IEEE TRANSACTIONS ON ELECTRON DEVICES AcKn IR Electron current incident on an electron-voltaic converte G.B. Shoop participated in the development of the I,= Current through the load of an electron voltaic the ac assembly of the batteries. J. Scott -Monck made the allium arsenide diodes and some of the silicon diodes. I The self-absorption limited electron current paport,who first conceived of this method of energy con- Io= Reverse saturation current of a diode. ss of a E. F. Pasierb, Jr, made the later silicon diodes. R. c. which would be emitted from the surface of a Hand designed the atomic battery assembly. P. Rap radioactive layer of semi-infinite thickne version, has provided helpful discussion 。 Short-circuit current of an electron-voltaic con- verter TABLE OF SYMBOLS Boltzmann's constant L. Diffusion length of electrons A, A'= Parameters used to describe the rate of de- L, Diffusion length of holes radation of semiconductors under electron ir- =q/ T. ao= Approximate self-absorption coefficient of Pm l47 "max Eleetron-voltaic conversion efficiency radiation q beta rays in cm tu= Radioisotope half-life. a/p Approximate absorption coefficient of Pm47beta T= Minority carrier lifetime rays in em /gm Minority carrier lifetime before electron ir- B Empirical constant to describe current voltage radiation relationship of a diode. Vmax Voltage across an electron-voltaic converter EB- Energy of an emitted electron under open circuit conditions EBsx Average energy of betas from a radioisotope. Vmp Voltage across an electron-voltaic converter at EBm Maximum energy of betas from a radioisotope a load which gives maximum power E. Energy gap of a semiconductor. Ionization energy required to generate and col- G Specific activity of a radioisotope in curies/gm lect an electron-hole pair in a diode. Analysis and Characterization of P-N Junction Diode Switching H. J. KUNOT, MEMBER, IEEE Summary--A new charge control model of a p-n junction diode Ⅰ NTRODUCTION is introduced in which the reverse current ig as well as the forward current IF are related to the charge Q stored in the base region by HE SWITCHING TIME of a semiconductor diode time constants Tg and Tp, respectively. The se switching is of great importance in computer circuit design transient is analyzed for normal switching operation where a con- Recently, in particular, development of very high exist, anrtent phase(storage phase )and a decaying current phase speed switching transistors made the switching time of current phase diodes much more significant than ever before. The switch equ ing time analysis of diodes have been carried out by some expressed in terms of measurable device para authors [1[8]. However, the switching tim he equations external circuit variables Ip and Ig; and an derived by most of the authors in the past, [1]-[41,[71, eter R. The proposed model is applicable to action diodes of [8), are expressed in terms of parameters which are ex- any type tremely difficult to measure and not practical for charac- ported. It is shown that the experimental results are in very good terizing the diode switching. It is, therefore, desirable for ent with the practical circuit design purposes to find new ways of characterizing the diode switching in terms of parameters s Received June 17. 1963 which can be measured simply 1968. Portions of this pa CON Con- In this paper the reverse switching transient is analyzed August, ational Cash Register Company, Electronics Division, Har and new diode switching time equations are derived thorne, calif. relating the reverse current ig as well as the forward Authorized licensed use limited to: IEEE Xplore. Downloaded on December 15, 2008 at 03 47 from IEEE Xplore. Restrictions apply
