The NED Lagrangian for Ref. <cit.> is

\begin{equation}\label{eq:a1}
\mathcal{L}(\mathcal{F}) = - \frac{\mathcal{F}}{1+\sqrt{2 \beta \mathcal{F}}}
\end{equation}

For the above Lagrangian equations for critical parameters are <cit.>

\begin{align}\label{eq:a2}
\Bigl( v_{c}^{2} +4Q_m \sqrt{\beta} \Bigl)^3 -
8 Q_m^2 \Bigl( 3v_c^4 +6Q_m \sqrt{\beta}v_c^2+8 \beta Q_m^2 \Bigl) =0, \\
T_{c} = \frac{1}{ \pi  v_{c}  }  -
\frac{8 Q_m^2 \bigl( v_c^2 +2Q_m \sqrt{\beta}\bigl)}{\pi v_c (4 Q_m \sqrt{\beta}+{v_{c}^{2}})^{2}}, \\
P_{c} = \frac{1}{2 \pi  v_{c}^2  }
- \frac{2 Q_m^2 \bigl( 3v_c^2 +4Q_m \sqrt{\beta}\bigl)}{\pi v_c^2 ({4} Q_m \sqrt{\beta}+{v_{c}^{2}})^{2}}.
\end{align}

Putting $x=v_c^2+4Q_m \sqrt{\beta}$ into the first equation of (<ref>)
we obtain

\begin{equation}\label{eq:a3}
\end{equation}

In order to satisfy $v_c \geq 0$, we must have

\begin{equation}\label{eq:a4}
\lvert x \rvert \geq 4Q_m \sqrt{\beta}.
\end{equation}

Three real roots of equation (<ref>) occur when
the discriminant is

\begin{equation}\label{eq:a5}
\Delta= 442368 Q_{m}^{8} \beta  (\sqrt{\beta}-Q_{m} ) (5 Q_{m} -4 \sqrt{\beta}) < 0.
\end{equation}

From above condition $\Delta < 0$ we obtain

\begin{equation}\label{eq:a6}
{Q_m}=\sqrt{\beta_0} < \sqrt{\beta} < \sqrt{\beta_2}= \frac{5Q_m}{4}.
\end{equation}

To find the solutions of equation (<ref>),
we will use the Tschirnhaus transformation method.
Putting $x=t+B$ into equation (<ref>)

\begin{equation}\label{eq:a7}
t^3 +pt +q=0,
\end{equation}

where we set coefficients of $t^2$ equal to zero &
$B=8Q_m^2$. Finally the solutions of equations (<ref>) is

\begin{equation}\label{eq:a8}
x_{j}=2 \sqrt{\frac{-p}{3}} \cos{\Biggr[\frac{1}{3} \arccos{\biggl(\frac{3q}{2p}\sqrt{\frac{-3}{p} } \biggl)}-\frac{2\pi j}{3} \Biggr]},
\end{equation}

where $j=0$, $1$ & $2$. The condition in the equation
(<ref>) was satisfied for $x_0$ and $x_1$ only, $x_2$ does not
satisfy condition (<ref>). Therefore we have two physical critical points.
The constant $p$ and $q$ are given by

\begin{align}\label{eq:a9}
p &=  144 \sqrt{\beta} Q_{m}^{3}-3 B^{2}, \\
q &=  -256 \beta  Q_{m}^{4}+144 B \sqrt{\beta} Q_{m}^{3}-2 B^{3}.
\end{align}

$\beta < \beta_{0}$ admits only one real critical point. For $\beta > \beta_2$ no critical points occur. Finally, the critical radius $v_c$ can be written as

\begin{equation}\label{eq:a10}
v_c=\sqrt{x-4Q_m \sqrt{\beta}} \text{,}  \:    x=
x_0 \; \& \; x_1, \: \text{where} \;  \beta_0  < \beta < \beta_2.
\end{equation}

For two critical pressures to be positive, we must have

\begin{equation}\label{eq:a13}
\sqrt{\beta} > \sqrt{\beta_1} = \frac{9Q_m}{8}.
\end{equation}

Excluding the range of $\beta$ from $\beta_0$ to $\beta_1$, we can say that
two critical points with positive critical pressures occur
for $\beta_1 < \beta <  \beta_2$. For $\beta_0 < \beta <  \beta_1$ two critical
points occur but with one negative critical pressure. Our table <ref> is
consistent with table 1 of Ref. <cit.>.

Case     CP     CP1     CP2

310em$\beta_1 < \beta=1.3 < \beta_2$     $v_c$     1.2996     3.0603

$T_c$     0.0458     0.0540

$P_c$     0.0013     0.0055

310em$\beta_0 < \beta=1.1 < \beta_1 $     $v_c$     0.8319     3.3416

$T_c$      0.0251     0.0524

$P_c$     -0.0115     0.0051

310em$0.50=\beta < \beta_0  $     $v_c$     $--$     4.0206

$T_c$     $--$      0.0483

$P_c$     $--$      0.0042

With $Q_m=1$.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Cambridge University Press

titleSpacetime and geometry

titleGravitation and cosmology: principles and applications of the general theory of relativity

journaltitleClass. Quant. Grav.
titleSingular hypersurfaces in scalar - tensor theories of gravity

journaltitlePhys. Rev. D
titleBlack holes in the Brans-Dicke-Maxwell theory

journaltitlePhys. Rev. D
titlePPN-limit of fourth order gravity inspired by scalar-tensor gravity

journaltitleClass. Quant. Grav.
titlef(R) gravity and scalar-tensor theory

titleScalar-tensor-vector gravity theory

journaltitlePhys. Rev. D
titlede Sitter space and the equivalence between f(R) and scalar-tensor gravity

journaltitleJ. Phys. Conf. Ser.
title6+1 lessons from f(R) gravity

journaltitleRev. Mod. Phys.
titlef(R) Theories Of Gravity

journaltitlePhys. Rept.
titleExtended Theories of Gravity

journaltitlePhys. Rev. D
note[Erratum: Phys.Rev.D 85, 089906 (2012)]
titleLinearized f(R) Gravity: Gravitational Radiation and Solar System Tests

journaltitlePhys. Rev. D
titlePolarizations of gravitational waves in $f(R)$ gravity

journaltitleEur. Phys. J. C
titleA new $f(R)$ gravity model and properties of gravitational waves in it

journaltitleJ. Math. Phys.
titleThe Einstein tensor and its generalizations

journaltitleJ. Math. Phys.
titleThe four-dimensionality of space and the einstein tensor

journaltitlePhys. Rev. D
titleThe Lovelock Gravitational Field Equations in Cosmology

journaltitlePhys. Rev. D
titleQuantum Gravity at a Lifshitz Point

titleOn the Extra Mode and Inconsistency of Horava Gravity

journaltitleJ. Phys. Conf. Ser.
titleHorava-Lifshitz gravity: a status report

journaltitleInt. J. Mod. Phys. D
titleHořava gravity at a Lifshitz point: A progress report

journaltitlePhysical review letters
titleEinstein-Gauss-Bonnet gravity in four-dimensional spacetime

journaltitlePhys. Lett. B
titleCharged black holes in AdS spaces in 4D Einstein Gauss-Bonnet gravity

titleThermodynamics, Phase Transition and Joule Thomson Expansion of novel 4-D Gauss Bonnet AdS Black Hole

journaltitlePhys. Rev. D
titleExtended thermodynamics and microstructures of four-dimensional charged Gauss-Bonnet black hole in AdS space

journaltitlePhys. Dark Univ.
titleHawking–Page phase transitions in four-dimensional Einstein–Gauss–Bonnet gravity

journaltitlePhys. Dark Univ.
titleClouds of strings in 4$D$ Einstein-Gauss-Bonnet black holes

journaltitlePhys. Lett. B
titleThermodynamics and P-v criticality of Bardeen-AdS Black Hole in 4$D$ Einstein-Gauss-Bonnet Gravity

journaltitleNucl. Phys. B
titleCharged 4D Einstein-Gauss-Bonnet-AdS black holes: Shadow, energy emission, deflection angle and heat engine

journaltitleAnnals Phys.
titlePhase transition of AdS black holes in 4D EGB gravity coupled to nonlinear electrodynamics

journaltitleAnnals Phys.
title4D AdS Einstein–Gauss–Bonnet black hole with Yang–Mills field and its thermodynamics

journaltitleEur. Phys. J. Plus
titleBlack hole solution and thermal properties in 4$D$ $AdS$ Gauss–Bonnet massive gravity

journaltitlePhys. Dark Univ.
titleStability of thin-shell wormhole in 4D Einstein–Gauss–Bonnet gravity

journaltitleEur. Phys. J. Plus
titleCharged AdS black holes in 4D Einstein–Gauss–Bonnet massive gravity

journaltitleInt. J. Geom. Meth. Mod. Phys.
titleQuasinormal modes and phase structure of regular AdS Einstein–Gauss–Bonnet black holes

journaltitleEur. Phys. J. Plus
titleExact solution of Bardeen black hole in Einstein–Gauss–Bonnet gravity

journaltitleProc. Roy. Soc. Lond. A
titleOn relativistic wave equations for particles of arbitrary spin in an electromagnetic field

journaltitleHelv. Phys. Acta
titleForce-free particles with any spin

journaltitlePhys. Rev. D
titleCan gravitation have a finite range?

journaltitlePhys. Rev. D
titleGeneralization of the Fierz-Pauli Action

journaltitlePhys. Rev. Lett.
titleResummation of Massive Gravity

journaltitlePhys. Rev. Lett.
note[Erratum: Phys.Rev.Lett. 121, 129902 (2018)]
titleTests of general relativity with GW150914

journaltitleClass. Quant. Grav.
titleThermodynamical Structure of AdS Black Holes in Massive Gravity with Stringy Gauge-Gravity Corrections

journaltitlePhys. Rev. D
titleCharged BTZ black holes in the context of massive gravity’s rainbow

titleThermal fluctuations of charged black holes in gravity’s rainbow

titleThermodynamic description and quasinormal modes of adS black holes in Born-lnfeld massive gravity with a non-abelian hair

journaltitleEur. Phys. J. Plus
titleExtended phase space thermodynamics of Bardeen black hole in massive gravity

journaltitlePhys. Scripta
titleThermodynamics, phase structure and quasinormal modes for AdS Heyward massive black hole

journaltitleProc. Roy. Soc. Lond. A
titleFoundations of the new field theory

journaltitleZ. Phys.
titleConsequences of Dirac's theory of positrons

journaltitleCIEA del IPN, Mexico City
titleNon-Linear Electrodynamics–a Study

journaltitlePhys. Rev. D
titleHigher order correction to the GHS string black hole

titleHigher-order corrections to mass-charge relation of extremal black holes

journaltitlePhys. Rev. D
titleShear Viscosity from Effective Couplings of Gravitons

journaltitlePhys. Rev. D
titleHigher derivative corrections to R-charged AdS(5) black holes and field redefinitions

titleThermodynamics of the Maxwell-Gauss-Bonnet anti-de Sitter Black Hole with Higher Derivative Gauge Corrections

journaltitlePhysics Letters B
titleThree dimensional black hole coupled to the Born-Infeld electrodynamics

journaltitlePhysical Review D
titleBorn-Infeld black holes in AdS spaces

journaltitleNuclear Physics B
titleNonlinearly charged dyonic black holes

journaltitleAnnals of Physics
titleBorn–Infeld-type electrodynamics and magnetic black holes

journaltitleEur. Phys. J. C
titleBorn–Infeld black holes in 4D Einstein–Gauss–Bonnet gravity

journaltitleMod. Phys. Lett. A
titleJoule–Thomson expansion of Born–Infeld AdS black holes in consistent 4D Einstein–Gauss–Bonnet gravity

journaltitlePhys. Lett. B
titleTopology of Born-Infeld AdS black holes in 4D novel Einstein-Gauss-Bonnet gravity

journaltitlePhys. Rev. D
titleCharged black points in general relativity coupled to the logarithmic U(1) gauge theory

titleAsymptotic charged BTZ black hole solutions

journaltitleAnnals Phys.
titleAsymptotic Reissner-Nordstroem black holes

journaltitleAnnals Phys.
titleThermodynamic properties of asymptotically Reissner-Nordstrom black holes

journaltitleGrav. Cosmol.
titleDyonic Black Holes with Nonlinear Logarithmic Electrodynamics

journaltitleInt. J. Geom. Meth. Mod. Phys.
titleMagnetic black holes in AdS space with nonlinear electrodynamics, extended phase space thermodynamics and Joule–Thomson expansion

title4D Einstein–Gauss–Bonnet gravity coupled to modified logarithmic nonlinear electrodynamics

journaltitleEur. Phys. J. C
titleOn Generalized Logarithmic Electrodynamics

journaltitlePhys. Scripta
titleDouble-logarithmic nonlinear electrodynamics

journaltitleAnnals Phys.
titleBlack hole as a magnetic monopole within exponential nonlinear electrodynamics

journaltitlePhys. Rev. D
titleExtended phase space thermodynamics and P-V criticality of black holes with a nonlinear source

journaltitleEur. Phys. J. C
titleGeometrical thermodynamics and P-V criticality of black holes with power-law Maxwell field

