Magnetization Dynamics Italian School on Magnetism Pavia - 6-10 February 2012 Giorgio Bertotti INRIM - Istituto Nazionale di Ricerca Metrologica, Torino, Italy Part I Free energy of a ferromagnetic body: relevant energy terms Energy minimization: Brownʼs equations Magnetization dynamics: Landau-Lifshitz-Gilbert (LLG) equation Energy relations Part II Switching mechanisms Ferromagnetic resonance Spin waves Spin-transfer-driven magnetization dynamics
Bibliography (textbooks only) D.D. Stancil and A. Prabakhar, Spin Waves, Springer, New York, 2009. H. Suhl, Relaxation Processes in Micromagnetics, Oxford University Press, Oxford, 2007. A. G. Gurevich and G.A. Melkov, Magnetization Oscillations and Waves, CRC Press, Boca Raton, 1996. Extensive discussion of microwave phenomena and devices, yet through a somewhat clumsy presentation of the material. G. Bertotti, I.D. Mayergoyz, and C. Serpico, Nonlinear Magnetization Dynamics in Nanosystems, Springer, Oxford, 2009. Advanced treatment of nonlinear magnetization dynamics, mainly for the case of uniformly magnetized nanomagnets, and of spin-wave instabilities. A.H. Morrish, The Physical Principles of Magnetism, IEEE Press, Piscataway, 2001. Classical text with a lot of information about resonance phenomena (nuclear, paramagnetic, ferromagnetic). R.C. OʼHandley, Modern Magnetic Materials, Wiley, New York, 2000. Comprehensive text with extensive information on behavior of various magnetic materials. A. Hubert and R. Schaefer, Magnetic Domains, Springer, Berlin, 1998. Text with useful chapters on micromagnetics, with discussion of the origin of magnetic domains. In addition, exceptional magnetic domain images. G. Bertotti, Hysteresis in Magnetism, Academic Press, San Diego, 1998. Pedagogical discussion of micromagnetics.
Magnetization Dynamics Italian School on Magnetism Pavia - 6-10 February 2012 Part I Free energy of a ferromagnetic body: relevant energy terms Energy minimization: Brownʼs equations Magnetization dynamics: Landau-Lifshitz-Gilbert (LLG) equation Energy relations
The micromagnetic approach Main contributions to the free energy of a ferromagnet: exchange energy magnetostatic energy magnetocrystalline anisotropy energy interaction with external field It is the competition between these energies that gives rise to magnetic domains and is eventually responsible for the hysteresis and switching phenomena observed in particles, films, etc. In micromagnetics, one is given the energy GL of the ferromagnet, defined with respect to certain configurational coordinates X (both GL and X will have to be defined in precise terms); then one looks for the set of local minima, characterized by G L / X =0and 2 G L / X 2 > 0, that represent possible metastable states for the system. The key complication is that X is not just a number, but represents the full magnetization vector field M(r) defined over the entire body volume. Thus, energy minimization has to be carried out in the infinite-dimensional functional space of all possible magnetization configurations (variational problem). The equations that express the condition of energy minimum for a given magnetization configuration are known as Brownʼs equations. This energy minimization program does not say anything about how the system will evolve if initially it is not in equilibrium; the Landau-Lifshitz-Gilbert (LLG) equation provides a suitable dynamic extension of micromagnetics for the description of out-of-equilibrium stuations.
