Journalof Luminescence47 (1990) 113—127 113North-Holland

Augerionizationof semiconductorquantumdropsin aglassmatrix

D.I. Chepic,A1.L. Efros, A.I. Ekimov, M.G. Ivanov, V.A. Kharchenko,l.A. Kudriavtsevand T.V. YazevaA.F. loffe Physico-TechnicalInstitute, 194021Leningrad USSR

Received1 October1989Revised23 April 1990Accepted26 April 1990

A microscopic model of degradationof both the nonlinearoptical and luminescentpropertiesof semiconductor-dopedglassesis proposed.In theframeworkof themodel thedegradationis dueto theprocessof Augerionizationof microcrystalswhensomeelectron—holepairsareexcited. The theoreticaldependencesof Augerionization rateon th; microcrystalsizeandenergybandparametersof the glass/semiconductorheterostructuresare obtained.The luminescencedegradationkineticshave beenstudied in CdS microcrystaldopedglasseswith different averagesize at variouspumping light intensities.Theexperimentaldependencesof theionization rateon themicrocrystalsizeandintensityarefoundto bein good agreementwiththe theoreticalcalculations.It is shownthat an abrupt heterointerfaceleadsto a strong increaseof theAugerprocessrate.Oscillationsof theAuger ionization probability andtheir dependenceson themicrocrystalsizearepredicted.

1. Introduction

Optical propertiesof semiconductormicrocrystalswith quantum size-governedspectrahave beenreceiving considerableattention recently [1—5].It is connectedwith the possibility to changeopticalpropertiesin a controlledmanner[1—6],sincethevariationof the sizeof muchmicrocrystals(the so-called‘quantumdrops’) leadsto themodification of energyspectraof theintrinsic [5,6] andimpurity [7] states.

The importantpeculiarityof quantumdropsdispersedin a glassmatrix is a substantialnonlinearityoftheir optical properties[8,9]. However, degradationof the luminescentand nonlinearproperties,whichoccursunderillumination, limits the applicationof thesestructures.This effectwasshown[11,12] to bedue to ionization of microcrystalsby optical photon excitation,followed by the captureof the ejectedelectronby long-livedcentersin theglass[11,12].The purposeof this paperis to clarify the microscopicnatureof the microcrystalionization resultingin the degradationof their optical properties.

Considerthis phenomenonqualitatively. The energy of optical photonsis smaller than the energyofdirect ejection of the electrons from the microcrystal. Therefore, the main source of ionization ofmicrocrystalsmustbe Augerannihilationof nonequilibriumelectron—holepairs,which are createdin themicrocrystalas a result of absorptionof morethan two photons.The chargedmicrocrystalswhich appearasa result of such Auger-ionizationexhibita low quantumefficiency~ It leadsto the degradationof theoptical propertiesof thesesemiconductorstructures.The rateof the degradationprocessesis determined

* Theobservedenhancementof thenonradiativechannelcanbearesultof effectiveAuger-recombinationin chargedmicrocrystals

since threequasiparticles(oneelectronandtwo holes)areformedwhenonly oneelectron—holepair is photoexcited.Thechargestateof amicrocrystaldoesnotchangeif theenergyobtainedby theholeasa resultof this Augerrecombinationis insufficient toeject it from the microcrystal.This means that theband offset hasto be larger for the valenceband than for the conduction band.

0022-2313/90/$03.50© 1990 — Elsevier SciencePublishers B.V. (North-Holland)

114 DI. Chepicci aL /Augerionization ofsemiconductorquantumdrops

by the Auger ionization rateandthe probabilityof creationof two electron—holepairs in the microcrystalunderillumination.

The intenmpurityluminescencedata, which include thedegradationratesmeasuredin microcrystalsofdifferent sizeat variousphotoexcitationintensitiesandtheir characteristicionization times,are presentedin the secondsectionof the paper.

Sections3 and4 are devotedto the theoreticalconsiderationof thedependenceof the Auger ionizationrateon the size of niicrocrystalsandon the energybandparameters.In the lastsection the experimentaldataarecomparedto the theoreticalcalculationandthe resultsarediscussed.

2. Experimental results

In this paperwe investigatedthe kineticsof photoluminescencedegradationin CdS microcrystalsgrownin a silicate glass matrix by the technologydescribedin ref. [13]. The valuesof the averageradius ofmicrocrystals~ were measuredfor eachsample by the method of small angleX-ray scatteringin theapproximationof sphericalparticlesandwereaccordinglya= 13, 16, 22, 33 and 39 A. Photoluminescencemeasurementsweremadeat temperatureT= 77 K.

The absorptionspectraof the investigatedsamplesare shown in fig. 1. A shift of the band edgetoshorter wavelengthsand oscillationsin the mterbandabsorptionspectraare observedwith decreasingmicrocrystalsize which, as was shown earlier [14], is due to the size quantizationof electronsin theconductionband.The arrowsin fig. 1 show the spectralpositions of the krypton laserlines A = 406 andA = 476 im~.Theline A = 406 nm is usedfor photoluminescenceexcitationin all investigatedsamples.Theline A = 476 nm is used for examinationof the dependenceof the degradationrate on the energyofexcitationlight quantafor the samplewith ~= 39 A. Detectionof photoluminescencewascarriedout inthe rangeof band maximum(D—A2), which is due to the transitions from the deepacceptor(bindingenergy EA2 = 0.9 eV) to the donor (ED = 0.075eV) [7].

