⇦Chinese Physics Letters, 2023, Vol. 40, No. 4, Article code 047301 Electronic Instability of Kagome Metal CsV$_{3}$Sb$_{5}$ in the $2 \times 2\times 2$ Charge Density Wave State Hongen Zhu (朱红恩)1†, Tongrui Li (李彤瑞)1†, Fanghang Yu (喻芳航)2, Yuliang Li (李昱良)1, Sheng Wang (王盛)1, Yunbo Wu (吴云波)1, Zhanfeng Liu (刘站锋)1, Zhengming Shang (尚政明)1, Shengtao Cui (崔胜涛)1, Yi Liu (刘毅)1, Guobin Zhang (张国斌)1, Lidong Zhang (张李东)1, Zhenyu Wang (王震宇)2,5, Tao Wu (吴涛)2,4,5, Jianjun Ying (应剑俊)2, Xianhui Chen (陈仙辉)2,3,4,5, and Zhe Sun (孙喆)1,2,4* Affiliations 1National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, China 2Department of Physics, CAS Key Laboratory of Strongly-coupled Quantum Matter Physics, University of Science and Technology of China, Hefei 230026, China 3CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei 230026, China 4Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China 5CAS Center for Excellence in Superconducting Electronics (CENSE), Shanghai 200050, China Received 20 January 2023; accepted manuscript online 6 March 2023; published online 2 April 2023 †Hongen Zhu and Tongrui Li contributed equally to this work.
*Corresponding author. Email: zsun@ustc.edu.cn Citation Text: Zhu H E, Li T R, Yu F H et al. 2023 Chin. Phys. Lett. 40 047301    Abstract Recently discovered kagome metals $A$V$_{3}$Sb$_{5}$ ($A$ = K, Rb, and Cs) provide an ideal platform to study the correlation among nontrivial band topology, unconventional charge density wave (CDW), and superconductivity. The evolution of electronic structures associated with the change of lattice modulations is crucial for understanding of the CDW mechanism, with the combination of angle-resolved photoemission spectroscopy (ARPES) measurements and density functional theory calculations, we investigate how band dispersions change with the increase of lattice distortions. In particular, we focus on the electronic states around $\bar{M}$ point, where the van Hove singularities are expected to play crucial roles in the CDW transition. Previous ARPES studies reported a spectral weight splitting of the van Hove singularity around $\bar{M}$ point, which is associated with the 3D lattice modulations. Our studies reveal that this “splitting” can be connected to the two van Hove singularities at $k_{z}=0$ and $k_{z}=\pi /c$ in the normal states. When the electronic system enters into the CDW state, both van Hove singularities move down. Such novel properties are important for understanding of the CDW transition.
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 DOI:10.1088/0256-307X/40/4/047301 © 2023 Chinese Physics Society Article Text The two-dimensional (2D) kagome lattice is a 2D network of corner-sharing triangles that could serve as a platform for frustration-driven exotic spin-liquid phases,[1,2] and the intrinsic band structure hosts a flat band, Dirac cone, and van Hove singularities (VHSs) as depicted in Fig. 1(a).[3-5] When the Fermi level $E_{\rm F}$ is close to the VHS, the sharp peak in the density of states could lead to various instabilities, including ferromagnetism, anti-ferromagnetism, charge density wave, and superconductivity.[6-13] In kagome metals $A$V$_{3}$Sb$_{5}$ ($A$ = K, Rb, and Cs), as shown by density functional theory (DFT) calculations in Fig. 1(b), a VHS (VHS1) is very close to the Fermi level and could cause electronic instabilities. Nontrivial band topology, unconventional charge density wave (CDW), and superconductivity coexist in $A$V$_{3}$Sb$_{5}$.[14-16] These novel properties make this system a desirable platform for studying the physics of kagome lattice with electronic instabilities driven by VHSs. For instance, many unusual properties have been observed in these materials, including unusual electron-phonon interaction without acoustic phonon anomaly,[17] unconventional anomalous Hall effect in the absence of local magnetic moments,[18-20] and double superconducting dome under high pressure.