Models for Pulsed-Mode IMPATT Diode Simulation

Transcription

1 Models for Pulsed-Mode IMPATT Diode Siulatio ALEXANDER ZEMLIAK Deartet of Physics ad Matheatics Puebla Autooous Uiversity Av. Sa Claudio s/, Puebla, 7570 MEXICO Abstract: - A grou of oliear odels for high-ower ulsed IMPATT diode siulatio ad aalysis is reseted. These odels are suitable for the aalysis of the differet oeratioal odes of the oscillator. Its tae ito accout the ai electric ad theral heoea i the seicoductor structure ad the fuctioal deedece of the equatio coefficiets o the electrical field ad teerature. The first odel is a recise oe, which describes all iortat electrical heoea o the basis of the cotiuity equatios ad Poisso equatio ad it is correct util 300 GHz. The secod aroxiate atheatical odel suitable for the aalysis of IMPATT diode statioary oeratio oscillator ad for otiizatio of iteral structure of the diode. This odel is based o the cotiuity equatio syste solutio by reducig the boudary roble for the differetial artial equatios to a syste of the ordiary differetial equatios. The teerature distributio i the seicoductor structure is obtaied usig the secial theral odel of the IMPATT diode, which is based o the uerical solutio of the o-liear theral coductivity equatio. The described odels ca be alied for the aalysis, otiizatio ad ractical desig of ulsed-ode illietric IMPATT diodes. Its ca be also utilized for diode theral regie estiatio, for the roer selectio of feed-ulse shae ad alitude, ad for the develoet of the differet tye of colex doig-rofile high-ower ulsed illietric IMPATT diodes with iroved characteristics. Key-Words: - Seicoductor icrowave devices, odelig ad siulatio, uerical ethods. Itroductio IMPATT (IMPact Avalache ioizatio ad Trasit Tie) diodes are ricial active eleets for use i illietric ulsed-ode geerators. Seicoductor structures suitable for fabricatio of cotiuos-ode IMPATT diodes have bee well ow for a log tie [-]. They have bee utilized successfully i ay alicatios i icrowave egieerig. The ossibilities of usig the sae structures for ulsedode icrowave geerators are very iterestig because the ulsed-ode IMPATT-diode geerators ca successfully oerate at high curret desities without deterioratio of reliability. The cross sectio of the ulsed-ode IMPATT diode ay be larger tha that of the cotiuous-ode diodes. Therefore, the ulsed-ode oscillator ca rovide a larger ower outut. Cosiderig, that the icrease of the outut ower of illietric geerators is oe of the ai robles of icrowave electroics; it is iortat to otiize the diode's active layer to obtai the geerator axiu ower outut. Oe of the ai sigularities i the oeratio of high-ower IMPATT-diode ulsed-ode geerator, is the large variatio of the diode adittace durig the ulse. This variatio is sigificatly durig each curret ulse due to the teerature chagig of the diode's seicoductor structure. Therefore, diffusio coefficiets, ioizatio rates ad charge obility exeriece large variatios durig the ulse. These chages strogly affect the alitude ad hase of the first haroic of the diode's avalache curret. Therefore, the adittace value also chages. This results i the istability of geerator's outut ower ad frequecy withi each geerated icrowave ulse. Pulsed-ode IMPATT diodes that are utilized i icrowave electroics are, ost frequetly, the sigledrift ad double-drift structures siilar to cotiuousode oes [-5]. The tyical diode structure is show

