Letter pubs.acs.org/NanoLett
Controlled Generation of Quantized Vortex−Antivortex Pairs in a Superconducting Condensate Jun-Yi Ge,*,† Vladimir N. Gladilin,†,‡ Jacques Tempere,‡ Jozef Devreese,‡ and Victor V. Moshchalkov† †
INPACInstitute for Nanoscale Physics and Chemistry, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium TQCTheory of Quantum and Complex Systems, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium
‡
S Supporting Information *
ABSTRACT: Quantized vortices, as topological defects, play an important role in both physics and technological applications of superconductors. Normally, the nucleation of vortices requires the presence of a high magnetic field or current density, which allow the vortices to enter from the sample boundaries. At the same time, the controllable generation of individual vortices inside a superconductor is still challenging. Here, we report the controllable creation of single quantum vortices and antivortices at any desirable position inside a superconductor. We exploit the local heating effect of a scanning tunneling microscope (STM) tip: superconductivity is locally suppressed by the tip and vortex−antivortex pairs are generated when supercurrent flows around the hot spot. The experimental results are well-explained by theoretical simulations within the Ginzburg−Landau approach. KEYWORDS: Superconducting vortex, STM, local heating, vortex generation n a superconducting quantum condensate, the magnetic field is only allowed to enter the superconductor in the form of quantized Abrikosov vortices with each of them carrying one flux quantum Φ0 = h/2e (h, the Plank constant and e, the electron charge). The study of vortex matter can shed the light on the nature of superconductor properties, such as the shape of the Fermi surface,1−4 the superconducting gap symmetry,5,6 and the crystal anisotropy.7,8 As the smallest magnetic bits in a superconductor, vortices also play a crucial role in designing superconductor based electronics and devices.9−11 Therefore, control and manipulation of vortex matter, especially at single vortex level, are of great importance. A lot of efforts have been devoted to the manipulation of individual vortices. By using the magnetic force microscope and the scanning tunnelling microscope, individual vortices can be dragged by magnetic and nonmagnetic tips.12−14 Optical control of vortices is also realized through the heating effect of the laser beam.15 However, these manipulated vortices were either trapped by pinning centers during the flux expulsion or entered the sample through the sample boundaries at high enough magnetic fields and current density. The controllable local generation of vortices from inside a superconducting area is still challenging.16 Recently, the observation of bound vortex dipoles in the Meissner state has been reported in superconducting films with naturally and artificially introduced pinning centers.17,18 Such magnetic dipoles are considered as precursors of quantized vortices. Theoretical calculations have also shown that the vortex dipoles can develop into quantized Abrikosov vortices when high enough supercurrent flows
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© XXXX American Chemical Society
around the pinning centers.17 This provides a possible way to generate single quantum vortices from inside a superconductor. However, for traditional pinning centers, once they are created, their position and pinning potential are already fixed. In addition, vortex generation from such a pinning center often requires a relatively high local supercurrent density. It may occur that vortices would rather penetrate into the superconductor through its edges where the current density reaches maximum. In this Letter, we propose the controllable generation of quantized vortex−antivortex (v−av) pairs by using a scanning tunneling microscope (STM) tip. Around the tip position the superconductivity is locally suppressed by the heating effect of the tunneling junction,14 and vortex−antivortex pairs are generated due to crowding of supercurrents which have to bypass the region with suppressed superconductivity. This region can be considered as a tip-induced pinning landscape well-controlled by tuning the STM bias voltage. This allows us to generate and launch vortex−antivortex pairs at opposite directions in the Meissner state of the superconductor. At low densities, vortices (antivortices) are arranged in a vortex street, while at high densities, they tend to form clusters. The experimental results are well-explained by theoretical simulations. Received: May 24, 2017 Revised: June 26, 2017 Published: July 11, 2017 A
