# Prevalence of non-standard collapsing of strong Langmuir turbulence in solar
corona plasmas

Yaokun Li  Institute of Frontier and Interdisciplinary Science, Shandong
University, Qingdao, Shandong, 266237, People’s Republic of China.  Institute
of Space Sciences, Shandong University, Shandong, 264209, People’s Republic of
China.  Haomin Sun  École Polytechnique Fédérale de Lausanne (EPFL), Swiss
Plasma Center (SPC), CH-1015 Lausanne, Switzerland  Hao Ning  Institute of
Frontier and Interdisciplinary Science, Shandong University, Qingdao,
Shandong, 266237, People’s Republic of China.  Institute of Space Sciences,
Shandong University, Shandong, 264209, People’s Republic of China.  Sulan Ni
School of Physics and Electronic Information, Yantai University, Shandong
264005, People’s Republic of China  Xiangliang Kong  Institute of Frontier and
Interdisciplinary Science, Shandong University, Qingdao, Shandong, 266237,
People’s Republic of China.  Institute of Space Sciences, Shandong University,
Shandong, 264209, People’s Republic of China.  Jiansen He  School of Earth and
Space Sciences, Peking University, Beijing 100871, People’s Republic of China
Yao Chen<EMAIL_ADDRESS>Institute of Frontier and Interdisciplinary
Science, Shandong University, Qingdao, Shandong, 266237, People’s Republic of
China.  Institute of Space Sciences, Shandong University, Shandong, 264209,
People’s Republic of China.

###### Abstract

We present a fully-kinetic simulation of the full life cycle of strong
Langmuir turbulence (SLT) excited by electron beams that are accelerated under
the solar corona conditions. We find that (1) most packets ($\sim$80%) are
affected by their neighbors during their collapse, as a result, their spatial
scale variations present non-standard evolutionary features, i.e., deviating
away from what was predicted by the Zakharov model; (2) the collapsing cavity
is too shallow to trap the wave packet due to the growth of the Coulomb force,
as a result a majority ($\sim$70%) of the packet energy runs away and a
secondary localization may occur. The study indicates that the non-standard
Langmuir collapse may play an important role in coronal plasmas interacting
with an intense electron beam, that may be eventually confirmed by humanity’s
first mission to fly through the corona.

Beams of energetic electrons, existing pervasively in astrophysical and space
plasmas, excite intense Langmuir waves and play a fundamental role in the
wave-particle interaction and energy transfer process in space [1, 2]. In
addition, nonlinear phenomena of beam-plasma interaction have long been
suggested to be the mechanism underlying some coherent radio bursts such as
the type-III and type-II solar radio bursts [3, 4, 5, 6, 7, 8]. Strong
Langmuir turbulence (SLT) develops when the energy density of the wave field
is comparable to the thermal energy density, leading to self-focusing and
localization of wave packets and formation of density cavities [9, 10, 11, 12,
13, 14]. Such process, termed as the Langmuir collapse, can transfer energy to
small scales within tens of the ion plasma oscillation periods, leading to
efficient damping of wave energy. As a result, plasmas can be heated via wave
dissipation and electrons can be accelerated coherently via the transit-time
damping (TTD) process [15, 16, 17, 18].

In 1972, Zakharov first introduced the concepts of Langmuir collapse and SLT
and developed the fluid model to describe such nonlinear behavior of wave
packets [19]. According to the Zakharov’s equations, the maximum of the
electric field energy and the scale of the wave packet follow the following
scaling laws:

$\displaystyle W_{\text{m}}\propto
E^{\text{2}}_{\text{m}}\propto|t_{\text{0}}-t|^{\text{-2}},L\propto|t_{\text{0}}-t|^{\text{2/d}}$
(1)

where $W_{\text{m}}$
$(=\epsilon_{0}E^{\text{2}}_{\text{m}}/4n_{0}k_{\text{B}}T_{\text{e}})$
represents the peak intensity relative to the thermal energy density,
$E_{\text{m}}$ represents the amplitude of the wave field, $L$ represents the
scale of the packet, $t_{0}$ represents the time required for the packet to
collapse to a singular point. For a two-dimensional ($d$ = 2) packet, the
inverse of the maximum amplitude ($E_{\text{m}}^{\text{-1}}$) and the scale of
the packet ($L$) decline linearly with $t_{0}-t$ due to the collapse. Wong &
Cheung (1984, 1985) [20, 21] did beam-plasma experiments and found both
$E_{\text{m}}^{\text{-1}}$ and $L$ of wave packets meet the above predictions,
thus verified the collapsing process of SLT for the first time in experiments.

Despite the great success, Zakharov model adopted the quasi-neutrality
approximation and small-amplitude assumption, thus this model does not
accurately describe processes faster than the ion response time
($\sim\omega_{\text{pi}}^{\text{-1}}$) in which the charge-neutrality
condition can be violated. Such highly-nonlinear phenomena can be simulated
with the particle-in-cell (PIC) method [22, 23, 24, 25, 26, 27], with which
Sun et al. (2022) [28, 29] modelled the beam-plasma interaction induced by
direct current discharges with a hot cathode. With the beam-injection
configuration, the authors found the large-amplitude pumping wave structure
with an almost fixed phase dominates the wave concentration and further
Langmuir collapse that takes place within tens of
$\omega_{\text{pe}}^{\text{-1}}$. This is too early for ions to move thus
charge separation and coulomb force set in to affect the SLT. They introduced
the concept of electron modulational instability (EMI) to describe such a fast
collapsing process of Langmuir waves.

Langmuir wave packets have been observed frequently in the solar wind using
data obtained by Ulysses and STEREO [30, 31, 32]. To evaluate whether they
collapse or not, Graham et al. analyzed 167 Langmuir wave packets in the solar
wind [33]. They concluded most packets are weaker than the threshold by 1-2
orders of magnitude. Therefore, little evidence of Langmuir collapse has been
found in the near-Earth solar wind plasmas.

In the solar corona, eruptions such as flares and coronal mass ejections occur
frequently, especially during active stages. Energetic electrons can be
accelerated efficiently via magnetic reconnection or coronal shocks [34, 35,
36, 37, 38], thus favoring the development of SLT and collapse. Combined
analysis of Hard-X ray and microwave data reveals that bulk acceleration of
electrons can take place in solar flares with their abundance reaching up to
tens of percents [2, 39]. This indicates that intense beams of energetic
electrons are not rare in the corona where SLT can develop and affect the
energy release and conversion during solar eruptions, as well as excitations
of solar radio bursts such as type-IIIs and type-IIs [40]. Yet, few studies
have investigated the kinetic evolution of SLT within the solar coronal
plasmas due to the limits of available observations.

In space, beams of energetic electrons can propagate over a long time and a
long distance [41, 42, 43]. Such situation should be better approximated with
periodic boundary conditions rather than the beam-injection configuration
since in which the system evolution is strongly affected by the pumping wave
near the injection point. Here we present a fully-kinetic PIC simulation of
the beam interaction with plasmas under the solar coronal conditions, with
periodic boundary conditions and the realistic mass ratio
($m_{\text{p}}/m_{\text{e}}$ = 1836). The purpose is to investigate the full
lifecycle of SLT in coronal plasmas. We found that the scale variation trend
of most SLT packets differ from the Zakharov prediction, and the cavity formed
during the collapsing stage is too shallow to trap efficiently the wave
packets whose majority of energy runs away and may undergo a secondary
concentration later. These findings improve our understanding of the nonlinear
collapse and further damping of SLT in space plasmas, and make an essential
step towards solving the established critical problem of SLT in solar corona
plasmas.