IEEE TRANSACTIONS ON ELECTRON DEVICES January ACKNOWLEDGMENT I, = Electron curent incident on an electron-voltaic 6. B. Shoop participated in the development of the source deposition technique and performed the actual assembly of the batteries. J. Scott-Monck made the gallium arsenide diodes and some of the silicon diodes. E. F. Pasierb, Jr., made the later silicon diodes. R. C. Hand designed the atomic battery assembly. P. Rappaport, who first conceived of this method of energy conversion, has provided helpful discussion. TABLE OF SYMBOLS Parameters used to describe the rate of degradation of semiconductors under electron irradiation. Approximate self-absorption coefficient of Pm14? beta rays in cm-l. Approximate absorption coefficient oiPm147 beta rays in cm2/gm. Empirical constant to describe current voltage relationship of a diode. Energy of an emitted electron. Average energy of betas from a radioisotope. Maximum energy of betas from a radioisotope. Energy gap of a semiconductor. Specific activity of a radioisotope in curies/gm. converter. = Current through the load of an electron voltaic converter. = The self-absorption limited electron current which would be emitted from the surface of a radioactive layer of semi-infinite thickness. = Reverse saturation current of a diode. = Short-circuit current of an electron-voltaic con- = Boltzrnann’s constant. = Diffusion length of electrons. = Diffusion length of holes. = q/kT. = Electron-voltaic conversion efficiency. = Electronic charge. = Radioisotope half-life. = Minority carrier lifetime. = Minority carrier lifetime before electron ir- = Voltage across an electron-voltaic converter = Voltage across an electron-voltaic converter at = Ionization energy required to generate and colverter. radiation. under open circuit conditions. a load which gives maximum power. lect an electron-hole pair in a diode. Analysis and Characterization of P-N Junction Diode Switching* a. J. KUNOt, MEMBER, IEEE Summary-A new charge control model of a p-n junction diode is introduced in which the reverse current iR as well as the forward current Zp are related to the charge Q stored in the base region by time constants 73 and T~, respectively. The reverse switching transient is analyzed for normal switching operation where a constant current phase (storage phase) and a decaying current phase exist, and for overdriven switching operation where no constant current phase exists. New switching time equations are derived. The equations are expressed in terms of measurable device parameters Tp, TR, and Cj; external circuit variables ZF and IR; and an external circuit parameter R. The proposed model is applicable to p-n junction diodes of any type. Experimental results using various types of diodes are also reported. It is shown that the experimental results are in very good agreement with the theory. * Received June 17, 1963; revised manuscript received August 20, 1963. Portions of this paper were presented at the WESCON Convention, San Francisco, Calif., August, 1963. t National Cash Register Company, Electronics Division, Hawthorne, Calif. INTRODUCTION HE SWITCHING TIME of a semiconductor diode is of great importance in computer circuit design. Recently, in particular, development of very high speed switching transistors made the switching time of diodes much more significant than ever before. The switching time analysis of diodes have been carried out by some authors [1]-[8]. However, the switching time equations derived by most of the authors in the past, [I]-[4], [7], [8], are expressed in terms of parameters which are extremely difficult to measure and not practical for characterizing the diode switching. It is, therefore, desirable for practical circuit design purposes to find new ways of characterizing the diode switching in terms of parameters which can be measured simply. In this paper, the reverse switching transient is analyzed and new diode switching time equations are derived by relating the reverse current iE as well as the forward Authorized licensed use limited to: IEEE Xplore. Downloaded on December 15, 2008 at 03:47 from IEEE Xplore. Restrictions apply
196 Kuno: P-N Junction Diode switching Fig. k-Planar junction diode model current Ip to the minority carrier charge Q stored in the such that base region of the diode by the reverse and forward time constants Ta and Tp, respectively. The equations are ex- 60,0+9=gO. pressed in terms of measurable diode parameters, via Te, TR, and Ci, external circuit resistor R, and the forward Then we get to reverse current ratio IP/IR ag=0,0-Q ANALYsis oF SWITCHING OPFRATION Charge Equation Considering the junction capacitance C, as shown in Fig. 1, the total current i(t) Alowing into the terminal 1 In order to simplify the analysis, let us consider a is given by planar junction