[21,22] Previous angle-resolved photoemission spectroscopy (ARPES) studies have shown that there are multiple VHSs around the $\bar{M}$ point with different vanadium $d$ orbitals.[23,24] VHS1 in Fig. 1(b) is close to $E_{\rm F}$, so its variation is expected to be closely related to the CDW transition. Recently, an unusual splitting of spectral weight around the $\bar{M}$ point has been observed in CsV$_{3}$Sb$_{5}$, and the possible origin was proposed in a structural model with alternating stacking star-of-David (SoD) and inverse-star-of-David (ISD) layers.[25,26] Although the origin of the spectral-weight splitting requires more investigation, this observation suggests that the crucial variation of electronic structures across the CDW transition in $A$V$_{3}$Sb$_{5}$ cannot be fully captured by the electronic instability of VHS1 around the Fermi level. More studies of the spectral-weight splitting could shed light on the underlying physics of the CDW transition in kagome metals. The evolution of electronic structures and the corresponding lattice modulations are important for understanding of CDW transitions, for which previous ARPES studies provided limited information. We aim to address this issue by studying how band dispersions change with the increase of lattice distortions. On the other hand, as depicted in Fig. 1(c), the cleaved surface varies with different coverages of alkali atoms as revealed by scanning tunneling microscopy (STM) studies.[27] The inhomogeneous coverage and migration of alkali atoms contribute to the change of electron doping at the top $A$V$_{3}$Sb$_{5}$ layer, and such extrinsic conditions partially lead to the controversy of surface-sensitive ARPES data. For example, distinct band reconstructions in CsV$_{3}$Sb$_{5}$ have been reported.[28] In this Letter, we investigate band structures of kagome metal CsV$_{3}$Sb$_{5}$ using ARPES on the cleaved surface of a low rate of Cs coverage with a focus on the evolution of the spectral-weight splitting across the CDW transition. In combination with DFT calculations, we show that there is a significant shift of spectral weight around the $\bar{M}$ point in the (001)-projected surface Brillouin zone to deeper binding energies in the CDW state. The “splitting” of spectral weight around the $\bar{M}$ point can be connected to the two van Hove singularities at $k_{z}=0$ and $k_{z}=\pi /c$ in the normal states. In the normal state, the former is below the Fermi level, the latter is above the Fermi level. When the system enters into the CDW state, both van Hove singularities move down and result in the splitting of spectral weight. Such novel properties could shed light on understanding of the CDW mechanism.
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Fig. 1. Crystal and band structure of kagome lattice. (a) Prototypical band structure and density of states of a kagome lattice. (b) Band structures of CsV$_{3}$Sb$_{5}$ from calculations without lattice distortion along the $\varGamma$–$M$–$K$–$\varGamma$ high symmetry directions. Three van Hove singularities are indicated by arrows around the $M$ point. (c) and (d) Crystal structure of CsV$_{3}$Sb$_{5}$ in the normal state ($T > T^*$, without lattice distortion) with space group $P6/mmm$ (No. 191), the pristine unit cell is outlined by black lines. (e) and (f) SoD and ISD distortion structures of $2 \times 2$ supercell in the CDW state ($T < T^*$). (g) Temperature dependence of resistivity of CsV$_{3}$Sb$_{5}$ single crystals. The superconducting transition is shown in the inset. (h) ARPES intensity along high symmetry directions in the (001)-projected surface Brillouin zone. The data were taken at 18 K using 23 eV photons.
