††thanks: These authors contributed equally

# Fermi surface mapping and the nature of charge density wave order in the
kagome superconductor CsV3Sb5

Brenden R. Ortiz†<EMAIL_ADDRESS>Materials Department and California
Nanosystems Institute, University of California Santa Barbara, Santa Barbara,
CA, 93106, United States  Samuel M. L. Teicher  Materials Department and
California Nanosystems Institute, University of California Santa Barbara,
Santa Barbara, CA, 93106, United States  Linus Kautzsch  Materials Department
and California Nanosystems Institute, University of California Santa Barbara,
Santa Barbara, CA, 93106, United States  Paul M. Sarte  Materials Department
and California Nanosystems Institute, University of California Santa Barbara,
Santa Barbara, CA, 93106, United States  Jacob P. C. Ruff  CHESS, Cornell
University, Ithaca, NY, 14853, United States  Ram Seshadri  Materials
Department and California Nanosystems Institute, University of California
Santa Barbara, Santa Barbara, CA, 93106, United States  Stephen D. Wilson
<EMAIL_ADDRESS>Materials Department and California Nanosystems
Institute, University of California Santa Barbara, Santa Barbara, CA, 93106,
United States

###### Abstract

The recently discovered family of AV3Sb5 (A: K, Rb Cs) kagome metals possess a
unique combination of nontrivial band topology, superconducting ground states,
and signatures of electron correlations manifest via competing charge density
wave order. Little is understood regarding the nature of the charge density
wave (CDW) instability inherent to these compounds and the potential
correlation with the accompanying onset of a large anomalous Hall response. To
understand the impact of the CDW order on the electronic structure in these
systems, we present quantum oscillation measurements on single crystals of
CsV3Sb5. Our data provides direct evidence that the CDW invokes a substantial
reconstruction of the Fermi surface pockets associated with the vanadium
orbitals and the kagome lattice framework. In conjunction with density
functional theory modeling, we are able to identify split oscillation
frequencies originating from reconstructed pockets built from vanadium
orbitals and Dirac-like bands. Complementary diffraction measurements are
further able to demonstrate that the CDW instability has a correlated phasing
between neighboring V3Sb5 planes. These results provide critical insights into
the underlying CDW instability in AV3Sb5 kagome metals and support minimal
models of CDW order arising from within the vanadium-based kagome lattice.

††preprint: APS/123-QED

##  I Introduction

While kagome insulators are traditionally sought as potential hosts of quantum
spin liquid states and laboratories for highly frustrated magnetism [1, 2, 3,
4, 5], kagome metals are equally interesting due to their potential to host
topologically nontrivial electronic states interwoven with local electronic
symmetry breaking. At a single-orbital tight binding level, the kagome
structural motif naturally gives rise to an electronic structure with Dirac
points and a flat band that together provide the potential for an interplay
between topologically nontrivial surface states and substantial electron
correlation effects. A wide array of instabilities have been predicted,
ranging from bond density wave order [6, 7], charge fractionalization [8, 9],
spin liquid states [10], charge density waves (CDW) [11] and superconductivity
[6, 12].

The electron filling within the kagome framework controls the formation of a
wide variety of predicted electronic instabilities. For band fillings near
$5/4$ electrons per band [6, 13, 14, 15, 16], a Van Hove singularity is formed
at the Fermi level due to the presence of saddle points along the zone edge.
Excitations between these saddle points can lead to CDW order, and, in some
limits, unconventional superconductivity. The recently discovered class of
AV3Sb5 (A: K, Rb Cs) kagome metals [17] are potential realizations of this
physical mechanism with each member exhibiting thermodynamic anomalies
associated with CDW order [18, 19, 20, 21, 22, 23] followed by the onset of
superconductivity at lower temperatures [20, 21, 24].

The CDW instability in AV3Sb5 compounds seemingly competes with
superconductivity [25, 26] and presages the formation of a potentially
unconventional superconducting ground state [21, 26]. However, the microscopic
origin of the CDW remains an open question. Concomitant to the onset of CDW
order, an exceptionally large anomalous Hall effect (AHE) appears [27, 28],
despite the absence of detectable local moments or magnetic correlations [29].
While the normal state electronic structure is a $\mathbb{Z}_{2}$ topological
metal [20, 21] and topologically-protected surface states are predicted close
to the Fermi level [20], below the CDW transition recent scanning tunneling
microscopy (STM) data [18] and theoretical proposals [16] have suggested the
formation of a chiral CDW order parameter. This chiral CDW, endemic to the
kagome lattice, is proposed to break time reversal symmetry and generate a
large Berry curvature, potentially accounting for the AHE. To date, however,
data directly linking the onset of CDW order with reconstruction of vanadium
orbitals associated with the kagome lattice in AV3Sb5 is lacking. Similarly,
the applicability of minimal, single orbital tight binding kagome models in
multiband AV3Sb5 compounds remains an open question.

Specifically, STM and diffraction experiments have observed charge order with
an in-plane q $=(0.5,0.5)$ wave vector in KV3Sb5 [18] and CsV3Sb5
corresponding to 3Q CDW order. A kagome “breathing” mode can give rise to
candidate distortions such as the “Star of David” (see Figure 1) and its
inverse structure [30], and recent studies have shown strong electron-phonon
coupling in KV3Sb5 promoting such a distortion [31]. Native electronic
instabilities promoting CDW order along this wave vector have long been
predicted in Kagome models at select fillings near Van Hove singularities [6,
13, 14, 15, 16], suggesting that a minimal model built around the kagome
planes of these materials may capture the essential physics governing their
unconventional electronic properties. Notably, additional features such as
unidirectional charge stripe order [19] also seemingly coexist with the 3Q CDW
state, further connecting the underlying interactions to stripe/nematic
instabilities predicted within a kagome network [16].

