Strength of correlations in a silver-based cuprate analog

${\mathrm{AgF}}_2$ has been proposed as a cuprate analog, which requires strong correlation and marked covalence. On the other hand, fluorides are usually quite ionic, and $4d$ transition metals tend to be less correlated than their $3d$ counterparts, which calls for further scrutiny. We combine valence band photoemission and Auger-Meitner spectroscopy of AgF and ${\mathrm{AgF}}_2$ together with computations in small clusters to estimate values of the Ag $4d$ Coulomb interaction $U_{4d}$ and charge-transfer energy ${\mathrm{\Delta }}_{pd}$. Based on these values, ${\mathrm{AgF}}_2$ can be classified as a charge-transfer correlated insulator according to the Zaanen-Sawatzky-Allen classification scheme. Thus, we confirm that the material is a cuprate analog from the point of view of correlations, suggesting that it should become a high-temperature superconductor if metallization is achieved by doping. We present also a computation of the Hubbard $U$ in density functional “$+U$” methods and discuss its relation to the Hubbard $U$ in spectroscopies.

Figure 1

${\mathrm{AgF}}_6$ clusters used in the computations (highlighted in gray) for AgF (a) and ${\mathrm{AgF}}_2$ (b). In the case of ${\mathrm{AgF}}_2$ we indicated some angles. Analogous angles in AgF are either ${90}^{\circ }$ or ${180}^{\circ }$.

Figure 2

Comparison between the valence XPS experimental spectrum (green) and cluster model calculation (black) of the AgF compound. In the computations we fixed hybridization matrix elements as explained in the text yielding $T_{pd}^{x^2-y^2}=1.38$ eV and $T_{pp}=0.29$ eV. The best fit to the experiment was found with ${\mathrm{\Delta }}_{pd}=1.2$ eV. We show the computation with a very narrow broadening to resolve the position of individual excitations and intensities and with a phenomenological Lorentzian broadening growing linearly in energy ($\eta =0.97$ eV $+0.14\omega$) to fit of the experiment. The inset shows the computed silver-$d$ (orange) and fluorine-$P$ (blue) spectral functions.

Figure 3

Comparison between Auger experimental spectrum (green) and Auger cluster model calculation spectrum (black) of the AgF compound. The black curve is given by the summation of the $M4$ and $M5$ singlet spectra. The two main structures are due to the SO splitting in the Ag $3d$ subshell (see Appendix pp2-s1). The broadening of the $M4$ component is $\eta =2.6$ eV and the one of the $M5$ component is $\eta =5.6$ eV. The fit between experimental and theoretical data fixes ${\overline{U}}_{dd}=6.0$ eV. Other cluster parameters are the same as in Fig. 2. In the inset, the self-convolutions of the experimental valence band XPS spectrum (red) and the experimental Auger spectrum (green) are shown. The energy splitting (dashed black vertical lines) between those two peaks is equal to 7.3 eV and represents a rough estimate of $U{(}^{1}G)$ in the Ag $4d$ subshell.

Figure 4

Valence band XPS spectra of ${\mathrm{AgF}}_2$. The green line is the experimental result and the black line is the result of the cluster computation with the crystal field parameters of Table 1 and ${\overline{U}}_{dd}=5.5$ eV ($U_{4d}=7.2$ eV), ${\overline{U}}_{pp}=6$ eV, $U_{pd}=1$ eV, ${\mathrm{\Delta }}_{pd}=4$ eV, and $T_{pd}=2.76$ eV. We used a Lorentzian broadening $\eta =0.83+0.17\omega$ eV and convolution with a Gaussian of FWHM = 1.0 eV (green line). We also show the theory with a very small broadening to show the energy of individual states in the cluster (black line). The inset shows the computed character of the spectra: $A_d\left(\omega \right)$ (orange dashed line) and $A_P\left(\omega \right)$ (blue dashed line). For the theory, the zero of the energy is set at the additional chemical potential. For the experiment, the spectrum was shifted in energy to align its main features to the cluster computation.

Figure 5

The crosses are the hole occupancy of the $d$ states (left scale) while the circles are the interatomic charge correlator (right scale) in the possible final states of the photoemission process. The colors of the symbols encode the singlet (red) or triplet (blue) character of the two-hole state. Parameters and binding energies are defined as in Fig. 4. At the bottom, we show the valence band spectrum in green to facilitate identification of the states. The horizontal lines are guide to the eyes separating large, small, and intermediate values of the expectation values.

Figure 6

One particle spectral function projected on the different orbital symmetries of the cluster for the same parameters as Fig. 4.

Figure 7

Same as Fig. 4 for the ad hoc crystal field parameters of the strongly bound ZR scenario. Other parameters are ${\overline{U}}_{dd}=6.5$ eV corresponding to $U_{4d}=8.2$ eV, ${\overline{U}}_{pp}=6.0$ eV, ${\mathrm{\Delta }}_{pd}=3.5$ eV, and $U_{pd}=1.5$ eV. The theoretical curve (black) is presented with a phenomenological broadening with a Lorentzian parameter varying linearly with energy as $\eta =0.45\mathrm{eV}+0.30\omega$ (green line) and with a very small broadening to show the energy of individual states in the cluster (black line). The inset shows the computed character of the spectra: $A_d\left(\omega \right)$ (orange dashed line) and $A_P\left(\omega \right)$ (blue dashed line).

Figure 8

The crosses are the hole occupancy of the $d$ states (left scale) while the circles are the interatomic charge correlator (right scale) in the possible final states of the photoemission process. The colors of the symbols encode the singlet (red) or triplet (blue) character of the two-hole state. Parameters and binding energies are defined as in Fig. 7. At the bottom, we show the valence band spectrum in green to facilitate identification of the states.

Figure 9

One particle spectral function projected on the different orbital symmetries of the cluster for the same parameters of Fig. 7.

Figure 10

The measured 3d core-hole spectra of AgF and ${\mathrm{AgF}}_2$. The splitting between the $M4$ and $M5$ states is due to spin-orbit coupling and amounts to ${\mathrm{\Delta }}_{\mathrm{SO}}=6.0$ eV for AgF.

Figure 11

The measured 3p core-hole spectra of AgF and ${\mathrm{AgF}}_2$.