Chiral Phonon Induced Spin-Polarization

The current understanding of chirality suggests the existence of a connection between structure and angular momentum, including spin. This is particularly emphasised in the chiral induced spin selectivity effect, where chiral structures act as spin filters. However, the recent discovery of chiral phonons have demonstrated that phonons too may carry angular momentum which also can be regarded as magnetic moments which add to the total moment. Here, it is shown that chiral phonons may induce a non-trivial spin-texture in an otherwise non-magnetic electronic structure. By considering a set-up in which electrons and phonons are interfaced with each other, it is shown that chiral phonons may transfer its angular momentum into the electron reservoir which, thereby, becomes spin-polarized. It is, moreover, shown that an equivalent mechanism does not exist whenever the electrons are interfaced with achiral phonons.

The existence of phonon angular momentum opens up the possibility to couple the electronic spin-degrees of freedom with the mechanical.It is well-established that spin and nuclear motion are coupled directly through, e.g., the Elliot-Yafet mechanism [29][30][31][32][33][34], but also indirectly via the electronic structure [15], and it has been demonstrated that such coupling opens for a viable explanation of the chiral induced spin selectivity effect [34][35][36][37].
Hitherto, however, the angular momentum of phonons and electrons have been considered as separate from one another, where the magnetic moment associated with the chiral phonons have been studied in its own right.While this is definitely pertinent, the effects angular momentum transfer between phonons and electrons has been discussed in a semi-classical model [38], in which a spin-dependent coupling between phonons and electrons is assumed and it is demonstrated that chiral phonons may give rise to the spin-Seebck effect, something that was also recently observed in experiments [39].Nonetheless, the mechanism that enables the angular momentum transfer is yet to be discussed.The purpose with this Letter is to present a coherent theory that ties the existence of phononic angular momentum to a broken electronic spin-degeneracy.It is shown that the mechanism is provided through a vibronically assisted spin-orbit interaction and while this coupling is always present, chiral phonons are required for inducing a spin-polarization.
Chirality is a geometrical property where the structure lacks both inversion and reflection symmetries.It is easy to demonstrate that chiral phonons must carry a non-vanishing angular momentum J ph .This can be seen directly from the definition where Q is the nuclear displacement, which is connected to phonons through the relation Q(r, t) = q l q q Q p (t)e iq•r , where Q q = b qµ + b † qµ , is the quantum phonon displacement operator.Here, b q and b † q denote the phonon destruction and creation operators, respectively, with q = (q, µ) ( q = ( q, µ) = (−q, µ)) comprising the wave vector q and normal mode µ, and l q = /2ρvω q which defines a length scale in terms of the phonon energy ω q , density ρ, and volume v, whereas q is the displacement polarization vector.
The expectation value of the angular momentum, hence, assumes the form where D > pq (t , t) = (−i) Q p (t )Q q(t) defines the correlations between the phonons Q p (t ) and Q p(t).The expression in Eq. (2) shows that a finite phonon angular momentum requires non-collinear polarisations p and q.
Non-collinear polarizations can on the one hand be achieved when there is a mechanism that mixes the phonon modes p and q.Such mode mixing may originate from, e.g., anharmonic effects or scattering off defects acting upon the otherwise orthogonal modes defined in the harmonic approximation.
In the harmonic approximation, the phonons can, up to a constant, be summarised the Hamiltonian form as H ph = q ω q b † q b q .In this form, it is assumed that the phonon modes do not mix which, therefore, leads to a vanishing angular momentum.However, the introduction of a component of the kind W qµν b † qµ b qν provides a mode mixing, which can be understood as chirality.