Figure 1: CsV3Sb5 is a layered, exfoliatable, kagome metal consisting of a
structurally perfect lattice of vanadium at room temperature. Upon cooling
below $T^{*}=93$ K, CsV3Sb5 exhibits charge density wave order. A concurrent
structural distortion emerges as well, which is suspected to be related to the
kagome “breathing mode.” Upon distortion and relaxation in both the positive
and negative displacements, the “breathing mode” gives rise to the “Star of
David (SoD)” and “Tri-Hexagonal (TrH)” candidate structures.

Here we investigate the origins of the 3Q CDW order in AV3Sb5 kagome compounds
via study of Shubnikov-De Haas (SdH) quantum oscillations in the
magnetotransport data of CsV3Sb5, which has the most pronounced CDW
instability within the family. By correlating quantum oscillation data with
DFT models of the breathing distortion of the kagome lattice, we are able to
observe the effect of the CDW on the electronic structure. Specifically, we
demonstrate that a series of low-frequency quantum oscillations originate from
CDW-reconstructed vanadium orbitals and exhibit transport consistent with the
Dirac-like features (high mobility, low cyclotron mass) of the kagome lattice.
The multiplicity and frequencies associated with these vanadium orbits are
shown to originate from a reconstructed Fermi surface with small pockets
linked to folded, vanadium-dominated bands. We further demonstrate that the
CDW instability is three-dimensional in nature with a resulting
2$\times$2$\times$4 superstructure. Together, our results provide direct
evidence that the in-plane CDW is derived from vanadium orbitals which
comprise the kagome lattice in AV3Sb5 and validate recent efforts to map the
core interactions in these materials to minimal tight-binding models built
from a two-dimensional kagome network.

##  II Methods

###  II.1 Synthesis

Single crystals of CsV3Sb5 were synthesized from Cs (liquid, Alfa 99.98%), V
(powder, Sigma 99.9%) and Sb (shot, Alfa 99.999%). As-received vanadium powder
was purified in-house to remove residual oxides. Due to extreme reactivity of
elemental Cs, all further preparation of CsV3Sb5 was performed in an argon
glovebox with oxygen and moisture levels $<$0.5 ppm. Single crystals of
CsV3Sb5 were synthesized using the self-flux method. The flux is a eutectic
mixture of CsSb Cs3Sb7[32] mixed with VSb2. Elemental reagents were milled in
a pre-seasoned tungsten carbide vial to form a composition which is 50 at.%
CsxSby eutectic and approximately 50 at.% VSb2. The fluxes were loaded into
alumina crucibles and sealed within stainless steel jackets. The samples were
heated to 1000°C at 250°C/hr and soaked there for 24 h. The samples were
subsequently cooled to 800°C at 100°C/hr and then further to 500°C at 2°C/hr.
Once cooled, the crystals are recovered mechanically. Crystals are hexagonal
flakes with brilliant metallic luster. Samples can range up to 1 cm in side
length and up to 1 mm thick. Elemental composition of the crystals was
assessed using energy dispersive x-ray spectroscopy (EDX) using a APREO C
scanning electron microscope.

###  II.2 Electrical transport measurements

Electronic transport measurements were performed using a Quantum Design 14 T
Dynacool Physical Property Measurement System (PPMS). For temperatures $>$1.8
K, the rotator option was used to collect angle-dependent and temperature-
dependent data. Crystals are exfoliated to remove any surface contaminants,
and electrical contacts were made in a standard 4-point geometry using gold
wire and silver paint. Crystals were initially mounted such that the c-axis
was parallel to the field (flat plates mounted flush on resistivity stage). An
alternating current of 8 mA and 12.2 Hz was driven in the $ab$-plane.

###  II.3 Electronic structure calculations

DFT simulations of the electronic structure of CsV3Sb5 unit cell were
performed in VASP v5.4.4 using identical parameters to several recently
reported studies [20, 21, 33]. We employed the PBE functional [34] with D3
corrections [35], a 500 eV plane wave energy cutoff, a $\Gamma$-centered
$11\times 11\times 5$ $k$-mesh, and the recommended PAW pseudopotentials for
$v$5.2. Spin orbit coupling was activated for all calculation steps except for
structural relaxation. All calculations were completed with an energy
convergence cutoff of 10-6 eV or better. The unit cell was relaxed, as
previously described [20], with final $a$ and $c$ lattice parameters of 5.45 Å
and 9.35 Å; in good agreement with the room-temperature values determined by
X-ray diffraction, 5.52 Å and 9.36 Å, respectively.

Wannier90 [36] was used to fit Wannier functions (Cs s, p; V s, p, d; Sb s, p;
with a frozen fitting window $E_{\text{F}}\pm 2$ eV) and interpolate unit cell
Fermi surfaces on a $101\times 101\times 101$ grid. Extremal orbits were
determined using Serendipity,111Serendipity is in development, pending an
open-source release. Interested parties can contact Samuel Teicher:
<EMAIL_ADDRESS>a new code that builds on the algorithms developed by Rourke
and Julian[38] with additional symmetry and interactive visualization tools
enabled by the python packages spglib,[39] trimesh,[40] and plotly[41].

