# Fileset [ms_v2.pdf](https://mdr.nims.go.jp/filesets/5646c0a4-fb7a-43e7-a3a3-3aa83c722e4b/download) ## Creator Wenxuan Qian, [Yash Chauhan](https://orcid.org/0009-0000-9371-2813), [Ryohei Nemoto](https://orcid.org/0000-0002-7343-7268), [Keisuke Sagisaka](https://orcid.org/0000-0002-5089-4271), [Shunsuke Yoshizawa](https://orcid.org/0000-0003-3380-5473), [Takashi Uchihashi](https://orcid.org/0000-0003-0811-5665) ## Rights ©2026 American Physical Society[In Copyright](http://rightsstatements.org/vocab/InC/1.0/) ## Other metadata [Anisotropic transport of Josephson vortices in atomic-layer superconductors on vicinal surfaces](https://mdr.nims.go.jp/datasets/341fed38-e327-4608-b47a-12fe86acb640) ## Fulltext Anisotropic transport of Josephson vortices in atomic-layer superconductors onvicinal surfacesWenxuan Qian,1, 2 Yash Chauhan,3, 4 Ryohei Nemoto,1, 5 KeisukeSagisaka,1 Shunsuke Yoshizawa,1 and Takashi Uchihashi1, 2, ∗1Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science, 1-1, Namiki, Tsukuba, Ibaraki 305-0044, Japan2Graduate School of Science, Hokkaido University,Kita-10 Nishi-8, Kita-ku, Sapporo 060-0810, Japan3School of Physical Sciences, National Institute of Science Education and Research, Jatni 752050, India4Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India5Department of Physics, School of Science, Institute of Science Tokyo,2-12-1, Ookayama, Meguro-ku, Tokyo 152-8551, Japan(Dated: June 18, 2026)Atomic steps have strong influences on surface two-dimensional superconductors. Josephson vor-tices formed at the atomic steps under magnetic fields may dominate transport phenomena at lowtemperatures, but its experimental verification is still lacking. Here, we report the vortex transportproperties of atomic-layer superconductor Si(111)-(√7 ×√3)-In with vicinal surfaces, for whichJosephson vortices are directly observed by scanning tunneling microscopy. A sharp drop in re-sistance with decreasing temperature T , detected under out-of-plane magnetic field B, reveals adistinctive anisotropy with respect to the atomic step direction. The anisotropy of sheet resis-tance, proportional to that of vortex mobility, amounts to the order of 103 at intermediate magneticfields. In the high-T and low-B region, Josephson vortices exhibit thermally excited creep mo-tions with anisotropic activation energy Uact. A further increase in B suppresses Uact toward zeroanisotropically, resulting in one-dimensional pinning-free vortex flow at 0.10 ≲ B ≲ 0.20 T. At thelowest temperatures, the vortex motion is governed by quantum tunneling. A B-T phase diagramconstructed based on these measurements reveals multiple regions characterized by directionallydependent vortex-transport mechanisms.I. INTRODUCTIONAtomic-layer superconductors, realized using diversesurfaces, interfaces, and two-dimensional (2D) materials,have attracted extensive attention due to their distinctiveand anomalous properties [1–14]. For example, they un-dergo superconductor-insulator transitions driven by dis-order, magnetic field, and carrier density, thus providingan ideal platform for studying quantum phase transitionsin 2D systems [10–12, 15, 16]. Another remarkable phe-nomenon is an anomalously high in-plane critical mag-netic field due to the breaking of the inversion symmetryand Rashba/Ising spin-orbit coupling [6, 7, 17].Important geometric features that may govern thephysics of atomic-layer superconductors are atomic stepsthat weakly couple the neighboring surface terraces.They work as Josephson junctions and allow supercur-rents to run over a macroscopic distance [3]. Under amagnetic field, such atomic steps accommodate Joseph-son vortices [5, 18–20]. They are anomalous in thatthe vortex core is elongated along