arXiv:2210.12722v1 [cond-mat.mes-hall] 23 Oct 2022
To see this, consider the phonon spinor Φ q = {b qµ i } N i=1 , for N modes, which enables us to write the phonon model as Written like this, the phonon spectrum is defined through the matrix ω q = ω 0q τ 0 + ω 1q • τ , where ω 0 and ω 1 represent the mode conservative and mode mixing components, respectively, whereas τ 0 and τ are the N-dimensional identity and vector of spin matrices.While in this model it is assumed that the mixing only takes place between modes µ i and µ j with the same momentum q, it is straight forward to generalise the model to also include mixing between different momenta.Considering a structure with two modes, such that τ reduces to the Pauli matrices, the phonon Green function D q (z) = Φ q |Φ q (z), can be written as where ω qs = ω 0q + sω 1q , ω 1q = |ω 1q | and ω1q = ω 1q /ω 1q .The form of this propagator written in Eq. (4b) explicitly describes two modes with opposite helicity, or, chirality.
On the other hand, the polarization for a chiral mode is neither reflection nor inversion symmetric, where the latter condition leads to that * qµ = qµ qµ , since equality in the last relation requires inversion symmetry.For, e.g., a helical structure with transversal and longitudinal lattice parameters a and c, respectively, the polarization may be expressed as q = (a cos φ q , a sin φ q , ¢φ q )/d(φ q ), where ¢ = c/2π and d(φ q ) = a 2 + ¢ 2 φ 2 q , which displays a variation of the mode that depends on the azimuthal angle φ q .Hence, for such a mode, the vector product q × q = (−a¢φ q sin φ q , a¢φ q cos φ q , −a 2 sin 2φ q )/d 2 (φ q ), which suggests that a free chiral phonon mode, for which where n B (ω) is the Bose-Einstein distribution function, carry the non-vanishing angular momentum It is important to notice that the off-diagonal components of the phonon propagator, Eq. (4b), carry the phase factors e ±iφ p , where φ p defines the azimuthal angle of the momentum vector p.This phase dependence reoccurs also in the helical polarization vector and is an essential feature of chirality.Due to this phase dependence, the phonons necessarily carry angular momentum.
The objective in this Letter, is to show that the phononic angular momentum can be transferred to the electronic subsystem and, hence, induce a spin-polarization.For this sake, consider the coupling between electrons and phonons, which generally can be written as [34] H where the coupling matrix U kq = U kq σ 0 + J kq • σ accounts for a spin-conservative electron-phonon coupling, U kq , and an electron-phonon assisted spin-orbit interaction, J kq .The spin-polarisation M k of the electrons are calculated using the identity M k = (−i)spσ G < kk (ω)dω/4π, where G < kk (ω) defines the lesser form of the general single electron Green function G kk (z) = ψ k |ψ † k (z), whereas sp denotes the trace over spin 1/2 space.To the second order (Hartree-Fock) approximation in the electron-phonon coupling, this Green function can be calculated from the Dyson equation where D qq = δ(ω)D a qq (ω)dω.While this equation should be self-consistently solved, for an analysis of the induced spin symmetries, it is sufficient to replace the electronic Green functions in these expressions with its unperturbed form g k (z) = σ 0 g k (z), where g k (z) = 1/(z − ε k ) and g < k (ω) = 2πσ 0 f (ω)δ(ω − ε k ), whereas f (ω) is the Fermi-Dirac distribution function.These replacements lead to the simplified self-energy The Hartree (first) contribution is to lowest order linear in the component J kq • σ which, hence, shows that the coupling between the electrons and phonons may break time-reversal symmetry.However, this contribution vanishes in this approximation since there are no phonons at p = 0.By contrast, the exchange (second) contribution does not only open for breaking the time-reversal symmetry, since in general but it also provides a correlation between the electronic and phononic degrees of freedom.However, before showing that chiral phonon leads to a breaking of the electronic spin-degeneracy, it is pertinent to discuss the origin of the phonon induced spin-polarization and spin-flip processes which are enabled by J kq .For this purpose, it is instructive to consider the spin-orbit coupling, which can be written as [40] where E is the total electric field acting on the electrons, whereas ξ = 1/4c 2 in atomic units.By introducing the field operator ψ(r) = ψ k e ik•r dk/Ω and its Hermitian conjugate, and assuming time-independent magnetic fields B, such that ∇ × E = −∂ t B = 0, the spin-orbit coupling contribution can in the second quantization be written where E S is the boundary surface contribution, and where the electric field has been identified with a the electronic confinement potential V(r) using the relation E = −∇V(r).
