††thanks: These authors have contributed equally to this work.††thanks: These
authors have contributed equally to this work.††thanks: Corresponding
<EMAIL_ADDRESS>
# Anisotropic longitudinal optical conductivities of tilted Dirac bands in
1T′-MoS2

Chao-Yang Tan  Department of Physics, Institute of Solid State Physics and
Center for Computational Sciences, Sichuan Normal University, Chengdu, Sichuan
610066, China  Chang-Xu Yan  Department of Physics, Institute of Solid State
Physics and Center for Computational Sciences, Sichuan Normal University,
Chengdu, Sichuan 610066, China  Yong-Hong Zhao  Department of Physics,
Institute of Solid State Physics and Center for Computational Sciences,
Sichuan Normal University, Chengdu, Sichuan 610066, China  Hong Guo
Department of Physics, Institute of Solid State Physics and Center for
Computational Sciences, Sichuan Normal University, Chengdu, Sichuan 610066,
China  Department of Physics, McGill University, Montreal, Quebec H3A 2T8,
Canada  Hao-Ran Chang  Department of Physics, Institute of Solid State Physics
and Center for Computational Sciences, Sichuan Normal University, Chengdu,
Sichuan 610066, China  Department of Physics, McGill University, Montreal,
Quebec H3A 2T8, Canada

###### Abstract

1T′-MoS2 exhibits valley-spin-polarized tilted Dirac bands in the presence of
external vertical electric field and undergoes a topological phase transition
between the topological insulator and band insulator around the critical value
of the electric field. Within the linear response theory, we theoretically
investigate the anisotropic longitudinal optical conductivities of tilted
Dirac bands in both undoped and doped 1T′-MoS2, including the effects of the
vertical electric field. The influence of the spin-orbit coupling gap, band
tilting, and vertical electric field on the optical conductivities of tilted
Dirac bands is revealed. A theoretical scheme for probing the topological
phase transition in 1T′-MoS2 via exotic behaviors of longitudinal optical
conductivities is proposed. The results for 1T′-MoS2 are expected to be
qualitatively valid for other monolayer tilted gapped Dirac materials, such as
$\alpha$-SnS2, TaCoTe2, and TaIrTe4, due to the similarity in their band
structures.

##  I Introduction

The discovery of graphene has led to extremely active research in two-
dimensional (2D) Dirac materials Science2004 ; RMP2009 -materials
characterized by Dirac points and linear and/or hyperbolic energy dispersions
in the momentum space, in clear contrast to traditional metals and
semiconductors Mahan . So far, many 2D Dirac materials have been
experimentally synthesized and theoretically studied, including
$\alpha$-(BEDT-TTF)2I3 JPSJ2006 , silicene PRBSilicene2007 ; PRLSilicene2009 ;
PRBSilicene2011 ; PRLSilicene2011 ; EzawaPRLSilicene2012 , 8-$Pmmn$ borophene
Science8Pmmn2015 ; PRB8PmmnRapid2016 ; PRB8Pmmn2016 , and various monolayer
transition metal dichalcogenides (TMDCs) PRLMoS2010 ; PRLMoS2012 ; Science2014
; AdvMater2016 ; NatPhys2017 ; NatCommun2017 ; NatPhysTang2017 ; ScienceWu2018
; ScienceFatemi2018 ; ScienceSajadi2018 . In Dirac materials, the Dirac
dispersion together with the band gap and/or band tilting can be exploited to
manipulate physical properties. For example, the valley-spin-polarized band
gap tuned by a vertical electric field in silicene generates a topological
phase transition between the topological insulator phase and the band
insulator phase EzawaPRLSilicene2012 ; the band tilting in 8-$Pmmn$ borophene
induces a strong anisotropy in physical properties PRB8PmmnRapid2016 ;
PRB8Pmmn2016 . These qualitatively influence important material physics
including plasmons PRBHRC2014 ; PRBTabert2014 ; PRBIurov2017 ; PRBAgarwal2017
; PRBJafari2018 ; JPSJNishine2011 ; JPSJNishine2010 , optical conductivities
JPSJNishine2010 ; PRBStille2012 ; PRBVerma2017 ; PRBIurov2018 ; PRBHerrera2019
; PRBIurov2020 , thermoelectric effects PRBGhosh2020 , Kondo effects
PRBSun2018 , and RKKY interactions JPCMXiao2014 ; SciRepWang2018 ; PRBPaul2019
; JMMMZhang2019 .

1T′ TMDCs, the monolayer TMDCs of the $\mathrm{T}^{\prime}$ structure phase,
were theoretically predicted to be quantum spin Hall insulators Science2014
and have been experimentally synthesized AdvMater2016 ; NatPhys2017 ;
NatCommun2017 ; NatPhysTang2017 ; ScienceWu2018 ; ScienceFatemi2018 ;
ScienceSajadi2018 . Excitingly, the quantum spin Hall effect AdvMater2016 ;
NatPhys2017 ; NatCommun2017 ; NatPhysTang2017 ; ScienceWu2018  and gate-
induced superconductivity ScienceFatemi2018 ; ScienceSajadi2018  were recently
observed in experiments. Interestingly, 1T′ TMDCs Science2014  possess tilted
Dirac bands around Dirac points similar to that in 8-$Pmmn$ borophene
PRB8PmmnRapid2016 ; PRB8Pmmn2016 , and undergo a topological phase transition
APLLi2009 ; PRLMiao2012 ; PRLZhang2013  induced by a vertical electric field
such as that in silicene EzawaPRLSilicene2012 . The tilted gapped Dirac
dispersions Science2014  are shared by other monolayer Dirac materials, such
as $\alpha$-SnS2 NPGMa2016 , TaCoTe2 PRBYang2019 , and TaIrTe4 PRBLu2020 .
Furthermore, the band structure of 1T′ TMDCs Science2014  differs strongly
from that of its polytypic structure the $1\mathrm{H}$ TMDCs PRLMoS2010 ;
PRLMoS2012 . These features make the 1T′ TMDC a very attractive monolayer
Dirac material and its physical properties are worthy of careful
investigations.

A particularly important physical property of any material is the optical
conductivity whose real part relates to absorption of photons and provides a
powerful approach for extracting band structures and optical properties. The
optical conductivity of Dirac materials has received intensive research both
theoretically and experimentally, for graphene PRBGusynin2007 ; PRBStauber2008
; PRLMak2008 , $\alpha$-(BEDT-TTF)2I3 JPSJNishine2010 , silicene PRBStille2012
, 8-$Pmmn$ borophene PRBVerma2017 , 1H-MoS2 PRBCarbotte2012 ; PRBCarvalho2013
; PRBAsgari2014 , and topological insulators PRBDiPietro2012 ; PRBCarbotte2013
; PRBXiao2013 .

It is the purpose of this work to investigate the optical conductivity of the
monolayer Dirac material 1T′ -MoS2 which is a typical material of 1T′ TMDCs.
This study, apart from further enriching the important materials physics of
Dirac materials, is focused on providing a fundamental understanding of at
least three outstanding issues. The first is to understand the influence of
the spin-orbit coupling (SOC) gap on optical conductivities of tilted Dirac
bands, by comparing 1T′-MoS2 (gapped) and 8-$Pmmn$ borophene (gapless). The
second is to establish the difference of optical conductivities originating
from polytypic structures of MoS2, by comparing 1T′-MoS2 with 1H-MoS2; or
originating from band tilting, by comparing 1T′-MoS2 (tilted) with silicene
(untilted). The third is to analyze the probing of the topological phase
transition in 1T′-MoS2 via longitudinal optical conductivities. To this end,
we theoretically investigate the anisotropic longitudinal optical
conductivities of tilted Dirac bands in both undoped and doped 1T′-MoS2 and
the influence of a vertical electric field acting on them.

Figure 1: (Color online) In the presence of SOC gap and vertical electric
field, parametrized respectively by $\Delta_{\mathrm{so}}$ and
$\alpha=E/E_{c}$, the energy bands and gaps are valley-spin-polarized. The
upper panels and lower panels stand respectively for $\kappa=-$ valley and
$\kappa=+$ valley. The red lines and blue lines denote spin-up ($s=+$) and
spin-down ($s=-$), respectively. The topological phase transition between the
topological insulator phase ($\alpha<1$) and the band insulator phase
($\alpha>1$) occurs at the phase termed by the valley-spin-polarized metal
($\alpha=1$).

This paper is organized as follows. In Sec. II, we briefly describe the model
Hamiltonian and formalism to calculate the longitudinal optical conductivity.
In Sec. III, we present our numerical results for the absorptive part of the
longitudinal conductivity in the undoped and doped 1T′-MoS2 when the vertical
electric field is absent and present, respectively. In addition, we provide a
detailed analysis of the essential physics. Our main results and conclusions
are summarized in Sec. IV. Finally, we give three appendices to provide key
steps of analytical calculation.

##  II Model and theoretical formalism

We begin with the low-energy $k\cdot p$ Hamiltonian for 1T′-MoS2, reported in
Ref. Science2014 . For simplicity, we further expand it in the vicinity of two
independent Dirac points located at $(0,\kappa\Lambda)$ with $\kappa=\pm$. The
linearized Hamiltonian around the $\kappa$ valley reads

$\displaystyle H_{\kappa}(k_{x},k_{y})=$ $\displaystyle\hbar
k_{x}v_{1}\gamma_{1}+\hbar k_{y}\left(v_{2}\gamma_{0}-\kappa v_{-}I-\kappa
v_{+}\gamma_{2}\right)$
$\displaystyle+\Delta_{\mathrm{so}}\left(\kappa\gamma_{0}-i\alpha\gamma_{1}\gamma_{2}\right),$
(1)

where the wave vector $\bm{k}=(k_{x},k_{y})$, the SOC gap
$\Delta_{\mathrm{so}}=0.042\mathrm{eV}$, and the Fermi velocities
$v_{1}=3.87\times 10^{5}m/s$, $v_{2}=0.46\times 10^{5}m/s$, $v_{-}=2.86\times
10^{5}m/s$, and $v_{+}=7.21\times 10^{5}m/s$. In addition, the $4\times 4$
unit matrix is given by $I=\tau_{0}\otimes\sigma_{0}$; the $4\times 4$ Dirac
matrices are defined as $\gamma_{0}=\tau_{1}\otimes\sigma_{1}$,
$\gamma_{1}=\tau_{2}\otimes\sigma_{0}$, and
$\gamma_{2}=\tau_{3}\otimes\sigma_{0}$, in which $\tau_{0}$ and $\tau_{i}$
stand for pseudospin space which indicates the conduction and valence band
degrees of freedom while $\sigma_{0}$ and $\sigma_{i}$ denote Pauli matrices
acting upon real-spin space. The ratio $\alpha$ is defined as
$\alpha=|E_{z}/E_{c}|$, where $E_{z}$ is the vertical electric field and
$E_{c}$ denotes its critical value. Hereafter, we set $\hbar=1$ for
simplicity.

A straightforward algebra leads to the eigenvalues of this Hamiltonian as

$\displaystyle\varepsilon_{\kappa s}^{\xi}(k_{x},k_{y})$
$\displaystyle=-\kappa v_{-}k_{y}+\xi\mathcal{Z}_{\kappa s}(k_{x},k_{y}),$ (2)

where

$\displaystyle\mathcal{Z}_{\kappa s}(k_{x},k_{y})$
$\displaystyle=\sqrt{\left[A_{\kappa
s}(k_{y})\right]^{2}+\left[B(k_{x})\right]^{2}+\left[C_{\kappa}(k_{y})\right]^{2}},$
(3)

with

$\displaystyle A_{\kappa s}(k_{y})$ $\displaystyle=v_{2}k_{y}+(\kappa-\alpha
s)\Delta_{\mathrm{so}},$ (4) $\displaystyle B(k_{x})$
$\displaystyle=v_{1}k_{x},$ (5) $\displaystyle C_{\kappa}(k_{y})$
$\displaystyle=-\kappa v_{+}k_{y}.$ (6)

The indices $s=\pm$ and $\xi=\pm$ denote the spins (up and down) and bands
(conduction and valence), respectively. It is noted that the dispersions are
tilted hyperboloids and asymptotically tilted cones in the region of large
wave vector.

