Determination of excess carrier lifetime ofp-i-n diodes from the r.f. resistance at

microwave frequenciesK.N. Bhat and J.M. Borrego

Indexing terms: Carrier lifetime, Semiconductor diodes, Semiconductor junctions

Abstract: A steady-state method for determining the excess carrier lifetime using p-i-n diodes is presented.The carrier lifetime is obtained from the r.f. resistance of a forward-biased p-i-n diode measured at microwavefrequencies. The carrier lifetime obtained with this method in diffused silicon p-i-n doides is compared withthe carrier lifetime obtained from other usual methods of measurement.

1 Introduction

The lifetime of excess carriers has a very important positionin solid-state-device theory, and so a knowledge of its be-haviour and magnitude is of great use to device designers. Itis also of considerable physical interest, as information onthe recombination mechanisms involved may be obtainedfrom an analysis of its characteristics. Although the statisticsof recombination has been analysed extensively by Shockley

et al.,1 the experimental investigation of the phenomenonhas been somewhat limited by the techniques used tomeasure carrier lifetimes.

The methods described in literature ,2~s to determinecarrier lifetimes using p-i-n diodes, are based on the assump-tion that the transient and steady-state values of the lifetimeare the same. However, since the carrier density in the /-region does not remain constant during the transientmeasurements, the interpretation of the results obtained bythese methods is rather difficult, because the carrier life-time depends upon the injection level. To overcome thisdifficulty associated with these transient methods, Colletet al.6 have proposed the 'double-pulse method' for carrier-lifetime measurement in p-i-n diodes. It is based on themeasurement of the voltage v(t) present across the diodeterminals when a second pulse is applied while excesscarriers are still present from a previous pulse. The lifetimeis related to v(t) directly. In practice, at high injectionlevels, the resistance of the heavily doped p- and ^-regionsof the diffused p-i-n diode can not be neglected in com-parison with the Mayer resistance, and the voltage v{t) mayalso include the drop in the heavily doped regions. Further,in this method a uniform carrier concentration over theentire /-region has to be assumed. This may not be true inthe case of diodes having wide /-regions unless the lifetimeis very high.

We have developed a new method to measure the carrierlifetime and it is based on the direct measurement of the r.f.resistance Rf of the forward-biased p-i-n diode at microwavefrequencies. The forward r.f. resistances Rf consists of aseries combination of current-independent resistors and acurrent-dependent resistor rf, the latter being a function ofthe width of the /-region, of the carrier lifetime and theforward-bias current IDC. Information about the carrierlifetime at different injection levels is obtained from the

Paper T38S, first received 6th September and in revised form 9thNovember 1976Dr. Borrego is, and Mr. Bhat was formerly, with the ESE Depart-ment, Rensselaer Polytechnic Institute, Troy, New York-12181,USA

measurement of this current-dependent resistor jy of the/-region. This is a steady-state method, and the measure-ments are made at fixed injection levels. This techniquepermits the exact determination of the lifetime for aspecific injection level. Since the resistances of the heavilydoped p- and n-regjons are taken care of easily in calcul-ations, there is no ambiguity in the value of the /-regionresistance. Furthermore, there is no need to put an extracontact as in the 'double-pulse method'. It is the purposeof this paper to describe this method for measuring excesscarrier lifetimes and to present the results obtained usingdiffused silicon p-i-n diode fabricated in our laboratory.

2 Theory of measurement

The idealised structure of a p-i-n diode and the r.f. equiv-alent circuit of the encapsulated diode under forward-biasconditions are shown in Fig. 1. The resistances rp and rn

represent, respectively, the ohmic resistance of the p- andw-regions, which include the regions of graded doping.rf is the injection-level dependent resistance of the /-region.The parasitic elements are approximately represented by aninductance Lc and a resistance rc (the contact resistance)in series with the diode, and a shunt capacitance Cc. Thecontribution of rp and rn to the total series resistance Rf

depends upon the thickness of the p- and n regions, theinjection level in the /-region and the /-region thickness.If the doping of the p- and ^-regions is high enough, appliedreverse and forward voltages do not change appreciably theelectrical conductivity or the thickness of these regions.

