~ HI 112 203 THS , ,.-,._“ A_AA,___.»:—-w-~v—-A-v 2,33%??? Miefieégm '1‘: r‘ - Ufifivgfififiw J a; I w. This is to certify that the thesis entitled THE FIELD AND CARRIER WAVES INTERACTION SEMI—INFINITE SEMICONDUCTOR presented by Chuck T. Hui has been accepted towards fulfillment of the requirements for M. degree in __SE_iE}£e__ 34w 5. 03m.” Major professor Date M g2 #32 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution MSU LIBRARIES m RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. THE FIELD AND CARRIER WAVES INTERACTION IN A SEMI-INFINITE SEMICONDUCTOR By Chuck T. Hui A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering 1982 ABSTRACT THE FIELD AND CARRIER WAVES INTERACTION IN A SEMI-INFINITE SEMICONDUCTOR By Chuck T. Hui An analytical treatment of the transverse magnetic field waves and carrier waves is presented without the quasistatic approximation. The effects of diffusion and an external magnetic field are included. To evaluate the properties of the carriers at the surface, Sumi's stiff boundary model as well as the "ripple" boundary condition are examined. The resulting dispersion relations are evaluated numeri— cally. Possible instabilities are analyzed, and gain shapes are studied under varying conditions. It is found that gain is possible theoretically under specific conditions. ACKNOWLEDGMENT The author wishes to thank his major professor, Dr. J.C. Freeman, for his guidance, helpful discussions, and advice. ii LIST OF FIGURES ........................ 1. 2. APPENDIX A. APPENDIX B. APPROXIMATION FOR exp [mAD(y2 - y1)] ....... LIST OF REFERENCES TABLE OF CONTENTS INTRODUCTION ..................... ANALYSIS OF THE STWA ................. Formulation of the Problem ........... The Stiff Boundary Case ............. 2 1 2.2. 2.3. The Stiff Boundary Case with Static 80 Field 2 4 The Ripple Boundary Condition .......... EVALUATION OF THE DISPERSION RELATIONS ........ 3.1. Procedure Used in Evaluation of the Roots . . . . 3.2. Evaluation of the Growing Roots ......... SUMMARY AND CONCLUSION ................ 4.1. Summary ..................... 4.2. Conclusion ................... APPROXIMATION FOR THE DIFFUSION WAVE ..... OOOOOOOOOOOOOOOOOOOO Page iv 12 16 22 3O 30 34 57 57 57 59 61 62 Figure #00 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. LIST OF FIGURES Semi-infinite semiconductor and meander line . . . . Rippled beam surface ................ Growing roots under the Bers and Briggs criteria . . . . k1 vs. operating frequency ............. ”C k. vs. -—- ..................... 1 w k vs. gg-(d = 0.1 u) ................ kvs.39(d=0) .................. v ki(max) vs. d ................... Navenumber at maximum gain vs. d .......... wc k1 vs. TS 0f case II ................ k (”C- i vs. a (I? — 5) of case II ............ w k vs. a (I? k vs. a (%% k1 vs. d (case II) ................. Wavenumber at maximum gain vs. d (case II) ..... U k vs. —9-of case II ................ v¢ k1 vs. operating frequency (case II) ........ w . ki vs. I? of the ripple boundary case ....... k vs. the depth factor m (ripple boundary case) wc wc . ki vs. m (ET = 1)"ZT = 1 (ripple boundary case) . . . . k vs. separation in the ripple boundary case . . . . u k vs.-V§ in the ripple boundary case ........ iv 1) of case II ............ 15) of case II ............ Page 6 25 35 36 37 39 4o 41 41 42 . . 44 45 46 48 48 49 so 51 52 53 55 56 1. INTRODUCTION The objective of this paper is to investigate analytically a device called the solid state traveling wave amplifier (STWA). This device is somewhat a solid-state analogue of the vacuum TNT where electromagnetic waves will be amplified if guided by a slow wave structure. The planar guiding structure is assumed to be a meander line. Amplification of waves is due to interaction of drifting carriers in semiconductor with the "external" slow waves. Since the device is so similar to the TWT it seems reasonable to extend the three-wave theory for the TNT to semiconductors as well. Such an idea was first proposed by Pierce and Suhl [1] in 1955. Since that proposal, wave interactions between the slow EM waves and drifting carriers in a semiconductor have been studied extensively. The STWA falls into the very broad category of wave instabili- ties in solids. The literature on this subject is quite extensive. Collectively, the theories of the STWA have been either one or two- dimensional analyses. The one-dimensional investigations follow very closely the successful theory developed by Pierce for the TWT. For reference purposes, the basic dispersion relation obtained by. Pierce in the analysis of the TWT is given as follows: 2 2 . 2 . 2 3 (v ' y1)(JBe - v) = 'JZBeY ch Numerous modifications were made to accommodate space charge effects [2, 3]. Further works in this area are: (1) In 1966, Solymar and Ash [4] furnished a one-dimensional analysis for an n-type STNA similar to the analysis for the vacuum TNT of Pierce. (2) Fujisawa (1968, 1970) worked out a transmission line analogue and a kinetic power theorem for carrier waves in semicon- ductors [5]. Similar works in transmission lines analogues were done by B. H0 in 1970 [6]. (3) Coupled mode theory for STNA was given by Fujisawa and Ishikawa in 1969 [7].’ However, it should be pointed out that though a one-dimensional analysis has been successful with TNT and ANA (acoustic-wave ampli- fier) and that it is much simpler than the many two-dimensional analyses used, it cannot be extended toliueSTNA without modification- One big difference between the TNT and the STNA lies in the fact that in the former the space charge waves are weakly damped and the carriers are inertia dominated. In the semiconductor, the stream is collision dominated and the normal modes are highly damped. Further discussion on these will be found in reference [8]. The two-dimensional models are basically field analyses and yield considerably more information about the device. In 1966, 1967 Sumi [9] published a two-dimensional analysis of the interactions between the surface TM waves on a semi-infinite semiconductor medium and the adjacent planar slow wave structure as an infinitely thin current sheet. An extension of his work was done by Vural and Steele (1969) and Freeman (1972) [10, 11]. In their work, as distinct from Sumi's stiff boundary model with zero transverse current, they have taken into account the effect of the surface charge arising on the semi- conductor by using the "ripple" boundary condition. Kino [12] and other authors have also applied the ripple boundary condition in the analysis of thin semiconductor slabs and wave instability in zero diffusion limit. In these two-dimensional analyses, many researchers have pre- dicted an extremely large gain of tens of decibels per mm length of circuit length. Yet the experimental results so far have been inconclusive though promising [13-15]; no net terminal gain has been achieved. This discrepancy between theoretical prediction and experimental result can be related to a simplification of model or imperfect experimental arrangement. From the experimental side: Nas gain unachievable solely due to the high transmission loss of the slow wave circuit? Or is the geometry of the meander line and its separation from the semiconductor a factor in achieving strong interaction between the drifting carrier and the slow waves? Two major centers of difference can be noted from some of the above theories, as suggested by Freeman [16]. The first point of disagreement centers about the importance of diffusion forces in the model of the carrier stream. Understandably, the omission of diffusion simplified the mathematics considerably. Yet, it has been shown that the effect of diffusion is important and must be included [17]. The other disagreement concerns the boundary condi- tions to be used at the surface of the semiconductor. This will ultimately determine the mechanism of coupling between the stream and the slow waves. In the analysis to be presented here, diffusion; will. be included. Also, a new model for the boundary condition essential to the founding of a dispersion relationship will be established. The analysis is essentially a two-dimensional field analysis follow- ing closely that of Sumi, Freeman, and that of Okamoto and Mizu- shima [18] especially, since a static B field (magnetic field) is included in the analysis. A computer is used to aid in the solutions of the dispersion relations. Theoretical results were obtained and an interpretation of them is presented. 2. ANALYSIS OF THE STNA 2.1. Formulation of the Problem In the analysis that follows here, we will consider a semi- infinite region which fills the lower half space x_: 0. The meander line is represented by a current sheet at distance d above the surface of the semiconductor. The semiconductor is assumed to be uniform, isotropic, and homogeneous. Uniform, static electric and magnetic fields E0 and B0 are applied in the -z and +y directions, respectively, as shown in Figure 1. The linearized forms of Maxwell's equations, along with Poisson's equation and the equation of motion, are used to describe the behavior of the carriers in the material. All the RF quantities are assumed to vary as exp. j(wt - kz). Furthermore, we make the following assumptions: (1) small signal assumption; (2) the carrier stream is collision-dominated; Iw - Buo|<- '* 2173 —>- JwV1--m* (E+v1>< [—,—E+j——v-E+v(v-E) -VE] 2 2 c c mm m U + + = -i —7§-E + C20 x (V x‘E) + i-g%-V(V°E) (7) c c c Assuming the fields do not vary with the y-coordinate, that is, §%-= O, we can separate them into TE and TM waves. This assumption IS Justified if the semiconductor extends far enough in the y direc- tion. Since the TE wave has only a y component of the electric field, we will study the TM mode here. 10 We assume the x variation to be eYX; thus, the coordinate dependence of all a.c. quantities is exp (-jkz + yx) where the time dependence is understood. The TM field is .+ _ . = vx-sz E(x,z) (Exax + Ezaz)e where Ex’ Ez are complex amplitudes. Then = _ . vx-jkz [7Ex JkEZJe .