S C I E N C E A D VA N C E S | R E S E A R C H A R T I C L E
C O N D E N S E D M AT T E R P H Y S I C S

Band-resolved Caroli–de Gennes–Matricon states ofmultiple-flux-quanta vortices in a multiband
superconductor
Thomas Gozlinski1*†, Qili Li1†, Rolf Heid2, Ryohei Nemoto3, Roland Willa4, Toyo Kazu Yamada3,5,
Jörg Schmalian2,4, Wulf Wulfhekel1,2

INTRODUCTION

The classical solutions in the Ginzburg-Landau theory allow a thermodynamic classification of superconductors into type I and type
II. Decisive for their behavior in magnetic field is the interface
energy between the superconducting and normal phase driven by
the ratio of the London magnetic penetration depth λL and the superconducting coherence length ξ. For Ginzburg-Landau parameters κ = λL/ξ < 1/√2, type-I behavior and for >1/√2, type-II
behavior was predicted (1). A type-I superconductor is characterized by a positive interface energy and an attractive vortex-vortex
interaction favoring an intermediate state with large normal
domains (2). A type-II superconductor is characterized by a negative interface energy and a repulsive vortex-vortex interaction thatfavors an Abrikosov lattice of single-flux-quantum vortices in themixed phase (3).
Flux quantization Φ = mΦ0 in units of the flux quantum Φ0 = h/2e is one of the most fundamental traits of the underlying off-diagonal long-range order of the superconducting condensate (4). m is
the integer winding number of the vortex. This number of confined
flux quanta is expected to affect the size and shape of the vortex.
More quantitatively, Volovik (5) demonstrated for superconductors
with isotropic Fermi surfaces that one can determine m from anindex theorem, provided one can measure the wave function of
the quasiparticle bound states that form in the vortex core. The vorticity is particularly ambiguous, and hence interesting near the Bogomol’nyi point, κ = 1/√2. Then, vortices near Tc behave likenoninteracting particles and the vortex configuration is infinitely
degenerate (6–9) such that multiple-flux-quanta (or giant) vortices
1

Physikalisches Institut, Karlsruhe Institute of Technology, Wolfgang-Gaede-Str.1,
76131 Karlsruhe, Germany. 2Institute for Quantum Materials and Technologies,
Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany. 3Department of Materials Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan. 4Institute for Theory ofCondensed Matter, Karlsruhe Institute of Technology, Wolfgang-Gaede-Str.1,
76131 Karlsruhe, Germany. 5Molecular Chirality Research Centre, Chiba University,
1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan.
*Corresponding author. Email: thomas.gozlinski@kit.edu†These authors contributed equally to this work.
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may be stabilized (10). This degeneracy is lifted below Tc and a transitional region, in which the superconductor cannot be categorized
into either of the types described above, emerges (11). When leaving
the Ginzburg-Landau limit toward lower temperatures and especially, when considering multiple-band (multiband) superconductors, microscopic interactions are predicted to become increasingly
important: The vortex-vortex interaction energy can become nonmonotonic in distance through the existence of multiple, distinct
superconducting coherence lengths ξi (12, 13) and topological hysteresis due to transitions between flux tubes and laminar pattern influences the flux patterns of the intermediate state (14).
In the past, the crucial parameter κ has been tuned toward thetype-II regime by either an increase of λL or reduction of the effective ξ, i.e., by using thin films below a critical thickness (15, 16), byincorporation of impurities [for relevant experiments see (11) andreferences within] or by interface scattering, e.g., Pb/Si(111) (17) or
Pb on black phosphorus (18). This approach, however, has the
drawback that the quasiparticle bound states in the vortex core
are considerably smeared out such that the index theorem cannot
be applied (17–22).
Here, we take the alternative approach and study the two-band
superconductor Pb in the clean limit in form of a bulk single crystal
Pb(111) at 45 mK. In this respect, the multiband superconductor
Pb, which is closest to the Bogomol’nyi point of all elemental superconductors, is a good candidate to study the transitional phase attemperatures well below Tc. We map the quasiparticle boundstates for the two superconducting gaps of Pb with high energy
and spatial resolution and use the index theorem including realistic
band structures to determine the winding number of single- andmultiple-flux-quanta vortices. Ultimately, we show that our observations allow to investigate the interband coupling of the two gaps.
RESULTS

