Instructions for use

Title Finite-element analysis of magnetostatic wave propagation in a YIG film of finite dimensions

Author(s) Koshiba, M.; Long, Y.

Citation IEEE Transactions on Microwave Theory and Techniques, 37(11), 1768-1772https://doi.org/10.1109/22.41043

Issue Date 1989-11

Doc URL http://hdl.handle.net/2115/6072

Rights© 1989 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material foradvertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists,or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.”IEEE, IEEE Transactions on Microwave Theory and Techniques, volume 37, issue 11, 1989, Page(s):1768 - 1772

Type article

File Information ITMTT37-11.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

1768 [EEE TRANSACftQNS ON ).IICROWAVE THEORY AND TEC It N[QUES, VOL. 37, NO. 11, NOVEMBER.

Finite-Element Analysis of Magnetostatic Wave Propagation in a YIG Film

of Finite Dimensions MASANORI KOSHIBA, SENIOR l>tEMBER, IEEE, AND YI LONG

Abstrucl -A unified numerical approach based on the finite-element method is described for the magnetostaric wafe propagation in a VIC f ilm of finite dimensions. Both mDgnefOSIDtic volume 1nl,·e Dnd magnelostatic: surface ",-a,·e modes are tTt'Dteci. The validity of tlit' method is confinnecl b)' calculating the magnetosiDlic "'1I,·e modes in a VIG-lo:Ided rectangular waveguide and in a VIC film of fin Ite width. "The numerical Il'SUlts of a VIC film with nonuniform bias Ileld along the film ",idth are also pre­sented, and the effects of bias field distributions on the delay characteris­tics and potential profiles are examined.

I. I NTRODUCTION

MAGNETOSTATIC WAVE (MSW) propagation in a YIG film of finite dimensions has been reported

in previous papers [1]- [11]. O'Keeffe el al. [2] and Bajpai et al. [4] have investigated the effects of finite sample widths on MSW propagation. Recently, inlerest in three­dimensional inhomogeneous MSW waveguides is increas­ing because high-precision dispersion control is making MSW device application possible [1], [3], [5]-[11]. Morgen­thaler et oJ. [1], [3], [6), [8], [9J have discussed the control of MSW propagation by means of a spatially nonuniform bias field. It is also found that control of important features of MSW modes is afforded through the use of bias field gradients and that magnetostatic forward volume waves can be forced to have strong field-displacement characteristics that are either nearly reciprocal or very strongly nonreciprocal. Such control may provide the basis for new forms of microwave signal processors [6] - [9]. In order to analyze these inhomogeneous MSW waveguides, the variational method [12), [13] and the finite-element method [14] have been introduced. These methods are valid for the solution of inhomogeneous waveguide struc­tures. In the former approach, however, great care is necessary in choosing the trial functions. The laller ap­proach, on the other hand, is applied only to planar structures of infinite wid th .

In this paper, a unified approach based on the fin ite-ele­ment melhod is described for the MSW propagation in a YIG film of fini te dimensions. Both magnetostatic forward volume wave (MSFVW) and magnetostatic surface wave (MSSW) modes are treated. In this finite-element ap-

Manuscript received February 23,1989; revised July 10, 1989. The au thors are with the Department of Electronic Engineering,

Hokkaido Universi ty. Sapporo. 060 Japan. IEEE Log Number 8930663.

r.

r, r.

z

y X r. Fi&- I. MSW wa,·eguide.

proach, the cross section of the waveguide is divided into triangular elements [15], [16J and it is easy to consider the inhomogeneities in the bias field and/ or the magnetiza_ tion. The validity of the method is confirmed by calculat­ing the MSW modes in a YIG-Ioaded rectangular wave­guide [10) and in a YIG film of fin ite width [4). Numerical resuhs on the delay characteristics and potential profiles of a YIG film with nonuniform bias fi eld distributions are examined.

