621.315.59 The Institution of Electrical EngineersPaper No. 3047 E

Mar. 1960

RECOMBINATION PROCESSES IN SEMICONDUCTORSBy R. N. HALL.

(The paper was received Wth April, 1959.)

SUMMARYRecombination of electrons and holes may take place in the

host crystal or at impurity centres, the energy being removed byradiation of a light quantum, by multiphonon emission, or by anAuger process. The probabilities for each of these six processes arediscussed. While the lifetime in semiconductors is usually determinedby multiphonon recombination at impurity centres, Auger recombina-tion in the host crystal can be expected to dominate in small-band-gapcrystals containing large concentrations of free carriers. Radiativerecombination in the host crystal may limit the lifetime in semicon-ductors where band-to-band transitions are direct, provided that thespecimens are reasonably free of recombination centres.

LIST OF PRINCIPAL SYMBOLSa = Optical absorption coefficient, cm"1.B = Recombination probability for radiative or multiphonon

recombination, cm3/sec.j8 = Recombination probability for Auger recombination,

cm6/sec.W8 = Band gap of a semiconductor.M = Number of recombination centres per unit volume, cm"3.n = Free electron concentration, cm"3.p = Free hole concentration, cm"3.r = Recombination rate, cm" 3 sec"1.

R = Net recombination rate, cm" 3 sec"1.T = Lifetime of excess free electrons or holes, sec.fx = Index of refraction.

(1) INTRODUCTIONSeveral mechanisms have been proposed to account for the

generation and recombination of electrons and holes in non-metallic crystals. These processes have been studied in thegreatest detail in germanium and silicon, but, in addition, con-siderable information has been obtained from investigations ofvarious other semiconductors. The paper reviews the behaviourto be expected from theoretical considerations of the basicrecombination mechanisms and compares these results with theavailable experimental information.

The following processes are examined:

(a) Radiative recombination, in which most of the energy iscarried away by a quantum of electromagnetic radiation, sometimeswith an accompanying lattice phonon.

(b) Multiphonon recombination, wherein the recombinationenergy is dissipated by the creation of a number of lattice phonons.

(c) Auger recombination, which results in the ejection of anenergetic electron or hole to help satisfy the laws of energy andmomentum conservation.

Each of these processes may be imagined to take place eitherin the host crystal itself or at an imperfection, making six casesin all to be discussed. Some characteristic features of each aresummarized in Table 1.

In the Sections that follow, an attempt is made to provideestimates of the probabilities for each of these processes. Thebehaviour with respect to variations in temperature and impurityconcentration, and such material parameters as energy gap and

Table 1

COMPARISON OF DIFFERENT RECOMBINATION PROCESSES

Process In host crystalRadiative . . Negligible compared with other

processes except with verylarge carrier concentrations.Not strongly dependent upontemperature and band gap.

Multiphonon Improbable process. Not yetseen experimentally.

At impurity centresProminent inmaterials

with large band-gaps, e.g. phosphors.

Probability increases rapidlywith temperature. Favouredby small band-gaps and largecarrier concentrations.

Mr. Hall is at the Research Laboratory of the General Electric Co., Schenectady.

Dominant process inmost semiconduct-ing materials.

Auger . . Probability increases rapidly Sometimes seen intrapping pheno-mena. Generallyrequires large car-rier concentrationsto be important.

effective mass are discussed. It is hoped that these estimateswill provide a convenient framework with which to compareexperimental results.

(2) RECOMBINATION RATES AND LIFETIMESWe shall use the symbol rn to denote the rate per unit volume

of disappearance of free electrons due to any given recombina-tion process. Our object is to obtain explicit expressions forthis quantity in terms of the electron and hole concentrations,n and p, for each of the processes that we wish to consider.From a knowledge of rn(n, p) and the corresponding expressionfor holes we can calculate all the other parameters that areused to describe the kinetics of the recombination of electronsand holes.

By the principle of microscopic reversibility, we know thatthe recombination and generation rates of each kind of carriermust be in balance at equilibrium for each of the recombinationprocesses taken separately. Therefore the rate of thermalgeneration of electrons must be given by rn(n0, p0), where n0 and

PQ represent the equilibrium carrier concentrations. The amountby which the recombination of electrons exceeds the thermalgeneration rate, and the corresponding expression for holes,may be written in the form

Rn = rn(n, p) — rn(n0, p0) . . . (la)

RP = rp(n, p) - rp(n0, p0) . . . (16)

Clearly, if several of these processes are simultaneously operative,the total net recombination rate is the sum of the individualrates.

