# Inversion and Tunability of Van Hove Singularities
in $A$V3Sb5 ($A$ = K, Rb, and Cs) kagome metals

Sangjun Sim  Department of Physics, Korea Advanced Institute of Science and
Technology (KAIST), Daejeon 34141, Korea  Min Yong Jeong  Department of
Physics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon
34141, Korea  Hyunggeun Lee  Department of Physics, Korea Advanced Institute
of Science and Technology (KAIST), Daejeon 34141, Korea  Dong Hyun David Lee
Department of Physics, Korea Advanced Institute of Science and Technology
(KAIST), Daejeon 34141, Korea  Myung Joon Han<EMAIL_ADDRESS>Department
of Physics, Korea Advanced Institute of Science and Technology (KAIST),
Daejeon 34141, Korea

###### Abstract

To understand the alkali-metal-dependent material properties of recently
discovered $A$V3Sb5 ($A$ = K, Rb, and Cs), we conducted a detailed electronic
structure analysis based on first-principles density functional theory
calculations. Contrary to the case of $A$ = K and Rb, the energetic positions
of the low-lying Van Hove singularities are reversed in CsV3Sb5, and the
characteristic higher-order Van Hove point gets closer to the Fermi level. We
found that this notable difference can be attributed to the chemical effect,
apart from structural differences. Due to their different orbital
compositions, Van Hove points show qualitatively different responses to the
structure changes. A previously unnoticed highest lying point can be lowered,
locating close to or even below the other ones in response to a reasonable
range of bi- and uni-axial strain. Our results can be useful in better
understanding the material-dependent features reported in this family and in
realizing experimental control of exotic quantum phases.

††preprint: APS/123-QED

##  I Introduction

Recently discovered superconducting kagome metals $A$V3Sb5 ($A$ = K, Rb, and
Cs) have attracted wide attention because they host multiple quantum phases
intertwined with each other, including charge density wave (CDW),
superconductivity, and nematic order [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The
CDW instability is commonly observed in the temperature range below
$T_{\text{CDW}}\sim$ 78–102K. Below $T_{\text{CDW}}$, other types of
unconventional breakings of time-reversal [12, 13, 14, 15, 16, 17] and
rotational symmetry [5, 6, 7, 18, 19, 20, 21] have been reported as well. The
superconducting phase is stabilized at $T_{\rm SC}\sim$ 1–3K [1, 3].

While these features are common to these metals with some degree of
quantitative differences, distinctive features or material dependence have
also been clearly observed. For example, scanning tunneling microscopy (STM)
studies have reported a $4\times 1$ charge order in CsV3Sb5 [4, 22, 8] and
RbV3Sb5 [13, 23, 24], whereas it is not observed in KV3Sb5 [12, 5, 8]. The CDW
modulation pattern along the $c$-axis direction varies depending on the type
of alkali metal. To the best of our knowledge, the additional $4c_{0}$
modulation has only been reported in CsV3Sb5 [25, 26, 27, 28, 29].
Discrepancies were also noted between recent nuclear quadrupole resonance
(NQR) [30, 31] and angle-resolved photoemission spectroscopy (ARPES)
experiments [32, 33, 34, 35]. The superconducting phase diagram as a function
of doping appears to be different as well: A ‘second dome’ above 35% Sn doping
was reported for $A$ = Cs [36, 33], whereas KV3Sb5 and RbV3Sb5 seem to exhibit
a single-dome superconductivity [36, 37, 33]. A qualitative difference in
superconducting gap symmetry was also noticed. Guguchia et al. [38] recently
reported that KV3Sb5 and RbV3Sb5 undergo the nodal-to-nodeless gap transition
as a function of pressure, but this was not observed for $A$ = Cs [39, 40,
41]. In the pressure phase diagram, $T_{\rm SC}$ approaches zero below 20–28.8
GPa for KV3Sb5 and RbV3Sb5 [42]. According to Yu et al. [43], on the other
hand, the second superconducting dome for CsV3Sb5 persists up to a hydrostatic
pressure of 150 GPa.