Figure 2: (a–d) 2-dimensional slices through reciprocal space on half-integer
Bragg planes at 12 K, highlighting superlattice peaks in CsV3Sb5. (e–h) Line
cuts through the 2D data, highlighting the periodicity of the superlattice
peaks. Blue data are background subtracted traces showing the difference
between the 12 K and 105 K data. Red data shows the raw 105 K data, confirming
that the superlattice peaks evolve as charge density wave order emerges at
$T^{*}\sim 94$ K. Our data indicates that the superlattice in CsV3Sb5 is
described by a wave vector of (0.5, 0.5, 0.25).

Supercell calculations were completed on a $2\times 2\times 1$ supercell
simulated using identical parameters and a $5\times 5\times 5$ $k$-mesh that
was distorted along the M${}_{1}^{+}$ P3 irreducible representation in
ISODISTORT.[42, 43] M${}_{1}^{+}$ P3 involves four distinct distortion modes,
including B3u and B2u V sublattice modes and Ea and A1a Sb2 (antimonene layer)
sublattice modes. Cs and Sb1 (kagome layer) atomic positions are unaffected.
Negative magnitudes of the B3u, B2u, and Ea modes recreate the “Star of David”
(SoD) while positive magnitudes recreate the inverse “Tri-Hexagonal” (TrH)
structure. We initialized super cell structural relaxations with four
different sets of mode magnitudes, (B3u, B2u, Ea, A1a)=0.3
Å$\cdot${($+$,$+$,$+$,$+$); ($+$,$+$,$+$,$-$); ($-$,$-$,$-$,$+$);
($-$,$-$,$-$,$-$)}, testing both the SoD and TrH structures, while
additionally trialing both positive and negative magnitudes of the A1a mode,
which corresponds to $c$-axial buckling of the antimonene lattice.

Relaxation proceeded in three steps: volumetric optimization, followed by
adjustment of the ionic positions, and finally a free relaxation of the super
cell lattice parameters and ionic positions simultaneously. Ultimately, the
two SoD configurations relaxed to a similar structure, and the same was true
for the two TrH structures. In the final SoD structure we find a positive A1a
mode, corresponding to Sb2 atoms moving further from the smaller triangles and
closer to the larger triangles of the SoD structure. In the final TrH
structure, we find that a negative A1a mode is favored, in which the Sb2 atoms
similarly move away from the smaller V-V triangle units. In each case,
$c$-axial buckling of the Sb2 layer is minuscule, 0.005$c$ and 0.002$c$ for
SoD and TrH, respectively.

The SoD and TrH distorted super cells are energetically favored over the
undistorted unit cell by 4.7 meV and 13.5 meV per formula unit, respectively,
consistent with recently reported simulations [30]. Super cell band unfolding
employed a modified version of VaspBandUnfolding [44]. Unfolded Fermi surface
slices in Fig. 7 were calculated on a $51\times 51$ BZ mesh. Cubic spline
interpolation was used for smoothing/upsampling prior to projecting onto the
larger display range.

Fermi levels for the electronic structure calculations in Fig. 3 and Fig. 7
were determined based on prior experiment. Additional discussion is provided
in the supporting material.[45]

###  II.4 X-ray diffraction measurements

High dynamic range x-ray diffraction maps were collected at the QM2 beamline
at CHESS. The incident x-ray energy was 50 keV, selected using a double-bounce
diamond monochromator. Temperature was controlled by bathing the small single
crystal samples inside a stream of cold flowing helium gas. Diffraction was
recorded in transmission though the sample using a 6 megapixel photon-counting
pixel-array detector with a silicon sensor layer. Full 360 degree sample
rotations, sliced into 0.1 degree frames, were indexed to the high-temperature
crystal structure and transformed to reciprocal space. Some elements of the
data reduction employed the NeXpy software package. Crystal structures were
visualized in VESTA.[46]

The x-ray diffraction dataset contains a temperature-independent fluorescence
background from the K edges of Sb and Cs. To increase contrast between
background and the superlattice peaks, the 2D cuts of data highlighting the
superlattice peaks (12 K) include a subtraction of the background (normal
state) data at 105 K. A small mosaic is present in the sample, resulting in
fine splitting of peaks at high HKL. To increase the clarity of the data for
linecuts, data points were binned and high-pass filter was applied to reduce
background noise.

##  III Experimental Results

###  III.1 Crystal and Electronic Structures

The AV3Sb5 (A: K, Rb Cs) family of kagome metals are layered, exfoliable
materials consisting of V3Sb5 slabs intercalated by alkali metal cations. The
vanadium sublattice forms a perfect kagome lattice under ambient conditions
(Figure 1). CsV3Sb5 is the terminal endpoint of the alkali-metal series, shows
the highest superconducting transition ($T_{c}=2.5$ K), and an onset of CDW
order below $T^{*}=94$ K [21]. The CDW is accompanied by a weak structural
distortion manifest as a superlattice of Bragg scattering in synchrotron x-ray
diffraction data.