We shall define the lifetime of an electron or hole to be theexcess carrier concentration divided by the corresponding netrecombination rate.

r n = ( n - n Q ) l R n = 8 n / R n . . . . ( 2 a )

No limitations need be placed upon the magnitudes of 8nand 8p, except that n and p must never be negative. This defini-tion of carrier lifetime is the one which is normally used in semi-

[923]

924 HALL: RECOMBINATION PROCESSES IN SEMICONDUCTORS

conductor electronics. Although it is generally used only todescribe the behaviour of minority carriers, it is equally applicableto the description of the properties of the majority-carrier con-centration. Note that it does not represent the time that anelectron or hole can expect to survive before being captured;this quantity (the capture lifetime) will be discussed later in thisSection.

In the calculation of lifetimes from eqn. (2), it is often neces-sary to specify the conditions under which they are to beevaluated. If electrons and holes are created within the sampleat a constant rate, L per unit volume (by irradiating with pene-trating light, for example), the excess carrier concentrations willincrease until steady-state values are reached with Rn = Rp = L.The quantities that are calculated by using these values in eqn. (2)are known as the steady-state lifetimes. The electron and holelifetimes may differ considerably in magnitude if the recom-bination process involves a sufficient concentration of impuritycentres.

On the other hand, the transient response following theremoval of an external stimulus is calculated from eqn. (1) bynoting that Rn = — dn\dt, and Rp= — dpjdt. Where the timedependence can be described by a simple exponential decay,the use of eqn. (2) leads to constant values for the carrier life-times. These transient lifetimes may, however, be quite differentfrom the steady-state lifetimes if charge storage at recombinationcentres is important. Quite often, a more complicated time-dependence is observed, and the concept of a carrier lifetimebecomes less meaningful.

The lifetime as defined by eqn. (2) is only indirectly related tothe average time interval, T, that a carrier spends as a freeentity between generation and capture processes. The latterquantity, which we shall call the 'capture lifetime', is given bythe ratio of the total electron or hole concentration to thecorresponding capture rate. The capture lifetime is more directlyrelated to the mechanics of the recombination process and isthe quantity that is generally used in analysing the behaviourof photoconductors.

These two definitions of lifetime, as applied to electrons, arecompared by the following equations:

n — n0Rn(n, p) rn(n, p) - rn(n0, pQ)

T =

• (3a)

rn(n,p)When the carrier concentrations have approximately theirequilibrium values, these two lifetimes will generally be quitedifferent, particularly in the case of the majority carrier. Inlarge band-gap materials, only a small stimulus is required tomake both the capture rate and the minority-carrier concen-

tration large compared with their equilibrium values. Underthese conditions, the two values of minority-carrier lifetime aresubstantially equal. However, in most instances it is necessaryto distinguish between these two definitions of lifetime. Thelifetime concept is discussed further by Schottky1 and Hoffmann.2

(3) RADIATIVE RECOMBINATION IN THE HOST CRYSTALRadiation which is presumably due to the recombination of

free electrons and holes has been observed in germanium,3- 4«5

silicon,5 gallium antimonide,6 gallium arsenide,6 indium phos-phide,6 indium antimonide7 and cadmium telluride.8 In a non-degenerate semiconductor, the rate at which electron and holesdisappear by this process is proportional to the product of theelectron and hole concentrations.

Thus r = Bnp (4)

and R = B(np - nj) (5)

The radiative lifetimes, calculated using eqn. (2), are

1—— I . , —

+ p0 + Sn) • • ( 6 )

Thus, to a good approximation, the radiative lifetime is inverselyproportional to the total concentration of free carriers under allcircumstances.

In intrinsic material, the low-level lifetime has a maximumvalue which is strongly temperature-dependent and variesgreatly from one semiconductor to another. Since one generallydeals with extrinsic semiconductors, however, a more significantquantity is the low-level lifetime in a sample having a givenmajority-carrier concentration. For samples having 1017

majority carriers/cm3 the room-temperature radiative lifetimemay range from a few milliseconds in a semiconductor such asgermanium or silicon where radiative transitions are indirect,to a few tenths of a microsecond in materials where the transi-tions are direct. It should be noted that internal absorptionwill increase the apparent radiative lifetime in samples whichare thick compared with the absorption distance for the recom-bination radiation.9

The capture probability, B, can be evaluated by the methodof van Roosbroeck and Shockley,10 using experimentallydetermined values of the optical absorption coefficient and theintrinsic carrier concentration. Table 2 summarizes values forB obtained in this manner for several semiconductors. Lifetimesand capture probabilities were calculated from optical data inthe references cited, corrected to 300° K where necessary. Forthe lead salts, «/ was deduced from Chasmar's work.64

Analytic expressions for B may be obtained by a direct cal-culation or by applying van Roosbroeck and Shockley's method

Table 2

FREE CARRIER RADIATIVE RECOMBINATION AT 300° K

Material

Si ..Ge ..GaSbInAsInSbPbS ..PbTePbSe

wt

eV1080-660-710-310180-410-320-29

"i

cm-3 x 10i"

0000150-240 043

16200

7 14062

B

cmVsec x 10-1*

00020 034

132140485240

T (intrinsic)

4-6h0-61 sec0-009 sec

15 /xs0-62 fxs

15 ps2-4)MS2-Ofj.s

r(for 101'majoritycarriers/crn3)

(AS

2500150

0-370-240120-210190-25

Reference

11, 1211,13

141516171717

HALL: RECOMBINATION PROCESSES IN SEMICONDUCTORS 925

to theoretical or semi-empirical formulae for the optical absorp-tion coefficient. We will make use of the latter procedure here,since a comparison between the calculated and observed absorp-tion coefficients can be helpful in testing the correctness of thecalculation.

