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NS(M71) acd rnmn,F320 or,FRANCE Tours, F-37200, Grandmont, de Parc (UMR7013), CNRS NS(M71) acd rnmn,F320 or,FRANCE Tours, F-37200, Grandmont, de Parc (UMR7013), CNRS yai antsrcinfrantiferromagnets for magnetostriction Dynamic ntttDnsPisn nvri´ eTus Universit´e Tours, Universit´e de Poisson, Denis Institut ntttDnsPisn nvri´ eTus Universit´e Tours, Universit´e de Poisson, Denis Institut E A/eRput P1,F320 ot,FAC and FRANCE Monts, F-37260, 16, BP Ripault, DAM/Le CEA 4 . E A/eRput P1,F320 ot,FRANCE Monts, F-37260, 16, BP Ripault, DAM/Le CEA 1 Therefore . aclThibaudeau Pascal hmsNussle Thomas tmNicolis Stam noprtdit osiuiennierlw felectro- of are laws me- bodies materials conductive nonlinear continuum of constitutive of properties into magnetic incorporated perspective the the where chanics, from either viewed o ant-lsi couplings magneto-elastic for models. a multiscale (where developing This for scales need limits). time the sig- its highlights and reaches become anyway length can description, short that continuum very strains; backreaction at mechanical a such nificant the is on it magnetization however the of action iemgei nstoismmcigtesai tan of strains solid static a the mimicking anisotropies magnetic tive nw,mr eeal,udrtehaigo “magneto– of heading the elasticity” under generally, more known, oprsnfrsnl rsasadplcytliesam- polycrystalline and crystals single for comparison ragmnso antcmmns hc nresponse in strains which mechanical non-local moments, produce magnetic to position of atomic response by arrangements mediated magnetic necessarily the is field level; electric in an classical allowed the is at components vacuum, field electromagnetic between separate to be must possible responses–all not elastic simultaneously. is and treated it electric magnetic, relevant, the are that scales ant-lsi hnmn aebe,typically, been, have phenomena Magneto-elastic vni utsaemdl aebe investigated been have models multiscale if Even backre- the investigate not do however, studies, Most ti,ide,ntwrh htn ietlclcoupling local direct no that noteworthy indeed, is, It ‡ ∗ opig h hrceitcoclain of oscillations characteristic the coupling, aaee yais hsi particularly is This dynamics. parameter ne hs pc n htalw sto us allows that and space phase ended 9 nyuigte oddc non-trivial deduce to them using only , erlvn o plctos o h case the for applications, for relevant re i yais sacneuneo such of consequence a As dynamics. tic ini lgtyfse n h amplitude the and faster slightly is tion † . yteculn otesri eeas here ; strain the to coupling the by d h prpit oso rce n a and bracket Poisson appropriate the esses. mnsa iier fanticommuting of bilinears as oments ut eotdi h ieaueadin and literature the in reported sults ecietemgei oet in moments magnetic the describe t 7 . ti osbet vi aigto having avoid to possible is it w oinfaeok htcntake can that framework, tonian so antcmmns that moments, magnetic of cs 8 rtruhteitouto feffec- of introduction the through or , d’Orl´eans, d’Orl´eans, 10 ihgo experimental good with , 5,6 hsrsos is response This . 2 ples11, insights from the framework of localization and ments in terms of anticommuting, Majorana, variables, homogenization are still the mainstays for deducing the whose salient properties are reviewed in appendix A. constitutive laws for polycrystalline media. Moreover, in In section III, we introduce a Poisson bracket, which this context, homogenization methods for multiscale me- allows us to obtain the equations of motion for the time- chanics assume the existence of well–separated scales and evolution of all the dynamical variables. These are, different relations among length scales, that lead to dif- then solved, in section IV using a symplectic integra- ferent effective equations, which, in turn, represent the tion scheme. Special attention is devoted to the stability corresponding different physical effects, appropriate at analysis of the schemes and to the order of the differ- each scale12. ent symplectic operators. This analysis is presented in Renormalization group ideas, of course, are instrumen- appendix B. tal in providing a description of how the dynamics de- Finally in section V we illustrate the formalism for the pends on the scale; what has been lacking, to date, is a case of a simple toy model for NiO, consisting of two unified construction of the dynamics of elastic and spin spins, interacting through antiferromagnetic exchange, degrees of freedom, within a common Hamiltonian ap- and switched by an external STT, as well as under an proach, at any particular scale. In particular, what has, external stress. It is here, in particular, that we show curiously, not been fully exploited is the resolution of the that the formalism is capable of handling the interaction constraints, that the spin degrees of freedom are subject between two domains, taking into account the coupling to, since they are their own canonically conjugate vari- between elastic and magnetic degrees of freedom. ables, as their equations of motion are first order. This means that they can be most usefully expressed as mul- tilinear combinations of anticommuting variables. As we II. THE HAMILTONIAN shall see, this can lead to useful insights, in practice, about the dynamics of the coupled system. In this section we propose a Hamiltonian for the com- While the generalization of the canonical formalism, bined system of elastic and spin degrees of freedom. that incorporates the constraints implied by spin degrees These parts have, of course, been considered before7, so of freedom has been worked out a long time ago13, it we shall review the salient features, before introducing has not been applied to concrete situations of physical our approach. interest, simply because these, up to now, were not of The starting point for the elastic degrees of freedom practical relevance. is the framework of mechanics for elastic solids, within