Robust determination of the superconducting gap sign structure via quasiparticle interference

Phase-sensitive measurements of the superconducting gap in Fe-based superconductors have proven more difficult than originally anticipated. While quasiparticle interference (QPI) measurements based on scanning tunneling spectroscopy are often proposed as definitive tests of gap structure, the analysis typically relies on details of the model employed. Here we point out that the temperature dependence of momentum-integrated QPI data can be used to identify gap sign changes in a qualitative way, and present an illustration for $s_{\pm }$ and $s_{++}$ states in a system with typical Fe-pnictide Fermi surface.

Figure 1

Integrated interband density of states $\delta {\rho }_{\text{inter}}/\left(t_3{\rho }_1{\rho }_2\right)$ for constant (weak) ${\tau }_3 t$ matrix. Gap magnitudes are $|{\mathrm{\Delta }}_1|/T_c=3, |{\mathrm{\Delta }}_2|/T_c=2$, and artificial broadening $\eta ={10}^{-3}$. Solid line: $s_{\pm }$ state. Dashed: $s_{++}$. All other components of $\delta \rho$ are zero for this case.

Figure 2

Thermally averaged interband $\mathbf{q}$-integrated LDOS change within constant density of states and constant $t$-matrix approximation for weak nonmagnetic scatterer with ${\tau }_3$ (Born limit) potential. The external frequency $\mathrm{\Omega }$ was taken to be 2.5 in units where $|{\mathrm{\Delta }}_1|$ and $|{\mathrm{\Delta }}_2|$ were 3.0 and 2.0, as in Fig. 1. Shown is antisymmetrized LDOS $\langle \delta {\rho }^{-}\left(\mathrm{\Omega }\right)\rangle$ for $s_{++}$ (dashed) and for $s_{\pm }$ state (solid). Symmetrized components are zero for both states.

Figure 3

Integrated density of states components $\delta {\rho }_{\mathrm{intra},\mathrm{inter}}^{\pm }$ for “realistic” screened Coulomb scatterers $U_{\mathrm{inter}}=0.2U_{\mathrm{intra}}$, for $U_{\mathrm{intra}}=0.01$ [(a) and (b)], $0.2$ [(c) and (d)], and 10 [(e) and (f)]. Dashed curves correspond to $s_{++}$ state, solid curves to $s_{\pm }$. Red and black are even and odd components of the LDOS, $\delta {\rho }^{+}$ and $\delta {\rho }^{-}$, respectively. Left panels represent intraband, right panels interband scattering channels, respectively.

Figure 4

Integrated density of states components $\delta {\rho }_{\mathrm{intra},\mathrm{inter}}^{\pm }$ for isotropic scatterers $U_{\mathrm{intra}}=U_{\mathrm{inter}}$ for $U_{\mathrm{intra}}=0.01$ [(a) and (b)], $0.2$ [(c) and (d)], and 10 [(e) and (f)]. Dashed curves correspond to $s_{++}$ state, solid curves to $s_{\pm }$. Red and black are even and odd components of the LDOS, $\delta {\rho }^{+}$ and $\delta {\rho }^{-}$, respectively. Left panels represent intraband, right panels interband scattering channels, respectively.

Figure 5

Thermally averaged antisymmetrized interband $\mathbf{q}$-integrated LDOS change $\langle \delta {\rho }_{\mathrm{inter}}^{-}\rangle \left(\mathrm{\Omega }\right)$ for nonmagnetic scatterers with parameters $U_{\mathrm{inter}}=0.2U_{\mathrm{intra}}$, with full $t$ matrix. The external frequency $\mathrm{\Omega }/T_c=2.5$, with $|{\mathrm{\Delta }}_1|/T_c$ and $|{\mathrm{\Delta }}_2|/T_c$ taken as 3.0 and 2.0, respectively. (a) refers to the weak potential $U_{\mathrm{intra}}=0.01$. Shown are antisymmetrized LDOS $\langle \delta {\rho }^{-}\left(\mathrm{\Omega }\right)\rangle$ for $s_{++}$ (dashed) and for $s_{\pm }$ state (solid). (b) shows the same for intermediate strength potential $U_{\mathrm{intra}}=0.2$. Here we have used the standard BCS type behavior for both superconducting gaps.