the atomic step andthe superconducting order parameter within the core islargely sustained [18, 21, 22]. These features are in strik-ing contrast to those of Pearl vortices formed on 2D flatsurfaces[23]. Such vortices can be regarded as the 2Danalogue of Josephson vortices induced within the layers∗ Corresponding author:UCHIHASHI.Takashi@nims.go.jpof cuprate superconductors under an in-plane magneticfield [24–28]If the atomic steps are perfectly straight, Josephsonvortices should flow freely along them because of trans-lational symmetry, while their motion may be hinderedby a pinning force in the perpendicular direction. Suchan anisotropic transport of vortices can potentially af-fect the quantum phase transitions in 2D superconduc-tors through the reduction of the effective dimensionality.They can also govern the thermal properties of the sys-tem, because an excess entropy is carried by a vortexcore [29–32]. Nevertheless, such a directional pinning ef-fect of the atomic step has never been probed, and evenits existence is not obvious because of a huge disparitybetween the typical length scales (atomic steps: < 1 nm,vortex core: ∼ 100 nm). Furthermore, the actual be-havior of Josephson vortices can be sensitive to the mor-phological details of atomic steps, which generally breakthe translational symmetry. This calls for an experimen-tal investigation of the transport properties of Josephsonvortices.Here, we report on highly anisotropic transport ofJosephson vortices in an atomic-layer superconductorSi(111)-(√7 ×√3)-In (referred to as (√7 ×√3)-In be-low) [2, 3, 33, 34], which is grown on a vicinal surface.The morphology of parallel atomic steps and the pres-ence of Josephson vortices are confirmed using a scan-ning tunneling microscope (STM). Transport propertiesof Josephson vortices in the directions parallel and per-2pendicular to the atomic steps are probed through four-terminal resistance measurements in a wide range of tem-perature T and magnetic field B. The superconductingtransition is detected as a sharp decrease in resistance un-der magnetic field B, revealing a distinctive anisotropywith respect to the atomic step direction. The in-planeanisotropy of sheet resistance, proportional to vortex mo-bility, amounts to the order of 103 at intermediate mag-netic fields. At relatively high T and low B, the vor-tices show thermally excited creep motions with a distinc-tive in-plane anisotropy of activation energy Uact. Uactvanishes for the parallel direction at B ≈ 0.10 T whileremaining finite for the perpendicular direction up toB ≈ 0.20 T, resulting in a one-dimensional (1D) pinning-free vortex flow in the intermediate fields. At lower T , thevortex motions are strongly suppressed and governed byquantum tunneling. A B-T phase diagram constructedfrom these results clarifies the coexistence of direction-ally dependent vortex-transport mechanisms in multipleregions.II. EXPERIMENTALA. InstrumentationAll experiments were conducted in ultrahigh vacuum(UHV) at a base pressure of P < 1 × 10−8 Pa. STMmeasurements were carried out at 0.4 K. To securebulk carriers and a tunneling current at this tempera-ture, highly doped n-type Si substrates with resistivitiesρ < 0.01 Ωcm were used. Transport measurements werecarried out separately in a home-built UHV instrumentbetween 0.4 and 4.2 K. For this purpose, non-doped Sisubstrates with resistivities ρ > 1000 Ωcm were used toquench the bulk carriers. Other parameters were identi-cal to those adopted for STM measurements. The effectof the substrate dopants on the superconducting prop-erties of (√7 ×√3)-In is negligible as revealed by thealmost identical values of Tc for highly-doped and non-doped Si substrates [2, 17]). To facilitate