In the present paper we provide a complete description the regime of the approximation of small deformations, of the combined Hamiltonian dynamics of the elastic and supplemented by the assumption of perfect mechanical 16 of the spin degrees of freedom in phase space and show micro-reversibility . These statements imply that an how the anticommuting nature of the spin degrees of free- elastic material is characterized by a collection of N time- dom can be–indirectly–tested. It is the refined control dependant elastic deformations, which are described by a i over these and the dynamical interplay between strain set of symmetric Cauchy strain tensors ǫIJ (t) where I,J and magnetization that are the particular novelties of are spatial indices, ranging from 1 to 3, and assigned to our approach. each lattice site, i ∈{1..N}, of the material. At equilib- The elastic (local mechanical strain) and spin degrees rium, each of these variables is independent of time and of freedom are thus assigned to sites of a lattice. We we assume that such a state not only exists, but can be 8 deduce the corresponding fully coupled equations of mo- relevant for the time scales of interest . tion from the corresponding Hamiltonian formalism, that The total internal energy of this system can be de- treats mechanical and spin degrees of freedom in a unified scribed in terms of the interactions between these vari- way and study the dynamics of this system, comparing it ables, that take into account the elastic mechanical part to previous studies, first for the purely magnetic part14 in the most general form and the interaction with ex- and then, also, for the magneto-elastically coupled sys- ternal stresses, which will be described by the magnetic tem15. degrees of freedom. Schematically, Of course to render the equations tractable we must H H H = mech + ext (1) resort to approximations. The approximation we employ V0 V0 V0 is the mean field approximation, that amounts, in the We shall now spell out each term. present case, to a coarse–graining of the lattice, which To lowest, non–trivial, i.e. quadratic, order, H /V means that the sites are, in fact, domains, within each ext 0 can be written as17 of which the properties are homogeneous. Within this N framework, our Hamiltonian formalism is, however, ex- Hmech 1 i i i act. = CIJKLǫIJ (t)ǫKL(t), (2) V0 2 The paper is organized as follows: In section II we con- Xi=1 struct a classical Hamiltonian for the mechanical, and the where an implicit sum on the repeated space indices is i spin degrees of freedom and discuss their couplings. We understood and where CIJKL are the elastic constants discuss the advantages of describing the magnetic mo- with V0 as the reference volume. 3
In this expression we have assumed that the different and leads to a “renormalization” of the effective elastic sites of the material do not interact through elastic defor- constants in the following way: If one compares the last mations (they will only interact through spin exchange term of equation (5) with equation (2), it is easy to see forces, to be spelled out below). that a pair of effective elastic constants can be introduced For an homogeneous material (i.e., C(i) = C for every that depends on the value of the spins on each site, i.e. site i), these elastic constants can be expressed in terms 16 1 of the two Lam´eparameters , designated by (C0, C1), eff,i,j ij (2)i,j i j CIJLK ≡ CIJLK + BIJKLMN SM (t)SN (t). (7) which in Cartesian coordinates read as follows: V0
CIJKL = C0δIJ δKL + C1 (δIK δJL + δILδJK ) . (3) This has been used to provide an interpretation of the “anomalous” temperature dependence in the elastic con- This mechanical system can be excited by an external stants in iron single crystals, for instance, as resulting ext stress σIJ , through the coupling from the competition between spin ordering and diffu- sion effects21. In the present work however, we will only N (1)i,j H consider BIJKL and set aside the non-linear mechanical ext = − σextǫi . (4) V IJ JI terms. Although there is no particular difficulty to take 0 Xi=1 the latter into account in this formalism, we will take B(2) ≡ 0 for the rest of this study and for the sake of While this stress can be, in general, a time- and space- simplicity. dependent quantity, it is, nonetheless, assumed to be uni- In Appendix A we recall that the most effective “clas- form over the sites of the material. sically equivalent description” of the spin degrees of free- The magnetic degrees of freedom, noted Si (t), are as- I dom, that can capture consistently their interaction with signed to the same lattice sites as the strain and describe the elastic degrees of freedom is not through the variables how their dynamics affects the mechanical degrees of free- S but through their “Doppelg¨anger” ξ, related to the Si dom, through the so-called magnetoelastic coupling. I through The contribution of the magnetoelastic coupling to the Hamiltonian can be deduced by assuming (global) ı Si ≡− ǫ ξi ξi . (8) Galilean invariance on the one hand and imposing in- I 2 IJK J K variance under the symmetries of the point group18 on the other. For an isotropic medium, these requirements The ξ can be identified with Majorana fermions, which lead to the expressions for the total energy and the to- have found many applications recently in condensed mat- tal magnetization19. The total internal energy is, thus, a ter systems22, where new methods for controlling spin sum over the lattice20. degrees of freedom have been developed. Consequently, in the approximation of small deforma- It is interesting to remark that the anticommuting i tions, the expansion of the magnetoelastic coupling en- variables, ξk are not Grassmann variables, satisfying i i j i i ij ergy will contain even powers of the magnetization S . {ξI , ξJ } = 0; but, rather, {ξI , ξJ } = δ δIJ , i.e. that 23 It will, however, contain all powers of the strain tensor, the ξI generate a Clifford algebra on each site . It is in i ǫ. this way that the SI , defined through eq. (8), satisfy the i 24 The linear part in ǫIJ defines the first-order, magne- angular momentum algebra . toelastic coupling, which is responsible for the magne- If one were tempted to simply replace S by ξ in eq.