# Orbital selective commensurate modulations of the local density of states in
ScV6Sn6 probed by nuclear spins

R. Guehne<EMAIL_ADDRESS>J. Noky  C. Yi  C. Shekhar  Max Planck
Institute for Chemical Physics of Solids, 01187, Dresden, Germany  M. G.
Vergniory  Max Planck Institute for Chemical Physics of Solids, 01187,
Dresden, Germany  Donostia International Physics Center, 20018 Donostia - San
Sebastian, Spain  M. Baenitz  C. Felser  Max Planck Institute for Chemical
Physics of Solids, 01187, Dresden, Germany

###### Abstract

The Kagome network is a unique platform in solid state physics that harbors a
diversity of special electronic states due to its inherent band structure
features comprising Dirac cones, van-Hove singularities, and flat bands. Some
Kagome-based non-magnetic metals have recently been found to exhibit favorable
properties, including unconventional superconductivity, charge density waves
(CDW), switchable chiral transport, and signatures of an anomalous Hall effect
(AHE). The Kagome metal ScV6Sn6 is another promising candidate for studying
the emergence of an unconventional CDW and accompanying effects. We use 51V
nuclear magnetic resonance (NMR) to study the local properties of the CDW
phase in single crystalline ScV6Sn6, aided by density functional theory (DFT).
We trace the dynamics of the local magnetic field during the CDW phase
transition and determine a loss in the density of states (DOS) by a factor of
$\sqrt{2}$, in excellent agreement with DFT. The local charge symmetry of the
V surrounding in the CDW phase reflects the commensurate modulation of the
charge density with wave vector
$q=\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)$. An unusual orientation
dependent change in the NMR shift splitting symmetry, however, reveals orbital
selective modulations of the local DOS.

Kagome lattice, charge density wave, nuclear magnetic resonance

## Introduction

Covalent two-dimensional metals with triangular motifs and confined electronic
states are a rich playground in modern solid states physics. Among the non-
magnetic materials, systems that are based on Kagome layers attract increasing
attention as they naturally feature Dirac cones, van-Hove singularities, and
flat bands in their electronic band structure, that may further chemically be
tuned to the vicinity of the Fermi level [1]. Lately, the V based Kagome metal
CsV3Sb5 appeared in the spotlight as it undergoes a charge density wave (CDW)
phase transition around $94\text{\,}\mathrm{K}$ before it becomes an
unconventional superconductor below about $2\text{\,}\mathrm{K}$, providing
the prospect to learn more about the interplay of both phenomena. Evidence for
a manipulable chiral transport as well as an anomalous Hall effect (AHE) in
the CDW regime further add to the rather diverse list of special electronics
of this Kagome metal [2, 3].

Recently, the closely related bilayer Kagome material ScV6Sn6 was reported to
undergo a CDW transition at about $92\text{\,}\mathrm{K}$, including evidence
for topologically non-trivial bands and signatures of an anomalous Hall
effect, while superconductivity was not observed at ambient pressure [4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. In
particular the unconventional nature of CDW phase with a 3-dimensional
wavevector $\mathbf{q}_{3}=\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)$
gives rise to an ongoing debate about the formation mechanism of the charge
order [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21]. Some
experiments and theory further suggest a high temperature short range CDW with
$\mathbf{q}_{2}=\left(\frac{1}{3},\frac{1}{3},\frac{1}{2}\right)$ that is
unstable at low temperatures [9, 10, 11, 13, 21].

The complexity of the CDW as well as numerous experimental reports of unusual
phenomena led us to employ nuclear magnetic resonance (NMR) to characterize
the CDW phase of single crystalline ScV6Sn6 and to search for signatures of an
unusual magnetism. Given its high sensitivity for chemical and electronic
properties, as well as the atomic resolution, NMR has a long tradition as a
method to investigate charge ordered systems, as e.g., in the case of the
famous dichalcogenides [25]. In the context of cuprates, NMR is used to deepen
the understanding of the interplay between charge order and superconductivity
[26, 27]. Similarly, NMR of CsV3Sb5 has led to crucial insights into the
charge modulation at ambient and elevated pressure [28, 29, 30, 31, 1].

With this paper we present a comprehensive single crystal 51V NMR study of the
Kagome metal ScV6Sn6 aided by density functional theory (DFT). We explore the
dynamic properties of the local magnetic field during the CDW phase transition
between $96\text{\,}$ and about $80\text{\,}\mathrm{K}$, and determine a drop
in the density of states (DOS) by a factor of $\sqrt{2}$. The CDW phase is
characterized as a commensurate, threefold modulation of the local V charge
symmetry in accordance with the reported reconstruction of the unit cell with
periodicity $\sqrt{3}\times\sqrt{3}\times 3$. Our NMR shift data further
reveal a peculiar phase shift $\Delta\varphi=\pi/2$ of the sinusoidal
modulation of the local magnetic field between the in-plane and the out-of-
plane field orientation. This observation can be explained by orbital
selective modulations of the local DOS resulting in a distinct wave vector for
the V 3$d_{z^{2}}$ orbitals with
$q_{z^{2}}=\left(\frac{1}{3},\frac{1}{3},\frac{2}{3}\right)$. Finally, we
neither find direct evidence of an unusual magnetism as related to time-
reversal symmetry breaking and an AHE, nor of an additional, high temperature
charge modulation with
$\mathbf{q}_{2}=\left(\frac{1}{3},\frac{1}{3},\frac{1}{2}\right)$.

Figure 1: One-dimensional commensurate CDWs and NMR (A) A chain of equidistant
atoms with evenly distributed electron density (gray circles) yields a single
NMR signal at $K_{0}$ (including intrinsic broadening). (B) For a CDW of
amplitude $\delta\rho$, periodicity $q=1/2$, and phase $\varphi=0$, local
charge density peaks and valleys alternate between neighboring atoms, yielding
a symmetric splitting into two resonance lines at $\pm\delta
K\propto\delta\rho$. The signal intensity is proportional to the number of
nuclei with the same local environment. (C) A CDW with $q=1/3$ yields 3
equally intense resonance lines. For $\varphi=\pi/2$, two of them are
symmetrically shifted to higher and lower frequencies, while the third one
remains un-shifted (no extra charge). (D) For the special case of
$\varphi=\pi$, two of the 3 lines overlap at $\delta K/2$, while the third one
is shifted by $\delta K$ in the opposite direction.

Charge density waves and NMR In the 1960s, Overhauser considered a sinusiodal
modulation of the local spin and charge density in metals which is translated
into characteristic spectral changes of the corresponding NMR signal through
the variation in the NMR shift or quadrupole interaction (as will be discussed
in more detail below) [32]. Consider a one-dimensional (1D) sinusiodal
modulation of the local electronic density with wavelength $\lambda=2\pi/q$,

$\rho=\rho_{0}+\delta\rho\cos(q\cdot x+\varphi)\ .$ (1)

Here, $\rho_{0}$ denotes the average or un-modulated charge in the CDW regime,
$\delta\rho$ the amplitude, $q$ the wave vector, $x$ the real-space position,
and $\varphi$ the phase of the CDW. That is, different to most other methods,
NMR can also determine the phase $\varphi$ of a charge modulation which can be
an important detail in terms of the dynamics of CDWs [33]. A variety of
commensurate CDW scenarios beside the corresponding NMR signals are
illustrated in Fig. 1. We assume a periodic lattice of atoms with lattice
parameter $a$ and a uniform local electron density $\rho_{0}$ (panel A). The
NMR signal appears at a $K_{0}$ that corresponds to the uniform local charge.
In panels (B) to (D) of Fig. 1 we show three different CDWs and their NMR
resonance patterns. The charge density variation is represented by the size of
the circles. The NMR line splits with a maximum of $2\delta
K\propto\delta\rho$. Here, NMR shifts to higher and lower frequencies display
the gain or loss of electron density for the respective crystal position. In
general, the sign of $\delta K$ does not allow a conclusion on whether the
corresponding nucleus sits in a valley or on a peak of the CDW, as it depends
on the type of hyperfine interaction [34]. For $q=1/2$ (as for 51V NMR in
CsV3Sb5), the resonance line always splits symmetrically, independent of the
CDW phase. Contrastingly, for $q=1/3$, as in case of 51V NMR of ScV6Sn6, the
relative position of the three resonances depends on $\varphi$ (cf. panels C
and D). This simple model is easily extended to incommensurate CDWs that give
rise to broadened NMR spectra with a characteristic double-peak structure at
$\pm\delta K$.[32, 27]

