# Possible quadrupole-order-driven commensurate-incommensurate phase
transition in B20 CoGe

S.-H. Baek<EMAIL_ADDRESS>Department of Physics, Changwon National
University, Changwon 51139, Korea  Department of Materials Convergence and
System Engineering, Changwon National University, Changwon 51139, Korea  V. A.
Sidorov  Vereshchagin Institute for High Pressure Physics, RAS, 108840 Moscow,
Troitsk, Russia  A. V. Nikolaev  Skobeltsyn Institute of Nuclear Physics
Lomonosov Moscow State University, 119991 Moscow, Russia  T. Klimczuk  Faculty
of Applied Physics and Mathematics, Gdansk University of Technology,
Narutowicza 11/12, 80-233 Gdansk, Poland  Advanced Materials Centre, Gdansk
University of Technology, Narutowicza 11/12, 80-233 Gdansk, Poland  F. Ronning
Los Alamos National Laboratory, MPA-CMMS, Los Alamos, New Mexico 87545, USA
A. V. Tsvyashchenko  Vereshchagin Institute for High Pressure Physics, RAS,
108840 Moscow, Troitsk, Russia

###### Abstract

The B20-type cobalt germanide CoGe was investigated by measuring the specific
heat, resistivity, and 59Co nuclear magnetic resonance (NMR). We observed a
phase transition at $T_{Q}=13.7$ K, evidenced by a very narrow peak of the
specific heat and sharp changes of the nuclear spin-spin ($T_{2}^{-1}$) and
spin-lattice ($T_{1}^{-1}$) relaxation rates. The fact that the entropy
release is extremely small and the Knight shift is almost independent of
temperature down to low temperatures as anticipated in a paramagnetic metal
indicates that the $T_{Q}$ transition is of non-magnetic origin. In addition,
we detected a crossover scale $T_{0}\sim 30$ K below which the resistivity and
the NMR linewidth increase, and $T_{1}^{-1}$ is progressively distributed in
space, that is, a static and dynamical spatial inhomogeneity develops. While
the order parameter for the $T_{Q}$ transition remains an open question, a
group-theoretical analysis suggests that the finite electric quadrupole
density arising from the low local site symmetry at cobalt sites could drive
the crystal symmetry lowering from the P213 symmetry that is commensurate to
the R3 symmetry with an incommensurate wavevector, which fairly well accounts
for the $T_{Q}$ transition. The quadrupole-order-driven commensurate-
incommensurate phase transition may be another remarkable phenomenon arising
from the structural chirality inherent in the noncentrosymmetric B20 family.

##  I Introduction

Transition-metal mono-silicides $M$Si and -germanides $M$Ge with the B20 cubic
structure, where $M$ denotes a transition metal such as Mn, Fe, and Co, are
noncentrosymmetric materials carrying inherent structural chirality (lack of
mirror symmetry) which has a drastic influence on the electronic and magnetic
properties of these materials, being of recent special interest from both
fundamental and technical points of view [1]. On one hand, the structural
chirality causes finite spin-orbit-driven Dzyaloshinskii-Moriya interaction,
which competes with the Heisenberg exchange coupling. As a result, a helical
spin ordered state whose chirality is determined by the structural one [2], is
stabilized as observed in MnSi [3, 4], FeGe [5], MnGe [6, 7, 8], and
Fe1-xCoxSi [9]. It has been shown that the application of a small magnetic
field in these materials generates peculiar particle-like localized spin
textures, such as magnetic skyrmion lattices in MnSi [10, 11] and Fe1-xCoxSi
[12, 13], a topological spin texture in MnGe [14], and chiral magnetic bobbers
in FeGe [15]. Furthermore, under hydrostatic pressure, chiral magnets undergo
a quantum phase transition into a non-Fermi liquid for MnSi [16, 17] or an
inhomogeneous chiral-spin state for FeGe [18]. On the other hand, for
nonmagnetic systems, e.g., CoSi and RhSi, the structural chirality results in
a three-dimensional chiral topological semimetal featured by unconventional
chiral fermions [19, 20, 21, 22, 23, 24] and a circular photo-galvanic effect
[25, 26]. Interestingly, chiral magnetism with skyrmionic spin texture could
be achieved in Co1+xSi1-x alloys for $x\geq 0.028$ [27].

Figure 1: (a) X-ray powder diffraction pattern of CoGe measured at $T=295$ K
at ambient pressure. The experimental points (red marks), the calculated
profile (green line) and their difference (blue line) are shown. The black
bars in the lower part of the graph represent the calculated Bragg reflections
that correspond to the B20 structure. (b) The specific heat divided by
temperature of CoGe between 2 K and 300 K. The inset enlarges the data near
the sharp peak found at $T_{Q}=13.7$ K. The peak is almost robust against the
magnetic field up to 9 T. (c) Temperature dependence of the resistivity of
CoGe, normalized at 300 K, which forms a minimum at $T_{0}\sim 30$ K. The
dotted data were taken from DiTusa et al. [28]. The inset shows $d\rho(T)/dT$
versus $T$ at low temperatures.