Superconducting gaps and intermediate state
After several cycles of sputtering and annealing, we obtained a clean
Pb(111) surface with wide terraces and monoatomic steps, as shown
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Superconductors are of type I or II depending on whether they form an Abrikosov vortex lattice. Although bulk
lead (Pb) is classified as a prototypical type-I superconductor, we show that its two-band superconductivity
allows for single-flux-quantum and multiple-flux-quanta vortices in the intermediate state at millikelvin temperature. Using scanning tunneling microscopy, the winding number of individual vortices is determined from thereal space wave function of its Caroli–de Gennes–Matricon bound states. This generalizes the topological index
theorem put forward by Volovik for isotropic electronic states to realistic electronic structures. In addition, thebound states due to the two superconducting bands of Pb can be separately detected and the two gaps close
independently inside vortices. This yields strong evidence for a low interband coupling.

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differential conductance at the coherence peak of ∆2 while
ramping the field down. Once a jump to the superconducting
state below the tip is detected, the ramp is stopped. This ensuresthat one typically finds both, superconducting and normal conducting, areas in the scan range of the scanning tunneling microscopy(STM) setup of 1.4 × 1.4 μm2. The intermediate state is characterized by large normal and superconducting domains. The shapes andsizes of these domains in the intermediate state of lead have been
extensively studied by magneto-optical methods revealing thestrong dependencies on temperature, sample shape, and magnetic
protocol (14, 28–30).
A typical domain wall in the intermediate state is shown in the
dI/dU map in Fig. 1B. At a tunneling bias of Ut = 1.3 mV, the normal
conducting domains show up as areas of low conductance (purple)
and the superconducting domains as areas of high conductance(green/yellow). Note that atomic step edges of the surface cause acontrast in dI/dU (here, visible as yellow-red lines) as typicallyfound in STM experiments and is illustrated by the overlay on theright, where we switch from dI/dU to topographic map at the white
dotted line. The contrast at step edges comes from a combination of
the Smoluchowski effect, locally reducing the work function at stepedges, and technical aspects of scanning like a finite speed of the
feedback mechanism and the tip geometry at the atomic scale.
For our purpose, these lines can be seen as an artifact. A cross-sectional line scan across the domain wall, as in Fig. 1D, shows how
both gaps change from zero to their maximum on the lengthscale of the coherence length. The local recovery of superconductivity agrees well with reported coherence lengths of ξ = 87 nm (23).

Fig. 1. Superconducting properties and intermediate state. (A) Topographic scan image of the Pb(111) surface after the cleaning procedure. (B) dI/dU map at Ut = 1.3
mVshowing a typical domain wall in the intermediate state at B = 23 mT. Right overlay: Continuation of the map showing the corresponding topography. (C) Differential
conductance in the superconducting state (or superconducting domain) (black points) including a two-gap BCS fit (orange) with ∆1 being the smaller and ∆2 being thelarger gap. Inset: 3D models of Fermi surface sheets 1 and 2 [from (64)] responsible for superconducting gaps ∆1 and ∆2 respectively. (D) dI/dU spectra along a crosssection from normal conducting to superconducting domain. The area is marked in white in (B) and the profile direction is indicated by a red arrow. Individual spectra areoffset with respect to each other by 0.7 μS. The spectra were locally averaged over a straight part of the domain boundary and recorded in distance increments of ∆d =9.19 nm.
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in Fig. 1A. Upon zero-field cooling the Pb(111) sample to 45 mK it
enters its superconducting state below Tc ≈ 7.2 K (23). Because of
our low electronic temperature of less than 100 mK (24), we are able
to resolve the two gaps (25) in the density of states by scanning tunneling spectroscopy, even with a normal conducting tip. Figure 1C
shows the differential conductance (dI/dU) in the superconducting
state as black dots, including a temperature broadened two-gap fit to
the density of states in the Bardeen-Cooper-Schrieffer (BCS) theory
in orange. We determine the superconducting gaps to be ∆1 = (1.26± 0.02) meV (smaller gap) and ∆2 = (1.40 ± 0.02) meV (larger gap)in good agreement with previous measurement of the difference ofthe two gaps (25). The electronic temperature of the fitted curve is T
= (121 ± 1) mK. The intensity difference of the two coherence peaks
has previously been attributed to the k-dependent tunneling matrix
elements and the larger gap has been assigned to the tubular Fermi
surface sheet (25). This is in contrast to Bogoliubov–de Gennes–
based Korringa-Kohn-Rostoker calculations (26), which deduced
an opposite band-to-gap assignment. As will be discussed below,
our study of the quasiparticle bound states in the vortices confirms
the band-to-gap assignment of Saunderson et al. (26). We will fromhere on index the bands and Fermi surfaces according to their superconducting gap, i.e., the tubular Fermi surface responsible for ∆1
as Fermi surface 1 (FS 1) and the compact Fermi surface responsiblefor ∆2 as Fermi surface 2 (FS 2) (Fig. 1C).
After applying a perpendicular magnetic field of B = 85 mT,
which is above the critical field μ0Hc ∼ 80 mT (27), magnetic flux
enters the sample from the sides and completely destroys superconductivity. Upon decreasing the field again below Hc, the Landau intermediate state is reached. It is detected by recording the

