Friedel Oscillations of Vortex Bound States Under Extreme Quantum Limit in KCa2Fe4As4F2
Xiaoyu Chen, Wen Duan, Xinwei Fan, Wenshan Hong, Kailun Chen, Huan Yang, Shiliang Li, Huiqian Luo, Hai-Hu Wen
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Friedel Oscillations of Vortex Bound States Under Extreme Quantum Limit inKCa Fe As F Xiaoyu Chen, , ∗ Wen Duan, , ∗ Xinwei Fan, , ∗ Wenshan Hong, , , ∗ KailunChen, Huan Yang, , † Shiliang Li, , , Huiqian Luo, , , ‡ and Hai-Hu Wen , § National Laboratory of Solid State Microstructures and Department of Physics,Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, ChinaBeijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, ChinaSchool of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China andSongshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
We report the observation of discrete vortex bound states with the energy levels deviating fromthe widely believed ratio of 1 : 3 : 5 in the vortices of an iron based superconductor KCa Fe As F through scanning tunneling microcopy (STM). Meanwhile Friedel oscillations of vortex bound statesare also observed for the first time in related vortices. By doing self-consistent calculations ofBogoliubov-de Gennes equations, we find that at extreme quantum limit, the superconducting orderparameter exhibits a Friedel-like oscillation, which modifies the energy levels of the vortex boundstates and explains why it deviates from the ratio of 1 : 3 : 5. The observed Friedel oscillationsof the bound states can also be roughly interpreted by the theoretical calculations, however somefeatures at high energies could not be explained. We attribute this discrepancy to the high energybound states with the influence of nearby impurities. Our combined STM measurement and theself-consistent calculations illustrate a generalized feature of the vortex bound states in type-IIsuperconductors. For a vortex in a type-II superconductor, it is gen-erally understood that the quantized magnetic flux ofΦ = h/ e = 2 . × Wb distributes in the regionwith the radius of penetration depth λ ; while the orderparameter ψ ramps up from zero at the core center tothe full value ψ in the scale characterized by the coher-ence length ξ . The vortex core region which has somefeatures of the normal-state can be regarded as a kind ofquantum well surrounded by the gapped superconduct-ing region, and naturally the vortex bound states (VBS)can appear based on the solutions to the Bogoliubov-de-Gennes (BdG) euqations [1–3]. An early simplified an-alytic solution to the BdG equations indicates that thediscrete energy levels of bound states should appear [1]at E µ = ± µ ∆ /E F , and it was argued later that µ =1/2, 3/2, 5/2, · · · [2], with ∆ the superconducting gapand E F the Fermi energy. Later on, the quantized VBSwere predicted based on the self-consistent calculations ofthe BdG equations [4, 5], which also yields the spatial de-pendence of the superconducting gap ∆( r ) ∝ tanh( r/ξ );here, ξ = v F / ∆ with v F the Fermi velocity and ∆ thesuperconducting gap at T = 0. In most superconductors,we have E F ≫ ∆, which is the basic requirement of theBCS theory in the weak coupling limit. Therefore, the en-ergy spacing for neighboring bound states ∆ /E F is toosmall to be discernible; what was shown by experimentalobservations is that a particle-hole symmetric VBS peak(assembled by many crowded bound states) locates atzero energy and then it splits and fans out when movingaway from the vortex center [4–8].The discrete bound states can be however distin-guished when the thermal smearing energy is smaller than the energy spacing of VBS [9], i.e., in the quantumlimit of T /T c ≤ ∆ /E F or T /T c ≤ / ( k F ξ ) with T c thecritical temperature and k F the Fermi wave vector. Thiscan be achieved in superconductors with a relatively largevalue of ∆ /E F . Furthermore, under the extreme quan-tum limit (EQL), T /T c ≪ ∆ /E F , it was shown that the∆( r ) should exhibit an oscillatory spatial variation witha period of about π/k F instead of the monotonic vari-ation behavior of tanh( r/ξ ) [9], and meanwhile Friedeloscillations of the charge profile which is related to thelocal density of states (LDOS) were predicted [9–11]. Ex-perimentally, some traces of discrete VBS were reportedas an asymmetric peak or two close peaks near zero-biasmeasured in vortex centers of cuprates [12], YNi B C[13], and some iron-based superconductors [14–18]. Itwas shown later that the two close peaks in cuprates arenot the VBS [19, 20]. Recently, discrete VBS were clearlyobserved in vortices measured in FeTe . Se . with thepeak-energy ratio near 1 : 3 : 5 [21]. Clear discrete boundstate peaks were also observed in the FeSe monolayer [22],and they were observed coexisting with the Majoranazero mode in FeTe . Se . [23], (Li . Fe . )OHFeSe[24], and KCaFe As [25]. Some preliminary signaturesof Friedel oscillations of the bound states were also re-ported around vortices in some iron based superconduc-tors [15, 18, 21], these were discussed as the consequenceof VBS in the quantum limit.In this Letter, we present scanning tunneling mi-croscopy/spectroscopy (STM/STS) results of the VBSin KCa Fe As F . Clear discrete VBS are observed withenergy ratio deviating from the expected 1 : 3 : 5, andFriedel oscillations are also observed in some vortices as FIG. 1. (a) Topographic image measured in KCa Fe As F .