Nonreciprocal phonon dichroism induced by Fermi pocket anisotropy in two-dimensional Dirac materials

Electrons in two-dimensional (2D) Dirac materials carry local band geometric quantities, such as the Berry curvature and orbital magnetic moments, which, combined with electron-phonon coupling, may affect the phonon dynamics in an 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-MoTe 2 on a EuO substrate. Our finding points to a different route towards electrical control of phonon nonreciprocity for acoustoelectronics applications based on 2D quantum materials.

An attempt has been made to study the circular phonon dichroism (CPD) in monolayer transition metal dichalcogenides [31], where there is a difference of absorption coefficients γ between left (L)-and right (R)-handed circularly polarized phonons [see Fig. 1 (d)]. Based on symmetry principles [32][33][34], such CPD was expected to be nonreciprocal, i.e., distinct γ when reversing q to −q, as a result of the simultaneously broken time-reversal T and space-inversion symmetryÎ. However, analyti-cal calculations suggested unexpected reciprocal behaviors γ L/R (q) = γ L/R (−q), and nonreciprocal behaviors γ L/R (M ) ̸ = γ L/R (−M ) under the reversal of magnetization M . More surprisingly, when further incorporating the Rashba spin-orbit interaction (SOI), the reciprocity in phonon attenuation between ±q survives despite the fact that the system becomes chiral due to the lack of mir- Figure 1. Schematics of phonon magnetochiral effect for (a) nonchiral and (b) chiral bulk crystals, (c) linear and (d) circular phonon dichroism forT -andÎ-broken 2D crystals. γ i (q) are the phonon absorption coefficients, with phonon wave vector q and polarization i. In (a)-(b), q ∥ (q ⊥ ) is parallel (perpendicular) to the external magnetic field B. In (c)-(d), 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 q to −q. Table I. Chirality and reciprocity for phonon magnetochiral effect (PMC), linear phonon dichroism (LPD) and circular phonon dichroism (CPD). H0, HZ , HX , HR and Hw are the Hamiltonian for intrinsic monolayer transition metal dichalcogenides, out-of-plane magnetization, in-plane (zigzag-direction) magnetization, Rashba SOI and trigonal warping, respectively. [11] no yes yes PMC (bulk) [12] yes no no LPD/CPD (H0 + HZ ) [31] no yes yes (LPD), no (CPD) LPD/CPD (H0 + HZ + HR) [31] yes ror 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) [11,12]. Here we adopt the concept of chirality from chiral crystals [19,35], 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 Fermisurface properties. To acquire the nonreciprocity, we find that the chirality is not a necessary condition [as summarized in Table I], 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 I]: (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-MoTe 2 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 readsĤ =Ĥ e +Ĥ e−ph , whereĤ e andĤ e−ph correspond to electronic and e-ph coupling term, respectively. The general form of e-ph coupling followŝ [36][37][38][39], where ψ(k) and ψ + (k) are the annihilation and creation operator of electrons. For acoustic phonons, u(q) is a Fourier transform of in-plane collective displacement u(r) = (u x , u y ) from the equilibrium position of ions, with phonon propagation vector q.T (q) can be regarded as a force operator acting on ions exerted by electrons.
For multivalley systems, the valley-resolved retarded response function reads [40] where α, β = x, y, τ = ±1 labels valley K ± , ρ is the 2D mass density, and m, n are band indices. The dynamical factor and geometric factor where E τ,m,k and |ψ τ,m,k ⟩ are eigen-dispersion and wave functions ofĤ e , respectively.
Geometric origin.-We now reveal the geometric origin of LPD and CPD effects. For 2D materials, acoustic phonon energy ω is usually much smaller than the electronic Fermi energy E F . In this sense, in the lowtemperature limit, the dynamical factor reduces to 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 τ αβ ) mn characterizes a connection between states with wave vector k and k ′ , triggered by the force operatorT τ (q). Here k and k ′ = k − q are not necessarily close [see Fig. 2 (c) and (d)], suggest-ing that (S τ αβ ) mn has a nonlocal nature in the momentum space. This is in stark contrast to the optical [5] or phononic response in the long-wavelength limit [12,13], 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 τ αβ ) mn is also discussed in the Supplementary Material [40].
The role ofT τ (q) in Eq. (3) can be viewed as a rotation of (pseudo-) spins of Bloch states. For conventional deformation potentialsT τ (q) = igq, χ αβ reduces to normal electron polarization function [41,42], corresponding to a vanishing CPD signal. In 2D hexagonal Dirac crystals, the e-ph coupling behaves more like pseudo-gauge potentials [37,43], e.g., , with the complex conjugation K. HereT τ x (q) andT τ y (q) act as a rotation of angle π in the (pseudo-) spin subspace about the axis along the direction (q x , −τ q y ) and (τ q y , q x ), respectively. In this sense, χ αβ can be regarded as a peculiar type of dynamical spin susceptibility [41,42], 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:Ĥ e = k ψ + (k)H e (k)ψ(k), with H e (k) = H 0 + H n and s and σ are Pauli matrices acting on spin and pseudospin (orbital) subspace, respectively. σ ± = 1 2 (σ 0 ± σ z ). H n refers to the magnetic exchange coupling. λ 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 [44], monolayer transition metal dichalcogenides [45,46] and other 2D Dirac materials [47]. Here we consider an out-of-plane magnetization H Z ≡ H n=ẑ and set the Fermi level in the valence band, whose dispersion is shown schematically in Fig. 2 (b).
