近代物理实验：反转恢复法测少子寿命（学习资料）_A Refined Step-Recovery Technique for Measuring

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-18, NO. 3, MARCH 1971 A Refined Step-Recovery Technique for Measuring Minority carrier Lifetimes and related Parameters in asymmetric p-n Junction Diodes RAYMOND H. DEAN, MEMBER, IEEE, AND CHARLES J NUESE metrical p-n junction diode can be measured by observing the time nature of the recombination processes [4], or electro- Abstract-Minority-carrier lifetime in a forward-biased response of the diode to a sudden reversing step voltage. An ap ther nescent junctions [2]. Krakauer [5]has considered impurity gradients is developed, and its results are within about 25 excitation for the special case of an exponentially graded percent of those previously obtained for the special cases of ideal impurity profile in the junction. In this paper we con step and exponentially graded junctions. A relatively simple experi- sider the transient response of an initially forward mental technique is described which is suita ble for measuring life- biased junction to a sudden reversing step. This tran are facilitated by the fact that the test diode is mounted at the end sient technique is ideally suited for a one-port measure of a single coaxial line which can be arbitrarily long. The raw data ment scheme, like the one we will describe, and for de- om the experiment are in the form of an oscilloscope trace, which termining lifetime as a function of injection level ovides an immediate qualitative and semiquantitative indication of The step recovery phenomenon was first studied by ie minority-carrier lifetime and the penetration length for the in- Pell [6], Lax and Neustadter [7], and Kingston [8] leads quickly to a more precise quantitative evaluation of these who developed physical and theoretical descriptions for arameters. In addition, the technique can be used to measure an the storage time T. and the recovery time f, in an ideal average junction depletion capacitance and the device series resis- step junction. For conventional germanium and silicon p-n junctions, this abrupt approximation is usually reasonable, since the impurity profile between the and p-sides of the junction usually becomes uniform HE effective lifetime and penetration depth of within a distance from the junction which is much less minority carriers injected across a forward- than a diffusion length. However, the possibility of an biased p-n junction play an important role in a impurity gradient extending to a significant fraction of variety of semiconductor devices, such as transistors, one diffusion length exists in many other semiconduc- lasers, cold-cathode"emitters, etc. Through the years, tors, particularly the Ill-V compounds where lifetimes a variety of techniques have been developed for the are on the order of 10-8 to 10-10 s and diffusion lengths determination of minority lifetimes, each of which has are often less than a micron. For such junctions, the its particular advantages and disadvantages and re- abrupt approximation would not be valid, and a graded quires its specific assumptions and approximations. impurity profile should be considered. Moll, Krakauer Such techniques include the external generation of and Shen [9] and Moll and Hamilton [10] have treated excess carriers near a reverse-biased junction [1, fre- junctions with an exponentially graded and p-i-n ap quency response and delay time measurements on elec- proximation, respectively. Particularly desirable, how troluminescent diodes 2, and analyses of the small- ever, would be a lifetime measurement procedure which signal impedance [3] and steady-state I-V character- could treat junction profiles intermediate to the step istics [4]of p-n junctions and graded junctions considered previously. Another approach is to make use of the time response The present paper treats the application of the step of a p-n junction to a large-signal idal or step recovery technique to p-n junctions with nearly arbi excitation. This approach is appropriate for asym- trary impurity distributions, and develops an approxi metric p-n junctions biased to intermediate current mate, but general, theory for such junctions. In addition levels. It does not require access to a surface perpen- a particular experimental procedure is described which dicular to the junction [1, knowledge of the specific is especially well-suited for measuring very short life- times(25x10-10 s)under a wide range of ambient script received October 16, 1970. The resear oratories, Princeton, N

