††thanks: These authors equally contributed to this work.††thanks: These
authors equally contributed to this work.

# Polarity-dependent charge-density wave in a kagome superconductor CsV3Sb5

Takemi Kato1 Yongkai Li2,3,4 Kosuke Nakayama1,5
<EMAIL_ADDRESS>Zhiwei Wang2,3,4<EMAIL_ADDRESS>Seigo Souma6,7 Miho Kitamura8 Koji Horiba8,9 Hiroshi Kumigashira10 Takashi
Takahashi1,7 Takafumi Sato1,6,7,11<EMAIL_ADDRESS>1Department
of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578,
Japan
2Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum
Architecture and Measurement (MOE), School of Physics, Beijing Institute of
Technology, Beijing 100081, P. R. China
3Beijing Key Lab of Nanophotonics and Ultrafine Optoelectronic Systems,
Beijing Institute of Technology, Beijing 100081, P. R. China
4Material Science Center, Yangtze Delta Region Academy of Beijing Institute of
Technology, Jiaxing, 314011, P. R. China
5Precursory Research for Embryonic Science and Technology (PRESTO), Japan
Science and Technology Agency (JST), Tokyo, 102-0076, Japan
6Center for Spintronics Research Network, Tohoku University, Sendai 980-8577,
Japan
7Advanced Institute for Materials Research (WPI-AIMR), Tohoku University,
Sendai 980-8577, Japan
8Institute of Materials Structure Science, High Energy Accelerator Research
Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan
9National Institutes for Quantum Science and Technology (QST), Sendai
980-8579, Japan
10Institute of Multidisciplinary Research for Advanced Materials (IMRAM),
Tohoku University, Sendai 980-8577, Japan
11International Center for Synchrotron Radiation Innovation Smart (SRIS),
Tohoku University, Sendai 980-8577, Japan

###### Abstract

Polar surface and interface play a pivotal role for realizing exotic
properties of materials, and a search for such polar states is of crucial
importance for expanding materials’ functionality. Here we report micro-
focused angle-resolved photoemission spectroscopy of CsV3Sb5, a member of
recently discovered kagome superconductors AV3Sb5 (A = K, Rb, Cs), and show
evidence for the polar nature of cleaved surface which is characterized by Cs-
and Sb-terminated surfaces with markedly different fermiology. The Cs-
terminated surface shows intriguing doubling of V-derived bands at low
temperature associated with the band folding due to the three-dimensional
charge-density wave (CDW), whereas the Sb-terminated one shows no band
doubling or resultant CDW-gap opening indicative of the suppression of bulk-
originated CDW due to polar charge. Such polar-surface-dependent band
structure must be incorporated for understanding the origin of unconventional
superconducting and charge order at the surface of AV3Sb5.

Surface and interface of crystals are a fertile playground where a variety of
exotic properties emerge owing to its symmetry, chemical-bonding, and
topological characteristics distinct from those of bulk counterpart. Sudden
change in the electrostatic potential at the surface generates conducting two-
dimensional (2D) electron gas called Shockley states ShockleyPR1939  which
evolve into spin-split Rashba states associated with the space-inversion
symmetry breaking BychkovJETP1984 , whereas the discontinuity of topological
index across the surface of topological insulators leads to spin-helical
Dirac-cone states HasanRMP2010 ; QiRMP2011 ; AndoJPSJ2013 . Another key factor
that gives a significant influence on physical properties at the
surface/interface is polarity; its impact is best demonstrated at the
interface of ionic crystals, where the electronic band structure is markedly
reconstructed so as to compensate the discontinuity of ionic polarity (known
as polar catastrophe). For instance, LaAlO3/SrTiO3 interface exhibits high
conductivity and even superconductivity despite the semiconducting nature of
host crystals OhtomoNature2004 ; CavigliaNature2008 ; ThielScience2006 .
Polarity also plays an essential role at polar surfaces to bring about various
exotic properties, as represented by Rashba-split 2D electron gas at SrTiO3
surface MeevasanaNM2011 ; SantanderNature2011 ; NakamuraPRL2012 ;
WalkerPRL2014 , extremely over-doped CuO2 planes at high-temperature
superconductor YBa2Cu3O7 surface NakayamaPRB2007 ; ZabolotnyyPRB2007 ,
ferromagnetic surface of antiferromagntic PdCoO2 MazzolaPNAS2018 , spin-
texture manipulation in giant Rashba compound BiTeI IshizakaNM2011 ;
MaassNCOM2016 , and surface-termination-dependent Fermi arcs in Weyl semimetal
NbP SoumaPRB2016 . As highlighted by these examples, understanding the physics
associated with the polar surfaces of quantum materials is a key topic of
modern condensed-matter physics.