DOI: 10.1021/acs.nanolett.7b02180 Nano Lett. XXXX, XXX, XXX−XXX
Letter
Nano Letters
In a vortex free state, the hot spot can serve as a vortex generator. Figure 1d shows the schematic image of the vortex− antivortex generation process. When a tunneling junction is established between the STM tip and the superconductor, a hot spot, playing the role of a pinning center, is created. The supercurrent, flowing in the vicinity of the hot spot, will be reoriented at the position of the hot spot thus creating a magnetic (vortex) dipole. For a fixed average supercurrent density, the intensity of dipoles mainly depends on the size of the pinning center.18 As the hot spot grows, the intensity of the magnetic dipole also increases, and eventually an Abrikosov vortex and antivortex are generated. Under the Lorentz force (F⃗L), vortices and antivortices will be pushed in opposite directions, perpendicular to the current lines. In the experiments described below, instead of a transport current, the Meissner current is used to produce vortex−antivortex pairs. Figure 2 presents the experimental observation of vortex− antivortex pairs generated by the STM tip. After zero-field
The sample we study consists of a multilayer structure of Au/ Ge/Pb with the size of 200 × 200 μm2, which is deposited on a SiO2/Si substrate (Figure 1a). The sample has clear and straight
Figure 1. (a) Schematic view of sample geometry and the experimental setup, not drawn to scale. (b) Scanning Hall probe microscopy image of the vortex pattern observed after field cooling at T = 4.2 K and B0 = 4.2 G. (c) After applying a pulse tunneling current to the vortex pattern in panel b, adjacent vortices around the tip are attracted to the hot spot at the STM tip position (marked by the cross), and a vortex cluster is formed after switching off the tunneling junction. (d) Schematic image demonstrating the generation mechanism of vortex−antivortex pairs. The dashed lines indicate the supercurrent flowing direction. The solid arrows indicate the moving direction of vortices and antivortices under the Lorentz force.
edges, which allow us to precisely estimate the flowing direction of the supercurrent along the edges. Sample fabrication details can be found elsewhere.14 This sample design has certain advantages when studying vortex generation. Contrary to pinning-free superconductors, the presence of quasi-homogeneously distributed, relatively strong natural pinning centers in our sample prevents the exit of the generated vortices from the sample as well as the recombination of the v−av pairs after switching off the tunneling current sent through the STM tip. Direct mapping of the vortex pattern is realized by using lowtemperature scanning Hall probe microscopy (SHPM), which combines an STM and a Hall cross to make a Hall sensor (schematically shown in Figure 1a).19 Figure 1b and c demonstrates the local heating effect by using the pulse tunneling current through the STM tip. Due to the randomly distributed pinning centers, vortices nucleated after field-cooling form a disordered lattice (Figure 1b). When applying a tunneling current at the position marked by the cross, a hot spot with locally suppressed superconductivity is created, and vortices are attracted to this normal region. After switching off the tunneling current by retracting the STM tip 200 nm away from the sample surface, a vortex cluster is formed14,20 as a result of the Kibble−Zurek symmetry breaking phase transition.21,22 This vortex cluster remains preserved due to the naturally existing pinning centers.
Figure 2. Evolution of vortex generation inside a superconductor by using the heating effect of an STM tip. (a) SHPM image taken after first performing zero-field cooling to 4.2 K and then increasing the external field to B0 = 3.9 G. The dashed line indicates the alignment of the sample edge (see Supplementary Figure S1). The arrow shows the direction of the Meissner current. (b) SHPM image observed after applying a current pulse through the STM tip at the position marked by the cross. (c−d) SHPM images taken after applying the second (c) and third (d) tunneling pulse at the same tip position. (e) Profiles of the absolute value of the magnetic field (squares) measured through the center of 10 vortices/antivortices generated as a result of the tunneling-current pulse. The solid line is the fit with the monopole model, yielding Φ = 1.16 ± 0.2h/2e.
