Inhomogeneous Knight shift in vortex cores of superconducting FeSe
I. Vinograd, S. P. Edwards, Z. Wang, T. Kissikov, J. K. Byland, J. R. Badger, V. Taufour, N. J. Curro
IInhomogeneous Knight shift in vortex cores of superconducting FeSe
I. Vinograd, S. P. Edwards, Z. Wang, T. Kissikov, J. K. Byland, J. R. Badger, V. Taufour, and N. J. Curro Department of Physics and Astronomy, University of California, Davis, California 95616, USA Department of Chemistry, University of California, Davis, California 95616, USA (Dated: February 19, 2021)We report Se NMR data in the normal and superconducting states of a single crystal of FeSe forseveral different field orientations. The Knight shift is suppressed in the superconducting state forin-plane fields, but does not vanish at zero temperature. For fields oriented out of the plane, little orno reduction is observed below T c . These results reflect spin-singlet pairing emerging from a nematicstate with large orbital susceptibility and spin-orbit coupling. The spectra and spin-relaxation ratedata reveal electronic inhomogeneity that is enhanced in the superconducting state, possibly arisingfrom enhanced density of states in the vortex cores. Despite the spin polarization of these states,there is no evidence for antiferromagnetic fluctuations. The iron-based superconductors have attracted broadinterest recently because they can host Majorana modeson the surface, at domain walls, and within vortex cores[1–4]. Fe(Se,Te), and Li(Fe,Co)As contain bands with p z and d xz /d yz character with non-trivial topologies, thatgive rise to both topological surface states as well as abulk Dirac point near the Γ point in k-space [5]. FeSe,although topologically trivial, is a particularly interest-ing case because the superconducting state emerges froma nematic phase that develops below T nem = 91K [6].Moreover, the Fermi energy, E F , in this system is usu-ally small, such that this system lies close to the BCS-BEC crossover regime [7, 8]. Evidence has emerged thatsuggests FeSe exhibits a Fulde-Ferell-Larkin-Ovchinnikov(FFLO) phase at high magnetic fields [9]. The possibilityof both FFLO and dispersive Majorana modes underliesthe importance of a detailed understanding of the natureof the vortices in these materials.To probe vortex matter it is important to first under-stand the underlying superconducting state. The spa-tial part of the superconducting wavefunction in FeSe isgenerally assumed to be either s ± or d -wave. The ne-matic normal state gives rise to twin domains and in-plane anisotropy, and the Fermi surfaces contain differ-ent orbital characters in the two domains. The super-conducting gap, ∆, appears to correlate with the orbitalcontent on the Fermi surface [10]. However, the presenceof the domains may mask intrinsic properties about thedensity of states below T c , and there are conflicting re-ports about the presence or lack of nodes and anisotropyof the superconducting gap function [11–14].Information about the spin component of the wave-function can be gleaned from nuclear magnetic resonance(NMR) Knight shift measurements. The spin susceptibil-ity of a condensate with singlet pairing vanishes, whereasthat with triplet pairing can remain unchanged through T c . Conventional magnetometry cannot discern thesechanges because the spin component is much smaller thanthe orbital component, however the Knight shift is usu-ally dominated by the former and is thus one of the onlyexperimental probes of the spin susceptibility of the con- FIG. 1. (a) Se NMR spectra (normalized) as a function oftemperature for H (cid:107) [110] (in tetragonal unit cell) at high rfpower (35.5 dBm). Below T nem = 91K, the single resonancesplits into two separate peaks, corresponding to domains with H (cid:107) a (upper peak) and H (cid:107) b (lower peak) in the nematicphase. (b) Spectra as a function of temperature for H (cid:107) [110]at low rf power (18.4 dBm). (c,d) Spectra in the supercon-ducting state as a function of radiofrequency pulse power for H (cid:107) [100] and H (cid:107) [110], respectively. Blue circles in (c)indicate the first moment of the spectrum. densate [7, 8]. To date, Knight shift measurements in thesuperconducting state have been inconclusive, revealinglittle or no change below T c [15–18]. A recent study