IEEE TRANSACTIONS ON ELECTRON DEVlCES, VOL. ED-18, NO. 3, MARCH 1971 151 A Refined Step-Recovery Technique for Measuring Minority Carrier Lifetimes and Related Parameters in Asymmetric b-n Tunction Diodes Absfracf-Minority-carrier lifetime in a forward-biased asym￾metrical p-n junction diode can be measured by observing the time response of the diode to a sudden reversing step voltage. An ap￾proximate but general theory for p-n junctions with almost arbitrary impurity gradients is developed, and its results are within about 25 percent of those previously obtained for the special cases of ideal step and exponentially graded junctions. A relatively simple experi￾mental technique is described which is suitable for measuring life￾times down to less than 1 ns. Measurements at extreme ambients are facilitated by the fact that the test diode is mounted at the end of a single coaxial line which can be arbitrarily long. The raw data from the experiment are in the form of an oscilloscope trace, which provides an immediate qualitative and semiquantitative indication of the minority-carrier lifetime and the penetration length for the in￾jected carriers. A graphical presentation of the theoretical results leads quickly to a more precise quantitative evaluation of these parameters. In addition, the technique can be used to measure an average junction depletion capacitance and the device series resis￾tance. INTRODUCTION HE effective lifetime and penetration depth of minority carriers injected across a forward￾biased p-n junction play an important role in a variety of semiconductor devices, such as transistors, lasers, “cold-cathode” emitters, etc. Through the years, a variety of techniques have been developed for the determination of minority lifetimes, each of which has its particular advantages and disadvantages and re￾quires its specific assumptions and approximations. Such techniques include the xternal generation of excess carriers near a reverse-biased junction [l], fre￾quency response and delay time measurements on elec￾troluminescent diodes [2], and analyses of the small￾signal impedance [3] and steady-state I-V character￾istics [4] of p-n junctions. Another approach is to make use of the time response of a p-n junction to a large-signal sinusoidal or step excitation. This approach is appropriate for asym￾metric p-n junctions biased to intermediate current levels. It does not require access to a surface perpen￾dicular to the junction [l], knowledge of the specific Manuscript received October 16, 1970. ‘The research reported herein was partially sponsored by the National aeronautics and Space Administration, Langley Research Center, Hampton, Va., under Contract NAS-12-2091, and RCA Laboratories, Princeton, N. J. The authors are with RCA Laboratories, Princeton, N. J. 08540. nature of the recombination processes [4], or electro￾luminescent junctions [2]. Krakauer [SI has considered the response of a quiescent junction to a large sinusoidal excitation for the special case of an exponentially graded impurity profile in the junction. In this paper we con￾sider the transient response of an initially forward￾biased junction to a sudden reversing step. This tran￾sient technique is ideally suited for a one-port measure￾ment scheme, like the one we will describe, and for de￾termining lifetime as a function of injection level. The step recovery phenomenon was first studied by Pel1 [6], Lax and Neustadter [7], and Kingston [8], who developed physical and theoretical descriptions for the storage time T, and the recovery time T, in an ideal step junction. For conventional germanium and silicon p-n junctions, this abrupt approximation is usually reasonable, since the impurity profile between the n￾and p-sides of the junction usually becomes uniform within a distance from the junction which is much less than a diffusion length. However, the possibility of an impurity gradient extending to a significant fraction of one diffusion length exists in many other semiconduc￾tors, particularly the 111-V compounds where lifetimes are on the order of lo-* to s and diffusion lengths are often less than a micron. For such junctions, the abrupt approximation would not be valid, and a graded impurity profile should be considered. Moll, Krakauer, and Shen [9] and Moll and Hamilton [lo] have treated junctions with an exponentially graded and p-i-n ap￾proximation, respectively. Particularly