Disorder-robust high-field superconducting phase of FeSe single crystals
Nan Zhou, Yue Sun, C. Y. Xi, Z. S. Wang, Y. F. Zhang, C. Q. Xu, Y. Q. Pan, J. J. Feng, Y. Meng, X. L. Yi, L. Pi, T. Tamegai, Xiangzhuo Xing, Zhixiang Shi
DDisorder-robust high-field superconducting phase of FeSe single crystals
Nan Zhou , , Yue Sun , ∗ C. Y. Xi , Z. S. Wang , Y. F. Zhang , C. Q. Xu , Y. Q. Pan , J.J. Feng , Y. Meng , X. L. Yi , L. Pi , T. Tamegai , Xiangzhuo Xing , † and Zhixiang Shi ‡ School of Physics and Key Laboratory of MEMS of the Ministry of Education, Southeast University, Nanjing 211189, China Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions,High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, China Department of Physics and Mathematics, Aoyama Gaguin University, Kanagawa 252-5258, Japan Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan and Institute for Solid State Physics (ISSP), The University of Tokyo, Kashiwa, Chiba 277-8581, Japan (Dated: February 5, 2021)When exposed to high magnetic fields, certain materials manifest an exotic superconducting (SC)phase that attracts considerable attention. A proposed explanation of the origin of the high-fieldphase is the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state. This state is characterized by inhomo-geneous superconductivity, where the Cooper pairs have finite center-of-mass momenta. Recently,the high-field phase has been observed in FeSe, and it was deemed to originate from the FFLO state.Here, we synthesized FeSe single crystals with different levels of disorders. The level of disorder isexpressed by the ratio of the mean free path to the coherence length and ranges between 35 and 1.2.The upper critical field B c2 was systematically studied over a wide range of temperatures, whichwent as low as ∼ ∼
38 T along the c axis and in the ab plane. In the high-field region parallel to the ab plane, an unusual SC phase was confirmed inall the crystals, and the phase was found to be robust to disorders. This result suggests that thehigh-filed SC state in FeSe may not be a FFLO state, which should be sensitive to disorders. The orbital and Pauli-paramagnetic pair-breaking ef-fects are two distinct mechanisms for destroying super-conductivity and limiting the maximum upper criticalfield in type-II superconductors [1, 2]. However, trig-gered by certain conditions, some unconventional super-conductors can easily overcome the Pauli limitation byforming an exotic SC phase [3–24]. Among them, an in-homogeneous SC state occurs when Pauli pair-breakingeffect dominates over orbital pair-breaking effect, whichis independently predicted by Fulde and Ferrell [3], andLarkin and Ovchinnikov [4] (FFLO state) half a centuryago. In the FFLO state, the Zeeman-split Fermi surfacescould drive the formation of Cooper pairs with a finitecenter-of-mass momenta, thus realizing a spatially mod-ulated SC state. The large Maki parameter α , clean limit(mean free path (cid:96) (cid:29) coherence length ξ ), and unconven-tional pairing symmetries [6] could drive a system easieraccess to the FFLO state. In previous reports, the FFLOstate was suggested to exist in heavy-fermion [5–7], or-ganic [8–18], and some iron-based superconductors [19–22]. In the heavy fermion superconductor CeCoIn , anadditional spin-density wave (SDW) has been observedto coexist with SC [23, 24], implying that the high-fieldphase does not simply originate from the FFLO state. Incontrast, in some layered organic superconductors, theFFLO state was observed without any magnetic order[8–18].The target material of this study is the iron chalco-genide superconductor FeSe [25], which has attractedconsiderable interest due to its unique electrical proper-ties, such