Magnetic oscillations induced by phonons in non-magnetic materials

An unexpected finding two decades ago demonstrated that Shockley electron states in noble metal surfaces are spin-polarized, forming a circulating spin texture in reciprocal space. The fundamental role played by the spin degree of freedom was then revealed, even for a non-magnetic system, whenever the spin-orbit coupling was present with some strength. Here, we demonstrate that, similarly to electrons in the presence of spin-orbit coupling, the propagating vibrational modes are also accompanied by a well-defined magnetic oscillation even in non-magnetic materials. Although this effect is illustrated by considering a single layer of the WSe2 dichalogenide, the phenomenon is completely general and valid for any non-magnetic material with spin-orbit coupling. The emerging phonon-induced magnetic oscillation acts as an additional effective flipping mechanism for the electron spin and its implications in the transport and scattering properties of the material are evident and profound. The authors show that, similarly to electrons, in the presence of spin-orbit coupling, phonons can induce an effective magnetic oscillation even in non-magnetic materials, and therefore can act as an additional spin-flipping mechanism in electron-phonon physics.

I n materials science, the most conventional point of view is to assume that propagating vibrational collective modes (phonons) are not associated with any magnetic property if the material itself is non-magnetic. In a magnetic material, however, it is natural to expect a magnetic oscillation associated to a phonon mode. Phonons are commonly understood as sinusoidal patterns of atomic displacements, which couple to electron states by the scalar potential induced by these atomic displacements. Electrons have a well defined spin-polarization under spin-orbit interaction, but when time reversal symmetry applies the spin polarizations for opposite momenta cancel each other and the material results to be non-magnetic. There is a clear evidence that the lattice thermal conductivity of diamagnetic materials couples to external magnetic fields 1 , which in principle might seem contrary to the idea that phonons do not have any associated magnetic property. Here we show that similarly to electrons, phonons are also accompanied by an induced effective magnetic oscillation when spin-orbit coupling is present even for nonmagnetic materials.

Results
In a solid with spin-orbit coupling, electron states are described by two-component spinor wave functions for each k point spanning the Brillouin zone (BZ), Ψ k;i ðrÞ ¼ u " k;i ðrÞ u # k;i ðrÞ e ikr . Each electron state has an associated spin-polarization defined as the expectation value of the Pauli vector m k;i ðrÞ ¼ hΨ k;i jσjΨ k;i i. Time reversed Kramers pairs at k and −k have opposite spinpolarizations that exactly cancel out when integrated, which therefore implies no magnetism. However, the electron spinpolarization is a crucial physical magnitude in many nonmagnetic systems with spin-orbit coupling, one of the most outstanding being probably its role in the topological properties of matter. A phonon excitation consists in a sinusoidal displacement of atoms and an induced (almost) static response of the electron gas which tries to weaken or screen out the electric perturbation generated by these displacements. Therefore, the question of whether a phonon perturbation creates a magnetic oscillation could be suitably treated considering a generalized dielectric theory of dimension 4 × 4 mixing the magnetic and electric components of the potential 2 . An alternative and more transparent way to see whether an overall magnetic property emerges is to consider the effect of the perturbation on each electron spinor wave function and then integrate over the BZ.
The key point is that if a phonon is excited with momentum q it couples differently with the electrons at k and −k, the result being that the spin-polarization of electrons with time reversed momenta do not cancel each other. Under these conditions the balance of the electron spin-polarizations within the BZ is broken and the BZ integral gives a finite value and, therefore, a net real-space magnetic oscillation with the same wave number as the phonon q.
Let us focus on a single frozen phonon-mode (ν) with momentum q, which produces a perturbation on both components of the periodic part of the electron spinor wave functions, δ q;ν u σ k;i ðrÞ, where δ q;ν denotes the self-consistent variation in the context of density-functional perturbation theory 3,4 and u σ k;i ðrÞ is the periodic part of each spinor component. The unit cell periodic part of the amplitude of the frozen charge-spin density wave is obtained by integrating the contributions from all occupied electron states (see Supplementary Note 1) Therefore, δñ σ;σ 0 q;ν ðrÞe iqr would represent the complete oscillation wave in real space. The absence of a magnetic component accompanying a phonon mode would require the off-diagonal elements (σ ≠ σ 0 ) to be zero and that both diagonal components are equal to each other, which, in general, are conditions only fulfilled at the Γ point (q = 0). Writing the 2 × 2 charge-spin matrix of Eq. (1) in terms of Pauli matrices allows to explicitly distinguish the electronic charge, δñ 0 q;ν ðrÞ, and the magnetic components, δ e M q;ν ðrÞ ¼ ðδñ 1 q;ν ðrÞ; δñ 2 q;ν ðrÞ; δñ 3 q;ν ðrÞÞ: In real space, the time-dependent charge-spin field is given by the real part of the above frozen complex amplitudes when accounting for the classical motion of atoms. For a single phonon mode (q, ν) of energy ω q;ν we have concluding that the appearance of an induced spin-density (or magnetization indistinctly) ðδ e M q;ν ðrÞÞ is completely general for crystals with spin-orbit coupling since the only requirement is a non-trivial pattern of the spin-polarization within the BZ associated to the absence of inversion symmetry 5 . The phonon modes break the symmetry of the BZ in a way that the electron spin-polarization is modulated within the BZ producing a net spin accumulation. The similarity of the phonon magnetism and the electron spin-polarization with timereversal symmetry (no net magnetism) is strengthened by the fact that time-reversed phonon momenta give strictly opposite magnetization exactly in the same way as for electrons δM q;ν ðr; tÞ ¼ ÀδM Àq;ν ðr; tÞ.
The above description of the spin-charge field induced by phonons is completely classical and focussed on a single phonon with a fixed momentum. Therefore, physically δM q;ν ðr; tÞ would be the time dependent magnetization associated with a single coherent phonon mode. In general the vector field defined by δM q;ν ðr; tÞ shows an interesting real-space and time dependent non-collinear pattern, which depends also on the particular atomic displacements (polarization vectors) associated with each phonon branch. Actually, it is the motion of the W atom, i.e., the atom for which the spin-orbit interaction is dominant, which determines the direction of the magnetization. For instance, for q = K and for a mode in which Se atoms rotate clockwise with opposite phase in the plane of the surface (x-y plane) and W atoms vibrate in the perpendicular direction (z) to the surface, a net circularly polarized induced magnetization appears in the surface plane around the W atoms (Fig. 1a-c and Supplementary Movie 1). However, for the same phonon propagation vector q = K, if W atoms rotate clockwise around their equilibrium positions in the plane of the surface, the net magnetization shows along the perpendicular direction to the plane (Fig. 2a-c

