Domain-wall and reverse-domain superconducting states of a Pb thin-film bridge on a ferromagnetic BaFe_{12}O_{19} single crystal
R. Werner, A.Yu. Aladyshkin, S. Guenon, J. Fritzsche, I.M. Nefedov, V.V. Moshchalkov, R. Kleiner, D. Koelle
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Domain-wall and reverse-domain superconducting states of a Pb thin-film bridgeon a ferromagnetic BaFe O single crystal R. Werner, A.Yu. Aladyshkin,
2, 3
S. Gu´enon, J. Fritzsche, I.M. Nefedov, V.V. Moshchalkov, R. Kleiner, and D. Koelle ∗ Physikalisches Institut – Experimentalphysik II and Center for Collective Quantum Phenomena in LISA + ,Universit¨at T¨ubingen, Auf der Morgenstelle 14, 72076 T¨ubingen, Germany INPAC – Institute for Nanoscale Physics and Chemistry,K.U. Leuven, Celestijnenlaan 200D, B–3001 Leuven, Belgium Institute for Physics of Microstructures RAS, 603950, Nizhny Novgorod, GSP-105, Russia (Dated: May 31, 2018)We report on imaging of the nonuniform superconducting states in a Pb thin film bridge on top ofa ferromagnetic BaFe O single crystal with a single straight domain wall along the center of thebridge by low-temperature scanning laser microscopy. We have visualized domain wall superconduc-tivity (DWS) close to the critical temperature of Pb, when the Pb film above the domain wall actsas a superconducting path for the current. The evolution of the DWS signal with temperature andthe external-field-driven transition from DWS to reverse domain superconductivity was visualized. PACS numbers: 74.25.F- 74.25.Sv 74.25.Op 74.78.-w 74.78.Na
It is well known that so-called surface or bound statescan be generated by the presence of boundaries in a ma-terial. For example, the formation of surface states fora single electron wave function in a semi-infinite crys-talline lattice due to the modification of the boundaryconditions was described by Tamm and by Shockley. Other examples of bound states are surface plasmons,propagating along the interface between a dielectric anda metal, and surface acoustic waves traveling alongthe surface of a material exhibiting elasticity.
In bothlatter cases these waves are confined in the direction per-pendicular to the wave vector, i.e. their amplitudes decayexponentially far from the interface/surface. The forma-tion of surface bound states for the superconducting or-der parameter wave function Ψ was first considered bySaint-James and de Gennes.
They showed that local-ized superconductivity at a superconductor (S)/vacuumor S/insulator interface can appear at an applied mag-netic field H ext above the upper critical field H c2 forbulk superconductivity. Similarly to this surface su-perconductivity, localized superconductivity can also nu-cleate near the sample edge in a thin semi-infinite su-perconducting film or in a thin superconducting diskof very large diameter in a perpendicular magneticfield. Such so-called edge superconductivity (ES), withtransition temperature T ESc , has the same phase tran-sition line as surface superconductivity, given by 1 − T ESc /T c0 ≃ . | H ext | /H (0)c2 . Here, T c0 is the supercon-ducting transition temperature in zero magnetic field, H (0)c2 = Φ / (2 πξ ) and ξ are the upper critical fieldand coherence length at temperature T = 0, respec-tively, and Φ = π ~ c/e is the magnetic flux quantum.This means that ES will survive up to the critical field H c3 = 1 . H c2 , while above H c2 = H (0)c2 (1 − T /T c0 ) bulksuperconductivity is totally suppressed.An alternative way to prepare localized states in su-perconducting films is to confine the order parameterwave function by a nonuniform magnetic field in hybrid S/ferromagnet (F) structures (see e.g. Ref. and ref-erences therein). Buzdin and Mel’nikov considered astep-like distribution b z ( x ) = B sgn( x ) of the perpen-dicular component of the magnetic field, B z = H ext + b z ,induced by domain walls in the ferromagnet (with the z -axis perpendicular