The crystal structure of CsV$_{3}$Sb$_{5}$ in the normal state is presented in Figs. 1(c) and 1(d), the vanadium layers are kagome lattices with two distinct Sb sublattices. Figures 1(e) and 1(f) shows two candidate distortions in the CDW state, SoD, and ISD [ISD, also named as tri-hexagonal (TrH)]. First-principles calculations suggest that the ISD model is energetically stable in the CDW state.[29] High-quality CsV$_{3}$Sb$_{5}$ single crystals were synthesized via a self-flux growth method as described elsewhere.[18] All the preparation processes were performed in an argon glove box to prevent the reaction of Cs with air and water. After a high-temperature reaction in a furnace, the excess flux was removed by using water, and single crystals were obtained as shown in Fig. 1(g). The CDW transition temperature of CsV$_{3}$Sb$_{5}$ is determined by the kink feature ($T^* \sim 94$ K) of the resistivity in Fig. 1(g). The residual-resistivity ratio (RRR) is 68, indicating the high quality of single crystals measured in our ARPES studies. In CsV$_{3}$Sb$_{5}$, DFT calculations indicate that the primary change of band structures in the vicinity of the Fermi level occurs around the $\bar{M}$ point,[30,31] so these electronic states are critical for understanding of the physics in this material. ARPES intensity of CsV$_{3}$Sb$_{5}$ in the CDW state is shown in Fig. 1(h), consistent with the previous ARPES reports.[32-40] The temperature dependence of ARPES data is shown in Fig. 2. Within the momentum resolution of our ARPES experiments, the detailed variations of the Fermi surface below and above the CDW transition cannot be resolved [Figs. 2(a) and 2(b)]. However, we will show later that band dispersions change drastically around the $\bar{M}$. In our experiments, we carefully monitored the aging effect to avoid extrinsic changes in ARPES data. Firstly, we examined band dispersions along the $\bar{\varGamma }$–$\bar{K}$ direction [see Figs. 2(c)–2(e)]. Our samples were cleaved at 130 K and cooled down to 18 K, and then warmed up to 130 K. Band dispersions around $\bar{\varGamma}$ remain the same throughout the temperature cycle. As shown in Figs. 2(f)–2(j), there are also no substantial changes in band dispersions around $\bar{\varGamma}$ along the $\bar{\varGamma }$–$\bar{M}$ as the temperature varies from 18 K to 130 K. Some reports show that the band bottom around $\bar{\varGamma}$ changes with temperature,[28,36] while our data suggest that such variations are minimal as indicated by the green dashed lines in Figs. 2(c)–2(e) and Figs. 2(f)–2(j). We argue that a substantial change in band dispersions could be due to the extrinsic aging effect of samples. The cleaved surface is covered by alkaline elements, which could escape and result in some changes in ARPES data. The cleaved surface of our samples should be at a low rate of Cs coverage and to a large extent avoid the variation caused by the variation of Cs coverage. By monitoring the aging effect, we can distinguish the intrinsic variations around the $\bar{M}$ point. As shown in Figs. 2(k)–2(n), a splitting of spectral weight occurs in the CDW state, as indicated by $\alpha$ and $\beta$ bands in Fig. 2(n). One may notice a gap-like feature at the $\bar{K}$ point, and we note here that it may arise from a slight misalignment. In order to reveal the evolution of the splitting feature across the CDW transition, we performed temperature-dependent measurements along the $\bar{K}$–$\bar{M}$–$\bar{K}$ direction [see Figs. 3(a)–3(j)]. Along the red line indicated in Fig. 3(a), we extract energy distribution curves (EDCs) at various temperatures and show them in Fig. 3(k). With the temperature decreasing, the splitting begins to show up as a peak-dip-hump feature in Fig. 3(k), and the hump gradually moves to deeper binding energies.
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Fig. 2. ARPES measurements of band structures of CsV$_{3}$Sb$_{5}$. (a) and (b) Fermi surface measured at $T =18$ K and 130 K, respectively. (c)–(e) and (f)–(j) Temperature dependence of band dispersions along the $\bar{\varGamma }$–$\bar{K}$ and $\bar{\varGamma }$–$\bar{M}$ directions, respectively. The data were taken using 23 eV photons. (k)–(n) ARPES intensity plots and the corresponding 2$^{\rm nd}$-derivative images along the $\bar{K}$–$\bar{M}$–$\bar{K}$ direction at $T =130$ K and 18 K, respectively.
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Fig. 3. Temperature dependence of electronic states along the $\bar{K}$–$\bar{M}$–$\bar{K}$ direction. (a)–(j) Temperature dependence of ARPES intensity along the $\bar{K}$–$\bar{M}$–$\bar{K}$ direction measured using 23 eV photons. (k) EDCs taken at different temperatures along the momentum indicated by the red line in (a).
