IEEE TRANSACTIONS ON ELECTRON DEVlCES, VOL. ED-18,NO. 3, MARCH 1971 151 A Refined Step-RecoveryTechnique for Measuring Minority Carrier Lifetimes and Related Parameters in Asymmetric b-n Tunction

Absfracf-Minority-carrier lifetimein a forward-biased asym- nature of therecombination processes [4], orelectro- metrical p-n junction can be measured by observing the time luminescent junctions [2]. Krakauer [SI has considered response of the diode to a sudden reversing step voltage. An ap- proximate but general theory for p-n junctions with almost arbitrary the response of a quiescent junction toa large sinusoidal impurity gradients is developed, and its results are within about 25 excitation for the special caseof 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- siderthe transient response of aninitially forward- mental technique is described which is suitable for measuring life- biased junction to a sudden reversing step. This tran- 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 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- from the experiment are in the form of an oscilloscope trace, which termining lifetime as a function of injection level. provides an immediate qualitative and semiquantitative indicationof The step recovery phenomenon was first studied by the minority-carrier lifetime and the penetration length for the in- Pel1 [6], Laxand Neustadter [7], andKingston [8], jected carriers. A graphical presentation of the theoretical results leads quickly toa more precise quantitative evaluation of these who developed physical and theoretical descriptions for parameters. In addition, the technique can be used to measure an the storage time T, and therecovery time T, in an ideal average junction depletion capacitance and the device series resis- step junction. For conventional germanium and silicon tance. p-njunctions, this abrupt approximation is usually reasonable,since the impurity profile between the n- INTRODUCTION and p-sides of thejunction usually becomes uniform HE effectivelifetime andpenetration depth of within a distance from the junction which is much less minoritycarriers 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 semiconductordevices, such as , one diffusion length exists in many other semiconduc- lasers, “cold-cathode” emitters, etc. Through the years, tors, particularly the 111-V compounds where lifetimes avariety of techniqueshave been developed for the areon the order of lo-* to s anddiffusion lengths determination of minority lifetimes, each of which has are often less than a micron. For such junctions, the itsparticular advantages and disadvantages and re- abrupt approximation would not be valid, and a graded quiresits specific assumptionsand approximations. impurity profile should be considered. Moll, Krakauer, Suchtechniques include the external generation of and Shen [9] and Moll and Hamilton [lo] have treated excess carriers near a reverse-biased junction [l],fre- junctions with an exponentially graded and p-i-n ap- quency response and delay time measurements on elec- proximation, respectively. Particularly desirable, how- troluminescentdiodes [2], andanalyses of thesmall- ever, would be a lifetime measurement procedurewhich signalimpedance [3] andsteady-state I-V character- couldtreat 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-njunction to a large-signalsinusoidal or step recovery technique to p-n junctions with nearly arbi- excitation.This approach is appropriate forasym- trary impurity distributions, and develops an approxi- metricp-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 dicularto the junction [l],knowledge of the specific is especially well-suited for measuring very short life- times (25XlO-’O s) under a wide range of ambient conditions. With this procedure, the step-recovery tech- Manuscript received October 16, 1970. ‘The researchreported herein was partially sponsoredby the National aeronautics and Spacenique and its interpretation is found to be remarkably Administration,Langley Research Center, Hampton, Va., under Contract NAS-12-2091, and RCA Laboratories, Princeton, N. J. straightforward. In most cases, a pairof closely related The authors are with RCA Laboratories, Princeton, N. J. 08540. measurements recorded on a single oscilloscope photo- j 52 IEEE TRANSACTIONS ON ELECTRON DEVICES, MARCH 1971 graph is sufficient determineto notonly the minority t carrierlifetime and penetration length, but also the no diode’s series resistance and average depletion capaci- E tance. A cursory examination of the oscilloscope photo- n wa graph allows an immediate qualitative and semiquan- a a titativeevaluation of theseparameters. Furthermore, s n1 byanalyzing the photograph with the aid of simplea Wn I-u chartcontained in this paper, one can refine there- W -J sultsandincrease accuracythe of quantitativethe DISTANCE INTO LIGHTLY-DOPEDMORE determinationswithinabout to 25 percent.way,this In 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- reversingpulse teaches the diode. Theshaded area indicates the charge remaining at the end of the storage time (T8).The dotted jection levels to obtain an extensive characterization of lines indicateour straight-line approximations. the junctions of interest.

