Domain wall motion in ferromagnetic nanowires driven by arbitrary time-dependent fields: An exact result

Domain w all motion in ferromagnetic nanowires driven b y arbitrary time-dep enden t fields: An exact result Arseni Goussev, JM Robbins, V aleriy Slastikov Scho ol of Mathematics, University of Bristol, Univers ity Walk, Bristol BS8 1TW, UK (Dated: N o vem ber 29, 2021) W e address the dynamics of magnetic domain walls in ferromagnetic nanowires under the influ ence of extern al time-dep endent magnetic fields. W e rep ort a new exact spatiotemp oral solution of the Landau-Lifshitz-Gilb ert equation for th e case of soft ferromagnetic wires and nanostruct ures with uniaxial anisotropy . The solution holds for applied fi elds with arbitrary strength and time dep endence. W e further extend this solution to applied fields slow ly v arying in space and to multiple domain w alls. P ACS n umbers: 75.75.-c, 75.78.Fg Intr o d uction.— The motion of magnetic do ma in walls (D Ws) in ferromagnetic nanowires has recent ly b ecome a sub ject of in tensive research in the condensed matter ph ysics communit y [1]. Manipulation of DWs by external magnetic fields , and in particular, the question of how the D W pro pagation velocity dep ends o n the applied field hav e drawn considerable atten tion [2 – 4]. In ferromagnetic na nowires, the dynamics o f the or i- ent ation of the magnetization dis tr ibution, m ( x, t ) (no r- malized so tha t | m | = 1 ), is describ ed b y the La ndau- Lifshitz-Gilb e rt (LLG) equatio n [5] ∂ m ∂ t + α m × ∂ m ∂ t = (1 + α 2 ) m ×  H ( m ) + H a  , (1) where x is the coo rdinate along the nanowire, t is time, α is the Gilb ert damping parameter , H a denotes the ap- plied magnetic field, and H ( m ) = − δ E /δ m , where E ( m ) = A 2 Z R     ∂ m ∂ x     2 d x + K 1 2 Z R  1 − ( m · ˆ x ) 2  d x + K 2 2 Z R ( m · ˆ y ) 2 d x. (2) is the reduced micromag netic energy . Her e, A is the ex- change constant of the mater ial, and K 1 , K 2 ≥ 0 are the anisotropy constants along the (ea sy) x - a nd (hard) y - axes. The anisotro p y constant along the z -axis is tak en to b e zero b y c onv en tion. T o date only one exact spatio tempor al [7] solution of the LLG equation has b een r epo rted in the literature , namely the so-called W alk er solution [6 ]. The analysis of Schry er a nd W alker [6] applies to the case wher e K 2 > 0, ie wher e the anisotropy consta n ts in the transverse plane are strictly unequal. This is appro priate for a thin film or thin strip ge o metry . The applied field is taken to b e uniform in space, co nstant in time, and dir ected along the nano wire, i.e., H a ( x, t ) = H a ˆ x . F or | H a | less than a certain threshold H W , the so-called W a lker breakdown field, a plana r domain wall pr opagates rigidly along the nanostrip with velo cit y depending nonlinearly on H a . In this Letter we presen t an exact spatiotemp or al so - lution of the LLG equation that, to our k nowledge, has not been previously repor ted in the literature. W e co n- sider the case of tra nsverse isotropy , ie K 2 = 0. This is appropria te for soft ferromag netic na nowires whose cross- sectional dimensions a re co mparable, as well as for uni- axial nanowires whose ea sy axis lie s along the wir e. W e take the applied field to lie along the nanowire, as in the case of the W alker solution, but allow for a rbitrary time depe ndence , i.e., H a ( x, t ) = H a ( t ) ˆ x . Exact solution of t he LLG e quation. — The boundar y conditions appropria te fo r a domain wall with finite mi- cromagnetic energy E ( m ) are given b y m ( x, t ) → ± ˆ x as x → ±∞ . F or K 2 = 0 the magnetiza tion-dep enden t field H is given by H ( m ) = A ∂ 2 m ∂ x 2 + K 1 ( m · ˆ x ) ˆ x . (3) W e now take in to account the fact that m ha s its v alues on S 2 , and pa rametrize m in terms of angles θ ( x, t ) and φ ( x, t ) according to m = (cos θ, sin θ cos φ, sin θ sin φ ). F rom Eqs. (1) and (3) we obtain the LLG equatio n in the e quiv alent form ˙ θ − α ˙ φ s in θ + A (1 + α 2 )  φ ′′ sin θ + 2 θ ′ φ ′ cos θ  = 0 , (4a) α ˙ θ + ˙ φ s in θ + (1 + α 2 )  − Aθ ′′ + A ( φ ′ ) 2 sin θ cos θ + K 1 cos θ sin θ + H a ( t ) sin θ  = 0 , (4b) where dot ˙ denotes ∂ /∂ t and prime ′ denotes ∂ /∂ x . W e now loo k for a so lution of Eq. (4) in the form θ ∗ ( x, t ) = θ 0 ( x − x ∗ ( t )) , φ ∗ ( x, t ) = φ ∗ ( t ) , (5) where θ 0 ( x ) = 2 arctan exp ( − x/d 0 ) , d 0 = p A/K 1 . (6) θ 0 ( x ) descr ibes the static domain wall in the absence o f an a pplied field. The magnetization dens it y deter mined by θ 0 ( x ) and φ 0 ( x ) = π / 2 minimizes the micromagne tic energy E ( m ) for the s pecified b oundary co nditions. Sub- stituting Eq. (6) into Eq. (4), a nd taking in to acco un t 2 that θ ′ 0 = − sin θ 0 /d 0 and θ ′′ 0 = sin 2 θ 0 / (2 d 2 0 ), w e find that θ ∗ and φ ∗ satisfy the LLG equation (4) pr ovided that x ∗ ( t ) and φ ∗ ( t ) sa tisfy ˙ x ∗ = − αd 0 H a ( t ) , ˙ φ ∗ = − H a ( t ) . (7) (In fact, (6 ) and (7 ) provide the only so lutio n of the form (5).) Equations (5-7) c o nstitute the main result of this Let- ter. They repr esent an exact solution o f the LLG equa- tion, and describe a D W, with profile indep endent of the applied field, propaga ting along the nanowire with veloc - it y ˙ x ∗ while prece ssing a bout the na nowire with angula r velocity ˙ φ ∗ . No r estrictions have b e en imp ose d on the str ength of the applie d magnetic fi eld and no assumptions have b e en made ab out its time dep endenc e. W e now compare the pr e c essing s olut ion Eqs. (5-7) with the W alker solution [6]. The W alker so lution is de- fined only for K 2 > 0 (the fully anisotro pic case) a nd time-indepe nden t H a less than the breakdown field H W = αK 2 / 2 . (8) It is given by θ W ( x, t ) = θ 0  x − V W t γ  , φ W ( x, t ) = φ W , (9) where sin 2 φ W = H a /H W (10) determines the (fixed) inclination of the DW plane and V W = γ 1 + α 2 α d 0 H a , γ =  K 1 K 1 + K 2 cos 2 φ W  1 / 2 (11) gives the DW velo city . There are several characteristic differences b etw een the W alker solution and the pr ecessing s olution which sho uld be distinguishable ex per imen tally . F oremost is the fact that the W alker solution exists only for co nstant applied fields whose s trength do es not exceed a certain thresh- old, so that the DW velo cit y is bounded. The preces sing solution is defined for time-dep endent a pplied fields o f ar- bitrary strength, so that the DW velo city , which for the precessing solution is pro po r tional to the field strength, can b e arbitra rily lar ge. Next, while for the W alker solu- tion the plane of the D W remains fixed, for the precessing solution it rotates ab out the nanowire at a rate prop or- tional to H a . Finally , w e observe that, for the W alker solution, the DW profile contracts ( γ > 1) or expands ( γ > 1) in resp onse to the applied field, whereas for the precessing solution the DW profile propa gates without distortion. Sp atial ly nonu niform applie d fields and multiple do- main wal ls.