Magnetic oscillations induced by phonons in non-magnetic materials
MMagnetic oscillations induced by phonons in non-magnetic materials
Idoia G. Gurtubay , , Aitzol Iturbe-Beristain , , Asier Eiguren , Condensed Matter Physics Department, Science and Technology Faculty, University of theBasque Country UPV/EHU, PK 644, E-48080 Bilbao, Basque Country, Spain. Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, E-20018 Donostia-San Sebasti´an, Spain. Correspondence and requests for materials should be addressed to A.E.(email:[email protected])
An unexpected finding two decades ago demonstrated that Shockley electron states in noblemetal surfaces are spin-polarized, forming a circulating spin texture in reciprocal space. Thefundamental 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. Herewe demonstrate that similarly to electrons in the presence of spin-orbit coupling, the propa-gating vibrational modes are also accompanied by a well-defined magnetic oscillation even innon-magnetic materials. Although this effect is illustrated by considering a single layer of theWSe dichalogenide, the phenomenon is completely general and valid for any non-magneticmaterial with spin-orbit coupling. The emerging phonon-induced magnetic oscillation actsas an additional effective flipping mechanism for the electron spin and its implications in thetransport and scattering properties of the material are evident and profound.Introduction In materials science, the most conventional point of view is to assume that propagating vibrationalcollective modes (phonons) are not associated with any magnetic property if the material itself isnon-magnetic. In a magnetic material, however, it is natural to expect a magnetic oscillation as-sociated to a phonon mode. Phonons are commonly understood as sinusoidal patterns of atomicdisplacements which couple to electron states by the scalar potential induced by these atomic dis-placements. Electrons have a well defined spin-polarization under spin-orbit interaction, but whentime reversal symmetry applies the spin polarizations for opposite momenta cancel each other andthe material results to be non-magnetic. There is a clear evidence that the lattice thermal conduc-tivity of diamagnetic materials couples to external magnetic fields [1], which in principle mightseem contrary to the idea that phonons do not have any associated magnetic property. Here weshow that similarly to electrons, phonons are also accompanied by an induced effective magneticoscillation when spin-orbit coupling is present even for non-magnetic materials.1 a r X i v : . [ c ond - m a t . o t h e r] J a n esults In a solid with spin-orbit coupling, electron states are described by two-component spinor wavefunctions for each k point spanning the Brillouin zone (BZ), Ψ k ,i ( r ) = (cid:32) u ↑ k ,i ( r ) u ↓ k ,i ( r ) (cid:33) e i kr . Eachelectron state has an associated spin-polarization defined as the expectation value of the Paulivector m k ,i ( r ) = (cid:104) Ψ k ,i | σ | Ψ k ,i (cid:105) . Time reversed Kramers pairs at k and − k have opposite spin-polarizations that exactly cancel out when integrated, which therefore implies no magnetism. How-ever, the electron spin-polarization is a crucial physical magnitude in many non-magnetic systemswith spin-orbit coupling, one of the most outstanding being probably its role in the topologicalproperties 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 perturbationgenerated by these displacements. Therefore, the question of whether a phonon perturbation cre-ates a magnetic oscillation could be suitably treated considering a generalized dielectric theory ofdimension × mixing the magnetic and electric components of the potential [2]. An alternativeand more transparent way to see whether an overall magnetic property emerges is to consider theeffect of the perturbation on each electron spinor wave function and then integrate over the BZ. Thekey point is that if a phonon is excited with momentum q it couples differently with the electronsat k and − k , the result being that the spin-polarization of electrons with time reversed momenta donot cancel each other. Under