- Email: [email protected]
Freezing vortex rivers
Physica C 470 (2010) 726–729
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Physica C journal homepage: www.elsevier.com/locate/physc
Freezing vortex rivers A.V. Silhanek a,*, R.G.B. Kramer a, J. Van de Vondel a, V.V. Moshchalkov a, M.V. Miloševic´ b, G.R. Berdiyorov b, F.M. Peeters b, R.F. Luccas c, T. Puig c a
INPAC – Institute for Nanoscale Physics and Chemistry, Nanoscale Superconductivity and Magnetism Group, K.U. Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium c Institut de Ciencia de Materials de Barcelona, CSIC, Campus de la UAB, 08193 Bellaterra, Spain b
a r t i c l e
i n f o
Article history: Available online 4 March 2010 Keyword: Superconductivity
a b s t r a c t We demonstrate experimentally and theoretically that the dissipative state at high current densities of superconducting samples with a periodic array of holes consist of flux rivers resulting from a short range attractive interaction between vortices. This dynamically induced vortex–vortex attraction results from the migration of quasiparticles out of the vortex core. We have directly visualized the formation of vortex chains by scanning Hall microscopy after freezing the dynamic state by a field cooling procedure at constant bias current. Similar experiments carried out in a sample without holes show no hint of flux river formation. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Superconductivity is generally regarded as a diamagnetic state of matter where dc electrical current can flow without dissipation. In type-II superconductors, however, a magnetic field can penetrate in form of flux tubes or vortices each of them bearing a quantum unit of flux U0. The repulsive interaction between vortices make them to distribute in a periodic triangular array known as Abrikosov lattice. The ultimate mechanism leading to this repulsive interaction between vortices in type-II superconductors is the increase of magnetic energy as vortices approach each other at distances of the superconducting penetration depth k. Interestingly, under certain circumstances, an attractive vortex– vortex interaction can appear even in type-II superconductors at equilibrium conditions. This is the case for vortices in an anisotropic superconductor tilted away from the principal symmetry axes [1,2]. Here, the net attractive interaction results from the change of sign of the component parallel to the vortex direction of the field generated by a vortex line. A similar field reversal in the magnetic field distribution of an isolated vortex results from the non-local relationship between supercurrents and vector potential in clean and low-j materials. It has been shown that this effect also leads to an attractive vortex interaction [3–6]. Another example can be found in the case of two components superconductors where two weakly coupled order parameters, each of which belonging to a different type of superconductivity, coexist in the same material [7,8]. * Corresponding author. E-mail address: [email protected] (A.V. Silhanek). 0921-4534/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2010.02.072
Although all this plethora of possible vortex arrangements correspond to equilibrium thermodynamical configurations, vortex– vortex attraction can also be found when the system is driven out of equilibrium. For instance, a fast moving vortex line creates an excess of quasiparticles behind its core thus generating a wake of depleted order parameter which attracts other vortices [10–12]. Time dependent Ginzburg–Landau calculations showed that the resulting direction dependent interaction between vortices may gives rise to the formation of vortex rivers in narrow transport bridges, somehow similar to the vortex stripes in the static case. These fast moving vortices, known as kinematic vortices, have a highly anisotropic vortex core along the direction of motion which eventually evolves into a phase slip line beyond certain critical velocity [12,13]. A way to promote the proliferation of these kinematic vortices at relative small currents can be achieved by introducing a periodic array of holes in the superconducting film. The reason is two fold, on the one hand the constriction imposed by the period of the hole array favors the formation of phase slip lines [13,14]. On the other hand the patterned sample leads to an inhomogeneous current distribution which magnifies the current density in between the holes. This picture indicates that the high current resistive state of samples with a periodic array of antidots should be dominated by kinematic vortices. It was anticipated by Reichhardt et al. [9] that under these circumstances the competition between a long range repulsive interaction and more local attractive force in presence of an applied dc drive can lead to the formation of conglomerates of vortices such as dimmers, labyrinths, or stripes. In the present work we directly visualize by scanning Hall microscopy the formation of vortex stripes perpendicular to the
A.V. Silhanek et al. / Physica C 470 (2010) 726–729
current direction in a conventional superconductor with a periodic array of holes. Due to the large integration time needed to acquire a single frame (about 3 min) the images are recorded after freezing the dynamic phase by quickly cooling the sample in presence of a bias current and applied field. The relevance of the periodic array of holes in stabilizing parallel vortex stripes is clearly evidenced by the lack of such stripes in a plain film without holes. Time dependent Ginzburg–Landau calculations support our interpretation and give further insight on the birth, growth and evolution of these flux rivers.