as multiband structure [26], electronic nematic-ity [27], extremely small Fermi energy [21], and stronglyorbital-dependent pairing mechanism [28]. Recently, a high-field SC phase has been observed in FeSe at temper-atures below 2 K and under the applied field close to theupper critical field [19, 20]. It has been suggested thatthe high-field SC phase may originate from the FFLOstate [19, 20].Since the phase transition from FFLO to the Bardeen-Cooper-Schrieffer (BCS) state is of the first order, theFFLO state is readily destroyed by disorders [6, 29, 30].Therefore, the disorder effect can be a useful method toclarify whether the high-field SC phase in FeSe stemsfrom the FFLO state. In this study, we probed the high-field SC phase in three FeSe single crystals with a disorderlevel (cid:96) / ξ ranging from 35 to 1.2. We found that the high-field SC phase exists in all the three crystals and is robustagainst disorders.The single crystals of FeSe studied here were synthe-sized by the vapor transport method [31, 32]. Singlecrystals with different amounts of disorders were selectedfrom different batches. The structure of FeSe single crys-tals was characterized by X-ray diffraction (XRD) withCu- K α radiation. The crystal composition was deter-mined by energy-dispersive X-ray spectroscopy (EDX).Temperature dependence of the resistivity up to 9 Twas measured by using a physical property measurementsystem (PPMS, Quantum Design). The magnetizationwas measured by a commercial SC quantum interferencedevice magnetometer (MPMS-XL5, Quantum Design).The high-field transport measurements were carried outusing standard a.c. lock-in techniques with a He cryo-stat and a dc-resistive magnet (up to 38 T) at the HighMagnetic Field Laboratory of Chinese Academy of Sci-ences.Figure 1(a) shows the zero-field resistivity ρ ( T ) for the a r X i v : . [ c ond - m a t . s up r- c on ] F e b S 3S 2S 1 S 1S 2 ( d )( c )( b ) ( a )
S 3S 2
Intensity (a.u.) (cid:2)(cid:1) ( d e g r e e )2 (cid:2)(cid:1) ( d e g r e e ) (103)(003) ( g )( f ) (004)(003)(002) Intensity (a.u.) (cid:2)(cid:1) ( d e g r e e ) (001) ( e ) S 1
S 3
S 1 S 2 S 3
S 1S 2S 3
S 1 S 2 S 3
F e S e F e S e F e S e F e S e F e S e d (cid:4) /d T (a.u.) T ( K ) F e S e (cid:4) (cid:1) ( mW cm ) T ( K ) (cid:4) ( mW cm ) T ( K ) c o h e r e n c e l e n g t h (cid:4) ab (cid:5) ab (nm) S 3S 2 m e a n f r e e p a t h l (cid:3) ( m W c m ) l (nm) S 1 l / (cid:5) ab (nm) (cid:3) (m W c m ) S 1 S 2 S 3 (cid:3)(cid:2) T ( K ) B = 5 O e B | | c S 2S 1
F e S e l / (cid:5) ab (nm) (cid:3) (m W c m ) (cid:3) ( m W c m ) l (nm) S 3S 2S 1 (cid:4) (cid:1) ( mW cm ) T ( K ) (cid:3) (m W c m ) l / (cid:5) ab (nm) F e S e
S 1 S 2 S 3 (cid:5) ab (nm) (cid:3) ( m W c m ) l (nm) S 3S 2
S 1
FIG. 1. Temperature dependencies of the resistivity at zero field in the temperature ranges of (a) 0-300 K, (b) below 16K for three selected FeSe single crystals with different amounts of disorders. The dashed lines correspond to the power-lawfitting ρ ( T ) = ρ + AT α . (c) Temperature dependence of magnetic susceptibility χ measured under 5 Oe field along c axis.(d) Residual resistivity ρ dependence of the ratio of the mean free path (cid:96) to the coherence length ξ ab . The inset displays the ρ dependence of (cid:96) (left axis) and the ξ ab (right axis). (e) The single-crystal XRD patterns for the three selected FeSe singlecrystals. The enlarged diffraction peaks of (f) (003) and (g) (103). three selected FeSe single crystals. As shown in Fig. 1(b),the SC transition temperature T c determined by the zeroresistivity is ∼ ρ is deter-mined by using the power-law fitting ρ ( T ) = ρ + AT α ( ρ , A , and α as the fitting parameters) from normal statedata to zero temperature as shown by the dashed lines inFig. 1(b). The obtained ρ is ∼ µ Ω cm (sample S1),24 µ Ω cm (sample S2) and 54.2 µ Ω cm (sample S3), re-spectively. The residual resistivity ratio (RRR), definedas ρ K / ρ , is estimated as ∼