‘* ___ A _./~, _. YX-jkz V(V E) [yaX Jkaz][YEx jkEZJe 2* V E = [y2 - k2]E Substituting the above equation into equation (7) and equating the a and a components, we obtain X Z 2 m w ku mu . wD 2 2 w . c . c o . o a : E [j ——'Y - k +-- (1 - J “—0 + J + Ja(k ' ‘——§YJ x x c2 c2 w C2 C2 we u0 w2 mu ok =EI-J'k(1-=1-)Y-—- ——2— oY-“(Yz +—2—-———)1 c2 c c c (8a) w2 . Dk ““0 2 a = E [Y(-Jk- w + j--) + a(- - k )]= z x C2 C2 C2 w2 u ok E ZIY2 + ---(l - ---i - j— %(w + k 2D) - iakY] (8b) C2 C2 Now using 11 the notation: 2 u k _ w 0 . w 2 ‘-—— (1 --———) - J-—— (m + Dk ) c2 w c2 C mu . . o . wD -Jk[1 - —- - J — kc2 c2 . mD J ___- C2 2 w 2 w ___ k -'—§ (1 ' J u)) c -J’k(1 - .i 9-2) C LOU . o Ja(k - —) c2 w _. a-——¢n c2 2 w 2 a(-—- - k ) C2 jak J wckuo C2 (1) U .. (:2 'Y , C then equations (8a) and (8b) become 12 [6Y2 - d + P + fv]Ex = [ev - avg - 9 + q)Ez (9a) [aw-rug = [Y2 + a — hy]EZ . (9b) 2.2. The Stiff Boundary Case If we drop the terms due to the magnetic fields, i.e., p, q and d = O, we have Sumi's case: [6Y2 - dJEx = [eYJEZ (lea) [butx = [92 + aJEZ (10b) In order to have a non-trivial solution, the determinant 6Y2 — d -eY = O by ->|Bij, + and if B0 = 0, Bi is still not of significance in the motion. Thus, equations (9a) and (9b) lead to: 4 3 3 Gy + (f - Gh + ab)y + (a0 - d - fh - eb + a£)y + (dh + af - e2 + gb)y - da + 92 = 0 . (14) The coefficient of y3 vanishes, and the only non zero term in- the coefficient of y is much smaller in magnitude when compared to coefficients of other terms. Thus, equation (14) reduces to If we assume that the constant term is still a product of the roots, the above equation can be rearranged as a bi-quadratic with the roots perturbed from the stiff-boundary case. Two possibilities exist here: either Y1 or Y2 is perturbed. After careful evaluation, it is found that v2 is perturbed. The 2 solutionscfiiy in equation (15) are given as follows: 17 Y2 2 w ”c i=d‘k "2‘1"”; C L . -a + 9—. w - 2 _ (i z 2 _2. ..9 2 Y2 - G k + I) + J D (1 + a ) ° Utilizing equations (9a) and (9b) again, we obtain the relationship between A1 and BI’ and A2 and B : 2 __°_ C Bl VIE-Y1 + jak(1' 8)] A1 (16a) 2 _ 'ak ; Y2 3 Y2 (16b) B2 ‘ wz 2 A2 -Jk(1 ' 5)Y2 + “(‘5 - k ) c wuo where B =-——§ and j-—§ is dropped when compared to unity. kc c For reasons stated previously, we 2 - wa -kY1(1 ' j 3') + jak le Hy - k . 2 Ale (Jak ' Y1) MkUw-YU) YX + 2 g 0 2e 2 (17) jky2 + ak Setting le = 0 at x = 0, i.e., the transverse component of the RF current being zero, we obtain the relationship between A1 and A2 , 18 Y1Y2&(Jak - Y1) A1 = -j 2 2 A2 (18) k (YlwC - ak uo) It is comforting to note that in the absence of the magnetostatic + field 30’ a = 0 the above equation reduces to the previous case as it should be. Substituting the above relationship into equa- tion (17) yields in _ .iweAzrjrzwhl - .iak)(1 - 17c) Y X Y U Y X 1 + 2 O 2 } Hy - 1 e e (19) k 2 w k[y1wc - ak uo] Finally, we will match the fields at x = 0 H5’s Hyc 5—." = 1:— (20) 2S zC where 2 2 _ Y u k (y w - ak u ) Hys .iweSIJ'YZMYl - iak)(1 - i 7?) + 2 ° lwfi ° EzS [k2(Y1wc - akzuo) + jrlrz 5(Y1 - jak)] and “yc jwec(282 - 283d - 83 E = 2 4 2C (880 - 28 d) with 19 .4 H I 7:. a 2 ° £%.[1 + j Eflirgg9LlJ (see Appendix A) (I) .< N II C where D wC After a lengthy algebraic manipulation, we arrive at a 6-degree complex polynomial dispersion relation for the stiff-boundary case, + with B0 = ayBo' For convenience it is represented here by 6 5 4 3 2 1 _ alk + blk + clk + dlk + elk + flk + g - 0 . Coefficient of k6: . 2dK¢u§o a1 = J w Coefficient of k5: 2duoKa . b1 = [2daxD + 2u0¢d(1 - K) + a) (owc - 1) - jZaouod] Coefficient of k4: 2dwCA uo c1 = {[2a(d — AD) + 4d¢w(K - 1) - 4dKo - 2u0¢ - D] 2de + j[2d(1 - K) + 4a¢wd(1 - K) + C (o - no) + 2a¢uo]} LU 20 Coefficient of K3: 2 . .Zw _ zoo d 2dwa __g; d1 - {F_U——— (1 - K) +-—Er— (K - 1) + 11 (ADi-Kd) 0 O 0 dwc aK + 244(2 + ) - 2a] + 11de (K - 1) u0 + ZdKUMP (w + w) - 2a¢w(2 + 999') " 2 uo C uO 2 2 2dKocwc - kOKouoaJ} (U u0 Coefficient of k2: 2 k Ku a e1 = {[kgouo(1 + K)+ °T ° (1 - iw ) +— “(a - 4w) uO + kgako] + 353% (1 + ooh) — kguoooJ} Coefficient of k1: (—- + 2km) = {[oK§(1 + K)- 2k §o¢(1 + K) + kow c ”o + j[k§ (1 + K) + 2q¢wk2(1 + K) + k3 ch (—-+ ¢)l} Coefficient of k0: 21 ¢w2k2 awkz k2 91=II u °<1+K)-—u-9<1+K)- ° °<1+¢w>1 0 O O kgK 2 wkg +J'I-J-(ounc-w-4wc-9wa)--l-‘-(1+awn} O O (21) 1 + a2 where ¢ = 2 “c It is interesting to note that in the case with no magnetic + field 30’ and the relationship between A1 and B1 being -1 the Y1 mode is called the solensoidual wave since 0 '* p V°E(Y1) = E- s 0 everywhere . '0 II + However, with the addition of the B0 field, this is no longer true. -"< >=—9- VEYI E: #0 S Therefore p f 0. The differencesbetween this dispersion equation and the one by Sumi are (1) the existence of an external magnetostatic field _). Bo represented by a, and (2) the approximations for a2 differ. Nhen 22 these significant differences are removed, the 6-degree polynomial here reduces to that of Sumi. It is important to point out that the above dispersion rela- tion, with or without the static magnetic field 66, is derived via the stiff boundary condition. However, it was point out by Dean and Robinson (1974) [23] that this condition is questionable when compared to experimental results and a free boundary condition must be used. Freeman in his dissertation used the ripple boundary con- dition, but only studied the dispersion relation for special cases. Ne will attempt to find a dispersion relation of the system by adding a new constraint. 