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a periodic background signal produced by the modulated gating of the Hall cross. Panels (b) and (c), obtained very close to the first and second matching condition, respectively, show a very ordered pattern mimicking the periodic array of antidots. In panel (d), a very disordered structure evidences the presence of interstitial vortices. From these images we can safely conclude that a maximum of two flux quanta can be trapped at each hole. This is in agreement with the results obtained by Grigorenko et al. [17] using similar samples. 4. Visualization of the vortex motion
2. Experimental details 4.1. Pulsed transport measurements The investigated sample consist of a 50 nm thick Pb film with a square array of square holes made by electron beam lithography and subsequent lift-off. The experimental procedure for sample preparation can be found in Ref. [16]. The period of the pattern is d = 1.5 lm and the size of the holes is a = 0.6 lm. Electrical transport measurements performed in a sister sample show clear commensurability effects at H/H1 = 1/2, 1, 1.5, and 2, where H1 = U0/d2 is the field at which the density of vortices equals the density of holes. From the normal/superconductor phase boundary obtained in an unpatterned Pb film we obtain a superconducting coherence length n(0) 33 nm and a critical temperature Tc = 7.2 K. All samples were patterned in a transport bridge 300 lm wide with voltage contacts which permitted us the simultaneous acquisition of four-points transport measurements and scanning Hall microscopy. The scanning Hall probe microscopy images were obtained using a modified Low Temperature Scanning Hall Probe Microscope (LT-SHPM) from Nanomagnetics Instruments. Most of the SHPM images shown in this work were recorded at 4.2 K with a scanning area of 13 13 lm2 after field cooling (FC) the sample. Measurements performed at higher temperatures (up to 6.8 K) show no difference with those obtained at 4.2 K, thus indicating that the vortex distribution is frozen at even higher temperatures close to the onset of the superconducting state. The images were recorded in lift-off mode with the Hall sensor at about 500 nm above the surface of the sample. Home-made xy positioners allows us to explore different regions of the same sample in order to avoid unwanted effects arising from the sample’s borders, defects, or small particles. 3. Static distribution of vortices A series of images obtained at 4.2 K after field cooling procedure with fields ranging from 4 mT to +4 mT in steps of 0.02 mT allowed us to determine the remanent field with high accuracy. In the patterned sample we clearly identify the different commensurate vortex states as described in previous reports [17–19]. Fig. 1 shows SHPM images obtained at 4.2 K with H = 0, H = H1 = 9.2 G, H = 2H1, and H = 3H1. Fig. 1a shows an isolated vortex with
An attempt to simultaneously record images while moving vortices with an external dc current for T < 6.8 K showed that vortices remain still until reaching the critical current beyond which a severe heat dissipation associated with the vortex motion drives the entire sample to the normal state. This effect becomes apparent in Fig. 2 where a voltage–current characteristic at T = 7.1 K has been recorded within 5 s time and 5 ls time. In this case, despite the fact that initially the system is under exactly the same thermodynamic conditions, the critical current above which the system shows dissipation differs by a factor of four. For standard dc current sweeps like the 5 s measurement, the transition from superconducting to normal state is very abrupt and hinders us from visualizing the actual vortex motion. In order to overcome this limitation it is imperative to apply short current pulses no longer than several microseconds thus minimizing the heating effects. Fig. 3a shows a typical voltage response for a pulse of 150 ls and amplitude 90 mA obtained at T = 7 K and three different fields. For H = 1.75 mT > Hc2 (dotted line), the voltage signal V(t) shows a square shape with a maximum value corresponding to the normal state resistance Vmax = IRn. For fields H = 1.15 mT Hc2/2, vortices are set in motion immediately after turning on the pulse. The dissipation associated with their motion give rise to a local heating and therefore, to a progressive increase in the voltage until eventually the temperature of the sample reaches Tc and an abrupt jump to the normal state appears. For very low fields, H = 0.01 mT Hc2, three clear regimes can be identified. At short times no dissipation is observed, at intermediate times the picked up signal monotonously increases as a function of time indicating once again a progressive heating of the sample [15], whereas at longer times the system eventually transits to the normal state and a large jump in the dissipation is observed. 4.2. Scanning Hall probe images after current pulses Based on the above described transport properties we established the following protocol for visualizing the vortex dynamics in each of these regimes, (1) the sample is cooled down at the desired field, (2) a short current pulse of duration 50 ls is then ap-
Fig. 1. Scanning Hall cross microscopy images of a Pb film with a periodic array of antidots after field cooling down to 4.2 K and (a) H = 0, (b) H = H1 = 9.2 G, (c) H = 2H1, and (d) H = 3H1. The size of each image is 13 13 lm2. Panel (a) shows an isolated vortex. Panels (b) and (c) obtained very close to the first and second matching condition, respectively, show a very ordered pattern mimicking the periodic array of antidots. In panel (d), a very disordered structure evidences the presence of interstitial vortices.
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A.V. Silhanek et al. / Physica C 470 (2010) 726–729
150
T = 7.1 K
V (mV)
100
t = 5 µs
t=5s
50
0
10
20
30
40
50
I (mA) Fig. 2. Current–voltage characteristics obtained at T = 7.1 K in a Pb bridge with a periodic array of antidots. The data collection has been done in two extreme different time scales, 5 s (solid line) and 5 ls (open circles).
