IIlII I I _ I I I n I J,- 1 I I I. I 4'." I‘“I”:‘III¥ ‘4’: {VIIII ‘nI' I III I III II . I III . II IIII.\.I‘.I1.I> .I ,I'I I I‘ I - , I'III .I I. II I III I I II I I, I' IIII.II I I I ‘ - .I I. | . - II‘III‘II II .I I I IIII'II ‘ I I V‘I) u‘HI. ‘ IIIIIIIIIII‘III L IIIIIIIIIIIIIINIFIIII .I. IIIIIyI II “ I I I I -‘I "II.1‘~I.III j IIII I ' I I III.‘. I , - ‘.' I'. .'. I I‘l ' I ‘3. I I- t II'Ifi‘IH II ,‘II'I‘IIH‘! ‘ ‘ I‘I II"‘I ‘ I I' I I I - I? I. I I I l I III I I I I . I . I I {I ‘I'III‘. \.'(I (1‘1, IIIII‘f' ‘vI‘I:I ‘I'I "in -. H II ‘ ‘ 1-}? :4: I. I I, .‘I‘I’IIIIIII ‘IIIII-IIII 'I!‘ u . .IIII‘II E‘III’II“ t - I . III I .;I 'l . I I. 2‘14": IAI . . III} fig “"0”!“ l" h "‘1?" A I ’f‘ g‘IIJIV'I” 'HIII‘EI‘ III‘I ‘ - IIII'I ’in ‘ “III‘EI:‘ ,IJI ’ In! It: I II IIIIIInIIIIfiIIII‘IIIIII Rug” In": I I . Iz‘fUqu‘ , .‘uyt‘t" {IIIIIII- 1 up IINL U I.'IIII “1|! .“ (Ht; “4 "I? .’ I ,‘IIIIIIIIu I‘lI‘J‘Af‘ V.’ "Ir :I L‘ I II I I.I I -~ L‘. J IIIIII.“‘I'.""”‘ (“‘15 I it“, ‘I‘II‘IIA CIJJI1I'I£¥II§Ij2|A~IHW$ 'I\ ”I. l.“ {381% 11%;, ”Jay“. 'II‘fi Itv‘un't‘ ‘rI-I‘ :5 ‘l'kkmi' 1' 5‘; HIV ‘1‘ {\v ”‘4‘ ‘ \I II; '{Eifi> 1 (a) (b) Figure 2. The effect of slab thickness on the mode character of the magnetostatic wave. The above sketch shows the transverse variation of wave field intensity. 6 performed by various researchers. Bini et al. have provided an "energetic approach" to describe the mechanism of energy transfer from drifting carriers to the slow waves. Analysis of a thin ferrite slab adjacent to a semiconductor half space was conducted. The develop- ment of a single surface model employing two semi-infinite half spaces was also presented. A bandwidth of 17 kilohertz for possible amplifi- cation was theoretically predicted; while for the thin slab, bandwidths should reach several hundred megahertz [17-20]. Awai and co-workers also performed numerical analyses of a finite YIG slab over InSb. Net amplification was predicted for the case of drifting carriers [21]. In bulk acoustic wave amplifiers a piezoelectric material adjacent to a semiconducting medium is used. The surface acoustic wave amplifier however, uses a thin film of semiconductor material in close proximity to the surface acoustic, or Rayleigh, wave. Because of the presence of an acoustic wave, close proximity without actual contact is required for the wave/carrier interaction to take place. This restriction can present difficulty in the manufacturing of these devices. This does not hold for the magnetostatic wave device. The YIG and semiconductor can be in direct physical contact. The device shall be referred to as a Magnetostatic Surface Wave Amplifier (MSSNA). It is anticipated that this device will operate up to about 10 gigahertz. Adam states however, that MSSN devices in general are narrowband at these frequencies, ex- hibiting approximately 300 megahertz of operating bandwidth [22]. This paper shall provide a thorough analysis of the MSSNA in the absence of the magnetostatic approximation. 0f major importance is the velocity relationship between the carriers and magnetostatic waves. The conditions for which amplification is observed are inconclusive at pre- sent. Many reserachers have stated that the drifting carriers must 7 travel faster than the propagating magnetostatic wave. Kawasaki et al. state however, that even when the drift velocity is less than the phase velocity of the wave a reduction in losses is observed [23]. It is the author's belief that eventhough the carriers may possess a velocity greater than the slow wave phase velocity, some degree of synchronism must exist for energy transfer to occur. This report shall also seek to determine the optimum geometry which will promote the desired synchronism for practical bandwidth and gain operation of the MSSWA. Some ideas regarding the implementation of this new geometry will also be discussed. 2. THE FERRITE SEMICONDUCTOR COMPOSITE STRUCTURE In the analysis of the MSSN amplifier an understanding of the fun- damental mechanisms on which potential amplification is based would be helpful. In this chapter the basic ideas behind ferrimagnetic reso- nance and the excitation of surface magnetostatic waves are discussed. The interaction of these waves with drifting carriers is also covered. The object is to present a qualitative description of this interaction as a foundation for detailed study of the MSSW amplifier dispersion re- lations in Chapter 3. 2.1. Ferrimagnetic Resonance Ferro - and ferrimagnetic resonance effects are based on interaction of magnetic dipoles with magnetic fields. A spinning electron constitutes such a dipole. The magnetic moment associated with this spin motion is given by E = Y(sh) (2 1-1) where Y = -IeI/m is the gyromagnetic ratio. The value sh is the electron spin angular momentum [24]. In the presence of a static magnetic field of magnitude Ho’ a torque is experienced by the spinning electron. This torque is due to the time rate of change of the spin angular momentum. This can be written as d(Sh) = + T - dt “0 “s x H0 (2.1 2) which when (2.1-1) is used becomes d-fis .) -> a? = YS(lJS X HO) (2.1-3) where Y5 = UOY and Do is in the z-direction [25]. The motion described by (2.1-3) is of a precessional nature. The spinning electron precesses 8 + .4 / (b) Figure 1. Illustration of the precessional motion for; (a) an electron in a static magnetic field and (b) a spinning top in a con- stant gravitational field. After B. Lax and K.J. Button. 10 about the static field in a manner similar to a top in the presence of a gravitational field. The torque relationship in the latter case is g% = to x B (2 1-4) where ”0 is the precession frequency and E is the angular momentum [26]. A similar term can be developed for the electron and is given by w = Y H (2.1-5) This expression shows that electron precession frequency is determined by the strength of the applied field. The development of we is given in [27]. In Figure 1 the two cases are shown, where the angle 60 is the precession angle. 0n the macroscopic level we are interested in the total magneti- zation of the ferrite system and the interaction it has with magnetic fields. The magnetization is defined as the magnetic moment per unit volume, so that M0 = NES where N is the density of dipoles. Now we may write (2.1-3) as + d + + __ = YS(MO X H ) (2.1‘6) 0 dt 0 with Do again in the z-direction. This equation describes the lossless case which is characterized by perpetual motion. The lossy case will be treated separately. It should be noted that M0 and BO are nearly parallel in the first approximation [28]. This becomes significant in the analysis to follow. If an alternating component of the magnetic field is introduced, the magnetization and field expressions become M = M0 + mert (2.1-7) A = A. + fieJmt (2.1-8) 1 where Hi and MO represent the static magnetic field and magnetization. ll 3+ 3+ + >h Figure 2. The rf and dc field and magnetization components. Note that h can be either circularly polarized or linearly polarized. 12 The rf components are D and i respectively. The field fii’ called the static internal magnetic field is actually the vector sum of the external dc field and the components due to anisotropy and demagnetizing effects. For simplicity, we will assume a single crystalline ferrite slab that is infinite in extent. Under these conditions the above effects can be neglected and mi = NO. + + The expressions for H and M are substituted into (2.1-6) to obtain 9. mi? = who x i + En’ x HO) (2.1-9) where time-harmonic dependence (ejwt) is implied. This is the linearized small signal approximation, where second order rf and dc terms are ig- nored [29]. The time variation of i as given by (2.1-9) is therefore a function of the first order interaction between dc and rf terms. The manner in which the rf magnetization changes with time determines the variation of the precession angle 90. In Figure 2 these field and mag- netization components are shown. It should be noted that since m is perpendicular to Do that 6 must be also in order to obtain the most ef- ficient energy transfer. If this spatial relationship between Ho and h is not satisfied, energy transfer does result, but at a reduced ef- ficiency. This occurs for values of m not equal to mo [30]. Figure 3 illustrates three cases; zero rf field, non-zero rf field at resonance and outside of resonance. Notice that an oscillatory motion occurs for M due to the nonresonant condition [31]. The oscillation shown is ex- aggerated for the purposes of illustration. In practice this "wobble" by the precessing magnetization would be noticeable only at very high values of the driving rf field and for frequencies not equal to do. In order to illustrate this precession in more detail, consider a rotating coordinate system. This system rotates about the field HQ with angular velocity Er. The equation of motion becomes l3 :5; +1 .coppzm .q.¥ ace xaa .m Laac< .mucmCOmmL to muwmgso .o x m on .wucm:0mmc ADV .o u ; va ”cowuoe Pacowmmmomca cw cowum_cm> egg cow mcowu_ccoo mwgcp .m mczmrm EV Al 3 +2 +5 ”r: l +2 +3: 1: dMO + a? = YSMO X (Y H + 1») (2.1-10) which is the same as (2.1-6) in the stationary reference frame if H0 is replaced by + + H = H - eff 0 (2.1-11) If an alternating field is introduced, e.g. one which is circularly polarized with frequency a? the equation of motion is 3% = YS (N x [fieff + h(t)]) (2.1—12) where h(t) = hert and the new effective field is ..) Heff ) = 2 (H - a ) + yh(t) (2.1-13) Ys with 2 and 9 indicating the z and y directions respectively. The field component h(t) is stationary in the rotating frame [32]. When the magnetization experiences this effective field there is precession about both Ho and h(t) in the stationary reference frame. In most cases, h(t) is small compared to H0 so that the precession due to the rf component is ignored. Figure 4 gives an illustration of this total motion and also shows the variation of 60 for precession about 86 due to the phase relationship between h(t) and M. The mechanism which governs the variation of and 90 is the torque 3+ 3+ created based on the spatial relationship between (t) and Do as men- tioned above. The phase relationship between h(t) and M is also very important in determining this torque. In order to promote growth in the precession angle the component h(t) must precede m(t) by a quarter period [33]. If this condition is not met then there are regions where zero or negative torque can result. The negative torque gives rise to l5 A —-—-—"—_'K K "\ + \ H R 0 \ 60 E l L >. g y + + Figure 4. The resultant motion of M due to the precession about Heff as shown by the dotted path. After R.F. Soohoo. 16 DIRECTION OF 602. l7 precessional damping. Note again Figure 4. When a positive torque is produced, it is directed in a downward fashion and tends to increase the precession angle 60. This effect is similar to that experienced by the precessing top as shown in Figure 5. The system described above does not account for losses which would relax the precessional motion. There are three forms for the phenome- logical equation of motion in which the losses are included [34]. They are: (1) The Landau - Lifshiftz form = y (M x H) - ———— [M x (M x H)] (2.1-14) where A is the damping coefficient with units of frequency (2) The Bloch - Bloembergen form __I = y (M x E) - ._;1 (2.1-15) dt 5 T T 2 dM M - M __z = t t z 0 (2.1-16) dt YS(M X H)z - T1 where MT and MZ are the transverse and z-directed components of the mag- netization. The constants 11 and 12 are related to the relaxation of the precessional motion. We cover this below in more detail. (3) The Gilbert form dM _ + + a + dM (2.1-17) d—t-YS(MXH)+ TMXa-E where a is the damping coefficient. The Landau - Lifshiftz form is used most often in ferrimagnetic resonance and is incorporated in this paper. It should be noted that the various forms are equivalent for the case of small rf signals pro- vided i2 = l/wa and a2<<1 [35]. Therefore, for the purposes of illus- tration we will examine the Bloch - Bloembergen form in more detail. l8 If we look at the case of zero dc and rf fields we obtain dMZ = - (MZ - M0) (2.1-18) EH7 T1 EMI = - fl. (2.1-19) dt T2 where M2 and MT.are as defined previously. The solutions to these equa- tions are _ -t/T MZ — M0 -AMe 1 (2.1-20) MT = MTOe't/T2 (2.1-21) where AM = MO-MZ is assumed positive at time t=0. Based on these ex- pressions we expect M0 to spiral inward to the z-axis in the absence of external excitation. This is shown in Figure 6. The losses which pro- duce this motion must be overcome to obtain a net gain in the system. The loss term is most commonly included in the development of the permeability tensor Hi This tensor relates magnetic induction 8 to the + + + field intensity H in ferrimagnetic materials. In general, B and H pos- sess both dc and rf components. One form for this tensor is u -JK 0 ‘3’: “o jK u 0 (2-1-22) 0 O 1 where ”0 “m (2 1 23) o and wwm K = - ——- (2.1-24) “02 _w2 where “nijs the saturation magnetization frequency, a constant of the system. This form for the permeability tensor results if the external l9 A\z T A : AM C M(t=0) ‘EF—TT'MTO fio [ 0 Figure 6. The Variation of the magnetization vector M in the absence of external excitation. After R.F. Soohoo. 