cooling followed by applying an external magnetic field, the Meissner state is observed with Meissner current flowing along the sample edges (Figure 2a). Here, the Meissner current will exert a Lorentz force on the vortices (with the direction depending on the magnetic field direction) and thus is used to separate the vortex and antivortex. The scanned area is chosen close to the sample edge, indicated by the dashed line in each SHPM image (see Supplementary Figure S1). The SHPM images are taken by first releasing the tip in tunneling and then retracting the tip 200 nm away from the surface. Figure 2b−c presents some snapshots of the magnetic field distribution after B
DOI: 10.1021/acs.nanolett.7b02180 Nano Lett. XXXX, XXX, XXX−XXX
Letter
Nano Letters
To study the effect of the supercurrent on the v−av generation, we have chosen four specific locations for the hot spot with different distances D from the sample edge (Figure 3a). For a constant distance D, as shown in Figure 3b, the
applying the tunneling current pulse. After applying a tunneling pulse at the position marked by the cross in Figure 2b, one vortex−antivortex pair is generated and then trapped by the pinning centers. The line connecting the vortex and antivortex is seen to be perpendicular to the Meissner current direction, which is normal to the Lorentz force. One can also notice that the distance between the tip position and the antivortex (blue) is larger than that for the vortex (red). This results from a stronger Lorentz force exerted on the antivortex (which is closer to the sample edge) since the Meissner current increases when approaching the sample edge. By applying the second tunneling pulse, another v−av pair is produced around the tip position as shown in Figure 2c. In both Figure 2b and c, the total vorticity inside the superconductor is zero; the vortex and antivortex are always generated as pair. However, since the tip is placed close to the sample edge, in some cases, the antivortices can be forced out of the superconductor. Starting with the vortex configurations in Figure 2c, we apply one more tunneling pulse at the same position. This time, no v−av pair is generated due to the strongly modified current density by the existing vortices and antivortices. However, the hot spot elevates the temperature at the positions where the vortices and antivortices are located. Therefore, the pinning becomes weaker, and under the Lorentz force exerted on them, vortices and antivortices are pushed further away from the STM tip position. The antivortex which was located closer to the sample edge is now expelled out of the sample. Only one antivortex and two vortices remain in the sample (Figure 2d). As a result, the vorticity inside the superconductor becomes nonzero even when the applied magnetic field is lower than the penetration field. Such effect enables us to inject vortices with only one polarity into the superconductor. Moreover, instead of injecting vortices, we are also able to locally erase vortices from the mixed state through vortex−antivortex annihilation and local flux expulsion (see Supplementary Figure S2). The magnetic profiles of 10 vortices/antivortices are displayed in Figure 2e. It should be noted that the deviation of vortex magnetic field distribution might arise from the difference of random pinning potentials. Also, to avoid the effect of Meissner current on the deformation of vortex profiles, the chosen vortices are all located far from the sample edge. It is also interesting to know whether these generated vortices/ antivortices are quantized as those penetrated into the superconductor from the edges. In superconductors, the magnetic field distribution of a fluxoid can be well simulated by the monopole model.23−26 Considering the active size s of the Hall cross, the model can be modified by integrating over the effective sensing area s2 and then dividing by s2: Bz (x , y) =
Φ 2πs 2 ×
y + s /2
∫y−s/2 2
critical magnetic field Bv−av, above which v−av pairs are generated, decreases with increasing temperature. This behavior can be attributed to a decrease of the vortex formation energy with temperature, so that a lower Meissner current appears sufficient for v−av pair generation. At the same time, at a constant temperature (4.2 K), higher values of Bv−av are needed to generate v−av pairs when increasing D (Figure 3c). This result reflects the behavior of Meissner current density in a thin superconducting film:27−29 J(x) = −(cB0 /2π )x / w 2 − x 2 , where w is the half width of the film and x = w − D is the distance to the center of the superconducting stripe. For the relevant distances D (larger than the penetration depth but still relatively close to the sample edge), the Meissner current density decreases with increasing D approximately as D−1/2, so that at larger D a stronger magnetic field should be applied to reach the Meissner current density required for v−av generation. To further clarify the mechanism of v−av generation, we performed simulations using the time-dependent Ginzburg− Landau (TDGL) equations (see Supplementary Note 1). A superconducting stripe with a width of Lx = 30 μm and a thickness of 85 nm is considered. Randomly distributed pinning centers are introduced in the superconducting film as regions with the reduced mean free path, as shown in Figure 4a. On the basis of the local heating model (see Supplementary Note 2), the temperature distribution in the superconductor at the STM tip location is given by
x + s /2
dy
∫x−s/2
Z [x + y + Z2]3/2 2
Figure 3. (a) SHPM image of the Meissner state observed at 4.2 K. Four tip positions, as marked by the crosses, are shown with different distances D from the sample edge (dashed line). (b) Temperature dependence of the critical magnetic field Bv−av, above which vortex− antivortex pairs are generated by the hot spot for D = 12 μm. (c) Critical magnetic field Bv−av as a function of D at T = 4.2 K.