Early measurements within the $L=0$ scattering plane resolved only q
$=(0.5,0,0)$ and $(0,0.5,0)$-type superlattice reflections [20], which is
seemingly at odds with recent STM reports of 2$\times$2 supercells associated
with 3$\bf{Q}$ charge order. To address this, an expanded exploration of
superlattice peaks was conducted at finite $L$-values with the results shown
in Fig. 2. In this higher resolution data, a more complex, three-dimensional
superlattice structure is observed that is best indexed by a q
$=(0.5,0.5,0.25)$ wave vector. (0.5, 0.5)-type superlattice reflections are
largely not resolvable in the $L=0$ plane, accounting for the initial failure
to index them. The in-plane component of the superlattice modulation agrees
with the 3Q structure observed in local probes. Crucially, the wave vector
identified in this expanded survey also indicates a modulation of the in-plane
distortion along the $c$-axis (interplane phasing).

Considering the in-plane distortions allowed on an idealized kagome lattice,
the kagome “breathing” mode is one likely candidate, and this mode matches
preliminary conclusions drawn from STM and DFT studies of KV3Sb5 [18, 30, 31].
As shown in Figure 1, the structure can distort between two potential
candidates: (1) the SoD distortion and (2) the TrH distortion. As the
interlayer interactions are expected to be weak in CsV3Sb5, we have neglected
the impact of the $c$-axis component of the superlattice on the electronic
structure. Reasonable candidate structures (M${}_{1}^{+}$ P3 irrep.) matching
the TrH and SoD distortions were selected for DFT relaxation and band
structure calculations. Our DFT studies find that both the SoD and TrH
distortions are slightly favored over the undistorted structure; by 4.7
meV/f.u and 13.5 meV/f.u, respectively. DFT-relaxed structures are shown
alongside the experimental, undistorted crystal structure in Figure 1.

Despite the low stabilization energy of the distorted structures relative to
the parent structure, the predicted vanadium lattice distortions are
significant. The V-V bond lengths are all equally 2.72Å in the parent
structure, and transform to 3 distinct lengths: 2.65Å 2.75Å and 2.84Å for the
SoD distortion, and 2.58Å 2.68Å and 2.82Å for the TrH distortion.
Concurrently, the Sb graphitic sublattice is fragmented into individual
hexagons and also hosts a slight buckling in the $c$-direction for both
distorted structure types.

Figure 3: (a) Unfolded electronic structure of the “Star of David (SoD)” and
“Tri-Hexagonal (TrH)” distortions in CsV3Sb5 with the undistorted electronic
structure superimposed for comparison (black). Largest perturbation to the
structure appears near the Dirac-like bands near M. (b) Close-up of the
changes near the M-point, highlighting the new bands that appear as a result
of the CDW and the associated structural distortion.

Figure 3 shows the effect of the two superlattice types (SoD and TrH) on the
ab initio electronic structure of CsV3Sb5. The resulting band diagrams were
unfolded for comparison to the undistorted electronic structure shown in
previous works [17, 20]. The heat map shows the relative projections of the
electronic states after the unfolding, and the black bands are the undistorted
structure of CsV3Sb5. In the low-temperature distorted state, the electronic
structure is largely unperturbed, particularly the central band about $\Gamma$
which derives from the Sb $p$-orbitals. However, the bands near the M-point,
which are the relevant Dirac-like bands associated with the vanadium
$d$-orbitals, are altered significantly.

Figure 3(b) shows expanded views of the SoD and TrH electronic structures in
the vicinity of the $M$-points. An orbital decomposed band diagram (“orbital
bands”) further identifies these states as originating primarily from vanadium
orbitals [45]. The emergence of the CDW and the resulting superlattice
therefore has a clear effect on the electronic structure near $E_{\text{F}}$
for the vanadium orbitals comprising the Dirac-like crossings. This effect has
a significant impact on the Fermi surface and is expected to impact transport
sensitive to topographical changes at $E_{\text{F}}$. Experimental detection
of these effects are discussed in the next section.

Figure 4: (a) Oscillatory component of the magnetoresistance extracted from
the temperature-dependent SdH data collected on single crystals of CsV3Sb5
mounted with the c-axis parallel to the magnetic field (0°). (b) A high-
resolution scan at 1.8 K shows significant contributions from high-frequency
modes, particularly at the peaks and troughs of the general oscillatory
behavior. (c) Magnified view of the high-frequency features, providing visual
confirmation for multiple high-frequency modes. (d,e) Fourier transformation
of the quantum oscillation data, showing the low- and high-frequency
components of the power spectrum. The 9 unique frequencies have been assigned
greek letters. (f) The magnitude of the Fourier coefficients is used in
conjunction with the Lifshitz-Kosevich (LK) formula to extract the cyclotron
“effective mass.” All low-frequency modes show very low effective masses,
consistent with transport originating from the Dirac-like crossings at M.

###  III.2 Quantum Oscillation Measurements

While early angle-resolved photoemission (ARPES) measurements were unable to
resolve a reconstruction of the Fermi surface below the CDW transition, an
alternate probe of the bulk band structure is the measurement of quantum
oscillations in high-field electron transport measurements. Crystals of the
AV3Sb5 kagome metals are high mobility metals with low residual resistivity
($\sim 0.1~{}\mu\Omega$-cm) values [17, 21, 27], rendering quantum oscillation
measurements an appealing probe for exploring the low temperature electronic
structure.

Figure 5: (a) Raw electronic resistivity as a function as field and angle,
where the angle is defined as between the c-axis and the magnetic field.
Oscillations are clearly visible $>$2 T. The oscillations appear dampened with
angle, vanishing for $\theta>$60°. (b) Fourier transform of angle-resolved SdH
data, showing primary low-frequency contributions to the power spectrum. The
high-frequency contributions are suppressed rapidly by rotation. (c) The most
prominent frequencies ($\beta$, $\delta$) are shown as a function of rotation
angle. The frequencies were estimated from (b) using Gaussian functions to
approximate both the broadening and shortening of peaks. These frequencies
exhibit a delayed onset of the angular dependence, consistent with Dirac-like
pockets.