Wiley Online Library

journaltitleAnnalen der Physik
titleNonlinear arcsin-electrodynamics

journaltitleAnnalen Phys.
titleNonlinear arcsin-electrodynamics and asymptotic Reissner-Nordström black holes

journaltitleInt. J. Mod. Phys. A
titleHolographic superconductor with nonlinear Born–Infeld-type electrodynamics

journaltitleAnnals Phys.
titleDyonic and magnetic black holes with nonlinear arcsin-electrodynamics

journaltitlePhys. Rev. D
titleAsymptotic Reissner-Nordström solution within nonlinear electrodynamics

Wiley Online Library

journaltitleAnnalen der Physik
titleNonlinear electrodynamics and magnetic black holes

journaltitleAnnals Phys.
titleEinstein  Gauss  Bonnet gravity with nonlinear electrodynamics

titleEinstein-Gauss-Bonnet gravity with rational nonlinear electrodynamics

title4D Einstein–Gauss–Bonnet Gravity Coupled with Nonlinear Electrodynamics

titleEinstein–Gauss–Bonnet Gravity with Nonlinear Electrodynamics: Entropy, Energy Emission, Quasinormal Modes and Deflection Angle

titleNew model of 4D Einstein–Gauss–Bonnet gravity coupled with nonlinear electrodynamics

journaltitleAnnals of Physics
titleNonlinearly charged AdS black holes, extended phase space thermodynamics and Joule–Thomson expansion

World Scientific

booktitleJACOB BEKENSTEIN: The Conservative Revolutionary
titleBlack holes and the second law

journaltitlePhysical Review D
titleExtraction of energy and charge from a black hole

journaltitlePhysical Review D
titleGeneralized second law of thermodynamics in black-hole physics

journaltitleCommunications in mathematical physics
titleThe four laws of black hole mechanics

journaltitleCommunications in mathematical physics
titleParticle creation by black holes

journaltitleCommun. Math. Phys.
titleThermodynamics of Black Holes in anti-De Sitter Space

journaltitleInternational journal of theoretical physics
titleThe large-N limit of superconformal field theories and supergravity

journaltitleAdv. Theor. Math. Phys.
titleAnti-de Sitter space and holography

journaltitlePhysics Letters B
titleGauge theory correlators from non-critical string theory

journaltitlePhys. Rept.
titleLarge N field theories, string theory and gravity

IOP Publishing

journaltitleClassical and Quantum Gravity
titleEnthalpy and the mechanics of AdS black holes

IOP Publishing

journaltitleClassical and Quantum Gravity
titleSmarr formula and an extended first law for Lovelock gravity

IOP Publishing

journaltitleClassical and Quantum Gravity
titleThe cosmological constant and black-hole thermodynamic potentials

IOP Publishing

journaltitleClassical and Quantum Gravity
titlePressure and volume in the first law of black hole thermodynamics

titleP-V criticality of charged AdS black holes

titleExtended phase space thermodynamics for charged and rotating black holes and Born-Infeld vacuum polarization

journaltitlePhysical Review D
titleP- V criticality of charged dilatonic black holes

journaltitleJournal of High Energy Physics
titleP- v criticality in quasitopological gravity

journaltitlePhys. Rev. D
titleP-V criticality in the extended phase space of black holes in massive gravity

journaltitleThe European Physical Journal C
titleExtended phase space thermodynamics and P–V criticality: Brans–Dicke–Born–Infeld vs. Einstein–Born–Infeld-dilaton black holes

journaltitleJournal of High Energy Physics
titleUniversality of P- V criticality in horizon thermodynamics

journaltitlePhysics Letters B
titlePV criticality of AdS black holes in a general framework

journaltitleThe European Physical Journal C
titleGeometrical thermodynamics and P–V criticality of the black holes with power-law Maxwell field

journaltitlePhysical Review D
titleP- V criticality of first-order entropy corrected AdS black holes in massive gravity

journaltitleEur. Phys. J. C
titleJoule–Thomson expansion of the charged AdS black holes

journaltitleEur. Phys. J. C
titleJoule–Thomson expansion of Kerr–AdS black holes

journaltitlePhys. Rev. D
titleJoule-Thomson expansion of $d$-dimensional charged AdS black holes

journaltitlePhys. Rev. D
titleThrottling process of the Kerr–Newman–anti-de Sitter black holes in the extended phase space

journaltitleEur. Phys. J. Plus
titleEffect of massive gravity on Joule–Thomson expansion of the charged AdS black hole

titleAdS Black Holes in the Framework of Nonlinear Electrodynamics, Thermodynamics, and Joule–Thomson Expansion

journaltitleNucl. Phys. B
titleNED-AdS black holes, extended phase space thermodynamics and Joule–Thomson expansion

IOP Publishing

journaltitleClassical and Quantum Gravity
titleHolographic heat engines

journaltitleJournal of High Energy Physics
titleOn heat properties of AdS black holes in higher dimensions

journaltitleGeneral Relativity and Gravitation
titlePolytropic black hole as a heat engine

IOP Publishing

journaltitleClassical and Quantum Gravity
titleGauss–Bonnet black holes and holographic heat engines beyond large N

IOP Publishing

journaltitleClassical and Quantum Gravity
titleBorn–Infeld AdS black holes as heat engines

journaltitleThe European Physical Journal C
titleHeat engines for dilatonic Born–Infeld black holes

IOP Publishing

journaltitleClassical and Quantum Gravity
titleHolographic heat engines: general considerations and rotating black holes

journaltitleJournal of High Energy Physics
titleHeat engine in the three-dimensional spacetime

journaltitlePhysics Letters B
titleBlack holes in massive gravity as heat engines

journaltitlePhysical Review D
titleReentrant phase transitions in rotating anti–de Sitter black holes

journaltitleJournal of High Energy Physics
titleMultiple reentrant phase transitions and triple points in Lovelock thermodynamics

titleReentrant phase transitions and van der Waals behaviour for hairy black holes

journaltitlePhys. Rev. D
titleRuppeiner geometry, reentrant phase transition, and microstructure of Born-Infeld AdS black hole

journaltitlePhys. Lett. B
titleTriple points and phase diagrams of Born-Infeld AdS black holes in 4D Einstein-Gauss-Bonnet gravity

titleEinstein-AdS Gravity Coupled to Nonlinear Electrodynamics, Magnetic Black Holes, Thermodynamics in an Extended Phase Space and Joule–Thomson Expansion

journaltitlePhys. Rev. D
titleThermodynamics of Black Holes in Massive Gravity

journaltitleGen. Rel. Grav.
titleHeat engine efficiency and Joule–Thomson expansion of nonlinear charged AdS black hole in massive gravity

journaltitleClass. Quant. Grav.
titleThe cosmological constant and the black hole equation of state

journaltitleMod. Phys. Lett. A
titlePhase transitions of black holes in massive gravity

titleCharged Black Hole Solutions in Gauss-Bonnet-Massive Gravity

journaltitleEur. Phys. J. C
titleReentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity

journaltitleEur. Phys. J. C
titleBehavior of quasinormal modes and Van der Waals-like phase transition of charged AdS black holes in massive gravity

journaltitleEur. Phys. J. C
titleNew perspective for black hole thermodynamics in Gauss–Bonnet–Born–Infeld massive gravity

De Gruyter Oldenbourg

journaltitleZeitschrift für Physikalische Chemie
titleDie gegenseitige löslichkeit von nikotin in wasser

journaltitlePhysics Reports
titleReentrant phase transitions in multicomponent liquid mixtures

journaltitleClass. Quant. Grav.
titleKerr-AdS analogue of triple point and solid/liquid/gas phase transition

titleThermodynamics of rotating black holes and black rings: phase transitions and thermodynamic volume

journaltitleClass. Quant. Grav.
titleThermodynamics of horizons: de Sitter black holes and reentrant phase transitions

titleMultiple Reentrant Phase Transitions and Triple Points in Lovelock Thermodynamics

journaltitlePhys. Rev. D
titleTriple points and phase diagrams in the extended phase space of charged Gauss-Bonnet black holes in AdS space

titleReentrant phase transitions and van der Waals behaviour for hairy black holes


# Nonlinear charge transport properties in chiral tellurium

Kazuki Nakazawa1, Terufumi Yamaguchi1, and Ai yamakage2 1RIKEN Center for
Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
2Department of Physics, Nagoya University, Nagoya 464-8602, Japan

###### Abstract

Extensive research has focused on phenomena arising from the chirality of
crystalline or magnetic structures. Recently, we have proposed the “Nonlinear
Chiral Thermo-Electric (NCTE) Hall effect,” in which current flows in the
direction of the cross product of the electric field and the temperature
gradient in a material with a chiral structure. Despite its importance for
device applications such as heat flow sensors and logic circuits, there are no
quantitative demonstrations in actual materials. We investigate the NCTE Hall
effect and the second-order response to the electric field in Te, which hosts
the chiral crystal structure, based on ab initio calculations and symmetry
argument. We clarify that the NCTE Hall current can flow parallel to the
chiral axis, whereas such behavior is absent for the second-order response to
the DC electric field. Quantitative estimation predicts the large NCTE Hall
current which is sufficiently detectable in experiments, and the orbital
magnetic moment will play an important role in the vicinity of the top of the
valence band.

Chirality is one of the fundamental degrees of freedom in nature Greed (2022);
Ozturk and Sasselov (2022). In particular, the chirality of crystals and
magnetic structures in condensed matter has attracted much attention because
of the novel phenomena they produce. For example, magnetochiral dichroism
Rikken and Raupach (1998), chirality induced spin selectivity (CISS) Ray  _et
al._ (1999); Naaman  _et al._ (2019), and chiral phonons Ishito  _et al._
(2023); Ueda  _et al._ (2023); Ohe  _et al._ (2024) are representative
examples of phenomena caused by the chirality of crystals, which are currently
under intensive study.