dx
(1)
where Φ is the total flux carried by the fluxoid and Z = λ + z0 with λ the penetration depth and z0 the distance from the sample surface to the two-dimensional electron gas of the Hall cross. In our measurements, a Hall probe with an active area of s2 = 0.4 × 0.4 μm2 is used. As shown by the solid lines, the fit of experimental data with eq 1 yields Φ = 1.16 ± 0.2 h/2e, which confirms the quantization of the generated vortices/antivortices.
T = T0 + αf (x , y) C
(2) DOI: 10.1021/acs.nanolett.7b02180 Nano Lett. XXXX, XXX, XXX−XXX
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Nano Letters
provide an efficient way to separate and stabilize the KZ vortices in superconductors. (2) At larger α and B0 or smaller D, Φ0 vortices are directly launched from the hot spot during tunneling current pulse (see Figure 4d). After the local heating, we observe four vortices and one antivortex in the simulated area. In this case, vortices, generated by flow past the normal hot spot, and those related to the KZ quenching process can coexist in the vortex pattern stabilized after switching off the tunneling current pulse. The reason why only one antivortex is observed on the lower panel of Figure 4d is that the other three antivortices are expelled out of the sample by the Lorentz force caused by the Meissner current. We also notice that the vortices have a nonzero vertical displacement. After generation from the hot spot, vortices are subsequently pinned by the pinning centers located around the hot spot. The pinned vortices exert a repulsive force on the new vortices launched from the hot spot, thus redirecting their motion.31 Our experimental results confirm that, instead of being arranged into one-dimensional vortex chains, the launched vortices tend to form clusters at high vorticities (see Supplementary Figure S3). Such a scenario is similar to the transition from regular to turbulent shedding observed recently in a BEC where a variety of vortex clusters form at large velocities of the moving obstacle.32 The observation of vortex/antivortex pairs is also reminiscent of the Karman vortex street generated when a fluid moves with respect to an obstacle.32−35 The Karman vortex street is manifested by a periodic shedding of vortices with alternating circulation, which has been observed in both classical and superfluid systems. Superconductivity, as a macroscopic quantum phenomenon, shows a remarkable similarity to superfluidity. However, a Karman vortex street has not been so far observed in superconductors. Our experimental design, that is, by sending a supercurrent through a hotspot (obstacle), and the corresponding results shed new light on the turbulence phenomena in superconductivity. To conclude, we have shown that the STM tip can be used to locally suppress superconductivity in a relatively small hot spot. By applying a Meissner supercurrent, we can create and preserve quantized vortex−antivortex pairs inside the superconductor in a controllable position, determined by the location of the hot spot. Such vortex−antivortex “launcher” can be easily switched on/off just by tuning the bias voltage. Besides the high resolution of the STM tip, one important advantage of our vortex launcher is that superconductivity fully recovers after retracting the STM tip. Moreover, the controllable generation of the vortex−antivortex pairs could also allow the manipulation of individual spins or charges by coupling the vortex to these objects, thus opening new ways for designing functional fluxon-based electronic devices.
Figure 4. Simulations. (a) Distribution of mean free path, showing the pinning landscape used in the TDGL simulations. (b) Temperature distribution in the hot spot generated by the STM tip in the local heating model. (c−d) Simulation results showing the magnetic field distribution before (upper panel), at the time (middle), and after applying the tunneling current. Vortex−antivortex pairs are generated only during the fast relaxation of the hot spot (c) or also in the course of local heating (d). The arrows indicate the averaged direction of the Meissner current. The scale bar is 2 μm.
