TECHNICAL PHYSICS LETTERS VOLUME 24, NUMBER 7 JULY 1998 Conditions for the existence of backward surface magnetostatic waves in a ferrite–insulator–metal structure V. I. Zubkov and V. I. Shcheglov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Fryazino ~Submitted December 10, 1996; resubmitted March 13, 1998! Pis’ma Zh. Tekh. Fiz. 24, 1–7 ~July 12, 1998! A theoretical analysis is made of the dispersion of surface magnetostatic waves ~SMSWs! at different frequencies propagating in ferrite–insulator–metal ~FIM! structures magnetized by a uniform magnetic field. It is established that depending on the ratio between the thicknesses of the ferrite and insulating layers in FIM structures, backward SMSWs exist in different frequency ranges and have different wave numbers and directions of propagation. The conditions for which backward SMSWs may be observed directly in FIM structures are determined. © 1998 American Institute of Physics. @S1063-7850~98!00107-4# Studies of the dispersion of forward and backward sur- with varying VH , whereas the angle w is considered to be a face magnetostatic waves ~SMSWs! in ferrite–insulator– free parameter which is defined by the antenna exciting the metal ~FIM! structures are of great interest because of their SMSW (w5w0) ~Refs. 1–6!. Below we determine the rela- possible applications for analog processing of information in tionship between the frequency range of existence, the wave the microwave range ~see Ref. 1 and the literature cited numbers, and directions of propagation of backward SM- therein!. The numerous studies on the dispersion of SMSWs SWs, and we show that the type of wave cannot be accu- in FIM structures with specific parameters1–6 create the im- rately determined merely using the dispersion law. pression that the problem of the existence of backward SM- It is known1–3,5 that SMSWs exist in a frequency range SWs in FIM structures has been well studied. However, the between a lower limit Vl and an upper limit Vu(s,w) and in absence of direct experimental evidence of their existence4–6 a range of angles w bounded by the ‘‘cutoff’’ angles is incomprehensible. Some possible reasons for this are sug- 6wc(s). We then have Vu(`,0),Vu(s,w),Vu(0,0) and gested below. wc(s),wc(0), where Vu(`,0) and Vu(0,0) and wc(`) and We shall consider an infinite FIM structure in the yz wc(0) are the upper frequencies and cutoff angles for a plane, consisting of a ferrite film of thickness d magnetized SMSW in the ferrite film (s5`) and in the ferrite–metal to saturation and an ideal conducting metal layer, separated structure (s50): by an insulating layer ~in magnetostatics, a vacuum layer! of thickness s ~subsequently, this layer is simply denoted by s!. Vl5AVH~VH11!, ~2! The x50 plane is the surface of the ferrite film closest to the Vu~`,0!5VH10.5, ~3! metal layer. The magnetizing field H0 is directed along the z 2 21 axis. In this FIM structure an SMSW propagates with the wc~`!5arccos$@V1AV 2VH~VH11!#~VH11! %, frequency v and wave vector k, whose group velocity vg is ~4! directed at the angles of w and c to the y axis. The dispersion relation for the SMSW in the FIM struc- Vu~0,0!5VH11, ~5! 7 ture is written in the form: 2 21 wc~0!5arccosA@V 2VH~VH11!#~VH11! . ~6! @b22ma coth~akd!#1~b1222pn cos w! Figure 1 shows dispersion curves Vs(ky ,d) for a SMSW 3exp~22ks!50, ~1! in an FIM structure with different values of s. It can be seen that there are three types of curves. First, in the frequency 21 2 2 1/2 2 2 2 where a5@m sin w1cos w# ; b5(n 2m 1m)cos w range Vu(`,0),V,Vu(0,0) the functions Vs(kyd) for 2 2 21 2 2 21 2m21; m511VH(VH2V ) ; n5V(VH2V ) ; SMSWs in an FIM structure with small s (0,s<sb1) have V5v(4puguM )21; V 5H (4pM )21;4pM is the a maximum at kd5k d and tend to V (`,0) for kd ` 0 H 0 0 0 max,s u → saturation magnetization of the ferrite film, g is the electron ~curves 2 and 3!. Each forward SMSW with k,kmax,s corre- gyromagnetic ratio, and p561. Only SMSWs propagating sponds to a backward SMSW with k in kmax,s,k,`. Second, in the x50 plane of the FIM structure (p51) can be both in the frequency range Vl,V,Vu(`,0) the function 1–3 forward and backward waves. Vs(kyd) for an SMSW in a FIM structure with large s