Figure 2: The crystal structures of ScV6Sn6 (A) On the left, side view of the
high temperature unit cell of ScV6Sn6, and below, top view of the V Kagome
lattice (red). In the CDW phase, the relative position of the Sn-Sc-Sn chains
(gray and purple) is altered along the crystal $c$-axis, yielding 3 different
Sc-Sn-Sc units as depicted in panel (B) containing one V Kagome double layer
each. This expands the unit cell by a factor of three. The displacement
affects the distance of the Kagome-close Sn atoms (1,2,3), creating 3 distinct
V crystal sites (V1, V2, V3).

The structural phases of ScV6Sn6 The high temperature phase of ScV6Sn6 has
P6/mmm symmetry. The unit cell consists of 13 atoms, where chemically
equivalent Vanadium atoms form the Kagome planes, cf. Fig. 2 (A). Below about
$96\text{\,}\mathrm{K}$, ScV6Sn6 undergoes a structural phase transition
yielding a change in the Sc and Sn positions along the crystal $c$-axis, while
the V lattice remains almost unperturbed [5]. In panel (B) of Fig. 2, we show
the relevant displacements in terms of Sn-Sc-Sn chains (gray-purple-gray)
containing two Kagome layers (red). Different to the symmetric arrangement
characteristic for the high $T$ phase, the low $T$ phase features 3 Sn-Sc-Sn
chains that vary in length and in relative distance to the V Kagome plane.
From the viewpoint of the V Kagome lattice, we identify 3 corresponding Sn
positions (1, 2, 3) that hover just above (below) the hexagonal V voids,
creating 3 non-equivalent V crystal sites (V1, V2, V3), each of them
neighbored by two different Sn sites as shown in the lower panel of Fig. 2
(B).

Figure 3:  The local Vanadium charge symmetry for high and low temperatures
(A) The Fourier transform of the 51V NMR spectra (from selective spin echoes)
for $8.73\text{\,}\mathrm{T}$ and $c\parallel B_{0}$. The spectra were
obtained at $110\text{\,}$ and $80\text{\,}\mathrm{K}$, confronting the
spectral differences between the high and the low temperature phase. The
central frequency has been subtracted for the clarity. (B) High temperature
orientation dependent quadrupole splittings for rotations about the crystal
$c$-axis (in-plane) reveals the six-fold rotation symmetry of the Kagome
lattice (cf. Fig. 9). Perspective view (C) and top view (D) of the equivalent
EFGs at V sites in the Kagome lattice. (E) Low temperature orientation
dependent quadrupole splittings for in-plane rotations confirms the three sets
of quadrupole patterns already observed in panel (A), while the six-fold
rotation symmetry of the Kagome lattice is conserved (cf. Fig. 11). (F) Thus,
for low temperatures, the three corners of each V triangle become chemically
non-equivalent while the EFGs keep the orientation of the high $T$ phase. (G)
The commensurate variation of the EFGs along the crystal $a$ or $b$ axis, can
be viewed as a 1D sinusoidal modulation with $q=1/3$ and
$\varphi\approx\pi/4$.

## Results

The local Vanadium charge symmetry The above discussed structural phases
feature different local V environments. NMR allows to differentiate them via
the electric quadrupole interaction, if such environments have non-cubic
symmetry and the nuclear spin satisfies $I>1/2$ (${}^{51}I=7/2$). In first
order approximation, the quadrupole interaction can be written as

$\displaystyle\mathcal{H}_{\mathrm{Q}}$
$\displaystyle=\frac{3I_{z}^{2}-I(I+1)}{4I(2I-1)}eQV_{ZZ}$ (2)
$\displaystyle\times\frac{1}{2}\left(3\cos^{2}\beta-1+\eta\sin^{2}\beta\cos\alpha\right)\
,$

where $eQ$ is the nuclear quadrupole moment and $V_{ZZ}$ is the principle
value of the electric field gradient (EFG). The EFG is derived from the charge
distribution at the nucleus’ position, and is thus tightly connected to the
above mentioned local environment. The EFG is represented as a traceless
($V_{ZZ}+V_{YY}+V_{XX}=0$), second rank tensor. It is conveniently expressed
in terms of its size ($|V_{ZZ}|$) and shape given by the asymmetry parameter
$\eta=(V_{XX}-V_{YY})/V_{ZZ}$ ($|V_{ZZ}|\geq|V_{YY}|\geq|V_{XX}|$). The Euler
angles $\alpha$ and $\beta$ determine the relative orientation of the EFG’s
principle axis system with respect to the external magnetic field (typically
laboratory $z$-axis), and thus with respect to the sample’s crystal lattice.

From (2) it follows that the V nuclear spin system splits into $2I=7$
equidistant resonance lines (note, second order shifts amount to about
$1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ in maximum), 1 central transition
(CT) and 3 pairs of satellites, as shown in the top spectrum of Fig. 3 (A).
The anisotropy of the interaction ($\alpha$ and $\beta$) is transferred to the
apparent quadrupolar line splitting $\tilde{\nu}_{Q}$, i.e., the splitting
depends on the sample orientation in the external field, as

$\tilde{\nu}_{Q}=\frac{3eQV_{ZZ}}{4I(2I-1)}\left(3\cos^{2}\beta-1+\eta\sin^{2}\beta\cos\alpha\right)\
,$ (3)

with the quadrupole splitting frequency defined as

$\nu_{Q}=\frac{3eQV_{ZZ}}{2I(2I-1)}\ .$ (4)

By evaluating $\tilde{\nu}_{Q}$ for various crystal orientations, size
($|V_{ZZ}|$), shape ($\eta$), and orientation of the EFG can be determined. In
Fig. 3 (B) we show an example of this procedure for the high temperature phase
(a detailed account is provided in the supplementary information SI), which
corresponds to $\tilde{\nu}_{Q}(\beta,\alpha=0)$, i.e., orientation dependent
measurements under crystal rotation about the $c$-axis (cf. Fig. 9 in SI). We
find that $V_{ZZ}$ points along to the crystal $a^{\ast}$-axis with
$\nu_{Q}=$475(2)\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ and
$\eta=$0.83(1)\text{\,}$$. The three identical data sets (triangles, squares,
and circles) shifted by $\pm$60\text{\,}\mathrm{\SIUnitSymbolDegree}$$ recover
the six-fold symmetry of the Kagome lattice (insets) and prove the equivalency
of the three corners of each V Kagome triangle in terms of the local Vanadium
symmetry. To visualize the V EFGs in the Kagome lattice, we display them as
ellipsoids formed by $|V_{ZZ}|$, $|V_{YY}|$, and $|V_{XX}|$, as shown in
panels (C) and (D) of Fig. 3.

