National Science Review

REVIEW 10: nwac199, 2023

https://doi.org/10.1093/nsr/nwac199
1
Beijing National Advance access publication 27 September 2022
Laboratory for
Condensed Matter
Physics and Institute of
Physics, Chinese PHYSICS
Academy of Sciences,
Beijing 100190, China;
2
School of Physical
Sciences, University of
Kagome superconductors AV3Sb5 (A = K, Rb, Cs)
Chinese Academy of
Sciences, Beijing Kun Jiang1,2 , Tao Wu3,4,∗ , Jia-Xin Yin5,∗ , Zhenyu Wang4 , M. Zahid Hasan6 ,
100190, China; 3 Hefei
National Laboratory for Stephen D. Wilson7 , Xianhui Chen3,4,∗ and Jiangping Hu1,8,∗

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Physical Sciences at the
Microscale, University
of Science and
Technology of China, ABSTRACT
Hefei 230026, China;
4 The quasi-two-dimensional kagome materials AV3 Sb5 (A = K, Rb, Cs) were found to be a prime example of
CAS Key Laboratory of
Strongly-coupled kagome superconductors, a new quantum platform to investigate the interplay between electron correlation
Quantum Matter effects, topology and geometric frustration. In this review, we report recent progress on the experimental
Physics, Department of and theoretical studies of AV3 Sb5 and provide a broad picture of this fast-developing field in order to
Physics, University of
Science and Technology
stimulate an expanded search for unconventional kagome superconductors. We review the electronic
of China, Hefei 230026, properties of AV3 Sb5 , the experimental measurements of the charge density wave state, evidence of
China; 5 Laboratory for time-reversal symmetry breaking and other potential hidden symmetry breaking in these materials. A
Quantum Emergence, variety of theoretical proposals and models that address the nature of the time-reversal symmetry breaking
Department of Physics,
Southern University of
are discussed. Finally, we review the superconducting properties of AV3 Sb5 , especially the potential pairing
Science and symmetries and the interplay between superconductivity and the charge density wave state.
Technology, Shenzhen
518055, China; Keywords: kagome superconductor, charge density wave, time-reversal symmetry breaking, topological
6
Laboratory for metal
Topological Quantum
Matter and Advanced
Spectroscopy (B7), INTRODUCTION Recently, fermionic models on kagome lattices
Department of Physics, have also become an important platform for study-
Princeton University, Unveiling new physics from simple lattice mod-
ing the interplay among electron-electron corre-
Princeton, NJ 08544, els plays a vital role in modern condensed matter
USA; 7 Materials lation effects, band topology and lattice geome-
physics. For instance, the exact solution of the two-
Department and try [13]. The point group of the kagome lattice is
California Nanosystems
dimensional (2D) Ising model on a square lattice by
the same as graphene [3], and a standard nearest-
Institute, University of Onsager revolutionized our view of phase transitions
neighbor tight-binding model on the kagome lat-
California Santa in statistical physics [1,2]; honeycomb lattice of
Barbara, Santa Barbara, tice exhibits Dirac cones at K points, as shown in
graphene can be used to mimic the physics of quan-
CA 93106, USA and Fig. 1(b). Many distinct properties associated with
8
Kavli Institute of
tum electrodynamics for Dirac fermions [3–5]. Mo-
Dirac√fermions [3] have been discussed, including
Theoretical Sciences, tivated by Onsager’s solution [1], the kagome lattice
the n B Landau level [14], tunable Dirac gaps
University of Chinese was introduced to statistical physics by Syozi [6],
Academy of Sciences, [15,16], Chern gaps [14] and the quantum anoma-
which serves as a rich lattice for realizing novel states
Beijing 100190, China lous Hall effect [17,18], etc. Besides its Dirac cones,
and phase behaviors [7–11]. As shown in Fig. 1(a),
a kagome lattice model can also display flat bands,
∗
a kagome lattice is formed by corner-sharing trian-
Corresponding as shown in Fig. 1(b). The flat band arises from the
authors. E-mails: gles. There are three sublattices labeled A, B, C, in-
destructive quantum interference of the wave func-
wutao@ustc.edu.cn; side each triangle forming the unit cell. Owing to
yinjx@sustech.edu; tions from each of the three sublattices. Studying ex-
this special lattice structure, the kagome lattice con-
chenxh@ustc.edu.cn; otic phenomena on flat bands, like fractional Chern
tains geometric frustration for spin systems, which
jphu@iphy.ac.cn insulator states, has been carried out both theoreti-
gives rise to extensively degenerate ground states in
cally and experimentally [19–25].
Received 23
the nearest-neighbor antiferromagnetic Heisenberg
In addition to these phenomena, supercon-
September 2021; model [12], as illustrated in Fig. 1(a). Accordingly,
ductivity in kagome lattice materials has also been
Revised 20 January the ground state of the kagome spin model is the
2022; Accepted 14 widely discussed. It has been argued that the kagome
most promising candidate for the long-sought quan-
February 2022 lattice can host a variety of unconventional pairing
tum spin liquid states [8–10].


C The Author(s) 2022. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. This is an Open Access article distributed under the terms of the Creative

Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original
work is properly cited.
 Natl Sci Rev, 2023, Vol. 10, nwac199

(a) (b
(b)
b)
as discussed later. Above and below the V-Sb plane,
out-of-plane Sb atoms form two honeycomb lattice
JSi·Sj planes respectively with lattice sites located above
and below the centers of the V triangles in the

E (k)
a2 kagome plane. A-site atoms form another triangular
C lattice above or below these Sb honeycomb or anti-
a1
monene planes.
A B We can first understand the electronic proper-
Γ K M Γ ties of AV3 Sb5 from the transport measurements.
The low-temperature electrical resistivity ρ(T) and
Figure 1. (a) The crystal structure for the kagome lattice, which originated from a its field dependence are plotted in Fig. 2(b) for