(5), i tostriction, while the quadratic terms in ǫIJ define the it is interesting to remark that, for N = 1, and because of (1) 7 second-order magnetoelastic coupling, that is responsi- the symmetries of B as recalled in reference , HME = ble for the “nonlinear response” in the elastic properties 0, which implies that the dynamics as it is cannot be of magnetic media, that have been, also, identified as encoded by this Hamiltonian. describing “morphic effects”7. On the other hand, this allows us to understand the Higher order terms are, usually, ignored, since their co- constraints on the allowed terms in the true Hamiltonian. efficients are assumed to be much smaller than the lower They must, necessarily, involve more than one spins. In- order terms, already discussed. deed, we can construct expressions that are multi-linear i Therefore, the contribution to the–total–Hamiltonian combinations of the SI on different sites, potentially, up of the magnetoelastic coupling can be written as to order N, the number of sites, since no two identical i ξI variables occur in the same monomial. This is, there- N (1)i,j i i j fore, a nice way of automatically organizing the multi- HME = BIJKLǫIJ (t)SK (t)SL(t) spin terms of the Hamiltonian. i,jX=1 In the formalism of Atomistic Spin Dynamics, since (2)i,j i j i j 25,26 + BIJKLMN ǫIJ (t)ǫKL(t)SM (t)SN (t)+ ... (5) the magnetic moments are localized , it is customary to consider spatial averages, around each site, defining The total Hamiltonian can, thus, be written as an effective macroscopic localized spin
eff i Htot = Hmech + Hext + HME (6) SI = hSI ii (9) 4 which implies that multi-linear expressions, SI SJ no III. THE POISSON BRACKETS AND THE longer vanish identically, as was imposed by the anti- EQUATIONS OF MOTION commuting nature of ξ before. We can, thus, understand the relevance of this averaging procedure, in terms of the To deduce from the Hamiltonian, discussed in the pre- description of the spin degrees of freedom in terms of the vious section, the equations of motion, we must define anticommuting variables. the appropriate pairs of canonically conjugate variables The justification for equation (5) has been advocated and consequently their Poisson bracket. a long time ago27, as stemming from a model for a First we recognize that ε acts as a tensor and one can two–body interaction, that is itself a pedagogical ver- understand the definition of the Poisson bracket of rank- sion of the quantum theory of interacting magnons and 2 symmetric tensors as an application of the DeDonder- phonons28. In eq. (5) what has been left unspecified are Weyl covariant hamiltonian formulation of field theory29. the properties of the tensor B(1) under exchange of the Although the context is different, the ADM procedure in indices i and j, that label the sites. What is customary is general relativity also provides such a Poisson bracket30 to assume that B(1), in fact, does not depend on i and j (with further relations, between the conjugate variables at all–it is homogeneous across the material. By stopping that are not relevant here). the expansion at first order in the mechanical deforma- Although no clear consensus has, in fact, emerged on tion in (5) and assuming homogeneity in the material, the properties of Poisson bracket of rank-2 tensors31, if we, therefore, end up with the following expression one focuses on the special case of strain tensors, that
(1) depend only on time, the following conjugate momentum HME = BIJKLǫIJ (t)SK (t)SL(t). (10) can be introduced L Again, in the case of an isotropic medium, the elements ∂ πIJ ≡ , (12) in B(1) enjoy the same symmetries than the elastic con- ∂ε˙IJ stants and can be expressed only in terms of two con- where L (εIJ (t), ε˙IJ (t)) is the unconstrained and free La- (1) (1) stants B0 and B1 by the following expression grangian.ε ˙IJ (t) are the components of the strain-rate tensor32. (1) (1) (1) BIJKL = B0 δIJ δKL + B1 (δIK δJL + δILδJK ) . (11) Thus, we can build the corresponding Hamiltonian H for the time evolution with tensor variables for mechani- In summary, the microscopic theory underlying Eq.(10) cal deformations ε and their conjugated momenta π, be- describes the interaction of three particles, two of which cause these quantities admit unbounded numerical val- describe spin states, and one the state of the elastic de- ues, as the corresponding Legendre transform formation. However the spin states, are, in fact, bound states of more “fundamental” entities, the anticommut- H(ε, π)= πIJ ε˙IJ − L , (13) ing ξ. B(1) can then be interpreted as the vertex of this up to a total time derivative for L . interaction, that describes a spinning particle, that does For the mechanical system only (i.e. for functions not directly interact with itself–since any such interac- A(ε, π) and B(ε, π)), the Poisson bracket can be defined tion is inconsistent with the fact that the ξ anticommute. in perfect analogy to that of any particle system, that Such a self-interaction can, however, appear on larger explores a given target space geometry (to which refer scales, when spatial averages can become meaningful for the indices I,J,K,L), by the usual relations describing the spin degrees of freedom. In conclusion to this section, the Hamiltonian we ∂A ∂B ∂A ∂B {A, B}P B = − . (14) have constructed describes the interaction of magnetic ∂εIJ ∂πIJ ∂πIJ ∂εIJ moments through their embedding within an elastic In our case, the dynamical variables are the real sym- medium. What remains to be done is to define the Pois- metric rank-two tensors, εIJ and πIJ (I,J = 1, 2, 3), son brackets, that can take into account the