Here, E S = ψ k S kk ψ k dkdk /Ω 2 , where S kk is given by where n denotes the outward normal on the boundary surface S.This contribution can be identified as the Rashba spin-orbit coupling.
The confinement potential V(r) = V(r − R)dR, where R = R 0 + Q(R) denotes the coordinate for the nuclei in terms of the equilibrium position R 0 and displacement Q(R).For small displacements, the potential can be expanded around the equilibrium positions where the first contribution along with the surface integral provide the static spin-orbit interaction.
The spin-orbit interaction with the lowest order coupling to the nuclear displacement is given by the Hamiltonian Using ∇ R V(r − R) = (−i) qV(q)e iq•(r−R) dq/Ω, leads to that the phonon assisted spin-orbit interaction can be written where J kq = iξU kq (k × q), where U kq = U q = il q q • qV(q) is the spin-conservative electron-phonon interaction.This gives the total electron-phonon coupling U kq = U q σ 0 + J kq • σ.
From an order of magnitude estimation, it can be deduced that the ratio exchange diagram originating from the spin-dependent and spin-conservative processes, that is, those which are proportional to U kq J k q and U kq U k q, respectively, is roughly 2ξq c /3 (×k), where q c is the momentum corresponding to the phononic high energy cut-off.More importantly, however, is that while J kq is linear in the spin-orbit coupling parameter ξ, that is, in the same order as the static spin-orbit interaction contribution H (0) SOC , the phonon assisted contribution may potentially have a deeply profound influence on the electronic structure.The reason is that the coupling to phonons by all means generates manybody effects, that is, electronic correlations, which may lead to spontaneous symmetry breaking of the electrons.
To see this, consider the influence from the chiral phonons on the magnetic moment M = M k dk/Ω.In the first order approximation of the lesser Green function, G < k ≈ g < k + g r k Σ (HF),< k g a k , the k-resolved spin-moment reduces to the expression (Ω = (2π) 3 ) where ω⊥q = |(ω (x)  1q , ω 1q , 0)|/ω 1q .By integrating out the phases φ q and θ q , defining the spintexture Kkq = π 0 2π 0 K kq dφ q sin θ q dθ q /(2π) 2 , under the assumption that V(q) = V(|q|), the resulting kernel is a regular function of the electronic momentum k.This is shown in Fig. 1, where Kkq is plotted as function of the polar and azimuthal angles θ k and φ k , respectively, for (a)-(c) achiral and (d)-(f) chiral phonons.
Both for achiral and chiral phonons, there is a vibrationally assisted spin-orbit coupling induced in the electronic structure, which follows on general grounds and has been discussed in different context, e.g., Ref. [41].The absence of a z-component for achiral phonons in the example given in Fig. 1 can be understood through the following observation.
The denominator in K kq , see Eq. (15b), depends on the difference between the phononic and electronic phases φ q and φ k , respectively, as [A kqss − B kq cos(φ q −φ k )] 2 , where A kqss = q 2 /2 + s ω qs − kp cos θ k cos θ q and B kq = kp sin θ k sin θ q .This is an even function of the phase difference φ q − φ k on the interval [0, 2π], while both | q • q| 2 and k × q can be written as sum of even and odd functions of φ q − φ k .For a the polarization q = (0, 0, 1), the numerator of the longitudinal spin-texture K (z)  kq is proportional to sin(φ q − φ k ), which is odd on the interval [0, 2π].Hence, 2π 0 K (z)  kq dφ q /2π = 0.For chiral phonons, ω⊥q 0, which leads to that the numerator of K (z)  kq also contains a term which is proportional to sin(φ q − φ k ) cos φ q = sin(φ q − φ k )[cos φ k sin(φ q − φ k ) + sin φ k cos(φ q − φ k )], where the first (second) contribution is an even (odd) function of φ q − φ k .Therefore, 2π 0 K (z)  kq dφ q /2π 0 for chiral phonons.The above discussion pertains to longitudinally polarized phonons.For transversely (xy-plane) polarized phonons, the vibrationally assisted spin-orbit coupling generates a nonvanishing spin-texture for all K(i) kq , i = x, y, z.However, the longitudinal spin-texture K(z) kq induced for achiral phonons polarized along some transverse axis, e.g., x, exactly cancels the spin-texture induced for phonons polarized along its orthogonal transverse axis, e.g., ŷ.Chiral phonons, on the other hand, breaks this symmetry such that there remains a net longitudinal spin-texture after summing over all phonon modes.