the intermediate length pulse, the vortex lattice is neither fully disorder nor similar to the equilibrium field-cooled image, but rather exhibits a clear tendency to conglomerate in branches along the principal directions of the underlying pinning potential. This result resemble the patterns early predicted by Reichhardt et al. [9] for a system of particles with competing long range repulsive and short range attractive interactions. Interestingly, the images obtained after applying pulses of long duration are identical to those obtained by a standard field cooling procedure (and also similar to Fig. 4a). This finding can be easily understood since long pulses produce an overall heating of the sample above Tc and therefore, once the bias current is turned off (nanoseconds process) the thermal relaxation of the sample towards the original temperature takes about 100 ms [20]. Naturally, this sort of field cooling procedure is very convenient and fast since the sample can be used as a micro-heater with no need to warm up the rest of the bulky addenda. 4.3. Scanning Hall probe images after field cooling with dc current
plied, (3) after that an image scan is performed, and (4) a new pulse, 10 ls longer than the previous pulse, is applied. Then, steps (3) and (4) are repeated until reaching the normal state. In Fig. 4, the SHPM images obtained at short times [panel (a)] and intermediate times [panel (b)] for H = 0.46 mT H1/2 are shown. The periodic distribution of vortices observed in Fig. 4a indicates that short pulses are unable to set the vortices in motion. In contrast to that, pulses of intermediate duration lead to a clear vortex motion as evidence in Fig. 4b. Strikingly, after applying
An alternative method to study the dynamics at high vortex velocities can be achieved by field cooling the sample while maintaining a dc current on it. In this way we ensure that the system will freeze the vortex structure with high vortex velocities at some current dependent temperature Tf. Due to the irreversible behavior of the current–voltage characteristics, differences with respect to the pulsed-current procedure might be expected. Fig. 5 shows images after field cooling without current applied [panel (a)] and with an applied current of 35 mA [panel (b)]. The most obvious feature of this figure is the formation of rivers of flux sitting exactly on top of the holes. This is the main result of this paper and indicates the existence of an attractive interaction between vortices when they move at high velocities. We have observed that the number of these vortex rivers inside the scanning area is always two, irrespective of the applied field. It is important to notice that since the sample is acting as a microheater, the control of cooling rates with an external thermometer is somehow meaningless since the actual temperature of the sample differs significantly from that obtained with standard thermometry. Measurements performed in a co-evaporated plain unpatterned Pb film under identical conditions show no hint of stripe formation (see Fig. 5c), thus pointing out to the key role played by the hole array in the formation and stabilization of flux rivers. Indeed, we have recently demonstrated [21] using the time dependent Ginzburg–Landau formalism that the effective freezing of these vortex rivers can only be achieved if (i) at low temperatures the pinning is strong enough to sustain the quenched metastable state and (ii) the cooling rate is above certain threshold value. It has been demonstrated experimentally [22] and theoretically [23–28] that vortices driven by an electrical current follow intricate paths rather than a well order and correlated lattice motion.
Fig. 4. Scanning Hall probe microscopy images obtained at T = 6.5 K and H = 0.46 mT after a short pulse (a) and intermediate length pulse (b).
Fig. 5. Scanning Hall probe microscopy images obtained at T = 6.5 K and H = 0.46 mT after a short pulse (a) and intermediate length pulse (b) in a Pb film with a periodic array of antidots. Panel (c) shows a SHPM image under the same condition than (b) but in a plain Pb film.
30 25
T=7K
V (mV)
20 15
1.75 mT 1.15 mT 0.01 mT
10 5 0 -5 -100
-75
-50
-25
0
25
50
75
100
t (µs) Fig. 3. Voltage vs. time during a current pulse of 150 ls duration and 90 mA amplitude for three different applied fields. The initial temperature of the system is 7 K. For H = 1.75 mT (dotted line) the system is in the normal state.
A.V. Silhanek et al. / Physica C 470 (2010) 726–729
In general, these vortex rivers flow around strongly pinned (immobile) vortices and therefore, corresponds to a plastic-like vortex motion occurring at currents close to the depinning current. It is important to point out that the flux rivers here investigated are of an entirely different nature than previously reported filamentary flux motion since all vortices are set in motion and no plastic regime is needed. An alternative way to obtain well ordered flux channels can be achieved in superconductors with weak pinning submitted to currents much higher than the critical current. In this case an initially imperfect vortex configuration can recrystallize in a perfect lattice which is oriented with one of its principal directions along the direction of motion or even form a smectic vortex flow with elastically coupled vortex channels [29–32]. In this scenario, it is expected that the separation between neighboring channels corresponds to the vortex lattice period. In contrast to this situation, in the present sample we have found that the number of channels seems to be rather insensitive to the applied field. In addition, the presence of steps in the IV characteristics at high currents indicates that in samples with antidots the vortex dynamics at high currents involves the formation of phase slips or kinematic vortices, thus departing from the assumption of point-like vortices used in molecular dynamic simulations [14]. In this work we claim that vortex rivers might have a different origin. Indeed, it has been theoretically proposed [12] that when vortices move at very high speed due to an external current, a directional attractive vortex-vortex interaction can appear. The ultimate reason for this behavior is the elongation of the vortex core (vortex tail) in the direction of motion as a result of the finite time needed for the superconducting order parameter to recover from the vortex passage. Since this intrinsic healing time of the superconducting condensate is at sub-nanoseconds scales, a moving vortex needs typically a very high speed in order to be able to see the wake of another vortex. Once this situation is reached, vortices like to align one behind the other forming stripes.