Direct transitions can occur in those semiconductors wherethe conduction-band minimum and the valence-band maximumboth occur at the same point in momentum space. For suchmaterials, the variation of the optical absorption coefficient,a^m"1), with energy has the form,18

m\/W_23/2mg2|~ memh ~|3/2/ m mw 3/A h2 lm{me + m^\ V + me

+ mkmhJ \ me1

(7)

for photon energies, W, slightly greater than the band gap, Wg.fj, is the index of refraction, and me and mh are the density-of-states electron and hole effective masses, me2 is the electronself-energy, 0-511 MeV.

Indirect transitions occur when the optical absorptionthreshold involves states of different ^-vector. In this case, theoptical absorption as analyzed by Bardeen et al.ls and byMacfarlane et a/.19*20 can be approximated by

^•indirect "•

- Wg-k6)2 (W-+

k6)2

(8)

The constant, A, is essentially independent of temperature. Ithas values of 2600, 3600, and 2500eV~2cm-1 for Ge, Si, andSiC as determined experimentally.12'13> 21 Corresponding valuesfor 0 are 260° K, 600° K, and 1044° K respectively.

In the method of van Roosbroeck and Shockley, the equili-brium rate of radiative recombination is set equal to the totalamount of black-body radiation absorbed by the crystal due toband-to-band processes:

L2KW2

- 1](9)

It should be noted that this formula is applicable without modi-fication to crystals in which the Franck-Condon shift is appre-ciable. Combining this equation with eqn. (7) gives an expressionfor the capture probability for direct transitions:

^direct —(2TT)3'2 he2

m'Cm

me

= 0-58 x 10 - 1 2 / /mh

sec . . . (10)

where the energy threshold for direct transitions, Wg, isexpressed in electron-volts. Values of B calculated for galliumantimonide, indium arsenide and indium antimonide using thisequation agree with those in Table 2 within the present experi-mental uncertainties of the effective masses.

A similar calculation of the capture probability for indirecttransitions gives

***** -m2 \ 3 / 2

t 9

—J *? tht 9

*? coth - (11)

= 2-17 x 10-

In this case, calculated values of the capture probability agreewell with those in Table 2, since the parameters A and 6 arechosen to fit the measured absorption data.

Examination of eqns. (10) and (11) shows that Bdirea shouldvary as T~3I2, whereas Bindirect can be expected to vary some-what more slowly than T+1. In neither case, however is astrong temperature dependence to be expected.* The mag-nitude of Bdirecl should not depend very strongly upon Wgin view of the tendency for small energy gaps to be asso-ciated with small effective masses. Thus, for semiconductorsin which recombination occurs by a direct radiative transition,B ~ 30 x 10-12cm3/sec. Most of the 1II-V, 1I-V1 and lead-salt compounds appear to fall in this category. For indirecttransitions, the capture probability is smaller by a factor of103 to 104, a quantity which may vary considerably from onematerial to another, being dependent upon the nature of theenergy surfaces. Similar remarks apply to the extrinsic lifetime,since this quantity is inversely proportional to B.

In the case of germanium, the calculated transition probabilityis several times too small because of failure to take into accountthe contribution made by vertical transitions between statesnear k = 0, where the energy gap is greater than the thermalband-gap by a rather small amount, 8Wg. For such materialsthe band gap appearing in eqn. (7) should be replaced byWg + 8 Wg, and the resulting expression for Bdlrect containsan additional factor, exp (— 8Wg/kT). As it stands, eqn. (10)represents the recombination probability for that fraction of thetotal electron concentration that has sufficient thermal energyto attain a zero momentum vector. The total recombinationprobability for a semiconductor having band edges at different^-values can therefore be written9

B = Bindirect + Bdlrectexv(-8lVglkT) . . (12)

The direct process tends to become dominant at high tem-peratures because of the effect of the exponential factor. Thisoccurs somewhat below room temperature in the case of ger-manium. Optical data for silicon22 show that direct transitionsare unimportant below the melting temperature.

Analysis of the electrical characteristics of p-i-n rectifiersshows that the high-level lifetime does not decrease belowlOmicrosec at carrier concentrations of 1018/cm3 in the case ofgermanium23 or 1017/cm3 in the case of silicon.24 Under theseconditions, the radiative lifetimes calculated from eqn. (6) are30 microsec for germanium and 5 000 microsec for silicon. Thus,in neither case is the lifetime limited by radiative recombination,although the margin is rather close in the case of germaniumrectifiers.

