EE TRANSACTIONS ON ELECTRON DEVICES, MARCH 1971 graph is sufficient to determine not only the minority carrier lifetime and penetration length, but also the diode's series resistance and average depletion capaci t5zu ance. A cursory examination of the oscilloscope photo- graph allows an immediate qualitative and semiquan tative evaluation of these parameters. Furthermore y analyzing the photograph with the aid of a simp chart contained in this paper, one can refine the re sults and increase the accuracy of the quantitativ DISTANCE RE LIGHTLY-DOPED determinations to within about 25 percent. In this way, it is practical to carry out a series of important junction Fig. 1. Density of injected minority carriers for times()after the measurements over a wide range of temperature and in- jection levels to obtain an extensive characterization of arge remaining at the end of the storage time(T,). The dotted es indicate our straight- line approximations. the junctions of interest. THEORY OF JUNCTION BEHAVIOR value. Then, with TI, the average capacitance C is de- Qualitative Description fined by the relation C=Ti/R In the extreme case of an ideal step junction in which the depletion width Minority-carrier density profiles for a cross section builds up from zero to a final value corresponding to a cutting through the plane of the junction are indicated final capacitance C,, one can show [9] that C=2.17 Cr by solid lines in Fig. 1. Carriers have been injected into Thus our "average"capacitance is somewhat higher the more lightly doped material to the right of the junc- than the final open-circuit value of the junction de- tion. The length L is the average"penetration length"pletion capacitance. Although the actual depletion of these carriers. This length is strongly affected by capacitance varies with applied voltage, for our de mpurity gradients in the region occupied by the in- velopment, we will approximate it by aconstant jected. carriers. For meaningful results, the built-in capacitance C whose value is determined in the above field due to such gradients must be either zero or di- fashion rected so as to retard injection (An injection-enhancing A current source is appropriate for representing the "drift"field pulls carriers away from the junction, extraction of the remaining minority carriers for t>T, where they cannot be retrieved by a reversing poten- since the extraction process is controlled by diffusion ideally abrupt step jur zero built-in field, the penetration length equals the across the depletion layer. By assuming a constant di classical“ diffusion length.”Fora“ graded” Junction from the maximum [91-[10 the penetration length is shortened by a re- minority carrier density, we will show later that the tarding field current source decays exponentially in time, with a time The top solid curve in Fig. 1 indicates the minority- constant equal to the recovery time carrier density profile before a reversing voltage is ap- The ratio of forward to reverse bias current Ip/I plied. Upon application of a reversing voltage, the den- determines the density profile at t=T,, and therefore sity at the junction starts dropping, and it continues to affects the magnitude of both T, and Tr. A relatively drop throughout a period of time defined as the storage large reverse current shortens the storage time T, by period [9]. During this period the excess carriers avail- extracting the carriers quickly and by producing a able at the junction make it effectively a short circuit density profile at t=T, which is skewed toward the unction. With such a profile, most of the previously When the carrier density at the junction reaches zero injected carriers are still present. During the recovery t=T,(storage time), a reverse-biased depletion layer period the density profile at t= T. determines the initial begins to form. The voltage across the diode builds up rate at which the remaining carriers are extracted. (A rapidly and approaches its steady-state value. The profile crowded close to the junction produces a high voltage buildup is retarded by the time required to extraction rate. ) When selecting the time constant for charge up the depletion capacitance and to extract the our current source it is appropriate to focus on the remainder of the carriers previously injected under density profile at t=T, since most of the voltage change forward bias(shaded area of Fig. 1). We will show below occurs early in the recovery transient. Ultimately, we that during this"recovery"period, the junction can be will test this time constant by comparing the results of schematically epresented b y a capacitor in parallel our derivation with the more rigorous results of others with a current source [6-10] in the special limiting cases which they treat. The capacitor represents an "average"depletion capacitance, whose value is determined by applying a Mathematical Mode reversing step voltage to the junction through a series Storage Period The storage time T, provides a rough resistance R and by measuring the time Ti required for indication of the minority-carrier lifetime T. The rela- the junction capacitance to charge to 1/ e of its final tion between lifetime and storage time has been derivedj 52 IEEE TRANSACTIONS ON ELECTRON DEVICES, MARCH 1971 graph is sufficient to determine not only the minority t carrier lifetime and penetration