and Supplementary Movie 2).
It is noteworthy that the magnitude of the induced magnetization is only an order of magnitude smaller than in the induced (scalar) charge (Fig. 2b). Since q ¼ K 0 is the timereversed momentum of q = K, as mentioned earlier, the realspace magnetization should be opposite in sign. For the first example given above but in the case in which q ¼ K 0 , Se atoms rotate anticlockwise in the plane and the W vibrates perpendicular to the plane (Supplementary Movie 3). Taking a snapshot in time for which the atomic positions coincide with those in Fig. 1 demonstrates that the magnetization shows exactly opposite chirality, and therefore the relation δM K;ν ðr; tÞ ¼ ÀδM ÀK;ν ðr; tÞ is fulfilled (Supplementary Fig. 1). Note that the propagation of the atomic displacements of W in the perpendicular direction to the plane along q = K (Fig. 1b) is exactly the same as the one along q = K′ when it is looked from right to left ( Supplementary Fig. 1b). A similar situation occurs for the second example, where the chirality of the atomic displacements reverses when changing q = K to its time-reversed value 6 , which again gives a magnetization opposite in sign (Supplementary Movie 4) and which can be compared to that of Fig. 2 when the atomic positions are frozen to be the same ( Supplementary Fig. 2). The non-collinear character in space and time of the magnetization is also observed when the atomic displacements are all linear. For instance, for an acoustic phonon with vector q = M at which Se atoms move along q and W atoms vibrate in the perpendicular direction to the plane (Supplementary Movie 5) a chiral magnetization pattern similar to that of Fig. 1 is found (Supplementary Fig. 3). Instead, when all atoms oscillate linearly in the plane at right angles to the propagation vector q = M (Supplementary Movie 6), the magnetization appears in the perpendicular direction to the plane as in Fig. 2 (Supplementary Fig. 4). Note that for this acoustic mode when all atoms go through their equilibrium positions both, the induced charge and spindensity fields, disappear. All the magnetization patterns show a periodicity in real space according to the wave number q of the propagation of the excited phonon, as they should. At this point it is worth mentioning that the magnetic polarization of the electron gas as described in this manuscript does not have a relation with the angular momentum of the atoms as described by Zhang et al. 7 . In our theory linearly polarized phonons with null angular momentum give a finite and meaningful contribution to the magnetization. It is therefore clear that the physics described in 7 is different and not connected to the spin response of the electron gas as described in the present work.
The size of the fluctuations of the real space unit-cell average of these oscillations gives an order of the magnitude of this effect, even though it does not capture all the detailed structure in real space. It is nevertheless physically meaningful, allowing to analyze the momentum and mode dependence at the same time, and making a connection with the possibility of experimental detection as it will be shown shortly. More specifically, for a given phonon q the root-mean-square (RMS) of the time dependence of the periodic part of this quantity reflects the overall amplitude of the spin-density associated to a single phonon mode. For each cartesian component α we have (see Supplementary Note 2): The above classical RMS amplitudes of the magnetization are directly connected to the charge-charge, spin-charge and spin-spin components of the dynamic structure factor, S α,β (ω, q + G) (see Supplementary Note 2), which is accessed by inelastic neutron scattering, inelastic X-ray spectroscopy and spin-polarized electron energy loss spectroscopy 8 . As van Hove first pointed out 9 , the dynamic structure factor is the space and time Fourier transform of the density-density correlation function hδn α ðr; tÞδn β ðr 0 ; 0Þi T . If we consider a 4-dimensional spin-charge quantized field δnðr; tÞ ¼ X qν a y q;ν e iω q;ν t δe n Ã q;ν ðrÞ þ a q;ν e Àiω q;ν t δe n q;ν ðrÞ ; ð4Þ where a q;ν and a y q;ν are creation and annihilation operators for a phonon mode (q; ν) with energy ω q;ν (Fig. 3a), then the dynamic structure factor can be written as S α;β ðω; q þ GÞ ¼ ½1 þ f B ðω q;ν Þδ ẽ n α q;ν ðGÞδ ẽ n βÃ q;ν ðGÞδðω À ω q;ν Þ þ f B ðω q;ν Þδ ẽ n αÃ q;ν ðGÞδ ẽ n β q;ν ðGÞδðω þ ω q;ν Þ; ð5Þ where δ ẽ n α q;ν ðGÞ indicates the Fourier transform or crystal field components of the real-space complex amplitudes δñ α q;ν ðrÞ, f B (ω q;ν ) denotes phonon occupation numbers and where we ignore the Debye-Waller factor 10 . Taking the G = 0 components (unit-cell average), it is easily seen that the classical RMS terms of Eq. ) and perpendicular (δM z q;ν ) to the WSe 2 layer shown in Fig. 3b, c, respectively, are basically the spin contributions to the structure factor connected to a given phonon mode (q; ν). Said in other words, Fig. 3b, c may be interpreted as the momentum/energy and phonon mode resolved contributions to the spin sector of the dynamic structure factor depicted along the high symmetry lines of the surface Brillouin zone. The magnetic character associated inherently to phonons as proposed in this work is therefore accessible by means of any experimental setup probing the spin components of the dynamical structure factor in the energy ranges corresponding to phonons.