Table 1: The electric quadrupole splitting frequency $\nu_{Q}$, the principle axis of the electric field gradient $V_{ZZ}$ from (3) (for the experiment, only the absolute value can be obtained, the sign cannot be accessed), and the asymmetry parameter $\eta$ as obtained from NMR and $DFT$ at V positions above (High $T$) and below (Low $T$) the CDW phase transition. |  |  High $T$ |  |  Low $T$
---|---|---|---|---
|  | NMR | DFT |  | NMR1 | DFT1 |  | NMR2 | DFT2 |  | NMR3 | DFT3
$\nu_{Q}$ $\text{\,}\mathrm{[}\mathrm{k}\mathrm{H}\mathrm{z}\mathrm{]}$ |  | 475(2) |  |  | 463(2) |  |  | 489(2) |  |  | 499(2) |
$V_{ZZ}$ $\text{\,}\mathrm{[}\mathrm{1}\mathrm{0}^{20}\mathrm{V}\mathrm{/}\mathrm{m}^{2}\mathrm{]}$ |  | 52.9(2) | -60.812 |  | 51.6(2) | -59.679 |  | 54.4(2) | -61.861 |  | 55.6(2) | -62.617
$\eta$ |  | 0.83(1) | 0.809 |  | 0.91(4) | 0.841 |  | 0.81(4) | 0.782 |  | 0.78(3) | 0.787

Upon cooling, between $96\text{\,}$ and about $80\text{\,}\mathrm{K}$, the
phase transition takes place and the low temperature structure replaces the
high $T$ one. From the NMR point of view (Fig. 3 A lower spectrum), the
changes are most prominently visible in a splitting of the low frequency
satellites (find a detailed account on the temperature dependent spectra in
Fig. 7 in the SI). The single quadrupole pattern observed for high temperature
is replaced by 3 sets of quadrupole spectra. In panel (E) of Fig. 3 we show
the results of the same orientation dependent measurements as in panel (B),
now performed at $80\text{\,}\mathrm{K}$ (cf. Fig. 11 in SI). The change to a
threefold quadrupole interaction is consistently found in any possible
direction. This proves that the local Vanadium environments have changed,
while the relative orientation of the 3 EFGs is same as at high temperatures.
The arrangement of the EFGs is illustrated in panel (F). When viewed along the
crystal $a$ or $b$ axis, panel (G), the 3 alternating EFGs create a
commensurate 1D sinusoidal variation with $q=1/3$, $\varphi=\pi/4$, and an
amplitude of about $20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ around the
average quadrupole splitting of
$483.7\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. Detailed DFT calculations as
implemented in VASP [35, 36] confirm the results shown in Fig. 3, cf. Tab.1.
Note that with the experiment we cannot determine the sign of $V_{ZZ}$ since
the quadrupole splitting is symmetric.

The CDW phase transition In Fig. 4 (A) we show the center line of the
quadrupole spectra from Fig. 3 (A), for temperatures between
$100\text{\,}\mathrm{K}$ and $20\text{\,}\mathrm{K}$. For the sake of clarity,
we have removed the (very small) temperature dependence of the NMR shift. At
$96\text{\,}\mathrm{K}$, first indications of an additional signal (blue)
appear about $15\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$
($\sim$$150\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$) next to the initial
resonance line (gray). As this shoulder grows in intensity with decreasing
temperature, the initial line decreases, until, for about
$80\text{\,}\mathrm{K}$ and below, the overall shape of the doublet remains
unchanged.

In order to track the volume fractions of the two competing phases during the
phase transition, we used Gaussian lines to fit the two resonances in panel
(A) of Fig. 4 and extracted the corresponding signal intensities. The results
are shown panel (B). The gray and blue data points denote the respective
signal intensities of panel (A) (for an analysis of the total signal
intensity, the reader is referred to Fig. 14 in the SI). The individual values
are normalized with respect to their sum for each temperature. In the course
of cooling, the low temperature resonance line (blue) grows to about 2/3 of
the total intensity, leaving about 1/3 (gray) to what appears to be the
initial high temperature line. With the colored dashed lines in Fig. 4 (B) we
show the true changes in the relative intensities, concluded from the
corresponding changes in the quadrupole splittings. The initial high
temperature CT disappears entirely between 96 and about
$80\text{\,}\mathrm{K}$ (black dashed line). It is replaced by three
resonances (purple, orange, and light blue) with intensity ratio 1:1:1 that
represent the 3 CTs of the low temperature quadrupole patterns identified in
the former section. Two of the 3 resonances (orange and light blue) form the
dark blue peak that appears during the phase transition in panel (A).

Figure 4: CDW phase transition and the local magnetic field (A) The central
region of the $c\parallel B_{0}$ spectrum as function of temperature at
$8.73\text{\,}\mathrm{T}$ (Fourier transform of selective spin echoes).
Highlighted is the temperature range of the phase transition. Gaussian lines
(gray and blue) were used to fit the spectra and to extract the intensity
changes. (B) The area of the Gaussians (proportional to signal intensity) from
panel (A) as a function of temperature. Dashed lines denote true relative
intensities. (C) and (D) The temperature dependence of the NMR shift for
$c\parallel B_{0}$ and $a^{\ast}\parallel B_{0}$. Note that
”$a^{\ast}\parallel B_{0}$” holds only for 1 of the 3 V nuclei per V triangle
as shown in the inset. Gray data points denote the average of the two and
three low temperature shift values.

In Fig. 4 (C) and (D) we show the temperature dependence of the NMR shift for
the out-of-plane orientation of the magnetic field ($c\parallel B_{0}$) and
for the in-plane orientation ($a^{\ast}\parallel B_{0}$). In the latter case,
a reliable determination of the shift values during the phase transition was
not possible because of the overlap of high and low temperature resonance
lines. For $80\text{\,}\mathrm{K}$ and below, the shift was determined via the
satellite transitions (see SI). We assigned the shift values between
$c\parallel B_{0}$ and $a^{\ast}\parallel B_{0}$ as shown by the colors based
on the DFT calculations. The assignment cannot be proven with experiment
because the superposition of many resonance lines for orientation dependent
measurements prevents a reliable tracing of individual signals. This, however,
has no effect on the conclusions of the present investigation.

The V shift in ScV6Sn6 is with more than about $0.6\text{\,}\%$ and
$0.8\text{\,}\%$ ($6000\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ to
$8000\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$) very large, strongly
anisotropic, and only very weakly dependent on temperature (2 to
$4\text{\,}\%$ of the total shift). From detailed orientation dependent
measurements (cf. Fig. 8 to 10 in the SI) it can be seen that the high
temperature shift tensor is axially symmetric with the main axis pointing
towards the center of the V triangle (along the $a^{\ast}$ direction) similar
to the EFG. Although it is very difficult to prove experimentally, it is
reasonable to assume that the CDW phase inherits the axial shift anisotropy.
Isotropic and axial shift components for the full range of temperatures are
depicted in Fig. 13 in the SI.

Figure 5: CDW phase transition and fluctuations of the local magnetic field
(A) The spin-lattice relaxation rate $1/T_{1}$ as a function of temperature
obtained from the CT ($c\parallel B_{0}$) for sample S1 and S2, and from the
first low frequency satellite for $a^{\ast}\parallel B_{0}$. The inset shows
the calculated DOS for V states at high (solid) and low (dashed line)
temperatures. At $E_{\mathrm{F}}$, the DOS drops by $\sqrt{2}$ due to the CDW
phase transition. (B) $1/T_{1}T$ as a function of temperature. The blue solid
line represents a fit for the high temperature phase data using
$\left(const.+\frac{\tilde{T}}{T}\times\exp\left(-\frac{\Delta
E/2}{k_{B}T}\right)\right)$ yielding $\tilde{T}\sim$730\text{\,}\mathrm{K}$$
and $\Delta E\sim$160\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$. Panel (C)
shows the corresponding Arrhenius plot for the high temperature region above
$150\text{\,}\mathrm{K}$.