20 dc field is in the z direction. Other forms for p are given in Appendix A. The losses are included by replacing mo in the above expressions with wo' = wO-j(1/T) (2.1-25) which can also be written as 610’ = wO-j%YSAH. (2.1-26) The parameters T and AH are the precession relaxation time and the ferri- magnetic resonance linewidth respectively. Notice that AH is inversely proportional to the relaxation time. Ferrites with lower values of AH are therefore not as lossy as materials with broader resonance linewidths. Typical values for AH in materials such as Yttrium Iron Garnet (YIG) are on the order of one oersted. This gives a relaxation time of approxi- mately 10'7 seconds. [36]. 2.2. Spin and Magnetostatic Waves Two forms of wave energy in ferrites will be discussed as they relate to propagation. These waves are distinguished by the relationship of their wavelengths to the magnetic dipole spacing. This will determine if exchange field effects between neighboring dipoles will be included. .The exchange interaction gives rise to the effect which causes adjacent electron spins to precess in phase with each other. The formulation for incorporating this effect into the analysis is covered in Appendix A. Magnetostatic waves are slow, dispersive waves which possess k values of less than 104 cm_1. This gives rise to large wavelengths with respect to the dipole spacing. Because of this, the exchange field interaction is small compared to the macroscopic magnetization field interaction [37]. 21 )( 10 10 4r w X10 WITH EXCHANGE " e = 90° 8" e = 0° I I _' "" ‘— 4 -1— 1 WITH UT —. qL fiL L 1 1 1 l 1 l 1 1 L 1 1 1 Li_l b 102 103 104 k (cm-1) Figure 7. Dispersion curve schematic of three principal wave regions; (a) Electromagnetic, (b) Magnetostatic, and (c) Spin-exchange. The angle a is for the direction of propagation in relation to the Z axis. After B.A. Auld. 22 / Ii FERRITE 2 Figure 8. The basic coordinate system for description of wave propaga- tion. The slab is assumed infinite in the xz plane. 23 The spin wave regime corresponds to wavenumber values larger than 4 cm'l. These are very high frequency waves so the exchange inter- 10 action is comparable to the macroscopic effects. For this reason, these waves are referred to as spin exchange waves [38]. A third major regime of wave propagation exists, the electromagnetic region. These waves are not examined here because their phase velocities are too great for interaction with drifting carriers [39]. Figure 7 shows a diagram of the dispersion relation which governs the electromagnetic, magnetostatic and spin wave regimes of propagation. To provide a conceptual picture of wave propagation in ferrites let us consider several cases. Suppose we are given the following co- ordinate system for our sample as shown in Figure 8. Assume the ferrite is magnetized in the y—direction as shown. A uniform rf field is applied in a system where exchange interaction is assumed strong. This effect causes all dipoles to precess in phase as shown in Figure 9. If h(t) is varied uniformly then all dipoles will experience the same variation in 60 but will continue to precess in phase. Now let us suppose the h(t) is nonuniform in the z direction. This causes a variation in 90 along the sample as shown in Figure 10. Notice that as time increases, there is no movement of the wave in the direction of propagation and that neighboring dipoles are still in phase. If exchange forces are ignored, i.e., we expand to the macroscopic level, two things occur; first, we must consider the precession of the magnetization vector instead of individual dipoles. Secondly, the phase of neighboring magnetization vectors is independent. This case is shown in Figure 11 where the rf field is assumed uniform in magnitude along the sample. The phase of adjacent M vectors differs depending on the position of the propagating wave. This is acceptable on the macroscopic 24 A A A A A '1 c is c < c < c ea (a) A A A A A A C ,C C C C C us 66 Hoy z ____..— 9b > ea (b) Figure 9. Magnetic dipoles in a ferrite slab under strong exchange interaction; (a) h = 0, (b) h f 0. All dipoles precess in phase. 25 Figure 10. A standing spin-exchange wave of wavelength A; (a) t = t , o (b) t = t1>tO and (c) t = t2>t1. 26 3 A Q) J v fir e . T i H 0 Figure 11. A travelling magnetostatic wave of wavelength A; t = to, (b) t = t1>tO and (c) t = t2>t1. Exchange is ignored here. 27 Figure 12. Generalized travelling magnetostatic wave of wavelength A; (a) t = t , o (b) t = t1 > to and (c) t = t2 > t]. 29 “’c — L. SURFACE (DB __ __ FORWARD VOLUME BACKWARD VOLUME (DA 1 _ _. .— i 10 L [a +017 130 (b) kd Figure 13. (a) Dispersion curves for the three forms of magnetostatic wave propagation in ferrites. (b) Orientation of the surface wave for a given H0. 30 level since the wavelength is so much larger than the dipole spacing. In practice an rf disturbance will give rise to both amplitude and phase variations in the precessing magnetization vectors as the wave propagates. The complete description of this wave motion has not been clearly stated in the literature for the macroscopic case. We will inter- pret this motion in the following manner for an rf disturbance which is nonuniform. The propagation is determined by variation of both 60 and the phase position of the vector M. These two components of the motion are in phase with each other. This means a particular value of 90 corre- sponds to a given phase poSition in the precession cycle [40]. A single wavelength will be defined using two corresponding points along the sample with the same value of 60 and relative phase positions. This is illustrated in Figure 12. The waves we are concerned with are surface magnetostatic waves. They are characterized by a concentration of wave energy on one face of the ferrite slab. Recall that the thickness of the sample is important in determining the degree of wave energy concentration on a single sur- face. The propagation of these waves is highly nonreciprocal. The di— rection of propagation, applied dc field and the outward pointing normal from the ferrite surface form a right hand coordinate system [41]. This means R x HO will point away from the ferrite surface. The dispersion curves for these waves along with the forward and backward volume modes are given in Figure 13 [42]. 2.3. The Semiconductor Carrier Wave System The drifting of carriers in semiconductor material is basic to the amplification mechanism for magnetostatic waves. A geometry contain- ing a slab adjacent to the ferrite is used. This is in contrast to mag- netic semiconductors such as CdCrZSe4. It has been determined that the 31 resistivity must be reduced in these materials by several orders of mag- nitude before they can be useful in amplifier applications [43]. The carriers drift at a constant velocity, uo, which is on the order of 2 x 107 cm/s. In order to keep the level of heating reasonable in these devices, high mobility semiconductors, such as GaAs, are used. Since the ferrite is adjacent to the semiconductor, the surface wave can pene- trate as long as the carrier density is not too great. Typical values 15 to 1020 cm'3. in the literature range from 10 The fields associated with this wave interact with the drifting carriers through the Lorentz force. This is given by + -> F : ”0 X hm(t), (2.1-27) where hm(t) is the rf component of the magnetostatic wave. The inter- action is a Hall effect motion which is transverse to the dc drift motion of the carriers. This is shown in Figure 14. The Hall field developed from this motion interacts with the magnet- ostatic wave, also shown in the figure. We will call this field compon- ent hs(t). This component must have the correct phase relationship with m(t) in order to promote the desired amplification. The amplification results if the system losses can be overcome by the torque created from the interaction of hs(t) and m(t). One must be careful to distinguish between hm(t), the rf component of the magnetostatic wave and hs(t) due to the rf Hall motion of the drifting carriers. The introduction of hm(t) causes the variation of 60 in phase with the wave propagation. The field hs(t) is introduced to provide for the wave amplification. Because it is the drifting carriers which drive the amplification in the MSSWA, a distinction between this device and the Traveling Wave Tube amplifier should be made. In the latter case the modulation of 32 m t VARIATION OF TORQUE ( ) ——fi- DUE TO hS AND hm. Figure 14. Interaction of a drifting electron in GaAs with the pre- cessing+M vector in the ferrite. Note that u0 >> Vyl (i.e., 00 not to scale). 33 carrier motion is collinear with the overall dc drift motion. This gives rise to the carrier bunching or space charge wave from which signal am— plification is obtained [44]. In the MSSWA the modulation is transverse to the dc carrier motion. Also of key importance is the fact that the space charge wave is weakly damped and inertia dominated in the TWT. The MSSWA, in contrast exhibits collision dominated waves whose normal modes are highly damped [45]. In this chapter the basic ferrimagnetic resonance interaction has been described. An illustration of carrier interaction with magneto- static waves has also been discussed in order to provide an overview of the mechanisms of wave amplification. This gives the reader a foundation for the study of different dispersion relationships which describe the amplifier geometries of interest in this report. 3. ANALYSIS OF THE MSSWA 3.1. Relevant Properties of Magnetostatic Surface Waves In this section the properties of magnetostatic surface waves are addressed in more detail. The literature on this topic is extensive and some controversy does exist [46, 47]. Here, we wish to present a detailed study of the effects of certain parameters such as slab thick- ness and metal plate spacing on the character of the dispersion rela- tions. The first case we examine is that of an unbounded ferrite slab of thickness d. The surface or (Damon-Esbach) mode has the property that wave energy is concentrated near one face of the ferrite as was stated in Chapter 2. The dispersion relation is affected by the thickness of the slab. It has been shown theoretically that the group velocity of the lowest order surface mode is dependent only on this parameter [48]. This effect tends to increase the velocity as d increases as shown in Figure 1(a). In order to maintain the velocity of the wave at reason- able values for amplification purposes, we conclude that a thinner slab is best. Recall that the drift velocity of carriers in GaAs reaches a practical maximum of 2 x 107 cm/s. A range of values for k in a ferrite of thickness d = 10pm is given in Figure 1(b). The second case involves a metal plate adjacent to one face of the ferrite slab. This is shown in Figure 2. The wave on the top face is the Damon-Esbach wave treated above. This wave is also referred to as a Ferrite-Air (FA) mode. The wave on the lower face is called the Ferrite-Metal (FM) mode or Seshadri wave [49]. Notice that the FM mode has a larger passband. It has also been speculated that this mode is more lossy than the FA mode [50—52]. The resonance for the FM mode 34 35 A (AC ._ _ —— T? d —-///////1A///// d2 > d1 > k (a) A HGHZ) A0 = 450 Oe d = 10 um AB | ‘2‘? = 3.8 __ __ 1 £2 d :11 = 2.9 _ - ® 2w ii Figure l. (a) Variation of dispersion curve for an unbound ferrite slab as a function of slab thickness. (b) a typical dispersion case. 36 (DC .__—————_———___— METAL Figure 2. Dispersion curves for the FA and FM modes. 37 (1) occurs at m0 + AM whereas the FA mode resonance is observed at m0 + -§—. These resonances are approached more quickly as d+m. Therefore, in order to support a propagating mode with a finite passband a slab of finite thickness is required. Now we consider the ferrite slab separated by a finite distance from the metal ground plane. The surface wave on the face opposite to the ground plane is just the FA mode of before. This wave is only slightly perturbed by the metal. We examine this more later. The wave on the bottom face is a hybrid mode referred to as a Ferrite-Air-Metal (FAM) mode. The spacing of the ground plane can create some interesting changes in the FAM dispersion curve. For spacings small compared to a wavelength the mode is basically the FM type. This wave is transformed into a FA mode as the ferrite to metal gap becomes very large. It is for intermediate values, such as Ad = 1pm, that the dispersion shows an interesting change. A region in the dispersion occurs where the group velocity goes to zero. The forward waves then become backward waves with negative group velocity as k is increased. This is shown in Figure 3. If the gap between ferrite and metal is held constant as slab thick- ness is varied, the effects on both FA and FAM modes can be assessed. In the FA case, the variation in thickness gives rise to the same modi- fications in the dispersion as when the metal plate is absent. The FAM mode shows a more significant change over the same range of thickness values. The variation in d creates a dispersion curve which has a zero group velocity regime similar to the case treated above for variations in the ferrite-metal gap. Note Figure 4. In the limit as d+w a complete backward wave results as shown in Figure 5 for the frequency range (AA, wB) [53]. This result is not conclusive however; some authors have 38 + 0 L T$ YIG U + _L_ f :11 A. V ® + H 0 ///7////7///WF7///// METAL l —————————————HIDC _________ (OB ___________ (”A -kd< ,e If : >kd 1.5 1.0 .5 Figure 3. Variation of the ferrite dispersion curve due to metal-air- ferrite spacing; (a) A = 01p, (b) A = 1.0p and (c) A = 7.0p. After Yukawa et al. 39 d YIG [\ <6 1 w 9 -k p ¢ 0 METAL f ._____.