Here, T0 is the equilibrium temperature in the absence of the tunneling current. The coefficient α is proportional to the power dissipated by the tunnelling current, while the spatial temperature distribution f(x, y) is determined by material and geometric parameters of the sample (see Supporting Information). Figure 4c (α = 0.25Tc, B0 = 0.015μ0Hc2(T0), D = 6 μm) and d (α = 0.2Tc, B0 = 0.015μ0Hc2(T0), D = 4.5 μm) shows the magnetic field distributions in the Meissner state before (upper panel), at the time of (middle panel), and after (lower panel) the local heating by the STM current. Before local heating, the Meissner current flows (on average) along the direction of the arrows. Because of the randomly distributed pinning centers (Figure 4a), many weak magnetic dipoles are formed.17,18 During the tunneling-current pulse, the area around the STM tip is heated to the normal state. As a result, the supercurrent density is redistributed, leading to the formation of a giant magnetic dipole. From the simulations, we have identified two mechanisms of the generation of the v− av pairs depending on the heating power and the Meissner current density at the hot-spot position. (1) In the case of a relatively low power α, or weak applied fields B0, or large distances D of the hot spot from the superconductor border, no Φ0 vortices are generated during the heating; they appear only as a result of the hot-spot relaxation. An example is shown in Figure 4c, where one pair of Φ0-vortex and antivortex is observed after switching the local heating off. In this case the pair production can be attributed to the Kibble−Zurek (KZ) symmetry breaking phase transition, which predicts the appearance of topological defects (vortex/ antivortex pairs for superconductors) when a system is quenched through a phase transition.21,22,30 Our estimates based on the GL modeling show that the quenching time (1− 10 ps) of the hot spot (normal state region) is comparable to the relaxation time (40−100 ps) of the superconducting condensate, bringing the system right into the KZ regime. The generated vortex and antivortex are kept apart by the Lorentz force, proportional to the Meissner current, and then eventually trapped by pinning centers. The KZ phase transition has rarely been observed in a controllable local area mainly due to the fast dynamical recombination of KZ vortices. Our experiments
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.7b02180. Details of the local heating model and the simulation process with the time-dependent Ginzburg−Landau approach (PDF) D
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(20) Shapiro, I.; Pechenik, E.; Shapiro, B. Y. Phys. Rev. B: Condens. Matter Mater. Phys. 2001, 63, 184520. (21) Kibble, T. Topology of Cosmic Domains and Strings, Cosmological Constants; Blackett Laboratory, 1986; p 277. (22) Zurek, W. Nature 1985, 317, 505−508. (23) Ge, J.; Gutierrez, J.; Cuppens, J.; Moshchalkov, V. V. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 174503. (24) Pearl, J. J. Appl. Phys. 1966, 37, 4139−4141. (25) Chang, A.; Hallen, H.; Harriott, L.; Hess, H.; Kao, H.; Kwo, J.; Miller, R.; Wolfe, R.; Van der Ziel, J.; Chang, T. Appl. Phys. Lett. 1992, 61, 1974−1976. (26) Wynn, J.; Bonn, D.; Gardner, B.; Lin, Y.-J.; Liang, R.; Hardy, W.; Kirtley, J.; Moler, K. Phys. Rev. Lett. 2001, 87, 197002. (27) Larkin, A.; Ovchinnikov, Y. N. Sov. Phys. JETP 1972, 34, 651− 655. (28) Huebener, R.; Kampwirth, R.; Clem, J. R. J. Low Temp. Phys. 1972, 6, 275−285. (29) Zeldov, E.; Larkin, A.; Geshkenbein, V.; Konczykowski, M.; Majer, D.; Khaykovich, B.; Vinokur, V.; Shtrikman, H. Phys. Rev. Lett. 1994, 73, 1428. (30) Maniv, A.; Polturak, E.; Koren, G. Phys. Rev. Lett. 2003, 91, 197001. (31) Gladilin, V.; Tempere, J.; Devreese, J.; Gillijns, W.; Moshchalkov, V. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 054503. (32) Kwon, W. J.; Kim, J. H.; Seo, S. W.; Shin, Y. Phys. Rev. Lett. 2016, 117, 245301. (33) Thoraval, M.-J.; Takehara, K.; Etoh, T. G.; Popinet, S.; Ray, P.; Josserand, C.; Zaleski, S.; Thoroddsen, S. T. Phys. Rev. Lett. 2012, 108, 264506. (34) Kobayashi, M.; Tsubota, M. Phys. Rev. Lett. 2005, 94, 065302. (35) Sasaki, K.; Suzuki, N.; Saito, H. Phys. Rev. Lett. 2010, 104, 150404.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]; tel.: +32(0)16327118. ORCID
Jun-Yi Ge: 0000-0001-5206-2930 Author Contributions
J.-Y.G. made the sample and performed the SHPM measurements. V.N.G., J.T., and J.D. did the TDGL simulations. All authors contributed to the discussion and analysis of the data. J.-Y.G. and V.N.G. wrote the manuscript. V.V.M. coordinated the whole work. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
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REFERENCES
We thank the support from the Methusalem funding by the Flemish government, the Flemish Science Foundation (3E150622), and the MP1201 COST action. J.T. also acknowledges support from the Research Council of Antwerp University (BOF) and from the Flemish Science Foundation (FWO) grant no. G.0429.15N.