Referring to the structure shown in Fig. la, with aforward bias applied across the diode, holes from the

p i n

-d

Cc

Lc R,

Cc

Fig. 1 p-i-n diodes

a Idealised structureb R.F. equivalent circuit of the encapsulated p-i-n diode underforward-bias conditions

SOLID-STATE AND ELECTRON DEVICES, APRIL 1977, Vol. l,No.3 69

p-region and electrons from the n-region are injected intothe /-region, and since recombination predominates, thecurrent of holes coming from the left is taken over by theelectrons coming from the right. Recombination takes placenot only in the /-region but, in principle, also in the highlydoped p- and ^-regions. However, the latter are not im-portant below current densities of about 20 A cm"2. Ratherit may be assumed as a very good approximation that atthe edges of the p- and ^-regions the total diode current iscarried by holes and electrons, respectively. With theseboundary conditions, the diffusion equation which deter-mines the injected carrier distribution is solved, and theohmic resistance drop in rf is given by7

JDCrf =kT Sb sinh(d/L)

q

tan"1 [y/{\ -B2tanh2(dlL)}smh(d/L)] (1)

where IDC is the d.c. bias current, q is the electronic charge,L is the ambipolar diffusion length equal to \/Dar, T is thelifetime of excess carriers, Da is the ambipolar diffusioncoefficient, b is the ratio of electron mobility nn to thehole mobility jup, and B is given by

B = (2)

k is the Boltzmann constant and T is the temperature inkelvins.

Eqn. 1 shows that the product Iocrf is independent ofthe cross-sectional area and depends only upon the ratio(d/L). Fig. 2 gives a plot ofIDCrf as a function of (<2/Z,) forT — 300° K. By measuring the resistanc rf at variousforward-bias currents IDC, the magnitudes of Iocrf a r e

obtained at different injection levels. The correspondingvalues of d\L are obtained from v\%. 2. Knowing themagnitude of d, the diffusion length L, and hence thelifetime T, is calculated for various injection levels. Thepresent method should give lifetimes values higher thanthe transient methods because the former is free from the

1OOOr

100

10

d/L-

secondary effects which may affect the transient methods.The excess carrier lifetime T is thus evaluated, assuming

a fixed value of ambipolar diffusion coefficient. It isassumed that the mobilities (jun and jup) and diffusionconstants Dn and Dp are independent of the injection level.This is the main limitation of the method because it isknown8 that at concentrations larger thanelectron-hole scattering decreases Dn and Dp.

1016cm"3

625 pm

jnOpm

Fig. 3 Physical structure of the p-i-n diodes fabricated for thepurpose of measurement

3 p-i-n diode fabrication

In the conventional p-i-n diodes fabricated using planartechniques, the carrier flow is three dimensional, resultingin carrrier flow in the lateral direction also. Experimentalresults on these structures should therefore be analysedusing a 3-dimensional analysis which would be tedious.To simplify the analysis and enable the use of the closed-form solution for Iocrf given by the 1 -dimensional analysisin eqn. 1, the structure shown in Fig. 3 was fabricated.The starting material for the fabrication work was a single-crystal Il-type silicon wafer with resistivity in excess of1000 £1 cm. Boron and phosphorus diffusions were carriedout on each side, respectively, of a well prepared wafer,and, after proper drive-in steps, the reverse leakage-currentreduction and carrier-lifetime improvement were achievedby suitable gettering and annealing steps. Ohmic contacts tothe heavily doped p- and ^-regions were made by chrome-gold evaporation and, finally, gold plating. Individualdiodes of 625 x 625 jum were separated using a wire saw.The devices were mounted on microwave packages bysoldering, and wire connections were made by nail-headbonding.