In panel (A) of Fig. 5 we show the spin-lattice relaxation rate of V nuclei in
the two different samples, for two orientations ($c\parallel B_{0}$ and
$a^{\ast}\parallel B_{0}$), and for a wide range of temperatures. After
continuously decreasing between room temperature and about
$100\text{\,}\mathrm{K}$, $1/T_{1}$ drops abruptly in the region of the phase
transition by about a factor of 2. Between $80\text{\,}$ and
$4\text{\,}\mathrm{K}$, the relaxation rate is proportional to the
temperature. Measurements for $a^{\ast}\parallel B_{0}$ (yellow circles) prove
the spin-lattice relaxation to be isotropic (see also Fig.16 in the SI). In
the low temperature phase, the data points represent the averaged relaxation
of the 3 components. In the inset we present the Vanadium DOS for both phases
obtained from DFT. The high temperature DOS are denoted by the solid line, the
CDW phase DOS by the dashed line. At the Fermi level, the difference between
high and low temperature DOS is $\sqrt{2}$, in agreement with the observed
changes in the relaxation (cf. discussion). The temperature dependence of
$1/T_{1}T$, as shown in panel (B), yields a constant value for temperatures
below $80\text{\,}\mathrm{K}$ and a pronounced step during the phase
transition. For temperatures above $T_{\mathrm{CDW}}$ and
$\leq$150\text{\,}\mathrm{K}$$, $1/T_{1}T$ appears to be rather constant as
well, while it clearly deviates from such a behavior for even higher
temperatures. This progressive increase implies a gain in available states as,
e.g., in the case of thermal excitation across an energy gap. An activation
type of fit (blue line) gives a gap of about
$160\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$. In panel (C) we show the
corresponding Arrhenius plot for the high temperature relaxation rate (above
$150\text{\,}\mathrm{K}$).

## Discussion

General remarks The temperature dependent 51V spectra of ScV6Sn6 as shown in
Fig. 3 (Fig. 7 in the SI) and 4 provide consistent evidence of a first order
phase transition between $96\text{\,}$ and about $80\text{\,}\mathrm{K}$ in
good agreement with literature [15]. The low $T$ phase is similarly
homogeneous as the high $T$ one, with no additional broadening, neither in the
NMR shift nor in the quadrupole interaction, proving any observed changes in
the static NMR at low temperatures are perfectly commensurate and sufficiently
long range such that domain effects are negligible. Hence, our observations do
not support the presence of an additional short-range intermediate charge
order [9, 10, 11, 13, 21]. The results also do not provide any direct evidence
of an unusual magnetism as related to TRS breaking and an anomalous Hall
effect. The $c\parallel B_{0}$-splitting in units of ppm as well as the line
shape, i.e., the relative intensity of this double-peak structure, are
independent of the applied magnetic field, and thus point at different local
spin densities as expected for a CDW (cf. Fig. 15 in SI). Similarly, total
signal intensity as well as excitation conditions of the NMR experiments do
not indicate any effect from magnetism, as there are no unusual losses or
enhancements. That is, any additional effect must be well below the observed
linewidth of our resonance lines which is in the order of
$100\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$.

Local properties of the CDW phase The EFG at V sites accessible through the V
quadrupole interaction gives a direct evidence of a characteristic charge
redistribution due to the CDW phase transition. As shown in the Fig. 3, the
single high temperature Vanadium EFG is replaced by 3 low temperature EFGs
differing in size and shape, but retaining the original orientation in the
crystal lattice. The 3 different EFGs mapped on the known low temperature
crystal structure correspond to a CDW with wave vector
$q=\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)$ which agrees with the
reported reconstruction of the unit cell with $\sqrt{3}\times\sqrt{3}\times 3$
periodicity. From the splitting frequencies we determine the phase to be
$\varphi\approx\pi/4$ for 1D sinusoidal modulations along any crystal
direction. The real-space charge distribution that leads to the different EFGs
for V1 to V3 cannot be extracted from the experimental results. Plausibly, the
EFGs could display the imbalence of Fermi level states in the five V $3d$
orbitals related to the following discussion of the NMR shift. But they may
also be affected by charges from bands below the Fermi energy and offsite
contributions such as neighbored ions (e.g., the 3 Sn sites in Fig. 2) [37].

We further determined the NMR shift for two crystal orientations ($a^{\ast}$
and $c\parallel B_{0}$) and for a wide range of temperatures as shown in Fig.
4 (C) and (D). For the analysis, we may separate the shift into two main
components: (1) a very large and highly anisotropic part with a slight
temperature dependence, and (2), the characteristic splitting as a consequence
of the CDW phase transition. Component (1) is most likely dominated by a very
large orbital shift (from $0.6\text{\,}\%$ to well above $0.8\text{\,}\%$)
stemming from the magnetic moment associated with the orbital motion of the
electrons (van Vleck paramagnetism) that was repeatedly found in metallic V
compounds [38, 39, 40, 41, 34, 42]. This shift term is typically independent
of or weakly dependent on temperature, was reported to be insensitive to the
CDW phase transition in 1T-VS2, and may thus explain why the total shift
including its temperature dependence seems not to be directly related to the
relaxation as expected from a Korringa relation (from $K^{2}T_{1}T=const.$ we
expect an isotropic shift of less than $0.1\text{\,}\%$ with a distinct
temperature dependence) [34, 42].

Figure 6: NMR Shift and the modulation of the local DOS (A) The shift values
from Fig. 4 (A) plotted as $K_{c}$ vs. $K_{a^{\ast}}$ with the temperature as
an intrinsic parameter. The red edged black square corresponds to
$96\text{\,}\mathrm{K}$, the red edged gray circle as well as the three 3d
objects to $80\text{\,}\mathrm{K}$. The dashed arrows indicate the direction
of $K_{i}(T)$ for decreasing temperatures. The black diagonal marks a line of
constant isotropic NMR shift of $0.75\text{\,}\%$. (B) Orbital and site
resolved DOS for high (black) and low (colored) temperatures from DFT. (C) The
resulting sinusoidal modulation of the DOS for $d_{x^{2}-y^{2}}+d_{xy}$
orbitals ($left$) and the crystal $a$ and $b$ directions (equivalent) as well
as for $c$, beside V hourglass building blocks to illustrate the structural
stacking of the V Kagome planes in the CDW phase ($right$). Colors denote
crystal sites, i.e., V1 to V3, while the size of the balls represent the local
DOS. The black solid and dashed boxes denote the unit cell for the high
temperature and the CDW structure, respectively. (D) Similar to panel (C) for
the $d_{z^{2}}$ orbitals and the corresponding modulation patterns of the
local DOS. Obviously, the wavelength of the DOS modulation for $d_{z^{2}}$
orbitals is half the CDW unit cell in the crystal $c$-direction, yielding an
${q}_{z^{2}}=\left(\frac{1}{3},\frac{1}{3},\frac{2}{3}\right)$.

The second shift component (2) then contains all the changes due to the CDW
phase. Obviously, the CDW phase emerges as an in good approximation
temperature independent splitting of the single high temperature resonance
line into a symmetric triplet of resonances for $a^{\ast}$ and an asymmetric
doublet for $c\parallel B_{0}$. The switching between the two types of
splittings appears unusual, but especially the two-fold splitting of _three_
components in the $c\parallel B_{0}$ orientation is surprising, because it is
not clear what causes such a symmetry breaking. To illustrate this puzzle, we
show in Fig. 6 (A) the shift data from Fig. 4 as a $K_{c}$ vs.
$K_{a^{\ast}}$-plot. The two shift patterns observed for $c$ and
$a^{\ast}\parallel B_{0}$ are sketched as projections on the two perpendicular
axes (the 3d objects represent shift data at $80\text{\,}\mathrm{K}$). They
are fingerprints of commensurate 1D CDWs with $q=1/3$ that have a phase
difference of $\Delta\varphi=\pi/2$ (cf. Fig. 1 C and D). The only difference
between the two shift splittings is the orientation of the external field and
it is rather unlikely that the field directly induces a phase change in the
corresponding CDW. This raises the question for what causes the observed phase
difference?