The degradationof photoluminescenceintensity ‘lum, measuredon the sample with a= 22 A atdifferent intensitiesof the excitinglight I, is shownin fig. 2. As is seenin fig. 2, the rateof luminescencedegradationand thefinal valueof the decreasedluminescenceintensitydependstrongly on the excitationintensity I.

1 10 I1um(reL u0.)

_ O5~

350 400 450 X(nm) 0 1000 2000

Fig. 1. Absorption spectraof CdS microcrystalswith different Fig. 2. Thedependenceof luminescenceintensity ‘lam on timeaveragesizes at T= 77 K. 1; a= 13 A, 2; 16 A, 3; 22 A, at different intensityof excitation: 1; 1, 2; 0.451, 3; 0.21. The

4; 39 A. inset shows the luminescencespectrumof the same samplewith ã=22A.

DI. Chepicci aL /Augerionization of semiconductorquantumdrops 115

To describethe experimentalresults,we considerthe simplestphenomenologicalmodel basedon thecompetitionof Auger-ionizationwith neutralizationprocessesin microcrystals.Neutralizationoccurs asaresult of the returnof the electroninto the microcrystalseitherthermallyor by tunneling.

Threetypesof excitedmicrocrystalsexist in glass(with a totalconcentrationof microcrystalsN) underillumination: microcrystalscontainingonenonequilibriumelectron—holepair (N1 is their concentration),two electron—holepairs (N2 is their concentration)* and ionized microcrystalswith concentrationN~.The equationswhich describethe kinetics of the filling of excitedmicrocrystalstateshavethe form:

dN0/dt= — W01N0+ N1/T~1),

dN1/dt= W01N0—(W12+1/To”))N, +N2/Td

2~+N±/T, (1)

dN2/dt= W12N1 — (1/TA +

dN±/dt=N2/TA —N~/T,

where I~= N — (N1 + N2 + N~) is the concentrationof nonexcitedmicrocrystals; W01, W12 are theprobabilities of photonabsorptionby a nonexcitedmicrocrystaland by microcrystalscontainingoneelectron—holepair, respectively.‘TA’ is the probability of Auger ionization in a microcrystalwith twoelectron—holepairs, T is the characteristictime of neutralization,and 7~1.2)are the lifetimes of excitedstateswith oneand two nonequilibriumelectron—holepairs,respectively.The luminescenceintensity ‘lurn

is proportionalto the concentrationof microcrystalswith one electron—holepair N1( t): ‘lum = constXN1(t). The microcrystalswith two electron—holepairs do not give a significant contribution to thelunmiescencebecauseat the excitation intensity used in our experimenttheir concentrationN2 ~ N1.Therefore, the processof the luminescencedegradationis representedby the slow part of the timedependenceN1(t)/N1(0).The fast part of the time dependenceN1(t) involves times of the order of

—. 10 ~ s anddoesnot revealthe degradationprocesses.The systemof linear equations(1) is solvedby the standardtechnique.Assumingan exponentialform

of the solution with characteristictimes ~y’ we obtain a 4th order equationfor y. At low excitationintensities I, when W01T~

1~‘~ 1, Wi2Td

2~<< 1, degradationis determinedby the smallest root of thisequationy~,whichcanbe easily found neglectingthe termsof order higherthan ~ The dependenceoftherelative intensityof luminescence~Ium = hium(t)/hium(0) obtainedfrom (1) hasthe form:

N1(t) T 11 1\ ____

~Ium = N,(0) = ~ + ~ exp — t — + —) + T~+ T’

(y~ (2)\ (2)— —1_I (1)\ k 12T0 j— T~ — ~ 0~T~ ~‘(~~)+ TA)

where ‘r~ is a typical time of ionization of the microcrystal. In the caseof fast Auger processes,when‘TA ‘~ ~ the rate of ionization is determinedonly by the rate of creationof microcrystalswith twoelectron—holepairs: ~ 1 = (WOITØ~

1))W12. As will be shownin the next sectionthe situationin which the

Auger annihilationrateis less than ~ is morerealistic. In that casetheionization rateis equal to theprobability of a microcrystalcontaining two electron—holepairs,multiplied by the Auger annihilationrate:

= (WolTd1~)(Wl

2Tj2~)/TA. (3)

At low intensity of pumping light I the probability of ionization is alwaysproportional to j2, sinceprobability W— I.

* Microcrystalscontainingmoreelectron—holepairswerenot consideredbecausetheir concentrationis muchsmallerthan N2 at low

excitation intensitiesI.

116 DI. ChepicetaL / Augerionization ofsemiconductorquantumdrops

‘ti(s)

3 7~•~

\ +

•\ S /•. ‘s • /

101 P 1mW) 10 (A)

100 101 10 20 30

Fig. 3. The dependenceof ionization time ; on absorption Fig. 4. The dependenceof ionization time T, on the micro-power P for niicrocrystals with different size. Black circles crystalsizeat fixed powerof absorptionlight P= I mW.show the experimentalpoints at excitation line A = 40 nrnandopencirclesat X = 476 nm. 1; a = 13 A, 2; 16 A, 3; 22 A,

4; 39 A.

In fig. 3 we show theexperimentaldependencesof ionization time ‘r1 on the absorbedpowerP obtained

by meansof eq. (2). P is connectedto the probabilityof light absorptionP = (W01N)hw,wherehw is theenergy of absorbedphotonsand N is the concentrationof the microcrystalsparticipatingin absorption.The curves (1—4) correspondto samplescontainingmicrocrystalswith different averagesize a. It is seenthat the dependencesof the ionization time T~on the absorbedpower P havethe form T~— P “ wherev = 1.80±0.3. Thecharacteristictimesof neutralization,T, obtainedfrom our experimentsare — 4.0 X 102

s, and within the experimentalerror of 25% they do not dependon the microcrystal size and on theintensityof excitinglight.

