# Nonreciprocal phonon dichroism induced by Fermi pocket anisotropy in 2D
Dirac materials

Wen-Yu Shan<EMAIL_ADDRESS>Department of Physics, School of Physics and
Materials Science, Guangzhou University, Guangzhou 510006, China

(August 28, 2024)

###### Abstract

Electrons in two-dimensional (2D) Dirac materials carry local band geometric
quantities, such as Berry curvature and orbital magnetic moment, which,
combined with electron-phonon coupling, may affect phonon dynamics in unusual
way. Here, we propose intrinsic nonreciprocal linear and circular phonon
dichroism in magnetic 2D Dirac materials, which originate from nonlocal band
geometric quantities of electrons and reduce to pure Fermi-surface properties
for acoustic phonons. We find that to acquire the nonreciprocity, the Fermi
pocket anisotropy rather than the chirality of electrons is crucial. Two
possible mechanisms of Fermi pocket anisotropy are suggested: (i) trigonal
warping and out-of-plane magnetization or (ii) Rashba spin-orbit interaction
and in-plane magnetization. As a concrete example, we predict appreciable and
tunable nonreciprocal phonon dichroism in 2H-MoTe2 on EuO substrate. Our
finding points to a different route towards electrical control of phonon
nonreciprocity for acoustoelectronics applications based on 2D quantum
materials.

Introduction.—Electrons in two-dimensional (2D) Dirac materials, including
graphene Castro Neto  _et al._ (2009) and transition metal dichalcogenides
Xiao  _et al._ (2012), carry local band geometric quantities, such as Berry
curvature Berry (1984); Haldane (1988); Xiao  _et al._ (2010) and orbital
magnetic moment Xiao  _et al._ (2007); Yao  _et al._ (2008), as a result of
broken time-reversal or inversion symmetry. These geometric quantities have
recently been recognized to play pivotal roles in various electronic
phenomena. When further taking into account the electron-phonon (e-ph)
coupling, the geometric information of electrons can be inherited by phonons,
leading to unusual phononic behaviors, e.g., phonon Hall viscosity Barkeshli
_et al._ (2012), anomalous phonon effective charge Rinkel  _et al._ (2017);
Ren  _et al._ (2021), phonon magnetochiral effect Spivak and Andreev (2016);
Sengupta  _et al._ (2020) and phonon helicity Hu  _et al._ (2021).

On the other hand, phonon nonreciprocity, referring to the asymmetric phononic
behaviors upon reversing the propagation vector $\bm{q}$, are attractive for
phononics applications Li  _et al._ (2012). The physical origins of phonon or
surface-acoustic-wave nonreciprocity have been ascribed to Lorentz force Heil
_et al._ (1982); Liu  _et al._ (2020), asymmetric magnon-phonon coupling or
magnon dispersion Emtage (1976); Sasaki  _et al._ (2017); Nomura  _et al._
(2019); Xu  _et al._ (2020); Shan  _et al._ (2020); Hernández-Mínguez  _et
al._ (2020); Tateno and Nozaki (2020), nonlinearity Liang  _et al._ (2010) and
parametric time dependence Xu  _et al._ (2019). In contrast, electrically
controlled phonon nonreciprocity based on e-ph coupling has been less explored
Sengupta  _et al._ (2020). One possible reason is that such phenomena of
nonreciprocity are generally weak in bulk materials. Given the enhanced e-ph
interaction and highly-tunable electronic band structure Ge and Liu (2013);
Navarro-Moratalla  _et al._ (2016); Choi and Choi (2018); Sohier  _et al._
(2019); Han  _et al._ (2021), 2D materials seem to be promising in the pursuit
of e-ph coupling-driven nonreciprocity.