G L (X) Energy landscapes apply field Barkhausen jump (meta)stable micromagnetic configurations M(.) X remove field The use of energy landscapes implies a separation of time scales: the relaxation time after which the system reaches equilibrium with respect to a particular value of X is much shorter than the time over which the system evolves from one value of X to another The number of stable magnetization configurations (local energy minima) can be very large due to structural disorder History dependence: the initial and final energy profiles are the same but the state occupied by the system is different depending on past history. The state occupied by the system is history-dependent if the temperature is low enough
Elementary volumes and saturation magnetization In micromagnetics, the ferromagnetic body is treated as a continuous medium with smooth magnetic properties. The smoothness comes from averaging over elementary volumes small enough with respect to the scale over which the magnetization varies significantly, but large enough with respect to atomic distances. ferromagnetic body M (r) elementary volume The local magnetization vector M(r) describes the magnetic state of the given elementary volume. We assume that its magnitude M(r) 2 is not affected by external fields (exchange dominates with respect to thermal fluctuations). M (r) = M s (T ) saturation magnetization The magnetic state of each elementary volume is thus defined by the vector: m(r) = M(r) M s constant unit modulus but variable orientation The state of the body is described by the magnetization vector field m(.) defined for each point inside the body. Although the magnetization magnitude is constant, its orientation can vary from point to point. It is the spatial variation of this orientation that defines the magnetic state of the body. X m(.) G L (X) V (f EX + f AN + f M +...) dv
Exchange energy Exchange energy is caused by the fact that whenever there is some misalignment of neighboring magnetic moments, there is an energy cost involved. The conclusion is evident if we consider for instance the Heisenberg Hamiltonian: m (r) H = <i,j> J ij S i S j Body volume is V If Si is not parallel to its neighbor Sj, the scalar product decreases and the energy ( under positive Jij ) increases. It is this non-uniformity energy that is usually meant when one speaks of exchange energy. in this sense, we have exchange energy only when the gradient of m takes non-zero values. If the variation from point to point is not too rapid, we can make a Taylor expansion of the exchange energy as a function of the magnetization gradients, and keep the lowest order terms. The exact form of the expansion will depend on the symmetry of the lattice hosting the magnetic moments. The leading term in the energy density consistent with cubic symmetry is the following: F EX = V A m x 2 + m y 2 + m x 2 dv The parameter A is the exchange stiffness constant. Its typical value is of the order of 10-11 J/m.
Magnetocrystalline anisotropy energy easy axis e AN m exchange is isotropic symmetry is broken: uniaxial anisotropy cubic anisotropy... Magnetocrystalline anisotropy energy depends on the relative orientation of the local magnetization with respect to certain preferred directions. In a perfect single crystal these directions will be the same everywhere inside the body. However, in a polycrystal they will vary from point to point. In all cases, anisotropy energy has a purely local character. this direction can be space-dependent F AN = V K 1 (m e AN ) 2 dv This is the energy expression for the particular case of uniaxial anisotropy. The parameter K1 is the anisotropy constant.
Magnetostatic energy m (r) Magnetostatic energy is potential energy of magnetic moments in the magnetic field they themselves have created. Magnetostatics permits one to compute this energy if the vector field m(.) is known: F M = µ 0M s 2 V H M m dv Equivalent expression is: F M = µ 0 2 all space H M 2 dv The magnetostatic field is solution of magnetostatic Maxwellʼs equations with the usual interface conditions at the surface of the body: H M = M s m, H M =0 H M =0, H M =0 inside the body outside The relation between HM and m is not local because magnetic charges even far away from a certain point may affect the value of the magnetostatic field at that point. Although the magnetostatic energy is expressed as a volume integral, it is not true that it comes from local contributions as it is the case, for example, for crystal anisotropy energy. It is only after defining the geometry of the problem and after selecting a particular magnetization configuration m(.) for the entire body that it is possible to calculate the total magnetostatic energy for the body.
Shape anisotropy Magnetostatic energy takes a particular form for uniformly magnetized ellipsoidal bodies. This leads to the notion of shape anisotropy. N z z Consider an ellipsoidal body with principal axes along x, y, z, and corresponding demagnetizing coefficients Nx, Ny, Nz. Assume that the body is uniformly magnetized, with normalized magnetization m. Then the magnetostatic energy is: F M = µ 0M 2 s V 2 Nx m 2 x + N y m 2 y + N z m 2 z N x x N y y ellipsoidal particle When the body has spheroidal shape with symmetry axis along z (i.e., N x = N y = N ), apart from inessential constant terms, the energy can be rewritten in a form identical to that for uniaxial anisotropy: F M = const µ 0M 2 s V 2 (N N z )(m e z ) 2 µ 0 M 2 s 2 (N N z ) K 1 Shape-anisotropy energy is just magnetostatic energy for a particular case.
Micromagnetic free energy G L (m(.); H a,t)= exchange energy V crystal anisotropy magnetostatic energy A( m) 2 + f AN (m) µ 0M s H 2 M external field interaction m µ 0 M s H a m dv exchange stiffness constant ~ 10-11 J/m ( m) 2 = m x 2 + m y 2 + m z 2 The vector m(r) has unit magnitude everywhere: m 2 =1 m = M M s The magnetostatic field HM is the solution of magnetostatic Maxwell equations under given m(.): H M = M s m, H M =0 H M =0, H M =0 inside the body outside The energy GL is not expressible in terms of a purely local energy density, because the magnetostatic field is known only after the magnetization configuration is specified for the entire body and magnetostatic Maxwell equations are solved.