5. Conclusion To summarize, in this manuscript we report on the first successful visualization of vortex stripe patterns formed by an unconventional vortex attraction when these entities are driven to very high velocities. These, so called, kinematic vortices attract each other forming stripe patterns that can be stabilized by literally freezing their motion via a fast thermal quench and a strong pinning potential produced by an array of antidots. Contrary to other systems exhibiting stripe phases, vortices in our samples have no attractive interaction in equilibrium conditions, and stripe formation results solely from the dynamic history of the superconducting condensate.
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Acknowledgements This work was supported by Methusalem funding by the Flemish government, the Flemish Science Foundation (FWO-Vl), the Belgian Science Policy, and the ESF NES network. A.V.S., G.R.B., and J.V.d.V. acknowledge support from FWO-Vl. R.F.L. acknowledge support from I3P CSIC program and MAT2008-01022. References [1] V.G. Kogan, N. Nakagawa, S.L. Thiemann, Phys. Rev. B 42 (1990) 2631. [2] A.I. Buzdin, A.Yu. Simonov, JETP Lett. 51 (1990) 191; A.M. Grishin, A.Yu. Martynovich, S.V. Yampolskii, Sov. Phys. JETP 70 (1990) 1089. [3] A. Jacobs, Phys. Rev. 34 (1971) 3029. [4] J. Auer, H. Ullmaier, Phys. Rev. B 7 (1973) 136. [5] K. Dichtel, Phys. Lett. A 35 (1971) 285. [6] E.H. Brandt, Phys. Lett. A 39 (1972) 193. [7] V.V. Moshchalkov, M. Menghini, Q.H. Chen, T. Nishio, A.V. Silhanek, V.H. Dao, L.F. Chibotaru, N.D. Zhigadlo, J. Karpinski, Phys. Rev. Lett. 102 (2009) 117001. [8] E. Babaev, M. Speight, Phys. Rev. B 72 (2005) 180502 (R). [9] C. Reichhardt, C.J. Olson Reichhardt, I. Martin, A.R. Bishop, Phys. Rev. Lett. 90 (2003) 026401. [10] A.I. Larkin, Yu.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 68 (1975) 1915; A.I. Larkin, Yu.N. Ovchinnikov, Sov. Phys. JETP 41 (1976) 960. [11] A. Andronov, I. Gordion, V. Kurin, I. Nefedov, I. Shereshevsky, Physica C 213 (1993) 193. [12] D.Y. Vodolazov, F.M. Peeters, Phys. Rev. B 76 (2007) 014521. [13] A.G. Sivakov, A.M. Glukhov, A.N. Omelyanchouk, Y. Koval, P. Müller, A.V. Ustinov, Phys. Rev. Lett. 91 (2003) 267001. [14] J. Gutierrez et al., Phys. Rev. B 80 (2009) 140514. [15] P. Tholfsen, H. Meissner, Phys. Rev. 185 (1969) 653. [16] S. Raedts, A.V. Silhanek, V.V. Moshchalkov, J. Moonens, L.H.A. Leunissen, Phys. Rev. B 73 (2006) 174514. [17] A.N. Grigorenko, G.D. Howells, S.J. Bending, J. Bekaert, M.J. Van Bael, L. Van Look, V.V. Moshchalkov, Y. Bruynseraede, G. Borghs, I.I. Kaya, R.A. Stradling, Phys. Rev. B 63 (2001) 052504. [18] S.B. Field, S.S. James, J. Barentine, V. Metlushko, G. Crabtree, H. Shtrikman, B. Ilic, S.R.J. Brueck, Phys. Rev. Lett. 88 (2002) 067003. [19] A.N. Grigorenko, S.J. Bending, M.J. Van Bael, M. Lange, V.V. Moshchalkov, H. Fangohr, P.A.J. de Groot, Phys. Rev. Lett. 90 (2003) 237001. [20] This time has been estimated by submitting the system to a train of pulses and determining the minimum separation time between consecutive pulses needed to obtain cumulative heating in the sample. [21] A.V. Silhanek et al., Phys. Rev. Lett. 104 (2010) 017001. [22] T. Matsuda et al., Science 271 (1996) 1393; T. Matsuda et al., Science 294 (2001) 2136. [23] N. Gronbech-Jensen, A.R. Bishop, D. Dominguez, Phys. Rev. Lett. 76 (1996) 2985. [24] S. Ryu, M. Hellerqvist, S. Doniach, A. Kapitulnik, D. Stroud, Phys. Rev. Lett. 77 (1996) 5114. [25] H. Jensen, A. Brass, A.J. Berlinsky, Phys. Rev. Lett. 60 (1988) 1676. [26] F. Nori, Science 271 (1996) 1373. [27] C.J. Olson, C. Reichhardt, F. Nori, Phys. Rev. Lett. 80 (1998) 2197. and references therein. [28] K.E. Bassler, M. Paczuski, G.F. Reiter, Phys. Rev. Lett. 83 (1999) 3956. [29] T. Giamarchi, P. Le Doussal, Phys. Rev. Lett. 76 (1996) 3408. [30] L. Balents, C.M. Marchetti, L. Radzihovsky, Phys. Rev. Lett. 78 (1997) 751. [31] F. Pardo, F. de la Cruz, P.L. Gammel, E. Bucher, D.J. Bishop, Nature 396 (1998) 348. [32] M. Marchevsky, J. Aarts, P.H. Kes, M.V. Indenbom, Phys. Rev. Lett. 78 (1997) 531.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Freezing vortex rivers A.V. Silhanek a,*, R.G.B. Kramer a, J. Van de Vondel a, V.V. Moshchalkov a, M.V. Miloševic´ b, G.R. Berdiyorov b, F.M. Peeters b, R.F. Luccas c, T. Puig c a
INPAC – Institute for Nanoscale Physics and Chemistry, Nanoscale Superconductivity and Magnetism Group, K.U. Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium c Institut de Ciencia de Materials de Barcelona, CSIC, Campus de la UAB, 08193 Bellaterra, Spain b
a r t i c l e
i n f o
Article history: Available online 4 March 2010 Keyword: Superconductivity
