# Breakdown of bulk-projected isotropy in surface electronic states of
topological Kondo insulator SmB6(001)

Yoshiyuki Ohtsubo<EMAIL_ADDRESS>National Institutes for Quantum Science and
Technology, Sendai 980-8579, Japan  Graduate School of Frontier Biosciences,
Osaka University, Suita 565-0871, Japan  Department of Physics, Graduate
School of Science, Osaka University, Toyonaka 560-0043, Japan  Toru Nakaya
Department of Physics, Graduate School of Science, Osaka University, Toyonaka
560-0043, Japan  Takuto Nakamura  Graduate School of Frontier Biosciences,
Osaka University, Suita 565-0871, Japan  Department of Physics, Graduate
School of Science, Osaka University, Toyonaka 560-0043, Japan  Patrick Le
Fèvre  François Bertran  Synchrotron SOLEIL, Saint-Aubin-BP 48, F-91192 Gif
sur Yvette, France  Fumitoshi Iga  Graduate School of Science and Engineering,
Ibaraki University, Mito 310-8512, Japan  Shin-Ichi Kimura
<EMAIL_ADDRESS>Graduate School of Frontier Biosciences, Osaka
University, Suita 565-0871, Japan  Department of Physics, Graduate School of
Science, Osaka University, Toyonaka 560-0043, Japan  Institute for Molecular
Science, Okazaki 444-8585, Japan

###### Abstract

The topology and spin-orbital polarization of two-dimensional (2D) surface
electronic states have been extensively studied in this decade. One major
interest in them is their close relationship with the parities of the bulk
(3D) electronic states. In this context, the surface is often regarded as a
simple truncation of the bulk crystal. Here we show breakdown of the bulk-
related in-plane rotation symmetry in the topological surface states (TSSs) of
the Kondo insulator SmB6. Angle-resolved photoelectron spectroscopy (ARPES)
performed on the vicinal SmB6(001)-$p$(2$\times$2) surface showed that TSSs
are anisotropic and that the Fermi contour lacks the fourfold rotation
symmetry maintained in the bulk. This result emphasizes the important role of
the surface atomic structure even in TSSs. Moreover, it suggests that the
engineering of surface atomic structure could provide a new pathway to tailor
various properties among TSSs, such as anisotropic surface conductivity,
nesting of surface Fermi contours, or the number and position of van Hove
singularities in 2D reciprocal space.

The close correspondence between symmetry operations in 3D bulk bands and the
topological character of the surface states lying on the edge of a crystal has
been one of the central topics of solid state physics in this decade [1]. Such
topological surface states (TSSs) are attractive not only for basic science
but also for spintronic applications due to their momentum-dependent spin-
orbital polarization [2, 3]. Since TSSs originate from the topological order
of bulk states, some qualitative characteristics of TSSs, such as the metallic
band dispersion across the bulk bandgap, are robust against any non-magnetic
external perturbations as long as the bulk states remain unchanged. This is a
strong point on the one hand because of the expected contamination-tolerant
working of devices, for example. On the other hand, the role of the surface
atomic structure in TSSs is regarded as rather unimportant because of the
robustness. Some theoretical and experimental studies reported modification of
TSSs by surface treatments, such as surface oxidization [4], chemical or
photochemical ageing [5, 6], and a small interlayer rotation forming a moiré
modulation on the surface [7]. However, such previous works were focused on
how to perturb the pristine TSS. Actually, the unit cells of the surface
lattice in these works were always the same length of basis vectors as the
ideally truncated bulk crystal, with the symmetry operations obtained by
simple projection of bulk 3D space group. Some studies focused on topological
electronic states localized at 1D edges or hinges, but they still suppose the
similar simple projection from 3D to 1D systems [8, 9]. Although some drastic
electronic phenomena, such as 1D charge-density-wave formation on 2D
semiconductor surfaces [10], obtained by forming a surface atomic structure
independent of the substrate are already known, the role of the surface
superstructures in TSSs has not been studied in detail yet, to the best of our
knowledge.

Samarium hexaboride (SmB6) is a long-known Kondo insulator, in which a bulk
bandgap opens at low temperature because of the Kondo effect [11]. It is the
first material proposed as a candidate for topological Kondo insulators
(TKIs), and it hosts a metallic TSS coexisting with strong electron
correlation [12, 13] and was recently confirmed as a TKI by angle-resolved
photoelectron spectroscopy (ARPES) experiments after a long debate [14, 15,
16, 1, 18], as summarized in ref. [19]. The (001) surface of SmB6 is also
known as a valuable example of a well-defined surface superlattice among TIs.
Although cleavage provides multiple surface terminations [3], in-situ surface
preparation, typically by cycling of Ar ion sputtering and annealing, results
in an uniform surface superstructure such as (1$\times$2) and (2$\times$2)
[14, 21, 4], making the SmB6(001) surface a good template to study the role of
the surface atomic structure in TSSs. The remaining barrier for such research
is that the obtained SmB6(001) surfaces have two or more equivalent domains
coexisting with nearly the same area. For example, double-domain (1$\times$2)
and (2$\times$1) surfaces could provide apparent fourfold rotation symmetry of
the TSS by overlapping of two twofold domains rotated 90∘ to each other, as
observed on the Si(001) surface [23].

In this work, we show the TSS on the uniform and semi-single-domain
SmB6(001)-$p$(2$\times$2) surface prepared in-situ. One surface domain is
dominantly grown by using a vicinal (001) substrate, confirmed by low-energy
electron diffraction (LEED). The dispersion and orbital-angular-momentum (OAM)
polarization of the SmB6(001) TSS is observed by ARPES without ambiguity from
the multidomain overlap for the first time. The ARPES data show that the TSS
is anisotropic and that the Fermi contour (FC) lacks the fourfold rotation
symmetry that is maintained in the bulk. This result emphasizes the important
role of the surface atomic structure even in the TSS.