Competing energies exchange uniform magn. magnetostatic zero magn. moment magnetocrystalline along easy axis external field along field
Energy minimization If the system energy is at a minimum when the magnetization is m(.), then, when m(.) is varied by the small amount m(r) m(r) +δm(r), the corresponding energy variation δg L is such that δg L =0 to the first order in δm and δg L > 0 to the second order in δm for every arbitrary variation δm that preserves the magnetization modulus, i.e., of the form δm(r) =m(r) δv(r), where δv(r) is a small arbitrary vector. G L (m(.); H a,t)= V A( m) 2 + f AN (m) µ 0M s H 2 M m µ 0 M s H a m dv δg L = µ 0 M s effective field V (H eff δm) dv +2A S m n δm ds H eff = 2A 2 m 1 f AN µ 0 M s µ 0 M s m + H + H M a effective field exchange field anisotropy field magnetostatic field external field δm(r) =m(r) δv(r) δg L = µ 0 M s V (m H eff ) δv dv +2A S m n m δv ds
Brownʼs equations δg L =0 for every arbitrary variation δv(r) m H eff =0 everywhere inside the body m n =0 everywhere at the body surface Brownʼs equations H eff = 2A 2 m 1 f AN µ 0 M s µ 0 M s m + H + H M a effective field exchange field anisotropy field magnetostatic field external field The effective field contains a Maxwellian part (applied and magnetostatic fields) and a non- Maxwellian part (exchange and anisotropy fields). They coexist and play identical roles. According to Brownʼs equations, at equilibrium there must be complete absence of internal magnetic torques in the body. Brownʼs equations are nonlinear because the effective field depends in general on m. The component of the effective field along m plays no role as a consequence of the fact that the magnetization modulus cannot change.
Characteristic parameters Micromagnetics is governed by a small number of fundamental parameters, which emerge once the micromagnetic energy is written in dimensionless form, by measuring energies in units of µ 0 Ms 2 V and fields in units of M s : g L = G L µ 0 M 2 s V, h a = H a M s, h M = H M M s The resulting dimensionless energy is (we consider for simplicity the case of uniaxial anisotropy): g L (m(.); h a,t)= V l 2 EX ( m) 2 2 κ (m e AN )2 2 1 2 h M m h a m dv V l EX = exchange length κ = 2K 1 µ 0 M 2 s 2A µ 0 M 2 s hardness (or quality) factor (dimensionless) The exchange length permits one to define what is large and what is small in micromagnetics. The hardness parameter permits one to introduce the notion of magnetically soft and magnetically hard material. The corresponding dimensionless effective field is: h eff = l 2 EX 2 m + κ (m e AN ) e AN + h M + h a
Characteristic parameters The hardness parameter permits one to introduce the notion of soft versus hard material: κ = 2K 1 µ 0 M 2 s κ 1 κ 1 soft material hard material For iron, where K 1 5 10 4 J/m 3 and µ 0 M s 2 T, one finds κ 0.03. There are three characteristic lengths in micromagnetics corresponding to different combinations of the exchange length and the hardness parameter: l EX = 2A µ 0 M 2 s The following order-of-magnitude estimate gives an idea of the typical values involved: l W = l EX = κ A K 1 A 10 11 J/m K 1 10 4 J/m 3 l EX 5 l W 60 nm nm l D = l EX κ = 2 AK 1 µ 0 M 2 s µ 0 M s 1 T l D 0.4 nm
G L (X) not in equilibrium Micromagnetics and hysteresis apply field Barkhausen jump (meta)stable micromagnetic configurations M(.) X remove field The use of energy landscapes implies a separation of time scales: the relaxation time after which the system reaches equilibrium with respect to a particular value of X is much shorter than the time over which the system evolves from one value of X to another The number of stable magnetization configurations (local energy minima) can be very large due to structural disorder History dependence: the initial and final energy profiles are the same but the state occupied by the system is different depending on past history. The state occupied by the system is history-dependent if the temperature is low enough
Dynamic considerations µ An isolated magnetic moment precesses around an external magnetic field Ha according to the equation: dµ dt = γ µ H a γ represents the absolute value of the gyromagnetic ratio. When the magnetic moment is due to electron spin: γ 2.2 10 5 m A -1 s -1. In micromagnetics, this equation is generalized in several respects: the magnetization M(r) takes the place of the individual magnetic moment; interactions inside the medium are taken into account by the micromagnetic effective field Heff, which takes the place of the external magnetic field; M t = γ M H eff Relaxation toward equilibrium is described by additional phenomenological damping terms. The resulting equation is the Landau-Lifshitz-Gilbert equation.
