$s+is$ superconductivity with incipient bands: Doping dependence and STM signatures

Motivated by the recent observations of small Fermi energies and comparatively large superconducting gaps, present also on bands not crossing the Fermi energy (incipient bands) in iron-based superconductors, we analyze the doping evolution of superconductivity in a four-band model across the Lifshitz transition including BCS-BEC crossover effects on the shallow bands. Similar to the BCS case, we find that with hole doping the phase difference between superconducting order parameters of the hole bands change from 0 to $\pi$ through an intermediate $s+is$ state, breaking time-reversal symmetry (TRS). The transition, however, occurs in the region where electron bands are incipient and chemical potential renormalization in the superconducting state leads to a significant broadening of the $s+is$ region. We further present the qualitative features of the $s+is$ state that can be observed in scanning tunneling microscopy (STM) experiments, also taking incipient bands into account.

Figure 1

Band structure of the considered 2D model. Two parabolic hole and electron bands are around the $\mathrm{\Gamma }$ and $M$ points, respectively. $E_{F_{h_1}}, E_{F_{h_2}}$($E_{F_e}$) correspond to the Fermi energies of hole (electron) bands. The position of the $k$ axis marks the overall chemical potential of the system. Three cases are illustrated. Undoped or moderate doping (black): both hole and electron bands cross the Fermi level. $E_{F_{h_i}}$ and $E_{F_e}$ are positive. Hole-doped (red dashed): hole bands cross the Fermi level while electron bands are incipient. $E_{F_e}$ is negative. Electron-doped (blue dashed-dotted): both hole bands are incipient. $E_{F_{h_i}}$ negative. Energy difference $E_D=E_{F_{h_1}}+E_{F_e}$ is independent from doping. Interactions are assumed to be frequency and momentum independent up to an upper energy cutoff $\mathrm{\Lambda }$.

Figure 2

$k_x,k_y$ cut through the Fermi surface of the folded Brillouin-zone at (a) moderate doping and (b) large hole doping. In panel (b), Fermi surface consists of hole pockets only while electron bands are incipient. Plus and minus signs represent the pairing symmetry.

Figure 3

Phase diagram as function of temperature ($T$) and doping ($E_{F_e}$). Color encodes the phase difference $\varphi$ between the order parameters of hole bands modulo $\pi /2$. Black dashed line marks the $s+is$ state boundaries if one ignores (7) and takes ${\mu }_i=E_{F_i}$. The parameters used are ${\lambda }_{hh}/{\lambda }_{eh}=0.95, {\lambda }_{eh}=0.15, {\vartheta }_{\alpha }=0, E_{h_1{h}_2}=0, \mathrm{\Lambda }=3000, E_D=110, E_{0e}=10.7$ and $E_{0h}=4.8\times {10}^{-3}$ in arbitrary energy units.

Figure 4

The discontinuous contribution to the specific heat $C_V-C_V^{\mathrm{reg}}$ [Eq. (9)] as a function of temperature for $E_{Fe}/E_{0e}=-0.467$. Marked are the critical temperatures for the transition into usual superconducting state $T_c$ and the $s+is$ state ($T_{s+is}$). The parameters of the model are taken to be the same as in Fig. 3.

Figure 5

The lower and the upper boundaries of the $s+is$ region for chemical potential ${\mu }_{e_{\text{min}}}$ and ${\mu }_{e_{\text{max}}}$ and doping $E_{F_e}^{max}$ and $E_{F_e}^{min}$ as functions of the ratio ${\lambda }_{hh}/{\lambda }_{eh}$ taking ${\vartheta }_{\alpha }=0$.

Figure 6

Interband density of states $\delta {\rho }_{ch}^{h1h2}\left(\omega \right)/t_3{\rho }_{h1}{\rho }_{h2}$ [(29) with $\delta ={10}^{-6}]$ for scattering between the hole bands. Inset shows ${\varphi }_1-{\varphi }_2$ as a function of $E_F/E_{0e}$ for $T=0$ obtained from (6) and (7) using the same parameters as in Fig. 3 except for $E_{h1h2}=90$ in arbitrary units. Arrows mark the dopings for which $\delta \rho$ curves are presented.

Figure 7

Doping evolution of the interband density of states $\delta {\rho }_{ch}^{h2e}\left(\omega \right)/t_3{\rho }_{h2}{\rho }_e$ [(31) with $\delta ={10}^{-6}]$ for scattering between the electron and the smaller hole (h2) bands. (a) Dopings close to the Lifshitz transition for the electron band. (b) Dopings around the $s+is$ region. Red dashed line is for the same doping as line 4 but with ${\mathrm{\Delta }}_e=0$. Inset in panel (a) shows ${\varphi }_1-{\varphi }_2$ as a function of $E_F/E_{0e}$ obtained from (6) and (7) using the same parameters as in Fig. 6. Arrows mark the dopings for which $\delta \rho$ curves are presented.

Figure 8

Panels (a) and (b): Eigenvalues of the second variation matrix of the action $\frac{{\partial }^{2}S}{\partial {\mathrm{\Delta }}_i^a\partial {\mathrm{\Delta }}_j^b}$ at $T=0$. Vertical dashed lines mark the boundaries of the $s+is$ state.

Figure 9

An example for $\mathbf{q}$-integration for a centrosymmetric two-band system. Green regions represent the extents of Green's function in momentum space.