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For a detailed analysis of the coherence length ξ1,2 of the two bands
measured inside vortices, we refer to the next section.

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Single-flux-quantum vortices
Inside the normal-core region of vortices, electronic bound statesthat lie within the superconducting gap are localized. These ingap states were first theoretically studied by Caroli, de Gennes,
and Matricon (CdGM) in 1964 (31). The CdGM states for isotropic
bands can be characterized by their orbital angular momentum
number μ and their energy E. The energy spacing of the discrete
CdGM states is of the order of ∆2/EF, where EF is the Fermi
energy, and the discrete states thus form a quasicontinuum or
branch of CdGM states for most superconducting materials. The
CdGM states with low μ are confined closer to the vortex corethan the ones with high μ, thus leading to a one-to-one correspondence between the angular momentum and the real-space behaviorof the bound-state wave function. The higher the angular momentum μ or energy E(μ), the further away from the vortex center are thestates localized (32). In case of an m-flux quanta vortex, m individual CdGM branches exist (5, 33–36), leading to a topological index
theorem: The number of zero-energy crossings of CdGM state
branches with varying angular momentum is directly related to
the vorticity (5, 33, 34). This theorem translates to real space andthe number of zero-energy crossings of CdGM state brancheswith varying radius from the vortex core is related to the vorticity.
STM allows to measure the variation of the local density of states(LDOS) inside the vortex and thus to determine the winding
number of the vortex using the index theorem. In 1989, Hesset al. (37) experimentally confirmed that CdGM states exist in
single-flux-quantum vortices by STM, but for vortices with multiple
flux quanta, although studied in thin films with electron holography(38) and scanning Hall probe microscopy (39), their predictedbound states still lack experimental verification (35, 36, 40, 41).
Using the detection method described in the previous section,
we are also able to find isolated, round normal conducting
domains (appearance at eU = ∆2) of ≈100 nm in diameter. An
example is shown in Fig. 2H. As will be shown later, these are vortices in the superconductor with integer number of flux quanta.
They appear alongside the normal intermediate state domain structure discussed in the previous section and without an orderedvortex pattern, as would be the case for a type-II superconductor(see the Supplementary Materials). The finding of such small
normal conducting domains is unexpected considering that thedomain wall energy in type-I superconductors is positive and thesystem thus tries to maximize its domain size. Even more unexpected is the fact that we found this shape after ramping the field downfrom the critical field. Magneto-optical measurements of cylindrical-shaped intermediate state lead samples at 4.5 K reveal that
normal domains are only tubular upon increasing magnetic field;
after ramping down from the critical field, the preferred structureis laminar (28). A deciding factor for the intermediate statedomain structure on a microscopic scale could be the effect offlux branching (42–44). Since the overall domain structure in our
experiment, however, consists of domains of various shapes andsizes, we argue that this finding is only consistent with circumstances under which vortices interact weakly and have a nonmonotonousinteraction energy causing hysteresis, i.e., a superconductor in thetransitional phase (close to the Bogomol’nyi point) at temperatures
well below Tc. The possibility of vortex pinning at large surface