THEORYOF JUNCTION BEHAVIORvalue. Then, ‘with TI,the average capacitance C is de- fined by the relation TI/R. In the extreme case of QualitativeDescription C= an ideal stepjunction in which thedepletion width Minority-carrierdensity profiles for a crosssection 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 Cf. by solid lines in Fig. 1. Carriers have been injected into Thusour “average” capacitance is somewhat higher the more lightly doped material to the right of the junc- thanthe final open-circuitvalue of thejunction de- tion. The length L is the average “penetration length” pletionCapacitance. Although the actual depletion of thesecarriers. This length is stronglyaffected by capacitancevaries with applied voltage, for our de- impuritygradients in the regionoccupied bythe in- velopment, we will approximate it by a constant jectedcarriers. 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 pullscarriers 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, tial.) For an ideally abrupt step junction [6]-[8] with and is thereforeindependent of the reversevoltage zero built-in field, thepenetration length equals the across the depletion layer. By assuming a constant dis- classical“diffusion length.” For a “graded”junction tance from the junction edge to the point of maximum [9]-[lO] the penetration length is shortened by a re- minoritycarrier density, wewill showlater 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 T,. carrier density profile before a reversing voltage is ap- The ratio of forwardto reverse bias current IF/IR 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 toaffects the magnitude of both T, and T,. A relatively drop throughout a period of time defined as the storage largereverse current shortens the storage time T, by period [9]. During this period the excess carriers avail- extractingthe carriers quickly and by producing a able at the junction make it effectively a short circuit density profile at t = T, which is skewedtoward the PI. junction.With such a profile, most of the previously When the carrier densityat the junctionreaches zero injected carriers are still present. During the recovery at t = T, (storage time), a reverse-biased depletion layer period, the densityprofile 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 rapidlyand approaches its steady-state value. The profile crowdedclose to the junction produces a high voltagebuildup is retardedby the time required to extraction rate.) When selecting the time constant for charge up the depletion capacitance and to extract the ourcurrent source, it is appropriateto focus onthe remainder of thecarriers previously injected under density profile at t = T,, since most of the voltage change forward bias (shaded areaof 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 schematicallyrepresented by a in parallel our derivation with the more rigorous results of others with a current source. [6]-[10] in the special limiting caseswhich they treat. Thecapacitor represents an “average” depletion capacitance, whose value is determined by applying a Mathematical Model 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 TIrequired for indication of the minority-carrier lifetime T. The rela- thejunction capacitance to charge to l/e of its final tion between lifetime and storage time has been derived theoreticallyfor three different idealized junction im- The triangle which approximates the actual initial dis- purity distributions. Ina graded p-n junction 193 and in tribution has an area which is exactly one half the area a p-i-n diode [ 101 the relationis under the corresponding exponential. Fora graded impurity distribution (L/LD< < 1) T,/T = log, (1 i-IF/IR). (1) with IF/IR > > (L/LD)2,the actual minority carrier In anideal step junction [7] the relation is density at t = T,can be expressed as [9] erf (T,/T)= (1 + IR/Ip)-’. (2) n x(TR/qAD) exp (-x/L). We will compare the results of our approximate theory We will approximate this distribution with a triangle with these formulas in the appropriate limits. formed by the original straightline and another one Following Moll, Krakauer, and Shen [9], we start by passing through the origin with a slope such that dif- writing the equation for the extraction of the total in- fusion exactly supplies the reverse currentIR.This slope jected charge Q: is

dQ A 1z -+-=Q -rR _- - IR/~AD= (IR/IF)(L/LD)’(%O/L) at T Ax 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 is been defined previously. Solving for the intersection of the two lines yields @(t) = [Q(O) + IRT]exp (-t/r) - IRT. ni = nO/[l (I~/I&)(LD/L) (5) At t = T,, the charge has decayed to that indicated by + ‘1 the shaded areaof Fig. 1, and has a value at

