Unusual charge density wave and superconductivity in kagome metals AV₃Sb₅ (A = K, Rb, Cs)

Abstract

The kagome metals AV₃Sb₅ (A = K, Rb, Cs) exhibit a host of intriguing phenomena, including a Z₂ nontrivial topological band structure, unconventional superconductivity, and charge density wave (CDW) states. These materials offer a unique platform to explore the interplay between lattice geometry, band topology, and electronic correlations. In this review, we focus on CDW, superconductivity, and the interactions between them in AV₃Sb₅. Additionally, we highlight several unresolved issues, such as the potential time-reversal symmetry breaking of 2×2 CDW, the nature of superconductivity (whether conventional or unconventional), and the underlying mechanism of CDW formation. We also summarize relevant experimental and theoretical evidence related to these topics. Our aim is to provide researchers with essential insights into CDW, superconductivity, and the outstanding questions surrounding AV₃Sb₅ for future investigations.

Graphical abstract

Keywords

kagome metal / charge density wave / superconductivity / pair density wave / vortex state

1 Introduction

The kagome lattice, composed of corner-sharing triangles, features a range of nontrivial characteristics, including van Hove singularities (VHSs), Dirac cones, and flat bands in its band structure [1-3]. These properties makes kagome lattice materials host a variety of rich and exotic electronic states, such as quantum spin liquid [4, 5], spin density wave [6], charge density wave (CDW) [6, 7], and superconductivity [6, 8, 9]. Due to these excellent properties, various kagome materials have been extensively studied, including Dirac Metals (CoSn [10], FeSn [11], and Fe₃Sn₂ [12]), Weyl semimetals (Mn₃Ge [13], Mn₃Sn [14], and Co₃Sn₂S₂ [15]) and strongly correlated insulating kagome compounds (ZnCu₃(OH)₆Cl₂ [16, 17]).

In recent years, the AV₃Sb₅ family (A = K, Rb, Cs), a new class of kagome metals, has garnered significant attention due to its excellent properties, such as the Z₂ nontrivial topological band structure [18], the anomalous Hall effect (AHE) [19, 20], unconventional superconductivity [21, 22] and CDW [23-27]. These intriguing characteristics make AV₃Sb₅ an ideal platform for studying correlated electronic states. However, the mechanisms of CDW and superconductivity in AV₃Sb₅ remain uncertain and require further investigation. The origin of the CDW in AV₃Sb₅ has mainly been attributed to two scenarios: one is an electronic instability driven by Fermi surface nesting associated with VHSs [26, 28], and the other is electron-phonon coupling [22, 29]. Regarding the nature of superconductivity in AV₃Sb₅, experimental evidence has revealed several possible unconventional superconducting signatures, including Majorana bound states [30], pair density wave (PDW) [22], time-reversal symmetry breaking (TRSB) [31, 32], and nodal gap characteristics [33, 34]. Nevertheless, some studies have also reported results consistent with conventional s-wave superconductivity in AV₃Sb₅ [35, 36].

Scanning tunneling microscopy (STM), a powerful experimental technique for investigating the properties of AV₃Sb₅, offers atomic-scale spatial resolution and high energy resolution. It enables direct visualization of the atomic lattice and the spatial distribution of electronic states. However, STM measurements are limited to surface information and are highly sensitive to the condition of the probe tip, which can significantly influence the experimental results. In this review, we discuss recent achievements in the crystal and band structures, CDW, superconductivity, and the interplay between CDW and superconductivity in AV₃Sb₅, primarily based on STM studies, followed by other experimental technique measurement results. First, we provide an overview of the fundamental aspects of the crystal and band structures in AV₃Sb₅. Next, we explore the 2×2 CDW and 1×4 CDW phases, summarizing the ongoing debates regarding the magnetic response and potential TRSB in the 2×2 CDW, along with controversial experimental evidence. Third, we discuss superconductivity in AV₃Sb₅, focusing on the evidence for superconductivity, PDW, and Majorana bound states. The nature of superconductivity (whether conventional or unconventional) remains uncertain, and we summarize the relevant experimental and theoretical evidence surrounding this controversy. Fourth, we introduce the interplay between CDW and superconductivity. Finally, we highlight key unresolved issues and provide essential information for researchers aiming to further investigate these phenomena.