(4) RADIATIVE RECOMBINATION AT IMPERFECTIONSThis Section will summarize and discuss experimental evidence

regarding the probability for radiative capture of free electronsand holes at impurity centres. In this case, both theory andexperiment are considerably less reliable than was the case inthe preceding Section. Nevertheless, it is felt that some usefulgeneralizations may be made. Transitions between localizedimpurity levels will not be considered, although this is animportant process in phosphor crystals.

The rate of capture of free carriers at an impurity centre isgiven by

r = BNn (13)

where B is the capture probability, N the concentration of emptycentres, and n the appropriate free-carrier concentration. The

* The strong temperature dependence for the radiative recombination cross-sectioncomputed by van Roosbroeck and Shockley'O can be attributed to lack of reliabledata regarding the temperature dependence of a and /»<.

926 HALL: RECOMBINATION PROCESSES IN SEMICONDUCTORS

kinetics of recombination due to this mechanism are similar tothe case for multiphonon recombination and will be discussedin Section 6.

There is very little direct experimental evidence regardingthe magnitude of B for the radiative process. Values in therange 10~13 to 10~14cm3/sec are suggested by work involvingsulphide phosphors.25 In germanium, an upper limit of2 x 10~12cm3/sec has been measured for the total captureprobability, including non-radiative transitions of electrons ationized aluminium centres.26

Optical absorption data may be used to deduce values for B,using detailed balance arguments analogous to those of vanRoosbroeck and Shockley:

, ? N l / 2 , .2 r00

B = (£) _ - £ _ _ aW2 exp [ - (W - WdlkT]dW\TTJ (mejtfkT)3l2c2Jw{

= 1 -58 X 10V(— ^ ) I oW1 exp [- {W\meff T / Jfn

W^\kT\dW

• • (14)

Wt is the threshold energy for ionization, o is the absorptioncross-section for ionization of the impurity centre by light ofenergy W, and meff is the effective mass of the free carrierinvolved. For numerical evaluation, W is expressed in electron-volts and a in cm2.

The optical data are frequently measured at temperatures lowenough for the integral in these equations to be determined to agood approximation by the threshold value of the absorptioncross-section, ah Under these circumstances, we obtain

' 2 \ ' / 2 / m \3/2

£ ~ f - ) (—)TJ \meff/

LCOs

m (15)

Measurements have been made27*28'29 of the optical absorp--tion due to the ionization of holes from column III acceptorsin silicon. Recombination probabilities calculated from thesedata assuming meff\m = 0 - 6 and a,- independent of temperatureare presented in Table 3. Also included are probabilities deducedfrom data by Kaiser and Fan30 for the first and second ionizationof holes from copper in germanium (meff\m = 0 • 36).

A comparison may be made between these recombinationprobabilities deduced from optical data and those listed in thefifth column of Table 3, which are calculated from a formuladue to Sclar and Burstein:31

5/2

w.' 3 m2c2 Z2\meff)

o.o69 x ,0-»£(J!L)«Y?»)I/Vl(,Z2\m.ffJ \T J

. (16)

where Z is the charge state of the ionized centre. This equationshould be applicable to the hydrogen-like donors and acceptors.For most of these impurities, it gives values that are in reasonablygood agreement with those deduced from absorption data.

Similar agreement is found for the two copper levels in ger-manium. For non-hydrogenic impurity centres such as these,however, the agreement must be regarded as fortuitious.Nevertheless, eqn. (16) can serve a useful purpose in providinga yardstick with which to compare experimental values ofradiative capture probabilities for deep-level impurities. Com-parison with eqn. (10) suggests that radiative recombination atimpurity centres should be somewhat less probable than free

Table 3

RADIATIVE RECOMBINATION OF HOLES AT NEGATIVE ACCEPTORSAT 300°K

Recombinationcentre

B" in SiAl- in Si..Ga~ in SiIn~ in Si..Cu~ in GeCu-~inGe

Wi

0 0460 06700710150 0550-25

cm2

x 10-161365

0-7121-6

B (from ot)

cm^/secx 10-1"

2-82-72-51-61130

S(eqn. 16)

cn13/secXlO-H

3-85-65-9121422

Reference

27,2827,2827,2828,29

3030

carrier recombination in crystals where direct transitions arepermitted.

The capture of electrons or holes at neutral hydrogenicimpurities involves the radiation of a light quantum of energynearly equal to the energy gap of the crystal. The probabilityfor this process should be approximately equal to that forradiative recombination of free electrons and holes, to theextent that the carrier which is bound at the neutral centre canbe regarded as a 'nearly free' carrier. This viewpoint should beapplicable to impurities having ionization energies that are notlarge compared with kT. Thus, eqn. (10) or (11)) may be used toestimate the probability for radiative recombination at neutralhydrogenic impurities in semiconductors where capture occursby direct or indirect transitions respectively. Haynes hasmeasured radiation resulting from recombination of free carriersat neutral donors and acceptors in silicon.32 His observationsare consistent with the foregoing remarks within the ratherlarge experimental uncertainties which are involved.

