eV})}=0.19925$ (e).

Figure 5

A comparison of electron density structure factors $F_{hkl}$ refined by CQCBED with different reference electronic structure models for NiO and ${\mathrm{CeB}}_6$. All points represent the refined $F_{hkl}$ normalized with respect to the corresponding model values. The horizontal lines represent each model where QCBED-DFT corresponds to DFT [$\mathrm{APW}+\text{lo}$: $\mathrm{GGA}(\mathrm{PBE})+U$, ($U=7.4$ and $J=0.95\mathrm{eV}$ for NiO), ($U=3\mathrm{eV}$, $J=0\mathrm{eV}$, and $x=0.19925$ for ${\mathrm{CeB}}_6$)], as determined by our QCBED-DFT refinements, and the IAM is the current International Union of Crystallography standard [107, 108, 109]. All $F_{hkl}$ correspond to the temperatures of the experiments and were converted from $V_{hkl}$ by the Mott-Bethe formula [83, 84]. See Table S11 in [62] for the actual values of $V_{hkl}$ and $F_{hkl}$.