__.__—t.———_____ d=50p 'k]-< { i i 3’ (cm ) 1000 100 1000 (cm H = ZOO Oe o 9:25p Figure 4. Variation of the dispersion curve due to a change in slab thickness in the presence of a metal plate. After W.L. Bongianni. 4o é YIG } T fi° = 50 0e C n _L . 49 U -k ® ///////////////> METAL -kd< > kd -k B e.g. the upper cutoff occurs for -k because the metal is closer to the lower face of the ferrite slab [56]. This is especially evident in the case of Figure 8 where B=0 and A is finite. A word should be said about the nonreciprocal nature of wave prop- agation in ferrite slabs. In the case of an ungrounded slab this shows up as one face being preferred for propagation from input to output based on the k x H0 relationship. Notice however, that the waves which propagate in the +k and -k directions have the same properties i.e. both are FA modes. In this sense the propagation might be thought of as re- ciprocal. If a metal plate is introduced, then one face will support an FA mode while the other side supports the FM mode. Recall that these modes are fundamentally different, e.g. the passband of the FM mode being larger. For a given transducer configuration and orientation of the static field Do, the propagation of the FA and FM modes is determined + -> using the k x H0 relationship. If the input ports are switched the 42 <0..- $ YIG- § © 4 1i i 1] E o 9 //////////////f///> METAL —_—_—_——_——“——_ =- k Figure 6. Summary of the dispersion curve variation for one metal plate; (a) g+0 (FM), (b) finite 9 (FAM), (c) g+w(FA), (d) small d and (e) large d. 43 METAL L/l/l/ILAU/ll/l/lll/ll 3 Ad [1 AIR <{Zl d 1:3:1\7:f::§?AIR Gig) It If/hU/lf/7/lfI/fl/II/f METAL 'kd YIG j; d .1. AIR 9 \\\\\\\\\VVV\ GaAs (C) GATE LLZ/////A//// d. AIR 1 {mt v : g MYLAR \\\\\\\\\\\\\V\ GaAs (e) Figure 11. GATE. YIG GaAs d YIG \\\\\\\\\\ GaAs (d) Illustration of all device model geometries. (a) YDS, (b) SINGLE-SURFACE, (C) YIGSLAB-GAP, (d) YIGSLAB, and (e) 51 3.3. Derivation of Ferrite Transverse Wavenumber Basic to the operation of the MSSWA is the rate of decay for the magnetostatic wave energy in the plane transverse to the direction of propagation. In the analysis developed here we are concerned with the variation of the field strength in the x-direction. This is assumed to have a general form of eYX where Y is the transverse wavenumber. The splane wave propagation variation is also used so that in general the field components have the form E, H a e'j(kz'wt)er (3.3-1) Note that Re(v) can be positive or negative depending on boundary con- ditions. The wavenumbers v and k are assumed to be complex in general. In this section the transverse wavenumber for the TE mode of mag- netostatic waves in the YIG is derived. This is denoted as Yf. We be- gin with Maxwell's curl equations -> . <—> + V X E = -pr0p°H (303'2) -> . -> V X H - JweocrfE. (3.3-3) The latter equation is used in the form shown because the YIG is assumed to have zero conduction current i.e. o=0. By substituting in the tensor form for u we obtain for a coordinate system in Figure 12 u 0 -jSK HX VXE =-dwu0 O 1 O H (3.3-4) JSK 0 u HZ where S = i 1 determines the orientation of HO along +y or -y. The above 3 expression upon expansion becomes with-gy— = 0 jkEy = -jwpn(pHX-jSKHZ) (3.3-5) -jkEX —YEZ = -jw110Hy (3.3-6) YEy = -jwu0(jSKHX + uHZ) (3.3-7) 52 —> HO (5 = +1) ‘//, H0 (3 = 4) IG k. x = 0 GaAs + X U 0 4w.— V Figure 12. The coordinate system for derivation of transverse wave- numbers Yf and VS. Note that 3/3y+0. 53 The expansion of the second curl equation yields with —%y— = 0. ijy = jwefEX (3.3-8) -ijX—YHZ = jweny (3.3-9) YHy = jwesz (3.3-10) Notice that the TE and TM modes are uncoupled for this case. This is observed because three of the equations relate to H2’ HX and E only, Y while the other equations relate to E2, EX and Hy. 0f further importance is the fact that the TM mode is not directly coupled to the spin para- meters u and K of the ferrite. This indicates that the TM mode of the magnetostatic wave does not interact directly with the spin system. Note also that due to the small signal assumption the component Hy is much smaller than the dc field HO and therefore has negligible effect. For these reasons we neglect the TM field components. Manipulation of the equations involving the TE components yields. . 2 2 2 - J(k -k0 p)Hx + (kY-kO SK)HZ - 0 (3.3-11) j(-Yk-k 2SK)H + (-Y2-k 2 )H = 0 (3 3-12) 0 x o u z ' which in determinantal form is . 2 2 2 3(k -k0 p) kY-kO SK = 0, (3.3-13) . 2 2 2 -J(Yk+ko SK) -(v +kO u) The solution of this determinant yields the transverse wavenumber which is: k 2 O (Kz-uz) (3.3-14) .< l _< _h I 7? + where k02 = wzpoef. For YIG, cf a 11. The expression obtained above for Yf agrees with that obtained by Bini et al. [59]. For reference purposes, if a similar analysis is carried through 54 using the TM component equations the following results: 2( 2 2 if TM) = k -k0 . (3.3-15) This shows us explicity that the transverse variation does not involve the spin system (note the absence of K and p.) Therefore, we see that the YIG appears as a simple dielectric for fields of TM mode configura- tion. 55 3.4. Derivation of Semiconductor Transverse Wavenumber In this section, vs, the transverse wavenumber for the TE mode in the semiconductor is derived. The electrons drift with a constant speed Lb in the +2 direction. Note again Figure 12. Starting with Maxwell's curl equations again, the manipulation proceeds as follows. vaxE = -jwu0VxH (3.4-1) which becomes 2+ v(v-E) - VZE = -jwp 3 + k E (3.4-2) 0 S where ks2 = wzpoes in the semiconductor and for GaAs the relative permitti- vity is Ers = 12. Continuing we have 2+ 2 V(V-E) - V E - ks E = -jwp03 (3.4-3) and if we write the electric field components, E = Exlx + Eyly + Ezlz' (3.4-4) Then we obtain v(v.E) = Y(YExl-jkEzl)x - jk(yEx1-jkEzl)2 (3.4-5) and 2* _ “ 2 “ 2 “ 2 _ 2 2 + V E _ XV EXI + yv Eyl + ZV E21 - (Y -k )E (3.4-6) so that the left hand side of (3.2-3) becomes .e. . " 22++ Y(YEX1-JkEzl)x -Jk(YEx1-JkEzl)Z + (k -Y )E-kSE. (3.4-7) If we note that charge density and velocity possess both dc and rf terms we obtain for the right side of (3.4-3) + A u _ o + -quOJ - -quo(oov1 + OIUOZ) (3 4-8) where the second order rf, and dc terms are ignored. Now using Poissons' equation we note that p1 = cS(V°E) = eS(YEX1-jkEzl) (3.4-9) so that (3.4-8) becomes 'quoJ = -3wuooov1-jwuoesu0(YEX1-jkEzl)z (3.4-10) Note that we can write the first term in the above expression as 56 -jprpOVl = -jpr(-|e|no)V1 (3.4-11) 0r 2 -jw 3 :' ks lelno V (3 4'12) “opo 1 3 w as 1' ° Therefore equation (3.4-3) becomes . k [eln 2+ . ks . A - o S ) 0 + 4- w This expression is written in final form by expressing V1 in terms of the electric field of the TE mode. This comes from the phenomelogical equation of motion as follows jwV + V-VV = -nE-n(V x B) -vV - VTZ -§9— (3.4-14) o where n = |e|/m* and v is the collision frequency. The term vT is the thermal velocity. Then we have a A A A _- -> + 3 V°VV -[vx1 ——— + (u0+vzl) 32 ] vX1 x + Vyly + (u0+vzl)z (3.4-15) 8X _—_-jkuovl (3.4-16) where V1 = V - u02. Thus we have for (3.4-14) by rearranging terms, 2 . + _ ++ VT [v + j(w-kuo)] v1 --nE - n(va) + TETfi; A (3.4-17) with A = [Yx-jkz]eS(YEX1-jkEzl). (3.4—18) We now work on the V x 8 term in Equation (3.4-17). The velocity and field terms can be written as A + v = uoz + v1 (3.4-19) B = SBOy + B1 (3.4-20) 57 for the case we are studying. This gives for the cross-product term in matrix form $2 9 2 v x B = vx1 vyl vzl (3.4-21) _Bx1 SBo+By1 B21_ Because we are considering only the TE case, only the terms involving Eyl in Equation (3.4-13) are retained. This gives us 2 A k |e|n . (kZ-yz-kSZ)Ey1y = j(—%—) (: 6 0 )vyly (3.4-22) S and thus only the y component of 31 is required from Equation (3.4-14). Which gives 2 A A + + V [V + JDIVyly = -nEy1y -n(v x B) I; + h T A|y (3.4-23) The third term on the right-hand side does not possess a y component and therefore will not contribute. The (V x 8) term becomes (V x s) (9 = yuoaxl (3.4-24) to give [v + JDIVYI = -nE -nuOBx1 (3.4-25) yl where D =w -kuo. Note that the term in brackets is sometimes referred to as 5. Therefore, in the semiconductor for the TE mode case we have . 2 _ JkS (lelnol) -nEy-nu0BX (3.4—26) E s (kZ-Yz-k52)E - y u) \) where the subscripts on field components have been dropped. Recall that the Ey and Bx terms represent small signal rf components. These are again related using Maxwell's equation as k 8x = - a Ey (3.4-27) so that (3.4-26) becomes 58 - 2 2 2 -k52 lelno k ()(k -Y —kS )Ey - J(—w-) Es [-nEy +r1uO I} Ey] (3.4-28) I where recall that n = 1%£. This equation is simplified to give the following , 2 k (I) .Y2 4,524.18 = -J-(S—o-) (3.4-29) (L) CIIEI 'k where w2 = n0|e[2/esm which is the square of the plasma frequency. P . 2 . Solv1ng for Y yields kw .-_ 2 = 2_ 2 2 .50 v-ww YSEMI ‘ Ys - k 'ks + J( m ) [v2 + a2] (3.4-30) This reduces to the final form of 2 2 2 k 2 (3431) ; ° S - .- TOF tYpical values of the variables involved. The term we = wpz/v is the dielectric relaxation frequency. The final Equation (3.4-31) is the same expression as stated by Bini et. al. [60] and used by Awai et. al. [61]. It is interesting to note that Awai obtained his expression from Vural [62], or the text by Vural and Steele [63]. This is important because the expressions given by Vural are for helicon waves which are TEM and propagate in the di- rection of the dc magnetic field (i.e. k is along 8 which is given by 302.) Therefore it appears that YSZ is the same for our TE case and the TEM case of Vural and Steele. This punctuates a general problem in this area of research i.e. the use of previously derived expressions without a thorough check of their validity and applicability to the investigator's special case. With the errors as mentioned in Appendix E, sign problems for instance, the situation can become confusing at best. The fact that our YSZ expression is the same as the TEM helicon case is somewhat coincidental. If the orientation of H0 is changed 59 this no longer holds true. It should also be noted that the presence or absence of the dc static field has an effect on the TM modes [64]. 60 3.5. Derivation of YIG-Dielectric-Semiconductor (YDS) Dispersion Relations In the previous sections the transverse wavenumbers for both ferrite and semiconductor were derived. We must now proceed to the development of the dispersion relations which are used to model the device geometry and propagation characteristics. This section will deal with the YDS model and the limiting case referred to as SINGLE-SURFACE, This latter case occurs in the limit as the dielectric thickness is reduced to zero. The ferrite and semiconductor regions are represented by semi-infinite half spaces. The geometries are as shown in Figure 13. In the analysis of TE waves it is possible to begin with either HZ or Ey as the assumed field generating component without a loss of generality. As long as Maxwell's equations are satisfied, the solutions obtained are acceptable. In the first two regions we have Region (1) st Ey = Ale (3.5-1) jv Y X _ _s S _ HZ — mu Ale (3.5 2) 0 Region (2) E = B ekx + B e‘kx (3 5-3) y 1 2 ' _ Jk kx -kx HZ - E Ble - Bze (3.5-4) where Equations (3.5-2) and (3.5-4) are obtained from Maxwell's v x E expression. The fields in Region (3) require more detail. In the ferrite recall that V X E = -jpr]-I°H (3.3-5) which yields 61 A YIG (3) 9 /////////// DIELECTRIC (2) (AIR) 0/////////////;z GaAs (1) (a) x /\ YIG (3) GaAs (1) (b) (a) YDS and (b) SINGLE-SURFACE geometries for development Figure 13. of dispersion relations. 62 -3E -—g% = —jwp0[uHx-jSKHZ] (3-5'6) BE 4:... . .- 32 quOLJSKHX+uHZI (3.5 7) ..). Whereas the equation V x H = jweoerfE Yields BHX 3H .___ -.__z = - - 32 3x Jweoerny (3.5 8) If the Expressions (3.5-6) and (3.5-7) are used to relate Ey and HZ we obtain 3E _ __y= _. 2_ 2 SKkEy—u ax jwu0(K u )Hz (3.5-9} which is derived by multiplying (3.5-6) by jSK and (3.5-7) by u. The above results when you subtract the two expressions. If we were to use (3.5-6), (3.5-7) and (3.5-8) the result which relates Ey and H2 appears different. We begin by differentiating (3.5-6) with respect to z. -92Ey _ aHx . aHZ _;;?_ - -quo u .3; - 35K 82 (3.5-10) If aHx/az is substituted for in (3.5-10) using (3.5-8) the following is obtained 2 2 . 