Zooming into the coexisting regime of ferromagnetism and superconductivity in ErRh4B4 single crystals
Ruslan Prozorov, Matthew D. Vannette, Stephanie A. Law, Sergey L. Bud'ko, Paul C. Canfield
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Zooming into the coexisting regime of ferromagnetism and superconductivity inErRh B single crystals Ruslan Prozorov, ∗ Matthew D. Vannette, Stephanie A. Law, Sergey L. Bud’ko, and Paul C. Canfield
Ames Laboratory and Department of Physics & Astronomy, Iowa State University, Ames, IA 50011 (Dated: 29 June 2007)High resolution measurements of the dynamic magnetic susceptibility are reported for ferro-magnetic re-entrant superconductor, ErRh B . Detailed investigation of the coexisting regimereveals unusual temperature-asymmetric and magnetically anisotropic behavior. The supercon-ducting phase appears via a series of discontinuous steps upon warming from the ferromagneticnormal phase, whereas the ferromagnetic phase develops via a gradual transition. A model basedon local field inhomogeneity is proposed to explain the observations. PACS numbers: 74.25.Dw; 75.50.Cc; 74.25.Ha; 74.25.-q; 74.90.+n
The coexistence of the long-range magnetic order andsuperconductivity was first discussed even before the ap-pearance of the microscopic theory of superconductivity[1]. Since then this topic remains one of the most inter-esting and controversial in the physics of superconduc-tors with many reviews and books devoted to the subject[2, 3, 4, 5, 6]. Despite significant effort in new materi-als design and discovery, there are only few, confirmed,ferromagnetic superconductors. Local, full-moment fer-romagnetic superconductors: ErRh B [7] ( T F M ≈ . T c ≈ . x Mo S [8] ( T F M ≈ . T c ≈ . B C [9] ( T F M ≈ . T AF M ≈ T c ≈
11 K); and more recent itin-erant superconducting ferromagnets, UGe ( T F M ≈ T c ≈ .
95 K at P ≈ . T F M ≈ . T c ≈ .
27 K). Whereas all these ma-terials are very interesting on their own, the coexistenceof growing, large, local moment ferromagnetism and su-perconductivity is most clearly presented in ErRh B ,which is the subject of the present Letter. In particular,we are interested in the details of the narrow tempera-ture interval ( ∼ . B was extensively studied over past 30 years[2, 3, 4, 5, 6]. The ferromagnetic phase is primitivetetragonal with the c − axis being the hard and a − axisbeing the easy magnetic axes. Detailed measurements ofanisotropic magnetization and upper critical filed, H c were done by Crabtree et al. [11, 12], who found that H ac (along a − axis) peaks at 5.5 K due to large param-agnetic spin susceptibility in that direction [3, 11]. In thecontrast, H cc collapses near the onset of the long-rangeferromagnetic order.Neutron diffraction studies have established the ex-istence of a modulated ferromagnetic structure at thelengthscale of ∼
10 nm [13, 14]. In single crystals, resultssuggested that coexisting phases consists of a mosaic ofnormal FM domains and SC regions larger than ∼ ∼
10 nm. These regions could beregular domains, spontaneous vortex lattices or laminar structures with ≥
200 nm periodicity and modulated SCdomains in between [14]. Thermal hysteresis is observedboth in the normal Bragg peak intensity and the small-angle peaks. For the small-angle peaks, the intensity ishigher on cooling than on warming. This is opposite tothe behavior of the regular Bragg peaks from the FMregions [13]. Furthermore, the first-order transition, ob-served in satellite peaks temperature dependence [14], isconsistent with the spiral state of Blount and Varma [15].However, a continuos transition was reported in otherneutron diffraction [13, 16] and specific heat experiments[17]. Such a transition can be realized in a modulatedstructure or via spontaneous vortex phase.Theoretically, some striking features of the coexist-ing phase include an inhomogeneous, spiral, FM struc-ture [15, 18] or a fine domain, ”cryptoferromagnetic”phase [2, 19], a vortex - lattice modulated spin struc-ture [20], type-I superconductivity [2, 20, 21], a gaplessregime and possibly, an inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [2]. Another interest-ing possibility is the development of superconductivity atthe ferromagnetic domain walls [22, 23].In this Letter we report precision measurements of thedynamic magnetic susceptibility of ErRh B with an em-phasis on the narrow temperature region where ferromag-netism and superconductivity coexist. We find that thetransition is highly asymmetric when FM → SC (heat-ing) and SC → FM (cooling) data are compared. TheFM ↔ SC transition proceeds via a series of discrete stepsfrom FM to SC phase upon warming and proceeds via asmooth crossover from the SC to FM state upon cooling.With this new information we analyze relevance of somepredictions made over years for the coexisting phase.Single crystals of ErRh B were grown at high temper-atures from molten copper flux as described in [24, 25].Resulting samples were needle shaped with crystallo-graphic c-axis along the needle. Transport measurementsgave residual resistivity ratio of about 8, consistent withprevious reports. The anisotropic H c ( T ) curves (see in-set to Fig.3 below) are consistent with earlier reports aswell [11, 12]. T (K)