## Results

### Vicinal (001) surface of SmB6

As depicted in Fig. 1a, the bulk-truncated (001) surface of SmB6 has fourfold
rotation symmetry. After cycles of Ar ion sputtering and annealing (see
methods for details), the vicinal SmB6(001) surface exhibited the LEED
patterns shown in Fig. 1b. In contrast to the most of the earlier studies
reporting (2$\times$1) surface periodicity [14, 21], the (1/2 1/2) fractional
spot indicated the $p$(2$\times$2) surface structure at the primary electron
energy ($E_{\rm p}$) of 65 eV. Moreover, other fractional spots at $E_{\rm p}$
= 45 eV, (1 -1/2) and (1/2 -1), show the different intensities from each
other, although they should be identical if the surface has fourfold rotation
and time-inversion symmetries. Note that different diffraction spots become
bright at $E_{\rm p}$ = 45 and 65 eV simply because to sweep LEED $E_{\rm p}$
corresponds to sweep the size of the Ewald sphere in reciprocal space. The
asymmetric intensities between (1 -1/2) and (1/2 -1) are quantitatively shown
by the LEED line profiles in Fig. 1c. This shows that the obtained surface has
an anisotropic atomic structure without fourfold rotation symmetry. The line
profile also shows that this surface hosts the mirror planes [100] and [010],
since the spots (1 $\pm$1/2) have nearly the same heights. Note that the
height difference between (1 -1/2) and (1/2 -1) does not directly reflect the
area ratios of surface domains; the obtained surface periodicity is
$p$(2$\times$2) and thus both (1 -1/2) and (1/2 -1) spots could appear even
from the perfect single-domain surface.

Figure 1d shows the B 1$s$ core level and Sm2+ 4$f$ valence bands obtained
from angle-integrated photoelectron spectra of SmB6(001)-$p$(2$\times$2).
Different from the cleaved SmB6(001) cases [1], we found no clear “surface”
term which appears at different energies from the main peaks. This would
occurs because the heating process during the surface preparation removes the
metastable Sm and B atoms which are the origin of the “surface” terms. The B
1$s$ peak has a tailed shape with respect to higher kinetic energies,
suggesting that some boron atoms are in a different surrounding environment
from the bulk case, but the difference is not as drastic as in the “boron-
terminated” case of the cleaved surfaces [3, 1]. This suggests that small
displacements of the surface boron atoms are the origin of the $p$(2$\times$2)
superlattice. While some cleaved surfaces of SmB6(001) were reported to be
inhomogeneous with various termination atoms [3, 1, 2], we found no such
signature from the current surface. Further discussion on this point is shown
in Supplementary note 1.

Based on these results, we propose a possible model of $p$(2$\times$2) surface
superstructure, as illustrated in Figs. 1e and 1f. In this model, no surface
defects nor adatoms are supposed and the $p$(2$\times$2) superlattice is
formed only by displacements of B6 and Sm atoms at the topmost 2 atomic
layers. Such picture agrees with the less drastic change of surrounding
conditions of the surface atoms, which was suggested by the core-level spectra
explained above. In addition, the number of neighboring atoms of the displaced
Sm is the same as those in the bulk, in contrast to the reduction for the
topmost B6. It rationalizes the core-level spectra showing the “surface” term
only for boron peaks. Note that this model is just a possible example to
satisfy what were observed by LEED and core-level spectra. Further experiments
are required to determine the surface atomic structure accurately, such as
dynamical LEED analysis [25] or Weissenberg reflection high-energy electron
diffraction [26], while the $p$(2$\times$2) surface unit cell without fourfold
rotation symmetry is enough for the following discussion in this article.

### TSS of the vicinal SmB6(001)-$p$(2$\times$2) surface

Figure 2a shows the FCs around the Fermi level ($E_{\rm F}$) measured with
circularly polarized photons at 35 eV. The spectra obtained by using both
right- and left-handed polarizations are summed to avoid the anisotropic
intensity distribution due to the circular dichroism (CD). Figure 2b shows a
schematic drawing of the obtained FCs S1, S2, and S3. The oval enclosing
$\bar{\rm X}$ (S1) has a similar shape to those observed in the earlier
studies [15, 16, 21, 1, 19]. The stark contrast to the earlier studies is that
the FC around the other $\bar{\rm X}$ point, S1’ depicted in Fig. 2b, is quite
weak, although they should be identical, if the surface has fourfold rotation
symmetry. To highlight this difference, the other $\bar{\rm X}$ point is named
as $\bar{\rm Y}$ (Fig. 2a). One possibility to explain this anisotropy is the
ARPES experimental geometry. However, as illustrated in Fig. 2c, the $\pm
k_{y//[010]}$ orientations are identical to each other, even when the
photoelectron incidence orientation is taken into account. Moreover, S1’
becomes as evident as S1 near the edge of the crystal surface where the
vicinal miscut is expected to be different from that at the centre of the
polished surface, indicating that the difference between S1 and S1’ is from
the surface domains (ARPES data are shown in SM). Together with the
anisotropic LEED patterns (Figs. 1b and c), the faint S1’ can reasonably be
assigned to the minor-area domains and to the fact that the
SmB6(001)-$p$(2$\times$2) surface lacks the fourfold rotation symmetry with
the S1 FC only around one $\bar{\rm X}$ point.