2 Atomic and band structures of AV₃Sb₅

Ortiz et al. [37] discovered a new family of kagome metals, AV₃Sb₅ in 2019, providing a new platform for investigating the rich physics of the kagome lattice. AV₃Sb₅ crystallizes in the P6/mmm space group, and its structure consists of alternating layers: A−Sb₂−VSb₁−Sb₂−A [Fig. 1(a)] [25, 38]. The central layer comprises vanadium (V) and antimony (Sb₁) atoms, where the V atoms form a kagome lattice interwoven with a hexagonal lattice of Sb₁. Each Sb₁ atom is located at the center of a V kagome lattice. The Sb₂ atoms form a honeycomb lattice positioned above and below the VSb₁ layer, while the outermost two layers consist of alkali metal atoms (K/Rb/Cs), forming hexagonal lattices. The kagome lattice exhibits key features such as VHSs, Dirac cones, and flat bands, as shown in the nearest-neighbor tight-binding band structure [Fig. 1(b)] [39]. The VHSs are located at the M points of the Brillouin zone, while the Dirac cones are at the K points. The flat band extends across the entire Brillouin zone. At VHSs, the electronic density of states becomes extremely high and can even diverge. When the Fermi level lies close to the VHS, the electronic system becomes highly unstable, thereby promoting the spontaneous emergence of both CDW and superconductivity [26]. In CsV₃Sb₅, two types of VHSs are present: p-type with filling fraction n = 5/12 and m-type with filling fraction n = 3/12, corresponding to pure and mixed sublattice character, respectively [28]. The sublattice configuration at van Hove filling plays a crucial role in understanding the unconventional many-body phases in the kagome lattice and determines the pairing symmetries of the coexisting CDW and superconducting states.

A more detailed band structure can be derived from density functional theory (DFT) calculations [Fig. 1(c)] [39]. In CsV₃Sb₅, there are four important bands across the Fermi level: G, K1, K2 and K2′ bands [39]. The G band is an electron pocket located at the center of the Brillouin zone (Γ), which has dominant Sb character. The K1 and K2 bands arise from the V d_{xy}/d_{x^2-y^2} and V d_xz/d_yz orbitals, respectively. Both bands exhibit VHSs near the Fermi level (E_F) and Dirac points at approximately –0.27 eV and –1.3 eV, respectively. The K2′ band, also originating from the V d_xz/d_yz orbitals like the K2 band, has the opposite parity. These band structures align closely with angle-resolved photoemission spectroscopy (ARPES) experiments [28, 29, 40]. In CsV₃Sb₅, three VHSs are present: two p-type VHSs originating from the K1 and K2 bands, and one m-type VHS arising from the K2′ band [28]. Among them, the p-type VHS associated with the K1 band and the m-type VHS from the K2′ band cross the Fermi level and thereby contribute to the formation of the CDW. In contrast, the p-type VHS derived from the K2 band lies below the Fermi level and thus does not directly participate in driving the CDW. Furthermore, the m-type VHS from the K2′ band exhibits nearly perfect Fermi-surface nesting and is considered as the primary driver of the CDW, whereas the p-type VHS from the K1 band promotes CDW formation via a higher-order nature and an enhanced density of states. Additionally, numerous studies indicate that Sb p_z-derived orbitals play an important role in the emergence of superconductivity, as supported by pressure [41] and doping [42] studies investigating the interplay between CDW and superconductivity.

3 CDW in AV₃Sb₅

3.1 2×2 CDW order and the magnetic response

2×2 CDW has been observed in AV₃Sb₅ (A = K, Rb, Cs) and reported in several studies [25, 30, 43, 44]. It can be detected on the K/Rb/Cs/Sb surfaces, with corresponding Fourier transforms obtained via STM [Figs. 2(a) and (b)] [25]. The CDW appears below the transition temperature T_CDW, which is approximately 80 K, 94 K and 103 K for KV₃Sb₅ [37, 45], CsV₃Sb₅ [18] and RbV₃Sb₅ [46], respectively. These transition temperatures are summarized in Table 1. At 4.2 K for KV₃Sb₅, an CDW gap opens in the vicinity of the Fermi level [Fig. 2(c)] [25]. Additionally, in the Fourier space, the 2×2 CDW vector peaks at 1/2 Q_Bragg do not change with energy, which suggests a static electronic order [Fig. 2(d)] [25].

2×2 CDW is not always symmetric in all three directions. At low temperatures, rotation symmetry can be broken due to phase shifts and the interlayer coupling between adjacent kagome layers [47-50]. As illustrated in the Sb surface of KV₃Sb₅ [Fig. 2(e)] and Cs surface of CsV₃Sb₅ [Fig. 2(g)], the intensity of 2×2 CDW differs significantly along one direction compared to the other two directions [Figs. 2(f) and (h)] [38, 51], leading to a change in the rotation symmetry of 2×2 CDW from C₆ to C₂. It is important to note that this rotation symmetry breaking has been observed on the surfaces of AV₃Sb₅ through STM. In the bulk, this symmetry breaking can be detected using nuclear magnetic resonance (NMR) and elastoresistance experiments in AV₃Sb₅ [38]. Besides, the c-axis resistivity in CsV₃Sb₅ shows twofold symmetry with in-plane rotating magnetic field, which may further suggest a rotation symmetry breaking of the electronic states [52]. They are all summarized in Table 1.