(b) A set of tunneling spectra measured under µ H = 0 T andalong a line with the length of 37 nm in the region near theshown area of (a). (c)-(k) Vortex images acquired at differentbiases and under µ H = 2 T; the mapping area is the sameas the one shown in (a). the VBS in the extreme quantum limit (EQL). We alsoobserve some Friedel oscillations surrounding the vortexcenter at energies near the superconducting gap, whichcannot be explained by the theory in the clean limit.Combining with the theoretical calculations, we concludethat the explicit evidence of the behaviors for the VBSunder EQL have been found.KCa Fe As F (K12442) is a newly found iron basedsuperconductor with T c = 33 . α pocket based on STM/STSmeasurements [28]. Hence, the VBS should be inter-esting in this multiple and shallow band superconduc-tor. Single crystals of KCa Fe As F were grown bythe self-flux method [29]. STM/STS measurements werecarried out in a scanning tunneling microscope (USM-1300, Unisoku Co., Ltd.). K12442 samples were cleavedat about 77 K in an ultrahigh vacuum with the basepressure of about 1 × − Torr, and then they weretransferred to the microscopy head which was kept at alow temperature. Electrochemically etched tungsten tipswere used for STM/STS measurements after the electron-beam heating. A typical lock-in technique was used inSTS measurements with an ac modulation of 0.1 mVand the frequency of 931.773 Hz. Setpoint conditionsare V set = 10 mV and I set = 200 pA; temperature is 0.4K for all STM/STS measurements.Figure 1(a) shows a typical topography which is com-monly obtained on the cleaved surface of K12442 [28]. The flat area is the √ ×√ . ± . I /d V ). Although the scanning area is not bigenough to see many vortices, one can still see that vor-tices are randomly but almost equidistant distributed.The expected hexagonal or square vortex lattice do notshow up in the mapping area, which suggests that thevortex pinning may be strong in the sample. In Fig. 1(c),we show an image of differential conductance measured at E = 0 meV, and one can see vortex cores with the diam-eters of about ξ ≈ . E = 3 . ≈ . Te /FeTe . Se . heterostructures[31], the dark disc here shows some internal structurein vortex images mapped at energies near ∆ in K12442[Figs. 1(h)-1(j)].The mapping of a single vortex is carried out undera small magnetic field of 0.2 T in order to minimizethe vortex-vortex interaction. Figure 2 shows vorteximages and tunneling spectra measured in two typicalvortices without (vortex 1, Fig. 2(b)) and with (vortex2, Fig. 2(f)) spatial oscillations in dark disc regions at E = +3 . FIG. 2. (a,b),(e,f) Vortex images measured at different ener-gies and under µ H = 0 . consistent with the theoretical prediction that the peakamplitude for a selected bound state will show spatialoscillation in the EQL [9, 13]. Here, the bound-state en-ergies are about 0 .
8, 1 .
3, 1 . . .
9, 2 .
1, and 2 . . . . . . E / : E / : E / = 1 : 3 : 5[2].Since there are many orders of VBS in line pro-files of tunneling spectra across vortex centers shown inFigs. 2(d) and 2(h), the peak energies of these boundstates can be extracted and they are plotted in Figs. 3(a)and 3(b). By ascribing the bound state peaks with sim-ilar energies to the same order of the VBS, we can de-rive averaged values of bound state energies and showthem in Figs. 3(c) and 3(d) for the two selected vortices.Obviously, the averaged bound-state energy is deviatingfrom the theoretical relationship of E µ = µ ∆ /E F . It -4 -3 -2 -1 0 1 2 3 4-15-10-505101520 E -5/2 E -4 -3 -2 -1 0 1 2 3 4-15-10-505101520-7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/2-4-3-2-101234 -5/2 -3/2 -1/2 1/2 3/2 5/2-4-3-2-101234 E -5/2 r ( n m ) Energy (meV)Vortex 1 E E -7/2 (a) E -3/2 E -1/2 E E E r ( n m ) Energy (meV)E E E -1/2 Vortex 2(b) E -3/2 E -3/2 EQL T/T c = 0.01 k F 0 = 10 Not EQL E E E -7/2 Vortex 1 E ( m e V ) E -1/2 E -5/2 E E E (c) EQL T/T c = 0.01 k F 0 = 6 Not EQL E
Vortex 2 E ( m e V ) E -1/2 E -3/2 E -5/2 E E E (d) FIG. 3. (a),(b) Spatial variations of the bound-state energyderived from the tunneling spectra shown in Figs. 2(d) and2(h) for vortex 1 and vortex 2, respectively. Peaks with similarenergies are indexed by the same order of the VBS. The darkfilled symbols represent the peak positions of many assembledVBS at high energies, thus the energy is not fixed. (c),(d)Averaged bound-state energy derived from experiments (fullsymbols), and theoretical results (open symbols) of boundstate energy calculated under and not under the quantumlimit. should be noted that the bound state energy does nothave to follow this relation at extreme low temperatures,namely in the EQL of
T /T c ≪ ∆ /E F , when there aresome oscillations in ∆( r ) which makes the bound-stateenergy deviate from the linear relation E µ = µ ∆ /E F [9]. Then we do self-consistent calculations of BdG equa-tions based on the routes in previous reports [4, 9]. Thedetailed calculations will be presented separately. Onecan see in Figs. 3(c) and 3(d) that the theoretical curvesof E µ match our experimental data well with differentvalues of k F ξ at the limit of T /T c = 0 .