Here we focus on $\rm{CsV}_{3}\rm{Sb}_{5}$, a member of kagome metals
$\rm{AV}_{3}\rm{Sb}_{5}$ (A = K, Rb, Cs) which were recently discovered to
show superconductivity with $T_{\rm{c}}$ = 0.9-2.5 K coexisting with charge-
density wave (CDW) below $T_{\rm{CDW}}$ = 78-103 K OrtizPRM2021 ; OrtizPRL2020
; YinCPL2021  with the in-plane 2$\times$2 periodicity ChenNature2021 ;
JiangNM2021 ; ZhaoNature2021 ; LiangPRX2021 ; WangPRB2021 ; ShumiyaPRB2021 ;
NieNature2022 ; LiarXiv2021 . The crystal structure of CsV3Sb5 consists of two
building blocks, Cs layer and V3Sb5 layer, and the cleaved surface is expected
to be terminated either by Cs or V3Sb5 (Sb2) layer [Fig. 1(a)]. Although the
bonding between V and Sb1/Sb2 atoms within the V3Sb5 layer may be
intermetallic, the coupling between the Cs and V3Sb5 layers is ionic because
Cs is fully ionized (Cs1+) to donate one electron to the V3Sb5 layer
[(V3Sb5)1-]. Such ionicity would lead to polar instability. In fact, two
different surface terminations have been identified by scanning-tunneling-
microscopy (STM) measurements, and intriguingly, there is growing experimental
evidence that such surface hosts peculiar quantum states distinct from those
of the bulk, as exemplified by the formation of unidirectional 4$\times$1 CDW
ChenNature2021 ; ZhaoNature2021 ; LiangPRX2021 ; WangPRB2021 ; ShumiyaPRB2021
; NieNature2022 ; LiarXiv2021 ; XuPRL2021 , surface-dependent vortex-core
states LiangPRX2021 , and pair-density wave ChenNature2021 . However, the band
structure associated with the polar surface remains totally unexplored in
$\rm{AV}_{3}\rm{Sb}_{5}$.

In this Letter, we report a spatially-resolved angle-resolved photoemission
spectroscopy (ARPES) study of $\rm{CsV}_{3}\rm{Sb}_{5}$. By utilizing the
micro-focused photon beam from synchrotron, we have succeeded in separately
observing two kinds of polar surfaces, and uncovered marked differences in the
band structures between them. The Cs-terminated surface shows the band
doubling associated with the 3D nature of CDW at low temperature, whereas the
Sb-terminated surface is strongly hole-doped, resulting in the suppression of
CDW. We discuss implications of the present results in relation to the
mechanism of CDW and unconventional surface properties.

High-quality $\rm{CsV}_{3}\rm{Sb}_{5}$ single crystals were synthesized with
the self-flux method OrtizPRM2019 . ARPES measurements were performed using
Scienta-Omicron DA30 and SES2002 spectrometers at BL-28A and BL-2A in Photon
Factory, KEK. We used energy tunable photons of $h\nu$ = 85-350 eV. ARPES data
were mainly obtained by circularly polarized 106-eV photons (corresponding to
the out-of-plane wave vector $k_{z}\sim 0$) NakayamaPRB2021  with the beam
spot size of 10$\times$12 $\mu$m2 KitamuraRSI2022 . The energy resolution was
set to be 25 meV. The samples were cleaved at the same temperatures at which
the ARPES measurements were performed (8 K and 120 K). All the data have been
recorded within 8 hours after the cleavage. Within this time period, we did
not observe discernible shifts in band energy. Thus, the doping effect
discussed later is not related to the time-dependent doping effect reported
earlier LuoarXiv2021 .

Figure 1:  (a) Crystal structure of $\rm{CsV}_{3}\rm{Sb}_{5}$. (b) Photograph
of a cleaved $\rm{CsV}_{3}\rm{Sb}_{5}$ single crystal. (c) Spatial map of the
$E_{\rm{B}}$ position of Sb-$4d_{5/2}$ core level measured with $h\nu$ = 106
eV at $T$ = 8 K in the spatial region enclosed by red dashed rectangle (area
size: 600$\times$800 $\mu$m2) in (b), obtained with the step size of 20
$\mu$m. (d), (e) EDC in the Sb-$4d$ and Cs-$4d$ core-level regions,
respectively, for the Cs-(red) and Sb-(blue) terminated surfaces. (f) Spatial
map of the intensity ratio of the surface/bulk Cs-$4d_{5/2}$ core-level peaks.
(g) Schematic view of the cleaved surface of $\rm{CsV}_{3}\rm{Sb}_{5}$.