For a clearer understanding of this intriguing kagome metal family, it is
therefore important to have detailed material-specific information as well as
further experimental clarification of the controversial issues. Here, we note
the principal roles of the Van Hove singularity (VHS) on which many previous
theoretical and experimental studies focused [44, 45, 46, 47, 48, 49, 50, 51,
52, 53]. As an important intrinsic characteristic of the kagome lattice band,
VHS is considered to be the main source of the wide variety of cascading
instabilities observed in this system. At least there are three VHSs have been
identified in the vicinity of the Fermi level ($E_{F}$) [54, 55, 56].
Apparently, several different phases can emerge out of VHSs in combination
with electronic correlation [44, 45, 46, 47, 48, 51, 52, 53]. Furthermore, a
detailed electronic structure analysis has revealed the intriguing nature of
VHSs, such as the unexpectedly important role of the out-of-plane Sb-$p$ state
and the different energy sequences depending on $A$ [54, 56, 57, 58, 59].
Still, this is far from being a clear understanding of the material’s specific
characters.

In this paper, we perform first-principles electronic structure calculations
based on density functional theory (DFT) together with a tight-binding
analysis. In particular, we focus on the different VHS structures in three
different materials of $A$ = K, Rb, and Cs. The energetic position and the
orbital character of the four VHSs are examined in a comparative manner. The
inverted sequence of two low-lying VHSs (one of which is known to have a high-
order nature and can possibly enhance the rotational symmetry-broken charge
order [49, 50, 52]) is found only in CsV3Sb5. This notable difference is
attributed solely to chemistry rather than the structural difference. Because
of the different orbital compositions, the four VHSs show qualitatively
different responses to uni- and bi-axial strain. The highest-lying VHS (named
VHS4), which was not previously noticed, can be lowered enough to be
comparable with other VHSs. Our results provide useful information to help
understand the intriguing $A$-cation dependence observed in this family and to
explore the possible control of their quantum phases.

##  II Computational details

We carried out first-principles DFT calculations using projector augmented-
wave (PAW) method [60] in the Vienna $ab$ $initio$ simulation package (VASP)
[61, 62, 63]. We used the generalized gradient approximation (GGA) as
parameterized by Perdew, Burke, and Ernzerhof (PBE) for the exchange-
correlation functional [64] and ‘Grimme’s DFT-D3’ method in the zero-damping
variant [65] for the van der Waals correction. Both lattice parameters and
internal coordinates were optimized until the residual forces became less than
1 meV/Å. For further electronic analysis, we also used the OpenMX [66, 67]
package. For OpenMX pseudo-atomic basis orbitals, we employed $s$2$p$2$d$1 for
K and Rb, $s$2$p$2$d$1$f$1 for Cs, $s$3$p$2$d$2 for V, and $s$2$p$1$d$2$f$1
for Sb. The atomic cutoff radii for the alkali atoms of K, Rb, and Cs were
10.0, 11.0, and 12.0 Bohr, respectively. The radii for V and Sb were chosen to
be 6.0, and 7.0 Bohr, respectively. The 12$\times$12$\times$6 $k$-point mesh
and the 400 Ry energy cutoff were used. We used our ‘DFTforge’ [68] code to
analyze the detailed electronic structure further. The charge analysis was
conducted with the Löwdin transformation [69]. Tight-binding analysis was
performed by constructing maximally localized Wannier functions using the
Wannier90 interfaced with OpenMX [70, 71]. In addition, the calculation
results were double-checked using the ‘HSE(Heyd-Scuseria-Ernzerhof)-06’
exchange-correlation functional [72]. For this calculation, 6 $\times$ 6
$\times$ 4 $\Gamma$-centered $k$-points and the energy cutoff of 500 eV were
used. The energy criterion was set to 10-7eV. As GGA gives rise to the better
agreement with ARPES data, we present it as our main result.