While the structural chirality is an essential feature of the B20 binary
compounds, the electron count in the $3d$ electron shell of the transition
element $M$ is another crucial factor for the ground state. On the other hand,
the structural chirality should have a considerable effect on an unfilled $3d$
shell of $M$, regardless of the detailed valence state which is not well-
defined in the B20 alloys (for example, see [29]). That is, since the local
site symmetry at $M$ is quite low in the B20 structure, the partial occupation
of the $3d$ shell implies that the density matrix elements
$\langle\psi_{lm}|\rho_{L,M}|\psi_{l^{\prime}m}\rangle$ ($l=l^{\prime}=2$ for
$d$-functions) have a significant quadrupole electron density at corresponding
crystal sites. The quadrupole contributions could arise also from the mixture
of $3d$ and $4s$ functions,
$\langle\psi_{lm}|\rho_{L,M}|\psi_{l^{\prime}=0}\rangle$ where the function
$\psi_{l^{\prime}=0}$ refers to the $4s$ states [30]. The appreciable
quadrupole density can lead to strong anisotropic electronic interactions
which can drive a structural or quadrupole phase transition [30, 31, 32]. Such
a situation is not uncommon for $4f$\- and $5f$-elements, where a quadrupole
moment is caused by mixing of $f$-states [31, 32], as observed in, for
example, CeB6 ($T_{Q}=3.3$ K) [31, 33], TmTe ($T_{Q}=1.8$ K) [34], and NpO2
($T_{Q}=25.5$ K) [31]. Quadrupole phase transitions are, however, quite rare
in $3d$ based compounds.

In this paper, we carried out an experimental and theoretical investigation in
B20 CoGe, one of the least studied B20 compounds partly due to the difficulty
in synthesizing B20-type CoGe which requires the application of high pressure
and temperature [35, 36]. So far, B20 CoGe has been known as a Pauli
paramagnet with a low carrier density without a phase transition [37, 38, 28],
similar to the silicide counterpart CoSi, while being distinguished from other
close germanides FeGe and MnGe that are helimagnets. In fact, the band
crossings in the $\Gamma$ and $R$ points of the Brillouin zone in CoGe [38,
28] bear remarkable resemblance to those in CoSi [22], suggesting that CoGe
could host unconventional chiral fermions.

While as yet little was known of a phase transition other than the magnetic
one in the B20 family, our experimental findings, such as anomalous changes of
the 59Co nuclear relaxation rates and a very sharp peak in the specific heat,
evidence the occurrence of a non-magnetic phase transition at $T_{Q}=13.7$ K
in CoGe. By performing a group-theoretical analysis, we propose that the
$T_{Q}$ phase transition may be a commensurate-incommensurate structural
transition driven by electric quadrupole density which arises from the
structural chirality. Our experimental data further suggest that some spatial
inhomogeneity develops at a higher temperature $T_{0}\sim 30$ K than $T_{Q}$.

##  II Experimental details

Polycrystalline samples of B20-cubic CoGe were synthesized under a pressure of
8 GPa and at high temperatures of 1500–1700 K using a toroidal high pressure
apparatus [39] by melting Co and Ge [40]. The purity was 99.9% for Co, and
99.999% for Ge. The B20 phase of high pressure synthesized samples remains
metastable after the pressure release at room temperature, while the
monoclinic phase appears only after heating the sample above 1000 K at ambient
pressure. The crystal structure was examined by x-ray diffraction (XRD), as
shown in Fig. 1(a). Measurements were performed at room temperature and
ambient pressure using the diffractometer STOE IPDS-II (Mo $K\alpha$) and
Guinier camera e G670, Huber (Cu $K\alpha 1$). Powder x-ray diffraction
revealed a single phase material with cubic B20-type structure. The lattice
parameter of CoGe (0.46392 nm) is close to those published earlier (0.4637 nm
[35] and 0.4631 nm [36] for CoGe). In the B20 structure both Co and Ge are
located at the Wyckoff positions (4a) with coordinates $(u,u,u)$,
$(u+0.5,0.5-u,-u)$, $(-u,0.5+u,0.5-u)$, and $(0.5-u,-u,0.5+u)$. Our refinement
determined $u$ to be 0.133 for Co and 0.838 for Ge. Also, the Rietveld fit of
the x-ray powder diffractogram shows that the site population ratio Co/Ge is
0.999(5), evidencing a negligibly small intermixing between the two ions.

Measurements of the specific heat $C_{p}$ were performed using the Quantum
Design Physical Properties Measurement System (PPMS). Resistivity $\rho$ was
measured by a four-terminal method using spot welded Pt wires as the
electrodes.

