Generalisation of Gilbert damping and magnetic inertia parameter as a series of higher-order relativistic terms
GGeneralisation of Gilbert damping and magneticinertia parameter as a series of higher-orderrelativistic terms
Ritwik Mondal ‡ , Marco Berritta and Peter M. Oppeneer Department of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-751 20Uppsala, SwedenE-mail: [email protected]
Abstract.
The phenomenological Landau-Lifshitz-Gilbert (LLG) equation of motionremains as the cornerstone of contemporary magnetisation dynamics studies, whereinthe Gilbert damping parameter has been attributed to first-order relativistic effects.To include magnetic inertial effects the LLG equation has previously been extendedwith a supplemental inertia term and the arising inertial dynamics has been relatedto second-order relativistic effects. Here we start from the relativistic Dirac equationand, performing a Foldy-Wouthuysen transformation, derive a generalised Pauli spinHamiltonian that contains relativistic correction terms to any higher order. Using theHeisenberg equation of spin motion we derive general relativistic expressions for thetensorial Gilbert damping and magnetic inertia parameters, and show that these ten-sors can be expressed as series of higher-order relativistic correction terms. We furthershow that, in the case of a harmonic external driving field, these series can be summedand we provide closed analytical expressions for the Gilbert and inertial parametersthat are functions of the frequency of the driving field.
1. Introduction
Spin dynamics in magnetic systems has often been described by the phenomenologicalLandau-Lifshitz (LL) equation of motion of the following form [1] ∂ M ∂t = − γ M × H eff − λ M × [ M × H eff ] , (1)where γ is the gyromagnetic ratio, H eff is the effective magnetic field, and λ is anisotropic damping parameter. The first term describes the precession of the local,classical magnetisation vector M ( r , t ) around the effective field H eff . The second termdescribes the magnetisation relaxation such that the magnetisation vector relaxes to thedirection of the effective field until finally it is aligned with the effective field. To include ‡ Present address: Department of Physics, University of Konstanz, D -78457 Konstanz, Germany a r X i v : . [ c ond - m a t . o t h e r] A p r large damping, the relaxation term in the LL equation was reformulated by Gilbert [2, 3]to give the Landau-Lifshitz-Gilbert (LLG) equation, ∂ M ∂t = − γ M × H eff + α M × ∂ M ∂t , (2)where α is the Gilbert damping constant. Note that both damping parameters α and λ are here scalars, which corresponds to the assumption of an isotropic medium. Both theLL and LLG equations preserve the length of the magnetisation during the dynamics andare mathematically equivalent (see, e.g. [4]). Recently, there have also been attempts M H eff PrecessionNutationDamping
Figure 1.
Sketch of extended LLG magnetisation dynamics. The green arrow denotesthe classical magnetisation vector which precesses around an effective field. The redsolid and dotted lines depict the precession and damping. The yellow path signifiesthe nutation, or inertial damping, of the magnetisation vector. to investigate the magnetic inertial dynamics which is essentially an extension to theLLG equation with an additional term [5–7]. Phenomenologically this additional term ofmagnetic inertial dynamics, M ×I ∂ M /∂t , can be seen as a torque due to second-ordertime derivative of the magnetisation [8–11]. The essence of the terms in the extendedLLG equation is described pictorially in Fig. 1. Note that in the LLG dynamics themagnetisation is described as a classical vector field and not as a quantum spin vector.In their original work, Landau and Lifshitz attributed the damping constant λ torelativistic origins [1]; later on, it has been more specifically attributed to spin-orbitcoupling [12–15]. In the last few decades, several explanations have been proposedtowards the origin of damping mechanisms, e.g., the breathing Fermi surface model[16, 17], torque-torque correlation model [18], scattering theory formulation [19], effectivefield theories [20] etc. On the other hand, the origin of magnetic inertia is less discussedin the literature, although it’s application to ultrafast spin dynamics and switchingcould potentially be rich [9]. To account for the magnetic inertia, the breathing Fermisurface model has been extended [11, 21] and the inertia parameter has been associatedwith the magnetic susceptibility [22]. However, the microscopic origins of both Gilbertdamping and magnetic inertia are still under debate and pose a fundamental questionthat requires to be further investigated.In two recent works [23, 24], we have shown that both quantities are of relativisticorigin. In particular, we derived the Gilbert damping dynamics from the relativisticspin-orbit coupling and showed that the damping parameter is not a scalar quantitybut rather a tensor that involves two main contributions: electronic and magneticones [23]. The electronic contribution is calculated as an electronic states’ expectationvalue of the product of different components of position and momentum operators;however, the magnetic contribution is given by the imaginary part of the susceptibilitytensor. In an another work, we have derived the magnetic inertial dynamics from ahigher-order (1 /c ) spin-orbit coupling and showed that the corresponding parameteris also a tensor which depends on the real part of the susceptibility [24]. Both theseinvestigations used a semirelativistic expansion of the Dirac Hamiltonian employing theFoldy-Wouthuysen transformation to obtain an extended Pauli Hamiltonian includingthe relativistic corrections [25, 26]. The thus-obtained semirelativistic Hamiltonian wasthen used to calculate the magnetisation dynamics, especially for the derivation of theLLG equation and magnetic inertial dynamics.In this article we use an extended approach towards a derivation of thegeneralisation of those two (Gilbert damping and magnetic inertia) parameters fromthe relativistic Dirac Hamiltonian, developing a series to fully include the occurringhigher-order relativistic terms. To this end we start from the Dirac Hamiltonian inthe presence of an external electromagnetic field and derive a semirelativistic expansionof it. By doing so, we consider the direct field-spin coupling terms and show thatthese terms can be written as a series of higher-order relativistic contributions. Usingthe latter Hamiltonian, we derive the corresponding spin dynamics. Our results showthat the Gilbert damping parameter and inertia parameter can be expressed as aconvergent series of higher-order relativistic terms and we derive closed expressionsfor both quantities. At the lowest order, we find exactly the same tensorial quantitiesthat have been found in earlier works.