In fig. 4 we show the experimental dependenceof the ionization time on the average size ofmicrocrystalsat a fixed valueof absorbedpowerP = 1 mW. It is seenthat the ionization time increasesstronglywhen a growsand this dependencehas the form ‘r1 — (a)

45. The dependencesfound reflect theparametricbehaviorof the Augerannihilationtime sinceT~— TAP 2N2 (seeeq.(3)). Thediscussionof thatdependencewill be carriedout in section5 after calculatingthe sizedependenceof the Auger ionizationprobability i/TA.

3. Energyspectraandwave functions of carriersin semiconductormicrocrystalsdispersedin glass

The calculationof the Auger annihilationprobability is a difficult problemevenin the bulk semicon-ductor as it demandsthe exact calculationof the small overlapintegralsof the electron—holestates[15].Therefore,the wave functionsof the electron—holepairsof such quasizero-dimensionalstructuresshouldbe consideredin the many-bandapproximation.The seconddifficulty is that the electronejectionfrom themicrocrystal(ionization) demandsconsiderationof quantumstructureswith finite barrierheight. Calcula-tions taking into accountthe finite barrierheight andmanybands(which leadto a nonparabolicenergyspectrumof carriers) havenot beenmadebefore. In this paperwe do this in the frameworkof a Kanemodel [16,17].

D.I. Chepicci aL / Augerionization ofsemiconductorquantumdrops 117

The basis functionsof the Kane model werechosenin the form: u~= I S), the Bloch function of theconduction band; Um. ±1= ±( X) ±ii Y>)/V’~and Um

0 = Z>, the Bloch functions of the valenceband. Then the electron wave function has the form:

,n=1

~ ‘1’m(r)U_m, (4)m= —i

where ~I’~(r)are envelopefunctions.Substituting~pe in Schrodinger’sequationwe obtain the expressionsfor “em’ which constitute the mixing of the valence band into the electron states:

V - V

~ ~‘±1=±E+Eg/2P±~~ (5)

where E is the energy of the state measuredfrom the middle of the energy gap with width

= —ih a/az, ~= —ih(8/ax±i 8/3y)/V~,V <S~P~IZ)/m0 is theKane matrix element, and m0

is the free electron mass. The wave function ‘It, has to satisfy the following equation:

V~2p2q~+[(Eg/2)2_E2fr

5+ ~ (6)

Forsemiconductormicrocrystalshaving the form of a spherewith radiusa, Eg(r) is given by:

Es, r<a,r>a. (7)

Taking into accountthe sphericalsymmetryof the investigatedsystemwe will find the solution of eq. (6)in the form of spherical waves:

(8)

where ~l,m are the normalizedsphericalfunctions.From eqs.(7) and(8) we obtainthe following equationfor the radial wave function 11(r):

~[~r2~_ 1(l±1)1() h2IVI2k(Eg/2)111(r)0. (9)

The intrasphericalf~<(r) (r < a) and interspherical1,> (r) (r> a) solutions of that equationhavetosatisfy the boundary conditions

(lOa)

1 af,<(a) — 1 3f,>(a)

E+E/2 ar — E+EI/2 ar (lob)For the electron states bound in the microcrystalthe solutionsto eq. (9) satisfying the boundary conditionshave the form:

A,j1(K5r), r<a

f~(r) = A j1(sc5a) K (11)

~ i(Kgr), r>ai k ~ ~

wherej~is a spherical_Besselfunction, and K1 is connectedto a modified Bessel function K,~1/2 by therelation: K1(z)= = .~/1r/2zKl÷l/2(z) [18]. A, is the normalizationconstantand

~= h2IV2[(Eg/2)EI, K~= h2I~I2FE2_(E~/2)2]. (12)

118 DI. Chepicci aL / Augerionization ofsemiconductorquantumdrops

Substitutingsolution (11) in the boundarycondition (10) we obtain an equation for determining theelectronenergylevels:

a E+E5/2 a~ mE ji(Ksr)I~a E+E/2 8r ln[Ki(wgr)~~ (13)

For further calculationsit is necessaryto havethe wave function of the lowest size-quantizationlevel(1= 0). It hasthe following evidentform:

af~/a~~(e) = — i’10 af

0/ar ‘ aS,5 = E0±ES~/2~ (14)

af~/a,~wheref0(r) is given by expressions(ii) and(12), E0 is theenergyof the size-quantizedelectrongroundstatewhich is determinedfrom eq. (13). ThenormalizationconstantA0 is:

E0 + ES/2A~= ~ (,c5a)

2X(K~, K

5),E0 a

sin(2ic5a) sin2(sc~a)E+2E ~

X(sc5, icg)= 1— 2,c5a + ic5a E+2E (15)

Furthermore,to take into accountthe electronscatteringat the semiconductor/glassheterointerfacewewrite down an analytical form of a free electron wave function:

(~J,<(k5, 1, m; r), r <a,~Pi,m(k, r) = 5 (16)

~, 4i> ~k5, 1, m; r, r> a,

where

~j.<(k5, 1, m; r)

ji(ksr)Yim

k [U—m+2)(1—m+l) . /(l+m)(l+m—1) .

SS V 2(21+ 3)(21+ 1) Ji±i 1+1,m-1 + V 2(21+ 1)(21—1) Ji~i t-1,m-1

k /(l+m+l)(l—m+1) . — / (l+m)(1—m) .