Figure 1:  Schematics of phonon magnetochiral effect for (a) nonchiral and (b) chiral bulk crystals, (c) linear and (d) circular phonon dichroism for $\hat{T}$\- and $\hat{I}$-broken 2D crystals. $\gamma^{i}(\bm{q})$ are the phonon absorption coefficients, with phonon wave vector $\bm{q}$ and polarization $i$. In (a)-(b), $\bm{q}_{\parallel}$ ($\bm{q}_{\bot}$) is parallel (perpendicular) to the external magnetic field $B$. In (c)-(d), $\bm{q}$ is in plane while magnetization $M$ is out of plane. $l/t$ labels phonons with longitudinal (transverse) polarization, and $L/R$ labels left- (right-) handed circular phonons. The polarization is indicated by double arrows in (c)-(d). Dash lines refer to the reversal process when changing $\bm{q}$ to $-\bm{q}$.  Table 1:  Chirality and reciprocity for phonon magnetochiral effect (PMC), linear phonon dichroism (LPD) and circular phonon dichroism (CPD). $H_{0}$, $H_{Z}$, $H_{X}$, $H_{R}$ and $H_{w}$ are the Hamiltonian for intrinsic monolayer transition metal dichalcogenides, out-of-plane magnetization, in-plane (zigzag-direction) magnetization, Rashba SOI and trigonal warping, respectively. | chiral | reciprocal | reciprocal
---|---|---|---
|  | $(\bm{q}\rightarrow-\bm{q})$ | $(\bm{B}/\bm{M}\rightarrow-\bm{B}/\bm{M})$
PMC (bulk) Spivak and Andreev (2016) | no | yes | yes
PMC (bulk) Sengupta  _et al._ (2020) | yes | no | no
LPD/CPD ($H_{0}+H_{Z}$) Shan (2022) | no | yes | yes (LPD), no (CPD)
LPD/CPD ($H_{0}+H_{Z}+H_{R}$) Shan (2022) | yes | yes | yes (LPD), no (CPD)
LPD/CPD ($H_{0}+H_{Z}+H_{w}$) | no | no | no (LPD), no (CPD)
LPD/CPD ($H_{0}+H_{X}$) | no | yes | yes (LPD), no (CPD)
LPD/CPD ($H_{0}+H_{X}+H_{R}$) | no | no | no (LPD), no (CPD)

An attempt has been made to study the circular phonon dichroism (CPD) in
monolayer transition metal dichalcogenides Shan (2022), where there is a
difference of absorption coefficients $\gamma$ between left ($L$)- and right
($R$)-handed circularly polarized phonons [see Fig. 1 (d)]. Based on symmetry
principles Szaller  _et al._ (2013); Cheong  _et al._ (2018); Tokura and
Nagaosa (2018), such CPD was expected to be nonreciprocal, i.e., distinct
$\gamma$ when reversing $\bm{q}$ to $-\bm{q}$, as a result of the
simultaneously broken time-reversal $\hat{T}$ and space-inversion symmetry
$\hat{I}$. However, analytical calculations suggested unexpected reciprocal
behaviors $\gamma^{L/R}(\bm{q})=\gamma^{L/R}(-\bm{q})$, and nonreciprocal
behaviors $\gamma^{L/R}(\bm{M})\neq\gamma^{L/R}(-\bm{M})$ under the reversal
of magnetization $\bm{M}$. More surprisingly, when further incorporating the
Rashba spin-orbit interaction (SOI), the reciprocity in phonon attenuation
between $\pm\bm{q}$ survives despite the fact that the system becomes chiral
due to the lack of mirror planes. This is in stark contrast to the phonon
magnetochiral effect in bulk crystals [see Fig. 1 (a)-(b)], which shows
nonreciprocal (reciprocal) behaviors when the underlying system is chiral
(nonchiral) Spivak and Andreev (2016); Sengupta  _et al._ (2020). Here we
adopt the concept of chirality from chiral crystals Nomura  _et al._ (2019);
Yu  _et al._ (2023), that is, a system is $chiral$ when distinguishable from
its mirror image. From the viewpoint of symmetries, an electron acquires
chirality when all mirror symmetries and time-reversal symmetry are broken,
otherwise it is nonchiral. The different nonreciprocal behaviors between CPD
and phonon magnetochiral effect raise a question: What kind of geometric
information (beyond the chirality) of electrons is inherited by phonons and
induce the phonon nonreciprocity? To address it, a deeper understanding of the
interplay of electronic band geometry, symmetry and phonon nonreciprocity is
required.

In this paper, we show that the linear and circular phonon dichroism arise
from nonlocal band geometric quantities of electrons in magnetic 2D Dirac
materials. For acoustic phonons, these effects reduce to pure Fermi-surface
properties. To acquire the nonreciprocity, we find that the chirality is not a
necessary condition [as summarized in Table 1], whereas the anisotropy of each
Fermi pocket is crucial, particularly in multivalley systems. We propose two
possible mechanisms to realize the Fermi pocket anisotropy [as shown in Table
1]: (i) trigonal warping and out-of-plane magnetization or (ii) Rashba SOI and
in-plane magnetization. To demonstrate our theory, nonreciprocal LPD and CPD
are discussed in 2H-MoTe2 deposited on EuO substrate. Our study uncovers a
connection between electronic band geometry and nonreciprocal phonon
attenuation, and paves the way towards electrical control of phonon
nonreciprocity for acoustoelectronics applications.