re-ported no change in the Knight shift along the c axis infields up to 16 T, which have been interpreted as evi-dence for highly spin-polarized Fermi liquid in the BCS-BEC regime [19]. A lack of suppression of the Knightshift may be evidence for spin-triplet pairing [20], butmay also reflect thermal instability of the sample dueto eddy-current heating from radiofrequency pulses [21].In fact, spin-orbit coupling can give rise naturally to aspin-triplet component [22, 23]. To fully characterize thesymmetry of the condensate, therefore, it is important tounderstand the full tensor nature of the Knight shift in a r X i v : . [ c ond - m a t . s up r- c on ] F e b the superconducting state.Here we report Se NMR on a high quality singlecrystal as a function of temperature and field. We findthat between 3.6 and 11.7 T, the spin part of the planarKnight shift is reduced by ∼ −
15% from their normalstate values below T c , whereas the out-of-plane compo-nent shows no change within the experimental resolution.These results are consistent with spin singlet pairing inthe presence of large orbital susceptibility and spin-orbitcoupling. Surprisingly, the NMR linewidths broaden in-homogenously by more than a factor of two below T c forplanar fields, but not for H (cid:107) c . Accompanying thisbroadening is a frequency-dependent spin-lattice relax-ation rate, T − , that reveals electronic inhomogeneity inthe superconducting state. This inhomogeneity cannotbe explained by the presence of a conventional vortex lat-tice, but may reflect an enhanced local density of stateswithin the vortex cores. FIG. 2. Linewidth (a) and Knight shifts (b) of the spectrain Fig. 1 as a function of temperature for the field along[110] ∼ a ( (cid:72) ), [1¯10] ∼ b ( (cid:78) ), [100] ( (cid:4) ), and [001] ∼ c ( (cid:7) ).(c) The in-plane anisotropy ∆ K = K a − K b as a function oftemperature. Solid (open) points were acquired at low (high)rf power, respectively, as discussed in the text. Single crystals of FeSe were grown by vapor transportwith a tilted two-temperature zone tube furnace [24].Several samples were characterized with magnetic sus- ceptibility and resistivity measurements, with the bestsamples having T c = 8 . × × H = 11 . T c issuppressed to ∼ . a ( b ) direction in the nematic phase [26]. Thespectra were measured at several temperatures down to2.1 K using low-power rf pulses ( π/ µ s), sweeping frequency and summing the Fouriertransforms. Our results are consistent with previous re-ports [15–18, 27], and reveal a splitting of the single Seresonance below T nem = 91 K due to twinning. The res-onance frequencies are given by f = γH (1 + K ), where γ = 8 .
118 MHz/T is the gyromagnetic ratio and K is theKnight shift. We fit each resonance to a Gaussian func-tion, and Figs. 2(a,b) shows the temperature dependenceof K and the full-width half-maxima, FWHM, for severaldifferent field directions. Below T c , the spectra exhibiteda strong dependence on the pulse power, as illustrated inFig. 1(c,d). The radiofrequency pulses induce eddy cur-rents around the sample, which can lead to Joule heating.As a result, the temperature may temporarily exceed T c immediately after the pulse. Similar effects have been ob-served in other superconductors, leading to misinterpre-tations about the temperature dependence of the Knightshift [21]. The shifts reported in Fig. 2(b) were measuredat 18.4 dBm, where there was no power-dependence tothe spectra.The Knight shift arises from the hyperfine interactionbetween the nuclear spin and the spin and orbital degreesof freedom of the electrons: H hf = I · A S · S + I · A L · L ,where A S,L are the hyperfine coupling tensors, and S and L are the spin and orbital angular momenta. TheKnight shift is given by K α = A Lαα χ orbαα + A Sαα χ spinαα +( A Sαα + A Lαα ) χ mixedαα , where χ spinαα , χ orbαα , and χ mixedαα arethe static spin, orbital, and mixed susceptibilities at zerowavevector [26, 28]. In the absence of spin-orbit cou-pling, the mixed term vanishes and the Knight shift isusually decomposed as K α = K α + A Sαα χ spinαα . K α isoften considered to be a temperature-independent shiftarising from a Van-Vleck orbital susceptibility, however,this decomposition breaks down in the presence of spin-orbit coupling [29]. Moreover, theoretical