3HZ 3Hz k Ey = (1) HOEOEY‘TUE)’ - quou “'8‘; ‘ (DUOSK _32- (3.5-11) The two expressions (3.5-9) and (3.5-11) seem to relate E and H2 in y in different ways. It will be shown later that both lead to identical results for the dispersion equations. The interim equations however do not seem to be equivalent. If we begin with (3.5-9) we obtain for the fields -fo Ey = Cle (3.5-12) SKk-py ‘Y X HZ = ; Cle f (3.5-13) qu (K -u ) 63 where Ey is again the assumed generating component. In order to derive the dispersion relation the boundary conditions must be satisfied at all interfaces. We equate the tangential components because they must be continuous at the interface in our case. Therefore we match (3.5-1) and (3.5-2) with (3.5-3) and (3.5-4) at x = 0. Equations (3.5-3) and (3.5-4) are matched with (3.5-12) and (3.5-13) at x = d. For brevity we do not match Hxl,for this condition will be satisfied if the previous boundary conditions (i.e. on Ey and H2) are satisfied. Also we approxi- mate Yf by k which simplifies the algebra but produces negligible error. The following equations result A1 - 31 - 32 = 0 (3.5-14) YSAl - kBl + kBZ = 0 (3.5-15) Blekg + 32e'kg - Cle‘kg = 0 (3.5-16) (KZ-p2)B1ekg - (KZ-p2)32e‘k9 + (SK-u)C1e"k9 = 0 (3.5-17) 2 If we let the term (K - p2) = D and express the system of equations as a matrix, the following results 1' " '1 1 -l -1 O 1 A1 Y - S k k 0 B1 - 0 0 ekg e-kg -e-kg B2 L. 0 Dekg -D€-kg (SK-“)9-kg ., _ C1 .1 (3.5-18) By setting the determinant equal to zero we obtain the following upon expansion e-Zkg . (rs+k) [SK-n+0] (VS-k) [SK-u-D] (3.5-19) In the limit as g + 0 we obtain (KZ-p2)YS + (SKk-kp) = 0 (3.5-20) 64 which is the dispersion relation for the single surface limit. At this point we demonstrate the apparent differences one obtains when choosing equations (3.5-11) versus (3.5-9). Let us consider the single surface case outright. Note again that the general field varia- tion is e'3(kz'wt)eYx for the problem under consideration. In Region (1) we assume YSX E = Be (3.5-21) y which gives JYS = -—— E (3.5-22) Z (1)110 y where the real part of Ys is greater than zero for x < 0 in order to satisfy boundary conditions. In Region (2) we have -fo Ey = Ae (3.5-23) which gives when equation (3.5-9) is used (SKk‘UYf) 'fo H = Ae (3.5-24) z . 2 2 qu0(K -u ) and for reference H = Ae (3.5-25) x 2 2 wuo(K -u ) If we match the tangential components at the boundary the following dispersion relation is obtained YS(K2-u2) + (SKk-urf) = 0 (3.5-26) which is just the result previously found in Equation (3.5-20) where yf was approximated as k. The dispersion relation can be determined in another manner as well. If we assume that HZ is the generating component. This gives 65 st HZ = ale (3.5-27) and -jwu Y X _ o s ‘ Ey - Y ale (3.5 28) s and in Region (2) ‘fo HZ = bze (3.5-29) which gives us Ey and HX from Equation (3.5-11) as -qu (uv +kSK) "Y X E = ————9———f—————— b e f (3.5-30) Y 2 2 2 (k0 u-k ) and . 2 j(kO SK+Yfk) -fo HX - 2 2 2e (3.5—31) (kO u-k ) where again k02 = wzuoeoerf' The tangential components are again matched at x = 0 to yield (kozu-kz) - ys(uvf+SKk) = 0 (3.5-32) At first glance, Equations (3.5-26) and (3.5-32) do not look equiva- lent. If we equate them, however, an identity results. This is most easily shown by solving both equations for Y5 and equating the resulting expressions. This gives 2 2 k0 U'k = UYf'kSK (3.5-33) pr+kSK K2_u2 OY‘ k02p(K2-p2) = pZYfz-pzk2 (3.5-34) k 2 k02u(K2-u2) = usz2+—%—(K2-u2)]-u2k2 (3-5-35) which yields after simplification 2 2 2 _ 2 2 2 k0 11(K '11) ' k0 “(K '11 ) Q.E.D. (3-5‘36) 66 Therefore we see that although the forms of the dispersion relation can appear to be different, they are actually identical results. This is especially important when comparing the various forms of the dispersion found in the literature. 67 3.6. Derivation of YIGSLAB and YIGSLAB-GAP Dispersion Relations In the last section the development of YDS and the limiting case SINGLE-SURFACE were covered. In this section the dispersion relations for the YIGSLAB-GAP and its limiting case YIGSLAB are derived. The ge- ometries are shown again in Figure 14. Listing the fields in all regions we obtain: Region (1) YSX Ey = “1‘e ‘ (3.6-1) -Y Y x _ J S S - HZ — (1)11 Ale (3.6 2) Region (2) y 1 2 . H = ‘ik— (B ekx - B e_kx) (3 6-4) 2 (11110 1 2 . Region (3) fo -fo Ey = C1e + (329 (3.6-5) — 'jk kX —kx HZ — wuoD [Cle (SK + u) + Cze (Sk-u)](3.6-6) where equation (3.5-9) was used to obtain Hz, and Yf 5 k. Region (4) Ey = Dle (3.6—7) D e (3.6-8) The field components are evaluated at x=0, x=g and x=g+d. Match- ing these components at the boundaries and solving as shown in previous sections yields the following dispersion equation 68 llx AIR (4) 9+d Y () g m 3 % MYLAR (2) 0//////////////>2 GaAs (l) (a) Xll AIR (4) d YIG (3) \‘ z /////////////’ GaAs (l) (b) Figure 14. (a) YIGSLAB-GAP and (b) YIGSLAB geometries for development of dispersion relations. 69 Y +k [e'Zkg - Y:_k ] [(5K+u)(sn<-u-o)e'2kd + (u-Sk)(SK+u-D)] Y +k -D [e'Zkg + 7—ST] [(SK-u-D)e'2kd-(3K+u-D)]= o (3.6-9) 5 Equation (3.6-9) is the dispersion relation for the YIGSLAB-GAP model. The limiting case for 9+0 is best obtained by returning to the system determinant and setting e -2kg = 1. A simplified version of this determinant is -2kg Ys+k -1 _e-2k9 e - Y -k s Y +k -2kg S ] _ _ -2k9 = 0 D[e + F (SK'HJ) (U SK)€ o (SK+u-D) (Sk-u-o)e'2k(9+d) (3°6‘10) The solution of the determinant is e-de : (SK-u-D) [D vs+k(u+SK)] Recall that D = (K2 -u2). We now show an alternative method for developing eqn. (3.6-11). This involves using the standard decomposition of Maxwell's equations in free space, H = ___l____ -k _EE£ _ ~ E 8E2 (3.6-12) x k2_k 2 J ax J“ 0 By J a _ 1 . 3H2 . aEZ Hy - _;§:;_§_ [.Jk.7i7.+-Jueo ax J (3.6-13) a aE 3H EX = k2 k12 [Jk 3X2 +jmu0 2] (3.6-14) 70 3E 8H =_1_ - _2-_~ _2 _ Ey 2 2 [ Jk 8y qu0 3X ] (3.6 15) where ka2 = “250“0 for air. For the YIGSLAB case i.e. 9+0, we find the fields in all regions as before. Starting with region (4), the air re- gion we have H = b e-Y4X (3.6-16) and this gives . _ 4 —Y x X - m2- b4e 4 (3-6'17) o _ where Re (v4) > O is assumed. The electric field component is then given by jwu Y _ E = ——‘fl b e Y4X (3.6-18) y 2 2 4 k -ka In order to determine Y4, Maxwell's curl equations are again used to yield JkEy = -jquHX (3.6-19) _Y4Ey = _jwu0HZ (3.6-20) and -ijX + Y4HZ = jweoEy (3.6—21) Setting the resulting determinant to zero yields 2 2 2 Y4 = k -ka (3.6-22) and since we are in the slow wave regime, k2 >> k 2, so that a (3.6-23) and thus for decay of the field intensity as x+m we choose the positive root i.e. Y4 = k. This gives for the fields in Region (4) _ -kx HZ — b4e (3.6—24) 7l _ . -kx HX — -Jb4e (3.6-25) qu _ o -kx _ Ey - k b4e (3.6 26) In Region (3) if H2 is the assumed generating component again Y x FY x _ f f HZ - a3e + b3e (3.6-27) If the Equation (3.5-11) is used as in the YDS section we find that jwuo fo -fo Ey = —2——2- [(mf - kSK)a3e -(uYf+kSK)b3e ] (3.6-28) k p-k o and a1$O (k 23K- fk) (k 2SK+ k) Y +Y H = a3efx [j( 2)3]+b H’Yf[(: 1‘ ](3.6-29) X 2"k2) k0 u In Region (1), the GaAs semiconductor, we have st HZ = ble (3.6-30) =J_k st HX Y5 b1e (3.6-31) and ~qu Y X = o S _ y Y5 ble (3.6 32) and are given by 2 k 2 2 yfz= five—(Kw) or 2 2 w -(w +w) ny = k2 + k02 2 m o 2 (3.6-33) w +w0wm-w and k YSZ = k2 - kSZ + j + (x1+1>1 [(A2_1) _ e2kdi(A2+1)] - 2e2kdie3kds [(A3+2) _ A3e‘2k5] [e-2k9(A2+1)+(A2-1)] [(A1+1) - edej(A1-1)] + e2k(d1.+d)ekdS (l-edes) [e'2k9(A1-1)+(A2+1)] [(x3+2)(A2-1) + edeiA3(1+A2)] {[(A3+2)-A3e-2k6] _ 2ekdS -2(ekd5e’2k6) }_ e2kd1ekdS (1-e2kds) [e'2k9(x2+1) + (AZ-1)] [(A1+1> (x3+2) + e2kdlx3 1.75 no slow wave was observed. This ties in with the previous discussion because the YIG again appears to be infinite. Notice in Figure 1 that as d is changed, the bandwidth over which a slow, magnetostatic wave is present also varies. The peak value is 10. Figure 1. OOOOOOOOOOOOOOOOOOOO r—amwbmmwooto 00...... 79 _1 H0 = 500 Oe ‘A lk1| (cm ) AH = .5 0e . u = 1 x 107cm/s ° 16 -3 N = 1 x 10 cm " O 1 = 2.5p l i f i .2 l i % lEh- 29 3.0 3.] 3.2 3.3 3.4 3.5 3.6 3.7 3.8 frequency (GHz) |k1| vs. frequency with YIG thickness as parameter for the YIGSLAB case. 80 = 1 x 1016cm' 500 0e 0 A |ki| (cm-1) AH = .5 0e 7 1 x 10 cm/s 3 I: ll t—‘P—ll—‘r-‘h-‘t—‘r—‘b—‘b—‘N Hmwhmmwoouoo 0000000000 0 C 0 II l—‘ O Hmwbmmwoouo 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 frequency (GHz) Figure 2. k1 vs. frequency with YIG thickness as parameter for YIGSLAB-GAP case. 8l at d = 1.5u as is shown. In all subsequent studies this value of YIG thickness is used. If a gap is present (e.g., g = 1.0“) we notice a small increase in loss as shown in Figure 2. It was also observed that for practical values of n0 an increase in the gap thickness does not significantly affect the k, value. The increase observed is based on the strength of interaction between the drifting carriers and the magnetostatic wave. For this reason the gap is set to zero in remaining cases. A word about the values of d and 9 that were selected. These values (d 1.5u and g = 0.0u) were picked because they provide the best com- promise for possible gain, bandwidth and accuracy of study. The concern for accuracy is based on the thin slab approximation where kd i 1.75. If d = 1.5u e.g., then the maximum value of kd obtained is 1.126 when k = 7500 at 3.7 gigahertz. Therefore, it is possible that values of d greater than 1.5u could be used. The problem is that as d is increased the bandwidth decreases because kr values are decreasing. This is evi- denced by the fact that at 1.5u the band of possible roots extends from 2.0 GHz to 3.7 GHz, whereas at 4.0u the band is from 3.3 to 3.7 GHz. The effect on k, is the major concern when 9 is varied. Once g is greater than 1.0u however, the values of k, are virtually unchanged. Therefore, a gap can be present for the purposes of signal excitation and detection but the gain will be improved if g = 0.0u. The variation of ki as a function of frequency with u0 as parameter is shown in Figure 3 for the lossless case. Notice that as U0 is in- creased ki increases as well. This also agrees with the work stated by Awai et al. The introduction of losses (i.e. AH # 0) produces a sig- nificant reduction however, when the carriers are drifted at the maximum 36 for GaAs of 2 x 107cm/s. This is shown in Figure 4. Since YIG films 82 -1 k. cm 10.0‘- l( ) 9.0.. 4 x 107 8.0-- 7.0.- 6.01- 3 x 107 5.0«- 4.0.. 2 x 107cm/s 3.0 - G 2.0 « i 1.0 . 1 4. i : i i i i‘ i 5- 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 frequency (GHz) Figure 3. k1 vs. frequency with u0 as parameter for the lossless YIGSLAB case. Figure 4. 83 = 1.5u H0 = 500 0e _1 AH = 0.2 0e A k,(cm ) N0 =1x1016 7 4 x 10 cm/s 7 . 3 x 10 cm/s *T"§:5”;:I”:T2 3:3 3:4 315 1 L r ' . I I 317 f(GHz) 2 x 107cm/s .O‘w k1 vs. frequency with u0 as parameter for the lossy YIGSLAB case. 84 d = 1.5u A ki(cm'1) nO = 1 x 1016cm.3 AH = 0.0 Ca 5.0 ‘” ”o = 1 x 107cm/s 4.0 " 3.0 *' 2.0 - 1.0 ‘- 3.0 3.5 -1.0 I. f(GHz) Figure 5. k1 vs. frequency with HO as parameter for the YIGSLAB case. 85 of highest quality may still possess AH values in the tenths of Oersteds, a geometry which provides for more gain at practical drift velocities is called for. The variation of k1 due to changes in 00, the static field is shown in Figure 5. Notice that as 00 is increased, the value of ki decreases slightly. The passband for wave propagation in the ferrite is shifted upward as RC is increased. It is important to note that for possible operation above 5.0 GHz the static field must be greater than 1000 0e, a fairly large field. 4.4. Evaluation of the GENERAL CASE The introduction of metal plates serves three purposes: (1) to tailor the magnetostatic dispersion relation for the improvement of gain, (2) to provide a close to uniform carrier density in the GaAs re- gion by counteracting the dc Hall-effect created by the static field 00, and (3) to provide a means for reducing the heat generated by the drift- ing carriers. The geometry for the optimized YIGSLAB case is used with the excep- tion that the GaAs region has a thickness dS = 1.0p. This value is used because we want the heat generated to be reasonable. It is also a prac- tical thickness from a fabrication standpoint. The variation of k1 versus frequency with d1 as parameter is shown in Figure 6. This graph shows that gain is improved if the metal plate is closer to the YIG region. This agrees with work as stated by Vashkovskiy et al. [71]. The lower metal plate is also close to the GaAs region. Studies show that if b is greater than 0.2 the root disap- pears. This result is especially significant because none of the litera- ture shows this case (thin semi region with metal plate) as having been examined in detail. 86 1.5u 1.0u 0.2u 500 Oe 0.2 0e 1 x 10 cm- 1 x 107cm/s 0.0u d d = b H 16 3 20.0 .. 10.0 l I l I 1 r 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 frequency (GHz) Figure 6. k1 vs. frequency with di as parameter for the GENERAL case. ’7 87 -1 A kj(Cm ) d = 1.511 dS = 1.0u 40 o -- _ ' b = .2 ,.AH=0 o O “ “ d1 = 1.0u 0 U0 = 1 x 107cm/s 0‘3" HO = 500 Oe 0'5 n0 = 1 x 1016cm-3 30.0 g z 0.0“ 20.0 —» i1- 4 10.0 --- ; : : 4* : 1 1 : 3» 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 frequency (GHz) Figure 7. k1 vs. frequency with AH as parameter for the GENERAL case. 88 = 1.5 S = 1.0 A ki (cm-1) H0 = 500 0e .1 AH = .5 0e d1 = 0.00 60.0 -- E = 0.211 n0 = 1 x 1016cm'3 50.0 "' g = 0.011 40.0 -- 30.0~~ 1 x 107cm/s 20.0-. 