length, but also the diode’s series resistance and average depletion capaci￾tance. A cursory examination of the oscilloscope photo￾E graph allows an immediate qualitative and semiquan￾titative evaluation of these parameters. Furthermore, by analyzing the photograph with the aid of a simple W n chart contained in this paper, one can refine the re￾sults andincrease the accuracy of the quantitative DISTANCE INTO MORE LIGHTLY-DOPED determinations to wihin about 25 percent. In this way, MATERIAL,X - it is practical to carry out a series of important junction Fig. 1. Density of injected minority carriers for times (t) after the lneasurements Over a wide range of temperature and in- reversing pulse teaches the diode. The shaded area indicates the charge remaining at the end of the storage time (T8). The dotted jection levels to obtain an extensive characterization of lines indicate our straight-line approximations. the junctions of interest. no n a w a a s n1 I- W u -J THEORY OF JUNCTION BEHAVIOR value. Then, ‘with TI, the average capacitance C is de￾Qualitative Description Minority-carrier density profiles for a cross section cutting through the plane of the junction are indicated by solid lines in Fig. 1. Carriers have been injected into the more lightly doped material to the right of the junc￾tion. The length L is the average “penetration length” of these carriers. This length is strongly affected by impurity gradients in the region occupied by the in￾jected carriers. For meaningful results, the built-in field due to such gradients must be either zero or di￾rected so as to retard injection. (An injection-enhancing “drift” field pulls carriers away from the junction, where they cannot be retrieved by a reversing poten￾tial.) For an ideally abrupt step junction [6]-[8] with zero built-in field, the penetration length equals the classical “diffusion length.” For a “graded” junction [9]-[lO] the penetration length is shortened by a re￾tarding field. The top solid curve in Fig. 1 indicates the minority￾carrier density profile before a reversing voltage is ap￾plied. Upon application of a reversing voltage, the den￾sity at the junction starts dropping, and it continues to drop throughout a period of time defined as the storage period [9]. During this period the excess carriers avail￾able at the junction make it effectively a short circuit When the carrier density at the junction reaches zero at t = T, (storage time), a reverse-biased depletion layer begins to form. The voltage across the diode builds up rapidly and approaches its steady-state value. The voltage buildup is retarded by the time required to charge up the depletion capacitance and to extract the remainder of the carriers previously injected under forward bias (shaded area of Fig. 1). We will show below that during this “recovery” period, the junction can be schematically represented by a capacitor in parallel with a current source. The capacitor represents an “average” depletion capacitance, whose value is determined by applying a reversing step voltage to the junction through a series resistance R and by measuring the time TI required for the junction capacitance to charge to l/e of its final PI. fined by the relation C= TI/R. In the extreme case of an ideal step junction in which the depletion width builds up from zero to a final value corresponding to a final capacitance C, one can show [9] that C=2.17 Cf. Thus our “average” capacitance is somewhat higher than the final open-circuit value of the junction de￾pletion Capacitance. Although the actual depletion capacitance varies with applied voltage, for our de￾velopment, we will approximate it by a constant capacitance C whose value is determined in the above fashion. A current source is appropriate for representing the extraction of the remaining minority carriers for t> T,, since the extraction process is controlled by diffusion, and is therefore independent of the reverse voltage across the depletion layer. By assuming a constant dis￾tance from the junction edge to the point of maximum minority carrier density, we will show later that the current source decays exponentially in time, with a time constant equal to the recovery time T,. The ratio of forward to reverse bias current IF/IR determines the density profile at t = T,, and therefore affects the magnitude of both T, and T,. A relatively large reverse current shortens the storage time T, by extracting the carriers quickly and by producing a density profile at t = T, which is skewed toward the junction. With such a profile, most of the previously injected carriers are still present. During the recovery period, the density profile at t = T, determines the initial rate at which the remaining carriers are extracted. (A profile crowded close to the junction produces a high extraction rate.) When selecting the time constant for our current source, it is appropriate to focus on the density profile at t = T,, since most of the voltage change occurs early in the recovery transient. Ultimately, we will test this time constant by comparing the results of our derivation with the more rigorous results of others [6]-[10] in the special limiting cases which they treat. Mathematical Model Storage Period: The storage time T, provides a rough indication of the minority-carrier lifetime T. The rela￾tion between lifetime and storage time has been derived