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Electron electron and electron hole interactions in small semiconductorcrystallites The size dependence of the lowest excited electronic stateL E BrusellLaboratories Murray Hill New Jersey 07974

(Received 26 October 1983; accepted 25 January 1984)We model, in an elementary way, the excited electronic states of semiconductor crystallitessufficiently small ( - 50A diam) that the electronic properties differ from those of bulk materials.In this limit the excited states and ionization processes assume a molecular-like character.However, diffraction of bonding electrons by the periodic lattice potential remains of paramountimportance in the crystallite electronic structure. Schrodinger's equation is solved at the samelevel of approximat ion as used in the analysis of bulk crystalline electron-hole states (Wannierexcitons). Kinetic energy is treated by the effective mass approximation, and the potential energyis due to high frequency dielectric solvation by atomic core electrons. An approximate formula isgiven for the lowest excited electronic state energy. This expression is dependent upon bulkelectronic properties, and contains no adjustable parameters. The optical/number for absorptionand emission is also considered. The same model is applied to the problem oftwo conduction bandelectrons in a small crystallite, in order to understand how the redox potential of excess electronsdepends upon crystallite size.

I. INTRODUCTIONThe band gap of a semiconductor is, by definition, the

energy necessary to create an electron e-) and hole h + , atrest with respect to the lattice and far enough apart so thattheir Coulomb attraction is negligible. If one carrier approaches the other, they may form a bound state (Wannierexciton) approximately described by a hydrogenic Hamiltonian

at an energy slightly below the band gap.1.2 Here mh m.) isthe effective mass of the hole {electron} and is the semiconductor dielectric constant. The effective masses of the twocharges are often only a small fraction of an electron mass.Small masses naturally imply that localization energies forthe hole and electron are large. The dielectric constant ininorganic semiconductors is typically in the range 5-12, implying that the Coulomb attraction is almost entirelyscreened. This combination of small masses and weak attraction causes the exciton wave function to extend over a largeregion. For example, the lowest IS exciton of CdS has adiameter of 6 A in the center of mass coordinate system.

In terms of a hydrogen atom analogy, the band gap isthe ionization limit of the hydrogenic electron-hole boundstates. Photon absorption at higher energies creates freeelectrons and holes with excess kinetic energies inside thesemiconductor.In this paper we consider crystallites sufficiently smallthat this bulk energy level scheme is not valid. We suggestthat the size of the IS exciton provides a natural, intrinsicmeasure of linear dimension at which crystallite size effectswill crea te a qualitatively different situation. As the crystallite approaches this size, the electron and hole interactionswith the crystallite surfaces will dominate the dynamics. Inthis molecular limit, the energy level scheme will dependupon the size and shape of the crystallite as well as upon the

nature of the material. There will be a series of excited, discrete bound states approaching an ionization limit corresponding to a positively charged crystallite and a free electron in vacuum. The crystal is essentially not large enoughfor the intrinsic band gap to form, i.e., to sustain noninteracting, Bloch-type plane wave electron and hole eigenstates atthe band edges.

We describe now elementary model calculations whichattempt to identify the principal electronic phenomenawhich should occur in these crystallites. We use experimental data from bulk crystalline materials to predict the majorchanges occurring as crystallite diameter decreases to 40-50A Our calculation is similar to a perturbation calculationwithout adjustable parameters. We neglect possible complications of the problem, such as size dependent structurallattice rearrangements, which may be important in somespecific cases. Our model treats the fate of bulk states inthe limit of small size, and we do not consider possible surface states. We work at the same level of approximation asinvolved in the analysis of the bulk bound excited states viathe Wannier Hamiltonian [Eq. 1)].

In an earlier paper labeled I, we considered the problemof the ionization limit itself as a function of crystallite size.3We modeled the elementary quantum mechanics of a crystallite containing one mobile charge. That paper discussesthe major assumptions of the model: a) the use of the effective mass in the kinetic energy, and b) the use of an interaction potential based upon high frequency dielectric solvation.