Figure 1: (a) Top view of kagome metal $A$V3Sb5 ($A$ = K, Rb, and Cs). The
yellow, red, gray, and black spheres represent $A$, V, Sb${}^{\text{out}}$,
and Sb${}^{\text{in}}$ atoms, respectively. The connecting red line shows the
V-kagome lattice. (b) Three-dimensional view of the $A$V3Sb5 crystal
structure. The magenta- and cyan-colored arrows indicate the local $x$\- and
$z$-axes, respectively. (c) The local structure of the VSb6 octahedron where
six Sb atoms (Sb${}^{\text{out}}$ and Sb${}^{\text{in}}$) surround a V atom.
The magenta- and cyan-colored arrows show the same local axes.

##  III Results and discussion

The crystal structure of $A$V3Sb5 belongs to the $P6/mmm$ space group (No.
191). As shown in Fig. 1, a V3Sb${}^{\text{in}}$ layer contains a V-kagome net
with Sb${}^{\text{in}}$ (in-plane Sb atoms) occupying the vacant center
positions of the V hexagons. Sb${}^{\text{out}}$ (out-of-plane Sb), forming a
honeycomb lattice, is located above and below V3Sb${}^{\text{in}}$. Alkali
metals are interposed between V3Sb5 layers so that the interlayer distance
depends on the size of $A$. Our geometry optimization gives rise to $a$ = $b$
= 5.423Å and $c$ = 8.886Å for K, $a$ = $b$ = 5.437Å and $c$ = 9.056Å for Rb,
and $a$ = $b$ = 5.453Å and $c$ = 9.308Å for Cs. These values are in good
agreement with experiments [1] as well as previous calculations [54, 56, 55].

Figure 2: (a–g) The orbital-projected band structures of (a) and (d) KV3Sb5,
(b) and (e) RbV3Sb5, and (c) and (f) CsV3Sb5. The shaded region around M in
(a)–(c) is highlighted (d)–(f). The Fermi level is set to zero. The dark
orange, purple, green, blue, and red colors represent the orbital characters
of Sb-$p$, V-$d_{3z^{2}\text{-}r^{2}}$, V-$d_{x^{2}\text{-}y^{2}}$,
V-$d_{xz\text{/}yz}$, and V-$d_{xy}$, sequentially. The orbital projection was
performed based on the local coordinates for all three kagome metals as shown
in Fig. 1(c). The arrows indicate the location of four Van Hove singularities
at M point. The labeling of each VHS follows the energetic sequence in
CsV3Sb5.

###  III.1 Electronic structure

The important common features of the three alkali metal variants in terms of
their electronic structure include the electron-like Fermi pocket at $\Gamma$,
multiple VHSs at M, and Dirac crossing at K point, which has been scrutinized
in a previous first-principles calculation and angle-resolved photoemission
spectroscopy (ARPES) [54, 73, 49, 74, 50, 55, 56]; see Figs. 2(a–c). Of
particular interest among them are the VHSs, as they are deemed to be
responsible for the multiple ordered phases cascading with lowering
temperature [44, 45, 47, 46, 48]. For convenience, we took CsV3Sb5 as a
reference and indexed the four VHSs in the increasing order of their energy;
VHS1, VHS2, VHS3, and VHS4 (see Fig. 2(c)). As shown in Figs. 2(a–c), VHS1 and
VHS2 are commonly located below $E_{F}$, whereas VHS3 and VHS4 are above. In
the literature, the highest-lying VHS4 has been on the periphery of attention,
as it is located far away from $E_{F}$.

