Electrical Properties of Multi p-n Junction Devices

Transcription

1 TRANSACTIONS IEEE ON ED-29, ELECTRON DEVICES, VOL. NO. 6, JUNE [ 5 J. G. Nash ad J. W. Holm-Keedy, Effect of electro-electro scatterig o hot-electro repopulatio i -Si at 77 K, hys. Rev. B, vol. 6, p. 284, 977. [ 6 J V. Faricelli, hysics of large-sigal respose of short-chael MESFET s, M.S. thesis, Corell Uiv., Ithaca, NY, 980. [ 7 K. Blotekjaer, Trasport equatios for two-valley semicoductors, IEEE Tras. Electro Devices, vol. ED-7, p. 8, 970. [8] M. S. Shur, Ifluece of o-uiform fiel distributio o frequecy limits of GaAs field-effect trasistors, Electro Lett., vol. 2, p. 65, 976. [ 9 C. Jacoboi et al., A review of some charge trasport properties of silico, Solid-state Electro., vol. 20, p. 77, 977. [20]. M. Smith, M. Ioue, ad J. Frey, Electro velocity i Si ad GaAs at very high electric fields, J. Appl. hys., vol. 7, p. 797, 980. [2] T. J. Maloey, No-equilibrium electro trasport i compoud semicoductors, h.d, dissertatio, Corell Uiv., Ithaca, NY, 977. [22] S. Kratzer, Computer simulatios of electro trasport i GaAs, M.S. thesis, Corell Uiv., Ithaca, NY, 978. [2]. N. Swartztrauber ad R. Sweet, Efficiet fortra subprograms for the solutio of elliptic partial differetial equatios, Natioal Ceter for Atmospheric Res., Tech. Note TN/IA-09, 975. [24] R. W. Hockey, OT4-A fast direct oisso solver for the rectagle allowig some mixed boudary coditios ad iteral electrodes, IBM Res. Rep., RC [25] D. L. Scharfetter ad H.K. Gummel, Large sigal aalysis of a silico Read diode oscillator, IEEE Tras. Electro Devices, VO~. ED-6, p. 64, 969. [26] M. Reiser, Large scale umerical simulatio i semicoductor device modelig, Computig Methods i Applied Mathematics ad Egieerig, vol., p. 7. [27] J. Ruch, Electro dyamics i short chael field effect trasistors, IEEE Tras. Electro Devices, vol. ED-9, p. 652, 92. [28] R. C. Ede ad B. M. Welch, GaAs digital itegrated circuits for ultra-high speed LSI/VLSI, i VLSI: Fudametals ad Applicatios, D. F. Barbe, Ed. Berli, Germay: Spriger-Verlag, 980. [29] T. Wada ad J. Frey, hysical basis of short-chael MESFET operatio, IEEE J. Solid-State Circuits, vol. SC-4, p. 98,979. Electrical roperties of Multi p- Juctio Devices JOSEH KATZ, SHLOMO MARGALIT, AND AMNON YARIV, FELLOW, IEEE Abstract-The electrical properties of multi p- juctio devices are aalyzed. It is foud that this type of device possesses bistable characteristics similar to that of a Shockley diode ad thus provides a alterative realizatio of devices for switchig applicatios. The iheretly greater curret gais ivolved i the operatios of such a device yield i pric,iple higher breakover voltages ad higher holdig currets. Furthermore, the icorporatio of heterostructures i this device itroduces a ew degree of freedom i tailorig their switchig characteristics. Multi p- heterojuctio devices operatig as SCR lasers were fabricated, ad the experimetal results are preseted, I. INTRODUCTION INCE THEIR itroductio, the Shockley diode [l] ad S other related devices have foud may applicatios i switchig ad regulatig circuits [2]. Recetly the operatio of Shockley diodes which fuctio also as AlGaAs ijectio lasers has bee demostrated []. Operatio of both homostructure [4] ad heterostructure [5] multi p- GaAs-GaAlAs Mauscript received November 6, 98; revised February, 982. This work was supported i part by the Jet ropulsio Laboratory, Califoria Istitute of Techology, uder NASA Cotract NAS7-00, the Office of Naval Research ad the Natioal Sciece Foudatio. J. Katz is with the Jet ropulsio Laboratory, Califoria Istitute of Techology, asadea, CA 909. S. Margalit ad A. Yariv are with the Departmet of Electrical Egieerig ad Applied hysics, Califoria Istitute of Techology, asadea. CA devices as ijectio lasers has also bee reported, but o aalysis of the electrical properties of such structures has bee published. This paper aalyzes the electrical properties of semicoductor devices cosistig of may layers of alteratig p- ad -type. Icorporatio of heterostructures i these devices makes the desig of their characteristics more flexible due to the itroductio of the additioal degreeof freedom of the eergy bad gap differece. It is foud that such devices