# Multiple energy scales and anisotropic energy gap in the charge-density-wave
phase of kagome superconductor CsV3Sb5

Kosuke Nakayama,1,2,#,∗ Yongkai Li,3,4,# Takemi Kato,1 Min Liu,3,4 Zhiwei
Wang,3,4,∗ Takashi Takahashi,1,5,6 Yugui Yao,3,4 and Takafumi Sato1,5,6,7
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of Physics, Faculty of Science,
Tohoku University, Sendai 980-8578, Japan
2Precursory Research for Embryonic Science and Technology (PRESTO), Japan
Science and Technology Agency (JST), Tokyo, 102-0076, Japan
3Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum
Architecture and Measurement (MOE), School of Physics, Beijing Institute of
Technology, Beijing 100081, China
4Beijing Key Lab of Nanophotonics and Ultrafine Optoelectronic Systems,
Beijing Institute of Technology, Beijing 100081, China
5Center for Spintronics Research Network, Tohoku University, Sendai 980-8577,
Japan
6WPI Research Center, Advanced Institute for Materials Research, Tohoku
University, Sendai 980-8577, Japan
7International Center for Synchrotron Radiation Innov1ation Smart (SRIS),
Tohoku University, Sendai 980-8577, Japan

###### Abstract

Kagome metals $A$V3Sb5 ($A$ = K, Rb, and Cs) exhibit superconductivity at
0.9-2.5 K and charge-density wave (CDW) at 78-103 K. Key electronic states
associated with the CDW and superconductivity remain elusive. Here, we
investigate low-energy excitations of CsV3Sb5 by angle-resolved photoemission
spectroscopy. We found an energy gap of 50-70 meV at the Dirac-crossing points
of linearly dispersive bands, pointing to an importance of spin-orbit
coupling. We also found a signature of strongly Fermi-surface and momentum-
dependent CDW gap characterized by the larger energy gap of maximally 70 meV
for a band forming a saddle point around the M point, the smaller (0-18 meV)
gap for a band forming massive Dirac cones, and a zero gap at the
$\Gamma$/A-centered electron pocket. The observed highly anisotropic CDW gap
which is enhanced around the M point signifies an importance of scattering
channel connecting the saddle points, laying foundation for understanding the
nature of CDW and superconductivity in $A$V3Sb5.

###### pacs:

71.18.+y, 71.20.-b, 79.60.-i

Kagome lattice, consisting of $3d$ transition metal ions with two-dimensional
(2D) network of corner-sharing triangles, provides an excellent platform to
explore novel quantum phenomena originating from electron correlation and
nontrivial band topology. While insulating kagome lattice has been intensively
studied in relation to geometric frustration and quantum magnetism
RamirezARMS1994 ; ShoresJACS2005 ; PollmannPRL2008 ; BalentsNature2010 ;
YanScience2011 ; HanNature2012 ; HanPRL2012 ; FuScience2015 , metallic kagome
lattice is currently attracting particular attention because of its unique
band structure characterized by the nearly flat band and Dirac-cone band that
promote strong-correlation and topological effects. Depending on the electron
filling of the kagome-lattice bands, various intriguing quantum states have
been predicted, e.g. Weyl magnet NayakSA2016 ; YangNJP2017 ; KurodaNM2017 ;
YeNature2018 ; LiuNP2018 ; MoraliScience2019 ; LiuScience2019 , density wave
orders IsakovPRL2006 ; GuoPRB2009 ; WangPRB2013 , charge fractionalization
OBrienPRB2010 ; RueggPRB2011 , and superconductivity WangPRB2013 ; KoPRB2009 ;
KieselPRB2012 ; KieselPRL2013 .

Recently, a family of $A$V3Sb5 ($A$ = K, Rb, and Cs) was discovered to be a
kagome superconductor OrtizPRM2019 ; OrtizPRL2020 ; OrtizPRM2021  with
superconducting transition temperature $T_{\rm c}$ of 0.93-2.5 K, despite the
fact that kagome metals rarely become a superconductor. $A$V3Sb5 crystallizes
in a layered structure consisting of alternately stacked V kagome-lattice
layer with hexagonally arranged Sb atoms (V1 and Sb1), graphene-like Sb layer
(Sb2), and hexagonal $A$ layer [see Fig. 1(a)]. Besides superconductivity,
$A$V3Sb5 commonly undergoes a charge-density wave (CDW) transition at $T_{\rm
CDW}$ = 78-103 K accompanied with three-dimensional (3D) 2$\times$2$\times$2
charge order LiAX2021 ; LiangAX2021 . Angle-resolved photoemission
spectroscopy (ARPES) clarified that the kagome-lattice bands participate in
the states near the Fermi level ($E_{\rm F}$), by observing the Dirac-cone-
like bands forming a large hexagonal Fermi surface (FS) centered at the
$\Gamma$ point and a saddle point near $E_{\rm F}$ at the M point, together
with an electron pocket at the $\Gamma$ point of 2D Brillouin zone (BZ),
consistent with the density-functional-theory (DFT) calculations OrtizPRL2020
; YangSA2020 ; LiAX2021 ; LiuAX2021 .