59Co (nuclear spin $I=7/2$) NMR experiments were carried out at the external
magnetic field of $H=3$ T in the temperature range $T=2.5-200$ K. The NMR
spectra and the spin-spin relaxation rate $T_{2}^{-1}$ were obtained using the
Hahn echo sequence with a typical $\pi/2$ pulse length of $\sim 3\;\mu$s. For
the full spectra including satellite transitions, we used a frequency-sweep
method. The spin-lattice relaxation rate $T_{1}^{-1}$ was measured by the
saturation method on the first satellite line and determined by fitting the
recovery curve of the nuclear magnetization $M(t)$ to the appropriate fitting
function for the first satellite transition in the $I=7/2$ case with the
inclusion of a stretching exponent $\beta$, i.e.,

$\begin{split}\textstyle 1-\frac{M(t)}{M(\infty)}=&\textstyle
A\left[\frac{1}{84}e^{-(t/T_{1})^{\beta}}+\frac{1}{84}e^{-(3t/T_{1})^{\beta}}+\frac{1}{33}e^{-(6t/T_{1})^{\beta}}\right.\\\
&+\textstyle\frac{9}{77}e^{-(10t/T_{1})^{\beta}}+\frac{1}{1092}e^{-(15t/T_{1})^{\beta}}\\\
&+\textstyle\left.\frac{49}{132}e^{-(21t/T_{1})^{\beta}}+\frac{196}{429}e^{-(28t/T_{1})^{\beta}}\right],\end{split}$
(1)

where $A$ is a fitting parameter that is ideally one.

Figure 2: (a) 59Co NMR spectra obtained at 3 T by a frequency sweep. They show
a typical spectral pattern found in a randomly oriented powder for $I=7/2$
nuclei with near axial symmetry. The vertical arrows are expected lines for
$H\perp z$ ($\theta=90^{\circ}$) in the absence of asymmetry ($\eta=0$). With
lowering temperature, the spectrum is moderately broadened with a small
reduction of the separation of satellites. (b) The spectrum at the central
transition which is determined by the second order quadrupole shift as a
function of temperature. The two maxima as well as the $\theta=0^{\circ}$
position are nearly unshifted with lowering temperature, indicating that the
Knight shift is unchanged. (c) Temperature dependence of the FWHM of the lower
central peak in (b). Anomalous increase of the FWHM at $T_{0}\sim 30$ K is
noticed. The dotted lines are a guide to the eyes to emphasize the anomaly at
$T_{0}$.

##  III Results

###  III.1 Specific heat and resistivity

The specific heat divided by temperature, $C_{p}/T$, as a function of
temperature is shown in Fig. 1(b). While the data are consistent with the
previous results reported in the literature [37, 28], we detected a weak but
very sharp peak at $T_{Q}=13.7$ K, as shown in the inset of Fig. 1(b), which
has been overlooked so far. There are two important features of this specific
heat peak that are prominent: (i) the peak position and amplitude are nearly
independent of external magnetic field strength up to 9 T, and (ii) an
extremely small magnetic entropy is released around the transition [$\Delta
S=0.89$ mJ/mol K = 0.00015($R\ln 2$)]. Because the uniform magnetic
susceptibility $\chi(T)$ does not exhibit any anomaly near $T_{Q}$ [37], these
features indicate that the transition is of non-magnetic origin. Below, we
will show that the low energy spin/charge dynamics probed by the 59Co
relaxation rates exhibit clear anomalous changes at $T_{Q}$, while the Knight
shift, or the local static spin susceptibility, is unchanged through $T_{Q}$,
demonstrating that the $T_{Q}$ transition is an intrinsic bulk property of the
material.

Resistivity measurements were performed on a few samples synthesized at high
pressure. General behavior of $\rho(T)$ for all samples is similar, while they
differ in the value of the residual resistivity. Representative
$\rho(T)/\rho(\text{300 K})$ data of CoGe are shown in Fig. 1(c). We also
compared the data from DiTusa et al. [28]. Metallic behavior in $\rho(T)$ is
observed at high temperatures between 30 and 300 K. However, our $\rho(T)$
data reveal a distinguished feature at low temperatures : $\rho(T)$ passes
through a minimum at a characteristic temperature $T_{0}\sim 30$ K and is
slightly enhanced upon further cooling. It appears that $\rho(T)$ is quite
symmetric around $T_{0}$, as evidenced by the linear temperature derivative of
the resistivity, $d\rho(T)/dT$, between $T_{Q}$ and 50 K [see red line in the
inset of Fig. 1(c)]. The enhancement of $\rho(T)$ at low temperatures
resembles the Kondo effect, but it should be noted that the $\gamma$-value
estimated from the $C/T$ data is only about 3 mJ/mol K2, which is comparable
to the free electron value. Also, there is no signature of magnetic screening
from the susceptibility measurements [37, 28]. Therefore, the enhanced
resistivity below $T_{0}$ may be related with a spatial inhomogeneity as
discussed below.