In the absence of vertical electric field ($\alpha=0$), the bands are spin-
degenerate. However, in the presence of vertical electric field ($\alpha>0$),
the spin-up band and spin-down band split at one valley, but reverse their
spins at the other valley, which leads to the valley-spin-polarized bands as
well as the valley-spin-polarized gaps. For convenience, we define the valley-
spin-polarized gaps as

$\displaystyle\Delta_{\kappa}^{s}(\alpha)=\mathcal{Z}_{\kappa
s}(0,0)=|\kappa-\alpha s|\Delta_{\mathrm{so}}.$ (7)

From these definitions, it is easy to find that the energy bands
$\varepsilon_{\kappa s}^{\xi}(k_{x},k_{y})$ and the gaps
$\Delta_{\kappa}^{s}(\alpha)$ do not change if we substitute
$(k_{y},\kappa,s)$ with $(-k_{y},-\kappa,-s)$. In addition, the dependencies
of valley-spin-polarized bands and valley-spin-polarized gaps on the vertical
electric field are explicitly shown in Fig.(1). Interestingly, at the
$\kappa=+$ valley, when the $E_{z}=E_{c}$ (or equivalently, $\alpha=1$), the
spin-up band become gapless and forms a Dirac point. Around this critical
electric field ($\alpha=1$), a topological phase transition occurs from the
topological insulator phase ($\alpha<1$) to the band insulator phase
($\alpha>1$). As the vertical electric field further increases ($\alpha>1$),
the gapless band reopens a gap.

Compared with other typical monolayer Dirac materials, 1T′-MoS2 exhibits
several attractive features in band structure. First, the valley-spin-
polarized behaviors are similar to that in silicene PRBStille2012  except for
a band tilting along the $k_{y}$ direction. Second, the tilted Dirac bands are
similar to that of 8-$Pmmn$ borophene PRBVerma2017  except for a finite SOC
gap. Furthermore, the band dispersion along the $k_{x}$ direction is upright,
similar to that in silicene PRBStille2012 . Third, the band structure of
1T′-MoS2 differs strongly from that of its polytypic structure 1H-MoS2
PRLMoS2010 ; PRLMoS2012 . These fascinating characteristics make 1T′-MoS2 a
very interesting monolayer Dirac material. In the following, we will
theoretically study the longitudinal optical conductivity in 1T′-MoS2.

Within the linear response theory, the longitudinal optical conductivity
$\sigma_{jj}(\omega,\alpha)$ at finite photon frequency $\omega$ is given by

$\displaystyle\sigma_{jj}(\omega)$
$\displaystyle=\sum_{\kappa=\pm}\sum_{s=\pm}\sigma_{jj}^{\kappa s}(\omega),$
(8)

where $\sigma_{jj}^{\kappa s}(\omega)$ denotes the longitudinal optical
conductivity at given valley $\kappa$ and spin $s$, whose explicit expression
can be found in Appendix A. It is noted that $\sigma_{jj}^{\kappa s}(\omega)$
does not vanish only when the bands contributed to the intraband/interband
transitions share the same spin index (see the Appendix A for details). To put
it equivalently, around the $\kappa$ valley, only when the photon energy
$\omega$ is the energy difference between the band $\xi$ and $\xi^{\prime}$ at
given wave vector $\bm{k}$ and spin index $s$,

$\displaystyle\Delta\varepsilon_{\kappa
s}^{\xi\xi^{\prime}}(k,\theta_{\bm{k}})$ $\displaystyle=\varepsilon_{\kappa
s}^{\xi^{\prime}}(k_{x},k_{y})-\varepsilon_{\kappa s}^{\xi}(k_{x},k_{y})$
$\displaystyle=(\xi^{\prime}-\xi)\mathcal{Z}_{\kappa
s}(k\cos\theta_{\bm{k}},k\sin\theta_{\bm{k}}),$ (9)

can an intraband/interband optical transition contribute to the optical
conductivity, where $k=|\bm{k}|$ and $\theta_{\bm{k}}=\arctan(k_{y}/k_{x})$.
It can also be proven that $\sigma_{jj}^{\kappa
s}(\omega)=\sigma_{jj}^{-\kappa-s}(\omega)$ (see Appendix B for details) such
that we are allowed to focus on either the $\kappa=+$ valley or the $\kappa=-$
valley. For convenience, we restrict our analysis to the $\kappa=+$ valley
hereafter.

After some tedious but straightforward algebra, we express the real part of
longitudinal optical conductivity as

$\displaystyle\mathrm{Re}\sigma_{jj}^{\kappa
s}(\omega)=\mathrm{Re}\sigma_{jj(\mathrm{intra})}^{\kappa
s}(\omega)+\mathrm{Re}\sigma_{jj(\mathrm{inter})}^{\kappa s}(\omega),$ (10)

where the intraband contribution and interband contribution are given
respectively as

$\displaystyle\mathrm{Re}\sigma_{jj(\mathrm{intra})}^{\kappa s}(\omega)$
$\displaystyle=\pi\int_{-\infty}^{+\infty}\frac{dk_{x}}{2\pi}\int_{-\infty}^{+\infty}\frac{dk_{y}}{2\pi}\Theta\left[\mu-\Delta_{\kappa}^{s}(\alpha)\right]\mathcal{F}_{jj;s,s}^{\kappa;+,+}(k_{x},k_{y})\left[-\frac{df\left[\varepsilon_{\kappa
s}^{+}(k_{x},k_{y})\right]}{d\varepsilon_{\kappa
s}^{+}(k_{x},k_{y})}\right]\delta(\omega)$
$\displaystyle+\pi\int_{-\infty}^{+\infty}\frac{dk_{x}}{2\pi}\int_{-\infty}^{+\infty}\frac{dk_{y}}{2\pi}\Theta\left[-\mu-\Delta_{\kappa}^{s}(\alpha)\right]\mathcal{F}_{jj;s,s}^{\kappa;-,-}(k_{x},k_{y})\left[-\frac{df\left[\varepsilon_{\kappa
s}^{-}(k_{x},k_{y})\right]}{d\varepsilon_{\kappa
s}^{-}(k_{x},k_{y})}\right]\delta(\omega),$ (11)
$\displaystyle\mathrm{Re}\sigma_{jj(\mathrm{inter})}^{\kappa s}(\omega)$
$\displaystyle=\pi\int_{-\infty}^{+\infty}\frac{dk_{x}}{2\pi}\int_{-\infty}^{+\infty}\frac{dk_{y}}{2\pi}\mathcal{F}_{jj;s,s}^{\kappa;-,+}(k_{x},k_{y})\frac{f[\varepsilon_{\kappa
s}^{-}(k_{x},k_{y})]-f[\varepsilon_{\kappa
s}^{+}(k_{x},k_{y})]}{\omega}\delta\left[\omega-2\mathcal{Z}_{\kappa
s}(k_{x},k_{y})\right],$ (12)

with $\Theta(x)$ the Heaviside step function, $\delta(x)$ the Dirac $\delta$
function, and $f(x)=\\{1+\exp{\left[(x-\mu)/k_{B}T\right]}\\}^{-1}$ the Fermi
distribution function in which $\mu$ denotes the chemical potential measured
with respect to the Dirac point, $k_{B}$ is the Boltzmann constant, and $T$
represents the temperature. It is emphasized that the physical behaviors of
optical conductivity are sensitive to the types of doping which are determined
by the criteria in terms of chemical potential $\mu$ and valley-spin-polarized
gaps $\Delta_{\kappa}^{s}(\alpha)$ as

$\displaystyle\begin{cases}$\emph{n}-doped for both $s=+$ and
$s=-$$,&\mathrm{Max}\\{\Delta_{\kappa}^{-}(\alpha),\Delta_{\kappa}^{+}(\alpha)\\}<\mu;\vspace{0.25cm}\\\
$\emph{n}-doped for $\kappa s=+$ but undoped for $\kappa
s=-$$,&\mathrm{Min}\\{\Delta_{\kappa}^{-}(\alpha),\Delta_{\kappa}^{+}(\alpha)\\}<\mu<\mathrm{Max}\\{\Delta_{\kappa}^{-}(\alpha),\Delta_{\kappa}^{+}(\alpha)\\};\vspace{0.25cm}\\\
$undoped for both $s=+$ and
$s=-$$,&\mathrm{Max}\\{-\Delta_{\kappa}^{-}(\alpha),-\Delta_{\kappa}^{+}(\alpha)\\}<\mu<\mathrm{Min}\\{\Delta_{\kappa}^{-}(\alpha),\Delta_{\kappa}^{+}(\alpha)\\};\vspace{0.25cm}\\\
$\emph{p}-doped for $\kappa s=+$ but undoped for $\kappa
s=-$$,&\mathrm{Min}\\{-\Delta_{\kappa}^{-}(\alpha),-\Delta_{\kappa}^{+}(\alpha)\\}<\mu<\mathrm{Max}\\{-\Delta_{\kappa}^{-}(\alpha),-\Delta_{\kappa}^{+}(\alpha)\\};\vspace{0.25cm}\\\
$\emph{p}-doped for both $s=+$ and
$s=-$$,&\mu<\mathrm{Min}\\{-\Delta_{\kappa}^{-}(\alpha),-\Delta_{\kappa}^{+}(\alpha)\\}.\end{cases}$
(13)

Especially, in the absence of electric field ($\alpha=0$), the energy bands
are spin-degenerate and the energy gaps
$\Delta_{\kappa}^{s}(0)=\Delta_{\mathrm{so}}$ are valley- and spin-
independent. It is undoped when
$-\Delta_{\mathrm{so}}<\mu<\Delta_{\mathrm{so}}$ where the valence bands are
fully occupied but the conduction bands are empty, $n$-doped when
$\mu>\Delta_{\mathrm{so}}$ where the conduction bands are partially filled
with electrons, and $p$-doped when $\mu<-\Delta_{\mathrm{so}}$ where the
valence bands are partially filled with electrons. It is further pointed out
that the zero-frequency part due to the intraband transitions contributes to
the Drude peak, whereas the frequency-dependent part is associated with the
interband transitions between a filled band to an empty band.

##  III Numerical results

In this section, we show the numerical results for the absorption peaks of
longitudinal conductivities in the undoped and doped 1T′-MoS2 when the
vertical electric field is absent and present, respectively. We set
$T=1\mathrm{K}$ throughout the numerical calculation in this paper.

###  III.1 In the absence of vertical electric field

Without applying the vertical electric field ($\alpha=0$), the energy bands of
1T′-MoS2 are fully gapped and spin-degenerate such that the total optical
conductivity
$\mathrm{Re}\sigma_{jj}(\omega)=g_{v}g_{s}\mathrm{Re}\sigma_{jj}^{+s}(\omega)$
with $g_{v}$ and $g_{s}$ the degeneracy factors of valley and spin. As shown
in Fig.(2), $\mathrm{Re}\sigma_{xx}(\omega)$ and
$\mathrm{Re}\sigma_{yy}(\omega)$ exhibit a strong anisotropy because the
dispersion is tilted along the $k_{y}$ direction but untilted along the
$k_{x}$ direction, similar to that in 8-$Pmmn$ borophene PRBVerma2017  but
unlike that in 1H-MoS2 PRBCarbotte2012 ; PRBCarvalho2013 ; PRBAsgari2014 .
When we set $v_{-}=v_{t}$, $v_{2}=0$, $v_{1}=v_{x}$, $v_{+}=v_{y}$, and
$\Delta_{\mathrm{so}}=0$, our results go back to that in 8-$Pmmn$ borophene
PRBVerma2017 , and when we set $v_{-}=v_{2}=0$ and $v_{+}=v_{1}=v_{F}$, our
results restore that in silicene PRBStille2012 .