2×2 CDW shows the chirality in defect-free surfaces of AV₃Sb₅ [25, 43, 44]. As shown in Figs. 3(a)−(c), Jiang et al. [25] found the intensity of 2×2 CDW vector peak is anisotropic in the Fourier transforms of the dI/dV(r,V) maps on the defect-free Sb surface of KV₃Sb₅ under magnetic fields of 0 T, +2 T and −2 T. This anisotropy suggests the presence of chirality. Chirality is defined as the counting direction from the lowest to highest vector peaks and the magnetic field can switch the chirality of 2×2 CDW. However, on the defect-rich Sb surface of AV₃Sb₅, the chirality and strong magnetic field response are suppressed due to the pinning effects of the defects [25, 43, 44]. Interestingly, another study of KV₃Sb₅ by Li et al. [51] reported that despite the Sb surface being defect-free, no magnetic field response was observed [Figs. 3(d)−(f)]. The two experiments discussed above were both conducted by STM, but they have distinct results. This discrepancy may arise from the tip artifacts. Li et al. [51] demonstrated that even a small change in the STM tip can artificially generate an apparent “magnetic-field-dependent rotation of the CDW”, closely resembling the observations reported by Jiang et al. [25]. In addition, differences in the sample surfaces used by the two groups may also contribute to the contrasting experimental results.

The magnetic field response implies the existence of TRSB, so the presence of TRSB of 2×2 CDW is uncertain in AV₃Sb₅ based on above two STM experiments. In addition to STM, there are various experiment techniques to explore TRSB of 2×2 CDW, which are all summarized in Table 1. Among them, muon spin spectroscopy (μSR) [31, 53, 54], Kerr effect [55], circular dichroism measurements [55], and tuning fork resonator measurements support the TRSB [56]. Besides, the TRSB is considered as unconventional and has an orbital origin [53, 57-59]. However, in another high-resolution polar Kerr effect measurements [60] and magnetic torque measurements [61], TRSB is not observed within the 2×2 CDW.

Next, we discuss these results in detail. μSR experiments [31, 53, 54] in AV₃Sb₅ have revealed that TRSB occurs below T_CDW, characterized by the emergence of weak and local magnetic fields. The advantages of μSR measurements lies in their high sensitivity to local magnetic fields and their ability to operate under zero external magnetic field. However, certain limitations should also be noted. In this case, the appearance of TRSB signal coincides with the onset of the 2×2 CDW, suggesting that the observed signal may originate from structural modifications that alter the muon stopping sites.

Xu et al. [55] performed Kerr effect and circular dichroism measurements in AV₃Sb₅ and found that both signals emerge at T_CDW, indicating TRSB associated with the 2×2 CDW. However, Saykin et al. [60] did not detect any signatures of spontaneous TRSB below the same transition temperature using high-resolution polar Kerr effect measurements in CsV₃Sb₅. One major advantage of Saykin et al.’s experiments lie in their high instrumental precision, with a noise floor of approximately 30 nanoradians. They utilize two different versions of zero-area loop Sagnac interferometers (ZALSI), designed under reciprocity constraint, ensuring that any detected Kerr rotation must originate from genuine TRSB in the sample. However, in this high-precision measurement, no Kerr signal at the microradian level is observed, which is obtained by Xu et al. [55]. Besides, it should also be noted that the ZALSI setup used by Saykin et al. [60] is sensitive only to the polar Kerr effect under normal incidence, but insensitive to the longitudinal or transverse Kerr effects that may arise at oblique incidence. Therefore, if the TRSB signal indeed originates from longitudinal or transverse Kerr effects, it could be missed in their measurements. These results suggest that further refinement of experimental design will be necessary to clarify the TRSB signatures detected by Kerr effect measurements.

Moreover, magnetic torque measurements in CsV₃Sb₅ by Asaba et al. [61] suggestted the TRSB appears above T_CDW. Magnetic torque measurements are highly sensitive probes capable of directly detecting magnetic anisotropy. However, TRSB signal detected by this technique reflects a macroscopic response. If local magnetic fields exist but cancel each other out such that the net magnetic moment is zero, magnetic torque measurements would be unable to detect them. Therefore, this experiment cannot rule out the possibility that TRSB also exists below T_CDW.

It is noteworthy that TRSB reported in μSR and optical measurements occurs simultaneously with the 2×2 CDW transition. However, several studies have also detected the onset of TRSB at temperatures below T_CDW [31, 54, 56]. For instance, Gui et al. [56] performed tuning fork resonator measurements of magnetotropic susceptibility in CsV₃Sb₅ and observed pronounced magnetic anisotropy below ~30 K, along with a small magnetic moment along the c-axis, indicative of TRSB. Tuning fork resonator measurements provide direct thermodynamic evidence of magnetic moments and are exceptionally sensitive to small susceptibility anisotropies. Nevertheless, similar to magnetic torque measurements, this technique is a macroscopic probe and cannot detect compensated magnetic signals. This limitation may explain why no TRSB signal was observed to emerge simultaneously with the 2×2 CDW transition.