these conditions the balance of the electron spin-polarizations withinthe BZ is broken and the BZ integral gives a finite value and, therefore, a net real-space magneticoscillation 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 pertur-bation on both components of the periodic part of the electron spinor wave functions, δ q ,ν u σ k ,i ( r ) ,where δ q ,ν denotes the self-consistent variation in the context of density-functional perturbationtheory [3, 4] and u σ k ,i ( r ) is the periodic part of each spinor component. The unit cell periodic partof the amplitude of the frozen charge-spin density wave is obtained by integrating the contributionsfrom all occupied electron states (see Supplementary Note 1) δ ˜ n σ,σ (cid:48) q ,ν ( r ) = occ (cid:88) k ,i (cid:104) u σ (cid:48) k ,i ( r ) | δ q ,ν u σ k ,i ( r ) (cid:105) . (1)Therefore, δ ˜ n σ,σ (cid:48) q ,ν ( r ) e i qr would represent the complete oscillation wave in real space. The absenceof a magnetic component accompanying a phonon mode would require the off-diagonal elements( σ (cid:54) = σ (cid:48) ) 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 = ). Writing the × charge-spin matrix of equa-tion (1) in terms of Pauli matrices allows to explicitly distinguish the electronic charge, δ ˜ n q ,ν ( r ) ,2nd the magnetic components, δ ˜M q ,ν ( r ) = ( δ ˜ n q ,ν ( r ) , δ ˜ n q ,ν ( r ) , δ ˜ n q ,ν ( r )) : (cid:110) δ ˜ n σ,σ (cid:48) q ,ν ( r ) (cid:111) → δ ˜n q ,ν ( r ) = δ ˜ n q ,ν ( r ) σ + δ ˜ n q ,ν ( r ) σ + δ ˜ n q ,ν ( r ) σ + δ ˜ n q ,ν ( r ) σ = δ ˜ n q ,ν ( r ) σ + δ ˜M q ,ν ( r ) σ . (2)In real space, the time-dependent charge-spin field is given by the real part of the abovefrozen complex amplitudes when accounting for the classical motion of atoms. For a single phononmode ( q , ν ) of energy ω q ,ν we have δ n q ,ν ( r , t ) = Re (cid:104)(cid:16) δ ˜ n q ,ν ( r ) σ + δ ˜M q ,ν ( r ) σ (cid:17) e i ( qr − ω q ,ν t ) (cid:105) . = δn q ,ν ( r , t ) σ + δ M q ,ν ( r , t ) σ , concluding that the appearance of an induced spin-density (or magnetization indistinctly) ( δ ˜M q ,ν ( r )) 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 sym-metry [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 ofthe phonon magnetism and the electron spin-polarization with time-reversal symmetry (no netmagnetism) is strengthened by the fact that time-reversed phonon momenta give strictly oppositemagnetization exactly in the same way as for electrons δ M q ,ν ( r , t ) = − δ M − q ,ν ( r , t ) .The above description of the spin-charge field induced by phonons is completely classicaland focussed on a single phonon with a fixed momentum. Therefore, physically δ M q ,ν ( r , t ) wouldbe the time dependent magnetization associated with a single coherent phonon mode. In generalthe vector field defined by δ M q ,ν ( r , t ) 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 forwhich 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 inthe plane of the surface ( x - y plane) and W atoms vibrate in the perpendicular direction ( z ) to thesurface, a net circularly polarized induced magnetization appears in the surface plane around theW atoms (Figs. 1 a - c and Supplementary Movie 1). However, for the same phonon propagationvector q = K , if W atoms rotate clockwise around their equilibrium positions in the plane of thesurface, the net magnetization shows along the perpendicular direction to the plane (Figs. 2 a - c and Supplementary Movie 2). It is noteworthy that the magnitude of the induced magnetization isonly an order of magnitude smaller than in the induced (scalar) charge (Fig. 2 b ). Since q = K (cid:48) isthe time-reversed momentum of q = K , as mentioned earlier, the real-space magnetization shouldbe opposite in sign. For the first example given above but in the case in which q = K (cid:48) , Se atomsrotate anticlockwise in the plane and the W vibrates perpendicular to the plane (SupplementaryMovie 3). Taking a snapshot in time for which the atomic positions coincide with those in Fig. 1demonstrates that the magnetization shows exactly opposite chirality, and therefore the relation3 M K ,ν ( r , t ) = − δ M − K ,ν ( r , t ) is fulfilled (Supplementary Figure 1). Note that the propagationof the atomic displacements of W in the perpendicular direction to the plane along q = K (Fig. 1 b )is exactly the same as the one along q = K (cid:48) when it is looked from right to left (SupplementaryFigure 1 b ). A similar situation occurs for the second example, where the chirality of the atomicdisplacements reverses when changing q = K to its time-reversed value [6] which again gives a mag-netization opposite in sign (Supplementary Movie 4) and which can be compared to that of Fig. 2when the atomic positions are frozen to be the same (Supplementary Figure 2). The non-collinearcharacter in space and time of the magnetization is also observed when the atomic displacementsare all linear. For instance, for an acoustic phonon with vector q = M at which Se atoms movealong q and W atoms vibrate in the perpendicular direction to the plane (Supplementary Movie5) a chiral magnetization pattern similar to that of Fig. 1 is found (Supplementary Figure 3). In-stead, 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 asin Fig. 2 (Supplementary Figure 4). Note that for this acoustic mode when all atoms go throughtheir equilibrium positions both, the induced charge and spin-density fields, disappear. All themagnetization patterns show a periodicity in real space according to the wave number q of thepropagation of the excited phonon, as they should. At this point it is worth mentioning that themagnetic polarization of the electron gas as described in this manuscript does not have a relationwith the angular momentum of the atoms as described by Zhang et al. in [7]. In our theory linearlypolarized phonons with null angular momentum give a finite and meaningful contribution to themagnetization. It is therefore clear that the physics described in [7] is different and not connectedto 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 anorder of the magnitude of this effect, even though it does not capture all the detailed structurein real space. It is nevertheless physically meaningful, allowing to analyze the momentum andmode dependence at the same time, and making a connection with the possibility of experimentaldetection 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 amplitudeof the spin-density associated to a single phonon mode. For each cartesian component α we have(see Supplementary Note 2): δM α q ,ν = (cid:118)(cid:117)(cid:117)(cid:116)(cid:42)(cid:18) Re (cid:90) Ω d r Ω (cid:104) δ ˜ M α q ,ν ( r ) e − iω q ,ν t (cid:105)(cid:19) (cid:43) T . (3)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-Rayspectroscopy and spin-polarized electron energy loss spectroscopy [8]. As van Hove first pointedout [9], the dynamic structure factor is the space and time Fourier transform of the density-densitycorrelation function (cid:104) δ ˆ n α ( r , t ) δ ˆ n β ( r (cid:48) , (cid:105) T . If we consider a 4-dimensional spin-charge quantized4eld δ ˆ n ( r , t ) = (cid:88) q ν (cid:0) a † q ,ν e iω q ,ν t δ ˜n ∗ q ,ν ( r ) + a q ,ν e − iω q ,ν t δ ˜n q ,ν ( r ) (cid:1) , (4)where a q ,ν and a † q ,ν are creation and annihilation operators for a phonon mode ( q , ν ) with energy ω q ,ν (Fig. 3 a ), then the dynamic structure factor can be written as S α,β ( ω, q + G ) = [1 + f B ( ω q ,ν )] δ ˜˜ n α q ,ν ( G ) δ ˜˜ n β ∗ q ,ν ( G ) δ ( ω − ω q ,ν ) (5) + f B ( ω q ,ν ) δ ˜˜ n α ∗ q ,ν ( G ) δ ˜˜ n β q ,ν ( G ) δ ( ω + ω q ,ν ) , where δ ˜˜ n α q ,ν ( G ) indicates the Fourier transform or crystal field components of the real-space com-plex amplitudes δ ˜ n α q ,ν ( r ) , f B ( ω q ,ν ) denotes phonon occupation numbers and where we ignore theDebye-Waller factor [10]. Taking the G = components (unit-cell average), it is easily seen thatthe classical RMS terms of equation (3) are proportional to the diagonal ( α = β ) spectral con-tributions to equation (5) for individual phonons: (cid:0) δM α q ,ν (cid:1) ∼ δ ˜˜ n α ∗ q ,ν ( ) δ ˜˜ n α q ,ν ( ) . This helps tophysically interpret the RMS of the induced magnetization defined as above because the terms par-allel ( (cid:113) ( δM x q ,ν ) + ( δM y q ,ν ) ) and perpendicular ( δM z q ,ν ) to the WSe layer shown in Fig. 3 b andFig. 