Figure 2: (Color online) Total longitudinal optical conductivities of 1T′-MoS2
in the absence of vertical electric field. In (a) and (b), the black lines
represent the optical conductivity in the undoped case in which the absorption
peak is at $\omega=\Delta_{\mathrm{so}}$ while the red lines stand for the
optical conductivity in the doped case where two spin-degenerate absorption
peaks are at $\omega=\omega_{1}^{s}$ and $\omega=\omega_{2}^{s}$. Panel (c)
shows the schematic diagram of interband transition for the $\kappa=+$ valley,
where the colored arrows and black arrows indicate the boundary of electron
transition for $n$-doped case and undoped case, respectively. The blue curve
(for doped case) and black line (for undoped case) in the panel (d) show the
lower boundaries of incident photon frequency $\omega$ for arbitrary direction
of wavevector $\theta_{\bm{k}}$ by using Eq.(9). Note that
$\theta_{\bm{k}}=\pi/2$ and $\theta_{\bm{k}}=3\pi/2$ correspond to
$\omega_{2}^{s}$ and $\omega_{1}^{s}$, respectively.

Specifically, for the undoped case
($-\Delta_{\mathrm{so}}<\mu<\Delta_{\mathrm{so}}$), the optical conductivity
is contributed entirely by the interband transitions, which can be seen from
Fig.(2c). As a consequence, $\mathrm{Re}\sigma_{xx}(\omega)$ and
$\mathrm{Re}\sigma_{yy}(\omega)$ share the same peak that appears at the
absorption edge where $\omega=\Delta_{\mathrm{so}}$, as shown in Fig.(2a) and
Fig.(2b). Hence, from this peak in experimental measurement the SOC gap
$\Delta_{\mathrm{so}}$ can be precisely determined. From Fig.(2a) and
Fig.(2b), $\mathrm{Re}\sigma_{xx}(\omega)$ and
$\mathrm{Re}\sigma_{yy}(\omega)$ decay respectively to different asymptotic
background values

$\displaystyle\mathrm{Re}\sigma_{xx(\mathrm{asyp})}$
$\displaystyle=\frac{v_{1}}{\sqrt{v_{2}^{2}+v_{+}^{2}}}\sigma_{0},$ (14)
$\displaystyle\mathrm{Re}\sigma_{yy(\mathrm{asyp})}$
$\displaystyle=\frac{\sqrt{v_{2}^{2}+v_{+}^{2}}}{v_{1}}\sigma_{0},$ (15)

in the regime of large photon energy, where $\sigma_{0}=e^{2}/4\hbar$ (we
restore $\hbar$ for explicitness). It is noted that the different values
originate from the strong anisotropy of Fermi velocities $v_{1}$ and
$\sqrt{v_{2}^{2}+v_{+}^{2}}$ instead of the tilting parameterized by $v_{-}$,
and that the asymptotic values result from the asymptotic linearity of Dirac
band in the high-energy regime. These properties can be well understood by the
analytical results of the gapless model ($\Delta_{\mathrm{so}}=0$) in Appendix
C. In addition, these two asymptotic values satisfy a universal relation
$\mathrm{Re}\sigma_{xx(\mathrm{asyp})}\times\mathrm{Re}\sigma_{yy(\mathrm{asyp})}=\sigma_{0}^{2}$.
It is emphasized that this kind of universal relation holds not only for the
undoped case presented above but also for the doped case to be shown below
(see Appendix C for more details and explanations), which has also been
reported in other tilted Dirac systems, such as the tilted gapless Dirac bands
in 8-$Pmmn$ borophene PRBVerma2017 .

In the $n$-doped case ($\mu>\Delta_{\mathrm{so}}$), the optical conductivities
$\mathrm{Re}\sigma_{xx}(\omega)$ and $\mathrm{Re}\sigma_{yy}(\omega)$ exhibit
more interesting physics, as shown in Fig.(2a) and Fig.(2b). First, both
$\mathrm{Re}\sigma_{xx}(\omega)$ and $\mathrm{Re}\sigma_{yy}(\omega)$ possess
a Drude peak around $\omega=0$ due to the intraband transitions. Second, they
share two absorption peaks depending on the chemical potential at
$\omega=\omega_{1}^{s}$ and $\omega=\omega_{2}^{s}$ due to the band tilting
along the $k_{y}$ direction. Third, the Pauli blocking prevents the optical
transition between valence band and conduction band entirely when
$0<\omega<\omega_{1}^{s}$ but partially when
$\omega_{1}^{s}<\omega<\omega_{2}^{s}$. Fourth, when $\omega>\omega_{2}^{s}$,
$\mathrm{Re}\sigma_{xx}(\omega)$ and $\mathrm{Re}\sigma_{yy}(\omega)$ approach
respectively to their asymptotic background values
$\mathrm{Re}\sigma_{xx(\mathrm{asyp})}$ and
$\mathrm{Re}\sigma_{yy(\mathrm{asyp})}$, due to the linearity of Dirac band in
the high-energy regime. All these interesting behaviors can also be directly
read out from either Fig.(2c) or Fig.(2d). It is further remarked that for the
$p$-doped case ($\mu<-\Delta_{\mathrm{so}}$), there is no qualitative
difference in the behaviors of longitudinal optical conductivity presented
here.

###  III.2 In the presence of vertical electric field

When the vertical electric field is applied ($\alpha>0$), the valley-spin-
polarized bands and gaps of 1T′-MoS2 result in many interesting changes in the
longitudinal optical conductivity. As shown in Figs.(3-5),
$\mathrm{Re}\sigma_{xx}^{+s}(\omega)$ and
$\mathrm{Re}\sigma_{yy}^{+s}(\omega)$ are generally anisotropic and valley-
spin-polarized, which is strongly different from that in 1H-MoS2
PRBCarbotte2012 ; PRBCarvalho2013 ; PRBAsgari2014 . When we set
$v_{-}=v_{2}=0$ and $v_{+}=v_{1}=v_{F}$, our results restore that in silicene
PRBStille2012 .

Figure 3: (Color online) Valley-spin-polarized optical conductivities
$\mathrm{Re}\sigma_{xx}^{\kappa s}(\omega)$ and
$\mathrm{Re}\sigma_{yy}^{\kappa s}(\omega)$ in the undoped 1T′-MoS2 when the
vertical electric field is present. The arrows indicate the moving tendency of
peaks as the electric field smoothly increases. The chemical potential is set
to be $\mu=0$ which corresponds to the undoped case
($-\Delta_{\mathrm{so}}<\mu<\Delta_{\mathrm{so}}$) when $\alpha=0$. We take
the case for the $\kappa=+$ valley as a demonstration.

For $-\Delta_{\mathrm{so}}<\mu<\Delta_{\mathrm{so}}$ which corresponds to the
undoped case when $\alpha=0$, the vertical electric field splits the spin-
degenerate absorption edge $\omega=\Delta_{+}^{s}(0)=\Delta_{\mathrm{so}}$
into two spin-polarized absorption edges $\omega=\Delta_{+}^{+}(\alpha)$ and
$\omega=\Delta_{+}^{-}(\alpha)$ as shown in Fig.(3), where the dependence of
valley-spin-polarized gaps $\Delta_{\kappa}^{s}(\alpha)=|\kappa-\alpha
s|\Delta_{\mathrm{so}}$ on the electric field was previously shown in Fig.(1).
In the region of topological insulator phase ($0\leq\alpha<1$), two spin-
polarized peaks $\omega=\Delta_{+}^{+}(\alpha)$ and
$\omega=\Delta_{+}^{-}(\alpha)$ for the $\kappa=+$ valley move oppositely as
the electric field smoothly changes, which can be seen in Fig.(3a) and
Fig.(3c). However, in the region of band insulator phase ($\alpha>1$), two
spin-polarized peaks $\omega=\Delta_{+}^{+}(\alpha)$ and
$\omega=\Delta_{+}^{-}(\alpha)$ separated by $2\Delta_{\mathrm{so}}$ shift in
concert towards the higher frequency as shown in Fig.(3b) and Fig.(3d).
Interestingly, at the critical electric field ($\alpha=1$), the gapped band
remains the larger absorption edge at $\omega=\Delta_{+}^{+}(1)$ while the
gapless band vanishes as the smaller absorption edge at
$\omega=\Delta_{+}^{-}(1)=0$. Moreover, when $0<\omega<\Delta_{+}^{+}(1)$, the
value of the step represented by the black line is half of the asymptotic
value, indicating that only the gapless band contributes to the optical
transition, whereas when $\omega>\Delta_{+}^{+}(1)$, the gapped band also
contributes to the optical transition such that the optical conductivity
approaches the asymptotic values in the large-$\omega$ regime. The total
optical conductivity is obtained by summing over valley index $\kappa$ and
spin index $s$, leading to a superposition of four valley-spin-polarized
optical conductivities $\mathrm{Re}\sigma_{jj}^{\kappa s}$. The exotic
behaviors of absorption edges can be used to probe the topological phase
transition induced by the vertical electric field, which has also been
reported in other similar systems, such as the untilted gapped Dirac bands in
silicene PRBStille2012 .

Figure 4: (Color online) Dependence of total optical conductivities
$\mathrm{Re}\sigma_{xx}$ and $\mathrm{Re}\sigma_{yy}$ on the vertical electric
field in 1T′-MoS2. The chemical potential is set to be
$\mu=0.15~{}\mathrm{eV}$ which corresponds to the $n$-doped case
($\mu>\Delta_{\mathrm{so}}$) when $\alpha=0$.

For $\mu>\Delta_{\mathrm{so}}$ which corresponds to the $n$-doped case when
$\alpha=0$, the total optical conductivities exhibit many richer features
after the vertical electric field is applied. It can be found from Fig.(4a)
that for different vertical electric fields $\mathrm{Re}\sigma_{xx}(\omega)$
share the same asymptotic values in the regime of large photon energy and that
the absorption peaks change with the electric field, which also holds for
$\mathrm{Re}\sigma_{yy}(\omega)$ as shown in Fig.(4b). Specifically, we show
two characteristic cases as a demonstration in Fig.(5). From the upper panels
of Fig.(5), when $\alpha=2$, the Fermi level cuts both spin-up and spin-down
conduction bands, which indicates that both spin-polarized energy bands are
$n$-doped according to the criteria in (13). Therefore two spin-degenerate
peaks $\omega=\omega_{1}^{s}$ and $\omega=\omega_{2}^{s}$ split further into
four spin-polarized peaks $\omega=\omega_{1}^{+}$, $\omega=\omega_{1}^{-}$,
$\omega=\omega_{2}^{+}$, and $\omega=\omega_{2}^{-}$. However, from the lower
panels of Fig.(5), when $\alpha=3.5$, the Fermi level cuts the spin-up
conduction band but lies in the gap of spin-down bands, which indicates that
one kind of spin-polarized energy band is $n$-doped but the other kind of
spin-polarized energy band is undoped according to the criteria in (13). As a
consequence, $\omega=\omega_{1}^{s}$ split further into two spin-polarized
peaks $\omega=\omega_{1}^{+}$ and $\omega=\omega_{1}^{-}$, but
$\omega=\Delta_{+}^{-}(3.5)$ does not split and hence contributes only to one
peak. These lead to an interesting dependence of total optical conductivities
on the magnitude of the vertical electric field. Therefore, one can select
some specific frequencies of absorbed photons by tuning the vertical electric
field. And all of these behaviors can also be directly read out from
Figs.(5c), (5d), (5g), and (5h). It is further remarked that for
$\mu<-\Delta_{\mathrm{so}}$ which corresponds to the $p$-doped case when
$\alpha=0$, the behaviors of longitudinal optical conductivities do not differ
qualitatively from those presented here.

Figure 5: (Color online) Valley-spin-polarized longitudinal conductivities
$\mathrm{Re}\sigma_{xx}^{\kappa s}(\omega)$ and
$\mathrm{Re}\sigma_{yy}^{\kappa s}(\omega)$ in 1T′-MoS2 for two specific
values of vertical electric field. The chemical potential is set to be
$\mu=0.15~{}\mathrm{eV}$ which corresponds to the $n$-doped case
($\mu>\Delta_{\mathrm{so}}$) when $\alpha=0$. We take the case for the
$\kappa=+$ valley as a demonstration.