Figure 4 presents a series of temperature-dependent quantum oscillation
measurements on stoichiometric crystals of CsV3Sb5 with RRR $\approx~{}80$ and
with the crystals’ $c$-axis parallel to the magnetic field and current driven
in the $ab$-plane. The normal “background” magnetoresistance (MR) was modeled
using a power function $\rho_{0}\approx\alpha H^{\beta}+\gamma$ fit over the
range from 4 T to 14 T. The oscillatory component of the MR was then isolated
by subtracting the background MR $\Delta\rho(H)=\rho(H)-\rho_{0}(H)$. Figure
4(a) shows the oscillatory component of the magnetoresistance as a function of
temperature and field, and oscillations are seen to persist up to $25$ K. The
oscillation pattern is relatively complex, with multiple harmonics visible by
inspection.

Quantum oscillation data collected at 1.8 K is isolated in Figure 4(b), where
multiple frequency components have been noted by Greek letters. At higher
fields, we further highlight several regions that show contributions from
additional, higher frequency oscillations in Figure 4(c). Due to the presence
of multiple closely spaced frequencies (discussed in the next paragraph), we
have grouped similar frequency components together in this initial inspection
of the subtracted data. All frequencies persist between different measurements
and different crystals.

Turning first to the low frequency spectrum, Figure 4(d) shows the Fourier
transform of the data at multiple temperatures with $H\leq 250$ T. Four well-
defined frequencies appear ($\alpha,\beta,\gamma,\delta$). At higher
frequencies with $400$ T$<H<2000$ T, the Fourier transform in Figure 4(e)
shows an additional five well-defined frequencies
($\epsilon,\xi,\eta,\kappa,\lambda$). The peak designated $\epsilon$
technically appears as two sharp peaks; however, this effect is likely
extrinsic, and we currently consider the $\epsilon$ peak as the average of
these two peaks.

While the modes above 250 T vanish quickly with increasing temperature above 2
K, the low frequency modes remain well-defined up to 25 K. The temperature
dependence of the Fourier coefficients of the $\alpha$, $\beta$, $\gamma$ and
$\delta$ orbits are shown in Figure 4(f). The cyclotron “effective mass”
($m^{*}_{\text{eff}}$) can be extracted using the approximate Lifshitz-
Kosevich (LK) form $a_{i}(T)\approx X/(B\sinh{X/B})$, where
$X=\alpha\,m^{*}_{\text{eff}}T$. Here, $B$ is the magnetic flux density and is
typically selected as the mean field within the FFT window. The parameter
$\alpha$ is a constant defined as 14.69 T/K. The resulting
$m^{*}_{\text{eff}}$ values are low for these low frequency orbits—nearly
$1/10$ of the free electron mass—consistent with transport originating from
the Dirac modes expected near the $M$-point.

Whereas the temperature-dependence of the quantum oscillations provides
information regarding the scattering, lifetime, and effective mass of the
carriers, the angular-dependence can provide information regarding the
topography of the Fermi surface. Figure 5(a) presents a series of angle-
dependent quantum oscillation measurements collected at 1.8 K where
$\theta=0^{\circ}$ denotes the $c$-axis parallel to the $H$-field. The Fourier
transforms of the data in Figure 5(a) are shown in Figure 5(b), and the
frequencies of the $\delta$ and $\beta$ orbits are plotted as a function of
angle in Figure 5(c). The $\alpha$ and $\gamma$ orbits shift and quickly
convolve into neighboring frequencies with increasing angle, precluding their
analysis at finite $\theta$.

Conceptually, the orbits that generate the oscillations can be imagined as
slices through the Fermi surface at different approach angles. ‘Extremal’
cross-sections with the largest and smallest cross-sectional area will
generate distinct oscillation frequencies. For example, a perfectly spherical
Fermi pocket exhibits no angular dependence and only one frequency from the
circular cross-section. A strongly anisotropic pocket (e.g. those from 2D
Dirac cones) would show a strong dependence with angle, as oblique slices
through a cylinder become progressively larger as the angle increases. The
sharp upturns seen above 40∘ in the $\delta$ and $\gamma$ orbits are
consistent with orbits derived from strongly anisotropic pockets, and—as we
will demonstrate in the next section—are best ascribed to electrons within
Dirac-like features associated with the vanadium kagome lattice.

###  III.3 Fermi Surface Topography and Frequency Correlation

The transport data shown in Figures 4 and 5 reveal a complex superposition of
quantum oscillations originating from multiple portions of the Fermi surface.
In order to identify how the CDW and the associated crystallographic
distortions perturb the Fermi surface, the oscillation frequencies (i.e.
enclosed Fermi surface pockets) seen experimentally must be correlated to the
DFT-calculated Fermi surfaces. To do so, we first examine the undistorted
Fermi surface in the context of the possible extremal orbits.