2.4. The Ripple Boundary Condition It is apparent that in Sumi's development of the stiff boundary condition, he assumed the carriers only have motion parallel to the physical surface of the semiconductor at x = 0. He was able to accomplish this by using the two basic modes Y1’ Y2 in such a way that the transverse component of the current is zero, i.e., le = O at the physical surface. This constraint can be somewhat relaxed if we assume a ripple boundary. Let us say the electrons have a transverse RF velocity and cause accumulation and depletion alternatively near the surface during each cycle. This change in carrier density results in an RF surface charge, p As demonstrated by Collin [24], this surface S. charge results from two causes: 23 (1) the transverse conduction current, Ox; (2) the charge being carried to a given point 2 due to the surface charge moving with the beam. Ops _ . _ Ops . at I prs — Jr ' mo '57:- ” Jr + Jkuops Thus _ Jr 'pon 0s ' in“ = in (22) Since this surface charge drifts along with the beam, it constitutes an RF surface current, K =up (23) We now proceed to model the system as such. (1) Near the surface of the semiconductor there exists a per- fect depletion region which extends about .Su into the semiconductor. (2) Since the surface of the semiconductor is rippled, standard boundary conditions cannot be applied. To remedy this we shall use the method described by Hahn [25], that is, to represent the ripple boundary by a static one with effective surface charge an current. At this static boundary, x = 0, which is not the physical surface of the semiconductor, but the unperturbed stream boundary. (i) E = EZ (tangential E must be continuous) c s 24 (ii) eCEX = ESEX + ps' (normal E must be discontinuous by c s and effective charge ps) (iii) H = H + K (tangential H must be discontinuous by yc ys S Ks) Conditions (i) and (iii) are useful for obtaining the dispersion relation. (3) Both the solenoid (yl) and diffusion (72) waves are highly damped and exist only on an extremely thin layer, with the former to within a skin depth and the latter to within a Debye length. Therefore at some depth from the surface vx(mAD) = 0 m = 1, 2, ..., 30 (24) (The geometry of the model is shown in Figure 2.) Assumptions (2) and (3) are of interest for obtaining the dispersion relation. Applying equation (24), the relationship between A1 and A2 YlmAD . ”(1)/2‘3“”k ' Y1) Y2on 1 ’ '3 2 2 2e k (Ylwc - ak uo) (25) or rewriting, A1 = -. Y1Y2&(iak - Y1) emAD(v2-Y1) — J A k2 with the approximation (see Appendix B) m4 (Y -Y ) , 2- - e D 21 mu)(. mu) 2w 2wc C 25 Figure 2. Rippled Beam Surface 26 Thus 51-: _j Y1Y2w(jak ' Y1) m2& (. _.flié) A 2 2 2w 2w 2 k (Ylwc - ak uo) c G Since _ -oovx(0) ps jw We must now evaluate vx at x = 0. . 2 2 vx(0) = 451:- A2 {[n2(k2 - fi) - ikv21+ [n1(k2 - 1Y1Y2w(Y1 - jak) . LAEQ-<' -.fléL)]} 2wc J 2wc k2(ylwC - akzuo) with _ . 2 mm -ka -av —a— B1 L 1 1 c2 n1 = A1 - Y1['Y1 + jak(1 ' 3)] YE - jakrz >|Ua N n 2 2 . w2 2 -Jk(1 - B)Y2 + 00-5 - k ) c The transverse admittances must match. Thus, I ___3 ye E E2 (25) 2 2L. - °k . C2) 3 Y1] (27) 27 The dispersion relation is at least thirteenth degree unless simplified. To avoid confusion and error, we will study the case where a = 0; no magnetostatic field. With this simplification, the dispersion relation will reduce to sixth degree. The quantities are -2Y d _ . c 2 2 Hyc - jwec[(1 + e )k - k0] -Zv d _ 2 c 2 E2 - Yc[k0 - (1 - e )k ] c juuoczpo 2 w2 . 2 w2 . KS = __m—c— A2 {[n2(k " c—Z') " JkYz] + [n1(k ' 2‘2‘) ' Jle] ° iv1Y25(Y1 - Jak) 2- - . 1 11M (i-flw—m k2(ylwC - akzuo) 20°C 2(”c iv Y 6(Y - Jak) 2- - E =A2{1+[212 1 1[%2(i-§fii;)11 Zs k (ylwC - akzuo) c A iv v 6(v - jak) 2- - Hy =5ug-Hkn1-1Y1N212 1 2 1%flu-gfln s o k (ylwC - ak uo) c c + (knz ‘ jY2)} 28 After a very lengthy algebraic manipulation, we arrive at the final dispersion relation 6 4 3 + e k2 + f 5 _ azk + b2k + czk + dzk 2 2k + 92 - 0 where dmu3 kmu3K a = ___O + j ___—0.. 2 Zwb Zwb 3dmou§ 2 . 2 m 11ng b2 = {(- amt "UodK) + JIdUO(1 +'§i ' TEE: ("o + 3dw)]} {[3dmw2uo 2 ( )1 [ 4duowE c = -—-————— + u K + u d 2wK - w + j - 2 Zwb o o c m2w¢ 3uomwK 4‘ T (U0 + dw) - dwu0(2 + 01):” 2 3 3 k mu _ dmw _ o 0 d2 - {[dwwC - wK(2uo + wd) - 2m 4w c c 2 3 2 k u m . 2 m w mK o o + j[dw [1 +-§] "7EET (3uo + wd) + 4wc ]} 2 2 2 2 k u 3 k u _ 2 _ o o 3mg . mw K _ o o .m_ 3mwK 92 " {[03 K 2 (zwc + K)] + 3':ch 2 (2 + 1 + 2mg )]} 29 2 .3kgmw2uo . 2_ . m f2 = {[kouo(wc + wK) - _—45;___] + j[kowuo(1 +-§] 218,132.; 3k2u maZK + 0 C O + 0 0 J} 2’ 4w m w¢ c k§w3m kgmw , kng m 92:“: 4Q) "' 2 (wC+wK)]+j[-T(l+§)- c We assume ¢ = 1 - kAD s 1 and w c o>>a . is assumed to be real instead of complex; i.e., 2 3 komw K 4wC ]} 3. EVALUATION OF THE DISPERSION RELATIONS In this chapter, the propagation constant k is examined by solv- ing the dispersion relations obtained in the previous chapter. In the past, many attempts to find k involved the use of the perturba- tion method. It is often assumed that the coupled modes differ from the normal modes by very small perturbations. There has been con- siderable disagreement in the literature concerning the validity of the perturbation approximations. With the advent of fast computers, numerical solution seems very convenient. This is the method we adopt in finding the roots for the dispersion relations. 