First, we demonstrate how to distinguish two types of polar surfaces. The
cleaved surface of $\rm{CsV}_{3}\rm{Sb}_{5}$ contains a flat shiny mirror-like
region [Fig. 1(b)]. By sweeping the micro-focused photon beam on the surface,
we have mapped out the spatial distribution of the binding energy
($E_{\rm{B}}$) of Sb-$4d_{5/2}$ core-level peak [see Fig. 1(c)], which is a
good measure of the surface doping level (note that the selection of
Sb-$4d_{3/2}$ peak reproduces the same result). As seen in Fig. 1(c), the
surface looks to consist of two regions (blue and red regions). The core-level
spectrum in the red region [red curve in Fig. 1(d)] is shifted as a whole
toward higher $E_{\rm{B}}$ by $\sim$130 meV with respect to that in the blue
region (blue curve), indicating that the red region is more electron doped
than the blue region (see Section 1 of Supplemental Material for detailed
discussion on the origin of the Sb-$4d$ core-level shift refSM ). Such a
difference in the doping level signifies different polarities of the two
regions. Here we assign the electron (hole)-rich red (blue) region to the Cs
(Sb)-terminated surface because Cs atoms on top surface do not need to donate
electrons to the missing upper V3Sb5 layer which is removed by the cleaving,
so that the V3Sb5 layer beneath the topmost Cs layer is more electron doped
for the Cs-terminated surface. Inversely, the V3Sb5 layer at the surface is
hole-doped compared with the bulk counterpart. This is also corroborated by
the behavior of the Cs-$4d$ core level. Figure 1(e) shows the Cs-$4d$ core-
level spectra measured at the same spatial location as in Fig. 1(d), where we
observe that each spin-orbit satellite of Cs-$4d$ core levels ($d_{5/2}$ or
$d_{3/2}$) consists of two peaks; a peak at lower $E_{\rm{B}}$ originates from
bulk Cs atoms whereas that at higher $E_{\rm{B}}$ is from surface Cs atoms, as
confirmed by the angular dependence of their intensity ratio (Section 1 of
Supplemental Material refSM ). One can recognize that the surface Cs peak is
stronger than the bulk one for the Cs-termination-dominant surface (red
curve), whereas it is weaker for the Sb-termination-dominant case (blue curve)
(see Section 2 of Supplemental Material for the analysis of peak position
refSM ). The spatial image of the intensity ratio between the surface and bulk
Cs peaks in Fig. 1(f) shows a good agreement with that of the Sb-core-level
$E_{\rm{B}}$ in Fig. 1(c). This supports our assignment of the Cs- and Sb-
termination-dominant surfaces (hereafter, we simply call them Cs- and Sb-
terminated surfaces, respectively), as schematically shown in Fig. 1(g).

We show in Figs. 2(a) and 2(b) the ARPES intensity at $T$ = 120 K (above
$T_{\rm{CDW}}$ = 93 K) along the $\Gamma$KM cut ($k_{z}\sim 0$) [red line in
Fig. 2(c)] for the Cs- and Sb-terminated surfaces, respectively. One can
recognize overall similarity in the band structure between the two, i.e. an
electron band at the $\Gamma$ point ($\alpha$) with the Sb-$5p_{z}$ character,
linearly dispersive V-$3d_{xz/yz}$ bands ($\beta,\gamma$) forming a Dirac cone
near $E_{\rm{F}}$ (marked by an arrow), the $\delta$ band with the
V-$3d_{xy/x^{2}-y^{2}}$ character forming a saddle point (SP) slightly above
$E_{\rm{F}}$, and the $\varepsilon$ band that intersects the $\delta$ band to
form a Dirac point at the K point at $E_{\rm{B}}\sim 0.3$ eV NakayamaPRB2021 ;
NakayamaPRX2022 ; JiangNM2021 ; WangPRB2022 ; TanPRL2021 ; FuPRL2021 ;
ZhaoNature2021 . A closer look further reveals some intrinsic differences; the
band structure of the Cs-terminated surface is shifted downward as a whole
with respect to that of Sb-terminated surface, as recognized by a direct
comparison of surface-termination-dependent ARPES spectra at representative k
points plotted in Figs. 2(d)-2(f), in which numerical fittings to the peak
position clarified the energy shift to be 30-90 meV (note that a finite
variation in the magnitude of the energy shift among different bands and
Sb-$4d$ core levels suggests a non-rigid-band-type energy shift; also see
Section 3 of Supplemental Material for the energy shift of the $\delta$ band
refSM ). Moreover, while the Sb-terminated surface exhibits a single $\alpha$
band bottomed at $E_{\rm{B}}\sim 0.5$ eV, it splits into two bands bottomed at
$E_{\rm{B}}\sim 0.5$ and 0.8 eV for the Cs-terminated surface. This suggests
that not only polarity but also the band structure above $T_{\rm{CDW}}$ is
different between the Cs- and Sb-terminated surfaces. We have performed ARPES
measurements more than three times with different samples for each surface
termination and confirmed the reproducibility (see Section 4 of Supplemental
Material refSM ).

Figure 2:  (a), (b) ARPES intensity at $T$ = 120 K for Cs- and Sb-terminated
surfaces, respectively, measured with $h\nu$ = 106 eV along the $\Gamma$KM
cut. (c) (top) Bulk Brillouin zone and (bottom) slice of Brillouin zone at
$k_{z}$ = 0 plane. (d)-(f) Comparison of the EDCs measured for the Cs (red)
and Sb (blue) surface terminations at k cuts A-C, respectively. The position
of k cuts is indicated by red dashed lines in (a) and (b).  Figure 3:  (a),
(b) FS map for the Cs-terminated surface measured at $T$ = 120 K and 8 K,
respectively. (c), (d) Enlarged view in the k region enclosed by dashed red
rectangle in (a) and (b), respectively. (e), (f) ARPES intensity plots at $T$
= 8 K measured along $k_{y}$ at $k_{x}=0$ ($\Gamma$KM cut) and $-\pi$ (KMK
cut), respectively. (g) Second-derivative intensity of (f). (h) Plot of
near-$E_{\rm{F}}$ ARPES intensity at $T$ = 8 K along the MK cut obtained with
higher resolution. (i) EDCs at $T$ = 120 K (red) and 8 K (blue) at the
$k_{\rm{F}}$ point along the MK cut [blue triangle in (h)]. The hump and peak
structures at $T$ = 8 K are indicated by blue arrows.