The other FCs enclosing $\bar{\Gamma}$, S2 and S3, are also clearly observed,
in contrast to the blurred FCs in earlier works [16]. This is due to the
uniform single-domain surface preparation over a wide area of the sample. The
shape of S2 is similar to that observed on the cleaved (001) surface. While
that state was claimed as the umklapp of S1 by (1$\times$2) surface
periodicity at first [1], this assignment has not yet reached a consensus, as
the following discussion made a counterargument on this assignment [19]. On
the other hand, as shown in Figs. 2a and b, the shapes of S1 and S2 observed
here are clearly different, indicating that S2 is another, independent FC.

All the three FCs observed here lacks the four-fold symmetry, as shown in Fig.
2. Such breakdown of the bulk-related symmetry in the TSS is observed for the
first time among TIs, to the best of our knowledge. It should be derived from
the anisotropic surrounding conditions of Sm atoms near the surface, since the
bulk states around the Kondo gap is mainly derived from Sm 4$f$ and 5$d$
orbitals. Such condition can be naturally satisfied by assuming
$p$(2$\times$2) surface superstructure; for example, minor displacements of Sm
below the topmost surfaces as depicted in Fig. 1f. On the other hand, no
folding of FCs at the SBZ boundary was observed, while it is rather common
case among surface states with superstructure, as discussed in supplementary
note 2.

Figure 3 shows the band dispersions near $E_{\rm F}$. The almost localized Sm
4$f$ band lies slightly below $E_{\rm F}$ ($\sim$20 meV) and the Sm 5$d$ bands
disperses in 150-50 meV, reflecting the itinerant character. Around the
crossing point between them, the bands bend, indicating the $c$-$f$
hybridization induced by the Kondo effect. Above the Sm 4$f$ band, the
metallic bands continuously disperses between $E_{\rm F}$ and Sm 4$f$ in Cuts
1-3. Their Fermi wavevectors ($k_{\rm F}$) are consistent with the FCs
observed in Fig. 2. Note that distinguishing S1, S2 and S3 in Fig. 3b is
difficult because they overlap with each other at $E_{\rm F}$ along this
orientation. While the surface bands corresponding to S2 and S3 are not very
clear from the 2D intensity plot in Fig. 3c, this is because of the too strong
Sm 4$f$ band and the momentum distribution curve (MDC) at $E_{\rm F}$ shows
distinct peaks corresponding to them. Moreover, Fig. 3d shows the dispersion
of S2 and S3 close to $E_{\rm F}$ without such difficulty. This is thanks to
the photon energy (18 eV) for Fig. 3d, where the photoemission cross section
for Sm 4$f$ becomes small. The $k_{\rm F}$ values for S2 and S3 obtained in
Figs. 3c and d show no difference depending on the photon energy, indicating
that their 2D nature arises from the surface. For further confirmation, we
also checked the lack of 3D dispersion for S2 and S3 by measuring a series of
MDCs at $E_{\rm F}$ with different photon energies (see Supplementary Fig. 2
in SM).

### OAM polarization of the TSS

One of the most prominent characteristics of TSSs is the helical spin and OAM
polarization, which is always perpendicular to the in-plane wavevector [1].
Figure 4 shows a CD-ARPES map corresponding to the region shown in Fig. 3. The
CD of ARPES reflects the OAM polarizations projected onto the photon-incident
orientation. As illustrated in Fig. 2c, CD in the current experimental
geometry corresponds to OAM polarization along the in-plane and normal to
$k_{y}$, or out-of-plane orientation. Figure 4a shows that the oval-shaped S1
around $\bar{\rm X}$ has helical OAM polarization consistent with the earlier
CD-ARPES [15] and spin-resolved ARPES [16] results. Figures 4b and c show that
the surface bands S2 and S3 also have OAM polarizations whose sign inverts
from $+k_{y}$ to $-k_{y}$, obeying the time inversion symmetry. This
behaviour, helical OAM polarizations with time-reversal symmetry, is
consistent with what is expected for TSSs [1, 19].

Figure 4d illustrates the OAM polarizations of each FC, assuming that they are
parallel to the helicities of the incident-photon polarizations: parallel
(anti-parallel) to the photon-incidence vector for right- (left-) handed
polarization. According to this, the helicities of the OAM polarizations for
S1 (counter clockwise) are opposite to those for S2 and S3 (clockwise). Such
behaviour is not expected from the analytical calculation with the winding
number [27]. However, note that the sign of the CD-ARPES signal is not
suitable information to be directly compared with such calculation of the
initial states. First, it is easily affected by the photoelectron excitation
process [28]. Second, the calculation does not include the surface atomic
structure obtained in this work. Finally, TSSs were recently revealed to often
contain multiple wavefunction components with different orbital and spin
polarizations even at single ($E$, $k$) point; thus, the polarization of
photoelectrons is not always the direct information of the initial state [29,
30]. Therefore, further datasets, especially spin-resolved ARPES with multiple
incident photon energies and polarizations, are required to conclude the
winding number of the TSS.

## Discussion

We have revealed that the TSS of SmB6(001)-$p$(2$\times$2) is highly
anisotropic. From this anisotropy, the other electronic properties, such as
the electron conductivity and net spin polarization via the surface currents,
are also suggested to be anisotropic on SmB6(001)-$p$(2$\times$2), as far as
they are derived from the TSS. Note that none of these properties are
topologically protected. Therefore, examining the rotation asymmetry from the
vicinal SmB6(001) sample with $in$-$situ$ surface preparation would be an
interesting new pathway to distinguish the origin and dimensionality (2D or
3D) of unconventional electronic phenomena observed in SmB6, such as the
thermodynamic properties [31] and quantum oscillations [32, 33, 34, 35, 36,
37]. If one could apply the surface preparation method similar to ours to
vicinal SmB6(001) sample and then insert it to a magnet equipment for de Haas
van Alphen oscillation measurements without breaking the vacuum environment,
such experimental setup would provide conclusive evidence about the role of
surface on the quantum oscillation.