3.1. Procedure Used in Evaluation of the Roots The evaluation of the dispersion equation involves three parts: (1) finding all the roots for the dispersion under a specific set of parameters, (2) evaluation of possible growing root(s), and (3) examination of the behavior of the growing root(s) under varying conditions. The first part is achieved through the use of a subroutine currently in the MSU computer library, called ZPOLY. This subrou- tine will find numerical solutions of complex polynomials up to 49th degree. We believe that this is far more accurate and convenient than the perturbation method. In the second part, we are entering the rather confused area of instability criteria. One approach for physically interpreting 3O 31 the solutions of the dispersion relation is based on the complex- mode method and the kinetic power theorem [26]. This method separates components of an interacting system into weakly interacting modes and studies them individually. As in the special case of ANA, one can decompose the system into a collision-dominant carrier stream and the sound waves. If the stream interacts with a sound wave with phase velocity less than the drift velocity of the electrons, it becomes "active"--it loses energy not because of collisions but due to interaction.with the "passive" sound wave. If the energy flows of these two modes are in the same direction, we have a con- vective "spatial" instability. If the energy flows are in opposite directions, then a nonconvective (temporal) instability occurs. An example of this nonconvective instability is the BNO, where the energy flows in the two subsystems are oppositely directed, thus only oscillation can occur. The above criteria set forth a necessary condition for convec- tive instability to occur in the system. To determine which mode(s) is actually unstable due to the interaction, we will utilize the Bers and Briggs criteria for convective instability. This approach is based on the principal of causality and the initial value problem. It essentially studies the asymptotic response of the system to a signal bounded both in space and time [27]. In application, it offers a way to test the system for convective instabilities. The procedure applicable to our case is outlined here [28]: 32 (1) Solve the dispersion relation D(w,k) = 0 for realu1(i.e., setting “i = O) and find complex k (k = kr + jki). The set of complex k gives forms of solutions for z > 0 and z < 0. (2) For the given geometry, we are only considering roots that correspond to solutions that exist for z > 0. This means only roots with ki greater than zero are possible candidates for insta- bility. The sign of kr is irrelevant. (3) If ki has a change of sign as we increase mi from zero to negative values, then a convective instability does occur and convects energy to the right, i.e., z > 0. From the previous chapter, there are three dispersion equations as a result of various boundary conditions and assumptions. Sumi's general dispersion relation is a fourth-degree complex polynomial with coefficients composed of the following parameters: (1) d is the distance between the plane of the meander line and the semiconductor surface. For the stiff boundary cases, this distance is just the thickness of the insulator. In the discussion to follow, d is to vary from O to In. (2) U0 is the drift velocity of the electron beam. The coupling mechanism of the system is affected as we vary uo. If the drift velocity is increased to a value near the phase velocity of the meander line, we hope to see a "growing root." 33 (3) w is determined by the carrier density. (no is assumed c to be able to vary from 1013 to 1014 cm'3). Thus wc can vary from 1010 to 1011. (4) v¢ is the phase velocity of the "slow" wave. Due to physical limitation of the "slow" factor, it is allowed to vary from 106 cm/s to 107 cm/s. (5) k0 is a measure of the "slow" factor of the line; k0 = w/v¢. (6) D is the diffusion constant; for GaAs it is about 220 cmz/s. (7) K is the ratio 5 13 c E— ___ c 4 III For the stiff boundary case with external transverse magnetic field, one more parameter is added. a is the product of the carrier mobility "m and the magnetic flux B The electron mobility of 0. GaAs is about 8500 cm2(vs)'1. The BC field is allowed to vary from 0 to 10 KGauss. In the final ripple boundary case, a is to be zero and the ”depth" of the ripple is represented by the dimensionless m. The above parameters will be varied within set limits and the movement of the root(s) will be plotted in the next section. 34 3.2, Evaluation of the Growing Roots Recall we are looking for wave solutions of the form exp [j(wt - kz)] which represent waves traveling in the z direction. If k1 changes sign as we vary wi from zero to negative values, we have an amplifying wave. Contrary to conventional belief, the sign of kr does not indicate the direction of wave propagation. For a more detailed discussion of this topic, see reference [27]. In Figure 3, movement of the roots increasingly negative. Only one of the possible four roots is a "growing" root. As described in the previous section, this wave grows as exp (kiz) as it propagates toward 2 > 0. The propagation constant of this root is near the value of the slowing parameter k0. This requirement is necessary because only the normal modes that can be excited by the meander line are of interest. Nith w being purely real, we step the excitation frequency of the line in 1 GHz increments and note the change of k1 of the growing wave. The result is shown in Figure 4. The gain in dB per wavelength is expressed by mfg—B) = 54.58lk1.