2 o the Fig. by curve, where N is the cocetratio of doors ad accetors, l is the legth of diode active layer. Figure. Doig rofile for two tyes of IMPATT diodes: - costat doig rofile; - quasi-readtye doig rofile. I this tye of diodes, the electrical field is strogly distorted whe the avalache curret desity is sufficietly high. This large sace charge desity is oe of the ai reasos for the shar electrical field gradiet alog the charge drift ath. Because of this field gradiet, the sace charge avalache ruis itself ad cosequetly the otiu hase relatios degrade betwee icrowave otetial ad curret. This factor is esecially iortat whe the IMPATT diode is fed at the axiu curret desity, which is exactly the case at the ulsed-ode oeratio. The idea to use a colex doig rofile seicoductor structure for icrowave diode was origially roosed i the first aalysis of IMPATT diode by Read [6]. This roosed ideal structure has ever bee realized till ow. However, a oder seicoductor techology rovides ew ossibilities for the fabricatio of sub icro seicoductor structures with colex doig rofiles. This stiulates the search for IMPATT-diode secial structure's otiizatio for ulsed-ode oeratio. The roosed ew tye of IMPATT diode doig rofile is show o the Fig. by the curve. This tye of seicoductor structure ca be aed as quasi- Read-tye structure. This tye of doig rofile rovides a cocetratio of electrical field withi the - uctio. This easure hels to decrease the destructio of the avalache sace charge ad therefore erits to irove the hase stability betwee the diode curret ad voltage. Historically, ay aalytical ad uerical odels have bee develoed for the various oeratioal odes of IMPATT diodes [, 7-0]. However, they are ot adequate for very high curret desity values ad differet teerature distributios iside the structure, which is exactly the case for the ulsed-ode IMPATT-diode oscillator. For this reaso, we have develoed a ew colex uerical odel of the IMPATT diode that is coosed of the advaced theral odel ad the odified local-field odel. The theral odel rovides the exact theoretical teerature distributio alog the diode active regio. The local-field electrical odel calculates the fuctioal deedece of equatio coefficiets fro electric field ad teerature, ad usig all these data fially derives the IMPATT-diode dyaic characteristics. Nuerical Models Three uerical odels are described i this sectio. Two differet electrical odels useful for the recise aalysis ad iteral structure otiizatio describe all iortat heoea ito the seicoductor structure. The theral odel describes the teerature distributio by eas of the theral coductivity equatio solutio.. Precise Nuerical Model The uerical odel develoed for the aalysis of various geerator oeratio odes. This odel is based o the syste of cotiuity equatios for seicoductor structure: (, ) J ( x, t) x t t = + α J x t + x (, ) J ( x, t) x t t (, ) α J ( x, t) = + α J x t + x ( ) ( ) J x, t = x, t V + D (, ) (, ) J x t = x t V D (, ) α J ( x, t) (, ) x t x (, ) x t x ()

3 Where, are the cocetratios of electros ad holes; J, J are the curret desities; α, α are the ioizatio coefficiets; V, V are the drift velocities; D, D are the diffusio coefficiets. Ioizatio coefficiets, drift velocities ad diffusio coefficiets are fuctios of two arguets; the saces coordiate x ad the ties coordiate t. This odel is differs fro the reviously roosed odels, described i [7-0] because i the reset odel the ioizatio coefficiets are fuctios of electric field ad teerature i all oits of seicoductor structure. The deedece of these coefficiets o teerature ca be aroxiated usig the aroach described i []: α α ( E, T) [ ( T) / E ]. 6 0 e 5 E < [ ( T) / E ] = 6. 0 e < E < [ (. +. T) / E ] e 5 E > [ ( T) / E ] e. 0 0 < E < ( E, T) = [ ( T) / E ] e E > The teerature T is exressed i o C ad electrical field E is exressed i V/c. The boudary coditios for the syste () ca be writte as follows: ( 0 ) = ( ) D 0 ( 0 ) = A( 0) (, ) = ; ( 0, ) =., t N ; l, t N l ; J l t J J t J 0 s s where J s (), J are electro curret ad hole curret s for iversely biased - uctio; N ( ), N ( l ) D 0 A 0 are cocetratios of doors ad accetors at two sace oits x = 0 ad x = l 0, where l 0 is the legth of the active layer of seicoductor structure. Electrical field distributio ito seicoductor structure ca be obtaied fro Poisso equatio. As electro ad hole cocetratios are fuctios of the tie, therefore, this equatio is tie deedet too ad tie is the equatio araeter. Poisso equatio for this roble