To determine the orbits accurately, the Fermi energy needs to be well-defined.
It is worth taking a moment to review the spread of $E_{\text{F}}$’s reported
in the current literature, as the Dirac-like nature of the bands near
$E_{\text{F}}$ renders rapid changes in the sizes of electron pockets with
relatively minor shifts in Fermi energy. Initial DFT studies [17, 21, 20, 18,
27] found Fermi levels slightly below those determined experimentally by ARPES
[10, 20] and STM measurements [18, 19]. For simplicity, we will refer to these
earlier results as $E_{\text{F,lit}}$ and $E_{\text{F,exp}}$ respectively.
Recent DFT studies have since provided self-consistent results closer to
experimental values [30]; we refer to this value as $E_{\text{F,DFT}}$, which
agree with the self-consistent DFT calculations presented in this work.

Figure 6 shows the calculated Fermi surface using the undistorted CsV3Sb5
structure where $E_{\text{F}}=E_{\text{F,exp}}$. Despite already motivating
that the underlying electronic structure is perturbed by the influence of the
CDW, it is nevertheless instructive to first understand the undistorted Fermi
surface. The orbits identified by Serendipity are shown in Figure 6(a) in
grey. 2D slices of the Fermi surface at $k_{z}=0$ and $k_{z}=0.5$ are provided
for a more convenient comparison, since all of the extremal orbits at
$E_{\text{F,exp}}$ occur on these two high-symmetry planes. The Fermi surface
maps reveal a variety of possible extremal orbits, confirming that the quantum
oscillations in CsV3Sb5 should contain multiple frequencies, though the 9329 T
and 12846 T frequencies are significantly above the experimental range of
detection in 4(d,e) and Figure 5(b).

Figure 6: (a) The Fermi surface for undistorted CsV3Sb5 calculated with
$E_{\text{F}}=E_{\text{F,exp}}$ shows a variety of potential orbits. Orbits
calculated by the Serendipity python package are shown highlighted in grey.
(b,c) For clarity, isoenergy contours at $k_{z}=0$ and $k_{z}=0.5$ for the
$E_{\text{F}}=E_{\text{F,exp}}$ surface are shown with all unique extremal
orbits marked. Several high- and low-frequency orbits can be identified. The
Fermi surface pocket of origin is indicated through the orbit color, which
will be used for comparison throughout our discussion. While there are clearly
multiple frequencies in the predicted spectrum, note that there is only 1
symmetry unique (39 T) low frequency-mode, which seems at odds with our
experimental observations. Further, for frequencies $400<f<2000$ T, there are
only 4 predicted modes, as opposed to the 5 experimentally observed
components.

First considering the measurable high frequencies (250 T $<f<$ 2000 T), we
find only four extremal orbits (B-I, 699 T; A-I, 1123 T; A-II, 1831 T; and
C-II, 1967 T) to match to five measured values ($\eta\approx$580 T,
$\mu\approx$860 T, $\lambda\approx$1370T, $\delta\approx$1700 T,
$\varepsilon\approx$1930 T). At lower frequencies, the agreement is much
worse: only one calculated low frequency orbit (C-I, 39 T) is found to compare
with four low frequency oscillations observed in experiments
($\alpha\approx$11 T, $\beta\approx$28 T, $\gamma\approx$74 T, and
$\delta\approx$90 T). A table of the calculated (DFT) frequencies at
$E_{\text{F}}=E_{\text{F,exp}}$, along with the associated cyclotron masses
$m_{\text{cyc}}$ has been included in the supplementary supporting material
(SFig. 3).[45]

Figure 7: (a) Fermi surfaces for the SoD and TrH distortions in CsV3Sb5
generated from the unfolded supercell band structures. Undistorted Fermi
surface (black lines) contours are shown for comparison. A significant
reconstruction of the V $d$ states occurs. Most high-frequency modes are
preserved, though the C-II orbit (dashed) is gapped in the SoD structure. (b)
Schematic description of Fermi surface reconstruction. Grey orbits are well
above measurable frequencies. Band reconstruction in the TrH supercell
introduces one additional triangular orbit around the $M$-point while
preserving higher frequency native V $d$ and Sb $p$ orbits. Notably, the
distortion also introduces 3 smaller, Dirac-like orbits (purple) consistent
with our measurements.

Given the significant CDW band reconstruction presented in Fig. 3, a low-
temperature Fermi surface modification in CsV3Sb5 is expected. Figure 7
displays unfolded Fermi surface slices for the SoD and TrH supercells at
$E_{\text{F,exp}}$, demonstrating this reconstruction. While the central Sb
$p$-orbits are largely unaffected by the CDW, V $d$-bands gap and change the
Fermi surface. On the $k_{z}=0.0$ plane, the single, large B-II orbit
reconstructs into small orbits in both structures, generating triangular
orbits around the $K$ points at the corners of the Brillouin zone. On the
$k_{z}=0.5$ plane, the A-II central pocket and the smaller B-I orbit around
$K$ are largely unaffected by the CDW; however, the larger C-II orbit (dashed
line) is strongly affected. In the SoD structure, this orbit is completely
gapped out at $E_{\text{F,exp}}$, while in the TrH structure, the C-II orbit
persists.

Supporting these models, prior ARPES results show the A-I and A-II orbits as
well as the B-I orbit in KV3Sb5 and CsV3Sb5 both above and below the CDW
transition [20, 18]. STM results at lower temperature also show the
preservation of the A-I and A-II orbits and are consistent with at least one
triangular orbit at the $K$-points [19]. Therefore, with regard to the
preservation of the A-I, A-II, and B-I orbits, CDW calculations in both SoD
and TrH structures appear consistent with experimental results to-date. STM
data also validate the reconstruction of the B-II orbit captured within the
DFT models; however our models also predict numerous other changes in the low
frequency (small orbit) regime.