— W e now extend o ur results to a pplied fields that dep end on b oth position along the nanowire and time, i.e, H a = H a ( x, t ) ˆ x . F or any (non-singular ) ap- plied field, Eq. (4) is satisfied at x outside the DW tran- sition lay er | x − x ∗ ( t ) | ≫ d 0 (up to exponentially small terms). Assuming now that the field v aries slowly across the tr ansition r egion,   H a  x, t  − H a  x ∗ ( t ) , t    ≪   H a  x ∗ ( t ) , t    for | x − x ∗ ( t ) | . d 0 , (12) we obtain an approximate solution of the LL G equation: the magnetization density is g iven b y Eqs . (5) and (6) with ˙ x ∗ = − αd 0 H a  x ∗ ( t ) , t  , ˙ φ ∗ = − H a  x ∗ ( t ) , t  . (13) The physical meaning of Eq. (13) is quite obvious: the D W is only sensitiv e to the applied field within the tra n- sition layer. FIG. 1: (Col or online) D ynamics of domain w alls. See text for discussion. This approximation can no w b e extended to the case of N non-overlapping DWs. Indeed, θ N ( x, t ) = N X n =1 θ 0  ( − 1) n +1 ( x − x n ( t ))  , (14a) φ N ( x, t ) = φ ¯ n ( t ) , n = ¯ n minimizes | x − x n ( t ) | , (14b) with x k +1 ( t ) − x k ( t ) ≫ d 0 for k = 1 , . . . , N − 1 , consti- tutes an appr oximate so lution o f the LLG equation given that ˙ x n = ( − 1) n αd 0 H a  x n ( t ) , t  , (15a) ˙ φ n = − H a  x n ( t ) , t  , (15b) for n = 1 , . . . , N . F or the ca se of a s pa tially uniform applied field Eqs. (14) a nd (1 5) describ e the time evolu- tion of N DWs such that a n y t wo adjacen t WDs tr av el in opp osite directions while rotating in the sa me direc- tion (and with the same a ngular velo city) ar ound the nanowire. Conclusions.— In this Letter we hav e presented an ex- act spatiotemp oral so lutio n o f the LLG equa tion that has not been previously rep orted in the litera ture. The v a- lidit y o f the new solution req uires no ass umptions ab out the time- de p endence or strength of the applied field. W e ha ve then provided a natural extension of the so- lution to physical situations in which the applied field 3 v aries slo wly in space. An approximate so lution of the LLG equa tion for the case of multiple domain w alls has also been obtained. A cknow le dgments.– A.G. ac knowledges the suppo rt by EPSRC under Gr ant No. EP/E 02462 9/1. [1] see e.g., S. S. P . Parkin, M. Haya shi, and L. Thomas, Science 320 , 190 (2008); R. P . Cow burn, Nature 448 , 544 (2007). [2] Z. Z. Sun and J. Schliemann, Ph ys. Rev. Lett. 104 , 037206 (2010). [3] X. R. W ang, P . Y an, and J. Lu, Europhys. Lett. 86 , 67001 (2009); X. R. W ang, P . Y an, J. Lu, C. H e, Ann. Phys. 324 , 1815 (2009). [4] M. C. Hick ey , Phys. Rev . B 78 , 180412(R) (2008). [5] see e.g., A. Hub ert and R. S c h¨ afer, Magnetic Domains: The A nalysis of Magnetic Micr ostructu r es (Springer, Berlin, 1998). [6] N. L. S c hryer and L. R. W alker, J. Appl. Phys. 45 , 5406 (1974). [7] The only other exact solution of the LLG equation re- p orted in t he literature [Z. Z. Sun and X. R. W ang, Ph y s. Rev. Lett . 97 , 077205 (2006)] appears in the problem of magnetization switc hing, where th e magnetization density is considered t o b e u niform in space and is a function of time only , i.e., m = m ( t ) .