Besides, the second-order response to an external field is linked to the
breaking of spatial inversion symmetry that resides in the materials. For
instance, both time-reversal and inversion symmetry breaking give rise to the
band asymmetry leading to the second-order response to the electric field
Rikken  _et al._ (2001); Yokouchi  _et al._ (2017); Ishizuka and Nagaosa
(2020); Dantas  _et al._ (2023), and the exhibition of the Berry curvature
dipole due to the inversion symmetry breaking also leads to the second-order
response Sodemann and Fu (2015); Kang  _et al._ (2019); Ma  _et al._ (2019);
Du  _et al._ (2021a, b). The inclusive understanding of such nonlinear
transport is summarized in terms of the band geometry Gao  _et al._ (2014);
Michishita and Nagaosa (2022), governed from the microscopic analyses
Michishita and Peters (2021).

Meanwhile, the “Nonlinear Chiral Thermo-Electric (NCTE) Hall effect,” in which
current flows in the direction of the cross product of the electric field and
the temperature gradient, have been studied theoretically Hidaka  _et al._
(2018); Nakai and Nagaosa (2019); Toshio  _et al._ (2020); Yamaguchi  _et al._
(2023), which is unique to the chiral materials without any mirror or
inversion symmetry. The microscopic calculation revealed that the NCTE Hall
current $\langle j_{z}^{\rm NCTE}\rangle=\sigma_{z}^{\rm
NCTE}\\{{\bm{E}}\times(-\nabla T/T)\\}_{z}$ is dominated by following two
terms Yamaguchi  _et al._ (2023);

$\displaystyle\sigma_{z}^{\rm
NCTE}\simeq\sigma^{\mathrm{BC}}_{z}+\sigma^{\mathrm{OM}}_{z},$ (1)
$\displaystyle\sigma^{\mathrm{BC}}_{z}=e^{2}\tau\sum_{n,\bm{k}}\left(\varepsilon_{n\bm{k}}-\mu\right)\left(-\frac{\partial
f}{\partial\varepsilon}\right)_{\varepsilon=\varepsilon_{n\bm{k}}}$
$\displaystyle\quad\quad\quad\times\left[v_{n\bm{k}}^{z}\Omega^{z}_{n{{\bm{k}}}}-\frac{1}{2}\left\\{v_{n\bm{k}}^{x}\Omega^{x}_{n{{\bm{k}}}}+v_{n\bm{k}}^{y}\Omega^{y}_{n{{\bm{k}}}}\right\\}\right],$
(2)
$\displaystyle\sigma^{\mathrm{OM}}_{z}=-\frac{1}{2}e\tau\sum_{n,\bm{k}}\left(\varepsilon_{n\bm{k}}-\mu\right)\left(-\frac{\partial
f}{\partial\varepsilon}\right)_{\varepsilon=\varepsilon_{n\bm{k}}}\bm{\nabla}_{\bm{k}}\cdot\bm{m}^{\perp}_{n\bm{k}},$
(3)

where ${\bm{E}}$ is the DC electric field, $T$ is the temperature, $e$,
$\tau$, and $\mu$ are the elementary charge, electron lifetime, and chemical
potential, respectively, $f$ is the Fermi-Dirac distribution function, and
${\bm{\Omega}}_{n{{\bm{k}}}}$ and ${\bm{m}}_{n{{\bm{k}}}}$ are the Berry
curvature and the orbital magnetic moment,

$\displaystyle{\bm{\Omega}}_{n{{\bm{k}}}}$ $\displaystyle=i\sum_{m\neq
n}\frac{\langle n{{\bm{k}}}|\hat{\bm{{{\cal
V}}}}_{{\bm{k}}}|m{{\bm{k}}}\rangle\times\langle m{{\bm{k}}}|\hat{\bm{{{\cal
V}}}}_{{\bm{k}}}|n{{\bm{k}}}\rangle}{({\varepsilon}_{n{{\bm{k}}}}-{\varepsilon}_{m{{\bm{k}}}})^{2}},$
(4) $\displaystyle{\bm{m}}_{n{{\bm{k}}}}$
$\displaystyle=\frac{ie}{2}\sum_{m\neq n}\frac{\langle
n{{\bm{k}}}|\hat{\bm{{{\cal V}}}}_{{\bm{k}}}|m{{\bm{k}}}\rangle\times\langle
m{{\bm{k}}}|\hat{\bm{{{\cal
V}}}}_{{\bm{k}}}|n{{\bm{k}}}\rangle}{({\varepsilon}_{n{{\bm{k}}}}-{\varepsilon}_{m{{\bm{k}}}})},$
(5)

respectively, with an eigenenergy ${\varepsilon}_{{{\bm{k}}}n}$ and an
eigenvector $|n{{\bm{k}}}\rangle$ of the Hamiltonian $\hat{H}_{{\bm{k}}}$, the
velocity operator $\hat{\bm{{{\cal
V}}}}_{{\bm{k}}}=\nabla_{{\bm{k}}}\hat{H}_{{\bm{k}}}$, and group velocity
${\bm{v}}_{n{{\bm{k}}}}=\nabla_{{\bm{k}}}{\varepsilon}_{n{{\bm{k}}}}$.

Figure 1: (a) Lattice structure of the tellurium. Black lines represent the
extended unit cell which contains eight primitive unit cells. (b) Top view of
(a). We denoted the definition of $x$, $y$, and $z$ axes. The crystal
structures are visualized by VESTA Momma and Izumi (2011).  Figure 2: (a) Band
structure of the tellurium obtained from the DFT calculation. The red line
specifies the highest-energy valence band V, which we focus on. The inset
shows the Brillouin zone, where the gray hexagon represents the $k_{z}=0$
plane. (b) Chemical potential dependence of NCTE Hall current density. Black,
red, and blue lines represent total, Berry curvature (BC), and orbital
magnetic moment (OM) contributions, respectively. (c) Longitudinal resistivity
with respect to $x$ and $z$ directions.

We here set the Dirac (Planck) constant as $\hbar=1\ (h=(2\pi)^{-1})$. This
NCTE Hall effect inverts its sign depending on the chirality Yamaguchi  _et
al._ (2023), hence it can detect the handedness of the lattice/magnetic
structure. Moreover, it also has the potential for device applications, such
as heat flow sensors or logic circuits. However, theoretical and experimental
demonstrations considering actual materials are yet to be proposed, and hence,
quantitative evaluation and characterization of the NCTE Hall current will be
important.

The tellurium (Te) crystal belongs to the space group $P3_{1}21$ (No. 152) or
$P3_{2}21$ (No. 154) which contains a three-fold chiral axis and does not have
any mirror or spatial inversion symmetry [Fig. 1]. While Te has various
applications as a compound, the physical properties of Te itself have recently
attracted attention such as a thermoelectric conversion due to the
semiconducting property Lin  _et al._ (2016), characteristic structure of the
Berry curvature related to the existence of Weyl points Hirayama  _et al._
(2015), and nonreciprocal transport phenomena under magnetic fields due to its
crystal chirality Hirobe  _et al._ (2022); Calavalle  _et al._ (2022).
Moreover, recently, the orbital moment and its consequences have attracted
attention Yoda  _et al._ (2015, 2018). In particular, the effective model
analysis predicts the hedgehog-like structure of the orbital moment at the top
of the valence band Maruggi  _et al._ (2023), which motivates us to study
whether the orbital magnetic moment plays a role in the NCTE Hall effect or
not.

In this paper, we investigate the NCTE Hall effect and the second-order
response to the electric field in tellurium based on ab initio calculations.
We clarify that the NCTE Hall current is finite while the second-order
response to the DC electric field is absent along the chiral axis. Our
quantitative estimation predicts that the NCTE Hall current is large enough to
observe experimentally, and the role of the Berry curvature and orbital
magnetic moment is clarified.

The OpenMX code website ; Ozaki (2003) is used to obtain the band structures
based on the density functional theory (DFT). The generalized gradient
approximation proposed by Perdew-Burke-Ernzerhof Perdew  _et al._ (1996) is
employed for the exchange correlation functional. The norm-conserving and
total angular momentum-dependent pseudopotentials are employed, and the
pseudo-atomic orbital Te7.0-s4p3d3f2 is chosen as a basis of the Kohn-Sham
wave functions. We perform the fully relativistic calculation to incorporate
the effect of spin-orbit coupling. We used the right-handed crystal of the
space group $P3_{1}21$, and set the lattice constant to
$a=4.6~{}{\mathrm{\mathring{A}}}$ and $c=5.9~{}{\mathrm{\mathring{A}}}$ to
reproduce the band gap; we will touch this point later. We set the cutoff
energy which specifies the fast Fourier transform grid to 1200 Ry and sampled
the Brillouin zone with $16^{3}$ $k$ points.

The band structure in the window [$-0.5$ eV, 0.5 eV] is shown in Fig. 2(a).
The band gap is present around the H point, where the lowest-energy conduction
band and highest-energy valence band exhibit exotic characteristics such as
Berry curvature (dipole) and orbital magnetic moments, as we will discuss
later. The overall band structure reproduces well that of previous study
Hirayama  _et al._ (2015). Note that Ref Hirayama  _et al._ (2015) takes the
experimental in-plane (perpendicular to chiral axis) lattice constant of
$a\sim 4.456~{}{\mathrm{\mathring{A}}}$ Keller  _et al._ (1977); Adenis  _et
al._ (1989), but considers the LDA+GW framework to implement the effect of
electron-electron interaction to obtain the band gap. On the other hand, if we
take the same $a$ within the GGA calculation, the band gap is closed and
cannot reproduce the semiconducting property. Instead, we just take the larger
in-plane lattice constant to reproduce the band gap. Actual Te samples are
slightly doped with holes Lin  _et al._ (2016), leading to an estimated
chemical potential of approximately $\mu\sim-0.1\ \mathrm{eV}$, which
corresponds almost to the top of the valence band. Henceforth we will focus on
this value frequently.

Figure 3:  Momentum space distributions of (a) Berry curvature
${\bm{\Omega}}_{n{{\bm{k}}}}$ and (b) orbital angular momentum
${\bm{m}}_{n{{\bm{k}}}}$ of the band V, which are calculated using Eqs. (4)
and (5), respectively. The dotted lines are projected boundaries of the
Brillouin zone, with the high-symmetry points H, L, A, $\Gamma$, K, shown in
the inset of Fig. 2(a). The horizontal axes in the second row of each panel
represent $k_{x,y}=\sqrt{3}k_{x}=k_{y}$. The pink circles represent the region
around the H point, which we mainly focus on in the main text.  Figure 4:
Chemical potential dependence of the normalized second-order nonlinear
conductivity $\tilde{\sigma}_{ijk}^{(2)}\equiv\sigma_{ijk}^{(2)}/(e^{3}/h)$.
The red, green, and blue lines correspond to the current along the $x$, $y$,
and $z$ directions, respectively. The small fluctuations seen in the
components except for $\tilde{\sigma}_{xxx}^{(2)}$,
$\tilde{\sigma}_{xyy}^{(2)}$, $\tilde{\sigma}_{yyx}^{(2)}$,
$\tilde{\sigma}_{xyz}^{(2)}$, and $\tilde{\sigma}_{yzx}^{(2)}$ come from
symmetry breaking due to the Wannierization.

To obtain the tight-binding model, we constructed maximally localized Wannier
functions Marzari and Vanderbilt (1997); Souza  _et al._ (2001) using OpenMX
Weng  _et al._ (2009). The eigenstates within the energy window of [$-6$ eV,
3.5 eV] are employed and projected to the $p$ orbitals for each atom, and
hence in total, 18 orbitals are considered, including spin degrees of freedom.
The Wannier model reproduces the original band structure obtained by DFT
calculations.

