IEEE TRANSACTIONS ON ELECTRON DEVICES. MARCH 1971 the basic response time of the measuring system, and can be as long as required for thermal isolation. The 老 【Rs/Ro)/(|+Rs/Ro basic response time of the apparatus is usually limited by the rise and fall times of the voltage pulse and sam pling scope, and it is on the order of 0.25 ns for the equipment we are now using vlL Initially, the voltage at the sampling probe is zere When the reversing step from the pulse generator first IIF Rol passes the probe, the probe voltage drops to the nega- Fig. 4. Idealized oscilloscope traces(v,) ure of the test diode, this voltage pe moderate reversing pulse reaches the diode and is reflected to the onse with small but finite Ip and probe. Thus, the first signal displayed on the sampling same V, at tax scope determines the magnitude of the reversing step Itage. Given the distance l from the probe to the with an exponentially decaying current source with an initial value of -Ie, one can write the total current as When the reflection from the diode reaches the probe I,=Cav/dt-IR exp(-t'/T)where t'is the time after (=O), the diode's response to the reversing signal begins T, and T is the time constant of the decaying current to appear. The diode exhibits a low resistance during source. Substituting V,=Ve-I,R, and then substituting the storage period Ta, and then approaches an open for I, and v, from the relations given above leads to a circuit during the subsequent recovery period. The first-order equation in V tance in series with the effectively shorted junction. Tidv/dr+ V,= 2V:+IpRo+ RRo exp(=t/T, The recovery toward an open circuit is approximately exponential. During this period we represent the diode by the above resistor in series with a parallel combina- T1=(R0+R)C tion of a capacitor and exponentially decaying current. With(17)and(18)for the initial value, the solution to source,as indicated previously. The remaining relationships shown in Fig. 4 can be this equation is derived by supposing that the current probe, blocking capacitor, and voltage probe are squeezed up tight +[T/(T,-T)exp(-/T)}.(19) against the diode, so that there are no transit-time de- lays between the diode and the voltage probe. This Equation (19)shows that the difference between 2Va simplification is possible because after the initial drop, and the steady-state probe voltage is IpRo. The am- all parts of the voltage waveform are retarded by the plitude of the recovery is simply IRRo. Thus, one can same time pick the ratio IF/IR=IFRo/iRRo directly off the oscil loscope trace During the storage time Te, the voltage drop across The capacitor time constant Ti is determined ex- the series resistance is Vr-V=IRe. Using the rela- tions in t able I to substitute for Vt and It, and per- perimentally by repeating the step-response measure forming simple algebraic manipulation, we ob ment with the forward current IF reduced to a small value, and the reversing volta /R0=(Vn/2V)/(1-V/2V (17) tain the same steady-state probe voltage. Equations(8) Thus, the fractional change in the probe voltage at and (15)show that T. and T, decrease rapidly with t=0 gives the series resistance relative to 500, as indi- IF/IR. (For L/Lp=1, T, and T, are proportional to cated in Fig 4. The magnitude of the reverse current is (IP/Ir)2.) The junction voltage, however, decreases IR=-II. Again, substituting for I, from Table I, and only as Vrloge (Ip/IE). Thus, one can reduce Ir/IR using(17) for Vp/2Vi leads to by two or three orders of magnitude and cause T and IR=(-2VvR)/(1+R/R0)-1 Ta to vanish, but still retain V a Vr. Since the forward biased junction voltage is approximately the same with sured value for the forward current, or as we mlay ea- small IF as it is with large IP, and the final voltages are To determine the value of IR, one can insert the m e made to be the same, the junction width changes be- shortly, employ another geometrical relation on the tween the same limits in the two experiments. Conse oscilloscope trace of Fig. 4 to determine the ratio IF/IR quently, the average depletion capacitance and the as sociated time constant Ti are nearly the same in the two The storage period ends at t=Ts when the probe experiments, and Ti can be measured by de termining voltage Vp starts changing again. Assuming that the the time required for the recovery in the low-current anction behaves like a constant capacitor C in parallel experiment to reach 1/e=0. 