4 Experimental results and discussions

The measurement of the r.f. resistance Rf under forward-bias conditions was carried out at 3 GHz. The encapsulateddiode was mounted at the end of a slotted line using a testfixture, and this was implemented with a standard im-pedance-measuring assembly. The reference plane was fixedby noting the position of the voltage minimum along theslotted line when a copper slug was mounted in the fixtureinstead of the diode. With the diode mounted in position,the shifts in the voltage minima and the standing-waveratio were noted for different values of the forward-biascurrent IDC, and the corresponding diode impedance ZL

calculated.Referring to Fig. \b, the diode impedance ZL is given,

approximately, by

Zr ^ + / •(l-w2LcCc)

(3)

Fig. 2 Plot of IDCrf product as function of d/L (T = 300° K) The package capacitance, C c = 0 - 3 3 p F , was measuredfor p-i-n diodes separately using a Boonton bridge and inductive reactance

70 SOLID-STATE AND ELECTRON DEVICES, APRIL 1977, Vol. l,No.3

wLc = 16-187 £2, and, hence, w2LcCc was estimated fromthe imaginary part of ZL in eqn. 3. The resistance Rf wascalculated using the relation

Rf = (i -w2LcCc)2x ReZL (4)

This resistance Rf is the sum of the contact resistancerc> rp> rn a nd rf. From eqn. 1 we find that rf is inverselyproportional to IDC. Therefore

= (rc rn)constant

lDC

(5)

O26n=rpTnTc

TO 20 30 40 50 601/lDC

Fig. 4 R.F. forward resistance against 1/IDC

2-Or

4 0 80 120 160 200forward current IQQ , mA

240

Fig. 5 Excess carrier lifetime against forward-bias current

I-I-I Present method•-•- Wilson's methodA-A Reverse recovery method

Fig. 4 is a plot of the measured Rf against 1//DC. The inter-cept on the resistance axis gives the resistance rc + rp + rn

= 0-26 Mfi. Thus, knowing Rf and the current-independentresistance (rc + rp + rn), rf is calculated for various valuesof IDC, and the Iocrf product is evaluated. Using Fig. 2the corresponding values of d/L are noted for each IDC.The /-layer width (2d) was obtained by measuring thepunchthrough capacitance Cp (Cp = eoerA/2rf). Thediodes fabricated for the present study have a squaregeometry of 625 jum x 625 jum and a punchthrough capaci-tance of 0-42 pF (1 MHz measurement using a Boontonbridge). The Mayer width estimated is 101 jum. The ambi-polar diffusion length L is thus known from the d/L ratio.Taking Da = 18-4 cm2 s"1 [Da = 2DpDn/(Dp + Dn) wherethe hole diffusion coefficient Dp is 12-4 cm2 s"1 and theelectron diffusion coefficient Dn is 34-8 cm2 s"1] , theexcess carrier lifetime T is calculated for different injectionlevels IDC-

The lifetimes obtained with this microwave measure-ment were compared with the lifetime values determinedwith the reverse-recovery and Wilson's methods, and theresults are plotted in Fig. 5. The lower values of T obtainedin the reverse-recovery method and Wilson's method incomparison with the microwave method, may be explainedfrom the fact that the carrier density in the /-region fallsduring the time when the transient measurements are made.From the Shockley-Read theory it is known that if thecarrier density is low then the lifetime is low. Thus, thetwo transient methods give an average lifetime during thetransient period. As shown in Fig. 5, the lifetime is constantat high injection levels, and this is in agreement with theShockley-Read theory. The observed fall in lifetime atlower injection levels is associated to the presence of deep-lying impurities introducing recombination levels and thiswas confirmed by thermally stimulated current (t.s.c.)and optical absorption measurements. Both methodsshowed the presence of a deep-lying level at 0-35 eV abovethe upper.edge of the valence band Ev. As reported in theliterature9, oxygen in silicon gives rise to an acceptor levelat 0-35 eV above Ev and this acts as an efficient recom-bination centre. It is quite possible that, in thep-i-w diodeswe have fabricated, oxygen was present in large quantitiesin the starting Fl-type silicon or else it is also likely thatoxygen has entered during diffusion.