In general, radiative recombination at impurities is lessprobable by factors of 104 to 106 than radiationless recombina-tion in typical semiconductors. Thus, it would appear unlikelythat an appreciable fraction of the recombination of free carriersat impurities can be caused to occur by radiative processes insuch materials. On the other hand, in large band-gap crystalssuch as phosphors, the difficulty of disposing of a large amountof energy by means of multiple phonon or other processes issufficiently great for radiative recombination to be competitive.

(5) MULTIPHONON RECOMBINATION IN THE HOSTCRYSTAL

In this process, an electron and a hole annihilate each other,with the simultaneous release of a sufficient number of latticephonons to account for the recombination energy. The kineticsare similar to the case of radiative recombination, being repre-sented by eqns. (4) and (5). Although estimates of the captureprobability for this process have not been made, there are goodtheoretical reasons for believing that it is very much smallerthan that of the radiative process. Up to the present time, thereare no experimental results which indicate that this kind ofradiationless recombination between a free electron and a holeis of importance.

(6) MULTIPHONON RECOMBINATION AT IMPURITYCENTRES

The mechanism which usually determines the recombinationrate in semiconductors is a non-radiative process at impuritycentres, the rate of capture being proportional to the productof the carrier concentration and the number of empty centres,as described by eqn. (13). Experimental values for the capture

HALL: RECOMBINATION PROCESSES IN SEMICONDUCTORS 927

probability, B, are usually in the range 10~6 to 10~9cm3/sec(corresponding to capture cross-sections of 10~13 to 10~16cm2),although much smaller values are sometimes observed. Thelatter correspond to centres which are often called 'traps'. Inmost cases, B is not strongly temperature dependent, except atvery low temperatures. In germanium and silicon, this is thedominant mechanism in the range from liquid air temperatureto at least 150°C, and for free carrier concentrations as high as1017/cm3, whether due to injection or to the presence of donoror acceptor impurities. Recombination having these charac-teristics will be called 'multiphonon recombination', since thereare good reasons to believe that the time-limiting step in thecapture process involves the capture or emission of one or morelattice phonons.

Impurities which make effective recombination centres usuallygive rise to energy levels that are near the middle of the forbiddenband. The capture of an electron or hole at such a centre releasesan amount of energy which is many times larger than that of alattice phonon. Since the probability for the simultaneousemission of a large number of phonons is small, calculationsbased upon a one-step capture process lead to values that aresignificantly smaller than those usually measured.33 On theother hand, Gummel and Lax34-35 have calculated values whichare comparable in magnitude to those found experimentally byassuming that the free carrier is first captured at shallow excitedstates of the impurity centre by one or more single-phonontransitions. Having thus become localized at the impuritycentre, the fate of the carrier is determined, and therefore thecapture probability is insensitive to the time required for furthertransitions to the ground state.

The time required for decay to the ground state can be animportant consideration under some circumstances. In orderto account for the behaviour of germanium and silicon rectifiers

at high injection levels, it is necessary to assume that the entirecapture process, including the final transition, takes place in atime less than 10~nsec. It is not easy to account for decaytimes as short as this by a multiphonon process. An alternativemechanism is that the energy is transferred to a nearby electronor hole by an Auger process. This idea is consistent with theobservation that crystals having high luminescence efficienciescontain small free-carrier concentrations, thereby suppressingthe Auger process sufficiently for radiative transitions to be ableto compete.

The kinetics of multiphonon recombination at impuritycentres has received considerable attention.2'3fr-41 We willreview briefly some of the characteristic features of this process.

The simplest case concerns the low-level behaviour of a semi-conductor where recombination takes place at a single set ofenergy levels. It is assumed that the equilibrium carrier concen-trations are controlled by donors and acceptors which play nopart in the recombination process. When the concentration, M,of the recombination centres is sufficiently small, the electronand hole lifetimes are equal and are given by

nr) + re(p + pr)n + p

. • (17)

In strongly n-type material, the lifetime has a 'plateau' value,Ty = (MBn)~

l, which is limited by capture of holes at centreswhich are full of electrons. A similar plateau lifetime, re, isfound in strongly p-type semiconductors, due to electron captureat empty centres. For intermediate concentrations of electronsand holes, the lifetime depends upon the free carrier concen-tration, generally showing a maximum for intrinsic material,as shown in Fig. 1. The quantity nr — n)lpr measures the posi-tion of the energy level of the recombination centre in the for-

oi -

0 0 1

3eV

10 10 10 10 10 10 10 I0T 10 10 10 10

Fig. 1.—Lifetime due to multiphonon recombination at a single set of energy levels.The dashed portions of the curves indicate regions where less than half of the recombination centres contain electrons.