has the followig for: (, ) U( x, t) E x t x ( ) ( ) (, ) (, ) = = N D x N A x + x t x t x (3) where N ( x), N ( x) are the cocetratios of the D A doors ad accetors accordigly, U(x,t) is the otetial, E(x,t) is the electric field. The boudary coditios for this equatio are follows: M ( 0, ) = 0; ( 0, ) = 0 + ( ω + ϕ ) U t U l t U U si t = (4) where U 0 is the DC voltage o diode cotacts, U is the alitude of the haroic uber, ω is the fudaetal frequecy, ϕ is the hase of haroic uber, M is the uber of haroics. Equatios ()-(4) adequately describe the hysical rocesses i the IMPATT diode i a wide frequecy bad. However, uerical solutio of this syste is very difficult because of the shar deedece of equatio coefficiets o electric field. The evidet uerical schees have oor stability ad require a lot of coutig tie for the good calculatio accuracy obtaiig. It is ore advatageous to use o-evidet uerical schee, that has a sigificat roerty of absolute stability. The coutatioal efficiecy ad the uerical algorith accuracy are iroved by alyig sace ad tie coordiates syetric aroxiatio. After the aroxiatio of the fuctios ad its differetials, the syste () is trasfored to the oevidet odified Cra-Nicholso uerical schee. This odificatio cosists of two uerical systes, each of the havig the three-diagoal atrix. These systes have the followig for: ( a b ) i ( a ) i ( a b ) i + ( ) ( ) = a + a + a + b + i i i+ i+ i ( ) α τ V + r D + α i i+ i ( ) τ V r D i i+ i

4 + + + ( a b ) i ( a ) i ( a b ) i + i ( ) i i+ ( i+ i ) = a + a + a b + α ( i+ i ) ( ) τ V r D + i α τ V + r D i i =,,... I ; i+ i = 0,,,... (5) D, where a, = τ V,, h b, = τ, r = τ, 4h h i is the sace coordiate ode uber, is the tie coordiate ode uber, h is the sace ste, τ is the tie ste, I is the sace coordiate ode uber. The aroxiatio of the Poisso equatio is erfored usig the ordiary fiite differece schee at every tie ste : ( D i A i i i ) U U + U = h N N + i i i+ (6) The uerical algorith for the IMPATT diode characteristics calculatio cosists of the followig stages: ) The voltage is calculated at the diode cotacts for every tie ste. ) The iitial charge distributio is calculated. 3) The electric otetial is calculated at every sace oit fro Poisso equatio by the factorizatio ethod []. The electrical field distributio alog the diode active layer is calculated. 4) The ioizatio coefficiets ad drift araeters are calculated i uerical et odes for the curret tie ste. 5) The syste (5) is solved by atrix factorizatio ethod ad electro ad hole cocetratio distributio is calculated for the ew tie ste. After this, the calculatio cycle is reeated for all tie stes fro the begiig to the ste 3. This rocess is cotiued util the covergece is achieved. The curret of the exteral electroic circuit is deteried. The all haroics of exteral curret are calculated by the Fourier trasforatio ( J ; J J ex 0 ( )) = φ ; the adittace is calculated for haroic uber ( Y = J / U ) ad the ower characteristics for all haroics ca be calculated by the followig forulas: P P = Re ( Y ) U ; η = J U 0 0. Aroxiate Nuerical Model Other uerical odel is ore suitable for the revious aalysis ad for the diode iteral structure otiizatio. This odel ca reduce the total couter tie of the structure otiizatio rocess. The uerical ethod for the solutio of the syste () is based o the classical Fourier series utilizatio. This aroach trasfors of the boudary roble for the syste of differetial artial equatios to a ordiary differetial equatio syste. The odel describes the hysical rocesses i IMPATT diode by the statioary-oeratio ode ad rovides the ossibilities to reduce the deads for a couter tie that is ecessary for the outut araeters calculatio. Let us assue that all fuctios of the syste () ca be reseted i a for of Fourier series: ( x, t) = ( x) ex( ω t) ; = x t x ω t ; (, ) = ( ) ex( ) = J x t I x ω t ; (, ) = ( ) ( ) ex( ) = J ( x, t) = ( I ) ( x) ex( ω t) ; = α α ω ; ( x, t) = ( ) ( x) ex( t) = α α ω ; ( x, t) = ( ) ( x) ex( t) = V ( x, t) = ( v ) ( x) ex( ω t ) ; = V x t v x ω t ; (, ) = ( ) ( ) ex( ) = D ( x, t) = ( d ) ( x) ex( ω t) ; = D ( x, t) = ( d ) ( x) ex( ω t). = I such a case the ricial syste () ca be reduced to a syste of the ordiary differetial equatios for the colex charge desity ad for the curret