|(;09) . Gain is possible over a range of 27 GHz, and limited by the carrier velocity uo. Interesting results occur when we vary the conduction fre- quency wc. It seems that wC must be near the operating frequency w if gain is to occur. As shown in Figure 5, the gain increases with increasing wC up to a maximum, but decreases gradually after 35 g I I 5 l L. k -104 -1o2 102 104 I _ -102 (P wai =" 109 Sumi's case d = 0.2u f = 2 GHz v o: 7—. 1.3 Figure 3. Growing Roots under the Bers and Briggs Criteria -1 ki(cm ) 36 Sumi's Case (1) C - .a-ZOO 'ZF - 3 d = 0.3 N u = 2x107 cm/s v- = 1.2x107 cm/s no = 2.95x1013 cm"3 J l j at 2 9H2L A] 2 4 8 12 I6 f(GHz) Figure 4. ki vs. Operating Frequency 37 ki(cm'1) .. 200 .. 150 Sumi's case _ 0 f = 2 GHz v¢ f 2.5x1o6 cm/s U —°-= 1.7 ”4 (”C Figure 5. k1 vs. 73- 38 that. Thus the carrier density no is an important factor in the coupling mechanism. Also, the threshold and the magnitude of the gain decreases in general as d increases. The reason for the decrease in gain magnitude is the less effective interaction between the circuit and the beam as the separation between them is increased. In Figures 6 and 7, the general variation of the real and imaginary part of k is shown as a function of the drift velocity. Once again near-synchronism is a criterion for gain. Maximum gain occurs when drift velocity is around 1.5 times that of the phase velocity of the line. A closer comparison shows a surprising fact: greater separation does not show a reduction in gain. This result was observed by Freeman [29] in his theoretical analysis of the behavior of the roots. Figure 8 reveals the existence of an optimal distance of separation for the case we study. Figure 9 shows that as the distance d increases kr approaches k0; the system decouples and only the normal mode(s) exists in the line. The above illustrations conclude our investigation on the stiff boundary case pioneered by Sumi. Ne now turn to the case with the transverse magnetic field and the modification in v2. As expected, the plot in Figure 10 resembles that of Figure 5 very closely. The only noted difference between the two is the lowering of the threshold for gain. This is understandable because if a = 0, then the only contrast remaining is the modification of Y2- 39 -1 - +ki(cm ) ' kr(cm 1) ,.80 e 6980 -’60 + . + - 4o ._ 6910 ~‘20 1 l 1 1 I L 6820 1 o 1.5 2 o 2.5 3.0 1‘9. V¢ . n . -20 Sum 5 case d e 0.1 u v¢ = 2.5x106 6 -40 f = 2 GHz U Figure 6. k vs. -9-(d = 0.1 a) 4 40 ki(cm-1) ' kr(cm'1) V- 32 1 7390 —24 r 16 --b 7360 .. 8 + l l I l I 1.0 1.5 2.0 2.5 3.0 7330 ”_o v¢ . - -8. Sum '5 case d = 0 _ 6 v¢ - 2.5x10 ”'15 f = 2 GHz uo Figure 7. k vs. —— (d = 0) ¢ 41 ki(max)(cm'1) C(%? T90 l I l J .2 .4 .6 .8 Separation (u) Figure 8. ki(max) vs. d kr(cm'1) Sumi's case v¢ = 2.5x106 7000 ’ u0 = 5.0x106 f = 2 GHZ x é 10.5 a 6000 _ l J I _1 4 500° .2 4 6 .8 Separation (u) Figure 9. Navenumber at maximum gain vs. d 42 ki(cm-1) F 600 - I \o d = .1 u b 400 d - .5 u l l l l _J 2 4 6 8 10 39. (U a = 0 v¢ = 5x106 U 79:18 ¢ f = 2 GHZ (D Figure 10. ki vs.-Z% of case II --- 43 The variation of the imaginary part of k as a function of a is shown in the next three figures. In Figure 11, ki increases to a maximum value, then decreases gradually afterward as we increase the magnetic flux. However, in Figure 12, the variation of ki does not follow the same trend as in the previous case. The magnitude of ki actually decreases as we increase a, reaching a minimum, then rebounding to a less negative value. Figure 13 shows still another variation. Note that the conduction frequency wc is the only sig- nificant parameter varying in these cases. It seems as if the intro- duction of magnetic field will disturb the synchronism of the system. If the system is already at synchronism (;%-= 1), an increase of magnetic flux has the effect of "lowering" wC to below synchronism, thus the decrease in gain. However, if the system is much beyond the synchronous condition, the applied 86 field will bring the system into synchronism. This effect becomes more apparent if we examine the expression for wc: nlelum we _ € + Due to the applied 80 field, the carrier mobility ”m is reduced to 1 ______j . 1 + 02 Thus, the "effective" conduction frequency is Physically, this decrease in mobility probably reduces and limits the amplitude of the a.c. motion of the carrier. 