Linear and circular phonon dichroism.—The Hamiltonian reads
$\hat{\mathcal{H}}=\hat{\mathcal{H}}_{e}+\hat{\mathcal{H}}_{e-ph}$, where
$\hat{\mathcal{H}}_{e}$ and $\hat{\mathcal{H}}_{e-ph}$ correspond to
electronic and e-ph coupling term, respectively. The general form of e-ph
coupling follows
$\hat{\mathcal{H}}_{e-ph}=\sum_{\bm{k},\bm{q}}\psi^{+}(\bm{k})[\bm{u}(\bm{q})\cdot\hat{\bm{T}}(\bm{q})]\psi(\bm{k}-\bm{q})$
lan ; Suzuura and Ando (2002); Liu and Shi (2017); Shan (2020, 2022), where
$\psi(\bm{k})$ and $\psi^{+}(\bm{k})$ are the annihilation and creation
operator of electrons. For acoustic phonons, $\bm{u}(\bm{q})$ is a Fourier
transform of in-plane collective displacement $\bm{u}(\bm{r})=(u_{x},u_{y})$
from the equilibrium position of ions, with phonon propagation vector
$\bm{q}$. $\hat{\bm{T}}(\bm{q})$ can be regarded as a force operator acting on
ions exerted by electrons.

For multivalley systems, the valley-resolved retarded response function reads
sup

$\begin{split}&\chi_{\alpha\beta}^{\tau}(\bm{q},\omega)=\sum_{n,m}\int\frac{\hbar
d^{2}\bm{k}}{\rho(2\pi)^{2}}F_{nm}^{\tau}(\omega,\bm{k},\bm{q})[S_{\alpha\beta}^{\tau}(\omega,\bm{k},\bm{q})]_{mn},\end{split}$
(1)

where $\alpha,\beta=x,y$, $\tau=\pm 1$ labels valley $K_{\pm}$, $\rho$ is the
2D mass density, and $m,n$ are band indices. The dynamical factor

$\begin{split}F_{nm}^{\tau}(\omega,\bm{k},\bm{q})&=\frac{f_{\tau,m,\bm{k}}-f_{\tau,n,\bm{k}^{\prime}}}{\omega+i\delta+E_{\tau,m,\bm{k}}-E_{\tau,n,\bm{k}^{\prime}}}\end{split}$
(2)

and geometric factor

$\begin{split}[S_{\alpha\beta}^{\tau}(\omega,\bm{k},\bm{q})]_{mn}&=\langle\psi_{\tau,m,\bm{k}}|\hat{T}_{\alpha}^{\tau}(\bm{q})|\psi_{\tau,n,\bm{k}^{\prime}}\rangle\\\
&\times\langle\psi_{\tau,n,\bm{k}^{\prime}}|\hat{T}_{\beta}^{\tau}(-\bm{q})|\psi_{\tau,m,\bm{k}}\rangle,\end{split}$
(3)

where $E_{\tau,m,\bm{k}}$ and $|\psi_{\tau,m,\bm{k}}\rangle$ are eigen-
dispersion and wave functions of $\hat{\mathcal{H}}_{e}$, respectively.
$\bm{k}^{\prime}=\bm{k}-\bm{q}$. $f_{\tau,m,\bm{k}}$
($f_{\tau,n,\bm{k}^{\prime}}$) is the Fermi distribution function and $\delta$
is a positive infinitesimal. $\hat{\bm{T}}^{\tau}(\bm{q})$ is the valley-
dependent force operator.

We mainly focus on the anti-Hermitian part of $[\chi_{\alpha\beta}]$ matrix,
that is, $-2i\omega\gamma(\bm{q},\omega)$, which physically describes the
dissipation of phonons. The form of the Hermitian matrix
$\gamma(\bm{q},\omega)$ reads sup

$\begin{split}\gamma(\bm{q},\omega)=\left[\begin{array}[]{cc}\gamma_{D}+\gamma_{\bar{D}}&\gamma_{\bar{A}}+i\gamma_{A}\\\
\gamma_{\bar{A}}-i\gamma_{A}&\gamma_{D}-\gamma_{\bar{D}}\\\
\end{array}\right],\end{split}$ (4)

