DDynamics of domain walls in weak ferromagnets
Zvezdin, A.K. Prokhorov General Physics Institute, Russian Academy of Sciences, Moscow 119991
Pis (cid:48) ma Zh. Exp. Teor. Fiz. , No. 10, 605-610 (20 May 1979)It is shown that the total set of equations, which determines the dynamics of thedomain bounds (DB) in a weak ferromagnet, has the same type of specific solutionas the well-known Walkers solution for ferromagnets. We calculated the functionaldependence of the velocity of the DB on the magnetic field, which is described by theobtained solution. This function has a maximum at a finite field and a section of thenegative differential mobility of the DB. According to the calculation, the maximumvelocity c ≈ × cm/sec in YFeO is reached at H m ≈ × Oe.
PACS numbers: 75.60.Ch I. The Landau-Lifshitz equations for a double sublattice weak ferromagnet can be repre-sented in the form ˙ M i = γ (cid:20) M i , δWδ M i (cid:21) + α (cid:104) M i , ˙ M i (cid:105) , i = 1 , W = a m + b l x + b l z + d m z l x − d m x l z − m · H + A ( ∇ l ) + A (cid:48) ( ∇ m ) m = M + M M , l = M − M M , δδq ≡ ∂∂q − ∇ ∂∂ ∇ q (1)To describe the dynamics of the domain bound, let us go over to the angular variables θ , φ , (cid:15) , and β in which ( (cid:15) (cid:28) β (cid:28) l x = sin θ cos φ, l y = sin θ sin φ, l z = cos φ, m z = − (cid:15) sin θm x = (cid:15) cos θ cos φ − β sin θ sin φ, m y = (cid:15) cos β sin φ + β sin θ cos φ. (2)To write Eqs. (1) we use the Lagrange formalism in the variables θ , φ , (cid:15) , and β . TheLagrange function L , the dissipative function F , and the corresponding Euler equations are L = Mγ (cid:104) ˙ φ(cid:15) sin θ − ˙ β cos θ (cid:105) − W ( θ, ψ, (cid:15), β ) (3) F = αM γ (cid:104) ˙ θ + sin θ (cid:16) ˙ φ + ˙ β (cid:17) + ˙ (cid:15) + 4 (cid:15) sin θ cos θ ˙ φ ˙ β (cid:105) (4) ∂∂t ∂L∂ ˙ θ = δLδθ − ∂F∂ ˙ θ , etc. (5) a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r is (cid:48) ma Zh. Exp. Teor. Fiz. , No. 10, 605-610 (20 May 1979) II. At H = (0 , , H ) we have a specific solution of the nonlinear equations (5) , : θ = π/ β = 0, φ ( r, t ), and (cid:15) ( r, t ). As a result of substituting this solution in Eqs. (5), two of theequations (obtained by varying θ and β ) become identities and the other two have the form˙ (cid:15) + α ˙ φ = c ω E ∇ φ + ω sin φ cos φ − ω d (cid:15) sin φ, (6a) α ˙ (cid:15) − ˙ φ = c (cid:48) ω E ∇ (cid:15) − ω E (cid:15) + ω d cos φ − ω H (6b)where ω = γb M , ω d = γd M ≡ γH d , ω H = γH,ω E = γaM ≡ γH E , c = 4 γ AH E /M, c (cid:48) = 4 γ A (cid:48) H E /M First, let us determine the approximate solution of Eqs. (6a) and (6b). In Eq. (6b) theterms (cid:0) c (cid:48) /ω E (cid:1) ∇ (cid:15) and α ˙ (cid:15) can be deleted in comparison to ω E . The parameters of smallnessof the deleted terms are ( a / ∆) and ( αa / ∆) , where a ( c/ω E ) = (2 A/a ) / = 10 − cmand ∆ is the thickness of the moving domain bound. Thus, we have from Eq. (6b) (cid:15) = 1 ω E (cid:16) − ω d cos φ + ω H + ˙ φ (cid:17) (7)Substituting it in Eq. (6a), we obtain¨ φ − c ∇ φ + ω A sin φ cos φ = ˙ ω H − ω d ω H sin φ − αω E ˙ φ (8)where ω A = ω d − ω E ω . At H = 0 and α = 0 this equation becomes the well-known Sine-Gordon equation. Its one-dimensional solution, which satisfies the boundary con- ditions φ ( x → −∞ ) = 0, φ ( x → + ∞ ) = π , has the form φ ( x, t ) = 2 arctan e x − vt ∆ , ∆ − = ω A /c (cid:112) − ( v/c ) , (9)where v < c . This function satisfies Eq. (8) at H = const (cid:54) = 0 and α (cid:54) = 0, but for aspecific value of v ( H ), satisfies the equation, ω d ω H = αω E δ − v , which can be easily verifiedby substituting Eq. (9) in Eq. (8). From the last equation using Eq. (9) we obtain , : v ( H ) = c HH d a (cid:2) H E H A + a − H H d (cid:3) − / . (10)The physical nature of such a dependence v ( H ) can be described if we use the mechanicalanalogy of the motion of DB. If the dependence of H and v on t is sufliciently small (thecharacterisitic frequencies of their variation are much smaller than ω d ), we can obtain fromEq. (8) the following equation for the velocity of DBdd t ( mv ) + ( mv ) r = 2 M s H, (11)2is (cid:48) ma Zh. Exp. Teor. Fiz. , No. 10, 605-610 (20 May 1979)where m = 2 M s H d γ ∆( v ) = m (cid:112) − ( v/c ) , τ = 1 αω E . All the terms in