We used the open-source Vector PIC (VPIC) code [44, 45] for the simulation
that is two-dimensional in space with three velocity components. We adopted
the usual coronal conditions assuming the background electrons and protons to
be isothermal with temperature of 1 MK. To excite SLT, according to the EMI
thresholds deduced by Sun et al. (2022) [29] we set the abundance of the beam
electrons to be 0.01. Initially, the beam propagates along the background
magnetic field ($\textbf{B}_{0}=B_{0}\vec{e}_{\text{z}})$ with a speed of
$v_{\text{b}}$ (= 0.2718 $c$), where $c$ is the speed of light in vacuum. The
plasma frequency to gyrofrequency ($\omega_{\text{pe}}/\Omega_{\text{ce}}$) is
taken to be 10. The simulation domain is taken to be $L_{\text{x}}\times
L_{\text{z}}$ = 120$\times$120 $c/\omega_{\text{pe}}$ ($\sim$9449$\times$9449
$\lambda_{\text{De}}$), the number of cells is 4096$\times$4096, and the total
simulation time is 1500 $\omega_{\text{pe}}^{\text{-1}}$. To deduce the
numerical noise, we take the number of background electrons in each cell to be
2000 and that of the beam and protons to be 1000. This gives 67 billion
particles within the domain.

Figure 1 presents the distribution of maximum intensity of wave packets (panel
(a)) and density cavities (panel (b)) in the domain, over the whole simulation
time. More than a thousand of localized packets and cavities exist. This is
distinct from the simulation based on the beam-injection model in which the
pumping packet dominates. According to Figure 1(c) and (d), both numbers of
packets and cavities exhibit exponential distribution with the packet
intensity ($W_{\text{m}}$) or the cavity depth ($\delta n_{\text{pm}}$). We
define $\alpha_{\text{w}}$ to be the fraction of the area with certain
intensity ($W$) and $\alpha_{\text{np}}$ to be the area fraction with certain
perturbation ($\delta n_{\text{p}}$). Both parameters present nice exponential
distributions (see the insets of panels (c) and (d)). For instance, in $\sim
1/3$ of the domain we have $W>1$, and in about 5% of the domain we have
$\delta n_{\text{p}}>0.1$.

According to Figure 1(e), the evolution of the system can be divided into four
stages: (1) Stage I (0 - 80 $\omega_{\text{pe}}^{\text{-1}}$), corresponding
to the linear development of the bump-on-tail instability and the growth of
the Langmuir waves, as a result, the electric field energy increases rapidly
to the saturation level of $\langle W\rangle\sim 0.13$; (2) Stage II (80 - 200
$\omega_{\text{pe}}^{\text{-1}}$), corresponding to the development of the EMI
process, in which the wave energy gets localized as evidenced by the rapid
rise of the energy fraction within the high-energy area with $W>1$ that
reaches up to 24% at the end of this stage. In the meantime, the total energy
of the system exhibits slight oscillation, indicating the dynamic nature of
this stage; (3) Stage III (200 - 400 $\omega_{\text{pe}}^{\text{-1}}$),
corresponding to the equilibrium stage during which the total energy and the
energy fraction in high-intensity area maintain a nearly constant level; (4)
Stage IV (400 $\omega_{\text{pe}}^{\text{-1}}$\- the end), corresponding to
the energy dissipation stage in which both the total energy and that within
the high-intensity area decline with time.

Figure 1(f) presents the power-spectra analysis of the electric-field
turbulence within the domain, according to which the peak of the energy
spectra shifts towards larger wave number (k). During stage II ($t=$ 160
$\omega_{\text{pe}}^{\text{-1}}$, 200 $\omega_{\text{pe}}^{\text{-1}}$), we
have $E^{\text{2}}\sim k^{\text{-7}}$, and during stage IV ($t=$ 400
$\omega_{\text{pe}}^{\text{-1}}$, 600 $\omega_{\text{pe}}^{\text{-1}}$), we
have $E^{\text{2}}\sim k^{\text{-6}}$, both spectra are steeper than those
obtained by Sun et al. for laboratory plasmas, indicating more-efficient wave
energy dissipation in our case.

Now we follow the evolution of the SLT packet using the strongest one (with
$W_{\text{m}}\sim 7$) as an example. According to Figure 2(a) and (b), the SLT
contains multi-collapsing packets along the beam direction, the transverse
scale $L$ of the packet and the inverse of the maximum of electric field
fluctuation intensity $E_{\text{m}}^{\text{-1}}$ follows the linear declining
trend from $t=$ 115 to 175 $\omega_{\text{pe}}^{\text{-1}}$ until $L$
decreases to $\sim 54$ $\lambda_{\text{De}}$. This agrees with the predicted
characteristics of Langmuir collapse.

Despite looking similar, the Langmuir collapse in our simulation is non-
standard and two significant features occur. The first feature is that the
majority of the packet energy escapes from the shallow cavity and undergoes a
secondary concentration later. According to Figure 2(d), the protons do not
respond actively during the collapse. As a result, Coulomb force due to the
charge separation arises to slow down the growth of the cavity by balancing
the ponderomotive force. When the packet intensity grows to the maximum, the
collapsing cavity is still too shallow
($\overline{\delta}n_{\text{em}}\sim\overline{\delta}n_{\text{pm}}<0.1$) to
trap the wave energy, thus the collapsing wave packet continue to run away,
with a propagation speed close to the electron thermal speed
($v_{\text{te}}=\sqrt{k_{\text{B}}T_{\text{e}}/m_{\text{e}}}=0.013$ $c$).
Eventually, a major fraction of the packet energy escapes from the cavity (90
$\sim$ 92 $\omega_{\text{pe}}^{\text{-1}}$, see Figure 2(c) and the
accompanying animation [46]). Around $t=330$ $\omega_{\text{pe}}^{\text{-1}}$,
the escaping energy is $\sim 70\%$ of the total wave energy, and there
presents a local bump on the $\overline{W}_{\text{m}}$ profile (Figure 2(d)),
indicating a secondary concentration during the escape of the wave packet.

After $t=200$ $\omega_{\text{pe}}^{\text{-1}}$, the protons respond in
accordance with the quasi-neutrality condition, resulting in deepening cavity
that gradually reaches a level of
$\overline{\delta}n_{\text{em}}\sim\overline{\delta}n_{\text{pm}}\sim 0.5$. In
the meantime, the dissipation of the wave packet sets in. This results in jets
of electrons via coherent acceleration of the TTD process and plasma heating
due to the Landau damping process. The jets can be observed from Figure 2(e)
that presents the $v_{\text{z}}-\text{z}$ phase space distribution after
detrending the oscillation profile of the Langmuir wave, and from Figure 2(f)
we can tell the heating of the background electrons from $t=400$ to $600$
$\omega_{\text{pe}}^{\text{-1}}$, with the temperature rising from 1 to 1.3
MK.

The second significant feature of the non-standard Langmuir collapse is the
dominance of wave packets exhibiting abnormal evolution in transverse scale
$L$. In a randomly-selected squared region with a size of 20$\times$60
$c/\omega_{\text{pe}}$, we identified 22 collapsing packets with
$W_{\text{m}}>3$. In Figure 3 we present intensity maps at a given moment and
profiles of $E_{\text{m}}^{\text{-1}}$ and $L$ for 3 of the 22 packets. We
found that $E_{\text{m}}^{\text{-1}}$ of all the 22 packets present a linear
declining trend during the collapse, while only 4 packets still present the
linear decreasing trend of $L$ with time. The variations of $L$ manifest the
following patterns: (1) $L$ increases first before a rapid decrease ($\\#1$);
(2) $L$ decreases first before a rapid increase ($\\#2$); (3) $L$ increases
linearly ($\\#3$). We found similar results within other regions of the
domain.

According to Figure 3(a1)-(a3) and data not presented, there always exists
another localized wave packet in regions within about one wavelength
($\lambda_{0}\sim 1.7$ $c/\omega_{\text{pe}}$) away from the packet with
abnormal pattern of size evolution, indicating such non-standard $L$ variation
is induced by the interaction of neighboring wave packets.

We highlighted that about 80% of the packets in the selected region have
abnormal (or non-standard) variation trends of $L$ that disagree with what
predicted using the Zakharov’s equations, due to the interaction between
neighboring wave packets.