While overall electronic structure of $A$V3Sb5 is almost established, the
mechanism of superconductivity and CDW is highly controversial. $Z_{2}$
topological invariant in the normal state suggested by the DFT calculation
OrtizPRL2020 ; OrtizPRM2021  may point to unconventional (topological)
superconductivity. Existence of a saddle point in the calculated band
structure was discussed to promote the $d$-wave pairing associated with the
scattering between saddle points via $Q$ = ($\pi$, 0) vector NandkishoreNP2012
. Weak electron-phonon coupling suggested by the DFT calculation TanAX2021
also supports an unconventional pairing. In the experiment, conventional vs.
unconventional nature of superconductivity is far from reaching a consensus,
as represented by contradictory reports on the presence/absence of gap nodes
DuanAX2021 ; CZhaoAX2021 . And very recently, a nematic electronic state and a
two-fold symmetry superconductivity was observed in CsV3Sb5 Xiang AX2021 .

The electronic states relevant to the CDW formation are also under intensive
debate. Apparent CDW-gap opening was not observed in previous ARPES studies on
RbV3Sb5 and CsV3Sb5 OrtizPRL2020 ; LiuAX2021 , whereas scanning tunneling
microscopy/spectroscopy (STM/STS) on KV3Sb5 and CsV3Sb5 reported a gap opening
JiangAX2020 ; ChenAX2021 ; HZhaoAX2021 . While the calculation suggested that
the CDW is triggered by the Peierls instability due to the FS nesting
TanAX2021 , x-ray scattering on RbV3Sb5 and CsV3Sb5 suggested an electronic-
driven mechanism because of the absence of expected phonon anomaly LiAX2021 .
In contrast, optical spectroscopy on CsV3Sb5 suggested the FS-nesting
mechanism by observing the reduction of Drude weight below $T_{\rm CDW}$ for
the saddle-point-related feature ZhouAX2021 . Thus, the mechanism of CDW and
superconductivity is still far from being established.

Figure 1: (a) ARPES-intensity maps at binding energies ($E_{\rm B}$’s) of
$E_{\rm F}$, 0.3 eV, and 0.6 eV plotted as a function of $k_{x}$ and $k_{y}$,
for CsV3Sb5 measured at $T$ = 20 K with 106-eV photons. (b) Photon-energy
dependence of the normal-emission energy distribution curves (EDCs). The inner
potential $V_{0}$ was estimated to be 10 eV. (c), (d) ARPES intensity as a
function of wave vector and $E_{\rm B}$ measured along the $\Gamma$KM ($k_{z}$
= 0) and AHL ($k_{z}$ = $\pi$) cuts, respectively. Solid curve in (d) is a
guide to the eyes to trace the experimental $\gamma$ band dispersion. (e)
Calculated band structure along the $\Gamma$KM$\Gamma$ (AHLA) cut (top)
without and (bottom) with SOC. Calculated $E_{\rm F}$ is shifted downward by
50 meV to obtain a better matching with the experimental data. (f) EDCs at the
Dirac-crossing points.

In this article, we report a high-resolution ARPES study of CsV3Sb5 single
crystals. We established the low-energy excitations in the CDW phase, and
found (i) the gap opening at the Dirac points due to the spin-orbit coupling
(SOC) and (ii) the strongly FS and momentum ($k$) dependent CDW gap
characterized by multiple energy scales. We discuss implications of the
present results in relation to the mechanism of CDW and superconductivity in
$A$V3Sb5.

We at first present the overall band structure of CsV3Sb5 (see Supplemental
Material for details of the experimental condition and the band-calculation
method SM ). Figure 1(a) shows the ARPES-intensity map at representative
binding energy ($E_{\rm B}$) slices as a function of $k_{x}$ and $k_{y}$,
obtained with 106-eV photons which probe the $k_{z}$ $\sim$ 0 plane of the
bulk BZ [the $k_{z}$ value was determined by the periodicity of band
dispersion as a function of photon energy ($h\nu$); $h\nu$ variation of the
band in the normal-emission measurement is highlighted in Fig. 1(b) and
Supplementary Fig. 5 SM ]. We found that the overall intensity pattern in Fig.
1(a) is consistent with the previous ARPES reports on $A$V3Sb5 OrtizPRL2020 ;
YangSA2020 ; LiAX2021 ; LiuAX2021 . One can identify a circular pocket
centered at the $\Gamma$ point. This pocket originates from an electron band
(called here the $\alpha$ band) as visible from ARPES-intensity plot along the
$\Gamma$KM cut in Fig. 1(c). According to the DFT calculations LiuAX2021 ;
JiangAX2020 ; JZhaoAX2021 , this band is attributed to the $5p_{z}$ band of
Sb1 atoms embedded in the kagome-lattice plane. One can recognize in Fig. 1(a)
triangular shaped intensity pattern centered at each K point which connects to
each other around the M point and forms a large hexagonal FS centered at the
$\Gamma$ point. As shown in Fig. 1(c), this pocket originates from a band
located at 1 eV at the K point (the $\beta$ band) that displays an overall
linear dispersion toward $E_{\rm F}$ on approaching the $\Gamma$ point and
crosses $E_{\rm F}$ at $k\sim 2/3$ of the $\Gamma$K interval. This band is
reproduced in the calculation in Fig. 1(e) and attributed to the kagome-
lattice band with mainly V-$3d_{xy}$ character LiuAX2021 ; JZhaoAX2021 . This
band intersects other two linearly dispersive bands ($\gamma$ and $\delta$
bands) at $\sim$0.1 eV and $\sim$0.5 eV, and the $\delta$ band also intersects
the linearly dispersive $\epsilon$ band at the K point, forming multiple Dirac
points (note that the Dirac point associated with the $\beta$-$\gamma$ band
crossing is predicted to form a nodal line along $k_{z}$ when the SOC is
neglected JZhaoAX2021 ). The $\delta$ band rapidly disperses toward $E_{\rm
F}$ on approaching the M point and participates in forming the saddle point.
At the M point, there is another holelike band ($\zeta$ band) with the top of
dispersion below $E_{\rm F}$.