defects can be discarded as the topographic image of the vortices
only shows the atomically flat Pb(111) surface, as shown in theinset of Fig. 2H. The role of pinning of vortices at invisible bulk
defects below the surface remains unclear. Repeating our magnetic
protocol several times, vortices may appear at similar positions,which can, however, be interpreted as consequence of the protocol,
i.e., stopping the magnetic ramp, when a vortex is under the tip. We
also find the vortices to be mobile when varying the magnetic field(see next section).
To determine the amount of flux carried by the small normal
domains, we record differential conductance (dI/dU ) maps at
subgap energies, which essentially show the LDOS of the CdGM
states. At zero bias voltage, we find a threefold symmetric state inform of a star with a maximum in the star’s center (Fig. 2A)
inside the normal domain similar to star-shaped CdGM states
seen by Hess et al. (45) in 2H-NbSe2. The quasiparticle density ofstates stretches over 100 nm in the ⟨211⟩ directions (blue/green). In
addition, three weak arms (dark blue) along the ⟨121⟩ directions arevisible. With increasing energy (independent of sign of bias
voltage), the star’s arms split into two with increasing splitting distance, while the central peak splits nearly isotropically to a ring
shape (Fig. 2, B to F). For E ≤ ∆1, the strong arms are still visible
and the ring reaches its maximal size (Fig. 2G). For E ∼ ∆2, thevortex shows as a relatively round area of low conductance with≈100 nm in diameter (Fig. 2H).
For bands with anisotropic Fermi velocity, like in our case, theindex theorem is not straightforward applicable since the radialsymmetry is removed and a radial-dependent measurement does
not ensure crossing all diabolical points (5, 33). Diabolical points
here mean that the points where the semiclassical particle andhole spectrum meet. The degenerate gapless fermionic excitations
or “zero modes” at these points carry the topological charge (singularity in the phase) and owe their name to the double-conical(diabolo) dispersion in parameter space (46). Here, a realistic
band structure needs to be considered. We, thus, simulated the quasiparticle trajectories inside a vortex carrying one flux quantum
within the quasiclassical Eilenberger theory, including the Fermi velocity of each band obtained from density functional theory (DFT)
calculations and compare the obtained LDOS maps to our experimental results (for details, see Materials and Methods). Right panels
in Fig. 2 (A and E) display the solutions reproducing the star shapefor the compact Fermi surface 2 and the ring-like structure of thetubular Fermi surface 1. The simulations confirm that the observed
states are the signature of a vortex in Pb(111) containing a single
flux quantum. The ring-shaped states, related to the tubular
Fermi surface (band 1) of nearly isotropic Fermi velocity, show
the expected behavior. At U = 0, it creates a sharp maximum inthe center of the vortex that splits into a ring of increasing diameter
upon variation of the voltage away from U = 0. For the compact
Fermi surface (band 2) with large flat structures in the Fermi
surface and anisotropic Fermi velocity, a star-shaped structure ispredicted whose arms split into two when going away from U =0. The observation of two sets of CdGM states in the experimentis in good agreement with the semi-classical treatment of separate
Fermi surfaces, which already indicates that the coupling between
the two Fermi surfaces is weak, i.e., the rate of scattering between
the two Fermi surfaces is lower than the inverse time for the electronround trip around the vortex.

