MAGNETIC CONTROL OF EXCHANGE TORQUES AND THE PENETRATION DEPTH OF MAGNETOSTATIC SURFACE SPIN WAVES

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MAGNETIC CONTROL OF EXCHANGE TORQUES AND THE PENETRATION DEPTH OF

MAGNETOSTATIC SURFACE SPIN WAVES

F. Morgenthaler

To cite this version:

F. Morgenthaler. MAGNETIC CONTROL OF EXCHANGE TORQUES AND THE PENETRATION DEPTH OF MAGNETOSTATIC SURFACE SPIN WAVES. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-1159-C1-1161. �10.1051/jphyscol:19711415�. �jpa-00214457�

LAMES MINCES (4 partie) : RESONANCE, EFFETS DE SURFACE-

MAGNETIC CONTROL OF EXCHANGE TORQUES AND THE PENETRATION DEPTH

OF MAGNETOSTATIC SURFACE SPIN WAVES (*) F. R. MORGENTHALER

Department of Electrical Engineering and Center for Materials Science and Engineering Massachusetts Institute of Technology

Cambridge, Massachusetts, 02139, U. S. A.

R&um6. ^- Un contrale magnetique du couple d'bhange agissant sur une onde de spin de surface de grande longueur d'onde aussi bien qu'un contr6le de sa profondeur de pknktration est possible car des changements d'orientation du champ magnktique peuvent modifier I'interaction dipolaire qui A son tour modifie la valeur de V2m. On pourrait donc modi- fier la constante d'bhange effective de certaines ondes de surface, permettant le contrBIe de la courbure d'kchange dans la relation de dispersion et donc celui de la vitesse de groupe des ondes. Les rQultats theoriques comprennent les effets des Cnergies Zeeman, dipolaires et d'echange aussi bien que ceux provenant des conditions aux limites applicables aux films ferromagnetiques minces et epais ayant des surfaces limit& par diverses combinaisons espace libre et plans conduc- teurs.

Abstract. - Magnetic control of the net exchange torque acting upon a long wavelength surface spinwave as well as control of its penetration depth is possible because changes in the magnetic field orientation can alter the dipolar inter- action which in turn modifies the value of Vziii. It should therefore be possible to vary the effective exchange constant of certain surface waves - allowing the exchange curvature of the dispersion relation and hence group velocity of the waves to be controlled. The theoretical results include the effects of the Zeeman, dipolar and exchange energies as well as those arising from the boundary conditions applicable to both thick and thin ferromagnetic films with surfaces bounded by various combinations of either free space and conducting planes.

1 . 0 Exchange-coupled TE inhomogeneous eigen- waves. - Recently, Morgenthaler [I] pointed out that the penetration depth of long wavelength magneto- static surface spin waves can be controlled indepen- dently of the wavelength by suitably altering the direc- tion and magnitude of the dc magnetizing field. In this paper quantum-mechanical exchange effects, pre- viously neglected, are considered. The analysis of exchange effects carried out by De Wames and Wol- fram [2] cannot be used directly because here the saturation magnetization does not in general lie either wholly in o r perpendicular to the plane of the surface.

Our starting point is consideration of the eigenwaves in a semi-infinite ferromagnet. Following Auld [3] and Soohoo [4] but allowing complex propagation cons- tants as in Morgenthaler [ 5 ] , we extend the concept of the Polder susceptibility tensor and include spatial as well as temporal dispersion in its components.

Consider a uniform magnetically saturated ferro- magnet and the coordinate system shown in figure 1.

The saturation d c magnetization vector M, is assumed t o lie in the x-y plane a t an angle B from the axis.

As is well known, the wave equation that results from combining the complex form of Maxwell's equations and the Poldcr susceptibility tensor F, has the form

When exchange effects are ignored, 2 is frequency dependent but independent ofspatialcoordinates ;when exchange is included, equation (1) may still be used for each plane wave component of h ^1.exp(- y.r) provided spatial dependence is added t o 2.

FIG. 1. - Coordinate system of obliquely magnetized fer- romagnet. Wave propagation is along either z or y thermed

respectively the ( T ) and (L) geometries.

Neglecting losses and magnetic anisotropy for simplicity and restricting solutions to quasi-transverse electric inhomogeneous waves propagating along

I t

\ I I

to eigenvectors with negligible! h, component. The propagation vector is taken as

and equation (1) reduces t o

. -

(*) This research was supported by the Advanced Research = of cp,, ( ^{1 *+ X [I + %(I - w2/m:) 0 )}

Projects Agency under Contract DAHC-IS-70-C-0190. I f %

1

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711415

F. R. MORGENTHALER C 1 - 1160

where

and

o: ⁼ o, + AoM(k2 - a2) . ( 5 ) The upper choice in { } is termed ( T ) because is transverse to k ; the lower choice (L) because M, has a longitudinal component along k. In these expressions, o, = - ypO Ho and co, = - ypo M, where y is the gyromagnetic ratio (negative), H, the internal magnetic field, Ms the saturation magnetiza- tion, 1 the exchange constant and 8 the permittivity of the ferromagnet.

When sin 8 # 0, equation (3) is nearly exact provided

I k2 - a2 I S ⁰² ep0 I sin 0 1 and I w I is not too close to I o; I ; when sin 8 = 0, equation (3) is exact, h, = 0 and the wave is precisely TE in character.

Within the magnetostatic limit the inequality is well satisfied ; therefore the right hand side of equation (3) may be set equal to zero. The remainder when combined with Eq. (5) yields a characteristic cubic equation in a2 where the coefficients depend upon o , o,, w,, 8 and lk2. Evidently there are six values of a for each real value of k. All may be real but two or more are generally imaginary or form complex conjugate pairs.

