Momentum dependence of superconducting gap, strong-coupling dispersion kink, and tightly bound Cooper pairs in the high-$T_c$ ${\left(\text{Sr},\text{Ba}\right)}_{1-x}{\left(\text{K},\text{Na}\right)}_x{\text{Fe}}_2{\text{As}}_2$ superconductors

We present a systematic angle-resolved photoemission spectroscopic study of the high-$\mathrm{T}c$ superconductor class ${\left(\text{Sr}/\text{Ba}\right)}_{1-x}{\text{K}}_x{\text{Fe}}_2{\text{As}}_2$. By utilizing a photon-energy-modulation contrast and scattering geometry we report the Fermi surface and the momentum dependence of the superconducting gap, $\Delta \left(\vec{k}\right)$. A prominent quasiparticle dispersion kink reflecting strong scattering processes is observed in a binding-energy range of 25–55 meV in the superconducting state, and the coherence length or the extent of the Cooper pair wave function is found to be about $20\text{Å}$, which is uncharacteristic of a superconducting phase realized by the BCS-phonon-retardation mechanism. The observed $40\pm 15\text{meV}$ kink likely reflects contributions from the frustrated spin excitations in a $J_1\text{−}{J}_2$ magnetic background and scattering from the soft phonons. Results taken collectively provide direct clues to the nature of the pairing potential including an internal phase-shift factor in the superconducting order parameter which leads to a Brillouin zone node in a strong-coupling setting.

Figure 1

(Color online) Phase transition, magnetization, and STM characterization. (a), (b) Bulk $T_c$ of crystalline $\left(\text{Ba},\text{K}\right){\text{Fe}}_2{\text{As}}_2$ and $\left(\text{Sr},\text{K}\right){\text{Fe}}_2{\text{As}}_2$ was determined based on the resistivity and magnetization profiles. $\left(\text{Ba},\text{K}\right){\text{Fe}}_2{\text{As}}_2$ samples exhibited $T_c=37\text{K}$ and $\delta T_c\sim 1\text{K}$ whereas $\left(\text{Sr},\text{K}\right){\text{Fe}}_2{\text{As}}_2$ samples exhibited a broad $\left(\sim 10\text{K}\right)$ transition with a $T_c\sim 26\text{K}$. (c) Surface quality was studied by atomic-resolution STM which revealed a high degree of flatness and confirmed the suitability for spectroscopic measurements. The derivative of a STM image is shown which was taken on a $500\times 500{\text{Å}}^2$ patch. The inset shows STM data in the superconducting state with a gap of $2\Delta \approx 30\text{meV}$ and kink structures around 40–50 meV loss energies.

Figure 2

(Color online) Fermi surface and quasiparticle behavior: (a) ARPES intensity integrated within $\pm 15\text{meV}$ of Fermi level in $\left(\text{Ba},\text{K}\right){\text{Fe}}_2{\text{As}}_2$. (b) Second-derivative image approximation of the Fermi-surface topology around the $\Gamma$ point. (c)–(e) Quasiparticle dispersion along cut-2 and its temperature evolution. (f) High-resolution fine-step binding-energy scans shown for some selected $k$-space points (A and B) near the [1,0] and [1,1] axes on the outer FS surrounding the $\Gamma$ point and (C) on a separate FS close to the $M$ point. (g)–(i) Quasiparticle intensity profiles along $k$-space cuts 1–3. The $k$-space cut-2 strongly suppresses the outer FS and provides a clear spectroscopic look at the quasiparticle that forms the innermost FS. Because of the spectral clarity this quasiparticle can be studied in quantitative detail. (j) Fermi-surface image $\left(\pm 15\text{meV}\right)$ taken on $\left(\text{Sr},\text{K}\right){\text{Fe}}_2{\text{As}}_2$. (k), (l) Wide $k$-range coarse-step scans are shown which were used for locating the Fermi crossings. (m) ARPES intensity map within $\pm 15\text{meV}$ of Fermi energy over the complete BZ measured with a photon energy of 18 eV. Hole-like (${\Gamma }_1$, ${\Gamma }_2$, $M_{\text{hl}}$) and electron-like $\left(M_{\text{el}}\right)$ Fermi sheets are labeled.

Figure 3

(Color online) Quasiparticle dispersion anomalies: (a) quasiparticle band dispersion along cut-4 at 15 K. The $k$-space cut-4 is approximately along the $Q_{\text{AF}}$ vector of the undoped compound, as defined in Fig. 2b. (b) Band dispersion for the ${\Gamma }_2$ quasiparticle along cut-5 shows a small kink near 18 meV. (c)–(e) Close-up view of dispersion along cut-4 near the 40 meV kink at 11 and 25 K. The MDCs can be fitted by Lorentzians with linear backgrounds. (d), (f) Quasiparticle line shapes for panels (c) and (e) are presented at selected energies (10–70 meV).

Figure 4

(Color online) Quasiparticle kink in the superconducting state: (a) by tracing the peak positions, quasiparticle band dispersion is plotted at $T=11$, 25, and 41 K. At all temperatures, the dispersion curve shows a “kinklike” feature at $40\pm 10\text{meV}$. The gray dashed line illustrates the “bare” band used for the self-energy estimation. The smaller kink at $18\pm 5\text{meV}$ on the ${\Gamma }_2$ band is shown in the inset measured at 11 K. (b) The real part of self-energy is obtained by subtracting the gray dashed line from the experimental band dispersion. Peak position is used to define the “kink” position. The kink-2 data were taken at 11 K. (c) Fitted MDC width as a function of binding energy for quasiparticles on ${\Gamma }_1$. A small $y$-axis offset in the $T=25\text{K}$ data is due to temperature-dependent (nonintrinsic) shifting of beam position on the sample and of no physical significance.

Figure 5

(Color online) Momentum and Fermi-surface dependence of superconducting gap: (a) temperature dependence of quasiparticles (cut-2) near the Fermi level through the superconducting transition. Below $T_c$ samples exhibit coherence-peak-like behavior similar to what is observed in some cuprates. Temperature dependence of the gap at the $k$-space location of (b) the innermost $\left({\Gamma }_1\right)$ FS and (c) the outer central $\left({\Gamma }_2\right)$ FS is estimated by symmetrization around $E_F$. (d) The temperature dependences of the gaps measured at different FS locations (${\Gamma }_1$ FS, blue; ${\Gamma }_2$ FS, red; $M_{\text{hl}}$, green; and $M_{\text{el}}$, black) are plotted along with the bulk resistivity curve (dotted green). A fluctuation regime above $T_c$ is observed. (e) The azimuthal $k$ dependences of gaps, $\Delta \left(k\right)$, are shown for different FS sheet locations on a polar plot. Blue and red solid lines are contours of the $\text{cos}\times \text{cos}$ function for the Fermi surface outlined in Fig. 2m.