The NCTE Hall current $j_{z}^{\rm NCTE}$ is calculated employing the Wannier
model. We use Eqs. (1)-(3) to calculate NCTE Hall current defining the $x$,
$y$, and $z$ axis parallel to ${\bm{a}}$, ${\bm{c}}\times{\bm{a}}$, and
${\bm{c}}$ directions, respectively [see Fig. 1(b)]. We take the $k$-mesh
number of $240^{3}$ for the momentum integral. The parameters are set as
$E_{x}=1100$ V/m, $k_{\rm B}T=0.03$ eV, ${\partial}_{y}T/T=100~{}{\rm
m}^{-1}$, $\tau=10$ fs. The chemical potential dependence of $j_{z}^{\rm
NCTE}$ is shown in Fig. 2(b), representing the finite NCTE Hall effect.
Comparing the value of $\sigma_{z}^{\rm BC}$ and $\sigma_{z}^{\rm OM}$, they
overall show a comparable value within the energy range of $-0.5$ to 0.5 eV.
Importantly, if we focus around the top of the valence band (at the vicinity
of $\mu=-0.1$ eV), $\sigma_{z}^{\rm OM}$ is dominant while $\sigma_{z}^{\rm
BC}$ is suppressed. This indicates that the emergence of the characteristic
structure of orbital magnetic moment and the cancellation of the total Berry
curvature dipole occurs around $\mu\sim-0.1$ eV simultaneously. We will later
discuss these points with the actual distribution of
${\bm{\Omega}}_{n{{\bm{k}}}}$ and ${\bm{m}}_{n{{\bm{k}}}}$. Moreover, the
obtained NCTE current density is valued around $0.04$ $\rm A/m^{2}$ at
$\mu\sim-0.1$ eV, which is already measurable. To consider the possible range
of the NCTE current, we calculate longitudinal resistivity $\rho_{ii}$ with
respect to $x$ and $z$ directions [Fig. 2(c)]. Under
$k_{\mathrm{B}}T=0.03~{}{\rm eV}$ and $\tau=10$ fs, the resistivity is valued
around $\sim 10^{-4}~{}\Omega{\rm m}$, which is comparable to the experimental
value. Depending on the sample quality, the typical value of the resistivity
is ranged between $10^{-3}$ to $10^{-6}~{}\Omega{\rm m}$ Ross (1992); Abad
_et al._ (2015); Lin  _et al._ (2016); Rabadanov  _et al._ (2021). As
electrical conductivity $\sigma_{ii}=1/\rho_{ii}$ and NCTE current both scale
as linear of $\tau$, the NCTE Hall current can be 10 to 100 times larger in
the more conductive sample.

$D_{3}$ | $E$ | $2C_{3}$ | $3C_{2}^{\prime}$ | Linear | Quadratic
---|---|---|---|---|---
$A_{1}$ | 1 | 1 | 1 |  |  $XX+YY$, $ZZ$
$A_{2}$ | 1 | 1 | $-1$ | $Z$ | $XY-YX$
$E$ | 2 | $-1$ | 0 | $(X,Y)$ |  $(XX-YY,-XY-YX)$,
|  |  |  |  |  $(YZ,-XZ)$, $(ZY,-ZX)$
Table 1: Character table for the point group $D_{3}$. $A_{1}$, $A_{2}$, and
$E$ represent irreducible representations, and $X$, $Y$, and $Z$ are linear
and quadratic basis functions corresponding to the coordinate $x$, $y$, and
$z$.

Figure 3 shows the momentum space distributions of the Berry curvature
${\bm{\Omega}}_{n{{\bm{k}}}}$ and the orbital magnetic moment
${\bm{m}}_{n{{\bm{k}}}}$ of the band V. Figure 3(a) is plots of each component
of ${\bm{\Omega}}_{n{{\bm{k}}}}$. We observe the dipole-like properties in
each component around the H point. However, due to its isotropy of the
momentum space distribution, Berry curvature (dipole) satisfies the relation
${\sum_{{\bm{k}}}v_{n\bm{k}}^{x}\Omega_{n\bm{k}}^{x}\sim\sum_{{\bm{k}}}v_{n\bm{k}}^{y}\Omega_{n\bm{k}}^{y}\sim\sum_{{\bm{k}}}v_{n\bm{k}}^{z}\Omega_{n\bm{k}}^{z}}$,
leading to the suppression of $\sigma_{z}^{\rm BC}$ around $\mu\sim-0.1$ eV.
On the other hand, ${\bm{m}}_{n{{\bm{k}}}}$ shown in Fig. 3(b) shows a similar
characteristic mapping around the H point. However, the NCTE Hall effect only
captures the properties of the in-plane component, namely
${\bm{m}}_{n{{\bm{k}}}}^{\perp}=(m_{n{{\bm{k}}}}^{x},m_{n{{\bm{k}}}}^{y},0)$
with the in-plane texture such that
$\nabla_{{\bm{k}}}\cdot{\bm{m}}_{n{{\bm{k}}}}^{\perp}\neq 0$ which prevent the
cancellation. This situation is quite similar to that in the minimal model
discussed in Ref. 24. Generally, $\sigma_{z}^{\rm BC}$ should be suppressed
and only $\sigma_{z}^{\rm OM}$ can be only visible in the isotropic and/or the
cubic systems.

For the comparison, the second-order current response to the DC electric field
$\langle j_{i}^{(2)}\rangle=\sigma_{ijk}^{(2)}E_{j}E_{k}$ are calculated in
the same Wannier model. We here consider the following expression of the
second-order DC nonlinear conductivity $\sigma_{ijk}^{(2)}$ Michishita and
Peters (2021); Michishita and Nagaosa (2022); not ;

$\displaystyle\sigma_{ijk}^{(2)}\simeq\frac{2e^{3}}{V}\int\frac{d{\varepsilon}}{2\pi}\left(-\frac{\partial
f}{\partial{\varepsilon}}\right)$ $\displaystyle\quad\times{\rm
Im}\sum_{{\bm{k}}}{\rm tr}\Biggl{\\{}\hat{{{\cal
V}}}_{i}\frac{\partial\hat{G}^{{\rm
R}}}{\partial{\varepsilon}}\left(\hat{{{\cal V}}}_{j}\hat{G}^{{\rm
R}}\hat{{{\cal V}}}_{k}+\frac{1}{2}\hat{{{\cal V}}}_{jk}\right)(\hat{G}^{{\rm
R}}-\hat{G}^{{\rm A}})\Biggr{\\}}$ $\displaystyle\quad+(j\leftrightarrow k),$
(6)

where $\hat{G}^{{\rm R}}=({\varepsilon}-\hat{H}_{{\bm{k}}}-\hat{\Sigma}^{{\rm
R}})^{-1}=(\hat{G}^{{\rm A}})^{\dagger}$ is retarded Green’s function with
self-energy $\hat{\Sigma}^{{\rm R}}$, and $\hat{{{\cal
V}}}_{ij}=\partial_{k_{i}}\partial_{k_{j}}\hat{H}_{{\bm{k}}}$. The trace runs
over all of the orbital/band indices. For simplicity, we here consider the
constant pure imaginary self-energy $\hat{\Sigma}^{\rm R}=-i/(2\tau)$. The
chemical potential dependence of normalized second-order nonlinear
conductivity $\tilde{\sigma}_{ijk}^{(2)}=\sigma^{(2)}_{ijk}/(e^{3}/h)$ is
shown in Fig. 4. We observe only 5 finite components out of 18 components and
found approximate relations
$\sigma_{xxx}^{(2)}=-\sigma_{xyy}^{(2)}=-\sigma_{yyx}^{(2)}$ and
$\sigma_{xyz}^{(2)}=-\sigma_{yzx}^{(2)}$; there are only two independent
tensor components. Moreover, strikingly, the nonlinear response along the $z$
direction (chiral axis) never arises, $\sigma_{zjk}^{(2)}=0$, if we only apply
DC electric fields, showing the peculiarity of the NCTE Hall current.

To verify the results of the NCTE Hall effect and second-order DC nonlinear
conductivity, we perform the symmetry analysis in the case of $D_{3}$ point
group in which the tellurium crystal belongs, summarized in Table 1. Due to
the symmetry, in the equation $J_{i}=\sigma^{(2)}_{ijk}E_{j}E_{k}$, the left
and right sides are represented by the same basis of irreducible
representation. As a result, the possible combinations of applied fields and
currents are

$\displaystyle(J_{x},J_{y},J_{z})$
$\displaystyle=\sigma_{1}(E_{x}^{2}-E_{y}^{2},-2E_{x}E_{y},0)$
$\displaystyle+\sigma_{2}(E_{y}E_{z},-E_{x}E_{z},0),$ (7)

for the second-order response to the electric field, and

$\displaystyle(J_{x},J_{y},J_{z})$
$\displaystyle=\sigma^{\prime}_{1}(0,0,E_{x}\partial_{y}T-E_{y}\partial_{x}T)$
$\displaystyle+\sigma^{\prime}_{2}(E_{x}\partial_{x}T-E_{y}\partial_{y}T,-E_{x}\partial_{y}T-E_{y}\partial_{x}T,0)$
$\displaystyle+{\sigma}^{\prime}_{3}(E_{y}\partial_{z}T,-E_{x}\partial_{z}T,0)$
$\displaystyle+{\sigma}^{\prime}_{4}(E_{z}\partial_{y}T,-E_{z}\partial_{x}T,0)$
(8)

for the response to the product of the electric field and the temperature
gradient. Here ${\sigma}_{i}$ and ${\sigma}^{\prime}_{i}$ are the
corresponding coefficients. $J_{z}$ only appears in
${\sigma}^{\prime}_{1}={\sigma}_{z}^{\rm NCTE}$, as a response to the cross
product of the electric field and temperature gradient applied to the
$xy$-plane, representing the NCTE Hall effect. The number of the independent
tensor components is two for the response to $E_{i}E_{j}$ as we already see in
the ab initio results, while that is four for the $E_{i}\partial_{j}T$
implying the variety of the response and motivating the analysis of the
general response to the product of the electric field and the temperature
gradient including symmetric components other than the NCTE Hall effect. A
more generalized implication from the symmetry argument is that the second-
order electric current response along the chiral axis requires two kinds of
applied field; for example, we could employ a chemical potential gradient
instead of the temperature gradient. Moreover, if we extend the consideration
to the AC applied field, the circularly polarized light also induces the
current as $J_{z}\propto\bm{E}\times\bm{E}^{*}$ for ${\bm{E}}=(1,i,0)$ Ma  _et
al._ (2022).

To summarize, we investigated the NCTE Hall effect, the flow of the current to
the direction of the cross product of the electric field and the temperature
gradient, in the tellurium utilizing the ab initio calculation and symmetry
arguments. The Berry curvature dipole and orbital magnetic moment will overall
equally contribute to the NCTE Hall effect. Especially, the orbital magnetic
moment is dominant around the band top of the valence band, while Berry
curvature contribution is canceled because of the isotropy around the H point.
The quantitative estimate of the NCTE Hall current density predicts that its
amount is sufficiently measurable in the experiment. Comparison to the second-
order response to the DC electric field revealed that the current along the
chiral axis is only expected for the NCTE Hall effect but does not occur for
the second-order nonlinear conductivity. This is verified by the symmetry
analysis, which predicts the diversity of the setups, e.g., applying chemical
potential gradient as well as circularly polarized light to obtain the
nonlinear response along the chiral axis.

The authors thank Y. Araki, H. Chudo, F. Kagawa, T. Kikkawa, J. Ieda, M. Imai,
T. Nomoto, Y. Michishita, T. Morimoto, T. Sato, and M. Umeda for fruitful
discussions. Parts of the numerical calculations have been done using the
Supercomputer HOKUSAI BigWaterfall2 (HBW2), RIKEN. This work is supported by
JSPS KAKENHI (Grant Nos. JP20K03835, JP21K14526, and JP21K13875).