##  IV Summary and Conclusions

In this work, we have theoretically studied the anisotropic longitudinal
optical conductivities of tilted Dirac bands in both undoped and doped
1T′-MoS2, including effects of an external vertical electric field. In the
absence of vertical electric field, the tilted Dirac bands are spin-
degenerate. For the undoped 1T′-MoS2, the longitudinal conductivities are
contributed entirely by the interband transitions and share the same peak at
the absorption edge, from which the SOC gap can be directly extracted from
experimental measurements. The longitudinal optical conductivities decay
respectively to different asymptotic background values originated from the
anisotropy and the asymptotic linearity of Dirac band. For the doped 1T′-MoS2,
two longitudinal conductivities possess a Drude peak around $\omega=0$ due to
the intraband transitions and share two absorption peaks depending on the
chemical potential at $\omega=\omega_{1}^{s}$ and $\omega=\omega_{2}^{s}$ due
to the band tilting along the $k_{y}$ direction. The Pauli-blocking prevents
the optical transition between the valence band and conduction band entirely
when $0<\omega<\omega_{1}^{s}$ but partially when
$\omega_{1}^{s}<\omega<\omega_{2}^{s}$. When $\omega>\omega_{2}^{s}$, two
longitudinal conductivities approach respectively to their asymptotic
background values due to the linearity of Dirac band in the high-energy
regime. In general, for both undoped and doped 1T′-MoS2, the asymptotic
background values always satisfy a universal relation
$\mathrm{Re}\sigma_{xx(\mathrm{asyp})}\times\mathrm{Re}\sigma_{yy(\mathrm{asyp})}=\sigma_{0}^{2}$
with $\sigma_{0}=e^{2}/4\hbar$.

In the presence of vertical electric field, the tilted Dirac bands in 1T′-MoS2
are valley-spin-polarized due to the interplay between SOC gap and electric
field, and hence a topological phase transition occurs between the topological
insulator phase and the band insulator phase with a valley-spin-polarized
metal state at a critical value in between. For the undoped 1T′-MoS2, when the
vertical electric field increases smoothly, the interband absorption edge
splits into two absorption edges marked by two peaks of longitudinal optical
conductivities. These two peaks move oppositely in frequency in the region of
the topological insulator phase and shift in concert to a higher frequency in
the region of the band insulator phase. The exotic moving of optical
conductivity due to the vertical electric field can be taken as a fingerprint
of topological phase transition in the undoped 1T′ -MoS2. For the doped
1T′-MoS2, the absorption peaks of longitudinal optical conductivity change
with respect to the competition among the SOC gap, electric field, and
chemical potential. By manipulating the vertical electric field, one can
select the specific frequencies of absorbed photons.

In conclusion, the influence of SOC gap, band tilting, and external vertical
electric field on the anisotropic longitudinal optical conductivities of
tilted Dirac bands in 1T′-MoS2 was revealed. A theoretical scheme for probing
the topological phase transition in 1T′-MoS2 via exotic behaviors of
longitudinal optical conductivities was proposed. This work provided critical
insights into understanding the tilted Dirac bands of 1T′ TMDCs in general,
and their contributions to optical properties of 1T′-MoS2 in particular. The
results for 1T′-MoS2 are expected to be qualitatively valid for other
monolayer tilted gapped Dirac materials, such as $\alpha$-SnS2, TaCoTe2, and
TaIrTe4, due to the similarity in their band structures.

## ACKNOWLEDGEMENTS

We are grateful to Xiang-Hua Kong, Yuanpei Lan, and Zhen-Bing Dai for valuable
discussions. This work is partially supported by the National Natural Science
Foundation of China under Grants No.11547200 and No.11874273, the China
Scholarship Council (Grant No.201608515061), and the Key Project of Sichuan
Science and Technology Program (2019YFSY0044). H.G. and H.-R.C. were also
supported by the NSERC of Canada and the FQRNT of Quebec (H.G.). We thank the
High Performance Computing Centers at Sichuan Normal University, McGill
University, and Compute Canada.

##  Appendix A Explicit expression of the longitudinal optical conductivity

Within the linear response theory, the longitudinal optical conductivity at
the $\kappa$ valley can be expressed as

$\displaystyle\sigma_{jj}^{\kappa}(\omega)$
$\displaystyle=\frac{i}{\omega}\frac{1}{\beta_{T}}\sum_{\Omega_{m}}\int_{-\infty}^{+\infty}\frac{dk_{x}}{2\pi}\int_{-\infty}^{+\infty}\frac{dk_{y}}{2\pi}\mathrm{Tr}\left[\hat{J}_{j}^{\kappa}G_{\kappa}(k_{x},k_{y},i\Omega_{m})\hat{J}_{j}^{\kappa}G_{\kappa}(k_{x},k_{y},i\Omega_{m}+\omega+i\eta)\right],$
(16)

where $\beta_{T}=1/k_{B}T$, $j=x,y$ refer to spatial coordinates, $\eta$
denotes a positive infinitesimal. The charge current operators read

$\displaystyle\hat{J}_{x}^{\kappa}$ $\displaystyle=e\frac{\partial
H_{\kappa}(k_{x},k_{y})}{\partial k_{x}}=ev_{1}\gamma_{1},$ (17)
$\displaystyle\hat{J}_{y}^{\kappa}$ $\displaystyle=e\frac{\partial
H_{\kappa}(k_{x},k_{y})}{\partial k_{y}}=e\left(v_{2}\gamma_{0}-\kappa
v_{-}I-\kappa v_{+}\gamma_{2}\right),$ (18)

and the Matsubara Green’s function takes the form

$\displaystyle G_{\kappa}(k_{x},k_{y},i\Omega_{m})$
$\displaystyle=\left[i\Omega_{m}+\mu-
H_{\kappa}(k_{x},k_{y})\right]^{-1}=\frac{1}{4}\sum_{s,\xi=\pm}\frac{\mathcal{P}_{\kappa
s}^{\xi}(k_{x},k_{y})}{i\Omega_{m}+\mu-\varepsilon_{\kappa
s}^{\xi}(k_{x},k_{y})},$ (19)

where $\mu$ is the chemical potential, and

$\displaystyle\mathcal{P}_{\kappa s}^{\xi}(k_{x},k_{y})$
$\displaystyle=\left[\tau_{0}+\xi\frac{-sA_{\kappa
s}(k_{y})\tau_{1}+B(k_{x})\tau_{2}+C_{\kappa}(k_{y})\tau_{3}}{\mathcal{Z}_{\kappa
s}(k_{x},k_{y})}\right]\otimes\left(\sigma_{0}-s\sigma_{1}\right),$ (20)
$\displaystyle\varepsilon_{\kappa s}^{\xi}(k_{x},k_{y})$
$\displaystyle=-\kappa v_{-}k_{y}+\xi\mathcal{Z}_{\kappa s}(k_{x},k_{y}),$
(21)

with

$\displaystyle A_{\kappa s}(k_{y})$ $\displaystyle=v_{2}k_{y}+(\kappa-\alpha
s)\Delta_{\mathrm{so}},$ (22) $\displaystyle B(k_{x})$
$\displaystyle=v_{1}k_{x},$ (23) $\displaystyle C_{\kappa}(k_{y})$
$\displaystyle=-\kappa v_{+}k_{y},$ (24) $\displaystyle\mathcal{Z}_{\kappa
s}(k_{x},k_{y})$ $\displaystyle=\sqrt{\left[A_{\kappa
s}(k_{y})\right]^{2}+\left[B(k_{x})\right]^{2}+\left[C_{\kappa}(k_{y})\right]^{2}}.$
(25)

After summing over Matsubara frequency $\Omega_{m}$, we express the
longitudinal optical conductivity at the $\kappa$ valley as

$\displaystyle\sigma_{jj}^{\kappa}(\omega)$
$\displaystyle=\frac{i}{\omega}\int_{-\infty}^{+\infty}\frac{dk_{x}}{2\pi}\int_{-\infty}^{+\infty}\frac{dk_{y}}{2\pi}\sum_{s,s^{\prime}=\pm}\sum_{\xi,\xi^{\prime}=\pm}\mathcal{F}_{jj;s,s^{\prime}}^{\kappa;\xi,\xi^{\prime}}(k_{x},k_{y})\mathcal{M}_{s,s^{\prime}}^{\kappa;\xi,\xi^{\prime}}(k_{x},k_{y},\omega),$
(26)

where

$\displaystyle\mathcal{F}_{jj;s,s^{\prime}}^{\kappa;\xi,\xi^{\prime}}(k_{x},k_{y})=\frac{\mathrm{Tr}[\hat{J}_{j}^{\kappa}\mathcal{P}_{\kappa
s}^{\xi}(k_{x},k_{y})\hat{J}_{j}^{\kappa}\mathcal{P}_{\kappa
s^{\prime}}^{\xi^{\prime}}(k_{x},k_{y})]}{16},$ (27)

and

$\displaystyle\mathcal{M}_{s,s^{\prime}}^{\kappa;\xi,\xi^{\prime}}(k_{x},k_{y},\omega)$
$\displaystyle=\frac{f[\varepsilon_{\kappa
s}^{\xi}(k_{x},k_{y})]-f[\varepsilon_{\kappa
s^{\prime}}^{\xi^{\prime}}(k_{x},k_{y})]}{\omega+\varepsilon_{\kappa
s}^{\xi}(k_{x},k_{y})-\varepsilon_{\kappa
s^{\prime}}^{\xi^{\prime}}(k_{x},k_{y})+i\eta},$ (28)

with $f(x)=1/\left\\{1+\exp[(x-\mu)/(k_{B}T)]\right\\}$ denoting the Fermi
distribution function.

Specifically, the explicit expressions of
$\mathcal{F}_{jj;s,s^{\prime}}^{\kappa;\xi,\xi^{\prime}}(k_{x},k_{y})$ are
given as

$\displaystyle\mathcal{F}_{xx;s,s^{\prime}}^{\kappa;\xi,\xi^{\prime}}(k_{x},k_{y})$
$\displaystyle=\frac{\mathrm{Tr}[\hat{J}_{x}^{\kappa}\mathcal{P}_{\kappa
s}^{\xi}(k_{x},k_{y})\hat{J}_{x}^{\kappa}\mathcal{P}_{\kappa
s^{\prime}}^{\xi^{\prime}}(k_{x},k_{y})]}{16}=\frac{e^{2}v_{1}^{2}}{2}\delta_{ss^{\prime}}\left\\{1-\xi\xi^{\prime}\frac{\left[A_{\kappa
s}(k_{y})\right]^{2}-\left[B(k_{x})\right]^{2}+\left[C_{\kappa}(k_{y})\right]^{2}}{\left[\mathcal{Z}_{\kappa
s}(k_{x},k_{y})\right]^{2}}\right\\},$ (29)

and

$\displaystyle\mathcal{F}_{yy;s,s^{\prime}}^{\kappa;\xi,\xi^{\prime}}(k_{x},k_{y})$
$\displaystyle=\frac{\mathrm{Tr}[\hat{J}_{y}^{\kappa}\mathcal{P}_{\kappa
s}^{\xi}(k_{x},k_{y})\hat{J}_{y}^{\kappa}\mathcal{P}_{\kappa
s^{\prime}}^{\xi^{\prime}}(k_{x},k_{y})]}{16}$
$\displaystyle=\frac{e^{2}}{2}\delta_{ss^{\prime}}\left\\{\left(v_{2}^{2}+v_{-}^{2}+v_{+}^{2}\right)-\xi\xi^{\prime}\kappa
v_{2}v_{+}\frac{4A_{\kappa
s}(k_{y})C_{\kappa}(k_{y})}{\left[\mathcal{Z}_{\kappa
s}(k_{x},k_{y})\right]^{2}}+2(\xi+\xi^{\prime})v_{-}\frac{v_{+}C_{\kappa}(k_{y})-\kappa
v_{2}A_{\kappa s}(k_{y})}{\mathcal{Z}_{\kappa s}(k_{x},k_{y})}\right.$
$\displaystyle\left.\hskip
14.22636pt+\xi\xi^{\prime}\frac{\left(v_{2}^{2}+v_{-}^{2}-v_{+}^{2}\right)\left[A_{\kappa
s}(k_{y})\right]^{2}-\left(v_{2}^{2}-v_{-}^{2}+v_{+}^{2}\right)\left[B(k_{x})\right]^{2}-\left(v_{2}^{2}-v_{-}^{2}-v_{+}^{2}\right)\left[C_{\kappa}(k_{y})\right]^{2}}{\left[\mathcal{Z}_{\kappa
s}(k_{x},k_{y})\right]^{2}}\right\\}.$ (30)

It is noted that $\delta_{ss^{\prime}}$ in Eqs. (29) and (30) implies that
both $\mathcal{F}_{xx;s,s^{\prime}}^{\kappa;\xi,\xi^{\prime}}(k_{x},k_{y})$
and $\mathcal{F}_{yy;s,s^{\prime}}^{\kappa;\xi,\xi^{\prime}}(k_{x},k_{y})$
vanish when $s^{\prime}\neq s$, which indicates that the bands contributed to
the intraband/interband transitions must share the same spin index.