Naturally, k may also be complex but in any case a linear combination of the six eigenwaves forms an eigenmode -provided it satisfies the pertinent boun- dary conditions.

2.0 Boundary conditions. - Because of the ex- change coupling, spatial dispersion exists and a new channel of power flow is created. This leads to addi- tional branches of the dispersion ^(o - y) diagram and the extra wave amplitudes must be determined by additional boundary conditions governing the non e x h power channel. Such conditions have been described and debated by various authors, many of whom are referenced in 121, but at an interface bet- ween a ferromagnet and a nonmagnetic medium they generally reduce to the form

where n is the coordinate normal to the boundary a n d 7 is a symmetric tensor. As discussed by Morgenthaler [6, 71 and implied by Sparks [8] unless the surface of the ferromagnet can store finite energy per unit area (infinite exchange energy density), 7 can only be a

1 aMs

scalar with value A = - - which is zero for a

M, an _-

uniform medium. On the other hand, if 2 has large enough components, partial or complete cr spin- pinning ^)> can result. In what follows we assume that

-

- A is a scalar of unspecified value. Of course, in addi- tion, the usual Maxwellian boundary conditions apply.

3.0 Quasi-TE surface eigenmodes. - 3 . 1 SEMI-

INFINITE SLAB. - AS a first example, again consider figure 1 and assume that a ferromagnet and free space extend respectively throughout the half-spaces y > 0

and y < 0. The surface is therefore the x-z plane and the propagation is again taken along either z or y ; however, for the sake of brevity, we here consider only the (7') geometry. Because we are interested in the magnetostatic limit, scalar potentials I) can be defined in each region. Also we tacitly assume no source (or reflection) of waves at I y I ^{-+ co, therefore}

the six values of a in the ferromagnet reduce to three and the two values in the free space region reduce to one. It follows that for real k

where the real part of each ^ai must be taken positive when a i is real or complex, and as + jki when wholly imaginary.

Continuity of n x h and n. b lead respectively to

and

-.

The exchange boundary conditions lead to (9)

and

where x i and ozi are given by equation (4) and (5) when ais replaced by a,. equation (1 la, b) determine the rela- tive amplitudes of the three waves -once k is known.

Provided l k 2 4 1 and a, 4 I I,

and the mode will be dominated by A , -- ^{1 unless} I A I

is comparable to I a, I and/or I a3 I. On the other hand if I A 1 is large enough, A2 and/or A,, although still numerically small compared to A,, will contribute importantly to equation (9) and may cause the mode to cc leak ⁾⁾ appreciable exchange power in the + ^{y direc-}

tion. In that case, k will become complex to signify the power loss and in equations (7) and (9), Ik I must be modified accordingly. Assuming that I A ( is small, we find that when

- k cos 8 < 0 and

I k l

O0 < c0s2 8 < 1 ,

W O + ^OM

the eigenfrequencies are given by

(I + ^COS' 0) (w, + ileff k2 uM) + whI C O S ~ 8

O N

2 1 cos 8 1 ⁽¹²⁾

where

MAGNETIC CONTROL OF EXCHANGE TORQUES AND THE PENETRATION DEPTH C 1 - ¹¹⁶¹

Notice that A,,, is a strong function of 8 and vanishes when For the general boundary value problem and either 6 = 0, .n because then a:

-

(T) or (L) geometry, as long as l k 2 < 1 and the two eigenwaves with a: (the order of k2) dominate the other four, the latter waves may be ignored - along with the exchange boundary conditions (except for abnor- mally large values of A). Exchange effects may then be taken into account by setting wZi equal to

wo + ^{l e f f} k2 ^{W M} where

A,,, = lo, o, sin2 8/

- wo(o0 + ^{O M 60s' 8)]}

[oo(oo + wd - ^o,2=0]

3 . 2 FINITE WIDTH SLAB. - Consider a ferromagne- tic slab of width D bounded with either : 1) free space on both sides, 2) perfect conductors on both sides, or 3) free space on one side and a perfect conductor on the other. The saturation magnetization is again biased at an angle 8 in either the (T) or (L) geometry.

The eigenmodes are assumed to contain predomi- nant + ^a, character and are governed in each instance by the secular equation given in table I.

For case 1) and sin 8 = 0, the dispersion is solely due to boundary effects because A,,

-

Approximate eigenmodes of finite width slab

Case 1) Case 2) Case 3)

Boundary conditions Secular equation

- -

free space both sides tanh aD(C+ C - + ¹⁾ + ^{C +} + ^{C - = 0}

perfect conductor both sides C+ C - = 0

free space one side ; perfect eZaD C + ( C - + ¹⁾ - C - ( C + - 1) = 0 conductor the other side

cooi * -- ^wo + ^{k2 03, ;} Aeff is given by equation (14)

References

[I] MORGENTHALER (F.), J. Appl, Phys., 1970, 41, 1014. [6] MORGENTHALER (F.), J. Appl. Phys., 1967, 38, 1069.

[2] DE WAMES (R.) and WOLFRAM (T.), J. Appl. Phys., [7] MORGENTHALER (F.), (( Magnetoelastic Wave Propa-

1970, 41, 987. gation in Time Varying Magnetic Fields N in

[3] AULD (B.), J. Appl. Phys., 1960, 31, 1642. Recent Advances in Engineering Science, 1970, [4] Soo~oo (R.), Phys. Rev., 1960, 120, 1978. 117.

[5] MORGENTHALER (F.), Electronics Letters. 1967, 3, 299. [8] SPARKS (M.), Phys. Rev. Letters, 1969, 22, 11 11.