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Japanese basket-weaving pattern. The translation vectors are labeled a1 and a2 . In CsV3 Sb5 [31]. One finds that the zero field ρ(T)
each unit cell, there are three sublattices, labeled A, B, C. For the nearest-neighbor shows a broad transition towards the SC ground
Heisenberg model JSi · S j , the kagome lattice faces geometric frustration. As illus-
state with Tc ≈ 2.3 K, which is continuously sup-
trated in the upper corner, if two adjacent spins are set antiparallel, the third spin will
pressed by applying a magnetic field. The magneti-
face a dilemma. (b) Band structures for the nearest-neighbor tight-binding model on
the kagome lattice. zation data in Fig. 2(c) also reveals a well-defined
Meissner effect, and heat capacity measurements
show a sharp entropy anomaly at the SC transition
superconducting states, including the d + id chiral
[31]. Therefore, the CsV3 Sb5 becomes the first ex-
superconductor (SC) [26–28] and f-wave spin-
ample of quasi-2D kagome SCs. The critical field Hc
triplet SC [29], among others. However, super-
for CsV3 Sb5 is relatively small with the c direction
conducting kagome materials are rare in nature.
Hc2 ≈ 0.4T [40,41]. Similarly, the ρ(T) of KV3 Sb5
Last year, the newly discovered kagome material
drops to zero with Tc ≈ 0.93 K shown in Fig. 2(e)
CsV3 Sb5 [30] was found to be a quasi-2D kagome
[32] and RbV3 Sb5 has a Tc ≈ 0.75 K [33]. Hence, all
SC with a transition temperature Tc ≈ 2.3 K
AV3 Sb5 compounds within the material family are
[31]. Subsequently, superconductivity was also
superconducting at low temperature.
found across the entire family of compounds
Above the SC ground state, the normal states
KV3 Sb5 (Tc ≈ 0.93 K) [32] and RbV3 Sb5
of AV3 Sb5 also show quite different behavior. The
(Tc ≈ 0.75 K) [33]. This discovery has stimulated
temperature-dependent resistivity of KV3 Sb5 can be
extensive research activity in this field [30–39].
modeled by a Fermi-liquid formula ρ(T) = ρ 0 +
In this review, we discuss the recent progress
aT2 [30], which shows a typical metallic behavior.
in studying this newly discovered AV3 Sb5 kagome
The in-plane and out-plane resistivity data show a
family. This paper is organized as follows. We first
large anisotropy with a ratio α = ρ c /ρ ab ≈ 600
discuss the crystal structure and the electronic prop-
in CsV3 Sb5 , as shown in Fig. 2(f) [31]. This large
erties of AV3 Sb5 (A = K, Rb, Cs). Second, we
anisotropy agrees well with the quasi-2D nature of
review both the experimental evidence and theo-
AV3 Sb5 , where the V kagome layers play a dominant
retical understanding of the unconventional charge
role in the electronic properties. Hence, the AV3 Sb5
density wave order that forms and reports of accom-
is a quasi-2D metal. The resistivity ρ(T) also con-
panying time reversal symmetry breaking. Third, we
tains a kink behavior around 94 K, which is related
report the current status of understanding the SC
to the long-range charge-density wave (CDW) or-
properties of AV3 Sb5 . Finally, we address other un-
der discussed later [31]. A sharp peak from the heat
conventional features in these compounds, such as
capacity data at this same temperature indicates that
pairing density wave order, and provide future
the CDW transition is a first-order phase transition
research perspectives.
[31], where the first derivatives of free energy are not
continuous. The lack of phonon softening near this
transition from the inelastic x-ray scattering also sug-
CRYSTAL AND ELECTRONIC gests that the transition is weakly first order [42,43].
STRUCTURES It is worth mentioning that this weak first-order tran-
The AV3 Sb5 materials crystallize into the P6/mmm sition is best characterized in CsV3 Sb5 , and the na-
space group and exhibit a layered structure of ture of the transition merits further study in other
V-Sb sheets intercalated by K/Rb/Cs, as shown in compounds.
Fig. 2(a) and (b) [30]. In the V-Sb plane, three To reveal the electronic nature of AV3 Sb5 ,
V atoms form the kagome lattice and an additional density functional (DFT) calculations and
Sb atom forms a triangle lattice located at the V angle-resolved photoemission spectroscopy
kagome lattice’s hexagonal center. This V kagome (ARPES) measurements have been performed
layer largely dominates the physics behind AV3 Sb5 , [30,31,42,44–53]. The DFT calculations show

Page 2 of 16
 Natl Sci Rev, 2023, Vol. 10, nwac199

CsV3Sb5
(a) (b) 0 H (Oe) >2500 (c) 5 H (Oe) >250

0.6 0.0

3.17 Å 0.5 –0.2

4πχν (emu Oe–1 cm–3 )

2.76 Å
2.75 Å 0.4 –0.4

ρ (μΩ cm)
0.3 –0.6
0.2 –0.8
0.1 –1.0
b c
0.0 –1.2
a a 2.0 2.5 3.0 3.5 2.0 2.2 2.4 2.6 2.8 3.0

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c b T (K) T (K)
0 H (Oe) >500
(d) (e) (f) (g)
0.30 30 CsV3Sb5 250 CsV3Sb5
0.25 0 T, ρab 0 T ZFC
25 0 T, α–1 ρc 200 1 T ZFC
1.5

Cρ (J mol–1 K–1 )
0.20 20
ρ (μΩ cm)

150
0.15 1.0 15 94 K 94 K
0.10 c
100
10
0.5 b
a
0.05 5 50
ρc
α = ρ ~ 600
ab
0.00 0.0 0 0
0.5 1.0 1.5 0.6 0.8 1.0 1.2 0 50 100 150 200 250 300 0 50 100 150 200
T (K) T (K) T (K) T (K)

Figure 2. (a) The crystal structure for CsV3 Sb5 . Adapted from [31]. (b) and (c) Field-dependent resistivity and magnetization at low temperatures, showing
the onset of superconductivity for CsV3 Sb5 with Tc ≈ 2.3 K. Adapted from [31]. (d) Field-dependent resistivity at low temperatures for KV3 Sb5 . Adapted
from [32]. (e) Resistivity at low temperatures for RbV3 Sb5 . Adapted from [33]. (f) and (g) The temperature-dependent electrical resistivity, and heat
capacity (zero field cooled (ZFC)) at higher temperature for CsV3 Sb5 showing a transition around 94 K. Adapted from [31].

multiple bands crossing the Fermi level (EF ) in are effectively modeled as weakly correlated sys-
CsV3 Sb5 , as shown in Fig. 3(a). Around the  tems [58]. For example, the high-resolution ARPES
point, there is an electron-like parabolic band, data from KV3 Sb5 find excellent matching between
which originates from the in-plane Sb pz orbital. The the measured and calculated FSs [51], as plotted in
bands around the Brillouin zone (BZ) boundaries Fig. 3(d) and (e).
are mainly attributed to the V d orbitals. Note that Besides the above electronic structures, CsV3 Sb5
there are two van Hove (VH) points close to EF also carries a non-trivial Z2 topological index [31].
around the M point, which play an important role For inversion symmetric and time-reversal sym-
in the symmetry breaking observed in AV3 Sb5 . The metric systems, the Z2 topological invariant can
upper VH point is further connected with the Dirac be obtained from time-reversal invariant momen-
cone around the K point, which reflects a typical tum points with their inversion operator eigenval-
feature of the kagome model described above. ues [59]. As listed in Fig. 3(a), the Z2 invariant is
ARPES measurements show that the electronic non-trivial for band numbers 131, 133, 135 enumer-
band structure of CsV3 Sb5 qualitatively agrees with ated in DFT calculations. The parity index for 133,
DFT calculations [31], as shown in Fig. 3(b), and 135 bands at the M point is different, which gives
DFT calculations provide qualitatively accurate rise to a band inversion at M. Therefore, the normal
descriptions of the electronic structures of AV3 Sb5 state of CsV3 Sb5 is a Z2 topological metal, and this
systems. Note that there are still discrepancies be- Z2 topological property leads to a surface state em-
tween quantum oscillations and DFT calculations bedding around the bulk FS at the M point. ARPES
[52,54], which calls for future studies. experiments have resolved this feature, as shown in
To confirm the quasi-2D nature of AV3 Sb5 , the Fig. 3(f).
three-dimensional Fermi surface (FS) of CsV3 Sb5
is calculated in Fig. 3(c). The FSs show the tra-
ditional cylinder behaviors as in copper-based and
CHARGE-DENSITY WAVE AND
iron-based superconductors [55–57], which is the
origin of large resistivity anisotropy. The excellent SYMMETRY BREAKING
agreement between DFT and ARPES indicates a As discussed in the previous section, a CDW phase
small band renormalization owing to correlation ef- transition occurs for all AV3 Sb5 materials ranging
fects in the lattice. Hence, the AV3 Sb5 materials from 78 to 103 K (TCDW ≈ 94 K for CsV3 Sb5 ,