5 alitudes: d dx d dx ( ) d I dx ( ) d I dx v = + = d = = v d ( I ) ( d ) ( I ) ( d ) {( ) ( ) ( ) ( ) } = = ω α I + α I {( ) ( ) ( ) ( ) } = =,,... = ω + α I + α I where ( α ) ( α ) (7), are the electro ad hole ioizatio coefficiet alitudes, ( v ) ( v ), are the electro ad hole velocity alitudes, ( d ) ( d ), are the electro ad hole diffusio coefficiet alitudes,, are the electro ad hole cocetratio alitudes, ( I ) ( I ), are the electro ad hole curret alitudes. A uber of haroics i these series ca be reduced dow to the uber M, which defies the accuracy of the solutio ad ecessary couter tie. The syste (7) ca be reseted i atrix for as: Y '= AY (8) The charge diffusio ad shar deedece of the ioizatio coefficiets o the electrical field deterie the great odule of eigevalues of the atrix A. For this case, a shootig ethod, which reduces a boudary roble to Cauchy roble, is ot suitable because coordiate basis degeerates i the solutio rocess ad therefore is ot stable. The boudary roble (8) is solved o the basis of the fuctioal atrix correlatio [3] : t B ( x) Y( x) G( x) = (9) where B t is the factorizatio atrix; G is the boudary coditio vector. The uow atrixes of equatio (9) are satisfied i the followig differetial equatio syste: B'+ A t B = 0 G'= 0 (0) The fudaetal atrix F is used to obtai the rocess stability of the itegratio of equatios (0). This atrix is deteried as F( x) { A t ( x ) h } = ex, where h is the sace ste. Trasitio to the ext coordiate ode is ade usig the ter B x + h = F x B x. The degradatio of ( ) ( ) ( ) coordiate basis B ca be overcoe usig the Gra- Schidt ortogoalizatio rocedure for equatio (9) o each itegratio ste. The algorith for the aalysis of IMPATT diode icludes the followig stes: ) the iitial charge distributio i the diode is calculated; ) the electric field haroics are deteried fro the Poisso equatio; 3) ioizatio ad drift araeters are deteried fro the Fourier aalysis, ad the atrix of the syste of equatios o the coordiate et is fored; 4) the boudary roble is solved for the syste of cotiuity equatios. Charge ad curret alitudes are deteried. The haroics of the exteral circuit curret are calculated. After this, the calculatio cycle is reeated fro the begiig to oit ) util the exteral curret is deteried with sufficiet covergece. The all outut araeters of the IMPATT diode are deteried. The ai advatage of this haroic ethod is the reducig the total couter tie for the calculatio of statioary ode of the IMPATT diode. I Fig. are show couter tie Tc i relative uits ad relative error Er as the fuctios of the haroic uber M. Figure. Couter tie Tc i relative uits ad relative error Er as the fuctios of the haroic uber M.

6 These data are corresoded to the oliear odes with average level of the o-liearity. For this case we deterie error as the relative differece of the diode adittace value that we obtai by this haroic ethod ad by ore recise uerical ethod of the sectio.. It is clear that the haroic uber M ore tha -5 is sufficiet to obtai a good accuracy of the diode araeters. At the sae tie we have a sigificatly reducig of the total couter tie. Couter tie for oe robe of diode aalysis is the ricial characteristic of the otiizatio rocedure. That is the ai reaso why this aroxiate odel is elaborated. For exale the total couter tie for the diode aalysis by recise uerical odel is corresoded to the uber of haroic M = Theral Model The data of the teerature distributio that is ecessary for the calculatio of the local-field electrical odel ay be obtaied fro IMPATT diode theral odel. This odel deteries the teerature distributio i all oits of the diodes active layer, for ay give oet i tie. The IMPATT