Q(TJ = [Q(o) + 1x71 exp (-T,/T) - IRT. (3) xi == L/[1 4-(IR/IF) (L/LD)’]. (6) Now we will define a new parameter a to be the frac- The area under our triangle is niL/2, which is exactly tion of initially injected carriers that remaint = at T, : half thearea under the morerigorously derived [9] graded-junctiondistribution indicated above. For an = Q(Ta)/Q(O). abrupt junction (L/LD= 1) with IF/IR<<1, the tri- This fractionCY can vary fromzero for a graded junction angulararea is again half thearea under the initial orlarge IF/IRto unity for a stepjunction or small exponential. This is as it should be since this limit cor- IF/IR.Setting (3) equalto Q(T,)=crQ(O), notingthe responds to the case in which almost none of the in- relation Q(0)= Ipr [9], and rearranging yields jected carriers are removed during the storage period. Since for the cases of graded and ideally abrupt p-n Ts/T = loge [(I IF/IR)/(l aIF/IR)]* (4) + junctions, the areas of our triangular density approxi- Equation (4) is not yet a useful solution, since 01 is a mations are one half those under the more rigorously function of both impurity grading and I~/IR.For an derived curves, we postulate that the ratio of the tri- estimate of a! we will use straight line approximations to angular areas (at t = 0 and t = T,) provides a reasonable the actual density distributions,as shown by the dotted general approximationto the area ratio for actual lines in Fig. 1. minority carrier distributions. Then, from simple geom- The initial density varies approximately exponentiallyetry, a= Q(T,)/Q(O) =ni/no,and from (j), with distance x according to the expression [7] O( = [I + (WI~(wLPJ-’. (7) n = lzo exp (-x/L). Substituting (7) for CY in (4) yields With an initial total chargegiven by Ts - loge { [I f IF/IR] [1 + (IF/IR)(LD/L)*] 187 = @(o) = qA ndx z qAnoL, T SoW /[1 + (IF/IIJ(LD/L)’ + IF/IR]1. (8) one obtains For a graded junction, a retarding field crowds the injected carriersclose to the junction,which leads to the no S IFT/qAL, condition L <

agreement is acceptable. The heuristic approach used in -(IF/IR) deriving (8) leads us to expect that intermediate values 54 3 2 1.0 .9 .8 .7 .6 .5 A of L/LDwill result in comparably small errors. Ts Recovery Period: The recovery time provides a rough (Ts+Tr'= indication of theminority carrier penetration length. 0.55 The relation between penetration lengthL and recovery time T,has been derived theoretically for the same three idealized impurity distributions considered above. In a 0.60 0.65 graded junction, the recovery can be represented by an exponential [9] with a time constant T,. This recovery 0.70 0.75 is associated with an average penetration length which 0.80 0.85 is given by the expression 0.90 0.95