Page 3 of 16
 Natl Sci Rev, 2023, Vol. 10, nwac199

(a) T.B. parity prod. (b)

1.0 Band δΓ δA δL δM 0.0
135 –1 –1 –1 +1
133 –1 –1 +1 –1
131 +1 –1 –1 –1 –0.2
E (eV)

0.0

E (eV)
–0.4
A- +
L
H –0.6
–1.0 Γ- M-
K
–0.8
M K Γ K M M K Γ K M
Γ M K Γ A L H A LM KH

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(c) (d) (e) (f)
L H L H
1.0 1.0
α α TSSs
K K
β β 0 0
0.5 γ 0.5 γ
δ δ
Ky (1 / Å)

Ky (1 / Å)
Γ Γ

E (eV)
0 M 0 M
–0.2 –0.2
High
–0.5 –0.5
–0.4 –0.4
–1.0 Low –1.0
–1.0 –0.5 0 0.5 1.0 –1.0 –0.5 0 0.5 1.0 M M
Kx (1 / Å) Kx (1 / Å)

Figure 3. (a) The band structure of CsV3 Sb5 calculated by DFT. The insert shows the parity eigenvalues for each band at the time-reversal invariant
momentum points. Adapted from [31]. (b) ARPES measured band structure (left) and its comparison with DFT (right) for CsV3 Sb5 . Adapted from [31]. (c)
FS calculated for CsV3 Sb5 at experimental EF . Adapted from [52]. (d) and (e) FSs measured by ARPES and calculated by DFT for KV3 Sb5 . Adapted from
[51]. (f) The ARPES measured (left) and DFT-calculated (right) topological surface states (TSSs) for CsV3 Sb5 . Adapted from [48].

TCDW ≈ 103 K for RbV3 Sb5 , TCDW ≈ 78 K for of Fig. 4(b). The splitting of Knight shift Kc be-
KV3 Sb5 ) [30–33]. In the first report of AV3 Sb5 crys- tween V(I) and V(II) sites shows a sudden jump at
tal growth, elastic neutron scattering measurements TCDW . Beyond the surface sensitive measurements,
ruled out the possibility of long-range magnetic or- the CDW state is found to be three dimensional and
der [30]. The absence of long-range magnetic or- be modulated along the c axis. This modulation is ei-
der was further confirmed by the muon spin spec- ther 2 × 2 × 2 or 2 × 2 × 4 for AV3 Sb5 materi-
troscopy, indicating the transition derives primarily als with 2 × 2 × 2 reported for KV3 Sb5 and both
from the charge degree of freedom [37]. Soon after 2 × 2 × 2 and 2 × 2 × 4 reported for CsV3 Sb5
SC was discovered in CsV3 Sb5 , scanning tunneling [38,42,43,52], as shown in Fig. 4(g) and (h). Disor-
microscopy (STM) measurements were performed der along the c axis impacts crystallinity in the direc-
on the Sb and K surfaces of KV3 Sb5 , revealing that tion of the out-of-plane modulation and potentially
the transition is a CDW transition with 2 × 2 super- accounts for this discrepancy. The 3D modulation
lattice modulation [38,60–66]. From the STM to- is also confirmed by the STM data collected across
pographic spectrum in Fig. 4(a), the charge modu- surface step edges [62] and a 133 Cs NMR spectrum
lation on the Sb surface is resolved [38]. By Fourier study [67]. Future studies are underway to fully un-
transforming the topographic image, there are six derstand the c-axis periodicity of the superlattice. On
additional ordering peaks Q3Q in addition to those the clean surface regions of CsV3 Sb5 and RbV3 Sb5 ,
from the primary lattice structure [38]. STM fur- STM detects real-space modulations of the CDW
ther shows an energy gap opened around the Fermi gap, as shown in Fig. 4(g). Interestingly, the Fourier
energy of ∼50 meV, which together with the 2 × transform of the gap map also shows the 2 × 2 vec-
2 superlattice modulation disappears above TCDW tor peaks with different intensities, thus revealing a
[38,60–66]. Across this gap, there is a real-space novel electronic chirality of the CDW order [38,66].
charge reversal for the 2 × 2 superlattice modulation In order to determine the gap structures in
[38], which is a hallmark of CDW ordering. momentum space, several high-resolution ARPES
Nuclear magnetic resonance (NMR) measure- measurements have been performed [42,49–51,53].
ments further support the absence of magnetic or- Based on ARPES data, we can find that different FSs
der and confirm that the CDW transition is in- in KV3 Sb5 exhibit diverse CDW gap structures, as
deed a first-order transition [67]. From the NMR shown in Fig. 4(e). The CDW gap vanishes for the α
spectrum, there are two V signals after the CDW FS around the BZ  point. Since the α FS stems from
transition, V(I) and V(II), as shown in the inset the pz band of the in-plane Sb, the pz orbital does

Page 4 of 16
 Natl Sci Rev, 2023, Vol. 10, nwac199

(a) (b) (-1/2,1/2) (c) 0.12

QBragg
0.10
84.20 K
Q3Q 88.06 K
89.10 K
90.20 K 0.08

Intensity (a.u.)
90.74 K
91.20 K

ΔKc (%)
91.44 K
91.66 K 0.06
92.00 K
92.30 K
92.60 K 0.04
92.75 K
93.00 K
93.25 K
93.59 K 0.02 TCDW~94K
Cooling 93.68 K
93.73 K
1 nm 94.10 K 0.00
94.38 K

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134.70 134.75 134.80 134.85 134.90 134.95 135.00 0 20 40 60 80 100 120
(d) (e) (f)
0

Δ (meV)
25

E (eV)
0
-0.5
MG3
Γ K
α
β
M
γ δ
-1.0

K M K K M K
(g) (h)
Q=(3.5,0,0) Q=(3.5,0,0.5 )
20 Fourier transform of gap map
Gap HWHM ~ 2.7×10-3 r.l.u. HWHM ~ 2.5×10-3 r.l.u.
Counts (%)

15
size 150
10 (meV) 40
Counts per second
5 105 K 105 K
0 103 K 103 K
48
30 40 50 102 K 100 102 K
Gap (meV) 44 101 K 101 K
40 20 100 K 100 K
99 K 99 K
36 98 K 50 98 K
RbV3Sb5 RbV3Sb5
0 0
-0.02 0.00 0.02 0.48 0.50 0.52
L (r.l.u.) L (r.l.u.)

