Landau-Lifschitz-Gilbert equation with applied electric current

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1 Landau-Lifschitz-Gilbert equation with applied electric current G. Bonithon To cite this version: G. Bonithon. Landau-Lifschitz-Gilbert equation with applied electric current. The 6th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Jun 006, Poitiers, France. pp.38-44, 007. <hal > HAL Id: hal Submitted on 3 Apr 0 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 DISCRETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS SUPPLEMENT 007 pp LANDAU-LIFSCHITZ-GILBERT EQUATION WITH APPLIED ELECTRIC CURRENT Gaël Bonithon University Bordeaux 35 cours de la Libération TALENCE cedex France Abstract. In this paper, we are concerned with a model of electric current effect in ferromagnetic materials, that is Landau-Lifschitz equation adding a transport term. We prove classical existence theorem in the general three dimensional case, and we justify a one dimensional approximation for wich we have the explicit behavior of the magnetisation.. Introduction. Landau-Lifschitz equation describe spontaneous magnetisation behavior in ferromagnetic materials and writes t u = H eff u + u H eff u).) where the magnetic moment u takes its value in S, and the effective field H eff is derived from micromagnetic energy see []), in particular in presence of an external magnetic field. In case of electric current injection see [3] and [] for some physical developmnents on electric current injection in ferromagnetic materials), Thiaville and Miltat propose in [4] a first transport term addition, that is simply v ) u considering Gilbert form of equation.) : t u = H eff u + u t u v ) u.) where v can be seen as the current carriers) speed. But because of undesired treshold effect in simulations, the same authors are led to introduce in [3] a global transport term that competes with the two dervatives in time in the last equation, that is t u = H eff u + αu t u v ) u + βu v ) u.3) where β is positive and α strictly positive the same model has been proposed independently in [9]). In the following, we consider equation.3) posed in a regular open set Ω of R 3, with where H is defined by H eff = u + Hu) { rot Hu)) = 0 div Hu) + ū) = Mathematics Subject Classification. Primary: 58F5, 58F7; Secondary: 53C35. Key words and phrases. Micromagnetism Landau-Lifschitz. 38

3 LANDAU-LIFSCHITZ-GILBERT EQUATION 39 with ū equals u in Ω and zero outside. We will use functions spaces notations like L p for applications with vectorial value or numerical value without difference in non ambigous cases, and hypothesis on v or u will be given precisely in the following, depending on the kind of solution we will consider. The paper is organized as follow : in a first part, we give classical existence results with an overview of the proofs, and a second part is dedicated to the study of a one dimensional approximation in nanowires.. General Results. Let us now give a first existence result for the global problem, that is with H defined as above. Theorem. For all u 0 in H Ω) with u 0, and for all v in Cb 0 R + Ω R 3) there exist a unitary vector field u in H R + Ω) satisfying equation.) and the following energy estimate for all positive t : t t u + Eut)) Eu 0 ) + It) exp It))) with and 0 Ω Eut)) = Ω It) = t 0 v L Ω R 3 ) s)ds ut) + Φut)) Hut)) ut) Ω Ω Remark. By simplicity, theorem is written for equation.), but as the proof shows it clearly, the same result is true for equation.3). In the estimate of the theorem, we see that v L R + ) ensures a bouded energy, and as a consequence u in H R + Ω). But in general, the estimate gives a time dependent bound, and this is the main difference with the corresponding result for Landau-Lifschitz equation. We now introduce two other formulations of equation.3), equivalent for sufficiently regular unitary vector field. The first one can be seen as derived from Landau-Lifschitz equation : +α ) t u = H eff u+αu H eff u) +αβ) v ) u α β)u v ) u.4) The second one is then deduced from.4) in order to construct strong solutions for equation.3), as made in [6] for Landau-Lifschitz equation. ) + α ) t u = α u + u u + H eff u + αu χu) u).5) + αβ) v ) u α β)u v ) u To clarify the part of the physical constraint u in the existence and uniquness of strong solutions, we define this term only from regularity point of view. Definition. A solution of equations.3),.4) or.5) is said strong if u C 0 0, t; H Ω) ) L 0, t; H 3 Ω) ) for all 0 < t < T