10.0-r L >- 3.7 40.04 3 x 107cm/s -20.0-. f(GHz) -30.0-v Figure 8. k1 vs. frequency with u as parameter for the GENERAL case. 0 89 If one examines the effect of losses we see a significant change from the YIGSLAB case. With values of AH as great as 0.5 0e the k1 is not significantly decreased over most of the frequency band from 3.2 to 3.7 GHz. This is shown in Figure 7. We believe the reason for this lies in the modification of the dispersion curve based on the presence of the two metal plates. In Figure 8, ki as a function of frequency with 00 as parameter is shown. Notice that in the GENERAL case, positive k, values now occur if 00 is less than 2 x 107cm/s. This contrasts with the YIGSLAB case. If 00 is increased from 1 x 107cm/s to 2 x 107cm/s, there is now a signi- ficant decrease in the ki values (and hence the gain). This would tend to support the argument that the carriers and wave must be near synchro- nism. If the carriers are moving much faster than the wave, the gain is decreased and eventually lost. Further illustration of this is shown in Figure 9 where k1 versus uO is shown. In Figure 10, the effect of the static field variation on the GENERAL case is shown. Notice that in this case a peak in the k1 values is ob- seved for H0 = 750 De. The GATE geometry was to be studied as a special case of the GENERAL configuration. This was to be achieved by allowing the GaAs region to be as large as possible for the thin slab approximation. It was dis- covered though that the lower plate plays a crucial part in obtaining net possible gain and therefore must be close. From a practical stand- point as well, the GaAs region should be as thin as possible to reduce heating loss. For these reasons the GATE geometry was not analyzed separately. Finally, one must test the observed root to see if it gives rise to a convective instability using the Bers-Briggs criteria. This is 90 ‘h N 3.0 GHz 500 0e .5 0e 1 x 10 0.0u 0.2p 1.50 1.00 0.00 I D I ll 16 LQQQU'IQZ u u (x 106cm/s) - o A kr(cm1) 1500.. 1200‘" 1100 4 i i i i i i i i i i i .. :=> 2 3 4 5 6 7 8 9 10 20 3O 4O u0(x 106cm/s) FIgure 9. k. and kr versus uO for GENERAL case. 1 33% ;.. _ 9l = 1.50 S = 1.00 1 n0 = 1 x 1016cm‘3 ’1‘ k,- (cm ) AH = 0.2 0e d1 = 0.00 60.0 _ B = 0.20 00 = 1 x 107cm/s ‘_ g = 0.00 50.0 . l 750 09 40.0 L HO = 500 Oe 30.0 i d 1000 De 20.0 ' 10.0 “ 1 a i : i i : +%3- 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 frequency (GHz) Figure 10. k. vs. frequency with H0 as parameter for the GENERAL C 58. 92 1 + f = 3.3 GHz 30 a. 20 .- 10 o d- 'P ‘l' r fih -IOO « -200 u “300 d» -400 "' -500 r Figure 11. . vs. log [w.| under the Bers and Briggs Criteria for k 1 GENERAL case. 93 shown in Figure 11. As indicated, the root is in fact “growing" since it experiences a change in sign. Therefore, energy is being convected in the +2 direction. This chapter has provided an evaluation of relevant dispersion re- sults for the MSSWA. The criteria for the selection, evaluation and testing of possible growing roots is covered. This includes the Bers- Briggs criteria for convective instabilities. The geometry which will provide for optimal gain and bandwidth is presented. Trends which show the effect of certain parameters on gain and bandwidth behavior are also discussed. 5. SUMMARY AND CONCLUSION 5.1. Summary The interaction of slow magnetostatic waves with drifting carriers in a layered ferrite-semiconductor structure has been studied in detail. The variation of the YIG-GaAs dispersion relation for different geome- tries has been modelled. The dispersion relations have been derived without the magnetostatic approximation. Using approximations for trans- verse wavenumber in the ferrite and semiconductor as well as an exponen- tial approximation the dispersion polynomials were derived. Numerical evaluation of these polynomials was performed using a Fortran subroutine. The solutions of the dispersion polynomials were optimized with respect to certain device parameters. Possible growing roots were tested under the Bers and Briggs criteria for convective instabilities. Gain is pre- dicted theoretically for the MSSWA. 5.2. Conclusion The dispersion relations for geometries from a simple SINGLE-SURFACE to a more complex GENERAL layered structure have been theoretically ana- lyzed. The results show that the YDS and SINGLE-SURFACE geometries will not support a slow magnetostatic wave. In the case of YIGSLAB and YIGSLAB-GAP, the following conclusions were drawn: (1) The YIG region must be of finite thickness with respect to a wave- length in the transverse direction. (2) Gain is improved if there is no gap between the YIG and GaAs regions, i.e., g = 0. (3) Gain is obtained for velocities less than 2 x 10 7cm/s only if carrier density is greater than 1018cm‘3. 94 95 For practical values of carrier density, Gain increases as YIG thickness increases in general. Bandwidth decreases as YIG thickness increases. The case of a GENERAL layered structure yields the following: The presence of the two metal plates is essential in order to ac- hieve gain for practical carrier densities and drift velocities. The GaAs region must be of finite thickness with respect to a wave- length in the transverse direction if metal plates are present. AH can be 0.5 0e and the gain is still significant over the fre- quency band of interest. The gain decreases significantly for 00 greater than 2 x 107cm/s. Maximum gain occurs for H0 = 750 De. The results of the Bers and Briggs criteria indicates that a con- vective instability is present. Due to the inconclusive nature of ex- perimental results presented in the literature, we conclude that further investigations of the MSSWA using optimized parameters should be con- ducted. APPENDICES APPENDIX A BASIC ANALYSIS AND DEFINITIONS The form for the permeability tensor 0*for a ferrite when the static magnetic field is along the +2 axis is: u jk 0 “xx l-ny “yy Uzy 0 . (A.1) “zz Often the signs of off-diagonal terms are different in the litera- ture. This issue and the errors which can result are addressed in [72]. If the orientation of 00 is changed, the tensor is also modified. As an example, if 00 = Hey the following cyclic change in subscripts is made; 2 replaces y, y replaces x and x replaces z to yield: u u u [ yy yz yx “ - Uzy uzz uzx - - uxy uxz uxx J -> A For H0 = Hox; z replaces x, x replaces y, y replaces z in the Equation (A.2) to give: 1 0 0 ‘3 = 0 0 JK 0 -jK 0 . 1 96 97 If we replace K by SK, then we can orient the static field along the :z axis by setting S = :1 in the above tensor. For our case, where H0 is directed along the :y axis, the permeability tensor is: p 0 451(— 1 0 . (A.4) fit 11 O jSK O 0 . J where recall that 0 and K are given by (Dowm (A) 2-032 O (A) O.) K = 2°23 (A.6) (A) '0) Some helpful definitions of important parameters are given below: 0.) 0 precessional angular frequency of spins about H0. w = Ys(4nMs), the saturation angular frequency. This is a constant of the system. The value is 3.096 x 1010 rad/sec if 4hus = 1760 0e. ._ g e = ' ' : = Ys - —§$;l— 00 gyromagnetic ratio 2.8 MHz/0e 2h(2.8 x 106) rad/0e for the electron. Notice that in general the precession angular frequency can be written as: .1 2 2 mo YsHi + JEYSAH + wexa k (A.7) OY‘ = Y H i--‘=Y AH + Y D “2 “ ‘(SK 2 (A 8) mo 5 i + ‘2 s s 2 ° 98 where again Hi is the static internal dc field given by: — H (A.9) i 0 DM A The terms H0, HDM and HA represent the external applied field, the de- magnetizing field and the anisotropy field respectively. The latter two components must be accounted for if the slab dimensions are finite with respect to a wavelength and the slab is not fabricated from single crystalline material. The parameter represented by AH is the ferrimagnetic resonance linewidth. This term gives a measure of the wave attenuation in the ferrite sample. It is inversely proportional to the precessional re- laxation time. This gives a measure also of the degree of heating that occurs within the ferrite due to energy dissipated by a wave in lossy material. As the value of AH is increased, the losses which tend to relax the spin precession increase. This explains the decrease in the relaxation time T. Typical values can range from .3 to 7 oersteds in YIG films. For our analysis we would like to use films with AH values of approximately 1 oersted. If the wave energy is of high enough frequency, the exchange field effects must be accounted for. This can take two forms: 2 2 (1) wexa k (A.10) 2 u-(SK) (2) YSD [_—7I—_—_] (A.11) where the parameter 0 is a phenomelogical, inhomogeneous exchange con- 9 Oe-cm for YIG. Recall 1 stant with a value of approximately 4.4 x 10- 4cm- that in the magnetostatic regime i.e. k g 10 exchange effects can be neglected. Another loss parameter can also be found in the litera- ture. This is designated as A and is related to the jéAH term in the Landau-Lifshitz loss formulation which is used in this report. ll 99 In this report, the units are those normally encountered in the literature and for reference we include some conversion factors 40M (in Gauss) x 79.5 = M 34E$Iflifl— (A.12) H(in 0e) x 79.5 = H Emflélflifl— (A.13) . -4 _ 2 8(10 Gauss) x 10 - B Wb/m (A.14) The above notation conforms to that given by Collin [73], Johnson [74], Ramo, Nhinnery and Van Duzer [75] and Beam [76]. It also appears to conform with the work due to Bini et al. [77] and also to that by Matsuo and Chang [78]. Finally, it should be noted that the analysis included in this re- port assumes that a/ay + 0 i.e. that the problem is a two dimensional one. This is done for simplicity; the TE and TM modes then become un- coupled [79]. APPENDIX B DERIVATION OF DISPERSION POLYNOMIALS B.1. Dispersion Polynomial for YDS From Section 3.3, the dispersion relation for the YDS case was de— veloped as Y _ e-2kg = ( S + k)[SK u + 0] (3.1-1) (YS ' k)[5K - u - D] where A1 = SK - u + D (B 1-2) A2 = SK - u - D (8.1-3) 0 = K2 - 112 (B 1-4) If an exponential approximation is used then, - - 2 2 N (k) e 2kg = 12 129k + 492k2 : 0l(k)' (8.1-5) 12 + 129k + 49 k 1 Solving for YS yields <81+1> Y = k B 1-6 5 (31-1) < > where A N1 81 = —. (B.1'7) D 1 1 The proper approximation for Y5 is, : §fl ys - k + k + SB . (B.1-8) where SA and SB are defined in Appendix C. This gives us the following for (8.1-6). lOO 101 + 1) - 1)’ SA (3 _ 1 k +-—E + $8 — k ( (B.1-9) Bl Since A1 and A2 are constant with respect to 9, Bl will vary as N1 and D The exponential approximation (see Appendix C) requires the con- 1. sideration of two cases. They are for kg 1 1.75 and kg 1 1.75. Case I kg 3 1.75 For this case 2 2 N1 = 12 - 129k + 49 k (0.1-10) 0 — 12 + 12 2 2 - 1 - gk + 49 k — N1 + 249k (3-1-11) We write A N N _ _§ 1 = 1 - B1 ’ A1 N1 + 24gk ‘ A3 N1 + 249k (3'1 12) so that 81 + 1 - A3N1 + N1 + 249k (8 1 13) B1 -1 A3N1 - N1 - 249k - or B1 + 1 = N1(A3 + 1) + 249k (8.1-14) B1 - 1 N1(A3 - 1) - 24gk This gives, for (B.1-9) after multiplying both sides by k: N (A + 1) + 249k 2 _ 2 1 3 + + - . ' k SBk SA k N (A - 1) _ 249k (8 1 15) l 3 If we cross-multiply and expand the LHS and RHS of (B.1-15), we obtain 2 3 (A3 - 1) le + (A3 - 1) leBk + (A3 - 1) NISA - 249k 2 _ 2 3 -24gSBk - 24gSAk — (A + 1) N k + 24gk . (B.1-16) 3 1 Further expansion yields 102 2 3) -2N1k2 + (A3 - 1) SBle + (A3 - 1) SAN1 - 48gk3 - 24gSBk - 24gSAk = 0 -2(12k2 - 12gk3 + 492k4) + (A3 - 1) 53 (12k - 129k2 + 49 k +(A3 -1) SA (12-1ng + 492k2) - 489k3 - 24gsek2 - 24gSAk = 0. This expression is rearranged to obtain (8.1-17) (-8gz)k4 + [249 + 492 (A3 - 1) so - 48g]k3 + 2 2 [-24 - 129 (A3 - 1) 50 + 49 (A3 - 1) SA - 249$B]k + [12 (A3 - 1) SB - 129 (A3 -1) SA - 24gSA]k + [125A (A3 -1)] = 0. which is the YDS dispersion for thin gap spacings. Case II kg 3 1.75 In this case we have N E8 = 0.11072 1 D1 = 2kg so that the expression for B1 becomes B1 = A3 2&3 and 01 + 1 A3EB + 2kg 01—i‘1 ' A3EB - 2kg or for (8.1-6), we have A3EB + 2kg A3EB - 2kg° 2 2 k + 58k + SA = k Upon expansion the following expression results (8.1-18) (B.1-19) (8.1-20) (8.1-21) (8.1-22) (B.1-23) 103 A3EBk2 + A3EBSBk + A3EBSA - 29k3 - ZgSBk2 - ZgSAk = A3EBk2 + 29k3 (3.1-24) The dispersion relation for the case of thick spacings is therefore, 3 (-49)k + (A 2 353 — 2953 - A EB)k + (A EBSB - ZgSA)k + 3 3 (A3EBSA) = 0. (8.1-25) 104 (B. B.2. Dispersion Polynomial for SINGLE-SURFACE (EXACT vs) The dispersion relation for the SINGLE-SURFACE case is given by 2 2 _ Ys (K - u ) + (SKk - UYf) - 0 or Y : “Yf - SKk s K2 _ 2 squaring both sides Y 2 :(“Yf - SKk)2 s 2 2 2 (K - ) Recall that k 2 2 _ 2 2 . 5 Y5 - k - ks + 1(;—) 0C (a - ku ) and 2 2 w2 - (mm + w0)2 K - 0 = 2 2 = D 0) “(.0 0 so that 02x 2 = quf2 - zuvaKk + (Skk)2 k 2 02) 2 = uztkz + —9- D] - ZUSKY k + (Kk) S 0 f let Yf 5 k so that 02x52 é 02k2 + ukOZD - ZuSKk2 + (Kk)2 = k2(02 + K2) + 0k020 - 208Kk2 = (02 - 20SK + K2)k2 + 0k020 again 2-1) .2-2) .2-3) .2-4) .2-5) .2-6) .2-7) .2-8) .2-9) .2-10) 105 2 2 2 DZYSZ = (u - SK) k + Pko D using the expression for Ys gives k 2 02 [k2- kSZ + j(--(-:-) 00C(u) -kuo)] = (U -SK) 2 . k 02k2 - 02k 2 + i( ) A (u) - kuo)02= (11 -SK) elm S C which yields k 2 k +0k00 k +0k00 (8.2-11) (8.2-12) (8.2-13) [02 - (11 - 5102])(2 -[j(—i) wcuoDZ k {021252 - j(k—:) Acaoz + 2 uko D]: O 8.3. Dispersion Polynomial for SINGLE—SURFACE (Approximate YS) Again for the SINGLE-SURFACE dispersion we have YSD = uYf - SKk and if Y Z k then f YSD = (u - SK)k The expression for YS is k 2 2 2 . s YS=k-kS+J(—E)wc(w‘kuo) é k2 + '(EEN; ( - ku ) J 0) (”cm 0 if k2 >> ks2 which is true for a slow wave 2 2 (1(2) 00C(w- kUO)k 2 . w ' Y5 = k + J k2 2 k 00 (01- ku) =k2[1+j(-§;) C k2 0] Then by the complex root approximation we have (8.2-14) (8.3-1) (8.3-2) (8.3-3) (8.3-4) (8.3-5) (8.3-6) 2 k w (w - ku ) L -_S_ C 0 .