T = 300° K. x« = lOOmicrosec. y = 1 microsec.

0-30-20 10 0 60

- 0 1- 0 - 2

1250002 500

5010

10 0 200004

928 HALL: RECOMBINATION PROCESSES IN SEMICONDUCTORS

bidden band. It is the free-electron concentration that wouldcause the Fermi level to coincide with this energy level.

Eqn. (17) is valid for M < nr + pr. For larger concentrationsof recombination centres, the electron and hole lifetimes differ.The steady-state lifetime for electrons, for example, is given by

^ Tf{n + Tlr) + pr) + Pr)M\{p\pr nlnr)

Two time-constants appear in the transient response of semi-conductors containing large concentrations of recombinationcentres.40-4t The larger of these reduces to eqn. (17) in thelimit M->0 .

Under high-level injection conditions, where the electronand hole concentrations are nearly equal and large comparedwith M, nr and pr, the carrier lifetime assumes a constant value

Tf (19)

Two interesting features of the steady-state lifetimes may bementioned. For an impurity concentration such that n = n/r} fre,the electron and hole lifetimes are both equal to the high-levellifetime, regardless of the injection level or the concentration ofrecombination centres. Furthermore, if n > nrTflre, theelectron lifetime is always equal to or greater than the holelifetime.

Real recombination centres generally give rise to more thanone level in the forbidden band of the semiconductor. Never-theless, the observed variation of lifetime with free carrier con-centration and temperature is similar to that of the one-levelmodel discussed above.42 This result is consistent with analysesof centres giving rise to more than one energy level.39'43 Severalreviews of the properties of recombination have recentlyappeared.44-46

Analyses of the response of photoconductors sometimes leadto conclusions regarding carrier lifetime which appear to be atvariance with some of the foregoing generalizations. Onereason for these differences is that it is customary to assume thatthe energy gap is sufficiently large compared with kT for thermalexcitation of either electrons or holes from the recombinationcentres to be negligible.47'48 In addition, attention is focusedupon cases where the free carrier concentration is small comparedwith the concentration of recombination centres, whereas theopposite situation is usually considered in dealing with semi-conductors. Lack of precise definition of the term 'lifetime' canlead to misinterpretation ,as discussed in Section 2.

(7) AUGER RECOMBINATION INVOLVING FREE CARRIERSCollisions between two free electrons and a hole, or vice versa,

may limit the lifetime in semiconductors having small energygaps, the recombination energy being carried off by the extraelectron or hole. The recombination rate for this process maybe written

or R = fin(n2p — n£pQ) + f3p(np2 — nop$) . . (20)

where the coefficient jSw(cm6sec~1) represents the probabilitywhen the extra carrier is an electron. Since it is necessary thatmomentum as well as energy be conserved in such an interaction,the three carriers which initiate the recombination process musthave kinetic energies which are significantly greater than thermal,unless the interaction also involves lattice phonons.

We can obtain an estimate of the magnitude of j8n by con-sidering the probability for the inverse process, impactionization. At thermal equilibrium, the rate at which carriersdisappear by Auger recombination is equal to the average over

the Boltzmann distribution of the rate at which pairs are createdby electrons that have the necessary thermal energy. Thus,

• r ^ d W . . . . ( 2 1 )

where P{W) is the probability per unit time that an electronhaving energy W will suffer an ionizing collision. P(W) hasbeen evaluated by Franz.49 It has the form

(- - lY • (22)

G is a dimensionless parameter somewhat smaller than unitywhich depends in a complicated way upon the band structureof the semiconductor. The exponent, v, is an integer which isdetermined by the symmetry in momentum space at the energythreshold, Wt. For the case analysed by Franz, v = 2, Wt is alsodependent upon the band structure; it is generally believed49'50-51

to be approximately ZWg\2. Evaluation of eqn. (21) gives

(23)

Assuming v = 2 and G = 0-1, we obtain

/kT\3l2

nfpn ~ 4-67 x 1015(—) exp ( - Wt\kT) sec"1 . (24)

It should be emphasized that the approximations made in thederivation of this expression could easily introduce errors ofone or two factors of ten. Nevertheless, it should be helpfulin providing a semi-quantitative estimate of the magnitude andtemperature dependence of the coefficient for Auger recom-bination. Calculations based upon estimates of the ionizationprobability by Wolff51 and Seitz52 give results that are in closeagreement with eqn. (24). A more detailed calculation for theprobability for Auger recombination in InSb has recently beenobtained by Beattie and Landsberg.53

Calculating the lifetime from eqns. (2) and (20) we find

T —1

(25)

If j8n = j8p, the lifetime has its maximum value at n = p = nfand decreases as the square of the majority-carrier concentrationin the extrinsic ranges. If these two coefficients are unequal, thelifetime will remain at approximately the intrinsic value ineither the n-type or the/>-type region as the carrier concentrationis increased until pjn = 2j8n/j8p, before entering the region ofquadratic decrease.