desirable, how￾ever, would be a lifetime measurement procedure which could treat junction profiles intermediate to the step and graded junctions considered previously. The present paper treats the application of the step￾recovery technique to p-n junctions with nearly arbi￾trary impurity distributions, and develops an approxi￾mate, but general, theory for such junctions. In addition, a particular experimental procedure is described which is especially well-suited for measuring very short life￾times (25 XlO-’O s) under a wide range of ambient conditions. With this procedure, the step-recovery tech￾nique and its interpretation is found to be remarkably straightforward. In most cases, a pair of closely related measurements recorded on a single oscilloscope photo-

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 derived

j 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

DEAN AND NUESE: STEP-RECOVERY TECHNIQUE FOR MEASURING PARAMETERS IN ASYMMETRIC p-n JUNCTION DIODES theoretically for three different idealized junction im- The triangle which approximates the actual initial dis- purity distributions In a graded p-n junction [9]and in tribution has an area which is exactly one half the area a p-i-n diode [10]the relation is under the corresponding exponential T/T=loge(1+IF/IR (1) For a graded impurity distribution(L/Lp<<1) with IF/IR>>(L/LD)?, the actual minority carrier n an ideal step junction [7] the relation is density at t= Ts can be expressed as [9] eri(T/r)=(1+I/I)-1 nt = x(Ir/qAD)exp(-x/L) We will compare the results of our approximate theory We will approximate this distribution with a triangle writing the equation for the extraction of the total in- fusion exactly supplies the reverse current I R. This slope jected charge Q is 2+=-1 Ir/QAD=(/IF)(L/Lp)(no/L) where the reverse current IR is considered to be con- where d is the minority carrier diffusion coefficient, LD stant for this derivation. The solution of this equation is the diffusion length, and the other variables have been defined previously. Solving for the intersection of e()=[(0)+ IrT]exp(t/)-IRt the two lines yields At t=T, the charge has decayed to that indicated by n;=n/[1+(In/)(LD/D)] the shaded area of Fig. 1, and ha as a value e(T)=le(0)+iRT]exp(Ts/T)-Iy x;=L/[1+(IR/1)(L/LD)2] Now we will define a new parameter a to be the frac- The area under our triangle is n L/2, which is exactl of initially injected carriers that remain att= Ta half the area under the more rigorously derived [9] a=Q(T)/Q(0 graded-junction distribution indicated above. For an abrupt junction (L/LD=1) with IF/IR<<l, the tri- This fraction a can vary from zero for a graded junction angular area is again half the area under the initial or large IF/IR to unity for a step junction or small exponential. This is as it should be since this limit cor- Ip/IR. Setting (3)equal to Q(T)=aQ(o), noting the responds to the case in which almost none of the in- elation Q(0)=IFT [9], and rearranging yields jected carriers are removed during the storage period T/r=loge [(1+Ip/IR)/(1+alp/Ir)]. (4) Since for the cases of graded and ideally abrupt p-n junctions, the areas of our triangular density approxi quation(4)is not yet a useful solution, since a is a mations are one half those under the more rigorous function of both impurity grading and Ip/IR. For an derived curves, we postulate that the ratio of the ti estimate of a we will use straight line approximations to ngular areas (at t=0 and t= T) provides a reasonable he actual density distributions, as shown by the dotted general approximation to the area ratio for actual minority tributions. Then, fro The initial density varies approximately exponentially etry a=Q(T /Q(0)an/no, and from(5) with distance a according to the expression [7] a s[1+(P/IE)(LD/L)2-1 Substituting(7)for a in(4)yie With an initial total charge given by T loge( [1 +IpIr][1+(IP/IR)(Lp/L)? (0)=q4|adx≈g4n0L /[1+(p/1)(L/D)2+Ip/l.(8) one obtains For a graded junction, a retarding field crowds the n0≈Ip/qAL condition L<<LD. Thus for L/LD=0,( 8)reduces to where q is the electronic charge and A is the junction (1), as expected. For an ideally abrupt junction,we area. We will approximate this distribution with a line have L=LD, and( 8)reduces to exponential ([1+Ip/l2/1+2r/l]l.(9) Values of T/T calculated from (9) are only about 25 L percent higher than those accurately calculated from