H = 0H=4000 Oe H c
FIG. 1: Dynamic magnetic susceptibility in ErRh B singlecrystal measured along the magnetic easy axis (perpendicu-lar to the needle-shaped sample). Each curve corresponds toa fixed value of the applied dc magnetic field in the rangeindicated in the figure. (color online) The AC magnetic susceptibility, χ , was measured witha tunnel-diode resonator (TDR) which is sensitive tochanges in susceptibility ∆ χ ∼ − . Details of the mea-surement technique are described elsewhere [26, 27, 28].In brief, properly biased tunnel diode compensates forlosses in the tank circuit, so it is self-resonating on itsresonant frequency, ω = 1 / √ LC ∼
10 MHz. A sampleis inserted into the coil on a sapphire rod. The effectiveinductance changes and this causes a change in the res-onant frequency. This frequency shift is the measuredquantity and it is proportional to the sample dynamicmagnetic susceptibility, χ [26, 27, 28]. Knowing geomet-rical calibration factors of our circuit, we obtain χ ( T, H ).Advantages of this technique are: very small AC excita-tion field amplitude ( ∼
20 mOe), which means that itonly probes, but does not disturb the superconductingstate; high stability and excellent temperature resolution( ∼ ∼
500 mK wide. Normal-stateskin depth is larger than the sample size, so we probe theentire bulk in the coexisting region, but when supercon-ducting phase becomes dominant, there is a possibilitythat some FM patches still exist, but are screened.Figure 1 shows the full temperature scale magnetic sus-ceptibility, χ , in single crystal ErRh B for an appliedfield oriented along the easy axis and perpendicular tothe needle-shaped crystal. The peak in χ ( T ) at the fer-romagnetic to superconducting boundary below 1 K is T (K)
H = 0 - 3 kOeH = 0
FIG. 2: Magnetic susceptibility in a ferromagnet-superconductor transition region measured at different ap-plied fields. Note the temperature scale and highly assymet-ric character of the FM-SC and SC-FM transitions. (coloronline). Blue - warming, red - cooling. a common feature observed in local moment ferromag-nets [29]. Clearly, superconductivity is fully suppressedin the ferromagnetic phase. Note that at elevated fields,the response is nonmonotonic on the SC side close, tothe FM boundary, indicative of enhanced diamagnetism(larger, negative χ ), which may be due to suppressedmagnetic pairbreaking or entering into another phase,such as FFLO [2].Figure 2 zooms into the SC ↔ FM transition region.Measurements were taken after zero-field cooling, apply-ing external field and warming up (ZFC-W) above T c and then cooling back to the lowest temperature (FC-C).There is striking asymmetry of the transitions - when thesuperconducting phase develops out of the FM state, theresponse proceeds with jumps in the susceptibility, whichare clearly associated with the appearance of supercon-ducting regions of finite size. The steps are present upto the largest field at which superconductivity survives.Decreasing temperature shows a completely different re-sult: the transition is smooth and gradual and proceedsto lower temperatures.To better understand the dynamics of the transition,the top frame of Fig. 3 shows measurements at H = 0 fordifferent temperature ramp rates. Temperature variationis shown in the inset. These data clearly demonstratethat this hysteresis is insensitive to heating/cooling rates.It should be noted that all the other data presented inthis Letter were taken with the slowest cooling rate of 60 µ K/s.Similar hysteresis and steps are also present in the an- T (K)
H = 0 -3 K/s1x10 -3 K/s2x10 -4 K/s6x10 -5 K/s T ( K ) time (s) ramp rates: FIG. 3: Hysteresis of the transition at zero applied field mea-sured at different ramp rates. Temperature sweep profiles areshown in the inset. Clearly, this hysteresis and jumps are notdynamic effects. T (K)
T (K)
H=0
H || c
H || c H ( k O e ) T (K) lines - TDRsymbols - transport