Therefore, the longitudinal optical conductivity at the $\kappa$ valley can be
expressed as

$\displaystyle\sigma_{jj}^{\kappa}(\omega)$
$\displaystyle=\sum_{s=\pm}\sigma_{jj}^{\kappa s}(\omega),$ (31)

where

$\displaystyle\sigma_{jj}^{\kappa s}(\omega)$
$\displaystyle=\frac{i}{\omega}\int_{-\infty}^{+\infty}\frac{dk_{x}}{2\pi}\int_{-\infty}^{+\infty}\frac{dk_{y}}{2\pi}\sum_{\xi,\xi^{\prime}=\pm}\mathcal{F}_{jj;s,s}^{\kappa;\xi,\xi^{\prime}}(k_{x},k_{y})\mathcal{M}_{s,s}^{\kappa;\xi,\xi^{\prime}}(k_{x},k_{y},\omega)$
(32)

denotes the valley-spin-polarized complex longitudinal conductivity at finite
photon frequency $\omega$.

##  Appendix B Relation between the valley-spin-polarized longitudinal
conductivities with opposite valley and spin

From the definitions in Eq.(21) and Eqs.(22-25), we have the following
relations

$\displaystyle A_{-\kappa-s}(-k_{y})$ $\displaystyle=-A_{\kappa s}(k_{y}),$
(33) $\displaystyle B(k_{x})$ $\displaystyle=B(k_{x}),$ (34) $\displaystyle
C_{-\kappa}(-k_{y})$ $\displaystyle=C_{\kappa}(k_{y}),$ (35)
$\displaystyle\mathcal{Z}_{-\kappa-s}(k_{x},-k_{y})$
$\displaystyle=\mathcal{Z}_{\kappa s}(k_{x},k_{y}),$ (36)
$\displaystyle\varepsilon_{-\kappa-s}^{\xi}(k_{x},-k_{y})$
$\displaystyle=\varepsilon_{\kappa s}^{\xi}(k_{x},k_{y}).$ (37)

Substituting these relations into Eqs.(28-30), we consequently get

$\displaystyle\mathcal{M}_{-s,-s}^{-\kappa;\xi\xi^{\prime}}(k_{x},-k_{y},\omega)$
$\displaystyle=\mathcal{M}_{s,s}^{\kappa;\xi\xi^{\prime}}(k_{x},k_{y},\omega),$
(38)
$\displaystyle\mathcal{F}_{xx;-s,-s}^{-\kappa;\xi\xi^{\prime}}(k_{x},-k_{y})$
$\displaystyle=\mathcal{F}_{xx;s,s}^{\kappa;\xi\xi^{\prime}}(k_{x},k_{y}),$
(39)
$\displaystyle\mathcal{F}_{yy;-s,-s}^{-\kappa;\xi\xi^{\prime}}(k_{x},-k_{y})$
$\displaystyle=\mathcal{F}_{yy;s,s}^{\kappa;\xi\xi^{\prime}}(k_{x},k_{y}).$
(40)

After substituting these relations into Eq.(32), we obtain the relation

$\displaystyle\sigma_{jj}^{-\kappa-s}(\omega)=\frac{i}{\omega}\int_{-\infty}^{+\infty}\frac{dk_{x}}{2\pi}\int_{-\infty}^{+\infty}\frac{dk_{y}}{2\pi}\sum_{\xi,\xi^{\prime}=\pm}\mathcal{F}_{jj;-s,-s}^{-\kappa;\xi\xi^{\prime}}(k_{x},-k_{y})\mathcal{M}_{-s,-s}^{-\kappa;\xi\xi^{\prime}}(k_{x},-k_{y},\omega)=\sigma_{jj}^{\kappa
s}(\omega).$ (41)

##  Appendix C Analytical results in the gapless model and asymptotic
background values

The asymptotic values of longitudinal optical conductivities denote the values
contributed completely by the interband transition in the high-energy regime
where the energy of incident photon $\omega$ is asymptotically infinite, which
is equivalent to that the energy of electron $\varepsilon_{\kappa
s}^{\xi}(k_{x},k_{y})$ is asymptotically infinite from the definition
$\omega=\varepsilon_{\kappa s}^{\xi^{\prime}}(k_{x},k_{y})-\varepsilon_{\kappa
s}^{\xi}(k_{x},k_{y})$. In this regime, the influence of the band gap can be
neglected such that the gapped model Hamiltonian reduces to its gapless
counterpart whose eigenvalues are given as

$\displaystyle\varepsilon_{\kappa s}^{\xi}(k_{x},k_{y})$
$\displaystyle=-\kappa
v_{t}k_{y}+\xi\sqrt{v_{x}^{2}k_{x}^{2}+v_{y}^{2}k_{y}^{2}},$ (42)

where $v_{x}=v_{1}$, $v_{y}=\sqrt{v_{2}^{2}+v_{+}^{2}}$, and $v_{t}=v_{-}$. It
is noted that they are the same as the energy bands of 8-$Pmmn$ borophene. In
the gapless model, the real part of the longitudinal optical conductivity is
given by

$\displaystyle\mathrm{Re}\sigma_{jj}(\omega)$
$\displaystyle=\sum_{\kappa=\pm}\mathrm{Re}\sigma_{jj}^{\kappa}(\omega)=\sum_{\kappa=\pm}\sum_{s=\pm}\mathrm{Re}\sigma_{jj}^{\kappa
s}(\omega).$ (43)

To make the following calculation more general, instead of focusing on the
asymptotic background values at $\omega\to\infty$, we proceed with the
longitudinal optical conductivity for arbitrary positive $\omega$, which is
contributed completely by the interband transition. The real part of the
interband longitudinal optical conductivity for valley $\kappa$ and spin $s$
reads

$\displaystyle\mathrm{Re}\sigma_{jj(\mathrm{inter})}^{\kappa s}(\omega)$
$\displaystyle=\pi\int_{-\infty}^{+\infty}\frac{dk_{x}}{2\pi}\int_{-\infty}^{+\infty}\frac{dk_{y}}{2\pi}\sum_{s^{\prime}=\pm}\mathcal{F}_{jj;ss^{\prime}}^{\kappa;-,+}(k_{x},k_{y})\frac{f[\varepsilon_{\kappa
s}^{-}(k_{x},k_{y})]-f[\varepsilon_{\kappa
s^{\prime}}^{+}(k_{x},k_{y})]}{\omega}\delta\left[\omega-2\sqrt{v_{x}^{2}k_{x}^{2}+v_{y}^{2}k_{y}^{2}}\right],$
(44)

where $\mathcal{F}_{xx;ss^{\prime}}^{\kappa;-,+}(k_{x},k_{y})$ and
$\mathcal{F}_{yy;ss^{\prime}}^{\kappa;-,+}(k_{x},k_{y})$ are given as

$\displaystyle\mathcal{F}_{xx;ss^{\prime}}^{\kappa;-,+}(k_{x},k_{y})$
$\displaystyle=4\sigma_{0}\frac{v_{x}^{2}v_{y}^{2}k_{y}^{2}}{v_{x}^{2}k_{x}^{2}+v_{y}^{2}k_{y}^{2}}\delta_{ss^{\prime}},$
(45) $\displaystyle\mathcal{F}_{yy;ss^{\prime}}^{\kappa;-,+}(k_{x},k_{y})$
$\displaystyle=4\sigma_{0}\frac{v_{y}^{2}v_{x}^{2}k_{x}^{2}}{v_{x}^{2}k_{x}^{2}+v_{y}^{2}k_{y}^{2}}\delta_{ss^{\prime}}.$
(46)

After introducing $\tilde{k}_{x}=v_{x}k_{x}$, $\tilde{k}_{y}=v_{y}k_{y}$,
$\tilde{k}=|\tilde{\bm{k}}|=\sqrt{\tilde{k}_{x}^{2}+\tilde{k}_{y}^{2}}$,
$\phi=\arctan(\tilde{k}_{y}/\tilde{k}_{x})$, and $0\leq\beta=v_{t}/v_{y}<1$,
one obtains

$\displaystyle\mathrm{Re}\sigma_{jj(\mathrm{inter})}^{\kappa s}(\omega)$
$\displaystyle=\int\frac{d\tilde{k}_{x}d\tilde{k}_{y}}{4\pi
v_{x}v_{y}}\sum_{s^{\prime}=\pm}\tilde{\mathcal{F}}_{jj;ss^{\prime}}^{\kappa;-,+}(\tilde{k}_{x},\tilde{k}_{y})\frac{f[\tilde{\varepsilon}_{\kappa
s}^{-}(\tilde{k}_{x},\tilde{k}_{y})]-f[\tilde{\varepsilon}_{\kappa
s^{\prime}}^{+}(\tilde{k}_{x},\tilde{k}_{y})]}{\omega}\delta\left[\omega-2\tilde{k}\right],$
(47)

where

$\displaystyle\tilde{\mathcal{F}}_{xx;ss^{\prime}}^{\kappa;-,+}(\tilde{k}_{x},\tilde{k}_{y})$
$\displaystyle=\mathcal{F}_{xx;ss^{\prime}}^{\kappa;-,+}(k_{x},k_{y})=4\sigma_{0}\frac{v_{x}^{2}\tilde{k}_{y}^{2}}{\tilde{k}^{2}}\delta_{ss^{\prime}},$
(48)
$\displaystyle\tilde{\mathcal{F}}_{yy;ss^{\prime}}^{\kappa;-,+}(\tilde{k}_{x},\tilde{k}_{y})$
$\displaystyle=\mathcal{F}_{yy;ss^{\prime}}^{\kappa;-,+}(k_{x},k_{y})=4\sigma_{0}\frac{v_{y}^{2}\tilde{k}_{x}^{2}}{\tilde{k}^{2}}\delta_{ss^{\prime}},$
(49) $\displaystyle\tilde{\varepsilon}_{\kappa
s}^{\pm}(\tilde{k}_{x},\tilde{k}_{y})$
$\displaystyle=\varepsilon_{\kappa}^{\pm}(k_{x},k_{y})=-\kappa\beta\tilde{k}_{y}\pm\tilde{k}=(-\kappa\beta\sin\phi\pm
1)\tilde{k}.$ (50)

Summing over $s^{\prime}$ and transforming to the polar coordinate, we have

$\displaystyle\mathrm{Re}\sigma_{xx(\mathrm{inter})}^{\kappa s}(\omega)$
$\displaystyle=\sigma_{0}\frac{v_{x}}{v_{y}}\int_{0}^{+\infty}\tilde{k}d\tilde{k}\int_{0}^{2\pi}\frac{\sin^{2}\phi
d\phi}{\pi}\frac{f[\tilde{\varepsilon}_{\kappa
s}^{-}(\tilde{k}\cos\phi,\tilde{k}\sin\phi)]-f[\tilde{\varepsilon}_{\kappa
s}^{+}(\tilde{k}\cos\phi,\tilde{k}\sin\phi)]}{\omega}\delta\left[\omega-2\tilde{k}\right],$
(51)

and

$\displaystyle\mathrm{Re}\sigma_{yy(\mathrm{inter})}^{\kappa s}(\omega)$
$\displaystyle=\sigma_{0}\frac{v_{y}}{v_{x}}\int_{0}^{+\infty}\tilde{k}d\tilde{k}\int_{0}^{2\pi}\frac{\cos^{2}\phi
d\phi}{\pi}\frac{f[\tilde{\varepsilon}_{\kappa
s}^{-}(\tilde{k}\cos\phi,\tilde{k}\sin\phi)]-f[\tilde{\varepsilon}_{\kappa
s}^{+}(\tilde{k}\cos\phi,\tilde{k}\sin\phi)]}{\omega}\delta\left[\omega-2\tilde{k}\right].$
(52)