2. Relativistic Hamiltonian Formulation
To describe a relativistic particle, we start with a Dirac particle [27] inside a material,and, in the presence of an external field, for which one can write the Dirac equationas i (cid:126) ∂ψ ( r ,t ) ∂t = H ψ ( r , t ) for a Dirac bi-spinor ψ . Adopting furthermore the relativisticdensity functional theory (DFT) framework we write the corresponding Hamiltonian as[23–25] H = c α · ( p − e A ) + ( β − ) mc + V = O + ( β − ) mc + E , (3)where V is the effective unpolarised Kohn-Sham potential created by the ion-ion, ion-electron and electron-electron interactions. Generally, to describe magnetic systems, anadditional spin-polarised energy (exchange energy) term is required. However, we havetreated effects of the exchange field previously, and since it doesn’t contribute to thedamping terms we do not consider it explicitly here (for details of the calculationsinvolving the exchange potential, see Ref. [23, 25]). The effect of the externalelectromagnetic field has been accounted through the vector potential, A ( r , t ), c definesthe speed of light, m is particle’s mass and is the 4 × α and β are theDirac matrices which have the form α = (cid:32) σσ (cid:33) , β = (cid:32) − (cid:33) , where σ = ( σ x , σ y , σ z ) are the Pauli spin matrix vectors and is 2 × O = c α · ( p − e A ), and the diagonal matrix elements can be written as E = V .In the nonrelativistic limit, the Dirac Hamiltonian equals the Pauli Hamiltonian,see e.g. [28]. In this respect, one has to consider that the Dirac bi-spinor can be writtenas ψ ( r , t ) = (cid:32) φ ( r , t ) η ( r , t ) (cid:33) , where the upper φ and lower η components have to be considered as “large” and “small”components, respectively. This nonrelativistic limit is only valid for the case when theparticle’s momentum is much smaller than the rest mass energy, otherwise it givesan unsatisfactory result [26]. Therefore, the issue of separating the wave functions ofparticles from those of antiparticles is not clear for any given momentum. This is mainlybecause the off-diagonal Hamiltonian elements link the particle and antiparticle. TheFoldy-Wouthuysen (FW) transformation [29] has been a very successful attempt to finda representation where the off-diagonal elements have been reduced in every step of thetransformation. Thereafter, neglecting the higher-order off-diagonal elements, one findsthe correct Hamiltonian that describes the particles efficiently. The FW transformationis an unitary transformation obtained by suitably choosing the FW operator [29], U FW = − i mc β O . (4)The minus sign in front of the operator is because of the property that β and O anticommute with each other. With the FW operator, the FW transformation of thewave function adopts the form ψ (cid:48) ( r , t ) = e iU FW ψ ( r , t ) such that the probability densityremains the same, | ψ | = | ψ (cid:48) | . In this way, the time-dependent FW transformedHamiltonian can be expressed as [26, 28, 30] H FW = e iU FW (cid:18) H − i (cid:126) ∂∂t (cid:19) e − iU FW + i (cid:126) ∂∂t . (5)According to the Baker-Campbell-Hausdorff formula, the above transformed Hamilto-nian can be written as a series of commutators, and the finally transformed Hamiltonianreads H FW = H + i (cid:20) U FW , H − i (cid:126) ∂∂t (cid:21) + i (cid:20) U FW , (cid:20) U FW , H − i (cid:126) ∂∂t (cid:21)(cid:21) + i (cid:20) U FW , (cid:20) U FW , (cid:20) U FW , H − i (cid:126) ∂∂t (cid:21)(cid:21)(cid:21) + .... . (6)In general, for a time-independent FW transformation, one has to work with ∂U FW ∂t = 0.However, this is only valid if the odd operator does not contain any time dependency. Inour case, a time-dependent transformation is needed as the vector potential is notablytime-varying. In this regard, we notice that the even operators and the term i (cid:126) ∂/∂t transform in a similar way. Therefore, we define a term F such that F = E − i (cid:126) ∂/∂t .The main theme of the FW transformation is to make the odd terms smaller in everystep of the transformation. After a fourth transformation and neglecting the higherorder terms, the Hamiltonian with only the even terms can be shown to have the formas [26, 30–33] H (cid:48)(cid:48)(cid:48) FW = ( β − ) mc + β (cid:18) O mc − O m c + O m c (cid:19) + E − m c [ O , [ O , F ]] − β m c [ O , F ] + 364 m c (cid:8) O , [ O , [ O , F ]] (cid:9) + 5128 m c (cid:2) O , (cid:2) O , F (cid:3)(cid:3) . (7)Here, for any