calculationshave revealed that χ orbαα (cid:29) χ spinαα , χ mixedαα due to the mul-tiorbital nature of the band structure and nematic in-stability [26]. As a result, the relationship between K α and the bulk susceptibility, χ = χ spin + χ orb + 2 χ mixed ,is complicated. Nevertheless, we find that K α varies lin-early with χ above T nem , as shown in the inset of Fig.3. Linear fits to the data yield parameters close to pre-viously reported values [18].For spin-singlet pairing, χ spin and χ mixed should van-ish in the superconducting state, giving rise to a suppres-sion of K below T c , as observed in Fig. 3. For planarfields, K a,b is suppressed by about 100 ±
15 ppm in bothdomains, as well as for the [100] direction oriented 45 ◦ tothe Fe-Fe bond direction. This magnitude of suppressiondoes not change significantly at lower fields. For out ofplane fields, any change in K c is within the noise, but isless than ∼
10 ppm. These results are also independentof applied field, and are consistent with previous reports[19, 30]. Note that K α ( T → (cid:54) = K α , or in other wordsthe low temperature limit of the shift does not equal theintercepts from the K − χ plot. In fact, χ orb is stronglytemperature dependent, so K α does not represent a tem-perature independent Van-Vleck term. The low temper-ature shift reflects a finite χ orb , since the spin componentvanishes for singlet pairing; however impurity states mayplay a role [31]. Determining how much χ spin , χ orb and χ mixed are suppressed below T c will likely require de-tailed theoretical calculations [26]. It is noteworthy thatthe difference K a − K b , shown in Fig. 2(c), exhibits asubtle enhancement below T c . This observation suggeststhat the superconductivity is slightly anisotropic in thetwo domains, and may reflect an anisotropy in the coher-ence lengths, ξ a,b .Below T c the spectra for both domains broaden andbecome asymmetric with a high frequency tail, as ob-served in Fig. 1(c,d) and 2(a). At room temperature,the FWHM of the spectrum ( ∼ .
08 kHz) is close tothe second moment of the nuclear spin dipole momentsof the lattice ( ∼ .
06 kHz). The excess inhomogeneousbroadening above T c may be due to either macroscopic ormicroscopic strain fields [32]. The crystals were initiallysecured to the goniometer with a light coat of super-glue and the coil fit loosely around the sample. In thiscase, the observed linewidths were smaller (open pointsin Fig. 2(a)), reflecting the high quality of this crys-tal. In later measurements, the crystal was remountedand we observed the linewidth increase by a factor ∼ T c ,which is unexpected. A vortex lattice certainly gives riseto a distribution of local magnetic fields, B ( r ), and invery low fields B (cid:28) B c , the second moment of the fielddistribution can be estimated as ∆ B ≈ . φ λ − , FIG. 3. K α versus temperature for fields and orientations.(INSET) K versus χ for in the normal state, using suscepti-bility data from [18]. The dotted lines indicate the best linearfits, with parameters K a = 0 . ± . A aa = 27 . ± . µ B , K c = 0 . ± . A cc = 29 . ± . µ B ,as described in the text. where φ is the flux quantum, and the penetration depthsare λ a = 446 nm, λ c = 1320 nm [33, 34]. There are im-portant corrections to this expression in the higher fieldswhere our measurements were conducted [35], howeverafter accounting for these we estimate that the normalstate spectra should broaden by only ∼ K α ( r ), that isequal to the normal state value within the vortex coresand decays to K α ( T →
0) outside. The spectrum isgiven by the histogram of the local resonance frequency, f ( r ) = γB ( r )(1 + K α ( r )). The exact shape of the spec-trum depends on microscopic details, but if the spatialvariation δK is equal to the 100 ppm suppression ob-served in Fig. 3, the spectrum will broaden by ∼
