Quasiparticle excitations in the photoemission spectrum of CuO from first principles: A $GW$ study

We present ab initio quasiparticle calculations for electronic excitations and the fundamental band gap of the strongly correlated transition-metal oxide CuO using the $GW$ approximation of many-body perturbation theory. Problems related to the suitability of the method for strongly correlated materials and issues of self-consistency are addressed. We explain why quasiparticle self-consistent $GW$ strongly overestimates the band gap of CuO. Apart from the band gap, electron addition and removal spectra in the quasiparticle approximation including lifetime and matrix-element effects are found to be in excellent agreement with the quasiparticle excitations in direct and inverse photoemission data.

Figure 1

Magnetic ordering and local magnetic moments in CuO. The magnetic unit cell illustrating the ground-state magnetic ordering is displayed in (a). Big white and blue balls represent Cu atoms of opposite spin directions. Small red balls depict O atoms. The approximately square-planar coordination of the Cu atoms (left panel) and the tetrahedral coordination of the O atoms (right panel) are highlighted. The figures have been produced with vesta [33]. The local magnetic moments $\mu$ that occur both at the Cu and O atoms are shown in dependence on the fraction of screened exchange $\alpha$ in the hybrid functional (b) or as a function of the on-site interaction $U$ in the $\mathrm{PBE}+U$ approach (c). Experimental values [9] including their error bars are displayed as shaded areas.

Figure 2

Kohn-Sham band structures calculated with the PBE (a), HSE06 (b), and $\mathrm{PBE}+U$ ($U=4\text{eV}$) (c) functionals. The VBM is set to zero and the band gap is highlighted as shaded area. The nomenclature of the high-symmetry points follows Ref. [42].

Figure 3

Orbital-projected DOS calculated with the hybrid functional for varying $\alpha$ (a) or with the $\mathrm{PBE}+U$ functional for varying $U$ (b). The spin-up (positive values) and spin-down (negative values) contributions of the individual Cu $3d$ orbitals and the O $2p$ orbitals are shown. The local coordinate system at the Cu atoms is oriented such that the $z$ axis is perpendicular to the Cu-O coordination square and the $x$ axis parallel to a Cu-O bond. For the O $2p$ partial DOS, the average contribution per O atom is shown. The VBM is set to zero and all DOS are convoluted with a Gaussian of 0.5 eV full width at half maximum (FWHM).

Figure 4

DOS as well as indirect (ind.) and direct (dir.) band gaps calculated in the $G_0{W}_0$ one-shot approach on top of the hybrid functional for varying $\alpha$ [(a),(c)] and the $\mathrm{PBE}+U$ functional for varying $U$ [(b),(d)]. The respective DFT DOS is shown as shaded area, whereas the $G_0{W}_0$ DOS is indicated by a thick solid line. The VBM is set to zero. A Gaussian broadening of 0.5 eV is applied to the DOS. In (c) and (d), the experimental direct band gap at zero temperature [12] (cf. Table 2), represented by a black horizontal line, is given for reference.

Figure 5

Convergence of the highest valence band and lowest conduction band at $\Gamma$ with the number of iteration steps $n$ in various flavors of the $GW$ approximation. Results are given for HSE06 and $\mathrm{PBE}+U$ starting electronic structures. The zeroth iteration step corresponds to the DFT eigenvalues.

Figure 6

Full (a) and orbital-resolved (b) DOS of CuO for various self-consistent $GW$ schemes. The screened Coulomb interaction $W$ is either fixed ($W_0$) to the subjacent DFT starting point (HSE06 and $\mathrm{PBE}+U$) or iterated to self-consistency ($W_n$). In the latter case, the resulting DOS is independent of the starting point. The DOS are broadened with a Gaussian of 0.5 eV (FWHM) and the VBM is set to zero.

Figure 7

Convergence of the macroscopic electronic static dielectric constant ${\varepsilon }_{\infty }$ (averaged over directions) with the number of iteration steps $n$ in various flavors of the $GW$ approximation. Results are given for HSE06 and $\mathrm{PBE}+U$ starting electronic structures. The zeroth iteration step corresponds to the macroscopic dielectric constant ${\varepsilon }_{\infty }$ deduced from $W_0$. The range of available experimental values as the high-frequency limit of infrared [60, 62] and the low-frequency limit of optical [61] spectroscopy is given as shaded area.

Figure 8

Frequency-dependent loss function of CuO for vanishing momentum transfer $\mathbf{q}=0$ averaged over Cartesian directions. The loss function $-Im{\varepsilon }^{-1}(\mathbf{q}=0,\omega )$ is calculated in RPA using the DFT electronic structures of the two $GW$ starting points (HSE06 and $\mathrm{PBE}+U$) and the self-consistent $\mathrm{sc}{G}_n{W}_n$ QP electronic structure. The direction-averaged loss function is compared to experimental data from spectroscopic ellipsometry [61] and electron-energy loss [68]. For completeness, the loss function calculated in RPA using the metallic PBE electronic structure is also given.

Figure 9

QP DOS of CuO compared to data from UPS, XPS, and BIS experiments. (a) QP DOS including intrinsic QP lifetimes are plotted together with XPS [4, 16] (taken at an incident photon energy of 1486.6 eV) and BIS [4] data. A Gaussian broadening of 1.0 eV (FWHM) is applied to the calculated spectra to mimic temperature and instrumental broadening effects. In (b), the QP DOS are additionally weighted with the photoionization cross sections [74] of each orbital to facilitate comparison with photoemission spectra for various incident photon energies. Experimental data from Ghijsen et al. [4] and Shen et al. [16] are shown as black and gray dots.