evolution in which are canonically conjugate in the sense that their phase space of commuting, as well as anticommuting de- Poisson brackets are deemed to satisfy the following prop- grees of freedom. Indeed, the construction procedure for erties: commuting canonical variables is well known, however, including anticommuting variables in a unified way has {εIJ , πKL}P B = δIJKL, (15) 13 remained a rather esoteric subject–known in theory , {εIJ ,εKL}P B = {πIJ , πKL}P B = 0 (16) not, however, implemented, in practice. In the following section, we shall construct the equa- where δIJKL is a δ “tensor”, reflecting the real, sym- 33 tions of motion in the phase space of the elastic and the metric, nature of the Poisson brackets and defined as spinning degrees of freedom, that implements these ideas a product of Kronecker δs. Very schematically, we may in practice. write In order to work directly with the anticommuting vari- δIJ δKL ables themselves, in combination with the mechanical de- δIJ δLK δIJKL = (17) grees of freedom requires implementing a “graded Poisson δJI δLK bracket”; one way to do this is discussed in Appendix A. δJI δKL 5 where each choice of the RHS corresponds to a choice and B(1) is the fully symmetric linear magnetostriction of indices in the Poisson brackets. This choice can be tensor. supplemented by any linear combination of these δs that The “kinetic” term, containing the conjugate mo- enforces the symmetries of the tensors. menta, can be written, schematically, as We now wish to include as phase space coordinates, the S 34 1 −1 components of the spin vector, . We follow reference Hkinetic = πIJ MIJKLπKL (22) and the details are summarized in appendix A. The “gen- 2 eralized” Poisson bracket, for the canonical variables of where M is a fully symmetric “mass” matrix, that de- our system, can be written as scribes the inertial response. For the case of isotropic materials, it is assuming that ∂A ∂B ∂A ∂B the M tensor has the form given by Lam´e, with only two {A, B}P B ≡ − ∂εIJ ∂πIJ ∂πIJ ∂εIJ (18) characteristic constants: 1 ∂A ∂B − ǫIJK SI , MIJKL = M0δIJ δKL + M1 (δIK δJL + δILδJK ) . (23) ~ ∂SJ ∂SK The tensors C and B are decomposed in the same way. where ~ is introduced to restore the physical dimensions Consequently, the inverse of these tensors of the Poisson bracket, since S is considered dimension- can then be deduced from M M −1 = less, for the sake of simplicity. IJKL IJMN 1 (δ δ + δ δ ). Thus It should be stressed that this ~ does not imply that 2 KM LN KN LM any quantum effects are present, since the dynamics is fully classical. It is simply a bookkeeping device for a −1 −M0 MIJKL = δIJ δKL quantity that has the dimensions of angular momentum– 2M1(3M0 +2M1) (24) i.e. of an area in phase space–and reflects the fact that 1 + (δIK δJL + δILδJK ) the equation of motion for SI is of first order. Quantum 4M1 fluctuations will introduce the “real” ~. Using this Poisson bracket, we can obtain the equations and the equations of motion become of motion for the phase space variables: −1 ǫ˙IJ = MIJKLπKL ∂H ε˙IJ = {εIJ , H}P B = ext ∂πIJ π˙ IJ = −V0CIJKLǫKL + V0σ − BIJKLSK SL (25) IJ ∂H (19) π˙ IJ = {πIJ , H}P B = − (1) ∂εIJ S˙ = ε ω − 2 B ǫ S S I IJK J ~ ABJC AB C K 1 ∂H S˙I = {SI , H}P B = εIJKSJ ~ ∂SK highlighting how the mechanical and magnetic subsys- tems are coupled. The consistency of this formalism can be checked by not- The last equation–as expected!–is a precession equa- ing that these equations preserve the volume in phase tion for the components of S around both the effective space field ω and an additional field, that depends on the strain tensor and the spin vector. ∂ε˙ ∂π˙ ∂S˙ IJ + IJ + K =0. (20) In the following section we shall show how to solve ∂εIJ ∂πIJ ∂SK these equations, in a way that preserves the symmetries That the dynamics preserves the volume in phase space of the extended phase space. does not, of course, imply anything about whether the system thus defined is integrable or shows Hamiltonian chaos. IV. GEOMETRIC INTEGRATION The internal energy U, involving the mechanical en- ergy for the deformed elastic medium16, the magnetic Solving the coupled system of equations (25) is the energy, defined by the Zeeman term35 and the magneto- next step. elastic energy7, that takes into account the interaction of Since, in the previous section, we have shown that the magnetic moment with the medium, takes the form these equations describe a volume preserving transfor- mation of the enlarged phase space, encompassing elastic V0 ext (1) and spin variables, it is natural to rewrite them, in terms U = CIJKLεIJ εKL − V0σ εIJ + B εIJ SK SL 2 IJ IJKL of the action of a Liouville operator. Therefore, we shall − ~ωI SI , (21) write eqs.(25) as where C is the fully symmetric tensor, defining the elastic ε˙ = Lǫε, response, σext is the external stress tensor, ω is the ef- π˙ = Lππ, (26) fective external magnetic field (expressed as a frequency) S˙ = LSS. 6 where L is the Liouville operator. This formulation al- A τ τ τ τ τ τ τ lows us to implement, manifestly, time-reversible, area 4 2 4 4 2 4 preserving algorithms, for solving these equations numer- 1 SπSǫSπS ically. 2 SǫSπSǫS The general scheme is as follows: Consider an arbitrary 3 πǫπSπǫπ function f of the canonically conjugate variables of our 4 πSπǫπSπ many-body system. This function, f(ε, π, S), depends 5 ǫπǫSǫπǫ on the time t implicitly; that is, through the dependence 6 ǫSǫπǫSǫ of (ε, π, S) on t. The time derivative of f is f˙ such as
∂f ∂f ∂f Table I. Decomposition table of symplectic integrators f˙ =ε ˙IJ +π ˙ IJ + S˙I ∂εIJ ∂πIJ ∂SI ≡ Lf. (27)
The last line defines the (total) Liouville operator It is important to keep in mind that the one–step evo- lution operators for ε and π describe shifts of the corre- sponding tensor components, whereas the one–step evo- ∂ ∂ ˙ ∂ L =ε ˙IJ +π ˙ IJ + SI . (28) S ∂εIJ ∂πIJ ∂SI lution operator for describes rotations. In equations Equation (27) can be integrated formally as an initial eLετ (ε(t), π(t), S(t)) = (ε(t)+ τε˙(t), π(t), S(t)) (32) value problem to obtain f at any time : π eL τ (ε(t), π(t), S(0)) = (ε(t), π(t)+ τπ˙ (t), S(t)) (33) f(ε(t), π(t), S(t)) = eLtf(ε(0), π(0), S(0)). (29) eLS τ (ε(t), π(t), S(t)) = (ε(t), π(t), S(t + τ)) (34)