Integrating over $\tilde{k}$ leads us to

$\displaystyle\mathrm{Re}\sigma_{xx(\mathrm{inter})}^{\kappa s}(\omega)$
$\displaystyle=\frac{\sigma_{0}}{4}\frac{v_{x}}{v_{y}}\int_{0}^{2\pi}\frac{\sin^{2}\phi
d\phi}{\pi}\left\\{f\left[(-\kappa\beta\sin\phi-1)\frac{\omega}{2}\right]-f\left[(-\kappa\beta\sin\phi+1)\frac{\omega}{2}\right]\right\\}$
$\displaystyle=\frac{\sigma_{0}}{4}\frac{v_{x}}{v_{y}}\int_{0}^{\pi}\frac{\sin^{2}\phi
d\phi}{\pi}\left\\{f\left[(-\kappa\beta\sin\phi-1)\frac{\omega}{2}\right]-f\left[(-\kappa\beta\sin\phi+1)\frac{\omega}{2}\right]\right.$
$\displaystyle\hskip
92.47145pt\left.+f\left[(\kappa\beta\sin\phi-1)\frac{\omega}{2}\right]-f\left[(\kappa\beta\sin\phi+1)\frac{\omega}{2}\right]\right\\}$
(53)

and hence

$\displaystyle\mathrm{Re}\sigma_{xx(\mathrm{inter})}(\omega)$
$\displaystyle=g_{s}\left[\mathrm{Re}\sigma_{xx(\mathrm{inter})}^{+}(\omega)+\mathrm{Re}\sigma_{xx(\mathrm{inter})}^{-}(\omega)\right]$
$\displaystyle=\sigma_{0}\frac{v_{x}}{v_{y}}\int_{0}^{\pi}\frac{\sin^{2}\phi
d\phi}{\pi}\left\\{f\left[(-\beta\sin\phi-1)\frac{\omega}{2}\right]+f\left[(\beta\sin\phi-1)\frac{\omega}{2}\right]\right.$
$\displaystyle\hskip
91.04872pt\left.-f\left[(-\beta\sin\phi+1)\frac{\omega}{2}\right]-f\left[(\beta\sin\phi+1)\frac{\omega}{2}\right]\right\\},$
(54)

where $g_{s}=2$ is the degeneracy factor of spin.

Parallel procedures give rise to

$\displaystyle\mathrm{Re}\sigma_{yy(\mathrm{inter})}(\omega)$
$\displaystyle=g_{s}\left[\mathrm{Re}\sigma_{yy(\mathrm{inter})}^{+}(\omega)+\mathrm{Re}\sigma_{yy(\mathrm{inter})}^{-}(\omega)\right]$
$\displaystyle=\sigma_{0}\frac{v_{y}}{v_{x}}\int_{0}^{\pi}\frac{\cos^{2}\phi
d\phi}{\pi}\left\\{f\left[(-\beta\sin\phi-1)\frac{\omega}{2}\right]+f\left[(\beta\sin\phi-1)\frac{\omega}{2}\right]\right.$
$\displaystyle\hskip
91.04872pt\left.-f\left[(-\beta\sin\phi+1)\frac{\omega}{2}\right]-f\left[(\beta\sin\phi+1)\frac{\omega}{2}\right]\right\\}.$
(55)

In order to obtain the analytical expressions, we perform the integrations
over $\phi$ at zero temperature where the Fermi distribution function $f(x)$
can be replaced by $\Theta[\mu-x]$. At zero temperature, we have

$\displaystyle\mathrm{Re}\sigma_{xx(\mathrm{inter})}(\omega)$
$\displaystyle=\sigma_{0}\frac{v_{x}}{v_{y}}\Gamma_{\beta}(\omega),$ (56)

where

$\displaystyle\Gamma_{\beta}(\omega)$
$\displaystyle=\Theta\left(\omega\right)\int_{0}^{\pi}\frac{\sin^{2}\phi
d\phi}{\pi}\left\\{\Theta\left[\mu+(\beta\sin\phi+1)\frac{\omega}{2}\right]+\Theta\left[\mu-(\beta\sin\phi-1)\frac{\omega}{2}\right]\right.$
$\displaystyle\left.\hskip
91.04872pt-\Theta\left[\mu+(\beta\sin\phi-1)\frac{\omega}{2}\right]-\Theta\left[\mu-(\beta\sin\phi+1)\frac{\omega}{2}\right]\right\\}.$
(57)

When $\beta=0$, we have

$\displaystyle\Gamma_{0}(\omega)$
$\displaystyle=\Theta\left(\omega\right)\int_{0}^{\pi}\frac{2\sin^{2}\phi}{\pi}d\phi\left[\Theta\left(\mu+\frac{\omega}{2}\right)-\Theta\left(\mu-\frac{\omega}{2}\right)\right]=\begin{cases}0,&\hskip
28.45274pt0<\omega<2|\mu|;\\\ 1,&\hskip 28.45274pt\omega\geq
2|\mu|.\end{cases}$ (58)

When $0<\beta<1$, we have

$\displaystyle\Gamma_{\beta}(\omega)$
$\displaystyle=\Theta\left(\beta\right)\Theta\left(1-\beta\right)\Theta\left(\omega\right)\int_{0}^{\frac{\pi}{2}}\frac{2\sin^{2}\phi\cos\phi
d\phi}{\pi\cos\phi}\left\\{\Theta\left[\mu+(\beta\sin\phi+1)\frac{\omega}{2}\right]+\Theta\left[\mu-(\beta\sin\phi-1)\frac{\omega}{2}\right]\right.$
$\displaystyle\left.\hskip
184.9429pt-\Theta\left[\mu+(\beta\sin\phi-1)\frac{\omega}{2}\right]-\Theta\left[\mu-(\beta\sin\phi+1)\frac{\omega}{2}\right]\right\\}$
$\displaystyle=\Theta\left(\beta\right)\Theta\left(1-\beta\right)\Theta\left(\omega\right)\int_{0}^{1}\frac{2x^{2}dx}{\pi\sqrt{1-x^{2}}}\left\\{\Theta\left[\mu+(\beta
x+1)\frac{\omega}{2}\right]+\Theta\left[\mu-(\beta
x-1)\frac{\omega}{2}\right]\right.$ $\displaystyle\left.\hskip
165.02606pt-\Theta\left[\mu+(\beta
x-1)\frac{\omega}{2}\right]-\Theta\left[\mu-(\beta
x+1)\frac{\omega}{2}\right]\right\\}$ $\displaystyle=\begin{cases}0,&\hskip
28.45274pt0<\omega<\frac{2|\mu|}{1+\beta};\\\
\frac{1}{2}-\frac{\chi\sqrt{1-\chi^{2}}-\arcsin\chi}{\pi},&\hskip
28.45274pt\frac{2|\mu|}{1+\beta}\leq\omega<\frac{2|\mu|}{1-\beta},\\\
1,&\hskip 28.45274pt\omega\geq\frac{2|\mu|}{1-\beta},\end{cases}$ (59)

where $\chi=\frac{\omega-2|\mu|}{\beta\omega}$. Therefore the analytical
expressions of $\mathrm{Re}\sigma_{xx(\mathrm{inter})}(\omega)$ at zero
temperature can be written in the untilted case ($\beta=0$) as

$\displaystyle\mathrm{Re}\sigma_{xx(\mathrm{inter})}(\omega)=\begin{cases}0,&\hskip
28.45274pt0<\omega<2|\mu|;\\\ \sigma_{0}\frac{v_{x}}{v_{y}},&\hskip
28.45274pt\omega\geq 2|\mu|,\end{cases}$ (60)

and in the tilted case ($0<\beta<1$) as

$\displaystyle\mathrm{Re}\sigma_{xx(\mathrm{inter})}(\omega)=\begin{cases}0,&\hskip
28.45274pt0<\omega<\frac{2|\mu|}{1+\beta};\\\
\sigma_{0}\frac{v_{x}}{v_{y}}\left[\frac{1}{2}-\frac{\chi\sqrt{1-\chi^{2}}-\arcsin\chi}{\pi}\right],&\hskip
28.45274pt\frac{2|\mu|}{1+\beta}\leq\omega<\frac{2|\mu|}{1-\beta},\\\
\sigma_{0}\frac{v_{x}}{v_{y}},&\hskip
28.45274pt\omega\geq\frac{2|\mu|}{1-\beta},\end{cases}$ (61)

where $\chi=\frac{\omega-2|\mu|}{\beta\omega}$.

Similarly, the analytical expressions of
$\mathrm{Re}\sigma_{yy(\mathrm{inter})}(\omega)$ at zero temperature can be
written in the untilted case ($\beta=0$) as

$\displaystyle\mathrm{Re}\sigma_{yy(\mathrm{inter})}(\omega)=\begin{cases}0,&\hskip
28.45274pt0<\omega<2|\mu|;\\\ \sigma_{0}\frac{v_{y}}{v_{x}},&\hskip
28.45274pt\omega\geq 2|\mu|,\end{cases}$ (62)

and in the tilted case ($0<\beta<1$) as

$\displaystyle\mathrm{Re}\sigma_{yy(\mathrm{inter})}(\omega)$
$\displaystyle=\begin{cases}0,&\hskip
28.45274pt0<\omega<\frac{2|\mu|}{1+\beta},\\\
\sigma_{0}\frac{v_{y}}{v_{x}}\left[\frac{1}{2}+\frac{\chi\sqrt{1-\chi^{2}}+\arcsin\chi}{\pi}\right],&\hskip
28.45274pt\frac{2|\mu|}{1+\beta}\leq\omega<\frac{2|\mu|}{1-\beta},\\\
\sigma_{0}\frac{v_{y}}{v_{x}},&\hskip
28.45274pt\omega\geq\frac{2|\mu|}{1-\beta},\end{cases}$ (63)

where $\chi=\frac{\omega-2|\mu|}{\beta\omega}$.

It is noted that two boundaries $\frac{2|\mu|}{1+\beta}$ and
$\frac{2|\mu|}{1-\beta}$ correspond to the position of peaks of optical
conductivities and are determined only by the tilt parameter
$\beta=v_{t}/v_{y}$ and the absolute value of chemical potential $|\mu|$,
indicating the tilt dependence and particle-hole symmetry in the gapless
model. In this sense, we analytically evaluate the real part of longitudinal
optical conductivities which agree exactly with the numerical results of
8-$Pmmn$ borophene PRBVerma2017  after setting $v_{-}=v_{t}$, $v_{2}=0$,
$v_{1}=v_{x}$, and $v_{+}=v_{y}$. Furthermore, in the untilted limit $\beta\to
0$, these two boundaries merge into one boundary $2|\mu|$, giving rise to the
result for the untilted case ($\beta=0$).