At $h\nu$ = 120 eV that corresponds to the $k_{z}$ = $\pi$ plane, there are
several similarities in the band dispersions with those at the $k_{z}$ = 0
plane, e.g. linear dispersions of the $\beta$, $\gamma$, $\delta$, and
$\epsilon$ bands and their Dirac-cone formation [compare Figs. 1(c) and 1(d)].
Besides such similarities, one can recognize a clear difference in energy
position of the $\alpha$-band bottom, i.e. it is located at $E_{\rm B}$ $\sim$
0.5 and 0.75 eV at $k_{z}$ = 0 and $\pi$, respectively. One can also find that
a shallow electron pocket centered at the L point appears only at the $k_{z}$
= $\pi$ plane. According to the calculation [Fig. 1(e)], this band is
connected to the $\gamma$ band as indicated by a guide in Fig. 1(d). The
observed band structures including finite $k_{z}$ dispersions reasonably agree
with the bulk band calculation, supporting their bulk nature. It is noted
that, while the existence of topological surface state at the $\bar{\rm M}$
point was predicted OrtizPRL2020 ; OrtizPRM2021 , we found no evidence for
such a surface state below $E_{\rm F}$.

As shown in Fig. 1(c), the intensity of the $\delta$ band is weakened at which
the $\epsilon$ band intersects at the K point at $\sim$0.3 eV. This is also
the case for the band-crossing point at the H point [Fig. 1(d)]. To better
visualize the band modulation associated with the intensity weakening, we plot
in Fig. 1(f) the EDCs at the K and H points. These EDCs commonly exhibit two-
peaked structure around the Dirac point, signifying an energy gap (called
Dirac gap; see also Fig. 6 of Supplemental Material SM ). From the energy
separation of two peaks, we have estimated the magnitude of Dirac gap to be
50-70 meV. The Dirac gap is associated with the SOC because the calculations
predict a spin-orbit gap at the Dirac point (note that, by referring to the
calculation without SOC, the gap at the K and H points is purely of SOC
origin, whereas those at other k points are additionally affected by the band
hybridization). Given the fact that some Dirac points are predicted to be
located at very close to $E_{\rm F}$ in $A$V3Sb5 OrtizPRM2019 , our
observation suggests that the SOC needs to be taken into account when the
connection between the Dirac-fermion and transport properties (such as
anomalous Hall effect YangSA2020 ; YuAX2021 ) is discussed.

Figure 2: (a), (b) ARPES intensity along the MK cut ($h\nu$ = 106 eV) measured
at $T$ = 120 K and 20 K, respectively. (c) EDCs at the kF point of the
$\delta$ band along the MK cut (magenta) and the $\gamma$ band along the LH
cut (green). (d) Same as (b) but measured along the LH cut ($h\nu$ = 120 eV).
(e), (f) Same as (a) and (b) but measured at 21.2-eV photons. (g)
Corresponding EDCs of (f). (h) Experimental band dispersion at $T$ = 10 and
120 K extracted by tracing the peak position of EDCs in (e) and (f). Filled
and open circles represent original and symmetrized data. (i) Temperature
dependence of the EDC at the kF point of shallow electron band measured across
$T_{\rm CDW}$. Inset shows the magnified view for the EDC at 10 K. Black arrow
indicates the peak position. (j) Same as (i) but symmetrized with respect to
$E_{\rm F}$.