The shift anisotropy as a function of temperature brings some light into this
problem. Beside the individual temperature dependencies drawn on the plane in
Fig. 6 (A) (color coding follows Fig. 4 (C) and D), the red edged data points,
representing $K$ just above (black - $96\text{\,}\mathrm{K}$) and the averaged
$K$ below (gray - $80\text{\,}\mathrm{K}$) the CDW phase transition, are both
located almost exactly on a line of constant isotropic shift of
$0.75\text{\,}\%$. Under the reasonable assumption of axially symmetric shift
tensors for both temperature regimes, the average isotropic shift thus remains
unaffected by the phase transition, while the average axial component
substantially changes (cf. Fig. 13 in the SI). Hence, the observed splittings
in $a^{\ast}$ and $c\parallel B_{0}$ are mainly caused by a splitting in the
local electronic spin susceptibility (Pauli paramagnetism) with an anisotropic
hyperfine interaction. Contact interaction and core polarization effects that
yield isotropic NMR shifts cannot be responsible for the peculiar shift
splittings. Spin dipolar interactions, on the other hand side, are highly
anisotropic and may offer a way to resolve the conundrum of the phase shift
between the two different field orientations [34, 43]. The shift measured for
different field orientations could represent the shift contribution from
orbitals with different geometry. In particular, the symmetric triplet in the
plane may predominately arise from different DOS in the planar orbitals, i.e.,
$d_{x^{2}-y^{2}}$ and $d_{xy}$, while the asymmetric doublet that appears with
the field along the crystal $c$-axis displays the DOS of the orbital with a
distinct component along that direction, i.e., $d_{z^{2}}$.

To gain independent insight, we examined the DOS of individual orbitals for
the 3 different V sites using DFT. The results are presented in Fig. 6 (B)
resolved for the sum of the $d_{x^{2}-y^{2}}+d_{xy}$ orbitals (for symmetry
reason both in-plane orbitals are equivalent), for $d_{z^{2}}$, and for the
$d_{xz}+d_{yz}$ orbitals (again, both orbitals are equivalent). The site-
selective DOS for the in-plane orbitals at the Fermi level splits
symmetrically into 3 different DOS (purple, orange, and light blue), while for
the out-of-plane orbital $d_{z^{2}}$ the splitting is two-fold, with V2 and V3
(orange and light blue) having almost exactly the same DOS. For
$d_{xz}+d_{yz}$ the three-fold splitting is again symmetric and visibly larger
than for the other two components. Thus, DFT calculations almost perfectly
reproduce the characteristic splittings observed with NMR, suggesting that the
distinct orbital geometry is dominating the shift for special field
orientations. Mapping the NMR/DFT data onto the low temperature crystal
structure with V1, V2, and V3, we find the periodic, orbital specific
modulations of the DOS as shown in panels (C) and (D) of Fig. 6. For
$d_{x^{2}-y^{2}}+d_{xy}$ orbitals, the corresponding wave vector is
${q}_{xy}=\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)$ with a phase
$\varphi\approx\pi/2$. The modulation for $d_{xz}+d_{yz}$ orbitals (not shown)
has the same wave vector but $\varphi\approx\pi/6$. Due to the twofold
splitting in the $d_{z^{2}}$ orbitals, the corresponding modulation is
characterized by
${q}_{z^{2}}=\left(\frac{1}{3},\frac{1}{3},\frac{2}{3}\right)$ and
$\varphi\approx 0$. Thus, NMR and DFT imply orbital specific modulations of
the DOS that differ in amplitude, phase, and, in case of the V $d_{z^{2}}$
modulation, even in periodicity of the out-of-plane component. In this
scenario, the switching between the in-plane orbitals and the out-of-plane
orbitals requires a highly selective hyperfine interaction. This should be
subject of a theoretical investigation in future studies. Finally, the
chirality of the crystal structure is transferred to the modulations of the
DOS where corner-sharing triangular columns (light and dark gray) have
opposite handedness.

The CDW phase transition and fluctuations of the local magnetic field For a
metallic system like ScV6Sn6, the spin-relaxation is typically governed by
magnetic relaxation, i.e., by fluctuations of the local magnetic field arising
from conduction electrons (cf. the corresponding section and Fig. 16 in the
SI). The temperature dependence of the relaxation rate $1/T_{1}$ should then
be proportional to the square of the DOS (Korringa relation). We evaluated the
following integral,

$\frac{1}{T_{1}}\propto A^{2}_{\mathrm{hf}}\int
f(E-\mu_{c})[1-f(E-\mu_{c})]D(E)^{2}\mathrm{d}E\ ,$ (5)

where $A_{\mathrm{hf}}$ represents the hyperfine interaction that we assume to
be temperature independent, $f(E)$ denotes the Fermi function, $\mu_{c}$ the
temperature dependent chemical potential, and $D(E)$ the energy dependence of
the DOS for the high and the low temperature phase from the inset of Fig. 5
(A). The results are shown by the dashed and the solid black line in Fig. 5
(A) after appropriate rescaling. We find a very good agreement between
experimental results and DFT. That is, the nuclear spins see the electronic
states at the Fermi level and relax via their excitations, while the
temperature limits the available states through the Fermi function ($\sim
k_{\mathrm{B}}T$). As mentioned in the results section, the relaxation rate
values in the CDW phase denote the average of the 3 individual resonance
lines, and similarly the low temperature DOS in panel (A) are not resolved for
the 3 V sites. It may thus be possible that the relaxation also shows
differences among the 3 V sites similar to the shift values. This, however,
could not reliably be resolved with experiment.

A different way to look at the relaxation data is presented in Fig. 5 (B)
where the $1/T_{1}T(T)$ is separated into 3 parts, Fermi liquid behavior below
and above the CDW phase transition separated by a change in DOS by a factor of
$\sqrt{2}$, and a high temperature dependence above $150\text{\,}\mathrm{K}$
that can very well be approximated by an activation-type of behavior. A single
exponential fit yields a gap of the size of about
$160\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ (cf. Arrhenius plot in panel C)
which might be relatable to band structure features, such as van Hove
singularities as suggested by others [44]. In the current case of ScV6Sn6,
however, the band structure is too complex to unambiguously relate the spin-
lattice relaxation to individual bands and their dispersion.

## Conclusions

We investigated the CDW Kagome metal ScV6Sn6 using single crystal 51V NMR and
DFT. The CDW phase transition occurs between $96\text{\,}$ and about
$80\text{\,}\mathrm{K}$ and takes place as a gradual replacement of the high
temperature phase by the CDW phase as expected from a first order transition.
The phase transition is accompanied by a change in the electronic band
structure and the corresponding changes in the total DOS by a factor of
$\sqrt{2}$ is very well reproduced with DFT to match the experimentally
observed evolution of the local magnetic field fluctuations for decreasing
temperatures. The CDW phase features an individual conversion of the three
formerly equivalent V environments per V triangle in agreement with DFT and
the reported reconstruction of the unit cell with
$\sqrt{3}\times\sqrt{3}\times 3$ periodicity. In the CDW phase, our findings
further comprise an unusual orientation dependent change in the NMR shift
splitting from a symmetric triplet of resonance lines for $a^{\ast}\parallel
B_{0}$ to a asymmetric doublet for $c\parallel B_{0}$, while the latter
reflects a symmetry that cannot be found in the crystal structure. When
regarded as one-dimensional sinusoidal modulations of the local magnetic
field, this observation implies a magnetic field induced phase shift of
$\Delta\varphi=\pi/2$. To resolve this discrepancy, we argue on the basis of
orbital selective modulations of the local DOS with in-plane wave vector
$q=\left(\frac{1}{3},\frac{1}{3}\right)$ but different out-of-plane wave
vectors and phases, driven by a field orientation selective hyperfine
coupling. The latter calls for a quantitative theoretical treatment of the
hyperfine interaction and possible implications for the compound’s transport
properties. Our work demonstrates that the combination of single crystal NMR
and DFT calculations leads to crucial local information about static and
dynamic electronic properties in the charge density wave system ScV6Sn6.