R(τ)S(t) It is not difficult to see that L = Lε + Lπ + LS . How- ever, these single Liouville operators do not commute | {z } two-by-two, as the reader may easily check by comput- where S(t + τ) = R(τ)S(t) is given by the Rodrigues’ rotation formula40 for a spin vector around a given ro- ing LuLvf − LvLuf 6= 0 for any function f and any ˜ combination (u, v) of the individual Liouville operator tation vector ω(t) where each of its components are 2 (1) Lε, Lπ, LS. This means that ω˜I (t) = ωI (t) − ~ BJKLI εJK(t)SL(t). These equations describe the phase space of one particle only. To de- eLt = eLεt+Lπt+LS t 6= eLεteLπteLS t. (30) scribe the dynamics of a continuum, we must deduce the equations for many particles. 36 According to the Magnus expansion , however, it is al- One way to generalize eqs.(25) for the case of many Lt ways possible to express e as a product of the indi- particles, labeled by an index, i =1,...,N, according to vidual operators at any given order in time, according the conventions of the previous sections, is the following to the so–called “splitting method”37. This ensures that the numerical algorithm preserves phase space volumes i i −1 i ǫ˙IJ = [M ]IJKLπKL (35) exactly. i i i i ext i i i For instance, for a fixed timestep τ, upon expanding π˙ IJ = −V0CIJKLǫKL + V0σIJ − BIJKLSK SL (36) up to the third order in time, the following sequence of i i 2 (1)i i i i S˙ = εIJK ω − B ǫ S S (37) products I J ~ ABJC AB C K
τ τ τ τ τ τ Lτ LS 4 Lπ 2 LS 4 Lετ LS 4 Lπ 2 LS 4 3 e = e e e e e e e + O(τ ), The case of a staggered AF state is treated simply by (31) letting N = 2. In order to simplify the mechanical part further, we can impose additional conditions pertaining can be generated; this sequence is, in fact, one of six pos- to the uniformity of the external stress, mechanical con- sible, that have the property of preserving the symplec- stants, mass matrices and magneto-elastic constants, at tic structure of the Poisson brackets. Therefore, any one each site. In what follows, we shall use the following of them can be chosen. The possible combinations are Ansatz: presented in table I. While these schemes are free from “global” errors, they are, of course, sensitive to “local” (1)1 (1)2 BIJKL = BIJKL, errors, due to the finite value of the timestep. It is, also, 1 2 not at all obvious that all six can be implemented with CIJKL = CIJKL, 1 2 (38) comparable efficiency. It is, therefore, useful to study the MIJKL = MIJKL, 1 ext 2 ext numerical stability and efficiency of these different com- σIJ = σIJ . binations, in particular, as former studies in molecular dynamics38 and magnetic molecular dynamics39 showed, The conservative part of the precession contains a local apparently, numerical differences between them. field, which is modified to include the antiferromagnetic A sampler of such a study is presented in appendix B. exchange between the sites and a single anisotropy axis 7 n with an intensity K. One has
i 1 ij j K i ωI = ~ J SI + ~ nJ sJ nI (39)
τ τ eLS τ = eLS1 2 eLS2 τ eLS1 2 + O(τ 3). (40)
For N = 2, this operator is numerically identical to the operator, obtained by permuting the site indices, 1 ↔ 2. The same reasoning is applied for the Liouville opera- Figure 1. (Color online) Switching scheme for antiferromag- tors for the elastic variables, that enter in equations (26) netic magneto-mechanically coupled toy model with external and the corresponding global geometric integrators are STT. constructed along the same lines. The system is then integrated by following one of the schemes displayed in new magnetic configuration. The resulting dynamics is a table I. switching of the two sub-lattices moments to the opposite With these tools, we can study a plethora of phenom- direction through the easy-plane. ena, that are sensitive to the coupling of magnetic, elec- This process defines the ultrafast antiferromagnetic tric and elastic degrees of freedom. In the following sec- switching effect. During the process, the N´eel vector, tion we shall apply this formalism for studying “switch- that probes the difference between the magnetic moments ing” effects of the magnetization in antiferromagnetic me- of the two sub-lattices, acquires a net value that can be dia. The numerical accuracy ensured by the geometric transferred as a so-called “spin accumulation”, to an ad- integrators is necessary to describe picosecond switching jacent non-magnetic normal material in order to pump times. the produced spin current via scattering of electrons. The reverse mechanism can be realized, as well. What was not considered before is the possibility to en- V. ANTIFERROMAGNETIC ULTRAFAST hance or inhibit such a switching, depending on tensile SWITCHING UNDER STRESS or compression effects, that are generated by an external stress, that couples to the internal strain, produced by In this section, we shall apply the formalism con- the intrinsic magneto-elastic interaction in such materi- structed above, to describe how it is possible to generate als, as depicted in figure 1. and manipulate picosecond switching of the magnetiza- In order to conform to the notation used in our pre- tion, induced by STTs, from a short pulse of electric cur- vious studies15, equation (37) is supplemented with a rent, in elastic media that exhibit staggered AF order. non-conservative part, labeled T , on the RHS, which in- Such a fast switching process in AFs is schematically cludes both a transverse damping and a damping-like displayed in figure 1 and can be realized, in practice, STT torque by a femtosecond laser excitation of the magnetic mo- i i ˙ i i i i i ments that generate a field-induced