Next we present the electronic states associated with the CDW. Figures 2(a)
and 2(b) show the ARPES-intensity plot near $E_{\rm F}$ measured along the MK
cut at temperature above (120 K) and below (20 K) $T_{\rm CDW}$, respectively.
In the normal state [Fig. 2(a)], the $\delta$ band crosses $E_{\rm F}$ at the
midway of the MK line and forms a saddle point slightly above $E_{\rm F}$ at
the M point. There also exists the $\zeta$ band which forms another saddle
point at the M point, while it does not cross $E_{\rm F}$ along the MK cut. In
the CDW phase [Fig. 2(b)], the intensity of the $\zeta$ band is modulated at
$E_{\rm B}$ $\sim$ 0.2 eV likely due to band hybridization by CDW. More
importantly, a hump structure at $E_{\rm B}$ $\sim$ 70 meV is observed at the
kF point of the $\delta$ band [see magenta curves in Fig. 2(c); see also Fig.
7 of Supplementary Material SM ]. Since the kF point of the $\delta$ band is
well isolated from the $\zeta$-band top, the 70-meV hump is unlikely
attributed to the saddle point of the $\zeta$ band but most probably reflects
a CDW-gap opening on the $\delta$ band. To clarify the 3D nature of CDW, the
near-$E_{\rm F}$ ARPES intensity at 20 K along the LH cut is plotted in Fig.
2(d). Here, a shallow $\gamma$ electron pocket is seen around $E_{\rm F}$ near
the kF point of the $\delta$ band. Intriguingly, unlike the MK cut, EDC near
kF of the $\gamma$/$\delta$ band does not show a clear hump structure [green
curve in Fig. 2(c)], suggesting the bulk origin of the hump structure and the
reduction of the CDW-gap size around the LH line.

To gain further insights into the electronic structure in the CDW phase, we
have performed higher-resolution ($\Delta E$ = 7 meV) measurements by
utilizing low-energy photons ($h\nu$ = 21.2 eV) [Figs. 2(e)-2(f)]. We found
that, while 21.2-eV photons are expected to probe $k_{z}$ $\sim$ 0.8$\pi$ by
referring to the estimated $V_{0}$ value, the ARPES data simultaneously
reflect the band structures at around $k_{z}$ = 0 and $\pi$ due to a $k_{z}$
broadening effect (for details, see Fig. 8 of Supplemental Material SM ). In
the CDW phase, one can see a broad hump in the EDCs at $\sim$150 meV around
the K/H point [dashed curve in Fig. 2(g)] which disperses toward $E_{\rm F}$
on approaching the M/L point and stays at $\sim$70 meV around the M/L point to
form a nearly flat band. This hump feature is attributed to the gapped
$\delta$ band at $k_{z}$ = 0 seen in Figs. 2(b) and 2(c). There exists another
sharp peak near $E_{\rm F}$, separated from the hump by a dip at 35 meV. This
band is assigned to the $\gamma$ electron pocket at $k_{z}$ = $\pi$ with a
finite mixture of the $\delta$ band ($k_{z}$ = $\pi$ component) near kF. In
fact, as shown by a plot of experimental band dispersion at 10 K obtained by
tracing the peak position of EDCs in Fig. 2(h), the low-energy branch has an
electronlike dispersion. Thus, the characteristic peak-dip-hump structure in
the EDCs is attributed to the existence of two types of bands at different
$k_{z}$’s, the hump and peak, called here the saddle-point band and Dirac
band, respectively. We found that the peak-dip-hump structure vanishes above
$T_{\rm CDW}$ [Fig. 2(e); for the experimental band dispersion, see red
circles in Fig. 2(h)]. To examine whether this change is associated with the
CDW, we show in Fig. 2(i) the temperature dependence of EDC at the kF point of
Dirac band. The hump survives up to 80 K (below $T_{\rm CDW}$) and disappears
at 100 K (above $T_{\rm CDW}$), demonstrating that the hump is indeed
associated with the formation of CDW and reflects the CDW gap for the saddle-
point band.

An important finding manifests itself by a careful inspection of the EDCs in
Fig. 2(i). The sharp peak at $T$ = 10 K is not located at $E_{\rm F}$ but at
slightly higher $E_{\rm B}$ ($\sim$15 meV) as seen from the magnified EDCs in
the inset (see arrow). Symmetrized EDC in Fig. 2(j) reveals spectral-weight
suppression in the narrow energy region of $\pm$15 meV within the sharp peaks,
indicative of the small gap opening at the Dirac band (note that this small
gap is not resolved in Figs. 2(b) and 2(c) because of insufficient energy
resolution for higher $h\nu$’s). This gap is associated with the CDW, because
it is gradually weakened on elevating temperature and disappears above $T_{\rm
CDW}$, in accordance with the behavior of saddle-point band at 70 meV. These
results indicate that the CDW of CsV3Sb5 is characterized by at least two
energy scales.

Figure 3: (a) Schematic FS in the surface BZ together with k cuts (cuts 1-6)
and k points (points 1-6) where the intensity in (b) and the EDCs in (c) were
obtained. Definition of the FS angle $\theta$ is also indicated. (b) ARPES
intensity measured at $T$ = 10 K along cuts 1-6 in (a). (c), (d) EDCs and
symmetrized EDCs at $T$ = 10 K, respectively, at points 1-6 in (a). (e) Plots
of the CDW-gap magnitude $\Delta_{\rm CDW}$ as a function of $\theta$.
Magenta, red, blue, and green circles represent the CDW gaps for the saddle-
point band, triangular FS centered at the K/H point, hexagonal FS centered at
the $\Gamma$/A point, and electron band at the $\Gamma$/A, respectively.