## Methods

Crystal synthesis High-quality single crystals of ScV6Sn6 were grown by the
flux method [5]. High-purity Sc, V, and Sn elements were loaded in an alumina
crucible in a molar ratio of 1 : 10 : 60 and then sealed in a quartz tube
under vacuum. The tube was then slowly heated to 1373 K, maintained for 10 h,
and cooled down to 973 K over 400 h. Hexagonal shape crystals with silvery
surfaces and a typical size of $2\times 2\times
1$\text{\,}\mathrm{m}\mathrm{m}^{3}$$ were obtained after centrifugation. The
crystal structure was refined from powder x-ray diffraction and the chemical
components were examined by using energy-dispersive X-ray spectroscopy The
crystal orientations were determined by using the Laue backscattering
diffractometer.

NMR experiments Measurements were carried out on a Janis sweepable
$9\text{\,}\mathrm{T}$ magnet and a Tecmag Apollo NMR console. Most
experiments were carried out on the hexagonal, plate-like Sample S2 with
dimensions
$$1.7\text{\,}$\times$1.1\text{\,}$\times$0.2\text{\,}\mathrm{m}\mathrm{m}^{3}$$
(cf. Yi et al. 2024 [15]). A few measurements were done on Sample S1 with
$$1.5\text{\,}$\times$0.45\text{\,}$\times$0.15\text{\,}\mathrm{m}\mathrm{m}^{3}$$.
In both cases, the rf-coil was wound directly around the sample and placed on
the single axis goniometer (accuracy
$\sim$$1\text{\,}\mathrm{\SIUnitSymbolDegree}$) of a home-built probe. Most
experiments were carried out at $8.73\text{\,}\mathrm{T}$ typically using
broad-band free induction decay (FID) measurements with pulse lengths of
$0.5\text{\,}\mu\mathrm{s}$ in combination with low $Q$-factors (in the order
of 10), as well as selective FID or spin-echo ($\pi/2-\tau-\pi$) experiments
with $\pi/2$-pulse lengths of 5 or $10\text{\,}\mu\mathrm{s}$ for individual
resonance lines ($Q$ between 16 and 32, cf. SI). Selective saturation recovery
pulse sequence ($\pi/2-\Delta-$FID/spin echo) was employed to measure the
spin-lattice relaxation time $T_{1}$ of individual transitions. The shifts
were obtained by referencing the 51V resonance frequencies versus VOCl3 using
the second reference method [45] and the omnipresent 63Cu resonance line of
the rf-coil.

Numerical calculations The simulated results were obtained by using ab initio
calculations in the framework of density-functional theory (DFT), as
implemented in the program VASP [35]. In this code, augmented plane waves are
used as a basis set together with pseudopotentials. To describe the exchange-
correlation potential, the generalized-gradient approximation (GGA) [46] was
used.

The EFGs and self-consistent calculations were carried out on a $18\times
18\times 9$ ($6\times 6\times 6$) k mesh for the high (low) temperature phase,
respectively. Convergence for total energy and EFGs was carefully checked. For
the DOS calculations, a k mesh of $33\times 33\times 33$ ($19\times 19\times
19$) was used for the high (low) temperature phase, respectively.

## Data availability

The data that support the ﬁndings of this study are available from the
corresponding authors upon reasonable request.

## Code availability

The codes that support the ﬁndings of this study are available from the
corresponding authors upon reasonable request.

## Acknowledgment

The authors thank O. Stockert, S. Wirth, J. Sichelschmidt, X. Feng, W.
Schnelle, and J. Haase (Leipzig) for helpful discussions. We acknowledge the
financial support by the Deutsche Forschungsgemeinschaft (DFG) under SFB1143
(Project No. 247310070), the Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter—ct.qmat (EXC 2147, Project No.
390858490). M.G.V. acknowledges support to the Spanish Ministerio de Ciencia e
Innovacion (grant PID2022-142008NB-I00), partial support from European
Research Council (ERC) grant agreement no. 101020833, the European Union
NextGenerationEU/PRTR-C17.I1, by the IKUR Strategy under the collaboration
agreement between Ikerbasque Foundation and DIPC on behalf of the Department
of Education of the Basque Government and the Ministry for Digital
Transformation and of Civil Service of the Spanish Government through the
QUANTUM ENIA project call - Quantum Spain project, and by the European Union
through the Recovery, Transformation and Resilience Plan - NextGenerationEU
within the framework of the Digital Spain 2026 Agenda. M.G.V. and C.F.
acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) for the project FOR 5249 (QUAST).

## Author contributions

R.G performed NMR experiments, analyzed data, and wrote the manuscript. M.B.
assisted in the NMR experiments and worked with R.G. on the interpretation of
the NMR data. J.N. performed theoretical calculations. J.N. and M.G. assisted
in data analysis. C.Y. and S.C. grew and characterized single crystals. C.F
supervised the project. All authors commented on the manuscript.

## Competing interest

The authors declare no competing interests.

## Additional information

The Supplementary information is attached at the end of this manuscript.

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SUPPLEMENTARY INFORMATION:
Orbital selective commensurate modulations of the local density of states in
$\textrm{ScV}_{6}\textrm{Sn}_{6}$ probed by nuclear spins

R. Guehne,1 J. Noky,1 C. Yi,1 C. Shekhar,1 M. G. Vergniory,1,2 M. Baenitz,1
and C. Felser1
1Max Planck Institute for Chemical Physics of Solids, 01187, Dresden, Germany
2Donostia International Physics Center, 20018 Donostia - San Sebastian, Spain

The following supplement comprises additional experimental data that
constitutes essential complementary evidence to the already documented results
which did not make it into the main file for reason of space. We will first
present temperature dependent spectra for the $c\parallel B_{0}$ orientation,
detailed orientation dependent NMR spectra that allows us to evaluate the
electric field gradient (EFG) at the V nuclei for high and low temperatures,
as well as the NMR shift anisotropy. We will next show how the NMR shift is
extracted for the $a^{\ast}\parallel B_{0}$ spectrum where the central
transitions are hidden. We will then provide evidence that the total signal
intensity can be brought into consistent agreement with changes in
temperature, circuit performance, and rf-penetration depth. We will further
provide some field dependent measurements, as well as a detailed analysis of
the high temperature spin-lattice relaxation to conclude on its magnetic
origin.

##  Appendix A Temperature dependent 51V spectra for $c\parallel B_{0}$

In Fig. 7 we show a detailed account of the spectral changes for temperatures
between $170\text{\,}$ and $20\text{\,}\mathrm{K}$ and $c\parallel B_{0}$. The
highlighted range of temperatures denotes the CDW phase transition where the
high temperature spectrum is progressively replaced by the 3 low temperature
spectra.

Figure 7:  Temperature dependent 51V NMR spectra for $c\parallel B_{0}$ and
$8.73\text{\,}\mathrm{T}$ (combined Fourier transformed selective spin
echoes). The highlighted temperature range marks the phase transition. The
peculiar low temperature intensity pattern, especially the pronounced first
high frequency satellite, reveal changes in the NMR shift in addition to the
obvious differences in the quadrupole splitting frequencies.