## References

  * Greed (2022) Stephanie Greed, “The dawn of asymmetry,” Nature Reviews Chemistry 6, 596–596 (2022).
  * Ozturk and Sasselov (2022) S. Furkan Ozturk and Dimitar D. Sasselov, “On the origins of life’s homochirality: Inducing enantiomeric excess with spin-polarized electrons,” Proceedings of the National Academy of Sciences 119, e2204765119 (2022).
  * Rikken and Raupach (1998) G. L. J. A. Rikken and E. Raupach, “Pure and cascaded magnetochiral anisotropy in optical absorption,” Phys. Rev. E 58, 5081–5084 (1998).
  * Ray  _et al._ (1999) K. Ray, S. P. Ananthavel, D. H. Waldeck, and R. Naaman, “Asymmetric scattering of polarized electrons by organized organic films of chiral molecules,” Science 283, 814–816 (1999).
  * Naaman  _et al._ (2019) Ron Naaman, Yossi Paltiel, and David H. Waldeck, “Chiral molecules and the electron spin,” Nature Reviews Chemistry 3, 250–260 (2019).
  * Ishito  _et al._ (2023) Kyosuke Ishito, Huiling Mao, Yusuke Kousaka, Yoshihiko Togawa, Satoshi Iwasaki, Tiantian Zhang, Shuichi Murakami, Jun-ichiro Kishine, and Takuya Satoh, “Truly chiral phonons in $\alpha$-HgS,” Nature Physics 19, 35–39 (2023).
  * Ueda  _et al._ (2023) Hiroki Ueda, Mirian García-Fernández, Stefano Agrestini, Carl P. Romao, Jeroen van den Brink, Nicola A. Spaldin, Ke-Jin Zhou, and Urs Staub, “Chiral phonons in quartz probed by X-rays,” Nature 618, 946–950 (2023).
  * Ohe  _et al._ (2024) Kazuki Ohe, Hiroaki Shishido, Masaki Kato, Shoyo Utsumi, Hiroyasu Matsuura, and Yoshihiko Togawa, “Chirality-induced selectivity of phonon angular momenta in chiral quartz crystals,” Phys. Rev. Lett. 132, 056302 (2024).
  * Rikken  _et al._ (2001) G. L. J. A. Rikken, J. Fölling, and P. Wyder, “Electrical magnetochiral anisotropy,” Phys. Rev. Lett. 87, 236602 (2001).
  * Yokouchi  _et al._ (2017) T. Yokouchi, N. Kanazawa, A. Kikkawa, D. Morikawa, K. Shibata, T. Arima, Y. Taguchi, F. Kagawa, and Y. Tokura, “Electrical magnetochiral effect induced by chiral spin fluctuations,” Nature Communications 8, 866 (2017).
  * Ishizuka and Nagaosa (2020) Hiroaki Ishizuka and Naoto Nagaosa, “Anomalous electrical magnetochiral effect by chiral spin-cluster scattering,” Nat. Commun. 11 (2020).
  * Dantas  _et al._ (2023) Renato M. A. Dantas, Henry F. Legg, Stefano Bosco, Daniel Loss, and Jelena Klinovaja, “Determination of spin-orbit interaction in semiconductor nanostructures via nonlinear transport,” Phys. Rev. B 107, L241202 (2023).
  * Sodemann and Fu (2015) Inti Sodemann and Liang Fu, “Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials,” Phys. Rev. Lett. 115, 216806 (2015).
  * Kang  _et al._ (2019) Kaifei Kang, Tingxin Li, Egon Sohn, Jie Shan, and Kin Fai Mak, “Nonlinear anomalous Hall effect in few-layer WTe2,” Nature Materials 18, 324–328 (2019).
  * Ma  _et al._ (2019) Qiong Ma, Su-Yang Xu, Huitao Shen, David MacNeill, Valla Fatemi, Tay-Rong Chang, Andrés M Mier Valdivia, Sanfeng Wu, Zongzheng Du, Chuang-Han Hsu, Shiang Fang, Quinn D Gibson, Kenji Watanabe, Takashi Taniguchi, Robert J Cava, Efthimios Kaxiras, Hai-Zhou Lu, Hsin Lin, Liang Fu, Nuh Gedik, and Pablo Jarillo-Herrero, “Observation of the nonlinear Hall effect under time-reversal-symmetric conditions,” Nature 565, 337–342 (2019).
  * Du  _et al._ (2021a) Z Z Du, Hai-Zhou Lu, and X C Xie, “Nonlinear Hall effects,” Nature Reviews Physics 3, 744–752 (2021a).
  * Du  _et al._ (2021b) Z Z Du, C M Wang, Hai-Peng Sun, Hai-Zhou Lu, and X C Xie, “Quantum theory of the nonlinear Hall effect,” Nat. Commun. 12, 5038 (2021b).
  * Gao  _et al._ (2014) Yang Gao, Shengyuan A. Yang, and Qian Niu, “Field induced positional shift of bloch electrons and its dynamical implications,” Phys. Rev. Lett. 112, 166601 (2014).
  * Michishita and Nagaosa (2022) Yoshihiro Michishita and Naoto Nagaosa, “Dissipation and geometry in nonlinear quantum transports of multiband electronic systems,” Phys. Rev. B 106, 125114 (2022).
  * Michishita and Peters (2021) Yoshihiro Michishita and Robert Peters, “Effects of renormalization and non-hermiticity on nonlinear responses in strongly correlated electron systems,” Phys. Rev. B 103, 195133 (2021).
  * Hidaka  _et al._ (2018) Yoshimasa Hidaka, Shi Pu, and Di-Lun Yang, “Nonlinear responses of chiral fluids from kinetic theory,” Phys. Rev. D 97, 016004 (2018), arXiv:1710.00278 .
  * Nakai and Nagaosa (2019) Ryota Nakai and Naoto Nagaosa, “Nonreciprocal thermal and thermoelectric transport of electrons in noncentrosymmetric crystals,” Phys. Rev. B 99, 115201 (2019), arXiv:1812.02372 .
  * Toshio  _et al._ (2020) Riki Toshio, Kazuaki Takasan, and Norio Kawakami, “Anomalous hydrodynamic transport in interacting noncentrosymmetric metals,” Phys. Rev. Res. 2, 032021 (2020), arXiv:2005.03406 .
  * Yamaguchi  _et al._ (2023) Terufumi Yamaguchi, Kazuki Nakazawa, and Ai Yamakage, “Microscopic theory of nonlinear Hall effect induced by electric field and temperature gradient,” (2023), arXiv:2305.05273 [cond-mat.mes-hall] .
  * Momma and Izumi (2011) Koichi Momma and Fujio Izumi, “VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data,” Journal of Applied Crystallography 44, 1272–1276 (2011).
  * Lin  _et al._ (2016) Siqi Lin, Wen Li, Zhiwei Chen, Jiawen Shen, Binghui Ge, and Yanzhong Pei, “Tellurium as a high-performance elemental thermoelectric,” Nature Communications 7, 10287 (2016).
  * Hirayama  _et al._ (2015) Motoaki Hirayama, Ryo Okugawa, Shoji Ishibashi, Shuichi Murakami, and Takashi Miyake, “Weyl node and spin texture in trigonal tellurium and selenium,” Phys. Rev. Lett. 114, 206401 (2015).
  * Hirobe  _et al._ (2022) Daichi Hirobe, Yoji Nabei, and Hiroshi M. Yamamoto, “Chirality-induced intrinsic charge rectification in a tellurium-based field-effect transistor,” Phys. Rev. B 106, L220403 (2022).
  * Calavalle  _et al._ (2022) Francesco Calavalle, Manuel Suárez-Rodríguez, Beatriz Martín-García, Annika Johansson, Diogo C. Vaz, Haozhe Yang, Igor V. Maznichenko, Sergey Ostanin, Aurelio Mateo-Alonso, Andrey Chuvilin, Ingrid Mertig, Marco Gobbi, Fèlix Casanova, and Luis E. Hueso, “Gate-tuneable and chirality-dependent charge-to-spin conversion in tellurium nanowires,” Nature Materials 21, 526–532 (2022).
  * Yoda  _et al._ (2015) Taiki Yoda, Takehito Yokoyama, and Shuichi Murakami, “Current-induced Orbital and Spin Magnetizations in Crystals with Helical Structure,” Sci. Rep. 5, 12024 (2015).
  * Yoda  _et al._ (2018) Taiki Yoda, Takehito Yokoyama, and Shuichi Murakami, “Orbital Edelstein Effect as a Condensed-Matter Analog of Solenoids,” Nano Lett. 18, 916–920 (2018).
  * Maruggi  _et al._ (2023) Gabriele P. Maruggi, Jaime Ferreira, Elisa Baggio-Saitovitch, Carsten Enderlein, and Marcello B. Silva Neto, “Hedgehog orbital texture in $p$-type tellurium and the antisymmetric nonreciprocal Hall response,” Phys. Rev. Mater. 7, 014204 (2023).
  * (33) OpenMX website, https://www.openmx-square.org/ , 35–39.
  * Ozaki (2003) T. Ozaki, “Variationally optimized atomic orbitals for large-scale electronic structures,” Phys. Rev. B 67, 155108 (2003).
  * Perdew  _et al._ (1996) John P. Perdew, Kieron Burke, and Matthias Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77, 3865–3868 (1996).
  * Keller  _et al._ (1977) R. Keller, W. B. Holzapfel, and Heinz Schulz, “Effect of pressure on the atom positions in Se and Te,” Phys. Rev. B 16, 4404–4412 (1977).
  * Adenis  _et al._ (1989) C. Adenis, V. Langer, and O. Lindqvist, “Reinvestigation of the structure of tellurium,” Acta Crystallographica Section C 45, 941–942 (1989).
  * Marzari and Vanderbilt (1997) Nicola Marzari and David Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847–12865 (1997).
  * Souza  _et al._ (2001) Ivo Souza, Nicola Marzari, and David Vanderbilt, “Maximally localized Wannier functions for entangled energy bands,” Phys. Rev. B 65, 035109 (2001).
  * Weng  _et al._ (2009) Hongming Weng, Taisuke Ozaki, and Kiyoyuki Terakura, “Revisiting magnetic coupling in transition-metal-benzene complexes with maximally localized Wannier functions,” Phys. Rev. B 79, 235118 (2009).
  * Ross (1992) Robert B. Ross, “Metallic Materials Specification Handbook,” Chapman & Hall, London (1992).
  * Abad  _et al._ (2015) Begoña Abad, Marta Rull-Bravo, Stephen L. Hodson, Xianfan Xu, and Marisol Martin-Gonzalez, “Thermoelectric properties of electrodeposited tellurium films and the sodium lignosulfonate effect,” Electrochimica Acta 169, 37–45 (2015).
  * Rabadanov  _et al._ (2021) M. R. Rabadanov, A. A. Stepurenko, A. E. Gummetov, and A. M. Ismailov, “Temperature dependence of the conductivity of tellurium whiskers,” Semiconductors 55, 551–556 (2021).
  * (44) The original formula actually contains the additional term Michishita and Peters (2021),  $\displaystyle\frac{2e^{3}}{V}\int\frac{d{\varepsilon}}{2\pi}f({\varepsilon}){\rm Im}\sum_{{\bm{k}}}{\rm tr}\left\\{\hat{{{\cal V}}}_{i}\frac{\partial\hat{G}^{{\rm R}}}{\partial{\varepsilon}}\left(\hat{{{\cal V}}}_{j}\hat{G}^{{\rm R}}\hat{{{\cal V}}}_{k}+\frac{1}{2}\hat{{{\cal V}}}_{jk}\right)\frac{\partial\hat{G}^{{\rm R}}}{\partial{\varepsilon}}\right\\}$ $\displaystyle+(j\leftrightarrow k).$ However, we checked that this term is zero under the time-reversal symmetry. This is also verified numerically.
  * Ma  _et al._ (2022) Junchao Ma, Bin Cheng, Lin Li, Zipu Fan, Haimen Mu, Jiawei Lai, Xiaoming Song, Dehong Yang, Jinluo Cheng, Zhengfei Wang, Changgan Zeng, and Dong Sun, “Unveiling Weyl-related optical responses in semiconducting tellurium by mid-infrared circular photogalvanic effect,” Nature Communications 13, 5425 (2022).


