the quantum mechanics of larger semiconductor clusters...

Annu. Rev. Phys. Chem. 1990. 41:477-96Copyright © 1990 by Annual Reviews Inc. All rights reserved

THE QUANTUM MECHANICSOF LARGER SEMICONDUCTORCLUSTERS ("QUANTUM DOTS")

Moungi G. Bawendi, Michael L. Steigerwald, and

Louis E. Brus

AT&T Bell Laboratories, Murray Hill, New Jersey 07974

KEY WORDS:solid state physics, quantum size effect, luminescence, excitedelectronic states, carrier dynamics.

INTRODUCTION

How can one understand the excited electronic states of a nanometer sizedsemiconductor crystallite, given that the crystallite structure is simply thatof an excised fragment of the bulk lattice?

This question is motivated by recent experiments on chemically syn-thesized "quantum crystallites," sometimes called "quantum dots," inwhich it is observed that the optical spectra are quite sensitive to size. Forexample, bulk crystalline CdSe is a semiconductor with an optical bandgap at 690 nm, and continuous optical absorption at shorter wavelengths.However, 3540/~ diameter CdSe crystallites containing some 1500 atomsexhibit a series of discrete excited states with a lowest excited state at 530nm (1-3). With increasing size, these states shift red and merge to formthe optical absorption of the bulk crystal. Electron microscopy and BraggX-ray scattering measurements show that these crystallites have the samestructure and unit cell as the bulk semiconductor. Such changes have nowbeen observed in the spectra of many different semiconductors.

This phenomenon is a "quantum size effect" related to the developmentof the band structure with increasing crystallite size (4). Smaller crystallitesbehave like large molecules (e.g. polycyclic aromatic hydrocarbons) their spectroscopic and photophysical properties. They are true "clusters"that do not exhibit bulk semiconductor electronic properties. In this review

4770066-426X/90/1101-0477502.00

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478 BAWENDI, STEIGERWALD & BRUS

we chart, as much as is presently known, the evolution from electrons indiscrete molecular orbitals in smaller crystallites, to electrons and holes incontinuous bands in the bulk crystal.

In the first section we discuss the development of band structure inpolymeric systems having strong chemical bonding. These ideas have heu-ristic application to three dimensional semiconductor crystallites. In thesecond section we treat the molecular orbital theory for the lowest severalexcited states of three dimensional semiconductor crystallites. In the sub-sequent section we improve upon this simple model by including electron-hole correlation. We then describe synthesis, structural characterization,and electronic spectroscopy of authentic nanoclustcrs. The first throughthird sections consider general questions of theory and modeling, andthe fourth section focuses on a major area of experimental work--thespectroscopy of CdSe crystallites.

BAND STRUCTURE IN POLYMERS OF VARYINGLENGTH

Strong and localized chemical bonding characterize crystalline semicon-ductors such as silicon and gallium arsenide. "Tight binding" or Huckel-type molecular orbital theories that neglect explicit electron-electron cor-relation provide an accurate picture of electronic structure in the bulkcrystals (5), and also provide a natural model for size evolution of bulkproperties (6). Semiconductors contain two electronic bands of interest: filled valence band and an empty conduction band. The following modelshows in general how individual bands develop discrete molecular orbitalsfor short polymerization lengths.

The Huckel description of a conjugated, linear polyene contains theessential physics of the electronic quantum size effect. The pi molecularorbitals (MOs) are modeled by a chain of N one-electron, one-orbitalatoms separated by a distance a. (A single atom on the chain correspondsto one unit cell in a semiconductor.) The orbital energy e isand the resonance integral beta is (~b,/H/(a,+ 1), where H is the electronicHamiltonian and q~, is the orbital on the nth atom. Neglecting orbitaloverlap, the one-electron eigenvalues for this system are given by

(m~]_)E(m,N) = e+2flcos m = 1,2,...,N

mT~= e+2flcos(kma) where km- (N+l)a" 2.


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QUANTUM DOTS 479

As N approaches infinity, k becomes a continuous variable running from0 to ~z/a (the Brillouin zone), and E(k) becomes continuous over the rangee + 2fl to e-2/~, as shown in Figure 1.