###  III.2 59Co NMR spectrum and Knight shift

Figure 2(a) shows the quadrupole-perturbed 59Co spectra in CoGe obtained by a
frequency sweep method at 3 T at three selected temperatures. For a nuclear
spin $I>1/2$ in an axial symmetric surrounding (asymmetry parameter $\eta=0$),
to first order, there are central ($-\frac{1}{2}\leftrightarrow\frac{1}{2}$)
and satellite transitions between the $m$th and $(m-1)$th levels
($m=-I,-I+1,\cdots,+I$) which are given by

$\begin{split}\nu(m\leftrightarrow m-1)=\;&\nu_{0}(1+\mathcal{K})\\\
&+\frac{1}{2}\nu_{Q}(3\cos^{2}\theta-1)\left(m-\frac{1}{2}\right),\end{split}$
(2)

where $\mathcal{K}$ is the Knight shift, $\nu_{0}$ is the unshifted Larmor
frequency, $\nu_{Q}$ is the nuclear quadrupole frequency, and $\theta$ is the
angle between the principal axis $z$ of the electric field gradient (EFG),
which is most likely along the diagonals, and an external field $H$.

The observed full NMR spectrum is roughly described by Eq. (2) in a randomly
oriented powder, in which the $m\leftrightarrow m-1$ transitions for
$\theta=90^{\circ}$ ($H\perp z$) form the sharp satellite peaks which are
equally spaced by $\nu_{Q}/2$, indicating that the local symmetry at 59Co is
close to axial. Nevertheless, we find that the positions of two outer pairs of
satellites slightly differ from the expected ones for $\eta=0$, implying a
finite asymmetry. In this case, the distance between the first satellites is
equivalent to the apparent quadrupole frequency $\nu_{Q}^{\prime}$ that is
related to the actual value by $\nu_{Q}=\nu_{Q}^{\prime}/(1-\eta)$ [41].

In addition to the first order quadrupole effect, there is also a second order
quadrupole effect which shifts the central transition
($-\frac{1}{2}\leftrightarrow\frac{1}{2}$) depending on $\theta$. For $I=7/2$,
it is written :

$\begin{split}\nu({\textstyle-\frac{1}{2}\leftrightarrow\frac{1}{2}})=&\nu_{0}(1+\mathcal{K})\\\
&+\frac{15\nu_{Q}^{2}}{16\nu_{0}}(1-\cos^{2}\theta)(1-9\cos^{2}\theta).\end{split}$
(3)

The powder pattern of the central transition as a function of temperature is
presented in Fig. 2(b). It reveals two peaks which correspond to the lines for
$\theta\sim 42^{\circ}$ and $90^{\circ}$ at which the second order quadrupole
shift is minimal and maximal, respectively, as can be easily verified in Eq.
(3). The separation between the two maxima $\Delta\nu$ depends on $\eta$ as
well as $\nu_{Q}$ [41]:

$\Delta\nu=\frac{15\nu_{Q}^{2}}{144\nu_{0}}(\eta^{2}+22\eta+25),$ (4)

where $\nu_{Q}=\nu_{Q}^{\prime}/(1-\eta)$. Since $\Delta\nu$ is only slightly
decreased at low temperatures without a noticeable anomaly, we compare the
$\nu_{Q}$ and $\eta$ values at the two temperatures, 160 K and 5 K, at which
the full spectra are available. By using the $\nu_{Q}^{\prime}$ values
extracted from the full spectra in Fig. 2(a), Eq. (4) yields $\eta=0.191$ at
160 K and $\eta=0.222$ at 5 K which, in turn, give rise to $\nu_{Q}=0.821$ MHz
at 160 K and 0.793 MHz at 5 K. This shows that the asymmetry is indeed finite
reflecting the low local site symmetry at 59Co with weak temperature
dependence.

Unlike the two sharp maxima for the central transition, the $\theta=0^{\circ}$
line, which often appears as a step-like anomaly in a clean sample as observed
at high temperatures [see the dotted line for $\theta=0^{\circ}$ in Fig.
2(b)], is unaffected by quadrupole effects up to second order, and thus
permits us to deduce the Knight shift $\mathcal{K}$. Although the step-like
anomaly gradually smears out at low temperatures due to line broadening, the
data indicate that the Knight shift is essentially independent of temperature
down to 2.5 K, evidencing that the system remains a Pauli paramagnet
regardless of the $T_{Q}$ phase transition.

Another interesting feature found in Figs. 2(a) and 2(b) is the broadening of
the NMR spectrum at low temperatures. As shown in Fig. 2(c), the full width at
half maximum (FWHM) measured on the left central peak for $\theta=42^{\circ}$
is more rapidly enhanced below $T_{0}\sim 30$ K, suggesting that a static
spatial inhomogeneity develops below $T_{0}$ and becomes even stronger at low
temperatures toward zero temperature.