provide a alterative for realizig bistable switchig characteristics. Compared to switchig devices fabricated from silico, GaAs devices are lessesitive to high temperatures because of their larger bad gap ad are iheretly faster because of their shorter carrier lifetime. Sice the commo-base curret gaiof the trasistors that model these devices (see ext sectio) is distributed amog all the regios of the structure, differet switchig coditios are obtaied. Maily it is foud that it takes more gai to perform the switchig, which results i a icrease i the breakover voltages ad i the holdig currets. The outlie of this paper is as follows: I Sectio, a qualitative aalysis of multi p- devices, based o a exteded trasistor model, is carried out. The results of this aalysis show that such structures have bistable characteristics similar to those of a Shockley diode. Sectios I ad IV aalyze /82/ $ IEEE

2 978 IEEE TRANSACTIONS ON ELECTRON DEVICES, VQL. ED-29, NO. 6, JUNE 982 (a) (b) (c) Fig.. Trasistor model of a (p-), device. (a) Schematic structure of the device. (b) Decompositio of the device ito idividual trasistors. (c) Equivalet circuit of the device. quatitatively the device i its two stable states: the forward blockig ( OFF ) ad the forward coductig ( ON ) states, respectively, Fially, Sectio V describes the fabricatio procedure ad the experimetal results of several types of such devices, ad compares the experimetal results with the theoretical calculatios.. MODIFIED TRANSISTOR MODEL FOR MULTI p- STRUCTURES Cosider a structure cosistig of 2m layers of alteratig p- ad -type, which is deoted by (p-),. I this structure the ith juctio separates the ith ad the (it )th layer. By a direct extesio of the two-trasistor model for the SCR, oe ca aalyze the structure usig a more complicated trasistor etwork. A example of a (p-), structure is show i Fig.. Geerally, it takes a 2 X (m - ) trasistor etwork to describe a (p-), structure. The X 2 X (m - ) equatios eeded to describe the etwork (three equatios for each trasistor) are IBj t ICj = IEi, i=,2,.,2(m - ) (la) Ici=aiI~i+Icoi, i=,2;..,2(m- ) (lb) ICi =IB,i- +IE,i-2 +IG,i- 9 i=2,4,6;**,2(m- ) (IC) IEi = -IB,i-l IC,i-2 2 IG,i-l > i=,5,7;-.,2m- (Id) where the trasistors are assumed to be iitially i the cutoff or active regio (i.e., the device is i the forward blockig state), Icoi is the collector to base reverse saturatio curret of the ith trasistor, ai is the commo-base curret gai of the ith trasistor, ad IG~ is the curret geerated at the ith gate of the device. The set of equatios () ca be cast i a matrix form For example, the (p-), structure is described by the follow- ig matrix equatio. A is give by r C Y, O l O O O - a c u O O - a, O l - ad Z ad Idfive are give by I = 7-l I-- -I IE -CIGi IB 0 IC IC0 IE2 IB 2 IC2. Idrive = IE IB IC IE4 I84 IG 0 IC702 IG2 0 IC0 IG 0 IC4 - - IC0 4 4 Equatio (2) ca be solved for la = IE~ with {ai} as a set of parameters. The particular case where IA approaches ifiity ()

3 KATZ et al.: MULTI p- JUNCTION DEVICES 979 O 7 a8 - a, / x f b // / / / a I -,/ I I / / 6 / / c,p JI El JZ J + Aode -bcothode + Aode -6 Cathode (a) (b) Fig.. Compariso betwee the geeric characteristics of (p-), ad (p-)z devices. (a) (p-), device. (b) Correspodig (p-)~ device (regular Shockley diode). I I I I I I r Fig. 2. Commo base curret gai (a) for switchig of a (p-), device versus m. (a) All the trasistors are idetical (ai = 0). (b) All the odd (or all the eve) umbered trasistors i the model have 0 = Show is a eeded from the other trasistors for switchig. (c) Same as i (b), but with a give AI of (i.e., the determiat of A equals zero) idicates the switchig coditio. Ispectio of the matrix A i (2) shows the followig: ) (p-)m structures, with m 2 2, caot possess more tha two stable states. This is deduced from the fact that for a give structure, oly oe set of {acui}, at most, yields, ) The coditio for switchig chages, depedig o the structure parameters. Some of the results are show i Fig. 2. I Fig. 2(a) all the trasistors are equal; i Fig. 2(b) all the odd (or all the eve) umbered trasistors i the model have (Y = 0.95, ad Fig. 2(c) is the same as Fig, 2(b) but with a give a = I Fig. 2(b) ad 2(c) show i the a eeded