L = (DT,)"2. (10) 1.0 IO For a p-i-n diode, one can compare the actual process [lo] with anexponential form to obtain the approxi- Fig. 2. Chart for determining T and L/Ln from T,, T,+Tr, and IP/IR. mate relation L = l.5(DTT)1/z. (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 X; (6) into the above expression forT, (14) gives TT/T (1 [(LD/L) (IR/IF)(L/LD)]a)-l.(15) but the relation between r and Tpand IF/IRis com- + + plicated[6]- [8]. We will compare the results of our For L/LD= 0 this reduces to the graded-junction result approximate theory with (10) for a graded junction and (lo), as it should. ForL/LD = 1.0, we have with the graphs in [6]-[8] for a step junction. T,/T = [ 1 f (1 f IR/IF)'I-'. (16) Thecarriers which contributemost of thereverse 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 xi from the junction higherthan the more rigorous values given in [Si, as (see Fig. 1). Minority carriers with x<%; have been re- IF/IRchanges from 0.5 to 5.0. moved previously, and those with x>~;have not yet Interpretation: 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 carriersin the vicinity of x; 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 theseprocesses is limits on the relation between storage time and lifetime. Then, for a typical forward-to-reverse current ratio of an -n a2n - -N _- + D-- unity, one finds the lifetime r to be 1.5 to 4 times longer at 7 dX2 than the storage time T,. Equation (15) divided by (8) gives a relation between T,/T, and the grading param- for %=xi. The recombination rate near X; is approxi- mately n(xi)/r.The diffusion loss rate canbe estimated eter L/LD.It turns out that for typical values of the by noting that in our straight-line approximation forward-to-reverse current ratio (IF/IE = I), thecase of to L/LD < and a dn/ax changes from n(Xi)/xiat x = 0 to zero at X u x;. T,/T8 Hence, one can writed2n/dx2= --n(x;)/x?, and the con- graded junction, whereas the case of T,/T, = 1 corre- sponds toL/LD = 1 and a step junction. tinuity equationfor the densityat xi becomes To improveaccuracy one can solve (8) and (15) dn(x;)__- simultaneously. It is most convenient to have the com- dt Fig. shownin (13) asform chart results binedplotted in 2. To use this figure, one must know T,, (T,+T,), and the Thisequation represents an exponential decay, With a ratio IF/IR.The curvesslanting do\.vn and left corre- timeconstant T,givenspondvalues by toof IF/IR.Thecurves slanting and left up correspond to values of T,/(T8+T,). The solution T, = (f f D/xi2)'. (14) occurs atthe intersection of theappropriate 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. I = qAD(dnz/dx) = qADn(x;)/x;, \;C7ith T and an independent knowledge of the diffusion the current also exhibits an exponential decay with a constant, one can determineL. time constant T,.This is our basis for simulating the An ambiguity arises if one allows the possibility of a DEAN AND NUESE 1 STEP-RECOVERY TECHNIQUE FOR MEASURING PARAMETERS IN ASYMMETRIC p-n JUNCTION DIODES 155

CURRENT hyperabrupt impurity distribution. Such a distribution SOURCE establishesaminority-carrier potential barrier very DIODE near thejunction, whichwould sweep carriers away I from thejunction during injection and prevent them from being retrieved by the reversing step. In this case, GENERATOR -LT the step recovery measurement would erroneously in- REVERSING vi+vr PULSE dicate a strongly graded profile with L/LDnearly zero. HI - FREO, 44 I PROBE E JUNCTION This relatively uncommon junction distribution would I INCIDENT B be indicatedby a capacitance-voltage measurement TRIGGER SCOPE which gives the form C-V-l’n, where n is significantly PULSE less than 2.0. With a hyperabrupt distribution the only Fig. 3. ’4pparatus for measuring short lifetimesat extreme ambients. conclusion that can be reached is that the lifetime is The polarities shown are appropriatefor p-side of diode connected longer than the value given by (9). to center conductor of coaxial line.