STT on the two sub- TI = αǫIJK SJ SK + G sI sJ pJ − pI sJ sJ . (41) lattices. During the pulse, energy is transferred from con- duction electrons to the sub-lattice magnetic moments The resulting equations of motion, are numerically inte- via STT (this is how the electric current affects the mag- grated, using the approach of section V. It is noteworthy netic response), and hence contributes to the exchange that, this equation, also, describes the “backreaction” of energy between the moments, since local moments can be the magnetic response on the spin transfer torque. On canted noncollinearly. The energy due to the strong ex- the other hand, we do not consider how this backreaction change interaction between neighboring moments, which affects the current pulse itself, that is assumed external. is commonly found in AFs, in particular due to magnetic In figures 2 and 3 we report the evolution of the average 1 1 2 anisotropy, appears as an effective inertia to the motion, magnetization m ≡ 2 S + S and the N´eel vector l ≡ 1 1 2 leading to the appearance of a time scale, much longer 2 S − S , in the presence as well as the absence of than that of the excitation pulse. Afterwards, the system magnetoelastic coupling, for a moderate external stress 2 follows a natural path along the easy-plane to circumvent of σxx = −2µ0Ms . We start the simulations using an the unfavorable anisotropy barrier and finally relaxes to a initial configuration, where spins are aligned along the xˆ- 8
In the presence of magnetoelastic coupling, however, as the mechanical system is undamped, the results are quite different. Because of the presence of a constant stress, a finite mass matrix and non-zero elastic constants, the me- chanical system is expected to be oscillating freely, which is, indeed, observed in figure 4, where we display the evo- lution of the strain components over time. Indeed, as one can see on figures 3 and 4 the characteristic oscillations of the mechanical system have repercussions on the N´eel order parameter dynamics. This is even more striking for the spin accumulation which is more than doubled by the coupling to the strain ; here as well, the mechanical oscil- lations are reflected on the magnetic dynamics. What is interesting is that these characteristic mechanical oscilla- tions are expected to be related to the sound velocity of 42 the medium , which is controlled by the constants M0 and M1. In the present study, we have chosen M0 = 0 Figure 2. (Color online) Average magnetization (upper panel) and M = 10000, but there should be a way to obtain and N´eel order parameter components (middle panel) for un- 1 these coefficients of the mass matrix more efficiently, in coupled switching. {K = 2πrad.GHz, ωE=172.16 rad.THz, 5 −1 1 2 order to describe these oscillations more accurately from Ms = 5.10 A.m }. Initial conditions: {s (0) = −s (0) = xˆ}. The lower panel displays the STT pulses. The figures experimentally easily accessible data. This, however is agree with the reference14 because the magnetoelastic con- not the focus of the present study and shall be studied stants are set to zero. more thoroughly elsewhere. As a consequence of such a
Figure 4. (Color online) Strain components as a function of Figure 3. (Color online) Average magnetization (upper panel) time with the magnetoelastic constants turned on with pa- and N´eel order parameter components (middle panel) for a rameters identical to figure 3. coupled switching. {K = 2πrad.GHz, ωE=172.16 rad.THz, 5 −1 1 2 Ms = 5.10 A.m }. Initial conditions: {s (0) = −s (0) = xˆ, i stress induced strain, the switching time of the magneti- ǫIJ (0) = 0}. The lower panel displays the STT pulses. Here (1) 2 (1) 2 zation is slightly faster and the amplitude of the magne- B = 7.7µ0Ms and B = −23µ0Ms . 0 1 tization enhanced, depending on the values of the stress and magnetoelastic constants, as depicted in figure 3. axis in an antiferromagnetic configuration and apply two We observe that a compression in one direction becomes tensile in the other one and vice-versa depending on the 10 ps electric pulses of 0.0034 rad.THz intensity, each (1) (1) separated by 50 ps, in the p = zˆ-direction. In addition sign of B1 . Conversely, if B1 is of the opposite sign, a to the exchange interaction, the spins are subjected to a tensile stress would slow the switching process down. global anisotropy, along the n = xˆ-axis. The numerical This leads to another interesting question which is how value for α in the following studies is 0.005. the switching time is affected by the values taken by ei- (1) (1) (1) In the absence of magnetoelastic coupling, our results ther B0 and B1 . We have verified that changing B0 are identical to those by Cheng et al.14 and to those has no influence on it because this coefficient does not we obtained under the same conditions in our previous enter in the precession equation. Its simply produces the work15. Joule magnetostriction phenomena as anticipated before 9
7 (1) in reference . However, changing B1 has a strong influ- be measured in real and numerical experiments, as we ence. The Figure 5 displays the switching time as a func- have shown. In particular, the ultrafast switching of the (1) magnetization, induced by a damping-like STT in NiO, tion of the values of varying B1 . By defining R as the (1) where both the duration and intensity can be modulated ratio of B1 divided by the its natural value in NiO, we perfom different simulations and report the time where by appropriate internal strains, thanks to the magnetoe- the Neel vector x-component crosses over to negative val- lastic coupling. ues. One can see that this time decreases by increasing It is, now, possible to set in perspective the limits of (1) the present framework and how they may be overcome. B which makes sense, as this is amplifying the effects 1 The strain can be modulated, in return, by the in- of the tensile stress. When R becomes lower than 1, the duced magnetization process by the same magnetoelastic effect is opposite. One aspect which is however surpris- coupling, so as to give rise to a modulated elastic wave, 10 whose properties, in extended systems can leave a char- acteristic imprint, that would be interesting to look for. It should be kept in mind, though, that, in real ex- 9 periments, because amplitude fluctuations of the electric pulses are inevitable, small damping values are not fa- vorable, since they may easily lead to overshoot, which is 8 amplified by the mechanical response. When Joule heat- ing in the normal underlying metal is taken into account, 7 a shorter pulse with stronger current intensity should be desirable, but a modulated undamped mechanical strain