207 for sample S1, ∼ ∼ T ∼
90 K related to the structuraltransition [33] could be seen more clearly in the plot of dρ / d T as shown in Fig. S1 [34]. The structural transi-tion temperature T s is found to be slightly suppressed bythe disorders, which is consistent with previous reports[32].Superconductivity was confirmed by the temperaturedependence of the susceptibility measurements as shownin Fig. 1(c). The obtained T c that is defined as the devia-tion of the zero-field-cooling and field-cooling susceptibil-ities is in good agreement with the resistivity data. Thesharp SC transition width indicates the homogeneous dis-tribution of disorders. We estimated the mean free path by assuming thatthe hole and electron pockets are compensated perfectly.According to the expressions (cid:96) = πc (cid:126) Ne k F ρ [20], where c isthe lattice parameter, N is the number of formula unitsper unit cell, k F = 1.07 nm − [20] is the Fermi wavevector, and ρ is the residual resistivity, (cid:96) is estimatedas ∼ (cid:96) to the in-plane coherence length ξ ab ispresented in the main panel of Fig. 1(d) and Table I.The estimation of ξ ab will be discussed later. Our resultsconfirm that sample S1 is in the clean limit with (cid:96) / ξ ab >
35. By contrast, considerable amounts of disordershave been successfully introduced into sample S2 and S3because (cid:96) / ξ ab is reduced to ∼ (cid:96) ) peaksare observed for all the three crystals, which can bewell indexed based on a tetragonal structure with the( P nmm ) space group. The positions of peaks werefound to be slightly shifted to a higher angle from S1 toS3, which can be seen more clearly in the enlarged partof (003) peaks shown in Fig. 1(f). To obtain the latticeconstant a / b , we measured (103) peaks by scanning the TABLE I. Summary of the experimentally derived parameters for samples S1, S2, and S3. T c , the temperature for the resistivitydeviates from the SC state; ρ , residual resistivity; RRR , residual resistivity ratio; α , Maki parameter; B c c2 (0) and B ab c2 (0) arethe estimated upper critical fields along c axis, and in the ab plane. (cid:96) , the mean free path; ξ c (0) and ξ ab (0) the coherencelength along c axis and in the ab plane are determined from B c c2 (0) and B ab c2 (0). (cid:96) / ξ ab (0), the ratio of the mean free path (cid:96) tothe coherence length ξ ab (0). T c (K) ρ ( µ Ω cm)
RRR α B c c2 (0)(T) B ab c2 (0)(T) (cid:96) (nm) ξ c (0)(nm) ξ ab (0)(nm) (cid:96) / ξ ab (0)S1 9.0 2.1 207 1.13 15.26 27.21 164 2.6 4.7 35.2S2 8.2 24 20.3 1.32 14.33 25.28 14.1 2.7 4.8 3.0S3 7.1 54.2 12.2 1.48 12.86 22.27 6.3 2.9 5.1 1.2 F e S e
S 3 B ^ c F e S e
S 2 B ^ c F e S e
S 3 B | | c F e S e
S 2 B | | c B ( T ) B ( T ) B ( T ) B ( T ) B ( T ) B ( T ) T ( K ) T ( K ) (cid:1) ( mW cm ) (cid:1) ( mW cm ) ( h )( g )( f ) (cid:1) ( mW cm ) ( e ) (cid:1) ( mW cm ) F e S e
S 1 B ^ c (cid:1) ( mW cm ) B e n d p o in tc 2 B B o n s e tc 2 ( d )( c ) (cid:1) ( mW cm )
0 T 0 . 5 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T (cid:1) ( mW cm ) ( b ) F e S e
S 1 B | | c F e S e
S 1 B ^ c T e n d p o in tc T T o n s e tc F e S e
S 1 B | | c