Since the folding of FCs by surface superstructure changes the apparent shape
of FCs drastically, one could have a question about its role on topological
order. In most cases including SmB6(001)-$p$(2$\times$2), surface
superstructure causes no change the energetic order of bulk bands nor magnetic
order. Therefore, there shouldn’t be any change in topological order before
and after the formation of surface superstructure. However, it would be
helpful to discuss how to determine the topological order with surface
superstructure, since it is often determined by counting the number of FCs to
be odd or even, around surface time-reversal invariant momenta (TRIM);
$\bar{\Gamma}$, $\bar{\rm X}$, $\bar{\rm Y}$, and $\bar{\rm M}$ for SmB6(001)
[38, 18, 19]. For this counting, we propose to double check the number of FCs
both by using (1$\times$1) bulk truncated SBZ as well as that with folding
according to the surface superstructure. Circled numbers in Figs. 4d and 4e
are the example of this method applied to SmB6(001)-$p$(2$\times$2), showing 3
FCs, odd number consistent with the non-trivial topological order [18, 19].
Note that S1 at the bottom side of Fig. 4d is identical to the top one because
of the translational symmetry and thus it should not be included to this FC
counting. This method emphasizes the importance to prepare the single-domain
surface to discuss the topological order of unknown material; for example,
surface Dirac cones doubled by two nonequivalent surface terminations, as
observed in ref. [39], could derive false counting of FCs.

The folding of TSS by surface superstructure could play further role on the
unconventional electronic phenomena of TSS. It duplicates FCs formed by TSS,
making the additional crossing points with van-Hove singularity (VHS, arrows
in Fig. 4f). Such VHS formation is discussed theoretically based on a moiré
modulation to expect enhanced topological superconductivity [40]. Moreover,
this work exhibited that the surface superstructure distorts surface FCs at
the same time. It suggests that engineering of surface superstructure could
tailor the number and position of VHS in the reciprocal space as well as the
other parameters of FCs, such as nesting vector dominating surface density-
wave formation [41]. Actually, the observed FC on SmB6(001)-$p$(2$\times$2)
does exhibit triple degenerate points of TSSs as guided by arrows in Fig. 4e.
On the other hand, the reconstruction of TSS in Kondo gap observed here is the
first case, to the best of our knowledge. In most cases, surface
superstructure are regarded to play major role in larger energy scale
typically in 0.1 to 1 eV, orders of magnitude larger than those of Kondo gap
(few tens of meV). The VHSs formed by surface superstructure possibly also
enhance the electron correlation effect expected to TSS of TKI, such as the
emergence of heavy surface states and non-Hermitian exceptional points [42,
43]. From these perspectives, detailed electron behaviour in smaller energy
and temperature scale, in orders of few meV and Kelvins, respectively, of
SmB6(001)-$p$(2$\times$2) would be an encouraging test peace to examine the
role of surface superstructure on TKI.

## Methods

### Sample preparation

Single crystalline SmB6 was grown by the floating-zone method by using an
image furnace with four xenon lamps [44, 21]. The sample cut along the (001)
plane was mechanically polished in air with a small angle offset towards [010]
until a mirror-like shiny surface was obtained with only a few scratches when
observed under an optical microscope (multiple 10x magnification). The miscut
angle is estimated to be $\sim$1∘ from the side-view images obtained by the
optical microscope. The polished sample was moved to ultra-high-vacuum (UHV)
chambers and cleaned $in$-$situ$ by repeated cycles of Ar ion sputtering (1
keV) and annealing up to 1400$\pm$50 K. The cleaned surface was transferred to
the measurement chambers for LEED or ARPES without breaking the UHV.

### LEED measurements

The LEED measurements were performed by using a conventional electron optics
(OCI, Model BDL800IR) at beamline 7U of UVSOR-III. The sample was mounted on a
homemade 6-axis goniometer maintained at 100 K. The electron incident angle
was adjusted to be parallel to the surface normal ([001]) by using the
rotation angles of the goniometers.

### ARPES experimental setup

The ARPES measurements were performed with synchrotron radiation at the
CASSIOPÉE beamline of synchrotron SOLEIL. The photon energies used in these
measurements ranged from 18 to 240 eV. The polarization of incident photons is
depicted in Fig. 2c, including linear polarization with the electric field
lying in the incident plane (H) and circular polarizations (R and L). The
photoelectron kinetic energy at $E_{\rm F}$ and the overall energy resolution
of the ARPES setup ($\sim$15 meV) were calibrated using the Fermi edge of the
photoelectron spectra from Ta foils attached to the sample.

## Data availability

The datasets generated and/or analysed during the current study are available
from the corresponding authors upon reasonable request.