This work is motivated by recent experimental interestin the transformation between molecular and bulk properties in small aggregates. Small semiconductor crystallites approaching the sizes considered here are used as catalysts andphotosensitizers.4-7 In the case ofCdS crystallites, moderatechanges in electronic absorption and resonance Raman excitation spectra have been reported and interpreted in terms ofthe quantum size effects we model here.8Related phenomena, generally of smaller magnitude,

J. Ch9m, Phys. 80 (9), 1 ay 1984 0021-9606/84/094403-07$02.10 1984 American Institut9 of Physics 4403Downloaded 09 Sep 2010 to 129.252.70.146. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions


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4404 L. E. Brus: Interactions in small semiconductor crystallites

have been observed and understood since 1974 in thin semiconductor layers made by molecular beam epitaxy techniques. 9 We treat kinetic energy via the same effective massapproximation used in layer exciton theory. to We treat electrostatic effects exactly, as there are large dielectric discontinuities in our problem. In the layer exciton theory, an effective dielectric constant approximation is used as thediscontinuities are smaller.

II THE POTENTIAL ENERGYThe fact that the Coulomb interaction is screened in Eq.

1) has interesting consequences in small crystallites. Significant screening necessarily implies a significant dielectric solvation energy. We model small crystallites where the electron and hole kinetic energies are several tenths of anelectron volt. At these frequencies ( - 5 X 1013 Hz), dielectricsolvation principally results from polarization of atomiccore electrons. We assume that this polarization, in responseto a mobile charge, will not saturate, in contrast to, e.g., thestatic orientational polarization of water molecules near ionic solutes. It appears physically valid to construct a potentialenergy for Schrodinger s equation from a consideration ofthe classical electrostatics involved. The potential will involve polarization charge at the crystallite surfaces.A sphere of radius R and dielectric coefficient Ez is surrounded by a medium of coefficient EI Two charges of magnitude eexist at pos itionsSIandS2 inside the sphere. Following the procedure employed in I, the work V (SI ,S2)classically necessary to assemble this charge distribution is_ _ eZ _V(SI,S2) = = +P(SI)EzlS I -S21

+P(S2) +PM (SI,S2) 2)As in I,P is defined [ S ]2n e2P S)= an - -R 2R 3)

wherean = E - l)(n + 1)1[E2 En + n + 1 ] and E= EzIE I 4)

5)

where (J is the angle between SI and S2 and Pn is a Legendrepolynomial. If the two charges have opposite signs, the -sign holds; the + sign holds if the charges have equal signs.Note that in the limit of a localized state in a largesphere, i.e.,R>SI andSz, we have V-e zIE2 ISI - S21. This isthe screened Wannier result ofEq. 1). As described in I, wehave dropped R-independent terms from V that mathematically represent the infinite polarization energies that classically exist near point charges. fEz> EI thenP (S) is a positiveterm increasing in magnitUde as R decreases. t representsloss of dielectric solvation energy as the volume of high dielectric constant Ezmaterial becomes small. P (S) describes arelatively weak radial F electric field pulling the charge toS = 0, the point of highest dielectric stabilization.

PM(SI,S2) is a mutual polarization term that can bethought of as the interaction of one charge with the surfacepolarization charge created by the other charge.PM dependsupon the absolute position ofSI andSz as well as the relativeseparation lSI - S21.Ifwe limit ourselves to S wave functions for e - and h + ,

simplifications occur in the three polarization energy terms.The Hamiltonian matrix elements of all PM terms for n> 1will be zero. The n = 0 term of PM is E - l)e2/EzR and isindependent of the S wave function. f he two charges haveopposite signs as appropriate for an electron and hole, thenthis n = 0 term will cancel the two n = 0 terms ofP (SI) andP(Sz). The potential energy Eq. 2) reduces to

where the subscript 0 indicates that the sphere has a netcharge of zero. Vo is the potential energy appropriate in modelling crystallite excited electronic states. f he two chargeshave the same sign, then the appropriate potential V2 ise2 2(E - l)e2Vz = + +E2 1S1 - Szl EzR

s ~ n + ~ n ) e+ L a n I R 2n I 2 7)This is the potential for a small crystallite with two electronsin the conduction band, or two holes in the valence band.III THE LOWEST EXCITED STATE

Schrodinger s equation for the crystallite excited states,within the confines of our elementary model, can be written

-=-V; + V ~ + VO(S.,Sh)


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L. E. Brus: Interactions in small semiconductor crystal lites 4405

where m is the appropriate effective mass.We shall find that the simple uncorrelated solution1

and CdS. As the kinetic and electrostatic energies scale differently with R, in each material the approximation is validfor sufficiently small R.