To analyze the orbital characters of the VHSs, we chose a local axis
coordinate considering the geometry of the VSb6 octahedron [56]; see Fig.
1(c). Each orbital component is represented by different colors in Fig. 2.
VHS1 and VHS2 are composed mainly of V-$d_{xz\text{/}yz}$ and V-$d_{xy}$,
respectively, with their weights more than 60$\%$ for all three cases of K,
Rb, and Cs (see Table 1). The contribution from Sbout-$p$ at VHS1 and VHS2 is
relatively small; less than 5%. For VHS3 and VHS4, on the other hand, the
Sbout-$p$ contribution is sizable, as partly discussed in a previous study of
$A$ = Cs [56]. For $A$ = Cs, the Sbout-$p$ portion is about 30% at VHS3 and
VHS4. For $A$ = K (Rb), it is 26% (28%) and 23% (30%) at VHS3 and VHS4,
respectively.

Another important point is the relative position of VHS1 and VHS2 depending on
$A$. Let us first note that, in $A=$ K and Rb (Figs. 2(a) and (b)), VHS2 is
lower in energy than VHS1, whereas their positions are reversed in CsV3Sb5.
This was noted by Labollita and Botana [54]. In fact, VHS2 gradually moves
upward from $-$132 meV for K to $-$108 and $-$62 meV for Rb and Cs. On the
contrary, VHS1 shows a decreasing trend; $E_{\rm VHS1}=$ $-$90, $-$104, and
$-$126 meV for K, Rb, and Cs, respectively. In KV3Sb5 and RbV3Sb5, $E_{\rm
VHS2}$ is lower than $E_{\rm VHS1}$, and as a result, VHS1 is located closer
to $E_{F}$, which is not the case for Cs.

It is noted that the ‘higher-order’ nature of VHS2 has been highlighted in
previous studies [49, 75, 50, 56, 76]. The faster divergence of its density of
states and the weaker Fermi surface nesting were presumed to be important for,
e.g., nematic charge order and nodal superconductivity [77, 78, 79, 52]. Thus,
the closer location in CsV3Sb5 to the Fermi energy can possibly be related to
the observed difference in characteristics from those of KV3Sb5 and RbV3Sb5.

###  III.2 Chemical effect

To elucidate the underlying origin of this intriguing material dependence of
VHSs, it is important to examine the effects coming from chemistry and
structure. Figure 3 summarizes the calculated results of VHS positions with
the fixed atomic structures. Figures 3(a–c) correspond to the structures
optimized with $A$ = K, Rb, and Cs, respectively. In each structure, we
performed three different calculations, with the insertion of $A$ = K, Rb, and
Cs. It was noted that VHS1 moves downward in its energy as the alkali ion gets
heavier (with the fixed geometry; see blue lines), whereas VHS2 goes upward
(red lines). In the case of $A$ = K structure (Fig. 3(a)), for instance, VHS1
was found at $-$90, $-$110, and $-$145 meV with $A$-site substitution of K,
Rb, and Cs, respectively. VHS2 was at $-$132, $-$119, and $-$78 meV for K, Rb,
and Cs, respectively. This feature is commonly observed in all three
structural cases, as presented in Figs. 3(b) and (c). Interestingly, with $A$
= Cs, the relative positions of VHS1 and VHS2 were always found to be
reversed; namely, $E_{\rm VHS1}<E_{\rm VHS2}$ regardless of the structure. It
is, therefore, solely attributed to the ‘chemical effect’ of the alkali metals
on the energetics of the VHSs [80].

This finding was partly supported by our tight-binding analysis. Our
parameterization based on maximally localized Wannier functions gives rise to
the $A$-site $s$-orbital on-site energies of $-$1.325, $-$1.209, and $-$1.003
eV, respectively, for KV3Sb5, RbV3Sb5, and CsV3Sb5. This is consistent with
the increasing trend observed for VHS2, which has the non-negligible portion
of $A$-$s$ contribution (see Table 1). The same was also true for VHS3 and
$A$-$p$ whose on-site energies were found to be 0.301, 0.537, and 0.758 eV for
$A$ = K, Rb, and Cs, respectively. Although the chemical effect of $A$ cation
cannot wholly be captured by two tight-binding parameters, and the real band
dispersion is quite complicated near $E_{F}$, it provides useful information.