Based on current experimental results, it is possible that local magnetic moments emerge alongside the 2×2 CDW in AV₃Sb₅, while the net magnetic moment remains very weak. This behavior could result from antiferromagnet-like ordering of orbital currents, such as an interlayer-coupled chiral flux phase [62]. Such a mechanism may account for why TRSB is detected in μSR experiments, which are sensitive to weak and local magnetic fields, but remains undetected in macroscopic probes like magnetic torque measurements. Currently, direct experimental evidence for TRSB associated with the 2×2 CDW state is still lacking. Further studies, including more precise Kerr effect measurements sensitive to longitudinal or transverse components, are needed. Additionally, performing multiple complementary measurements on the same sample would help eliminate discrepancies due to sample variations and allow cross-verification across different experimental techniques.

3.2 1×4 CDW order

In addition to the 2×2 CDW, a unidirectional 1×4 CDW can also be observed on the Sb surfaces of AV₃Sb₅ (A = Rb, Cs) [Figs. 4(a) and (b)] [63]. Unlike the 2×2 CDW, which can be observed on both Sb and K/Rb/Cs surfaces, 1×4 CDW is only present on the Sb surfaces of RbV₃Sb₅ [43] and CsV₃Sb₅ [26]. The presence or the absence of 1×4 CDW in different members of the AV₃Sb₅ family may be explained by the chemical potential changes or small band-structure variations [26]. Besides, once 1×4 CDW appears, it tends to remain stable and is not easily affected by step edges, strains and point defects [64]. Like 2×2 CDW, the 1×4 CDW vector also shows a non-dispersive nature [Fig. 4(c)] and has a transition temperature [26]. For example, 1×4 CDW in CsV₃Sb₅ only appears below approximately 50 K [Fig. 4(d)] [26]. To sum up, 1×4 CDW has rich properties, which naturally promotes the study of its mechanism. Some works suggest that 1×4 CDW is related to the interlayer modulation of the CDW along the c axis and breaks the C₆ rotational symmetry of the crystal AV₃Sb₅ [22, 26, 65]. Other studies propose that 1×4 CDW originates from the in-plane shifting of V atoms relative to surface Sb atoms [44].

The twofold symmetry observed in magnetoresistance measurements by Chen et al. [22] and the 4a₀ stripes seen in STM images by Zhao et al. [26] both emerge at the same temperature of ~ 50 K. This suggests that 1×4 CDW could either be a quasi-3D form of the 4a₀ stripes with interlayer coupling or a different state that manifests as the 4a₀ stripes on the Sb surface of CsV₃Sb₅. While magnetoresistance measurements provide insights into the bulk properties, they cannot directly resolve the period of the 1×4 CDW, which is observable via STM. Furthermore, Ratcliff et al. [65] have found that the 2×2 CDW results from the simultaneous condensation of three optical phonon modes at one M and two L points, as revealed by ultrafast coherent phonon spectroscopy and first-principles DFT calculations. This condensation breaks the C₆ rotational symmetry of CsV₃Sb₅ and may contribute to the 1×4 CDW. Coherent phonon spectroscopy, though sensitive to phonon modes, is an indirect method, measuring phonons and inferring the structure of the electronic order based on phonon behaviors. Additionally, Wang et al. [44] suggested the 1×4 CDW originates from the in-plane shifting of V atoms relative to surface Sb atoms, based on DFT calculations and STM experiments in CsV₃Sb₅. STM provides a clear and intuitive visualization of the 1×4 CDW, but the conclusion about the in-plane shifting of V atoms is drawn from theoretical calculations, not directly observed via STM. For two mechanisms of 1×4 CDW mentioned above, there is a possibility that interlayer coupling might induce structural changes on the surface, which may be verified by multiple experiment techniques and DFT calculations on the same samples in the future.

4 Superconductivity in AV₃Sb₅

4.1 Superconducting states in AV₃Sb₅

In addition to CDW, superconductivity is another fascinating property of AV₃Sb₅. Superconductivity has been extensively studied in various materials [66-69], particularly in unconventional superconductors such as cuprates [70], Ni-based superconductors [69, 71] and iron-based superconductors [68, 72]. As a member of the kagome lattice family, AV₃Sb₅ presents a new and promising platform for investigating superconductivity [31, 73, 74].

The existence of superconductivity in AV₃Sb₅ has been confirmed by STM [22] and transport measurements [18, 46, 75]. The superconducting transition temperature T_C is ~ 0.93 K [75], ~ 2.5 K [18], and ~ 0.92 K [46] in KV₃Sb₅, CsV₃Sb₅, and RbV₃Sb₅, respectively, which is summarized in Table 1. Here, we focus on how STM has been used to demonstrate superconductivity in AV₃Sb₅. Chen et al. [22] reported the unconventional superconductivity in CsV₃Sb₅ using STM. To confirm the existence of superconductivity, they performed two sets of experimental measurements on CsV₃Sb₅ surface using normal and superconducting tips, respectively [Figs. 5(a) and (d)]. As demonstrated in Figure 5b, gaps appear in the dI/dV(r,V) spectra obtained on the Cs and Sb surfaces with a normal tip. The gap gradually gets smaller as the temperature increases, and at around 2.3 K, the gap vanishes [Fig. 5(c)].