The abnormal evolution of the spatial size of wave packets affects the
strength of the ponderomotive force and the depth of the cavity and the
accompanying wave-particle interaction. According to Figure 3(d3),
$W_{\text{m}}$ maintains an almost constant level after the collapse ($t=$ 200
- 400 $\omega_{\text{pe}}^{\text{-1}}$), and at $t=$ 600
$\omega_{\text{pe}}^{\text{-1}}$ only a shallow cavity with
$\overline{\delta}n_{\text{pm}}<0.2$ forms at the location of the Langmuir
collapse (see Figure 3(c3)). This can be explained with a relatively weak
ponderomotive force since the packet size is still large with $L\sim 70$
$\lambda_{\text{De}}$ after the collapse (see Figure 3(b3)). So the protons do
not respond fully from $t=200$ to $400$ $\omega_{\text{pe}}^{\text{-1}}$, and
more energy of the wave packet escapes. In addition, one nearby cavity that is
two-times deeper with $\overline{\delta}n_{\text{pm}}\sim 0.4$ forms due to
the secondary concentration of the escaping wave energy (see Figure 3(c3)).

To understand the full lifecycle of SLT excited by beam of energetic electrons
in space, we carried out fully-kinetic PIC simulation with periodic boundary
conditions. We found that most wave packets ($\sim 80\%$) manifest non-
standard pattern of the size variation. Such deviations from theoretical
predictions of Langmuir collapse are likely due to the interaction of nearby
packets, affecting further dissipation of wave energy and cavity formation. We
also found that the cavity out of the Langmuir collapse is in general shallow
with the depth being less than 0.1. This is due to the hindering effect of the
Coulomb force in response to the charge separation during the fast EMI
process. Such cavities cannot trap wave packets efficiently thus a major part
of the wave energy propagates away and may undergo a secondary concentration
to form another density cavity. This study improves our understanding of the
SLT and Langmuir collapse in space plasmas and is helpful to understand the
role of beam-plasma interaction in the origin of density structures, electron
acceleration and plasma heating, and the origin of coherent radio bursts. The
results presented here may be eventually confirmed by Parker Solar Probe, the
humanity’s first mission to fly through the corona.

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Figure 1: (a) Spatial distribution of wave packets with peak intensity
$W_{\text{m}}>2$. The solid rectangle outlines the location of the strongest
wave packet shown in Fig. 2, and the dashed one shows the sampling region for
wave packets plotted in Fig. 3. (b) Spatial distribution of proton cavities
with depth $\delta n_{\text{pm}}>0.2$. (c) The histograms for the number of
wave packets versus peak intensity $W_{\text{m}}$. (d) The histograms for the
number of cavities versus cavity depth $\delta n_{\text{pm}}$. The variation
curves of the corresponding filling factor
($\alpha_{\text{w}},\alpha_{\text{np}}$) are shown in the top right corner of
(c) and (d). The red lines represent the exponential fitting curves. (e)
Temporal profiles of the average energy density of electric field $\langle
W\rangle$ and the ratio of energy in the $W>1$ region to the total field
energy. The vertical lines (at $t=$ 80, 200 and 400
$\omega_{\text{pe}}^{\text{-1}}$) indicate four evolution stages I, II, III
and IV. (f) Energy spectrum $E^{\text{2}}(k)$ versus time, where the vertical
line represents $k_{0}(=\omega_{\text{pe}}/v_{\text{d}})$.  Figure 2: (a)
Spatial distribution of the normalized energy density $W$ of wave packets,
superposed by the $1/e$ contours of $W$. The vertical lines present the
location of the collapsing wave packets (90 - 93 $c/\omega_{\text{pe}}$). (b)
The transverse scale ($L$) and the inverse of $E$ amplitude
($E_{\text{m}}^{-1}$) versus time. The vertical lines present the collapsing
stage of the packet (115 - 175 $\omega_{\text{pe}}^{\text{-1}}$), the green
bars represent the measurement error of $L$. (c) Spatial distribution of the
time-averaged energy density $\overline{W}$ and proton density perturbation
$\overline{\delta}n_{\text{p}}$ over 20 $\omega_{\text{pe}}^{\text{-1}}$, with
the $\overline{W}$ contours spaced by 0.5. (d) Temporal profiles of the time-
averaged peak intensity of wave packet and depth of cavity
($\overline{W}_{\text{m}}$,
$\overline{\delta}n_{\text{em}},\overline{\delta}n_{\text{pm}}$). The vertical
lines represent $t=$ 200, 330 and 600 $\omega_{\text{pe}}^{\text{-1}}$. (e)
Normalized phase space distribution of the background electrons. The
oscillation components induced by the Langmuir waves have been removed. The
vertical lines show the region used to calculate the EVDF plotted in (f).
Overplotted in panel (f) are the curves (dashed) for the Maxwellian velocity
distribution function with $T_{\text{e}}=$ 1 MK and 1.3 MK.  Figure 3:
(a1)-(a3) Spatial distribution of the normalized energy density $W$ of wave
packets ($\\#1-\\#3$) at a given moment. (b1)-(b3) The transverse scale ($L$)
and the inverse of $E$ amplitude ($E_{\text{m}}^{-1}$) versus time. The
vertical lines present the collapsing stage of the packet. (c3) Spatial
distribution of the time-averaged energy density and proton density
perturbation ($\overline{W}$, $\overline{\delta}n_{\text{p}}$). Two vertical
lines indicate the location of the collapse and the secondary concentration.
(d3) The temporal profiles of the time-averaged peak intensity of the wave
packet $\overline{W}_{\text{m}}$ and the time-averaged cavity depth
($\overline{\delta}n_{\text{em}},\overline{\delta}n_{\text{pm}}$). The three
vertical lines represent $t=$ 260, 400 and 600
$\omega_{\text{pe}}^{\text{-1}}$.


# Fermi surface topology and electronic transport properties of a chiral
crystal NbGe2 with strong electron-phonon interaction

Yoshiki J. Sato1,2<EMAIL_ADDRESS>Ai Nakamura1 Rei Nishinakayama2
Ryuji Okazaki2 Hisatomo Harima3 Dai Aoki1 1Institute for Materials Research,
Tohoku University, Oarai, Ibaraki 311-1313, Japan
2Department of physics, Faculty of Science and Technology, Tokyo University of
Science, Noda, Chiba 278-8510, Japan
3Graduate School of Science, Kobe University, Kobe 657-8501, Japan

###### Abstract

We report the electronic structures and transport properties of a chiral
crystal NbGe2, which is a candidate for a coupled electron-phonon liquid. The
electrical resistivity and thermoelectric power of NbGe2 exhibit clear
differences compared to those of NbSi2 even though both niobium ditetrelides
are isostructural and isoelectronic. We discuss the intriguing transport
properties of NbGe2 based on a van Hove-type singularity in the density of
states. The analysis of de Haas-van Alphen oscillations measured by the field
modulation and magnetic torque methods reveals the detailed shape of the Fermi
surface of NbGe2 by comparison with the results of energy band structure
calculations using a local density approximation. The electron and hole Fermi
surfaces of NbGe2 split into two because of the anti-symmetric spin-orbit
interaction. The temperature dependence of quantum oscillations indicates that
the effective mass is isotropically enhanced in NbGe2 due to strong electron-
phonon interaction.

††preprint: APS/123-QED

##  I INTRODUCTION

Chiral crystals have attracted great attention as a platform for a wide
variety of interesting physical phenomena including magnetochiral dichroism
Rikken and Raupach (1997), chiral magnetism Mühlbauer  _et al._ (2009); Togawa
_et al._ (2012); Sato  _et al._ (2022, 2023), non-reciprocal electronic
transport Rikken  _et al._ (2001), and topologically nontrivial band
structures Tsirkin  _et al._ (2017); Chang  _et al._ (2018). In particular, a
topological electronic structure, which is called Kramers-Weyl fermion, is
well defined at time-reversal invariant momenta in non-magnetic chiral
crystals Chang  _et al._ (2018). The unique electronic structures in non-
magnetic chiral crystals, such as B20-type silicides Chang  _et al._ (2017);
Tang  _et al._ (2017); Zhang  _et al._ (2018) and elemental telluriumFurukawa
_et al._ (2017); Sakano  _et al._ (2020); Gatti  _et al._ (2020), have been
intensively studied. When the topological degeneracies exist in the vicinity
of the Fermi level, exotic electronic transport and quantized responses are
expected.