Other Effects TABLE I Minority-carrier traps can have two effects. 1) Traps DEFIKITIONS capture injected minority charge in the region near the Z=length of line from probe to sample edge of the junction. The (additional) trapped charge c =velocity of electromagnetic wave in line repulses minority free carriers and reduces the minority Td= 21/c =delay time Tl =ROC=depletion capacitance time constant free-carrier density at the junction edge. This shortens T, =storage time thestorage time for given forward and reverse cur- T, = recovery time rents. 2) Traps continue to release carriers for a long 7 =lifetime L = injected carrier penetration length time after T,9and thus inordinately increase the mea- LO = injected carrier diffusion length sured recovery time T,. In fact, the observed recovery Ra = characteristic impedance of line R,= series resistance of contact or sample timeconstant provides a direct measure of thetrap Vi = incident reversing step voltage depth. These phenomena make it possible to identify V, = reflected reversing step voltage V, = Vi+ V, = probe voltage trappihg by associating it with the condition T,> > T, Vp=initial value of lorward-biased junction voltage Vi Ii = Vi/Ro = incident current (positive toward sample) for IF/IRof order unity or larger. I,= V7/Ro=reflected current (positive away from sample) Large-scalerecombination inhomogeneities produce IF = magnitude of forward current 112 =magnitude of initial reversing current effects similar to those produced by impurity grading. It=Ip-tZi-ZI.= total sample current Gibbons [ll]considers a step junction and shows that V,= V,.f Vf +IF& =total sample voltage recombination inhomogeneities weaken the dependence of r/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 r/Ts 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 junctionwould can argue that carriers in short-lifetime material con- not be observed by the C-V measurement, and in gen- tributeonly to T,, andthus 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 in- creases the ratio T,/(T,+ T,). One concludes that a low EXPERIMENTALTECHNIQCE L/LD obtainedfrom our step-recoveryexperiment is A schematicrepresentation ofap- the evidencefor either recombination inhomogeneity, or paratus is shown in Fig. 3, and the impuritygrading, or both.From a phenomenological the ensuing discussion are defined in TableI. The setup point of view,both of thesemechanisms reduce the is verysimilar to that employed in time domain re- averagepenetration of electronsinjected across the flectometry measurements [12]. The diode is mounted junction: grading piles up the stored chargeclose to the coaxially at the end of a single 5042 line. Simple pres- junction;short lifetimeregions prevent charge from surecontacts allowrapid 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- ing 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 diodebehaves as though it werein series with a puritygradient extends to a significantfraction of a purely resistive504 load up to veryhigh frequencies. diffusion length, the measure of the retarding field de- To forward-bias the sample diode, a direct current termined by the ratio L/LDcan be compared with the IFis fed into the 5042 line through a high-impedance measure of junction abruptness determined from C-V tee. A high-impedance sampling probealso is connected characteristics. It should be noted,however, that the to the line,with a blockingcapacitor separating this C-V measurement probes only that portion of the im- probe from the dc connection. The lengthof the purity distribution supporting the space-charge layer, line between the probes and the sample does not affect 156 IEEE TRANSACTIONS ON ELECTRONDEVICES. MARCH 1971