0 T 0 . 5 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T (cid:1) ( mW cm ) ( a ) FIG. 2. Temperature dependence of the resistivity for sample 1 with (a) B (cid:107) c (b) B ⊥ c under fields from 0 to 9 T. Magneticfield dependence of the resistivity with B (cid:107) c and B ⊥ c at several temperatures for (c-d) sample 1, (e-f) sample 2, and (g-h)sample 3. crystal angle ω independently of 2 θ (angle between inci-dent and scattered X-rays), as shown in Fig. 1(g). Thelattice constants are estimated as a = (3.777 ± c = (5.524 ± a = (3.765 ± c = (5.521 ± a = (3.761 ± c = (5.519 ± a and c decrease with increasing disorders, indicating lat-tice shrinkage. The EDX measurements show that themolar ratio of Fe : Se is ∼ B c2 by the criteria of onset, 50%, andthe end point of the SC transition are shown in Figs. 2(a) and 2(c). In the following main text, B c2 determined bythe criterion of 50% of the SC transition is adopted fordiscussion. While, the B c2 obtained by the criteria of theonset and end point of the SC transition were presentedin Fig. S3 [34]. Here, we want to emphasize that ourconclusion is not affected by the criteria of B c2 . B c2 of the three crystals are shown in Figs. 3(a)–(c).The anisotropy γ = B ab c2 / B c c2 is found to be slightly in-creased in crystals with more disorders (see Fig. S5(d)[34]). The zero-temperature coherence length was esti-mated by using ξ ab (0) = (Φ /2 π B c c2 (0)) / and ξ c (0) =Φ /2 πξ ab (0) B ab c2 (0) (Φ = 2.07 × − Wb is the mag-netic flux quantum). Herein, we extract the B c c2 (0) fromthe two-band fitting for estimating the coherence lengths,with ξ ab (0) ∼ B ab c2 (0) is estimated by linearly ex-trapolating the low temperature upper critical field datato zero temperature, obtains the ξ c (0) ∼ α = B c | T ) |d) α = |c)|b) WHH fittwo-band fit α = α = |a) S1 α = α = B c | T ) S2 T |K) α = α = α = B c | T ) S3 T / T c B c / B W HH c | ) T. Terashima et al .S. Kasahara et al .J. M. Ok et al .S1S2S3T. Terashima et al .S. Kasahara et al .J. M. Ok et al .S1S2S3 B c / B W HH c | ) T / T c B ⊥ cB || cB ⊥ cB ⊥ cB || cB || c B ⊥ c FIG. 3. [(a)–(c)] Temperature dependencies of the uppercritical field B c2 ( T ) for the three selected FeSe single crystals.Symbols of the diamond and circle represent the case of B (cid:107) c and B ⊥ c , respectively. The two-band model (green lines)and WHH fitting curves (black lines) are shown for B (cid:107) c and B ⊥ c . Meanwhile, the WHH model predictions with α = 0are also presented for comparison. (d) The critical field B c2 normalized by the B WHHc2 (0 K) for the crystals used in thispaper and the previous works [19, 20, 36]. The solid linesrepresent the WHH fitting. The inset shows the enlarged lowtemperature part. equations (see Fig. S5 [34]).For B (cid:107) c , the experimental data B c c2 for the three crys-tals is well above the predicted upper limit based on theWerthamer-Helfand-Hohenberg (WHH) theory [1] withMaki parameter α = 0, and spin orbit interaction λ =0 (shown as the black lines), and manifests a convex in-crease at T >
T < B (cid:107) ab , the exper-imental data B ab c2 falls below the WHH prediction at lowtemperatures with α = 0 and λ so = 0, indicating thatthe spin-paramagnetic effect cannot be ignored. By con-sidering a finite α ( λ keeps as 0), the WHH model canfit the B ab c2 well in the temperature range of T > α extracted from the WHH fittingis plotted in Fig. S5(e) [34], which is increased with ρ (Table I).Interestingly, at T < B ab c2 normal-ized by the B WHHc2 , ab (0 K) (WHH fitting with finite α ) areshown in Fig. 3(d). The upper critical field B ab c2 re-ported by T. Terashima et al . (open squares) [36], S.Kasahara et al . (open triangles) [20] and J. M. Ok etal . (open diamonds) [19] are also incorporated for com-parison. Crystals used in those reports are supposed tocontain different amounts of disorders. Obviously, theupturn behavior was observed in the B ab c2 for all the crys-tals. Furthermore, the enhancements of B ab c2 above theWHH fitting for different crystals are almost identical,which