5a5 V (21+ 3)(21+ 1) Ji+i 1+1,m V (21+ 1)(21—1) li-i 1-1,m

k [O+m+2)(l+m+1) . + /(l—m)(l—m+i) .

5a5 V 2(21+ 3)(21+ 1) Ji+i /+1,,n+1 V 2(21+l)(2/— 1) J/-1 /-1,m±1

(17)

Here k~= [E~ — (E/2)2 ]/h2 I V I 2 E~is the energy of a free electronandthe coefficientsa are

a55(E~)=—ihV/(E1+E~/2). (18)

The wave function ili,(k5, 1, m; r) is obtained by replacementin eq. (17) of k~ by k~= [E~ —

(E/2)2]/h2 I V I 2; of a~by a

5, and of all spherical Bessel functions j~(k5r) by the following expressions:

J~(k5r) =k~a2[L~+Jp(kgr) +L~_yp(kgr)I, (19)

DI. Chepicci aL /Augerionization of semiconductorquantumdrops 119

wherey,, are sphericalBesselfunctionsof the secondkind [18].

L~÷(p,kg) = [ jp(ksa)yj;(kga) — (ks/kgy)3(ksa)yp(kga)], (20)

&(p, k5) = [(k5/k5y) j~(k5a)j,(k5a) —j~(k5a)j~’(.k5a)},

and y = (Ef + E~/2)/(Ef + E/2). The normalizationcoefficient A[ is determinedin accordancewiththeusualnormalizationmethodfor the sphericallysymmetricwave functions:

f~Pi’~m(k~r)~P,’m’(k’,r)r2drdQ=~i’,6m’m6(k—k’). (21)

As a result A{ is

A12=! (1+E:/2E~) (22)1 ~ a4k~[zV~(l, kg) +~(i, kg)]

For the calculationof the rateof Augerprocessesit is necessaryto find the form of the hole wavefunction ,lJh(r). In the investigatedphenomenathe lowest optically activestatesof the hole play the mainrole. In thesestatesthe hole energy is mainly due to the heavyhole effectivemassmh andis much lessthan the height of the barrier at the microcrystalboundary.The hole wave function vanisheson theinterfaceif the barrierhasinfinite height: ‘I’ h(r = a) = 0. On theotherhandtheconditionmh >> m~(me isthe electron mass) permits us to neglect the nonparabolocityof the hole energyspectrumand to use athree-band Luttinger model [19] to describe the hole quantization. In this case we neglect the terms oforder (me/mb)<< 1 in the overlapintegralsof electron and hole wave functions.

In a spherical potential the hole states are characterizedby the total momentumJ and its projection M.In the framework of the three-band Luttinger model the wave functions of these states can be representedin the form [20]:

J+1 m=+1

~Ph~~PJ.M~ ~ R1(r) ~ Y1MmUm, (23)

l=J—1

where ~1,M,m are the components of the spherical vector VIM [21] and R,(r) are the radialwavefunctions.For each value of J, M therearetwo solutionswith differentparity. As canbe shownthe groundstateofthe hole in a quantum well has J = 1. Therefore, we write down the wave function of one of the even*optical-active and Auger-active states (J = 1, M = 0, R 1) which is necessary for further calculations:

o 00 y3/10 Y2,_1

SPJ=l,M=o(r) = Ro(r) — ~ R2(r). (24)

0

The zeros in the upperline correspondto the neglectof the conduction band contribution to the heavy

* The analysisof the selectionrules shows that theodd stateof a hole with J= 1 doesnot give the contributionto the Auger

processesandtheoptical transitionsin thedipole approximationareforbidden.

120 D.I. Chepicci aL / Augerionization ofsemiconductorquantumdrops

hole wave function under the condition me/rn h << 1. The radial wave functions R0, R2 satisfy theequations

+ -~(-~ + ~R~+2(~ — + + -~)R2= (6/h2)(Eh— U(r))Ro,

2(1 — — 4~)R0_ (~ — + -~_ — -.~)j~= (6/h

2)(Eh— U(r))R2,

(25)

wherem1 is the light hole effectivemass,U(r) is the sphericallysymmetricpotential actingon the hole.Inthe simplestcaseof “strong” sizequantizationweneglectthe Coulombinterparticleinteractionandobtainfor radial wavefunctions

R0= C0a3”2[jo(4

2r/a) JO(~2)]’ 26)

R2= —~C0a3~j

2(~2r/2),

where ~2 5.76 is the first root of the Bessel function j2(x), I C0 I 2 _ 48.5 is determinedfrom thenormalizationcondition. In this approximationthe energyof the hole quantumsize level is Eh =

2mha2

4. TheAuger ionizationrateof quantumdrops

Auger disintegration is a many-electronprocessin which the energy of the electron—hole pairannihilationis transferredto oneelectron.In thecasewhenthis energyis larger than the barrierheight theelectron ejects into the glass and ionization of the microcrystal takes place. We will consider theprobabilityof Augerdisintegrationwhenbothelectronsandbothholesare situatedon the lowestquantumsize level. If weneglectthe Coulombinteractionthe initial antisymmetricwave function of the two-elec-tron statemaybe written in the form:

K~P’I = lp(e)(r),p(e)(r) a(l)/3(2)—a(2)/3(l) (27)

where ~ is the wave function of the electronground state(seeeq.(14)), a and /3 are the componentsofthe spin wave function with projection ±~. As a result of the Auger processone of the electronsannihilateswith thehole and the otheroneejectsinto the continuousspectrum.The antisymmetricwavefunctionof suchfinal statehas the form:

\ — ~PJ=l,M(rl)%P/,m(k, r2) + sPJ=l,M(r2)’~P/,m(k,r1) a(l)/3(2) — a(2)/3(i) (28)

k,I,m/ —

Herewetakeinto accountthat the Coulomb interactionleadingto the Augerdisintegrationcannotchangethe spin stateof the system.Sincethe initial andfinal wave functions are known we calculatethe Augerionization rateof microcrystals‘TA to be

~ J~Iv(r1, r2)I,m)~2~(Ei—Et)dk, (29)

A 1,m,M

wherethe sumis over the quantumnumbersof sphericallysymmetricfinal statesof the electron, 1, m, k,and the hole M [22],v(r

1, r2) = e2/sc I r

1 — r2 I is the Coulombrepulsivepotentialfor the electrons,sc is thedielectric coefficient. The angle integration in eq. (29) gives the nonzero matrix elements only for final

D.I. Chepicci aL / Augerionization ofsemiconductorquantumdrops 121

states with 1= 1 and m = M. Such “selectionrules” for transitionsare due to the conservationof totalmoment. Weobtain

= ~ K~I v(rl, r2)I~k 11mM-O> ( a~k) ~-1 (30)

whereE(k5) is the dispersionlaw for a free electronin glass,andkg is the quasi-momentumof the ejectedelectron. The matrix element in eq. (29) is independentof M becauseof the spherical symmetry.Therefore,we calculateonly the matrix elementfor the statewith M = 0, and the factor 2 in eq.(30) takesinto accountthe probabilityof annihilationof the electronwith two holesof the initial state.

The calculation of the matrix element in eq. (30) is carried out by the standardtechniqueof themultipoleexpansionof theCoulombpotential.Introducingdimensionlessvariablesof integrationx’ = r1/aand x = r2/a we obtain theAuger ionization rateby thecumbersomebut trivial calculation:

1 E 2 11 81C0I

2 2 — 2E0 +2E (K5a)

4 e2 2E~ 2

TA 27 X (~,Kg) ~ k5)+~(l, k5) (k5a)

3 hIVIK TIMAI , (31)wherematrix elementMA is the sumof two integrals

_______ ~ \2 ~‘

MA=J<(k$, K~)+ K ~sc~ J>~k5, K5). (32)

0~ g I

In the first one, J.<, the integration is carried out over the volume of the microcrystal:

J<(k5, ic5)=ff dx ~X>

x (ji(k5ax’)fo(K5ax’) — (k5sc5/3)a5(E1)a5(E0)[j0(k5ax’)

(33)

where x> = max{x, x’}, x< = min(x, x’}. In the other one it is carried out over the volume of the glassmatrix surrounding of the microcrystal:

J>(kg, Kg) ~“j dx{Ko(Kgax)[/.~+(1, kg)Ji(kgax) +L~_(i, k5)y1(k5ax)]

_(kgK~/3)a’(Ef)ag(Eo)Ki(Kgax)[2z~+(2, kg)j2(kgax)+~1+(0, k5)j0(k5ax)

+~_(2,kg)y2(kgax)+2z.1_(0,kg)yo(kgax)]}, (34)

= 101 dx’x’~/h(x’)jl(K5ax’), (35)

wherefh(x)=j0(~2x)—j2(~2x)—Jo(~2).In expressions(33) and (34) the rangeof integrationis dividedinto two regions.

The expressionfor the probabilityof the Auger ionization (31) hasa rathercomplicatedstructure.Torevealthephysicalnatureof the Augerdisintegrationprocesseswe considerfirst the casewhenthe ejectedelectronshave largefinal momentakga>> 1, as is the casefor t~sE= I E — E I <<E. At this condition

<<E, where ~ = h2/2m~a2is a typical energy scalefor the size-quantizedelectron, kg k

5 sinceL~E/Eg~ 1 and eq. (20) reducesto ~_(l, kg) 0, ~ kg) l/(kga)

2. The energy of an electronejectedfrom the microcrystalis E

1 3/2E. The abovechoice of heterostructureparametersleads to

122 DI. Chepicci aL / Augerionization ofsemiconductorquantumdrops

a~(E0)/a2 a~(E

0)/a2 h2 IV I 2/(aE)2

0/E <<1 and X—1. Substituting all these values intoexpression(31) for the Augerdisintegrationprobability weobtain

1 — 64v~ C 2 5 e2 2 ~q 1~2E~M 2 36

i~ 27 I ol ‘T~ hIVIK ~ ~I Al ‘ ( )where = h2rr2/2mea2 is the energy of an electron ground state in a well with infinitely high walls.Underthe condition k~<<a theintegrandin MA canbe written as the product of a rapidly oscillatingfunction and a smooth function. The integrals then can be expandedas a power series in the smallparameter(akg,s)1<< 1, the coefficientsof which are determinedby the specialpoints of the smoothfunction [23]. In our casethesepoints are x = 0 and x = 1 (the boundariesof a microcrystal).It can beshownthat the largesttermin thispower seriesis determinedby the regionof the specialpoint x = 1. Thecoefficient of (ak

5~y2and (ak

5,~Y3vanish becauseof the requirementsof continuity of the wave

functionand its derivativeat the microcrystalboundaries.The first nonzeroterm is (ak55) ~. The region

of theotherspecialpoint x = 0 contributesto the termsof higherorder in (akgs)~. This calculationgivesfor the matrix elementMA

sin(k5a)sin(K5a)MA=~(KSa) 4

K~a(k~a)2 2

(Ka) +(Ka)x [3(KSa+1)(Kga+1)+(Kga)2_(KSa)21 + g S (37)