When $\omega>\frac{2|\mu|}{1-\beta}$ is satisfied, the real part of
longitudinal optical conductivities becomes constant, namely,

$\displaystyle\mathrm{Re}\sigma_{xx(\mathrm{asyp})}$
$\displaystyle=\frac{v_{x}}{v_{y}}\sigma_{0}=\frac{v_{1}}{\sqrt{v_{2}^{2}+v_{+}^{2}}}\sigma_{0},$
(64) $\displaystyle\mathrm{Re}\sigma_{yy(\mathrm{asyp})}$
$\displaystyle=\frac{v_{y}}{v_{x}}\sigma_{0}=\frac{\sqrt{v_{2}^{2}+v_{+}^{2}}}{v_{1}}\sigma_{0},$
(65)

which are nothing but the asymptotic background values. It is noted that they
are related only to the ratio between $v_{x}$ and $v_{y}$, irrelevant with the
tilt parameter $\beta=v_{t}/v_{y}$. In addition, they satisfy a universal
relation

$\displaystyle\mathrm{Re}\sigma_{xx(\mathrm{asyp})}\times\mathrm{Re}\sigma_{yy(\mathrm{asyp})}$
$\displaystyle=\frac{v_{x}}{v_{y}}\sigma_{0}\times\frac{v_{y}}{v_{x}}\sigma_{0}=\sigma_{0}^{2}.$
(66)

It is remarked that Eqs.(64-66) hold for arbitrary chemical potential such
that the $n$-doped ($\mu>0$), $p$-doped ($\mu<0$), and undoped ($\mu=0$) cases
share the same asymptotic background values. Furthermore, from the Fermi
distribution function
$f\left[\varepsilon_{\kappa}^{\xi}(k_{x},k_{y})\right]=1/\left\\{1+\exp\left[\left(\varepsilon_{\kappa}^{\xi}(k_{x},k_{y})-\mu\right)/(k_{B}T)\right]\right\\}$,
when $\varepsilon_{\kappa}^{\xi}(k_{x},k_{y})$ is much greater than chemical
potential $\mu$ and temperature $k_{B}T$, we are allowed to set $T=0$, which
indicates that finite temperature does not affect the asymptotic background
values. The result that asymptotic background values of longitudinal optical
conductivity are not affected by finite temperature was also found by Carbotte
in the numerical calculation of type-I and type-II Weyl semimetals
PRBCarbotte2016 . In summary, both the asymptotic values in Eqs.(64) and (65)
and the universal relation in Eq.(66) hold for arbitrary band gap, arbitrary
chemical potential, and arbitrary temperature since physically the band gap,
chemical potential, and temperature are relatively small compared to the
energy in the asymptotic regime.

As a further comparison, we adopt the notations
$\epsilon_{1}=\frac{2\mu}{1+\beta}$ and $\epsilon_{2}=\frac{2\mu}{1-\beta}$
PRBVerma2017  and can express the ratio $\beta=v_{t}/v_{y}$ as
$\mu\left(\frac{1}{\epsilon_{1}}-\frac{1}{\epsilon_{2}}\right)$ or
$\frac{\epsilon_{2}-\epsilon_{1}}{\epsilon_{2}+\epsilon_{1}}$. Note that the
relation $\beta=\mu\left(\frac{1}{\epsilon_{1}}-\frac{1}{\epsilon_{2}}\right)$
is nothing but the result given in Ref. PRBVerma2017 .

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# Optimal magnetization switching via spin-orbit torque on the surface of a
topological insulator

Grzegorz J. Kwiatkowski  Science Institute, University of Iceland, 107
Reykjavík, Iceland  Ivan P. Miranda  Department of Physics and Astronomy,
Uppsala University, SE-75120 Uppsala, Sweden  Department of Physics and
Electrical Engineering, Linnaeus University, SE-39231 Kalmar, Sweden  Cecilia
M. Holmqvist  Department of Physics and Electrical Engineering, Linnaeus
University, SE-39231 Kalmar, Sweden  Carlo M. Canali  Department of Physics
and Electrical Engineering, Linnaeus University, SE-39231 Kalmar, Sweden  Igor
S. Lobanov  Department of Physics, ITMO University, 197101 St. Petersburg,
Russia  Valery M. Uzdin  Department of Physics, ITMO University, 197101 St.
Petersburg, Russia  Andrei Manolescu  Department of Engineering, Reykjavík
University, 102 Reykjavík, Iceland  Pavel F. Bessarab<EMAIL_ADDRESS>Science Institute, University of Iceland, 107 Reykjavík, Iceland  Department
of Physics and Electrical Engineering, Linnaeus University, SE-39231 Kalmar,
Sweden  Sigurður I. Erlingsson  Department of Engineering, Reykjavík
University, 102 Reykjavík, Iceland

###### Abstract

An optimal protocol for the current-induced switching of a perpendicularly
magnetized nanoelement placed on the surface of a topological insulator is
presented. The time dependence of both in-plane components of the surface
current that induces the magnetization reversal via Dirac spin-orbit torque
with minimal Joule heating is derived analytically as a function of the
required switching time and material properties. It is demonstrated that a
particularly energy-efficient switching is realized for vanishing dampinglike
torque. The optimal reversal time providing a tradeoff between the switching
speed and energy efficiency is derived. The obtained switching protocol is
contrasted with the one realized in heavy-metal systems. Topological
insulators provide a highly tunable platform for the realization of energy-
efficient magnetization switching.

##  I Introduction

Spin-orbit torque (SOT)[1], a torque relying on the conversion of charge
current to spin via spin-orbit interaction, provides an efficient mechanism
for electrical control of magnetization [2, 3, 4, 1]. Magnetization switching
is one important application of the SOT [5, 6] as it can be assigned to
represent logical operations in nonvolatile information technologies. The
challenge is to minimize the energy cost of the switching. A SOT-induced
magnetization reversal can be realized by applying an in-plane current in a
heavy-metal (HM) layer on which a switchable ferromagnetic (FM) element is
placed. In this case, the SOT originates from the inverse spin galvanic effect
[7, 8, 9] and the spin Hall effect [10, 11, 12, 13, 14] in the HM, both giving
rise to fieldlike (FL) and dampinglike (DL) torques [1]. Conventional
switching protocols rely only on the DL SOT and involve a unidirectional
current [15, 16], but switching with much lower current densities can be
achieved by using both in-plane components of the current [17]. Analysis based
on the optimal control theory has previously revealed a theoretical limit for
the minimum energy cost of the SOT-induced magnetization reversal in the FM/HM
system, identified the corresponding optimal switching current pulse as a
function of the reversal time and relevant material properties, and uncovered
a sweet-spot ratio of the FL and DL torques which permits for a particularly
efficient switching by a down-chirped rotating current pulse [18].
Optimization of magnetization switching protocols is both fundamentally
interesting and technologically important as it enables identification of the
right combination of materials properties for energy-efficient applications.

Topological insulators (TIs) [19, 20, 21, 22], i.e., materials characterized
by an insulating bulk but conducting surface states, represent a promising
alternative to HM systems in the context of electrical control of
magnetization [23]. The spin-momentum locking of the Dirac electrons at the
surface of a TI results in a large charge-spin conversion, thereby enabling a
significant SOT on an adjacent FM [23]. Remarkably, such Dirac SOT has a
different geometrical form compared to the SOT found in HM systems [24],
affecting the magnetization dynamics. It is therefore interesting to explore
how much Dirac SOT-induced magnetization dynamics can be optimized and whether
using TI systems offers benefits for achieving energy-efficient magnetization
switching.

In this article, we present a complete analytical solution to the problem of
energy-efficient magnetization switching in a nanoelement with perpendicular
magnetic anisotropy (PMA) by means of Dirac SOT realized in TI systems. We
show that the optimal switching protocol, i.e. the protocol that minimizes the
energy cost associated with Joule heating, involves both in-plane components
of the surface current whose time dependence is determined by the materials
properties and the switching time. We obtain noteworthy exact dependencies
concerning the optimal switching and compare them with the results obtained
earlier for the HM systems [18]. We demonstrate that the best scenario for
magnetization switching is realized for vanishing dampinglike Dirac SOT.
Finally, we discuss various possibilities to tune the SOT coefficients to
achieve the most efficient switching. We conclude that TI systems offer a
convenient platform for energy-efficient magnetization control, providing
several advantages over other systems.

Figure 1:  Energy-efficient switching of a PMA nanoelement (cylinder) by a
Dirac spin-orbit torque induced by an optimal 2D electric current pulse
$\vec{j}_{m}$ flowing at the surface of a topological insulator substrate. The
calculated optimal control path for the switching for $\alpha=0.1$ and
$\xi_{D}=\xi_{F}$ and $T=50\tau_{0}$ is shown with the green line. The
direction of the normalized magnetic moment $\vec{s}$ of the element (optimal
current $\vec{j}_{m}$) is shown with the blue (red) arrow.

##  II Model

Figure 1 shows the simulated PMA nanoelement placed on the surface of a TI.
The magnetic moment of the nanoelement is reversed by a surface current via
Dirac SOT. The nanoelement is assumed small enough to be treated essentially
as a single-domain particle at any time of the reversal process. The energy
$E$ of the system is defined by the anisotropy along the $z$ axis which is
perpendicular to the surface of the TI (see Fig. 1),

$E=-Ks_{z}^{2},$ (1)

where $s_{z}$ is a $z$ component of the normalized magnetic moment $\vec{s}$
and $K>0$ is the anisotropy constant. The task is to identify the optimal
current pulse that reverses the magnetic moment from $s_{z}=1$ at $t=0$ to
$s_{z}=-1$ at $t=T$, with $T$ being the switching time. Both the amplitude and
the direction of the current $\vec{j}$ at the surface of the TI are allowed to
vary in time. The efficiency of the reversal is naturally defined by the
amount of Joule heating generated in the resistive circuit during the
switching process [25]. In particular, the optimal reversal is achieved when
the cost functional

$\Phi=\int_{0}^{T}|\vec{j}|^{2}dt,$ (2)

is minimized. This optimal control problem is subject to a constraint imposed
by the zero-temperature Landau-Lifshitz-Gilbert (LLG) equation describing the
dynamics of the magnetic moment induced by the Dirac SOT [24]:

$\displaystyle\dot{\vec{s}}=$ $\displaystyle-$
$\displaystyle\gamma\vec{s}\times\vec{b}+\alpha\vec{s}\times\dot{\vec{s}}$ (3)
$\displaystyle+$
$\displaystyle\gamma\xi_{F}\vec{s}\times(\vec{j}\times\vec{e}_{z})+\gamma\xi_{D}s_{z}(\vec{s}\times\vec{j}).$

Here, $\gamma$ is the gyromagnetic ratio, $\alpha$ is the damping parameter,
$\vec{e}_{z}$ is the unit vector along the $z$ axis, and $\vec{b}$ is the
anisotropy field: $\vec{b}\equiv-\mu^{-1}\partial E/\partial\vec{s}$, with
$\mu$ being the magnitude of the magnetic moment. The third and the fourth
terms on the rhs of Eq. (3) represent FL and DL components of the SOT,
respectively. The coefficients $\xi_{F}$ and $\xi_{D}$ can be written in terms
of microscopic system parameters [24, 1]. We will assume that both $\xi_{D}$
and $\xi_{F}$ are independent of $\vec{s}$, which is valid for
$J_{\mathrm{xc}}/\varepsilon_{F}\ll 1$ [24], where $J_{\mathrm{xc}}$ is the
exchange coupling and $\varepsilon_{F}$ is the Fermi energy, see Section III.3
for details.

In what follows, we will focus on how the ratio of the SOT coefficients
affects the switching cost. For this we introduce the following notation:

$\xi_{F}=\xi\cos\beta,\qquad\xi_{D}=\xi\sin\beta$ (4)

where $\xi=\sqrt{\xi_{F}^{2}+\xi_{D}^{2}}$ is the magnitude of spin-orbit
torque and $\beta$ is a parameter characterizing the proportion between its
components.

##  III Results

To find the optimal switching current $\vec{j}_{m}(t)$ that makes $\Phi$
minimum, we first express the energy cost in terms of the switching trajectory
and then minimize it so as to find the optimal control path (OCP)
$\vec{s}_{m}(t)$ for the switching process; after that, the optimal switching
pulse $\vec{j}_{m}(t)$ is derived from the OCPs. Similar procedure was applied
earlier in the context of magnetization switching induced by applied magnetic
field [26] and SOT in HM systems [18]. Here, we only sketch briefly the
derivations, but a complete analytical solution will be published elsewhere.