##  Appendix B Evaluation of the EFG tensor at high and low temperatures

In the following Figures 8 to 11 the results of detailed orientation dependent
measurements are provided. The experiments were carried out for
$170\text{\,}\mathrm{K}$ (Figs. 8 to 10) and $80\text{\,}\mathrm{K}$ (Fig. 11)
at a magnetic field at $8.73\text{\,}\mathrm{T}$ using broad band FIDs
($0.5\text{\,}\mu\mathrm{s}$) and stepwise rotation of the single crystal
placed on a single axis goniometer. When rotating the EFG about one of its
principle axis ($X,Y$ or $Z$) the apparent quadrupole splitting
$\tilde{\nu}_{Q}$ of the observed spectrum changes according to

$\tilde{\nu}_{Q}=\frac{\nu_{Q}}{2}(3\cos^{2}\beta-1+\eta\sin^{2}\beta\cos
2\alpha)$ (6)

where $\nu_{Q}$ denotes the quadrupole splitting frequency defined in the main
text and $\alpha$ and $\beta$ are the Euler angles in the PAS of the EFGs.

Note, the various orientation dependent measurements are carried out to
consistently determine the orientation of the EFG with respect to the lattice.
Size ($V_{ZZ}$) and shape ($\eta$) are obtained from measurements with the
single crystal being well adjusted along the $Z$ and $Y$ direction and making
use of the EFG being traceless and the definition of the asymmetry parameter.

Figure 8:  Orientation dependent broad band NMR spectra at
$170\text{\,}\mathrm{K}$ and $8.73\text{\,}\mathrm{T}$ for crystal rotations
around the crystal $b$-axis which corresponds to a rotation about the $X$-axis
of the EFG’s PAS (gray ellipsoid). Thus, the rotation connects $V_{ZZ}$ and
$V_{YY}$. Note, only the black resonances belong to this well defined
rotation, while unmarked resonances represent the other two V nuclei per V
triangle that undergo a rotation that leaves the PAS and can thus only be
evaluated with much more difficulty. In addition, just above
$97\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and at about
$98.75\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ the isotropic 27Al (light
green) and 63Cu (light red) resonance lines, respectively, can be seen. The
signal stem from a small piece of Al metal inside, and from the Cu wire of the
rf-coil. The plot on the right hand side shows the quadrupole splitting as
function of angle, including the typical fit using the formula above and
keeping $\alpha=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$. Figure 9:
Orientation dependent broad band NMR spectra at $170\text{\,}\mathrm{K}$ and
$8.73\text{\,}\mathrm{T}$ for crystal rotations around the crystal $c$-axis
which corresponds to a rotation about the $Y$-axis of the EFG’s PAS (gray
ellipsoid). Thus, the rotation connects $V_{ZZ}$ and $V_{XX}$, the values of
which are easily extracted. Note, since the PAS of the EFG’s of each of the
three V nuclei per V triangle share the $Z$-axis, all of the resonances
(black, blue, and red) can be used to evaluate the EFGs as shown in the plot
on the right hand side. Again, the isotropic 27Al and 63Cu resonance lines are
visible. This time, $\alpha=$0\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and the
phase shift $d\approx\pm$60\text{\,}\mathrm{\SIUnitSymbolDegree}$$ determines
the relative orientation of the three EFG’s and the sixfold symmetry of the
Kagome lattice.  Figure 10:  Orientation dependent broad band NMR spectra at
$170\text{\,}\mathrm{K}$ and $8.73\text{\,}\mathrm{T}$ for crystal rotations
around the crystal $a^{\ast}$-axis which corresponds to a rotation about the
$Z$-axis of the EFG’s PAS (gray ellipsoid). Thus, the rotation connects
$V_{YY}$ and $V_{XX}$. Note, only the black resonances belong to this well
defined rotation, while unmarked resonances belong to the other two V nuclei
per V triangle that undergo a rotation that leaves their PAS and can thus only
be evaluated with much more difficulty. The plot on the right hand side shows
the quadrupole splitting as function of angle, including the typical fit using
the formula above while keeping
$\beta=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$. Figure 11:  Orientation
dependent broad band NMR spectra at $80\text{\,}\mathrm{K}$ and
$8.73\text{\,}\mathrm{T}$ for crystal rotations around the crystal $c$-axis
similar to the one shown in Fig. 9. In the CDW phase, the NMR spectrum visible
splits into 3 individual spectra for each of the 3 V per V triangle (green,
red, and blue $-$ in light, regular, and dark color) while the sixfold
symmetry of the Kagome lattice is retained. The plot on the right documents
the orientation dependence and sixfold symmetry of the underlaying lattice.

##  Appendix C Determination of the NMR shift for the in-plane orientation

For $c\parallel B_{0}$ the central transitions and thus the NMR shift are
directly accessible as can be seen in Fig. 4 (A). For the $a^{\ast}\parallel
B_{0}$ orientation, the CTs (for high and low temperatures) are hidden behind
the broad quadrupole spectrum of the 2 other V nuclei in each V triangle. In
order to obtain the NMR shift, therefore, the satellites can be used, and the
first order (equidistant) quadrupole pattern. The procedure is shown in Fig.
12. The relevant low temperature spectrum consists of 3 equally intense
quadrupole patterns differing in shift and quarupole splitting (purple,
orange, and light blue).

Figure 12:  Fourier transform of a broad band NMR signal for
$80\text{\,}\mathrm{K}$ and $8.73\text{\,}\mathrm{T}$ with the single crystal
aligned as $a^{\ast}\parallel B_{0}$. The four spectra below represent the
three individual quadrupole patterns (colored) and their sum (black).

##  Appendix D NMR shift anisotropy

For the high temperatures, the NMR shift was found to be axially symmetric, as
there is essentially no change in the resonance frequency of the CT in Fig.
10. It is a reasonable assumption that the CDW phase inherits this symmetry.
The corresponding isotropic ($K_{\mathrm{iso}}=(2K_{c}+K_{a^{\ast}})/3$) and
axial ($K_{\mathrm{axial}}=2(K_{c}-K_{a^{\ast}})/3$) shift components are
shown in Fig. 13. The colors follow Fig. 4 (C) and (D) and are identified with
the high single temperature resonance V and three chemically non-equivalent
low temperature V sites V1, V2, and V3. In the CDW phase, the isotropic shifts
assume 3 different values, from about $0.743\text{\,}\%$ for V1 to about
$0.76\text{\,}\%$ for V3. Surprisingly, their mean value (gray column in panel
C) agrees almost exactly with $K_{\mathrm{iso}}$ for the high temperature
resonance just above the onset of CDW phase transition at
$96\text{\,}\mathrm{K}$ (black column). Contrastingly, the anisotropy of all 3
low temperature V sites increases, yielding values between $-0.14\text{\,}\%$
(V3) to almost $-0.17\text{\,}\%$ (V2). The mean axial shift component (gray)
deviates clearly from the high temperature $K_{\mathrm{axial}}$ at
$96\text{\,}\mathrm{K}$, with a difference in the order of
$190\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ ($0.019\text{\,}\%$). Thus, one
way to look at the CDW transition is through the shift anisotropy, with a net
change in the axial part of the shift ($\Delta K_{\mathrm{axial}}$), while the
average isotropic shift remains unaltered in comparison to the high
temperature phase.

Figure 13:  The isotropic and axial shift component as functions of
temperature. Note that for low temperatures an axial symmetry cannot be
verified with the experiment. The values during the phase transition are
missing because for $a^{\ast}\parallel B_{0}$ it is not possible to identify
individual resonances due to the overlap of resonances.