quantitativefour-terminal resistance measurements, a current pathwas defined using a shadow mask technique [3, 17]. Fordetails, see Methods in Supplemental Material [35].B. Sample DesignFigure 1(a) shows the side view of the schematic sam-ple geometry in the present work. The Si substrate hasa vicinal (111) surface with the normal direction tiltedby 0.2 ◦ toward the [1̄1̄2] orientation. Assuming thatthe atomic steps are mostly of the monatomic height(h0 = 0.31 nm), the surface is composed of parallel ter-races running in the [11̄0] direction with an average widthof 89 nm. Such a regular atomic step array can be real-ized by adopting a special protocol of substrate cleaning(a)0.31 nm89 nm SiIn0.2°112111110(b)current || steps+I -I+V -Vcurrent ⊥ steps+V +I-V -I 112110Lorentz force|| stepsLorentz force⊥ steps𝑱𝑱FIG. 1. Schematic drawing of the sample design. (a) Sideview of the sample geometry. The Si(111) substrate has avicinal surface with the normal direction tilted by 0.2 ◦ to-ward [1̄1̄2] orientation. The surface is composed of parallelterraces running in the [11̄0] direction, separated mostly bymonatomic steps. The Si(111) surface is covered by In atomicbilayers to form (√7 ×√3)-In. (b) Top view of the samplegeometry adopted for transport measurements. The upperpanels illustrate linear current paths in two configurations,which are connected to voltage/current electrodes for four-terminal resistance measurements. The lower panels are mi-croscopic views within the current paths. The cores of Joseph-son vortices are represented by white-shaded ovals, which areaccompanied by circulating supercurrents (yellow lines) andquantum magnetic fluxes (red crossed circles). The red arrowsshow the directions of the Lorentz force exerted by externalcurrents Jext. In the left panels, the Lorentz force is appliedparallel to the atomic steps (parallel configuration), while per-pendicular in the right panels (perpendicular configuration).[35, 36]. The Si(111) surface is covered by In atomic bi-layers to form (√7×√3)-In, the structure model of whichhas been established previously [33, 34]. It is a repre-sentative atomic-layer superconductor with a transitiontemperature of Tc ≈ 3.1 K [2, 3, 17]. The emergenceof Josephson vortices at atomic steps on this surface hasbeen established in previous work [18].Figure 1(b) shows the top view of the sample designadopted for transport measurements. The upper panelsillustrate current paths with a size of 0.3×1.2 mm2 in twoconfigurations, which are connected to voltage/currentelectrodes for four-terminal resistance measurements.The lower panels show microscopic views within the cur-rent paths, where Josephson vortices are created alongthe steps under an out-of-plane magnetic field. The coresize of the Josephson vortex in (√7×√3)-In perpendicu-lar to the steps is about 80–100 nm [18], which is approxi-mately equal to the average terrace width of 89 nm. Thisscale matching is intended to enhance the pinning effecton Josephson vortices and simultaneously to reduce thearea of flat terraces available for Pearl vortices. In the3left panels, a current runs in the direction perpendicularto the atomic steps. Since the Lorentz force is exertedon vortices along the steps, vortices are expected to moveeasily in this direction. This vortex motion will be de-tected by a large sheet resistance Rsheet of the samplethrough a relation [32]Rsheet = BΦ0µϕ, (1)where Φ0(= h/2e) is the magnetic flux quantum andµϕ is the mobility of vortices. µϕ is defined here byµϕ = −⟨vϕ⟩/fL, where ⟨vϕ⟩ is the average velocity of avortex and fL is the Lorentz force. We note that Eq. (1)is conventionally expressed using viscosity η(= µ−1ϕ ) fora continuous vortex flow. In the right panel of Fig. 1(b),by contrast, a