Next we analyze the electronic structure of the Cs-terminated surface in more
detail, particularly in relation to the CDW transition. Figures 3(a) and 3(b)
show a comparison of the Fermi-surface (FS) mapping at the $k_{z}\sim 0$ plane
between $T$ = 120 K (above $T_{\rm{CDW}}$) and 8 K (below $T_{\rm{CDW}}$). At
$T$ = 120 K, one can recognize a small circular and a large hexagonal pocket
centered at the $\Gamma$ point, together with a triangular pocket at the K
point LiPRX2021 ; LouPRL2022 ; NakayamaPRB2021 ; LiuPRX2021 ; ChoPRL2021 ;
LuoNCOM2022 ; KangNP2022 ; KatoCOMMAT2022 ; HuNCOM2022 . On the other hand, at
$T$ = 8 K, the hexagonal pocket splits into two pockets as more clearly seen
in Figs. 3(c) and 3(d), where the enlarged FS image is displayed. Such a band
splitting is observed in a wide ($E,\textbf{k}$) region, as highlighted by the
ARPES intensity at $T$ = 8 K measured along the representative $k_{y}$ cuts at
$k_{x}$ = 0 and $-\pi$ in Figs. 3(e) and 3(f). For example, the $\delta$ and
the $\varepsilon$ bands show an energy splitting as better visualized in Fig.
3(e) [also see the energy distribution curve (EDC) in Sec. 5 of the
Supplemental Material [35]]. Also, the $\beta$ and $\gamma$ bands show
splitting as seen in Fig. 3(f) [note that there are two peaks in the momentum
distribution curve [(MDC); red curves] in the Supplemental Material [35]] and
in the corresponding second-derivative intensity in Fig. 3(g). Thus, in the
CDW phase, all the V-derived bands show splitting. It is noted, however, that
although the Sb-derived $\alpha$ band appears to split, it is unclear whether
it is of the CDW origin, because the splitting was also observed at $T$ = 120
K [Fig. 2(a)]. The origin of the $\alpha$-band splitting in the normal state
has been discussed in connection with the formation of quantum well states
CaiarXiv2021  and the $k_{z}$ broadening effect NakayamaPRB2021 .

Now, we discuss the origin of the band splitting for the V-derived bands.
Apparently, it is not associated with the exchange splitting, because no
signature of ferromagnetism was reported in $\rm{AV}_{3}\rm{Sb}_{5}$
OrtizPRM2019 . The existence of surface states near the main bulk bands and
the influence of the $k_{z}$ broadening would be also unlikely, because the
splitting is absent at $T$ = 120 K. A plausible explanation is the CDW-
potential-induced band folding along the $k_{z}$ direction, which is naturally
expected from the reported 3D nature of CDW accompanied by the out-of-plane
unit-cell doubling (2$\times$2$\times$2) LiangPRX2021 ; LiPRX2021  (see Fig.
S8 in Supplemental Material for details refSM ). It is noted that we also
found a signature of the in-plane band folding, as detailed in Section 6 of
Supplemental Material refSM .

We measured the $\delta$ band dispersion along the MK cut which forms the SP
at $T$ = 8 K [Fig. 3(h)] and revealed that it disperses back toward higher
$E_{\rm{B}}$ without reaching $E_{\rm{F}}$ due to the CDW-gap opening. This
gap is well seen from the EDC at the Fermi wave vector ($k_{\rm{F}}$)
[indicated by triangle in Fig. 3(h)] at $T$ = 8 K, where a characteristic hump
structure appears at $E_{\rm{B}}\sim 70$ meV in agreement with previous
studies NakayamaPRB2021 ; LiuPRX2021 . Besides the 70-meV hump, the EDC at $T$
= 8 K shows a small peak at $E_{\rm{B}}\sim 20$ meV, which may be due to the
CDW-gap opening in the folded band from the $k_{z}=\pi$ plane.

Figure 4:  (a) FS map for the Sb-terminated surface measured at $T$ = 8 K with
$h\nu$ = 106 eV. (b) ARPES-intensity plot at $T$ = 8 K measured along the
$\Gamma$KM cut. (c) Comparison of the MDCs between the Sb- and Cs-terminated
surfaces, obtained along the k cut indicated by brown dashed arrow in (a). (d)
Comparison of EDC between the two surface terminations, obtained in the
$E_{\rm{B}}$ region shown by dashed green arrow in (b). Arrows in (c) and (d)
indicate peak positions of the MDC/EDC. (e) Plot of near-$E_{\rm{F}}$ ARPES
intensity for the Sb-terminated surface, measured along the MK cut with higher
energy resolution at $T$ = 8 K. (f) Comparison of experimental band
dispersions between the Sb- and Cs-terminated surfaces in the ($E,\textbf{k}$)
region enclosed by dashed green rectangle in (e). (g) Comparison of EDC at the
$k_{\rm{F}}$ point of the $\delta$ band [blue triangle in (e)]. Red arrows
show hump-and-peak structure associated with the CDW gap for the Cs-terminated
surface.