##  Appendix E The circuit’s quality factor, rf-penetration depth, and the
total signal intensity

Figure 14:  (A) The CT for $c\parallel B_{0}$ as a function of temperature
reproduced from Fig. 4. The gray areas correspond to the total signal
intensity. (B) The rf-circuit’s quality factor $Q$ as a function of
temperature. Clearly, a bend in the dependence is visible at the onset of the
CDW phase transition, most potentially due to the corresponding change in the
sample resistivity and the thus changing rf-penetration depth that affects the
inductance of the rf-coil. (C The CT signal intensity from panel (A) after
correction for the temperature and $Q$ fits matches almost perfectly with the
$T$ dependence of the rf-penetration depth (rescaled to match the
$100\text{\,}\mathrm{K}$ value of the intensity) implying the loss in signal
intensity is due to a loss in accessible nuclei from a reduced penetration
depth.

To evaluate the total signal intensity we investigate the central transition
of Fig. 4 (A). This allows us to eliminate some uncertainties that are
connected with the excitation conditions in terms of selective excitations,
power levels, band widths etc. The CTs are nevertheless representative for the
total signal intensity. We reproduce the spectra in Fig. 14 (A). These spectra
are corrected for signal averaging as well as for the temperature. An NMR
signal is further proportional to $\sqrt{Q}$, where $Q$ is the quality factor
of the rf-circuit. We plot $Q$ as a function of temperature in panel (B). $Q$
changes with temperature since it is connected to the resistivity of the
sample. We thus corrected the signal intensity given by the gray area under
the curves in panel (A) by the changes in $Q$, and plot it in panel (C) as
gray stars. Finally, since the resistivity of the sample changes significantly
as a function of temperature [15], we estimated the corresponding penetration
depth of the rf-field at $f=$100\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ due
to the skin effect, using $\delta(T)=\sqrt{2\rho(T)/(2\pi f\mu_{0})}$, where
$\rho$ denotes the sample resistivity, $f$ the frequency, and $\mu_{0}$ the
vacuum magnetic permeability. The results are plotted as purple triangles in
panel (C). Consistently, as the resistivity decreases with decreasing
temperature, the skin depth decreases as well, thus, reducing the number of
nuclear accessible with NMR. Obviously, the total signal intensity’s
dependence on temperature follows very well the changes of the skin depth
expected for a thick sample ($200\text{\,}\mu\mathrm{m}$). This analysis
confirms that we have no unusual intensity loss in the present system.

Figure 15:  Fourier transform of spin-echo measurements of the central double-
peak for $c\parallel B_{0}$ and $50\text{\,}\mathrm{K}$ for external fields of
4.8, $6.7\text{\,}$, and $8.8\text{\,}\mathrm{T}$ in units of ppm. The red
dashed lines denote the approximate center frequencies of the two peaks. A
constant splitting in units of ppm proves the origin of the splitting is due
to different NMR shifts (paramagnetic spin susceptibility) rather then an
unusual magnetism.

##  Appendix F Magnetic field dependent measurements

The double-peak central transition structure from Fig. 4 (A) was measured for
two more magnetic fields. The spectra are shown in Fig. 15 in units of ppm.
Evidently, the two peaks do not change in relative position
($150\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$), nor in relative intensity.
This is a strong evidence that the origin of this peak system is related to
different Knight shift values, i.e., different DOS as expected from a CDW.
Second order quadrupole effects ($\propto 1/B_{0}$) obviously do not affect
the spectrum, while, on the other hand, unusual magnetism as potentially
related to orbital currents, are not evident.

##  Appendix G Magnetic relaxation and determination of $T_{1}$

In a quadrupolar split system, it can be difficult to determine $T_{1}$, as it
is defined to be the relaxation time of the unsplit spin system, i.e., as if
no quadrupole interaction is present. For 51V, the quadrupole spectrum covers
almost $3\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and thus, the relaxation of
individual transitions is the only way to measure $T_{1}$ for fairly well
defined conditions. The relaxation of individual transition, however, depends
on the transition and the relaxation mechanism, i.e., whether its driven by
magnetic fluctuations due to free carriers or lattice vibrations via the
quadrupole interaction.

In order to check the relaxation mechanism in the current system, we assume
the relaxation to be of magnetic origin, and apply the corresponding recovery
equations to extract $T_{1}$. We used selective saturation recovery pulse
sequences and analyzed the individual recovery curves with:

$M(t)=M_{0}\left[1-f\Biggl{\\{}\sum_{i=1}^{7}a_{i}\cdot\exp\left(-\frac{\lambda_{i}t}{T_{1}}\right)\Biggr{\\}}\right]\
,$ (7)

where $M_{0}$ is the equilibrium signal intensity, and $f$ the inversion
factor ($f=1$ saturation, $f=2$ inversion). The coefficients $a_{i}$ vary for
each transition, while the exponents $\lambda_{i}$ are shared. We summarize
the various values in Tab. 2.

Table 2: Coefficients $a_{i}$ and exponents $\lambda_{i}$ for magnetic relaxation and recovery measurements of selectively saturated or inverted transitions of a spin $7/2$ system. $a_{i}$ | CT | 1st Sat | 2nd Sat | 3rd Sat | $\lambda_{1}$
---|---|---|---|---|---
$i=1$ | 1/84 | 1/84 | 1/84 | 1/84 | 1
2 | 0 | 1/84 | 1/21 | 3/21 | 3
3 | 3/44 | 1/33 | 1/132 | 3/11 | 6
4 | 0 | 9/77 | 25/308 | 25/77 | 10
5 | 75/364 | 1/1092 | 100/273 | 75/364 | 15
6 | 0 | 49/132 | 49/132 | 3/44 | 21
7 | 1225/1716 | 392/858 | 98/858 | 8/858 | 28

The results are shown in Fig. 16. Black denotes the recovery for the CT, red
for the first, petrol for the second, and pink for the third lower satellite.
The results provide strong evidence for the assumption, because each of the 4
transition selective recoveries, i.e, CT and the three satellites, yield the
same $T_{1}$ of $24(1)\text{\,}\mathrm{m}\mathrm{s}$ at
$250\text{\,}\mathrm{K}$ and $c\parallel B_{0}$. We further repeated the
measurement for the 1st satellite at the same temperature for
$a^{\ast}\parallel B_{0}$ (blue data), which yields the same result, and
hence, the relaxation is isotropic.

Finally, there are a few crystal orientations where one of the 3 V quadrupole
splittings disappears, and therefore, the quadrupole interaction is
eliminated. We used
$\angle(c,B_{0})\approx$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$ (cf. Fig. 8)
to measure the pure $T_{1}$ of the un-split spin system (Fig. 16 yellow data)
and applied the simple exponential relationship

$M(t)=M_{0}\left[1-f\cdot\exp\left(-\frac{t}{T_{1}}\right)\right]$ (8)

which, evidently, gives the same $T_{1}$ as for selective excitation in the
other orientations. We have thus confirmed the relaxation to be driven by
magnetic fluctuations from free electrons which is the prerequisite for our
treatment of $1/T_{1}(T)$ using the DOS.

Figure 16:  Evaluation of the spin-lattice relaxation at
$250\text{\,}\mathrm{K}$ and $8.73\text{\,}\mathrm{T}$. The plot shows the
results from individual saturation recovery measurements carried out for the
CT (black), and the first (red), second (petrol), and third (pink) satellite
of the V spectrum for $c\parallel B_{0}$, as well as for the first satellite
(blue) for $a^{\ast}\parallel B_{0}$, and for the non-selective spectrum at
$75\text{\,}\mathrm{\SIUnitSymbolDegree}$ (yellow). The solid curves denote
the best fit using equations (7), (8), and Tab. 2.