current runs in the direction parallel to thesteps. In this case, vortices feel the Lorentz force in thedirection perpendicular to the atomic steps, and thus areimpeded by them. This should result in a low vortex mo-bility µϕ, resulting in a low resistance Rsheet according toEq. (1). In the following, the former and latter setups arecalled parallel and perpendicular configurations, respec-tively; i.e., the corresponding directions refer to those ofvortex motions.III. RESULTS AND DISCUSSIONA. STM MeasurementsBefore carrying out transport measurements, we firstcharacterized the surface morphology of a (√7×√3)-Insample using an STM. As mentioned above, the samplewas prepared using a highly doped Si substrate with thesame protocol as adopted for transport measurements.Figure 2(a) shows a representative STM topograph withan area of 1000× 1000 nm2. The image reveals an arrayof atomic steps running approximately in the [11̄0] direc-tion. A line profile taken along the red dashed line shows12 terraces with an average width of about 80 nm, ac-companied by seven monatomic steps (height: h0 = 0.31nm) and four diatomic steps (height: 2h0 = 0.62 nm)(Fig. 2(b)). These observations confirm that the surfacemorphology of the actual sample approximately followsthe design in Fig. 1(a). The inset of Fig. 2(a) shows amagnified image with an area of 10 × 10 nm2, reveal-ing a perfectly ordered In atomic layers [3, 37]. The redparallelogram indicates the unit cell of (√7×√3)-In.Superconductivity and vortex formation in (√7×√3)-In were also confirmed through STM measurements. Fig-ure 2(d) shows a dI/dV spectrum acquired on a surfaceterrace at T = 0.4 K and B = 0 T (blue dots), signifyingthe appearance of a superconducting energy gap. Ouranalysis using the Dynes formula gives a nearly perfectfit (black line), giving an energy gap of ∆ = 0.552 meV[38]. Assuming Tc = 3.1 K, the ratio of ∆/kBTc = 2.06is almost equal to a previously reported value of 2.08 [2],while it is slightly larger than ∆/kBTc = 1.76 predicted(b)(c)110200 nm 200 nmAB(d)(a) 112dI/dV(arb.)Bias voltage (mV)-2 -1 0 212.01.00.00.51.5FIG. 2. STM characterization of the sample surface. (a) STMtopograph image with a size of 1000× 1000 nm2 with a sam-ple bias voltage V = 100 meV and a tunneling current I = 5pA. The arrows indicate the crystalline orientations of the sur-face. The inset is a magnified image with a size of 10×10 nm2(V = 500 mV and I = 50 pA). The red parallelogram indi-cates the unit cell of (√7 ×√3)-In. (b) Height profile takenalong the dashed line in Fig. 2(a). (c) ZBC image acquiredover the same area of Fig. 2(a) (Setpoint: V = 20 meV andI = 500 pA). The orange-colored and yellow lines indicate thelocations of monatomic and diatomic steps, respectively. Abrighter color corresponds to a higher intensity of ZBC sig-nal. The vertical arrow indicates the sub-scanning directionof STM imaging. (d) dI/dV spectrum acquired on a terraceof (√7 ×√3)-In (blue dots, Setpoint: V = 10 mV and I =500 pA), signifying the appearance of a superconducting en-ergy gap. The solid line is a fit to the data using the Dynesformula. All measurements were conducted at T = 0.4 K.Magnetic field was set at B = 0.04 T (a,c) and B = 0.00 T(d).by the BCS theory. The zero-bias conductance (ZBC)mapping taken at B = 0 T shows that ZBC is uniformlysuppressed nearly to zero including at the step edges, in-dicating homogeneous development of superconductingenergy gap (see Fig. S1 of Supplemental Materials [35]).The absence of significant disorder helps establish su-perconductivity at least down to T = 0.4 K, which willalso be shown by transport measurements below. Fig-ure 2(c) displays a ZBC image acquired at T = 0.4 Kand B = 0.04 T over the same area of Fig. 