Having established the electronic states of the Cs-terminated surface, now we
turn our attention to the Sb-terminated surface. Figure 4(a) shows the typical
FS mapping for the Sb-terminated surface at $T$ = 8 K well below
$T_{\rm{CDW}}$. By comparing with that of the Cs-terminated counterpart [Fig.
3(b)], we observe shrinkage of the circular pocket centered at the $\Gamma$
point and the triangular pocket at the K point due to the excess hole doping
in the Sb-terminated surface (see Section 7 of Supplemental Material refSM
for quantitative comparison). All the energy bands observed in this study have
a surface character (specifically, the energy bands of the topmost V3Sb5
layer) because of a relatively short escape depth of photoelectrons ($\sim$10
Å). For the Sb-terminated surface, the hole-doped character of the
$\Gamma$-centered pocket derived from the Sb1 atoms suggests that the hole-
doping effect extends to the Sb1-V plane. Taking into account these points, it
is natural to observe an energy shift of the V-derived bands. It is noted
that, besides the polarity-induced charge doping, the inversion symmetry is
broken at the Sb-terminated surface. But the band splitting, e.g. Rashba-type
spin splitting, expected by the inversion-symmetry breaking has not been
observed within our experimental uncertainty, suggesting that such an effect
may not be strong in CsV3Sb5.

More importantly, the band dispersion is basically temperature independent as
seen from the ARPES intensity at $T$ = 8 K in Fig. 4(b) which is similar to
that at $T$ = 120 K [Fig. 2(b)]; one can see no doubling of the Sb and
V-derived bands. The absence of band doubling for the Sb-terminated surface is
more clearly visualized in Figs. 4(c) and 4(d), where the representative MDCs
and EDCs are compared between the Cs- and Sb-terminated surfaces (red and blue
curves, respectively). We also found that the Sb-terminated surface shows no
clear CDW-gap opening. As shown in Fig. 4(e), the $\delta$ band simply crosses
$E_{\rm{F}}$ (see triangle) without bending back behavior, in contrast to the
case of the Cs-terminated surface [see the band dispersion extracted from the
peak position of EDCs in Fig. 4(f)]. Also, the EDC at the $k_{\rm{F}}$ point
for the Sb-terminated surface [blue curve in Fig. 4(g)] exhibits no hump
structure unlike the case of the Cs-terminated counterpart (red curve).

The band diagram as a function of surface Cs coverage derived from the present
study and our previous work on Cs-dosed CsV3Sb5 NakayamaPRX2022  is summarized
in Fig. 5. The excess hole doping realized in the Sb-terminated surface pushes
up the energy bands, whereas the electron doping in the Cs-terminated or Cs-
dosed surfaces pushes down (note that $E_{\rm F}$ for the bulk band structure
would be obtained by averaging $E_{\rm F}$ of the Cs and Sb terminations
because the electron concentration identical to the bulk is realized at the Cs
coverage of 50 $\%$). Despite the surface character of observed band
structures in the present study, a signature of 2$\times$2$\times$2 CDW is
found in the Cs-terminated surface at low temperatures, suggesting that the
2$\times$2$\times$2 CDW and the resultant band folding along $k_{z}$ can
survive even at the surface under a certain condition. On the other hand, the
2$\times$2$\times$2 CDW is suppressed in the Sb-terminated and heavily Cs-
dosed surfaces. Understanding the relationship between this contrasting CDW
behavior and the difference in surface charge is a next important step toward
clarifying the mechanism of unconventional CDW in CsV3Sb5.

Figure 5:  Schematic of key band structure in CsV3Sb5 as a function of surface
Cs coverage. Inset shows the crystal structure at the surface of Sb-terminated
(left), Cs-terminated (middle), and Cs-dosed $\rm{CsV}_{3}\rm{Sb}_{5}$
(right).