4 40 GAËL BONITHON u depends continiously on u 0 0 < t < T for the topology of C 0 0, t; H Ω) ) for all local in time if T < and global in time otherwise. Theorem. For v in Cb R + Ω R 3), and for all u 0 in H Ω) sutch that ν u 0 0 on Ω there exist a unique strong solution u of equation.5). Moreover, if u 0 takes its value in S, then u and u is the unique strong solution of formulations.4) and.3). Proof of Theorem. Proof is based on a construction of Alouges and Soyeurs for Landau-Lifschitz equation see []). We first consider a relaxed problem t u λ = H eff u λ ) u λ ) t u λ T u λ ) v ) u λ in R + Ω 4 λ u λ u λ ν u λ = 0 on R +.6) Ω u λ t = 0) = u 0 in Ω where T is identity in the unitary ball, and projection on S otherwise. We prove existence of weak solutions for equation.6) by a Galerkine approximation, using a bound for equation.6) similar to the theorem s one and independent of Galerkine approximation s degree. Aubin Lemma permits us to take the limit in non linear terms of equation.6), and by the same arguments, we take the limit in lambda to obtain by construction u. To prove that u satisfy equation.), or its integral form in fact, we choose a function test of the form u λ φ to refind equation.3) by cross product properties. Remark. As said in remark, v in L R + ) gives u in H R + Ω) but obviously similar result with u in H ]0, T [ Ω) for all positive T is true for v only bouded and weakly continous in its third variable. The truncate term permits us to obtain energy estimate of theorem but makes this bound be time dependent too, and more generaly, it s not easy to integrate transport term contribution in micromagnetic energy, as it s the case for the terms of the effective field. Proof of theorem. Proof of existence and uniquness of strong solutions for equation.5) is given in [6], because non linear transport term estimations can be seen as particular cases of the estimations made in the reference above for the term u u. Let now u 0 be with value in S. Multiplying equation.5) by ut) H Ω) L Ω), we obtain with α = β = ) : but t u ) = u u + u u v ) u u u ) = u +u u et v ) u u = u.v u = v t u).u = v u ) so with b := u,

5 LANDAU-LIFSCHITZ-GILBERT EQUATION 4 t b = b + u b v b in [0, T [ Ω ν b = 0 on [0, T [ Ω bt = 0) = 0 in Ω.7) Now, because of the dissipative term, we can conclude directly that b is zero by classical energy estimates using Young inequality and Gronwall Lemma. We have then directly, by cross product properties, that u is strong solution of equations.3) and.4). By equivalence of the three formulations for regular solutions with value in S, uniquness of strong solutions for equations.3) and.4) is equivalent to pointwise norm conservation by these equations. Adapting equation.7) to equation.3) or equation.4) with different multiplicative constants) we obtain : t b = v b in [0, T [ Ω ν b = 0 on [0, T [ Ω.8) bt = 0) = 0 in Ω We are going to prove that b is in fact solution in classical sense, and by caracteristics method we will obtain that b is zero. Remark first that : H Ω) is an algebra, so b C 0 0, t; H Ω) ) b = t u.u C 0 0, t; H Ω) ) C 0 0, t; H Ω) ) C 0 0, t; H Ω) ) by.8), t b C 0 0, t; H Ω) ) so b C 0, t; H Ω) ) b L 0, t; H 3 Ω) ) L 0, t; H 3 Ω) ) L 0, t; C Ω )) so finally b C [0, T [ Ω ) and equation.8) is quasilinear without second member so b is constant along caracteristics. Let now t,x) be in [0, T [ Ω, if b is nonzero there exist a unique caracteristic γ passing by x at time t, and defined by : { γ s) = vs, γs), λ) γt) = x where λ = bt, γt)). Let ]t 0, t [ be the existence intervale of γ, then if t 0 = 0 and γ0) Ω, bt, x) = bt, γt)) = b0, γ0)) = 0, otherwise γt 0 ) Ω and bt, x) = bt 0, γt 0 )). So we have to show that b is zero on the boundary. As Ω is regular, we can decompose equation.8) in tangencial and normal directions in some neyborwood V of Ω : t b = v ν ν b v τ τ b in [0, T [ V Ω) ν b = 0 on [0, T [ Ω bt = 0) = 0 in V Ω And taking the limit in x, we obtain as v ν is suposed bounded on Ω) : { t b = v τ τ b on [0, T [ Ω bt = 0) = 0 on Ω Now, by regularity and hypothesis on v, we can conclude using classical methods that b 0.

6 4 GAËL BONITHON 3. One Dimensionnal Approximations. In this section, we follow a work of Carbou and Labbé on Bloch Wall motion in nanowires submitted to an external magnetic field. These authors use in [4] and [5] a one dimensional approximation of Landau-Lifschitz equation, that is a one dimensional approximation of the demagnetising field H defined at the end of section, given by Sanchez in [0] for perfectly cylindrical wires. Following its method, we justify a model used by Thiaville and Miltat in [] for Bloch wall motion in nanowires to an external magnetic field magnetic field, and also used by the same authors in [3] and [4] for wall motion caused by electric current injection. We consider an infinite wire in direction e, of section E k defined by { x ) E k x 0 ) = X = x, x ) {x = x 0 }, + x k < } 3.9) and we recall that in this case, magnetostatic equations writes div Hu)) = 0 dans Ω k c Ω k rot Hu) = 0) dans Ω k c Ω k 3.0) [Hu) ν] = u ν sur Ω k The following theorem include the result given by Sanchez in [0]. Theorem 3. The problem 3.0) as a unique solution in R for all k R + \ {}, given by Hu)X) = ) k + u k D0,) z ) u k z u k z z z ) 3.) D0,) z u k z u k z z z where D0, ) = {z C, z < }, and denoting also X the complex number associated to x, x ), ) u u k = and z j = X + )j X k ), for j =, ku k Moreover, Hu) is constant in Ω k for all k R + and we have Hu)X) = k + u k Proof of theorem 3. As rot Hu)) = 0 in R, there exists a potential φ such that Hu) = φ and { φ = 0 in Ωk c Ω k this gives and ν φ = u ν on Ω k φx) = ln X Y ) u ν)y )dσy ) Ω k π Hu)X) = Ω k π u ν)y ) X Y X Y dσy )