2(a). Theorange-colored and yellow lines indicate the locations ofmonatomic and diatomic steps, respectively. A brightercolor in the image corresponds to a higher intensity ofZBC signal and thus to a higher local density of states atthe Fermi level. The elongated bright regions straddlingatomic steps are assigned to the cores of Josephson vor-tices. Vortices prefer to take these positions because therecovery of superconducting order parameter at a Joseph-son vortex core leads to an increase in superconductingcondensation energy gain [18, 21, 22].4Interestingly, most of the Josephson vortices inFig. 2(c) appear cut in the middle of scanning (e.g., seeFeature A). This means that Josephson vortices emergeand vanish in the scanning area abruptly, suggesting ahigh vortex mobility along the steps as expected. Here,the STM image was acquired by scanning horizontally,while the scanning line was translated vertically (indi-cated by the arrow). The fact that several vortices onthe same line moved simultaneously indicates a collec-tive motion of vortices triggered by a single event. Themechanism of the vortex motion cannot be determinedfrom the STM measurements; it could be driven by quan-tum tunneling at the lowest temperatures, as indicatedby the transport measurements (see Sec. III-C), or by avortex-STM tip interaction exerted during scanning.We note that Fig. 2(c) includes bright-colored regionslocated within terrace constrictions, which are not dis-turbed by scanning (e.g. see Feature B). These are as-cribed to vortex cores immobilized at these constrictions.As such, they should not affect transport properties of theoverall system, at least at low magnetic fields where vor-tices are spatially separated. In the following discussions,we will focus on the effects of Josephson vortices.B. Magneto-transport MeasurementsWe now move on to the results of transport measure-ments under out-of-plane magnetic fields. Four sampleswere prepared in total; Samples A1 and A2 for the paral-lel configuration (left panels of Fig. 1(b)) and Samples B1and B2 for the perpendicular configuration (right panelsof Fig. 1(b)). The sheet resistances for the former and thelatter are denoted as Rsheet∥ and Rsheet⊥, respectively.The symbols for the parallel (∥) and perpendicular (⊥)configurations will be omitted when the description ap-plies to both of them. The normal-state values measuredat T = 4.2 K are denoted as Rsheet,N.Figure 3(a) shows the sheet resistance Rsheet∥ of Sam-ple A1 as a function of temperature T at different mag-netic fields from B = 0.00 to 0.16 T with an incrementof 0.02 T. For B = 0.00 T, Rsheet∥ plummets toward zeroaround T = 3.0 K, signifying a superconducting tran-sition [3, 17]. A small but finite resistance of ∼ 0.1 Ωremains even at sufficiently low temperatures, probablydue to a stray magnetic field within the cryostat. It couldalso be due to vortices and antivortices created duringthe superconducting transition through the Kibble-Zurekmechanism [39]. The determination of the exact cause is,however, beyond the scope of the present study.This transition is rapidly suppressed by applying mag-netic field. At the lowest temperature of 0.4 K, Rsheet∥starts to deviate from zero around B = 0.06 T, finallyreaching 160 Ω (≈ Rsheet∥,N) at B = 0.16 T. The sameexperiment was carried out for Sample B1 from B = 0.00to 0.40 T (Fig. 3(b)). In this case, Rsheet⊥ is less sensi-tive to magnetic field; at the lowest temperature of 0.4K, Rsheet⊥ stays at less than 0.1 Ω up to B = 0.16 T.