II. B ASIC EQUAT IONS

Fig. 1 shows a MSW waveguide, where the boundaries f1 and f 2 are assumed to be perfect electric conductors (PEe's) or perfect magnetic conductors (PMC's), and rl

and f4 are assumed to be PEC's. The constitutive relations are

B - .ol. , ]H B = p.oH

for ferrite

for dielectric

(I.) (Ib)

where B is the magnetic nux density, H is the magnetic field, P.o is the permeability in free space, and [J£r] is the relative permeability tensor [14J.

For the MSW propagating along the y direction, the governing equations can be written by

aBx aBz ax + a; - Js{lBy - 0 (2)

a. H -- - (3.) • ax Ny-JsM (3b)

a. H, - - a; (3e)

0018-9480/89/ 1100-1768$01.00 <>1989 IEEE

'

SA "NO 1.01'10: FINlTE-[ llMENT ANA LYSIS Of MAONETOST" n c WAVE PROPAGATION .,"" 1769

z

x y

Fig. 2. Second-order triangular element.

where fJ is the phase constant in the y direction, s - ± 1 is a directional parameter [14), and ~ is the magnetostatic potcntial.

III. M ATHEMATICAL F ORMULATIONS

Dividing the region enclosed by the boundaries f 1 to f4 into a number of second-order triangular elements [15), (16} as shown in Fig. 2, the magnetic potential 4> within each element is defined in terms of the magnetic potential ~k at the nodal point k (k = 1,2,···, 6) :

where

~ ~ (N ) T(¢) . exp ( -j,P, ) (4)

() [ " " "IT ~ .~ ~, ., ~, ~, ~, .... ( N) - [ N, N, N, N, N, N,IT.

(Sa)

(5b)

Here shape functions NI to N6 are given by

N, ~ L ,(2L , - 1) (6a)

N, ~ L , (2 L , -1) (6b)

N, ~ L , (2L, - I) (60)

N,~4L,L, (6d)

N~ = 4LIL3 (6e)

N, ~ 4L,L, (6f)

with the area coordinates L1• L2• and L3 [151, [16}. The relation between the area coordinates and the Cartesian coordinates is expressed in the foUowing form:

x XI x 2 X3 L\ L, L,

(7)

where (xi' Zj ) are the Cartesian coordinates of the vertex j (j = 1, 2, 3) of the triangle. Also, tbe diagonal component p, and the off-d iagonal component " of the relative perme­ability tensor within each element are approximated by

.~ (N}T( . ) . (8a)

K ~ {N )T (K }. (8b)

where

{.} . - [., ., ., ., ., p" f (9a)

(K }._ [ K, K, K, K, K, K,f (9b)

and P, k and K k ( k = 1, 2, · . · ,6) are, respectively, the JL and K values at the nodal point k . Using a Galerkin procedure on (2), we obtain

f (aB, aB_ )

(N ) -a-+ - - - j,PB, dO - (O) , x az (10)

where the integration is carried out over the element subdomain n" and {OJ is a null vector.

Integrating by parts, we obtain for (10)

f[ ( N, ) B, + (N,} B, + j ' P( N ) By] dO •

- f(N} B" dr~ (0 ) •

(11)

where

(12)

Here (N,) - a( N )/ax ; ( N, ) ~ a(N) / az; 'he second integration on the left-hand side in (11) is carried out over the contour r, of the region 11,; and nx and n: are the x and z components of an outward normal unit vector to r,. respecti vel y.