In this study, we focus on a transition metal digermanide NbGe2 with a chiral
crystal structure [space group $P6_{2}22$ (No. 180, $D^{4}_{6}$) or $P6_{4}22$
(No. 181, $D^{5}_{6}$)]. NbGe2 belongs to C40-type transition metal
ditetrelides $TX_{2}$ ($T$ = V, Cr, Nb, Ta and $X$ = Si, Ge). The C40-type
hexagonal crystal structure of NbGe2 is shown in Fig. 1(a). NbGe2 is a
superconductor with $T_{\rm c}$ = 2 K Remeika  _et al._ (1978), and the
prediction that NbGe2 belongs to the Kramers-Weyl semimetalChang  _et al._
(2018) promoted detailed studies of its superconducting properties including
the superconducting phase diagram, full-gap structure, and crossover from
type-${\rm I}$ to type-${\rm II}$/1 superconductivity Lv  _et al._ (2020);
Emmanouilidou  _et al._ (2020); Zhang  _et al._ (2021). Several previous
studies have reported energy band structures obtained by first-principles
calculations taking into account the spin-orbit coupling (SOC), and doubly
degenerate Kramers-Weyl points appear near the Fermi energy level at M and L
points Chang  _et al._ (2018); Emmanouilidou  _et al._ (2020); Lv  _et al._
(2020); Garcia  _et al._ (2021).

Recent theoretical and experimental studies of NbGe2 have shown that strong
electron-phonon interaction exists, and NbGe2 has been deemed to be an ideal
candidate to observe phonon-mediated hydrodynamic flow Garcia  _et al._
(2021); Yang  _et al._ (2021). When the momentum-conserving scattering process
is dominant compared to the momentum-relaxing one, the hydrodynamic electron
flow Levchenko and Schmalian (2020), as seen in a metallic delafossite PdCoO2
Moll  _et al._ (2016) and Weyl semimetals Gooth  _et al._ (2018); Jaoui  _et
al._ (2018); Vool  _et al._ (2021), can be observed. A previous theoretical
study using the first-principles calculation and the Boltzmann transport
equation has pointed out that the momentum-conserving phonon-mediated
electron-electron scattering lifetime is dominant, and an anisotropic
scattering time leads to an anisotropic electronic transport in NbGe2 Garcia
_et al._ (2021). A subsequent experimental study using quantum oscillations
and Raman scattering experiments has demonstrated the strong electron-phonon
interaction and strong suppression of momentum-relaxing phonon-phonon
processes in NbGe2 Yang  _et al._ (2021). However, little is known about the
topology of bulk Fermi surfaces and anisotropy of the effective masses due to
the anisotropic electron-phonon scattering time in NbGe2.

Figure 1: (a) Crystal structure of NbGe2. (b) Single crystal of NbGe2. (c)
Temperature dependence of resistivity $\rho_{\rm exp}$ of NbGe2 for $J$ $||$
$[10\bar{1}0]$ and $[0001]$ (left axis). The dashed lines show calculated
temperature dependence of resistivity $\rho_{\rm calc}$ of NbGe2 (right axis)
from Ref.Garcia  _et al._ (2021).

In this paper, we report on the topology of the Fermi surface of NbGe2 and the
transport properties closely related to the characteristic electronic
structure. NbGe2 and NbSi2 exhibit a notable difference in the electronic
transport and thermoelectric properties due to the characteristic electronic
structure of NbGe2 although both compounds are isostructural and
isoelectronic. Then, we report the de Haas-van Alphen (dHvA) experiments of
NbGe2. In spite of rather complicated Fermi surfaces, a limited number of dHvA
branches has been observed experimentally Emmanouilidou  _et al._ (2020); Yang
_et al._ (2021). To fully reveal the Fermi surface topology of NbGe2, we have
performed two different techniques: magnetic field modulation and magnetic
torque methods. By comparing experimental results with energy band structure
calculation, the detailed shape of the Fermi surface is clarified. The
enhancement of effective masses in NbGe2 has been demonstrated by the analysis
of temperature dependence of quantum oscillations. The effective mass
enhancement and electronic transport properties indicate the presence of
strong electron-phonon interaction in NbGe2.

##  II EXPERIMENTAL DETAILS

The polycrystalline samples of NbGe2 and NbSi2 were synthesized via arc
melting using an arc furnace under an argon atmosphere. The starting materials
were Nb (99.9 %), Ge (99.999 %), and Si (99.9999 %) with a stoichiometric
ratio, namely Nb:Ge = 1:2 and Nb:Si = 1:2. Single crystals of NbGe2 were grown
using the chemical vapor transport (CVT) technique. The NbGe2 polycrystal was
crushed into small pieces and sealed in a quartz tube with 80 mg iodine as a
transport agent. The sealed quartz tube was placed in a furnace with a
temperature gradient of 950 ∘C/850 ∘C for two weeks. The NbGe2 single crystals
were grown at the high-temperature side. The typical size of the grown crystal
is up to the dimensions of 0.6 mm $\times$ 0.6 mm $\times$ 0.6 mm, as shown in
Fig. 1(b). Single crystals of NbSi2 were grown using the Czochralski method in
a tetra-arc furnace. The crystal structure of NbGe2 single crystals grown by
CVT was confirmed using a single-crystal x-ray diffractometer (Rigaku XtaLAB
mini ${\rm II}$) with Mo $K\alpha$ radiation ($\lambda$ = 0.71073
$\mathrm{\mathring{A}}$). The crystal structure was solved with SHELXT and
then refined with SHELXL. The refined crystallographic parameters and atomic
positions of NbGe2 are summarized in Tables 1 and 2. The single crystals were
oriented using a Laue camera (Photonic Science Laue x-ray CCD camera).

Table 1:  Crystallographic and structural refinement data of NbGe2 obtained from single-crystal XRD. Empirical formula | NbGe2
---|---
Formula weight | 238.09
Crystal system | hexagonal
Space group |  $P6_{4}22$ (#181)
$a$ ($\mathrm{\mathring{A}}$) | 4.9727(7)
$c$ ($\mathrm{\mathring{A}}$) | 6.7872(10)
Volume ($\mathrm{\mathring{A}}^{3}$) | 145.35(4)
Formula units per cell (Z) | 3
Number of measured reflections (total) | 1177
Number of measured reflections (unique) | 191
Cut off angle (2$\theta_{\rm max}$) | 65.9∘
$R1$ ($I$ $>$ 2.00$\sigma$($I$)) | 0.0205
$R$ (All reflections) | 0.0291
$wR2$ (All reflections) | 0.0470
Goodness of fit | 1.165
Flack parameter | 0.01(3)
Max Shift/Error in Final Cycle | 0.000
Table 2: Atomic positions and displacement parameters of NbGe2. NbGe2
---
Atom | Site | x | y | z | $B_{eq}$
Nb | 3$c$ | 1/2 | 0 | 0 | 0.37(3)
Ge | 6$i$ | 0.16444(8) | 0.32888(8) | 0 | 0.31(2)

The electrical resistivity was measured with a Quantum Design physical
property measurement system (QD PPMS). The thermopower was measured by a
steady-state technique using a manganin-constantan thermocoupleYamanaka  _et
al._ (2022) in a GM refrigerator. The temperature gradient ($\sim 0.5$ K/mm)
was applied along the in-plane direction using a resistive heater. The dHvA
effects were detected using the magnetic field modulation and magnetic torque
methods. The magnetic field modulation measurements were performed using a
top-loading dilution refrigerator in magnetic fields up to 14.7 T at low
temperatures down to 30 mK. The magnetic torque was measured using a membrane-
type surface stress sensor in magnetic fields up to 9 T at low temperatures
down to 1.7 K.

The band structure calculations were performed using the KANSAI code based on
the full-potential linearized augmented plane wave (FLAPW) method within the
local density approximation (LDA). In the band calculations, the scalar
relativistic effect is considered for all electrons. The spin-orbit coupling
is included by means of a second variational procedure for all valence
electrons. The lattice parameters and atomic positions used for the
calculations are listed in Tables 1 and 2.