To determine whether this gap is a superconducting gap, dI/dV(r,V) spectrum measurements on CsV₃Sb₅ surface are implemented with a superconducting tip [Fig. 5(e)]. The appearance of the Josephson effect proves the existence of superconductivity in CsV₃Sb₅. Furthermore, as shown in Fig. 5(f), the inner peaks relating to the difference (Δ_tip−Δ_sample) disappear at around 2.3 K, leaving the two remaining peaks from Δ_tip (Δ_tip and Δ_sample are superconducting gap sizes from tip and sample respectively). Here, this temperature (around 2.3 K) is consistent with the normal tip case. Additionally, the superconducting gap of the sample is in accordance with the value measured by the normal tip. Thus, the above gaps with the normal tip [Figs. 5(b) and (c)] are superconducting gaps and T_C of CsV₃Sb₅ is around 2.3 K.

Moreover, the V-shaped superconducting gap with nonzero local density of states at E_F suggests the unconventional superconductivity [76, 77]. Interestingly, the gap-to-T_C ratio 2Δ/(k_BT_C) (Δ is superconducting gap size) is about 5.2, implying the strong-coupled superconductivity, which contrasts with the weak-coupled superconductivity shown in the heat capacity measurements of KV₃Sb₅ in another article [75]. The opposite result here may be due to the multigap nature of superconductivity in AV₃Sb₅. Furthermore, STM probes the surface of the sample, providing more surface information, whereas heat-capacity measurements reflect bulk properties. Therefore, the nature of superconductivity in AV₃Sb₅ needs a deeper exploration.

4.2 PDW and the spatial periodic modulation

In a conventional superconductor, the two electrons of a Cooper pair have opposite momentum and total momentum is zero. When an external magnetic field is applied, the momentum of the Cooper pair may become non-zero, resulting in the formation of the superconducting condensate with a wave function modulated in space. This is known as the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state [78, 79]. However, in an unconventional superconductor, the non-zero momentum of the Cooper pair can also be formed in the absence of a magnetic field, giving rise to PDW. Evidence for PDW has been observed in Cu-based high temperature superconductors [80, 81] and AV₃Sb₅ [22].

Next, we discuss the PDW in AV₃Sb₅. In 2021, Chen et al. [22] reported the observation of PDW and its spatial periodic modulation in CsV₃Sb₅. They observed new peaks at Q_{3q-4a/3}=\frac{3}{4}{Q}_{\mathrm{B}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{g}}^{(\mathrm{a},\mathrm{b})}, in addition to peaks corresponding to the 2×2 CDW and 1×4 CDW in Fourier transform of dI/dV(r,V) map on the Sb surface, which suggests the appearance of a new electronic order [Figs. 6(a) and (b)]. Furthermore, the order vector at Q_3q−4a/3 is non-dispersive [Fig. 6(c)], indicating the possible formation of a 3Q density wave.

To visualize the 3Q density wave more clearly, the atomic Bragg peaks and incoherent background of a dI/dV(r,V) map were filtered out, leading to a clear 4a₀/3 checkerboard modulation in Fig. 6(d). More importantly, the 4a₀/3 spatial modulation of the superconducting gap 2Δ(r) is shown in plot of 2Δ(r) along the linecut in the q_b direction [Figs. 6(e) and (f)] and Δ(r) map [Figs. 6(g) and (h)], which is consistent with the 4a₀/3 period of the above 3Q density wave. So the above 3Q density wave with 4a₀/3 period is a PDW. Lastly, it is noteworthy that the PDW can only be observed in the Fourier transform of the dI/dV(r,V) map on the Sb surface, not in the Fourier transform of the topographic Sb surface.

4.3 Vortex in superconducting states

AV₃Sb₅, as Z₂ topological kagome metals, host nontrivial topological Dirac surface states near the Fermi energy [18, 40, 75]. When the Z₂ topology combines with superconductivity in AV₃Sb₅, the natively proximitized superconducting topological surface states may cause the appearance of Majorana bound states within the superconducting vortex states [22, 82-84], which is crucial for realizing topological quantum computation. Therefore, exploring vortex states in AV₃Sb₅ is vital [22, 30, 36, 85].

In 2021, Liang et al. [30] explored the vortex states in the superconducting states of CsV₃Sb₅ under the magnetic field applied vertically to the Sb and Cs surfaces [Figs. 7(a) and (b)]. The vortex states in the Sb and Cs surfaces exhibit distinct behaviors. On the Sb surface, an X-type splitting is observed in the distribution of the dI/dV(r,V) spectra across a vortex [Fig. 7(c)]. This X-type splitting indicates the presence of conventional Caroli–de Gennes–Matricon (CdGM) bound states, which are commonly observed in conventional s-wave superconductors, such as NbSe₂ [86]. However, on the Cs surface, the distribution of the dI/dV(r,V) spectra crossing a vortex manifests a Y-type splitting [Fig. 7(d)], in which the zero-bias conductance peak begins to split at a large distance away from the center of the vortex. This unusual Y-type splitting behavior may indicate the presence of Majorana bound states [82, 87], pointing to the unconventional nature of superconductivity in CsV₃Sb₅. However, it is important to note that in another article by Xu et al., the vortex states on the Sb and Cs surfaces of CsV₃Sb₅ are similar and the spatial evolution of the dI/dV(r,V) spectra across a vortex is consistent with conventional CdGM bound states [Figs. 7(e−h)] [36]. Both studies employed via STM but obtained different results. Due to the complexity of the sample growth and cleaving process, the cleaved surfaces prepared by different research groups may exhibit subtle variations, which may influence topological surface states on the Cs surface. Besides, Xu et al. [36] observed multiband superconductivity in CsV₃Sb₅, revealing different superconducting gap spectra (V-shape and U-shape) on the same surface. These different gaps may be preferentially probed in different measurements with different tip conditions, potentially leading to the above different phenomena.