Noting that BIf is continuous across f, (boundary condi­tions at the interface between two different media) and B" = 0 on f) and r., from (1), (3), and (4) the following globaJ matrix equation is derived: ,

[A ]{ ~ ) -- [ (- 1) '[/.1 {N },B,.,I /. , dz (13) I - I " ,

where

[ Aj - [ flp ', ,( N )( N ) T - P' K. • •

'(( N, }(N )T +(N}( N, )T)

+ (N,)( N, ) T + • • ( N, )( N, } T] dxdz

for bias field applied parallel 10 the x axis (14a)

[A j - [ flp( N}( N} T + j K. • •

and

((N, }(N,} T _ (N,}( N, }T)

+ • • ( N,)(N,} T + ( N, }( N,j') ] dxdz

for bias £ieid applied parallel to the y axis (1 4b)

[A j - [fl p' •• ( N}( N} T + Ps<. • •

' ((N,}(N }T + (N }( N, )T)

+ ( NJ (NJ T + • • ( N,)( N,) T] dxdz

for bias field applied paraJlei to the Z axis. (14c)

Here Il , and "r are expressed by (8a) and (8b), respec­tively; {$ } is the nodaJ magnetic potential; L, and L r •

1770 IEEE TJl.ANSACTIONS ON MICROWAVE THEORY ANI> TECHNIQ UES. VO L. 31. NO. l1,NO"': ::-;;

extend over all elements and the elements related to rj

(i - 1,2), respectively; B". ; is the magnetic nux density on r j; and {N}; is the shape function vector on rj , namely, (N L = ( N(x ;, z) }.

Combining the boundary conditions on the planes r l

and r2, (13) becomes

where

[A] -

[A] [A]oo

[[ A] oo [A] "

[ [A]oo [A] "

(.) (. ),

[(.),] (. ),

[(.),] ( . ),

[AI{~} = (0) (15)

for B" 1- B" 2=0 , ,

fm(. ),~ (. },~(O )

[A] " ] [A] "

forB", - Oand (.J,- (O)

[A] ,, ] [A ]" for B" ,-O and (.;), - (0)

(16)

for B",I = Bx •2 = 0 for (.},-(.}, - (O}

for B", - 0 and (. J, - (O) (17)

for B", = 0 and (.), - (O )

(. ), (.)- (. ),

( . ), (18)

[A ]00 [A] - [A] "

[ A],o

[A]" [A]" [A ],.

[A]" [A]" [A]"

(19)

Here [A )00' [A 101" . " [A122 are the submatrices of [A 1, and { <P }; is the nodal magnetic potential on rj ( i co 1,2). Using (15), onc can determine the dispersion characteristics of MSW waveguides.

IV. CoMPlITED REsULTS

First, we consider a YIG-Ioaded rectangular waveguide [10) as shown in Fig. 3, where the planes x = 0, x'" a, z = 0, and z = b are assumed to be PEC's, the bias field is along the x axis, and there exist MSSW modes. In this case, the potential profile is symmetric with respect to the plane x = a /2, Taking advantage of this symmetry, we subdivided one half of the cross section into second-order triangular elements, where the plane x - a / 2 is assumed to be a PMC. Table I exhibits the computed results, where a = 20 mm, b = 10 mm, 1=1 mm, the saturation magneti­zation 4'7rM. = 1750 G, the bias field Ho - 1800 Oe, fJuae!

is the exact solution (10J, and fJ FEM is the finite-element solution. It is apparent from Table I thai the finite-element solutions are in good agreement with the exact solutions. The convergence behavior of the phase constant f3 is shown in Fig. 4. With an increasing numbcr of elements, the phase constant f3 converges monotonically. Also, we

• Dielectric do

YIC b

' 0 • Dielec tric do

y x

Fig. 3. A YIG·ioaded rectangular waveguide (the planes x .. O. x .. a, z .. 0, and z - b are assumed to be PEC's) or a YIG film of finite width (the planes x " 0 and x - a arc assumed to be PMC's and ~ planes z - 0 and l " b arc assumed to be PEC's).

~

c 0 c c • ~ 0 • 0 c • ~

10-' ,----------,

10-2

20-5 .0 l1li

F"7.1 GHz

Z0-5.0 "II F-7.4 GH-;z:

Zoa9.5 l1li

F-9.5 GHz

500 '000

Number of elements NE Fig. 4. Convergcnce behavior of phase constant /l

rind from Table I and Fig. 4 that the accuracy of the finite-element solutions fa lls off as the film position Zo is lowered or the frequency F increases.