H c FM FIG. 4: Transition region measured for magnetic field ori-ented along the needle and c-axis. Inset: summary phase di-agram for two orientations measured by transport (symbols)and TDR. other orientation, when external magnetic field is appliedalong the c − axis. This is shown in Fig. 4. Note that peakin χ ( T ) at the FM boundary is not present, which isconsistent with the behavior of anisotropic local-momentferromagnet [29]. The inset to Fig. 4 shows the phase di-agram obtained from resistivity and TDR measurementsfor both orientations. There is excellent agreement be-tween the two techniques and, as noted earlier, this dia-
10 9 7 6,84 3,52 T (K)
H = 660 Oe1 0.1 K H || c T (K)H = 0
FIG. 5: Details of the hysteresis with partial scans as ex-plained in the text. Inset shows anothe partial loop on acooling part of the curve. gram is consistent with previous reports [11, 12].Finally, Fig. 5 shows so called minor hysteresis loops(not as function of field, but temperature). The labelsshow the evolution of the susceptibility. It starts fromlow temperature at (1) when sample was warmed up tofirst signs of superconductivity that appeared as smalljump at (2), then warmed further and reaching almostfull superconductivity at (3), but then cooled back downto (4) as indicated by arrow and warmed back to (5).Note that along (3) → (4) χ is significantly differentfrom (4) → (5). Another similar minor cooling-warmingloop follows (6) → (7) → (8) after which the sample wascooled down to return to (10) = (1) via (9). Interestingly,there are no steps or jumps observed on the minor loopseven on warming. Also, the slope dχ/dt is similar on cool-ing and warming and is very different from the originalsteep slope (2) → (3). This is consistent with the presentof vortices, probably pinned by the modulated FM/SCstructure. The inset in Fig. 5 shows a small minor loopon a cooling part. This loop has small slope comparableto the larger loops described in the main frame.Let us now turn to the interpretation of these results.Clearly, the FM → SC transition proceeds via a series ofjumps in diamagnetic screening due to formation of su-perconducting regions of macroscopic volume, roughly5% −
20% of the sample volume depending on the ap-plied field and temperature. Indeed, each observed stepmay be a result of simultaneous formation of many in-dividual superconducting domains of similar size. Thesesteps in χ are present both for H || c and H || a axes, al-though in the latter case the steps are smaller and aremore pronounced, possibly due to magnetic and shapeanisotropies. The number of steps increases with the in-creasing field and the first step (the first sign of supercon-ductivity) occurs at a higher temperature for larger ap-plied field. Overall, FM → SC transition is apparently ofthe first order and exhibits behavior consistent with type-I superconductivity as predicted theoretically [2, 20, 21].From our point of view, the first jump occurs at a tem-perature where internal field is equal to a supercoolingfield of a type-I superconductor. (Note that apparent su-perheating of the FM → SC transition [17] corresponds tosupercooling of a regular type-I superconductor [17, 30],because we enter normal phase on cooling). When firstsuperconducting domains appear, effective magnetic fieldaround them increases due to the flux expulsion and in-ternal field becomes more inhomogeneous. This net in-crease in the internal field in the remaining FM regionsstabilizes them to higher temperatures. The system nowneeds to get farther away from the initial FM boundary,deeper into the SC state to produce more superconduct-ing patches. In this scenario, the observed jumps in χ cor-respond to a cascade of supercooling transitions. If thetemperature is lowered before the transition is completedomains stay stable to lower temperatures due to physicssimilar to superheating of a type-I superconductor. It isalso quite possible that superconducting domains havethe modulated spin structure seen in neutron scattering[14]. Finally, it seems that ferromagnetic domains are notdirectly related to the observed steps, because at higherfields, the number of these domains decrease and domi-nant domains (along the applied field) grow in size.In a striking contrast with FM → SC transition, theSC → FM transition is smooth and proceeds to much lowertemperatures. Yet the transition is hysteretic as evidentfrom the minor loops shown in Fig. 5. It is possiblethat Abrikosov vortices are being spontaneously createdas the temperature is lowered and the systems crossesover into the normal state when vortex cores overlap.The vortex state is also compatible with long-range co-herence observed in neutron scattering experiments [14].At the same time ferromagnetic modulation with a pe-riod of ∼