two operators A and B the commutator is defined as [ A, B ] and theanticommutator as { A, B } . As already pointed out, the original FW transformationcan only produce correct and expected higher-order terms up to first order i.e., 1 /c [26, 30, 33]. In fact, in their original work Foldy and Wouthuysen derived only theterms up to 1 /c , i.e., only the terms in the first line of Eq. (7), however, notablywith the exception of the fourth term [29]. The higher-order terms in the original FWtransformation are of doubtful value [32, 34, 35]. Therefore, the Hamiltonian in Eq. (7)is not trustable and corrections are needed to achieve the expected higher-order terms.The main problem with the original FW transformation is that the unitary operators intwo preceding transformations do not commute with each other. For example, for theexponential operators e iU FW and e iU (cid:48) FW , the commutator [ U FW , U (cid:48) FW ] (cid:54) = 0. Moreover, asthe unitary operators are odd, this commutator produces even terms that have not beenconsidered in the original FW transformation [26, 30, 33]. Taking into account thoseterms, the correction of the FW transformation generates the Hamiltonian as [33] H corr . FW = ( β − ) mc + β (cid:18) O mc − O m c + O m c (cid:19) + E − m c [ O , [ O , F ]]+ β m c {O , [[ O , F ] , F ] } + 364 m c (cid:8) O , [ O , [ O , F ]] (cid:9) + 1128 m c (cid:2) O , (cid:2) O , F (cid:3)(cid:3) − m c [ O , [[[ O , F ] , F ] , F ]] . (8)Note the difference between two Hamiltonians in Eq. (7) and Eq. (8) that are observedin the second and consequent lines in both the equations, however, the terms in thefirst line are the same. Eq. (8) provides the correct higher-order terms of the FWtransformation. In this regard, we mention that an another approach towards the correctFW transformation has been employed by Eriksen; this is a single step approach thatproduces the expected FW transformed higher-order terms [34]. Once the transformedHamiltonian has been obtained as a function of odd and even terms, the final formis achieved by substituting the correct form of odd terms O and even terms E in theexpression of Eq. (8) and calculating term by term.Since we perform here the time-dependent FW transformation, we note that thecommutator [ O , F ] can be evaluated as [ O , F ] = i (cid:126) ∂ O /∂t . Therefore, following thedefinition of the odd operator, the time-varying fields are taken into account throughthis term. We evaluate each of the terms in Eq. (8) separately and obtain that theparticles can be described by the following extended Pauli Hamiltonian [24, 26, 36] H corr . FW = ( p − e A ) m + V − e (cid:126) m σ · B − ( p − e A ) m c + ( p − e A ) m c − (cid:18) e (cid:126) m (cid:19) B mc + e (cid:126) m c (cid:40) ( p − e A ) m , σ · B (cid:41) − e (cid:126) m c ∇ · E tot − e (cid:126) m c σ · [ E tot × ( p − e A ) − ( p − e A ) × E tot ] − e (cid:126) m c (cid:26) ( p − e A ) , ∂ E tot ∂t (cid:27) − ie (cid:126) m c σ · (cid:20) ∂ E tot ∂t × ( p − e A ) + ( p − e A ) × ∂ E tot ∂t (cid:21) + 3 e (cid:126) m c (cid:110) ( p − e A ) − e (cid:126) σ · B , (cid:126) ∇ · E tot + σ · [ E tot × ( p − e A ) − ( p − e A ) × E tot ] (cid:111) + e (cid:126) m c ∇ · ∂ E tot ∂t + e (cid:126) m c σ · (cid:20) ∂ E tot ∂t × ( p − e A ) − ( p − e A ) × ∂ E tot ∂t (cid:21) . (9)The fields in the last Hamiltonian (9) are defined as B = ∇ × A , the external magneticfield, E tot = E int + E ext are the electric fields where E int = − e ∇ V is the internal fieldthat exists even without any perturbation and E ext = − ∂ A ∂t is the external field (onlythe temporal part is retained here because of the Coulomb gauge). It is clear that as theinternal field is time-independent, it does not contribute to the fourth and sixth linesof Eq. (9). However, the external field does contribute to the above terms wherever itappears in the Hamiltonian.The above-derived Hamiltonian can be split in two parts: (1) a spin-independentHamiltonian and (2) a spin-dependent Hamiltonian that involves the Pauli spin matrices.The spin-dependent Hamiltonian, furthermore, has two types of coupling terms. Thedirect field-spin coupling terms are those which directly couples the fields with themagnetic moments e.g., the third term in the first line, the second term in the thirdline of Eq. (9) etc. On the other hand, there are relativistic terms that do not directlycouple the spins to the electromagnetic field - indirect field-spin