Discussion
We conclude that in any material with a non-trivial spin-pattern within the Brillouin zone, even if non-magnetic, phonon modes are connected inherently to a magnetic property analogous to the electron spin-polarization and it can be stated quite generally that phonon modes are accompanied by an induced spin-density (magnetization), which is rich in real space details. It is also shown that this magnetic modulation is only one order of magnitude weaker than the purely electrostatic (spin-diagonal) terms. All the above physics is illustrated convincingly for WSe 2 in a mode by mode analysis where the real space and time dependence of the induced magnetization is revealed for the most relevant modes. The implications are extensive and profound because phonons, which are now intrinsically attached to an effective magnetic moment, should be understood as an additional spinflip mechanism even for materials without a net magnetic moment. This means that the whole electron-phonon physics is modulated in every system with spin-orbit coupling and, for instance, even electron backscattering events may be aided by the phonon magnetic moment. Experimental detection of magnetic oscillations for coherent phonons should be done by ultrafast probes and our calculated details of these fields may indicate a detection strategy. However, we also show that probing the spin components of the dynamical structure factor may be an alternative route to measure what we could name as the spin polarization of phonons.  Fig. 3a, c. This mode is composed by clockwise rotations of the W atoms (circles) around their equilibrium positions and in-phase anticlockwise rotations of the Se atoms located above (triangle up) and below (triangle down) the W plane. The color code represents the magnetization in the perpendicular direction. The displacements of the atoms have been scaled to make them visible. b Induced electronic charge for the same atomic configuration as in panel a. Note that the induced magnetization is only an order of magnitude smaller than the induced (scalar) charge. c Side view of the WSe 2 formula-unit in the lower left corner unit-cell. The names of the atoms display their displacements from the equilibrium positions, denoted as in a. This figure is a snapshot of the time evolution of the induced magnetization for this mode (Supplementary Movie 2).