Characteristic length, time, and field scales Equation for magnetization dynamics before introducing damping effects: M t = γ M H eff H eff = 2A µ 0 M s 2 m + H AN + H M + H a H AN = 1 f AN µ 0 M s m m = M M s The above equation is nonlinear, of partial differential (due to exchange) as well as integral (due to magnetostatic interactions) character. In the problem, there exist a characteristic field scale, given by the saturation magnetization M s (a typical value is µ T, i.e., A/m) and a characteristic time scale, given by (γm s ) 1 0 M s 1 M s 10 6 ( (γm s ) 1 6 ps when µ 0 M s 1 T ) By measuring time, magnetization, and fields in these units one obtains the dimensionless equation: m t = m h eff m = M/M s, h eff = H eff /M s, time is measured in units of (γm s ) 1. If in addition one measures lengths in units of the exchange length l EX = 2A/µ 0 Ms 2, one obtains for the effective field the dimensionless expression: h eff = 2 m + h AN + h M + h a lengths are measured in units of the exchange length l EX = 2A/µ 0 M 2 s
G L (X) not in equilibrium Relaxation toward equilibrium GL is a thermodynamic potential with the property that it is a decreasing function of time for any transformation taking place under constant external field Ha and temperature T. (meta)stable micromagnetic configurations δg L = V (h eff δm) dv V X dg L dt = V this expression is valid under constant external field Ha h eff m t dv V m t = m h eff dg L dt =0 the system never relaxes toward equilibrium! something is missing m t = m h eff + (dissipation) + (fluctuation) constant energy decreasing energy random fluctuations
Landau-Lifshitz equation Energy relaxation mechanisms can be taken into account by suitable phenomenological terms. Relaxation must favor the progressive alignment of the magnetization to the effective field. In their 1935 paper, Landau and Lifshitz introduced a contribution to the magnetization rate of change proportional to the effective field component perpendicular to the magnetization: m t = m h eff + α [h eff (m h eff ) m] h eff α [h eff (m h eff ) m] m m h eff This equation can be written in the equivalent form: m t = m h eff α m (m h eff ) this is the form in which the equation is most often written nowadays. α The dimensionless parameter measures the importance of damping effects. Usually α 1. m m/ t =0 Writing the equation in this form makes it evident that under all circumstances. This means that the micromagnetic condition of constant magnetization magnitude is preserved by the dynamics. m/ t =0 m h eff =0 The equation is consistent with Brownʼs equations, because when.
Gilbert form If one heuristically thinks of the effective field as the driving force and the magnetization rate as the velocity, one is led to consider the typical viscous relaxation law: h eff α m t =0 This simple law is not completely satisfactory because it affects also the magnetization modulus. Since we know that the magnetization modulus will stay constant irrespective of the forces acting on the system, we should restrict the validity of the relaxation law to the component of the effective field perpendicular to the magnetization rate: m h eff α m t =0 We would like to further modify this law in order to make it consistent with the precessional law,, that should be recovered in the limit of no relaxation. This suggest: m/ t = m h eff m t = m or equivalently: h eff α m t m t α m m t = m h eff This is the Gilbert form of the Landau-Lifshitz equation (Landau-Lifshitz-Gilbert (LLG) equation)
Equivalence of Landau-Lifshitz and Gilbert forms If one takes the vector product m x... of both members of the equation: m t α m m t = m h eff and one combines the resulting equation with the original equation, one obtains the following result: 1+α 2 m t = m h eff α m (m h eff ) which coincides with the Landau-Lifshitz equation apart from a renormalization of the time unit. In this sense the Landau-Lifshitz and Gilbert forms of the dynamic equation are mathematically equivalent. It should be noted that there are no strict reasons why the damping parameter should be a simple constant. In general it may be expected to be a function of the dynamical state of the system. α
Energy relations Rate of change of the system energy: dg L dt = V h eff m dv t V + g L t this term is present if energy explicitly depends on time (e.g., because the external magnetic field is time-dependent) From the dynamic equation expressed in Gilbert form, one finds: h eff m t = α h eff m m t = α m m t t α m m t = α m t 2 In the case when the energy explicitly depends on time only through the external magnetic field, one finally obtains: dg L dt = V α m 2 dv m dh a t V dt m = 1 V V m dv assuming that the field is uniform in space Under constant field the energy can only decrease. Consequently, the only admissible processes are those of relaxation toward micromagnetic configurations corresponding to local energy minima.