Figure 3: (a) Decay of the nuclear magnetization $M_{\text{n}}$ of 59Co with
time $2\tau$ at selected temperatures. At low temperatures, the Lorentzian
rate $T_{2L}^{-1}$ becomes dominant over the Gaussian one $T_{2G}^{-1}$. (b)
Temperature dependence of $T_{2L}^{-1}$ and $T_{2G}^{-1}$. The sharp anomalous
changes of both rates are observed at $T_{Q}=13.7$ K.

###  III.3 59Co nuclear relaxation rates

Figure 3(a) shows the decay of the nuclear magnetization $M_{\text{n}}$ of
59Co at three selected temperatures, being normalized at zero time.
$M_{\text{n}}$ is described by both the Lorentzian and Gaussian spin-spin
relaxation rate, $T_{2L}^{-1}$ and $T_{2G}^{-1}$:

$M_{\text{n}}(2\tau)=M_{0}\exp\left(-\frac{2\tau}{T_{2L}}\right)\exp\left[-\frac{1}{2}\left(\frac{2\tau}{T_{2G}}\right)^{2}\right],$
(5)

where $\tau$ is the delay time between the echo pulses. In general, the two
rates result from different relaxation processes : while $T_{2L}^{-1}$ is the
contribution from the spin-lattice relaxation process, $T_{2G}^{-1}$ describes
the indirect coupling between neighboring nuclear spins through electronic
excitations [42]. Clearly, as shown in Fig. 2(b), both $T_{2L}^{-1}$ and
$T_{2G}^{-1}$ significantly change at $T_{Q}=13.7$ K. Below $T_{Q}$,
$T_{2L}^{-1}$ is enhanced forming a peak centered at $\sim 7$ K. One can see
that its temperature dependence is quite similar to that of $T_{1}^{-1}$ [see
Fig. 4(a)], supporting the direct relationship between $T_{2L}^{-1}$ and
$T_{1}^{-1}$. In contrast, the Gaussian contribution $T_{2G}^{-1}$ is nearly
temperature independent at high temperatures, but rapidly decreases with
decreasing temperature below $T_{Q}$, contrasting sharply with the upturn of
$T_{2L}^{-1}$.

The measurement of the spin-lattice relaxation rate $T_{1}^{-1}$ was carried
out in the relatively low temperature region $T\leq 60$ K. The results are
shown in Fig. 4(a). $T_{1}^{-1}$ is proportional to $T$ down to 30 K, or the
constant $(T_{1}T)^{-1}$, as shown in Fig. 4(b). Together with the
$T$-independent Knight shift, this indicates that the so-called Korringa law
$(T_{1}T)\mathcal{K}^{2}=\text{const.}$, which is expected in a normal metal,
holds. Below $T_{0}\sim 30$ K, however, the Korringa behavior breaks down. At
the same time, we find that the stretching exponent $\beta$ [see Eq. (1)],
shown in the inset of Fig. 4(b), decreases from a constant below $T_{0}$,
indicating that $T_{1}^{-1}$ is progressively distributed in space.
Remarkably, the temperature at which $\beta$ begins to decrease is very close
to $T_{0}$ at which the resistivity $\rho(T)$ forms a minimum [see Fig. 1(c)]
and the NMR linewidth increases more rapidly [see Fig. 2(c)]. These findings
strongly suggest that $T_{0}\sim 30$ K indeed represents a crossover scale
below which a both static and dynamical spatial inhomogeneity of electronic
phases begins to develop. Also, we note that $\beta$ is a constant but still
notably smaller than unity above $T_{0}$, suggesting that a temperature-
independent weak spatial inhomogeneity already exists at high temperatures.

Upon further cooling, we find an additional increase of $(T_{1}T)^{-1}$ just
below $T_{Q}$. Since the system remains a paramagnetic metal down to very low
temperatures, the increase of $(T_{1}T)^{-1}$ below $T_{Q}$ suggests that the
$T_{Q}$ transition is intimately related to a slowing of collective
spin/charge fluctuations. Regardless of the underlying mechanism, the clear
changes of the nuclear relaxation rates, $T_{2L}^{-1}$, $T_{2G}^{-1}$, and
$(T_{1}T)^{-1}$ at $T_{Q}$ demonstrate that the $T_{Q}$ transition is the
intrinsic bulk property of B20 CoGe.

Figure 4: Temperature dependence of (a) $T_{1}^{-1}$ and (b) $(T_{1}T)^{-1}$.
The data deviate from the linear $T$-dependence roughly below $T_{0}\sim 30$ K
and reveal a small kink at $T_{Q}=13.7$ K followed by a rapid increase of
$(T_{1}T)^{-1}$. The inset shows the temperature dependence of the stretching
exponent $\beta$, which is a measure of a dynamical spatial inhomogeneity.

