Ab initio studies on interactions in ${\mathrm{K}}_3{\mathrm{C}}_{60}$ under high pressure

Fullerene solids doped with alkali metals (${\mathrm{A}}_3{\mathrm{C}}_{60}$, A = K, Rb, Cs) exhibit a superconducting transition temperature $T_c$ as high as 40 K, and their unconventional superconducting properties have been a subject of debate. With the application of high pressure on ${\mathrm{K}}_3{\mathrm{C}}_{60}$ and ${\mathrm{Rb}}_3{\mathrm{C}}_{60}$, experiments demonstrate the decrease of $T_c$. In this paper, we focus on ${\mathrm{K}}_3{\mathrm{C}}_{60}$ and derive the structure of ${\mathrm{K}}_3{\mathrm{C}}_{60}$ under different pressures based on first-principles calculations, exploring the trends of Coulomb interactions at various pressures. By utilizing the maximally localized Wannier function approach, constrained density functional perturbation theory, and constrained random phase approximation, we construct a microscopic low-energy model near the Fermi level. Our results strongly indicate that, in the ${\mathrm{K}}_3{\mathrm{C}}_{60}$ system, as pressure increases, the effect of phonons is the key to intraorbital electron pairing. The phonon-driven superconducting mechanism is dominant at high pressure.

Figure 1

(a) The atomic structure of ${\mathrm{K}}_3{\mathrm{C}}_{60}$, which is set at a size of $2\times 2\times 2$, and the real-space Wannier function displayed using vesta software [57]. K atoms are depicted as purple spheres, while C atoms are represented by brown spheres. Positive isosurfaces are highlighted in yellow, while negative isosurfaces are shown in blue. (b) Brillouin zone of the fcc lattice; the path is $\mathrm{\Gamma }\text{−}X\text{−}U\text{−}K\text{−}\mathrm{\Gamma }\text{−}L\text{−}K\text{−}W\text{−}X$, as described in Ref. [58].

Figure 2

The pressure dependence of (a) lattice constant $a$, (b) bandwidth $W$, and (c) Fermi velocity $v_{\mathrm{F}}$. We fix the lattice constants to the experimental values [37] and calculate the bandwidth of the $t_{1u}$ band using quantum espresso [59, 60, 61]. We combine Eq. (14) to obtain the Fermi velocity from the energy band results.

Figure 3

Calculated ab initio electronic band structure of fcc ${\mathrm{K}}_3{\mathrm{C}}_{60}$ at different pressures. The pressure parameters are set as 0, 0.53, 1.42, 2.05, 2.81, 3.34, 3.74, and 5.03 GPa. The horizontal axis is labeled with the special points in the Brillouin zone: $\mathrm{\Gamma }$ (0, 0, 0), $X$ (0.5, 0, 0.5), $U$ (0.625, 0.25, 0.625), $K$ (0.375, 0.375, 0.75), $L$ (0.5, 0.5, 0.5), and $W$ (0.5, 0.25, 0.75). Fermi energy is depicted as blue dashed lines.

Figure 4

Our calculated density of states (DOS) for the $t_{1u}$ band of fcc ${\mathrm{K}}_3{\mathrm{C}}_{60}$ at different pressures. The pressure parameters are set as 0, 0.53, 1.42, 2.05, 2.81, 3.34, 3.74, and 5.03 GPa.

Figure 5

(a)–(c) $U$, $U^{\prime }$, and $J$ with different screening levels [unscreened (bare) and constrained RPA (cRPA)]. For bare and cRPA $U$, $U^{\prime }$, and $J$ values, the units are given in eV. (d) ${\epsilon }_M$, the cRPA-macroscopic-dielectric constant in Eq. (15).

Figure 6

Diagonal terms of the inverse dielectric matrix for all $\mathbf{q}$ points ${\epsilon }_{\mathbf{G}\mathbf{G}}^{-1}(\mathbf{q},\omega )$ at the first frequency $\omega$ = 0. The horizontal axis is limited within the range of 0 to 0.5 ${\mathrm{bohr}}^{-1}$, where it represents $|\mathbf{q}+\mathbf{G}|$. Inset: The horizontal axis represents the entire range of $|\mathbf{q}+\mathbf{G}|$.

Figure 7

Pressure dependence of (a) the average of the on-site effective Coulomb repulsion $\overline{U}$, (b) the on-site effective exchange interaction $J$, (c) the average of the off-site effective Coulomb repulsion between neighboring sites $\overline{V}$, and (d) the correlation strength $(\overline{U}-\overline{V})/W$, which are derived using the cRPA method.

Figure 8

Partially renormalized electron-phonon coupling parameter for different phonon modes at the $\mathrm{\Gamma }$ point calculated using the cDFPT method. The phonon modes are as follows: (a) $H_g$(1)–$H_g$(8) and (b) $A_g$(1) and $A_g$(2). The mode decomposition of phonon-mediated interactions (c) $U_{\mathrm{ph}}\left(\nu \right)$ and (d) $U_{\mathrm{ph}}^{\prime }\left(\nu \right)$ as a function of pressure.

Figure 9

Pressure dependence of the effective phonon-mediated interaction. $U_{\mathrm{ph}}$ is the intraorbital density-density-type interaction, $U_{\mathrm{ph}}^{\prime }$ is the interorbital density-density-type interaction, and $J_{\mathrm{ph}}$ is the exchange-type interaction.

Figure 10

Pressure dependence of effective interactions $U_{\mathrm{eff}}(U+U_{\mathrm{ph}})$, $U_{\mathrm{eff}}^{\prime }(U^{\prime }+U_{\mathrm{ph}}^{\prime })$, and $J_{\mathrm{eff}}(J+J_{\mathrm{ph}})$. $U\left[U_{\mathrm{ph}}\right]$, $U^{\prime }\left[U_{\mathrm{ph}}^{\prime }\right]$, and $J\left[J_{\mathrm{ph}}\right]$ are the intraorbital, interorbital, and exchange components of the cRPA Coulomb (cDFPT phonon-mediated) interactions, respectively.

Figure 11

Pressure dependence of the comparison of effective interactions. $\overline{U}\left[{\overline{U}}_{\mathrm{ph}}\right]$ and $J\left[J_{\mathrm{ph}}\right]$ are the on-site effective Coulomb repulsion and exchange components of the cRPA Coulomb (DFPT phonon-mediated) interactions, respectively.

Figure 12

The evolution of Coulomb interaction parameters with increasing energy cutoff. It can be observed that beyond 36 Ry, the changes in interaction parameters are minimal. To conserve computational resources and time, we utilized a cutoff of 36 Ry in our calculations.

Figure 13

(a) The on-site Coulomb repulsions at different coordinates of the Brillouin zone are presented in 2D plots for various pressures. (b) A 3D contour plot is used to depict the on-site Coulomb repulsions for a pressure of 2.81 GPa.