Throughout the whole 2D BZ, we have investigated the CDW gap for the saddle-
point and Dirac bands, and found that the gap is strongly k dependent. Figure
3(b) shows the ARPES intensity plotted along several k cuts shown as cuts 1-6
in the FS mapping in Fig. 3(a) which covers the kF points of the shallow
electron band (cut 1), the hexagonal pocket associated with the Dirac band
(cuts 2-4), and the electron pocket at $\Gamma$/A (cuts 5-6). The dip-
originated spectral weight suppression, which is a measure of the CDW gap
opening for the saddle-point band, is most prominent along cut 1 (MK/LH cut)
and gradually becomes less prominent on moving away from the M/L point, as
visible from a systematic change in the ARPES intensity along cuts 2-4. We
have extracted the EDCs at representative kF points on the Dirac band (points
1-4) in Fig. 3(c), and found that the hump gradually approaches $E_{\rm F}$ on
moving away from the M point. This signifies the strong k dependence of the
CDW gap. It is noted that the temperature-induced band energy shift in Fig.
2(h) is another measure of the gap size for the saddle-point band; one can
confirm that the band shift becomes smaller (i.e., the gap becomes smaller) as
moving away from the M point. The symmetrized EDC at $T$ = 10 K in Fig. 3(d)
further reveals that the Dirac band near $E_{\rm F}$ also shows anisotropic
CDW gap; the gap is $\sim$16 meV at point 1 but it vanishes at point 4 (along
the $\Gamma$K/AH line). Another important finding is that there exists no
obvious spectral anomaly for the $\Gamma$/A-centered electron pocket [see cuts
5 and 6 in Fig. 3(b)]. Also, the hump is absent at points 5 and 6 in Figs.
3(c) and 3(d). These results indicate that the CDW gap is absent in the
electron pocket at $\Gamma$/A. We have estimated the size of CDW gap
$\Delta_{\rm CDW}$ at $T$ = 10 K at various kF points and plotted the
$\Delta_{\rm CDW}$ as a function of FS angle $\theta$ in Fig. 3(e) (see Fig. 9
of Supplemental Material for a full data set SM ). As clearly seen, the CDW
gap is larger for the saddle-point band (maximally 70 meV) and smaller for the
Dirac band (maximally 18 meV), whereas it is absent on entire electron pocket
at $\Gamma$/A. Intriguingly, the CDW gap has a strong anisotropy and takes a
maximum around $\theta$ = 0∘ (along the $\Gamma$M/AL cut) and a minimum around
$\theta$ = $\pm$30∘ (along the $\Gamma$K/AH cut) for both the saddle-point and
Dirac bands.

The ARPES-determined CDW gap has a good correspondence to the unusual behavior
in the DOS observed by other spectroscopic techniques. STM/STS reported a
broad hump feature at $\sim$70 meV and V-shaped DOS within $\pm$10-20 meV of
$E_{\rm F}$ at low temperatures JiangAX2020 ; ChenAX2021 ; HZhaoAX2021 . These
features likely correspond to the CDW gaps for the saddle-point and Dirac
bands, respectively. STM/STS also found residual DOS at zero bias voltage, and
it could be contributed from the electron pocket at $\Gamma$ and the gapless
portion of the Dirac band. Recent optical spectroscopy of CsV3Sb5 revealed a
marked suppression of the Drude weight below $T_{\rm CDW}$ ZhouAX2021 . This
would be also related to the CDW gap for the saddle-point band because its
energy scale ($\Delta_{\rm CDW}$ $\sim$86.5 meV) is similar to the hump energy
in ARPES. Thus, our ARPES data definitely help interpreting characteristic
spectroscopic signatures seen by other experiments.

Our observation also sheds light on the mechanism of CDW and superconductivity
in $A$V3Sb5. Theoretical studies predicted that the inter-band scattering
between saddle points promote CDW or unconventional density waves TanAX2021 ;
FengAX2021 , but direct experimental proof was elusive. The observed large CDW
gap at the saddle-point band around the $\bar{\rm M}$ point supports this
prediction and suggests that the electron scattering via $Q$ = ($\pi$, 0)
connecting saddle points contributes largely to the energy gain for
stabilizing the CDW with in-plane 2$\times$2 component. To understand this
point in more detail, we plot a schematic of the reconstructed FS in Fig.4.
Original and folded FSs are indicated by solid and dashed curves,
respectively, with orbital-dependent colorings [original FSs were determined
by tracing the ARPES intensity at $E_{\rm F}$ in Fig. 4(a)]. Also, original
and folded saddle-point regions of the $\delta$ band are shown by magenta and
light magenta paintings, respectively. One can see that original and folded
saddle points around the M point overlap well, whereas the FSs along the
$\Gamma$K cut show a finite mismatch as marked by red arrows. Such a
difference in the band/FS overlaps via the 2$\times$2 $Q$ vector qualitatively
explains the observed CDW-gap anisotropy; the gap size is larger around the M
point due to the overlap of the original and folded saddle points. A remaining
issue is the origin of additional 3D component of CDW (with
2$\times$2$\times$2 periodicity), which is puzzling because the CDW vector
connects the M point to the L point where no saddle point exists. This
necessitates further $k_{z}$ selective experiments. It is also predicted that
the scattering between saddle points may promote unconventional
superconductivity, depending on the degree of competition with CDW
NandkishoreNP2012 . However, the observed large CDW gap suggests substantial
reduction of DOS responsible for the superconducting pairing on the saddle-
point band. Microscopic theories of superconductivity must be constructed
based on such a largely gapped saddle-point band and the CDW-gap nodes for the
Dirac bands. In this regard, one can even suggest that major superconducting
carriers could originate from the Sb $5p_{z}$ bands at the $\bar{\Gamma}$
point rather than the V $3d$ bands participating in the 2D kagome network.