# Impact of a minority relativistic electron tail interacting with a thermal
plasma containing high-atomic-number impurities

Nathan A. Garland  Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Hyun-Kyung Chung  National Fusion Research Institute (NFRI), 169-148 Gwahak-
ro, Yuseong-gu, Daejeon 34133, Korea  Christopher J. Fontes  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  Mark C. Zammit  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  James Colgan  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  Todd Elder  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  Columbia University, New York,
NY 10027, USA  Christopher J. McDevitt  Los Alamos National Laboratory, Los
Alamos, NM 87545, USA  Timothy M. Wildey  Sandia National Laboratories,
Albuquerque, NM 87185, USA  Xian-Zhu Tang  Los Alamos National Laboratory, Los
Alamos, NM 87545, USA

(August 27, 2024)

###### Abstract

A minority relativistic electron component can arise in both laboratory and
naturally-occurring plasmas. In the presence of high-atomic-number ion
species, the ion charge state distribution at low bulk electron temperature
can be dominated by relativistic electrons, even though their density is
orders of magnitude lower. This is due to the relativistic enhancement of the
collisional excitation and ionization cross sections. The resulting charge
state effect can dramatically impact the radiative power loss rate and the
related Bethe stopping power of relativistic electrons in a dilute plasma.
(Approved for release under LA-UR-19-28749)

Major disruptions must be adequately mitigated in a tokamak for a viable
fusion power reactor, with the current approach for ITER’s Disruption
Mitigation System (DMS) based on injecting large quantities of high-atomic-
number impurities such as argon or neon Lehnen and Maruyama (2018); Lehnen
_et al._ (2015). This can serve the dual purpose of thermal quench mitigation
by spreading the plasma power load over the reactor first wall through
radiative cooling of the bulk plasma Whyte  _et al._ (2003); Granetz  _et al._
(2006); Hollmann  _et al._ (2013), and current quench mitigation via enhanced
runaway electron current dissipation due to increased collision rates Granetz
_et al._ (2006); Paz-Soldan  _et al._ (2014); Boozer (2015). The resulting
plasma mixture of hydrogenic fuel, helium ash, and high-Z impurities, at a
total atomic number density of $10^{20-21}$ m${}^{-3},$ leads to a radiatively
cooled bulk electron population, along with a minority runaway electron
component at a number density of $10^{16-17}$ m${}^{-3},$ which is sufficient
to carry a plasma current of a few to 10 mega-amperes (MA) in ITER discharges.
This is a most unusual plasma that emphasizes a variety of radiative processes
for plasma power loss, some combination of which point to the surprising role
of minority relativistic runaway electrons. Although the presentation here
focuses on the conditions of a disrupting tokamak plasma, the issues uncovered
apply to a range of plasma applications in which a minority population of
relativistic electrons co-exists with a cold thermal bulk that contains high-
atomic-number ion species. Some non-fusion examples include atmospheric
lightning Gurevich  _et al._ (1992), planetary radiation belts Fletcher  _et
al._ (2011), and any synchrotron emitting astrophysical plasma that is also
metal rich Israelian and Meynet (2008).

In a mitigated tokamak disruption discharge where $n_{\rm Ar}=n_{\rm
D}=10^{20}$ m${}^{-3},$ line radiation by argon would dominate the plasma
radiative power loss, which is around a few GW/m3 over the electron
temperature range of 10 eV to 1 keV. For an ITER plasma of 300 m3 in volume
and $300$ MJ in thermal energy, this loss mechanism alone would ensure a
thermal collapse (electron temperature, $T_{e}$, from a few keV to below 10
eV) over $1$ ms or less. Once the electron thermal temperature reaches a few
eV, radiative loss in a thermal plasma of comparable argon and deuterium
number density becomes negligibly inefficient. The minority runaway electron
population, despite the tiny number density of $10^{16-17}$ m-3 required to
carry a few to 10 MA of current in ITER, would dominate the radiative loss.
The underlying cause for runaway electrons aiding efficient radiative power
loss is fundamentally due to a quantum electrodynamics (QED) effect dating
back to Møller, Breit, and Bethe.

At electron energies significantly above the threshold energy, the excitation
and ionization integrated cross-section (ICS) would normally decrease with
increasing impacting electron energy. However, this trend reverses when the
impacting electron reaches relativistic energies, usually at a threshold
around one MeV. This is a lowest-order QED correction associated with the
Møller or generalized Breit interaction Llovet  _et al._ (2014); Fontes  _et
al._ (2015); Scofield (1978); Bostock  _et al._ (2013); Fontes  _et al._
(2014). Much of these physical effects are captured by the Bethe formula for
electron stopping  Bethe (1932); Inokuti (1971); Berestetskii  _et al._
(1982); Breizman  _et al._ (2019)

$\displaystyle-\frac{d{E}}{dx}=\sum_{\alpha}\frac{2\pi
e^{4}n_{e\alpha}}{m_{e}v^{2}}\left[\log\left(\frac{m_{e}^{2}c^{4}(\gamma^{2}-1)(\gamma-1)}{2\left\langle
I_{\alpha}\right\rangle^{2}}\right)\right.$
$\displaystyle-\left.\left(\frac{2}{\gamma}-\frac{1}{\gamma^{2}}\right)\log
2+\frac{1}{\gamma^{2}}+\frac{(\gamma-1)^{2}}{8\gamma^{2}}\right],$ (1)

where $c$ is the speed of light, $\epsilon_{0}$ the vacuum permittivity,
$\beta=v/c,$ and $e$ and $m_{e}$ the electron charge and rest mass
respectively. For each ion species in a specific charge state, which is
denoted by subscript $\alpha$, there is a unique mean excitation energy
$\left\langle I_{\alpha}\right\rangle$ and bound electron density
$n_{e\alpha}.$ In fact, the standard treatment in runaway modeling is to
incorporate the runaway slowing down due to excitation and ionization via a
friction in the Fokker-Planck collision operator using the Bethe formula
Hesslow  _et al._ (2018a, 2017). The effect of ion charge state distribution
(CSD) on runaway and thermal bulk plasma evolution is through a collisional-
radiative (CR) model that deals with only the background Maxwellian population
at given temperature.

In a dilute plasma such as those in tokamak disruption, the aforementioned
decoupled treatment of background thermal electrons and runaways Hesslow  _et
al._ (2018a, 2017) has a straightforward prediction for the radiative power
loss as the bulk electrons are cooled to below a few eV. Namely, the
background electron density drops precipitously due to cold thermal electron
recombination with ions. Along with a mean charge number $\left<Z\right>$
approaching zero, the radiative power loss by thermal electrons can decrease
by 5 orders of magnitude between $T_{e}=5$ eV to $T_{e}=1$ eV. Meanwhile, as
$\left\langle Z\right\rangle\rightarrow 0$ with $T_{e}\rightarrow 0,$ the
simultaneous increase in bound electron density $n_{e\alpha}$ and decrease in
mean excitation potential $\left\langle I_{\alpha}\right\rangle$ leads to
enhanced radiative loss by runaways, according to the Bethe formula.

Unlike in a solid, inelastic collisions between runaways and high-atomic-
number impurities in a dilute plasma can directly alter the charge state of
the plasma ions. The change in ion CSD, in turn, modifies both the radiative
power loss by thermal electrons and the runaways. The former is through
modified line emissions as the dominant bound-bound transitions have explicit
charge state dependence, while the latter, using the Bethe stopping power
formula, is by way of modified bound electron density $n_{e\alpha}$ and mean
excitation potential $\left\langle I_{\alpha}\right\rangle$. This Letter
provides CR modeling taking into account both the thermal bulk and the
runaways with enhanced excitation and ionization scattering processes, which
allows us to not only elucidate the new qualitative trends enabled by the
coupling between the minority runaways and bulk thermal population through
high-atomic-number ions, but also quantify the discrepancies between a full CR
treatment and a decoupled treatment of thermal bulk and minority runaways,
which has been the state of the art in runaway modeling. Supplementary
material gar (a), contrasts the inclusion of both excitation and ionization
collision with a recent analysis in the context of impurity transport that
models the ionization balance using the Bethe stopping power formula for the
runaway ionization rate Hollmann  _et al._ (2019).

Our CR model is a fork (called FLYCHKLite) of the FLYCHK model/code, which is
widely used in dense plasma applications Chung  _et al._ (2005), for its
demonstrated applicability of the super-configuration model to sufficiently
replicate the required physics in a computationally efficient manner Chung
_et al._ (2013); Piron  _et al._ (2017). The Los Alamos suite of atomic
physics codes Fontes  _et al._ (2015) have been employed to calibrate and
verify the model, especially the contribution from $\Delta n=0$ transitions in
line emissions. The robustness of our physics findings gar (b) is further
assessed via an uncertainty quantification analysis on the relativistic ICS
for both excitation and ionization.

Relativistic inelastic scattering: To account for the collisional processes of
an arbitrary electron energy distribution function (EEDF), we need the ICS for
electron impact excitation and ionization, in both the non-relativistic and
relativistic energy regimes. Our approach is to employ non-relativistic
analytic fits used in FLYCHK, augmented by the addition of an analytic
relativistic correction following a Møller-Bethe-like functional form Bretagne
_et al._ (1986); Scofield (1978); Fontes  _et al._ (2014). For electron-impact
excitation, the analytic form of van Regemorter van Regemorter (1962) is used
for the non-relativistic regime,

$\displaystyle\sigma_{i\rightarrow
j}^{\text{NR}}=\frac{8\pi^{2}a_{0}^{2}}{\sqrt{3}}\left(\frac{Ry}{\Delta
E_{ij}}\right)^{2}\frac{f_{ij}g(U)}{U},$ (2)

where $U=E/\Delta E_{ij}$ is the scaled incoming electron energy, $Ry$ is the
Rydberg energy, $f_{ij}$ is the collision oscillator strength and $a_{0}$ is
the Bohr radius. The Gaunt factor is

$\displaystyle g(U)=A\log U+B+C(U+a)^{-1}+D(U+a)^{-2},$ (3)

where coefficients are found via empirical fits of (3) to averaged hydrogenic
ICS computed by Chung et al. Chung  _et al._ (2007) via plane-wave Born
calculations augmented by near-threshold scaling of Kim Kim (2001).

Using (2) as a basis, a Møller-Bethe-like relativistic correction Bretagne
_et al._ (1986); Bethe (1932); Møller (1932) is applied so the total ICS is

$\displaystyle\sigma_{i\rightarrow
j}^{\text{TOT}}=(1-S(E))\sigma_{i\rightarrow
j}^{\text{NR}}+S(E)\sigma_{i\rightarrow j}^{\text{R}},$ (4)

where

$\displaystyle\sigma_{i\rightarrow j}^{\text{R}}=\frac{8\pi
a_{0}^{2}Ry^{2}}{m_{e}c^{2}\beta^{2}}\frac{f_{ij}}{\Delta
E_{ij}}\left[\log\left(\frac{\beta^{2}}{1-\beta^{2}}\frac{m_{e}c^{2}}{2\Delta
E_{ij}}\right)-\beta^{2}\right],$ (5)

and the sigmoid smoothing function is defined as

$\displaystyle S(E)={\left[1+\exp(10^{-5}(10^{5}-E))\right]}^{-1},$ (6)

with $E$ in units of eV. The form of (6) is chosen to adequately describe
known relativistic ICS for inelastic processes.