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Star-shaped CdGM states have first been found by Hess et al.
(45) in 2H-NbSe2, which due to the crystal symmetry have sixfold
rotational symmetry. Similarly, CdGM states of vortices in Nb(110)were shown to exhibit twofold rotational symmetry (47). Pb crystallizes in the face-centered cubic structure and belongs to the point
group Fm3m. The electronic structure therefore carries a threefold
rotational symmetry about the [111] axis. Because of the discrete
rotational symmetry, the angular momentum is not a goodquantum number and the states mix. In general, the mixing
results in states with different lateral confinement for different
angles. At zero bias, the low angular momentum states carry thelargest weight and lead to a maximum in the center of the vortex.
At higher bias, states of larger angular momentum become increasingly important, leading to a movement of the maxima away fromthe center, i.e., the star splits. The large variation in quasiparticle
localization depending on the angle from ∼10 nm to over 100 nm
agrees with the strong anisotropy of the three-dimensional (3D)
Fermi velocity in the compact band. The extension of the CdGM
states beyond the vortex core illustrates that the sample is in theclean limit.
Star-shaped vortices with long arms have been recently observedin the Abrikosov lattice of La(0001) (48), owing to the large anisotropy of the responsible band’s in-plane Fermi velocity. In Pb with its
two bands, one may expect that the individual gaps will close independently when approaching the vortex center. Figure 3 displays theevolution of the gaps and in-gap states in the second derivative of
dI/dU with respect to U, as function of the distance from the vortex
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center and direction (indicated in the inset). The energy of the coherence peaks appear as dark lines and energy of the CdGM states as
faint dark lines. To clarify, we plot gap evolution functions ∆(r) =∆∞ tanh(r/ξ) for band 1 (red line) and band 2 (cyan line), which we
fitted to the coherence peaks in the leftmost panel. In the case ofband 1, we find that the gap opening can only be described when
considering a nonuniversal ξ, i.e., ξ1 for the behavior far from thecore and ξ1 for the behavior deep in the core of the vortex. Dashedlines in red [cyan] are guides to the eye for the spatially anisotropic
CdGM states of band 1 [2], i.e. the faint star arms in the ⟨121⟩ [pronounced star arms in the ⟨211⟩] directions. For large distances fromthe center, the gaps ∆1 and ∆2 decrease toward the center in parallelwith roughly the same length scale ξ1 ∼ ξ2 ∼ 45 nm. In the whole
range, ∆2 follows a simple tanh function. At 50 nm, however, ∆1crosses ∆2 and stays larger than ∆2. Thus, for the band 1, the gap
size deviates from the tanh function near the center and decreases
on a shorter length scale. This observation for a single-band superconductor is known as the Kramer-Pesch effect (49). It was quantified by numerical calculations by Gygi and Schlüter (32) for thevortex core size of type-II superconductors. The slope of ∆1 near
h
i 1
ðcÞ
the core corresponds to a core size ξ1 ¼ Δ1 ð1Þ limr!0 dΔdr1 ðrÞof only ∼ 10 nm. In our self-consistent calculation of the pair-potential ∆(r) for an isotropic vortex in the quasiclassical theory (seethe Supplementary Materials), this Kramer-Pesch shrinking effect isalso present and leads to substantial deviation from a tanh function
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Fig. 2. Single-flux-quantum vortex signature. (A to H) dI/dU maps (B = 0 mT) of a single-flux-quantum vortex at different bias voltages Ut (indicated in the bottom left
corner of each image) displaying the quasiparticle density at the surface. Right panels in (A) and (E) show the simulated LDOS of the bands 1 and 2 for the respectiveenergy. The inset in (A) shows a 2D fast Fourier transform–filtered topographic image of the Pb(111) lattice. Bulk crystal directions (red/yellow arrows) have been determined from glide planes. The inset in (H) shows the topographic image of the same area recorded at Ut = 1.4 mV.

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Fig. 3. Coherence lengths in the normal single-flux-quantum vortex. Angle-dependent radial measurement of the vortex core states from the vortex in Fig. 2. Displayed is the second derivative of the differential conductance −∂2σ/∂U 2 to highlight maxima in the LDOS. The point distance of single spectra is ∆d = 2.57 nm. Marked
are the opening of ∆1 (red line), the opening of ∆2 (cyan line), and the CdGM states of band 1/2 (red/cyan dashed line). The inset shows the direction of the y axis.