Landau-Lifshitz-Gilbert equation (summary) Landau-Lifshitz equation: 1+α 2 m t = m h eff α m (m h eff ) Gilbert form: m t α m m t = m h eff Effective field: h eff = 2 m + h AN + h M + h a m m/ t =0 when m/ t =0 m h eff =0 Rate of change of system energy: dg L dt = V α m 2 dv m dh a t V dt m = M/M s, h eff = H eff /M s, time is measured in units of (γm s ) 1, α describes damping effects ( ) α 1 lengths are measured in units of the exchange length l EX = 2A/µ 0 Ms 2 the dynamics preserve the magnetization magnitude condition m(r,t) 2 =1 the dynamics preserve the micromagnetic equilibrium condition m h eff =0 under constant external field, the energy is always a decreasing function of time
Magnetization Dynamics Italian School on Magnetism Pavia - 6-10 February 2012 Part II Switching mechanisms Ferromagnetic resonance Spin waves Spin-transfer-driven magnetization dynamics
Disorder versus dynamical nonlinearities g L (m(.); h a,t)= V l 2 EX ( m) 2 2 κ (m e AN )2 2 1 2 h M m h a m dv V m t α m m t = m h eff g L (X) magnetization processes dominated by disorder domain nucleation domain wall motion rate-independent hysteresis... micromagnetic configurations m(.) ( ) X Easy Axis M magnetization processes dominated by dynamical nonlinearities particle switching spin-wave instabilities spin-transfer-driven dynamics... h a H
LLG dynamics for uniformly magnetized magnets ( ) The state of a uniformly magnetized nanomagnet is described by a single vector m; magnetization dynamics take place on the surface of the unit sphere m 2 = 1 m sphere easy axis Easy Axis M In the case of an ellipsoidal particle with principal axes along x,y,z, and crystal anisotropy characterized by the same symmetry as shape anisotropy, the system energy takes the simple quadratic form: h a external field g L (m; h a ) = 1 2 Dx m 2 x + D y m 2 y + D z m 2 z ha m H (D x,d y,d z ) describe shape + crystal anisotropy if the dynamics do not explicitly depend on time, e.g., the external field is constant, (autonomous dynamics): ellipsoidal particle easy e x e z hard e y dynamics can be given a powerful geometrical representation through the associated phase portrait D x <D y <D z If, then x axis is the easy axis and z axis is the hard axis
Phase portraits of magnetization dynamics phase portrait under ha = 0 and α =0 hard axis e z e z hard u easy e x e y D x <D y <D z easy axis e x s d e y dm dt = m h eff h eff = g L m Dx m 2 x + D y m 2 y + D z m 2 z g L (m) = 1 2 stereographic projection e z d 1 w y u s 2 u 1 s 1 s d w x e x e y energy minimum d 2 energy minimum
Effect of damping α =0 d 1 w y e z u s 2 u 1 s 1 w x e x s d e y d 2 w y wy α = 0 d 1 d 1 s 2 u 1 s 1 w x u 1 s 2 s 1 wx d 2 d 1
Magnetization switching In the micromagnetic approach, the central notion is that of stable equilibrium state: when stability is lost, it is tacitly understood that the system will jump to some other stable state. What occurs to the system when stability is lost is a dynamical aspect that must be studied by considering the equations for the magnetization dynamics (LLG equation). By the term switching one means situations where there are only two states involved in the process: the initial state which becomes unstable and a second stable state (typically with reversed magnetization) which the system reaches (switching mechanism) after a certain characteristic time (switching time). The key problem is to predict under what field conditions and on what time scale switching will take place. A number of questions of interest to applications can be made: Which are the conditions that will make the reversal particularly fast? How important is the role of damping? Are there cases where the magnetization remains spatially uniform during the reversal? Is the initial presence or lack of spatial uniformity a feature that significantly affects the conditions for magnetization reversal?