Figure 1 also shows the discrete eigenvalues for finite chains of 11 and13 atoms. These discrete eigenvalues fall on the same dispersion curve asthe infinite chain; a finite chain simply selects out those MOs of the infinitechain that have nodes at the ends of the chain. The lowest MO has noadditional nodes within the chain. As N increases the lowest level becomesmore stable, asymptotically approaching the lower band edge energye+ 2/L However, since this MO has nodes on the chain ends, its energy isshifted higher than the band edge: This is in essence the "quantum sizeeffect." Higher MOs have an increasing number of internal nodes. Statesnear e have a node on ever) other atom and are essentially nonbonding

2’8

POLYENE CHAINE = e + 2/~cos(Kja)

j~rKj= (N+l)a

oN=11oN=15

-2’80 Tr/ak WAVE VECTOR

Figure 1 Polyene chain band structure (adapted from Ref. 6). Discrete eigenvalues forN - 13 and N = 11 chains are also shown. (e = ~b)


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MOs. All these states are discrete, standing electron waves formed becausethe electron coherently senses both ends of the chain.

For kma << 1, Eq. 2 expands to give

E(km) = (e + 2fl)

where

L = (U+ 1)a.

2~k~a2 _ rc2

~ E°+(~7)2~ a2m2

The energy spectrum of a particle of mass m* in an (infinitely high)potential well of dimension L is

Ep(n) - 2 m*" 4.

Comparison of Eqs. 3 and 4 indicates that the MO eigenspectrum of theN atom chain, over a limited range of energy near the lower band edge, isthe same as that of a particle of mass m* = h~-/(2fla~) in a potential wellof length L.

This analogy is simple yet still instructive. The limited ka << 1 cor-responds to the electron wavelength being much larger than the inter-nuclear separation a. The line of atoms is essentially continuous, and theinternal potential felt by the electron is a constant. The effective massapproximation fails when the wavefunction develops more nodes in higherstates; now the electron samples the potential frequently enough to sensemodulation at the atomic scale. (Also note that, since beta is a measure ofthe strength of the covalent bonds in the chain, the effective mass is ameasure of bond energy in a specific band. In real materials m* is often asmall fraction of the real electron mass.)

MOLECULAR ORBITALS FOR sp 3 HYBRIDIZED

CRYSTALLITES

These ideas have been extended to three dimensional semiconductor crys-tallites, in a model originally outlined by Slater & Koster (7) and Coulsonand coworkers (8) for four-electron sp3-hybridized atoms (Figure 2). this model the basis orbitals are not atomic orbitals as above, but bondorbitals between nearest neighbor atoms: a set of bonding orbitals sigmaand a set of antibonding orbitals sigma star. As for the polyene, discussedabove, each of these localized bond orbital sets forms MOs extending overthe three dimensional crystallite as it grows. The highest energy orbital inthe sigma (i.e. valence) band is the crystallite HOMO, and the lowest


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QUANTUM DOTS 481

LOCALIZED ~Si ATOM ORBITAL CLUSTER ~ATOMIC DEGENERATE MOLECULAR DENSITY OF

ORB I TALS BAS IS ORBITALS STATES ~

Figure 2 Molecular orbital model for evolution of discrete MOs into continuous bands(adapted from Ref. 6).

energy orbital in the sigma star (i.e. conduction) band is the crystalliteLUMO. In the limit of the bulk crystalline solid, the HOMO-LUMOspacing becomes the band gap, and the bands become separately con-tinuous in energy.

In a II-VI or III-V zinc blende crystal such as GaAs, each Ga atom issurrounded tetrahedrally by four As atoms, and vice versa. With a minimalatomic basis of three P and one S atomic orbital (AO) on each cation andeach anion, there will be four occupied electronic bands (analogous tosigma above) and four unoccupied electronic bands (9). The bands havethe Bloch form

eie’~Z,,(F)

where %~ is a symmetrized combination of bond orbitals within one unitcell. The bands have a complex structure as a function of K inside theBrillouin zone and cannot be characterized by one scalar effective massvalid over the entire zone. However, for crystallites of diameter D, theHOMO and LUMO are principally formed from regions of IKI < ~/Daround the top of the highest occupied band and the bottom of lowest


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unoccupied band, respectively (10). If D is much larger than a unit celldimension, then the region of the Brillouin zone involved is relativelysmall, and effective mass approximations can be used.