The PDW and Y-type splitting of vortexes in superconducting states of CsV₃Sb₅ both suggest the unconventional superconductivity. However, the nature of superconductivity in AV₃Sb₅ remains under debate, as different experimental techniques yield different results, summarized in Table 1. The μSR measurements of RbV₃Sb₅ and KV₃Sb₅ [31] as well as thermal conductivity measurements of CsV₃Sb₅ [33] reveal evidence of nodal superconductivity. In addition, the observation of dynamic superconducting domains with boundary supercurrents in intrinsic CsV₃Sb₅ flakes by superconducting diode effect measurements suggests TRSB [32]. Moreover, Wu et al. [34] demonstrated via DFT calculations that the sublattice interference mechanism plays a key role in understanding the superconductivity of AV₃Sb₅ by DFT, further supporting its unconventional nature.

However, both the decrease of Knight shift and a Hebel−Slichter coherence peak observed just below T_C in NMR measurements indicate that CsV₃Sb₅ behaves as an s-wave superconductor [35]. Besides, Xu et al. [36] reported the vortex states on both the Sb and Cs surfaces of CsV₃Sb₅ exhibit X-type splitting, and magnetic impurities can induce in-gap states in CsV₃Sb₅ by STM measurements. These findings further support the characterization of CsV₃Sb₅ as an s-wave superconductor. Furthermore, μSR measurements of magnetic penetration depth suggest anisotropic two-gap s-wave superconductivity in CsV₃Sb₅ [88]. More importantly, impurity effect measurements using electron irradiation point to a non-chiral, anisotropic s-wave superconductor with no sign change in CsV₃Sb₅ [89].

Based on the experiment results above, it remains challenging to determine which experimental measurements provide the most definitive characterization. Although a majority of studies support a nodeless superconducting gap with s-wave pairing symmetry in AV₃Sb₅, numerous reports of unconventional features, such as PDW, Majorana bound states, nodal behavior, TRSB, and pronounced anisotropy, cannot be ignored. These apparently conflicting observations may reflect different aspects of superconductivity arising from the multiband nature of AV₃Sb₅. Moreover, the superconductivity of AV₃Sb₅ is highly sensitive to external factors like pressure [31] and doping [90]. For example, in μSR measurements of RbV₃Sb₅ and KV₃Sb₅ [31], an increase in pressure causes the superconducting state to progressively evolve from nodal to nodeless. A definitive resolution of this issue will likely require more systematic and comprehensive investigations in the future.

5 The interplay between CDW and superconductivity

The interplay between CDW and superconductivity arises in many cases [91-93] due to the struggle for the same density of states at the Fermi level. Understanding them in AV₃Sb₅ is crucial for gaining a deeper understanding of their underlying physics [94-98]. The interplay between the CDW and superconductivity in AV₃Sb₅ can be tuned by applying external pressure or chemical doping. As shown in Fig. 8(a), the interplay between CDW and superconductivity can be observed in the pressure-dependent electronic phase diagram of CsV₃Sb₅ [99]. The CDW transition temperature monotonically decreases with increasing pressure and is completely suppressed at P_c2 (approximately 2.0 GPa). Below P_c1 (approximately 0.58 GPa), the leading CDW is the 3Q CDW and an electronic nematicity appears at lower temperatures, with its transition temperature remaining independent of pressure. Just above P_c1, a coexisting phase emerges, characterized by both stripe-like CDW and nematic 3Q CDW states. From 0.9 GPa, the pure stripe-like CDW appears and at P_c2, this phase terminates, restoring the original kagome lattice.

For T_C, a double dome evolution is observed with the suppression of the CDW. T_C increases as pressure rises, peaking at P_c1, where it begins to decrease due to strong competition between CDW and superconductivity, forming the first dome. Afterward, T_C increases again because of the suppressed CDW, reaching its maximum value of around 8 K at P_c2 [94, 100], forming the second dome. The suppression of superconductivity between P_c1 and P_c2 is related to the stripe CDW state, suggesting a complicated interplay between the stripe CDW state and superconductivity. Besides, the monotonic suppression of CDW and the double dome evolution of superconductivity in CsV₃Sb₅ can also be observed in KV₃Sb₅ and RbV₃Sb₅ [97, 101, 102]. They have different maximum of T_C, which are 3 K, 4 K, and 8 K for KV₃Sb₅, RbV₃Sb₅, and CsV₃Sb₅, respectively. The corresponding data are summarized in Table 1.