Figure 2:  Comparison of $\rho(T)$ of NbGe2 and NbSi2 for (a) in-plane and (b)
out-of-plane current directions. The solid lines are fitting results based on
Eq. (1). (c) Temperature dependence of the Seebeck coefficient $S$ of NbGe2
and NbSi2. The temperature gradient is along the in-plane direction. (d)
$S$/$T$ of NbGe2 and NbSi2 on a logarithmic $T$ scale. Electronic band
structure of (e) NbGe2 and (f) NbSi2 along the $\Gamma$-K-M-$\Gamma$ cut.
Calculated total and partial density of states for (g) NbGe2 and (h) NbSi2.

##  III RESULTS and DISCUSSION

We have measured the temperature dependence of electrical resistivity
$\rho(T)$ of a NbGe2 single crystal for the electrical current $J$ $||$
$[10\bar{1}0]$ (in-plane) and $||$ $[0001]$ (out-of-plane), as shown in Fig.
1(c). At room temperature, $\rho_{\rm 300\,K}$ = 54 ${\rm\mu\Omega\,cm}$ for
$J$ $||$ $[10\bar{1}0]$ and 29 ${\rm\mu\Omega\,cm}$ for $J$ $||$ $[0001]$. The
anisotropy of $\rho$ is approximately 2 between the in-plane and out-of-plane
current directions at 300 K. The residual resistivity ratio (RRR $\equiv$
$\rho_{\rm 300\,K}$/$\rho_{0}$) is 320 ($\rho_{0}$ = 0.17
${\rm\mu\Omega\,cm}$) for $J$ $||$ $[10\bar{1}0]$ and 640 ($\rho_{0}$ = 0.05
${\rm\mu\Omega\,cm}$) for $J$ $||$ $[0001]$. The small values of $\rho_{0}$
and the large values of RRR indicate the high quality of the single crystals
of NbGe2 in this study.

The calculated electrical resistivity obtained using the Boltzmann transport
equation within the relaxation time approximation Garcia  _et al._ (2021) is
also shown in Fig. 1(c). Except for a difference in the absolute values,
$\rho_{\rm exp}(T)$ and $\rho_{\rm calc}(T)$ show similar behavior including
the convex upward curvature. As shown in Fig. 1(c), previous theoretical
calculations have predicted much larger and more anisotropic electrical
resistivity of NbGe2 at room temperature because of the strong electron-phonon
scattering. The discrepancy between $\rho_{\rm exp}(T)$ and $\rho_{\rm
calc}(T)$ may be due to the dominance of momentum-conserving scattering
processes in NbGe2 at high temperatures, as discussed in the previous
experimental work Yang  _et al._ (2021).

Here, we show a notable difference in the temperature dependence of electrical
resistivity of NbGe2 and NbSi2, both of which are isostructural and
isoelectronic. Figures 2(a) and 2(b) show a comparison of $\rho(T)$ between
NbGe2 and NbSi2 for in-plane and out-of-plane current directions,
respectively. NbGe2 exhibits distinct anisotropy in electrical resistivity,
while NbSi2 shows isotropic electrical resistivity. The electrical resistivity
of NbSi2 and NbGe2 are well fitted by the Bloch–Grüneisen–Mott formula Mott
(1936); Wilson (1938); Mott and Jones (1958) as:

$\rho(T)=\rho_{0}+A\left(\frac{T}{\theta_{\rm
D}}\right)^{5}\int_{0}^{\theta_{\rm D}/T}\frac{x^{5}}{(e^{x}-1)(1-e^{-x})}dx-
KT^{3},$ (1)

where the second term and the third term describes the electron-phonon
intraband scattering and the electron-phonon interband ($s$-$d$) scattering,
respectively. We obtained the Debye temperature $\theta_{\rm D}$ of NbSi2 as
$\theta_{\rm D}=460$ K ($\theta_{\rm D}=485$ K) for $J$ $||$ in-plane (out-of-
plane) and $\theta_{\rm D}$ of NbGe2 as $\theta_{\rm D}=315$ K ($\theta_{\rm
D}=330$ K) for $J$ $||$ in-plane (out-of-plane). The Debye temperature is
isotropic in both NbSi2 and NbGe2. The obtained coefficients for intraband
($A$) and interband ($K$) electron-phonon scattering of NbSi2 are as follows:
$A=280$ ${\rm\mu\Omega cm}$ ($A=330$ ${\rm\mu\Omega cm}$ ), and $K=-6\times
10^{-8}$ ${\rm\mu\Omega cm/K^{3}}$ ($K=-5\times 10^{-8}$ ${\rm\mu\Omega cm}$)
for in-plane (out-of-plane) current direction. Similarly, the coefficients for
NbGe2 are as follows: $A=290$ ${\rm\mu\Omega cm}$ ($A=160$ ${\rm\mu\Omega cm}$
), and $K=4\times 10^{-7}$ ${\rm\mu\Omega cm/K^{3}}$ ($K=2\times 10^{-7}$
${\rm\mu\Omega cm}$) for in-plane (out-of-plane) current direction. The
coefficients obtained from fitting the Eq.(1) are summarized in Table 3. The
intraband electron-phonon scattering is comparable and rather isotropic in
both NbSi2 and NbGe2. Meanwhile, the coefficients for the interband electron-
phonon scattering in NbGe2 differ by an order of magnitude from those in
NbSi2, indicating the interband $s$-$d$ scattering process is dominant in
NbGe2. This significant interband electron-phonon scattering is the origin of
the pronounced convex temperature dependence in the resistivity of NbGe2. In
addition, the sign of the coefficient $K$ is reversed between NbSi2 and NbGe2,
reflecting the difference in $d$-band electronic structure Mott and Jones
(1958).

Table 3:  Coefficients obtained from fitting the Bloch–Grüneisen–Mott formula for NbSi2 and NbGe2. |  |  $\rho_{0}$ (${\rm\mu\Omega\ cm}$) |  $\theta_{\rm D}$ (K) |  $A$ (${\rm\mu\Omega\ cm}$) |  $K$ (${\rm\mu\Omega\ cm/K^{3}}$)
---|---|---|---|---|---
NbSi2 | in-plane | 0.322 | 460 | 280 | $-6\times 10^{-8}$
| out-of-plane | 0.167 | 485 | 330 | $-5\times 10^{-8}$
NbGe2 | in-plane | 0.165 | 315 | 290 | $4\times 10^{-7}$
| out-of-plane | 0.050 | 330 | 160 | $2\times 10^{-7}$

In addition, intriguing differences in thermoelectric properties were observed
between NbGe2 and NbSi2. Figure 2(c) shows the temperature dependence of
Seebeck coefficient $S$ of NbGe2 and NbSi2 for $\Delta T$ $||$ in-plane.
$S(T)$ varies linearly with temperature above 150 K. Figure 2(d) shows a
comparison of $S(T)/T$ between NbGe2 and NbSi2. $S(T)/T$ of NbGe2 showed a
metallic behavior ($S(T)/T=$ cosnt.) at high temperatures, while $S(T)/T$ is
enhanced with a logarithmic temperature dependence ($S(T)/T\propto-{\rm ln}T$)
at low temperatures. In a high temperature region, $S(T)$ of NbGe2 is smaller
than that of NbSi2, but exceeds the $S(T)$ of NbSi2 below approximately 9 K.

To discuss the characteristic electronic transport properties of NbGe2, we
performed LDA calculations for energy band structures of NbGe2 and NbSi2.
Figures 2(e) and 2(f) show the energy band structures obtained by LDA
calculations of NbGe2 and NbSi2. The parity violation, namely the lack of
inversion symmetry in the chiral crystal, leads to the spin-splitting of
energy bands. However, two-fold degenerate points, namely the Kramers-Weyl
points, exist at high-symmetry points of the Brillouin zone, such as $\Gamma$
and M points. The calculated energy band structure is essentially consistent
with the energy band structure reported in previous studies Lv  _et al._
(2020); Emmanouilidou  _et al._ (2020); Yang  _et al._ (2021).