diode theral odel is based o the uerical solutio of the o-liear theral coductivity equatio for silico crystal, cotact laes ad heat sis. It deteries the istataeous seicoductor structure teerature at ay oit withi the device for ay "log" tie oet t. The theral equatio is solved i the regio that is show o the Fig.3. Figure 3. The scheatic diode costructio with heat si. R d - the diode radius, R hs - the heat si radius. The theral equatio has the followig for: T t = ρ C T + ρ C Q x t T (,, ) () where t is the tie () coordiate (this tie scale differs fro the scale i the syste () ), r is the radial coordiate, x is the logitudial coordiate, T is the Kelvi teerature; ρ is the aterial desity, C is the secific therocaacity, is the therocoductivity coefficiet, ad Q ( x, t, T) is the iteral heat source that, i the geeral case, has a deedecy o the electrical field, curret desity, ad teerature, is the two-diesioal Lalace oerator ad for the cylidrical coordiate syste T has a for: T T = + r. The x r r r equatio () is solved withi a volue that icludes the silico crystal; the gold cotact lae deosited o the crystal; a itegrated theral cotact ad the sei-ifiite coer heat-si. The boudary coditios for the syste () are follows: T r = 0 T o the vertical axis of syetry, = λ ( T θ ) o r T all vertical boudaries facig the air, = λ ( T θ ) x o all horizotal boudaries facig the air, T q( t ) = o the iteral boudary with a seiifiite coer heat si. The variable λ is the heat x trasissio coefficiet o the etal-air boudary, θ is the air teerature, q( t ) is the theral flux eterig sei-ifiite coer heat si. The ricial differece betwee the equatio () ad the syste () is that: the fuctio T i () deeds o two saces coordiates x ad r. O cotrary, all fuctios of the syste () deed oly o oe sace coordiate x. The deedece of all fuctios of syste () o r ca be eglected, because of aroxiatios which result i egligible error. However, the sae deedece ca ot eglect for equatio (), because it corresods really to the twodiesioal case (Fig. 3). We eed to deterie the fuctioal deedecy of the iteral heat source Q ( x, t, T) o the teerature to elaborate the IMPATT-diode theral odel. This odel ay be silified sigificatly by the followig iortat

7 aroxiatios: the role of soe etal layers (e.g., chroiu, gold, alladiu) i the diode theral balace ad the ifluece of gold cotact wire ad of ceraic housig of IMPATT-diode crystal ay be eglected. Also, the heat exchage betwee diode eleets ad the atoshere ay be eglected. These silificatios do ot seriously affect the accuracy of the odel. The iteral heat source is defied for all oits withi the odel volue as follows: π Q ( x, t, T) = J( ϕ, t, T) E( x, ϕ, t, T) dϕ () π 0 where ϕ = ωt, J( ϕ, t, T) is the istataeous IMPATT diode structure curret desity value, E( z, ϕ, t, T) is the electric field itesity i the oit x at the tie t ; This odel is essetially differet fro the odel described i [4] because the heat source is described ow as the fuctio of the electric field itesity iside the diode structure (). This iroveet is esecially iortat for icreasig the accuracy of the teerature distributio calculatio of the active layer of the IMPATT diode. The uerical solutio of equatio () is erfored by the fiite differece ethod. Equatio () is solved by the alteratig directio iteratio ethod for each coordiate directio. The secod order of the uerical aroxiatio schee is used ore frequetly i this case. The alteratig directio ilicit ethod ca be exressed i coact for as: T i T s i s = Λ T T C C Q + Λ + τ ρ ρ T s + i T i s T T C C Q s = Λ + Λ + τ ρ ρ s + + i =,,... I ; =,,... J ; s = 0,,,... ; (3) where i, are the sace coordiate ubers, s is the tie coordiate uber, Λ is the artial uerical Lalace oerator o the directio r, Λ is the artial uerical Lalace oerator o the directio x. Two of these oerators are defied i the stadard five-oits uerical atter: Ti+, Ti, Ti+, Ti, + Ti, Λ T = +, hi h h Ti, + Ti, + Ti, Λ T =. The uerical schee h (3) has