Nematicity as a Probe of Superconducting Pairing in Iron-Based Superconductors

In several families of iron-based superconducting materials, a $d$-wave pairing instability may compete with the leading $s$-wave instability. Here, we show that when both states have comparable free energies, superconducting and nematic degrees of freedom are strongly coupled. Whereas nematic order causes a sharp nonanalytic increase in $T_c$, nematic fluctuations can change the character of the $s$-wave to $d$-wave transition, favoring an intermediate state that does not break time-reversal symmetry but does break tetragonal symmetry. The coupling between superconductivity and nematicity is also manifested in the strong softening of the shear modulus across the superconducting transition. Our results show that nematicity can be used as a diagnostic tool to search for unconventional pairing states in iron pnictides and chalcogenides.

Figure 1

Schematic Fermi surfaces of three different systems where competing $s^{+-}$ and $d$-wave instabilities have been proposed [21, 22, 23, 28, 29, 50]. Thick red (thin blue) lines denote electron (hole) pockets. (a) In $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Mn}}_x{)}_2{\mathrm{As}}_2$, the $s^{+-}$ state arises from $(\pi ,0)/(0,\pi )$ stripe-type fluctuations, whereas the $d$-wave state comes from ($\pi$, $\pi$) Néel-type fluctuations [23]. (b) In $A_{1-y}{\mathrm{Fe}}_{2-x}{\mathrm{Se}}_2$ chalcogenides, a $d$-wave state appears due to the direct $XY$ interaction [50], whereas $s^{+-}$ is favored by FeAs hybridization [29]. (c) In strongly doped $({\mathrm{Ba}}_{1-x}{\mathrm{K}}_x){\mathrm{Fe}}_2{\mathrm{As}}_2$, the $s^{+-}$ state appears when small electron pockets emerge with doping, whereas a $d$-wave state can appear due to the $M$ intrapocket interaction [22, 28].

Figure 2

Schematic phase diagrams as function of temperature ($T$) and doping ($x$) for the interplay between $s^{+-}$-wave and $d$-wave superconductivity in iron pnictide materials. Dotted (solid) lines denote second- (first-)order phase transitions. Panel (a): no nematic order and weak nematic fluctuations (${\chi }_{\mathrm{nem}}<2\alpha /{\lambda }^2$). The $s$-wave and $d$-wave states are separated by an intermediate TRSB $s+id$ state. Panel (b): preexisting nematic order. $T_c$ is enhanced with respect to the tetragonal case (dashed line), and the superconducting order parameter is characterized by the real combination $s+d$ and evolves smoothly with $x$ with no TRSB. Panel (c): no nematic order, but larger nematic fluctuations ($2\alpha <{\lambda }^2{\chi }_{\mathrm{nem}}<{\beta }_{sd}+\alpha +\sqrt{{\beta }_s{\beta }_d}$). The coexistence region is enhanced, but the intermediate state is of $s+d$ character, spontaneously breaking rotational but not time-reversal symmetry. Panel (d): no nematic order, but even larger nematic fluctuations (${\lambda }^2{\chi }_{\mathrm{nem}}>{\beta }_{sd}+\alpha +\sqrt{{\beta }_s{\beta }_d}$). The $s$-wave to $d$-wave transition becomes first order.

Figure 3

Dependence of $T_c$ on the Néel-type (${\xi }_{\mathrm{N}\acute{\mathrm{e}}\mathrm{el}}$) and stripe-type (${\xi }_{\mathrm{stripe}}$) magnetic correlation lengths obtained from Eliashberg calculations as described in the text. Panel (a) shows the evolution of $T_c$ (in units of ${\gamma }_{\mathrm{stripe}}/2\pi$) as function of ${\xi }_{\mathrm{N}\acute{\mathrm{e}}\mathrm{el}}/{\xi }_{\mathrm{stripe}}$ in the absence (dashed line) and presence of nematic order (solid line, $\phi =1.0$). Panel (b) presents the variation of $T_c$, $\Delta T_c=T_c-T_{c,0}$, as function of the nematic order parameter $\phi =\ln ({\xi }_X/{\xi }_Y)$, for three fixed values of the ratio ${\xi }_{\mathrm{N}\acute{\mathrm{e}}\mathrm{el}}/{\xi }_{\mathrm{stripe}}$ indicated by the arrows in panel (a): ${\xi }_{\mathrm{N}\acute{\mathrm{e}}\mathrm{el}}/{\xi }_{\mathrm{stripe}}=0.1$ (dot-dashed blue line), ${\xi }_{\mathrm{N}\acute{\mathrm{e}}\mathrm{el}}/{\xi }_{\mathrm{stripe}}=0.26$ (dashed green line), and ${\xi }_{\mathrm{N}\acute{\mathrm{e}}\mathrm{el}}/{\xi }_{\mathrm{stripe}}=0.33$ (solid red line).