The conduction band is nondegenerate (ignoring spin) and shows isotropic E(K) near the bottom at K = 0, and therefore the effective massme is a scalar. For a crystallite in the shape of a sphere, the discrete(particle-in-a-box) lowest few unoccupied MOs are characterized by radialN and angular L quantum numbers (11), in partial analogy with thehydrogen atom. For example, calculated MOs for a 45 A diameter ZnSecrystallite appear in Figure 3 (12). The lowest state is N = 1 and L = and is designated a "Is" state. Note that the allowed quantum numbers,and energetic ordering of states, are different from those of hydrogenbecause the radial potential is a square well and not Coulombic.

Discrete occupied MOs near the top of the valence band have a far morecomplex structure. Three of the four occupied valence bands of the bulkcrystal are degenerate at K = 0, due to the underlying three-fold degener-acy of p AOs. Away from K= 0 these bands split and can be describedby a 6 x 6 tensor "hole" Hamiltonian (13) that, in a spherical harmonic

-1

MOLECULAR ORBITAL DIAGRAM FORZnSe CLUSTERS 22,& RADIUS

ID

--tP -- -- tS LUMO

£9 ....... CONDUCTION EDGE

I’~: ~. V..,~__AL, ENCE EDGE -

SP’N~°RB’TI £~’~:,S: [ ~ Is HOMO,~. - Ip -~tD

Figure 3 Molecular orbital energy levels for a 45 ~. diameter ZnSe crystallile (from ReC12). The lowest two allowed transitions are indicated.


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QUANTUM DOTS 483

basis, can be written

~s is a diagonal matrix containing s-like hole momentum operators andthe spin-orbit energy./~ contains d-like operators and is nondiagonal. Ifg~ is ignored, there are two series of occupied discrete MOs offset by aspin-orbit energy A, as shown in Figure 3 (12). (In materials containingheavier elements, delta can be a significant fraction of an eV.) A singleisotropic hole mass describes the size dependence of these levels. Spin, L, N,and total angular momentum J(= L+ S) are individually good quantumnumbers. The HOMO-LUMO transition corresponds to the Is(electron)-Is(hole) excited state. In this review, we always designate the electron statefirst.

If the complete hole Hamiltonian is included, then only J remains agood quantum number. A recently reported calculation (14) includingmajor parts of gu shows a slight splitting of initially degenerate levelscontaining different Js, mixing of S and D wavefunctions, and mixing ofdifferent values of N. Two independent scalar "masses" are required todescribe all hole levels. The highest "s" hole state (the crystallite HOMO)behaves as a J = 3/2 particle in (cubic) zinc blende crystallites. Quantitativecalculations for uniaxial materials such as wurtzite have not been reported.

Mixing induced by Ha has consequences for optical selection rules. Withno mixing, where N and L are good quantum numbers, allowed transitionsexhibit AN = 0 and AL = 0. The lowest transitions are ls-ls and lp-lp,each split by the spin-orbit coupling. These transitions conserve the num-ber of nodes in the wavefunctions and are equivalent to K = 0 transitionsin the bulk crystal. When the full hole Ha~niltonian is used, only J is agood quantum number, and some forbidden transitions acquire oscillatorstrength. In ZnSe, for example, the low lying ls-2s transition becomesabout one third as strong as the ls-ls (14).

This pattern of MOs applies to spherical crystallites. The shape of thecrystallite obviously affects angular momentum degeneracies. In the limitof platelet shape, for example, the valence band MO pattern is differentand has a distinctive pattern similar to that of an infinite semiconductorsheet (a "single quantum well" in solid state physics).