Anomalous single-flux-quantum vortices
Besides the regular vortex situation, we find vortices, in which thetwo sets of CdGM states are laterally displaced. Figure 4 shows suchan anomalous vortex. Both the ring center and the star center of the
CdGM states are independently movable by a change in magnetic
field and their relative displacement can be manipulated into different configurations, even back to the normal configuration from
Fig. 2 (see the Supplementary Materials). We explain this by
two effects.
First, a change in magnetic field laterally moves the vortex andwith it, the two sets of CdGM states. Second, a change in magnetic
field can lead to a tilting or bending of the flux lines away from anormal direction to the surface. This leads to a breaking of the cylinder symmetry and can displace the two sets of CdGM states relative to each other. The individual sets of CdGM states behave as
those of the regular vortices, except for their relative displacement(see Fig. 4, A to H).
The lateral displacement allows an independent probing of thestate sets and thus, an independent identification of their bands. By
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looking at single bias spectra at the star center (black) and the ring
center (red) in Fig. 4I, we find that the amplitude of the peak at zero
bias is three times larger for the ring than for the star, which is supported by our separate band simulations from earlier and can be explained by the larger lateral confinement of low angular momentum
states in band 1 compared to band 2. Figure 4 (J and K) reveals
maxima in the LDOS along the cross sections marked in the rightpanel of Fig. 4I in the form of heatmaps of the differential conductance’s second derivative with respect to bias voltage. It becomesapparent that ∆2 (cyan line) closes entirely, whereas ∆1 (red line)
does not completely close in the star’s center (black circle).
Instead, ∆1 closes about 20 nm away from the star center (redcircle). Consequently, the superconducting gap ∆1 is linked to thering and ∆2 to the star. The fact that the quasiparticles of ∆1 and ∆2
can be independently displaced with respect to each other rules out
rigid interband coupling.
Multiple-flux-quanta vortices
We also observed larger vortices in the experiments. Figure 5 displays two examples. Figure 5 (A to C) shows a vortex with two
flux quanta. Our semiclassical calculations indicate that for each
flux quantum and each band, a branch of CdGM states is present.
As a result, for the vortex with two flux quanta, the band 2 causes a
structure with two arms per direction at zero bias (see Fig. 5A) that
individually split into two arms with bias voltage (see Fig. 5B) just as
in the single-flux-quantum vortex. An analogous behavior is observed in the vortex with three flux quanta shown in Fig. 5 (D to
F). Both vortices are larger than the single-flux-quantum vortex(compare Fig. 2H with Fig. 5, C and F). Further, they deviate
from a round shape and laterally grow with the number of flux
quanta. The number of flux quanta in the vortices does not seemto be limited. We observed giant vortices with over 10 flux quanta(see the Supplementary Materials). For band 1, the problem of multiple-flux-quanta vortices is similar to that of a single-band superconductor with a spherical Fermi surface and has been studied indetail by Volovik (5, 33). In essence, an m-flux quanta vortex
results in m branches of circular CdGM states of different radii.
Because of symmetry, at zero bias and for odd m, a central spot isformed by one branch of the CdGM states and the other states form
pairs of increasing ring diameters. The CdGM states at energies
away from the Fermi level evolve by changing the radii. Thus, thecentral CdGM state turns from a spot of zero radius to a ring of
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predicts a core shrinking proportional to T/Tc when lowering thetemperature. At very low temperatures T ≪ Tc the slope of theorder parameter d∆/dr at the vortex center is even predicted to
become infinite, which would show as a jump of ∆(r) at r = 0
that is smoothed out over the distance ξT/Tc (33). Experimentally,
we do not find this extreme shrinking. Meanwhile, the “squeezing”
of low angular momentum states in the vortex core and thus the
Kramer-Pesch effect is absent for the star, i.e., CdGM states ofðcÞðcÞband 2. Consequently, ξ1 deviates from ξ2 , which indicates that
intraband coupling dominates over interband coupling, i.e., thetwo bands are sufficiently decoupled from each other, despite thegap sizes being not too different (50). For an isotropic two-band
superconductor, the shrinking of the core region for only one of
the bands has also been predicted in the case of rather weakly
coupled bands (51). In this theoretical prediction, the KramerPesch effect for the two weakly coupled bands depends on theratio of the Fermi velocities. The band with the larger Fermi velocity
is expected to show a smaller Kramer-Pesch effect. This fully agrees
with our observation of a lower Kramer-Pesch effect for band 2,
which has a larger Fermi velocity according to our calculations
and the band calculations by Saunderson et al. (26). Again, thishints toward a low interband coupling in Pb.