In addition to pressure, chemical doping in CsV₃Sb₅ provides another way to explore the interplay between CDW and superconductivity [42, 103-105]. Small variations in the electronic structure induced by chemical doping, such as shifts of the Fermi level and orbital band reconstructions, can exert a significant influence on both superconductivity and CDW. Moreover, different dopant elements and substitutional sites lead to distinct impacts on the interplay between CDW and superconductivity. Here, we categorize chemical doping into three types of electron doping, hole doping, and isovalent doping. We find that for hole doping and isovalent doping, in most cases, within the doping concentration range where T_CDW is gradually suppressed, T_C generally exhibits an upward trend [42, 104, 106, 107]. This behavior suggests a competitive relationship between T_CDW and T_C. However, in some cases, such as CsV_3−xTi_xSb₅ [108] (hole doping) and Cs(V_1−xCr_x)₃Sb₅ [109] (electron doping), where T_CDW and T_C are simultaneously suppressed. These two cases of AV₃Sb₅ are summarized in Table 1 and discussed in detail below.

For the competitive relationship between CDW and superconductivity, a notable example is CsV₃Sb_5−xSn_x [42]. As shown in Fig. 8(b), T_C exhibits a double dome evolution, while the CDW transition temperature decreases as the Sn doping content increases. The two maxima of the T_C are at x = 0.03 and x = 0.35 (where x represents the Sn doping content), and the CDW transition temperature disappears at x = 0.06. We observe that within the doping concentration range where T_CDW is suppressed (x < 0.06), T_C exhibits an overall upward trend. Moreover, as presented in Fig. 8(c), electronic phase diagram of Ti-doped CsV₃Sb₅ [108] reveals that with the increasement of hole doping, T_CDW is gradually suppressed and completely suppressed at x ≈ 0.07, where the second superconductivity dome appears. Notably, within the region where T_CDW is suppressed (x < 0.07), T_C also decreases.

6 Outlook

Although extensive experimental and theoretical studies have been conducted on AV₃Sb₅, several key questions remain unresolved.

Firstly, the origin of CDW is still elusive. Various theoretical models have been proposed to explain its mechanism [110-112]. For the 1×4 CDW, different explanations have been suggested, including interlayer CDW modulation along the c axis [22, 26, 65] and in-plane shifting of V atoms relative to surface Sb atoms [44]. Magnetoresistance measurements [22] and ultrafast coherent phonon spectroscopy [65] support an interlayer CDW modulation, whereas STM experiments suggest an in-plane shift of V atoms relative to surface Sb atoms [44]. Since magnetoresistance probes bulk properties, it cannot directly resolve the real-space periodicity of the 1×4 CDW, which is accessible by STM. Coherent phonon spectroscopy is sensitive to lattice vibrational modes; however, it provides only indirect information on the electronic order through phonon dynamics. Moreover, the conclusion regarding the in-plane displacement of V atoms is primarily based on theoretical calculations rather than direct measurements. For the two proposed mechanisms of the 1×4 CDW, it remains possible that interlayer coupling could induce surface structural distortions. Future studies combining multiple experimental techniques with DFT calculations on identical samples will be essential to resolve this issue.

Secondly, the magnetic field response of 2×2 CDW by STM implies the existence of TRSB [25], which also has been confirmed by μSR [31, 53, 54], Kerr effect [55], circular dichroism measurements [55], and tuning fork resonator measurements [56]. However, in another STM measurement [51], high-resolution polar Kerr effect measurements [60] and magnetic torque measurements [61], TRSB is not observed within the 2×2 CDW. μSR measurements [31, 53, 54] are highly sensitive to local magnetic fields and can be performed under zero external field. However, the emergence of TRSB signal coincides with the onset of the 2×2 CDW, suggesting that the observed signal may originate from structural modifications that alter the muon stopping sites. Kerr effect measurements offer high instrumental precision with reciprocity constraint [60], ensuring that any detected Kerr rotation reflects genuine TRSB in the sample. Nevertheless, these high-precision measurements detect no Kerr signal, in contrast to the results reported by Xu et al. [55]. Moreover, the experimental configuration used by Saykin et al. [60] is insensitive to both longitudinal and transverse Kerr effects.

Magnetic torque measurements [61] serve as highly sensitive probes of magnetic anisotropy. However, TRSB signal detected by this technique reflects a macroscopic response. Tuning fork resonator measurements [56] revealed a pronounced magnetic anisotropy below ~30 K, yet no TRSB signal emerged concurrently with the 2×2 CDW transition. While tuning fork resonators provide direct thermodynamic evidence of magnetic moments and exceptional sensitivity to small anisotropies in susceptibility, they are also macroscopic probes and cannot detect compensated magnetic signals, which may explain the absence of TRSB signals concurrent with the CDW transition.