$\gamma_{D}=(\gamma_{xx}+\gamma_{yy})/2$ and
$\gamma_{\bar{D}}=(\gamma_{xx}-\gamma_{yy})/2$ refer to the symmetric and
antisymmetric normal (longitudinal) absorption.
$\gamma_{\bar{A}}=\mathrm{Re}[\gamma_{xy}]$ and
$\gamma_{A}=\mathrm{Im}[\gamma_{xy}]$ refer to the symmetric and antisymmetric
anomalous (Hall) absorption. When linearly polarized longitudinal or
transverse phonons are injected, i.e.,
$|u_{l}(\bm{q})\rangle=[\cos\phi_{\bm{q}},\sin\phi_{\bm{q}}]^{T}$ and
$|u_{t}(\bm{q})\rangle=[-\sin\phi_{\bm{q}},\cos\phi_{\bm{q}}]^{T}$, the
damping (absorption) coefficients $\gamma^{l/t}=\gamma_{D}\pm\cos
2\phi_{\bm{q}}\gamma_{\bar{D}}\pm\sin 2\phi_{\bm{q}}\gamma_{\bar{A}}$, where
the azimuthal angle $\phi_{\bm{q}}=\tan^{-1}(q_{y}/q_{x})$. The difference
between $\gamma^{l}$ and $\gamma^{t}$ defines the $linear$ $phonon$
$dichroism$ (LPD) [see Fig. 1 (c)]. On the other hand, when circularly
polarized phonons are considered,
$|u_{L/R}(\bm{q})\rangle=\frac{1}{\sqrt{2}}[1,\pm i]^{T}$, the damping
coefficients $\gamma^{L/R}=\gamma_{D}\mp\gamma_{A}$. The difference between
$\gamma^{L}$ and $\gamma^{R}$ defines the $circular$ $phonon$ $dichroism$
(CPD) [see Fig. 1 (d)]. As LPD and CPD effects arise from different parts of
$\gamma$ matrix, the two phenomena may behave rather differently [see Table
1].

Figure 2:  Schematics of (a) the lattice and (b) electronic band structure of
2H-MoTe2 on EuO substrate. Pseudospin structure on the Fermi surface
(intersected by $E_{F}$ in (b)) at (c) valley $K_{+}$ and (d) valley $K_{-}$.
In (b), red (green) arrows label spin-up (-down) bands. In (c) and (d), purple
arrows denote the in-plane projection of pseudospin for states lying on the
Fermi surface. Phonon wave vector $\bm{q}$ is assumed to be along the
$\hat{x}$ (zigzag) direction, connecting two electronic states with wave
vector $\bm{k}$ and $\bm{k}^{\prime}=\bm{k}-\bm{q}$. $k_{0}$ is a small
deviation of wave vector from the Fermi surface due to energy offset $\omega$.

Geometric origin.—We now reveal the geometric origin of LPD and CPD effects.
For 2D materials, acoustic phonon energy $\omega$ is usually much smaller than
the electronic Fermi energy $E_{F}$. In this sense, in the low-temperature
limit, the dynamical factor reduces to

$\begin{split}\mathrm{Im}[F^{\tau}_{nm}]\approx-\pi\omega\delta(E_{\tau,m,\bm{k}}-E_{F})\delta(\omega+E_{\tau,m,\bm{k}}-E_{\tau,n,\bm{k}^{\prime}}).\end{split}$
(5)

Physically, this means that acoustic phonons only introduce transitions
between electrons on the Fermi surface. If a single band is intersected by the
Fermi level at each valley [see Fig. 2 (b)], only intraband transition $m=n$
is allowed.

The geometric factor $(S_{\alpha\beta}^{\tau})_{mn}$ characterizes a
connection between states with wave vector $\bm{k}$ and $\bm{k}^{\prime}$,
triggered by the force operator $\hat{\bm{T}}^{\tau}(\bm{q})$. Here $\bm{k}$
and $\bm{k}^{\prime}=\bm{k}-\bm{q}$ are not necessarily close [see Fig. 2 (c)
and (d)], suggesting that $(S_{\alpha\beta}^{\tau})_{mn}$ has a $nonlocal$
nature in the momentum space. This is in stark contrast to the optical Xiao
_et al._ (2010) or phononic response in the long-wavelength limit Sengupta
_et al._ (2020); Hu  _et al._ (2021), where the occurrence of Berry curvature
and orbital magnetic moment all relies on the local operations in the momentum
space. As a comparison, the long-wavelength limit of
$(S_{\alpha\beta}^{\tau})_{mn}$ is also discussed in the Supplementary
Material sup .