from the other trasistors for switchig. It is clearly see that as m icreases, the device must have more gai i order to possess two stable states. Structures with isufficiet gai remai i the forward blockig state, ad whe the applied voltage is icreased they evetually udergo either avalache or zeer breakdow. Oe simple explicit expressio is obtaied for the case where all the trasistors have the same gai, i.e., a = az =, *., a. I this case switchig occurs i a (p-), structure whe &=I-- m For m = 2 we get the well kow result for the SCR (a = a2 = 0.5). As a fial remark, it is iterestig to ote that the above aalysis ca be also carried out for multi p- structures i which the first layer ad the last layer are of the same type (e.g., a p--p--p structure). I this case it is foud that the device behavior does ot show bistability, ad thus it is similar to that of a trasistor. It seems that all geeric types of oe port low frequecy low field semicoductor devices are described by oe of the followig structures: p (or ), p-, p--p (or -p-) ad p--p-. Fig. helps to explai this fact. I particular, it describes why both (p-), ad (p-);, devices have the same basic electrical characteristics, Fig. (a) depicts the device i the forward blockig ( OFF ) state. The crosshatched areas represet the depletio regios of the reverse biased juctios (J2 ad J4). The juctio J, is, i priciple, forward biased. However, sice the curret that flows through the device is very small, there is also a very small voltage drop i the regio betwee Jz ad J4. Sice a regio with virtually o curret ad voltage has a little effect o the device, to the exteral world the device appears basically as if it had the structure depicted i Fig. (b), which is a device. I the forward coductig ( ON ) state, all the iteral regios i the (p-), device are i saturatio, which is the same situatio as i the (p-)z device. Of course, the quatitative aalysis is differet for the two cases, as will be see i sectios which follow.. SOLUTION OF DIFFUSION EQUATION IN THE FORWARD BLOCKING ( OFF ) STATE I this sectio the (p-), structure i the forward blockig state is aalyzed. I this sectio ad i the ext oe, the idexes o the various parameters refer to either the juctios (e.g., voltages, depletio regio recombiatio currets) or to the layers betwee the juctios (e.g., diffusio legths, widths of the layers). The miority-carrier distributio i the forward blockig ( OFF ) state is show i Fig. 4. All the eve umbered juctios are reverse biased so thatheir miority-carrier cocetratios are effectively zero. The equatio for the curret desity through the reverse biased juctios is J=JZz =MZi[JG,Zcui+Jp,Zi+l(W) -t J,Zi(o)l i =,2;..,(m- ). (5) Mzi is the avalache multiplicatio i the depletio regio of the 2ith juctio, which, for GaAs, is the same for both electros ad holes ad ca be approximated by the followig empirical formula [6, p. : where VBD,~~ is the breakdow voltage of the 2ith p- juctio ad c is a empirical costat, J G, is ~ the ~ curret desity geerated i the depletio regioof the 2ith juctio (e.g., thermal or light geeratio) which ca-to the first orderbe approximated as a costat. Jp,2i+l(W) is the hole dif-

4 980 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-29, NO. 6, JUNE 982 fusio curret desity eterig the juctio from the left ad is give by where JRO ad NE are empirical costats. I the followig calculatios it will be assumed that NE = 2, which is a good approximatio for practical devices [6, p Sice the curret is the same throughout the device, the curret desities through the eve ad odd umbered juctios ca be equated, so the curret desity is give by where A = qjkt ad Vi is the voltage across the ith juctio ad J,zi(0), the electro diffusio curret desity eterig the juctio from the right, is give by where po, W,, D,, L, ad po, W,, D,, L, are the equilibrium cocetratio of the miority carriers, the width, the diffusio coefficiet ad the diffusio legth of the miority carriers i the appropriate p ad regios, respectively. The advatage of usig heterostructures i the multi p- devices ca be uderstood from (8) ad (0). Sice the itrisic carrier cocetratio i i a material is proportioal to exp (- Eg2kT) where Eg is a bad gap, ad sice, for a give dopig