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>(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 > (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 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 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

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 value

156 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

DEAN AND NUESE: STEP-RECOVERY TECHNIQUE FOR MEASURING PARAMETERS IN ASYMMETRIC P-n JUNCTION DIODES fraction of the diodes tested gave values of L/Lp sig nificantly less than unity attests to the need for a theory such as ours which includes the effects of unknown im purity CONCLUSIONS An efficient Ital technique has been de veloped for carrying out step-recovery measurements of very short lifetimes. The outstanding feature of this that only one electrical line is contact the test diode, and the distance from the diode to the measuring equipment may be arbitrarily long. This simplicity makes it possible to minimize stray circuit reactances for definitive high-speed measure ments, It also facilitates subjecting the test diode trace for p*n GaAs Time scale is 2 ns wide range of ambient conditions, including temper oltage scale is 0.5 v/di ature. The raw data from the experiment provide an im The recovery in the high-current experiment is more mediate qualitative and semiquantitative indication of complicated because it is the sum of two exponentials. the key parameters, and a large amount of informa Fortunately, the coefficients of the exponentials are tion is available in a single oscilloscope display. The such that the sum can be approximated by a single various features on the oscilloscope display have been exponential having a time constant equal to the sum of quantitatively characterized reaches 1/e of its final value in a time given by oigy' the time constants of the two original expo Using simple heuristic arguments based on the key numerical evaluation of (19)shows that the recovery physical phenomena in the experiment, we have de- veloped an approximate theory which makes direct use of the salient features in a typical oscilloscope display. t=T1+(1+b)T The results are presented in an easy-to-use graphical where 8 has a maximum value of +0.17 at T/T, =1.75 form, and fit closely the results of more rigorous deriva and approaches zero in the two limits of large and small tions previously done by others, in the appropriate Ti/Tr. Thus, with an error of les than 20 percent(which limits. The present technique is more general, however,, tends to cancel the previous errors), one can evaluate t, and provides more information from a single experi- as the difference between the times required for the volt- ment than could have been obtained previously. In age to reach 1/e of their final values in the high- and formation conveyed includes series resistance, depletion low-current experiments described above. In the limit of capacitance, injected-carrier lifetime, and the effect of small series resistance, one has R,<<Ro, and the de- ossible impurity gradients or recombination in termination becomes especially simple. The time be- homogeneities on the penetration length of the injected tween the two curves measured 1/e of the way from the arriers steady-state value to zero is simply ( T,+T,). This AcknOWledgment determination is depicted in Fig 4 The technique we have described has been used to The authors are indebted to A. M. Garofalo and measure lifetime and penetration length in a large J J. Gannon for experimental assistance number of GaAs and GaAs-P2 p-n junctions, a typical oscilloscope trace is shown in Fig. 5. Results of these REFERENCES measurements will be published later, but some general [11 H: Higuchi and H. Tamura, Measurement of the lifetime of observations are in order here. Series resistances deter roscope, mined from the step-recovery oscilloscope trace are (21 Oku, The average depletion capacitance determined from the step-recovery oscilloscope trace is typically slightly higher than the depletion capacitance measured on a bridge with the bias voltage equal to the steady-state (515. M. Krakauer, "Harmonic ge lifetimes range between 1 and 60 ns and values of L/Lp (6]E,M.61962, pp. 1665-1676p recovery diode, "Proc.IRE,vol voltage in the step-recovery measurement. Measured range between 0.25 and 1.00. The fact that a reasonable of pulsed reverse characteristic 1953 p