switching time (ps) is then produced. For the moment, a major conceptual 6 issue is how to describe, in a consistent way, the mechan- ical damping effects, that are induced by a differential thermal stress, produced by such a Joule heating. We 5 0 1 2 3 4 5 hope to return to this issue in future work. Hints may R parameter be gleaned from our previous work on the stochastic dy- namics of the magnetization44. (1) Figure 5. Switching time (in ps) as a function of varying B1 For the case of undamped dynamics we have devel- over its natural value in NiO. oped the mathematical and computational framework, that consistently takes into account elastic and magnetic ing is that there is a quick drop of the switching time for degrees of freedom and that provides a starting point for (1) approximately 4 times the value of B1 over its natural designing future AFs devices for spintronics applications value in NiO. This is due to the oscillations of the magne- that combine both mechanical and magnetic responses tization induced by the elastic coupling. As this coupling and their interactions. And we have, also, shown how becomes stronger, one of the mechanical oscillatory peaks to take into account a particular class of damping terms, dips below 0 and thus the switching time appears to drop those that can describe spin–transfer–torque sources. in a discontinuous fashion. This is another interesting It is this framework that is the starting point for taking consequence to the interplay of the magnetization and into account the stochastic effects mentioned above. the strain, that could be captured in real experiments.
ACKNOWLEDGMENTS VI. DISCUSSION
All authors contributed equally to the manuscript. In the present paper we have presented the details for a T.N. acknowledges financial support through a joint doc- Hamiltonian framework that can describe the dynamics, toral fellowship “CEA–R´egion Centre” under grant num- as well as the coupling, between magnetism and elasticity ber 201600110872. by a careful definition of the corresponding dynamical variables in an extended phase-space. The magnetoelastic coupling is taken into account at the linear level only of the strain tensor, by the coef- Appendix A: A model for precession through anticommuting variables ficients, B0 and B1, that can be derived either experi- mentally from magnetostriction constants7 or by ab ini- tio calculations43. It should be stressed that the elastic We review some issues when dealing with classical spin constants, C0 and C1 can be deduced from similar exper- variables and explain some limitations which are met by iments, also. using the framework of commuting variables45. A com- Notwithstanding the presence of only the linear term mon starting point to build precessional models is choos- of the strain, it does describe non-trivial effects, that can ing the appropriate representation of the su(2) algebra. 10
In the Heisenberg picture the Ehrenfest theorem im- the “antibracket” of any two functions on a flat super- plies that the equations of motion for the average of the manifold, satisfies all necessary properties for a graded spin operator Sˆ components read bracket, namely (graded) Leibniz rule, (graded) anti- symmetry and (graded) Jacobi identity46. dhSˆi ı~ = h[Sˆ, H]i (A1) By taking this graded Poisson bracket for any two of dt these anti-commuting variables, we get where H is the Zeeman Hamiltonian operator for an ex- ı ternal magnetic field H such that {ξI , ξJ }P B = ~δIJ . (A6)
H = −gµ0µBSˆ · H. (A2) This implies that any two such variables are canonically conjugate, since {ξ , −ı~ξ } = δ and π ≡ −ı~ξ ˆ ˆ ˆ I J P B IJ I I From the commutation relations [Si, Sj ] = ıǫijkSk, one defines the canonical conjugate13. can quite straightforwardly show that By using the Grassmann properties of ξ, one proves that dhSˆi gµ µ = B 0 hSˆi× H, (A3) dt ~ 1 {SI ,SJ }P B = ~ǫIJKSK (A7) which corresponds to the well-known Larmor precession of the expectation value of the spin in an external mag- which is a consistency check that the SI as defined in netic field H. (A4) do realize a representation of the rotation group24. Interestingly, a different–but equivalent–approach can In eq.(A5), if the functions of the ξ are chosen to con- be followed, by considering the Majorana representation, tain only quadratic terms, then one can identify the pre- in terms of anticommuting variables, ξI , ξI ξJ + ξJ ξI =0 vious bracket as a regular Poisson bracket on a Rieman- 22 of the vector S: nian manifold for the commuting variables S ı S = − ǫ ξ ξ , (A4) ←− −→ I 2 IJK J K ı ∂f ∂ ∂ ∂g {f(S),g(S)}P B = SI SJ ~ ∂SI ∂ξK ∂ξK ∂SJ which can easily be shown to commute, as SI SJ = 0. (A8) This last relation is, of course, not satisfactory. It can be ∂f ∂g 1 ∂f ∂g = {SI ,SJ }P B = ~ǫIJK SK , easily avoided, however, by imposing that the ξI satisfy ∂SI ∂SJ ∂SI ∂SJ ij the anticommutation relations {ξI , ξJ } = δ –that they which is, precisely, the “spinning part” of the bracket generate a Clifford algebra (up to a constant normaliza- introduced by Yang and Hirschfelder34 for magnetized tion)23. 