The accuracy of the present method of excess-carrier-lifetime measurement depends mainly on the following twofactors:

(i) first, the accuracy with which the /-layer width (2d)can be measured. For diffused junction p-i-n diodes, thedepletion layer widens slightly even after the entire Il-layerhas been depleted. Although the change in capacitance afterpunchthrough has occurred is quite small, for a diode of100/zm /-layer width, an error of 2—3% is quite possible inthe estimated value of d.

(ii) Secondly, the accuracy with which Rf can bemeasured determines the accuracy of the estimated valueof T. The low forward-bias specification indicates that thefixture and equipment be particularly sensitive to thisparameter. In our experiments, with the coaxial short-circuit terminating the slotted line, a standing-wave ratioof 84-4 dB is measured. When the fixture, with a copperslug mounted in place of the diode, terminates the coaxialline, a standing-wave ratio of 68-8 dB is measured. There-fore it is concluded that an uncertainty of 0-018 Q existsin our measurements. This is the termination resistance

SOLID-STATE AND ELECTRON DEVICES, APRIL 1977, Vol. l,No. 3 71

corresponding to 68-5 dB s.w.r. on a 50 £1 transmission line.The use of a d.c. microvoltmeter as a null indicator permitsstanding-wave ratios greater than 50 dB to be measuredreproducibly.

5 Conclusions

A new method was developed for determining the excesscarrier lifetime r under steady-state injection-level con-ditions by measuring the r.f. resistance of a p-i-n diode.A 1-dimensional theory correlating the r.f. resistance andthe diffusion length is described. The p-i-n structure fabri-cated for the experimental purpose enables the results tobe analysed using this 1-dimensional theory. The carrierlifetime has been determined over a fairly wide range ofinjection levels and the results have been compared withthe transient methods, namely, the reverse-recovery andWilson's method. The present method gave lifetime valueshigher than the transient methods, as predicted.

The excess carrier lifetime determined by the presentmethod is found to be independent of injection level, oncethe injection level is sufficiently high. This is in agreementwith Shockley-Read recombination theory. The fall in theestimated value of carrier lifetime at lower injection levelshas been explained on the basis of the presence of a deep-lying level. The presence of oxygen giving rise to a recom-bination centre was confirmed by both the t.s.c. methodand the optical method.

6 Acknowledgment

The authors are thankful to Prof. R.J. Gutmann, RensselaerPolytechnic Institute, Troy, whose comments and sugges-tions were helpful in carrying out the microwave measure-ment.

References

1 SHOCKLEY, W., and READ, W.T.: 'Statistics of the recom-bination of holes and electrons', Phys. Rev., 1952, 87, pp.835-842

2 MOLL, J.L., KRAKAUER, S., and SHEN, R.: 'P-N junctioncharge storage diodes', Proc. IRE, 1962, 50, pp. 43-53

3 WILSON, P.G.: 'Recombination in silicon p-rrn diodes', Solid-State Electron., 1967,10, pp. 145-153

4 LEDERHANDLER, S.R., and GIACOLETTO, L.J.: 'Measure-ment of the minority carrier lifetime and surface effects injunction diodes', Proc. IRE, 1955,43, pp. 473-483

5 BROUSSEAU, M., and SCHUTTLER, R.: 'Use of microwavetechniques for measuring carrier lifetime and mobility in semi-conductors', Solid-State Electron., 1969, 12, pp. 417-423.

6 COLLET, J., BAILON, L., BRABANT, J.C., BAURRAU, J.,and BROUSSEAU, M.: 'New methods for carrier lifetimemeasurements in pnn sturctures', ibid. 1973, 16, pp. 999—1005

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