Figure 4: (a) ARPES intensity at $E_{\rm F}$ measured at the $k_{z}$ $\sim$ 0
plane, together with the experimental FSs (solid curves) and the momentum
region where the saddle-point of the $\delta$ band is present (magenta). (b)
Original FSs extracted in (a) (solid curves) and their folded counterparts by
the 2$\times$2 CDW wave vector (dashed curves). Original and folded saddle-
point regions are shown by magenta and light magenta, respectively.

In conclusion, the present ARPES study of CsV3Sb5 has revealed two key
signatures in the band structure, (i) the energy gap on the multiple Dirac
points due to the SOC, and (ii) the modification of band structure associated
with the CDW-gap opening. We found that the CDW-induced band modification is
categorized into three types, the larger CDW gap for the saddle point around
the M point, the smaller CDW gap for the Dirac band, and the absence of CDW
gap for the electron band at $\Gamma$/A. The CDW gap also displays a strong
anisotropy. The present result opens a pathway toward understanding the
mechanisms of CDW and superconductivity in $A$V3Sb5.

Note Added: During the preparation of this manuscript, we became aware of
WangAX2021 , which also reports a CDW gap opening on CsV3Sb5.

###### Acknowledgements.

This work was supported by JST-CREST (No. JPMJCR18T1), JST-PRESTO (No.
JPMJPR18L7), and Grant-in-Aid for Scientific Research (JSPS KAKENHI Grant No.
JP17H01139, JP18H01160). The work at Beijing was supported by the National Key
R&D Program of China (Grant No. 2020YFA0308800), the Natural Science
Foundation of China (Grant No. 92065109), the Beijing Natural Science
Foundation (Grant No. Z190006), and the Beijing Institute of Technology (BIT)
Research Fund Program for Young Scholars (Grant No. 3180012222011). Z.W thanks
the Analysis & Testing Center at BIT for assistances in facility support.

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##  I Supplemental Material

###  I.1 Experimental condition and band-calculation method

High-quality single crystals of CsV3Sb5 were synthesized with the self-flux
method OrtizPRM2019 ; ZWangAX2021 . ARPES measurements were performed using
Scienta-Omicron SES2002, DA30, and SES2002 spectrometers at Tohoku University,
BL-28A, and BL-2A in Photon Factory (PF), KEK, respectively. We used the He
discharge lamp ($h\nu$ = 21.218 eV) at Tohoku University and energy tunable
photons of the vacuum ultraviolet (VUV) region ($h\nu$ = 85-125 eV) at BL-28A
and the soft-x-ray (SX) region ($h\nu$ = 230-350 eV) at BL-2A in PF. The
energy resolution ($\Delta E$) was set to be 7-30 meV at Tohoku University,
and 35-100 meV at PF. Specifically, the ARPES data shown in Figs. 2(e)-2(j),
3(b)-3(d), and Supplementary Figs. 8(e) and 9 were obtained at Tohoku
University with $\Delta E$ = 7 meV, those in Supplementary Figs. 8(a)-8(d)
were obtained at Tohoku University with $\Delta E$ = 30 meV, those in Figs. 1,
2(a)-2(d), 3(a), and Supplementary Figs. 5(a), 5(b), 6, 7, 8(f), and 8(g) were
obtained at BL-28A in PF with $\Delta E$ = 35 meV, and those in Figs. 5(c) and
8(h) were obtained at BL-2A in PF with $\Delta E$ = 100 meV. The angular
resolution was set to be 0.2∘ at Tohoku University as well as BL-2A in PF and
0.3∘ at BL-28A in PF. Crystals were cleaved in-situ in an ultrahigh vacuum of
$<$1$\times$10-10 Torr. $E_{\rm F}$ of the sample was referenced to that of a
gold film evaporated onto the sample holder. The first-principles band-
structure calculations were carried out using the full-potential linearized
augmented plane-wave method implemented in the WIEN2K code Blaha2013  with
generalized gradient approximation (GGA) PerdewPRL1996  and the Perdew-Burke-
Ernzerhof (PBE) PerdewPRL1997  type exchange-correlation potential. Spin-orbit
coupling was included self-consistently, and the lattice parameters were
directly obtained from experiments OrtizPRM2019 . k-points mesh for the
irreducible BZ is 17$\times$17$\times$10\. The Muffin-tin radii (RMT) are 2.50
a.u. for Cs and V and 2.60 a.u. for Sb, respectively. The maximum modulus for
the reciprocal vectors Kmax was chosen to satisfy RMT$\times$Kmax = 8.0.

