Unconventional superconductivity protected from disorder on the kagome lattice

Motivated by the recent discovery of superconductivity in the kagome ${A\mathrm{V}}_3{\mathrm{Sb}}_5$ $(A=\mathrm{K}, \mathrm{Rb}, \mathrm{Cs})$ metals, we perform a theoretical study of the symmetry-allowed superconducting orders on the two-dimensional kagome lattice with focus on their response to disorder. We uncover a qualitative difference between the robustness of intraband spin-singlet (even-parity) and spin-triplet (odd-parity) unconventional superconductivity to atomic-scale nonmagnetic disorder. Due to the particular sublattice character of the electronic states on the kagome lattice, disorder in spin-singlet superconducting phases is only weakly pair-breaking despite the fact that the gap structure features sign changes. By contrast, spin-triplet condensates remain fragile to disorder on the kagome lattice. We demonstrate these effects in terms of the absence of impurity bound states and an associated weak disorder-induced $T_c$ suppression for spin-singlet order. We also discuss the consequences for quasiparticle interference and their inherent tendency for momentum-space anisotropy due to sublattice effects on the kagome lattice. For unconventional kagome superconductors, our results imply that any allowed spin-singlet order, including for example $d+id$-wave superconductivity, exhibits a disorder-response qualitatively similar to standard conventional $s$-wave superconductors.

Figure 1

(a) Illustration of the kagome lattice with lattice vectors ${\mathbf{t}}_1$ and ${\mathbf{t}}_2$, and sublattice vectors ${\mathbf{a}}_1$, ${\mathbf{a}}_2$, and ${\mathbf{a}}_3$. The grey area denotes the unit cell with the three different sublattice sites denoted A (blue), B (orange), and C (green). (b) Tight-binding energy bands plotted along the high-symmetry path shown in the inset. Here, $n$ is a band label, introduced for later convenience. (c) Distribution of the sublattice weights across the Brillouin zone for the second band ($n=2$). The Fermi surface for $\mu =0$, corresponding to the upper van Hove singularity in (b), is shown in black.

Figure 2

DOS near the Fermi energy for systems with either singlet (left column) or triplet (right column) superconducting order. The singlet cases are plotted for $\mu =0$ while the triplet cases are plotted for $\mu =0.08$. (a) Singlet $A_1$, resulting in the usual fully gapped $s$-wave spectrum. (b) Singlet $E_2$, which is 2D and corresponds to $d$-wave at lowest order in momentum. In black we illustrate the $d+id$ TRSB combination. This combination results in a fully gapped spectrum. In orange and blue we show the results for the two individual components, $d_{x^2-y^2}$ (orange) and $d_{xy}$ (blue). Both exhibit nodes on the Fermi surface and lead to V-shaped gaps. They also break rotational symmetry and the LDOS differs between the different sublattice sites. Here we show the case corresponding to the $A$ sublattice. (c) Triplet $B_1$, which is commonly denoted $f$ wave. This has nodes on the Fermi surface (see Table 1) resulting in a V-shaped gap. (d) Triplet $E_1$, which is also 2D and corresponds to $p$ wave at lowest order in momentum. We show the TRSB $p+ip$ variant in black, while the $p_x$ and $p_y$ components are shown in orange and blue, respectively. Both $p_x$ and $p_y$ break lattice rotational symmetry. Here, we stress that we only plot the DOS on the $A$ sublattice, which is why the $p_x$ component has a U-shaped gap; the $A$ sublattice has vanishing spectral weight in the regions of the BZ where $p_x$ has nodes, compare Fig. 1 and Table 1.

Figure 3

Energy dependence of the real (a) and imaginary (b) parts of $\text{det}\left[\widehat{D}\left(\omega \right)\right]$ as given by Eq. (23) for a range of nonmagnetic scattering potentials from $V=-10$ to $V=10$, denoted by the colors, for a singlet $d_{x^2-y^2}+i{d}_{xy}$ gap at $\mu =0.0$ with the DOS in the clean case shown by the red line. The absence of impurity bound states inside the gapped region is evident from panel (a). We note that the finite $\text{Im}\left(\text{det}\left[\widehat{D}\left(\omega \right)\right]\right)$ inside the hard gap in (b) is due to the finite smearing $\eta =0.002$. Vanishing values of $\eta$ lead to a vanishing imaginary part inside the hard gap.

Figure 4

Momentum dependence of the real [(a)–(d)] and imaginary [(e)–(h)] parts of the anomalous free Green's function components at $\omega =0$ for $d_{x^2-y^2}+i{d}_{xy}$ superconductivity at $\mu =0.0$. Panels (a) and (e) display the Green's function entries for the second band ($n=2$), reflecting the $d_{x^2-y^2}+i{d}_{xy}$ gap structure at the Fermi surface. The lower six panels refer to the anomalous entries in sublattice space. Importantly, the individual components of the Green's function in sublattice space, when integrated over the Brillouin zone, do not vanish, implying that Eq. (23) may have no real solutions.