3 c , respectively, are basically the spin contributions to the structure factor connected to a givenphonon mode ( q , ν ) . Said in other words, Fig. 3 b and Fig. 3 c may be interpreted as the momen-tum/energy and phonon mode resolved contributions to the spin sector of the dynamic structurefactor depicted along the high symmetry lines of the surface Brillouin zone. The magnetic char-acter associated inherently to phonons as proposed in this work is therefore accessible by meansof any experimental setup probing the spin components of the dynamical structure factor in theenergy ranges corresponding to phonons. Discussion
We conclude that in any material with a non-trivial spin-pattern within the Brillouin zone, evenif non-magnetic, phonon modes are connected inherently to a magnetic property analogous to theelectron spin-polarization and it can be stated quite generally that phonon modes are accompaniedby an induced spin-density (magnetization) which is rich in real space details. It is also shown thatthis 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 in a mode by modeanalysis where the real space and time dependence of the induced magnetization is revealed for themost relevant modes. The implications are extensive and profound because phonons, which arenow intrinsically attached to an effective magnetic moment, should be understood as an additionalspin-flip mechanism even for materials without a net magnetic moment. This means that the wholeelectron-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. Experimentaldetection of magnetic oscillations for coherent phonons should be done by ultrafast probes andour calculated details of these fields may indicate a detection strategy. However, we also show5hat probing the spin components of the dynamical structure factor may be an alternative route tomeasure what we could name as the spin polarization of phonons. Data availability
The data that support the findings of this study are available from the correspondingauthor upon reasonable request. [1] Jin, H. et al.
Phonon-induced diamagnetic force and its effects on the lattice thermal conduc-tivity.
Nature Materials , 601 (2015).[2] Lafuente-Bartolome, J., Gurtubay, I. G. & Eiguren, A. Relativistic response and novel spin-charge plasmon at the Tl/Si(111) surface. Phys. Rev. B , 035416 (2017).[3] Baroni, S., de Gironcoli, S., Dal Corso, A. & Giannozzi, P. Phonons and related crystalproperties from density-functional perturbation theory. Rev. Mod. Phys. , 515–562 (2001).[4] Dal Corso, A. Density functional perturbation theory for lattice dynamics with fully rela-tivistic ultrasoft pseudopotentials: Application to fcc-Pt and fcc-Au. Phys. Rev. B , 054308(2007).[5] LaShell, S., McDougall, B. A. & Jensen, E. Spin splitting of an Au(111) surface state bandobserved with angle resolved photoelectron spectroscopy. Phys. Rev. Lett. , 3419–3422(1996).[6] Zhu, H. et al. Observation of chiral phonons.
Science , 579–582 (2018).[7] Zhang, L. & Niu, Q. Angular momentum of phonons and the Einstein–de Haas effect.
Phys.Rev. Lett. , 085503 (2014).[8] Sturm, K. Dynamic Structure Factor: An Introduction.
Zeitschrift fr Naturforschung A ,233–242 (1993).[9] Van Hove, L. Correlations in space and time and Born approximation scattering in systemsof interacting particles. Phys. Rev. , 249–262 (1954).[10] Jancovici, B. Infinite susceptibility without long-range order: The two-dimensional harmonic“solid”. Phys. Rev. Lett. , 20–22 (1967). Acknowledgements
The authors acknowledge the Department of Education, Universities and Research ofthe Basque Government and UPV/EHU (Grant No. IT756-13), the Spanish Ministry of Economy and Com-petitiveness MINECO (Grant No. FIS2016-75862-P) and the University of the Basque Country UPV/EHU(Grant No. GIU18/138) for financial support. Computer facilities were provided by the Donostia Interna-tional Physics Center (DIPC). uthor contributions A.E. conceived the ideas. A.I. contributed to the early stages of the calculations.I.G.G. and A.E. carried out the calculations, discussed the results and contributed to the writing of themanuscript.