In order to correctly describe the larger diameter sizesof each material, it is necessary to use a more general wavefunction. Physically, the Coulomb term will introduce correlation into the wave function, in that the electron and holewill attempt to reside near each other in order to maximizethe Coulomb attraction. Also, the polarization termpushes both e and h + towards the sphere center, whichis the point of maximum dielectric stabilization. These ef

fects occur at the cost of ncreased kinetic energy of localization.We perform a simple variational calculation incorpor

ating radial but not angular correlation between electronand hole. Consider the function


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4406 L. E. Brus: Interactions in small semiconductor crystallites

4.5

4.0

3.5

_ 3.01->.

2.5a::liJ

2.0

1.5

1.00.5

: ~ _ z n o~ _ C d S\ \ ~ _ G a S

~ ~ InSbO L -_ _ L__ ~ L i ~ ~ ~ I _ ~ l _ ~ ~ . ~ I ~ i _ __LI_ _

20 30 40 50 60 80 100 150 200 300 500DIAMETER A)

FIG. 1. Energy of he lowest excited electronic state as a function of crystallite diameter, as calculated via wave function 13). Short horizontal solidlines are the bulk band gap energies of he indicated materials. Dot-dashedline for InSb incorporates surface carrier charge density as described in thetest.ergy dominates, and the lowest excited state shifts blue of thebulk band gap.

Excited state energies as a function of R for ZnO, CdS,GaAs, and InSb appear in Fig. 1. The two larger gap materials have moderate Coulomb terms and relatively comparedwith GaAs and InSb) large effective masses. This combination keeps the excited state energy near the bulk bandgap fordiameters larger than - 60A. The small gap materials GaAsand InSb must reach remarkably large crystallite sizes, however, before the excited state approaches the bulk bandgap.This occurs because me is extremely small, and the Coulombattraction is strongly shielded.

At small R our model will fail in a number of ways. Theeffective mass approximation is no longer valid when theelectron i.e., lighter particle) kinetic energy becomes substantial. Inspection of band structure calculations , I2 forthese materials indicates that beyond -0.5 eV, the kineticenergy increases with K substantially less steep than K ;2m . n Fig. 2 we have indicated these regions with a dashedcurve. One could incorporate a more complex anisotropic ifnecessary) representation of E K).

At high kinetic energy there is a second potential errorin that the basis functions [Eq. 9)] go to zero at the crystallitesurface. We demonstrated in Sec. C of I that, for smallerdiameter crystallites in materials with small effectivemasses, the calculated surface carrier charge density is substantial ifone assumes the Ben Daniel and Duke 13 boundarycondition at a finite potential energy step. The present calculation overestimates the kinetic energy if the surface chargedensity is important. This should occur principally insmaller gap materials.n InSb the electron effective mass is extremely small.The Coulomb interaction is almost entirely screened, andthe equation 12) energy Eis essentially electron localization

energy. n this case the Ben Daniel and Duke boundary con-

W:I:ZV>

-2

-1

o

> +1

+2

+3

, ~ t - . - . iOd,r[cds f +e-+[cds]CONDUCTION- BAND

[CdS] -e- [Cds t

[CdS] _h+ [Cds r- VALENCEBAND

CdS] _ h + + [CdS]n

20 40 60 80 100 120DIAMETER AlFIG. 2. Calculated size dependence of the indicated redox processes.

dition gives a size dependent localization energy indicated bythe dot-dashed curve in Fig. 1. We suggest this curve betaken as a semiquantitative estimate. The validity of the BenDaniel and Duke boundary condition in this physical situation needs to be more carefully investigated. I tmay be necessary to go beyond the effective mass approximation. Theseambiguities should be less important in the larger gap materials.