## References

  * [1] Hasan, M. Z. & Kane, C. L.  Colloquium: Topological insulators.  _Rev. Mod. Phys._ 82, 3045–3067 (2010).
  * [2] Manchon, A., Koo, H. C., Nitta, J., Frolov, S. M. & Duine, R. a.  New perspectives for Rashba spin$–$orbit coupling.  _Nature Materials_ 14, 871–882 (2015).
  * [3] Han, W., Otani, Y. & Maekawa, S.  Quantum materials for spin and charge conversion.  _npj Quantum Materials_ 3, 27 (2018).
  * [4] Wang, X., Bian, G., Miller, T. & Chiang, T.-C.  Fragility of Surface States and Robustness of Topological Order in ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ against Oxidation.  _Phys. Rev. Lett._ 108, 096404 (2012).
  * [5] Bianchi, M., Hatch, R. C., Mi, J., Iversen, B. B. & Hofmann, P.  Simultaneous Quantization of Bulk Conduction and Valence States through Adsorption of Nonmagnetic Impurities on ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$.  _Phys. Rev. Lett._ 107, 086802 (2011).
  * [6] Sakamoto, K. _et al._ Spatial Control of Charge Doping in n-Type Topological Insulators.  _Nano Letters_ 21, 4415–4422 (2021).  PMID: 33978424.
  * [7] Cano, J., Fang, S., Pixley, J. H. & Wilson, J. H.  Moiré superlattice on the surface of a topological insulator.  _Phys. Rev. B_ 103, 155157 (2021).
  * [8] Xu, Y. _et al._ Connection topology of step edge state bands at the surface of a three dimensional topological insulator.  _New Journal of Physics_ 20, 073014 (2018).
  * [9] Schindler, F. _et al._ Higher-order topology in bismuth.  _Nature physics_ 14, 918–924 (2018).
  * [10] Yeom, H. W. _et al._ Instability and Charge Density Wave of Metallic Quantum Chains on a Silicon Surface.  _Phys. Rev. Lett._ 82, 4898–4901 (1999).
  * [11] Cohen, R. L., Eibschütz, M., West, K. W. & Buehler, E.  Electronic Configuration of SmB6.  _Journal of Applied Physics_ 41, 898–899 (1970).  URL https://doi.org/10.1063/1.1659008.
  * [12] Dzero, M., Sun, K., Galitski, V. & Coleman, P.  Topological Kondo Insulators.  _Phys. Rev. Lett._ 104, 106408 (2010).
  * [13] Takimoto, T.  SmB6: A Promising Candidate for a Topological Insulator.  _Journal of the Physical Society of Japan_ 80, 123710 (2011).
  * [14] Miyazaki, H., Hajiri, T., Ito, T., Kunii, S. & Kimura, S.-i.  Momentum-dependent hybridization gap and dispersive in-gap state of the Kondo semiconductor SmB6.  _Phys. Rev. B_ 86, 075105 (2012).
  * [15] Jiang, J. _et al._ Observation of possible topological in-gap surface states in the Kondo insulator SmB6 by photoemission.  _Nature Communications_ 4, 3010 (2013).
  * [16] Xu, N. _et al._ Direct observation of the spin texture in SmB6 as evidence of the topological Kondo insulator.  _Nature Communications_ 5, 4566 (2014).
  * [17] Hlawenka, P. _et al._ Samarium hexaboride is a trivial surface conductor.  _Nature Communications_ 9, 517 (2018).
  * [18] Ohtsubo, Y. _et al._ Non-trivial surface states of samarium hexaboride at the (111) surface.  _Nature Communications_ 10, 2298 (2019).
  * [19] Li, L., Sun, K., Kurdak, C. & Allen, J.  Emergent mystery in the Kondo insulator samarium hexaboride.  _Nature Reviews Physics_ 2, 463–479 (2020).
  * [20] Rößler, S. _et al._ Hybridization gap and fano resonance in smb6.  _Proceedings of the National Academy of Sciences_ 111, 4798–4802 (2014).
  * [21] Ellguth, M., Tusche, C., Iga, F. & Suga, S.  Momentum microscopy of single crystals with detailed surface characterisation.  _Philosophical Magazine_ 96, 3284–3306 (2016).
  * [22] Miyamachi, T. _et al._ Evidence for in-gap surface states on the single phase SmB6(001) surface.  _Scientific Reports_ 7, 12837 (2017).
  * [23] Aizaki, N. & Tatsumi, T.  In situ RHEED observation of selective diminution at Si (001)-2$\times$ 1 superlattice spots during MBE.  _Surface Science_ 174, 658–665 (1986).
  * [24] Kotta, E. _et al._ Metallic chemical potentials in an insulating topological state.  _preprint at Research Square_ (2022).  URL https://doi.org/10.21203/rs.3.rs-1239795/v1.
  * [25] Noguchi, A., Takamura, Y., Nakagawa, T., Rokuta, E. & Mizuno, S.  Structural analysis of clean LaB6(100), (111), and (110) surfaces via quantitative low-energy electron diffraction.  _Surface Science_ 701, 121686 (2020).
  * [26] Aoyama, H. & Abukawa, T.  Complete three-dimensional structure of Bi-adsorbed Si(110) surface: Discovery of heavily reconstructed Si(110) substrate.  _Phys. Rev. Research_ 3, 043164 (2021).
  * [27] Baruselli, P. P. & Vojta, M.  Spin textures on general surfaces of the correlated topological insulator ${\mathrm{SmB}}_{6}$.  _Phys. Rev. B_ 93, 195117 (2016).
  * [28] Ärrälä, M., Nieminen, J., Braun, J., Ebert, H. & Lindroos, M.  Photon energy dependence of circular dichroism of the au(111) surface state.  _Phys. Rev. B_ 88, 195413 (2013).
  * [29] Zhu, Z.-H. _et al._ Photoelectron Spin-Polarization Control in the Topological Insulator ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$.  _Phys. Rev. Lett._ 112, 076802 (2014).
  * [30] Yaji, K. _et al._ Spin-dependent quantum interference in photoemission process from spin-orbit coupled states.  _Nature Communications_ 8, 14588 (2017).
  * [31] Phelan, W. A. _et al._ Correlation between Bulk Thermodynamic Measurements and the Low-Temperature-Resistance Plateau in ${\mathrm{SmB}}_{6}$.  _Phys. Rev. X_ 4, 031012 (2014).
  * [32] Li, G. _et al._ Two-dimensional Fermi surfaces in Kondo insulator SmB6.  _Science_ 346, 1208–1212 (2014).
  * [33] Tan, B. S. _et al._ Unconventional fermi surface in an insulating state.  _Science_ 349, 287–290 (2015).
  * [34] Hartstein, M. _et al._ Fermi surface in the absence of a Fermi liquid in the Kondo insulator SmB6.  _Nature Physics_ 14, 166–172 (2018).
  * [35] Xiang, Z. _et al._ Bulk Rotational Symmetry Breaking in Kondo Insulator ${\mathrm{SmB}}_{6}$.  _Phys. Rev. X_ 7, 031054 (2017).
  * [36] Xiang, Z. _et al._ Unusual high-field metal in a Kondo insulator.  _Nature Physics_ 17, 788–793 (2021).
  * [37] Millichamp, T. E. _et al._ Direct measurement of a remnant Fermi surface in SmB6.  _arXiv:2111.07727_ (2021).
  * [38] Teo, J. C. Y., Fu, L. & Kane, C. L.  Surface states and topological invariants in three-dimensional topological insulators: Application to ${\text{Bi}}_{1-x}{\text{Sb}}_{x}$.  _Phys. Rev. B_ 78, 045426 (2008).
  * [39] Okuda, T. _et al._ Experimental Evidence of Hidden Topological Surface States in ${\mathrm{PbBi}}_{4}{\mathrm{Te}}_{7}$.  _Phys. Rev. Lett._ 111, 206803 (2013).
  * [40] Wang, T., Yuan, N. F. Q. & Fu, L.  Moiré Surface States and Enhanced Superconductivity in Topological Insulators.  _Phys. Rev. X_ 11, 021024 (2021).
  * [41] Fu, L.  Hexagonal Warping Effects in the Surface States of the Topological Insulator ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$.  _Phys. Rev. Lett._ 103, 266801 (2009).
  * [42] Peters, R., Yoshida, T., Sakakibara, H. & Kawakami, N.  Coexistence of light and heavy surface states in a topological multiband Kondo insulator.  _Phys. Rev. B_ 93, 235159 (2016).
  * [43] Michishita, Y., Yoshida, T. & Peters, R.  Relationship between exceptional points and the Kondo effect in $f$-electron materials.  _Phys. Rev. B_ 101, 085122 (2020).
  * [44] Iga, F., Shimizu, N. & Takabatake, T.  Single crystal growth and physical properties of Kondo insulator YbB12.  _Journal of magnetism and magnetic materials_ 177, 337–338 (1998).