to the film surface). They demon-strated that superconductivity will survive in vicinityalong the step, even if the amplitude of the nonuniformmagnetic B > H c2 . The dependence of the transitiontemperature T DWSc ( H ext ) for domain-wall superconduc-tivity (DWS) in a plain superconducting film (i.e. infinitein lateral direction) can be estimated as 1 − T DWSc /T c0 ≃{ . − . H ext /B ) + 0 . H ext /B ) } B /H (0)c2 . For flux-coupled S/F structures of finite lateral size thelocalized states of ES and DWS may compete as illus-trated in Fig. 1 for the case of a thin film S strip of finitewidth above a F substrate with a domain wall along thecenter of the bridge, for H c < B < H c . For a domainstructure with step-like b z ( x ) profile and H ext = 0, ESand DWS nucleate simultaneously in the S strip as shownin Fig. 1(a). Figure 1(b) shows the case of a domain wallwith finite width and H ext = 0. Here, DWS becomesenergetically more favorable compared to ES and onlyDWS nucleates. For H ext = 0, the local field is com-pensated above the domain with magnetization directionopposite to H ext . If || H ext | − B | < H c2 superconduc-tivity is turned on above this reverse domain while it isstill suppressed above the parallel domain [c.f. Fig. 1(c)].This effect is termed reverse-domain superconductivity (RDS) . We note that when H c2 becomes larger than | H ext | + B (e.g. upon cooling) above the parallel domain,superconductivity may also nucleate there and the entirestrip will be in the superconducting state, which we call complete superconductivity (CS).First fingerprints of RDS and DWS have been foundby electric transport measurements on S/F hybrids witha rather complex domain structure in BaFe O (BFO)crystals and multilayered CoPt films with perpendic- DWS(b)DWS M c2 ES M (a) -H c3 B x b z | | Y x | | Y b z M RDS M H ext b +H z ext | | Y b z M H c3 H c2 (c) x0 FIG. 1: (color online). Illustration of DWS, ES and RDSacross a thin film S strip on top of a F substrate with twodomains with perpendicular Magnetization M . Magnetic fieldprofiles B z ( x ) = H ext + b z ( x ) inside the S strip generatedby the domains underneath and modulus of superconductingorder parameter | Ψ( x ) | are shown for (a) step-like b z ( x ) for H ext = 0, (b) field profile with finite width for H ext = 0and (c) finite H ext ≈ B . White dashed lines indicate uppercritical fields ± H c2 and ± H c3 . ular magnetic anisotropy. Using low-temperature scan-ning laser microscopy (LTSLM), RDS has been visual-ized in a hybrid Nb/PbFe O system. However, dueto the complex domain structure and relatively small do-main size, visualization of DWS was not possible. Re-cently, significant improvements have been achieved, re-garding the fabrication of specially polished BFO crys-tals, characterized by a well defined and stable domainstructure with straight domain walls separated by typi-cally 30 µ m. Here we report on the direct imaging ofthe development of DWS and RDS in a hybrid S/F struc-ture, consisting of a superconducting Pb film on top of aferromagnetic BFO crystal by means of LTSLM.
We prepared a 40 nm thick and 30 µ m wide Pb micro-bridge on top of a BFO substrate, so that only a singledomain wall is running along the center of the Pb bridgeparallel to the current flow. The BFO substrate and thePb thin film were separated by a 4 nm thick insulatingGe layer so that the system is only flux-coupled. Fromresistance R vs H ext measurements at variable T of areference sample with several domain walls oriented per-pendicular to the long side of the bridge, we composethe H ext − T phase diagram shown in Fig. 2(b).For imaging by LTSLM, the sample was mounted onthe cold finger of a Helium gas flow cryostat, with an opti-cal window to enable irradiation of the sample surface inthe ( x, y ) plane by a focused laser beam with beam spotdiameter ∼ . − µ m. The amplitude modulatedlaser beam (at frequency f ≈