a b s t r a c t We demonstrate experimentally and theoretically that the dissipative state at high current densities of superconducting samples with a periodic array of holes consist of flux rivers resulting from a short range attractive interaction between vortices. This dynamically induced vortex–vortex attraction results from the migration of quasiparticles out of the vortex core. We have directly visualized the formation of vortex chains by scanning Hall microscopy after freezing the dynamic state by a field cooling procedure at constant bias current. Similar experiments carried out in a sample without holes show no hint of flux river formation. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Superconductivity is generally regarded as a diamagnetic state of matter where dc electrical current can flow without dissipation. In type-II superconductors, however, a magnetic field can penetrate in form of flux tubes or vortices each of them bearing a quantum unit of flux U0. The repulsive interaction between vortices make them to distribute in a periodic triangular array known as Abrikosov lattice. The ultimate mechanism leading to this repulsive interaction between vortices in type-II superconductors is the increase of magnetic energy as vortices approach each other at distances of the superconducting penetration depth k. Interestingly, under certain circumstances, an attractive vortex– vortex interaction can appear even in type-II superconductors at equilibrium conditions. This is the case for vortices in an anisotropic superconductor tilted away from the principal symmetry axes [1,2]. Here, the net attractive interaction results from the change of sign of the component parallel to the vortex direction of the field generated by a vortex line. A similar field reversal in the magnetic field distribution of an isolated vortex results from the non-local relationship between supercurrents and vector potential in clean and low-j materials. It has been shown that this effect also leads to an attractive vortex interaction [3–6]. Another example can be found in the case of two components superconductors where two weakly coupled order parameters, each of which belonging to a different type of superconductivity, coexist in the same material [7,8]. * Corresponding author. E-mail address: [email protected] (A.V. Silhanek). 0921-4534/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2010.02.072
Although all this plethora of possible vortex arrangements correspond to equilibrium thermodynamical configurations, vortex– vortex attraction can also be found when the system is driven out of equilibrium. For instance, a fast moving vortex line creates an excess of quasiparticles behind its core thus generating a wake of depleted order parameter which attracts other vortices [10–12]. Time dependent Ginzburg–Landau calculations showed that the resulting direction dependent interaction between vortices may gives rise to the formation of vortex rivers in narrow transport bridges, somehow similar to the vortex stripes in the static case. These fast moving vortices, known as kinematic vortices, have a highly anisotropic vortex core along the direction of motion which eventually evolves into a phase slip line beyond certain critical velocity [12,13]. A way to promote the proliferation of these kinematic vortices at relative small currents can be achieved by introducing a periodic array of holes in the superconducting film. The reason is two fold, on the one hand the constriction imposed by the period of the hole array favors the formation of phase slip lines [13,14]. On the other hand the patterned sample leads to an inhomogeneous current distribution which magnifies the current density in between the holes. This picture indicates that the high current resistive state of samples with a periodic array of antidots should be dominated by kinematic vortices. It was anticipated by Reichhardt et al. [9] that under these circumstances the competition between a long range repulsive interaction and more local attractive force in presence of an applied dc drive can lead to the formation of conglomerates of vortices such as dimmers, labyrinths, or stripes. In the present work we directly visualize by scanning Hall microscopy the formation of vortex stripes perpendicular to the
A.V. Silhanek et al. / Physica C 470 (2010) 726–729
current direction in a conventional superconductor with a periodic array of holes. Due to the large integration time needed to acquire a single frame (about 3 min) the images are recorded after freezing the dynamic phase by quickly cooling the sample in presence of a bias current and applied field. The relevance of the periodic array of holes in stabilizing parallel vortex stripes is clearly evidenced by the lack of such stripes in a plain film without holes. Time dependent Ginzburg–Landau calculations support our interpretation and give further insight on the birth, growth and evolution of these flux rivers.