However, the strong three-dimensionality of band dispersions around the $\bar{M}$ point could mislead our attention. In the process of photoemission, when the inelastic mean free path is small, the corresponding uncertainty of momentum is large. In this case, the strong 3D character of the electronic states around the $\bar{M}$ significantly broadens the ARPES intensity with a large range of spectral weight from various $k_{z}$. Though band calculations indicate that there is a VHS around the $\bar{M}$ closed to $E_{\rm F}$, this feature could be smeared out by the $k_{z}$ broadening in ARPES data. In addition, the cutoff at the Fermi level can also give rise to a peak-like feature in the ARPES intensity, which resembles a flat band dispersion. From this perspective, the data in Fig. 3(k) may miss some intrinsic changes across the CDW transition. On the other hand, the splitting of spectral weight around the $\bar{M}$ point at low temperatures is a key to the understanding of electronic structures in the CDW state. To reveal its origin, we performed DFT calculations. There are two possible lattice distortions (SoD, ISD) in the CDW state when concerning the in-plane deformation. Based on measurements from the same batch of crystals, nuclear magnetic resonance and STM indicated a $2 \times 2\times 2$ CDW configuration of ISD layers with a $\pi$ phase shift between adjacent layers [see Fig. 4(b)] at low temperatures.[41] Moreover, ISD + ISD (with $\pi$ phase shift) in the CDW state has the lowest total energy (relative to the pristine phase) in $A$V$_{3}$Sb$_{5}$ ($A$ = Cs, Rb, K) according to the DFT calculation of the $2 \times 2\times 2$ CDW phases (Table S2 in Ref. [42]). Therefore, we calculated band dispersions of CsV$_{3}$Sb$_{5}$ in the CDW state based on this type of microscopic structure.
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Fig. 4. Three-dimensional CDW deformation structure and DFT calculations. (a) and (c) Two kagome layers in the normal state ($T > T^*$) and the corresponding band structure. The Cs and Sb atoms are not shown. (b) and (d) The $2 \times 2\times 2$ supercell with alternating stacking of ISD distortion with a $\pi$ phase shift ($T < T^*$) and the corresponding band structure. (e)–(h) DFT calculations for band structures at $k_{z} = 0$ ($\varGamma$–$K$–$M$–$K$–$\varGamma$) with various distortion amplitudes (from zero to three) for a $2 \times 2$ supercell. (i)–(l) DFT calculations for band structures at $k_{z} = 0$ ($\varGamma$–$K$–$M$–$K$–$\varGamma$) and $k_{z}=\pi /c$ ($A$–$H$–$L$–$H$–$A$) with various distortion amplitudes (from zero to three) for $2 \times 2\times 2$ supercell.
By comparing band structures of the normal state [Fig. 4(c)] and the CDW state [Fig. 4(d)], one may notice that a large portion of electronic states around the $\bar{M}$ point in the vicinity of the Fermi level moves toward deeper binding energies. Comparing band calculations of the CDW state in Fig. 4(d) with ARPES data in Fig. 3(a), we argue that the splitting of spectral weight in Fig. 3(a) is related to the high density of states at $k_{z} \sim 0$ and $k_{z} \sim \pi /c$ of the distribution of spectral weight along $k_{z}$, the dispersions of which are corresponding to the $\beta$ and $\alpha$ bands, respectively. In fact, in addition to the VHS1 as already shown in Fig. 1(b), there is another van Hove singularity (VHS4) at $k_{z}=\pi /c$ above the Fermi level as indicated in Fig. 4(c).