Here,the first andseconditemsin bracescorrespondto the contributionfrom theconductionandvalenceband.It is seenthat thesecontributionsmay becomeof the sameorderof magnitudeat a largevalueof(ak55)>>1. If the energy of the size quantization Cq is much smaller than the depth of the well

= (E — E)/2 (Eq <<z.~)expression(36) is greatsimplified. Then sc5a = ‘rr(l — 1/K5a) and Kga (L~/

()‘/2 Averaging the fast changingvalue sin2(k

5a)in eq. (36) ((sin2(k

5a))= ~)and taking into account

that i~(ir) = 2.15 X 10-2 we obtaini 2

2E~ ~ 7/2

K~~)=1.8xbo3(hI~IK) -~(~) (~). (38)

It is seenthat theAuger ionization ratedecreasesas the sizeof microcrystalgrowsand follows the law(i/TA) — a~. Carrying out the numericalcomputationwe obtain for a semiconductorwith CdS energybandparameters,mentionedlater

K~)=2.9xbo71(~a~~)7. (39)We want to note that the asymptoticexpression(39) is valid only for microcrystalswith large enoughradiusandfor quantumstructuresin which k

5a kga>> 1.The other limit is of interestwhen the momentumof the electronejectedinto glass k5 is much smaller

than the typical momentumof the boundelectron — a1: k

5a<< 1. This is possibleunderthe followingcondition:

E~/2— 3E/2 <<

2Eq~ (40)

andcanbe realizedby a specialchoiceof the energybandparameters.In this case,as earlier, k5a>> 1. At

k5a << 1 thevaluesenteringin eq. (31)—(33) are E1 i.5E E/2 and

~÷(l, kg)= ~ 3[2j1(k5a)+k5a/y.j~(k5a)],

(akg) (41)

~.(i, k5) = ~[k5a/y .j1’(k5a) —j1(k5a)].

D.I. Chepicet aL / Augerionization ofsemiconductorquantumdrops 123

At kga ~ 1 the integrand in MA oscillatesfastonly within the inner rangeof the integration(x < 1). As aresult the main term in the parameter (k5a)’ <<1 yields the following expressionfor J<:

1— ~ (42)

The condition kga << 1 permits us to neglect the contribution of the valence band states to J>.Furthermore,the valueof J> is determined only in the neighborhood of the microcrystalboundaries,since

>> 1/a. As a result,usingeq. (41)we obtainfor J>.:

1 2fi(ksa)Ko(Kga)~(Ksa). (43)

(kga)

From eqs. (41) and (42) it follows:

1 cos(k a)

MA = — ~—jo(ksa) +j1(k5a) jo(ic5a)’~i(’rr) — (Kga)(ksa) (44)

In obtainingeq. (44) we use the relationship ~5a ~r(1 — l/Kga). Substituting eq. (44) into eq. (31) andtaking into account that X 1 we find

1 e2 2 ~ 3ES

_~~ICol2n2(.IT)(hIvIK)(~) -t

xcos2(k5a) (kga)

3 2 (45)(sin(k

5a) + 2 cos(k5a)/(3k5a))2+ ((kga) sin(k

5a)) /9

It is seenfrom eq. (45) that the probability of Auger disintegration is a strongly oscillating function. Theproportionalityof i/’TA to cos

2(k5a)is connectedwith quantuminterferenceof electronstatestaking part

in Auger disintegration. As in the case of k5a>> 1 and kga>> 1, it resultsin the vanishingof the matrixelementMA at some values of k5a.Of more interest is the increase of the Auger disintegration rate whichis due to resonance effects and which is described by the last factor in eq. (45). The value of this factorchanges, over a wide range, from (kga)

3 to (kga) ~. Such a resonance increase of i/TA occurs whensin(k

5a)+ 4 cos(k~a )/ (k~a)= 0*, correspondingto the merging of the electron quantum-size level with1= 1 into thecontinuousspectrum.As is seenfrom eq.(45) thewidth of suchresonanceis F — (kga)

3.Thedependenceof F on the freeelectronmomentumk

5 is typical for resonancescatteringof p-waves[22].Averaging the oscillations of the Auger disintegrationprobabilitygives a smoothdependence(i/’TA) on

the radiusof a microcrystala:

I 1 \ 16~’~1T 2 2 e2 2 Eq \5/2 E

~ ICoI 11(’TT) hIVIK ~‘) -h--, (46)

whichbehavesas <i/TA) — a ~. In that case the rate of Augerdisintegrationasa result of the resonanceeffect is substantiallylarger than it is in the case of large momentum of the ejectedelectron(k

5a>> 1).The band parameters of the CdS microcrystalsinvestigatedfall into theintermediaterangebetweentwo

limiting cases.Therefore,describingthe experimentalresultsit is necessaryto usethe full expressionforthe probability of Auger disintegration (31). In fig. 5 we show the dependence of the Auger ionizationprobability on the microcrystal radius. The computation was carried out for the CdS microcrystal

* This equationcoincideswith dispersionrelationship(13) under the conditions: k5a~ 1, E = 3E.