Switching strategies Going over energy barriers: damping-dominated switching slow small initial torque m ha long switching time switching is guaranteed if the field is applied for a sufficiently long time ha damping plays a dominant role dg L dt = α m h eff 2 Going around energy barriers: precessional switching m ha large initial torque short switching time switching occurs only if a field pulse of suitable duration is applied damping plays no role (quasi-conservative dynamics) e x e x e z m (1) m (0) h a e z m (2) m (1) h a h M e y e y dm dt = m h eff fast ha
Damping-dominated switching ( ) Easy easy Axis axis Spheroidal particle, i.e., D x = D y D M Easy axis is along z axis, i.e., D z <D Initially, m = ez, field ha is applied along -z direction h a Switching occurs when h a < κ eff 1 dm z κ eff dt = α m z + h a 1 m 2 z κ eff θ H m h eff tan θ = α κ eff = D D z h a < κ eff αm (m h eff ) dm/dt When the initial state becomes unstable, the system starts to precess around the anisotropy axis and gradually approaches the reversed state. This approach is controlled by the damping constant. Trajectories traversed under nonzero damping form a fixed angle with respect to the trajectories of the conservative dynamics. The magnetization motion during switching is the same for all systems if one uses as normalized time coordinate and as normalized field intensity. h a /κ eff κ eff t α
Precessional switching zero field d 1 w y Γ2 d 1 w y Γ 1 nonzero field along y axis s 2 u 1 s 1 w x 1 e x s 2 2 s 1 +e x L w x switching trajectory reversed magnetization u 1 initial magnetization d 2 d 2 Precessional switching is fast, so it is a good approximation to neglect the effect of damping and assume that the magnetization motion is conservative during the switching process. Precessional switching is possible when there exists a conservative trajectory under nonzero field visiting both of the states (initial and reversed magnetization) that are stable under zero field. Since the field induces a persistent precessional motion, precessional switching occurs only if the field is switched off at the right time, that is, when the system is within the zero-field energy region associated with reversed magnetization. Therefore, there exists a regular sequence of time windows inside which the field should be switched off in order to produce switching. This ceases to be true for large times, when the cumulative effect of damping starts to be important. After the field is switched off, the magnetization reaches the final stable state of reversed magnetization by oscillations of decreasing amplitude controlled by damping ( ringing effect).
Ferromagnetic resonance Magnetization tends to precess around the effective field. Because of stiffness brought about by exchange interactions, magnetization precession in ferromagnets tends to be coherent in space. There are situations, typically uniformly magnetized particles with convenient shape, subject to some external dc magnetic field, in which uniform and coherent precession of the magnetization around its equilibrium orientation is one of the natural oscillation modes for the system ( Kittel mode). The associated precessional angular frequency ω 0 depends on the dc magnetic field as well as on the geometrical shape of the particle. Resonance occurs when a small rf magnetic field is applied transversally to the equilibrium magnetization and its frequency is equal to ω 0. When the rf field amplitude is increased, nonlinear effects appear in the form of distortions in the dynamic susceptibility curve ( foldover effects).