[24,25] There two van Hove singularities contribute the dominant spectral weight at $k_{z} \sim 0$ and $k_{z} \sim \pi /c$. From this perspective, the splitting of spectral weight can be directly connected to the two van Hove singularities. We note here that, in our data the folded bands in the $2 \times 2\times 2$ CDW phase is very weak, and the significant variation of electronic states should be attributed to the physics in the main bands. To further illustrate this finding, we performed band calculations with $k_{z} = 0$ ($\varGamma$–$K$–$M$–$K$–$\varGamma$), and show the evolution of band structures with various distortion amplitudes (from zero to three) in Figs. 4(e)–4(h). This approach is able to catch the effect as the CDW phase becomes stronger with decreasing temperature. This model contains band dispersions in the $2 \times 2$ CDW state in individual ISD layers and ignores the 3D structure of the $2 \times 2\times 2$ CDW state. For comparison, Figs. 4(i)–4(l) show the effect of the 3D structure of ISD stacking with both $k_{z} = 0$ ($\varGamma$–$K$–$M$–$K$–$\varGamma$) and $k_{z}=\pi /c$ ($A$–$H$–$L$–$H$–$A$) included. We note here that, for those calculations with lattice distortions, all bands must be unfolded to the primary Brillouin zone in order to make a comparison with the “normal” band structure. To keep a concise image, we only select bands at $k_{z} = 0$ and $k_{z}=\pi /c$, and other bands between the van Hove singularities remain in between.[34] In Fig. 4(i), the bands around the two van Hove singularities at $k_{z} = 0$ and $k_{z}=\pi /c$ are denoted by $\beta$ and $\alpha$, respectively. With the increase of lattice distortions, $\alpha$ and $\beta$ bands move down gradually. As shown in Fig. 4(l), it is evident that $\alpha$ and $\beta$ bands in Fig. 3(a) of the CDW state can be connected to the high spectral weight of $\alpha$ and $\beta$ bands in the $2 \times 2\times 2$ CDW state. On the other hand, it has been shown that the spectral weight splitting around the $\bar{M}$ could be reproduced by band calculations with ISD + SoD layer by layer stacking,[25,26] suggesting that 3D lattice modulations are important for interpreting the novel behavior of electronic states around the $\bar{M}$ point. To further reveal electronic structures near the $\bar{M}$ point, we deposited potassium on sample surfaces at low temperatures to investigate band structures above the Fermi level as well as the change of electronic states associated with the CDW transition. In Figs. 5(a)–5(d), it is evident that with the increase of K deposition, all bands move down to deeper binding energies. The corresponding EDCs are shown in Figs. 5(f)–5(i). To confirm K was deposited onto the surface of samples, we monitored the increase of K $3p$ core level as shown in Fig. 5(j). In Fig. 5(e), using s-polarization, the Dirac dispersion (indicated by the red arrow) also clearly shows the same shift in energy, and the top of the valence band shows up at $k_{\scriptscriptstyle{||}} \sim 0.6$ Å$^{-1}$. In addition to the shift of band dispersions in energy, the peak-dip-hump feature originated from $\alpha$ and $\beta$ bands in Fig. 5(a) disappears with the deposition of K. This is due to the fact that the charge doping destroys the original condition that favors the CDW state, thus inhibiting the CDW phase. Under such a situation, band structures are similar to those at high temperatures without CDW. The continuum of spectral weight around the $\bar{M}$ point is related to the 3D spectral weight spreading over a large energy scale with $k_{z}$ varying from 0 to $\pi /c$.
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Fig. 5. Evolution of electronic states along the $\bar{K}$–$\bar{M}$–$\bar{K}$ direction upon potassium deposition. (a)–(d) ARPES intensity with increasing electron doping taken at $T \sim 18$ K using 23 eV photons. (e) Band dispersion measured with s-polarized light before and after electron doping. (f)–(i) EDC plots around the $\bar{M}$ point along the $\bar{K}$–$\bar{M}$–$\bar{K}$ direction. (j) Core level of K $3p$ measured using 91.5 eV photons. (k) EDCs for various potassium deposition taken from the red dashed lines in (a)–(d).