The use of the effective mass model for MO energies is an approximationthat is correct only asymptotically in the limit of large crystallites. Thisoccurs because typically the bands are parabolic only for small regions ofthe Brillouin zone near the band extrema. The approximation over-estimates the shift, and can be improved by considering thc finite potentialstep at the crystallite surface (15, 16). Alternately, in PbS an improvedeffective-mass-like model appropriate for hyperbolic bands has beendeveloped (17). Quite recently, a version of the electronic "tight binding"


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484 BA’~rENDI, STEIGERWALD & BRUS

method has been applied to ZnO and CdS, with the result that the lselectron shift is only about half that estimated by the effective mass modelfor intermediate sizes (18). The independence of this important result surface bonding and model approximations needs to be examined; theeffective mass model is known to be a quite adequate approximation forzinc blende materials in superlattice physics.

ELECTRON-HOLE CORRELATION

The preceding section describes an independent particle MO model forcrystallite excited states, and therefore electron and hole motion are uncor-related. In bulk crystals, a simple physical idea accurately describes elec-tron-hole interaction for distances greater than a unit cell (ca. 5 ~): Theelectron and hole attract each other via (shielded) Coulomb forces. Thissuggests a model Hamiltonian for the lowest ls-ls excited state (19):

_h2 h2 e2/~ 2 2= ~Vc - ~Vh 6.

zrnc Zmh ~IL--~hI"

Equation 6 describes internally correlated, bound "excitons" in bulk crys-tals and in superlattices quite well. In crystallites, the Coulomb terminduces electron-hole correlation and is relatively more important at largediameters, because the kinetic energies vary as (1/R2) and the Coulombterm as (I/R) (19), where R is the crystallite radius. The Coulomb mixes higher lying states into the Is-Is state. In the limit of small R, mixingis minor due to large state splittings in zero order, and the wavefunctionapproaches ls-ls. This is the molecular limit.

Only in the limit of large R does an internally correlated, hydrogenicelectron-hole "exciton" form. The electronic polarizability becomes quitehigh as the eigenspectrum becomes dense. In the limit of continuousband structure near the band edges, such large crystallites could conductelectricity in an excited state with an electron and hole present in thepresence of an electric field.

In the small crystallite, Is-Is limit, the energy is (19, 20)

h2rc2 F 1 1 I 1"8e27.¯ E(R) -- Eg+ ~-[_~ + m,

GoR"

Here the Coulomb energy is simply evaluated in first order perturbationtheory. If the kinetic energies are a few tenths of an eV, then the opticaldielectric coefficient is used. We have neglected in Eq. 7 a smaller termdue to the dielectric discontinuity at the crystallite surface. This surfacepolarization term has the effect of making the Coulomb energy modestly


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QUANTUM DOTS 485

less shielded. The predicted ls-ls energies for several semiconductorsappears in Figure 4. Note that the approach to the bulk band gap is quiteslow in GaAs and InSb where the electron effective masses are very small,since the Ga-As and In-Sb bonds are strong.

Coulombic correlation can become significant at intermediate sizes. Ifmo << .mh, then for some range of R the Coulomb energy will be intermediatebetween the electron and hole kinetic energies. In an adiabatic approxi-mation, the hole will be attracted to the time average electron densitytoward the crystallite center (21a,b). The hole wavefunctions are approxi-mately those of a three dimensional harmonic oscillator, and the AN = 0selection rule is no longer obeyed. In real crystallites, however, this simplemodel is likely to be complicated by the previously mentioned tensorvalence band Hamiltonian, and a non-negligible hole mass.

If mo ~- mh then the correlation is principally angular, and not radial asin the previous paragraph. With correlation present, the total orbitalmomentum of the electron/hole pair remains a good quantum number,even though individual angular momenta mix. The lp-lp state has L = 0,1, and 2 components. Calculation (M. G. Bawendi, unpublished result)

5.0

4.5--

w 2.0

APPROACH OF CLUSTER LOWEST EXCITEDELECTRIC STATE TO THE BULK BAND GAP

Figure 4

\~ -- ZnOBAND GAP

1.5 ~ GaAs

0.5 ~ InSb-

O I I I I ~ I30 5o too 200300 50o

DtAWETER (~)Lowest allowed electronic transition energy versus size (from Ref. 19).