theoretically for three different idealized junction im￾purity distributions. In a graded p-n junction 193 and in a p-i-n diode [ 101 the relation is T,/T = log, (1 i- IF/IR). (1) In an ideal step junction [7] the relation is erf (T,/T) = (1 + IR/Ip)-’. (2) We will compare the results of our approximate theory with these formulas in the appropriate limits. Following Moll, Krakauer, and Shen [9], we start by writing the equation for the extraction of the total in￾jected charge Q: dQ Q -+-= -rR at T where the reverse current IR is considered to be con￾stant for this derivation. The solution of this equation is @(t) = [Q(O) + IRT] exp (-t/r) - IRT. At t = T,, the charge has decayed to that indicated by the shaded area of Fig. 1 , and has a value Q(TJ = [Q(o) + 1x71 exp (-T,/T) - IRT. (3) Now we will define a new parameter a to be the frac￾tion of initially injected carriers that remain at t = T, : = Q(Ta)/Q(O). This fraction CY can vary from zero for a graded junction or large IF/IR to unity for a step junction or small IF/IR. Setting (3) equal to Q(T,) =crQ(O), noting the relation Q(0) = Ipr [9], and rearranging yields Ts/T = loge [(I + IF/IR)/(l aIF/IR)]* (4) Equation (4) is not yet a useful solution, since 01 is a function of both impurity grading and I~/IR. For an estimate of a! we will use straight line approximations to the actual density distributions, as shown by the dotted lines in Fig. 1. The initial density varies approximately exponentially with distance x according to the expression [7] n = lzo exp (-x/L). With an initial total charge given by 187 = @(o) = qA ndx z qAnoL, SoW one obtains no S IFT/qAL, where q is the electronic charge and A is the junction area. We will approximate this distribution with a line through no having a slope equal to the initial slope of the exponential : An Ax -_ - - no/L. The triangle which approximates the actual initial dis￾tribution has an area which is exactly one half the area under the corresponding exponential. For a graded impurity distribution (L/LD < < 1) with IF/IR > > (L/LD)2, the actual minority carrier density at t = T, can be expressed as [9] n x(TR/qAD) exp (-x/L). We will approximate this distribution with a triangle formed by the original straight line and another one passing through the origin with a slope such that dif￾fusion exactly supplies the reverse current IR. This slope is A 1z Ax _- - IR/~AD = (IR/IF)(L/LD)’(%O/L) where D is the minority carrier diffusion coefficient, LD is the diffusion length, and the other variables have been defined previously. Solving for the intersection of the two lines yields ni = nO/[l + (I~/I&) (LD/L) ‘1 (5) at xi == L/[1 4- (IR/IF) (L/LD)’]. (6) The area under our triangle is niL/2, which is exactly half the area under the more rigorously derived [9] graded-junction distribution indicated above. For an abrupt junction (L/LD = 1) with IF/IR< <1, the tri￾angular area is again half the area under the initial exponential. This is as it should be since this limit cor￾responds to the case in which almost none of the in￾jected carriers are removed during the storage period. Since for the cases of graded and ideally abrupt p-n junctions, the areas of our triangular density approxi￾mations are one half those under the more rigorously derived curves, we postulate that the ratio of the tri￾angular areas (at t = 0 and t = T,) provides a reasonable general approximation to the area ratio for actual minority carrier distributions. Then, from simple geom￾etry, a= Q(T,)/Q(O) =ni/no, and from (j), O( = [I + (WI~ (wLPJ-’. (7) Substituting (7) for CY in (4) yields - loge { [I f IF/IR] [1 + (IF/IR)(LD/L)*] Ts T /[1 + (IF/IIJ (LD/L)’ + IF/IR] 1. (8) For a graded junction, a retarding field crowds the injected carriers close to the junction, which leads to the condition L < <LO. Thus for L/LD =O, (8) reduces to (l), as expected. For an ideally abrupt junction, we have L =LD, and (8) reduces to T,/T = log, { [1 + IP/’TR]~/[~ + ~IF!JR]). (9) Values of T,/r calculated from (9) are only about 25 percent higher than those accurately calculated from (2) over a wide range of IF/IR. For most purposes, such