coupling terms. Theseterms include e.g., the second term of the second line, the fifth line of Eq. (9) etc. Thedirect field-spin interaction terms are most important because these govern the directlymanipulation of the spins in a system with an electromagnetic field. For the externalelectric field, these terms can be written together as a function of electric and magneticfield. These terms are taken into account and discussed in the next section. The indirectcoupling terms are often not taken into consideration and not included in the discussion(see Ref. [36, 37] for details). In this context, we reiterate that our current approach ofderiving relativistic terms does not include the exchange and correlation effect. A similarFW transformed Hamiltonian has previously been derived, however, with a generalKohn-Sham exchange field [23, 25, 26]. As mentioned before, in this article we do notintend to include the exchange-correlation effect, while mostly focussing on the magneticrelaxation and magnetic inertial dynamics. The aim of this work is to formulate the spin dynamics on the basis of the Hamiltonianin Eq. (9). The direct field-spin interaction terms can be written together as electric ormagnetic contributions. These two contributions can be expressed as a series up to anorder of 1 /m [36] H S magnetic = − em S · (cid:34) B + 12 (cid:88) n =1 , , , (cid:18) iω c (cid:19) n ∂ n B ∂t n (cid:35) + O (cid:18) m (cid:19) , (10) H S electric = − em S · (cid:34) mc (cid:88) n =0 , (cid:18) i ω c (cid:19) n ∂ n E ∂t n × ( p − e A ) (cid:35) + O (cid:18) m (cid:19) , (11)where the Compton wavelength and pulsation have been expressed by the usualdefinitions λ c = h/mc and ω c = 2 πc/λ c with Plank’s constant h . We also have usedthe spin angular momentum operator as S = ( (cid:126) / σ . Note that we have droppedthe notion of total electric field because the the involved fields ( B , E , A ) are externalonly, the internal fields are considered as time-independent. The involved terms in theabove two spin-dependent Hamiltonians can readily be explained. The first term in themagnetic contribution in Eq. (10) explains the Zeeman coupling of spins to the externalmagnetic field. The rest of the terms in both the Hamiltonians in Eqs. (11) and (10)represent the spin-orbit coupling and its higher-order corrections. We note that thesetwo spin Hamiltonians are individually not Hermitian, however, it can be shown thattogether they form a Hermitian Hamiltonian [38]. As these Hamiltonians describe asemirelativistic Dirac particle, it is possible to derive from them the spin dynamics ofa single Dirac particle [24]. The effect of the indirect field-spin terms is not yet wellunderstood, but they could become important too in magnetism [36, 37], however, thoseterms are not of our interest here.The electric Hamiltonian can be written in terms of magnetic contributions withthe choice of a gauge A = B × r /
2. The justification of the gauge lies in the factthat the magnetic field inside the system being studied is uniform [26]. The transverseelectric field in the Hamiltonian (10) can be written as E = 12 (cid:18) r × ∂ B ∂t (cid:19) . (12)Replacing this expression in the electric spin Hamiltonian in Eq. (11), one can obtain ageneralised expression of the total spin-dependent Hamiltonian as H S ( t ) = − em S · (cid:104) B + 12 ∞ (cid:88) n =1 , ,... (cid:18) iω c (cid:19) n ∂ n B ∂t n + 14 mc ∞ (cid:88) n =0 , ,... (cid:18) i ω c (cid:19) n (cid:18) r × ∂ n +1 B ∂t n +1 (cid:19) × ( p − e A ) (cid:105) . (13)It is important to stress that the above spin-Hamiltonian is a generalisation of the twoHamiltonians in Eqs. (10) and (11). We have already evaluated the Hamiltonian formsfor n = 1 , , , only the linear interaction terms,that is we neglect the e A term in Eq. (13). Here, we mention that the quadratic termscould provide an explanation towards the previously unknown origin of spin-photoncoupling or optical spin-orbit torque and angular magneto-electric coupling [38–40].The linear direct field-spin Hamiltonian can then be recast as H S ( t ) = − em S · (cid:104) B + 12 ∞ (cid:88) n =1 , ,... (cid:18) iω c (cid:19) n ∂ n B ∂t n + 14 mc ∞ (cid:88) n =0 , ,... (cid:18) i ω c (cid:19) n (cid:26) ∂ n +1 B ∂t n +1 ( r · p ) − r (cid:18) ∂ n +1 B ∂t n +1 · p (cid:19)(cid:27) (cid:105) . (14)This is final form of the Hamiltonian and we are interested to describe to evaluate itscontribution to the spin dynamics.