Figure 5

LDOS along a cut through the impurity site, shown in the inset, for the case with $d_{x^2-y^2}+i{d}_{xy}$ superconductivity at $\mu =0.0$ and an impurity potential $V=-4.0$. The different curves correspond to different sites along the cut with the black curve denoting the impurity site, while blue and green curves denote $A$ and $C$ sites, respectively, as shown in the inset. The impurity is located at an $A$ site.

Figure 6

Energy dependence of the real (a) and imaginary (b) parts of $\text{det}\left[\widehat{D}\left(\omega \right)\right]$ as given by Eq. (23) for a range of nonmagnetic scattering potentials from $V=-10$ to $V=10$, denoted by the colors, for a triplet $p_x+i{p}_y$ gap at $\mu =0.08$ with the DOS for the clean case shown by the red line. In this case, the gapped region exhibits impurity bound states, in contrast to the case shown in Fig. 3. As in Fig. 3, it is the finite value of $\eta$ that causes a finite imaginary part inside the hard gap in (b).

Figure 7

Momentum dependence of the real [(a)–(d)] and imaginary [(e)–(h)] parts of the anomalous free Green's function components at $\omega =0$ for $p_x+i{p}_y$ superconductivity. Panels (a) and (e) display the Green's function entries in band space, simply reflecting the $p_x+i{p}_y$ gap structure at the Fermi surface. The lower six panels refer to the anomalous entries in sublattice space. All plots here are for the $p_x+i{p}_y$ phase at $\mu =0.08$. In contrast to the $d_{x^2-y^2}+i{d}_{xy}$ case, the individual components of the sublattice Green's function, when integrated over the Brillouin zone, all vanish. As a consequence Eq. (23) always has real solutions.

Figure 8

LDOS along a cut through the impurity site, shown in the inset, for the case with $p_x+i{p}_y$ superconductivity at $\mu =0.08$ and an impurity potential $V=-4.0$. The different curves correspond to different sites along the cut with the black curve denoting the impurity site, while blue and green curves denote $A$ and $C$ sites, respectively, as shown in the inset. The impurity is located on an $A$ site. In contrast to the $d_{x^2-y^2}+i{d}_{xy}$ case shown in Fig. 5, an impurity bound state appears inside the gap for the $p_x+i{p}_y$ order parameter, and extends along the ${\mathbf{a}}_1$ direction as shown.

Figure 9

LDOS along a horizontal cut (same as Figs. 5 and 8) near the impurity site (black) for the case of a (a) $B_1$ ($f$ wave), (b) $E_1^{\left(1\right)}$ ($p_x$), (c) $E_2^{\left(1\right)}$ ($d_{x^2-y^2}$), and (d) $E_2^{\left(2\right)}$ ($d_{xy}$) superconducting order parameter with an impurity strength of $V=-4$ in all cases. The spin-singlet cases are obtained with a chemical potential $\mu =0$, while the triplet cases are calculated with $\mu =0.08$, i.e., both near the upper van Hove filling. The LDOS for spin-triplet order parameters is visibly affected by the impurity and features low-energy resonant states whereas the singlet $d$-wave order parameters are protected from in-gap resonant states.

Figure 10

LDOS $\rho (\mathbf{r},0)$ plotted for a fixed energy $\omega =0$ with $d_{x^2-y^2}+i{d}_{xy}$ superconductivity and an impurity of strength $V=-4$ located at an A sublattice site. The breaking of several point group symmetries by the impurity position is reflected in the associated LDOS modulations in the vicinity of the impurity.

Figure 11

(a) The temperature dependence of the $d_{x^2-y^2}+i{d}_{xy}$ order parameter $\mathrm{\Delta }\left(T\right)$ on the kagome lattice for a range of $n{V}^2$. The AG results (circles) have been fitted with the solutions to the mean-field equation $x=tanh(x/T)$ (full lines) to obtain critical temperatures $T_c$. (b) The critical temperatures $T_c/T_{c0}$ as a function of $n{V}^2$, where $T_{c0}=T_c(n{V}^2=0)$, for an $s$-wave (blue), $d_{x^2-y^2}$ (red), $d+id$ (purple), $p_x$ (dark green), and $p+ip$ (light green) order parameter. Note that the $p_x$ and $p+ip$ results for the kagome lattice nearly coincide. The results are for $\mu =0$ in the kagome lattice and $\mu =-1$ for the square lattice. In the case of the rotational symmetry breaking order parameters the critical temperatures were calculated with the impurities located on sites A, B, and C, respectively, and an average of the three values is plotted. The critical temperatures have been matched in the $n{V}^2=0$ case by self-consistently solving the BdG gap equation. The solid line in panel (b) shows the $T_c$ suppression from the universal AG curve [58, 77].

Figure 12

LDOS near an impurity potential extended equally to the $A$, $B$, and $C$ sublattice sites, as shown in the inset, for a $d_{x^2-y^2}+i{d}_{xy}$ superconductor at $\mu =0.0$ and an impurity potential $V=-2.0$. In the present case impurity bound states emerge inside the gap and the in-gap bound state position is tuned by $V$.