Theoretical study of the physical properties of ${\mathrm{BiS}}_2$ superconductors

Recent angle-resolved photoemission reveals that the doping level in some ${\mathrm{BiS}}_2$-based superconductors is much lower than the nominal one. Here we determine the pairing function at such a realistic low doping level by a functional renormalization group (FRG) and study the resulting physical properties by an FRG-based effective mean field theory. We find that the pairing is a mixture of singlets and triplets due to spin-orbital coupling, and the triplet components dominate. The gap function transforms as a $d_{x^2-y^2}^{*}$ wave under simultaneous rotation of spin and lattice, but is nodeless on the Fermi pockets which avoid the nodal lines. The specific heat, superfluid density, and Knight shift are similar to that in a fully gapped conventional $s$-wave superconductor, in agreement with available experiments. However, the absence of the Hebel-Slichter peak in spin-lattice relaxation rate (in the dirty limit), the impurity induced in-gap states, and the gapless edge states (due to the weak topological nature of the pairing) are defining properties of the nodeless $d_{x^2-y^2}^{*}$-wave pairing to be measured in future experiments.

Figure 1

(a) FRG flow of leading $S_{\mathrm{ph},\mathrm{pp}}$ versus the running energy scale $\Lambda$. (b) $ln|S_{\mathrm{ph}}\left(\mathbf{q}\right)|$ versus q at the divergence energy scale (color scale). (c) Fermi surface and gap function thereon (color scale).

Figure 2

(a) The electronic specific heat $C$ versus the temperature $T$. Here ${\gamma }_n=C/T$ $(T>T_c$) is a constant in the normal state. (b) Temperature dependence of the superfluid density from our theory (solid line) and from experimental data for ${\mathrm{Bi}}_4{\mathrm{O}}_4{\mathrm{S}}_3$ [17] (squares) and ${\mathrm{LaO}}_{0.5}{\mathrm{F}}_{0.5}{\mathrm{BiS}}_2$ [18] (circles).

Figure 3

(a) $1/T_{1}T$ versus temperature, normalized by the value in the normal state. (b) The direction-resolved Knight shift versus temperature. The results are presented for both the nodeless $d_{x^2-y^2}^{*}$-wave pairing (solid lines) and a trivial $s$-wave pairing (dashed lines).

Figure 4

$lnD\left(\omega \right)$ (color scale) versus $\omega$ and $V_0$ in the presence of a nonmagnetic impurity for (a) nodeless $d_{x^2-y^2}^{*}$-wave pairing and (b) $s$-wave pairing. The vertical black lines in panel (a) highlight the gap edges.

Figure 5

$lnD\left(\omega \right)$ (color scale) versus $\omega$ and $V_0$ in the presence of magnetic impurity along the $z$ axis for (a) nodeless $d_{x^2-y^2}^{*}$-wave pairing and (b) $s$-wave pairing. The vertical black lines in panel (a) highlight the gap edges.

Figure 6

$lnD\left(\omega \right)$ (color scale) versus $\omega$ and $V_0$ in the presence of magnetic impurity along the $x$ axis for (a) nodeless $d_{x^2-y^2}^{*}$-wave pairing and (b) $s$-wave pairing. The vertical black lines in panel (a) highlight the gap edges.

Figure 7

Energy spectrum of a ribbon of the nodeless $d_{x^2-y^2}^{*}$-wave superconductor versus the transverse momentum. The thick lines indicate the edge states.