Figure 4. (a) A topographic image of a large Sb surface and its Fourier transformation showing a 2 × 2 modulation for
KV3 Sb5 from STM. Besides the Bragg peaks QBragg , there are additional charge modulation peaks Q3Q . Adapted from [38].
(b) The temperature dependence of the central transition lines of 51 V NMR with the temperature cooling across TCDW for
CsV3 Sb5 . Adapted from [67]. (c) Temperature dependence of the splitting of Knight shift Kc for CsV3 Sb5 . Adapted from [67].
(d) The STM scanning of the step edge in CsV3 Sb5 . The dashed lines track the chains with CDW modulation on the upper
side. A π -phase jump can be observed between the upper and lower sides. The illustration of the CDW patterns near a
single-unit-cell step is plotted in the lower panel. Adapted from [62]. (e) The CDW gap structures for each FSs in KV3 Sb5
measured by ARPES. Adapted from [51]. (f) ARPES measured band structures (right) and their second derivatives along K¯ -
M̄ - K¯ . There is one additional gap MG3 away from EF . Adapted from [51]. (g) Real-space CDW gap map for RbV3 Sb5 and its
Fourier transform. The 2 × 2 vector peaks show different intensities, defining a kind of electronic chirality. Adapted from
[66]. (h) The temperature-dependent CDW peaks of RbV3 Sb5 at Q = (3.5, 0, 0) and (3.5, 0, 0.5). The CDW peak at half-integer
L demonstrates a 3D CDW with 2 × 2 × 2 superstructure. Adapted from [42].

not participate in the CDW formation [50,51]. In that the structural transition plays an important role
contrast, the V-derived FSs around the BZ boundary in this CDW transition. It is also clear that the struc-
exhibit highly momentum-dependent CDW gaps, tural transition mostly affects the V kagome network,
which are dominated by quasiparticles around the while the out-of-plane coupling involving Sb pz or-
van Hove singularities at the M points [50,51]. bitals is hardly changed.
Quantum oscillation measurements also support the
dominant role of vanadium orbitals within the CDW
order [52]. Hence, the V kagome layer dominates Time-reversal symmetry breaking
the CDW gaps and the VH quasiparticles deeply Interestingly, accumulated evidence for time-
influence the gap structure in AV3 Sb5 . In addition reversal symmetry breaking (TRSB) signals was
to the gaps resolved around the FSs, ARPES data found in the CDW phases of AV3 Sb5 compounds.
in KV3 Sb5 have also observed a large CDW gap Since charge is a quantity to preserve time-reversal
opening away from EF [51]. For instance, at the M symmetry, the emergence of this TRSB becomes
point, a 125-meV gap opens at MG3 at 20 K, as one of the more intriguing phenomena in these
shown in Fig. 4(f). This feature strongly indicates otherwise non-magnetic AV3 Sb5 materials. The first
Page 5 of 16
 Natl Sci Rev, 2023, Vol. 10, nwac199

108
(a) (b) (c) Nd2(Mo1-xNbx)2O Gd film
Mn3Sn La1-xSrxCoO3
B=2 T B=-2 T 107 Mn3Ge MnSi
0.20 Fe3Sn2
106 Fe film

xx
~σ 2
High High Co3Sn2S2 Co film

Relaxation rates (ms-1)

105 K1-xV3Sb5 Ni film
TCDW

HE
σAHE (Ω-1 cm-1)
CsV3Sb5

σA
Low Low
Δ34 104 CsV3Sb5
(1.42 GPa)
0.10 Γ34 103 Cu1-xZnxCr2Se4
Γ12
102
101

100 Localized hopping Intrinsic Skew scattering

0
0 50 100 150 200 250 300 10-1 1
10 102 103 104 105 106 107 108
T (K) σxx (Ω-1 cm-1)

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Figure 5. (a) Spectroscopic 2 × 2 vector peaks for KV3 Sb5 taken at B = 2 T and B = −2 T. The highest vector peaks shift their positions under magnetic
field. Adapted from [38]. (b) The temperature-dependent muon relaxation rates in KV3 Sb5 . The  12 measures the rates collected in the forward and
backward detectors, while the  34 and 12 measure the rates collected in the up and down detectors. The relaxation rates start to increase below the
CDW transition. Adapted from [69]. (c) Plot of σ AHE versus σ xx for a variety of materials compared with CsV3 Sb5 spanning various regimes from the
localized hopping regime to the skew scattering regime. Adapted from [39].

evidence for TRSB was found in magnetic-field- temperature is slightly lower than TCDW ≈ 90 K. We
dependent STM measurements [38]. As discussed return to discuss the physical origin of this TRSB in
above, there are six CDW ordering vectors Q3Q the next section.
from the STM topographic spectrum. However, Moreover, a giant anomalous Hall effect (AHE)
the intensities of these three pairs of vectors are dif- has also been observed in AV3 Sb5 [35,39], and the
ferent in the clean regions for all AV3 Sb5 materials onset of this AHE was found to be concurrent with
[38,61,66], thus defining a chirality of the CDW the CDW order [39]. Normally, there are two ori-
order (counting direction from the lowest intensity gins of the AHE: intrinsic Berry curvature and extrin-
peak pairs to highest intensity peak pairs). The chi- sic impurity scattering [71]. As shown in Fig. 5(c),
rality of the CDW order further shows an unusual by comparing transverse σ AHE and longitudinal σ xx
response to the perturbation of external magnetic conductivity, both the intrinsic Berry curvature and
field B. As shown in Fig. 5(a), the chirality switch the impurity-induced skew scattering contribute to
from anticlockwise to clockwise when the magnetic the giant AHE in KV3 Sb5 and CsV3 Sb5 . However,
fields changes from +2 to −2 T applied along the compared to conventional spontaneous AHE with
c axis. Owing to the Onsager reciprocal relation, ferro- or ferrimagnetic ordering, the AHE in AV3 Sb5
the response functions of a time-reversal preserving exhibits σ AHE (B → 0) = 0 without a hysteresis be-
system under +B and −B must relate to each other havior. The σ AHE (B → 0) = 0 feature might orig-
by a time-reversal operator. This non-reciprocal inate from the anti-phase TRSB between adjoining
relation under magnetic field breaks the Onsager kagome layers or domain walls [70]. The origin of
relation indicating the TRSB in this non-magnetic this non-hysteresis anomalous Hall effect is still un-
kagome system [38]. However, we still want to clear, which deserves further careful study.
mention that this chirality signal is missing in recent We want to emphasize that the conclusive proof
STM and spin-polarized STM reports [63,68], of TRSB in AV3 Sb5 is still lacking. Besides the above
which deserves further investigations. μSR, and magnetic-field-dependent STM measure-
The straightforward evidence for TRSB comes ments, results from other TRSB sensitive techniques
from the zero-field muon spin relaxation/rotation like the polarized neutron diffraction and Kerr effect
(μSR) spectroscopy [69,70]. The spin-polarized are highly desired.
muons were implanted into the AV3 Sb5 single crys-
tals. The muon spin will rotate and relax under the
influence of local magnetic fields. The μSR tech- Spatial symmetries
nique is highly sensitive to the extremely small mag- Besides the translation symmetry breaking and
netic fields, capable of detecting of the order of time-reversal symmetry breaking associated with
0.1 Gauss fields experienced by the implanted the CDW state, an interesting question is: what
muons. As shown in Fig. 5(b), the relaxation rates are the remaining symmetries within the CDW
of KV3 Sb5 start to increase below the CDW transi- state? The point group of the AV3 Sb5 P6/mmm
tion temperature TCDW , which strongly suggests the space group is D6h , which can be generated by the
emergence of a local magnetic field owing to TRSB C6 rotation, inversion operator I and the mirror
[69]. Similar measurements on CsV3 Sb5 also found operator σ x about the y–z plane [72]. Although
TRSB signals [70]. However, the TRSB transition there is still some debate on what kind of spatial

Page 6 of 16
 Natl Sci Rev, 2023, Vol. 10, nwac199

(a) (0,1) (d) 200 Max (e)

4.1 THz 150 K
175
30
(1,0) 150

Temperature (K)