The notable difference in the electrical resistivity and the Seebeck
coefficient between NbGe2 and NbSi2 may be attributed to the difference in the
energy dependence of the density of states $D(\varepsilon)$ near the Fermi
energy $E_{\rm F}$. The calculated density of states is shown in Figs. 2(g)
and 2(h). A van Hove-type distinct peak structure appears in $D(\varepsilon)$
of NbGe2 near $E_{\rm F}$. In addition, a contribution from Nb-4$d$ electrons
is significant in the $D(E_{\rm F})$ of NbGe2. Since the density of states in
the $d$ band is much larger than in the $s$ ($p$) state, the final density of
states is higher when the $s$ ($p$) electrons scatter to the $d$ state. The
dominant interband electron-phonon scattering process in $\rho(T)$ of NbGe2 is
most likely due to the large contribution of Nb-4$d$ electrons in the vicinity
of $E_{\rm F}$. In the archetypal example, the electrical resistivity of Pd
exhibits a convex temperature dependence due to the peak structure in the DOS
resulting from the contribution of the Pd-4$d$ electrons Mott and Jones
(1958); Mueller  _et al._ (1970); Fradin (1974).

The enhancement of low-temperature $S(T)/T$ of NbGe2 is also attributed to
$D(\varepsilon)$. The Mott formula that describes the Seebeck coefficient in
metals is expressed as Cutler and Mott (1969):

$S=\frac{\pi^{2}}{3}\frac{k_{\rm B}}{q}(k_{\rm B}T)\left.\frac{\partial{\rm
ln}(D)}{\partial\varepsilon}\right|_{\varepsilon=E_{\rm F}},$ (2)

where $k_{\rm B}$ and $q$ are the Boltzmann constant and the carrier charge,
respectively. The enhancement of low-temperature $S/T$ of NbGe2 is probably
due to the large value of $\partial{\rm
ln}(D)/\partial\varepsilon|_{\varepsilon=E_{\rm F}}$, while the flat
$D(\varepsilon)$ near $E_{\rm F}$ results in a small low-temperature $S/T$ of
NbSi2. Note that a peak in $S(T)/T$ due to the phonon-drag effect has been
observed at around 20 K in the previous study Yang  _et al._ (2021), while the
phonon-drag effect is absent in $S(T)/T$ of the single crystal used in this
study. The difference in $S(T)/T$ may be attributed to the surface scattering
effect depending on the sample size Takahashi  _et al._ (2016).

The dHvA measurements of NbGe2 single crystals were performed to reveal the
topology of Fermi surfaces of NbGe2. The dHvA frequency $F$ is proportional to
the extremal cross-sectional area of the Fermi surface $S_{\rm F}$, that is,
$F$ = $(\hbar c/2\pi e)S_{\rm F}$. Figure 3(a) shows the typical dHvA
oscillations of NbGe2 detected using the field modulation technique, and the
corresponding fast Fourier transform (FFT) spectra are shown in Fig. 3(b).

Figure 3:  Typical dHvA oscillations (a) and corresponding FFT spectra (b) of
NbGe2 at $T$ = 40 mK. The dHvA oscillations were detected using the field
modulation technique, and the magnetic field range of FFT is 7.0 to 14.7 T.
The FFT spectrum reported in a previous work (Ref.Yang  _et al._ (2021)) is
also shown in panel (b).

Several dHvA frequencies appeared in the FFT spectra. The FFT peaks $\zeta$,
$\zeta^{\prime}$, $\kappa^{\ast}$, $\varepsilon$, $\delta_{1}$,
$\delta_{1}^{\prime}$, $\beta$, $\beta^{\ast}$, $\gamma_{1}$, $\gamma_{2}$,
$\alpha$, and $\alpha^{\prime}$ are fundamental signals in Fig. 3(b). The
harmonic signals $n\zeta$ ($n$ = 1, 2, and 3), 2$\varepsilon$, and 2$\delta$
were also observed. The dHvA branches $\zeta$, $\zeta^{\prime}$, and
$\varepsilon$ have been observed in magnetic torque measurements in previous
studies Emmanouilidou  _et al._ (2020); Yang  _et al._ (2021). The critical
field of NbGe2 is $H_{\rm c2}$($T$ = 0) $\sim$ 0.04 T Lv  _et al._ (2020);
Emmanouilidou  _et al._ (2020), and dHvA measurements in this study were
performed in the normal state.

In the field modulation technique, the Bessel function of the first kind in
the detected dHvA signal makes it difficult to observe small Fermi surfaces.
The magnetic torque measurement is a suitable method to detect the small Fermi
surface since the dHvA oscillations in the magnetic torque are not affected by
the Bessel function. Therefore, we have performed dHvA experiments of NbGe2
using the magnetic torque method. Figure 4(a) shows the FFT spectra of the
dHvA oscillations for the magnetic field directions from $[0001]$ to
[10$\bar{1}$0].

Figure 4:  The dHvA oscillations (a) and corresponding FFT spectra (b)
obtained from magnetic torque measurements. The field-angle is from $[0001]$
to $[10\bar{1}0]$ in steps of 5∘. (c) Angular dependence of experimental
$\zeta-\zeta^{\prime}$ and $\lambda_{1}-\lambda_{1}^{\prime}$ branches. (d)
Angular dependence of calculated dHvA frequencies of NbGe2. Figure 5:
Comparison of (a) experimental dHvA frequencies of NbGe2 with (b) calculated
dHvA frequencies obtained by the LDA calculations. The solid lines in panel
(a) are guides to the eyes. The cyclotron effective mass and the band masses
are denoted for the principal directions. (c)–(f) Calculated Fermi surfaces of
NbGe2. (c) and (d): Spin-split hole Fermi surfaces. (e) and (f): Spin-split
electron Fermi surfaces. Solid and dashed lines on the Fermi surfaces indicate
the extremal cross-sections. (g) Brillouin zone of the hexagonal lattice. The
symmetry points and axes are denoted by their common names.

Figure 4(b) shows the field angle dependence of the experimental dHvA
frequencies of NbGe2. In addition to the $\zeta$ and $\zeta^{\prime}$ branches
observed in the field modulation method, dHvA branches $\lambda_{1}$ and
$\lambda_{1}^{\prime}$ have been observed in the magnetic torque measurement.
Note that the labels of dHvA branches are different from that in a previous
study: $\lambda$ and $\zeta$ correspond to $\alpha$ and $\beta$ in
Ref.Emmanouilidou  _et al._ (2020), respectively. As discussed below, the
existence of $\lambda_{1}$ and $\lambda_{1}^{\prime}$ branches and the angular
dependence provide important insights into the topology of the Fermi surface.

Here, we compare the experiment and calculated angular dependence of dHvA
frequencies. Figure 5(a) shows the experimental angular dependence of the dHvA
frequencies obtained using the field modulation and magnetic torque methods.
To interpret the observed dHvA frequencies, we calculated the Fermi surface
using the result of LDA calculations. Figure 6 shows the calculated energy
bands of NbGe2.

Figure 6: Calculated band structure of NbGe2. The dashed line indicates the
Fermi level.

The calculated Fermi surfaces of NbGe2 are shown in Figs. 5(c)–5(f). The Fermi
surfaces of NbGe2 consist of spin-split electron and spin-split hole Fermi
surfaces. Then, we constructed the angular dependence of the dHvA frequency
from the calculated Fermi surface, as shown in Fig.5(b). The angular
dependence of the experimental dHvA frequencies [Fig. 5(a)] can be interpreted
based on the calculated result [Fig. 5(b)] although some dHvA branches, such
as $\beta^{\ast}$, $\eta^{\ast}$, and $\kappa^{\ast}$, appeared only in the
experimental result.