154 EEE TRANSACTIONS ON ELECTRON DEVICES, MARCH 1971 agreement is acceptable. The heuristic approach used in deriving(8)leads us to expect that intermediate values of L/Lp will result in comparably small errors. Recovery Period The recovery time provides a rough (4+) indication of the minority carrier penetration length The relation between penetration length L and recovery idealized impurity distributions considered above. In a Ep 05 graded junction, the recovery can be represented by an o exponential [9]with a time constant T,. This recovery is associated with an average penetration length which a2 is given by the expression . L≈(DT)12 (10) For a p-i-n diode, one can compare the actual process +T) [10 with an exponential form to obtain the approxi Fig. 2. Chart ate relation from Ta I+r ning r and L/Lp Ip/IE L≈1.5(DT) (11) extraction process with an exponentially decaying cur- In an ideal step junction the average penetration length rent source is the same as the diffusion length Substituting the previously derived expression for xi L=LD =(Dr)/2, (12) (6)into the above expression for T,(14)gives but the relation between T and T, and Ip/IR is com T/r≈{1+【(LD/L)+(m/r)(L/LD)]-.(15) plicated [6[8]. We will compare the results of our For L/Lp=0 this reduces to the graded-junction result approximate theory with(10)for a graded junction and (10), as it should. For L/Lp=1.0, we have with the graphs in [6]-[8]for a step junction The carriers which contribute most of the reverse T/r≈[1+(1+Ia/I)2 (16) current during the dominant early part of the recovery Values of T,/r given by(16)range from 5 to 20 percent period are those located a distance x from the junction higher than the more rigorous values given in [8],as (see Fig. 1). Minority carriers with x <x have been re- IF/IR changes from 0.5 to 5.0 moved previously, and those with x>xi have not yet Inter pretation: Equation ( 8)relates the storage time been perturbed by the reversing diffusion gradient. As to lifetime. For a first approximation one can set L/LD recovery proceeds, the carriers in the vicinity of xr either equal to its two limiting values, zero and one, and use recombine or diffuse back across the junction. The con- the two resulting expressions to obtain upper and lower tinuity equation describing these processes is limits on the relation between storage time and lifetime. Then, for a typical forward-to reverse current ratio of unity, one finds the lifetime r to be 1.5 to 4 times longer (15) divided by(8) for x sx. The recombination rate near x, is approxi ives a relation between T/ T, and the grading param- mately n(x/. The diffusion loss rate can be estimated eter L/Lp. It turns out that for typical values of the an/ax changes from n(a)/xi at a=0 to zero at x sxi T/T,<<l corresponds to L/Lp<<1 and a steeply Hence, one can write a'n/dx%s-n(xi)/x2, and the con- graded junction, whereas the case of T./T. =1 corre- tinuity equation for the density at xi becomes sponds to L/Lp=1 and a step junction To improve accuracy one can solve(8)and(15) dn(ai (13) simultaneously. It is most convenient to have the com- bined results plotted in chart form as shown in Fig. 2. To use this figure, one must know Ta,(T,+T), and the This equation represents an exponential decay, with a ratio Ir/I. The curves slanting down and left corre- spond to values of Ip/IR. The curves slanting left and up correspond to values of T/(T,+T-). The solution +D/ (14)occurs at the intersection of the appropriate pair of curves. The abscissa below gives the ratio T/(T+T) Since the current supplied by the extraction process is With(T+T,) one can immediately determine T. The adin I=qAD(dn/d)gA Dn(=i)/xi, L/Li With T and an independent knowledge of the diffusion he current also exhibits an exponential decay with a constant, one can deter time constant Tr. This is our basis for simulating the An ambiguity arises if one allows the possibility of a

154 IEEE TRANSACTlONS ON ELECTRON DEVICES, MARCH 1971 agreement is acceptable. The heuristic approach used in deriving (8) leads us to expect that intermediate values of L/LD will result in comparably small errors. Recovery Period: The recovery time provides a rough indication of the minority carrier penetration length. The relation between penetration length L and recovery time T, has been derived theoretically for the same three idealized impurity distributions considered above. In a graded junction, the recovery can be represented by an exponential [9] with a time constant T,. This recovery is associated with an average penetration length which is given by the expression L = (DT,)"2. (10) For a p-i-n diode, one can compare the actual process [lo] with an exponential form to obtain the approxi￾mate relation L = l.5(DTT)1/z. (11) In an ideal step junction the average penetration length is the same as the diffusion length: but the relation between r and Tp and IF/IR is com￾plicated [6]- [8]. We will compare the results of our approximate theory with (10) for a graded junction and with the graphs in [6]-[8] for a step junction. The carriers which contribute most of the reverse current during the dominant early part of the recovery period are those located a distance xi from the junction (see Fig. 1). Minority carriers with x<%; have been re￾moved previously, and those with x>~; have not yet been perturbed by the reversing diffusion gradient. As recovery proceeds, the carriers in the vicinity of x; either recombine or diffuse back across the junction. The con￾tinuity equation describing these processes is an -n a2n at 7 - + D-- dX2 - N _- -(IF/IR) 54 3 2 1.0 .9 .8 .7 .6 .5 A Ts (Ts+Tr'= 0.55 0.60 0.65 0.70 0.75 0.80 0.90 0.85 0.95 1.0 IO Fig. 2. Chart for determining T and L/Ln from T,, T,+Tr, and IP/IR. extraction process with an exponentially decaying cur￾rent source. Substituting the previously derived expression for X; (6) into the above expression for T, (14) gives TT/T (1 + [(LD/L) + (IR/IF)(L/LD)]a)-l. (15) For L/LD = 0 this reduces to the graded-junction result (lo), as it should. For L/LD = 1.0, we have T,/T = [ 1 f (1 f IR/IF) 'I-'. (16) Values of T,/r given by (16) range from 5 to 20 percent higher than the more rigorous values given in [Si, as IF/IR changes from 0.5 to 5.0. Interpretation: Equation (8) relates the storage time to lifetime. For a first approximation one can set L/LD equal to its two limiting values, zero and one, and use the two resulting expressions to obtain upper and lower limits on the relation between storage time and lifetime. Then, for a typical forward-to-reverse current ratio of unity, one finds the lifetime r to be 1.5 to 4 times longer than the storage time T,. Equation (15) divided by (8) for %=xi. The recombination rate near X; is approxi￾mately n(xi)/r. The diffusion loss rate can be estimated by noting that in our straight-line approximation Hence, one can write d2n/dx2= --n(x;)/x?, and the con￾tinuity equation for the density at xi becomes gives a relation between T,/T, and the grading param￾eter L/LD. It turns out that for typical values of the forward-to-reverse current ratio (IF/IE = I), the case of graded junction, whereas the case of T,/T, = 1 corre￾sponds to L/LD = 1 and a step junction. To improve accuracy one can solve (8) and (15) simultaneously. It is most convenient to have the com￾dt (13) bined results plotted in chart form as shown in Fig. 2. To use this figure, one must know T,, (T,+T,), and the time constant T, given by spond to values ofIF/IR. The curves slanting left and up correspond to values of T,/(T8+T,). The solution (14) occurs at the intersection of the appropriate pair of curves. The abscissa below gives the ratio r/(T,+ Tp). Since the current supplied by the extraction proc:ess is With (T,+T,) one can immediately determine 7. The ordinate to the left gives the grading parameter L/LD. \;C7ith T and an independent knowledge of the diffusion the current also exhibits an exponential decay with a constant, one can determine L. time constant T,. This is our basis for simulating the An ambiguity arises if one allows the possibility of a dn/ax changes from n(Xi)/xi at x = 0 to zero at X u x;. T,/T8 to L/LD < and a dn(x;) __- This equation represents an exponential decay, With a ratio IF/IR. The curves slanting do\.vn and left corre￾T, = (f f D/xi2)'. I = qAD(dnz/dx) = qADn(x;)/x;

DEAN AND NUESE: STEP-RECOVERY TECHNIQUE FOR MEASURING PARAMETERS IN ASYMMETRIC P-n JUNCTION DIODES hyperabrupt impurity distribution. Such a distribution establishes a minority-carrier potential barrier very near the junction, which would sweep carriers away e junction during injection and prevent them from being retrieved by the reversing step. In this case, GENERATOR the step recovery measurement would erroneously in- dicate a strongly graded profile with L/Lp nearly zero JUNCTION This relatively uncommon junction distribution would be indicated by a capacitance-voltage measurement which gives the form CV-1/n, where n is significantly TRIGGER SCOPE less than 2.0. With a hyperabrupt distribution the only conclusion that can be reached is that the lifetime is Fig 3. Apparatus for measuring short life longer than the value given by(9) Other Efects TABLE I Minority-carrier traps can have two effects. 