Next, we consider a YIG film of finite width [4] as shown in Fig. 3, where the planes x - 0 and x = a arc assumed to be PMC's, the planes z",. 0 and z = bare assumed to be PEC's [2], [4], the bias field is along the z axis, and there exist MSFVW modes. Table II exhibits the computed results, where a = 3 mm, dl = 50 p.m, d 2 = 178 p.m, t = 9.1 p.m, 4'lTMs = 1750 G, and the effective slatic field Hi = 3000 Oe (H1 - Ho -4'lTMs for MSFVW modes). The finite-element solutions agree well with the exact solutions [4]. Also, larger errors are observed as the fre· quency mcreases.

Lastly, we consider the MSW waveguide with a bias field of parabolic distribution where the planes x = 0 and x - a in Fig. 3 are assumed to be PMC's, the planes z - 0 and z = b are assumed to be PEC's, and the bias field is

f;:OSHIEIA AND t.ONG: FINITE-ELEMENT ANALYSIS OF MAONETOSTATIC WAVE PROPAGATION 1771

TABLE 1 COMPARISON B£IWEES ExACT SoLUTIONS AND FEM

Sot.UTIONS FOR MSSW MODES .. f Phase constsnt 0/ •• )

( .. ) (6Hz) 11 ..... D ,c"

7 . , O. (/4190H 0.041 9052

7.3 0.1 193626 0.1193620

••• 7.' 0.1633299 0.1633295

7. 5 0.2235310 0 .2235304 7. , (/.3563842 0.3563856

7., O.UOU45 0.U04&39

7. 2 0. 2253411 0.2253469 5. '

7.3 0_4832040 0.4832315

7.4 0.90U406 0.9019049

/1 -20 mm, b-10 mm. 1- 1 mm, 4'ITM, - 1750 G, Ho-18000e.

TABLE II COMPARISON BETWEEN ExACT SoLUTIONS AN D

FEM SOt.UTIONS FOR MSFVW MODES

f Phase constant 0/ •• )

(GHz )

8.11

8.9 1

9. 0 1

9. 11

9. 21 9. 31

9. 41

9. 51

B • •• ••

3.200592

16.25161

28.5 9313

41.05768

53.96416

67. 73419

112 .69318

99.12166

' 3.200420

16.25179

28.59392

41.06009

53.91119 61.14865

SZ.11910

99.11226

d[-SO /lm, d2 - 178 /lm. 1-9.1 /lm, 0-3 mm, 4"11"M. - 1750 G , and Hi -

3000 O.

~.x.-------~~~------·

XI. 1

Fig. 5. Bias field distribution along the width of a YIG film.

along the z axis. The two parabolic distributions consid­ered here are shown in Fig. 5. The bias field distribution is given by

H,- H,-(H,- H,)X[(a-2x)/aj' (20) where HI and H2 are the minimum and the maximum value or the errective field for a hollowed parabolic distri­bution (case A), respectively. and HI and H2 are the maximum and the minimum value for a projecting parabolic distribution (case B), respectively.

I , • <

~

~ • -~ I ~

" C.~e A. IHollowed parabolic dl,trlbullonl

Ho' .... 3110 Oe

H.I ..... J050~ HO':lIXJO oej Iv , ,

./ /

.......... ' - 3,0 1111 V \/ \ /

/ ... ---- dl- 50 ~II dz-178 ~. V t-9.1 r H-c-3 10 Oe

22

20

18 8,5 8.7 8.'

Frequency F (10Hz) (., 4rN~"1750 G .. , ' .3

",------------------------,

22

20

18

1101"..3000 Oe

1-1.1..-3050 De

H.17"0 Oe

, , /-t;;::'1,=---/ ........ ':;0 ..

\ _---- dl-SO,. \ / \./ dz-178 ~.