44 ki(cm'1) t' 500 ”400 P300 P200 d = 0.3 11 U0 = 2x107 cm/s u —° = 1.7 V4) L' 100 95. = 5 (.0 f = 2 GHZ 2 4 6 .8 10 t (I u’c Figure 11. k. vs. (1(z;-= 5) of case II 45 -1 .. +ki(cm ) ' kr(cm 1) l l 4200 4 if- 600 ”' 400 3600 . L 200 3000 ' - ~200 * -4OO - -600 (D Figure 12. k vs. 01 (Uc- = 1) of case 11 46 300 250 (cm-1) 7800 200 7700 150 -»7600 1 I 1 l _ 1 # .2 .4 .6 .8 1.0 (I. f = 10 GHZ u0 = 2x107 cm/s Uo _ va-- 1.7 w —c = 15 w d = 0.3 u (A) Figure 13. k vs. a (]§-= 15) of case II 47 Quite different from the Sumi case, the imaginary part of k decreases monotonically as the distance d increases (as shown in Figure 14). One possible explanation could be the effect of the magnetic field on the "diffusion" wave YZ' Ascxincreases, this. mode may become more "damped" and cancel any benefit from an optimal spacing. This is merely a speculation and the problem is open to discussion. Figure 15 shows that as separation increases, kr approaches the normal mode k0. Near-synchronism as a necessary condition for growth is demon- strated in Figure 16. Figure 17 shows the growth rate as a function of operating frequency. The above illustrations show some resemblance to the Sumi case as well as some marked differences. ‘One thing to remember is that both cases assume a stiff boundary condition. It would be interesting to see how the "ripple" boundary case compares with these results. As shown in Figure 18, the "window" for gain is much narrower and the decay much more abrupt than in the previous cases. Gain is possible only when wt is about 2m to 4m, though the gain can reach as high as 20 dB per wavelength. Figures 19 and 20 show the variation of ki as a function of the ripple "depth" m. The gain increases as m increases. This result might not be as meaningful if we consider the physical viewpoint. Assuming the "ripple" is 48 ki(max)(cm'1) G(%? 400 . 300 200 100 l l J l .2 .4 .6 .8 Separation (u) Figure 14. ki vs. d (case II) J kr(cm'1) 4000 = 0 _ 6 - 5x10 cm/s uo _ 3500 "“17 = 2 GHZ - 1 3000 = 17.9 1 l I I , .2 .4 .6 .8 Separation (u) Figure 15. Wavenumber at maximum gain vs. d (case II) 9 HH ammo mo.|p..m> x .mH assure 49 o 7% m\su oH u e> 5. OOH-- - a m o u u and .e|> Nxm OH u c o: . o.m mwm o.N im-H _ _ . . T com“ OOH 1 - ooom com . - oofiw com - - comm my» " L AH-EUV x ace L 50 -1 ki(cm ) on = 0 d = 0.3 .150 u0 = 2x107 cm/s U 79 = 1.7 ¢ III—c. = 3 w I I I I I J A 4 6 8 1o 12 14 16 f (GHZ) Figure 17. ki vs. operating frequency (case 11) 51 ki(cm‘1) 41—105 F104 -103 -102 ~10 I J J 1 L 1 1 L 1 A o 1 2 3 4. 5 .6 7 8 9 10 (I) ._2 _ (I) : L— :j _ w = 10 GHZ u0 = 2x107 cm/s U I '52 = 1.7 (I m = 4 d = 0.6 u m Figure 18. ki vs.-Z% of the ripple boundary case 52 -1 - +ki(cm ) b kr(cm 1) -2500 6000 - r2000 5000 a *-1500 4000 - L'1000 3000 4 F500 xki(stiff) 2200 g I g I l 1 '2 4 6 10 m f = 10 GHZ v¢ = 1.18x107 cm/s u0 = 2x107 cm/s d = 0.6 u w _E.= 3 m Figure 19. k vs. the depth factor m (ripple boundary case) -1 _ o kr(cm ) ki(cm-1) r 6700 - 6600 4 — 1340 ' b I 53 6500 - 1290 6400 f 1230 6300 ; 1180 1 l l I 1 ' 0 2 4 6 8 10 f 2 GHZ d 0.3 0 ¢ 5x106 cm/s u0 = 9x106 cm/s V we we Figure 20. ki vs. m (I? = 1)"07 = 1 (ripple boundary case) 54 symmetrical about the x = 0 line, the "ripple" might extend well into the insulator region if m is large. This would imply that carriers are injected into the insuIator! In Figure 21, k1 decreases if the separation is increased while kr increases to an optimal value of 2800 cm"1 before decreasing. In Figure 22, we confirm the necessity of near-sychronism for gain. 1000 800 600 400 200 -200 55 -1 -1 dB +ki(cm ) o kr(cm ) G (7f0 1 ‘ l — 2800 ‘ 21.0‘ _ 2700 - 16.8- ~ 2600 .. 12.6‘ . 2500 .. 8.4.. - 240' - 4.24 1 I L L I J I I .2 .4 .6 .8 1.0 1.2 1.4. 1.- 8 Separation (u) L f = 2 GHZ m = 3 u0 = 2x107 cm/s v¢ = 1.1x107 cm/s (I) _£.= 1 (A) Figure 21. k vs. separation in the ripple boundary case 56 -1 +k. cm .. F 1( ) o kr(cm 1) 6000.. 10001 0 5500 - 500, o 5000 .. I I I 0.5 1.0 1 2.0 2.5 32 v 4 4500 - -500 _ 4000 - -1000 - 1 f = 10 GHZ m = 4 q 3 0 _ d = 0.6 11 5 0 -1500 m __C_ = 3 w l- v¢ = 1.17x107 cm/s 3000 -2000 - J . u '1 Figure 22. k vs. 79 in the ripple boundary case (b 4. SUMMARY AND CONCLUSION 4.1. Summary An analysis of the interaction between slow circuit waves guided by a meander line and drifting carrier in a semiconductor has been given. The system is decomposed into its components: the circuit and the carrier stream in the semiconductor. The investi- gation emphasizes the importance of the properties of the carriers at the surface and explores different models for its behavior. The theoretical treatment follows that of Sumi, Freeman, Okamoto, and Mizushima closely. Sumi's "stiff boundary" case is studied, and extended to include an external magnetic field. Secondly, the "rippled boundary" is applied and both modes are kept in our analysis. A new constraint is introduced to treat the ripple boundary condi- tion. Evaluation of the dispersion relations is carried out numeri- cally through the use of a Fortran subroutine. Possible convective instabilities are examined through criteria adopted from Bers and Briggs. Gain ranges and gain shapes are studied under varying condi- tions. Theoretical gain for STWA is predicted. Also, we find the frequency range for STWA operation to be around 27 GHz. This band- width is limited by the maximum carrier velocity obtainable in GaHs; 2 x 107 cm/sec. Theoretical gain for STWA is predicted. 