level, the miority-carrier cocetratio (i.e., po or po) is proportioal to ;, the diffusio currets ad hece the curret gais a of the trasistors that model the device ca be modified idepedetly of the dopig levels. Thus it is possible to achieve low a trasistors i devices with thi layers, i.e., to icrease the breakover voltage without sacrificig its temporal respose. The odd umbered juctios are slightly forward biased, so th effect of the depletio regio recombiatio currets must be icluded. These currets ca be approximated by [6, pp or J= J;zitl(e AV2it -, Js,zi+l tjro,ziti emitl/~ i=o,,2;..,(m- ) () where Jp, 2i+l (0) is the hole diffusio curret desity eterig the juctio from the right, J,,zi+2(W) is the electro diffusio curret desity eterig the juctio from the left, Jsp ad Js are as defied before i (8) ad (0) ( 5) ad J ~ o, ~ is i + the ~ recombiatio currefit costat of the (2i t )th juctio. Note that for i = 0, Jp(0) = 0, ad for i = m -, J(Wp) = 0, sice it ca be assumed that the diffusio currets i the two extreme layers are egligible, e.g., these layers are AlGaAs layers with high Al cotets, ad thus their values of po ad po are much lower tha those of GaAs, because of the differeces i the bad gap eergies. Usig (5) to (), a closed form expressio for the J-V curve of the device ca be obtaied i the followig way. For a give value of J, () is a quadratic i

5 KATZ et al.: MULTI p- JUNCTION DEVICES 98 - x Fig. 5. Miority-carrier distributio across a (p-), ward coductig ( ON ) state. device i the for- Oce a solutio is obtaied for all the odd umbered juctios, (6) ca be solved for Vi, i= 2,4, - -, (m - 2) usig (5) ad (7) to (0). The sum of all of the juctio where voltages thus obtaied is the total voltage Vacross the device, correspodig to the assumed value of J. The particular form of the resultig expressios is quite complicated, buthe calculatios are straightforward, as described above. The importat parameter J-V curve i the OFF state is the breakover voltage (VBo), which is defied i the same way as for the Shockley diode. At this poit dv/d = 0. As the curret is further icreased, the voltage across the device decreases. This is a egative resistace [dv/dj< 0 regio ad thus ustable, leadig to the ON state. I this ew situatio the assumptio about the juctio voltages are o loger valid, ad ew calculatios eed to be doe. The breakover voltage of the (p-), deviceis the sameas,avzr -2 - the voltage across (m- ) Shockley diodes operatig i series. This is because the structure cosists of distict sectios, ad each isolated betwee two reverse biased juctios at which the carrier cocetratio is virtually zero (see Fig. 4). Thus is see that by icreasig m devices ca be foud with higher - breakover voltages. J=.. IV. SOLUTION OF THE DIFFUSION EQUATION IN THE FORWARD CONDUCTING ( ON ) STATE The matrix E, give by I the forward coductig state all the juctios are forward J&l -Js2 biased I Vil >> kt/q, i =,2, * - *, (2m - ) i.e., all the trasistors that model the device are i saturatio. B = This is similar to the behavior of the commo Shockley diode i the ON state. The distributio of the miority-carrier cocetratio i thi state is show i Fig. 5. (Note the chage of sig i the otatio for the eve umbered juctio voltages; ow all the juctios are forward biased). The resultig diffusio equatios of the etire structure ca be writte i the form Bu+Ew=J ( 6) I

6 982 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-29, NO. 6, JUNE 982 TABLE I ARAMETERS OF A (p-)7 STRUCTURE I I-0 i TABLE I ARAMETERS OF A (p-)s STRUCTURE.Is Layer Number Type + A Dopig Width Cotets Cocetratio (X) x 08 Lcr- [uml = T l I 2 Layer Number (b) Fig. 6. ON state characteristics of the (p-), device described i Table I. (a) Excess miority carriers. (b) Recombiatio curret distributio i the various regios. E 2.4~0 ~ z cotais the diffusio cotributio to the total curret ad JROl JR02 0 c=. JR0 0 (2) 0.. JRO, 2r - cotais the depletio regio recombiatio cotributio to to total curret. Jsp,i, Js, i, J2i ad J ~ oare i give i (8), (lo), (5) ad ( l), respectively. Also ad J$, i = Jsp, i C O ( w~, JLp, i) Jz, i = Js, i cash (wp, i/l, i>. The derivatio of (6) to (2) is based o solvig the diffusio equatio i each regio separately ad