the basic response time of the measuring system, and can be aslong as required for thermalisolation. The basic response time of the apparatus is usually limited vp =o} by the rise and fall times of the voltage pulse and sam- pling scope,and it is on theorder of 0.25 nsfor 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- Fig. 4. Idealized oscilloscope traces (V,) for a reversing step volt- tive potential Vi,as shown in Fig. 4. Regardless of the age. Zero time occurs whenthe reflected voltage step just reaches nature of the test diode, this voltage persists until the the probe. The solid curve indicates the response witha moderate forward biasing current IF. The dotted curve indicates the re- reversing pulse reaches the diode and is reflected to the sponsewith small but finite IF andwith Vi adjustedfor the probe. Thus, the first signal displayed on the sampling same V,att=m. scope determines the magnitude of the reversing step voltage.Given the distance 1 from theprobe to the with an exponentially decaying current source with an diode, it also calibrates the time scale. initial value of -IR, one can write the total current as When the reflection from the diode reaches the probe It = Cd Vj/dt - IRexp( -t’/T,) where t‘ is the time after (t = O), the diode’s response tothe reversing signal begins T,, and T,is the time constant of the decaying current to appear. The diode exhibits a low resistance during source. SubstitutingVi = V,- ItR8and then substituting thestorage period T,, andthen approaches an open for It and Vt from the relations given above leads to a circuitduring the subsequent recovery period. The first-order equation in V,: initial resistance is due to contact, bulk, or lead resis- tancein series with the effectively shorted junction. TldV,/dt’ 4- V, = 2Vi + IpRo + IRR~exp (-t’/T,) The recovery toward an open circuit is approximately where exponential. During this period we represent the diode by the above in series with a parallel combina- TI = (Ro + RJC. tion of a capacitor and exponentially decaying current With (17) and (18) for the initial value, the solution to source, as indicated previously. this equationis The remaining relationships shown in Fig. 4 can be derived by supposing that the current probe, blocking V, = 2Vi + IF& -I- I&( [TI/(TI - TT)]exp TI) capacitor,and voltage probe are squeezed up tight + [Tr/(Tr - TI)] exp (-t’/Tr)]e (19) against the diode, so that there are no transit-time de- laysbetween the diode and the voltage probe. This Equation (19) shows that the differencebetween 2Vi simplification is possible because after the initial drop, andthe steady-state probe voltage is IFRo. The am- all parts of the voltage waveform are retarded by the plitude of the recovery is simply IBRo. Thus, one can same time. pick the ratio IF/IR= I.PRo/IRR~directly off the oscil- During the storage time T,, the voltage drop across loscope trace. Thecapacitor time constant TI is determinedex- the series resistance is Vi-V, = IiR,. Using the rela- tions in Table I to substitute for V, and It, and per- perimentally by repeating the step-response measure- forming simple algebraic manipulation, we obtain ment with the forward current IF reduced to a small value, and the reversing voltage, Vi readjusted to ob- R,/Ro = (Vp/2VJ/(1- VP/2Vi). (17) tain the same steady-state probe voltage. Equations(8) Thus,the fractional change in the probe voltage at and (15) showthat T, and T, decreaserapidly with t = 0 gives the series resistance relative to 50 Q,as indi- IF/IR. (For L/LD=1, T, and T, areproportional to cated in Fig. 4. The magnitude of the reverse current is (IF/IR)’.) Thejunction voltage, however, decreases IR= -It. Again, substituting for It from Table I, and only as V,-log, (IFIIR).Thus, one can reduce IF/IR using (17) for Vp/2Vi leads to by two or three orders of magnitude and cause T,and T,to vanish, butstill retain Vi = VF.Since the forward- IR = (-2Vi/Ro),/(l + R,/RO) - IF. (18) biased junction voltage is approximately the same with To determine the value of IR,one can insert the mea- small IFas it is with large IF, and the final voltages are sured value for the forward current, or as we shall see 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- directly. sociated time constantTI are nearly the same in the two The storage periodends at t = T, when theprobe experiments, and TI can bemeasured by determining voltage V, startschanging again. Assuming that the the time required for the recoveryin the low-current junction behaves like a constant capacitor C in parallel experiment to reachl/e = 0.368 of its final value. DEAN AND NUESE: SmP-RECOVERY TECHNIQUEMEASURINO FOR PARAMETERS IN ASYMMETRIC p-n JUNCTION DIODES 157

fraction of the diodes tested gave values of LILD sig- nificantly less than unity attests to theneed for a theory such as ours which includes the effects of unknown im- purity grading.

CONCLUSIONS Anefficient experimentaltechnique has been de- veloped for carrying out step-recovery measurementsof veryshort lifetimes. Theoutstanding feature of this technique is that only one electrical line is required to contact the test diode, and the distance from the diode tothe measuring equipment may be arbitrarilylong. Thissimplicity makes it possible tominimize stray circuitreactances fordefinitive high-speed measure- ments. It also facilitates subjecting the test diode to a Fig. 5. Typical oscilloscope trace for p+-n GaAso,&Po,l~ diode. wide range of ambientconditions, including temper- Time scale is 2 ns/div. Voltage scale is 0.5 V/div. 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. thekey parameters, and a large amount of informa- Fortunately,the coefficientsof theexponentials are tion is availablein a singleoscilloscope display.The suchthat the sum can be approximatedby a single various features on the oscilloscope display have been exponential having a time constant equal to the sumof quantitatively characterized. the time constants of the two original exponentials. A Using simple heuristic arguments based on the key numerical evaluation of (19) shows that the recovery physicalphenomena in the experiment, we havede- reaches l/e of its final value in a time given by veloped an approximate theory which makes direct use of the salient features in a typical oscilloscope display. t’ = TI + (1 + S)TT The results are presented in an easy-to-use graphical form, andfit closely the resultsof more rigorous deriva- where 6 has a maximum value of +0.17 at T1/Tr= 1.75 tionspreviously done by others, in the appropriate and approacheszero in the two limitsof large and small limits. The present technique is more general, however, T1/TT.Thus, with an errorof less than 20 percent (which tends to cancel the previous errors), one can evaluateT, andprovides more information from a singleexperi- mentthan couldhave been obtained previously. In- as the difference between the times required for the volt- formation conveyed includes series resistance, depletion age to reach l/e of their final values in the high- and capacitance, injected-carrier lifetime, and the effect of low-current experiments described above. In the limitof possible impuritygradients orrecombination in- small series resistance, one has R, <

[71 B. Lax a,:d S. F. Neustadter,“Transient response of a P-N recoverydiode for pulse andharmonic generation circuits,” junction, J. AppZ. Phys., vol. 25, 1954, p. 1148. Proc. IEEE, VO~.57, July 1969, pp. 1250-1259.