Intensity (a.u.)
125
20
100
(0.5,0) 75 1.3 THz
25.4 27.5
(-0.5,-0.5)
(0,-0.5) 50 10
25
3.1 THz
3K
(b) (c) 0
1.5 2.0 2.5 3.0 3.5 4.0 4.5
Min
Frequency (THz) 10 20 30
10 K 10 Energy (meV)
120 7K
4K
3K 0.4 T
(f) (g)
90
ρc (μΩ cm)

2.5 K

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H = 0.4 T

ρc (θ=90°)-ρc (θ=0°) (μΩ cm)

60 0
Q1Q
Q3Q
30 2.2 K

0 2K
-10 a H
5T
540 High
40 K q1
θ
ρc (μΩ cm)

520 H = 5 T
4a0
300 15 K -20
10 K QBragg
280 2K
qa
0.4 Å-1 qb qc
260 Low
-30
0 60 120 180 240 300 360 0 20 40 60 80 100
θ (degree) T (K)

Figure 6. (a) Spectroscopic 2 × 2 vector peaks for KV3 Sb5 taken at zero external field. Adapted from [38,43]. (b) Angular
dependent c-axis resistivity for CsV3 Sb5 measured at different temperatures under magnetic fields of 0.4 T (upper panel) and
◦
5 T (lower panel). Adapted from [40]. (c) Temperature dependence of nematicity of c-axis resistivity between θ = 0 and
◦
90 . Adapted from [40]. (d) Temperature dependence waterfall map of coherent phonon spectroscopy for CsV3 Sb5 . Adapted
from [76]. The 4.1-THz coherent phonon is present at all temperatures through phase change. The 1.3-THz phonon can only
be detected below TCDW , while the 3.1-THz phonon only shows up at temperatures below 30 K–60 K. (e) Raman spectroscopy
for KV3 Sb5 . Below 30 K, two new phonon modes at 25.4 and 27.5 meV are observed. Adapted from [42]. (f) and (g) The 1 ×
4 charge modulation and its Fourier transformation found in the Sb surfaces of CsV3 Sb5 . In (g), there are two Q1Q peaks in
addition to QBragg and Q3Q . Adapted from [61].

symmetry is broken at low temperatures, knowledge in Fig. 6(a) as a simulation of 2 × 2 vector peaks
of these generators provides a general outlook of the on the surface based on bulk 2 × 2 × 2 CDW.
remaining symmetries. Magnetoresistance measurements in CsV3 Sb5 also
To test the inversion symmetry I, second- reveal the nematic nature of the CDW state per-
harmonic generation (SHG) optical data were col- sisting into the superconducting phase [40,41], as
lected for CsV3 Sb5 [70]. SHG measures the second- shown in Fig. 6(b). Therefore, the CDW state is
order non-linear optical response P = 0 χ (2) EE, electronically nematic with only C2 rotation symme-
where P is the electric polarization induced by the try at low temperature. Note that the z-direction-
incident light with electric field E and 0 is the vac- modulated CDW reduces the point group symme-
uum permittivity. Since P and E are odd under in- try from D6h down to D2h [43,72]. However, from
version symmetry I, the rank-three non-linear op- the magnetoresistance data in Fig. 6(c), the onset
tical susceptibility tensor χ (2) is only finite when of electronic nematicity is around 15 to 60 K de-
parity is broken. Only negligibly small SHG signals pending on the magnetic field strength [40]. Hence,
(likely originating from the surface) were detected the electronic nematic transition seems to be sepa-
from 120 K down to 6 K. Hence, inversion symme- rated from the CDW transition at least in CsV3 Sb5 .
try I remains a valid symmetry for AV3 Sb5 at all tem- More than that, the signature of this nematic tran-
peratures, which constrains the CDW order and will sition can also be found in μSR, coherent phonon
also be important for the superconducting pairing spectroscopy and Raman spectroscopy [42,75,76].
possibilities discussed in the following section. The muon spin relaxation rate has a second fea-
Rotational symmetry breaking without transla- ture around T = 30 K in addition to the onset of
tional symmetry breaking, namely nematicity, is an- the primary TRSB transition [70]. Optical data per-
other important issue for understanding unconven- forming coherent phonon spectroscopy show that a
tional electron liquids [73,74]. For KV3 Sb5 , low- 3.1-THz peak appears below 30 K–60 K in addi-
temperature STM data above SC Tc at zero field tion to the 1.3-THz peak coupled to the onset of the
showed that the CDW peak intensities at Q3Q show CDW and 4.1-THz normal peaks [75,76], as shown
a C6 rotation broken feature [38,43,63], as shown in Fig. 6(d). Raman spectroscopy also reveals addi-

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 Natl Sci Rev, 2023, Vol. 10, nwac199

(a) (b) (c) (d)

6
4
Frequency (THz)

2
0
–2
–4
–6
Г M K Г A L H A
(e) (f) 1.5 (g) (h)
M1

0.1
1

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g2
M3 M2 0.5 2 1 Phase II
1

0.05
Qa Phase I
0
g3 g1
E (eV)

g3(0)
-0.5

0
Qb Qc 3 4 3
Γ DFT
–1 TB

–0.1 -0.05
M2 M3 Phase III
–1.5 1 2 1
z V Sbout
M1 –2
Y –0.1 -0.05 0 0.05 0.1
g4 –2.5
X
Sbin
g1(0)
Г M K Г

Figure 7. (a) Phonon spectrum calculated for CsV3 Sb5 . (b) and (c) Star of David and tri-hexagonal distortions for CsV3 Sb5 . Adapted from [52]. (d) 3D
structure distortion for AV3 Sb5 with a π shift between the adjacent kagome layers. (e) The low-energy effective theory of three VH points M1–3 for
AV3 Sb5 . The arrows denote the scattering processes described by interactions g1–4 . (f) Band structure for the minimal model for CsV3 Sb5 . (g) The flux
configuration for the chiral flux phase. (h) Renormalization group phase diagram for the effective model. Adapted from [82].

tional peaks below 30 K [42], as plotted in Fig. 6(e). perature is D2h . Since I is a good symmetry, there
A similar 40-K transition was also identified from the are only three possible point groups, D2h , C2h and Ci ,
NMR measurement [77]. Hence, it is highly pos- which calls for further experimental investigations to
sible that there is an electronic nematic transition determine the remaining symmetries, especially the
around 30 K–40 K in CsV3 Sb5 . bulk sensitive measurements.
Additionally, STM experiments show an in-plane
1 × 4 charge modulation below 50 K ∼ 60 K
[61,62,64], as shown in Fig. 6(f). From the Fourier
THEORIES AND MODELS
transform of STM topographs shown in Fig. 6(g),
there is one additional CDW peak (Q1Q ) appear- Theoretically, how one models and describes the
ing alongside the structural Bragg peaks (QBragg ) and AV3 Sb5 materials, especially their unconventional
2 × 2 CDW peaks (Q3Q ) [61]. Since similar 1 × CDW states, becomes a crucial question. As dis-
4 charge orders have been widely found in cuprates cussed above, DFT calculations qualitatively agree
[78–80], this 1 × 4 charge order has attracted con- with the electronic structures of AV3 Sb5 from
siderable attention. To date, however, bulk measure- ARPES measurements. Therefore, DFT calculations
ments such as x-ray scattering and NMR still fail to could provide a reasonable starting point for the un-
confirm this 1 × 4 order [81]. As it depends on the derstanding of AV3 Sb5 . Since the structural transi-
cleaved surface environment [42,43,52,67], this 1 × tion is found to play a vital role in the CDW for-
4 charge order may come from a surface manifesta- mation, the most stable structural distortion can be
tion of the intermediate 30–60 K transition, which probed by DFT. For example, in CsV3 Sb5 , phonon
is supported by the DFT calculations [66]. On the dispersion relations are calculated from the ab initio
other hand, we should note that observing diffuse DFT calculations shown in Fig. 7(a) [83]. From the
quasi-1D correlations in a system that has three such phonon modes, one finds that there are two nega-
domains is very challenging in conventional x-ray tive energy soft modes around the M and L points.
measurements, which calls for further exploration. The structural instabilities led by these soft modes,
For the mirror symmetry, there is still a lack of the ‘Star of David’ (SoD) and ‘tri-hexagonal’ (TrH)
conclusive evidence for its existence or absence at structure configurations are proposed to be the likely
low temperatures. For example, the STM data in candidates for CDW structures [52,75,83], as illus-
[38] breaks all the mirror symmetries, while another trated in Fig. 7(b) and (c). Note that TrH is also
measurement shows one remaining mirror symme- named the ‘inverse Star of David’ in the literature.
try in [63]. However, using the knowledge discussed Based on XRD data, STM and quantum oscillation
above, the largest point group of AV3 Sb5 at low tem- measurements, the TrH state is suggested to be the