We discuss implications of the present results in relation to the exotic
surface properties recently reported by STM of $\rm{AV}_{3}\rm{Sb}_{5}$
ChenNature2021 ; ZhaoNature2021 ; LiangPRX2021 ; WangPRB2021 ; ShumiyaPRB2021
; NieNature2022 ; LiarXiv2021 . One of the intriguing surface properties is
that the superconducting-gap structure shows a higher density of states at
$E_{\rm{F}}$ for the Cs-terminated surface compared to the Sb-terminated one
ChenNature2021 . This could be interpreted in terms of the enhanced gap
anisotropy in the Cs-terminated surface. In light of our observations, such a
difference is suggested to be related to the strength of CDW at the surface;
namely, the Cs-terminated surface is more strongly perturbed by CDW so that
the superconducting-gap structure becomes more complex and unconventional.
Another exotic surface property is the emergence of surface charge order with
a unidirectional 4$\times$1 periodicity that appears only for the Sb-
terminated surface ChenNature2021 ; ZhaoNature2021 ; LiangPRX2021 ;
WangPRB2021 ; ShumiyaPRB2021 ; NieNature2022 ; LiarXiv2021 . This surface
charge order may be associated with the suppression of the 2$\times$2 CDW in
light of the present ARPES result (in fact, one may see a finite change in the
intensity of 2$\times$2 superspots in the Fourier transform image of the STM
pattern LiangPRX2021 ). Also, the mechanism of the 4$\times$1 charge order
cannot be easily explained from the observed FS topology; it is hard to infer
from the FS mapping in Figs. 3(b) and 4(a) that the Sb-terminated surface
possesses a good nesting condition with the 4$\times$1 periodicity. This
suggests that other possibilities beyond the simple FS nesting such as
electron correlation and surface instability WangPRB2021 ; ChenNature2021 ;
ZhaoNature2021  must be taken into account to understand the unconventional
surface charge order; we leave this issue as an open question. In any case,
the present study strongly suggests that the polarity-dependent band
characteristics, such as the critical difference in the carrier doping levels
and CDW properties, must be properly taken into account to understand the
microscopic mechanism behind the intriguing surface electronic anomalies in
$\rm{AV}_{3}\rm{Sb}_{5}$.

In conclusion, the present micro-ARPES study of $\rm{CsV}_{3}\rm{Sb}_{5}$ has
revealed the polar nature of the cleaved surface and self-doping effects. We
uncovered that the Sb-terminated surface is characterized by excess hole
doping and resultant suppression of CDW, whereas the Cs-terminated surface
shows the band doubling and the resultant CDW-gap opening below $T_{\rm{CDW}}$
which support the 3D CDW. The present result opens a pathway toward
understanding and manipulating exotic properties in kagome superconductors
$\rm{AV}_{3}\rm{Sb}_{5}$.

###### Acknowledgements.

This work was supported by JST-CREST (No. JPMJCR18T1), JST-PRESTO (No.
JPMJPR18L7), Grant-in-Aid for Scientific Research (JSPS KAKENHI Grant Numbers
JP21H04435 and JP20H01847), KEK-PF (Proposal number: 2021S2-001), and the
Sasakawa Scientific Research Grant from the Japan Science Society. The work at
Beijing was supported by the National Key R&D Program of China Grant No.
2020YFA0308800), the Natural Science Foundation of China (Grants No.
92065109), the Beijing Natural Science Foundation (Grant No. Z210006), and the
Beijing Institute of Technology (BIT) Research Fund Program for Young Scholars
(Grant No. 3180012222011). T.K. acknowledges support from GP-Spin at Tohoku
University and JST-SPRING (No. JPMJSP2114). Z.W. thanks the Analysis & Testing
Center at BIT for assistance in facility support.

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##  I SUPPLEMENTAL MATERIAL

###  I.1 Section 1: Photoelectron emission angle dependence of core-level
spectra

Figure 6 shows a comparison of core-level spectra measured at different
photoelectron emission angles $\theta=90^{\circ}$ and $60^{\circ}$ [see Fig.
6(a) for the definition of $\theta$]. Because of the finite escape depth of
photoelectrons ($\lambda$) in the sample, the photoemission spectrum taken at
a smaller $\theta$ is more surface sensitive; an effective escape depth
$\lambda_{\rm eff}=\lambda\sin{\theta}$ is reduced by $\sim$15% by changing
$\theta$ from $90^{\circ}$ to $60^{\circ}$ [see Fig. 6(a)]. If the Sb-$4d$
core levels had surface and bulk components at higher and lower binding
energies ($E_{\rm{B}}$’s), respectively, and the surface component became
stronger in intensity by changing $\theta$ from $90^{\circ}$ to $60^{\circ}$
as observed for the Cs-$4d$ core levels (see below), the peak position of
Sb-$4d$ core levels at $\theta=60^{\circ}$ would be located at higher
$E_{\rm{B}}$ than that at $\theta=90^{\circ}$. However, as shown in Fig. 6(b),
such an energy shift is not seen within the present experimental uncertainty
of 10 meV, which is much smaller than the 130-meV energy shift observed by the
termination-dependent ARPES study. This result is not consistent with the two-
components scenario, but supports the doping scenario as the origin of the
energy shift in the Sb-$4d$ core level [Fig. 1(d)]. In contrast, the Cs-$4d$
core levels show surface and bulk components. As shown in Fig. 6(c), each
spin-orbit satellite of Cs-$4d$ core levels ($d_{\rm{5/2}}$ or $d_{\rm{3/2}}$)
consists of two peaks. A peak at lower $E_{\rm B}$ originates from bulk Cs
atoms, whereas that at higher $E_{\rm B}$ is from surface Cs atoms, as
supported by the angular dependence of their intensity ratio; the shorter
$\lambda_{\rm eff}$ at $\theta=60^{\circ}$ leads to the stronger surface-
derived (weaker bulk-derived) peak intensity of the Cs-$4d$ core levels.