7 LANDAU-LIFSCHITZ-GILBERT EQUATION 43 Using natural parametrisation of Ω k = E k x 0 ) and complex numbers, we have Hu)X) = where z = u z + ) iku z )) dz z z X k+ z k )z iz = iπ T + T ) T = z = u + iku zp z) with P z) = k )z Xz + k + ). We now use Residus Theorem to obtain and T = z = u iku )z P z) ) T = iπu + iku ) k + + D0,)z ) z z z ) + D0,)z ) z z z ) and so ) z z T = iπu iku ) D0,) z ) + D0,) z ) z z z z Hu)X) = ) k + u k D0,) z ) u k z u k z z z ) D0,) z u k z u k z z z By classical optimisation methods, we show that that gives the result. min z j >, j =, E k x 0) We then obtain an one dimensional approximation for H : Hu) = xu+u e u e, where u is the wire s direction and u is transverse to the Bloch s Wall. For this approximation, we can extend results given in [4] and [5] for equation.).we obviously have stability of the following static profile M 0 = thx) 0 chx) and with a magnetic field injection δe, we use local inversion theorem to show wall motion for all non zero applied magnetic field, and magnetic structure rotation toward the wire. In the case of an applied electric current, equation.) has a static solution for positive v, that is we have a treshold effect : ux) = R θ M 0 ex)

8 44 GAËL BONITHON with θ = arcsin e ), and e = v ) + v ) for v <. In the general case of equation.3), we also have an explicit solution given by with c = β α ve, and e = 3 ut, x) = R θ M 0 ex ct) 9 8 ) + α) β v ) + α) β v for v < ) β α. In the particular case α = β, we have ut, x) = M 0 x vt), stable for all v R. REFERENCES [] F. Alouges and A. Soyeur, On global weak solutions for Landau-Lifschitz equations: existence and non uniqueness, Nonlinear Anal., Theory Methods Appl., 8) 99), [] Amikam Aharoni, Introduction to the Theory of Ferromagnetism Second Edition), International Series of Monographs on Physics, Oxford University Press, ). [3] L. Berger, Motion of a magnetic domain wall traversed by fast-rising current pulses, J. Appl. Phys., 76) 99), [4] Gilles Carbou and Stéphane Labbé, Stability for Static Walls in Ferromagnetic Nanowires, to appear in Discrete and Continuous Dynamical Systems. [5] Gilles Carbou and Stéphane Labbé, Stability for Walls in Ferromagnetic Nanowires, to appear in Discrete and Continuous Dynamical Systems. [6] Gilles Carbou and Pierre Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded domain, Differential Integral Equations, 4), 00) 3 9. [7] Gilles Carbou and Pierre Fabrie, Time average in micromagnetism, J. Differential Equations, 47) 998), [8] R. Dautray, J.L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Tomes, et 3, Coll. du CEA, Masson. [9] Z. Li, J. He and S.Zhang, Effects of spin current on ferromagnets, cond-mat/ ). [0] D. Sanchez, Behaviour of the Landau-Lifschitz Equation in a Ferromagnetic Wire, to appear in Math. Methods Appl. Sci. [] Gen Tatara, Hiroshi Kohno, Theory of Current-Driven Domain Wall Motion: Spin Transfer versus Momentum Transfer, Phys. Rev. Lett., 9 004), [] A. Thiaville, J.M. Garcia, J. Miltat, Domain Wall Dynamics in Nanowires, J. Magn. Magn. Mat., ), [3] A.Thiaville, Y.Nakatani, J.Miltat and Y.Susuki, Micromagnetic understanding of currentdriven domain wall motion in patterned nanowires, Europhys. Lett., 696) 005), [4] A.Thiaville, Y.Nakatani, J.Miltat and N.Vernier, Domain wall motion by spin-polarized current: a micromagnetic study, J. Appl. Phys., Part, 95) 004), [5] N. Vernier, D. A. Allwood, D. Atkinson, M. D. Cooke and R. P. Cowburn, Domain wall propagation in magnetic nanowires by spin-polarized current injection, Europhys. Lett., 654) 004), Received September 006; revised February address: gael.bonithon@laposte.net