Momentum dependence of the superconducting gap in ${\text{Ba}}_{1-x}{\text{K}}_x{\text{Fe}}_2{\text{As}}_2$

The precise momentum dependence of the superconducting gap in the iron-arsenide superconductor ${\text{Ba}}_{1-x}{\text{K}}_x{\text{Fe}}_2{\text{As}}_2$ (BKFA) with $T_c=32\text{K}$ was determined from angle-resolved photoemission spectroscopy (ARPES) via fitting the distribution of the quasiparticle density to a model. The model incorporates finite lifetime and experimental resolution effects, as well as accounts for peculiarities of BKFA electronic structure. We have found that the value of the superconducting gap is practically the same for the inner $\Gamma$ barrel, $\mathrm{X}$ pocket, and “blade” pocket, and equals 9 meV, while the gap on the outer $\Gamma$ barrel is estimated to be less than 4 meV, resulting in $2\Delta /k_B{T}_c=6.8$ for the large gap and $2\Delta /k_B{T}_c<3$ for the small gap. A large $\left(77\pm 3\%\right)$ nonsuperconducting component in the photoemission signal is observed below $T_c$. Details of gap extraction from ARPES data are discussed in Appendixes A,B.

Figure 1

(Color online) (a) Distribution of the photoemission intensity at the FL with superimposed FS contours (white lines). (b) Momentum-energy cut through the $\Gamma$ point [cut1 in panel (a)] taken at 10 K. (c) Same cut, taken at 45 K. [(d) and (e)] MDC taken at the FL and symmetrized EDC from cuts (b) and (c), respectively. Maxima of the symmetrized EDC are marked by dots. (f) $k_F$ EDC referring to the inner $\Gamma$ barrel recorded at 10 and 45 K. (g) Near-$k_F$ EDC emphasizes onset of the superconductivity even better. (h) Energy dependence of the inner $\Gamma$-barrel intensity extracted from the fit of MDC. (i) The same for the outer $\Gamma$ barrel.

Figure 2

(Color online) Temperature and momentum dependence of the superconducting gap on the inner $\Gamma$ barrel. Reliability of the results. (a) Evolution of the integrated EDC from cut1 [see Fig. 1a] with temperature and fits to formula (1). EDC are shifted along left axis for clarity. Panel (b) shows the extracted magnitude of the gap plotted against temperature. Inset to (b) shows temperature dependence of resistivity (with and without magnetic field) confirming high quality of the crystals and emphasizing equality of bulk and surface $T_c=32\text{K}$. (c) Integrated EDC from cut2 [see Fig. 1a] measured at different excitation energy at 11 K and corresponding fits to formula (1) reveal reproducibility of the data and robustness of the fitting procedure. EDC are shifted along left axis. The inset in (c) shows a single EDC recorded with $h\nu =80\text{eV}$, demonstrating high resolution at high excitation energies.

Figure 3

(Color online) Superconducting gap on the $\mathrm{X}$ pocket. (a) Energy-momentum cut through the $\mathrm{X}$ point [cut3 on Fig. 1a] taken at 36 K. (b) Same cut taken at 11 K. (c) Evolution of the IEDC with temperature and fits to formula (2) and symmetrized $k_F$ EDC. (d) Shows temperature dependence of the gap. (e) Comparison of IEDC referring to the $\mathrm{M}$ pocket and to the blades reveals virtually the same values of the superconducting gap.

Figure 4

(Color online) Momentum dependence of the superconducting gap in ${\text{Ba}}_{1-x}{\text{K}}_x{\text{Fe}}_2{\text{As}}_2$ $\left(T_c=32\text{K}\right)$ is shown as a three-dimensional plot with underlying FS intensity map for orientation. The solid green lines in the basal plane denote the boundaries of the Brillouin zone.