Current evidence suggests that local magnetic moments may appear with the 2×2 CDW in AV₃Sb₅, while the net moment remains weak. This behavior could arise from antiferromagnet-like ordering of orbital currents, such as an interlayer-coupled chiral flux phase [62]. This may explain why TRSB is observed in μSR — sensitive to weak local fields — but absent in macroscopic probes like magnetic torque measurements. In conclusion, direct evidence for TRSB associated with the 2×2 CDW state remains lacking. Further studies, including more precise Kerr effect measurements sensitive to the longitudinal or transverse components, as well as complementary experiments on the same samples, are needed to reconcile discrepancies and enable cross-verification among different techniques.

Thirdly, the distribution of the dI/dV(r,V) spectra crossing a vortex on the Cs and Sb surfaces in CsV₃Sb₅ is controversial. In one study by Liang et al. [30], the dI/dV(r,V) spectra across a vortex on the Cs surface is different from that on the Sb surface. An X-type splitting on the Sb surface shows the presence of CdGM bound states, while a Y-type splitting on the Cs surface indicates the presence of Majorana bound states. But in another study by Xu et al. [36], the dI/dV(r,V) spectra across a vortex state on the Sb surface is similar to that on the Cs surface, which both show the X-type splitting and are consistent with conventional CdGM bound states. Different results may be due to sample differences or multiband nature of superconductivity in CsV₃Sb₅.

Fourthly, there is no unified explanation for the nature of superconductivity in AV₃Sb₅. The presence of PDW [22], Majorana bound states [30], TRSB [32], nodal [31, 33] and anisotropic [88, 89] characteristics all point to unconventional superconductivity of AV₃Sb₅. However, both the decrease of Knight shift and a Hebel-Slichter coherence peak just below T_C in NMR measurements indicate that CsV₃Sb₅ exhibits an s-wave superconductor [35]. Besides, X-type splitting [36] and magnetic impurity-induced in-gap states [36] further support an s-wave pairing symmetry in CsV₃Sb₅. Therefore, the nature of superconductivity in AV₃Sb₅ remains elusive and deserves more studies. Based on the experimental results above, while most studies favor the nodeless and s-wave superconductivity in AV₃Sb₅, the unconventional features cannot be ignored. The multiband nature of superconductivity in AV₃Sb₅ and sample-dependent variations may account for these discrepancies. To date, most research has focused on CsV₃Sb₅, therefore, further research on KV₃Sb₅ and RbV₃Sb₅ are essential. Moreover, external parameters such as pressure or chemical doping can influence the nature of superconductivity and its interplay with the CDW. Systematic investigations under these conditions may provide crucial insights into the underlying mechanisms of superconductivity in AV₃Sb₅.

Fifthly, the rich physical phenomena observed in AV₃Sb₅ motivate the exploration of new materials containing the V-kagome lattice such as RV₆Sn₆ (R = Gd, Ho) [113], AV₆Sb₆ [114-116], AV₈Sb₁₂ [114], V₃Sb₂ [117], and V₆Sb₄ [118]. Further study of these new kagome materials may lead to the discovery of novel physical phenomena and help address existing questions related to kagome lattice materials.

Sixthly, chromium-based kagome metal, such as CsCr₃Sb₅, represent an emerging class of materials with structures similar to those of AV₃Sb₅, yet they exhibit several distinct properties. On the one hand, CsCr₃Sb₅ exhibits stronger electron correlations and larger effective mass renormalization, whereas AV₃Sb₅ displays relatively weak electron correlations [119, 120]. On the other hand, CsCr₃Sb₅ shows only a stripe-like 4a₀ modulation, in contrast to the 2×2 modulation observed in AV₃Sb₅ [121, 122]. More importantly, at ambient pressure, CsCr₃Sb₅ is not superconducting. Its superconductivity emerges at 3.65 GPa and reaches a maximum T_C of 6.4 K at 4.2 GPa [121]. In contrast, AV₃Sb₅ is superconducting even at ambient pressure and maintains its superconductivity under higher pressures. These differences suggest that further investigation of CsCr₃Sb₅ is essential, as it may provide deeper insights into the underlying properties of AV₃Sb₅.

Finally, AV₃Sb₅ has recently emerged as a highly prominent frontier research system in condensed matter physics and materials science. Although these materials have not yet been widely applied in commercial or mainstream technological devices, their unique and diverse physical properties endow them with tremendous potential for future high-tech applications. The coexistence of Z₂ topology and superconductivity in AV₃Sb₅ may give rise to Majorana bound states [22, 82-84], which is crucial for realizing topological quantum computation. Moreover, giant AHE [19] exhibited in AV₃Sb₅ holds promise for future applications in quantum Hall devices. In addition, the interplay between superconductivity and CDW can be precisely tuned by doping [42, 103, 104, 106-109] and pressure [97, 99, 101, 102] on AV₃Sb₅. This remarkable tunability gives this material great potential for development into highly sensitive quantum sensors, capable of detecting tiny variations in pressure and doping.

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