1 fluid dynamics, which reads as These relations imply that the SI define a spin– 2 rep- resentation of the su(2) algebra. Higher spin representa- ∂A ∂B ∂A ∂B 1 ∂A ∂B 13 {A, B}P B ≡ − − ǫIJKSI . tions can be defined by multilinear combinations , that ∂qI ∂pI ∂pI ∂qI ~ ∂SJ ∂SK are relevant for describing magnetic properties of com- (A9) posite objects; since we can work with the SI instead of Similar expressions have been found already by Casal- the ξI , however, this complication will not affect us here. buoni a long time ago13 without having attracted the This representation highlights that the spin degrees of attention they deserve. Following the inverse path freedom, SI , are, in fact, “composite” objects and that of canonical quantization, we use this graded Poisson the “fundamental degrees of freedom” are the ξI . There- Bracket on any commuting quantity, which allows us to fore, it is useful to develop the description of the dynam- compute directly ics, directly, in terms of the ξI themselves. We shall recall the salient features below. gµBµ0 {S, H}P B = ~ S × H (A10) The Poisson brackets for the anticommuting variables ξI are related to those of SI , in order that the dynamics thereby highlighting that the description of S in terms be, indeed, equivalent, in a way that was set forth many of ξ is an equivalent description of the dynamics. years ago, through the construction of a corresponding For any time-dependent commuting functions of S(t), graded Poisson bracket, which generalizes Poisson brack- we can use this Poisson bracket to deduce a Liouville 13 ets from manifolds to super-manifolds . equation In terms of any functions of anti-commuting variables ξ this bracket reads dF (S(t)) = {F (S(t)), H}P B. (A11) ←− −→ dt ı ∂ ∂ {f(ξ),g(ξ)} ≡ f(ξ) g(ξ), (A5) PB ~ ∂ξ ∂ξ Consequently, the equations of motion for the variables K K ξ read much more simple expressions as with the corresponding definition of the left and right −→ derivative of any function of the anti-commuting vari- dξ(t) ı ∂ H = {ξ, H} = , (A12) ables. One can show that this bracket, also known as dt P B ~ ∂ξ 11 which are known to form a non-relativistic pseudoclassi- responds to the label A = 1 in table I. The numerical cal mechanics13. precision is controlled by the “quality factor Q”, chosen One reason why using the representation of S in terms at the beginning of the simulation, which produces vari- of ξ is useful is that it is intrinsically difficult to build able timesteps τ according to the relation τ = Q/kωk. a Lagrangian model for the commuting spin variable S, Simulation conditions produce a total energy equal to 2 since its canonically conjugate variable cannot be unam- π11(0)/4M1, which has to stay constant over time. We biguously identified. As is, by now, well known, the observe that this is, indeed, the case, whatever the value conjugate of the dynamical variable ξ is proportional of the quality factor Q. We remark that, as Q → 0, to itself47. Therefore, the dynamics of spinning degrees the variations about the average value of the energy are of freedom can be described, either through a vector of suppressed, as should be expected. commuting variables on a curved manifold, or by a vec- Figure 7 displays some of the non-vanishing compo- tor of anti-commuting variables on a flat–though non– nents of the strain tensor over time (here ε22(t)= ε33(t) Riemannian–manifold. Finally, it has been found that and this third component is not reported), when using 1 the Majorana-fermion representation of 2 -spin opera- different splitting algorithms, among those displayed in tors is also a powerful tool to straightforwardly compute Table I. We observe less than 1% of numerical relative spin-spin correlators48, which represents an advantage difference between the two algorithms on the strain and for computing magnetic response functions in many-body its conjugate variables, and no difference (up to the ma- systems. chine precision) on the magnetization with a “coarse” quality factor Q, which cannot be detected by visual in- spection on the figure. This difference falls to 0.1% when Appendix B: Numerical accuracy of the splitting the quality factor is divided by 4. The same procedure algorithms can be repeated for all the splitting combinations in Ta- ble I, the conclusions previously drawn apply, also, for The accuracy of the numerical schemes represented in the magnetization, strain and its conjugate variable, de- table I depends on the relative amplitude of the velocity pending on the frequency of appearance of the splitted terms (ε, ˙ π,˙ S˙), and can be monitored by checking the operator. stability of equation (20) over time36,37. ε SπSεSπS 2.506 11 2e-05 ε π ε π Q=0.025 22 S S S S ε ε π ε Q=0.0125 11 S S S S 2.505 Q=0.00625 ε ε π ε 22 S S S S 1e-05 2.504 4 0
2.503 strain Energy x10 2.502 -1e-05
2.501 -2e-05
2.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 time (ns) time (ns) Figure 7. (Color online) Single-site strain components as a Figure 6. (Color online) Single-site total energy as a func- function of time for various numerical schemes. Conditions tion of both time and variable timestep Q. Conditions of of the simulation are identical to those for figure 6 and the 2 the simulation are expressed in reduced units: 2C0/µ0Ms = results are produced for Q = 0.0025 only. 5 2 5 2 2 5.1 × 10 , 2C1/µ0Ms = 3.5 × 10 , M1V0µ0Ms /2~ = 1000, ωDC = (0, 0, 2π) rad.GHz, π11(0) = 1, s(0) = (1, 0, 0). All the other parameters not reported, included initial conditions As expected, once a splitting algorithm is selected, the are zero. more frequently an operator is evaluated, the smaller the error, though, of course, the time required, also, grows. Figure 6 displays the total energy in one domain (given This opens the possibility to select optimally a proper by the sum of eqs.(21) and (22)) upon varying the numer- splitting algorithm, depending on the relative intensity ical precision. The splitting algorithm considered cor- ofε ˙ij , π˙ ij , S˙i over time. 12
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