##  IV Discussion

Combining our experimental findings, it is clear that a spatially
inhomogeneous phase appears below a crossover scale $T_{0}\sim 30$ K and a
non-magnetic long-range phase transition occurs at $T_{Q}=13.7$ K in B20 CoGe.
In order to obtain further insights into the possible order parameter for the
$T_{Q}$ transition and the underlying origin of the spatial inhomogeneity
developed below $T_{0}$, we employ group theory methods as follows.

As the B20 structure with the P213 space group symmetry ($\Gamma_{c}$ T4,
#198) lacks inversion symmetry, the local site symmetry in CoGe is quite
complicated. There are four different 3-fold rotational axes directed along
the cube diagonals, but they intersect at different cobalt and germanium
atoms. This symmetry imposes a certain restriction on the electron density
component expressed in terms of spherical harmonics $Y_{L}^{M}$ about atoms.
In particular, the expansion of electron density at cobalt sites has the
$Y_{L=2}^{0}$ quadrupole ($L=2$) and even the $Y_{L=1}^{0}$ dipole ($L=1$)
component lying along the cube diagonals [30].

An electric quadrupole interaction can lead to an effective attractive
interaction between electrons at a high symmetry point of the Brillouin zone
[43]. The effective interaction between electrons can lower the crystal energy
by driving a structural phase transition from P213 to a phase with lower space
symmetry. This implies a breaking of equivalence between four 3-fold
rotational axes. In this case, there is only one relatively high symmetry for
the quadrupole ordering: the R3 space symmetry ($\Gamma_{\text{rh}}$
$C_{3}^{4}$, #146). In R3 symmetry, there is only one 3-fold rotational axis
(e.g., [111] in terms of cubic system) and there are two different channels
for the P213 $\rightarrow$ R3 symmetry lowering [44]. The first involves the
quadrupole active mode at the $X$ point [$q_{X}=\pi/a(1,0,0)$] of the
Brillouin zone, while the second is connected with the quadrupole mode at the
$R$ point [$q_{R}=\pi/a(1,1,1)$]. Which mode actually drives the
transformation depends on the energy lowering associated with the mode and
requires an additional study. [Both modes ($X$ and $R$) and the related
expansion of the Landau free energy are described in detail in the Appendix.]

As described in the Appendix, the theory shows that the Lifshitz condition for
both $q_{X}$ and $q_{R}$ modes fails, i.e., in the expansion of the Landau
free energy in powers of the order parameter components $\eta_{i}$, the
antisymmetrical terms, $\eta_{k}\frac{\partial\eta_{i}}{\partial
x}-\eta_{i}\frac{\partial\eta_{k}}{\partial x}$ etc, are not zero [45].
Consequently, compared to the high-temperature P213 symmetry that is
commensurate, the low-temperature phase is expected to be of the R3 symmetry
but with an incommensurate wave vector $q$ lying close to $q_{X}$ or $q_{R}$.
That is, both discussed schemes of the P213 $\rightarrow$ R3 symmetry lowering
is a commensurate-incommensurate phase transition induced by quadrupole
ordering. Note that one should expect a weak anomaly of the specific heat at
the transition because the transformation due to the gradient contribution to
the free energy is smoother, which is indeed the case at $T_{Q}$ in the
specific heat measurement [see the inset of Fig. 1(b)]. In this case, the weak
anomalous changes of low energy spin/charge dynamics at $T_{Q}$, probed by all
the 59Co relaxation rates ($T_{2L}^{-1}$, $T_{2G}^{-1}$, and $T_{1}^{-1}$),
may be understood by the coupling between the quadrupole electron density and
the spin moment.

The static and dynamical spatial inhomogeneity below $T_{0}\sim 30$ K, as
inferred from the reduction of the stretching exponent $\beta$, the upturn of
the resistivity $\rho(T)$, and the additional NMR line broadening, may be
understood within the quadrupole ordering scenario, if the incommensurate
modulation of the quadrupole electron density exists even above $T_{Q}$ which
could cause a spatial inhomogeneity. It may be worthwhile to recall that an
intermediate chiral fluctuating region exists above $T_{N}$ in MnGe [46],
which could be explained by the peculiar band structure.

##  V Summary

We have established that the sharp specific heat peak with a negligibly small
release of the entropy at $T_{Q}=13.7$ K represents a non-magnetic phase
transition, an unprecedented one in the isostructural B20 family of materials.
The bulk collective nature of the phase transition at $T_{Q}$ was verified by
probing the clear changes of the 59Co spin-spin and spin-lattice relaxation
rates.

The application of group theory suggests that quadrupole electron density
arising from the low local site symmetry of the transition metal element in
the B20 structure could drive a structural symmetry lowering from P213 to R3.
For this symmetry transformation, the Lifshitz condition is violated and
therefore the low-temperature R3 phase should be incommensurate. Indeed, the
commensurate-incommensurate structural phase transition accounts for the tiny
sharp peak in the specific heat observed at $T_{Q}$. Whereas the (long-range)
phase transition takes place at $T_{Q}$, the resistivity, the stretching
exponent, and the NMR linewidth data point that a static and dynamical spatial
inhomogeneity develops at a crossover scale $T_{0}$ considerably above
$T_{Q}$.