Given that VHS(s) in the vicinity of $E_{F}$ is the key to inducing multiple
quantum phases in this family [77, 78, 79, 52], the characteristic features of
CsV3Sb5 distinctive from KV3Sb5 and RbV3Sb5 can possibly be related to its VHS
energetics. It is also noteworthy that VHS2 is the only one presumed to be
higher-order in nature [49, 50]. A detailed and systematic further study
focusing on this relation could be a useful future direction.

Figure 3: (a-c) The calculated VHS positions in the fixed crystal structures. The optimized (a) KV3Sb5, (b) RbV3Sb5, and (c) CsV3Sb5 structures were used with different $A$ element substitutions. The Fermi level is set to zero for all cases. The blue, red, brown, and orange colors represent VHS1, VHS2, VHS3, and VHS4, sequentially.  Table 1: The calculated orbital portions of the four VHSs in $A$V3Sb5 ($A$ = K, Rb, and Cs). The orbital projections were performed with atomic radii of $R_{\rm V}=6.0$, $R_{\rm Sb}=7.0$, $R_{\rm K}=10.0$, $R_{\rm Rb}=11.0$, and $R_{\rm Cs}=12.0$ Bohr. The other orbital portions not shown here are approximately 30–35%. Unit (%) |  $A$-$s$ | V-$d_{xy}$ | V-$d_{x^{2}-y^{2}}$ | Sbin-$p$
---|---|---|---|---
VHS1 | VHS2 | VHS3 | VHS4 | VHS1 | VHS2 | VHS3 | VHS4 | VHS1 | VHS2 | VHS3 | VHS4 | VHS1 | VHS2 | VHS3 | VHS4
KV3Sb5 | 0.00 | 3.21 | 0.00 | 0.00 | 0.00 | 64.72 | 0.00 | 0.00 | 0.00 | 0.00 | 1.31 | 31.88 | 0.00 | 0.00 | 4.96 | 0.00
RbV3Sb5 | 0.00 | 2.84 | 0.00 | 0.00 | 0.00 | 64.49 | 0.00 | 0.00 | 0.00 | 0.00 | 0.77 | 32.82 | 0.00 | 0.00 | 5.45 | 0.00
CsV3Sb5 | 0.00 | 2.50 | 0.00 | 0.00 | 0.00 | 64.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.26 | 34.62 | 0.00 | 0.00 | 6.19 | 0.00
Unit (%) |  $A$-$p$ | V-$d_{xz/yz}$ | V-$d_{3z^{2}-r^{2}}$ | Sbout-$p$
VHS1 | VHS2 | VHS3 | VHS4 | VHS1 | VHS2 | VHS3 | VHS4 | VHS1 | VHS2 | VHS3 | VHS4 | VHS1 | VHS2 | VHS3 | VHS4
KV3Sb5 | 0.00 | 0.00 | 1.13 | 0.00 | 74.80 | 0.01 | 30.22 | 0.00 | 0.00 | 0.92 | 0.00 | 0.00 | 5.07 | 3.77 | 26.69 | 30.46
RbV3Sb5 | 0.00 | 0.00 | 1.09 | 0.00 | 74.77 | 0.01 | 29.31 | 0.00 | 0.00 | 1.23 | 0.00 | 0.00 | 4.89 | 4.25 | 27.99 | 30.77
CsV3Sb5 | 0.00 | 0.00 | 1.82 | 0.00 | 74.59 | 0.01 | 28.17 | 0.00 | 0.00 | 1.95 | 0.00 | 0.00 | 4.89 | 5.42 | 28.94 | 29.64