###  III.1 Optimal magnetization switching

We first notice that the direction $\psi$ of the current can be optimized
relative to the azimuthal angle $\varphi$ of the magnetic moment (see Fig.1)
before the OCP is actually known. This gives us:

$\psi=\varphi-\arctan\left[\frac{\cos\theta\left(\alpha\cos\beta+\sin\beta\right)}{\cos\beta-\alpha\cos^{2}\theta\sin\beta}\right],$
(5)

where $\theta$ is the polar angle of the magnetic moment $\vec{s}$. It follows
from Eqs. (3) and (5) that the amplitude of the current is given by the
following equation:

$j=\frac{2K}{\mu}\frac{\left(1+\alpha^{2}\right)\tau_{0}\dot{\theta}+\alpha\sin\theta\cos\theta}{2\xi\sqrt{(1+\alpha^{2}\cos^{2}\theta)(1-\sin^{2}\theta\sin^{2}\beta)}},$
(6)

where $\tau_{0}=\mu\left(2K\gamma\right)^{-1}$ defines the timescale. It is
seen from Eq. (6) that the current amplitude is the lowest for $\beta=0$ and
$\beta=\pi$ leading to most energy-efficient switching. On the other hand, for
$\beta=\pi/2$ and $\beta=3\pi/2$ the switching is impossible as the current
amplitude is infinite at the top of the energy barrier ($\theta=\pi/2$). This
is a consequence of the geometrical form of the Dirac SOT, for which the
dampinglike (DL) torque vanishes for an in-plane orientation of the magnetic
moment. It is also of note that the current amplitude does not depend on the
azimuthal angle $\phi$, so the cost functional $\Phi$ is only dependent on the
polar angle $\theta$, which makes the minimization problem straightforward.

We are interested in minimizing the cost functional $\Phi$ defined in Eq. (2)
with the boundary conditions of $\theta(0)=0$, $\theta(T)=\pi$. The solution
to the corresponding Euler-Lagrange equation for $\theta(t)$ is expressed in
terms of Jacobi elliptic functions [27]. Subsequently, current amplitude $j$
is obtained from Eq. (6). After that, $\varphi_{m}(t)$ is calculated via
direct integration of the equation of motion and $\psi_{m}(t)$ is obtained
using Eq. (5). Regardless of the system parameters, the OCP always crosses the
energy barrier $\theta=\pi/2$ at $t=T/2$ which is a result of a more general
symmetry
$\theta_{m}\left(T/2+t^{\prime}\right)=\pi-\theta_{m}\left(T/2-t^{\prime}\right)$,
$0\leq t^{\prime}\leq T/2$.

According to Eq. (5), the current rotates following the magnetic moment as it
precesses around the anisotropy axis, gradually changing its frequency.The
exact expressions for the frequency and amplitude are rather complex, but an
analytic expression for the average current can be obtained:

$\langle
j_{m}\rangle=\frac{2\left(1+\alpha^{2}\right)\mathcal{K}\left[\sin^{2}\beta+\alpha^{2}\cos^{2}\beta\right]}{\xi\gamma
T},$ (7)

where $\mathcal{K}[\ldots]$ is the complete elliptic integral of the first
kind. Noteworthy, the average current is independent of $K$, although
$\vec{j}_{m}(t)$ does depend on the strength of the anisotropy field.
Similarly to the results we obtained earlier for the average current [18] and
magnetic field [26], this is connected to the fact that the contribution of
relaxation due to internal magnetic field to the switching process cancels
out: it hinders the switching for $\theta<\pi/2$ but it helps the system reach
the final state for $\theta>\pi/2$. In that sense, it is instructive to
analyze both limiting cases of $K$. For a large anisotropy, the current pulse
tends to be large and localized in time, while in the weak anisotropy regime,
it becomes extended in time, characterized by an almost constant value. For
$\alpha=0$, Eq. (7) coincides exactly with the one obtained in [18] for the
Slonczewski-like SOT, while for non-zero damping the average optimal current
for the Dirac SOT is slightly larger.

Figure 2 shows the optimal switching current as a function of time for the
optimal ratio of the SOT coefficients ($\beta=0$) and various values of the
switching time $T$ and damping parameter $\alpha$.

Figure 2: (Color online) (a-c) Optimal current magnitude $j_{m}(t)$ as a
function of the time for several values of the damping parameter and switching
time, as indicated in the legend; (d) Magnitude and in-plane components of the
optimal switching current as functions of time for $T=10$ and $\alpha=0.4$. In
all cases, $\beta=0$ which corresponds to the lowest energy cost of switching.

###  III.2 Minimum energy cost of magnetization switching

Figure 3 shows the minimum energy cost as a function of inverse switching time
for various parameter values. As in the previously studied cases [26, 18],
$\Phi_{m}$ monotonically decreases with $T$ and demonstrates two distinct
asymptotic behaviours. For short switching times,
$T\ll\left(\alpha+1/\alpha\right)\tau_{0}$, the contribution from internal
dynamics becomes insignificant and the solution approaches that of a free
magnetic moment resulting in

$\Phi_{0}=\frac{4(1+\alpha^{2})^{2}\mathcal{K}^{2}[1-(1+\alpha^{2})\cos^{2}\beta]}{\xi^{2}\gamma^{2}T}$
(8)

On the other hand, for $T\rightarrow\infty$ the energy cost approaches a lower
limit of

$\Phi_{\infty}=\frac{4\alpha
K(1+\alpha^{2})\ln\left[(1+\alpha^{2})\cos^{2}(\beta)\right]}{\gamma\mu\xi^{2}\left[(1+\alpha^{2})\cos^{2}(\beta)-1\right]}$
(9)

The intersection of those asymptotics gives an optimal switching time
$T_{\textnormal{opt}}$ for which a balance between switching speed and energy
efficiency is obtained.

In a scenario where the system parameters (i.e., $\xi,K$) are the same, the
energy efficiencies of switching protocols based on the Dirac SOT and
Slonczewski-like SOT are found to be very close to each other. Physically,
this can be understood by the fact that the optimal switching relies mostly on
the FL torque in both cases and its form is the same for the Dirac and
Slonczewski-like SOT. Furthermore, typical values of the Gilbert damping
parameter for $3d$ transition metals, when in contact with TI substrates, are
greatly enhanced due to the strong spin-orbit coupling of the TI, but
generally predicted to be of the order of $10^{-1}$ or less [28]. For these
values of $\alpha$ and in the assumption of same model parameters the
difference in switching cost between the Dirac and Slonczewski-like SOTs is
less than $\sim 1\%$. At the same time, topological insulators have the
advantage of channeling the whole current through the surface instead of the
bulk increasing the efficiency of spin-orbit coupling.

$\Phi_{m}$ depends strongly on the $\beta$ parameter as shown in Fig. 4. As
predicted from Eq. (6), we obtain the lowest energy cost for $\beta=0,\pi$,
while for $\beta=\pi/2,3\pi/2$ the cost diverges to infinity indicating that
switching is not possible using just the spin-orbit torques (no-switching
regime). The main difference from the Slonczewski-like SOT case is that both
the optimal value of $\beta$ as well as the one that prevents switching to
occur do not depend on $\alpha$.

Figure 3: (Color online) Minimum energy cost $\Phi_{m}$ of the magnetization
switching induced by the Dirac-SOT as a function of $\frac{1}{T}$ for various
values of the damping parameter as indicated in the legend. Dotted and dashed
lines show the long and short switching time asymptotics, respectively. Inset:
comparison of $\Phi_{m}$ for the Dirac and Slonczewski-like spin-orbit
torques, considering $\alpha=0.4$ and same model parameters ($\xi,K$). In all
cases, $\beta$ is chosen to represent the lowest cost functional $\Phi_{m}$,
i.e., $\beta=0$ for the Dirac SOT and $\beta=-\arctan{\alpha}$ for the
Slonczewski-like SOT. Figure 4: (Color online) Minimum energy cost of
magnetization switching $\Phi_{m}$ as a function of $\beta$ for $T=10$ (black
lines) and $T\rightarrow\infty$ (blue lines), considering two different spin-
orbit torques: Dirac (solid lines), and Slonczewski-like (dashed lines), for
$\alpha=0.3$ and same model parameters ($\xi,K$). The red dotted line
represents the damping-like and field-like parameters ratio in which no
switching occurs in the Slonczewski-like spin-orbit torque.

###  III.3 Tunability of spin-orbit torque coefficients

Having identified the optimal ratio of the Dirac SOT coefficients, $\beta=0$,
we now link $\beta$ to the physical parameters of an FM/TI system and explore
their potential tunability.

The coefficients of the SOT acting on a magnet when due to current flowing at
the surface of a TI are given by [24]

$\xi_{F}=e\rho\frac{J_{xc}\tau\varepsilon_{F}}{\pi\hbar^{2}v_{F}\gamma},\,\textnormal{and
}\xi_{D}=-e\rho\frac{2J_{xc}^{2}}{\pi\hbar v_{F}\varepsilon_{F}\gamma},$ (10)

where $e>0$ is the elementary charge, $\hbar$ the reduced Planck constant,
$J_{xc}$ the exchange coupling in the TI, $\tau$ the relaxation time, and
$\varepsilon_{F}$ and $v_{F}$ are the Fermi energy and velocity, respectively.
Here we assume that $\vec{E}=\rho\vec{j}$, where $\rho$ is some relevant
resistivity. It is interesting to note that for a normal 2D semiconductor with
Rashba spin-orbit interactions in the limit $J_{xc}\ll\alpha_{R}k_{F}$, the
SOT coefficients take the same form as in Eq. (10) [29]. The parameter in
front of the two terms, field-like and damping-like, are of course different
but accounting for the differences in the density of states for the 2D Rashba
system and the 2D Dirac system – they can be matched perfectly.

The angle $\beta$ can be related to the microscopic model parameters via

$\tan(\beta)=\frac{-2J_{xc}\hbar}{\varepsilon_{F}^{2}\tau},$ (11)

and the magnitude of the total SOT is

$\xi=\left|\frac{J_{xc}}{\pi\hbar
v_{F}}\frac{e\rho}{\gamma}\right|\sqrt{\left(\frac{\varepsilon_{F}\tau}{\hbar}\right)^{2}+\left(\frac{2J_{xc}}{\varepsilon_{F}}\right)^{2}}.$
(12)

The angle $\beta$ does not explicitly depend on $v_{F}$, but $\xi$ is,
however, inversely proportional to $v_{F}$. By applying strain [30], or
external gate voltages [31] the value of the Fermi velocity can be modified,
and in some cases made asymmetric [30]. The torque coefficients in Eq. (10)
only contain surface contribution, relevant when the Fermi energy is in the
gap. For $\varepsilon_{F}$ closer to bulk bands additional terms appear due to
e.g. hexagonal warping [32, 33], and spin-transfer torque due to bulk states
[34, 35, 33], but these are beyond the scope of the present paper.

##  IV Conclusions

We investigated the energy-efficient switching process in a single macrospin
controlled by external currents when subject to the so-called Dirac SOT, that
is predicted to arise in ferromagnet/topological insulator systems [24]. With
this aim, we used the optimal control theory to study the energy limits of the
Dirac SOT-induced magnetization switching in a PMA nanoelement at zero applied
magnetic field, deriving the optimal time-dependent current pulse. We
established that the FL component of the Dirac SOT is the only responsible for
switching and the energy cost is the lowest for vanishing DL torque. This
represents a fundamental difference to the Slonczewski-like SOT [1, 18],
arising typically in FM/HM systems, in which the DL torque may compensate for
the relaxation and reduce the overall energy cost of the switching.

The obtained minimal switching cost is comparable to that obtained for
Slonczewski-like SOT. Results for respective types of SOTs coincide in the
undamped case and the difference between them increases with larger Gilbert
damping. For damping constant values expected in the PMA nanoelement on a TI
substrate, the energy cost difference is estimated to be less than $\approx
1\%$ under the assumption of the same model parameters and respective optimal
ratio of torque coefficients for both SOT models. This shows that magnetic
systems exhibiting Dirac SOT are a viable candidate for memory elements.

###### Acknowledgements.

The authors would like to thank T. Sigurjónsdóttir for helpful discussions and
useful comments. This work was supported by the Icelandic Research Fund (Grant
No. 217750 and No. 217813), the University of Iceland Research Fund (Grant No.
15673), the Faculty of Technology and the Department of Physics and Electrical
Engineering at Linnaeus University (Sweden), the Swedish Research Council
(Grants No. 2020-05110 and No. 2021-046229), the Russian Science Foundation
(Grant no. 23-72-10028), and Carl Tryggers Stiftelsen (CTS20:71) .

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