## Acknowledgements

We acknowledge W. Breton and F. Deschamps for their support during the
experiments on the CASSIOPÉE beamline at the SOLEIL synchrotron. We also thank
S. Ideta and K. Tanaka for their support for LEED measurements at UVSOR with
proposal No. 20-784. The ARPES experiments were performed under SOLEIL
proposal No. 20191629. This work was also supported by JSPS KAKENHI (Grants
Nos. JP20H04453, JP19H01830, and JP20K03859).

## Author contributions

Y.O. conducted the ARPES experiments with assistance from P.L.F. and F.B..
Y.O., T.Nakaya, and T.Nakamura performed the LEED experiments. F.I. grew the
single-crystal samples. Y.O. and S.-i.K. wrote the text and were responsible
for the overall direction of the research project. All authors contributed to
the scientific planning and discussions.

## Competing interests

The authors declare no competing interests.

Figure 1:  Surface preparation of the vicinal SmB6(001) surface. (a) Crystal
structure of SmB6. (b) LEED patterns of the cleaned SmB6(001) surface taken at
100 K. The distortions in the patterns are due to the flat microchannel plate
in the LEED electron optics. Two-headed arrows indicate the lines along which
the profiles in (c) were taken. (c) LEED line profiles taken from the pattern
shown in (b). (d) Angle-integrated photoelectron spectra taken at 13 K. 6F and
6H are the Sm2+ 4$f^{5}$ final states after photoexcitation. (e) A surface
atomic structure of SmB6(001) without any surface superstructure. (f) The same
as (e) but with hypothetical $p$(2$\times$2) surface reconstruction. Arrows
guide the displacements of Sm and B6 near the surface.  Figure 2:  Fermi
contours (FCs) obtained by ARPES. (a) ARPES FCs taken with circularly
polarized photons ($h\nu$ = 35 eV) at 13 K. The ARPES intensities from the
left- and right-handed polarizations are summed up to show all the states
without any influence of circular dichroism (CD). The photon incident plane is
($\bar{1}$10). The arrows indicate the positions where the $E$-$k$ dispersions
shown in Figs. 3, 4, and Supplementary Fig. 1 were taken. (b) Schematic
drawing of the observed FCs together with the border of surface Brillouin
zones; solid lines for bulk-truncated (1$\times$1) and dashed for
(2$\times$2). (c) Experimental geometry and definition of the in-plane
wavevectors $k_{x}$ and $k_{y}$. $k_{x}$ and $k_{y}$ are always in the photon
incident and photoelectron detection planes, respectively. In this work, two
cases, the incident planes of (010) and ($\bar{1}$10) were measured. The
relationship between $k_{x,y}$ and the crystal orientation is shown by
subindexes such as $k_{y//[\bar{1}10]}$ and $k_{y//[010]}$.  Figure 3:  Band
dispersions of SmB6(001)-$p$(2$\times$2). ARPES intensity plots with the same
condition as in Fig. 2 together with the momentum distribution curves cut at
$E_{\rm F}$, measured along the (a) $\bar{\rm X}$-$\bar{\rm M}$ (Cut 1), (b)
$\bar{\Gamma}$-$\bar{\rm X}$ (Cut 2), and (c, d) $\bar{\Gamma}$-$\bar{\rm M}$
(Cut 3) orientations.  Figure 4:  CD-ARPES band dispersions. (a-c) CD-ARPES
plots taken at 13 K. The measurement geometries are the same as those in Figs.
2 and 3. (d) Schematic drawing of the OAM polarizations corresponding to
surface FCs, assuming that they are parallel to the helicities of the
incident-photon polarizations: parallel (anti-parallel) to the photon-
incidence vector for right- (left-) handed polarization. Fat lines indicate
the cut orientations shown in a-c. Circled digits indicate numbers of surface
FCs in SBZ. (e) Surface FCs folded by (2$\times$2) SBZ boundary. Arrows guide
the possible singularity points (see main text for details). (f) Schematic
drawings of TSS with folding and distortion from anisotropic surface
superstructure.