3. Spin dynamics
Once we have the explicit form of the spin Hamiltonian in Eq. (14), we can proceed toderive the corresponding classical magnetisation dynamics. Following similar proceduresof previous work [23, 24], and introducing a magnetisation element M ( r , t ), themagnetisation dynamics can be calculated by the following equation of motion ∂ M ∂t = (cid:88) j gµ B Ω 1 i (cid:126) (cid:68)(cid:2) S j , H S ( t ) (cid:3)(cid:69) , (15)where µ B is the Bohr magneton, g is the Land´e g-factor that takes a value ≈ ∂ M (1) ∂t = − γ M × B , (16)with the gyromagnetic ratio γ = g | e | / m . Here the commutators between two spinoperators have been evaluated using [ S j , S k ] = i (cid:126) S l (cid:15) jkl , where (cid:15) jkl is the Levi-Civitatensor. This dynamics actually produces the precession of magnetisation vector aroundan effective field. To get the usual form of Landau-Lifshitz precessional dynamics, onehas to use a linear relationship of magnetisation and magnetic field as B = µ ( M + H ).With the latter relation, the precessional dynamics becomes − γ M × H , where γ = γµ defines the effective gyromagnetic ratio. We point out that the there are relativisticcontributions to the precession dynamics as well, e.g., from the spin-orbit coupling dueto the time-independent field E int [23]. Moreover, the contributions to the magnetisationprecession due to exchange field appear here, but are not explicitly considered in thisarticle as they are not in the focus of the current investigations (see Ref. [23] for details).The rest of the terms in the spin Hamiltonian in Eq. (14) is of much importancebecause they involve the time-variation of the magnetic induction. As it has been shownin an earlier work [23] that for the external fields and specifically the terms with n = 1in the second terms and n = 0 in the third terms of Eq. (14), these terms togetherare Hermitian. These terms contribute to the magnetisation dynamics as the Gilbertrelaxation within the LLG equation of motion, ∂ M (2) ∂t = M × (cid:18) A · ∂ M ∂t (cid:19) , (17)where the Gilbert damping parameter A has been derived to be a tensor that has mainlytwo contributions: electronic and magnetic. The damping parameter A has the form[23, 24] A ij = − eµ m c (cid:88) (cid:96),k (cid:2) (cid:104) r i p k + p k r i (cid:105) − (cid:104) r (cid:96) p (cid:96) + p (cid:96) r (cid:96) (cid:105) δ ik (cid:3) × (cid:0) + χ − (cid:1) kj , (18)where is the 3 × χ is the magnetic susceptibility tensor that can beintroduced only if the system is driven by a field which is single harmonic [26]. Notethat the electronic contributions to the Gilbert damping parameter are given by the0expectation value (cid:104) r i p k (cid:105) and the magnetic contributions by the susceptibility. We alsomention that the tensorial Gilbert damping tensor has been shown to contain a scalar,isotropic Heisenberg-like contribution, an anisotropic Ising-like tensorial contributionand a chiral Dzyaloshinskii-Moriya-like contribution [23].In an another work, we took into account the terms with n = 2 in the second termof Eq. (14) and it has been shown that those containing the second-order time variationof the magnetic induction result in the magnetic inertial dynamics. Note that theseterms provide a contribution to the higher-order relativistic effects. The correspondingmagnetisation dynamics can be written as [24] ∂ M (3) ∂t = M × (cid:18) C · ∂ M ∂t + D · ∂ M ∂t (cid:19) , (19)with a higher-order Gilbert damping tensor C ij and inertia parameter D ij that have thefollowing expressions C ij = γ (cid:126) m c ∂∂t ( + χ − ) ij and D ij = γ (cid:126) m c ( + χ − ) ij . We notethat Eq. (19) contains two fundamentally different dynamics – the first term on theright-hand side has the exact form of Gilbert damping dynamics whereas the secondterm has the form of magnetic inertial dynamics [24].The main aim of this article is to formulate a general magnetisation dynamicsequation and an extension of the traditional LLG equation to include higher-orderrelativistic effects. The calculated magnetisation dynamics due to the second and thirdterms of Eq. (14) can be expressed as ∂ M ∂t = em M × (cid:104) ∞ (cid:88) n =0 , ,... (cid:18) iω c (cid:19) n +1 ∂ n +1 B ∂t n +1 + 14 mc ∞ (cid:88) n =0 , ,... (cid:18) i ω c (cid:19) n (cid:26) ∂ n +1 B ∂t n +1 (cid:104) r · p (cid:105) − (cid:68) r (cid:18) ∂ n +1 B ∂t n +1 · p (cid:19) (cid:69)(cid:27) (cid:105) . (20)Note the difference in the summation of first terms from the Hamiltonian in Eq. (14).To obtain explicit expressions for the Gilbert damping dynamics, we employ a generallinear relationship between magnetisation and magnetic induction, B = µ ( H + M ).The time-derivative of the magnetic induction can then be replaced by magnetisationand magnetic susceptibility. For the n -th order time-derivative of the magnetic inductionwe find ∂ n B ∂t n = µ (cid:18) ∂ n H ∂t n + ∂ n M ∂t n (cid:19) . (21)Note that this equation is valid for the case when the magnetisation is time-dependent.Substituting this expression into the Eq. (20), one can derive the general LLG equationand its extensions. Moreover, as we work out the derivation in the case of harmonicdriving fields, the differential susceptibility can be introduced as χ = ∂ M /∂ H . Thefirst term ( n -th derivative of the magnetic field) can consequently be written by the1following Leibniz formula as ∂ n H ∂t n = n − (cid:88) k =0 ( n − k !( n − k − ∂ n − k − ( χ − ) ∂t n − k − · ∂ k ∂t k (cid:18) ∂ M ∂t (cid:19) , (22)where the magnetic susceptibility χ − is a time-dependent tensorial quantity andharmonic. Using this relation, the first term and second terms in Eq. (20) assumethe form ∂ M ∂t (cid:12)(cid:12)(cid:12) first = eµ m M × ∞ (cid:88) n =0 , ,... (cid:18) iω c (cid:19) n +1 n (cid:88) k =0 n ! k !( n − k )! ∂ n − k ( + χ − ) ∂t n − k · ∂ k ∂t k (cid:18) ∂ M ∂t (cid:19) , (23) ∂ M ∂t (cid:12)(cid:12)(cid:12) second = eµ m c M × ∞ (cid:88) n =0 , ,... (cid:18) iω c (cid:19) n n (cid:88) k =0 n ! k !( n − k )! (cid:104) ∂ n − k ( + χ − ) ∂t n − k · ∂ k ∂t k (cid:18) ∂ M ∂t (cid:19) (cid:104) r · p (cid:105)− (cid:68) r (cid:18)(cid:26) ∂ n − k ( + χ − ) ∂t n − k · ∂ k ∂t k (cid:18) ∂ M ∂t (cid:19)(cid:27) · p (cid:19) (cid:69)(cid:105) . (24)These two equations already provide a generalisation of the higher-order magnetisationdynamics including the Gilbert damping (i.e., the terms with k = 0) and the inertialdynamics (the terms with k = 1) and so on.