10 kHz) induces a local in-crease of temperature δT ( x − x , y − y ) centered at thebeam spot position ( x , y ) on the sample surface. Dur-ing imaging, the Pb bridge is biased at a constant current I , and the beam-induced change of voltage ∆ V ( x , y ) isrecorded with lock-in technique. The LTSLM voltage sig-nal can be interpreted as follows: If the irradiated partof the sample was in the normal state with resistivity ρ n , the laser beam induces a very small voltage signal∆ V ∝ ∂ρ n /∂T . However, if the irradiated region took part in the transfer of a substantial part of the supercur-rents, the beam-induced suppression of superconductiv-ity might switch the sample from a low-resistive state toa high-resistive state. This effect should be maximal if I is close to the overall critical current I c = I c ( T, H ext ) ofthe sample. In this case LTSLM allows one to map outthe ability of the sample to carry supercurrents.In order to trace out the evolution of DWS with tem-perature, we recorded a series of LTSLM voltage images∆ V ( x, y ) at H ext = 0 and different T across the resistivetransition of the Pb bridge.Figure 2(a) shows the R ( T ) curve of the Pb/BFOmicrobridge; the labels 1–8 indicate the bias pointsfor which LTSLM images and line scans are shown inFigs. 2(d) and 2(c), respectively. The dots in the H ext − T phase diagram shown in Fig. 2(b) indicate the bias pointsfor which LTSLM data are shown. The LTSLM voltageimages 1–8 in Fig. 2(d) show the evolution of the super-conducting properties of the Pb/BFO bridge upon cool-ing through T c (from left to right) at H ext =0; accordingto Fig. 2(b), these should cover the transitions from thenormal state to DWS and finally to CS. The graph on theright shows an optical LTSLM image, in order to indicatesize and position of the bridge in the voltage images. For a more quantitative analysis, in Fig. 2(c) we show linescans ∆ V ( x ) across the bridge [along the white dashedlines in Fig. 2(d)].Starting with the highest temperature T = 6 . T to 6.4 K [entering theresistive transition shown in Fig. 2(a)], the voltage imagegives a small homogeneous signal with a broad maximumcentered above the bridge [red line in Fig. 2(c)]. For a(still) resistive Pb bridge with homogeneous conductivitybut finite ∂R/∂T , this behavior can be simply explainedby the finite width of the beam-induced δT ( x, y ) pro-file, i.e. its tails will induce a voltage signal, even if thebeam spot is positioned outside the bridge. This is con-firmed by numerical simulations [c.f. red data points inFig. 2(c)], which solve the heat diffusion equation for anabsorbed laser power of 25 µ W, a beam spot diameterof 2 µ m and thermal conductivity of the BFO substrateof 0 . − K − . These simulations yield a maximumincrease in beam-induced temperature ∆ T = 0 .
14 K.Upon further cooling (see voltage images and corre-sponding line scans for T = 6 . K and T = 6 . K ), a clearLTSLM signal develops, running along the domain wall[green and blue lines, respectively, in Fig. 2(c)]. This ob-servation can be interpreted as an evidence that a channelabove the domain wall with higher conductivity than theregions above the domains has formed, and therefore thecurrent density j ( x ) has a maximum above the domainwall. We note that, although according to the H ext − T phase diagram the sample should be in the the DWSstate, the overall resistance of the bridge is close to thefull normal resistance. This is consistent with numeri-cal simulations based on the time-dependent Ginzburg-Landau equations, which indicate that for our experimen- ΔV (µV) (d) x (µm) optical image (a) (b) (c) FIG. 2: Evolution of DWS upon cooling through T c and H ext − T phase diagram. (a) R ( T ) curve ( I = 100 µ A); dots indicatebias points of LTSLM voltage images 1–8 in (d) and corresponding line scans in (c). (b), H ext − T phase diagram, constructedfrom experimentally determined values T c0 = 7 .