## Supplementary Information for: Breakdown of bulk-projected isotropy in
surface electronic states of topological Kondo insulator SmB6(001)

Supplementary Fig. 1:  (a) Schematic drawing of the surface Fermi contours
(FCs) observed by angle-resolved photoelectron spectroscopy (ARPES) (the same
setup as that shown in Fig. 2c in the main text) together with the arrow
indicating the position where the ARPES data in b and c were taken. (b) ARPES
intensity plot taken with linearly polarized photons ($h\nu$ = 30 eV) at 14 K.
(c) Momentum distribution curves (MDCs) of the ARPES intensity plots at
$E_{\rm F}$ along Cut 4 indicated in (a) at the sample centre (lower curve)
and corner (upper curve). The peaks correspond to S1 and S1’ defined in the
main text. At the corner of the sample, the vicinal miscut angle is different
from that in the rest of the area because of the edge rounding effect during
the polishing process. Since the other measurement conditions are the same for
the MDCs, this intensity difference between S1 and S1’ should come from the
area difference between surface domains, with the majority (minority) domains
corresponding to S1 (S1’).  Supplementary Fig. 2:  (a) ARPES MDCs along
$\bar{\Gamma}$-$\bar{\rm M}$ (Cut 3 defined in the main text) taken with
linearly polarized photons at 13 K. Fat lines indicate the measured peaks
without significant dispersion along $k_{z//[001]}$, corresponding to surface
FCs S2 and S3. The incident photon energies range from 20 to 45 eV. (b)
Schematic drawing of the bulk and (001) surface Brillouin zones (BZ) of SmB6.
(c) Measured $k_{z//[001]}$ range of (a) in the bulk BZ assuming the inner
potential of 10 eV. Photon energies of 18-45 eV (data shown in (a) and those
with $h\nu$ = 18 eV shown in the main text (Fig. 3d)) are used for calculating
$k_{z//[001]}$ values. The initial and final positions of the measured range
(the arrowheads) are placed as the worst case, where the inversion at the
centre of the BZ doubles the measured range. The current energy range
corresponds to the wavevector of 1.2 Å-1 along the surface normal ([001]) with
this assumption. The reciprocal lattice vector along [001] of SmB6 is 1.5 Å-1,
and thus, the measured range covers $\sim$80 % of the bulk BZ, even in this
worst case. Together with the metallic character, in contrast to the
insulating bulk electronic states at this temperature, the observed bands are
quite likely to be the 2D surface states.

## Suppl. Note 1: Homogeneity of the SmB6(001)-$p$(2$\times$2) surface

It is known that the cleaved SmB6(001) surfaces exhibit numerous different
surface domains with various topmost elements and inhomogeneous atomic
structures, as observed by PES and STM [1, 2, 3]. However, we think that such
patchy surface structure on cleaved surfaces is not dominant and is playing
just a minor role on the SmB6(001)-$p$(2$\times$2) surface we prepared in this
work. Let us discuss this issue here.

Firstly, it is already known that the inhomogeneous nature of the SmB6(001)
surface is mostly removed by in-situ cleaning process by cycles of sputtering
and annealing. For example, STM images and LEED pattern taken before and after
the cleaning process exhibited a significant improvements of surface flatness
and uniform surface structure (ref. [4]). While the miscut angle and annealing
temperature of ours are different from those of ref. [4], it would be a
natural expectation that the similar improvement to remove the patchy surface
morphology also occurred in our samples.

Secondly, the sharp FCs around G-bar also supports the small (or nearly zero)
area of inhomogeneity on our surface. Most of the ARPES studies performed on
the cleaved surface exhibited just a blurred feature around there. This could
be understood by supposing that the patchy surface structures behave as a
random perturbation to TSS and thus the averaged ARPES data results in a broad
hump as observed. The oval-shaped S1 is always observed even in such
occasions, perhaps because it is not sensitive to surface perturbation
comparing with the other TSSs around G-bar. This assumption is not so strange,
if the wavefunction of S1 does not have large amplitude around the topmost
surface but rather deeper layers. In contrast, in our case, all the three FCs
were observed sharply. The peak widths of S2 and S3 around G-bar are nearly
the same as that of S1, as shown in Fig. 3. It suggests that the broadening of
S2 and S3 from surface patches discussed above has almost negligible
contribution and the peak width of surface bands there are simply determined
by $k$ resolution of the ARPES equipment.