Figure 6:  (a) Schematic of experimental setup and the definition of
photoelectron emission angle $\theta$, escape depth $\lambda$, and effective
escape depth $\lambda_{\rm eff}$. (b) Comparison of EDCs in the Sb-$4d$ core-
level region measured at $\theta=90^{\circ}$ (black) and $60^{\circ}$ (red)
for the Cs-terminated surface. (c) Same as (b) but for the Cs-$4d$ core-level
region.  Figure 7:  Comparison of energy distribution curves (EDCs) in the
Cs-$4d$ core-level region measured for Cs- and Sb-terminated surfaces (red and
blue curves, respectively).

###  I.2 Section 2: Peak position of the Cs-$4d$ core levels

The termination-dependent Cs-$4d$ core-level data in Fig. 7 shows that the
surface components of the Cs-terminated surface shift toward lower
$E_{\rm{B}}$ by $\sim$30 meV with respect to those of the Sb-terminated
surface. Since this direction is opposite from that observed in the Sb-$4d$
core levels, the observed Cs-$4d$ core-level shift cannot be understood in
terms of the carrier doping, but probably reflects a difference in the density
of Cs atoms at the surface. It is known that the core level of alkaline-metal
atoms shifts toward lower $E_{\rm{B}}$ with increasing the number of alkaline-
metal atoms per area. Such a characteristic energy shift is theoretically
explained by a Born-Haber cycle PirugSurfSci1985  and provides a reasonable
explanation on our observation. Namely, the Cs-terminated surface has the
higher density of Cs atoms compared to that of the Sb-terminated surface and
hence the Cs-$4d$ core level appears at lower $E_{\rm{B}}$, whereas the
Sb-$4d$ core levels shift toward higher $E_{\rm{B}}$ because of the doping-
induced chemical potential shift. A similar behavior was reported in K-dosed
Fe(Se,Te) ZhangAPL2014  in which gradual K deposition on the surface leads to
the shift of the Te-$4d$ core level toward higher $E_{\rm{B}}$ due to the
electron doping to the Fe(Se,Te) layer, whereas the K-$3p$ core level shifts
toward lower $E_{\rm{B}}$.

Figure 8:  (a), (b) ARPES intensity measured along the KM cut at $T=120$ K in
the Cs- and Sb-terminated surfaces, respectively. Green dashed curves are a
guide for the eyes to trace the SP-band dispersions.

###  I.3 Section 3: Surface-termination dependence of the saddle-point
position

Figure 8 shows the ARPES data that supports the SP-band energy shift, in which
a comparison of the ARPES intensity along the KMK cut between the Cs- and Sb-
terminated surfaces is made. One can directly recognize that the saddle point
(SP) band (green dashed curve) is located at lower $E_{\rm{B}}$ in the Sb-
terminated surface compared to that in the Cs-terminated surface, signifying
the difference in the doping level.

###  I.4 Section 4: Reproducibility of polar-surface formation

We have performed ARPES measurements more than three times with different
samples for each surface termination and confirmed the reproducibility, e.g.
the termination-dependent band energy shifts, as seen by a comparison of the
band dispersions measured with different samples in Fig. 9. We also performed
ARPES measurements with some different photon energies between 70 and 130 eV
which have comparable probing depths, and obtained qualitatively the same
results.

###  I.5 Section 5: Band splitting observed in the Cs-terminated surface

Figure 10 shows representative EDCs and momentum distribution curves (MDCs) at
$T=8$ K measured for the Cs-terminated surface. One can clearly see a two-
peaked structure in EDCs [Figs. 10(c) and 10(d)] and MDCs [Figs. 10(e) and
10(f)] which supports the band splitting associated with the CDW-induced band
folding.

Figure 9:  (a), (b) Comparison of the band dispersions at $T=8$ K for the Sb-
terminated surface between two different $\rm{CsV}_{3}\rm{Sb}_{5}$ samples
(samples A and B). (c), (d) Same as (a) and (b) but for the Cs-terminated
surface.

###  I.6 Section 6: A signature of in-plane band folding

Figures 11(a) and 11(b) show a comparison of the second-derivative ARPES
intensity along the $\Gamma$KMK$\Gamma$ cut at $T=8$ K and 120 K for the Cs-
terminated surface. We observed the discontinuity of the $\beta$ band
dispersion along the $\Gamma$K line of the Cs-terminated surface at $T=8$ K,
as indicated by white arrows in Fig. 11(a). This anomaly is of CDW origin
because it disappears at $T=120$ K [Fig. 11(b)] and is well reproduced by the
density-functional theory that incorporates only the in-plane $2\times 2$
periodicity [Fig. 11(c)]. Therefore, the Cs-terminated surface shows the unit-
cell doubling not only along the $c$ axis but also in the $a-b$ plane. On the
other hand, for the Sb-terminated surface, the discontinuity of the $\beta$
band is absent even at 8 K, so both the c-axis and in-plane unit-cell
doublings are suppressed.

Figure 10:  (a), (b) ARPES intensity plots at $T=8$ K measured along $k_{y}$
at $k_{y}=0$ ($\Gamma$KM cut) and $-\pi$ (KMK cut), respectively. (c), (d)
EDCs at representative energy slices indicated by red lines in (a). (e), (f)
MDCs at representative k slices shown by red lines in (b). Blue curves in
(c)-(f) are the result of numerical fittings.