Eq. (11) have a clear mechanical meaning; mv/τ is the frictional forceacting on the domain bounds, 2 M s H is the pressure exerted on the domain bounds, etc.At ( d/dt ) ( mv ) = 0 Eq. (11) gives Eq. (10). Thus, the velocity of the domain boundssaturates as H → ∞ because of the relativistic dependence of the mass m of the DB on itsvelocity. Chetkin et al. observed experimentally and investigated the effect of saturationof the velocity of DB in YFeO ; they as well as Baryakhtar et al. theoretically estimatedthe limiting velocity of the DB in orthoferrites. III. At v ∼ c the deleted terms in Eq. (6b) should be taken into account. Let us analyzeasymptotically Eqs. (6a) and (6b) by the method proposed and developed in Refs. [9] and[10]. Let us linearize Eqs. (6a) and (6b) near the stationary points φ = O and π , whichcorrespond to the domains, and let us find solutions of the linear equations in the formexp [ ± ( ω E /c ) k ( x − v ∓ t )] at ( x − v ∓ t ) → ∓∞ . The conditions for the existence of nontrivialsolutions have the form (cid:16) vc (cid:17) (cid:0) a (cid:1) − avc k (cid:20) − k − (cid:18) ω A ω E ∓ ω H ω d ω E (cid:19)(cid:21) − k − (cid:0) − k − (cid:1) (cid:18) ω A ω E ∓ ω H ω d ω E (cid:19) = 0 . (12)Let us assume that there is a solution of the nonlinear equations (6a) and (6b) in theform φ ( x − vt ), (cid:15) ( x − vt ) and that the function φ ( x − vt ) is symmetric; thus the equality v + ( k ) = v − ( − k ) = v , where v + ( k ) and v − ( k ) are determined by Eq. (12), gives (in the linearapproximation of α and H/H E ): v = c (cid:0) p − pk − − k (cid:1) / , (13a) H = H MC k (cid:0) − pk − (cid:1) / (cid:0) − k (cid:1) − / (cid:0) p − k (cid:1) , (13b)where H MC = a H E H d , p = (cid:18) ω A ω E (cid:19) . These equations determine the function v ( H ) in the parametric form. The character-istic shape of this curve is given in Fig. 1. The maximum of this curve, which has thecoordinates : v m = c (1 − √ p ), H m = H MC p / (cid:0) − √ p (cid:1) , corresponds to k m = p / ( p (cid:28) H = 0, v = 0 corresponds to k m = p / ( p (cid:28) k m /k ≈ p / characterizes the thickness ratio of the DB at v = v m and v = 0. The function (13) coincideswith Eq. (10) at k < k < k m or at 0 < H < H m . The last inequality is the condition ofapplicability of Eqs. (8) and (10). The motion of the DB, in which l rotates in the ac plane,is determined by more complicated equations than (6a) and (6b), but the function v ( H ),which is determined by Eqs. (10), (13a), and (13b), remains valid in this case (if d = − d ).In them it must be assumed that ω a = ( b − b ) /M .3is (cid:48) ma Zh. Exp. Teor. Fiz. , No. 10, 605-610 (20 May 1979) FIG. 1. A plot of the v ( H ) function constructed according to Eq. (13) at p = 10 , a part of the v ( H ) curve [( H/H MC ) > .
2] requires further study since here the condition (cid:15) (cid:28)
IV.
We give the numerical estimates. In YFeO A ≈ × − erg/cm, H E = 6 . × Oe, H d = 10 Oe, and p ≈ − . The scale of the field H MC can be expressed in terms ofthe mobility µ when H → µ = ( c/H MC ) p / . Hence, H MC = (cid:0) c √ p/µ (cid:1) . Accordingto Ref. [12], µ (cid:39) × cm/sec · Oe. Using these values, we obtain c ≈ × cm/sec, H MC ≈ × Oe, V m = 0 .
99 s, and H m ≈ × Oe. For definiteness, we examine a crystal of rhombic symmetry. L. Walker and J. Dillon, Magnetism , 450 (1963). It has the same meaning as the well-known Walkers solution for ferromagnets, although itsequations are more complex. E. Gyorgy and F. Hagedorn, Journal of Applied Physics , 88 (1968). A similar dependence v ( H ) was obtained in Ref. [4], where the authors assumed that the dy-namics of the weak ferromagnetic moment are described by the same equations as the dynamicsof the ferromagnet and the magnetization remains constant during the motion of the domainbounds. M. Chetkin, A. Shalygin, and A. Kampa, Fizika Tverdogo Tela , 3470 (1977). M. Chetkin and A. de La Campa, JETP Letters , 157 (1978). V. Baryakhtar, B. Ivanov, and A. Sukstanskii, Fizika Tverdogo Tela , 2177 (1978). E. Schl¨omann, Applied Physics Letters , 274 (1971). V. Eleonskiy and N. Kirova, Problems of Solid State Physics UNTs , 184 (1975). This velocity coincides with that obtained in Ref. [ ]. R. Uait, Usp. Fiz. Nauk , 593 (1971)., 593 (1971).