###  III.3 Structural effect

The structure effect on the VHS positions can also be seen in Figs. 3(a–c).
Let us consider the case of $A$ = K, for example. The energy location of VHS2
becomes higher in the optimized structure with a heavier alkali-metal; $E_{\rm
VHS2}$ = $-$132, $-$120, and $-$106 meV in the optimized structure of KV3Sb5,
RbV3Sb5, and CsV3Sb5, respectively. The increasing feature is also observed in
VHS1. On the other hand, the behavior of VHS3 and VHS4 is the opposite;
namely, their energies get lower in the geometry optimized with heavier alkali
atoms. The downward shifting of VHS4 is significantly greater, and indeed, the
Sbout-$p$ portion is greatest in VHS4. For detailed quantitative information
on the orbital components in each VHS, see Table 1. Further analysis based on
maximally localized Wannier functions also showed that the tight-binding
hopping parameter between V-$d_{x^{2}\text{-}y^{2}}$ and Sbout-$p$ exhibits
sizable dependence on the corresponding structural change.

Figure 4: (a–c) The energetic positions of four VHSs in $A$V3Sb5 ($A$ = K, Rb,
and Cs) in response to compressive (negative sign) and tensile (positive sign)
strain up to $\pm$3% along $c$-axis. (d–f) The calculated positions of the
VHSs under biaxial compressive and tensile strains along the $a$-$b$ axis. The
blue, red, brown, and dark orange colors represent VHS1, VHS2, VHS3, and VHS4,
respectively. The Fermi energy is set to be zero.

###  III.4 Tuning the energy levels of VHSs by strain

The above result motivated us to investigate the possible use of strain to
control the VHS positions. Figures 4(a–c) show the calculated VHSs in response
to the $c$-lattice parameter change. While this uniaxial strain effect is
relatively weak for VHS1 and VHS2, VHS3 and VHS4 were significantly affected,
and both gradually shifted down toward $E_{F}$ as the $c$ lattice parameter
increased. While this result is consistent with a previous study by Consiglio
et al. [55], our analysis provides additional information and insight,
especially for VHS4. Taking KV3Sb5 as an example, VHS4 is located at +288 meV
at zero strain, and it decreases to $\sim$ +244 meV at +3% tensile strain (see
Fig. 4(a)). On the other hand, VHS1 and VHS2 are originally located at $-$90
meV and $-$132 meV, respectively, and they move to $-$101 meV and $-$132 meV
at +3% strain. Once again, this notably different response between the two
lower (VHS1 and VHS2) and higher energy VHSs (VHS3 and VHS4) is attributed to
the different orbital characters involved in their formation; see Table 1.
Further control may be feasible by employing biaxial strain. Figs. 4(d–f)
shows that a qualitatively different response in VHS positions can be obtained
with biaxial strain. From compressive to tensile strain (along the $a$ and
$b$-axis), VHS1, VHS2, and VHS3 moved toward the higher energy, whereas the
VHS4 position got lower, just as in the uniaxial case. The response of VHS3 is
so sizable that its relative position to VHS4 is eventually reversed under a
large tensile strain limit in RbV3Sb5 and CsV3Sb5; see Figs. 4(e) and (f).

##  IV Conclusions

In summary, we investigated the alkali-metal-dependent electronic structure of
$A$V3Sb5 ($A$ = K, Rb, and Cs) to understand the previously reported
intriguing material dependence. Our calculation results show that the
energetic positions of low-lying Van Hove singularities in CsV3Sb5 are
distinctive from those of $A$ = K and Rb, and the characteristic higher-order
Van Hove point is located closer to $E_{F}$. Detailed electronic analyses
revealed that this notable difference can be attributed to chemical effects
apart from structures. Additionally, the different orbital compositions lead
the four VHSs to exhibit qualitatively different responses to strain. In
particular, the highest-lying VHS4 can be lowered enough to be comparable with
other VHSs. This distinct response of VHS4 to strain demonstrates the useful
potential for strain engineering of these materials. Our findings provide
valuable insights into understanding the $A$-cation dependence observed in
experiments and shed new light on understanding recent experiments and
approaches to controlling the intertwined quantum phases in this material
family.

##  V Acknowledgements

We thank Sergey Savrasov for the useful discussion. This work was supported by
the National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIT) (Grant Nos. 2021R1A2C1009303 and RS-2023-00253716).

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