DEAN AND NUESE: SmP-RECOVERY TECHNIQUE FOR MEASURINO PARAMETERS IN ASYMMETRIC p-n JUNCTION DIODES 157 Fig. 5. Typical oscilloscope trace for p+-n GaAso,&Po,l~ diode. Time scale is 2 ns/div. Voltage scale is 0.5 V/div. The recovery in the high-current experiment is more complicated because it is the sum of two exponentials. Fortunately, the coefficients of the exponentials are such that the sum can be approximated by a single exponential having a time constant equal to the sum of the time constants of the two original exponentials. A numerical evaluation of (19) shows that the recovery reaches l/e of its final value in a time given by t’ = TI + (1 + S)TT where 6 has a maximum value of +0.17 at T1/Tr= 1.75 and approaches zero in the two limits of large and small T1/TT. Thus, with an errorf less than 20 percent (which tends to cancel the previous errors), one can evaluate T, as the difference between the times required for the volt￾age to reach l/e of their final values in the high- and low-current experiments described above. In the limit of small series resistance, one has R, < <Rot and the de￾termination becomes especially simple. The time be￾tween the two curves measured lle of the way from the steady-state value to zero is simply (T8+ TT). This determination is depicted in Fig. 4. The technique we have described has been used to measure lifetime and penetration length in a large number of GaAs and GaAsl-,P, p-n junctions. A typical oscilloscope trace is shown in Fig. 5. Results of these measurements will be published later, but some general observations are in order here. Series resistances deter￾mined from the step-recovery oscilloscope trace are typically about 30 percent higher than the saturation resistance indicated on a forward-biased I-V curve. The average depletion capacitance determined from the step-recovery oscilloscope trace is typically slightly higher than the depletion capacitance measured on a bridge with the bias voltage equal to the steady-state voltage in the step-recovery measurement. Measured lifetimes range between 1 and 60 ns and values of L/LD range between 0.25 and 1.00. The fact that a reasonable fraction of the diodes tested gave values of LILD sig￾nificantly less than unity attests to the need for a theory such as ours which includes the effects of unknown im￾purity grading. CONCLUSIONS An efficient experimental technique has been de￾veloped for carrying out step-recovery measurements of very short lifetimes. The outstanding feature of this technique is that only one electrical line is required to contact the test diode, and the distance from the diode to the measuring equipment may be arbitrarily long. This simplicity makes it possible to minimize stray circuit reactances for definitive high-speed measure￾ments. It also facilitates subjecting the test diode to a wide range of ambient conditions, including temper￾ature. The raw data from the experiment provide an im￾mediate qualitative and semiquantitative indication of the key parameters, and a large amount of informa￾tion is available in a single oscilloscope display. The various features on the oscilloscope display have been quantitatively characterized. Using simple heuristic arguments based on the key physical phenomena in the experiment, we have de￾veloped an approximate theory which makes direct use of the salient features in a typical oscilloscope display. The results are presented in an easy-to-use graphical form, and fit closely the results of more rigorous deriva￾tions previously done by others, in the appropriate limits. The present technique is more general, however, and provides more information from a single experi￾ment than could have been obtained previously. In￾formation conveyed includes series resistance, depletion capacitance, injected-carrier lifetime, and the effect of possible impurity gradients orrecombination i￾homogeneities on the penetration length of the injected carriers. ACKNOWLEDGMENT The authors are indebted to A. M. Garofalo and J. J. Gannon for experimental assistance. REFERENCES minority carriers in semiconductors with a scanning electron H. Higuchi and H. Tamura, “Measurement of the lifetime of m~croscope,” Japan. J. Appl. Phys., vol. 4, 1965, p. 316. T. Nakano and T. Oku, (‘Temperature dependence ofrecombina￾tion lifetime in gallium arsenide electroluminescent diodes,” Japan. J. Appl. Phys., vol. 10, 1967, p. 1212. and recombination in P-A’ junctions and P-N junction char￾C.-T. Sah, R. N. Noyce, and W. Shockley, “Carrier generation acteristics,” Proc. IRE, vol. 45, Sept. 1957, pp. 1228-1243. W. Shockley, ‘(The theory of P-N junctions in semiconductors and P-N junction transistors,” Bell Syst. Tech. J., vol. 28, 1949, p. 435. S. M. Krakauer, ‘Harmonic generation, rectification, and life￾time evaluation with the step recovery diode,” Proc. I=, vol. 50, July 1962, pp. 1665-1676. E. M. Pell, “Recombination rate ;,n germanium by observation of pulsed reverse cliaracteristic, Phys. Rev., vol. 90, 1953, p. 278

