Scanning SQUID microscopy of vortex clusters in multiband superconductors Taichiro Nishio, 1 Vu Hung Dao, 1 Qinghua Chen, 1 Liviu F. Chibotaru, 1 Kazuo Kadowaki, 2 and Victor V. Moshchalkov 1,$_{∗}$ $^{1}$INPAC-Institute for Nanoscale Physics and Chemistry, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium $^{2}$Institute of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan (Dated: July 13, 2018) In type-1.5 superconductors, vortices emerge in clusters, which grow in size with increasing mag- netic field. These novel vortex clusters and their field dependence are directly visualized by scanning SQUID microscopy at very low vortex densities in MgB 2 single crystals. Our observations are elu- cidated by simulations based on a two-gap Ginzburg-Landau theory in the type-1.5 regime. PACS numbers: 74.25.Qt, 74.70.Ad microscope images and electron diffraction patterns for the crystals show that the crystals have no grain bound- aries. No impurity contamination was detected within 0.1% accuracy by an electron probe microanalyzer. In magnesium diboride $^{1}$, MgB$_{2}$ , the superconducting gaps open for both the two-dimensional (2D) σ -band (gap size Δ$_{σ}$=7.1 meV $^{2,3}$) and the 3D π -band (Δ$_{π}$=2.2 meV $^{3,4}$). The possibility of type-1.5 superconductivity5 has been suggested for clean single crystals of MgB$_{2}$, which lie in the type-1.5 regime: κ$_{σ}$ =3.7 > 1 $_{/}$√$_{2 (type-}$ 2) and κ$_{π}$ =0.66 < 1 $_{/}$√$_{2 (type-1), where}$ κ$_{α}$ ( α = σ, π ) is the Ginzburg-Landau (GL) parameter for each band esti- mated from the band structure calculations $^{6}$. In contrast to conventional type-2 superconductors $^{7}$, in type-1.5 su- perconductors, vortex stripes and gossamer-like vortex patterns, i.e., vortex clusters emerge at relatively low applied fields, 5 which is due to a competition between attractive (type-1) and repulsive (type-2) vortex inter- actions governed by a two-gap GL theory $^{8–12}$. Interest- ingly, a substantial difference in vortex structure between type-1.5 and type-2 superconductors, as a fingerprint of type-1.5 superconductivity, is expected at very low vor- tex densities because the intervortex distances in clusters are likely to be almost independent of the applied field in type-1.5 superconductors, whereas the intervortex dis- tances in type-2 superconductors follow the conventional dependence ( φ$_{0}$/B ) 1 / $^{2}$, where φ$_{0}$ (=2.07 × 10 $_{−}$15 T m) is the flux quantum and B the magnetic field. This mo- tivates strongly direct vortex visualization experiments aimed at investigating vortex structure at very low vor- tex densities in MgB$_{2}$ single crystals. Additionally, it is important to verify the existence of type-1.5 supercon- ductivity in high quality MgB$_{2}$ crystals different from the ones used in Ref. 5. We model vortex systems in a type-1.5 superconduc- tor in the same way as in Ref. 5. We invoke the fol- lowing two-gap GL theory to derive the vortex-vortex interaction numerically. A two-gap GL free energy functional 8–12 is the sum of two single-band GL func- tionals with the Josephson coupling term corresponding to the interaction between two bands: Our scanning SQUID microscope is based on prototype SQM2000 15 (SII NanoTechnology Inc.). We fabricated a SQUID sensor, which is a SQUID magnetometer linked with a circular Nb pickup loop. The Nb pickup loop with an inner diameter of 8 µ m scans the surface of a sample, keeping a few micrometers away from the surface. A magnetic flux sensitivity is 5 µφ$_{0}$ /Hz 1 / $^{2}$. A field noise referred to the pickup loop is ∼ 10 $_{−}$10 T/Hz 1 / $^{2}$. A spatial resolution is ∼ 2.5 µ m. Before scans, as-grown crystals were cooled from a temperature higher than T$_{c}$ to 4.2 K under the magnetic field parallel to the c -axis and then scans were made on the ab -plane at 4.2 K. Scanning steps are 0.5 µ m and 1.0 µ m for MgB$_{2}$ single crystals and a Nb film, respectively. where F$_{α}$ ( α = σ, π ) is the free energy of each band and h (= | h | ) is a magnetic field ( h = rot A , where A is the vector potential): 16 In order to study vortex structure in MgB$_{2}$, we made scans with a scanning superconducting quantum inter- ference device (SQUID) microscope on single crystals of MgB$_{2}$ which were grown by a pressure synthesis technique 13 (a crystal grown by this technique was used elsewhere $^{14}$). The superconducting transition temper- atures T$_{c}$ of the crystals measured by a SQUID mag- netometer are 38.5 K [the transition width Δ T$_{c}$ =0.8 K (10-90 % criterion)]. The surface of crystals was con- firmed not to have any cracks or holes by using a field emission scanning electron microscope in the secondary electron regime. High-resolution transmission electron where E$_{α}$ is the condensation energy 17 and C =( φ$_{0}$/ 2 π ) √ | E$_{α}$ | /µ$_{0}$κ 3 $_{α}$. F$_{σπ}$ is the Josephson coupling term: F$_{σπ}$ = E$_{γ}$ 2 (Ψ ∗ $_{σ}$Ψ$_{π}$ + Ψ ∗ $_{π}$Ψ$_{σ}$ ), where E$_{γ}$ is the coupling energy, where we take a value used in Ref. 17. The vortex-vortex interaction energy between two vortices is the sum of increased energies of each vortex caused by the presence of the other. It can be F$_{α}$ = 2 E$_{α}$ | Ψ$_{α}$ | $^{2}$+ | E$_{α}$ || Ψ$_{α}$ | $^{4}$+ C | ( − i ∇ + 2 π φ$_{0}$ A )Ψ$_{α}$ | $^{2}$, (2) F = ∫ dr $^{3}$( F$_{σ}$ + F$_{π}$ + F$_{σπ}$ + h 2 2 µ$_{0}$ ) , (1) arXiv:1001.2199v1 [cond-mat.supr-con] 13 Jan 2010