01. It should benoted that ξ/ξ ≈ ( k F ξ ) − in quantum limit [9], so here ξ used in our calculations is much larger than the coresize ξ determined from the experiments.Based on self-consistent calculations of BdG equations,the line profile of LDOS across a vortex core under theEQL is shown in Fig. 4(a). The result clearly shows dis-crete bound state peaks and spatial oscillations of LDOS.These Friedel oscillations can be clearly seen at fixed en-ergies, two typical examples are given in Fig. 4(c) for dif-ferent energies. For the low energy one at E = 0 . E = 0 . E = 3 . E = +3 . E ± / and E ± / ) also have spatialoscillations as marked by the red circle in Fig. 4(b). Butinterestingly, even with these different features in vortex1 and 2, the oscillation periodicity seems to be similarto each other. Since Friedel oscillations should have theperiodicity[9] of about π/k F , we just calculate the valueno matter where the Friedel oscillations locate in thesetwo vortices. The obtained period of π/k F is about 3.6nm for vortex 1 and 3.0 nm for vortex 2.In the vortices measured on the K12442 sample, dis-crete VBS are observed with the energy levels deviatingfrom the ratio of 1 : 3 : 5. This is due to a relatively high T c and large value of ∆ /E F , which makes the extremequantum limit condition T /T c ≪ ∆ /E F easily satisfied.The Friedel oscillation can be clearly seen around the vor-tices also because the EQL is satisfied. The smaller k F or k F ξ makes the relation of E µ more deviating from thelinear relation of E µ = µ ∆ /E F although the quantumlimit condition T /T c ≤ ∆ /E F are all satisfied [10]. InK12442, the dominant scattering is the intra-band scat-tering of the hole-like α pocket which has a small k F .As mentioned above, the feature of VBS in vortex 2 ata high energy is not compatible with current theoreticalcalculations. In Fig. 4(d), one can see that the back-ground differential conductance at the vortex center ismuch higher for vortex 2 when compared with the onefor vortex 1. A reasonable explanation is that the impu-rity scattering is strong in core area of vortex 2, whichwill bring in more complex into the calculation to the FIG. 4. (a) Line profile of local DOS across a vortex corefrom theoretical calculations under the EQL (
T /T c = 0 . k F ξ = 6). (b) Three-dimensional plot of the partial of tun-neling spectra shown in Fig. 2(h). (c) Theoretical results ofthe spatial variation of LDOS at two selected energies. Theinset shows the two-dimensional color plot of the LDOS of avortex core calculated at E = 0 . BdG equations. Thus we believe that the discrepancybetween the features of vortex 2 at a high energy and therelated theoretical calculations is induced by the effectof impurities in the core region, which modifies the totalHamiltonian involved in the calculations. Unfortunatelytheoretical considerations on the VBS due to both thevortex confinement and the impurity effect are still lack-ing and thus highly desired. Our results of the boundstate energies deviating from the ratio of 1 : 3 : 5 and theobservations of the Friedel oscillations clearly indicatethat the extreme quantum limit condition is satisfied inpresent system.In conclusion, we have observed discrete vortex boundstates with energies deviating from 1 : 3 : 5 inKCa Fe As F . Friedel oscillations of the vortex boundstates are also observed. These two unique features areconsistent with our self-consistent calculations on theBdG equations under the extreme quantum limit. How-ever, in some vortices at energies close to the gap, weobserve the Friedel oscillations staring from the vortexcore center, this cannot be explained by the present the-oretical frame. We attribute this discrepancy to the co-operative effect by both the vortex confinement and im-purity scattering. Our results inspire a more completetheoretical treatment to include also the impurity scat-tering when solving the BdG equations, and should shednew light on a generalized understanding on the vortexcore state in a type-II superconductor.We appreciate very useful discussions with ChristopheBerthod and Da Wang. This work was supportedby National Key R&D Program of China (GrantsNo. 2016YFA0300401, No. 2018YFA0704200, No.2017YFA0303100, and No. 2017YFA0302900), Na-tional Natural Science Foundation of China (GrantsNo. 12061131001, No. 11974171, No. 11822411, No.11961160699, No. 11674406, and No. 11674372), andthe Strategic Priority Research Program (B) of Chi-nese Academy of Sciences (Grants No. XDB25000000,and No. XDB33000000). H. 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