diode shown in Fig. 1. ( We do not lose generality by this simplification. )We shall also assume that the conductivity of the p-type material o, is much i(0=ir(,t+ at greater than that of the n-type material on, so that the hole current at the junction may be considered to be the i, (0, t)+C total current. The continuity equation in the n-type (base)region can be written in terms of the excess hole whe nsity as [8] P C (v)dvi where Thus we can relate the total current i(t) that flows into p(, t=P(, t-Po he terminal l to the charge Q()stored in the base region of the diode and the junction voltage V, by the following namely, the charge equation Q 十+C; 1) P(, t)=hole density Integrating this equation over the region <a <w Now let us consider the switching circuit shown in and defining the total charge stored in the base region Q Fig. 2. Initially, the switch S is in position 1, and the such that diode is forwardly biased and conducting current IF We shall assume that the switch s has been in position 1 Q(0=9, for a long time so that a steady state may be assumed the time t=0. Then the initial condition can be ob- we get tained by substituting steady-state condition do Q dt i, (0, t)-i,(w, t) Noting that the term Q/T, represents the recombination dQ rate in the base region where a w and that i,(w, t) represents the recombination rate at the boundary a =wo let us define the total recombination rate constant, tp to i, be subscript F is chosen since, for steady state, Q is related into(1) Authorized licensed use limited to: IEEE Xplore. Downloaded on December 15, 2008 at 03: 47 from IEEE Xplore. Restrictions apply
1964. Kuno: P-N Junction Diode Switching 5 Fig. 1-Planar junction diode model. current IF to the minority carrier charge Q stored in the base region of the diode by the reverse and forward time constants rR and TF, respectively. The equations are expressed in terms of measurable diode parameters, vix., rP, T~, and Ci, external circuit resistor 8, and the forward to reverse current ratio IF,/IR. ANALYSIS OF SWITCHING OPERATION Charge Equation In order to simplify the analysis, let us consider a planar junction diode shown in Fig. 1. (We do not lose generality by this simplification.) We shall also assume that the conductivity of the ptype material a, is much greater than that of t,he n-type material o,, so that the hole current at the junction may be considered to be the total current. The continuity equation in the n-type (base) region can be written in terms of the excess hole density as [8] where = hole current P(x, t) = hole density P,(x) = hole density in thermal equilibrium. Integrating this equation over the region 0 _< x 5 w, and defining the total charge stored in the base region Q such that Q(t) = P \m P dx, -0 we get Noting that the term Q/T~ represents the recombination rate in the base region where 0 < x < w and that ip(w, t) represents the recombination rate at the boundary x = w, let us define the total recombination rate constant, TF,~ to IF by The subscript F is chosen since, for steady state, Q is related such that Then we get Considering the junction capacitance Cj as shown in Fig. 1, the total current i(t) flowing into the terminal 1 is given by where Vi = junction voltage Thus we can relate the total current i(t) tlhat flows into the terminal 1 to the charge Q(t) stored in the base region of the diode and the junction voltage Vi by the following, namely, the charge equation Initial Condition Now let us consider the switching circuit shown in Fig. 2. Initially, the switch X is in position 1, and the diode is forwardly biased and conducting current IF. (We shall assume that the switch S has been in position 1 for a long time so that a steady state may be assumed at the time t = 0-). Then the initial condition can be obtained by substituting steady-state conditions i(O-) = IF into (1). Authorized licensed use limited to: IEEE Xplore. Downloaded on December 15, 2008 at 03:47 from IEEE Xplore. Restrictions apply