124 DI. Chepicci aL / Augerionization ofsemiconductorquantumdrops

—t —1

~A )s10~

10 20 3040

Fig. 5. The theoreticaldependenceof theAugerionizationprobability of CdSmicrocrystalson theirradiusa. The crossesshowtheexperimentaldata.

parameters:E = 2.58 eV, me= E/2 I V I 2 = 0.2m0 and e2/h V K = 0.15. The value of an effective

energygap of glass E = 7.02 eV was obtainedfrom direct experimentalphotoionizationdataon CdSmicrocrystals dispersed in glass [12].

As canbeseenfrom fig. 5 the probabilityof Augerdisintegrationhasa strongoscillatorycharacter.Thecomputedmaximaof i/TA correspondto microcrystalsfor which the 1= 1 quantum-sizelevelseithercrossthe boundaryof the continuousspectrumor are closeto it. This situation is in qualitative agreementwiththe effect of the resonanceAuger disintegrationconsideredanalytically in the limit kga << 1. Minimalvaluesof i/TA are due to the vanishingof the matrix elementMA, caused by the quantum mechanicalinterferenceof the electronwave functionstalking part in the Augerprocess.

In real structuresthe oscillations i/TA are averagedout. This is causedby numerousreasons,such asdisordering of glass matrix, dispersionof the microcrystal sizes and shapes,and fluctuations of thesemiconductor/glassbandoffset. We took into accounttheseeffectswhen wecarriedout the averagingofeq. (31) to obtain eqs. (38) and (46) at large values of the ejected electron momentum (k

5a>> 1).Nevertheless,the possibility existsof observingquantumoscillationsin i/TA when the momentaof theelectrons k5,5 taking part in the Auger processare comparableto the inverse size of a microcrystal(k55a — 1).

5. Discussionof the results

In this part we comparetheexperimentaldependencesof ionization time T~on the absorbed power P(see fig. 3) and averagemicrocrystalradiusa (seefig. 4) with our theoreticalcalculationof the probabilityof theAugerdisintegration‘TA’S So long as ‘TA is independentof P, ~ hasto beproportionalto P” withv = 2, as follows from section2. Sucha degradationprocessis evidencefor a two-photonmechanism.Inour model ‘r’ is proportionalto the steadystatevalueof the fraction of the microcrystalscontainingtwononequilibrium electron—holepairs (W01TJ’~)(W12TJ

2~).Though this fraction is small at low excitationintensity, Auger ionization leading to degradationis possibleonly in thesemicrocrystals.It is clear that

D.I. ChepicetaL /Augerionization ofsemiconductorquantumdrops 125

this kind of dependenceis notconnectedwith the size of microcrystals.As wasmentionedin section2, theexperimental dependence yields v = i.8 ±0.3. Thisvalueagreeswith thepredictionof the Augerionizationmodel (v = 2) within the experimental accuracy. The observed systematic deviation to a smaller value of v

may be connectedto the method of measurement of the luminescencedegradation,which wasobtainedfrom the time dependentquantumefficiency of the interimpurity luminescence.It is well known that thebands due to donor—acceptorpair radiative recombinationin bulk semiconductorshift to shorterwavelengthswith increasingexcitation intensity. The sameeffect was observedin semiconductormicro-crystals [7]. Here, however, the dependenceof the shift of the luminescenceband maxima on I iscomplicated by the fact that thereexistsa dispersionin the sizeof microcrystalscontainedin the sample.Therefore, the observed long wavelengthedgeof the interimpurity bandis dueto thelargermicrocrystals,whereasthe short wavelengthedgeis formedby the smaller microcrystals.With increasingthe excitationlevel at fixed frequency of the emitted light we detect luminescence from microcrystals of largerandlargersize a. The probability of Auger ionization decreasesas a grows, leadingto somedecreasein v.

The experimentaldependenceof the ionization time i~ on the averagemicrocrystal radius a was

approximatedas T~— (a)45 at fixed P (seefig. 2). This dependenceis weakerthan the theoreticalone

— ‘TA — a5 (seeeq. (46)). It is necessaryto note that thereal parametersof our structuresdo notpermitthe use of the asymptotic expression (46). This meansthat dependenceof ‘r

1 on a cannot be approximatedby the power function and has a more complicatedform (see fig. 5). It is seenin fig. 4 that theexperimentaldependenceof ~(a)is actually not a power of a. Therefore,it is necessaryto directlycompare the experimental data to the results of a numerical calculation of TA(a) from eq. (3i). Weobtainedthe experimentalvaluesof TA(a) from the experimentallyobservedi~,(a)by usingeq. (3). To dothis it is necessaryto estimatethesteadystatefractionof microcrystalscontainingtwo electron—holepairs.Estimating that as the square of the steadystatefraction of a singly excitedmicrocrystalW01T0,we obtainin our experiment that ‘TA = (W01T~

1~)2’T1i0~

0’r~(the value W01T0(l) — iO~ was estimated from the

absorbedpower).

~ 30 20 15 ö (Â)

h~(eV)

ao is

2.5 , 1/a’ (A’.104)

10 20 30 40 50

Fig. 6. The dependenceof maximum positionsin interbandabsorptionspectraon CdS microcrystalsize. Experimentaldataareshown by the horizontal segmentswhich lengths reflect the accuracyof the measurementsof the averagemicrocrystalradiusa.Dashedlines showthetheoreticaldependencesof transitionenergiesto thelowest electronquantum-sizelevel (is, ip, id, 2s)on a.The shading shows the overlap region of these transitions. The solid lines displays the dependencesof the lowest electron

quantum-sizelevelon a, calculatedin theframeworkof theKanemodel.