In the intrinsic range the lifetime is given by

1

~ 3 • 6 x 10-17( —1) exp (WJkT) sec (26)

We note that the intrinsic lifetime is strongly dependent upontemperature and energy gap. The temperature dependenceshould be more rapid than nf2, in most cases varying approxi-mately as nf3. This behaviour offers a convenient method fordetermining experimentally whether the lifetime is limited byAuger recombination or by other processes. Room-temperatureintrinsic lifetimes calculated from eqn. (26) are shown in Fig. 2.

For samples which are sufficiently extrinsic, eqn. (25) showsthat the lifetime decreases as the square of the majority-carrier

HALL: RECOMBINATION PROCESSES IN SEMICONDUCTORS 929

io-4

10"'L>\n

UJ

LIFE

TIN

5NS

IC

cc

= lO-'o

10""

-

1 / 1 1 1

/

/

1 10-1 0-2 0-3 0-4 0-5

THRESHOLD ENERGY. eV

Fig. 2.—Lifetime due to Auger recombination of free electrons andholes in an intrinsic semiconductor at 300° K.

concentration. The temperature dependence in this range isless rapid than in the intrinsic region. Nevertheless, the lifetimeshould still show a moderately rapid decrease with increasingtemperature to the extent that Wt exceeds the energy gap.

Lifetimes which are apparently limited by Auger recombinationhave been reported for a number of semiconductors. Measure-ments of InSb by Zitter et a/.54 indicate an intrinsic lifetimewhich is approximately 2 x 10~8sec at room temperature andwhich increases rapidly with decreasing temperature. Thebehaviour of the lifetime in Te in the intrinsic and extrinsicranges is likewise characteristic of Auger recombination.55

Moss56 has reported measurements of lifetime in extrinsiccrystals of PbS which show an inverse-square-law dependenceupon the majority-carrier concentration. These results are con-sistent with the behaviour indicated by Fig. 2, within the ratherlarge uncertainties which are involved.

At high injection levels, where H =/>>/*/, the lifetimebecomes

1= -1T. . . (27)

electrons in material having a concentration, N, of emptycentres is

nppnp) . (28)

jS ,̂ and f$np represent the capture probabilities (cm6/sec) forelectrons when the energy is carried off by another electron orhole respectively. A similar expression, involving coefficientsfipn and flpp> describes the capture of holes at filled centres.

The kinetics of recombination due to Auger recombinationat impurities can be worked out using procedures analogous tothose which have been applied to multiphonon recombina-tion.36- 37 The resulting expression for the net recombinationrate under steady-state conditions is given by

Since this lifetime decreases as the square of the carrier concen-tration, Auger recombination is likely to become the dominantprocess when large concentrations of carriers are injected intoa semiconductor. In the case of germanium at room tem-perature, eqns. (26) and (27) indicate that the lifetime in thepresence of 1018/cm3 injected holes and electrons should be ofthe order of 1 microsec, assuming W, = 3 J*y2. This is com-parable with the lifetime which is observed under these condi-tions, as noted at the end of Section 3, and indicates that Augerrecombination may be of importance in germanium under condi-tions of very-high-level injection, particularly at elevated tem-peratures. In semiconductors with smaller band gaps, thismechanism can be expected to be dominant at smaller injectionlevels.

(8) AUGER RECOMBINATION AT IMPURITIESThe energy released when an electron or hole is captured at

an impurity centre can be transferred to another electron orhole which happens to be in the neighbourhood. The capturerates for such an Auger process will be proportional to theproduct of the two carrier concentrations. The equationanalogous to eqn. (13) which describes the rate of capture of

pn

P +Pr"Pnn+PP,

- 1(29)

np'

M, nr and pr have the same significance as in Section 6. WhenM is small enough for charge storage to be neglected, the free-carrier lifetime under low-level conditions is given by eqn. (17),except that the plateau lifetimes, instead of being independentof the carrier concentration, are given by

'pit

. . . . (30)

Thus, the variation of lifetime with carrier concentration issimilar to that for multiphonon recombination, except that itdecreases indefinitely with increasing concentration of freeelectrons or holes.

In the simple situation where all the jS-values are equal, thelow-level lifetime becomes

+ nr + p + pr

Mfcn+p)2 • - (31)

The high-level lifetime is given by

1Mnfi

(32)

Values for j8 have been calculated by Pincherle,57 Bess58

and Nagae.59 The latter estimates a value of 2 x 10-27cm6/secfor finn or fipp for impurities giving rise to energy levels of approxi-mately ieV from the conduction or valence band respectively.j8wp and fipn should be of this same general magnitude in semi-conductors such as germanium and silicon, in which band-to-band transitions are indirect, but may be larger by a factor of104 in materials characterized by direct transitions.