TEEE TRANSACTIONS ON ELECTRON DEVICES Fig.sTeady state hole density distributions in the base region Fig. 2-Diode switching circuit. The initial hole density distribution can be found by At this time, the total charge in the base region is given by solving the diffusion equation. Typical steady-state dis- tributions are shown in Fig 3 Q(t,=9 P(a, t,)dx at t=t, Normal Switching Operation Noting that Q(t )is related to IR, let us define TR,re In normal switching operation, switching transient con- lating the minimum amount of charge remaining in the sists of constant current phase(storage phase) and de- base region which is required to support the reverse cur- caying current phase as shown in Fig. 4. Now suppose rent, such that that att=0 the switch S is quickly thrown from position 1 to position 2. Then the current starts to fow in the Q()= IRTR opposite direction. The maximum reverse current is Although it is not immediately obvious why the mini- limited, by the external resistor R, to mum amount of charge required to support the reverse V,-V V current is linearly related to the reverse current, the (3) justification is made empirically as shown in a later Then the stored charge Q starts to decay in a manner Substituting (8)into(4), and solving for the storage For0<t<t,,(see Fig. 4), the constant reverse cur time t,w rent Ie flows and the junction voltage does not change (Strictly speaking, it changes slightly as shown in Fig 4 m(+分-m(+2)] but the term C,dv/dt)is negligible compared with other terms in (1)during the constant current phase. If we plot t, vs In (1+ Ir/Ie), we will get a straight Then(1) reduces to line with a slope Tr and offset by In (1+ TR/Tp),as 妲Q旦 wn in Fig. 6. dt (4) For t> t,, the hole density decays further, and current tarts to decay he stored charg With the initial condition given by(2), we can solve for gradient at z=0 necessary to support I g. Let us assume Q(0)and get that the reverse current ig(o is related to the charge Q(0), as we have defined TR, by the relationship Q(0=(I,+IxTp exp t)-IRTP Q(0= TRin(O (10 During the constant current phase, i.e., 0< t<t the where reverse current I n is given by t> t Ia=9 uEP-D, for 0<t<t,.(6) Considering the circuit shown in Fig. 2, the junction oltage V, is related to in by Att= t,, the hole density at the junction(r=0)becomes zero, and(6)become a Le Can et al, [51, and Moll et al. thus obtaining t,=T In (l+IriG (7) t correct, sinc orted by the hole density gradient at the junction as given by (7) Authorized licensed use limited to: IEEE Xplore. Downloaded on December 15, 2008 at 03: 47 from IEEE Xplore. Restrictions apply
10 S IEEE TRANXAGTIONS ON ELECTRON DEVICES January h 0 W 0 W - I- I-x c-x - Fig. 3-Steady state hole density distributions in the base region Fig. 2-Diode switching circuit. of various types of diodes. The initial hole density distribution can be found by solving the diffusion equation. Typical steady-state distributions are shown in Fig. 3. Normal Switching Operation In normal switching operation, switching transient consists of constant current phase (storage phase) and decaying current phase as shown in Fig. 4. Now suppose that at t = 0' the switch S is quickly thrown from position 1 to position 2. Then the current starts to flow in the opposite direction. The maximum reverse current is limited, by the external resistor R, to Then the stored charge Q starts to decay in a manner shown in Fig. 5. For 0 < t < t,, (see Fig. 4), the constant reverse current I, flows and the junction voltage does not change. (Strictly speaking, it changes slightly as shown in Fig. 4, but the term Ci(dVi/dt) is negligible compared with other terms in (1) during the constant current phase.) Then (1) reduces to (4) With the initial condition given by (2), we can solve for &(t) and get Q(t) = (I, 3- I,)T~ exp (-"> - IRTF. (5) During the constant current phase, Le., 0 < t < t,, the reverse current I, is given by 7F At 2 = t,, the hole density at the junction (x = 0) becomes zero, and (6) becomes At this time, the totaI charge in the base region is given by2 Q(t8) = q 1 P(x, t.) dx at t = t,. Noting that &(&) is related to I,, let us define 7x7 relating the minimum amount of charge remaining in the base region which is required to support the reverse current, such that W 0 Q(tJ = IRTR. (8) Although it is not immediately obvious why the minimum amount of charge required to support the reverse current is linearly related to the reverse current, the justification is made empiricalIy as shown in a later section. Substituting (8) into (4), and solving for the storage time t,, we get If we plot t, vs In (1 + IF/'IE), we will get a straight line with a slope rF and off set by In (I 3- ~R/TF), as shown in Fig. 6. For t > t,, the hole density decays further, and current starts to decay, since the stored charge cannot keep the gradient at x = 0 necessary to support I,. Let us assume that the reverse current iR(t) is related to the charge &(t), as we have defined rR, by the relationship Ut) = TRiR(t) (10) where t > t,. Considering the circuit shown in Fig. 2, the junction voltage Vi is related to iR by thus obtaining t, = T~ In (1 + IF/IR), but their assumptions are LeCan et al., [5], and Moll et al., [6], assume that Q(t.) = 0, not correot, since the current.1, at t = t, is supported by the hole density gradient at the junctlon as given by (7). Authorized licensed use limited to: IEEE Xplore. Downloaded on December 15, 2008 at 03:47 from IEEE Xplore. Restrictions apply
1964 no: P-N Junction Diode Switching 11 Fig. 4-Current and voltage switching characteristics of a diode Fig 5-Hole density distributions during the switching Fig. 6-Storage time 4, of a diode as a function of Ip/IR iRR 4=232+R (16) Differentiation of this with respect to time t gives 1+_8 d (12) Overdriven Switching Operation Then(1) becomes In normal switching operation, we have derived the expression for the storage time t, given by(9). It can be di ir=(TR+RC+-iR (13)seen that, as Is is inereased keeping Ir constant, t, de- creases and becomes zero when with the boundary condition given by( 8); i.e IRE-IE Let us define I we can solve for in(t) for t>t, We get Ro as the critical reverse current beyond which t, becomes zero, b.e., 1+ (17 i(6-1s(-(x+Z(-)·10 When i,>IRo, viz., overdriven switching, the charge Defining t, as the time that in takes to decay from I2 stored in the base region cannot support the reverse to 10 per cent of Ir as shown in Fig. 4, i.e current. Thus, the reverse current immediately starts to iR(n=0.1IN lo=I, (18) t, can be obtained from(14) as shown in Fig. 7, and assume that to is very gmall so tr (15) that the charge stored in the base region does not decay appreciably during this interval of time. Then the all Authorized icensed use limited to: IEEE Xplore. Downloaded on December 15, 2008 at 03: 47 from IEEE Xplore. Restrictions appl
1964 Kuno: P-N Junction Diode Switching 11 owt "(t) Fig. &Current and voltage switching characteristics of a diode. (cl t = tS (a) t > tg Fig. 5-HoIe density distributions during the switching. Fig. 6-Storage time t, of a diode as a function of IF/IR. Vj = - V, + iRR. Differentiation of this with respect to time t gives Then (1) becomes with the boundary condition given by (8); i.e., &(is) = IRTR we can solve for iR(t) for t > t,. We get 7 (12) Overdriven Switching Operation In normal switching operation, we have derived the expression for the storage time t, given by (9). It can be (13) seen that, as I, is increased keeping I, constant, ts decreases and becomes zero when TF TR 1, = --I,. Let us define IRo as the critical reverse current beyond which t, becomes zero, Le., I20 = - I,. TP TR (17) (14) When i, > IRO, vix., overdriven switching, the charge Defining tf as the time that iR takes to decay from I, stored in the base region cannot support the reverse to 10 per cent of I, as shown in Fig. 4, i.e., current. Thus, the reverse current immediately starts to decay to IRo. Let us define to such that iR(tf) = 0.1 I,, i,(t3 = IRO, (18) tf can be obtained from (14). as shown in Fig. 7, and assume that to is very small so (15) that the charge stored in the base region does not decay appreciably during this interval of time.3 Then the all 1+- I TF J or thus to, is very small compared with rg in most cases. This assumption is justifiable since the time constant RCi, Authorized licensed use limited to: IEEE Xplore. Downloaded on December 15, 2008 at 03:47 from IEEE Xplore. Restrictions apply