We find that γ τ A = 0 exactly on the Fermi surface, whereas γ τ D/D/Ā are generally nonzero [40]. The vanishing γ τ A 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, γ τ A depends on the information of band geometries of purely valence bands, and vanishes for Dirac model. The vanishing γ τ A , however, does not imply that CPD vanishes since there is a small energy offset ω in Im[F τ vv ] which drives electrons slightly away from the Fermi surface (that is, k 0 in Fig. 2 (c)). As a result, we find that Im[S τ xy ] vv ∝ ω [40], which gives rise to a finite, linear-in-ω contribution to γ τ A . Nonreciprocity and symmetry.-When reversing the phonon propagation vector q, we find that γ i (q, ω) = γ i (−q, ω), 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 (k) = H 0 + H Z at each valley [40]: mirror reflectionσ h (z → −z), three-fold rotationĈ 3 , and two hidden "inversion" symmetriesP 1 = s 0 σ z and P 2 = s z σ z . Note that onlyP 1 andP 2 symmetries relate electrons with ±k at a single valley K τ , that is, P + 1(2) H e (k)P 1(2) = H e (−k). By contrast,σ h relates (τ, k) to itself andĈ 3 relates (τ, k) to (τ,Ĉ 3 k), neither of which is relevant to the reciprocal relations between ±q.
The role of chirality can be examined by incorporating Rashba SOI into H e (k). Such term naturally breaksσ h symmetry, meaning that no mirror plane exists. Given that time-reversal symmetry is already broken in Eq. (6), the system becomes chiral. Detailed calculations show that LPD and CPD are still reciprocal [see Table I], 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 preservesP 2 symmetry, which establishes a link between ±k. The role ofĈ 3 symmetry can be further examined by replacing the out-of-plane magnetization H Z with the inplane magnetization H X ≡ H n=x . We find that LPD and CPD are reciprocal [see Table I] despite the fact that H X breaks theĈ 3 symmetry. This implies thatĈ 3 symmetry is also not the origin of nonreciprocal behaviors. By contrast, H X preservesP 1 symmetry [40], which still provides a link between ±k.
Therefore to acquire the nonreciprocity, the simulta-neousP 1 andP 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 M → −M can be derived by the generalized Onsager reciprocal theorem γ(q, M ) = γ T (−q, −M ) [34,40,50,51] [see Table I].
(a) Trigonal warping and out-of-plane magnetization.-One scheme to acquire the nonreciprocity is to consider the Hamiltonian H e (k) = H 0 + H w + H Z , with trigonal warping term [52][53][54] This leads to the Fermi pocket anisotropy at each valley, as shown in Fig. 3 (a). The q-dependent γ l/t and γ L/R exhibit two peaks for both positive and negative q [see Fig. 3 (b)], labeled by X A/Ā and X B/B . These peaks have distinct physical origins. Peaks , which can be regarded as a new type of joint density of states between wave vector k and k − q of valence band. This is a natural consequence of Fermi pocket anisotropy. On the other hand, peaks X B/B at q/k F = ±0.70 are due to a sudden vanishing of contributions from valley K + [see Fig.  3 (a)]. According to the dynamical factor Im[F τ vv ] from Eq. (5), the critical value q c can be derived by solving the condition ω(q c ) + E τ =1,v,k = E τ =1,v,k−qc 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 where i = l, t, L, R label different types of phonon polarization. In Fig. 3 (c), δγ i is tunable by q and becomes pronounced around peaks X A/Ā and X B/B . The behaviors at peaks X B/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 (k) = H 0 + H R + H X , whose results are shown in Fig. 4. Under a magnetization 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/B in γ l/L/R since the critical value q c is always the same between valley K + and K − . The occurrence of peaks X A/Ā still arises from the divergent [dE τ =±,v,k,k ′ /dk] −1 , as characteristics of Fermi pocket anisotropy. γ t is vanishingly small [see Fig. 4 (b)], since γ t = γ D + γD ∝ geometric factor [S τ xx ] vv ∝ |⟨ψ τ,v,k |q y σ y |ψ τ,v,k−q ⟩ kx=0 | 2 according to Eq. (3). Such matrix element almost vanishes since the pseudospin of two states |ψ τ,v,kx=0,ky ⟩ and |ψ τ,v,kx=0,ky−qy ⟩ 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 q is along x direction, i.e., q ∥ 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-MoTe 2 . Interestingly, similar proposals have been given in electronic systems in order to acquire the nonreciprocal current in noncentrosymmetric Rashba superconductors [55][56][57], nonreciprocal resistivity in BiTeBr [58] and unidirectional valley-contrasting photocurrent [59]. 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 |Σ s |/ω ∼ 10 −6 when |q|/k F = 0.1; |Σ s |/ω ∼ 10 −3 when |q|/k F = 1.8 [40]. Σ s is the real part of self-energy. Therefore the correction from Σ 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 [60]. Alternatively, we can use the Raman spectroscopy analysis of phonon polarization [61]. 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.