57 58 4.2. Conclusion The results from the different models are similarly encouraging. Amplification of signal due to a "collision dominated" carrier stream is possible theoretically if the conditions are proper. This concep- tual problem can be viewed in light of the success of the AHA where the carrier stream is "collision dominated." Also, the interpreta- tion of gain is supported by the Bers and Briggs criteria used con- sistently in the investigation. The importance of synchronism in velocities and frequencies demonstrates itself in our study and cannot be understated. Gain 0.) is predicted whenever-—9-and'3§-are near unity throughout our inves- v tigation. Other factors which might enhance the possibility of gain include: (1) the separation between circuit and semiconductor; (2) the application of a transverse magnetic field to "tune" wc, thereby bringing-gg-toward unity when interaction occurs; and (3) the carrier density of the semiconductor and ultimately the material itself. In light of the results we gathered, I propose further inves- tigation in the applicability of STWA. APPENDICES APPENDIX A APPROXIMATION FOR THE DIFFUSION WAVE The expression for Y2 is . 2 _ 2 we + 3(1 + a )(w - kuo) £ Y2 - [k + D. ] . Since w c 2 'D—>>k , then 2 . we . (Id-a )(w - kuo) % Y2=['E'+J D ] Assume k is purely real for the time being, and utilize the relationship [x + jy]é = r% (cos-%-+ j sin-g) where r = [x2 + yZJI, cos B = é-and sin 6 = %-. Then 02 (1 + a2)2(w ku )2 1 _ C 0 2 r - L—— + ] 0" D2 60 Assuming that in many cases, 0) >(1+012)(w-ku) c o ’ then (.0 rét—Dfifi. Also, cos-g = E% (1 + cos 6)]% = [g— (1 + 91* 4w2 + (1 + a2)2(w - ku )2 = I c o I 2 2 2 4wc + 2(1 + a ) (w - kuo) s 1 and . 0 $111 2 = % (1 - %)]If 2 2 2 = (1 + a ) (m - kuo) I 402 + 2(1 + az)z(w - ku )2 c o (1 + a2)(w - kuo) = 2wc Thus, w 2 - Y2 t-Dfiiiu + 1' W1 . C APPENDIX B APPROXIMATION FOR exp [mAD(v2 - 71)] We have :._~.1_ -__§)_ 2 YZ-XD[1+JZ(DC (1+a)] Y1 5 k . Therefore, - 2 . + exp [mob/2 - 11)] = exp [m(1 - kADHexp m [—‘—H 12w °‘ . C Now, exp (x) é 1 + x + 2T’+ ... We will take a linear approximation for simplicity even though error will be introduced. Thus, eXp [mXD(Y2 - Y1)] 5 2 - 2 U+m-mMflU-[mu C C 2 Assume m to be no less than three; exn [mx(v2 - Y1)] 51%39 (j --§EL) . C C 61 LIST OF REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] LIST OF REFERENCES J.R. Pierce and P. Suhl, U.S. Patent 2, 743, 322. J.W. Jewartoski and H.A. Watson, Principal of Electron Tube. New Y0rk: D. Van Nostrand, 1955. J.R. Pierce, Traveling Wave Tubes, Chapter VII. New York: 0. Van Nostrand, 1950. L. Solymar and E.A. Ash, "Some Traveling Wave Interaction in Semiconductors, Theory and Design Considerations," Int. J. Electronics, Vol. 20, No. 2, 1966. K. Fujisawa, "Transmission Line Analog and Kinetic Power Theorems for Space Charge Waves in Semiconductors," Electronics and Communications in Japan, Vol. 51-C, No. 5, 1968. B. Ho, "Transmission Line Analog of Electron Stream in Solid State Plasma," IEEE Trans. on Elec. Devices, Vol. ED-17, No. 11, Nov. 1970. K. Fujisawa and H. Ichikawa, "A Couple Mode Theory for Wave Amplification in Semiconductors," Electronics and Communications in Japan, Vol. 52—C, No. 6, 1969. J.C. Freeman, B. Ho, and C. Hui, "Traveling Wave Interaction. in Semiconductors," to be published. M. Sumi, "Traveling-Wave Amplification by Drifting Carriers in Semiconductors," Appl. Physics Letters, Vol. 9, No. 6, 1966. M. Steele and B. Vural, Wave Interactions in Solid State Plasma, New York: McGraw-Hill, Chapter II. J.C. Freeman, Traveling-Wave Amplification in Semiconductors, Ph.D. Dissertation, Univ. Microfilms, A.A. M1, 1972. 6.5. Kino, "Carrier Waves in Semiconductors--I: Zero Tempera- ture Theory," IEEE Transactions on Electron Devices, Vol. ED-17, No. 3, 1970. M. Sumi and T. Suzuki, "Evidence for Directional Coupling between Semiconductor Carriers and Slow Circuit Waves, Appl. Physics Letters, Vol. 13, Nov. 1968. 62 [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] 63 J.C. Freeman, as in Reference 11. S. Lefeuvre and V. Hanna, "Solid State Traveling Wave Tubes," L'Onde Electrique, Vol. 56, pp. 341-344, 1976 (trans. J.C. Freeman). J.C. Freeman, as in Reference 11. J.C. Freeman, as in Reference 11. H. Okamoto and Y. Mizushima, "Interface-Wave Instability in a Parallel Arrangement of Two Semiconductor Sheets With Trans- verse Magnetic Field," Japanese J. Appl. Physics, Vol. 9, No. 9, Sept. 1970. ‘ J.C. Freeman, as in Reference 11. J.C. Freeman, as in Reference 11. Y.C. Wang, "The Field and Carrier Waves in a Semi-Infinite Semiconductor", Dept. of Electrical Engineering, Howard Univer- sity, Nov. 5, 1975. J.C. Freeman, as in Reference 11. R.H. Dean and B.R. Robinson, IEEE Trans. Electron Devices, Vol. 21, p. 61, 1974. R.E. Collins, Foundation for Microwave Engineering, New York: McGraw Hill, 1966. W.C. Hahn, "Small Signal Theory of Velocity Modulated Electron Beams," Gen. Elec. Rev., Vol. 42, p. 258, 1939. M. Steele and B. Vural, as in Reference 10, Chapter V. A. Bers and R.J. Briggs, "Criteria for Determining Absolute Instability and Distinguishing between Amplifying and Evanescent Waves," Quarterly Progress Report, No. 71, Research Laboratory of Electronics, M.I.T., Cambridge, MA, Oct. 15, 1963. J.C. Freeman, private communication. J.C. Freeman, as in Reference 11, p..110. STATE UNIVE TY L BR “WIN IIINIIIES 61 8 3 40 'MmeAN 3 1293 030