Competing interests
The authors declare no competing interests. ) b)z displacement of W along q=K − . − . − . − . . . . . . z d i s p l a ce m e n t o f a t o m s ( a ) c) Figure 1:
Induced magnetization for the second lowest energy acoustic phonon mode at q = K involving pure out of plane displacements of W atoms a) Real-space representation of themagnetization in the plane of the W atoms for × unit cells along the hexagonal axes of WSe for the second lowest energy mode at q = K (in orange in Figs. 3 a and 3 b ). In this modethe W atoms (filled circles) displace along the perpendicular direction (see colour-bar) and theSe atoms above (filled triangle up) and below (filled triangle down) the W plane rotate clockwisewith opposite phase around their equilibrium positions (crosses) in their respective planes. Thecoloured vector-field is proportional to the in-plane magnetization at each point in real space, withyellow/light (blue/dark) arrows representing the largest (smallest) values. These arrows as wellas the displacements of the Se atoms have been scaled to make them visible. b) The colouredarrows give the z displacement of the W atoms along the q = K direction (dotted magenta line inpanel a) ) according to the colour-bar. The dashed line describes the propagation of the vibrationalong several unit-cells in real space. Note that K = [1 / , / , in crystal axes, and hence theperiodicity of the wave. c) Side view of the WSe formula-unit in the lower left corner unit-cell.The names of the atoms display their displacements from the equilibrium positions, denoted as inpanel a) . This figure is a snapshot of the time evolution of the induced magnetization for this mode(Supplementary Movie 1). 8 ) − . − . − . − . . . . . . M a g n e t i z a t i o n a l o n g z i n t h e W p l a n e ( − µ B ) b) − − − − Charge fluctuation in the W plane ( − au) c) Figure 2:
Induced magnetization for the highest energy acoustic phonon mode at q = K involving in-plane displacements of the W atoms a) Real-space representation of the perpendic-ular component of the magnetization in × unit cells along the hexagonal axes of WSe for thehighest energy acoustic mode for q = K (represented in green in Figs. 3 a and 3 c ). This mode iscomposed by clockwise rotations of the W atoms (circles) around their equilibrium positions andin-phase anticlockwise rotations of the Se atoms located above (triangle up) and below (triangledown) the W plane. The colour code represents the magnetization in the perpendicular direction.The displacements of the atoms have been scaled to make them visible. b) Induced electroniccharge for the same atomic configuration as in panel a) . Note that the induced magnetization isonly an order of magnitude smaller than the induced (scalar) charge. c) Side view of the WSe formula-unit in the lower left corner unit-cell. The names of the atoms display their displace-ments from the equilibrium positions, denoted as in panel a) . This figure is a snapshot of the timeevolution of the induced magnetization for this mode (Supplementary Movie 2).9 q , ( m e V ) a) | M x q , | + | M y q , | ( B ) b) M K M K' | M z q , | ( B ) c) KMK q=K q=Kq=K q=K q=K q=K Figure 3:
Mode and momentum resolved magnetization induced by lattice vibrations inmonolayer WSe a) Phonon energy spectrum along high symmetry lines in the surface Brillouinzone (inset). b) and c) Plane ( x , y ) and out-of-plane ( z ) components of the magnitude of the unit-cell average magnetization for each phonon mode in a) and with the same colour convention. Insetsshow the corresponding polarization vectors for q = K , the length of the arrows being proportionalto the magnitude of the atomic displacements. Vertical arrows represent linear displacements in theperpendicular direction to the plane, and semicircular arrows show circular displacements of theatoms around their equilibrium positions. The direction of the magnetization is determined by themotion of the W atom. When W vibrates in the perpendicular direction of the plane, it induces a netmagnetization in the plane (panel b) ). However, when W atoms move on the plane (with circularpolarization for q = K ), the induced magnetization appears in the perpendicular direction (panel c) ). For the three middle modes in the phonon spectrum W atoms move significantly less, yielding asmaller magnetization. Panels b) and c)c)