IV OSCILL TOR STRENGTHS AND THE ABSORPTIONSPECTRUM

n bulk materials, if one assumes that the valence toconduction band optical matrix element is independent ofK,then the oscillator strength density is uniform throughout Kspace. The absorption coefficient is proportional to the density of states in K space and increases rapidly with energyabove the band gap. Almost all of the oscillator strengthexists above the gap in transitions creating free electrons andholes.The absorption strength of the bound electron-hole exciton can be considered to derive from a Fourier transform ofthe exciton wave function into a distribution of plane wave Kstates, for both electron and hole. 14 n heuristic terms, themore compact the exciton wave function, the higher the values of necessary IK I and the greater the exciton oscillatorstrength. The oscillator strength scales as the volume of Kspace necessary to form the exciton.Even though the exciton oscillator strength is small relative to the above gap region, it can be large in comparison tofree molecules. For example, the IS exciton in CdS, whenweakly bound at a lattice defect, has a radiative lifetime of

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L. E. Brus: Interactions in small semiconductor crystallites 4407

~ ns. IS Henry and Nassau have theoretically consideredthe oscillator strengths of arbitrary (nonhydrogenic) electron-hole bound wave functions J 7. ; h) in bulk CdS. 5 Following the earlier work of Rashba and Gurgenishvili,16 theyderived an expression for the f number in terms of ex, theintrinsic hydrogenic IS number for crystalline bulk CdS:

(14)Here (/>ex is the hydrogenic IS wave function and nM is thevolume of one CdS diatomic unit. lex = 0.00256 from thework of Thomas and Hopfield. 7 Wex / W is the ratio of emission frequencies of bulk CdS to the small crystallite.

This expression is applicable to the lowest excited stateof small crystallites, for those sizes where the effective massapproximation and the K independence of the transition dipole remain valid. As a general property, the integral in Eq.(14) is unity if both the electron and hole are in the samespatial wave function. As we have seen, this physically happens in small crystallites when carrier confinement dominates the electrostatic terms. Equation (14) predicts that fhas only a weak size dependence through w.

This result physically represents a cancellation of twodifferent effects. The absolute interbandfnumber density inK space is proportional to the number of CdS molecules, andthus to the crystallite volume. At any size this f number isdistributed over all the excited electron-hole quantum states.The fraction taken by the lowest bound sta te varies stronglywith size. If he crystallite diameter is halved, then the diameter of the lowest wave function is halved and the absolutevalues of IK Ineeded to synthesize the bound wave functionis doubled in both electron and hole space. f IK I doubles,then the volume K space devoted to the lowest exciton increases by a fac tor of 8. This cancels the factor of 8 decreasein absolute oscillator strength in the factor of 2 smaller crystallite.

The net result is that relative intensity shifts into thelowest bound state as crystallite size decreases. The excitonbecomes more prominent in the absorption spectrum. Inthose regions where the electron and hole wavelengths aresmaller than the crystallite dimension, the spectrum remainsrelatively unchanged. The absolute f number calculatedfrom Eq. (14) is f 0.3 if we take W = Wex ' This is the sameorder of magnitude as thef number of IS hydrogenic excitons d ~ 6 0 A) bound at lattice defects in bulk CdS.V. COMPARISON WITH EXPERIMENT

In the related problem of an exciton trapped in a thinsemiconductor layer, extensive spectroscopic work has beenperformed with small band gap materials. In this situation,the wave function is compressed in one dimension, yet remains extended in the other two. Quantitive agreement isgenerally obtained with simple models based upon the effective mass approximation. 10

However, almost no experimental work has been reported on well character ized crystallites of the sizes we consider here. We report a ~ 0 . 2 5 e V shift in the absorption edgeof in situ colloid CdS crystallites (zinc-blend structure) of

average diameter ~ 4 5 A.8 This shift is in approximateagreement with Fig. 1. More recently larger shifts have beenobserved in colloidal crystallites of smaller diameters; theseexperiments will be described in a forthcoming publication.Shifts in the lowest excited state can also be observed via theresonance Raman excitation spectra of LO crystallite phonons.8VI. ELECTRON AFFINITIES AND REDOX POTENTIALS

Consider the following charge transfer or ionizationprocesses involving crystalline CdS:[CdS]n=-e- [CdS]n-' (15)[CdS]; [CdS] , (16)

[ C d S ] ~ e [CdS]n+' (17)[ CdS] n+ h + [CdS] , (18)