Figure 5: (a) $h\nu$ dependence of EDCs for CsV3Sb5 measured in the normal-
emission set up at $T$ = 20 K. (b) Second-derivative intensity of EDCs plotted
as a function of $k_{z}$ and binding energy ($E_{\rm B}$) obtained with VUV
photons ($h\nu$ = 94-125 eV). (c) Same as (b) but obtained with the SX photons
($h\nu$ = 230-350 eV). (d) The calculated band structure along the $k_{z}$ cut
obtained from first-principles band-structure calculations.

###  I.2 Normal-emission measurement

Figure 5(a) shows energy distribution curves (EDCs) measured with synchrotron
radiation in the VUV region at the normal-emission set up, reproduced from
Fig. 1(b). One can recognize that the energy location of the $\alpha$-band
bottom shows a systematic variation as a function of $h\nu$ and such variation
is also visualized in the second-derivative intensity plot in Fig. 5(b). From
the periodicity of band dispersion together with the normal-emission ARPES
data obtained with SX photons [Fig. 5(c)], we have estimated the inner-
potential value to be $V_{0}$ = 10.0 eV.

###  I.3 SOC-induced gap opening on a Dirac band

Figures 6(a) and 6(b) show the magnified view of the APRES intensity around
the Dirac gap and the corresponding EDCs, respectively, in which the gapped
nature of Dirac cones is clearly visible.

Figure 6: (a) ARPES intensity for the Dirac-cone dispersions at the K (left)
and H (right) points. (b) Corresponding EDCs of (a). Blue triangles are a
guide for the eyes to trace the peak position of EDCs. Figure 7: (a)
Comparison of EDCs at $T$ = 20 K at the $k_{\rm F}$ point of the $\delta$ band
($k_{z}$ = 0; magenta) and the $\gamma$ band ($k_{z}$ = $\pi$; green). Black
and light blue lines at the binding energy of 0.2-0.3 eV are the linear fit
and those with an offset of $\pm$2$\sigma$, respectively. Inset shows the
ARPES intensity along the KMK cut at $T$ = 20 K. (b) EDCs measured along the k
cut crossing the $k_{\rm F}$ point of the $\delta$ band [blue arrow shown in
the inset to (a)]. The peak derived from the $\delta$ band is highlighted with
blue shade.

###  I.4 Hump structure in the CDW phase

In Fig. 7(a), we reproduce two EDCs from Fig. 2(c) in the main text; the
magenta and green curves were recorded at 20 K at $k_{z}$ = 0 and $\pi$,
respectively. The peak position of the magenta curve ($\sim$70 meV) is shifted
toward higher binding energy compared with that of the green curve ($\sim$40
meV), indicative of a CDW-gap opening at $k_{z}$ = 0 (note that the CDW gap at
$k_{z}$ = $\pi$ is not visible due to the limited resolution of synchrotron-
based ARPES). To confirm that the observed peak-energy shift is not due to
noise, we have evaluated the noise level of the spectrum. We have estimated
the standard deviation ($\sigma$) in the linear fit to a feature-less energy
region (black line) and obtained the noise level of $\pm$2$\sigma$ (see light
blue lines; note that the linear fit would give the upper limit for the noise
level because the inclusion of additional parameters commonly gives a better
fit). It is remarked here that the peak height of the magenta curve [roughly
10$\sigma$, see Fig. 7(a)], as defined by the difference in the spectral
intensity at 70 meV and 40 meV, is more than twice larger than the noise level
(4$\sigma$). This demonstrates that the observed peak-energy shift is not an
artifact due to noise, but reflects the CDW-gap opening. To further
corroborate the CDW-gap opening, we plot in Fig. 7(b) the EDCs measured at 20
K along a k cut including the $k_{\rm F}$ point [blue arrow in the inset to
Fig. 7(a)]. One can see that the $\delta$ band (blue shade) dispersing from
the high binding energy side reaches the binding energy of 70 meV at $k_{\rm
F}$ (magenta curve) and then almost flattens, in agreement with the CDW-gap
opening in the $\delta$ band.

Figure 8: (a) ARPES intensity measured at 25 K along the $\Gamma$KM/AHL line
with $h\nu$ = 21.2 eV. (b) Magnified view of the EDCs around the $\alpha$-band
bottom at the $\Gamma$/A point. (c) Second-derivative intensity around the
$\Gamma$/A point. Dashed curves in (a)-(c) are a guide for the eyes to trace
the $\alpha$-band dispersion. (d) Comparison of the normal-emission EDCs
measured with $h\nu$ = 106, 120, and 21.2 eV. (e)-(g) ARPES-intensity plots
measured along the MK/LH cut obtained with $h\nu$ = 21.2 eV, 120 eV, and 106
eV, respectively. Red and blue dashed curves are a guide for the eyes to trace
the band dispersions at $k_{z}$ = 0 and $\pi$, respectively. (h) ARPES
intensity measured along the $\Gamma$KM cut with $h\nu$ = 250 eV.

