Displacement and annihilation of Dirac gap nodes in $d$-wave iron-based superconductors

Several experimental and theoretical arguments have been made in favor of a $d$-wave symmetry for the superconducting state in some Fe-based materials. It is a common belief that a $d$-wave gap in the Fe-based superconductors must have nodes on the Fermi surfaces centered at the $\mathrm{\Gamma }$ point of the Brillouin zone. Here we show that, while this is the case for a single Fermi surface made out of a single orbital, the situation is more complex if there is an even number of Fermi surfaces made out of different orbitals. In particular, we show that for the two $\mathrm{\Gamma }$-centered hole Fermi surfaces made out of $d_{xz}$ and $d_{yz}$ orbitals, the nodal points still exist near $T_c$ along the symmetry-imposed directions, but are are displaced to momenta between the two Fermi surfaces. If the two hole pockets are close enough, pairs of nodal points can merge and annihilate at some $T<T_c$, making the $d$-wave state completely nodeless. These results imply that photoemission evidence for a nodeless gap on the $d_{xz}/d_{yz}$ Fermi surfaces of ${\mathrm{KFe}}_2{\mathrm{As}}_2$ does not rule out $d$-wave gap symmetry in this material, while a nodeless gap observed on the $d_{xy}$ pocket in ${\mathrm{K}}_x{\mathrm{Fe}}_{2-y}{\mathrm{Se}}_2$ is truly inconsistent with the $d$-wave gap symmetry.

Figure 1

The Fermi surfaces (FSs) and the location of the nodal points near the two $\mathrm{\Gamma }$-centered $d_{xz}/d_{yz}$ pockets. Panel (a) shows the two FSs in the normal state, highlighting the orbital that gives the largest spectral weight at each point along the FSs (yellow for $d_{xz}$ and green for $d_{yz}$). Panel (b) illustrates the location of the $d$-wave nodes on the two FSs (blue and red lines) if the band off-diagonal gap term was absent. Panel (c) presents the actual location of the nodal points (red and blue dots) for the case $\mathrm{\Delta }=0.8{\mathrm{\Delta }}_{\mathrm{cr}}$. The dispersions are given by $E_{a,b}=\sqrt{{\mathrm{\Delta }}^2{cos}^{2}2\theta +{\epsilon }_{a,b}^2}$ and the terms ${\epsilon }_{a,b}$ vanish along the two gray lines adjacent to the FSs.

Figure 2

The evolution of the $d$-wave nodes as $\mathrm{\Delta }$ increases beyond the critical value ${\mathrm{\Delta }}_{\mathrm{cr}}$. The blue and red lines are the normal-state FS. The gray lines denote the locations of ${\epsilon }_{a,b}=0$, and the dispersions are given by $E_{a,b}=\sqrt{{\mathrm{\Delta }}^2{cos}^{2}2\theta +{\epsilon }_{a,b}^2}$. The nodal points are marked by the red and blue dots. Four pairs of nodal points are present for $\mathrm{\Delta }<{\mathrm{\Delta }}_{\mathrm{cr}}$ and disappear for $\mathrm{\Delta }>{\mathrm{\Delta }}_{\mathrm{cr}}$. In this figure, we used circular band dispersions with $m_2=3m_1$.

Figure 3

The diagrammatic representation of the linearized gap equations, Eqs. (8). Blue and red lines denote fermions from bands $c_1$ and $c_2$.

Figure 4

The dispersion of the $d$-wave gap along the two FSs [Eq. (14)] for two values of $\mathrm{\Delta }$. Blue (red) lines denote band $c_1$ ($c_2$). There is substantial angular variation but no nodes. Note also that the minimum value of the gap is different in both bands.

Figure 5

The quasiparticle dispersion in the $d$-wave superconducting state for the two-orbital model of Raghu et al. [34]. The dispersion has the form $E_{\mathbf{k}}^2={\epsilon }_{\mathbf{k}}^2+{\mathrm{\Delta }}^2{cos}^{2}2{\theta }_{\mathbf{k}}$, where ${\epsilon }_{\mathbf{k}}={({\xi }_{+}^2+{\mathrm{\Delta }}^2{sin}^{2}2{\theta }_{\mathbf{k}})}^{1/2}\pm \left|\vec{B}\right|$ (see Eqs. (12) and (13) and Ref. [16]). Reference [16] used the parameters from Ref. [34]: ${\xi }_{+}=-0.3t(cos{k}_x+cos{k}_y)+3.4tcos{k}_{x}cos{k}_y-1.45t, |\vec{B}|=t{[2.3^2{(cos{k}_x-cos{k}_y)}^2+3.4^2{sin}^2{k}_x{sin}^2{k}_y]}^{1/2}$, and $sin2{\theta }_{\mathbf{k}}=3.4tsin{k}_{x}sin{k}_y/\left|\vec{B}\right|$, where $t$ sets the overall energy scale. Nodal points of the dispersion are the ones for which $cos2{\theta }_{\mathbf{k}}=0$ and ${\epsilon }_{\mathbf{k}}=0$. We plot ${\epsilon }_{\mathbf{k}}$ as a function of momentum $\mathbf{k}$ along the direction $k=k_x=k_y$ (for which $cos2{\theta }_{\mathbf{k}}=0$) for different gap values $\mathrm{\Delta }$. A pair of nodal points is observed unless $\mathrm{\Delta }$ exceeds the critical value ${\mathrm{\Delta }}_c\approx 3.08t$, which is about a quarter of the bandwidth.