For electron impact ionization, the non-relativistic ICS is modeled by the
semi-empirical expression of Burgess and Chidichimo Burgess and Chidichimo
(1983)

$\displaystyle\sigma_{(Z,i)\rightarrow(Z+1,j)}^{\text{NR}}=\pi
a_{0}^{2}C\xi\left(\frac{Ry}{\Delta
I_{Z}^{i}}\right)^{2}\frac{1}{U}\log(U)W(U),$ (7)

where $\Delta I_{Z}^{i}$ is the ionization threshold energy for the level $i$
of charge state $Z$, $U=E/\Delta I_{Z}^{i}$, $\xi$ is the number of electrons
in the shell being ionized, $C$ is an arbitrary constant nominally around 2
Burgess and Chidichimo (1983), and

$\displaystyle W(U)=(\log(U))^{\frac{\beta^{*}}{U}},$ (8)
$\displaystyle\beta^{*}=0.25\sqrt{\frac{100Q_{n}+91}{4Q_{n}+3}}-1.25,$ (9)

where $Q_{n}$ is the screened charge Chung  _et al._ (2005).

Again using the same form as (4) for the total ionization ICS, we add a
relativistic correction Scofield (1978); Bethe  _et al._ (1933) on top of (7)
following the form

$\displaystyle\sigma_{(Z,i)\rightarrow(Z+1,j)}^{\text{R}}=$ $\displaystyle\pi
a_{0}^{2}C\xi\left(\frac{Ry}{\Delta I_{Z}^{i}}\right)$
$\displaystyle\left[\log\left(\frac{\beta^{2}}{1-\beta^{2}}\frac{m_{e}c^{2}}{2\Delta
I_{Z}^{i}}\right)-\beta^{2}\right].$ (10)

Figs. 1 and 2 compare the proposed fits with calculated ICS for excitation and
ionization in the literature. We note agreement within a factor of two of
recent R-matrix theories at the peak of the cross section, and that unlike
those reported in Ref. Wang  _et al._ (2018), normally such calculations are
not extended into the relativistic regime of relevance here. The plasma
physics impact established in this paper certainly points to the importance
and urgency of such systematic first-principle cross section calculations.

Figure 1: ICS for excitation of Cl-like Ar+ $n=3\rightarrow n=4$ transition.
Comparison of our proposed approximate ICS is made against the non-
relativistic results of Chung et al. Chung  _et al._ (2007), van Regemorter
van Regemorter (1962) and the relativistic result of Bretagne et al. Bretagne
_et al._ (1986). Figure 2: ICS for lumped-average L-shell ionization of Ar.
Comparison of our proposed approximate ICS is made against the non-
relativistic result of Burgess and Chidichimo Burgess and Chidichimo (1983),
and the relativistic results of Scofield Scofield (1978) and Wang et al. Wang
_et al._ (2018).

Simulation problem setup: To demonstrate the impact of using ICS that describe
enhanced scattering observed at relativistic energies we perform a comparison
between CR model results computed with and without relativistic ICS effects
applied to conditions representative of a tokamak disruption mitigation
scenario. Without losing generality, the electron distribution model is
defined as a bulk Maxwellian at given temperature $T_{e}$ and a Gaussian
runaway tail that has a mean energy $\left<E_{0}\right>$ and full width at
half maximum spread of $\Delta E_{\rm FWHM},$ at runaway number density
$n_{\rm RE}.$ The ion population is fixed with $(n_{\rm D},n_{\rm Ar}).$ The
thermal bulk then has an electron density ($n_{e}$) from the quasi-neutral
condition $n_{e}=\left\langle Z\right\rangle_{\rm Ar}n_{\rm Ar}+\left\langle
Z\right\rangle_{\rm D}n_{\rm D}-n_{\rm RE}.$ As $T_{e}$ varies from 10 keV to
1 eV, $n_{e}$ can vary greatly as a function of $\left\langle Z\right\rangle$,
but $n_{\rm RE}$ is a fixed number that only depends on how much current the
runaways need to carry. For example, to carry $10$ MA of runaway current in
ITER, $n_{\rm RE}\approx 10^{17}$m${}^{-3}.$ Our scan over $T_{e}$ is
performed with fixed $\left<E_{0}\right>=10\ \text{MeV},\Delta E_{\rm FWHM}=5\
\text{MeV},n_{\rm RE}=10^{17}\ \text{m}^{-3},$ and $n_{\rm D}=n_{\rm
Ar}=10^{20}\ \text{m}^{-3}.$ The CR model iteratively solves for $\left\langle
Z\right\rangle$ and $n_{e}$ when everything else is fixed.

Runaway enhanced ionization at low $T_{e}$: With the addition of a
relativistic tail, and the inclusion of enhanced inelastic scattering by this
relativistic electron tail, a clear increase in $\left\langle Z\right\rangle$
and broadening of the CSD is observed for the argon discharge at low $T_{e}$,
exemplified by Fig. 3 for $T_{e}=2$ eV. Here, we see the addition of
relativistic electrons sampling non-enhanced ICS increases $\left\langle
Z\right\rangle$ slightly, and then the addition of enhanced relativistic ICS
to the model further increases $\left\langle Z\right\rangle$ and promotes CSD
broadening. This is a direct result of the relativistic tail of the EEDF being
able to sample an increasing ICS at higher energies. A wider spread of charge
states shown in Fig. 3 will also lead to very different radiative cooling
pathways, discussed in the next section. Contrasting the yellow line with blue
and green lines in Fig. 3, one can also see that the complex excitation-
ionization pathways enabled by QED enhancement of the cross sections can
deplete the Ar+ population.

Figure 3: Argon ion CSD at $T_{e}=2$ eV with a relativistic Gaussian tail
supporting a 10 MA current compared to a thermal discharge. The impact of
including relativistic ICS effects ($\blacktriangle$) compared to using ICS
without the Møller/generalized Breit interaction ($\blacksquare$) or simply
assuming a thermal plasma ($\bullet$) is apparent.

Considering $\left\langle Z\right\rangle$ variation with $T_{e}$ Fig. 4
demonstrates a clear increase in $\left\langle Z\right\rangle$ at $T_{e}<50$
eV, when compared to a Maxwellian EEDF typically used in decoupled CR
modeling. For benchmarking purposes the result for a thermal plasma computed
with the LANL CR modeling code, ATOMIC Fontes  _et al._ (2015), is included in
Fig. 4, and agrees with the present thermal model.

Figure 4: Average argon charge state, $\left\langle Z\right\rangle$, over
thermal $T_{e}$ demonstrating enhancement produced by relativistic electrons
sampling enhanced relativistic ICS. $\left\langle Z\right\rangle$ for a
thermal plasma computed by LANL ATOMIC code Fontes  _et al._ (2015) shown for
benchmarking purposes. Average charge state of deuterium (…) shown for
reference, indicating a fully stripped ion for nearly all plasma temperatures
considered.

Runaway-induced ion charge state effects: The runaway-induced ionization at
low $T_{e},$ which is expected for the current quench phase, can impact
disruption physics and its modeling in a number of ways. These are generally
known as ion charge state effects, and here we give a few most prominent
examples. The first is the enhanced pitch angle scattering of runaways due to
the partial screening effect, which has a direct dependence on the ion charge
state distribution, according to Hesslow et al. Hesslow  _et al._ (2017,
2018b). The physical importance is that enhanced pitch angle scattering would
limit the energy of the O-point of the runaway vortex, which is responsible
for a lower runaway energy but broader pitch distribution McDevitt  _et al._
(2018); Guo  _et al._ (2017). The runaway-induced ionization would then
modulate the pitch angle scattering rate due to partial screening, impacting
both the avalanche threshold and runaway energy distribution, as well as the
spatial transport of the runaways McDevitt  _et al._ (2019).

The second example is the slowing down of the runaways by inelastic collisions
with high-atomic-number impurities. The runaway-induced ion charge state
effect can be seen from the ICS for both excitation and ionization, Eqs. (5)
and (10). Both ICS tend to decrease with higher ion charge number since the
excitation energy $\Delta E_{ij}$ and ionization threshold energy $\Delta
I_{Z}^{i}$ are generally higher with more electrons stripped from the atom. In
the case of ionization ICS, the bound electron density $n_{e\alpha}$ would
also decrease linearly with higher ion charge number. The impact of decreasing
ICS with higher charge state is demonstrated by contrasting the Bethe stopping
power. In Fig. 5, we contrast the Bethe stopping power using (i) a CSD from
the CR prediction of a background Maxwellian alone, and (ii) CSD from the CR
prediction that includes the runaway contribution. There is significant over-
estimate of the Bethe stopping power if the runaway-induced ionization is not
taken into account at low $T_{e}$, which is crucial in a post-thermal-quench
plasma.

Figure 5: Inelastic collision stopping power experienced by relativistic
runaway electrons colliding with argon ions. Values computed with the Bethe
stopping power formula Eq. (1) demonstrate the impact of coupling a runaway-
enhanced CSD ($\bullet$) compared to using a thermal plasma CSD
($\blacksquare$).

The third example is more conventional, and it concerns the effect of Coulomb
collisions on thermal bulk electrons. This comes from the strong $Z_{\rm eff}$
dependence for thermal plasma conductivity, and the collisional damping of
both the externally injected Guo  _et al._ (2018) and self-induced Liu  _et
al._ (2018) fast waves, by thermal electrons. The runaway-induced ionization,
in the plasma regime of a mitigated tokamak disruption, can dramatically
increase $Z_{\rm eff},$ and hence increase dissipation of the remnant Ohmic
current and the fast waves. The latter is particularly unfortunate since
resonant wave-particle interaction offers a valuable tool for controlling the
runaway energy Guo  _et al._ (2018).

Runaway-induced radiative power loss (RPL) effects: While it is expected that
radiative power loss, shown in Fig. 6, would be dominated by collisional
excitation and ionization by runaways at low $T_{e}$ ($T_{e}<4$ eV), by virtue
of a higher charge state, there is also a subtler runaway effect when RPL
rates by thermal electrons and runaways are comparable. This can be seen in
Fig. 6, in the range of $5\leq T_{e}\leq 100$ eV, where the RPL of the
combined thermal plus runaway plasma is up to 40% lower than that of the pure
thermal plasma. The reduced RPL of the runaway enhanced discharges is found to
be a result of the relativistic ICS enhancement diffusing and decreasing both
the relative population of ion charge states and excited levels, resulting in
a smaller RPL in this temperature range.

Figure 6: RPL of argon over thermal $T_{e}$ demonstrating enhanced RPL
produced by relativistic electrons sampling relativistic inelastic ICS. RPL of
thermal plasma computed with LANL ATOMIC code Fontes  _et al._ (2015) is shown
for benchmarking purposes.

At higher temperatures, all RPL results effectively converge above
approximately 100 eV, which is also seen in the result of the average charge
state. Convergence occurs once the thermal population begins to dominate the
ionization balance of the discharge, negating the excitation and ionization
channels introduced by the relativistically enhanced scattering.