The role of $\hat{\bm{T}}^{\tau}(\bm{q})$ in Eq. (3) can be viewed as a
rotation of (pseudo-) spins of Bloch states. For conventional deformation
potentials $\hat{\bm{T}}^{\tau}(\bm{q})=ig\bm{q}$, $\chi_{\alpha\beta}$
reduces to normal electron polarization function mah ; giu , corresponding to
a vanishing CPD signal. In 2D hexagonal Dirac crystals, the e-ph coupling
behaves more like pseudo-gauge potentials Suzuura and Ando (2002); Bhalla  _et
al._ (2022), e.g.,
$\hat{\bm{T}}^{\tau=-1}(\bm{q})=ig[\bm{q}\cdot\bm{\sigma},(\bm{q}\times\bm{\sigma})_{z}]$
and
$\hat{\bm{T}}^{\tau=1}(\bm{q})=\mathcal{K}[\hat{\bm{T}}^{\tau=-1}(-\bm{q})]$,
with the complex conjugation $\mathcal{K}$. Here $\hat{T}_{x}^{\tau}(\bm{q})$
and $\hat{T}_{y}^{\tau}(\bm{q})$ act as a rotation of angle $\pi$ in the
(pseudo-) spin subspace about the axis along the direction $(q_{x},-\tau
q_{y})$ and $(\tau q_{y},q_{x})$, respectively. In this sense,
$\chi_{\alpha\beta}$ can be regarded as a peculiar type of dynamical spin
susceptibility mah ; giu , which, however, only depends on states on the Fermi
surface.

Model.—To facilitate the discussion, we consider a generic model for magnetic
2D Dirac materials:
$\hat{\mathcal{H}}_{e}=\sum_{\bm{k}}\psi^{+}(\bm{k})H_{e}(\bm{k})\psi(\bm{k})$,
with $H_{e}(\bm{k})=H_{0}+H_{\bm{n}}$ and

$\begin{split}&H_{0}=\frac{\Delta}{2}\sigma_{z}+\hbar
v(\tau\sigma_{x}k_{x}+\sigma_{y}k_{y})+\tau
s_{z}(\lambda_{c}\sigma_{+}+\lambda_{v}\sigma_{-}),\\\
&H_{\bm{n}}=-\bm{s}\cdot\bm{n}(M_{c}\sigma_{+}+M_{v}\sigma_{-}).\end{split}$
(6)

$\bm{s}$ and $\bm{\sigma}$ are Pauli matrices acting on spin and pseudospin
(orbital) subspace, respectively.
$\sigma_{\pm}=\frac{1}{2}(\sigma_{0}\pm\sigma_{z})$. $H_{\bm{n}}$ refers to
the magnetic exchange coupling. $\lambda_{c/v}$ and $M_{c/v}$ characterize the
intrinsic SOI and Zeeman field in the sublattice space, respectively. This
model can be applied to graphene Kane and Mele (2005), monolayer transition
metal dichalcogenides Qi  _et al._ (2015); Norden  _et al._ (2019) and other
2D Dirac materials Cui  _et al._ (2021). Here we consider an out-of-plane
magnetization $H_{Z}\equiv H_{\bm{n}=\hat{z}}$ and set the Fermi level in the
valence band, whose dispersion is shown schematically in Fig. 2 (b).

We find that $\gamma_{A}^{\tau}=0$ exactly on the Fermi surface, whereas
$\gamma_{D/\bar{D}/\bar{A}}^{\tau}$ are generally nonzero sup . The vanishing
$\gamma_{A}^{\tau}$ is a result of Dirac model rather than specific
symmetries. This is in contrast to the non-vanishing optical conductivities of
Dirac model. Such difference originates from the different optical and
phononic processes. For optical processes, Hall conductivities are contributed
by the band geometric quantities of both conduction and valence bands since
the interband transitions of electrons are optically induced. For acoustic
phonons, electronic transitions are within valence bands with different wave
vectors since phonons are unable to induce the transitions between conduction
and valence bands [see Fig. 2 (b)]. As a result, $\gamma_{A}^{\tau}$ depends
on the information of band geometries of purely valence bands, and vanishes
for Dirac model. The vanishing $\gamma_{A}^{\tau}$, however, does not imply
that CPD vanishes since there is a small energy offset $\omega$ in
$\mathrm{Im}[F^{\tau}_{vv}]$ which drives electrons slightly away from the
Fermi surface (that is, $k_{0}$ in Fig. 2 (c)). As a result, we find that
$\mathrm{Im}[S_{xy}^{\tau}]_{vv}\propto\omega$ sup , which gives rise to a
finite, linear-in-$\omega$ contribution to $\gamma_{A}^{\tau}$.