the secod aroxiatio order oly. I this case, it is ecessary to develo the uerical et with a large uber of cells to obtai sufficiet accuracy. That is the reaso why the total couter tie that is ecessary for the solutio of the otiizatio roble is too great. I this wor, we roose the other tye of theral equatio uerical aroxiatio schee for the acceleratio of the theral equatio solutio ad for the reductio of the couter aalysis tie. The total aalytic Lalace oerator T ca be aroxiated with the uerical Lalace oerator Λ T as: + h ΛT = Λ + Λ + Λ Λ T h (4) I that case, we ca aroxiate the right art of equatio () by the followig uerical forula: ρ C ρ C h Λ T + E + Λ Q (5) where E is the idetity oerator. The oerator Λ is defied i the ie-oit uerical atter. The aroxiatio (5) is ore colicated, but it has the fourth aroxiatio order. I such a case, we ca use the uerical et that is sigificatly ore thi to obtai accuracy that is equal to the schee (3) described above. For the solutio of the ricial equatio () by aroxiatios (4)-(5), we used oe odificatio of the Peacea-Rachford uerical schee that had bee develoed by [5]: ( E b ( τ χ ) Λ ) T = ( E + b ( τ + χ ) Λ ( E b ( τ χ ) Λ ) T = ( E+ b ( τ + χ ) Λ ) T ) T s + τ b ( E + χ Λ ) Q + τ b ( E + χ Λ ) Q s (6)

8 where b = ( / ρc), χ = /). We solve the (, h, syste (6) by the tridiagoal algorith for radial ad logitudial directios. This uerical schee rovides a sigificat gai of couter tie i coariso with the schee (3). The grou of the odels reseted i sectios.-.3 serves as a basis for the recise ad colete aalysis of the IMPATT diodes with the differet doig rofiles for the various oeratio odes. 3 Nuerical Results The odels described above have bee utilized for the ivestigatio of teerature distributio i ulsed ode IMPATT diode havig differet doig rofiles. Also, the diode adittace characteristics have bee aalyzed. This aalysis has bee erfored for two tyes of diode structures: for the diode havig a traditioal costat doig rofile, ad for the diode havig the ew secial colex doig rofile aed the quasi-read tye structure. Both of these structures are ade of silico. The first structure has the doig 7 3 value N = c for active layer. The 0 regio legth is 0.4 µ; the regio legth is 0.36 µ. The secod structure has two levels of the active layer doig rofile: Ni =. 0 c ; Nax=.0 0 c ad the, +,, + regio's legths are 0.0 µ, 0.8 µ, 0.8 µ, 0.6 µ resectively. The diode uerical siulatio has bee erfored for the followig oeratioal araeters: electrical curret ulse has a square for, ulse duratio τ =00 sec, eriod T=0 µ sec. The calculated teerature distributio alog the diode active layer obtaied for ulse curret desity of 00 KA / c is show o a Fig. 4 by the solid lie for a IMPATT-diode havig a eve doig rofile. The results obtaied for the quasi- Read-tye structure are also reseted i Fig.4 by the dashed lie. The data obtaied for the diode havig colex doig rofile deostrate that the ier art of this diode is hotter tha that of the costat doig rofile diodes. This occurs because i the colex rofile diode electric field itesity axiu is located further fro the diode cotacts tha i the costat doig rofile diode. It eas that the heat source is located further fro the cotact regios too ad for a such case the colex-doig-rofile structure Figure 4. The teerature distributio. theral flow dissiates slower tha i the first structure. This also exlais the existece of a larger teerature gradiet alog the active layer i the colex doig rofile diode. For the structures studied, the teerature differece betwee the - uctio ad the cotacts has bee 8,8 C for the colex rofile diode ad 4,4 C for eve doig rofile oe. Data o the diode active layer teerature, obtaied for all tie ste fro 0 to 00 sec has bee used for the calculatio of the diode dyaic adittace characteristics eloyig the oliear electrical odel. Several exales of the calculated adittace characteristics are show i Fig. 5. Figure 5. Adittace frequecy characteristics

