Friedel Oscillations of Vortex Bound States under Extreme Quantum Limit in ${\mathrm{KCa}}_2{\mathrm{Fe}}_4{\mathrm{As}}_4{\mathrm{F}}_2$

We report the observation of discrete bound states with the energy levels deviating from the widely believed ratio of $1:3:5$ in the vortices of an iron-based superconductor ${\mathrm{KCa}}_2{\mathrm{Fe}}_4{\mathrm{As}}_4{\mathrm{F}}_2$ through scanning tunneling microscopy (STM). Meanwhile Friedel oscillations of vortex bound states are also observed for the first time in related vortices. By doing self-consistent calculations of Bogoliubov–de Gennes equations, we find that at extreme quantum limit, the superconducting order parameter exhibits a Friedel-like oscillation, which modifies the energy levels of the vortex bound states and explains why it deviates from the ratio of $1:3:5$. The observed Friedel oscillations of the bound states can also be roughly interpreted by the theoretical calculations, however some features at high energies could not be explained. We attribute this discrepancy to the high energy bound states with the influence of nearby impurities. Our combined STM measurement and the self-consistent calculations illustrate a generalized feature of vortex bound states in type-II superconductors.

Figure 1

(a) Topographic image measured in ${\mathrm{KCa}}_2{\mathrm{Fe}}_4{\mathrm{As}}_4{\mathrm{F}}_2$. (b) Atomically resolved topography measured in another flat area near the region shown in (a). (c) A set of tunneling spectra measured under ${\mu }_{0}H=0\mathrm{T}$ and along the arrowed line in (a); the spectra plotted in black are measured in hollows. (d) Topographic image of an area near the region shown in (a). (e)–(m) Vortex images acquired at different biases and under ${\mu }_{0}H=2\mathrm{T}$; the mapping area is the same as the one shown in (d). The yellow arrows in (f) and (g) indicate clear Friedel-like oscillations surrounding some vortex cores. (n) Spatial dependence of the differential conductance along the dashed line in (f)–(h) and across one vortex.

Figure 2

(a),(b),(e),(f) Vortex images measured at different energies and under ${\mu }_{0}H=0.2\mathrm{T}$. (c),(g) Tunneling spectra measured close to vortex core centers in (b) and (f), respectively. (d),(h) The line profiles of tunneling spectra across vortex centers for vortex 1 and 2, respectively. Friedel oscillations can be observed in the dark-disc region of vortex 2 at $+3.8\mathrm{meV}$ (f), but they are absent in vortex 1 (b). The vortex core center is determined by the geometric center of the dark disc in (b),(f) at $+3.8\mathrm{meV}$. The yellow dashed lines in (d) and (h) mark the vortex-core center with the same position in (a),(b) and (e),(f), respectively.

Figure 3

(a),(b) Spatial variations of the bound-state energy derived from the tunneling spectra shown in Figs. 2 and 2 for vortex 1 and vortex 2, respectively. Peaks with similar energies are indexed by the same order of the VBS. The black filled symbols represent the peak positions at high energies. The high-energy peak may consist of assembled bound states of several energy levels because the level spacing of the bound states is very small at high energies. (c),(d) Averaged bound-state energy derived from experiments (solid symbols), and theoretical results (open symbols) of bound state energy calculated under and not under EQL.

Figure 4

(a) Line profile of local DOS across a vortex core from theoretical calculations under EQL ($T/T_c=0.01$, $k_F{\xi }_0=6$). (b) Theoretical results of the spatial variation of LDOS at two selected energies. The inset shows the two-dimensional color plot of the LDOS of a vortex core calculated at $E=0.48\mathrm{\Delta }$. (c) Three-dimensional plot of the partial of tunneling spectra shown in Fig. 2. (d) Experimental results of the spatial variation of differential conductance measured across two vortex cores along the blue and red arrowed line in Figs. 2 and 2, respectively.