$\alpha$, $\gamma$, $\varepsilon$, and $\zeta$ branches correspond to the
pairs of spin-split electron Fermi surfaces [Figs. 5(e) and 5(f)]. The field
angle dependence of these dHvA branches corresponding to the electron Fermi
surfaces is in good agreement with the calculations. On the other hand, dHvA
branches corresponding to the spin-split hole Fermi surfaces, such as
$\theta$, $\beta$, $\delta$, and $\lambda$, show discrepancies between the
results of experiments and LDA calculations. The $\theta$ and
$\theta^{\prime}$ branches corresponding to the inner paths of the hole Fermi
surfaces were not observed in the experiment. In the LDA calculation, we have
obtained the relatively large curvature factors, namely the term of
($\partial^{2}S_{\rm F}/\partial k^{2}$)-1/2 in the dHvA amplitude, for
$\theta$ and $\theta^{\prime}$ branches. The large curvature factors for the
inner orbits indicate that these branches are difficult to observe
experimentally. Instead of the $\theta$ and $\theta^{\prime}$ branches, the
$\lambda$ branches corresponding to the small neck near the $M$ point provide
evidence for the existence of the connected hole Fermi surfaces. As mentioned
above, the small dHvA branches $\lambda_{1}$ and $\lambda_{1}^{\prime}$ have
been detected by using the magnetic torque method. Figure 4(d) shows the
angular dependence of the calculated dHvA frequencies of NbGe2 in the region
of low dHvA frequencies. The experimental result [Fig. 4(c)] and calculated
angular dependence [Fig. 4(d)] show a good agreement, indicating the existence
of the spin-split hole surfaces with the shapes shown in Figs. 5(c) and 5(d).

Furthermore, the experimentally observed $\beta$ branch shows a different
angular dependence from the calculated one. As shown in Fig. 5(a), the $\beta$
and $\beta^{\prime}$ branches have been observed for all field-angle in the
basal plane, namely $[10\bar{1}0]$ to $[11\bar{2}0]$, whereas the calculated
$\beta$ and $\beta^{\prime}$ branches were absent for $H$ $||$ $[11\bar{2}0]$
[Fig. 5(b)] because of the small neck of hole Fermi surfaces. Moreover, the
$\beta^{\ast}$ branch has been observed for the field directions from $[0001]$
to $[11\bar{2}0]$ in the field modulation measurements although the
$\beta^{\ast}$ branch does not appear in the LDA calculation. This difference
between experiments and calculations is likely due to the magnetic breakdown
on the small neck orbit at around the M point. We note that many dHvA
frequencies have been observed between $\beta$ and $\beta^{\prime}$ branches.
The many dHvA frequencies can be interpreted as being caused by the magnetic
breakdown effect in the two spin-split Fermi surfaces.

To obtain the cyclotron effective mass, we have measured dHvA oscillations at
several constant temperatures in the range from 0.06 to 0.8 K for three field
directions: $[0001]$, $[10\bar{1}0]$, and $[11\bar{2}0]$. Figure 7(a) shows
the dHvA oscillations for $H$ $||$ $[0001]$, demonstrating that the dHvA
amplitude decays as the temperature increases.

Figure 7: (a) The dHvA oscillations measured at several constant temperatures
for $H$ $||$ $[0001]$. (b) Mass plot of $\alpha$, $\alpha^{\prime}$,
$\gamma_{1}$, and $\gamma_{2}$ branches.

Based on the Lifshitz-Kosevich (LK) formula, the temperature dependence of
dHvA amplitude can be written as follows Shoenberg (1984):

${\rm ln}\left\\{\frac{A}{T}\left[1-{\rm exp}\left(-\frac{2\alpha m_{\rm
c}^{\ast}T}{H}\right)\right]\right\\}=-\frac{\alpha m_{\rm
c}^{\ast}}{H}T+const.,$ (3)

where $A$ is the dHvA amplitude, $m_{c}^{\ast}$ is the cyclotron effective
mass, and $\alpha=2\pi^{2}ck_{\rm B}/e\hbar$. The mass plot, namely, a plot of
${\rm ln}\\{A[1-{\rm exp}(-2\alpha m_{\rm c}^{\ast}T/H)]/T\\}$ against $T$, is
shown in Fig.7(b). The dHvA amplitudes of the $\alpha$, $\alpha^{\prime}$,
$\gamma_{1}$, and $\gamma_{2}$ branches are found to decay according to the LK
formula, and the cyclotron effective masses were obtained from the slope of
the fitting curves. The dHvA frequencies and the ratios of cyclotron effective
mass to band mass are listed in Table 4.

Table 4: Experimental and calculated Fermi surface parameters of NbGe2. $F$ and $F_{\rm b}$ are experimental and calculated dHvA frequencies, respectively. $m^{\ast}_{\rm c}$, $m_{0}$, and $m_{\rm b}$ are the cyclotron effective mass, rest mass, and band mass, respectively. | Experiment | Calculation |
---|---|---|---
Branch | $F$ | $m^{\ast}_{\rm c}$ | $F_{\rm b}$ | $m_{\rm b}$ | $m^{\ast}_{\rm c}/m_{\rm b}$
| (kT) | ($m_{0}$) | (kT) | ($m_{0}$) | (-)
$H$ $||$ $[0001]$
$\alpha$ | 11.7 | 1.89 | 11.0 | 1.23 | 1.54
$\alpha^{\prime}$ | 11.4 | 2.18 | 10.7 | 1.19 | 1.83
$\gamma_{1}$ | 6.89 | 4.48 | 6.12 | 3.00 | 1.49
$\gamma_{2}$ | 6.21 | 3.63 | 5.74 | 2.10 | 1.73
$\zeta$ | 0.34 | 0.96 | 0.49 | 0.57 | 1.68
$\zeta^{\prime}$ | 0.25 | 0.82 | 0.27 | 0.62 | 1.32
$\beta^{\ast}$ | 4.07 | 3.07 | - | - | -
$\kappa^{\ast}$ | 3.45 | 2.67 | - | - | -
$H$ $||$ $[10\bar{1}0]$
$\beta$ | 4.74 | 2.31 | 5.10 | 1.68 | 1.38
$\beta^{\prime}$ | 4.16 | 2.48 | 4.53 | 1.50 | 1.65
$\gamma$ | 2.79 | 1.96 | 2.24 | 0.63 | 3.11
$\gamma^{\prime}$ | 3.00 | 2.25 | 2.36 | 1.19 | 1.89
$\lambda_{1}$ | 0.14 | 0.35 | 0.21 | 0.27 | 1.30
$\lambda_{1}^{\prime}$ | 0.12 | 0.36 | 0.17 | 0.25 | 1.44
$\eta^{\ast}$ | 2.25 | 1.40 | - | - | -
$H$ $||$ $[11\bar{2}0]$
$\beta^{\ast}$ | 4.26 | 2.21 | - | - | -
$\gamma$ | 3.33 | 2.26 | 2.86 | 1.29 | 1.75

The cyclotron effective masses of NbGe2 were approximately 1.5 times larger
than the band masses obtained by the LDA calculations, namely $m_{\rm
c}^{\ast}$/$m_{\rm b}$ $\sim$ 1.5. Note that the calculated electronic
specific heat coefficient $\gamma_{\rm calc}$ is obtained to be 4.3 mJ/(K2
mole) from the LDA calculation. From the reported experimental value of
$\gamma_{\rm exp}\sim 6.3$ mJ/(K2 mole) Lv  _et al._ (2020); Emmanouilidou
_et al._ (2020), we estimated the ratio of $\gamma_{\rm exp}$/$\gamma_{\rm
calc}$ to be approximately 1.5. The $m_{\rm c}^{\ast}$/$m_{\rm b}$ ratio is
consistent with the $\gamma_{\rm exp}$/$\gamma_{\rm calc}$ ratio.

Such mass enhancement is generally caused by the electronic correlations
Stewart (1984); Taillefer and Lonzarich (1988), magnetic
fluctuations/frustrations Wada  _et al._ (1987); Kondo  _et al._ (1997); Urano
_et al._ (2000), and electron-phonon interactions Gomersall and Gyorffy
(1974); Miyake  _et al._ (1989). NbGe2 is a non-magnetic compound without
$f$-electrons. Thus, the possibility that strong electronic correlations or
magnetic contributions cause mass enhancement is ruled out. The results of
dHvA measurements and first-principles calculations support that the strong
electron-phonon interaction is responsible for the enhancement of the
effective mass. Although the anisotropic electron-phonon scattering time has
been demonstrated in the previous theoretical work Garcia  _et al._ (2021),
rather isotropic effective mass enhancement was observed in NbGe2 in this
study.