Nonreciprocity and symmetry.—When reversing the phonon propagation vector
$\bm{q}$, we find that $\gamma^{i}(\bm{q},\omega)=\gamma^{i}(-\bm{q},\omega)$,
with $i=l,t,L,R$, indicating that both LPD and CPD are reciprocal [see Fig. 1
(c) and (d)]. To understand it, we check the remaining symmetries for
electronic Hamiltonian $H_{e}(\bm{k})=H_{0}+H_{Z}$ at each valley sup : mirror
reflection $\hat{\sigma}_{h}$ $(z\rightarrow-z)$, three-fold rotation
$\hat{C}_{3}$, and two hidden “inversion” symmetries
$\hat{\mathcal{P}}_{1}=s_{0}\sigma_{z}$ and
$\hat{\mathcal{P}}_{2}=s_{z}\sigma_{z}$. Note that only
$\hat{\mathcal{P}}_{1}$ and $\hat{\mathcal{P}}_{2}$ symmetries relate
electrons with $\pm\bm{k}$ at a single valley $K_{\tau}$, that is,
$\hat{\mathcal{P}}_{1(2)}^{+}H_{e}(\bm{k})\hat{\mathcal{P}}_{1(2)}=H_{e}(-\bm{k})$.
By contrast, $\hat{\sigma}_{h}$ relates $(\tau,\bm{k})$ to itself and
$\hat{C}_{3}$ relates $(\tau,\bm{k})$ to $(\tau,\hat{C}_{3}\bm{k})$, neither
of which is relevant to the reciprocal relations between $\pm\bm{q}$. Here
$\bm{k}$ is a relative wave vector away from the valley center. Furthermore,
we find that $\hat{\mathcal{P}}_{1(2)}$ transforms the force operator
$\hat{\bm{T}}^{\tau}(\bm{q})$ by
$\hat{\mathcal{P}}_{1(2)}^{+}\hat{\bm{T}}^{\tau}(\bm{q})\hat{\mathcal{P}}_{1(2)}=\hat{\bm{T}}^{\tau}(-\bm{q})$.
As a result, we recognize that either $\hat{P}_{1}$ or $\hat{P}_{2}$ symmetry
is the origin of reciprocal relations for phonon attenuation between
$\pm\bm{q}$ sup .

Figure 3:  (a) Anisotropic (Isotropic) Fermi pockets of monolayer MoTe2 with
(without) trigonal warping, denoted by solid (dashed) lines and
$\lambda_{w}\neq 0$ ($\lambda_{w}=0$). $k_{F}$ is the Fermi wave vector at
valley $K_{-}$ without trigonal warping. $\bm{q}=q\hat{x}$ and
$\bm{M}=M\hat{z}$. (b) Linear and circular phonon dichroism $\gamma^{l/t}$ and
$\gamma^{L/R}$ (in units of $\gamma_{0}$) as functions of phonon wave vector
$q$ (in units of $k_{F}$). (c) Degree of nonreciprocity $\delta\gamma^{i}$
(with $i=l,t,L,R$) as functions of phonon wave vector $q$. Parameters:
$\Delta=1.05$ eV, $\lambda_{v}=0.11$ eV, $\lambda_{c}=0.029$ eV, $\hbar
v=2.33$ eV$\cdot\AA$, $M_{c}=0.206$ eV, $M_{v}=0.17$ eV Qi  _et al._ (2015),
$\lambda_{w}=-1.0$ eV$\cdot\AA^{2}$, $E_{F}=-0.5$ eV, $g=0.32$ eV Chen  _et
al._ (2016), $\gamma_{0}=2.23\times 10^{9}$ s-1 Shan (2022). Phonon dispersion
$\omega=\hbar c_{l/t}|\bm{q}|$, with longitudinal (transverse) sound velocity
$c_{l}=3.64\times 10^{3}$ m/s ($c_{t}=2.21\times 10^{3}$ m/s) Rano  _et al._
(2020).

The role of chirality can be examined by incorporating Rashba SOI

$\begin{split}H_{R}=\lambda_{R}(\tau
s_{y}\sigma_{x}-s_{x}\sigma_{y})\end{split}$ (7)

into $H_{e}(\bm{k})$. Such term naturally breaks $\hat{\sigma}_{h}$ symmetry,
meaning that no mirror plane exists. Given that time-reversal symmetry is
already broken in Eq. (LABEL:basic), the system becomes $chiral$. Detailed
calculations show that LPD and CPD are still reciprocal [see Table 1], thereby
excluding a possible origin of nonreciprocity due to nonzero chirality. On the
other hand, such result can be explained by recognizing that an introduction
of $H_{R}$ still preserves $\hat{\mathcal{P}}_{2}$ symmetry, which establishes
a link between $\pm\bm{k}$.