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finite radius, and the pairs of CdGM states split in their radius. Foran even m, no central spot is present at the Fermi energy but only
pairs of CdGM states with distinct radii exist that again split whenmoving away from the Fermi energy. This is the essence of the topological index number theory of Volovik. In our case, we find acentral spot of high differential conductance at zero bias voltage
for the vortices with m = 1 and m = 3, while there is no central
spot for the vortex with m = 2 in agreement with the topologicalindex number theory. When going away from the Fermi energy,
the ring-shaped and spot-like CdGM states overlap with the starshaped states such that their distinction becomes impractical. Tunneling spectra at selected locations inside the vortex showing thevery same effects are shown in Fig. 5 (G and H).
While the original formulation of the Volovik’s index number
theorem is not directly applicable in real space (that is, counting
the number of zero modes in the vortex core as function ofradius), in our case of an anisotropic Fermi surface, we still
believe that the number of parallel arms of our star-shaped
CdGM state pattern at zero energy exactly reflects the winding
number m of the vortex. The reason for our confidence lies in thefact that the diabolical points in k space, that need to be counted,
still lie at zero energy, it is just not obvious, which path in real
space crosses each of them exactly once. Comparing the axisymmetric problem with the problem at hand, it becomes obvious that aparametrization in terms of the set of quantum numbers (kr, kϕ,
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and kz) has to be replaced by an irreducible representation of thecrystallographic point group. The flat parts of Fermi surface 2
focus the quasiparticles into high-symmetry directions andinstead of a localization of different CdGM state branches at different radii, they now localize at different impact parameters.
In total, we studied 99 vortices below 40 mK, which all show the
behavior presented here. At last, we tested whether vortices would
also be present at temperatures much closer to Tc. Our measurements at 4.3 K revealed that it is substantially harder to trap a
vortex in our scan frame, but we managed to in two cases (see theSupplementary Materials).
DISCUSSION

We report the observation of single-flux-quantum and multipleflux-quanta vortices in a traditional type-I bulk superconductor,
i.e., single-crystal Pb(111), by low-temperature STM. We also demonstrate a robust determination method for the winding number ofthe vortex by usage of a topological index theorem, which relates thenumber of flux quanta and CdGM state branches. The spatial anisotropy of the quasiparticle states inside the vortex reflects the
crystal symmetry, and its shape is governed by the anisotropic
Fermi velocity in the superconducting bands. An influence ofneighboring flux lines can be ruled out due to the absence of anordered flux pattern. In addition, we could show how CdGM
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Fig. 4. Anomalous vortex signature. (A to H) dI/dU maps (B = 19 mT) at different bias voltage for an anomalous vortex (compared to the normal vortex configurationfrom Fig. 2). (I) Right: Enlarged zero bias dI/dU map of the anomalous vortex including the position of line spectra (white arrows) and single spectra locations (red/black
circle). Left: Single dI/dU spectra in the star center (black) and ring center (red) revealing zero bias peaks of different amplitude. (J and K) Heatmaps of the second
derivative of the differential conductance −∂2σ/∂U 2 along the cross sections marked in (I). Red and cyan lines follow ∆1 and ∆2, respectively.

S C I E N C E A D VA N C E S | R E S E A R C H A R T I C L E

states from two weakly coupled bands interact with each other in aflux line. While the generically unavoidable interband couplingleads to the formation of a single coherence length and order parameter right at Tc (52), the physics below Tc is considerablyricher, in particular in the regime of weak interband coupling (50,51). Of particular interest is the emergence of mixed gap modes (51)or the possibility of the observation of the rather elusive Leggett
mode in the limit of weak interband coupling (53), i.e., a collective
fluctuation of the relative phase of the two bands, for superconducting Pb. We experimentally demonstrate that for systems with multiband pairing, the transitional region of the hysteretic behavior
expands due to microscopic interactions and also allows for interesting vortex configurations, like vortex clusters or multiple-fluxquanta vortices (54). The emergence of such vortices in the intermediate state of a prototypical type-I superconductor is very unexpected and demonstrates that the low-temperature flux patterns of
two-band superconductors rank on a spectrum between types I and
II. Not only does the existence of vortices in Pb present an alternative test ground of vortex physics in multicomponent superconducting systems, the variability of the vortex winding number alsosuggests a combination with topological crystal defects that was recently predicted to result in topological quasiparticle states, like Majorana zero modes, under certain circumstances (55, 56).
MATERIALS AND METHODS

Experimental details
The experiments were performed with a home-built STM with dilution refrigeration, which can reach a base temperature of 25 mK ina magnetic field of up to 7.5 T (24). In our setup, the bias voltage Ut
is applied between sample and common machine ground so that a
Gozlinski et al., Sci. Adv. 9, eadh9163 (2023)