As we have discussed, the Fermi surface of NbGe2 is comprised of spin-split
electron and spin-split hole Fermi surfaces. Such spin-split Fermi surfaces
have been widely observed in non-centrosymmetric metals. We can estimate the
energy scale of spin-splitting $\Delta\varepsilon$, namely the magnitude of
anti-symmetric spin-orbit interaction, from the splitting of dHvA frequencies
$\Delta F$ and the cyclotron effective mass $m_{\rm c}^{\ast}$. Using the
Onsager relation $\Delta F=(c\hbar/2\pi e)\Delta S$ and the definition of the
cyclotron mass $m_{\rm c}^{\ast}=(\hbar^{2}/2\pi)(\Delta
S/\Delta\varepsilon)$, we obtain a simple expression as follows:

$\Delta\varepsilon=\frac{\hbar e}{m_{\rm c}^{\ast}c}\Delta F.$ (4)

Note that the energy scale of the anti-symmetric spin-orbit interaction should
be estimated using experimentally obtained effective masses in strongly
correlated electron systems with enhanced effective masses Mineev and Samokhin
(2005); Terashima  _et al._ (2008); Maurya  _et al._ (2018); Sato  _et al._
(2020). $\Delta\varepsilon$ of NbGe2 is estimated to be 182 K for
$\alpha$-$\alpha^{\prime}$ branches (electron Fermi surface, $H$ $||$
$[0001]$) and 360 K for $\beta$-$\beta^{\prime}$ branches (hole Fermi surface,
$H$ $||$ $[10\bar{1}0]$). The estimated energy scale of spin-splitting is
comparable to an isostructural compound NbSi2 Ōnuki  _et al._ (2014) and a
non-centrosymmetric metal LaRhGe3 with Rh-4$d$ electrons Kawai  _et al._
(2008).

Finally, let us compare the Fermi surface properties of NbGe2 and NbSi2. The
Fermi surface properties of both compounds are summarized in Table 5.

Table 5:  Comparison of Fermi surface properties of NbGe2 and NbSi2. $\Delta
F_{\rm exp}$ and $\Delta\varepsilon_{\rm exp}$ are the splitting of
experimental dHvA frequencies and the corresponding energy scale of spin-
splitting, respectively. The reported values of $\Delta F_{\rm exp}$,
$\Delta\varepsilon_{\rm exp}$, $m_{\rm c}^{\ast}$, and $m_{\rm
c}^{\ast}$/$m_{\rm b}$ of NbSi2 are from ref.Ōnuki  _et al._ (2014). NbGe2
---
Branch |  $\Delta F_{\rm exp}$ (T) |  $\Delta\varepsilon_{\rm exp}$ (K) |  $m_{\rm c}^{\ast}$ ($m_{0}$) |  $m_{\rm c}^{\ast}$/$m_{\rm b}$ (-)
$\alpha$ & $\alpha^{\prime}$ | 270 | 182 | 1.89 & 2.18 | 1.54 & 1.83
(electron) |  |  |  |
$\beta$ & $\beta^{\prime}$ | 586 | 360 | 2.31 & 2.48 | 1.38 & 1.65
(hole) |  |  |  |
NbSi2 Ōnuki  _et al._ (2014)
Branch |  $\Delta F_{\rm exp}$ (T) |  $\Delta\varepsilon_{\rm exp}$ (K) |  $m_{\rm c}^{\ast}$ ($m_{0}$) |  $m_{\rm c}^{\ast}$/$m_{\rm b}$ (-)
$\alpha$ & $\alpha^{\prime}$ | 120 | 209 | 0.73 & 0.70 | 0.78 & 0.78
(electron) |  |  |  |
$\gamma$ & $\gamma^{\prime}$ | 240 | 334 | 0.98 & 0.91 | 1.12 & 1.10
(hole) |  |  |  |

The Fermi surface of NbSi2 has been investigated by dHvA experiments, and the
hole Fermi surfaces consist of pocket Fermi surfaces around the K point in the
Brillouin zone Ōnuki  _et al._ (2014). In contrast, the (121st and 122nd) hole
Fermi surfaces of NbGe2 have a connected shape along the boundary of the
Brillouin zone, as shown in Figs. 5(c) and 5(d). The difference in the
topology of hole Fermi surfaces as well as the density of states between NbGe2
and NbSi2 is arising from the different energy dispersion near the M point. As
pointed out by the previous study Emmanouilidou  _et al._ (2020), the energy
band dispersion of NbGe2 exhibits saddle-point-like dispersions near $E_{\rm
F}$. Pairs of saddle-point-like dispersions exist at around the H and M points
near $E_{\rm F}$ [see Fig. 6], resulting in a distinct peak structure in the
density of states. The van Hove-type singularity and the significant
contribution of Nb-4$d$ electrons in the DOS near the Fermi level affect to
the electronic transport and thermoelectric properties of NbGe2. Note that the
$\lambda_{1}$ and $\lambda_{1}^{\prime}$ branches correspond to orbits
circulating near the saddle-point at the M point. The $m_{\rm
c}^{\ast}$/$m_{\rm b}$ ratios for $\lambda_{1}$ and $\lambda_{1}^{\prime}$
branches are comparable in magnitude to the mass enhancement observed in the
other dHvA branches despite the presence of saddle-point-like band dispersion.

The energy scales of spin-splitting $\Delta\varepsilon_{\rm exp}$ are
comparable between NbGe2 ($\Delta\varepsilon_{\rm exp}=$ 182 K for electron
Fermi surfaces and $\Delta\varepsilon_{\rm exp}=$ 360 K for hole Fermi
surfaces) and NbSi2 ($\Delta\varepsilon_{\rm exp}=$ 209 K for electron Fermi
surfaces and $\Delta\varepsilon_{\rm exp}=$ 334 K for hole Fermi surfaces
Ōnuki  _et al._ (2014)). In NbGe2, the effective mass is enhanced by the
strong electron-phonon interaction compared to NbSi2. As a result, while the
spin-splitting of dHvA frequencies $\Delta F$ differs significantly between
NbGe2 and NbSi2, the energy scale of spin-splitting remains comparable.

##  IV Conclusions

We investigated the detailed Fermi surface topology and electronic transport
properties of NbGe2. NbGe2 and NbSi2 exhibit a notable difference in the
temperature dependence of electrical resistivity and Seebeck coefficient due
to the strong interband electron-phonon scattering. The characteristic
behavior in $\rho(T)$ and $S(T)$ of NbGe2 are attributed to the van Hove-type
peak structure and the significant contribution of Nb-4$d$ electrons in the
DOS near the $E_{\rm F}$. The detailed topology of the Fermi surfaces in NbGe2
is revealed by combining the magnetic field modulation and magnetic torque
methods. The angular dependence of the experimental dHvA frequencies can be
interpreted based on the results of LDA calculations. The dHvA measurements
also revealed the isotropic enhancement of the cyclotron effective mass
($m_{\rm c}^{\ast}/m_{\rm b}\sim 1.5$), indicating the existence of strong
electron-phonon interaction in NbGe2. Our results provide valuable insights
into the intriguing electronic structure of NbGe2, which is a candidate for a
coupled electron-phonon liquid.

###### Acknowledgements.

We would like to thank Y. Haga, Y Ōnuki, N. Kimura, F. Honda, Y. Shimizu, Y.
Homma, and D. X. Li for discussion. We acknowledge all the support from the
International Research Center for Nuclear Materials Science at Oarai
(Institute for Materials Research, Tohoku University). This work was supported
by KAKENHI (JP22K20360, JP22H04933, JP21K03448), Grant-in-Aid for JSPS
Research Fellow (JP19J20539), and DIARE research grant.

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