(1) Guillamon, I.; Suderow, H.; Guinea, F.; Vieira, S. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 77, 134505. (2) Hess, H.; Robinson, R.; Waszczak, J. Phys. Rev. Lett. 1990, 64, 2711. (3) Miranović, P.; Nakai, N.; Ichioka, M.; Machida, K. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 68, 052501. (4) Wang, Y.; Hirschfeld, P. J.; Vekhter, I. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85, 020506. (5) Nakai, N.; Miranović, P.; Ichioka, M.; Machida, K. Phys. Rev. Lett. 2002, 89, 237004. (6) Shan, L.; Wang, Y.-L.; Shen, B.; Zeng, B.; Huang, Y.; Li, A.; Wang, D.; Yang, H.; Ren, C.; Pan, S. H.; Wang, Q.-H.; Wen, H.-H. Nat. Phys. 2011, 7, 325−331. (7) Yethiraj, M.; Mook, H.; Wignall, G.; Cubitt, R.; Forgan, E.; Lee, S.; Paul, D.; Armstrong, T. Phys. Rev. Lett. 1993, 71, 3019. (8) Grigorenko, A.; Bending, S.; Tamegai, T.; Ooi, S.; Henini, M. Nature 2001, 414, 728−731. (9) Golod, T.; Iovan, A.; Krasnov, V. M. Nat. Commun. 2015, 6, 8628. (10) Nagasawa, S.; Hinode, K.; Satoh, T.; Kitagawa, Y.; Hidaka, M. Supercond. Sci. Technol. 2006, 19, S325. (11) Hastings, M.; Reichhardt, C. O.; Reichhardt, C. Phys. Rev. Lett. 2003, 90, 247004. (12) Auslaender, O. M.; Luan, L.; Straver, E. W.; Hoffman, J. E.; Koshnick, N. C.; Zeldov, E.; Bonn, D. A.; Liang, R.; Hardy, W. N.; Moler, K. A. Nat. Phys. 2009, 5, 35−39. (13) Kremen, A.; Wissberg, S.; Haham, N.; Persky, E.; Frenkel, Y.; Kalisky, B. Nano Lett. 2016, 16, 1626−1630. (14) Ge, J.-Y.; Gladilin, V. N.; Tempere, J.; Xue, C.; Devreese, J. T.; Van de Vondel, J.; Zhou, Y.; Moshchalkov, V. V. Nat. Commun. 2016, 7, 13880. (15) Veshchunov, I. S.; Magrini, W.; Mironov, S.; Godin, A.; Trebbia, J.-B.; Buzdin, A. I.; Tamarat, P.; Lounis, B. Nat. Commun. 2016, 7, 12801. (16) Kirtley, J.; Tsuei, C.; Tafuri, F. Phys. Rev. Lett. 2003, 90, 257001. (17) Ge, J.-Y.; Gutierrez, J.; Gladilin, V. N.; Devreese, J. T.; Moshchalkov, V. V. Nat. Commun. 2015, 6, 6573. (18) Ge, J.-Y.; Gladilin, V. N.; Xue, C.; Tempere, J.; Devreese, J. T.; Van de Vondel, J.; Zhou, Y.; Moshchalkov, V. V. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 93, 224502. (19) Bending, S. J. Adv. Phys. 1999, 48, 449−535. E
DOI: 10.1021/acs.nanolett.7b02180 Nano Lett. XXXX, XXX, XXX−XXX