Page 8 of 16
 Natl Sci Rev, 2023, Vol. 10, nwac199

promising ground-state configuration below TCDW in Cu square plaquettes. Both states break the time-
in a single-layer model. To accomplish the 2 × 2 × 2 reversal symmetry and are candidates for the pseu-
structure modulation, a π shift between the adjacent dogap in cuprates [57,89–92].
kagome layer TrH distortions is needed [43,75], as For kagome lattices and other hexagonal lattices,
illustrated in Fig. 7(d). On the other hand, recent the 3Q electronic instabilities at VH filling have been
studies have suggested that the average structure widely discussed [26,28,29,93–99], including chi-
shows signatures of both TrH and SoD structures ral spin density wave order, charge bond orders,
in the staggered layer sequence [52], which calls for intra-unit cell CDW and d+id SC, etc. Based on
further investigation. the minimal model and the 3Q electronic instabil-
Beyond the structural transition, a model that ities, several TRSB flux states have been proposed
captures the electronic properties of AV3 Sb5 is im- to explain the TRSB. The most promising candidate

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portant. DFT calculations and ARPES measure- is the chiral flux phase among the 18 flux classes
ments show that multiple bands cross the Fermi level [72,82,84,100,101]. In this chiral flux state shown in
[30,31]. As discussed above, the in-plane Sb pz or- Fig. 7(g), there are two special flux loops. The two
bital forms one electron pocket around the  point anti-clockwise triangle current flux loops (red cir-
and the V d orbitals form multiple FSs around the cles) form a honeycomb lattice and the clockwise
M points, as illustrated in Fig. 7(e) [84]. It is very dif- hexagonal current flux (blue circle) forms a triangu-
ficult to capture such a complicated Fermi surface to- lar lattice. The charge order of the chiral flux phase
pography in a simplified tight-binding model. How- coincides with 2 × 2 charge order and the TrH lat-
ever, the essential electronic structure of AV3 Sb5 is tice configuration [84].
widely believed to be dominated by the quasiparti- Microscopically, how to stabilize the flux state
cles around the VH points based on the following is still under debate. Starting from the VH points,
facts. First, the VH points are very close to the Fermi the low-energy effective theory of AV3 Sb5 can be
level as obtained from DFT calculations and ARPES constructed by projection [72,82], as illustrated
measurements [31,50,51]. Second, the quasiparti- in Fig. 7(g). Using the parquet renormalization
cle interference spectrum shows that the dominant group, various leading and subleading instabilities
scattering momenta are 3Q (Qa , Qb , Qc ) related to have been determined, including superconductivity,
three M points as well as the -point FS-induced charge order, orbital moment and spin density waves
q1 scattering [61,62], as illustrated in Fig. 7(e). Fi- [82]. For example, a renormalization group phase
nally, the CDW gap size is at maximum around the diagram is shown in Fig. 7(h) when the bare inter-
VH points while its vanishes at the  pocket [50,51]. action is g2 > 0. There are three possible phases, I,
Therefore, a minimal model capturing the VH points II and III. Although both the leading and sublead-
and -point FS could faithfully describe the physics ing instabilities have been discussed in this work, we
behind AV3 Sb5 [52]. Following this spirit, a mini- only focus on the leading one. Among these three
mal four-band model based on the V local d X 2 −Y 2 phases, the leading instability of phase II is the ‘imag-
orbital and in-plane Sb pz orbital is proposed, as inary charge-density wave’, which is the low-energy
shown in Fig. 7(f) [85]. And the V local d X 2 −Y 2 or- version of the flux phase. In this case, we find that
bital model is adiabatically connected to the nearest- the TRSB phase can be stabilized if the bare interac-
neighbor tight-binding model in the kagome lattice. tion g1 is negative and g2 , g3 , g4 are positive. But how
This model provides a solid ground for further theo- to achieve attractive interactions needs to be further
retical investigation. explored [82]. An extended Hubbard model with
The most intriguing property of the AV3 Sb5 on-site Hubbard interaction U and nearest-neighbor
CDW is its TRSB. However, neutron scattering, Coulomb interaction V is also proposed to stabilize
NMR and μSR experiments have already ruled out the TRSB order [84,100]. However, the TRSB or-
the possibility of long-range magnetic order with der has not been found in the realistic parameter re-
conventional moments in the resolution of the mea- gion in this type of model. Phenomenologically, the
surements [30,37,69,70]. This feature is reminiscent various Ginzburg Landau theory approaches have
of long-discussed flux phases in condensed matter, also been discussed to describe the TRSB phases
such as the Haldane model on the honeycomb lat- [82,100,101].
tice [86]. Moreover, the flux phases breaking TRSB
are also widely discussed in cuprate superconduc-
tors after the seminal study by Affleck and Marston SUPERCONDUCTIVITY
in t–J models [87,88]. Generalizing this idea, Varma Superconductivity remains an important property
[89] proposed a loop-current phase formed in the of AV3 Sb5 materials. We focus on discussing the
Cu-O triangles and Chakravarty et al. [90] pro- superconducting mechanism and pairing symme-
posed the d-density wave state with staggered flux try. Whether an SC is driven by electron-phonon

Page 9 of 16
 Natl Sci Rev, 2023, Vol. 10, nwac199

(a) 1 (b) (c) 1.0 1.0

0.8
0.24
0.8

ρs
0.9

(1/T1T) (s–1 K –1 )
0.7

(1/T1T) (s–1 K–1)

0.21
0
ΔKc (%)

0.6 0.8
0.6 TC 0.18 #A-3 H // c 0.0 0.1 0.2 0.3 0.4

ρs
λ(0)=387 nm T/TC
0.4
0.5

121

123
0.15 One gap
–1 Tc
Two gap
0.2 p wave
0.4 0.12 d wave
0 2 4 6 8 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0
T (K) T (K) 0.2 0.4 0.6 0.8 1.0 1.2
(d) 1.2 (e) T/TC
2.0