1)Traps DEFINITION capture injected minority charge in the region near the edge of the junction. The (additional)trapped charge I=length of line from probe to sample repulses minority free carriers and reduces the minority veloeity of electromagnetic wave in line free- carrier density at the junction edge. This shortens -RoCedepletion capacitance time constant the storage time for given forward and reverse cur- ents. 2)Traps continue to release carriers for a long relifetvery time time after T, and thus inordinately increase the mea- njected carrier rler sured recovery time Tr. In fact, the observed recovery time constant provides a direct measure of the trap R,=series resistance of contact or sample depth. These phenomena make it possible to identify V, - reflected reversing step voltage trapping by associating it with the condition T,>>T. for IP/IR of order unity or larger. arge-scale recombination inhomogeneities produce /r=magnitude of forar d current Gibbons [11] considers a step junction and shows that Vi= Vp+V+IpR, total sample voltage recombination inhomogeneities weaken the dependence of T/T, on IR/Ip. Our development above shows that impurity grading similarly weakens the dependence of and at the maximum reverse voltage, this may extend T/T, on IR/Ip. By noting that injected carriers are re- only a relatively short distance from the junction. Thus, moved more quickly from short-lifetime material, one a change in the grading further from the junction would can argue that carriers in short-lifetime material con- not be observed by the C-V measurement, and in gen tribute only to T and thus recombination inhomo- eral the "average"grading indicated by these two geneities increase the ratio T,/(T,+T.). Our develop. methods need not closely agree ment above shows that impurity grading similarly creases the ratio T,/(T,+T,). One concludes that a low EXPERIMENTAL TECHNIQUE L/Lp obtained from our step-recovery experiment is A schematic representation of the experimental ap. impurity grading, or both. From a phenomenological the ensuing discussion are defined in Table I. The setup point of view, both of these mechanisms reduce the is very similar to that employed in time domain re average penetration of electrons injected across the flectometry measurements [12]. The diode is mounted junction: grading piles up the stored charge close to the coaxially at the end of a single 50-42 line. Simple pres- junction; short lifetime regions prevent charge from sure contacts allow rapid sample changes. The one piling up in the first place port electrical connection makes it easy to subject the In some cases, the ambiguity between impurity grad- sample to a variety of temperatures and ambient con- ng and recombination inhomogeneity can be lifted by ditions. In addition, the stray reactances are low, and performing C-V measurements. When the junction im- the diode behaves as though it were in series with a purity gradient extends to a significant fraction of a purely resistive 50-02 load up to very high frequencies. diffusion length, the measure of the retarding field de- To forward-bias the sample diode, a direct current termined by the ratio L/ Lp can be compared with the Ir is fed into the 50-4 line through a high-impedance measure of junction abruptness determined from C-v tee. a high-impedance sampling probe also is connected characteristics. It should be noted, however, that the to the line, with a blocking capacitor separating this C-V measurement probes only that portion of the im- probe from the de biasing connection. The length of the purity distribution supporting the space-charge layer, line between the probes and the sample does not affect

DEAN AND NUESE 1 STEP-RECOVERY TECHNIQUE FOR MEASURING PARAMETERS IN ASYMMETRIC p-n JUNCTION DIODES 155 hyperabrupt impurity distribution. Such a distribution establishes aminority-carrier potential barrier very near the junction, which would sweep carriers away from the junction during injection and prevent them from being retrieved by the reversing step. In this case, the step recovery measurement would erroneously in￾dicate a strongly graded profile with L/LD nearly zero. This relatively uncommon junction distribution would be indicated by a capacitance-voltage measurement which gives the form C-V-l’n, where n is significantly less than 2.0. With a hyperabrupt distribution the only conclusion that can be reached is that the lifetime is longer than the value given by (9). Other Effects Minority-carrier traps can have two effects. 