Application of cDFPT to Alkali-Doped Fullerides

Abstract

To derive phonon-related terms in the realistic low-energy Hamiltonians for fcc $\mathrm{A}_3\mathrm{C}_{60}$ systems, we apply a newly developed ab initio downfolding scheme, constrained density-functional perturbation theory (cDFPT). The effective interaction between the low-energy electrons is given by the sum of the Coulomb and phonon-mediated retarded interactions. By evaluating the latter contribution by the cDFPT and comparing it to a previously estimated Coulomb contribution, we show that an effectively negative exchange interaction is realized on each molecule in fcc $\mathrm{A}_3\mathrm{C}_{60}$. This is because a contribution from the coupling to the Jahn-Teller phonons surpasses a small Hund’s coupling, turning the sign of the exchange interaction from positive to negative. We also present a comparison between the cDFPT and the conventional DFPT to get more insight into the phonon properties of the fullerides.

Appendix: Frequency Dependences of Partially Screened Coulomb Interactions

In this thesis, we assume that the partially screened Coulomb interactions are static (see Sect. 2.4.1.2). In order to check the validity of this assumption, we calculate the frequency dependences of the partially screened Coulomb interactions within the cRPA [31] for a representative material, fcc $\mathrm{Cs}_3\mathrm{C}_{60}$ with $V_{{\mathrm{C}_{60}}^{3-}} = 762$ $\AA ^3$. We find that the assumption is justifiable because the frequency dependences are weak from zero frequency to a frequency well beyond the bandwidth.

We performed the calculation, following the conditions employed in Ref. [2]. We expanded the dielectric function in Eq. (2.33) in the plane waves with the energy cutoff of 7.5 Ry. The total number of bands considered in the calculation of the polarization function was set to 335 (120 occupied+3 target+212 unoccupied). The Brillouin-zone integral with respect to the wave vector was evaluated by the generalized tetrahedron method [32, 33]. We employed the Lorentzian smearing for the delta function with the full width at half maximum of 0.3 eV.

Fig. 3.7

Frequency dependences of $U (\omega )$, $U' (\omega )$, and $J_\mathrm{H} (\omega )$ for fcc $\mathrm{Cs}_3\mathrm{C}_{60}$ with $V_{{\mathrm{C}_{60}}^{3-}} = 762$ $\AA ^3$. $U(\omega )$, $U'(\omega )$, and $J_\mathrm{H}(\omega )$ are the intraorbital, interorbital, and exchange components of the cRPA Coulomb interactions, respectively. The frequency dependences for a wider frequency region are shown in the inset. For comparison, we also plot the frequency dependence of the phonon-mediated intraorbital and exchange-type interactions [$U_\mathrm{ph}(\omega )$ and $J_\mathrm{ph}(\omega )$, respectively]. Adapted from Nomura et al., Ref. [21]

Figure 3.7 shows the frequency dependences of $U (\omega )$, $U' (\omega )$, and $J_\mathrm{H} (\omega )$ for fcc $\mathrm{Cs}_3\mathrm{C}_{60}$ with $V_{{\mathrm{C}_{60}}^{3-}} = 762$ $\AA ^3$. We find that $U (\omega )$, $U' (\omega )$, and $J_\mathrm{H} (\omega )$ are almost flat in the frequency region where the phonon-mediated interactions such as $U_\mathrm{ph} (\omega )$ and $J_\mathrm{ph}(\omega )$ are active ($\omega \lesssim 0.2$ eV). Therefore, it can be justified to neglect the frequency dependences of the Coulomb interactions in investigating the subtle balance between the Coulomb and phonon-mediated interactions. Furthermore, the values of the cRPA interactions at a finite frequency are within 15 percent difference from that at $\omega =0$ up to, at least, $\omega = 3$ eV, which is larger than the bandwidth $\sim$ 0.5 eV. This smallness of the frequency dependence suggests that the assumption of the static Coulomb interaction is valid.

Finally, we discuss the origin of small frequency dependence between $\omega =0$ eV and $\omega = 3$ eV. In general, the values of the screened Coulomb interactions change drastically around the plasmon frequency [34]. In $\mathrm{C}_{60}$ compounds, the most prominent plasmon peak is seen in the loss function around $\omega = 27$ eV, which is related to the collective excitation involving all the valence electrons [35–37]. We expect that this plasmon gives a substantial reduction of the Coulomb interactions from their bare values $U=3.32$ eV, $U'=3.12$ eV, and $J_\mathrm{H} = 0.10$ eV, which are about 3-4 times larger than those at $\omega =0$ ($U=0.94$ eV, $U'=0.87$ eV, and $J_\mathrm{H} = 0.035$ eV) [2]. There also exists a less prominent plasmon peak around $\omega = 6.5$ eV, which is relevant to the collective excitation of the $\pi$ electrons [35–37]. These plasmon frequencies (27 eV and 6.5 eV) are away from the frequency regime ($\omega < 3$ eV) in Fig. 3.7. Therefore, we see only a small frequency dependence in Fig. 3.7.