###  I.7 Section 7: Comparison of Fermi surface between Cs- and Sb-terminated
surfaces

Figures 12(a) and 12(b) show a comparison of the ARPES intensity map at the
Fermi level ($E_{\rm{F}}$) around the K point between the Cs- and Sb-
terminated surfaces. The peak position of MDC at $E_{\rm{F}}$ is overlaid for
the triangular-shaped electron Fermi surface (FS; black circles and red
triangles, respectively). Direct comparison of the MDC peak position in Fig.
12(c) shows the shrinkage of the triangular FS in the Sb-terminated surface,
consistent with the hole-doped character.

Figure 11:  (a), (b) Second-derivative ARPES-intensity plots for the Cs-
terminated surface at $T=8$ K and 120 K, respectively. White arrows in (a)
highlight the appearance of band hybridization by the in-plane band folding.
(c) Calculated band structure of $\rm{CsV}_{3}\rm{Sb}_{5}$ with the $2\times
2\times 1$ unit cell, reproduced from TanPRL2021 .  Figure 12:  (a), (b) ARPES
intensity plots around the K point measured at $T=8$ K for the Cs- and Sb-
terminated surfaces, respectively. Black and red triangles in (a) and (b)
represent the Fermi wave vectors determined by numerical fittings to the MDCs
at $E_{\rm{F}}$. (c) Comparison of the Fermi wave vectors between the Cs- and
Sb-terminated surfaces. The data points were symmetrized by assuming the
$C_{3}$ symmetry centered at the K point.  Figure 13:  (a), (b) ARPES-
intensity plots at $T=120$ K and 8 K, respectively, measured along the KM cut.
(c), (d) Second-derivative ARPES-intensity plots of (a) and (b), respectively.
(e), (f) Calculated band structures with SOC for (e) $k_{z}=0$ (red) and (f)
$k_{z}=0$ (red) and $\pi$ (blue) (reproduced from NakayamaPRB2021 ). The
calculations were carried out using the full-potential linearized augmented
plane-wave method implemented in the WIEN2K code with generalized gradient
approximation (GGA) and the Perdew-Burke-Ernzerhof (PBE) type exchange-
correlation potential. Spin-orbit coupling was included self-consistently,
while the lattice parameters were directly obtained from experiments. The
$k$-points mesh for the irreducible Brillouin zone is $17\times 17\times 10$.
The Muffin-tin radii are 2.50 a.u. for Cs and V and 2.60 a.u. for Sb,
respectively. The maximum modulus for the reciprocal vectors Kmax was chosen
to satisfy $R_{\rm{MT}}\times\rm{K}_{\rm{max}}=8.0$.

###  I.8 Section 8: Comparison between the experimental and calculated band
structures

To clarify the origin of band doubling in the charge-density-wave (CDW) phase
for the Cs-terminated surface of $\rm{CsV}_{3}\rm{Sb}_{5}$, we compare the
ARPES results with the first-principles band-structure calculations. Figures
13(a) and 13(b) show the ARPES intensity along the KM cut at $T=120$ K (above
$T_{\rm{CDW}}$) and 8 K (below $T_{\rm{CDW}}$), respectively, measured with
$h\nu=106$ eV photons corresponding to the out-of-plane wave vector of
$k_{z}\sim 0$ NakayamaPRB2021 . The second-derivative intensity is displayed
in Figs. 13(c) and 13(d). At $T=120$ K, we observe five dispersive bands
(labelled as $\beta,\gamma,\delta,\varepsilon,$ and $\eta$) near $E_{\rm{F}}$
[Figs. 13(a) and 13(c)], in agreement with the first-principles calculations
for the KM cut ($k_{z}=0$) [Fig. 13(e)]. On the other hand, at $T=8$ K, these
bands show doubling as marked by green arrows in Fig. 13(d) [doubling of the
$\delta$ band is seen in the data obtained with higher energy resolution in
Figs. 3(h) and 3(i)]. This band doubling is explained in terms of the band
folding along the $k_{z}$ direction due to the three-dimensional (3D) CDW
LiangPRX2021 . Namely, the doubling of unit-cell along the c-axis causes the
folding of energy bands from $k_{z}=\pi$ to 0, and vice versa. The calculated
band dispersions superimposed with those of $k_{z}=0$ and $\pi$ [Fig. 13(f)]
is in good agreement with the experimental results, supporting the 3D-CDW
origin of the band doubling for the Cs-terminated surface. It is noted that to
obtain a better matching between the experiment and calculation, additional
level splitting associated with the repulsion between the original and folded
bands (especially for the $\varepsilon$ band) as well as the CDW-gap opening
near $E_{\rm{F}}$ needs to be taken into account. It is also noted that,
besides the $2\times 2\times 2$ periodicity, the $2\times 2\times 4$
periodicity has been also reported in $\rm{CsV}_{3}\rm{Sb}_{5}$ OrtizPRX2021 .
We naively expect the quadruple band splitting when the charge order has the
$2\times 2\times 4$ periodicity. But our data show only doubling. In this
respect, our results are more consistent with the $2\times 2\times 2$
periodicity.