Spatially uniform precession (Kittel mode) Ellipsoidal particle with principal axes along x, y, z; crystal anisotropy is neglected Demagnetizing factors are: D x,d y,d z Demagnetizing field is spatially uniform if particle is uniformly magnetized There exist exact spatially uniform solutions to the LLG equation External dc magnetic field is applied along z axis m 0 e z is a stable equilibrium orientation for m m m 0 + δm ω 0 h az m 0 e z dm dt = m h eff + (damping) h eff =(h az D z m z ) e z D x m x e x D y m y e y m e z + δm x e x + δm y e y h eff (h az D z ) e z D x δm x e x D y δm y e y d dt δmx δm y = 0 (h az + D y D z ) (h az + D x D z ) 0 we consider small motions around the equilibrium orientation m 0 e z δm is a solution of the equation obtained by linearizing the LLG equation around δm lies in x-y plane because the magnitude of m must be preserved we neglect damping effects in the computation of δm δmx δm y δm m 0 d 2 x dt 2 + ω2 0 x =0 ω 2 0 =(h az + D x D z )(h az + D y D z ) Kittel frequency
Kittel frequency ω 2 0 =(h az + D x D z )(h az + D y D z ) Kittel frequency depends on the geometrical shape of the particle, but it is independent of its absolute linear dimensions Examples of Kittel frequency for a soft material with no crystal anisotropy D x + D y + D z =1 ω 0 = h az 1 ω 0 = h az +1/2 ω 0 = h az ω 0 = h az (h az + 1) ω 0 = h az (h az 1/2)
Ferromagnetic resonance curves spheroidal symmetry: D x = D y D circularly polarized rf magnetic field h a (t) is applied in addition to the dc field magnetization response to the rf field is computed from the linearized equation for the forced motion under h a (t) damping must be included in the computation m h az dm dt α m dm dt = m h eff h eff =(h az D z m z ) e z D m + h a (t) h a (t) =h a (e x cos ωt + e y sin ωt) h a lorentzian shape ω αωχ ω 0 = h az + D D z αωχ absorbed power = αω 2 h 2 a (ω ω 0 ) 2 + α 2 ω 2 (ω 0 ω)/αω
dm dt α m dm dt = m h eff Nonlinear effects (foldover) dm dt = ω 0 (m z ) e z m m h a (t) α dm dt m h az ω 0 (m z )=[h az +(D D z ) m z ] h a ω increasing h a absorbed power foldover ω =0.5 D z =1 D =0 α =0.1 h az
Spin waves Given the static, spatially uniform, saturated magnetization state, it is natural to consider the nature and properties of small-amplitude perturbations of the saturated state. These perturbations will be solutions of an equation that is obtained by linearizing the LLG equation around the saturation state. The Kittel mode is just one of these solutions, with the property of being spatially uniform; spatially non-uniform solutions analogous to the Kittel mode, in the sense that they too depend on the geometrical shape of the particle but are independent of absolute dimensions, are termed magnetostatic modes. There also exist wave-like solutions to the linearized equations, characterized by a wave-vector and a well-defined dispersion relation: these solutions are termed spin waves. The simplest situation occurs for bulk spin waves in extended media, where boundary conditions play a negligible role. Substantial deviations from this behavior occur in confined media (e.g., thin films, multilayers, etc.) due to the dominant role played by boundary conditions.
Spin waves Spheroidal object with polar axis along the z direction; crystal anisotropy is neglected Demagnetizing factors are: D x = D y D,D z External dc magnetic field is applied along z axis m 0 e z is a stable equilibrium orientation for the magnetization h az m m 0 e z θ q q m t = m h eff + (damping) this a matrix that h eff = 2 m + h az e z + h M (m) operates on the vector δm m e z + δm exp (iq r) h eff (h az D z ) e z q 2 I + q q q 2 δm exp(iq r) d δmx 0 h = az D z + q 2 dt δm y haz D z + q 2 +sin 2 θ q 0 We consider plane-wave perturbations with wave-vector q: m e z + δm exp (iq r) Boundary effects are neglected (this is an acceptable approximation if 1/q << L, where L is typical linear dimension of the object) Damping effects are neglected We assume that q lies in the x-z plane (this is no restriction since problem is invariant with respect to rotations around z axis) δmx δm y d 2 x dt 2 + ω2 0 x =0 ω 2 q = h az D z + q 2 h az D z + q 2 +sin 2 θ q spin-wave dispersion relation
Spin-wave manifold θ q = π/2 magnetostatic modes θ q =0 ω q Kittel mode with frequency: ω 0 = h az + D D z ω q = q (h az D z + q 2 ) h az D z + q 2 +sin 2 θ q
Magnetostatic modes
Magnetoresistance and spin transfer magnetoresistance electron transport is affected by magnetic state M M M M! -! -! +! + R low! -! +! +! - R high spin transfer spin-transfer torque magnetic state is affected by electron transport fixed layer free layer
Including spin transfer into LLG equation J e m e p free layer free layer spacer spacer fixed layer fixed layer Current density is positive when electrons flow from the free layer into the fixed layer bp and cp are model-dependent parameters smaller than 1 in magnitude h a spin-transfer term dm dt α m dm dt = m h eff + β m (m e p) 1+c p m e p current density β = b p J e J p J p = µ 0 M 2 s e d free-layer thickness damping is here current density is here J p 10 9 A cm 2 m dm/dt =0 magnetization magnitude is preserved (dynamics takes place on the unit sphere) α 1, β 1 damping and spin transfer act as small perturbations of the conservative dynamics dm/dt = m h eff
Admissible dynamical regimes The key aspects of spin-transfer-driven phenomena are revealed by studying spatially uniform magnetization dynamics as a nonlinear dynamical system evolving on the surface of the unit sphere. α 1 β 1 dm dt α m dm dt = m h eff + β m (m e p) 1+c p m e p m 2 =1 : magnetization dynamics take place on the unit sphere Under constant field and constant current: chaos is precluded only two situations are possible: stationary modes self-oscillations α 1 and β 1 Dynamics can be viewed as a perturbation of the conservative dynamics corresponding α =0 β =0 dm/dt = m h eff to and :.