Investigating the band reconstruction in the distorted state further, the data
shown in Figure 7(a) clearly show additional features. Figure 7(b) provides a
simplified, pictorial representation of the Fermi surface and closed orbits in
the undistorted and TrH structures to aide discussion. All possible orbits
that exist in the undistorted structure 7(a) are depicted at
$E_{\text{F,exp}}$. Orbits which are too large to be experimentally observed
with our current data are shown in grey. The remaining orbits are color-coded
consistent with the pocket designations shown previously in Figure 6. For this
qualitative comparison, we focus on the orbits within the TrH structure for
two reasons; 1) the additional modes are less obvious in the 2D data for the
TrH (but no less relevant), and 2) as we will show, the TrH structure
ultimately shows higher correlation with experiment and is energetically
favored in DFT.

Focusing on comparison of the high-frequency orbits, we see that there are a
total of 4 experimentally accessible orbits at $E_{\text{F,exp}}$ in the
undistorted structure. Upon introducing the TrH distortion, several key
changes occur. In the $k_{z}=0$ plane, the distortion generates 3 distinct
vanadium $d$-orbits at $E_{\text{F,exp}}$ by shifting and gapping bands around
the $M$ and $K$ points near the corners and sides of the zone. Notably, band
reconstruction about the $M$ point forms a new medium-frequency triangular
orbit at $E_{\text{F,exp}}$. The $k_{z}=0.5$ plane, in contrast, is largely
preserved in the undistorted configuration, with the exception of the addition
of one Dirac-like orbit at the $L$ point.

Thus, in this high frequency regime, the TrH distortion has several effects:
1) the generation of 3 additional Dirac-like modes, 2) the preservation of
high frequency orbits primarily comprised of Sb-states, and 3) the
introduction of a new triangular orbit from the band reconstruction. It is
worth noting that other, smaller Dirac-like orbits are likely present, but the
resolution of the present supercell calculation limits our search to orbits
$>$30 T.

Figure 8 summarizes all the numerical data characterizing orbits in the
undistorted and distorted SoD and TrH Fermi surfaces and overlays these with
the experimentally observed quantum oscillations. An exhaustive search was
performed for all extremal orbits within a range of $E_{\text{F}}$ spanning
from above and below the Fermi levels reported throughout the literature thus
far ($E_{\text{F,lit}}$–$E_{\text{F,exp}}$–$E_{\text{F,DFT}}$). Orbits are
again colored to remain consistent with their pocket designations in previous
figures. For completeness, we also show orbits up to 15000 T, though these
orbits are not resolvable in the current experiments. The experimentally
observed frequencies are overlaid as horizontal grey bars.

Figure 8: Frequencies extracted as a function of Fermi level in undistorted
CsV3Sb5 allow us to directly compare the experimental SdH data (grey bars)
with the DFT models. The undistorted oscillation frequencies are shown as a
function of Fermi level (left). Colored frequencies have been grouped by the
pocket and orbit of origin. A conservative numerical estimate for the
supercell orbits for the SoD and TrH structures are shown (left) at
$E_{\text{F}}=E_{\text{F,exp}}$. While 5 mid-frequency orbits are indeed
expected in the range $E_{\text{F}}\pm 0.1$ eV, no singular choice of
$E_{\text{F}}$ produces complete agreement with experiment. However, the TrH
distortion introduces a new high-frequency V $d$ orbit, similar to B-III,
which provides a consistent hypothesis. The DFT resolution limit (pink shaded)
indicates the frequency range below which we cannot accurately resolve closed
orbits.

First examining comparisons of the models in the high-frequency regime
($200<f<3000$ T), there are a total of four possible (symmetry unique) orbits
at $E_{\text{F,exp}}$. These are the same four shown in the schematic
representation. There is one orbit (B-III) which appears at slightly lower
$E_{\text{F}}$, though it is gapped by $E_{\text{F,exp}}$. Comparison at
$E_{\text{F}}=E_{\text{F,exp}}$, shows that, while precise quantitative
agreement between all the frequencies varies between orbits, the TrH is the
only qualitative match to 5 frequencies in this regime. In contrast, due to
the loss of the C-II orbit, the SoD distortion only recovers 4 of the 5 modes.
More detailed comparisons for $E_{\text{F}}$ shifted away from
$E_{\text{F,exp}}$ will be discussed in the Discussion section.

Next, examining the low-frequency ($f<200$ T) regime, the data show at least 4
well-defined frequencies about $E_{\text{F,exp}}$. However, at
$E_{\text{F,exp}}$ there is only 1 orbit in the undistorted structure. This
orbit (C-I) is predominantly comprised of vanadium $d$-orbitals associated
with the Dirac-like crossings along the $\Gamma$-$K$ line. However, as shown
throughout this manuscript, substantial reconstruction is expected about the
$M$ point. Characterization of the well-defined, closed orbits in the
supercell models is shown to the right of the low-frequency (undistorted)
structure panel. There are 4 orbits in the SoD model, and 3 in the TrH model.
However, it is important to note that our unfolded supercell calculations are
unable to resolve orbits with frequencies $<30$ T, thus providing a “lower
bound” on the number of possible frequency components. Regardless, the CDW-
mediated reconstruction of the Fermi surface near $M$ provides excellent
qualitative agreement with the experimentally observed frequencies.