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TI-2201 CsV3Sb5 (f) (g)
1.0 NbSe2 InBi
Nb Δ3
1.5
0.8
[κ0/T]/[κN0/T]

dI/dV (a.u.)
1.0

dI/dV (a.u.)
0.6 1.0
Δ1 Al
0.4 0.8
Δ2 Half Cs
0.5 (V gap)
0.2 Half Cs 100 nm 0.6
(U gap)
–1 0 1
0.0 0 L H
0.0 0.2 0.4 0.6 0.8 1.0 –1.0 –0.5 0 0.5 1.0 E (meV)
H/Hc2 E (meV)

Figure 8. (a) Temperature dependence of the Knight shift K of 121 Sb for CsV3 Sb5 with H//c. Adapted from [102]. (b) Temperature dependence of
121
(1/T1 T) (left axis) and 123 (1/T1 T) (right axis). A Hebel-Slichter coherence peak appears just below Tc for CsV3 Sb5 . Adapted from [102]. (c) The normal-
ized superfluid density ρ s for CsV3 Sb5 as a function of the reduced temperature T/Tc . Adapted from [107]. The dash-dot-dot, solid, dashed and dash-dot
lines respectively represent fits to models with a single s-wave gap, two s-wave gaps, a p-wave gap and a d-wave gap. The inset is en enlargement
of the low-temperature region. (d) The normalized residual linear term κ 0 /T of CsV3 Sb5 as a function of H/Hc2 . Similar data for Nb, InBi, NbSe2 and
an overdoped d-wave cuprate superconductor Tl-2201 are shown for comparison. Adapted from [34]. (e) Two kinds of superconducting gap spectra
observed on the half-Cs surface for CsV3 Sb5 . Adapted from [60]. (f) The dI/dV map showing a superconducting vortex on the Cs surface for CsV3 Sb5 .
Adapted from [62]. (g) Tunneling spectra obtained in the vortex core (red) with zero-bias peak and outside the vortex (dark blue). Adapted from [62].

coupling, or unconventionally driven by electron- The superconducting gap structure can also pro-
electron correlation, is the central issue we need to vide information about the pairing symmetry. A
address. To find clues for this hard-core question, Hebel-Slichter coherence peak appears just below
we first focus on the superconducting pairing sym- Tc in CsV3 Sb5 from the spin-lattice relaxation mea-
metries of AV3 Sb5 . Since the inversion symmetry surement of the 121/123 Sb nuclear quadrupole reso-
I is always a good symmetry for AV3 Sb5 , as found nance [102], as shown in Fig. 8(b). This coherence
in SHG measurements [70], the spin-singlet pairing peak is widely known as a hallmark for a gapped
and spin-triplet pairing must be separated. conventional s-wave SC [105,106]. Moreover, an
To reveal the pairing properties, multiple exper- exponential temperature dependence of magnetic
imental techniques have been applied. The first task penetration depth is found at low temperatures, sug-
is to determine whether the Cooper pairs form a sin- gesting a nodeless superconducting gap structure for
glet or triplet, which can be determined through the CsV3 Sb5 [104,107], as shown in Fig. 8(c). No sub-
temperature-dependent spin susceptibility. From gap resonance state is found near non-magnetic im-
the NMR spectrum shown in Fig. 8(a), one finds purities, while the magnetic impurities destroy the
that the temperature-dependent z-direction Knight SC quite efficiently from STM measurements [60].
shift of 121 Sb drops below the SC transition Tc Hence, the SC of AV3 Sb5 is a conventional spin-
in CsV3 Sb5 [102]. The Knight shifts in the other singlet SC. This feature is also consistent with the
two directions also show a similar drop below Tc weakly correlated nature of AV3 Sb5 and remarkable
[102]. Therefore, the ground state of AV3 Sb5 be- electron-phonon coupling of the V-derived bands
longs to a spin-singlet SC. Additionally, the μSR found from ARPES [51].
measurements fail to detect any additional TRSB However, this simple picture is complicated by
signals below Tc , compared to the distinct in- experimental observation of nodes or deep min-
crease in the Sr2 RuO4 SC [103], suggesting a time- ima in the superconducting gap. From thermal
reversal invariant superconducting order parameter transport measurements, a finite residual thermal
[69,70,104]. Therefore, the SC order parameter of conductivity κ 0 at T → 0 has been found in
AV3 Sb5 belongs to the time-reversal preserved spin CsV3 Sb5 , which suggests a nodal feature of the pair-
singlet. ing order parameter [34,108]. This residual thermal

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 Natl Sci Rev, 2023, Vol. 10, nwac199

conductivity κ 0 also shows a similar magnetic field In addition, CsV3 Sb5 may host an intriguing
dependence found in a d-wave cuprate, as shown in electronic state, known as the pair density wave
Fig. 8(d). Additionally, a multiple-gap feature is re- (PDW), in which the Cooper-pair density modu-
solved from the millikelvin STM measurements, as lates spatially at a characteristic wave vector. A low-
shown in Fig. 8(e). The multi-gap behavior agrees temperature STM study on CsV3 Sb5 found that
with the multiple FSs revealed from the DFT calcula- both the height of the superconducting coherence
tions and the ARPES measurement. Interestingly, in peak and the zero-energy gap depth show spatial
different regions of CsV3 Sb5 , both the U-shaped and modulations with a distinct periodicity of 4a/3, sug-
V-shaped suppression of the density of states have gesting a PDW state [64]. In the Fourier transforms
been observed at the Fermi level with a relatively of the differential conductance maps taken inside the
large residual density of state that can hardly be ex- superconducting gap, six additional Q4/3a modula-