10 kHz,which agrees well with the excess linewidth below T c inFig. 2(a). These results suggest that the local spin sus-ceptibility within the vortex cores is identical to that inthe normal state.This interpretation is supported by T − measure-ments. Fig. 4(a) shows ( T T ) − versus temperature.The data in the normal state agree well with publishedresults [15, 18, 37]. This quantity drops due to the super-conductivity, and becomes inhomogeneous in the mixedphase. Fig. 4(b) shows that T − increases by nearlya factor of two in the high frequency tails of the spec-tra in the superconducting state, which correspond withthe vortex cores. Localized Caroli-deGennes-Matricon(CdGM) electronic states normally exist within isolatedcores [38]. At higher fields quasiparticles from differentcores can propagate coherently across multiple vortices,and the energy spectrum becomes dispersive, with gap-less excitations remaining within the vortex cores thatgive rise to a finite local density of states (LDOS) whichshould be manifest in any technique sensitive to low en-ergy excitations [39–42]. Indeed NMR studies have iden-tified enhanced spin-lattice-relaxation rate within thevortex cores of both conventional [43] and unconventionalsuperconductors [44, 45]. FIG. 4. (a) ( T T ) − versus temperature (symbols defined inFig. 2). The dotted lines are best fits to a Curie-Weiss form,as described in the text. (b) ( T T ) − versus frequency at3.05K in the mixed phase, revealing an enhanced rate withinthe vortex cores. The spectrum is shown in gray. T − andspectra were acquired with high power rf pulses. There are, however, important differences betweenFeSe and previous observations on other superconduc-tors. In the cuprates, the excess relaxation rate has beenattributed to antiferromagnetic fluctuations from a com-peting ground state to superconductivity [44, 45], as wellas from Doppler-shifted quasiparticles associated with d-wave nodes [46]. In such cases ( T T ) − exhibits a strongCurie-Weiss divergence within the cores, whereas out-side the cores ( T T ) − remains temperature independent.In the s-wave superconductor LaRu P , ( T T ) − in the cores is also strongly temperature dependent, and evenexceeds the value in the normal state [43]. In the case ofFeSe, ( T T ) − exhibits Curie-Weiss behavior in the nor-mal state (dotted lines in Fig. 4a), but drops below T c .This behavior has been attributed to antiferromagneticspin fluctuations that are gapped by the superconduc-tivity [47, 48]. The open circles (squares) in Fig. 4(a,c)show the temperature dependence at the upper end ofthe spectra in the superconducting state. ( T T ) − in thevortex cores changes only by a factor of two from thebackground rate, remaining well below the normal statevalue, and exhibits the same trend with temperature asthe background. These results suggest the absence of anyspin fluctuations within the normal cores of FeSe.FeSe appears unique in that there is a q = 0 spin re-sponse in the vortex cores. It is unclear whether thisbehavior could be related to either a proximity to theBCS-BEC crossover, or either a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase [8, 9] or a field-induced spindensity wave [49] for parallel fields above H ∗ = 24 T.A true FFLO phase should exhibit both segmented vor-tex lines and normal planes where the LDOS reachesthe normal state values, giving rise to inhomogeneouslybroadened NMR spectra. Although H ∼ . H ∗ in ourexperiments, the inhomogeneity we observe already indi-cates the presence of large spin polarization in spatialregions where the superconducting order vanishes. Itis noteworthy that the q = 0 susceptibility in FeSe isdominated by orbital contributions, whereas the finite q response is dominated by spin fluctuations [26]. Conden-sation of singlet pairs enables us to probe the small spinresponse at q = 0. In Ba(Fe,Co) As , finite q spin fluc-tuations can freeze and exhibit long range antiferromag-netism in vortex cores [50]. In FeSe we see no evidencefor such behavior, which may be due to the presence ofnematic order and the different contribution of orbitalversus spin susceptibility. The absence of such fluctua-tions suggests that the high field phase is unrelated toSDW order [49, 51].In summary, we find that the Knight shift is suppressedbelow T c for in-plane fields, but see little to no sup-pression for field along the c -axis. The spectra are in-homogeneously broadened below T c , and T − becomesfrequency-dependent. These observations are consistentwith a finite LDOS within the vortex cores. We find noevidence of competing antiferromagnetic fluctuations inthe vortex cores. Further studies at higher fields or withTe doping should shed light on the unusual nature of thevortex states in this system. Acknowledgment.