###  I.5 $k_{z}$-broadening effect

Figure 8(a) shows the ARPES intensity measured along the $\Gamma$KM/AHL cut
with the He-I$\alpha$ photons ($h\nu$ = 21.2 eV). One can identify electron-
like band ($\alpha$ band) centered at the $\Gamma$/A point. A closer look at
the EDCs around the $\Gamma$/A point reveals that the $\alpha$ band consists
of two branches each bottomed at $E_{\rm B}$ of $\sim$0.5 and $\sim$0.75 eV
[see Fig. 8(b); also see the second-derivative intensity around the $\Gamma$/A
point in Figs. 8(c)]. These branches are attributed to the $\alpha$ band at
the $k_{z}$ = 0 ($\Gamma$KM) plane and at the $k_{z}$ = $\pi$ (AHL) plane,
respectively, as supported by the observation of the $\alpha$-band bottom at
$E_{\rm B}$ $\sim$ 0.5 eV at $k_{z}$ = 0 and 0.75 eV at $k_{z}$ = $\pi$ [Fig.
8(d); also see Figs. 1(c) and 1(d)] which agrees well with the energy position
of the two peaks at $h\nu$ = 21.2 eV. Considering short photoelectron escape
depth (5-10 Å) of 21.2-eV photons and a resultant large $k_{z}$ broadening
effect, the ARPES data reflect the photoelectron signal averaged over a wide
$k_{z}$ area in the bulk BZ so that the band structures at $k_{z}$ = 0 and
$\pi$ planes which have a large contribution to the density of states are
simultaneously observed. Such a $k_{z}$ broadening effect is also seen around
the Brillouin-zone corner. Namely, the band dispersions observed along the
MK/LH cut with 21.2-eV photons can be explained by a simultaneous observation
of the bulk bands at $k_{z}$ = 0 and $\pi$ [see Figs. 8(e)-8(g)]. It is noted
that overall band dispersions in a wide energy range and their ARPES-intensity
distributions obtained with He-I$\alpha$ photons [Fig. 8(a)] also agree with
those for the bulk bands observed with SX photons [Fig. 8(h)] except for the
appearance of the second $\alpha$-band branch, supporting the bulk origin of
the observed bands. Here, we briefly comment on the amount of the $k_{z}$
dispersion. Since the $\alpha$-band bottom is located at $\sim$0.5 eV and
$\sim$0.75 eV at $k_{z}$ = 0 and $\pi$, respectively, the total $k_{z}$
dispersion is $\sim$0.25 eV in the experiment. This is smaller than the
calculated value of $\sim$0.4 eV, suggesting that the $k_{z}$ dispersion is
overestimated in the calculation.

Figure 9: (a) Plots of the CDW-gap magnitude $\Delta_{\rm CDW}$ as a function
of $\theta$. Red, blue/light blue, and green circles represent $\Delta_{\rm
CDW}$ for the saddle-point band, Dirac band, and the $\alpha$ band. (b)
Schematic of the Fermi surface together with the k locations where
$\Delta_{\rm CDW}$ was estimated (red, blue, and green circles). Definition of
$\theta$ is also shown. (c) ARPES intensity at 10 K along cuts A-L in (b).
(d), (e) EDCs and symmetrized EDCs at 10 K, respectively, at points 1-5. (f),
(g) Same as (d) and (e), respectively, but for points 6-16.

###  I.6

###  I.7 Full data set for the CDW-gap measurements

Figure 9(a) shows the gap plot reproduced from Fig. 3(e) but additionally
includes the numbering of the k points at which the gap size has been
estimated [see Fig. 9(b)]. Figure 9(c) shows the ARPES intensity plots at $T$
= 10 K along various k cuts with different Fermi surface angles ($\theta$)
[orange lines A-L in Fig. 9(b)]. One can see a general trend that the band
dispersion around the M point ($\theta$ = 0∘) is flattened by an opening of a
large CDW gap (e.g, cuts G, H, and J), while a relatively steep dispersion is
retained around the $\Gamma$K cut ($\theta$ = 30∘; e.g., E, F, and L) due to a
small CDW gap. Figures 9(d)-9(g) display EDCs measured at various $k_{\rm F}$
points indicated by red triangles in Fig. 9(c), for the $\alpha$ band [Fig.
9(d)] and the Dirac band [Fig. 9(f)], as well as the corresponding symmetrized
EDCs [Figs. 9(e) and 9(g) for the $\alpha$ and Dirac band, respectively]. For
the $\alpha$ band, one can recognize the absence of a clear gap opening
irrespective of $\theta$. For the Dirac band, blue curves in Figs. 9(e) and
9(f) were obtained on the large hexagonal Fermi surface centered at the
$\Gamma$/A point, whereas red ones were obtained on the triangular Fermi
surface centered at the K/H point. The gap size estimated on these Fermi
surfaces is almost the same in magnitude and shows a similar anisotropy; it
shows the maximum of 16-18 meV along the $\Gamma$M/AL line ($\theta$ = 0∘),
while it shows the minimum of 0-5 meV along the $\Gamma$K/AH line ($\theta$ =
30∘). EDCs for the saddle-point band are displayed in Fig. 2(g) of the main
text.