IEEE TRANSACTIONS ON ELECTRON DEVICES January current flowing during the interval to may be considered Then(1)becomes to discharge the junction capacitance, and for the over- driven switching, where ts to, (1) reduces to iR(O=(TR+RC3+ -in(o-I,+rc, die (19) with the boundary condition ro>in(to=i the solution of (26)becomes Q(0=IPT, a constant for t< to 1+ The initial condition is ia(t)=IRo exp L-TR +RC4-6 where Ik is defined as the overdriven peak current which is determined by the external circuit such that t=2.38+BC 1 where t, is defined as the time required for iz to decay as shown in Fig. 7 from IRo to 10 per cent of IRo:i.e The solution of (19) with the initial condition(20) is then given by i( 0.1 Ino. (0=(+12)c(-BC)-2) EXPERIMENTAL RESULTS Various types of diodes were selected for switching time measurement. also, current waveforms during the iRO>IRO, switching transient were analyzed using the set-up shown n Fig. 8. Diodes tested are grouped into four general types: 1) germanium gold bonded, 2)germanium expitaxial 0<<t mesa, 3)silicon planar, and 4)silicon diffused mesa. Fig. 9(a)-9(d) shows typical results of the storage time of the overdriven phase; i.e., t to, we have measurement of each group. I, was varied from 1 ma r(lo=iRo (23)to 20 ma and Ir/In up to 100 It can be seen that each curve shows straight line Substituting (23)into (22)and solving for to, we obtain relationship between t, and In(1 +Ir/Ix), as predicted by( 9). The results also indicate that Tp and TR are very t= rc In(1+e-In(1+ Ro (24) constant over the wide range of I, and Ig, as we have assume Fig. 10(a)-10(b) shows typical switching transient characteristics observed with a sampling scope driving a to=RC In (1+ 分h(+2) (25) X-Y recorder. Fig. 10(a) shows normal switching opera- tion, and Fig 10(b)overdriven switching operation. These For t> to, we have ig Ino and results show very good agreement with the switching characteristics given by(14),(22), and (27) TRiR t o(t AY工树篇 Fig. 7-Current switching characteristics of a diode overdriven in reverse directi Fig. 8--Set-up for diode switching test. Authorized licensed use limited to: IEEE Xplore. Downloaded on December 15, 2008 at 03: 47 from IEEE Xplore. Restrictions apply
12 IEEE TRANSACTION# ON ELECTRON DEVICES January current flowkg during the interval to may be considered Then (1) becomes to discharge the junction capacitance, and for the overdriven switching, where t < to, (I) reduces to iR(t) = (712 f RCj) + iR diR (26) TF -i~(t) = I, + RC; - diR dt (19) where iR(t> > ix(to) = 1x0 Q(t) = Ip~p = constant for t < to. The initial condition is iR(0) = I; (20) where I; is defined as the overdriven peak current which is determined by the external circuit such that as shown in Fig. 7. then given by The solution of (19) with the initial condition (20) is for iR(t) > IRo t %.e., 0 < t < to. At the end of the overdriven phase; Le., t = to, we have iR(tO) = IZO. (23) Substituting (23) into (22) and solving for to, we obtain to = h%,[ In (1 + 2) - In (I + ?)] (24) or 3- OVER-DRIVEN Fig. 7-Current switching characteristics of a diode overdriven in the reverse direction. with the boundary condition iR(tO) = the solution of (26) becomes and where t, is defined as the time required €or i, to decay from IBo to 10 per cent of IRO; i.e., i(tf) = 0.1 IRO. EXPERIMENTAL RESULTS Various types of diodes were selected for switching time measurement. Also, current waveforms during the switching transient were analyzed using the set-up shown in Fig. 8. Diodes tested are grouped into four general. I types: 1) germanium gold bonded, 2) germanium expitaxial mesa, 3) silicon planar, and 4) silicon diffused mesa. Fig. 9(a)-9(d) shows typical results of the storage time measurement of each group. IF was varied from 1 ma to 20 ma and IF/IR up to 100. It can be seen that each curve shows straight line relationship between t, and In (1 + IF/IR), as predicted by (9). The results also indicate that rF and rR are very constant over the wide range of IF and IE, as we have assumed. Fig. lO(a)-lO(b) shows typical switching transient characteristics observed with a sampling scope driving a X-Y recorder. Fig. 10(a) shows normal switching operation, and Fig. 10(b) overdriven switching operation. These results show very good agreement with the switching characteristics given by (14), (2.29, and (27). I 2.4 X 1 5M 500n POWER SUPPLY mTTPA- ~ S4MPLIUG TRIOGER SCOPE PROBE c yr DIODK 0-50v I m - - x-Y REMRDER Fig. 8-Set-up for diode switching test. Authorized licensed use limited to: IEEE Xplore. Downloaded on December 15, 2008 at 03:47 from IEEE Xplore. Restrictions apply