8 September 2023

positive bias voltage probes the unoccupied states of the sample. The
STM chamber is kept at a base pressure of 1 × 10−10 mbar. The
single-crystal Pb(111) (miscut angle, ±0.1°; purity, 99.999%) has
been purchased from MaTecK GmbH. It has cylindrical (hat)shape with a diameter of 8 mm and a thickness of 2 mm. At abase pressure of 1 × 10−10 mbar, the Pb crystal was prepared incycles of sputtering with Ar+ ions of 3 keV and subsequent annealing at 190°C and directly transferred into the STM in situ. A tungsten tip was prepared by high-temperature flashing and soft dipping
into a Au(111) surface to avoid picking up Pb atoms. The measurements (except for the one at 4.3 K) were all performed below 45 mK.
After zero-field cooling the Pb crystal, the vortices were formed byramping the perpendicular magnetic field from 0 to 85 mT and backdown to a constant value below the critical field. Note that 80 mT is
the critical magnetic field for Pb (27). The ramping rate was 5
mT/min.
The differential conductance was measured using a lock-in amðPKÞ
plifier at a frequency of 3.4 to 3.6 kHz and AC peak amplitude Uac
between 10 and 100 μV (for details, see the Supplementary Materials). dI/dU maps at subgap energies were recorded in a multipass
configuration: In the “record” phase the tip records the z-profileat the feedback condition (constant tunneling current of It at eUt> ∆) and in the “play” phase the z-profile is repeated at a different
bias voltage. To increase the signal for subgap energies an offset isoften added to the z-profile bringing the tip closer to the surface.
This record and play phase alternation is performed line for line
until the entire area has been scanned.
Calculation details
To obtain the simulated LDOS for vortices containing an arbitrary
number of flux quanta, we used the Riccati parametrization of the
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Fig. 5. Multiple-flux-quanta vortex signature. (A to F) dI/dU maps of an m = 2 vortex at B = 0 mT (A to C) and an m = 3 vortex at B = 33 mT [(D) to (F)] at selected bias
voltages. Right panels in (A) and (D) show the simulated LDOS of the bands 1 and 2 for the respective m quanta vortex and energy. (G and H) Bias spectroscopies (bottom)
recorded with the tip at specific locations marked in the zero bias maps (top) of the m = 2 (G) and m = 3 (H) vortex. Individual spectra are offset by 0.4 μS for clarity. Blackdashed lines indicate their respective zero conductance baselines.

S C I E N C E A D VA N C E S | R E S E A R C H A R T I C L E
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This PDF file includes:
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Supporting Data 1 and 2
Table S1
Figs. S1 to S10
Legends for data S1 to S6
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Acknowledgments: We thank A. Ustinov and S. Rex for helpful discussions. Funding: We
acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG) with grant Wu 394/12-1 and through CRC TRR 288 - 422213477 “ElastoQMat,” projects A07 and B06. R.W.
acknowledges support by the state of Baden-Württemberg through bwHPC. This work was
supported by JSPS KAKENHI under the grant number 17K19023. Author contributions: T.G.
and Q.L. conducted the experiments, performed the quasiclassical (Eilenberger) calculations,and analyzed the data. T.G. was responsible for presentation of the data. R.H. performed the DFT
calculations. R.N. and T.K.Y. performed initial experiments and provided the crystal. R.W. and J.S.
gave theoretical support. W.W. headed the study. T.G. and W.W. wrote the manuscript including
input from all authors. Competing interests: The authors declare that they have no competinginterests. Data and materials availability: All data needed to evaluate the conclusions in thepaper are present in the paper, the Supplementary Materials, or in the Supplemental Dataset
published at https://doi.org/10.35097/1624.
Submitted 22 March 2023
Accepted 7 August 2023
Published 8 September 2023
10.1126/sciadv.adh9163

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Band-resolved Caroli–de Gennes–Matricon states of multiple-flux-quanta vortices
in a multiband superconductor
Thomas Gozlinski, Qili Li, Rolf Heid, Ryohei Nemoto, Roland Willa, Toyo Kazu Yamada, Jörg Schmalian, and Wulf
Wulfhekel

Sci. Adv., 9 (36), eadh9163.
DOI: 10.1126/sciadv.adh9163

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