##  IV Discussion

Comparison of the experimentally observed quantum oscillation frequencies with
DFT models fixed at $E_{\text{F,exp}}$ suggests that only the TrH low
temperature distortion provides a qualitative match to the data. Furthermore,
the observation of multiple low frequency orbits provides direct evidence of a
CDW-derived reconstruction of vanadium bands endemic to the underlying Kagome
lattice in CsV3Sb5.

It is worthwhile to consider the impact of choosing alternative Fermi levels
in the DFT models (i.e. away from $E_{\text{F,exp}}$), in the comparison
between models and the data. In the low frequency regime for the undistorted
structure, multiple orbits appear when moving both above and below
$E_{\text{F,exp}}$, mimicking the multiple modes found in the experiment;
however, this scenario can be precluded with the following arguments: (1) For
the case where $E_{\text{F}}$ lies below $E_{\text{F,exp}}$—in the regime
where multiple low frequency orbits appear—the multiplicity of the high-
frequency modes does not match the data, as the C-II mode is absent. (2) For
the case where $E_{\text{F}}$ lies above $E_{\text{F,exp}}$, the C-II and B-I
orbits are quickly gapped out in the distorted structures, leaving no
explanation for the five mid-frequency oscillations.

While precise numerical agreement between quantum oscillation frequencies and
DFT-derived oscillation frequencies is lacking in all structures explored, the
TrH structure at $E_{\text{F,exp}}$ is the only structure which captures the
multiplicity of orbits within the experimentally-accessible frequency windows.
When this fact is combined with the TrH structure being significantly favored
in DFT, our data strongly suggest that the TrH structure is the dominant
distortion in CsV3Sb5. More broadly, when combined with DFT models, our data
demonstrate that the reconstructed electronic states near the Fermi level are
intimately tied to the vanadium $d$-orbitals. This is particularly true for
the $M$ points, which are relevant for their contributions to the
topologically protected surface states.

Looking beyond the core distortion within the kagome planes, single-crystal
synchrotron x-ray diffraction data further reveal a three-dimensional phasing
between planes in the CDW state. This manifests itself via superlattice
reflections with a propagation wave vector q $=(0.5,0.5,0.25)$ indicating a
2$\times$2$\times$4 superstructure. While the in-plane wave vector ($h$,
$k$)=(0.5, 0.5) matches the 3Q structure observed in STM, the out-of-plane
component of q implies a four unit cell phasing along the $c$-axis. Current
DFT models do not capture this modulation along $c$, due to the
2$\times$2$\times$1 supercells used. More detailed, normalized single crystal
diffraction data will be required to fully refine the low-temperature unit
cell structure. Although, we note here that this analysis may be nontrivial
due to the potential for twinning and the low relative intensities of the CDW
superlattice reflections.

##  V Conclusion

Combined DFT modeling, high-resolution x-ray scattering, and quantum
oscillation measurements demonstrate that the CDW state in CsV3Sb5 derives
from the reconstruction of the kagome-plane vanadium orbitals with an
accompanying out-of-plane modulation of the distorted structure. The in-plane
component of the resulting 2$\times$2$\times$4 superstructure is best modeled
using the kagome “breathing mode”, with the SoD and TrH patterns emerging as
energetically favorable structures. Quantum oscillation data interpreted via
DFT models of the Fermi surface topography suggest that the TrH state is the
likely ground state configuration below the CDW distortion. More broadly, our
quantum oscillation measurements provide a bulk probe of the electronic
structure that demonstrates the CDW’s reconstruction of the Fermi surface.
They show the dominant role of vanadium orbitals within the kagome planes in
CsV3Sb5 in the CDW, and support theoretical approaches drawn from minimal
models focused on the kagome substructure in AV3Sb5 superconductors.

##  VI Notes Added

While this submission was in preparation, another manuscript appeared by Li et
al. [47] who instead report a 2$\times$2$\times$2 superstructure in CsV3Sb5.
We note here that our data are in agreement in the momentum space regions
reported in that work; however in these regions the $q_{z}=0.25$-type
superlattice peaks are weak and below our experimental resolution. Larger
surveys of reciprocal space reveal the 1/4-type $c$-axis superlattice
reflections reported here. Furthermore, a work by Chen et al. [28] focusing on
the anomalous Hall effect in CsV3Sb5 also appeared during the submission of
this work. They report the Fourier spectrum of the low-frequency orbits in
quantum oscillation data, but do not observe the high-frequency orbits
reported in our data. This could be a consequence of crystal quality, or
smoothing/aliasing considerations during data collection.

##  VII Acknowledgments

Stephen Wilson gratefully acknowledges discussions with Leon Balents, Binghai
Yan, and Ziqiang Wang. This work was supported by the National Science
Foundation (NSF) through Enabling Quantum Leap: Convergent Accelerated
Discovery Foundries for Quantum Materials Science, Engineering and Information
(Q-AMASE-i): Quantum Foundry at UC Santa Barbara (DMR-1906325). The research
made use of the shared facilities of the NSF Materials Research Science and
Engineering Center at UC Santa Barbara (DMR- 1720256). The UC Santa Barbara
MRSEC is a member of the Materials Research Facilities Network.
(www.mrfn.org). B.R.O. and P.M.S. also acknowledge support from the California
NanoSystems Institute through the Elings Fellowship program. S.M.L.T has been
supported by the National Science Foundation Graduate Research Fellowship
Program under Grant No. DGE-1650114. This work is based upon research
conducted at the Center for High Energy X-ray Sciences (CHEXS) which is
supported by the National Science Foundation under award DMR-1829070.

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