25 K, B = 480 G and H (0)c2 = 2 .
25 kOe. The phase diagram contains separateregions of DWS, ES, RDS and CS. Dots label bias points for LTSLM data shown in (c), (d) and Fig. 3. (c), line scans ∆ V ( x )across the bridge for different T , taken from voltage images in (d). Red dots show simulation results for T = 6 . V ( x, y ) (1–8 from left toright) taken at different T during cooling the Pb bridge through its resistive transition ( I = 10 µ A). White dashed lines indicatethe position of line scans in (c). The graph on the right shows a corresponding optical LTSLM image. tal situation the critical current density j c , DWS along thedomain-wall channel is too small, i.e. the bias currentmight be above the critical current of this channel. Thisalso explains why, upon decreasing T , the LTSLM signalat the domain wall increases, as j c , DWS increases, and thepeak in ∆ V ( x ) becomes sharper (see below). We did notfind a similar enhancement of the LTSLM voltage signalat the edges of the bridge, i.e. we do not find any signa-ture of ES. We attribute this to the finite width of thedomain wall, which stabilizes DWS compared to ES.For T < T decreases,and the maximum of the LTSLM signal shifts towardsthe edges of the bridge; see magenta and orange linesin Fig. 2(c) for T = 6 . K and T = 5 . K , respectively,and the corresponding voltage images in Fig. 2(d). Weinterpret this observation as the transition from DWS toCS, which is consistent with the phase diagram shown inFig. 2(b). At this transition, CS spreading over the wholesample becomes favorable and the sample is turned intothe mixed state. The onset of CS can explain the appear-ance of two pronounced maxima in ∆ V ( x ) at the sampleedges: In the mixed state the current distribution de-pends on the edge energy barrier for vortex entry. Uponlaser irradiation, the edge energy barrier is locally sup-pressed, which in turn opens a gate for vortex entry/exit.Hence one can expect that irradiation at the edges of thebridge should strongly affect the vortex pattern and theresulting current distribution. In contrast, laser irradia-tion of the interior of the bridge does not change the ex- isting energy barrier, and the modification of the currentpattern is probably less pronounced, and therefore thebeam-induced voltage change is much smaller. Finally,at T =5.0 K the LTSLM signal is zero [c.f. Fig. 2(d) andbrown line in Fig. 2(c)], which indicates that the bridgeis in the CS state and the beam-induced perturbation isnot strong enough to suppress superconductivity and toinduce a voltage signal.Finally, we investigated the effect of finite perpen-dicular field | H ext | ≤
165 Oe on superconductivity inour system. The measurements were carried out at T = 6 . H ext = 0. Fig-ure 3(a) shows the evolution of the LTSLM voltage signal (b) -165 Oe 165 Oe D V ( m V ) x (µm) H ext (Oe)0336699132165 (a) FIG. 3: (color online). Switching from DWS to RDS: variable H ext at T = 6 . V ( x ) along white dashedline in (b) for different H ext ≥
0; dashed grey lines indicateedges of the bridge. (b) LTSLM images for maximal | H ext | . ∆ V ( x ) with increasing external field for positive polar-ity. For H ext = 0 the DWS signal is clearly visible asdescribed above. With increasing H ext the amplitude ofthe domain-wall signal decreases monotonously while itswidth stays roughly constant. Simultaneously a signalabove the reverse (right) domain appears. In the RDSstate, for H ext > ∼
70 Oe, the voltage signal shows a peak atthe right edge of the bridge, which can be explained inthe same way as for the edge signal discussed in the con-text of the T -series shown in Fig. 2. Figure 3(b) showsLTSLM voltage images taken at H ext = −
165 Oe (leftimage) and H ext = +165 Oe (right image), which clearlydemonstrate switching between the RDS states above thetwo domains upon reversing the external field polarity.In conclusion, we have clearly identified the formationof the spatially inhomogeneous superconducting state ina superconducting Pb thin film induced by the stray fieldof the domains in the ferromagnetic substrate BFO un-derneath. The crucial feature of the investigated systemis that the superconducting Pb bridge was fabricated ex-actly above a single straight domain wall, which is run- ning along the center of the bridge. Such a well-definedgeometry of the hybrid Pb/BFO sample makes it possibleto directly visualize the localized and delocalized super-conductivity by means of low-temperature scanning lasermicroscopy. We imaged the evolution of DWS with de-creasing temperature. Using the external field as a con-trol parameter, we demonstrated that superconductivityin a wide superconducting bridge can be switched fromthe DWS to RDS state. This opens up interesting per-spectives for the creation of spatially nonuniform super-conducting states and for their manipulation by externaland ”internal“ magnetic fields.This work was funded by the DFG via Grant No. KO1303/8-1, the Methusalem Funding of the Flemish Gov-ernment, the NES – ESF program, the Belgian IAP,the Fund for Scientific Research – Flanders (F.W.O.–Vlaanderen), the RFBR, RAS under the Program“Quantum physics of condensed matter”, and FTP “Sci-entific and educational personnel of innovative Russia in2009–2013”. R. W. gratefully acknowledges support bythe Cusanuswerk, Bisch¨ofliche Studienf¨orderung. ∗ Electronic address: [email protected] I. Tamm, Physik. Zeits. Sowjetunion , 733 (1932). W. Shockley, Phys. Rev. , 317 (1939)). L.D. Landau and E.M. Lifshitz,
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