9 These characteristics obtaied for the sae two tyes of IMPATT diode doig rofiles (solid lies for the eve doig rofile structure ad dashed lies for the quasi-read structure), ad for the sae oeratioal odes as described reviously. These diagras rovide colete iforatio o diode adittace variatio with tie, durig the feed ulse. Durig the first 5 sec the diode adittace varies very sigificatly. This variatio is due to the strog teerature deedece of the hysical araeters of silico, i the teerature rage of C. Durig the iitial art of the feed ulse, the istataeous active layer teerature falls exactly withi this teerature rage. Durig the ext 85 sec, the diode adittace has a stable value. It is obvious, that it is iossible to obtai adequate coditios for a stable icrowave frequecy durig the iitial art of the curret feed ulse, if oly secial frequecy stabilizatio ethods are ot utilized. Durig this iitial eriod of the feed ulse, it is ossible to obtai soe accetable frequecy stability oly by utilizig a secial exteral assive circuit, or by the sychroizatio of the sigal. Durig the eriod fro 0 to 00 sec, the iagiary art of diode adittace I(Y) is chagig ore raidly tha the real art Re(Y). Relative variatios of I(Y) are aroud 50% of its average value. However, variatios of Re(Y) are oly withi 5% of the average value. This deostrates that the ai cause of the geerator's sigal istability is the violatio of reactive eergy balace durig the feed ulse. Cocurretly, the active eergy balace is coserved to a large degree of accuracy. I order to rovide soe eas for irovig the frequecy ad alitude stability of the ulsed IMPATT diode geerator, it ay be of iterest to ivestigate the ifluece of the feed curret ulse shae o hase relatioshi betwee diode curret ad voltage. Usig the reset odel, the otiu shae of the feed ulse, ad the iitial ad the fial feed curret desity values ca be deteried for the articular case as fuctios of the diode aterial, doig rofile, diode structure diaeter, etc. However, the coariso of the adittace characteristics for the two tyes of IMPATT diodes leads to soe iortat coclusios. The data o the diode adittace characteristics reseted i the Fig. 5 show that the variatios of adittace value durig the ulse are uch less for the secod structure, (dash lies) tha for the first (solid lies). This ca be observed i Fig. 5, where the adittace curves for the quasi-read structure lie closer to each other, tha the curves dislayig the traditioal eve-rofile diode. This ca be exlaied by the larger electric field itesity i the cetral art of active layer. The larger field itesity results i ore coact ad dese ad avalaches roagatig i the seicoductor structure ad, therefore, i better hase stability betwee curret ad voltage. Therefore, the use of the quasi-read structure, secures better frequecy ad alitude stability of the geerated electroagetic oscillatios. This is the ricial ad very iortat advatage of the ew quasi-read structure i coariso to the eve doig rofile structure. 4 Coclusio The uerical odels of the IMPATT diode, which are reseted i this wor, have iortat advatages whe coared to other odels. The electric odels together with the theral odel tae ito accout the teerature distributio i the seicoductor structure ad the deedece of all ricial hysical araeters of the seicoductor structure o teerature ad the electrical field. The other iortat advatage is the high stability of the calculatio rocess by eas of the o-evidet differece schee that is used for the solutio of the ai syste of equatios. The aroxiate o-liear IMPATT diode odel ca be used successfully for the iteral structure otiizatio. I such a case a great acceleratio of the otiizatio rocess ca be obtaied. The roosed theral ad local-field electrodyaic odels for ulsed ode IMPATT-diode aalysis icreases the accuracy of diode iteral ad exteral characteristics calculatio. The ethod reseted here ca be alied for ractical desig of ulsed-ode illietric IMPATT diodes. It ca also be utilized for diode theral regie estiatio ad for the selectio of feed curret-ulse shae ad alitude. The ethod is suitable for the desig of IMPATTdiode based oscillators, alifiers ad ixers. The ost roisig alicatio of the odels is the develoet of the ulsed-ode colex-doigrofile high-ower illietric IMPATT diodes with iroved characteristics. Coarative aalysis of IMPATT diode theral ad electro-dyaic roerties erfored for two tyes of the differet doig rofiles shows that diodes with colex quasi-read doig rofile have better ersectives for the ulsed feed