can be seen clearly in the inset of Fig. 3(d).Since the FFLO state is very sensitive to disorders[6, 29, 30], it can only exist in crystals in the cleanlimit, which is in stark contrast to our observation ofthe disorder-robust high-field phase. Therefore, the ob-served high-field SC phase seems not to be a FFLO state,although in some theoretically predicted special cases,the FFLO state can survive under moderate disorderstrength [39] or is a second-order phase transition beingmore robust against disorders [40].Another possible origin of the high-field SC phase isthe coexistence of the SDW order, which is triggered bythe nesting effect around the nodal position of SC gap[41]. The SDW order has been discussed as anotherpossible mechanism for the high-field phase in heavyfermion superconductor CeCoIn beside the FFLO state[23, 41]. Theoretical calculation proposed that the SDWorder close to B c2 is a direct result of the strong spin-paramagnetic pair-breaking effect and nodal gap struc-ture [42]. In FeSe, the gap nodes or deep minima in boththe electron-type (cid:15) -band with small gap size and the hole-type α -band with large gap size have already been con-firmed [21, 28, 43], which makes the field-induced SDWorder possible. Such a kind of SDW order coexisting withsuperconductivity should be sensitive to the gap struc-ture. If the nodes or gap minima are smeared out, sucha SDW order will disappear spontaneously. Our previ-ous work has observed that the nodes or gap minima inthe small (cid:15) -gap can be smeared out by disorders [44].However, the nodes or gap minima in the large α -gapshould be more robust against disorders because the v-shape spectrum was observed on the STM measurementsin the FeSe crystals with high level of S-doping [45]. Sincethe high-field SC phase in FeSe is only observed at fieldsclose to B c2 , it should be attributed to the larger α -gap.Therefore, our observation of the disorder-robust behav-ior is not contradictory to the SDW order induced in thehigh-field phase. Further efforts such as nuclear magneticresonance and neutron diffraction measurements are re-quired to clarify the origin of the high-field SC phase inFeSe. Nevertheless, our observations suggest that FeSeprovides an intriguing platform to study the interplaybetween multiple phases such as superconductivity, ne-maticity, and the SDW or FFLO state.To conclude, we studied the upper critical fields of FeSesingle crystals with different amounts of disorders. Ahigh-field SC phase was observed in all the crystals, andit was found to be robust against disorders. These resultssuggest that the high-field SC state in FeSe may not befrom the FFLO state. Therefore, the high-field phaseshould be related to a disorder-robust order, which mayprovide new clues to understand the exotic properties ofFeSe.The authors would like to thank Shigeru Kasaharafor stimulating discussions. A portion of this work wasperformed on the Steady High Magnetic Field Facili-ties, High Magnetic Field Laboratory, Chinese Academyof Sciences, and supported by the High Magnetic FieldLaboratory of Anhui Province. This work was partlysupported by the National Key R&D Program of China(Grant No. 2018YFA0704300), the Strategic Priority Re-search Program (B) of the Chinese Academy of Sciences(Grant No. XDB25000000), the National Natural Sci-ence Foundation of China (Grant No. U1932217, No.11674054, and No. 11874359), and JSPS KAKENHI (No.JP20H05164, No. JP19K14661, and No. JP17H01141).N. Z. and Y. S. contributed equally to this paper. ∗ Corresponding author:[email protected] † Corresponding author:[email protected] ‡ Corresponding author:[email protected][1] N. 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