The dispersion relation ~1! is considered to be the dis- (s>sb1) has a maximum and a minimum ~curves 4–7!.A persion law Vs(k) for given parameters w, d, s, and VH and backward SMSW exists for sb1,s,sb2 and in the frequency is used to determine the type of SMSW ~forward or range dV(s,0)!Vl . The forward SMSWs have wave num- 1–5 backward!. The form of the dispersion law Vs(k) depends bers k in the ranges 0,k,kmax,s and kmin,s,k,` while the on the ratio between d and s and does not vary qualitatively backward SMSWs have wave numbers in the range 1063-7850/98/24(7)/3/$15.00 499 © 1998 American Institute of Physics 500 Tech. Phys. Lett. 24 (7), July 1998 V. I. Zubkov and V. I. Shcheglov FIG. 1. Dispersion curves Vs(kyd) for SMSWs in an 4 FIM structure with different s for VH50.25 ~sb1' 3d and 1 sb2'2d!. Curves: 1—s50; 2—s5 3d; 3—s5d; 4—s51.467d; 5—s51.6d; 6—s51.733d; 7—s51.867d; 8—s52d; 9—s54d, and 10—s5`. For curves 2 and 6 the points kmax,s8 , kmin,s8 , Vu(s,0), and dVs are indicated. kmax,s,k,kmin,s ~curves 4–7!. Third, the functions Vs(kyd) points. If these are numbered according to increasing ky , for SMSWs in an FIM structure with sb2,s,` have a point forward SMSWs exist at the first and third points with back- of inflection ~curve 8!, and only forward SMSWs could exist ward SMSWs at the second. in such structures. It can be seen from the curves kz(ky) ~Fig. 2! that in FIM However, relation ~1! defines the dispersion surface structures with different s, backward SMSWs in different Vs(ky ,kz) ~ky and kz are the projections of the SMSW wave frequency ranges propagate in different ranges of variation of vector k on the y and x axes! and the dispersion law Vs(ky) the angle w and have different wave numbers. is its intersection with the plane kz50 and does not give We shall analyze SMSWs in FIM structures with small s complete information, which is obtained by additionally ~the line of direction intersects the kz(ky) curves at two studying its intersections with the planes Vi5const(ky0kz), points!. In the frequency range between Vu(`,0) and 8,9 i.e., the curves kz(ky) ~as for an SMSW in a ferrite film !. Vu(0,0) backward SMSWs exist for angles w between Figure 2 gives the curves kz(ky) for an SMSW wave with the 2wc(0) and 1wc(0), since the curves kz(ky) resemble non- frequencies V50.608, 0.695, 0.745, and 0.775 in an FIM canonical ellipses lying at ky.0 ~dotted curve 2! and the line structure with different s. In the ferrite film and the ferrite– of direction can only intersect these for uw0u,uwc(0)u. In the metal structure the curves kz(ky) resemble canonical frequency range between Vl and Vu(`,0) backward SM- 2,8,9 hyperbolas ~curves 1 and 4!. We denote these as qH(`) SWs exist in the range of angles uwc(`)u,uw0u,uwc(0)u for s5` and qH(0) for s50. The direction of propagation since the curves kz(ky) are strictly increasing functions ~solid of the SMSWs on the k y0kz plane is given by a straight line curves 2 and 3! situated between qH(0) and qH(`), and the emerging from the origin at the angle w0 ~subsequently line of direction can only intersect them in this range of called the line of direction; when discussing Fig. 2 we shall angles. The curves kz(ky) at the point with the lowest value draw this line mentally!. For uw0u,uwc,su this line intersects of ky ~ky,inf,s for kz50! are close to qH(0); ky,inf,s increases qH(`) and qH(0) at a single point at which the projection with increasing s. of the group velocity on the direction of the phase velocity We shall now analyze SMSWs in an FIM structure with shows that the SMSWs in the ferrite film and in the ferrite– large s ~the line of direction intersects the curves kz(ky)at metal structure are forward waves. In the FIM structure the three points! where two forward and one backward SMSW line of direction intersects the curves kz(ky) at two or three exist. In an FIM structure with sb1,s,sb2 the curves kz(ky) Tech. Phys. Lett. 24 (7), July 1998 V. I. Zubkov and V. I. Shcheglov 501 FIG. 2. Curves of kz(ky) for SMSWs at frequencies V50.695 ~solid curves!, 0.745 ~dashed curves!, and 0.775 ~dotted curves! in an FIM structure with different s for VH50.25. Solid and dotted curves: 1—s50; 2—s5d; 3—s52d; 4—s54d; 5—s5`.