- Ys - k[],+'J( m ) , 2 ] (8.3 7) 2k so that the original dispersion becomes 2 k u) (w-ku) k [1 + j(—§-) C 2 ° ] 0 = (A - SK)k (8.3-8) (Ll 2k kS 2 wc(w - kuo) ( _ SK [1 + j1‘5 ) 2 1: u 0 (3'3‘9) 2k ks 2 A (A - ku ) + 2k2 = £—:—§E— 2k2 (8 3-10) J (- ) c O D ' . ks 2 “c 0 - SK‘ 2 J(-; ) —7 (w - kUO) =[:*——5-— -1] k (8.3-11) Which in final form is k 200 k 2010) 0 - SK 2 .[ s 011 ] . s c _ 1. -11). +1(-,-,) —. . .- (7)—.- -o W) 8.4. Dispersion Polynomials for YIGSLAB and YIGSLAB-GAP The dispersion relation for YIGSLAB-GAP was found to be -2kg Y5 + k -2kd e -(Y _ k (SK + u)(Sk - u -D)e + 5 Y + k _ _ _ -2kg ( s ) (ll SK)(SK + U 0)] 0|}: i" W] [SK - u - 0)e'2kd - (SK + h -0)] (8.4-1) Using the exponential approximation for the thin slab/gap case, after considerable algebra we obtain, 107 5 [2f 4(b3- b1+b2 -b 0)]k6 + [2f3(b1-b3) + 293(b0-b2) -2$Bf4(b0+b1)]k {2[f2(b3-b1) + 92(b2-b0) - SAf4(b0+b1)] - S8[f3 (b o+b2+b3 bl) + g3(b1+b3+b2-b0)]}k4+ {2[f1(b1-b3) + 91(80 b ()1 - SA[f3 (b o+b2+b3- -b 3 _ I + g3(b1+b3+b2-O -b )] SB[f2 (b O+b2- b3+b1) + gZ,b1+b3— b2+bO )]}k + {2[f0 (- -b1+b3+b2 -b O)] - SA[f2 (b 0+82- b3+b1) + 92(b1+b3-b2+bo)] 2 -SB[f1(b0+b +b -b1) + 91(b +b +b -b O)]} k + {SAN1 (b O+b2+b3- b1) + 91 2 3 1 3 2 (b +b +b -b O)] + 2[SBfO(bO+b 1 3 2 )1} k + [254i0(b0+b1)1 = 0 (8.4-2) 1 In the case of kd and kg being greater than 1.75 we obtain [8gd(b3-b1)]k4 + [4gb(b -b0) + 4SBgd(b -b 2 + [258(b -b O)gEB 3 1)1k 2 dEB + 4SA(b -b1)gd]k2 + [25A(b -b0)gEB -2$A(b +b -2$B(b +b 3 2 1 3) 1 3) dEB - SB(bo+b2)(EB)2]k - [SA(EB)2(bO+b2)] = 0 (8.4-3) where again the following definitions hold E8 = 0.1172 d = YIG slab thickness 9 = spacing between YIG and GaAs The terms b0, b1, b2, b3 are expressions which are related to the YIG magnetostatic wave parameters. They are defined as (8.4-4) (8.4-5) 108 b2 = -a2a4N1N2 (8.4-6) b3 = -a3a5 (3-4-7) where a1 = SK - u + D (8.4-8) a2 = SK - u - D (8.4-9) a3 = SK + u - D (3.4-10) a4 = SK + p (8.4-11) a5 = u - SK . (8.4-12) We represent the exponentials as N e‘de = D—2 (8.4-13) 2 N e'2kg = fil- (3.4-14) 1 similar to the previous section. The terms f1, f2, f3 and f4 are re- lated to the exponential terms N1, D1, N2 and D2 from multiplication operations that result from the simplification of (8.4-1). The same holds true for 91’ 92 and 93. These are not included for brevity since they can be readily obtained. In order to obtain the polynomial for the YIGSLAB case the spacing between the YIG and GaAs regions goes to zero. This produces the follow- ing conditions. The following polynomials result for the two separate cases, Case I kd i 1.75 109 {2tf2(b3-b1) + 92(b2-b0)]} k4 + {2Ef1(b1-b3) + 91(b0-b2)] 3 -53[f2(b0+b2-b3+b1) + 92(b1+b3-b2+b0)]} k +{_2[f0(-b1+b3+b2-b0)] -SA[f2(bO+b2—b3+b1) + 92(b1+b3-b2+b0)] - SB[f1(bO+b2+b3-b1) + 91 2 (”1+b3+b2‘bo)] )k T 1SA[f1(bo+b2+b3‘b1) +91(b1+b3+b2'bo)] + ZSBfO(bo+b1) )k + [23Af0(b0+b1)] = 0. (3.4-15) Cast II kd 3 1.75 The dispersion polynomial in this case is, [-2$BEB(b +b3)d]k2 - [ZSAEB(b +b ) + SB(EB)2(b0+b2)]k + 1 1 3 [SA(EB)2(b0+b2)] = 0. (8.4-16) 8.5. Dispersion Polynomial for the GENERAL case The algebraic manipulations for the GENERAL case are exceedingly long. For this reason only the dispersion in its final symbolic form will be given. The method used in the development of previous polynomials was used here, as well as the following approximations: y 5 k (only in exponents) (8.5-1) 5 k (8.5-2) Using the above information along with the dispersion relation for GENERAL developed in Chapter 3, the following polynomial results for the case of thin slabs and gap spacings, 110 + M 15 17 16 k k + M k + M T17 T16 T15 11 10 9 M k8 + k +M k + T8 + M T9 k + M k T11 T10 7 6 5 4 3 2 1 _ MT7k + MT6k + MTsk + MT4k + MT3k + MTzk + Mle - 0 (3.5-3) The coefficients of this polynomial depend in a complicated way on YIG parametersS, K and u, as well as the thicknesses of the various layers which make up the device. A separate dispersion polynomial for GATE was not developed due to the similarity of this case with others which were studied. In order to analyze the GATE model, the following conditions were applied to the GENERAL case: (1) The semiconductor region is allowed to be as thick as possible for the thin region approximation. (2) The gap between the GaAs and the heat sink is as large as the ap- proximation will allow. These conditions provide for a semiconductor region that appears semi- infinite in the transverse plane which is the basic difference between the GATE and GENERAL geometries. APPENDIX C COMPLEX RO0T AND EXPONENTIAL FUNCTION APPROXIMATION In the derivation of certain dispersion polynomials, it is necessary to use vs or yf which cannot be expressed exactly since they depend on k, which is to be found. It is possible, however, to obtain YSZ and Yfz, and these are in general complex quantities. Thus, an approximation to the complex square root is required in order to obtain workable dis- persion polynomials. The exponential approximation is also discussed. This is because the general form of the more complex relations is e-X. An appropriate choice for this approximation is important because we want to restrict the function e-X to positive values. C.1. Complex Root Approximation Here we look at the square root of a complex number 2 = A + jB MAG z = (A2 + 32)% (c.1-1) ANG z = tan-1(B/A) (c.1-2) If B << A, then tan'l(B/A) é B/A (c.1-3) and _ 2 2 % 1 MAG z - (A + B ) - A (c.1-4) ANG z é B/A (c.1-5) so for the /7, /A (C.1-6) B/2A (C.1-7) MAG /7 ANG /Z 111 112 In rectangular form you have _ B -- __B__ - /Z — /A cos(fi)+ JSln(2A) (C.18) and for B << A again, we have v7&5"|:1+3(‘2'%‘)] (c.1-9) because we recall that cos (%)é cos 0 = 1 (c.1-10) sin (§%—) 5 _%A (c.1-11) Now we obtain Yf the transverse wavenumber in the ferrite. k 2 = k2 + —9— (k2 - 112) (c.1-12) Recall that when losses are included in the YIG, mo becomes complex and therefore (K2 - p2) is also complex. We write (Kt - U2) = D, + 10, (c.1-13) Thus - k 2 Yfz = k2 1 + -9§ (K2 - p2) (C.1-14) pk - 2 2 k D k D. = k2 1 + 0 r + j 0 ‘ (c.1-15) 2 2 _ pk pk hence kOZDr kOZDi 1/2 Yf = k 1 + 2 + J 2 (C.1-16) pk pk 1/2 ' kozor Di ; k (1+ > <1+j—) (C.1-17) . 2 2D _ pk Y‘ k D o. . g k (1+ 0 2r)(1 + J- 751)] (C.1-18) 211k Y‘ so that finally Yf D k[:1 + j 751 ] Y‘ 113 For the semiconductor we have, 2 Y5 then k2 ' k52 + MES—1)“) k2[1 +j(—(B 2 (m C - kuo) Upon taking the square root we have 75 5 W[:1 + J or is é W{: +J [()Z 5 k[:1 + é? + k : §fl YS - k + k + $8 where 2 k s ”cm 5A=J(‘t) ‘2— 2 k w __- _5 .. A.) In order to k = kr + jki and that C.2. k r w (w-ku ) \ C 0 >> k., 1 Description of the Exponential Approximation obtain (C.1-19) and (C.1—25), we assume that (Cl-19) .1-20) .1-21) .1-22) .1-23) .1-24) .1-25) .1-26) .1-27) The exponential function is approximated by the first few terms of the continued fraction expansion [79] and has the form 114 e“x = 1 + 12X 2 (0.2-1) 12 - 6x + x or 2 e-x = 12 + 6x + x = p(x) (C.2-2) 12 - 6x + x2 This approximation does not allow e'x to become negative as e'x = 1 - x would when [x] > 1. Because we are searching for x = 2kd, we cannot check the magnitude of x in any step of the analysis. A sketch of Equation (C.2-1) and l/x is shown in Figure C-l. We will use (C.2-2) when x i 3.5 (i.e., kd, kg i 1.75). Thus, the thicknesses of concern are less than .279A for reasonable accuracy. For larger values, we let e.X = E2. The solution of this equation for x = 3.46 gives a value for EB of 0.11072. With the above approximations, we can determine the dispersion poly- nomials for the various geometries of concern in this report. 115 e.X ‘\\ \ \ ‘\\\\\ \\\ \\\ \ \ ‘\~“‘ -- \\ i i) 2 /§ = 3.46 Figure C-l. Comparison of the_§ontinued fraction expansion approxima- tion to l/x for e APPENDIX D YIGSLAB PASSBAND CALCULATION The starting equation is the dispersion relation for this model. We will take the limits of n0+0, n +w, to model the extreme cases of o the GaAs being a dielectric or a prefect conductor. In the first limit (i.e., the dielectric case), if we let vf = Ys é k, then e-de = [u + SK + 1][u - SK + 1] [u - SK - 1][u + SK - 1] (D.1) Notice that the expression is invariant with the sign of S; thus, -2kd _ [u + K + 1][p ‘ K + 1] e ' [u - K - 1][u + K - 1] (D.2) We now take the limit of the above expression as k+0 and k+w. If k is assumed pure real (i.e., we examine the lossless case), then for k+O after considerable algebra, we have NIH A O (A, V w E wA Y[HO(HO + 4WMS)] ll r'fi E E The terms mm, MS and H0 are defined in Appendix A. For a representative bias field HO = 500 De, where 41rMS = 1760 Oe we find wB = 2.43 x 1010rad/sec wA = 1.87 x lOlorad/sec 116 117 which gives a frequency passband f8 5 fA of 888 megahertz. In the limit as n0+w, ys+jm, the original dispersion gives [YS(U + $K)][u - SK + 1] e'de = (0.6) [Ys(u - SK)][u + SK - 1] (u + SK)(u - SK + 1) ‘ (u - SK)(u + SK -1) (0'7) 02 - K2 + 0 + SK (0 8) 02 - K2 - 0 + SK ° For k+0, w2 = w 2 + w m (D.9) O O m 0)" 0 = (DA (0.10) Therefore, in the limit as k+0, the or the metal cases. For k+0 we set let 02 - K2 - 0 + SK # 0 Using 2 2 _ K2 = m2 - (mm + wO) “ 2 _ 2 (.00 (.0 which for positive w implies that S Thus, value of m is ”A for the dielectric the numerator of (0.8) to zero and (0.11) (0.12) (0.13) (0.14) = +1. 118 w E w = w + m (0.15) for the case of S = +1 only. As a check for Equation (0.15), if this is substituted into Equation (0.11) we have 000 f o (0.16) which always holds. Therefore, (0.15) is correct. Therefore, the mode for which I x Hoy points away from the YIG exists up to mo + mm. Our geometry is given in Figure 01. For the geometry shown, this wave must propagate in the +Z-direction, and its energy is concentrated along the YIG-GaAs interface. If the wave is to travel in -Z (for S = +1 still) it must propagate along the YIG-air surface. In this report we will restrict ourselves to the k >0 case. We examine only the wave moving in the +Z-direction in detail. If we allow S = -1, the mode of concern propagates on the YIG-air surface. Since S f -1 for the passband edge given in (0.15), we must investigate the complete dispersion equation for -k. Setting k = -k, the dispersion yields 2 2 ede = “2 ' K2 + u + SK (0.17) p - K - p + SK For k+oo we have 2 2 u - K - u + SK = O (0.18) and 02 - K2 + 0 + SK f 0 (0.19) Using Equation (0.18) gives 2 Sm 2 3w (0 + 431.) = (00 + Tm) (D20) Now the -k wave must travel on the upper surface. Since E x fio must 119 point away from the YIG, we must let S = +1. If this is used in (0.20), we obtain wm 3mm 00 + _4 = :(wo + T) (0.21) or (L) - : _m. w : wB mo + 2 Therefore, the -k wave at the YIG-semiconductor surface must have S = -1. Therefore, for S = -1, the +k wave must be concentrated on the YIG-air surface. For wave propagating in the +Z direction we assume the genera- tor is at the left edge in Figure 01 and a matched termination exists on the right edge so that no reflections are possible. For S = -1, the passband TS (wA, 08) and the wave is on the YIG-air surface. When S = +1, the passband is (wA, wC) and the wave is on the YIG-GaAs interface. Figure 01 provides a summary for the n0+0 case. Figure 02 gives a sum- mary for n0+m along with representative numerical values. The reader should be careful to distinguish between ”C in the YSZ expression (the conductivity frequency) and the upper cutoff wc referred to in this ap- pendix. 120 — — — — — — - — _ -_ — — — — _— __ _ YIG Figure 0-1. Passband of dispersion behavior and propagation orienta- tion for the dielectric case. 121 S: S: -k -<: 2: k S = -1 \Jfa: <%E> YIG <§>> {UL} ’2 :+1 n0+w GaAs H = 500 0e 0 = 8.796 x 109 0 10 0 10 “A = 1.87 x 10 rad/sec mm = 3.096 x 10 08 = 2.43 x 1010 f0 = 1.4 GHz “c = 3.98 x 1010 fn = 0.49 GHz fC = 6.33 GHz fB = 3.86 GHz fA = 2.9 GHz Figure 0-2. Passband summary for the metal case. APPENDIX E REVIEW OF PREVIOUS RESEARCH E.1. Review of work by Bini, et al. This work is presented in references [80-83]. Bini obtains the growth factor k, by using an approximate formula that iS the ratio of power and/or energy terms; thus, the name "energetic analysis approach". In the first paper [84], the fields were obtained by first assuming the carrier density in the semiconductor is zero. This is his "cold mode" approach. The introduction of carriers in subsequent calculations was assumed to be a small perturbation. In the subsequent papers, it was not clear when nO was finite or zero during particular calculations. They predicted active coupling only for MSSW (surface waves) stating that volume waves did not look promising. These surface waves are TE and £3: 0 i.e., no space charge. The interaction is derived from the G x 8 term in the Lorentz force equation. They assume a single boundary (YIG-SEMI interface) supports a slow surface wave. The passband is be- tween fA and f8 and it occurs when k x RO points away from the YIG. Their eq. (14) in [85] is the basic "single-surface“ dispersion which they study in detail. The letter [86] in 1977 did not use the ”cold mode” concept. The YIG was a thin film with thickness h~x. Bini assumes YIG losses can be directly subtracted from the gain. In the third paper [87] in 1978, a slab is considered at the beginning of the paper. The complexity of the resulting dispersion was noted and a mapping technique to track roots was discussed. Subsequently, they returned to the llsingle surface" case and obtained approximate results in this limit. For no YIG loss (AH = 0), they find both weak and strong interaction regions. 