Zombory, L.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

SELF -RESONANT LSA OSCILLATOR DIODE OF RECTANGULAR CROSS·SECTION By L. ZOc\IBORY Department of Theoretical Electricity Technical University, Budapest Received :?IIay 9, 1972 Presented by Prof. Dr. G. FODOR In two recent publications [1, 2] Rode has proposed an interesting operating mode of the LSA diodes. In this self-resonant mode the semiconduc-tor bulk of negative conductivity operates as an open cavity resonator and therefore the oscillation frequency is determined by the device geometry rather than by an external resonator. The configuration investigated by Rode was a circular cylinder of linear homogeneous, isotropic material with negative conductivity. The exact cal-culation to obtain the lowest eigenfrequency of this geometry in the E mode is possible and the result is fa =~o' - , (1) ::r where d is the diameter of the cylinder, c the velocity of the light in free space and q' the smallest real root of the transcendental equation: 1'81'(1/80')'1 (1180') = H(l)' (o')IH(l)(O') f Ol'-iOt_ 0 _,0 ... " (2) where we have the relative permittivity C = CL + j( (j wco). Unfortunately, the cylindrical geometry is unusual for practical appli-cations. The usual geometry for the active layer of the LSA diodes is a rec-tangular prism. Among the planar geometries the active planparallel layer was investigated in detail only [3-9]. Chawla and Coleman [6] (whose letter was quoted by Rode [2]) as well as Giannini et al. [9] fixed the real part of the relative permittivity as CL = 12. * For the lowest real eigenfrequency they got the frequency by thickness product as fL = 4.86 . 109 cm/so "This is the value of the relative permittivity of GaAs. 2* 20 L. ZOJIBORY This result is far from Rode's fd product. Using Fig. 2 of [1] we have the value of 1/ "-' 0.24 with CL = 12. Therefore fd cc", 2.3 . 109 cm;s. The exactly calculable lowest real eigenfrequencies of the rectangular but open (planparallel) layer and the closed but edgeless cylinder are relatively far from each other. This fact is unconvenient for estimating the eigenfrequency of the rectangular prism from the results given above. In the following we show that the lowest eigenfrequency of a rod with square cross-section and of nega--I -.--~f---., 20 -------Fig. 1 tive conductivity is far nearer to Rode's result than to the result in [6] and [9]. Thus, we shall apply the geometrical optical considerations of Vainshtein [10]. Let the cross-section of an infinite rod be as given in Fig. l. The only component of E inside the rod is parallel 'with the y axis. Let it be chosen inside the rod in the foIlo'wing form: Ey = Aj" (kx x)f: (k: z)e -jwt (3) where (4) k being the wave number in free space. Outside the rod the electromagnetic field is represented hy an outward travelling plane wave. For example in the region x > a, z < I E _.1 j"~xf (k "') -jwt y-"':1. o e z .:~e (5) where ko -1ik~ - k2 -1./ ---'----'--'-----'-x _.. z -cp (6) From lVIaxwell's equations and the continuitv conditions of the fields SELF-RESOSAST LSA OSCILLATOR DIODE on the surface x = a, !z[ < 1 'we have the equation: and similarly IS also valid where f;(k z 1) fz(k z 1) . k~ =JP k~ k~ = V_k-='~ __ k-,,-~(,-,c,,-U __ 1),-cf! 21 (7) (8) (9) Looking for the most homogeneous distribution of the field corresponding to the lowest eigenfrequency we may suppose that Ix (kx x) = cos (kx x) (10) and fz (kz z) = cos (kz z) . (11) Supposing furthermore that a = 1 and because of the symmetry kx = kz is valid, from Eqs (6) to (11) the dispersion equation has the following form (Il = 1). tan ( 1 ; ka) = ] 2 c (12) The deduction given above neglects the diffraction. This fact involves that Equ. (12) has real k solutions substituting real .c > 2. This phenomenon is the consequence of the total reflection on the inner surfaces. Taking into consideration this fact -we look for the f . 2a product with c = CL = 12. With this value we get 1 1/10 ka = arc tan 1.1 - = 0.302 1 12 and c cm f· 2a = - ka=2.89 .109 :T S This result shows that Rode's exact lowest eigenfrequency IS a good approximation in the case of a rectangular parallelepiped geometry of the LSA diode, if the diameter d is chosen to be equal to the length of the sides of the square cross-section. 22 L. ZO.1IBORY Summary Several results on the eigenfrequency of the self·resonant linear active open cavity resonator with circular cylindrical geometry were applied to the LSA diode operation in recent papers. It is shown here that the exact value of frequency obtained in this way is a good approximation for the real LSA diode geometries that, however, are inaccessible to exact treatment. References 1. RODE, D. L.: Self·resonant LSA oscillator diode. Proc. lEEK 57, 1216-1217 (1969). 2. RODE. D. L.: Dielectric·loaded self·resonant LSA diode. IEEE Trans., ED·17, 47 -52. (1970). 3. KHAPALYUK, A. P., STEPANOV, B. 1., and SOTSKIY, B. A.: Electromagnetic field inside a planparallel layer in the self·resonant operation (in Russian). Optika i Spektr., 13, 282-285. (1962). 4. STEPANOV, B. 1.. KH..-I.PALYUK, A. P.: Transmission and reflection of planparallel layer in the amplifier and oscillator operation (in Russian). Optika i Spektr. 13, 714-720 (1962). 5. LEWIN, L.: Amplifying properties of bulk negative resistance material. Electron. Lett. 4, 145-147. (1968). 6. CH..-I.WLA, B. R., COLE)IAiS', D. J. jun.: Critical conductiyity.length product for electro· magnetic instability in negatiye.differential conductivity media. Electron Lett. 5, 31-32 (1969). 7. LEWliS', L.: Stability of bulk negatiye·resistance material. Electron Let!. 5, 184-185 (1969). 8. ZOllIBORY, L.: Propagation and reflection of electromagnetic waves in the presence of substances with arbitrary complex permittivity. Periodica Polytechnica El. Eng. 14, 419-434 (1970). . 9. GIANNliS'I, F., OTTAVI, C. M., SALSAiS'O, A.: Wave propagation in negative conductivity media. Electron. Lett. 7, 65-66 (1971). 10. VAINSHTEliS', L. A.: Open resonators and open wayeguides. The Golem Press. Boulder, Colorado. 1969. Dr. Liisz16 ZOl\IBORY, 1502 Budapest, P.O.B. 91. Hungary 

SELF-RESONANT LSA OSCILLATOR DIODE OF RECTANGULAR CROSS-SECTION

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SELF-RESONANT LSA OSCILLATOR DIODE OF RECTANGULAR CROSS-SECTION

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Periodica Polytechnica (Budapest University of Technology and Economics)