4. Discussion
It is obvious that, as Gilbert damping dynamics involves the first-order time derivative ofthe magnetisation and a torque due to it, k must take the value k = 0 in the equations(23) and (24). Therefore, the Gilbert damping dynamics can be achieved from thefollowing equations: ∂ M ∂t (cid:12)(cid:12)(cid:12) first = eµ m M × ∞ (cid:88) n =0 , ,... (cid:18) iω c (cid:19) n +1 ∂ n ( + χ − ) ∂t n · ∂ M ∂t , (25) ∂ M ∂t (cid:12)(cid:12)(cid:12) second = eµ m c M × ∞ (cid:88) n =0 , ,... (cid:18) iω c (cid:19) n (cid:104) (cid:18) ∂ n ( + χ − ) ∂t n · ∂ M ∂t (cid:19) (cid:104) r · p (cid:105)− (cid:68) r (cid:18)(cid:26) ∂ n ( + χ − ) ∂t n · ∂ M ∂t (cid:27) · p (cid:19) (cid:69)(cid:105) . (26)Note that these equations can be written in the usual form of Gilbert damping as M × (cid:0) G · ∂ M ∂t (cid:1) , where the Gilbert damping parameter G is notably a tensor [2, 23]. The2general expression for the tensor can be given by a series of higher-order relativisticterms as follows G ij = eµ m ∞ (cid:88) n =0 , ,... (cid:18) iω c (cid:19) n +1 ∂ n ( + χ − ) ij ∂t n + eµ m c ∞ (cid:88) n =0 , ,... (cid:18) iω c (cid:19) n (cid:104) ∂ n ( + χ − ) ij ∂t n ( (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) ) (cid:105) . (27)Here we have used the Einstein summation convention on the index l . Note that thereare two series: the first series runs over even and odd numbers ( n = 0 , , , , · · · ),however, the second series runs only over the even numbers ( n = 0 , , , · · · ). Eq. (27)represents a general relativistic expression for the Gilbert damping tensor, given as aseries of higher-order terms. This equation is one of the central results of this article. Itis important to observe that this expression provides the correct Gilbert tensor at thelowest relativistic order, i.e., putting n = 0 the expression for the tensor is found to beexactly the same as Eq. (18).The analytic summation of the above series of higher-order relativistic contributionscan be carried out when the susceptibility depends on the frequency of the harmonicdriving field. This is in general true for ferromagnets where a differential susceptibilityis introduced because there exists a spontaneous magnetisation in ferromagnets evenwithout application of a harmonic external field. However, if the system is driven by anonharmonic field, the introduction of the susceptibility is not valid anymore. In generalthe magnetic susceptibility is a function of wave vector and frequency in reciprocal space,i.e., χ = χ ( q , ω ). Therefore, for the single harmonic applied field, we use χ − ∝ e iωt andthe n -th order derivative will follow ∂ n /∂t n ( χ − ) ∝ ( iω ) n χ − . With these arguments,one can express the damping parameter of Eq. (27) as (see Appendix A for detailedcalculations) G ij = eµ m c (cid:20) (cid:126) i + (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) (cid:21) ( + χ − ) ij + eµ m c (cid:34) (2 ωω c + ω ) (cid:126) i + ω ( (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) )4 ω c − ω (cid:35) χ − ij . (28)Here, the first term in the last expression is exactly the same as the one that has beenderived in our earlier investigation [23]. As the expression of the expectation value (cid:104) r i p j (cid:105) is imaginary, the real Gilbert damping parameter will be given by the imaginarypart of the susceptibility tensor. This holds consistently for the higher-order termsas well. The second term in Eq. (28) stems essentially from an infinite series whichcontain higher-order relativistic contributions to the Gilbert damping parameter. As ω c scales with c , these higher-order terms will scale with c − or more and thus theircontributions will be smaller than the first term. Note that the higher-order terms willdiverge when ω = 2 ω c ≈ sec − , which means that the theory breaks down at thelimit ω → ω c . In this limit, the original FW transformation is not defined any morebecause the particles and antiparticles cannot be separated at this energy limit.3 Magnetic inertial dynamics, in contrast, involves a torque due to the second-order time-derivative of the magnetisation. In this case, k must adopt the value k = 1 in theafore-derived two equations (23) and (24). However, if k = 1, the constraint n − k ≥ n ≥