arragig the idividual solutios with the appropriate idexig. Several calculated results for the devices described i Tables I ad I are show i Figs. 6 ad 7, respectively. art (a) of these figures shows the distributio of the excess miority carriers across the devices. The distributio across the (p-)5 device is much more balaced tha i the (p-), device. This fact is also clearly demostrated i part (b) of the figures, which shows the distributio of the recombiatio curret (b) II 8 9 Fig. 7. ON state characteristics of the (p-)s device described i Table. (a) Excess miority carriers. (b) Recombiatio curret distributio i the various regios. across the device. As will be discussed i Sectio V, it is sometimes desirable to have as uiform distributio i the GaAs regios whe possible. Whe desigig a structure for a particular carrier profile, the parameters at our disposal are the umber of the layers, their types ad widths, ad the dopig cocetratio. All these parameters appear i the solutio of

7 KATZ et al.: MULTI p- JUNCTION DEVICES 98 Fig. 8. Calculated depedece of the holdig curret desity (JH) o the umber of layers i the device. (a) W, = Wp = 0.5 pm. (b) W, = W, = pm. m the diffusio equatio, ad thus ca affect the performace of the device. The holdig curret of the device (i.e., the miimum forward curret which is required to sustai the ON state) ca be estimated i the followig way. We kow that if the curret is reduced below the holdig value, the device exhibits egative resistace ad is ustable. This leads to the OFF state described i the precedig sectio, so the value of J for which dv/dj = 0 is the holdig curret desity (JH). This parameter ca be foud by solvig (6) umerically. Calculated depedece of JH o several parameters is show i Fig. 8. I Fig. 8(a) the width of all the layers is 0.5 pm, ad i Fig. 8(b) the width of all the layers is pm. The device cosists of (m- ) p- sectios of GaAs sadwiched betwee two layers of high Al cotets AlGaAs. As expected, the holdig curret desity icreases with icreasig the umber of the p- sectios of the device, with icreasig the widths of the layers ad with icreasig the dopig levels. The basic cause for this icrease is the eed to repleish recombied carriers i more ad more regios while still maitaiig all the layers i saturatio. Because of the basic expoetial depedece betwee the curret ad the voltage i p- devices, the icrease i the holdig curret with icreasig the umber of sectios i the device is larger tha the correspodig icrease i the breakover voltages. It should be oted that the above aalysis ca be easily exteded to ay arbitrary structure, ot ecessarily oe which cosists of alteratig p ad y regios (e.g., p-p--... ). V. FABRICATION ROCEDURE AND EXERIMENTAL RESULTS The layered structures of AlGaAs described i this paper were grow by liquid phase epitaxy at 800 C. arameters of two typical device structures (e.g., layers types, widths, dopig) are described i Tables I ad. Sice the umber of solutio chambers i the graphite boat is smaller tha the required umber of layers i the structure, the periodic parts of the structure were grow by movig the slide-bar of the boat i both directios betwee the solutios. I that case, two dummy wafers were used, oe o each side of the ac- Fig. 9. Depedece of the I-V curve of (p-), devices o the A co. tets (x) i the waveguide layers. (a) x =. (b) x =. (c) x = (o bistability). tual growth wafer. The dopats used were Ge (p-type), S (-type, forregioswithno 5 X 0l8 ~m-~) ad Te (-type, for regios withnd 2 X 0 ~m-~). Devices tested oly for electrical parameters were etched dow to a 00 X 00 pm2 mesas, while devices which operated also as ijectio lasers were etched dow to a 00 pm mesa i oe directio ad cleaved to m00 pm legth i the perpedicular directio. Cr-Au was used o the p-type cotact ad AuGe/Au (with a post-depositio alloyig at 60 C) was used for the -type cotact. The first types of devices tested were (p-), structures whose order ad type of layers, dimesios ad dopig (but ot the Al cotets (x)) are described i Table I. I particular, the depedece of the I-V curve of the structure o the Al cotets (x) i the layers betwee the GaAs regios were ivestigated. Curves of devices with x =,, ad are show i Fig. 9. Whe the Al