Figure 6

Absolute magnitude of thermally averaged unsymmetrized interband (black) and intraband (red) $\mathbf{q}$-integrated LDOS changes $|\langle \delta {\rho }_{\mathrm{inter},\mathrm{intra}}\rangle \left(\mathrm{\Omega }\right)|$ for $s_{++}$ (a) and $s_{\pm }$ (b) vs $\mathrm{\Omega }/T_c$ for nonmagnetic scatterers with parameters $T=0.2T_c$ and $U_{\mathrm{inter}}=0.2U_{\mathrm{intra}}$, with full $t$ matrix, and $|{\mathrm{\Delta }}_1|/T_c$ and $|{\mathrm{\Delta }}_2|/T_c$ taken as 3.0 and 2.0, respectively. $\mathrm{\Omega }=|{\mathrm{\Delta }}_1|$ and $|{\mathrm{\Delta }}_2|$ are indicated by dashed lines.

Figure 7

Thermally averaged LDOS changes for antisymmetric, interband channel, $|\langle \delta {\rho }_{\mathrm{inter},\mathrm{intra}}\rangle \left(\mathrm{\Omega }\right)|$ vs $\mathrm{\Omega }/T_c$. Curves shown are for nonmagnetic scatterers with parameters $T/T_c=0.1$ (a), 0.3 (b), 0.6 (c), 0.9 (d) and $U_{\mathrm{inter}}=0.2U_{\mathrm{intra}}$, with full $t$ matrix, and $|{\mathrm{\Delta }}_1|/T_c$ and $|{\mathrm{\Delta }}_2|/T_c$ taken as 3.0 and 2.0, respectively. $\mathrm{\Omega }=|{\mathrm{\Delta }}_1|$ and $|{\mathrm{\Delta }}_2|$ are indicated by dashed lines. Solid curve: $s_{\pm }$. Dashed curve: $s_{++}$.

Figure 8

Integrated density of states components $\delta {\rho }_{\mathrm{intra},\mathrm{inter}}^{\pm }$ for isotropic scatterers $U_{\mathrm{inter}}=0.2U_{\mathrm{intra}}$ for $U_{\mathrm{intra}}=0.01$ [(a) and (b)], $0.2$ [(c) and (d)], and 10 [(e) and (f)] (all in units of energy as in the main text), computed for momentum resolved Green's functions, Eq. (A14). As in Fig. 3, the dashed curves correspond to $s_{++}$ state, solid curves to $s_{\pm }$. Red and black are even and odd components of the LDOS, $\delta {\rho }^{+}$ and $\delta {\rho }^{-}$, respectively. Left panels represent intraband, right panels interband scattering channels, respectively.

Figure 9

Thermally averaged antisymmetrized interband $\mathbf{q}$-integrated LDOS change $\langle \delta {\rho }_{\mathrm{inter}}^{-}\rangle \left(\mathrm{\Omega }\right)$ for nonmagnetic scatterers with parameters $U_{\mathrm{inter}}=0.2U_{\mathrm{intra}}$, with full $t$ matrix and momentum resolved Green's function, Eq. (A14). The external frequency $\mathrm{\Omega }/T_c=2.5$, with $|{\mathrm{\Delta }}_1|/T_c$ and $|{\mathrm{\Delta }}_2|/T_c$ taken as 3.0 and 2.0, respectively. (a) refers to the intermediate potential $U_{\mathrm{intra}}=0.2$. Shown are antisymmetrized LDOS $\langle \delta {\rho }^{-}\left(\mathrm{\Omega }\right)\rangle$ for $s_{++}$ (dashed) and for $s_{\pm }$ state (solid). (b) shows the same for the larger strength potential $U_{\mathrm{intra}}=10$. As in Fig. 5, here we have used the standard BCS type behavior for both superconducting gaps.

Figure 10

Thermally averaged LDOS changes for antisymmetric, interband channel, $|\langle \delta {\rho }_{\mathrm{inter},\mathrm{intra}}\rangle \left(\mathrm{\Omega }\right)|$ vs $\mathrm{\Omega }/T_c$. Curves shown are for nonmagnetic scatterers with parameters $T/T_c=0.1$ (a), 0.3 (b), 0.6 (c), 0.9 (d) and $U_{\mathrm{inter}}=0.2U_{\mathrm{intra}}$ ($U_{\mathrm{intra}}=0.2$), with full $t$ matrix, momentum-resolved Green's functions, Eq. (A14), and $|{\mathrm{\Delta }}_1|/T_c$ and $|{\mathrm{\Delta }}_2|/T_c$ taken as 3.0 and 2.0, respectively. $\mathrm{\Omega }=|{\mathrm{\Delta }}_1|$ and $|{\mathrm{\Delta }}_2|$ are indicated by dashed lines. Solid curve: $s_{\pm }$. Dashed curve: $s_{++}$.