TEEE TRANSACTIONS ON ELECTRON DEVICES, VOL ED-18, NO. 3, MARCH 1971 recovery diode for pulse and harmonic 259 g time in junction diodes and junc- [(11) purities in semiconductors [12] See, for example, "Time domain reflecto [10]J. L. Moll and S. A. Hamilton, "Physical modeling of the step ard Company, Application Note b52 19(aetry, "Hewlett-Pack- Temperature in Gunn Diodes with Inhomogeneous Power Dissipation NILS O. JOHNSON, STUDENT MEMBER, IEEE, KJELL O. I OLSSON AND S. JORGE WILDHEIM Abstract-The output power of Gunn oscillators is limited by the under the assumption that the dissipation power density increased temperature which affects the velocity-field curve and is homogeneous throughout the diode volume. This assumption is certainly good for some cases, e. g, diodes tensity is more or less inhomogeneous, depending on the kind of operated in the LSA mode [6]. However, when diodes scillation mode. The present paper makes use of simple models of are operated in the usual transit-time dipole-domain the dipole domain mode and the accumulation mode in an analysis mode [6 and the transit-time accumulation mode [6] neous dissipation power density. The temperature the dissipation is much larger near the anode contact temperature dependence on diode dimensions and mounting is also than near the cathode contact [7]. In this paper simple is turned towards the heat sink than when the cathode is turned the operation are used to calculate the inhomogeneity of the ame way. This difference amounts to 150K for an accumulation dissipation. From this the temperature is calculated mode diode with 500K maximum temperature in the former case. Gunn diodes are typically mounted with only one con tact acting as an efficient heat sink. The present theor thus tells the difference in temperature when the diodes N HE Gunn diode is a solid-state microwave power are mounted with the cathode to the heat sink and with source with small dimensions. Today more than the anode to the heat sink. Obviously the temperature is 1-kW peak power is available from single diodes higher in the former case and the difference may be as under pulsed operation [1] but less than 1 W from Cw large as 40 K for domain mode diodes and 150K for diodes. The dissipation power density and the cor accumulation mode diodes when the maximum tem- nected temperature increase limit the production o perature with the anode to the heat sink is 500%K microwave power at high duty cycles. The allowable temperature of a diode has an upper limit. The contac HEAT CONDUCTION material is usually gold or silver alloys, which form an First we will give the equations by which the tem eutetic with GaAs at about 700 to 900K. The diode can perature is calculated from the dissipation power density thus be permanently damaged in this temperature re- P(=time average of the scalar product of electrical gion. The negative differential mobility in GaAs dis- field and current density). The geometry assumed appears at very high temperatures and there exists some shown in Fig. 1. The diode and the contact layer are evidence that it is too small to cause oscillations at assumed to be of circular shape when looked upon from a temperatures above 600 to 700K perature increase in diodes with different mountings ordinate a only. The heat conductivity of the heat sink is of GaAs. It is therefore ot received June 30, 1970 typically much larger than 6 ch Laboratory of Electronics, reasonable to approximate the heat How in the Gaas to be in the direction only. The equation of heat oratory of Electronics, tion then becomes(in the stationary state) niversity of Technology, Gothenburg, Sweden. He Chalmers university of Technology, Gothenburg, sweden. he is now +P(x) with STAL-LAV