cotets is too high, the curret gai of the p--p trasistors i the device model (these are the trasistors which have the -AlGaAs layers as their base regios) becomes too small to maitai the device i the ON state. For x = eve reductio i the umber of layers, e.g., (p-k is ot eough. I this case the obtaied I-Vcurve is that of a trasistor i avalache. The calculated carriers distributio i the device is show i Fig. 6(a) for x =. The carrier cocetratio is foud to be highest i the upper GaAs regio, with fewer carriers i the subsequet regios. This result was qualitatively verified i the followig way. Istead of etchig the devices ito the 00 X 00 pmz mesas, they were etched oly i oe directio ad cleaved i the

8 984 IEEE TRANSACTIONS ELECTRON DEVICES, VOL. ED-29, NO. 6, 982 JUNE of recombiatio currets i the differet active regios is uiform to withi 0 percet. The breakover voltage (V&) of the device is 9 V. Devices with breakover voltages of more tha 5 V were also observed. The value of VBO i each particular device also depeds o the amout of leakig due to imperfectios. The holdig curret desity (JH) is about.5 A cm-2, Lasers made of the (p-)s devices had threshold curret desity of about ka cm-2, which is comparable to covetioal large optical cavity lasers of the same dimesios. More details o the optical characteristics of multi p- structures ca be foud i [5] [7]. VI. CONCLUSIONS The electrical properties of multi p- juctio devices (both homostructures ad heterostructures) were aalyzed. By usig a modified trasistor model it was foud that devices of this type possess bistable characteristics similar to that of a Shockley diode, ad thus they are potetially suitable for switchig applicatios. Amog these applicatios are semicoductor cotrolled rectifiers with higher breakover voltages ad potetially shorter switchig times, semicoductor laser Fig. 0. I-V curve of a device (horizotal scale: V/div; vertical devices which also have itrisic electrical switchig capascale: ma/div). bilities, ad large optical cavity lasers with multiple active regios. Quatitative aalysis idicated thathe iheretly other dimesio (with legths about 00 pm) thus formig greater curret gais ivolved i the operatio of such a the commo Fabri-erot cavity of semicoductor ijectio device yield higher breakover voltages ad higher holdig lasers. Below the lasig threshold curret of the device, the currets. Experimetal results verified the basic geeric distributio of the amout of light emitted from each GaAs characteristics ad showed a good fit with the calculated ( active ) regio via the spotaeous emissio, which is pro- performace. portioal to the carrier cocetratio i it, was observed to be i agreemet with Fig. 6(b). As the curret was further i- REFERENCES creased, it was foud that at, or slightly above the threshold [ J. L. Moll et al., p--p- trasistor switches, roc. IRE, vol. (Jth 5.5 ka - cm-2) oly the upper active regio is lasig, 44, pp ,956. while all the other active regios emitted oly spotaeous [2] F. E. Getry et al., Semicoductor Cotrolled Rectifiers. Eglewood Cliffs, NJ: reticehall, 964. emissio. Oly whe the curret was raised to about.4 X [] C.. Lee et al., Barrier-cotrolled low-threshold p--p- GaAs Ith did the ext active regio lase. heterostructure laser, Appl. hys. Lett., vol. 0, pp , 977. Devices of a secod type were grow with the goal of [4] W. F. Kosooclcy, R. H. Comely, ad I. J. Hegyi, Multilayer equalizig the carrier distributio i all the GaAs layers of GaAs ijectio laser, IEEE J. Quatum Electro., vol. QE-4, the device so that the light emitted will be more evely dis- pp ,968. [5] J. Katz et al., Large optical cavity AlGaAs lasers with multiple tributed, The parameters resultig the of structure active regios, J. Appl. hys., vol. 5, pp , 980. are described i Table, ad its typical I-V curve is show [6] S. M. Sze, hysics of Semicoductor Devices. New York: Wileyi Fig. 0. The calculated results o this structure are give Itersciece, 969, pp ,. [7]. Yeh, A Yariv, ad C. S. Hog, Electromagetic propagatio i Fig From Fig. - 6(b).. it is see that most of the carriers i periodic stratified media. I. Geeral theory, J. Opt. Soc. (42 percet) recombie i the active regios, ad the level Am&., vol. 67, pp , 977.