We have extracted EDCs along red dashed lines in Figs. 5(a)–5(d) and shown them in Fig. 5(k) with various K doping concentrations. For the pristine condition, there are two peaks near the Fermi level, and peak A is higher than peak B. With the increase of K dopants, the whole spectrum shifts to deeper binding energies. In this process, peak B remains, while peak A quickly disappears. If the CDW remains upon K deposition, the spectral weight splitting should be maintained by lattice modulations, and the whole feature shifts to deeper binding energies. Cs deposition on the surface of CsV$_{3}$Sb$_{5}$ has been performed by Nakayama et al.,[43] suggested that the saddle point pinned at the Fermi level as the CDW was killed by Cs deposition. Our data, however, show that the CDW feature is suppressed while electronic states around the van Hove singularities shift down with an electron doping effect caused by K deposition. The remarkable difference between Cs and K depositions on CsV$_{3}$Sb$_{5}$ suggests a different carrier-injection mechanism, which demands further studies. Moreover, an extra band of potassium-covered surface appears at the $\bar{M}$ point as indicated by the black dashed rectangles in Figs. 5(h)–5(i). As shown in Fig. 1, the ARPES measurements of CsV$_{3}$Sb$_{5}$ are highly consistent with the band calculations, and we can refer those to examine the origin of the extra band. Theoretical calculations indicate that there are topological surface states slightly above the $E_{\rm F}$ at the $\bar{M}$ point.[15] In Fig. 5(d), the Fermi level shifts up about 0.08 eV after electron doping, and the energy and momentum of the extra band are in line with the topological surface states in band calculations, though we could not distinguish its dispersion. Our data show that electronic states around the $\bar{M}$ point in the vicinity of $E_{\rm F}$ show a spectral-weight splitting, which comes from the modulation of band dispersions associated with the $2 \times 2\times 2$ CDW state. Such behavior removes partial spectral weight around the Fermi level and decreases the system energy, which is corresponding to the instability of the density of states that is associated with the CDW formation in CsV$_{3}$Sb$_{5}$. The overall trend is consistent with band calculations of alternating stacking layers of the ISD distortion with a $\pi$ phase shift between adjacent layers in the CDW phase. In addition to the evident shift of spectral weight of the VHS at $k_{z} = 0$, electronic states around the Fermi level from $k_{z} = 0$ to $\pi /c$ near the zone boundary also move down to deeper binding energies. Our data suggests that the instability of electronic states is far beyond the VHS around the $M$ point in the vicinity of the Fermi level. These electronic states should be taken into account for better understanding of the mechanisms of CDW in CsV$_{3}$Sb$_{5}$. On the other hand, the splitting of the spectral weight around the $\bar{M}$ point shows up in the $2 \times 2\times 2$ supercell structure, suggesting that the coupling between electronic states and lattice distortions cooperates in the change of band dispersions and the development of the CDW phase. Our studies provide a spectroscopic test for future models of the CDW physics in kagome metal $A$V$_{3}$Sb$_{5}$. We note here that the mechanism of CDW in $A$V$_{3}$Sb$_{5}$ system could be similar to the physics in some transition-metal-dichalcogenide systems, such as 2H-NbSe$_{2}$ because they have similar saddle points near the Fermi level.[11] Meanwhile, electron doping allows us to detect electronic states above $E_{\rm F}$, that is, the Fermi level could be tuned to the position of nontrivial topological surface states. This approach could realize more manipulations, including the coupling between nontrivial topological states and superconductivity. As a matter of fact, different carrier concentrations have different effects on electronic states in CsV$_{3}$Sb$_{5}$. STM studies have shown that high Cs coverage on the surface of CsV$_{3}$Sb$_{5}$ could host Majorana bound states below $T_{\rm c}$,[27] which could be related to extra states around the $\bar{M}$ point in Figs. 5(h) and 5(i). In summary, we have investigated the band structure of kagome metal CsV$_{3}$Sb$_{5}$ and its temperature evolution using ARPES, in combination with DFT calculations for the ISD lattice distortion in the alternating stacking layers with a $\pi$ phase shift. Our data suggest that the spectral-weight splitting around the $\bar{M}$ point in the vicinity of $E_{\rm F}$ arises from the high intensity of spectral weight of $k_{z} \sim 0$ and $\pi /c$ in the $2 \times 2\times 2$ CDW state, which can be connected to the van Hove singularities at $k_{z} =0$ and $\pi /c$ in the normal state. Besides the shift of VHS spectral weight at $k_{z} = 0$ below the Fermi level, electronic states around $k_{z}=\pi /c$ above the Fermi level also move down to deeper binding energies. Such a change of spectral weight over a large energy scale above and below the $E_{\rm F}$ plays a crucial role in the CDW physics of kagome metal CsV$_{3}$Sb$_{5}$. Acknowledgements. This work was supported by the National Key R&D Program of China (Grant No. 2017YFA0402901), the National Natural Science Foundation of China (Grant No. U2032153), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB25000000), and the Users with Excellence Program of Hefei Science Center of the Chinese Academy of Sciences (Grant No. 2021HSC-UE004). 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