10 curret odulatio ode. This secial seicoductor structure has better hase correlatio betwee curret ad voltage ad has the saller variatios i the diode adittace. Therefore the colex-doig-rofile diodes have iroved frequecy stability i ulsedode oeratio coared to the traditioal IMPATT diodes havig a costat doig rofile. Refereces: [] D.L. Scharfetter, ad H.K. Guel, Large-Sigal Aalysis of a Silico Read Diode Oscillator, IEEE Tras., Vol.ED-6, No., 969, [] Edited by M.J.Howes, ad D.V.Morga, Microwave Devices.Devices Circuit Iteractios. Joh Wiley & Sos, 976. [3] T.T. Fog, ad H.J. Kuo, Millieter-Wave Pulsed IMPATT Sourse, IEEE Tras., Vol.MTT- 7, No.5, 979, [4] Edited by Kai Chag, Hadboo of Microwave ad Otical Cooets, Joh Wile & Sos, Vol., 990. [5] M. Tscheritz, ad J. Freyer, 40 GHz GaAs Double-Read IMPATT Diodes, Electro. Letters., Vol.3, No.7, 995, [6] W.T. Read, A Proosed High-Frequecy Negative-Resistace Diode, Bell Syste Tech. J., Vol. 37, 958, [7] A.S. Tager, ad V.M. Vald-Perlov, Avalache Diodes ad Alicatio o Microwave Edieerig, Sov. Radio, 968. [8] A.M. Zelia, Differece Circuit Stability Aalysis for IMPATT-Diode Desig, Izvestiya VUZ Radioelectroica, Vol.4, No.8, 98, [9] A.M. Zelia, ad S.A. Zicheo, No-Liear Aalysis of IMPATT Diodes, Vesti K.P.I., Radiotechia, Vol.6, 989,.0-4. [0] M.A. El-Gabaly, R.K. Mais, ad G.I. Haddad, Effects of Doig Profile o GaAs Double-Drift IMPATT Diodes at 33 ad 44 GHz Usig the Eergy-Moetu Trasort Model, IEEE Tras., Vol.MTT-3, No.0, 984, [] A.M. Zelia, ad A.E. Roa, IMPATT Diode for the Pulsed-Mode, Izvestiya VUZ Radioelectroica, Vol.34, No.0,99,.8-3. [] M. Curow, Proosed GaAs IMPATT Devices Structure for D-bad Alicatios, Electro. Lett., Vol.30, 994, [3] K.V. Vasilevsii, Calculatio of the Dyaic Characteristics of a Silico Carbide IMPATT Diode, Sov. Phys. Seicod., Vol.6, 99, [4] R.P. Joshi, S. Patha, ad J.A. Mcadoo, Hot- Electro ad Theral Effects o the Dyaic Characteristics of Sigle-Trasit SiC Iact- Ioizatio Avalache Trasit-Tie Diodes, J. Al. Phys., Vol.78, 995, [5] H.J. Kafa, ad K. Hess, A Carrier Teerature Model Siulatio of a Double-Drift IMPATT Diode, IEEE Tras., ED-8, No.7, 98, [6] A.M. Zelia, ad S.A. Zicheo, No-liear Aalysis of IMPATT Diodes, Vesti K.P.I., Radiotechia, Vol.6, 989, [7] C. Dalle, ad P.A. Rollad, Drift-Diffusio Versus Eergy Model for Millietric-Wave IMPATT Diodes Modellig, It. J. Nuer. Modellig, Vol., 989, [8] V. Stoilovic, M.J. Howes, ad V. Postoyalo, Noisotheral Drift-Diffusio Model of Avalache Diodes, J. Al. Phys., Vol.7, 99, [9] O. Torblad, U. Lidefelt, ad B. Breitholtz, Heat Geeratio i Si Biolar Power Devices: the Relative Iortace of Various Cotributios, Solid State Electroics, Vol.39, No.0, 996, [0] A. Zelia, S. Khotiaitsev, ad C. Celaya, Colex Noliear Model for the Pulsed-Mode IMPATT Diode, Istruetatio ad Develoet, Vol.3, No.8, 997, [] W.N. Grat, Electro ad Hole Ioizatio Rates i Eitaxial Silico at High Electric Fields, Solid- State Electroics, Vol.6, No.0, 973, [] V.I. Krylov, V.V. Bobov, ad P.I. Moastyrsi, Nuerical Methods, Naua, 977. [3] N.S. Bahvalov, N.P. Zhidov, ad G.M. Kobelov, Nuerical Methods, Naua, 987. [4] L.H. Holway, Trasiet Teerature Behavior i Pulsed Double-Drift IMPATT Diodes, IEEE Tras., Vol.ED-7, No., 980, [5] A.A. Saarsy, About the Choice of Iteratio Paraeters for Alteratig Directio Method for Dirichlet High order Accuracy Differetial Proble, Dolady Acad. Nau USSR, Vol.79, No.3, 968,