Uncertainty quantification: To validate the qualitative trends observed in
this study, we employed UQ software Dakota Adams et al. (2017) to perform an
epistemic global interval estimation uncertainty analysis with relativistic
electron ICS. In this UQ analysis, prefactors, $0.5\leq C_{E},C_{I}\leq 2$,
were applied to the relativistic excitation and ionization ICS formulas (5)
and (10) to describe uncertainty in the approximate formulas used in this
work. For $\left\langle Z\right\rangle$ and RPL appearing in Figs. 4 and 6,
uncertainty bounds are shown. The confidence intervals computed via 500 model
evaluations with Latin hypercube sampling over the range of ICS uncertainty
supports our observation that there is a definitive enhancement in both
$\left\langle Z\right\rangle$ and total RPL of argon ions considered as a
direct result of relativistic electrons sampling relativistically enhanced
ICS.

Conclusion: A minority relativistic electron component, at a number density
three orders of magnitude smaller than the low-temperature thermal bulk, can
dominate the CSD in the presence of high-atomic-number impurities, due to the
relativistic enhancement of the collisional excitation and ionization ICS. The
resulting charge state effects can impact (1) radiative power loss rate and
related Bethe stopping power on the runaways, and (2) collisional dissipation
of Ohmic current and externally-injected and self-excited plasma waves. These
physics findings suggest the necessity of a coupled CR model that takes into
account both the bulk electron population and the minority relativistic
electron component, if high-atomic-number impurities are present.

This work was jointly supported by the U.S. Department of Energy through the
Magnetic Fusion Theory Program and the Tokamak Disruption Simulation (TDS)
SciDAC project at both Los Alamos National Laboratory (Contract No.
89233218CNA000001) and Sandia National Laboratory (Contract No. DE-
NA-0003525). Additional support is provided by Laboratory Directed Research
and Development program of Los Alamos National Laboratory under project number
20200356ER. T. Elder was also supported by the Science Undergraduate
Laboratory Internships (SULI) program.

## References

  * Lehnen and Maruyama (2018) M. Lehnen and S. Maruyama,  _Executive report of the Disruption Mitigation Workshop_ , Tech. Rep. ITR-18-002 (ITER Organization, St-Paul-lez-Durance, 2018).
  * Lehnen  _et al._ (2015) M. Lehnen, K. Aleynikova, P. B. Aleynikov, D. J. Campbell, P. Drewelow, N. W. Eidietis, Y. Gasparyan, R. S. Granetz, Y. Gribov, N. Hartmann, E. M. Hollmann, V. A. Izzo, S. Jachmich, S. H. Kim, M. Kočan, H. R. Koslowski, D. Kovalenko, U. Kruezi, A. Loarte, S. Maruyama, G. F. Matthews, P. B. Parks, G. Pautasso, R. A. Pitts, C. Reux, V. Riccardo, R. Roccella, J. A. Snipes, A. J. Thornton, and P. C. de Vries, Journal of Nuclear Materials PLASMA-SURFACE INTERACTIONS 21, 463, 39 (2015).
  * Whyte  _et al._ (2003) D. G. Whyte, T. C. Jernigan, D. A. Humphreys, A. W. Hyatt, C. J. Lasnier, P. B. Parks, T. E. Evans, P. L. Taylor, A. G. Kellman, D. S. Gray, and E. M. Hollmann, Journal of Nuclear Materials Plasma-Surface Interactions in Controlled Fusion Devices 15, 313-316, 1239 (2003).
  * Granetz  _et al._ (2006) R. Granetz, D. G. Whyte, V. A. Izzo, T. Biewer, M. L. Reinke, J. Terry, A. Bader, M. Bakhtiari, T. Jernigan, and G. Wurden, Nuclear Fusion 46, 1001 (2006).
  * Hollmann  _et al._ (2013) E. M. Hollmann, M. E. Austin, J. A. Boedo, N. H. Brooks, N. Commaux, N. W. Eidietis, D. A. Humphreys, V. A. Izzo, A. N. James, T. C. Jernigan, A. Loarte, J. Martin-Solis, R. A. Moyer, J. M. Muñoz-Burgos, P. B. Parks, D. L. Rudakov, E. J. Strait, C. Tsui, M. A. V. Zeeland, J. C. Wesley, and J. H. Yu, Nuclear Fusion 53, 083004 (2013).
  * Paz-Soldan  _et al._ (2014) C. Paz-Soldan, N. W. Eidietis, R. Granetz, E. M. Hollmann, R. A. Moyer, J. C. Wesley, J. Zhang, M. E. Austin, N. A. Crocker, A. Wingen, and Y. Zhu, Physics of Plasmas 21, 022514 (2014).
  * Boozer (2015) A. H. Boozer, Physics of Plasmas 22, 032504 (2015).
  * Gurevich  _et al._ (1992) A. V. Gurevich, G. M. Milikh, and R. Roussel-Dupre, Physics Letters A 165, 463 (1992).
  * Fletcher  _et al._ (2011) L. Fletcher, B. R. Dennis, H. S. Hudson, S. Krucker, K. Phillips, A. Veronig, M. Battaglia, L. Bone, A. Caspi, Q. Chen, P. Gallagher, P. T. Grigis, H. Ji, W. Liu, R. O. Milligan, and M. Temmer, Space Science Reviews 159, 19 (2011).
  * Israelian and Meynet (2008) G. Israelian and G. Meynet, eds.,  _The Metal Rich Universe_ , (Cambridge University Press, Cambridge, UK, 2008).
  * Llovet  _et al._ (2014) X. Llovet, C. J. Powell, F. Salvat, and A. Jablonski, Journal of Physical and Chemical Reference Data 43, 013102 (2014).
  * Fontes  _et al._ (2015) C. J. Fontes, H. L. Zhang, J. A. Jr, R. E. H. Clark, D. P. Kilcrease, J. Colgan, R. T. Cunningham, P. Hakel, N. H. Magee, and M. E. Sherrill, J. Phys. B: At. Mol. Opt. Phys. 48, 144014 (2015).
  * Scofield (1978) J. H. Scofield, Physical Review A 18, 963 (1978).
  * Bostock  _et al._ (2013) C. J. Bostock, C. J. Fontes, D. V. Fursa, H. L. Zhang, and I. Bray, Phys. Rev. A 88, 012711 (2013).
  * Fontes  _et al._ (2014) C. J. Fontes, C. J. Bostock, and K. Bartschat, EPJ H 39, 517 (2014).
  * Bethe (1932) H. Bethe, Z. Physik 76, 293 (1932).
  * Inokuti (1971) M. Inokuti, Rev. Mod. Phys. 43, 297 (1971).
  * Berestetskii  _et al._ (1982) V. Berestetskii, E. Lifshitz, and L. Pitaevskii,  _Quantum Electrodynamics: Volume 4 (Course of Theoretical Physics)_ (Butterworth-Heinemann, 1982).
  * Breizman  _et al._ (2019) B. N. Breizman, P. Aleynikov, E. M. Hollmann, and M. Lehnen, Nuclear Fusion 59, 083001 (2019).
  * Hesslow  _et al._ (2018a) L. Hesslow, O. Embréus, M. Hoppe, T. C. DuBois, G. Papp, M. Rahm, and T. Fülöp, Journal of Plasma Physics 84 (2018a), 10.1017/S0022377818001113.
  * Hesslow  _et al._ (2017) L. Hesslow, O. Embréus, A. Stahl, T. DuBois, G. Papp, S. Newton, and T. Fülöp, Physical Review Letters 118, 255001 (2017).
  * gar (a) The online supplementary material shows the critical role of collisional excitation in ionization balance and hence CSD, which is absent in Hollmann et al. Hollmann  _et al._ (2019) (a).
  * Hollmann  _et al._ (2019) E. M. Hollmann, N. W. Eidietis, J. L. Herfindal, P. B. Parks, A. Y. Pigarov, D. Shiraki, M. E. Austin, L. Bardoczi, L. Baylor, I. Bykov, T. N. Carlstrom, D. Kaplan, C. J. Lasnier, A. Lvovskiy, A. Moser, R. A. Moyer, C. Paz-Soldan, D. L. Rudakov, C. Samuell, M. Shafer, M. V. Zeeland, A. Welander, and R. Wilcox, Nuclear Fusion 59, 106014 (2019).
  * Chung  _et al._ (2005) H. K. Chung, M. H. Chen, W. L. Morgan, Y. Ralchenko, and R. W. Lee, High Energy Density Physics 1, 3 (2005).
  * Chung  _et al._ (2013) H. K. Chung, C. Bowen, C. J. Fontes, S. B. Hansen, and Y. Ralchenko, High Energy Density Physics 9, 645 (2013).
  * Piron  _et al._ (2017) R. Piron, F. Gilleron, Y. Aglitskiy, H. K. Chung, C. J. Fontes, S. B. Hansen, O. Marchuk, H. A. Scott, E. Stambulchik, and Y. Ralchenko, High Energy Density Physics 23, 38 (2017).
  * gar (b) Both FLYCHK Chung  _et al._ (2005) and ATOMIC Fontes  _et al._ (2015) treat Auger processes by including multiply-excited states and inner-shell excited states. More exotic multi-Auger events Salop (1974), such as ”shake off,” are not included in either codes. Their correction to single ionization cross section is a few percent for Ar. (b).
  * Bretagne  _et al._ (1986) J. Bretagne, G. Callede, M. Legentil, and V. Puech, J. Phys. D: Appl. Phys. 19, 761 (1986).
  * van Regemorter (1962) H. van Regemorter, The Astrophysical Journal 136, 906 (1962).
  * Chung  _et al._ (2007) H. K. Chung, R. W. Lee, and M. H. Chen, High Energy Density Physics 3, 342 (2007).
  * Kim (2001) Y.-K. Kim, Phys. Rev. A 64, 032713 (2001).
  * Møller (1932) C. Møller, Annalen der Physik 406, 531 (1932).
  * Burgess and Chidichimo (1983) A. Burgess and M. C. Chidichimo, Monthly Notices of the Royal Astronomical Society 203, 1269 (1983).
  * Bethe  _et al._ (1933) H. Bethe, F. Hund, N. F. Mott, W. Pauli, A. Rubinowicz, G. Wentzel, and A. Smekal,  _Quantentheorie_ , 2nd ed., Handbuch der Physik (Springer-Verlag, Berlin Heidelberg, 1933).
  * Wang  _et al._ (2018) X. J. Wang, J. Seuntjens, and J. M. Fernández-Varea, Journal of Physics B: Atomic, Molecular and Optical Physics 51, 145201 (2018).
  * Hesslow  _et al._ (2018b) L. Hesslow, O. Embréus, G. J. Wilkie, G. Papp, and T. Fülöp, Plasma Physics and Controlled Fusion 60, 074010 (2018b).
  * McDevitt  _et al._ (2018) C. J. McDevitt, Z. Guo, and X.-Z. Tang, Plasma Physics and Controlled Fusion 60, 024004 (2018).
  * Guo  _et al._ (2017) Z. Guo, C. J. McDevitt, and X.-Z. Tang, Plasma Physics and Controlled Fusion 59, 044003 (2017).
  * McDevitt  _et al._ (2019) C. J. McDevitt, Z. Guo, and X.-Z. Tang, Plasma Physics and Controlled Fusion 61, 024004 (2019).
  * Guo  _et al._ (2018) Z. Guo, C. J. McDevitt, and X.-Z. Tang, Physics of Plasmas 25, 032504 (2018).
  * Liu  _et al._ (2018) C. Liu, E. Hirvijoki, G.-Y. Fu, D. P. Brennan, A. Bhattacharjee, and C. Paz-Soldan, Physical Review Letters 120, 265001 (2018).
  * Adams et al. (2017) B. Adams et al., Dakota: A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.0 User’s Manual, Sandia Technical Report SAND2014-4633, July 2014\. Updated November 2017 (Version 6.7) (2017).
  * Salop (1974) A. Salop, Phys. Rev. A 9, 2496 (1974).