1) Traps capture injected minority charge in the region near the edge of the junction. The (additional) trapped charge repulses minority free carriers and reduces the minority free-carrier density at the junction edge. This shortens the storage time for given forward and reverse cur￾rents. 2) Traps continue to release carriers for a long time after T,9 and thus inordinately increase the mea￾sured recovery time T,. In fact, the observed recovery time constant provides a direct measure of the trap depth. These phenomena make it possible to identify trappihg by associating it with the condition T,> > T, for IF/IR of order unity or larger. Large-scale recombination inhomogeneities produce effects similar to those produced by impurity grading. Gibbons [ll] considers a step junction and shows that recombination inhomogeneities weaken the dependence of r/T, on IR/IP. Our development above shows that impurity grading similarly weakens the dependence of r/Ts on IR/IP. By noting that injected carriers are re￾moved more quickly from short-lifetime material, one can argue that carriers in short-lifetime material con￾tribute only to T,, and thus recombination inhomo￾geneities increase the ratio T,/(T, + T?). Our- develop￾ment above shows that impurity grading similarly in￾creases the ratio T,/(T,+ T,). One concludes that a low L/LD obtained from our step-recovery experiment is evidence for either recombination inhomogeneity, or impurity grading, or both. From a phenomenological point of view, both of these mechanisms reduce the average penetration of electrons injected across the junction: grading piles up the stored charge close to the junction; short lifetime regions prevent charge from piling up in the first place. In some cases, the ambiguity between impurity grad￾ing and recombination inhomogeneity can be lifted by performing C-V measurements. When the junction im￾purity gradient extends to a significant fraction of a diffusion length, the measure of the retarding field de￾termined by the ratio L/LD can be compared with the measure of junction abruptness determined from C-V characteristics. It should be noted, however, that the C-V measurement probes only that portion of the im￾purity distribution supporting the space-charge layer, CURRENT SOURCE DIODE I GENERATOR -LT REVERSING vi+vr PULSE HI - FREO, 44 I PROBE E JUNCTION I INCIDENT B TRIGGER SCOPE PULSE Fig. 3. ’4pparatus for measuring short lifetimes at extreme ambients. The polarities shown are appropriate for p-side of diode connected to center conductor of coaxial line. TABLE I DEFIKITIONS Z=length of line from probe to sample Td= 21/c =delay time c =velocity of electromagnetic wave in line T, =storage time Tl =ROC= depletion capacitance time constant T, = recovery time 7 =lifetime L = injected carrier penetration length LO = injected carrier diffusion length R, = series resistance of contact or sample Ra = characteristic impedance of line Vi = incident reversing step voltage V, = Vi+ V, = probe voltage V, = reflected reversing step voltage Vp=initial value of lorward-biased junction voltage Vi Ii = Vi/Ro = incident current (positive toward sample) IF = magnitude of forward current I,= V7/Ro=reflected current (positive away from sample) 112 =magnitude of initial reversing current It=Ip-tZi-ZI.= total sample current V, = V,.f Vf +IF& =total sample voltage and at the maximum reverse voltage, this may extend only a relatively short distance from the junction. Thus, a change in the grading further from the junction would not be observed by the C-V measurement, and in gen￾eral the “average” grading indicated by these two methods need not closely agree. EXPERIMENTAL TECHNIQCE A schematic representation of the ap￾paratus is shown in Fig. 3, and the the ensuing discussion are defined in Table I. The setup is very similar to that employed in time domain re￾flectometry measurements [12]. The diode is mounted coaxially at the end of a single 5042 line. Simple pres￾sure contacts allow rapid sample changes. The one￾port electrical connection makes it easy to subject the sample to a variety of temperatures and ambient con￾ditions. In addition, the stray reactances are low, and the diode behaves as though it were in series with a purely resistive 504 load up to very high frequencies. To forward-bias the sample diode, a direct current IF is fed into the 5042 line through a high-impedance tee. A high-impedance sampling probe also is connected to the line, with a blocking capacitor separating this probe from the dc biasing connection. The length of the line between the probes and the sample does not affect