122 123 The weak regime is for low conductivity; the strong for large conduc- tivities. In the weak regime, the gain increases with 0, whereas in the strong one, it decreases with increasing 0. Thus, an optimum carrier 14cm'3) . density was predicted. (For GaAs, this corresponds to n0”10 When loss was introduced, the gain, and bandwith for gain, decreased. A lower limit on 0 occurs now, which means that practically only the strong regime is feasible. Thus, the optimum value for no calculated when AH = 0, will not be realized in practice. We now summarize in more detail. 1. They say all experiments conducted have used thick YIG samples, thus, the "single-surface” model should apply. The semiconductors used were Ge and InSb. 2. For single-surface, the gain occurs quite close of f8, where wb = 2"f8 = 00 + 43—. The bandwidth is about 17 kHz when AH = o and about 9 kHz when AH = 10 0e. 3. We question this result since Bini says AH gets very large near f or f ; so is AH = 10 0e a realistic value to assume? With no loss, the gain goes to infinity at f8. 4. He calculates loss contributions under the condition nO = O i.e., ”cold mode” again. 5. In the first paper, Equations (11) and (12) are in error; it is not clear how, or if this jeopardizes some of the results. 6. Says one cannot use the magnetostatic limit V x H = O as this yields zero net power flow. 7. For single-surface, the approximate growth constant is V 00 . 0 C Bi‘Br(Vp ') w A 8. For a finite slab 00 V _ C 0 81—h 10(Vp -1)A1 9. For the film thin (h i A) Bl~ h 10. 11. 12. 13. 14. 15. 16. 17. 18. 124 Notice the approximations in Items 7-9 are independent of h, directly proportional to h, then inversely respectively. Thus, a trend is not at all evident. The carriers and wave energy are in the same direction for gain, and v > v . 0 C Says loss reduction of Bi is _ AH Bim 'Y 2v U) 2-102 _ 8 v9 - h m but how is this developed? They did not use the Bers-Briggs criteria as it would be complicated due to their numerical scheme. Instead, they used the direction of the Poynting vector. It was not clear if actual or cold mode fields were used; and no calculations were presented. They pointed out several serious errors in previous theoretical treatments. a) Vural [88], Robinson [89], had vp and v0 in opposite directions, so only an evanescent mode was studied! b) Kawasaki et al. [90] chose the wrong root; i.e., it becomes unbounded at infinity. After studying the thick slab case (h>>x) in detail, they conclude slabs Should not be used; only thin films seem feasible. However, only the letter [91] addressed this case! The bandwidth for thin films (~30) was ~200 MHz at 3-4 GHz. Their comments on the existing experimental results were quite in- formative. a) Vural [92] saw electronic gain; whereas Szustakowski [93] under very similar conditions observed no interaction! b) The experimental data (obtained from pulsed conditions) is not suitable for comparison with theory. c) Vaskovskii's [94] data are unclear Since the measurement fre- quency is out of the passband for the proposed modes! The predicted 10 kHz bandwidth (for h>>x, single surface) clashes with reported data, since the spectrum is about 10 MHz wide about the carrier (0.205ec pulse widths). 19. 20. 21. 22. 23. 24. 25. 125 Surprisingly they claini”all the theoretical work has now been com- pleted"; and "subsequent analysis is not needed." They state all reported experimental results are questionable; and we agree. Concludes Shat the device must be a thin film operated in pulsed mode at 77 K with excellent heat sinking. Since the magnetic saturation occurs at tens of microwatts, low power applications are the only foreseeable ones. Surprisingly, the Hall-effect that would accumulate the surface was not mentioned at all. No balancing bias plate was envisioned. 2 2 We agree with their expressions for Yf and Y5 , but not necessarily with some of the approximations for YS. The llidle modes" introduced are not clear at all; are they evane- scent modes? 126 E.2. Review of Awai et al. 1. The case analyzed is YIGSLAB; an optimum carrier density is predicted. The gain increases for increasing frequency [95]. 2. They mention that there is no mode (slow) at the interface of an in- finite half-space of YIG on a dielectric. They reference Damon and Eshbach for this result. Therefore, the work by Robinson et al. is in jeopardy. 3. Notice Bini mentioned Robinson's error of u0 and vp in opposite di— rections. However, now we see the single-surface concept of Bini is also in jeopardy, since no "cold mode" actually exists. 4. Therefore, Awai used a finite slab. A slow mode exists with or with- out the carriers. Assumes AH = 0 always. 5. Used u0 = 8 x 107cm/sec, which is not realizable. Finds ki~1-1OO -1 . 17 cm . Says optimum n0 occurs for n0~10 . 6. Used the BerS-Briggs criteria to verify actual convective instability. E.3. 127 Review of Lukomskii et al. Finds gain for 40 YIG on 200 InSb at 3.5 GHz. AH = 0.4 0e, no = 3 x 1015. Says heating reduces operation to pulse mode only, and pulse width should be less than 50 sec, for a temperature rise of 10°C [96]. They state for a film of YIG of thickness a, 2 U.) (1)2 = 1002 + -2—(1-e'2ka) 2 _ mm -2ka v — ——— ae 9 4w 2 (A) Notice if at00 (for Single surface), wztwoz + —%— and vg+0; Thus no passband. States YIG films ~50 can have AH~0.5-1.0 Oersted, but T = 3000K. Says AH>1 reduces gain to zero, in general. Lower HO widens the bandwidth and shifts it to lower frequencies. ‘ States the experimental data are probably due to current filament formation; thus, the apparent amplification may be just reduced damping. The gain calculated assumes ki~10cm-1. E.4. 10. 11. 128 Review of Vashkovskii et al. They are one the experimentalists. They found reduced loss by about 8 dB with Ge during the pulse. The YIG was 10000 thick [97]. They say that measurements of delay time vs Ho show that a surface wave is present. This does not agree with our results, however. Bini mentions the passband descrepancy! ' When uO and vp are in opposite directions, only attenuation occurs. The results are not very clear or conclusive. They analyze the case we called GATE. They state the presence of the metal gate always enhances the gain. The separation between the YIG and bias plate alters the gain and bandwidth. The dispersion is practically independent of the carrier density no, but is sensitive to the plate position. They employed the Bers-Briggs test, as well as including losses via AH. They chose u0 = 5 x 107cm/sec. which is not realizable. Best gain occurs when the gate is directly against the YIG. Best gain-BW for b = 0, d f 70 (d = YIG thickness) (b = gate sepa- ration). The ki~1 to 2 cm Says absolute (oscillation) instability is possible by using the Bers-Briggs conditions. 129 E.5. Review of Yukawa et al. 1. They analyze a YIG slab between two semiconductor Slabs and this structure between two metal plates [98]. 2. They find gain for the special case of a 100 semiconductor and a 1000 YIG slab with metal on both sides. For n0 1015-1016, ki~1-10. (But u0 = 8 x 107cm/sec) 3. They predict absolute instability in some conditions. 130 E.6. Review of Schlomann 1. Explains in detail the Hall-effect action that causes the magneto- static wave to grow [99]. 2. Analyses a "single surface model" almost (the semiconductor is a slab). Assumes real I and complex 3. 131 E.7. Review of Shapiro 1. He assumes a single surface geometry [100]. + 2. Assumes k is real but 3 complex; hence only oscillation is sought. 132 Recall Bini mentions their incorrect root choice. They point out that the additional H0 due to the current pulse was not included in analysis of previous experiments; thus, the variation of delay with or without the current pulse does not necessarily imply The analysis used a YIG slab, and a semiconducting slab and metal They obtain the dispersion for this case, but simplify it, and ul- The experiment used YIG with AH : 0.5 Oersted, and 9000 thick. The input power was less than 0.1 mW to prevent saturation. The measurements were pulsed at 4.1 GHz. Room temperature conditions The measured attenuation implied AH = 0.7 Oersted. A peculiar situation occurs in Figure 2 of the paper. For the value of HO given, the lower cutoff frequency for the surface waves is about 4.65 GHz. However, the measurement frequency (center) was 4.1 GHz. This implies that a volume wave was being generated and detected. No other author has mentioned this problem. Bini men- tioned a similar situation occurred in Vashkovskii's work. Electronic gain occurred when the holes drifted with the wave. At- tenuation when drifted in opposite direction. The maximum magnitude of this gain was 3 dB at field strength of 2 KV/cm. Was there real gain or reduced damping due to current filament pro- duction? Lukomskii made this statement in reference to Vashkovskii's The initial insertion loss when the Ge was placed on the YIG was The experimental values for interaction were only poorly reproduc- In defense of Comment #11. If the interaction was due to filament production, then why does the direction of drift make a difference? Item #10 shows the interaction is not reciprocal. E.8. Review of Kawasaki et al. 1. 2. gain [101]. 3. Their experiment using Ge showed gain. 4. on the YIG face. 5. timately reach the single surface limit. 6. The semiconductor was 3000 thick. 7. prevailed. 8. 9. 10. 11. experiments with Ge. 12. 25-30 dB. 13. able. 14. 15. In response to the previous item, we can easily see that holes drift- ing to the right (With the wave) are pushed away from the YIG, and 16. 17. 18. 133 de-damping is possible. When drifted to the left, they are pushed toward the YIG and cause more loss. A verification of interaction was instituted by replacing the Ge with a copper plate. No change in attenuation was observed for either polarity of potential applied to the copper. They state reduced insertion loss even for carriers drifting slower than wave. (This again sounds like filament production). They mention most authors find gain for 00 about 108cm/sec; an un- realizable condition. E.9. 134 Chang, et al. 10. 11. 12. The first paper [102] looked at the intersection of dispersion curves for a semiconductor rod in an annular YIG wrapped in metal. The U0 i 5 x 108cm/sec was used. A backward branch of the dispersion was used. The thermal velocity is set to zero. The next paper finds dispersion curves versus carrier density. The carriers do not drift. AH = 0 [103]. A fifth order polynomial is found in w. Anticipates lowest damping occurs for n0~1017cm'3. The YIGSLAB model is used here. The next several papers analyze special cases of the GENERAL model. They supposedly let various gaps+m; but with assumed terms such as -fx 0- = a. e + bje+fx the bi term blows up! How this is handled is not clear [104-105]. They obtain ki~1-10 near 4 GHz but u0~5 x 107 to 1 x 108cm/sec, which is unrealizable. They get best gain when the gap between the semiconductor and YIG goes to zero. The next paper covers the GATE case. They use u0 = 108cm/sec. The values found for Ki~1o‘Z-1o1 [106]. Increasing AH (from 0 to 1 0e) reduces gain and raises the value of the low frequency wherein gain starts. This shift was about 2 GHz. The next paper develops an energy analysis for the interaction. They use a GATE model [107]. Their energy expression is applicable for the case of small loss (nearly transparent media). We question this transparency assump- tion, since the carrier stream is very lossy. During the change from loss to gain, however, the system can be considered low loss; hence transparent. 15 7 Here they use nO = 10 and U0 = 2 x 10 cm/sec. This is one of the few calculations where realizable parameter values were used. Gain is lost when AH = 4 0e, but here u0 = 108cm/sec. For u0 = 2 x 107, AH = 0.05 apparently reduces the gain to zero; this is for the YIG energy concentrated on the face away from the semi- conductor. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 135 The next paper treats the GENERAL model but without the bias plate. Their signs for S = :1 appear to be confused [108]. They plot the threshold velocity versus AH. If we assume 00 j 2 x 107cm/sec, then AH :- 1.0 0e or the gain is lost. The values for k. before AH reaches 0.4 are about 1 to 3 cm'l. The gain increases mdnotonically as 00 increases. States this is unique to the MSSWA. This condition of ki increasing monotonically with 00 was also found by Awai et al. They finally perform a calculation wherein most parameters are reasonable. 