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a periodic background signal produced by the modulated gating of the Hall cross. Panels (b) and (c), obtained very close to the first and second matching condition, respectively, show a very ordered pattern mimicking the periodic array of antidots. In panel (d), a very disordered structure evidences the presence of interstitial vortices. From these images we can safely conclude that a maximum of two flux quanta can be trapped at each hole. This is in agreement with the results obtained by Grigorenko et al. [17] using similar samples. 4. Visualization of the vortex motion
2. Experimental details 4.1. Pulsed transport measurements The investigated sample consist of a 50 nm thick Pb film with a square array of square holes made by electron beam lithography and subsequent lift-off. The experimental procedure for sample preparation can be found in Ref. [16]. The period of the pattern is d = 1.5 lm and the size of the holes is a = 0.6 lm. Electrical transport measurements performed in a sister sample show clear commensurability effects at H/H1 = 1/2, 1, 1.5, and 2, where H1 = U0/d2 is the field at which the density of vortices equals the density of holes. From the normal/superconductor phase boundary obtained in an unpatterned Pb film we obtain a superconducting coherence length n(0) 33 nm and a critical temperature Tc = 7.2 K. All samples were patterned in a transport bridge 300 lm wide with voltage contacts which permitted us the simultaneous acquisition of four-points transport measurements and scanning Hall microscopy. The scanning Hall probe microscopy images were obtained using a modified Low Temperature Scanning Hall Probe Microscope (LT-SHPM) from Nanomagnetics Instruments. Most of the SHPM images shown in this work were recorded at 4.2 K with a scanning area of 13 13 lm2 after field cooling (FC) the sample. Measurements performed at higher temperatures (up to 6.8 K) show no difference with those obtained at 4.2 K, thus indicating that the vortex distribution is frozen at even higher temperatures close to the onset of the superconducting state. The images were recorded in lift-off mode with the Hall sensor at about 500 nm above the surface of the sample. Home-made xy positioners allows us to explore different regions of the same sample in order to avoid unwanted effects arising from the sample’s borders, defects, or small particles. 3. Static distribution of vortices A series of images obtained at 4.2 K after field cooling procedure with fields ranging from 4 mT to +4 mT in steps of 0.02 mT allowed us to determine the remanent field with high accuracy. In the patterned sample we clearly identify the different commensurate vortex states as described in previous reports [17–19]. Fig. 1 shows SHPM images obtained at 4.2 K with H = 0, H = H1 = 9.2 G, H = 2H1, and H = 3H1. Fig. 1a shows an isolated vortex with
An attempt to simultaneously record images while moving vortices with an external dc current for T < 6.8 K showed that vortices remain still until reaching the critical current beyond which a severe heat dissipation associated with the vortex motion drives the entire sample to the normal state. This effect becomes apparent in Fig. 2 where a voltage–current characteristic at T = 7.1 K has been recorded within 5 s time and 5 ls time. In this case, despite the fact that initially the system is under exactly the same thermodynamic conditions, the critical current above which the system shows dissipation differs by a factor of four. For standard dc current sweeps like the 5 s measurement, the transition from superconducting to normal state is very abrupt and hinders us from visualizing the actual vortex motion. In order to overcome this limitation it is imperative to apply short current pulses no longer than several microseconds thus minimizing the heating effects. Fig. 3a shows a typical voltage response for a pulse of 150 ls and amplitude 90 mA obtained at T = 7 K and three different fields. For H = 1.75 mT > Hc2 (dotted line), the voltage signal V(t) shows a square shape with a maximum value corresponding to the normal state resistance Vmax = IRn. For fields H = 1.15 mT Hc2/2, vortices are set in motion immediately after turning on the pulse. The dissipation associated with their motion give rise to a local heating and therefore, to a progressive increase in the voltage until eventually the temperature of the sample reaches Tc and an abrupt jump to the normal state appears. For very low fields, H = 0.01 mT Hc2, three clear regimes can be identified. At short times no dissipation is observed, at intermediate times the picked up signal monotonously increases as a function of time indicating once again a progressive heating of the sample [15], whereas at longer times the system eventually transits to the normal state and a large jump in the dissipation is observed. 4.2. Scanning Hall probe images after current pulses Based on the above described transport properties we established the following protocol for visualizing the vortex dynamics in each of these regimes, (1) the sample is cooled down at the desired field, (2) a short current pulse of duration 50 ls is then ap-
Fig. 1. Scanning Hall cross microscopy images of a Pb film with a periodic array of antidots after field cooling down to 4.2 K and (a) H = 0, (b) H = H1 = 9.2 G, (c) H = 2H1, and (d) H = 3H1. The size of each image is 13 13 lm2. Panel (a) shows an isolated vortex. Panels (b) and (c) obtained very close to the first and second matching condition, respectively, show a very ordered pattern mimicking the periodic array of antidots. In panel (d), a very disordered structure evidences the presence of interstitial vortices.