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plained by thermal excitations [60,64]. These find- tion peaks were found in addition to the 2 × 2 CDW
ings, on the other hand, prefer a superconducting peaks Q3Q , 1 × 4 CDW peaks Q1Q and Bragg peaks
gap with nodes. shown in Fig. 9(a) and (b). Four of these additional
This leads to a seeming dichotomy between gap- Q4/3a vectors cannot be obtained by linear combi-
less excitations in the SC state and a convention- nations of Q3Q and Q1Q peaks, which provides evi-
ally gapped s-wave SC for AV3 Sb5 . However, if we dence for the PDW in AV3 Sb5 [64].
take the TRSB normal states into account, the gap- As the superconductivity in AV3 Sb5 arises within
less excitations arise within a fully opened supercon- the pre-existing CDW states, exploring the corre-
ducting gap [85]. There are two key discrete sym- lation between these two states can help to reveal
metries in SCs to guarantee the presence of Cooper the underlying physics [122–128]. By applying ex-
pairing: time-reversal T and inversion symmetry I ternal pressure to CsV3 Sb5 , CDW order becomes
[109–111]. For the even-parity spin-singlet pairing destabilized quickly and vanishes at 2 GPa, while
formed by (ck, ↑ c−k, ↓ − ck, ↓ c−k, ↑ ), the system at the SC state shows a double-peak behavior with a
least contains time-reversal symmetry T because T maximum of 8 K around 2 GPa [122,123], as plot-
maps a |k, ↑ state to a | −k, ↓ state. Similarly, ted in Fig. 9(c). The competition between the CDW
the odd-parity, spin-triplet pairing needs inversion and SC is a common feature of all AV3 Sb5 ma-
symmetry I owing to the fact that I maps a |k, ↑ terials, while the double-peak behavior is clearest
state to a | −k, ↑ state. These two symmetry condi- in CsV3 Sb5 [127]. Hence, the CDW order highly
tions are known as Anderson’s theorem [109–111]. correlates with the SC in the low-pressure region,
For AV3 Sb5 SC cases, the normal state before the known as SC I. By further increasing the pressure, a
SC transition breaks the T symmetry as discussed new SC dome, named SC II, appears for all AV3 Sb5
above. Therefore, the edge modes on CDW do- materials, as shown in Fig. 9(d). A recent DFT cal-
main walls or other places where the TRSB domi- culation with electron-phonon coupling shows that
nates cannot be gapped out by the SC pairing. These the Tc calculated from the McMillan-Allen-Dynes
gapless excitations could contribute a finite residual formula qualitatively agrees with the experimental
thermal conductivity. values obtained above 20 GPa [129], as shown
Although SC seems to be conventional, the in Fig. 9(e). Hence, the SC-II state at high pres-
non-trivial band structure of AV3 Sb5 could lead to sure likely stems from the electron-phonon cou-
non-trivial excitations. Based on Fu-Kane’s semi- pling. However, the Tc calculated based on electron-
nal proposal, if the helical Dirac surface states of a phonon coupling in the low-pressure range is far
topological insulator are in proximity to an s-wave above the experimental values, which cannot give
SC, Majorana zero modes (MZMs) may arise in- rise to a reliable conclusion. The underlying pairing
side the vortex cores of the superconducting Dirac mechanism for AV3 Sb5 needs more experimental ex-
surface states [112]. The proposal has been widely ploration and theoretical analysis.
used in Bi2 Te3 /NbSe2 heterostructures, and in the
iron-based SC Fe(Te,Se), (Li1 − x Fex )OHFeSe, etc. SUMMARY AND PERSPECTIVE
[113–121]. Similar to these aforementioned mate-
rials, AV3 Sb5 hosts Dirac surface states near the In this article, we have reviewed the physical prop-
Fermi energy [31] that can open a superconduct- erties of the newly discovered kagome materials
ing gap below Tc . Therefore, MZMs are theorized AV3 Sb5 . Owing to tremendous efforts during the
to emerge inside the vortex core. Using STM, zero- past years, we have achieved considerable under-
bias states with spatial evolution similar to the zero- standing of AV3 Sb5 , which can be summarized as fol-
bias peaks in Bi2 Te3 /NbSe2 heterostructures have lows.
been resolved in the vortex cores of CsV3 Sb5 [62], r AV3 Sb5 is a quasi-2D electronic system with
as shown in Fig. 8(f) and (g). cylindrical Fermi surfaces, where the electronic

Page 11 of 16
 Natl Sci Rev, 2023, Vol. 10, nwac199

6
(a) (c) (f) CsV3Sb5 Tc
10%
High 90 5 10%
RbV3Sb5 Tc
80 210
KV3Sb5 Tc10%
Q3Q MR (%) Metal 4
60

T (K)

TC (K)
TCDW S1 (PCC)

Intensity (a.u.)
3
30 CsV3Sb5
Q1Q CDW
SC 2
0 1
qx (d) RbV3Sb5
10 onset zero
KV3Sb5
P1 P2 Tc Tc S1 (PCC) 0
Low onset zero 0 10 20 30 40 50
qy QBragg π/a0 Tc Tc S2 (DAC) P (GPa)
M1 M2
15

Tc (K)
Tc Tc S3 (MPMS)

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5 (g) Cal. (μ* = 0.05)
(b)
High Cal. (μ* = 0.10)
SC Exp. Zhang et al.
10
Q4a/3 0 Exp. Chen et al.
(e) Exp. Zhu et al.

TC (K)
Intensity (a.u.)

4 S1 (PCC)
μ0Hc2(0) (T)

S2 (DAC)
5
2

qx SC I SC II
0 0
Low 0 2 4 6 8 0 10 20 30 40
qy π/a0 P (GPa) P (GPa)

Figure 9. (a) Fourier transformation of atomically resolved STM topography of the Sb surface for CsV3 Sb5 . (b) The dI/dV map at −0.25 meV for CsV3 Sb5
at Telectron = 300 mK. Comparing to (a), there are additional peaks at Q4/3a . Adapted from [64]. (c) Phase diagram for CsV3 Sb5 with pressure. CDW
transition temperature TCDW gradually suppressed with increasing pressure. The color inside the CDW represents the magnitude of magnetoresistance
measured at 9 T and 10 K. (d) Pressure dependence of superconducting transition temperatures showing two dome behavior. (e) Pressure dependence
of the upper critical field at zero temperature. Adapted from [123]. (f) Temperature-pressure phase diagram of AV3 Sb5 . Adapted from [127]. (g) Electron-
phonon calculated Tc for CsV3 Sb5 and its comparison with experiments. Adapted from [129].

properties are dominated by the V-Sb kagome excitations inside the vortex core. The CDW order
layers. is intertwined with the SC in an unconventional
r AV3 Sb5 is a multi-band system with at least four way, inducing multiple SC domes under pressure.
bands crossing the Fermi level. The FS around the The discovery of the AV3 Sb5 SC opens a new
 point is attributed to the Sb pz bands, while FSs route towards realizing unconventional orders
around the BZ boundary mainly consist of V d or- within 2D kagome metals, which brings us a new
bitals. The VH points at the M points play an im- platform to investigate the interplay between
portant role in the unconventional properties of correlation, topology and geometric frustration.
AV3 Sb5 . We hope that this review provides a broad picture
r Owing to band inversions at M points, AV3 Sb5 is a of the recent progress on AV3 Sb5 kagome mate-
Z2 topological metal with unconventional surface rials and stimulates new research frontiers within
states. kagome-related physics.
r The correlation strength of AV3 Sb5 is weak based
on DFT calculations and ARPES measurements.
r AV3 Sb5 undergoes a first-order phase transition ACKNOWLEDGEMENTS
into charge density wave order around 80 to
We thank Hechang Lei, Hu Miao, Jianjun Ying, Xingjiang Zhou,
104 K, depending on the A-site cation. Within the
Junfeng He, Shancai Wang, Li Yu, Xiaoli Dong, Fang Zhou, Yan
kagome layer, the CDW enlarges the unit cell to 2
Zhang, Nanling Wang, Huan Yang, Haihu Wen, He Zhao, Ilija
× 2 accompanied by a c-axis modulation. Zeljkovic, Binghai Yan, Ziqiang Wang, Zheng Li, Jianlin Luo, Yu
r There is evidence for the emergence of time- Song, Huiqiu Yuan, Shiyan Li, Yajun Yan, Donglai Feng, Hui
reversal symmetry breaking inside the CDW Chen, Geng Li, Hongjun Gao, Rui Zhou, etc. for useful discus-
state. Besides translational symmetry breaking sions. We also thank Yuhao Gu and Yuxing Wang for help with
and time-reversal symmetry breaking, inversion the DFT calculations.
symmetry perseveres while C6 rotation symmetry
is broken.
r The superconducting order parameter of the FUNDING
AV3 Sb5 SC is a spin singlet with Tc around 1–3 K, This work is supported by the National Key Basic Re-
depending on the A-site cation. The SC appears search Program of China (2017YFA0303100), the
to be a conventional s-wave with unconventional National Natural Science Foundation of China (NSFC-

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11888101) and the Strategic Priority Research Program 16. Ye L, Kang M and Liu J et al. Massive Dirac fermions in a fer-
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