I58 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL ED-18, NO. 3, MARCH 1971 [71 B. Lax a,:d S. F. Neustadter, “Transient response of a P-N recovery diode for pulse and harmonic generation circuits,” [SI R. H. Kingston, “Switching time in junction diodes and junc- ill] P. E. Gibbons, “Problems concerning the soatial distribution of junction, J. AppZ. Phys., vol. 25, 1954, p. 1148. Proc. IEEE, VO~. 57, July 1969, pp. 1250-1259. tion transistors,” Proc. IRE, vol. 42, May(1954, pp. 829-834. deep impurities in semiconductors,” Solid-State Electron., vol. [Uj J. 1,. Moll, S. ,f(rakauer, and R. Shen, P-hT junction charge￾storage diodes, Proc. IRE, vol. s!, 1962, p. 43. 13, 1969, p. 989. [lo] J. L. Moll and S. A. Hamilton, Physical modeling of the step [12] See, for example, “Time domain reflectometry,” Hewlett-Pack￾ard Company, Application Note 62, 1964. .. Temperature in Gunn Diodes with Inhomogeneous Power Dissipation Abstract-The output power of Gunn oscillators is limited by the increased temperature which affects the velocity-field curve and which may also permanently damage the diode or the metal contacts. The behavior of these oscillators indicates that the dissipation power density is more or less inhomogeneous, depending on the kind of oscillation mode. The present paper makes use of simple models of the dipole domain mode and the accumulation mode in an analysis of the inhomogeneous dissipation power density. The temperature increase caused by this dissipation power is then calculated. The temperature dependence on diode dimensions and mounting is also discussed. It is shown that the temperature is lower when the anode is turned towards the heat sink than when the cathode is turned the same way. This difference amounts to 150’K for an accumulation mode diode with 500’K maximum temperature in the former case. INTRODL-CTION T HE Gunn diode is a solid-state microwave power source with small dimensions. Today more than 1-kW peak power is available from single diodes under pulsed operation [l] but less than 1 W from CW diodes. The dissipation power density and the con￾nected temperature increase limit he production of microwave power at high duty cycles. The allowable temperature of a diode has an upper limit. The contact material is usually gold or silver alloys, which form an eutetic with GaAs at about 700 to 900’K. The diode can thus be permanently damaged in this temperature re￾gion. The negative differential mobility in GaAs dis￾appears at very high temperatures and there exists some evidence that it is too small to cause oscillations at temperatures above 600 to 700’K. A number of authors [2]- [5] have calculated the tem￾perature increase in diodes with different mountings Manuscript received June 30, 1970. Chalmers University of Technology, Gothenburg, Sweden. N. 0. Johnson is with the Research Laboratory of Electronics, K. 0. I. Olsson was with the Research Laboratory of Electronics, Chalmers University of Technology, Gothenburg, Sweden. He is now with Philips Teleindustri, Jakobsberg, Sweden. S. J. Wildheim was with the Research Laboratory of Electronics, Chalmers University of Technology, Gothenburg, Sweden. He is now with STAL-LAVAL Turbine Co., Finsping, Sweden. under the assumption that the dissipation power density is homogeneous throughout he diode volume. This assumption is certainly good for some cases, e.g., diodes operated in the LSA mode [6]. However, when diodes are operated in the usual transit-time dipole-domain mode [6] and the transit-time accumulation mode [6] the dissipation is much larger near the anode contact than near the cathode contact [7]. In this paper simple models of the domain mode and the accumulation mode operation are used to calculate the inhomogeneity of the dissipation. From this the temperature is calculated. Gunn diodes are typically mounted with only one con￾tact acting as an efficient heat sink. The present theory thus tells the difference in temperature when the diodes are mounted with the cathode to the heat sink and with the anode to the heat sink. Obviously the temperature is higher in the former case and the difference may be as large as 40’K for domain mode diodes and 150’K for accumulation mode diodes when the maximum tem￾perature with the anode to the heat sink is 500’K. HEAT CONDUCTION First we will give the equations by which the tem￾perature is calculated from the dissipation power density P( = time average of the scalar product of electrical field and current density). The geometry assumed is shown in Fig. 1. The diode and the contact layer are assumed to be of circular shape when looked upon from a direction perpendicular to the heat sink. The dissipation P is assumed to be a function of the longitudinal co￾ordinate x only. The heat conductivity of the heat sink is typically much larger than that of GaAs. It is therefore reasonable to approximate the heat flow in the GaAs to be in the x direction only. The equation of heat conduc￾tion then becomes (in the stationary state) dx \ dx/