126 D.I. Chepicet aL / Augerionization ofsemiconductorquantumdrops

Theprobabilityof AugerdisintegrationTA I obtainedin sucha wayis shownin fig. 5. It is seenthat theexperimentaldataare in good agreementwith the theoretical dependence.However, a more detailedanalysisof the experimentaldata is difficult to make. There are many reasonsfor that, such as thedispersionof microcrystalsize, disorderingof the glassmatrix structure,andfluctuationsof relativebandoffset in the semiconductor/glassheterostructure.All of them lead to the smoothing of quantumoscillations.It seemsto us that oscillationsof ‘TA I might possiblybe observedexperimentallyin samplescontaining small size microcrystalswith low dispersion.

Let us discuss the applicability of the Kane model to the descriptionof energyandwave function ofsemiconductor/glassheterostructureelectronstates.This problem includes the question of boundaryconditions, taking into accountthe different characterof atomic (Bloch) functionsin semiconductorsandin glass.It is especiallyimportantsincethe abruptheterointerfacewhich is shownin section4 significantlyincreasesthe rate of Auger processes.Nevertheless,the Kane model gives quantitativeand qualitativeagreementwith experimentnot only for the timesof Auger ionization but for the energydependenceofquantum-size electron levels on the size of microcrystals.Thesedependencesobtainedfrom eq. (13) areshown by solid lines in fig. 6. It also gives the experimentalenergiesof interbandtransitionsto thequantum-sizeelectronlevels. The dashedlineshereshowthe theoreticaldependenceof the totalenergyofthe optical transitions on a, obtainedon the assumptionof “strong” sizequantizationof holes[2]. Theenergiesof thequantum-sizeholelevels are calculatedtaking into accountthe six-fold degeneracyof thevalenceband(seesection3). In thatcasethe selectionrulespermit the transitionfrom somequantum-sizehole levels and the dashedlines show the two strongesttransitions from them to every quantum-sizeelectron level: ip, id, 2s. The regions betweenthese lines are shadedsince the dispersionof themicrocrystalsize leadsto substantialbroadeningof every transition, resulting in their overlap. It is seenthat experimental and theoretical results are in good agreement.The experimentallyobserved andtheoreticallypredicteddeviationof behaviorof the optical transitionenergyfrom the 1/a2 law is mainlydueto the nonparabolicityof the conductionband.The calculationcarriedout shows that the finite depthof the quantumwell doesnotplay a significant role in this effect.

The authors are grateful to M.E. Raikh, A.E. Cherednichenkoand S.A. Fillippova for valuablediscussionandhelp.

References

[11A.I. Ekimov andA.A. Onushchenko,Soy. Phys.Semicond.16 (1982)775.[2] Al.L. Efros and A.L. Efros, Soy. Phys.Semicond.i6 (1982)772.[3] R. Rosetty,S. NakaharaandL.E. Brus, J. Chem. Phys.79 (1983) 1086.[4] D.S. Chemlaand D.A.B. Miller, J. Opt. Soc. Amer. B 2 (1985) 1i55.[5] Y. Kayanuma,Phys.Rev. B 38 (1989)9797.[6] A.I. Ekimov, A1.L. Efros and A.A. Onushchenko,Solid StateConunun.56 (1985)921.[7] Al. Ekimov, l.A. Kudriavtsev,M.G. Ivanov and Al.L. Efros, Soy. Phys.Solid State 31 (1989)1385.[8] G.L. Olbrigbt and N. Peyghambarian,Appi. Phys.Lett. 48 (1986) 1148.[9] U.V. Vandichev,VS. Dneprovskii,At. Ekimov, D.K. Okorokov, L.B. PopovaandAl.L. Efros,JETP Lett. 46 (1987)495.

[10] P. Roussignol,D. Ricard,J. LukasikandG.C. Flitzams,J. Opt. Soc. Amer. B 4 (1987) 5.[11] A.!. Ekimov, Al.L. Efros, Phys.Stat. Sol. (b) 150 (1988) 627.[12] V.!. Grabovskis,Y.Y. Dzenis, A.!. Ekimov, IA. Kudriavtsevand M.N. Tolstoii, Soy. Phys.Solid State 31 (1989)149.

[13]V.V. Golubkov,A.I. Ekiniov, A.A. Onushchenkoand V.A. Tsekhomskii, Fiz. Khim. Stekla 7 (1981) 397.[i4] A.!. Ekimov and A.A. Onushchenko, JETP Lett. 40 (i984) 1136.

1151 B.L. Gel’mont, V.A. KharchenkoandI.N. Yasievich, Soy. Phys. Solid State 29 (1987) 1355.[i6] E.O. Kane, J. Phys. Chem. Sol. 1 (1956) 249.[17] R.A. Suns,Fiz. Tekhn. Poluprov. 20 (1986) 2008.

D.I. Chepicci aL / Augerionization ofsemiconductorquantumdrops 127

[i8] M. Abramowizand l.A. Stegun, eds., Handbook of Mathematical Functions with Graphs and Mathematical Tables (Dover,New York, i964).

[19] J.M. Luttinger, Phys. Rev. 102 (1956) 1030.[20] B.L. Gel’mont and M.I. D’yakonov, Soy. Phys.Semicond. 5 (1972) 1905.[21] A.I. Akhiezerand V.B. Berestetskii, Quantum Electrodynamics(Interscience,New York, i965).[22] L.D. LandauandE.M. Lifchitz, QuantumMechanics,(Pergamon,Oxford, i974).[23] A.B. Migdal, Thequalitativemethodsof quantumtheory (Nauka,Moskya,1975).