1. Therefore, the magnetic inertial dynamics can be described withthe following equations: ∂ M ∂t (cid:12)(cid:12)(cid:12) first = eµ m M × ∞ (cid:88) n =1 , ,... (cid:18) iω c (cid:19) n +1 n !( n − ∂ n − ( + χ − ) ∂t n − · ∂ M ∂t , (29) ∂ M ∂t (cid:12)(cid:12)(cid:12) second = eµ m c M × ∞ (cid:88) n =2 , ,... (cid:18) iω c (cid:19) n n !( n − (cid:104) (cid:18) ∂ n − ( + χ − ) ∂t n − · ∂ M ∂t (cid:19) (cid:104) r · p (cid:105)− (cid:68) r (cid:18)(cid:26) ∂ n − k ( + χ − ) ∂t n − k · ∂ M ∂t (cid:27) · p (cid:19) (cid:69)(cid:105) . (30)Similar to the Gilbert damping dynamics, these dynamical terms can be expressedas M × (cid:16) I · ∂ M ∂t (cid:17) which is the magnetic inertial dynamics [8]. The correspondingparameter has the following expression I ij = eµ m ∞ (cid:88) n =1 , ,... (cid:18) iω c (cid:19) n +1 n !( n − ∂ n − ( + χ − ) ij ∂t n − + eµ m c ∞ (cid:88) n =2 , ,... (cid:18) iω c (cid:19) n n !( n − (cid:104) ∂ n − ( + χ − ) ij ∂t n − ( (cid:104) r l p l (cid:105) − (cid:104) r i p l (cid:105) ) (cid:105) . (31)Note that as n cannot adopt the value n = 0, the starting values of n are different inthe two terms. Importantly, if n = 1 we recover the expression for the lowest ordermagnetic inertia parameter D ij , as given in the equation (19) [24].Using similar arguments as in the case of the generalised Gilbert dampingparameter, when we consider a single harmonic field as driving field, the inertiaparameter can be rewritten as follows (see Appendix A for detailed calculations) I ij = − eµ (cid:126) m c ( + χ − ) ij − eµ (cid:126) m c (cid:18) − ω + 4 ωω c (2 ω c − ω ) (cid:19) χ − ij + eµ m c (cid:126) i ( (cid:104) r l p l (cid:105) − (cid:104) r i p l (cid:105) ) (cid:18) ωω c (4 ω c − ω ) (cid:19) χ − ij . (32)The first term here is exactly the same as the one that was obtained in our earlierinvestigation [24]. However, there are now two extra terms which depend on thefrequency of the driving field and that vanish for ω →
0. Again, in the limit ω → ω c ,these two terms diverge and hence this expression is not valid anymore. The inertiaparameter will consistently be given by the real part of the susceptibility.4
5. Summary
We have developed a generalised LLG equation of motion starting from fundamentalquantum relativistic theory. Our approach leads to higher-order relativistic correctionterms in the equation of spin dynamics of Landau and Lifshitz. To achieve this, we havestarted from the foundational Dirac equation under the presence of an electromagneticfield (e.g., external driving fields or THz excitations) and have employed the FWtransformation to separate out the particles from the antiparticles in the Dirac equation.In this way, we derive an extended Pauli Hamiltonian which efficiently describes theinteractions between the quantum spin-half particles and the applied field. The thus-derived direct field-spin interaction Hamiltonian can be generalised for any higher-orderrelativistic corrections and has been expressed as a series. To derive the dynamicalequation, we have used this generalised spin Hamiltonian to calculate the correspondingspin dynamics using the Heisenberg equation of motion. The obtained spin dynamicalequation provides a generalisation of the phenomenological LLG equation of motionand moreover, puts the LLG equation on a rigorous foundational footing. The equationincludes all the torque terms of higher-order time-derivatives of the magnetisation (apartfrom the Gilbert damping and magnetic inertial dynamics). Specifically, however, wehave focussed on deriving an analytic expression for the generalised Gilbert dampingand for the magnetic inertial parameter. Our results show that both these parameterscan be expressed as a series of higher-order relativistic contributions and that theyare tensors. These series can be summed up for the case of a harmonic driving field,leading to closed analytic expressions. We have further shown that the imaginary partof the susceptibility contributes to the Gilbert damping parameter while the real partcontributes to the magnetic inertia parameter. Lastly, with respect to the applicabilitylimits of the derived expressions we have pointed out that when the frequency of thedriving field becomes comparable to the Compton pulsation, our theory will not be validanymore because of the spontaneous particle-antiparticle pair-production.
6. Acknowledgments
Appendix A. Detailed calculations of the parameters for a harmonic field
In the following we provide the calculational details of the summation towards the resultsgiven in Eqs. (28) and (32).