Effect of damping and current dg L dt = α dm dt 2 m e p β dm 1+c p m e p dt Energy balance equation governing deviations from conservative trajectories Under zero current, the energy can only decrease, so the only admissible process is relaxation toward local energy minima and the only possible persistent states are stationary modes. Under nonzero current, the rate of change of energy has no longer a definite sign and stable limit cycles (i.e., persistent magnetization oscillations) become possible. α = 0 Stationary mode α = 0 β =0 β = 0 Self-oscillation (limit cycle) dg L dt 0 dg L dt =?
Energy balance dg L dt = α dm dt 2 β m e p 1+c p m e p dm dt Self-oscillation (limit cycle) In a limit cycle the energy is periodic in time T dm dt 2 + β α m e p 1+c p m e p dm dt dt =0 oscillation period Since α 1 and β 1, the limit-cycle trajectory is approximately equal to a certain trajectory of the conservative dynamics C(g 0 ) conservative trajectory m h eff + β α m e p dm 0 1+c p m e p dm along dt = m h eff C(g 0 )
Existence of self-oscillations M (g 0 ; h a,β/α)= C(g 0 ) m h eff + β α m e p dm 1+c p m e p z y Existence of limit cycles: In the limit, the equation represents the necessary and sufficient condition for the existence of a periodic solution (limit cycle) of the equation for the magnetization dynamics under spinpolarized current. The limit cycle is asymptotically coincident with a certain trajectory of energy α 0,β 0,β/α const g 0 when. M (g 0, h a,β/α)=0 of the conservative dynamics and is stable (unstable) M (g 0, h a,β/α) / g 0 > 0(< 0) C (g 0 )
Uniaxial symmetry φ D x = D y = D D D z = κ eff θ m An inessential constant has been dropped h a free layer spacer e p fixed layer g L (m; h a ) = κ eff 2 cos2 θ h a cos θ Φ(m; h a,β/α)= κ eff 2 cos2 θ h a cos θ + β α ln (1 + c p cos θ) c p J e dm dt α m dm dt = m h eff + β m (m e p) 1+c p m e p dθ dt = α Φ θ dφ dt = 1 sin θ g L θ This is pure viscous motion in a potential This determines the precession frequency
Uniaxial symmetry Φ(m; h a,β/α)= κ eff 2 cos2 θ h a cos θ + β α ln (1 + c p cos θ) c p J e = αj p b p β α h a =1.1, κ eff = 1, c p =0.5 Φ β/α =0 Φ β/α =0.75 Φ β/α =1.5 P O A 0.5!! 0.5!! 0.5!! P state: m is parallel to the polarization axis ep θ θ θ O state: m precesses around the polarization axis ep A state: m is antiparallel to the polarization axis ep
Φ θ = sin θ κ eff cos θ + h a β α Uniaxial symmetry 1 1+c p cos θ θ =0 and θ = π are always solutions: they can be either stable or unstable depending on field and current. Φ θ =0 Precession around θ =0becomes possible when: β α = (1 + c p)(h a + κ eff ) Precession around θ = π becomes possible when: β α = (1 c p)(h a κ eff ) When the system is in a current-induced precessional state and the angle of precession is θ 0, the corresponding frequency of precession is: f 0 = 1 2π (h a + κ eff cos θ 0 ) remember that dφ dt = 1 sin θ g L θ
Stability diagrams β α 4 3 2 A O β α 4 3 2 A/O A P/A 1 P 1 O P 1 2 3 h 4 a 1 2 3 h a 4-1 -1 κ eff = 1, c p =0 κ eff = 1, c p =0.5 These diagrams identify the dynamical regimes that are realized under different combinations of field and current. In dynamical system theory language, the separating lines represent bifurcation conditions: continuous lines: Hopf bifurcation; dashed line: semi-stable limit-cycle bifurcation.