Figure 5

(Color online) Superconducting and nonsuperconducting constituents of the spectrum. (a) Energy distribution of the intensity corresponding to superconducting and nonsuperconducting parts of the spectrum. (b) Second derivatives of the data and fit. Structure of the second derivative confirms presence of superconducting and nonsuperconducting components. (c) Sketch, illustrating presence of two different components in the same spectrum.

Figure 6

(a) Spectral function in the superconducting state for the case of linear normal-state dispersion ${\xi }_k$. According to formula (A4), the spectral weight above the Fermi level is governed by $u_k^2$ and by $v_k^2$ below. (b) Integration in our case is performed along one energy-momentum cut (gray stroke), which intersects Fermi surface at only one point. (c) In the momentum-integrated techniques integration is naturally performed over the whole momentum space.

Figure 7

(Color online) Integration along the contour on the complex plane. According to Cauchy’s residue theorem, the integral along the contour ${\Gamma }_{R,\eta }$ equals to the residue in the pole of the integrand inside, ${\omega }_0+i{\Sigma }^{″}$.

Figure 8

(Color online) Influence of the energy resolution on the determination of the gap via symmetrization, leading edge, and fit. First column: energy resolution for the corresponding row. Second column: simulated energy-momentum cut above $T_c$ (${\Sigma }^{″}=3\text{meV}$, $kT=3\text{meV}$, and $\Delta =0\text{meV}$). Third column: simulated energy-momentum cut below $T_c$ (${\Sigma }^{″}=3\text{meV}$, $kT=1\text{meV}$, and $\Delta =10\text{meV}$). Fourth column: determination of the gap with symmetrization. Fifth column: determination of the gap with leading edge. Sixth column: determination of the gap with fit to formula (1) and ${\chi }^2$ criterion as insets to some panels. Symmetrization and leading edge provide acceptable results for good resolution and fail when the resolution becomes worse. The fitting procedure always provides the correct result.

Figure 9

(Color online) Influence of the momentum resolution on the determination of the gap via symmetrization, leading edge, and fit. First column: energy resolution for the corresponding row. Second column: momentum resolution for the corresponding row. Third column: simulated energy-momentum cut above $T_c$ (${\Sigma }^{″}=3\text{meV}$, $kT=3\text{meV}$, and $\Delta =0\text{meV}$). Fourth column: simulated energy-momentum cut below $T_c$ (${\Sigma }^{″}=3\text{meV}$, $kT=1\text{meV}$, and $\Delta =10\text{meV}$). Fifth column: determination of the gap with symmetrization. Sixth column: determination of the gap with leading edge. Seventh column: determination of the gap with fit to formula (1) and ${\chi }^2$ criterion as insets to some panels. Symmetrization and leading edge provide acceptable results for good resolution and fail when the resolution becomes worse. The fitting procedure always provides the correct result.

Figure 10

(Color online) Influence of the small band depth on the determination of the gap via symmetrization, leading edge, and fit. First column: momentum resolution for the corresponding row. Second column: simulated energy-momentum cut above $T_c$ (${\Sigma }^{″}=3\text{meV}$, $kT=3\text{meV}$, $\Delta =0\text{meV}$, and ${\epsilon }_0=20\text{meV}$). Third column: simulated energy-momentum cut below $T_c$ (${\Sigma }^{″}=3\text{meV}$, $kT=1\text{meV}$, $\Delta =10\text{meV}$, and ${\epsilon }_0=20\text{meV}$). Fourth column: determination of the gap with symmetrization. Fifth column: determination of the gap with leading edge. Sixth column: determination of the gap with fit to formula (2) and ${\chi }^2$ criterion as insets to some panels. First row: no resolution effects added. Second row: small resolution effects are added. Third row: moderate resolution effects are added (resolution effects are comparable and may be even smaller to those in the real data, which is easy to see comparing these energy-momentum cuts to those directly measured). For simplicity, only momentum resolution is added. Symmetrization and leading edge provide acceptable results for good resolution and fail when the resolution becomes worse. The fitting procedure always provides the correct result.