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150
T = 7.1 K
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t=5s
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I (mA) Fig. 2. Current–voltage characteristics obtained at T = 7.1 K in a Pb bridge with a periodic array of antidots. The data collection has been done in two extreme different time scales, 5 s (solid line) and 5 ls (open circles).
the intermediate length pulse, the vortex lattice is neither fully disorder nor similar to the equilibrium field-cooled image, but rather exhibits a clear tendency to conglomerate in branches along the principal directions of the underlying pinning potential. This result resemble the patterns early predicted by Reichhardt et al. [9] for a system of particles with competing long range repulsive and short range attractive interactions. Interestingly, the images obtained after applying pulses of long duration are identical to those obtained by a standard field cooling procedure (and also similar to Fig. 4a). This finding can be easily understood since long pulses produce an overall heating of the sample above Tc and therefore, once the bias current is turned off (nanoseconds process) the thermal relaxation of the sample towards the original temperature takes about 100 ms [20]. Naturally, this sort of field cooling procedure is very convenient and fast since the sample can be used as a micro-heater with no need to warm up the rest of the bulky addenda. 4.3. Scanning Hall probe images after field cooling with dc current
plied, (3) after that an image scan is performed, and (4) a new pulse, 10 ls longer than the previous pulse, is applied. Then, steps (3) and (4) are repeated until reaching the normal state. In Fig. 4, the SHPM images obtained at short times [panel (a)] and intermediate times [panel (b)] for H = 0.46 mT H1/2 are shown. The periodic distribution of vortices observed in Fig. 4a indicates that short pulses are unable to set the vortices in motion. In contrast to that, pulses of intermediate duration lead to a clear vortex motion as evidence in Fig. 4b. Strikingly, after applying
An alternative method to study the dynamics at high vortex velocities can be achieved by field cooling the sample while maintaining a dc current on it. In this way we ensure that the system will freeze the vortex structure with high vortex velocities at some current dependent temperature Tf. Due to the irreversible behavior of the current–voltage characteristics, differences with respect to the pulsed-current procedure might be expected. Fig. 5 shows images after field cooling without current applied [panel (a)] and with an applied current of 35 mA [panel (b)]. The most obvious feature of this figure is the formation of rivers of flux sitting exactly on top of the holes. This is the main result of this paper and indicates the existence of an attractive interaction between vortices when they move at high velocities. We have observed that the number of these vortex rivers inside the scanning area is always two, irrespective of the applied field. It is important to notice that since the sample is acting as a microheater, the control of cooling rates with an external thermometer is somehow meaningless since the actual temperature of the sample differs significantly from that obtained with standard thermometry. Measurements performed in a co-evaporated plain unpatterned Pb film under identical conditions show no hint of stripe formation (see Fig. 5c), thus pointing out to the key role played by the hole array in the formation and stabilization of flux rivers. Indeed, we have recently demonstrated [21] using the time dependent Ginzburg–Landau formalism that the effective freezing of these vortex rivers can only be achieved if (i) at low temperatures the pinning is strong enough to sustain the quenched metastable state and (ii) the cooling rate is above certain threshold value. It has been demonstrated experimentally [22] and theoretically [23–28] that vortices driven by an electrical current follow intricate paths rather than a well order and correlated lattice motion.
Fig. 4. Scanning Hall probe microscopy images obtained at T = 6.5 K and H = 0.46 mT after a short pulse (a) and intermediate length pulse (b).
Fig. 5. Scanning Hall probe microscopy images obtained at T = 6.5 K and H = 0.46 mT after a short pulse (a) and intermediate length pulse (b) in a Pb film with a periodic array of antidots. Panel (c) shows a SHPM image under the same condition than (b) but in a plain Pb film.
30 25
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t (µs) Fig. 3. Voltage vs. time during a current pulse of 150 ls duration and 90 mA amplitude for three different applied fields. The initial temperature of the system is 7 K. For H = 1.75 mT (dotted line) the system is in the normal state.
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In general, these vortex rivers flow around strongly pinned (immobile) vortices and therefore, corresponds to a plastic-like vortex motion occurring at currents close to the depinning current. It is important to point out that the flux rivers here investigated are of an entirely different nature than previously reported filamentary flux motion since all vortices are set in motion and no plastic regime is needed. An alternative way to obtain well ordered flux channels can be achieved in superconductors with weak pinning submitted to currents much higher than the critical current. In this case an initially imperfect vortex configuration can recrystallize in a perfect lattice which is oriented with one of its principal directions along the direction of motion or even form a smectic vortex flow with elastically coupled vortex channels [29–32]. In this scenario, it is expected that the separation between neighboring channels corresponds to the vortex lattice period. In contrast to this situation, in the present sample we have found that the number of channels seems to be rather insensitive to the applied field. In addition, the presence of steps in the IV characteristics at high currents indicates that in samples with antidots the vortex dynamics at high currents involves the formation of phase slips or kinematic vortices, thus departing from the assumption of point-like vortices used in molecular dynamic simulations [14]. In this work we claim that vortex rivers might have a different origin. Indeed, it has been theoretically proposed [12] that when vortices move at very high speed due to an external current, a directional attractive vortex-vortex interaction can appear. The ultimate reason for this behavior is the elongation of the vortex core (vortex tail) in the direction of motion as a result of the finite time needed for the superconducting order parameter to recover from the vortex passage. Since this intrinsic healing time of the superconducting condensate is at sub-nanoseconds scales, a moving vortex needs typically a very high speed in order to be able to see the wake of another vortex. Once this situation is reached, vortices like to align one behind the other forming stripes.
5. Conclusion To summarize, in this manuscript we report on the first successful visualization of vortex stripe patterns formed by an unconventional vortex attraction when these entities are driven to very high velocities. These, so called, kinematic vortices attract each other forming stripe patterns that can be stabilized by literally freezing their motion via a fast thermal quench and a strong pinning potential produced by an array of antidots. Contrary to other systems exhibiting stripe phases, vortices in our samples have no attractive interaction in equilibrium conditions, and stripe formation results solely from the dynamic history of the superconducting condensate.