It should be emphasized that our group-theoretical arguments could be almost
equally applied in the other B20 compounds, and thus the quadrupole-order-
driven commensurate-incommensurate phase transition may be ubiquitous in them.
However, the tiny anomaly in the specific heat could be easily missed or
overwhelmed by other effects, explaining why such a phase transition has
hardly been found in these compounds despite many decades of study.

###### Acknowledgements.

We thank D.A. Salamatin for the analysis of x-ray powder diffractograms. This
work was supported by the National Research Foundation of Korea (NRF) grant
funded by the Korea government(MSIT) (Grant No. NRF-2020R1A2C1003817). T.K.
was supported by the National Science Centre (Poland; Grant
UMO-2018/30/M/ST5/00773). Work at Los Alamos National Laboratory was performed
under the auspices of the US DOE, Office of Basic Energy Sciences, Division of
Material Sciences and Engineering.

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*

##  Appendix A Group-theoretical analysis

The $X$ point of the B20 Brillouin zone has three rays [at
$q_{X,1}=2\pi/a(1,0,0),q_{X,2}=2\pi/a(0,1,0),q_{X,3}=2\pi/a(0,0,1)$], and the
active mode belongs to the two-dimensional irreducible representation $E$ of
the little group of $X$. Therefore, the P213 space group irreducible
representation at the $X$ point (the $X_{1}$ irreducible representation) has
six components (two components for each ray). A possible mechanism for
symmetry lowering from P213 to R3 involves condensation of the component in
the form ($a,b,a,b,a,b$) in the six-dimensional space of $X_{1}$ [44]. Thus,
there are two different order parameter amplitudes: $a$ and $b$. Condensation
of a single component of one ray [for example at $2\pi/a(1,0,0)$] implies that
this component changes sign in real space in going from one crystallographic
plane (perpendicular to the $x$ axis) to another. Condensation of all three
components means that the sign changes in going from one plane to another
along the $x$, $y$, and $z$ axes. Such structures are called
triple-$q$-antiferroquadrupolar (3-q-AFQ) [43]. In real space the new
quadrupole electron density is given by

$\begin{split}\rho(\mathbf{R}_{n},\mathbf{r})&=a\left(\rho_{1x}(\mathbf{r})e^{iq_{1}\mathbf{R}_{n}}+\rho_{1y}(\mathbf{r})e^{iq_{2}\mathbf{R}_{n}}+\rho_{1z}(\mathbf{r})e^{iq_{3}\mathbf{R}_{n}}\right)\\\
&+b\left(\rho_{2x}(\mathbf{r})e^{iq_{1}\mathbf{R}_{n}}+\rho_{2y}(\mathbf{r})e^{iq_{2}\mathbf{R}_{n}}+\rho_{2z}(\mathbf{r})e^{iq_{3}\mathbf{R}_{n}}\right)\end{split}$

where $\rho_{1}$ and $\rho_{2}$ are active quadrupole functions while
$\mathbf{R}_{n}$ define the $n$-th cubic unit cell of the crystal and
$\mathbf{r}$ is the local radius vector, defining the electron density within
the $n$-th cubic unit cell.

The rhombohedral (trigonal) quadrupole phase has three basis vectors
($\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}$), which in terms of the B20
cubic lattice vectors ($e_{x},e_{y},e_{z}$) are given by [44]

$\mathbf{v}_{1}=2e_{x}-2e_{y},\mathbf{v}_{2}=2e_{y}-2e_{z},\mathbf{v}_{3}=2e_{x}+2e_{y}+2e_{z}.$

The main (hexagonal) axis therefore is along the (1,1,1) main cube diagonal,
see Fig. 5(a).

Figure 5: Rhombohedral (hexagonal) unit cell of R3 ($\Gamma_{\text{rh}}$
$C_{3}^{4}$) for the $X_{1}$ mode (a) and $R_{1}R_{3}$ mode (b).
$\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are perpendicular to $\mathbf{v}_{3}$.
The angle between $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ is $2\pi/3$.