These calculations of Auger recombination coefficientsshould be regarded as order-of-magnitude estimates, since theydepend rather critically upon the wave functions that are usedto describe the recombination centre. In general, they indicatethat the j8-values should vary approximately as the inversecube of the capture energy, and that they should not be stronglytemperature dependent.

The values for the high-level lifetimes in germanium andsilicon discussed at the end of Section 3 provide upper limitsfor the Auger recombination coefficients in these materials.Assuming a concentration of recombination centres ofM = 10n/cm3, eqn. (32) indicates j8 < 10-24cm6/sec, a valuewhich is consistent with the theoretical estimates. Values assmall as 10~32 to 10~34cm6/sec are indicated by trappingexperiments in silicon.60-61 The smallness of the latter quan-tities may be attributed to the effects of coulomb repulsion if thetrapping centres are suitably charged.62

The foregoing discussion indicates that Auger recombination

930 HALL: RECOMBINATION PROCESSES IN SEMICONDUCTORS

at impurity centres in semiconductors can be expected to benegligible in comparison with multiphonon recombination underalmost all circumstances. Exceptions to this statement mayresult from impurities which do not give rise to the shallowexcited states that seem to be necessary for multiphonon recom-bination. Since these shallow states are not to be expected inthe case of centres which have a repulsive coulomb field, it isreasonable to suppose that they should behave as traps (i.e.that the capture probability for one type of carrier should bevery much smaller than for the other). For such centres, Augercapture of one type of carrier should be observable undersuitable circumstances, while the other kind of carrier would becaptured by a multiphonon process.

Under certain circumstances, an Auger process having alarge capture probability can be expected to occur at impuritieswhich give rise to several charge states.63

(9) CONCLUSIONSTwo of the important recombination processes are intrinsic

properties of the semiconductor material. While radiativerecombination is well understood and has been unambiguouslyobserved in a number of semiconductors, the experimentalevidence regarding Auger recombination is quite meagre. Thelatter can be expected to become the dominant process in semi-conductors with small energy gaps when large carrier concen-trations are present. A characteristic feature which can be usedto identify it experimentally is a strong temperature dependence,even in the extrinsic range.

With the concentrations of recombination centres that arenormally found in most semiconductors, the lifetime is almostinvariably limited by multiphonon recombination. Radiativeand Auger processes involving impurity centres can sometimescontribute significantly to the recombination rate in large-band-gap materials. In particular, it appears that they can sometimesplay an important part as one of the intermediate steps inmultiphonon recombination.

The relative magnitudes of these processes at room tem-perature are compared in Figs. 3 and 4, for germanium and fora hypothetical semiconductor having the same energy gap andintrinsic carrier concentration, but in which band-to-bandtransitions are direct. A material which closely approximatesthe latter is GaSb. In these diagrams, (a) represents free carrierradiative recombination with B = 3-4 x 10~14 and3-0 x 10~ncm3/sec respectively; (b) represents the Augerprocess in the host crystal as calculated from eqns. (24) and (25)with Wt — 1-0eV. Multiphonon recombination at impurities,(c), is assumed to be the same in both materials and is patternedafter the behaviour which is typical of germanium; (d) repre-sents Auger recombination at impurities, as calculated fromeqns. (17) and (30). For this process, it is assumed that thereare 10n/cm3 recombination centres lying in the middle of theenergy gap and that all the §-values are equal to 10~27cm6/sec,except that pnp and #,„ are larger by a factor of 104 in the'direct' semiconductor. Fig. 4 indicates that radiative recom-bination is likely to be the dominant process in crystals whichpermit direct band-to-band transitions, provided that the concen-tration of recombination centres can be made as small as it isin germanium.

It is evident that there are several areas where our under-standing of recombination processes is unsatisfactory. This isparticularly true for semiconductors having very large concen-trations of free electrons or holes. Studies of recombination inlarge-band-gap semiconductors containing few free carriersshould shed light upon some of the mechanisms involved inmultiphonon recombination.

- - I 0 4

10" 10'* 10M K)'6 10"

MAJORITY CARRIER CONCENTRATION,cm"3

Fig. 3.—Comparison of recombination processes at room temperaturein germanium.

Estimated lifetimes as limited by (a) radiative and (6) Auger recombination in thehost crystal, and (c) multiphonon and (</) Auger (4) recombination at impurities.

10" 10" 10" 10" 10" 10"MAJORITY CARRIER CONCENTRATION, c m ~ 3

Fig. 4.—Comparison of recombination processes at room temperaturein a semiconductor similar to germanium but having direct band-to-band transitions.

Estimated lifetimes as limited by (a) radiative and (ft) Auger recombination in thehost crystal, and (c) multiphonon and (</) Auger recombination at impurities.

HALL: RECOMBINATION PROCESSES IN SEMICONDUCTORS 931

(10) ACKNOWLEDGMENTS

The author is indebted to H. Ehrenreich and E. O. Kane ofthis laboratory for numerous helpful discussions.

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