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Larsen, B. M. Uranga, G. Stieper, S. L. Holm, C. Bern-hard, T. Wolf, K. Lefmann, B. M. Andersen, and C. Nie-dermayer, Competing superconducting and magnetic or-der parameters and field-induced magnetism in electron-dopedBa(fe1-xCox)2as2, Physical Review B , 024504(2015).[51] B. L. Young, R. R. Urbano, N. J. Curro, J. D. Thompson,J. L. Sarrao, A. B. Vorontsov, and M. J. Graf, Micro-scopic evidence for field-induced magnetism in CeCoIn ,Phys. Rev. Lett. , 036402 (2007). !""! ’ &(" ) &("&"" (" ""("" !*+),- . /*0"*)+1, 2"! "2"!" " *’* * + ! ! !’ , & " ’ &(" * ) /*8! /* / FIG. 5. (a) Temperature dependent resistivity curve show-ing the structural transition, T s ≈
90 K and a T c = 8 . SUPPLEMENTAL INFORMATIONSample Characteristics
Single crystals of FeSe were grown using a vapor trans-port technique with an angled two-zone furnace [24]. Sin-gle crystal samples were characterized via resisitivity andmagnetic susceptibility measurements (Fig. 5). The resis-tivity data (
R/R K ) clearly shows the structural tran-sition T s ≈
90 K and has a superconducting transition T c = 8 . K ) value (18 .
8) show that our samplesare of similar quality to the ones previously reported. Asfirst noted by B¨ohmer, the offset of resistivity supercon-ducting transition corresponds well with the initial down-turn of the magnetic susceptibility. We also note that wefound a slight inverse relationship between sample sizeand transition temperature (Fig. 5(b)). This relation-ship combined with the slight oxidation of storing thesample in an Ar-glovebox for a year explains the slightlylower superconducting transition of the sample measuredin NMR.
Knight shift at Different Power Levels
The left panel of Fig. 6 shows the Knight shift versustemperature for different power levels. In the first case,the spectra were measured at high power and there isno obvious change below T c due rf heating effects. Atlow power, the shifts are reduced in the superconductingstate. Vortex Lineshape and Knight shift Inhomogeneity
Figure 7 compares the spectrum in the normal and su-perconducting state for H || [110]. In order to modelthe lineshape in the mixed phase, we use a Monte Carloapproach to compute the histogram, P ( f ), of local res- FIG. 6. (Knight shifts along the a and b directions measuredat low and high power. The reduction below T c is only evidentwith low power pulses, such that there is no Joule heating ofthe sample. Differences in the shift above T c between highand low power spectra are due to slightly worse alignmentwith [110]/[1¯10] for the low power spectra. onance frequencies in a hexagonal vortex lattice, wherethe frequency is given by f ( r ) = γB ( r )(1 + K ( r )). Thelocal field is given by the London model with a Gaussiancutoff [35]: B ( r ) = H (cid:88) G e − i G · r λ · G ) e −| G | ξ / , (1) G are the reciprocal lattice vectors for a hexagonal vortexlattice, λ = ( λ a , λ c ) with λ a = 446 nm, λ c = 1320 nm,and ξ = 3 . K ( r ) = 0, then P ( f ) is given by the green curve in Fig. 7. Although thisspectrum does have a high frequency tail, it remains toosmall to explain the observed inhomogeneity.To model the spatial dependence of K ( r ), we assumethat it exhibits the periodicity of the vortex lattice withmaxima within the cores and vanishing outside, with theexpression: K ( r ) = δK (cid:88) G e −| G | ξ / (cid:0) e − i G · r − e − i G · r (cid:1) /K , (2)where K = (cid:80) G (cid:48) exp( −| G (cid:48) | ξ / (cid:16) − e − i G (cid:48) · r (cid:17) , r = { a/ , √ a/ } is the location of the field minimum be-tween the vortex cores, a is the unit cell length for thehexagonal vortex lattice, and δK is the shift within thecores. This distribution is illustrated in the inset of Fig.7 for δK = 100 ppm. In this case, P ( f ), shown in blue inFig. 7, is broader and the high frequency tail is extendedup to the normal state resonance frequency. FIG. 7. Se NMR spectra in the normal and superconduct-ing states at 12 T, measured at low power for field along the[110] direction. The solid green and blue regions indicate thetheoretical spectra computed for a vortex lattice with andwithout a finite Knight shift, δK , in the cores, respectively,as discussed above. The inset shows how the resonance fre-quency varies as a function of position with and without δKδK