10 nm may also develop between the vortices[2, 15, 18, 19]. This would be also be similar to FFLOstate in the presence of vortices [31]. Moreover, thiswould explain different intensities of small-angle satel-lite peaks, because coherence volume in the vortex statemust be much larger compared to domain-like state onwarming. We also note that unusual enhancement of dia-magnetism in the vicinity of the FM boundary from theSC side, could be due to an FFLO pocket as predicted byBulaevskii for ErRh B [2]. If we plot temperature of aminimum in χ ( T ) as function applied field, and also H c ,we obtain phase diagram remarkably similar to Fig.7 ofRef.[2].Overall, we conclude that we are witnessing an unusualtransition. It is definitely first order on warming withsigns of type-I superconductor ”supercooling”. However, it is smooth upon cooling and exhibits smooth minorloops similar to a type-II superconductor. A second or-der transition occurs between normal and FFLO phasesas well as between normal and SC for type-II supercon-ductor. However, transition from SC to FFLO state isfirst order as well as from SC to spiral state. It is possi-ble that size of the new phase nuclei is so small that wecannot resolve it upon cooling.Discussions with Lev Bulaevskii, Alexander Buzdin,Vladimir Kogan and Roman Mints are appreciated.Work at the Ames Laboratory was supported by the De-partment of Energy-Basic Energy Sciences under Con-tract No. DE-AC02-07CH11358. R. P. acknowledgessupport from NSF grant number DMR-05-53285 and theAlfred P. Sloan Foundation. ∗ Electronic address: [email protected][1] V. L. Ginzburg, Soviet Phys. JETP , 153 (1957).[2] L. N. Bulaevskii, A. I. Buzdin, M. L. Kuli, and S. V.Panjukov, Adv. Phys. , 175 (1985).[3] O. Fischer, Magnetic superconductors , vol. 5 (Elsevier,1990).[4] M. L. Kulic, Comptes Rendus Physique , 4 (2006).[5] M. B. Maple, Physica B , 110 (1995).[6] K. P. Sinha and S. L. Kakani, Magnetic Superconduc-tors: Recent Developments (Nova Science Publishers,New York, 1989).[7] W. A. Fertig, D. C. Johnston, L. E. DeLong, R. W. Mc-Callum, M. B. Maple, and B. T. Matthias, Phys. Rev.Lett. , 987 (1977).[8] M. Ishikawa and O. Fischer, Solid State Commun. , 37(1977).[9] P. C. Canfield, S. L. Bud’ko, and B. K. Cho, Physica C , 249 (1996).[10] S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche,R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R.Walker, S. R. Julian, P. Monthoux, et al., Nature ,587 (2000).[11] G. W. Crabtree, F. Behroozi, S. A. Campbell, and D. G.Hinks, Phys. Rev. Lett. , 1342 (1982).[12] G. W. Crabtree, R. K. Kalia, D. G. Hinks, F. Behroozi,and M. Tachiki, J. Magn. Magn. Mater. , 703(1986).[13] D. E. Moncton, D. B. McWhan, P. H. Schmidt, G. Shi-rane, W. Thomlinson, M. B. Maple, H. B. MacKay, L. D.Woolf, Z. Fisk, and D. C. Johnston, Phys. Rev. Lett. ,2060 (1980).[14] S. K. Sinha, G. W. Crabtree, D. G. Hinks, and H. Mook,Phys. Rev. Lett. , 950 (1982).[15] E. I. Blount and C. M. Varma, Phys. Rev. Lett. , 1079(1979).[16] D. E. Moncton, D. B. McWhan, J. Eckert, G. Shirane,and W. Thomlinson, Phys. Rev. Lett. , 1164 (1977).[17] J. M. DePuydt, E. D. Dahlberg, and D. G. Hinks, Phys.Rev. Lett. , 165 (1986).[18] H. Matsumoto, H. Umezawa, and M. Tachiki, Solid StateCommun. , 157 (1979).[19] P. W. Anderson and H. Suhl, Phys. Rev. , 898 (1959). [20] M. Tachiki, H. Matsumoto, and H. Umezawa, Phys. Rev.B , 1915 (1979).[21] K. E. Gray, J. Zasadzinski, R. Vaglio, and D. Hinks,Physical Review B , 4161 (1983).[22] A. I. Buzdin, L. N. Bulaevskii, and S. V. Panyukov, Zh.Exp. Teor. Fiz. , 299 (1984).[23] A. I. Buzdin and A. S. Mel’nikov, Phys. Rev. B ,020503/1 (2003).[24] S. Okada, K. Kudou, T. Shishido, Y. Satao, andT. Fukuda, Jpns. J. Appl. Phys., Part 2: Letters ,L790 (1996).[25] T. Shishido, J. Ye, T. Sasaki, R. Note, K. Obara, T. Taka-hashi, T. Matsumoto, and T. Fukuda, J. Solid StateChem. , 82 (1997).[26] R. Prozorov, R. W. Giannetta, A. Carrington, and F. M. Araujo-Moreira, Phys. Rev. B , 115 (2000).[27] R. Prozorov, R. W. Giannetta, A. Carrington,P. Fournier, R. L. Greene, P. Guptasarma, D. G. Hinks,and A. R. Banks, Appl. Phys. Lett. , 4202 (2000).[28] R. Prozorov and R. W. Giannetta, Supercond. Sci. Tech-nol. , R41 (2006).[29] M. D. Vannette, A. Safa-Sefat, S. Jia, S. A. Law,G. Lapertot, S. L. Bud’ko, P. C. Canfield, J. Schmalian,and R. Prozorov, J. Mag. Mag. Mater. in print (2007).[30] J. Feder, S. R. Kiser, and F. Rothwarf, Phys. Rev. Lett. , 87 (1966).[31] M. Ichioka, H. Adachi, T. Mizushima, and K. Machida,J. Magn. Magn. Mater.310