Cue B taS.1 ~. (P~Jecting par.hol ic distr ibutIon) H...-3110 Oe

Lo-~_-C~ ___ -::-___ ~,.'"'",O·,"75~.~' 8.5 8.7 8,9 9.1 9.3

FrllqJency F !GHzl

(b'

Fig. 6. Time delay cbaracteristics for parabolic bias field distribution along the film width_ (a) Hollowed parabolic dislribution. (b) Project· ing parabolic distribution.

The time delay characteristics are illustrated in Fig. 6(a) and (b) for cases A and B, respectively. where the dotted line represents the delay curve for the homogeneous case [4]. These figures show that the nondispersive bandwidth is extended in the case of the parabolic distribution of the bias field. Comparing case A with case B, it is also found that the projecting parabolic bias field offers a wider nondispersive bandwidth.

The effect of parabolic bias field on the potential prorile is investigated, Fig. 7(a), (b), and (c) shows the potential profiles ror the uniform, hollowed parabolic, and project­ing parabolic bias fields, respectively. The phase constant fJ is independent of the direction of propagation. In the case of parabolic bias fields, the potential has a stronger localiz.ation. Comparing case A with case B, it is found that the projecting parabolic bias field creates stronger poten tial- displacemen t nonreci proci t y.

A similar potential- displacement nonreciprocity was re­ported in (9].

V. CONCLUSIONS

A finite-element method is developed for the analysis of the MSW propagation in a YIG film or finite dimensions. Both MSVW and MSSW modes are treated in a unified

1772 IEEE TRANSACTIONS ON MICROWAVE THEO RY AN D T(CHN IQUES, VOL 37, NO. 11 , NO,",""" 119"

. ., , .. _ 1. 01 _ .,

• _ 3I1t o.

• ••

o. )0

, •

(.) . ., _1.,5 _" , .

•

o. 10 O. 30

•

" , • 'b) •

. ., - , , .. _ t. U _ . , _ •. U _ .,

. .. JfU 0- , _lOU 0.

• • ... ...

• -lJ'fttt O. JO

'." ',II

• (0)

Fig. 1. Potential pro riles. (a) Homogeneous distribution. (b) Hollowed parabolic d istribution. (c) Projecting parabolic distribution.

manner. In order to examine the validi ty of the present method, MSW modes in a YIG-Ioaded rectangular wave­guide and in a YIG film of finite width are calculated. We also present the application of this approach by analyzing the MSFVW propagation in a YIG film with the bias field of parabolic distribution along the film width. It is found that the parabolic bias field offers an improvement in the nondispersive bandwidth.

REfERENCES

(I] F. R. Morgenthaler. " Magnetostatic waves bound to a DC rield gradient:' IEEE TrallJ. Magll., vol. MAG-D, PI'. 1252- 1254, Sept. 1977.

[2] T . W. O'Keeffe and R W. Patterson, " Magnetostatic surface-wave propagation in rinite samples," J. Appl. Pirys ., vol. 49, PI' . 4886-4895, Sept. 1978.

[3] F. R. Morgenthaler. "Bound magnetostatic waves controlled by rield gradients in YIG single crystal and epitaxial films," IEEE TraIlS. Magll .. vol. MAG-14, pp. 806- 810, Sept. 1978.

(4) S. N. Bajpai and N. C. Srivastava, " Ma&Detostatic bulk wave propagation in multilayered structure," EI«fTOn. UII., vol. 16, PI'. 269- 270, Mar. 1980.

[5] M. Tsutsumi, Y. Masaoka, T. Ohira, and N. Kumagai. "A new technique ror m3gnetos tatic wave delay lines," IEEE Trans. Mi­Cr OWl/III' Theory Tech., vol. MIT-29, PI'. 583-587, June 1981.