The role of $\hat{C}_{3}$ symmetry can be further examined by replacing the
out-of-plane magnetization $H_{Z}$ with the in-plane magnetization
$H_{X}\equiv H_{\bm{n}=\hat{x}}$. We find that LPD and CPD are reciprocal [see
Table 1] despite the fact that $H_{X}$ breaks the $\hat{C}_{3}$ symmetry. This
implies that $\hat{C}_{3}$ symmetry is also not the origin of nonreciprocal
behaviors. By contrast, $H_{X}$ preserves $\hat{\mathcal{P}}_{1}$ symmetry sup
, which still provides a link between $\pm\bm{k}$.

Therefore to acquire the nonreciprocity, the simultaneous
$\hat{\mathcal{P}}_{1}$ and $\hat{\mathcal{P}}_{2}$ symmetry breaking is
required. This naturally leads to an anisotropy of Fermi pocket at each
valley. To realize it, two possible schemes are proposed in the following.
Moreover, the nonreciprocal responses under the reversal of magnetization
$\bm{M}\rightarrow-\bm{M}$ can be derived by the generalized Onsager
reciprocal theorem $\gamma(\bm{q},\bm{M})=\gamma^{T}(-\bm{q},-\bm{M})$ Onsager
(1931); Rikken  _et al._ (2001); Tokura and Nagaosa (2018); sup  [see Table
1].

Figure 4:  (a) Anisotropic (Isotropic) Fermi pockets of monolayer MoTe2 with
Rashba SOI in the presence (absence) of in-plane magnetization, denoted by
solid (dashed) lines. $k_{F}$ is the Fermi wave vector at valley $K_{\pm}$
without magnetization. $\bm{q}=q\hat{y}$ and $\bm{M}=M\hat{x}$. (b) Linear and
circular phonon dichroism $\gamma^{l/t}$ and $\gamma^{L/R}$ (in units of
$\gamma_{0}$) as functions of phonon wave vector $q$ (in units of $k_{F}$).
(c) Degree of nonreciprocity $\delta\gamma^{i}$ as functions of phonon wave
vector $q$. Parameters: $\lambda_{R}=0.2$ eV, $E_{F}=-0.6$ eV.

Trigonal warping and out-of-plane magnetization.—One scheme to acquire the
nonreciprocity is to consider the Hamiltonian
$H_{e}(\bm{k})=H_{0}+H_{w}+H_{Z}$, with trigonal warping term Liu  _et al._
(2013); Kormányos  _et al._ (2013); Battilomo  _et al._ (2019)

$\begin{split}H_{w}=\lambda_{w}[(k_{x}^{2}-k_{y}^{2})\sigma_{x}-2\tau
k_{x}k_{y}\sigma_{y}].\end{split}$ (8)

This leads to the Fermi pocket anisotropy at each valley, as shown in Fig. 3
(a). The $q$-dependent $\gamma^{l/t}$ and $\gamma^{L/R}$ exhibit two peaks for
both positive and negative $q$ [see Fig. 3 (b)], labeled by $X_{A/\bar{A}}$
and $X_{B/\bar{B}}$. These peaks have distinct physical origins. Peaks
$X_{A/\bar{A}}$ at $q/k_{F}=\pm 0.22$ arise from the divergent
$[dE_{\tau=-1,v,k,k^{\prime}}/dk]^{-1}$ (valley $K_{-}$ is dominant over
$K_{+}$), which can be regarded as a new type of joint density of states
between wave vector $\bm{k}$ and $\bm{k}-\bm{q}$ of valence band. This is a
natural consequence of Fermi pocket anisotropy. On the other hand, peaks
$X_{B/\bar{B}}$ at $q/k_{F}=\pm 0.70$ are due to a sudden vanishing of
contributions from valley $K_{+}$ [see Fig. 3 (a)]. According to the dynamical
factor $\mathrm{Im}[F^{\tau}_{vv}]$ from Eq. (5), the critical value $q_{c}$
can be derived by solving the condition
$\omega(q_{c})+E_{\tau=1,v,\bm{k}}=E_{\tau=1,v,\bm{k}-\bm{q}_{c}}$ for valley
$K_{+}$. When $q$ becomes larger than $q_{c}$, electronic scattering within
valley $K_{+}$ by phonons is inhibited, and only scattering within valley
$K_{-}$ exists and contributes to phonon absorption. The degree of
nonreciprocity can be defined as
$\delta\gamma^{i}=[\gamma^{i}(\bm{q})-\gamma^{i}(-\bm{q})]/[\gamma^{i}(\bm{q})+\gamma^{i}(-\bm{q})]$,
where $i=l,t,L,R$ label different types of phonon polarization. In Fig. 3 (c),
$\delta\gamma^{i}$ is tunable by $q$ and becomes pronounced around peaks
$X_{A/\bar{A}}$ and $X_{B/\bar{B}}$. The behaviors at peaks $X_{B/\bar{B}}$
are complex due to a competition of $l$ and $t$ phonon modes, which have
different critical value $q_{c}$.