368 of its final value156 IEEE TRANSACTIONS ON ELECTRON DEVICES. MARCH 1971 the basic response time of the measuring system, and can be as long as required for thermal isolation. The basic response time of the apparatus is usually limited by the rise and fall times of the voltage pulse and sam￾pling scope, and it is on the order of 0.25 ns for the equipment we are now using. Initially, the voltage at the sampling probe is zero. When the reversing step from the pulse generator first passes the probe, the probe voltage drops to the nega￾tive potential Vi, as shown in Fig. 4. Regardless of the nature of the test diode, this voltage persists until the reversing pulse reaches the diode and is reflected to the probe. Thus, the first signal displayed on the sampling scope determines the magnitude of the reversing step voltage. Given the distance 1 from the probe to the diode, it also calibrates the time scale. When the reflection from the diode reaches the probe (t = O), the diode’s response to the reversing signal begins to appear. The diode exhibits a low resistance during the storage period T,, and then approaches an open circuit during the subsequent recovery period. The initial resistance is due to contact, bulk, or lead resis￾tance in series with the effectively shorted junction. The recovery toward an open circuit is approximately exponential. During this period we represent the diode by the above resistor in series with a parallel combina￾tion of a capacitor and exponentially decaying current source, as indicated previously. The remaining relationships shown in Fig. 4 can be derived by supposing that the current probe, blocking capacitor, and voltage probe are squeezed up tight against the diode, so that there are no transit-time de￾lays between the diode and the voltage probe. This simplification is possible because after the initial drop, all parts of the voltage waveform are retarded by the same time. During the storage time T,, the voltage drop across the series resistance is Vi- V, = IiR,. Using the rela￾tions in Table I to substitute for V, and It, and per￾forming simple algebraic manipulation, we obtain R,/Ro = (Vp/2VJ/(1 - VP/2Vi). (1 7) Thus, the fractional change in the probe voltage at t = 0 gives the series resistance relative to 50 Q, as indi￾cated in Fig. 4. The magnitude of the reverse current is IR = -It. Again, substituting for It from Table I, and using (17) for Vp/2 Vi leads to IR = (-2Vi/Ro),/(l + R,/RO) - IF. (1 8) To determine the value of IR, one can insert the mea￾sured value for the forward current, or as we shall see shortly, employ another geometrical relation on the oscilloscope trace of Fig. 4 to determine the ratio IF/IR directly. The storage period ends at t = T, when the probe voltage V, starts changing again. Assuming that the junction behaves like a constant capacitor C in parallel vp =o} Fig. 4. Idealized oscilloscope traces (V,) for a reversing step volt￾age. Zero time occurs when the reflected voltage step just reaches the probe. The solid curve indicates the response with a moderate forward biasing current IF. The dotted curve indicates the re- sponse with small but finite IF and with Vi adjusted for the same V,att=m. with an exponentially decaying current source with an initial value of -IR, one can write the total current as It = Cd Vj/dt - IR exp( -t’/T,) where t‘ is the time after T,, and T, is the time constant of the decaying current source. Substituting Vi = V, - ItR8 and then substituting for It and Vt from the relations given above leads to a first-order equation in V,: TldV,/dt’ 4- V, = 2Vi + IpRo + IRR~ exp (-t’/T,) where TI = (Ro + RJC. With (17) and (18) for the initial value, the solution to this equation is V, = 2Vi + IF& -I- I&( [TI/(TI - TT)] exp TI) + [Tr/(Tr - TI)] exp (-t’/Tr)]e (19) Equation (19) shows that the difference between 2Vi and the steady-state probe voltage is IFRo. The am￾plitude of the recovery is simply IBRo. Thus, one can pick the ratio IF/IR = I.PRo/IRR~ directly off the oscil￾loscope trace. The capacitor time constant TI is determined ex￾perimentally by repeating the step-response measure￾ment with the forward current IF reduced to a small value, and the reversing voltage, Vi readjusted to ob￾tain the same steady-state probe voltage. Equations (8) and (15) show that T, and T, decrease rapidly with IF/IR. (For L/LD= 1, T, and T, are proportional to (IF/IR)’.) The junction voltage, however, decreases only as V,-log, (IFIIR). Thus, one can reduce IF/IR by two or three orders of magnitude and cause T, and T, to vanish, but still retain Vi = VF. Since the forward￾biased junction voltage is approximately the same with small IF as it is with large IF, and the final voltages are made to be the same, the junction width changes be￾tween the same limits in the two experiments. Conse￾quently, the average depletion capacitance and the as￾sociated time constant TI are nearly the same in the two experiments, and TI can be measured by determining the time required for the recovery in the low-current experiment to reach l/e = 0.368 of its final value