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Acknowledgements This work was supported by Methusalem funding by the Flemish government, the Flemish Science Foundation (FWO-Vl), the Belgian Science Policy, and the ESF NES network. A.V.S., G.R.B., and J.V.d.V. acknowledge support from FWO-Vl. R.F.L. acknowledge support from I3P CSIC program and MAT2008-01022. References [1] V.G. Kogan, N. Nakagawa, S.L. Thiemann, Phys. Rev. B 42 (1990) 2631. [2] A.I. Buzdin, A.Yu. Simonov, JETP Lett. 51 (1990) 191; A.M. Grishin, A.Yu. Martynovich, S.V. Yampolskii, Sov. Phys. JETP 70 (1990) 1089. [3] A. Jacobs, Phys. Rev. 34 (1971) 3029. [4] J. Auer, H. Ullmaier, Phys. Rev. B 7 (1973) 136. [5] K. Dichtel, Phys. Lett. A 35 (1971) 285. [6] E.H. Brandt, Phys. Lett. A 39 (1972) 193. [7] V.V. Moshchalkov, M. Menghini, Q.H. Chen, T. Nishio, A.V. Silhanek, V.H. Dao, L.F. Chibotaru, N.D. Zhigadlo, J. Karpinski, Phys. Rev. Lett. 102 (2009) 117001. [8] E. Babaev, M. Speight, Phys. Rev. B 72 (2005) 180502 (R). [9] C. Reichhardt, C.J. Olson Reichhardt, I. Martin, A.R. Bishop, Phys. Rev. Lett. 90 (2003) 026401. [10] A.I. Larkin, Yu.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 68 (1975) 1915; A.I. Larkin, Yu.N. Ovchinnikov, Sov. Phys. JETP 41 (1976) 960. [11] A. Andronov, I. Gordion, V. Kurin, I. Nefedov, I. Shereshevsky, Physica C 213 (1993) 193. [12] D.Y. Vodolazov, F.M. Peeters, Phys. Rev. B 76 (2007) 014521. [13] A.G. Sivakov, A.M. Glukhov, A.N. Omelyanchouk, Y. Koval, P. Müller, A.V. Ustinov, Phys. Rev. Lett. 91 (2003) 267001. [14] J. Gutierrez et al., Phys. Rev. B 80 (2009) 140514. [15] P. Tholfsen, H. Meissner, Phys. Rev. 185 (1969) 653. [16] S. Raedts, A.V. Silhanek, V.V. Moshchalkov, J. Moonens, L.H.A. Leunissen, Phys. Rev. B 73 (2006) 174514. [17] A.N. Grigorenko, G.D. Howells, S.J. Bending, J. Bekaert, M.J. Van Bael, L. Van Look, V.V. Moshchalkov, Y. Bruynseraede, G. Borghs, I.I. Kaya, R.A. Stradling, Phys. Rev. B 63 (2001) 052504. [18] S.B. Field, S.S. James, J. Barentine, V. Metlushko, G. Crabtree, H. Shtrikman, B. Ilic, S.R.J. Brueck, Phys. Rev. Lett. 88 (2002) 067003. [19] A.N. Grigorenko, S.J. Bending, M.J. Van Bael, M. Lange, V.V. Moshchalkov, H. Fangohr, P.A.J. de Groot, Phys. Rev. Lett. 90 (2003) 237001. [20] This time has been estimated by submitting the system to a train of pulses and determining the minimum separation time between consecutive pulses needed to obtain cumulative heating in the sample. [21] A.V. Silhanek et al., Phys. Rev. Lett. 104 (2010) 017001. [22] T. Matsuda et al., Science 271 (1996) 1393; T. Matsuda et al., Science 294 (2001) 2136. [23] N. Gronbech-Jensen, A.R. Bishop, D. Dominguez, Phys. Rev. Lett. 76 (1996) 2985. [24] S. Ryu, M. Hellerqvist, S. Doniach, A. Kapitulnik, D. Stroud, Phys. Rev. Lett. 77 (1996) 5114. [25] H. Jensen, A. Brass, A.J. Berlinsky, Phys. Rev. Lett. 60 (1988) 1676. [26] F. Nori, Science 271 (1996) 1373. [27] C.J. Olson, C. Reichhardt, F. Nori, Phys. Rev. Lett. 80 (1998) 2197. and references therein. [28] K.E. Bassler, M. Paczuski, G.F. Reiter, Phys. Rev. Lett. 83 (1999) 3956. [29] T. Giamarchi, P. Le Doussal, Phys. Rev. Lett. 76 (1996) 3408. [30] L. Balents, C.M. Marchetti, L. Radzihovsky, Phys. Rev. Lett. 78 (1997) 751. [31] F. Pardo, F. de la Cruz, P.L. Gammel, E. Bucher, D.J. Bishop, Nature 396 (1998) 348. [32] M. Marchevsky, J. Aarts, P.H. Kes, M.V. Indenbom, Phys. Rev. Lett. 78 (1997) 531.