Rashba SOI and in-plane magnetization.—Another scheme to acquire the
nonreciprocity is to consider the Hamiltonian
$H_{e}(\bm{k})=H_{0}+H_{R}+H_{X}$, whose results are shown in Fig. 4. Under a
magnetization $\bm{M}$ along $x$ direction, the anisotropic Fermi pocket of
each valley shifts collectively along $y$ direction [see Fig. 4 (a)]. This
excludes the possibilities of peaks $X_{B/\bar{B}}$ in $\gamma^{l/L/R}$ since
the critical value $q_{c}$ is always the same between valley $K_{+}$ and
$K_{-}$. The occurrence of peaks $X_{A/\bar{A}}$ still arises from the
divergent $[dE_{\tau=\pm,v,k,k^{\prime}}/dk]^{-1}$, as characteristics of
Fermi pocket anisotropy. $\gamma^{t}$ is vanishingly small [see Fig. 4 (b)],
since $\gamma^{t}=\gamma_{D}+\gamma_{\bar{D}}\propto$ geometric factor
$[S_{xx}^{\tau}]_{vv}\propto|\langle\psi_{\tau,v,\bm{k}}|q_{y}\sigma_{y}|\psi_{\tau,v,\bm{k}-\bm{q}}\rangle_{k_{x}=0}|^{2}$
according to Eq. (3). Such matrix element almost vanishes since the pseudospin
of two states $|\psi_{\tau,v,k_{x}=0,k_{y}}\rangle$ and
$|\psi_{\tau,v,k_{x}=0,k_{y}-q_{y}}\rangle$ points oppositely along $y$
direction [see Fig. 2 (c) and (d)]. These lead to nonreciprocal LPD and CPD as
shown in Fig. 4 (c). On the other hand, when $\bm{q}$ is along $x$ direction,
i.e., $\bm{q}\parallel\bm{M}$, the nonreciprocity vanishes.

Discussion.—We have studied the nonreciprocal phonon dichroism induced by
Fermi pocket anisotropy in 2D Dirac materials. We find that these effects have
the origin of nonlocal electronic band geometries and are determined by the
Fermi-surface properties for acoustic phonons. In multi-valley systems, the
nonreciprocity is driven by the Fermi pocket anisotropy rather than the
chirality. Two possible schemes are proposed to realize the Fermi pocket
anisotropy in 2H-MoTe2. Interestingly, similar proposals have been given in
electronic systems in order to acquire the nonreciprocal current in
noncentrosymmetric Rashba superconductors Wakatsuki  _et al._ (2017);
Wakatsuki and Nagaosa (2018); Hoshino  _et al._ (2018), nonreciprocal
resistivity in BiTeBr Ideue  _et al._ (2017) and unidirectional valley-
contrasting photocurrent Asgari and Culcer (2022). Our findings suggest that
the geometric information of electrons can be inherited by phonons through the
pseudo-gauge e-ph coupling, thereby inducing the phonon nonreciprocity.

In the above analysis, we have only considered the imaginary part of self-
energy and neglected the real part. The treatment is appropriate since we
mainly focus on the acoustic phonons, whose sound velocity is about two order
of magnitude smaller than the Fermi velocity of electrons. This leads to
$|\Sigma_{s}|/\omega\sim 10^{-6}$ when $|\bm{q}|/k_{F}=0.1$;
$|\Sigma_{s}|/\omega\sim 10^{-3}$ when $|\bm{q}|/k_{F}=1.8$ sup . $\Sigma_{s}$
is the real part of self-energy. Therefore the correction from $\Sigma_{s}$ to
the phonon dispersion can be safely neglected.

Experimentally, the circular phonon dichroism can be detected by the pulse-
echo technique based on the different absorption rates between left- and
right-handed circular phonons tru . Alternatively, we can use the Raman
spectroscopy analysis of phonon polarization lut . Our studies can be further
generalized to other situations, such as discussing the role of disorder,
superconducting states and nonlinear effect in phonon dichroism. These will be
the subject of future work.

Acknowledgments. This work is supported by the National Natural Science
Foundation of China (NSFC, Grant No. 11904062), the Starting Research Fund
from Guangzhou University (Grant No. RQ2020076) and Guangzhou Basic Research
Program, jointed funded by Guangzhou University (Grant No. 202201020186).

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