[SI R. H. Kingston, “Switching time in junction diodes and junc- ..ill] P. E. Gibbons, “Problems concerning the soatial distribution of 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-hTjunction charge- 13, 1969, p. 989. storage diodes, Proc. IRE, vol. s!, 1962, p. 43. [12] See, for example, “Time domain reflectometry,” Hewlett-Pack- [lo] J. L. Moll and S. A. Hamilton, Physical modeling of the step 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 under the assumption that the dissipation power density increasedtemperature whichaffects the velocity-field curveand is homogeneousthroughout the diode volume. This which may also permanently damage the diode or the metalcontacts. The behavior of these oscillators indicates that the dissipation power assumption is certainly good for some cases, e.g., diodes density is more or less inhomogeneous, depending on the kind of operated in the LSA mode [6].However, when diodes oscillation mode. The present paper makes use of simple models of areoperated in theusual transit-time dipole-domain the dipole domain mode and the accumulation mode in an analysis mode [6] and the transit-time accumulation mode [6] of the inhomogeneous dissipation power density. The temperature the dissipation is much larger near the contact increase caused by this dissipation power is then calculated. The than near the cathode contact [7].In this paper simple temperature dependence on diode dimensions and mounting is also discussed. It is shown that the temperature is lower when the anode models of the domain mode and the accumulation mode is turned towards the heat sink than when the cathode is turned the operation are used to calculate the inhomogeneityof the same way. This difference amounts to 150’K for an accumulation dissipation.From this the temperature is calculated. mode diode with 500’K maximum temperature in the former case. Gunn diodes are typically mounted with only one con- INTRODL-CTION tact acting as anefficient heat sink. The present theory thus tells the difference in temperature when the diodes HE is a solid-state 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 temperatureis T 1-kW peak power is available from single diodes higher in the former case and the difference may be as under pulsed operation [l]but less than 1 W from CW large as 40’K for domain mode diodes and 150’K for diodes. The dissipationpower density and the con- accumulationmode diodes when the maximum tem- nectedtemperature increase limit the production of perature with the anode to the heat sinkis 500’K. microwavepower at high duty cycles. The allowable temperature of a diode has an upper limit. The contact 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 about700 to 900’K. 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. Thenegative differential mobility in GaAsdis- field andcurrent density). The geometry assumed is appears at veryhigh temperatures and there exists some shown in Fig. 1. The diode and the contact layer are evidence that it is toosmall to cause oscillations at assumed to beof circular shape whenlooked upon from a temperatures above600 to 700’K. direction perpendicular to the heat sink. The dissipation A number of authors [2]- [5]have calculated the tem- P is assumed to be a function of the longitudinal co- peratureincrease in diodes with different mountings ordinate x only. The heat conductivityof the heat sinkis typically much larger than that of GaAs. It is therefore Manuscript received June 30, 1970. N. 0. Johnson is with the Research Laboratory of , reasonable to approximate the heat flow in the GaAs to Chalmers University of Technology, Gothenburg, Sweden. be in the x direction only. The equationof heat conduc- K. 0. I. Olsson was with the Research Laboratoryof Electronics, ChalmersUniversity of Technology,Gothenburg, Sweden. He is tion then becomes (in the stationary state) now with Philips Teleindustri, Jakobsberg, Sweden. S. J. Wildheim was with the Research Laboratoryof Electronics, Chalmers University of Technology, Gothenburg, Sweden. He nowis with STAL-LAVAL Turbine Co., Finsping, Sweden. dx \ dx/