[ C d S ] ~ h + [CdS]n-' (19)Here [CdS] n represents a crystallite of n CdS diatomicunits, represents the lowest excited state (effective band gapstate), and or - represents an excess hole or electron. Inlarge crystals when individual h + and are in thermalequilibrium with the lattice, one has the simple situation thatthe energies of e- and h + are those of the conduction andvalence band edge, respectively. In addition, the binding energies of Wannier excitons are negligible, and interelectronrepulsion is negligible as the e- are far apart. In this limit theenergy required for the first three processes is the same-it issimply ,1E between the conduction band edge and the ereference state in vacuum (conventionally called the electronaffinity E.A.). In a similar fashion, the energy for the last twoprocesses is the same-Eg E.A., the difference from thevalence band edge.In small crystallites, all five processes require differingamounts of energy. This occurs because a) there is a sizedependent interaction for two e-, or for a h + and e-, confined inside a crystallite, and b) there are size dependentenergies of localization and polarization for a single h ore . The size dependence of [CdS] n+ and [CdS] n- was considered in I. Section III considers [CdS] . We now consider[CdS ] =.A. Two conduction band electrons

Using the electrostatic potential V2 Schrodinger's equation becomes[ If (Vi V2] (/> (SI s2) = E /> (SI, S2) 20)2m,

whereas beforeV2 = e2 2 E - l e2E21S1 - S21 E R

co a S2n S2n e2+ I 2 (21)2R 2n IThe Coulomb term is now repUlsive, and the two electronswill try to minimize their overlap. There are two polarizationterms. The second one, the smaller of the two, is identical tothe polarization term in Vo. The first term is independent ofthe wave function, and is the classical charg ing energy of a

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4408 L. E. Brus: Interactions in small semiconductor crystallitesTABLE IV. Energy te rms in e V for the two e- problem [Eqs. 20) and 21)].{3 is the best fit parameter in Eq. 22) for CdS crystallites of the indicateddiameter.

Diameter(A)Parameter 120 100 80 60 40{3 -0 .1 - 0.1 -0.1 0 0Kinetic 0.11 0.16 0.24 0.42 0.61Coulomb 0.Q7 0.08 0.11 0.15 0.18Charging 0.20 0.24 0.30 0.40 0.59Polarization 0.03 0.03 0.04 0.05 0.06Total shift 0.41 0.51 0.69 1.02 1.45

dielectric sphere.There is an interesting comparison between [CdS];

and [CdS] n- for the charging term 2 - I e2/ 2 above. In I,we have seen that the equivalent term for [CdS] n= is onefourth not one-half ) this term in [CdS] n= This occurs, heuristically, because the stored polarization energy scales as 2,so that a doubling of average field leads to a factor of 4 increase in polarization energy.

We take correlated wave functionP SI,S2) = IPISdlPl S2) + 3 [1P2SdlPl S2)+ PI{SI)1P2S2)], 22)

which is symmetric in space coordinates. The two spins mustcouple into a singlet state, so that the overal wave functionwill be antisymmetric, and will satisfy the Pauli Principle forindistinguishable electrons. As a numerical example we consider CdS crystallites in water. Table IV shows an energybreakdown of he variational solution for Eq. 20). As beforethe reference energy is (twice) the conduction band edge inthe bulk crystal. The dominant term for diameters greaterthan 40 A is the classical charging term. Large shIfts arepredicted, in comparison with [CdS] n- and [CdSJ Notealso that all terms here are positive, i.e., act to destabilize thesmall charged sphere case with respect to the large spherecase. The principal limitation of this calculation is the possibility ofcarrier charge density on the crystallite surface. Thiseffect will lower the kinetic energy contribution. The smallnegative values of f in Table III indicate the two electronshave moved nearer the surface in order to minimize theiroverlap.The formulation described here could be alternativelyused to describe two holes in the conduction band.B. Redox potentials

In order to demonstrate the different size dependenciesof reactions ISH 19), we have performed numerical calculations for CdS crystallites in water. We subtract the calculated shifts for the initial and final states of each process. Theresults are plotted in Fig. 2.