(b)(a) parallel(c) (d)B10.00 T0.28 T|T-Tc(B)|/(BT)1/2 (K1/2 T-1/2)Gfl(B/T)1/2(Ω-1T1/2 K-1/2)T trans(K), T c2(K)B (T)Ttrans ||Tc2A1 A2Ttrans ⊥ B1 B2B1 B20.00 T0.16 TRsheet||(Ω)T (K)Rsheet ||,N/20.06 TA1T (K)0.00 TRsheet⊥(Ω)Rsheet ⊥,N/20.16 T0.40 T0.22 Tperpendicular B1FIG. 3. Temperature-dependent sheet resistance Rsheet,which is directly proportional to the mobility of vortices, mea-sured at different magnetic fields. (a) Rsheet∥ (Sample A1) formagnetic fields from B = 0.00 to 0.16 T with an increment of0.02 T. The dashed line indicates the Rsheet∥ = Rsheet∥,N/2,from which Ttrans∥ is determined. (b) Analogous plot ofRsheet⊥ (Sample B1) for magnetic fields from B = 0.00 to0.40 T with an increment of 0.02 T. (c) Ullah-Dorsey scalinganalysis applied to the Rsheet⊥ (Sample B1) from B = 0.00 to0.28 T with an increment of 0.02 T. Tc2 is determined fromthe analysis. (d) Ttrans∥, Ttrans⊥, and Tc2 plotted as a func-tion of B for all samples.This clearly indicates that a large portion of the sampleis still covered by superconducting regions while vorticescreated by magnetic field can hardly move in the perpen-dicular direction. By contrast, under the same condition,the vortices are highly mobile in the parallel direction,as signified by Rsheet∥ ≈ Rsheet∥,N of Sample A1. Ac-cording to Eq. (1), Rsheet∥/Rsheet⊥ > 103 at T = 0.4K and B = 0.16 T means that µϕ∥/µϕ⊥ > 103, whereµϕ∥ and µϕ⊥ are vortex mobilities in the parallel andperpendicular directions, respectively. Rsheet⊥ exhibitsa marked increase only above B = 0.22 T, reaching 170Ω (≈ Rsheet⊥,N) at B = 0.40 T. The same experimentswere conducted using Samples A2 and B2, confirming thereproducibility of the general behaviors (see Fig. S2 ofSupplemental Material [35]).Here, we define the transition temperature Ttrans∥ forthe parallel configuration at a magnetic field B from arelation Rsheet∥(Ttrans∥) = Rsheet∥,N/2 (see the dashedline in Fig. 3(a)). We note that Ttrans∥ for B ̸= 0 re-flects temperature-induced changes in vortex mobilityµϕ∥ rather than a thermodynamic superconducting tran-sition. Likewise, Ttrans⊥ is defined for the perpendic-ular configuration from Rsheet⊥(Ttrans⊥) = Rsheet⊥,N/2(see the dashed lines in Fig. 3(b)). In this case, Ttrans⊥corresponds to a thermodynamic transition, because su-5perconductivity is broken at the mean-field level nearTtrans⊥.The mean-field transition temperature under a mag-netic field B can be determined more systematically us-ing a scaling analysis based on the Ullah-Dorsey theory,which is denoted as Tc2(B) below [16, 40, 41]. Accordingto the theory, Rsheet⊥ measured as a function of T andB satisfies the following scaling relation:Gfl(B/T )1/2 = f(T − Tc2(B)(BT )1/2), (2)where Gfl ≡ R−1sheet⊥ − R−1sheet⊥,N is a conductance dueto superconducting fluctuations near Tc2, and f is ascaling function. A series of Tc2(B) was determinedby manually adjusting them in such a way that thecurves of Gfl(B/T )1/2 plotted as a function of (T −Tc2(B))/(BT )1/2 showed the best collapse for the wholedata set. Figure 3(c) shows the result of such an anal-ysis, showing that all data roughly collapse on a singlecurve. The deviation from the ideal scaling behavior maybe attributed to the in-plane anisotropy of the system,while the theory assumes an isotropic 2D superconductor.Nevertheless, the convergence of the curves is excellentnear Tc2, probably because significant overlap of vorticesin both directions makes the system less anisotropic nearTc2.Figure 3(d) plots the results on Ttrans∥, Ttrans⊥, and Tc2as a function of B for all samples. As expected, Ttrans⊥and Tc2 show similar dependencies. In the following, weuse Tc2 to represent the thermodynamic phase transitionbecause it is uniquely determined from Eq. (2), whileTtrans∥ is subject to the numerical factor in its definition.We find that Ttrans∥ is significantly lower than Tc2. Thismeans that Josephson vortices are highly mobile only inthe parallel direction within a wide range of Ttrans∥