2 x 107cm/sec 00 = AH = 0.2 0e Ho = 600 Oe semiconductor thickness, 50 insulator between semiconductor and heat Sink, .010 YIG thickness, 100 nO a 2 x 1015 They found ki~0'5 to 3 cm—l. Finally, the experimental results were published in 1982; the first paper was published in 1968 [109]. The experimental parameters are Ga As: no = 8.1 x 1015 YIG: AH = 0.5 0e “e = 6 x 103 thickness = 900 thickness = 800 They state the previous experiments used YIG slabs that were too thick 500-10000; and Ge, which has a low saturated drift velocity. The references are to Vashkovskii, Szustakowski, and Kawasaki. Therefore, only four groups have performed experiments. Surprisingly, they used spacers between the slabs. No metal plates were used. The parameters for the spacers are, thickness = 65-1550, polyimide, tan a = 10'3 Using pulsed signals at 4.2 GHz, they were able to completely recover (to within 2-3 dB) the insertion loss. The I.L.~10-38 dB. Pin = 7 dBm. 24. 25. 26. 27. 136 The reported data was for the 650 spacer. The GaAs loading loss was ~17 dB. Apparently, the spacers were used. The repetition rate was several tens of hertz to keep heating low. They state the next experiments will use thin epitaxial GaAs YIG films. Note they obtained interaction when the electrons were closest to the YIG by the Hall deflection. Thus, the concept of de-damping definitely was not occuring as could be assumed in the previous experiments. 137 E.10. Review of Spector 1. It appears he treats the forward and backward volume mode cases only. The only conclusion is that gain occurs when uo>vo. A rather inter- esting conclusion is drawn about the role of collisions. Whenx2+0, the carriers and spin system cease to interact in a manner wherein energy exchange occurs. The material is a ferromagnetic semicon- ductor [110]. 138 E.11. Review of Vural 1. This paper appeared in 1966. Here spin and helicon waves are consid- ered. H0 and drift are collinear; thus, the system is considerably different from our cases [111]. 2. They point out several descrepancies between their work (with Bloom) and that of Akhiezer et al. REFERENCES [11 [21 [31 [41 [51 [61 [7] [8] [9] [101 [11] [12] LIST OF REFERENCES R.W. Damon and J.R. Esbach, "Magnetostatic Modes of a Ferromagnet Slab," J. Phys. Chem. Solids, Vol. 19, p. 308, 1961. L.K. Brundle and N.J. Freedman, "Magnetostatic Surface Waves on a YIG Slab,” Electronics Letters, Vol. 4, p. 132, 1968. S.R. Seshadri, "Surface Magnetostatic Modes of a Ferrite Slab," Proc. IEEE (Lett.), Vol. 58, pp. 506-507, Mar. 1970. J.B. Merry and J.C. Sethares, "Low Loss Magnetostatic Surface Waves at Frequencies up to 15 GHz," IEEE Trans. on Magnetics, Vol. MAG-9, No. 3, p. 527, Sept. 1973. K. Kawasaki and M. Umeno, "Influence of Surface Metallization on the Propagation Characteristics of Surface Magnetostatic Waves in an Axially Magnetized Rectangular YIG Rod," IEEE Trans. on Microwave Theory and Techniques, Vol MTT-22, No. 4, Apr. 1974. 0.F. Vaslow, ”Group Delay Time for the Surface Wave on a YIG Film Backed by a Grounded Dielectric Slab," Proc. IEEE (LettL), p. 142- 143, Jan. 1973. J.0. Adam and J.H. Collins, "Microwave Magnetostatic Delay Devices Based on Epitaxial Yttrium Iron Garnet,” Proc. IEEE, Vol 64, No. 5, p. 794, May 1976. J.M. Owens and C.V. Smith, I'Beyond SAW Filters: Magnetostatics Show Promise,” MSW, p. 44, June 1979. J.B. Merry and J.C. Sethares, as in Reference 4. W.L. Bongianni, ”Magnetostatic Propagation in a Dielectric Layered Structure," Journal of Applied Physics, Vol. 43, No. 6, p. 2541, June 1972. T. Yukawa et al., ”Effects of Metal on the Dispersion Relation of Magnetostatic Surface Waves," Japanese Journal of Applied Physics, Vol. 16, No. 12, p. 2187, Dec. 1977. K. Kawasaki et al., "Passband Control of Surface Magnetostatic Waves by Spacing a Metal Plate Apart from the Ferrite Surface," IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-22, No. 11, p. 924, Nov. 1974. 139 [13] [14] [15] [16] [17] [181 [191 [201 I21] [22] [23] [24] [25] [261 [27] [28] [291 [301 140 A.V. Vashkovskii et al., "Instability of Surface Magnetostatic Waves in a Semiconductor-Ferromagnetic-Dielectric Metal Structure," Radio Electronics and Electron Physics, 1974. M.R. Daniel et al., "A Linearly Dispersive Magnetostatic Delay Line at X-Band," IEEE Trans. on Magnetics, Vol. MAG-15, No. 6, p. 1735, Nov. 1979. W.S. Ishak and K.W. Chang, "Magnetostatic-Wave Devices for Micro- wave Signal Processing," Hewlett Packard Journal, p. 10, Feb. 1985. J.D. Adam and J.H. Collins, as in Reference 7. M. Bini et al., "Energetic Derivation of the Amplification of Mag- netic Waves Interacting with a Flow of Charges in a Semiconductor," Journal of Applied Physics, Vol. 47, No. 7, P. 3209, July 1976. M. Bini et al., "Thin-Film Magnetostatic Amplifier: Analytical Expressions of Dispersion and Gain Properties," Electronics Letters, Vol. 13, pp. 114-115, Feb. 1977. M. Bini et al., "Amplification of Surface Magnetic Waves in Trans- versely Magnetized Ferrite Slabs," Journal of Applied Physics, Vol. 49, No. 6, p. 3554, June 1978. M. Bini et al., "Interaction of Magnetic Waves with Drifting Charges," IEEE Trans. on Magnetics, Vol. MAG-14, No.5, p. 811, Sept. 1978. I. Awai et al., "Interaction of Magnetostatic Surface Waves with Drifting Carriers," Japanese Journal of Applied Physics, Vol. 15, No.7, p. 1297, July 1976. J.D. Adam and J.H. Collins, as in Reference 7, p. 796. K. Kawasaki et al., "The Interaction of Surface Magnetostatic Waves with Drifting Carriers in Semiconductors," IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-22, No. 11, p. 918, Nov. 1974. R.F. Soohoo, Microwave Magnetics, New York, Harper and Row, 1985, Chapter 5. Ibid, p. 107. B. Lax and K.J. Button, Microwave Ferrites and Ferrimagnetics, New York, McGraw-Hill, 1962, Chapter 4. Ibid, pp. 146-147. B. Lax and K.J. Button, as in Reference 26, p. 149. Ibid. B. Lax and K.J. Button, as in Reference 26, p. 148. [311 [321 [331 [341 [351 [36] [37] [381 [39] [40] [41] [42] [43] [44] [45] [46] [471 [48] [491 [50] 141 Ibid. R.F. Soohoo, as in Reference 24, pp. 109-113. E. Schlomann, "Amplification of Magnetostatic Surface Waves by Interaction with Drifting Charge Carriers in Crossed Electric and Magnetic Fields," Journal of Applied Physics, Vol 40, No. 3, p. 1422, Mar. 1969. R.F. Soohoo, as in Reference 24, pp. 119-122. Ibid, p. 120. B. Lax and K.J. Button, as in Reference 26, pp. 167-169. W.S. Ishak and K.W. Chang, as in Reference 15, p. 14. B.A. Auld, "Walker Modes in Large Ferrite Samples,” Journal of Applied Physics, Vol. 31, No.9, p. 1642, Sept. 1960. J.C. Freeman, private communication. J.C. Freeman, private communication. E. Schlomann, as in Reference 33. W.S. Ishak and K.W. Chang, as in Reference 15, p. 11. B.B. Robinson et al., I'Spin-Wave/Carrier—Wave Interactions," IEEE Trans. on Electron Devices, Vol. ED-17, No.3, p. 224, Mar. 1970. S.F. Adam, Microwave Theory and Applications, New Jersey, Prentice- Hall, 1969, Chapter 4, pp. 118-122. C.T. Hui, The Field and Carrier Waves Interaction in a Semi-Infinite Semiconductor, M.S. Thesis, Michigan State University, 1982. S.R. Seshadri, as in Reference 3. R.E. 0e Wames and T. Wolfram, "Characteristics of Magnetostatic Surface Waves for a Metallized Ferrite Slab," Journal of Applied Physics, Vol. 41, No. 13, p. 5243, Dec. 1970. P. Young, "Effect of Boundary Conditions on the Propagation of Surface Magnetostatic Waves in a Transversely Magnetized Thin YIG Slab," Electronics Letters, Vol. 5, No. 18, p. 429, Sept. 1969. S.R. Seshadri, as in Reference 3. J.D. Adam and G.A. Bennett, "Identification of Surface-Wave Res- onances on a Metal-Backed YIG Slab," Electronics Letters, Vol. 6, No. 24, p. 789, Nov. 1970. [51] [52] [53] [54] [551 [56] [571 [58] [59] [60] [61] I62] I63] [64] [65] [66] [67] [681 I69] [70] [71] [72] 142 K. Kawasaki et al., as in Reference 12. R. Pauchard et al., "Electromagnetic Surface Waves in a Metallized Ferrite Slab," Electronics Letters, Vol. 7, No. 24, p.428, July 1971. H. Van De Vaart, "Influence of Metal Plate on Surface Magnetostatic Modes of Magnetic Slab," Electronics Letters, Vol. 6, No. 19, p. 601, Sept. 1970. W.K. Bongianni, as in Reference 10. W.K. Bongianni, as in Reference 10. T. Yukawa et al., as in Reference 11. R. Pauchard et al., as in Reference 52. K. Kawasaki et al., as in Reference 12. M. Bini et al., as in Reference 19, p. 3554. Ibid, p. 3555. I. Awai et al., as in Reference 21. B.Vural, "Interaction of Spin Waves with Drifted Carriers in Solids," Journal of Applied Physics, Vol. 37, No. 3, Mar., 1966. M.C. Steele and B. Vural, Wave Interactions in Solid State Plasmas, New York, Mc Graw Hill, 1969. C.T. Hui, as in Reference 45. Ibid. A. Bers and R.J. Briggs, ”Criteria for Determining Absolute Insta- bility and Distinguishing Between Amplifying and Evanescent Waves,” Quarterly Progress Report, No. 71, Research Laboratory of Electron- ics, M.I.T., Cambridge, MA, Oct. 15, 1963. J.C. Freeman, private communication. J.C. Freeman, private communiction. I. Awai et al., as in Reference 21. J.C. Freeman, private communication. A.V. Vashkovskiy et al., as in Reference 13. P.M. Bolle and L. Lewin, l'0n the Definition of Parameters in Ferrite-Electromagnetic Wave Interactions," IEEE Trans, on Micro- wave Theory and Techniques, Vol. MTT-21, p. 118, Feb. 1973. [73] [74] [75] I76] [77] [781 [79] [80] [811 [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] 143 R.E. Collin, Foundations for Microwave Engineering, New York Mc- Graw-Hill, 1966. C.C. Johnson, Field and Wave Electrodynamics, New York, McGraw- Hill, 1965. S. Ramo, J.R. Whinnery and T. Van Duzer, Field and Waves in Com- munication Electronics, New York, John Wiley and Sons, 1965. W.R. Beam, Electronics of Solids, New York. McGraw-Hill, 1965. M. Bini et al., as in Reference 17. N.S. Chang and Y. Matsuo, "Ferromagnetic Loss Effect on Magneto- static Surface Wave Amplification by YIG-Semiconductor Coupled System," IEEE Trans. on Magnetics, Vol. MAG-13, No. 5, Sept. 1977. J.C. Freeman, private communication. M. Bini, et al., as in Reference 17. M. Bini, et al., as in Reference 18. M. Bini, et al., as in Reference 19. M. Bini, et al., as in Reference 20. M. Bini, et al., as in Reference 17. Ibid. M. Bini, et al., as in Reference 18. M. Bini, et al., as in Reference 19. M.C. Steele and B. Vural, as in Reference 63, Chapter 10. B.B. Robinson, et al., as in Reference 43. K. Kawasaki, et al., as in Reference 23. M. Bini, et al., as in Reference 18. B. Vural, "Interaction of Spin Waves with Drifting Carriers in Solids,“ Journal of Applied Physics, Vol. 37, No.3, March 1966. M. Szustakowski and B. Wecki, "Amplification of Magnetostatic Sur- face Waves in the YIG-Ge Hybrid System," Proc. of Vibr. Probl., Vol. 14, p. 155, 1973. A.V. Vashkovskiy et al., Interaction of Surface Magnetostatic Waves with Carriers on a Ferrite-Semiconductor Interface,” ZhETF Pis. Red., Vol. 16, No. 1, p. 4, July, 1972. [95] [961 I971 [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] 144 V.P. Lukomskii and Yu. A. Tavirko, "Amplification of Magnetostatic Waves in Ferromagnetic Films Due to the Drift Current of Carriers, Fiz. Tverd. Tela., Vol. 15, p. 700, Mar. 1973. I. Awai, as in Reference 21. A.V. Vashkovskii, et al., as in Reference 94. T. Yukawa et al., "Absolute Instability of Magnetostatic Surface Waves Interacting with Drifting Carriers in Closed Ferrite- Semiconductor Structure," Trans. on Magnetics, Vol. MAG-14, No. 5, Sept. 1978. E. Schlomann, as in Reference 33. R. Kh. Shapiro, ”Spectrum of Surface Electromagnetic Waves in Ferromagnet-Semiconductor Structures,"Soviet Physics-Solid State, Vol. 14, No. 11, May 1973. K. Kawasaki, et al., as in Reference 23. N.S. Chang and Y. Matsuo, "Possibility of Utilizing the Coupling Between a Backward Wave in YIG and Waves Associated with Drift Carrier Stream in Semiconductors," Proc. IEEE (Lett.), p. 765, April 1968. M. Masuda, N.S. Chang, Y. Chang, ”Magnetostatic Surface Waves in Ferrite Slab Adjacent to Semiconductor," IEEE Trans. on Micro- wave Theory and Techniques, p. 132, Feb. 1974. N.S. Chang, et al., "Characteristics of Magnetostatic Surface Waves in a Layered Structure Consisting of Metals, Dielectrics, a Semi- conductor and YIG," Electronics Letters, No. 11, p. 83, Jan. 1975. N.S. Chang, et al., "Amplification of Magnetostatic Surface Wave Propagation in a Layered Structure Consisting of Metals, Dielec- trics, a Semiconductor and YIG,“ Journal of Applied Physics, Vol. 47, No. 1, Jan. 1976. N.S. Chang, et al., as in Reference 104. N.S. Chang, et al., "Energy Analysis for the Amplification Pheno- mena of Magnetostatic Surface Waves in a YIG-Semiconductor Coupled System" IEEE Trans. on MIcrowave Theory and Techniques, Vol. MTT-25, No. 7, July 1977. N.S. Chang, et al., as in Reference 78. N.S. Chang, et al., "Experimental Investigation of Magnetostatic Surface Wave Amplification in GaAs-Yttrium Iron Garnet Layered Structure," Journal of applied Physics, Vol. 53, No. 8, Aug. 1982. H.N. Spector, ”Amplification of Spin Waves in Ferromagnetic Semi- conductors," Solid State Communications, Vol. 6, p. 811, 1968. 145 [111] B. Vural, as in Reference 92. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 00000001