[6\

\1\

[8\

[9\

[10]

[Il]

[12]

[13]

[14J

(15]

[16)

F. R. Morgenthaler. "Nondispersive magnetostatic forward vollilnt waves under field gradient control:' J . Appl. Phy~., vo\. 53, Jlp 2652-26S4,Mar.1982. . N. P. V1annes, " Investigation or mag.netostalic waves in unifoto:J and nonuniform magnetic bias fields wi th a new induction probe' presented at Third Joint Intermag-Magnetism and Magnetic Mat~ rials Cont., Montreal , Que., Canada, luly 1982. D. D. Stancil and F. R. Morgenthaler, "Guiding magnetostatic surface waves with nonuniform in-plane fields," J . Appl. Phys., vol 54, PI" 1613- 1618, Mar. 1983. F. R. Morgenthaler, "Control or magne tos tatic waves in thin rilms by means of spa tially nonuniform bias fields," Circuits SYJf. SignQ} Process., voL 4, PI'. 63-88, 1985. M. Radmanesh, C. M. Chu, and G . I. Haddad, " Magnetostatic wave propagation in a rinite YIG-loaded rectangular waveguide,' IEEE TraM. MicrOWI/~ Tht()f)' Tech., voL MlT-34, pp. 1317-1381 ~19U. ' M. Radmanesh, C. M Cbu, and G . I. H3ddad, " Magneloslatic waves in a normally magnetized waveguide structure," fEEE Tr4lu. Microwave 77rt()f)' Tech ., vol. MlT-3S, pp. 1226- 1229, Dec. 1987 . D . D. Stancil, "Variational rormulation of magnetostatic W31"C

dispersion relations," I EEE Tr(mJ . Magn .. vol. MAG-19, pp. 1865- 1867, Sept. 1983. E. Saw3do and N. S. Chang, "Variational 3pproach to analysis 01 propagation of magnetostatic waves in highly inhomogencously ma8netized media," J. Appl. PhYJ., vol. 55, PI'. 1062-1067, Feb. 1984. Y. Long, M. Koshiba, and M. Suzuki, " F inite-element solution of planar inhomogeneous waveguides ror magnetost3tic waves," IEEE TrollS. MicroMl(lve Theory T«h., vol. MTI-35, pp. 731-736, Au&-1987. P. E. Lagasse, " Higher-order rini t~lemen t analysis of topographic guides supporting elastic surface waves," J . Arol<St. Soc. Amer., voL 53, PI'. 1116-1122, Apr. 1973. O. C. Zienkiewilz, The Finite Elemenf Method, 3rd ed. London: McGraw·HiIl, 1977.

Masanori Koshiba (SM'84) was born in Sapporo, lapan, on November 23, 1948. He received !be B.S., M.S., and Ph.D. degrees in electronic engi· neering from Hokkaido University, Sapporo, lapan, in 1971 , 1973, and 1976, respectively.

In 1976, he joined the Department of E1ee­tronic Engineering, Kitami Institute of Technol· ogy, Kitami, Japan. From 1979 to 1987, he was an Associate Professor or Electronic Engineering at Hokkaido University, and in 1987 he became a Professor. He has been Cn8a8ed in research on

lightwave technology, surrace acoustic waves, magnetostatic waves, mi· crowave field theory, and applic3tions of finite-element and boundary· element methods to field problems.

Dr. Koshiba is a member of the Insti tu te o r Electronics, Inrormation and Communication Engineers (IEICE), the Institute of Television Engi· neers of bpan, the Institute of E1ectrical Engineers of l apan, the Japan Society for Simulation Technology, and the lapan Society for Computa· tional Methods in Engineering. He received the Paper Award in 1981 rrom t~ IEICE.

.. Vi Long was born in N3nchang. China. on De­cember I. 1955. She received the B.S. and M.S. degrees in electronic engineering rrom Huazhong Universi ty of Science and Technology, Wuhan, China. in 1982 and 1984, respectively, and the Ph.D. degree in electronic engineering from Hokkaido Universi ty, Sapporo, Japan. in 1989.