Reactions 15) and 16) are cathodically shifted, as afunction of decreasing diameter, from the valence band. Reaction 15) has the larger ~ i t due to the large charging termin Eq. 21). Reaction 18) shows almost no shift, as the is

stabilized in the excited state by attraction to the hole. Notethat Henglein 18 has argued, from the point of view of surfacecapacitance, that negative charge buildup on small crystallites will give a large shift in electron redox potential.The conduction band reaction 18) shows an anodicshift, as discussed in I, due to the localization energy of thehole in the crystallite. The excited state conduction bandprocess [reaction 19)] shows a cathodic shift as the attraction for the is greater than the localization energy.VII. DISCUSSION AND SUMMARY

Our elementary calculations model an intermediatesize regime for small crystallites. The excited states and ionization potentials are a strong function of size; bulk properties have not been reached even though the crystallite contains thousands of individual molecular units.Nevertheless, the bulk band structure, through the effectivemasses, is ofcritical importance in determining the deviationfrom bulk properties. The model implies that size dependentstanding wave diffraction of valence electrons, occurring asan electron moves through the interior and senses the periodic lattice potential and the crystallite boundaries, is thedominant physical effect for these materials.These materials have strong chemical bonds betweenindividual molecular units, in contrast to dielectric crystalsof organics or rare gas atoms, for example. Strong bondingimplies large widths for the valence and conduction bands,and a strong tendency to delocalize electrons involved in thebonding, as embodied numerically in the small effectivemasses. In an Ar crystal, by way of contrast, excited electronic states would localize on individual Ar atoms (Frenkelexcitons). The lattice would distor t around the excited state,tending to trap the excited state, and leading to a slow hopping rate for the excitation.

In the excited state problem, electrostatic effects are lessimportant. The field originates on one charge and terminates on the other within the crystallite. The field mainly resides inside the crystallite, with the result that the Coulombattract ion is shielded by 2 with small polarization terms. Inthe two problem Sec. VI), the field originates on the twoand terminates at infinity on two positive charges. Thefield is strong outside the crystallite, and size dependent polarization terms are more significant.

Themodel predicts that small gap materials must reacha large crystallite size before optical absorption occurs at thebulk band gap. Large gap materials achieve the bulk bandgap at intermediate sizes -60 Adiam) through an approximate cancellation of quantum localization and electrostaticeffects. At the smaller diameters where the simple result [Eq.12)] predicts large blue shifts, it will be necessary to go be-yond the effective mass approximation and to considercharge density at the crystallite surface in order to obtainaccurate results.

I t is interesting to note that metallic crystallites appearto behave like bulk metals in the same size range we considerhere. ) .20 Kreibig concluded that normal metallic propertiesare present in a silver cluster of 400 atoms 22 Adiam). 19 Hesuggested that clusters would have to be substantiallysmaller to show quantum effects.

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L E Brus: Interactions in small semiconductor crystallites 4409

CKNOWLEDGMENTSWe thank our colleagues M.Schluter, M. Sturge, J Tul-ly, D. Aspnes, and especially D. Kleinman for discussionand useful suggestions. We also thank the referee for usefulcomments.

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61 Kuczynski and J K. Thomas, Chem. Phys. Lett. 88,445 (1982).7M A. Fox, B Lindig, and C. C. Chen, I Am. Chem. Soc 104, 5828 (1982).8R. Rossetti, S Nakahara, and L. E. Brus, I Chern. Phys. 79,1086 (1983).9R Dingle, W. Wiegmann, and C. H. Henry, Phys. Rev. Lett. 33, 827(1974).

C. Miller, D. A. Kleinman, W. T. Tsang, and A. C. Gossard, Phys. Rev.B 24, 1134 (1981), and references contained therein.

I I I R. Chelikowsky and M. L. Cohen, Phys. Rev. B 14,556 (1976).12A Zunger and A. I Freeman, Phys. Rev. B 17,4850 (1978).130. I Ben Daniel and C. B Duke, Phys. Rev. 152,683 (1966).14R I Elliot, Phys. Rev. 108, 1384 (1957).I'C. H. Henry and K. Nassau, Phys. Rev. B I, 1628 (1970).16E I. Rashbaand G. E. Gurgen ishviIi, Sov. Phys. Solid Sta te 4, 759 (1962).170. G. Thomas and I I Hopfield, Phys. Rev. 116, 573 (1959).18Z Alfassi, O. Bahnermann, and A. Henglein, I Phys. Chem. 86, 4656

(1982).I U. Kreibig, I Phys. 4, 999 (1974).20M A. Smithard, Solid State Comm. 13, 153 (1973).

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