From the condensation scheme one can obtain invariant polynomials, which can
be used to construct the Landau expansion of the free energy. Limiting
ourselves to the polynomials of the fourth degree in order parameters, we
obtain

$\displaystyle\Phi(P,T,a,b)$
$\displaystyle=\Phi_{0}+A(a^{2}+b^{2})+B_{1}(a^{2}+b^{2})^{2}$
$\displaystyle+B_{2}(a^{4}+b^{4})+B_{3}a^{2}b^{2}+B^{4}ab(a^{2}-b^{2}).$

This expansion can be rewritten in the compact form in terms of the order
parameter amplitudes $\eta$ and $\xi$ :

$\Phi(P,T,a,b)=\Phi_{0}+A\eta^{2}+B_{1}^{\prime}\eta^{4}+B_{2}^{\prime}\eta^{4}\sin
2\xi\sin 2(\xi+\xi_{0})$

Here $\eta=\pm\sqrt{a^{2}+b^{2}}$ and $\xi=\arccos(a/\eta)$, and
$B_{1}^{\prime}$, $B_{2}^{\prime}$ are expressed through $B_{1}$, $B_{2}$,
$B_{3}$, and $B_{4}$. According to the Landau theory of the phase transitions
above a certain critical temperature $T_{Q}$ which we call the quadrupole
transition temperature, we have $A=\alpha(T-T_{Q})>0$, which leads to
$\eta=0$. Below $T_{Q}$ the parameter $A$ becomes negative, which gives a
nontrivial solution $\eta\neq 0$, $\xi_{1}\neq 0$, signaling the appearance of
the new quadrupole phase. Notice that if we find one solution with $\xi_{1}$,
we obtain the other solutions with the values $\xi_{2}=\xi_{1}+\pi/2$,
$\xi_{3}=\xi_{1}+\pi$, $\xi_{4}=\xi_{1}+3\pi/2$. Different values of $\xi$
imply interchange between $\pm a$ and $\pm b$ with different signs (domains of
R3). In terms of $\eta$ and $\xi_{i}$ (here $i$ = 1 or 2, 3, 4) we have

$a=\eta\cos(\xi_{i}-\xi_{0}),\;b=\eta\sin(\xi_{i}-\xi_{0})$ (6)

From the expansion of the Landau free energy it follows that the phase
transition would be of the second order if the Lifshitz condition [45] had
been fulfilled. The Lifshitz condition requires that the coefficient $A$ in
the expansion of the free energy as a function of $q$ should have a minimum
for $q=q_{1}$ (and therefore for all other rays of the star, i.e. for
$q=q_{2}$ and $q=q_{3}$). If the Lifshitz condition is violated, the Landau
free energy depends on the quantity

$\eta\nabla\xi-\xi\nabla\eta\neq 0,$

and the phase transition leads to the appearance of an incommensurate phase
with a long wavelength modulation ($\lambda=2\pi/k\gg a$) of the order
parameter amplitudes $\eta$ and $\xi$,

$\eta=\eta_{0}\cos(\mathbf{k}\cdot\mathbf{R}+\varphi_{0}),\;\xi=\xi_{0}\sin(\mathbf{k}\cdot\mathbf{R}+\varphi_{0}),$
(7)

where $k\ll 1/a$. In this case, the macroscopic modulations of the order
parameter amplitudes depend continuously on the thermodynamical parameters, in
particular on temperature and pressure. From Eq. (6) we see that the variation
of $\eta$ causes a gradual change of the quadrupole moments of Co, while the
variation of $\xi$ implies a gradual rotation of the quadrupole component.
Therefore, Eq. (7) describes a helical charge density wave, which is
presumably directed along the (1,1,1) crystal axis.

The consideration of the $R_{1}R_{3}$ active mode at the $R$ point of the B20
Brillouin zone leads to the same qualitative conclusions. The $R$ point has
one ray and the mode belongs to the four-fold irreducible representation
$R_{1}R_{3}$ of the P213 space group. The condensation scheme in the four
dimensional space is given by the vector $(a,a,a,b)$. In real space new
quadrupole electron density is given by

$\rho(\mathbf{R}_{n},\mathbf{r})=[a\rho_{1}(\mathbf{r})+a\rho_{2}(\mathbf{r})+a\rho_{3}(\mathbf{r})+b\rho_{4}(\mathbf{r})]e^{iq_{R}\mathbf{R}_{n}},$

where $\rho_{1}-\rho_{4}$ are four active basis functions. In this case three
basis vectors are written as $\mathbf{v}_{1}=-e_{x}+e_{y}$,
$\mathbf{v}_{2}=-e_{y}+e_{z}$, $\mathbf{v}_{3}=2e_{x}+2e_{y}+2e_{z}$. The unit
cell is smaller than for the $X_{1}$ mode and the trigonal axis is again along
the (1,1,1) direction, see Fig. 5(b).

As before, the Landau free energy expansion in terms of the order parameter
amplitude $a$ and $b$ is characteristic of a second order transition (since
the Landau free energy for this case is quite complicated, we do not quote it
here), but since the Lifshitz condition is violated, the transformation occurs
to an incommensurate phase with a long wave-length modulation
($\lambda=2\pi/k\gg a$) of $a$ and $b$,

$a=a_{0}\cos(\mathbf{k}\cdot\mathbf{R}+\varphi_{0}),\;b=b_{0}\sin(\mathbf{k}\cdot\mathbf{R}+\varphi_{0}),$

This relation implies a rotational dependence of the quadrupole density about
the vector $\mathbf{k}$ in the Brillouin zone.