Appendix A.1. Gilbert damping parameter
Eq. (27) can be expanded as follows G ij = eµ m iω c ( + χ − ) ij + eµ m c ( (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) ) ( + χ − ) ij + eµ m ∞ (cid:88) n =1 , ,... (cid:18) iω c (cid:19) n +1 ( iω ) n χ − ij + eµ m c ∞ (cid:88) n =2 , ,... (cid:18) iω c (cid:19) n ( (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) ) ( iω ) n χ − ij = eµ m iω c ( + χ − ) ij + eµ m c ( (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) ) ( + χ − ) ij + eµ m iω c ∞ (cid:88) n =1 , ,... (cid:18) ω ω c (cid:19) n χ − ij + eµ m c ∞ (cid:88) n =2 , ,... (cid:18) ω ω c (cid:19) n ( (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) ) χ − ij = eµ m c (cid:20) (cid:126) i + (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) (cid:21) ( + χ − ) ij + eµ m c (cid:34) (cid:126) i ∞ (cid:88) n =1 , ,... (cid:18) ω ω c (cid:19) n + ( (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) ) ∞ (cid:88) n =2 , ,... (cid:18) ω ω c (cid:19) n (cid:35) χ − ij = eµ m c (cid:20) (cid:126) i + (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) (cid:21) ( + χ − ) ij + eµ m c (cid:20) (cid:126) i ω ω c − ω + ( (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) ) ω ω c − ω (cid:21) χ − ij = eµ m c (cid:20) (cid:126) i + (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) (cid:21) ( + χ − ) ij + eµ m c (cid:34) (2 ωω c + ω ) (cid:126) i + ω ( (cid:104) r l p l (cid:105) − (cid:104) r l p i (cid:105) )4 ω c − ω (cid:35) χ − ij . (A.1)We have used the fact that ωω c < x + x + x + ... = 11 − x ; − < x < . (A.2) EFERENCES Appendix A.2. Magnetic inertia parameter
Eq. (31) can be expanded as follows I ij = eµ m (cid:18) iω c (cid:19) ( + χ − ) ij + eµ m ∞ (cid:88) n =2 , ,... (cid:18) iω c (cid:19) n +1 n !( n − ∂ n − ( + χ − ) ij ∂t n − + eµ m c ∞ (cid:88) n =2 , ,... (cid:18) iω c (cid:19) n n !( n − (cid:104) ∂ n − ( + χ − ) ij ∂t n − ( (cid:104) r l p l (cid:105) − (cid:104) r i p l (cid:105) ) (cid:105) = eµ m (cid:18) iω c (cid:19) ( + χ − ) ij + ∞ (cid:88) n =2 , ,... (cid:18) iω c (cid:19) n +1 n !( n − iω ) n − χ − ij + eµ m c ∞ (cid:88) n =2 , ,... (cid:18) iω c (cid:19) n n !( n − (cid:104) r l p l (cid:105) − (cid:104) r i p l (cid:105) ) ( iω ) n − χ − ij = − eµ (cid:126) m c ( + χ − ) ij + eµ m (cid:18) iω c (cid:19) ∞ (cid:88) n =2 , ,... (cid:18) ω ω c (cid:19) n − n !( n − χ − ij + eµ m c (cid:18) iω c (cid:19) ∞ (cid:88) n =2 , ,... (cid:18) ω ω c (cid:19) n − n !( n − (cid:104) r l p l (cid:105) − (cid:104) r i p l (cid:105) ) χ − ij = − eµ (cid:126) m c ( + χ − ) ij − eµ (cid:126) m c ∞ (cid:88) n =2 , ,... (cid:18) ω ω c (cid:19) n − n !( n − χ − ij + eµ m c (cid:126) i ∞ (cid:88) n =2 , ,... (cid:18) ω ω c (cid:19) n − n !( n − (cid:104) r l p l (cid:105) − (cid:104) r i p l (cid:105) ) χ − ij = − eµ (cid:126) m c ( + χ − ) ij − eµ (cid:126) m c χ − ij (cid:34) (cid:18) ω ω c (cid:19) + 3 (cid:18) ω ω c (cid:19) + 4 (cid:18) ω ω c (cid:19) + ... (cid:35) + eµ m c (cid:126) i ( (cid:104) r l p l (cid:105) − (cid:104) r i p l (cid:105) ) χ − ij (cid:34) (cid:18) ω ω c (cid:19) + 4 (cid:18) ω ω c (cid:19) + 6 (cid:18) ω ω c (cid:19) + ... (cid:35) = − eµ (cid:126) m c ( + χ − ) ij − eµ (cid:126) m c (cid:18) − ω + 4 ωω c (2 ω c − ω ) (cid:19) χ − ij + eµ m c (cid:126) i ( (cid:104) r l p l (cid:105) − (cid:104) r i p l (cid:105) ) (cid:18) ωω c (4 ω c − ω ) (cid:19) χ − ij . (A.3)Here we have used the formula1 + 2 x + 3 x + 4 x + 5 x + ... = 1(1 − x ) ; − < x < . (A.4) References [1] Landau L D and Lifshitz E M 1935
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