The conclusion is, that a vibrationally assisted spin-orbit coupling exists for any type of phonons which, in turn, gives rise to a transversal spin-texture.However, a net longitudinal spin-texture only arises in presence of chiral phonons.Nevertheless, as is illustrated in Fig. 1, no magnetic moment is generated from this spin-texture since its angular dependencies integrate to zero.
A manifestation of the chiral phonon induced spintexture is the presence of spin-currents.The spin-current can, in the current approximation, be written as J(r) = (−i)spσ kG < k (ω)dkdω/4πΩ = k M k dk/Ω, where the last expression vividly suggests a drift of the spin-texture.This drift is a consequence of the spin-orbit coupling and exists under general conditions, including equilibrium.However, a net spin flow can only be induced by chiral phonons, not by achiral phonon.This is illustrated in Fig. 2, where the non-zero components of the spin-current tensor are plotted as function of the mode mixing parameter ω 1 for different phonon polarizations.Achiral phonons are represented by ω 1 = 0, at which condition there is no induced spin-current, regardless of the phonon polarization.
By contrast, chiral phonons induces spin-currents for which the spatial distribution depends on the polarization.As can be seen in Fig. 2, for phonons polarized solely in the longitudinal or the transverse directions, the spin-currents flow between these directions only.For phonons polarized along some direction with both longitudinal and transverse components, the spin-currents flow between all these.However, in general, the diagonal spin-currents, J ii , i = x, y, z, are negligible which is expected for a non-collinear spin-texture.
The induced spin-currents would be measurable as a spin accumulation at, e.g, a terminating surface, by exposing the Non-zero components of the spin-current tensor as function of the mixing ω 1 for phonons polarized along (a) q = (0, 0, 1), (b) /2.Other parameters are as in Fig. 1.
system to an accelerating force.For instance, a spatially weakly inhomogenous temperature, such that T = T (r) ≈ T 0 + (r − r 0 ) • ∇ r T (r) |r 0 , where T 0 = T (r 0 ) represents the reference temperature, provides an accelerating field acting on the electrons via the temperature gradient ∇T .Then, to linear order in this field, the longitudinal spin-current partitions into two contributions.The first of these derive from the electronic temperature variations k B β 2 ([ε k − ε F ]/4) cosh −2 (β[ε k − ε F ]/2)(r − r 0 )•∇T .This contribution gives the expected drift of electrons around the Fermi level ε F .
The second contribution derives from the thermal variations in the phonon distribution, providing accelerating forces involving the processes ss k B β 2 ω qs (1 + sω ⊥q cos φ q )[2 sinh(s βω qs /2)(ε q + s ω qs − k • q)] −2 (r − r 0 ) • ∇T .The factor s makes reference to the absorption (s = −1) and emission (s = 1) processes, whereas s refers to the effective chirality.Ultimately, the phonon driven contribution to the drift current results from the net of the two opposite chiralities, c.f. Eq. ( 4).The theoretical model predicts that the induced spin-currents should result in a spin accumulation at some interface surface intersecting the current, in agreement with recent experimental observations [39].
In conclusion, it has been demonstrated that the angular momentum carried by chiral phonons may be transferred to electrons via a phase exchange.The phonon chirality translates to the electron spin in the sense that it generates a nontrivial spin-texture including circulating spin-currents.
The author thanks the CISSors for helpful and construc-