Thirdly, the $p$(2$\times$2) surface superstructure could be reproduced three
times, even after a few months under ambient condition. It means that the
current surface preparation method removes surface contamination as well as
disorders from ambient contamination repeatedly. It is unrealistic to assume
that all the surface “patches” are robust against such ambient pressure as
well as the removal of surface layers by Ar ion sputtering in our preparation
method. Therefore, we think the “patches” could exist on the surface only
after the first polish of the substrate crystal, but should be removed during
the repeated surface contamination (in the air) and cleaning processes.

Based on these points, we are sure that the SmB6(001)-$p$(2$\times$2) surface
prepared in this work is free from the surface inhomogeneity.

## Suppl. Note 2: Photoexcitation process of ARPES

It is known that ARPES intensity often differs from the simple distribution of
the density of initial (ground) states, because the observed electron belongs
to the final state, free electron in vacuum and thus the photoexcitation
process plays an important role upon it. Actually, evident intensity
modulations at ARPEC FCs were reported in both bulk [5] and surface [6, 7]
systems. This effect is called photoemission structure factor (PSF), as an
analogy from those among diffraction experiments. Therefore, it is worth to be
discussed whether the observed different intensities of S1 and S1’ could be
attributed to such artificial effects or not.

To consider this point, it should be noted that PSF appears between different,
nonequivalent SBZs. For example, a metallic surface state on
Ag/Si(111)-($\sqrt{3}\times\sqrt{3}$)$R$30∘ was observed at the 2nd SBZ but
not at the 1st zone [7]. In addition, the photon-incident and photoelectron-
detection planes also play a significant role, as the two-fold, not six, ARPES
constant energy contours observed in graphite [5]. However, even after
including such effect, ARPES intensities from the equivalent SBZ with the
common photon-incident and photoelectron detection planes must be identical,
as shown both from computational and experimental results [5, 7].

In the current case, SmB6(001)-$p$(2$\times$2), PSF rationalizes the absence
of the FCs around $\bar{\rm M}$ folded by $p$(2$\times$2) surface
superstructure for example, because the $p$(2$\times$2) SBZ around $\bar{\rm
M}$ is the 3rd zone, different from the 1st (around $\bar{\Gamma}$) or 2nd
(around $\bar{\rm X}$) zones. On the other hand, tiny intensity from S1’
cannot be derived from it, because the SBZs around $\bar{\rm X}$ and $\bar{\rm
Y}$ are equivalent 2nd zones, if one suppose four-fold rotation symmetry. Note
that the experimental geometry for S1 and S1’ is also equivalent with the
common photon-incident and photoelectron-detection planes as depicted in Fig.
2c in the main text. For further evidence, we measured the ARPES FC of
SmB6(001)-$p$(2$\times$2) with different ARPES geometry as shown in
supplementary Fig. 3. Here, the common photon-incident and photoelectron-
detection planes are rotated 45∘ from those of Fig. 2. From supplementary Fig.
3, one can find that the S1 FC is still intense while those of S1’ are very
weak. Intensity modulations for $\pm k_{y}$ is because of circular dichroism
(CD), since we measured this FC only by right-handed circularly polarized
photons; CD does not justify the absence of S1’, since it disappears at both
signs of $k_{y}$. It is a smoking-gun evidence that PSF plays just a minor
role on the disappearance of S1’, strongly supporting that the difference
between S1 and S1’ are from the surface atomic structure.

Supplementary Tab. 1:  Fermi velocities ($v_{\rm F}$) and areas of FCs ($A$) of each TSS. $v_{\rm F}$ for S1 (S2 and S3) is estimated from Cut 1 (Cut 3) in Fig. 3 taken with $h\nu$ = 35 (18) eV. $A$ are listed in two units so that they could be easily compared with the data from de Haas van Alphen measurements.  |  $v_{\rm F}$ (eV Å) |  $A$ (Å-2) |  $A$ (kT)
---|---|---|---
S1 | -0.25 | 0.30 | 3.1
S2 | 1.9 | 0.42 | 4.4
S3 | -0.27 | 0.24 | 2.5
Supplementary Fig. 3:  Fermi contours (FCs) obtained by ARPES taken with
circularly polarized photons (right handed, $h\nu$ = 35 eV) at 13 K. The
photon incident plane is around (010).

## References

  * [1] Hlawenka, P. _et al._ Samarium hexaboride is a trivial surface conductor.  _Nature Communications_ 9, 517 (2018).
  * [2] Kotta, E. _et al._ Metallic chemical potentials in an insulating topological state.  _preprint at Research Square_ (2022).  URL https://doi.org/10.21203/rs.3.rs-1239795/v1.
  * [3] Rößler, S. _et al._ Hybridization gap and fano resonance in smb6.  _Proceedings of the National Academy of Sciences_ 111, 4798–4802 (2014).
  * [4] Miyamachi, T. _et al._ Evidence for in-gap surface states on the single phase SmB6(001) surface.  _Scientific Reports_ 7, 12837 (2017).
  * [5] Nishimoto, H. _et al._ Two-dimensional angular distribution of photoelectrons of single-crystal graphite.  _Journal of Physics: Condensed Matter_ 8, 2715–2732 (1996).
  * [6] Rotenberg, E. _et al._ Indium $\sqrt{7}\times{}\sqrt{3}$ on Si(111): A Nearly Free Electron Metal in Two Dimensions.  _Phys. Rev. Lett._ 91, 246404 (2003).
  * [7] Hirahara, T., Matsuda, I. & Hasegawa, S.  Photoemission Structure Factor Effect for Fermi Rings of the Si(111)$\sqrt{3}\times\sqrt{3}$-Ag Surface.  _e-Journal of Surface Science and Nanotechnology_ 2, 141–145 (2004).