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shows that the Coulomb term mixes the L = 0 component (which, in fact,carries all the absorptive oscillator strength of lp-lp) into ls-ls. Thismixing is stronger than S type radial mixing. The hole tends to be on thesame side of the crystallite as the electron.

In CdS the hole mass is about four times larger than the electron mass.A simple variational calculation shows that the Coulomb term by itselfcauses the ls-ls state to have approximately 1% admixture of bothlp-lp and ls-2s (M. G. Bawendi, unpublished result), for a 50 ]k diametercrystallite. In order to calculate absorption oscillator strengths and ener-gies accurately, however, the ihole tensor Hamiltonian should be included.

In a bulk semiconductor, the diameter of the Coulombic "exciton" is(by definition) twice the Bohr radius b. This diameter is about 60/~ i nCdS. In crystallites, the HOMO-LUMO state transforms from ls-ls like,to exciton like, at R = 2-3 ab. Calculations on large, internally correlatedcrystallites have been done directly via the quantum Monte Carlo method(24) or via the variational method using a single particle basis or theHylleraas-type wavefunction (19, 25-27). [The transformation has alsobeen studied in rectangular shaped crystallites (28).] At this diameter, thelowest excited state occurs quite close to the band gap energy; however,the spectrum remains discrete. An accurate calculation, including allimportant phenomena, of the discrete yet dense excited state spectrumexpected at this size would be quite complex. Both electromagnetic (i.e.polariton) and quantum size effects are equally significant at this size (29,30).

Large crystallites with significant electron-hole correlation are fasci-nating. For example, the summed oscillator strength for all valence toconduction band discrete transitions will scale as the number of unit cells.The "Is-Is" oscillator strength per crystallite, however, is predicted tobe independent of size (excluding a wavelength factor) as long as thewavefunction shows no correlation (19). As correlation develops, the oscil-lator strength per crystallite increases, and the purely radiative lifetimedecreases from about a nanosecond to a few picoseconds ("giant oscillatorstrength" effect) (26, 31-33). The electron-hole correlation in such largecrystallites is due to a weak, long range shielded Coulomb attraction, andtherefore this large correlated state should be especially sensitive to thermal(i.e. vibration), inhomogeneous, and/or electromagnetic dephasing in realcrystallites (34).

Electron-hole correlation is also important if the crystallite contains twoholes and two electrons ("bie×citon"), such as would result from sequentialphoton absorption. The Pauli principle allows the electron Is orbital tobe doubly occupied; the hole J = 3/2 orbital could in principle containfour holes. Neglecting the Coulomb interaction and Franck-Condon


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QUANTUM DOTS 487

effects, this state occurs at twice the Is-Is energy. If the Coulomb inter-action is included, then in most cases it appears that the sequential photontransition creating the biexciton is redshifted from the initial transition toIs-Is (35, 36). These calculations neglect possible complications causedby hole MO degeneracy. Also, if the effective mass approximation over-estimates MO energies, then the mixing due to the Coulomb term will beeven greater than calculated.

The splitting between Is-Is and biexciton transitions is predicted tobecome larger if the medium outside the crystallite has a low dielectricconstant (36). This occurs because the Coulomb interaction is enhancedby fringing electric fields (i.e. surface polarization) (37). As correlationdevelops, this sequential transition also develops giant oscillator strength,similar to the initial transition.

The presence of close-lying exciton and biexciton transitions with pre-dicted giant oscillator strengths suggests that large third order nonlinearoptical effects will occur. In all theories, the linewidths of these transitionsare critical and uncertain parameters. The resonant nonlinear coefficientsare predicted to be quite large if the linewidths are small (31, 33, 36). Thequestion of a large nonresonant, nonlinear coefficient, at energies belowthe Is-Is transition, remains unclear at this time (38, 39).

EXPERIMENTAL ELECTRONIC SPECTRA

A 40 ~, diameter crystallite of CdSe has some 1500 atoms, about one thirdof which are on the surface and interact with adsorbed species. Structuraland compositional characterization is absolutely critical in understandingthe spectra of such real crystallites. Characterization of large clustersrequires the development of new physical and synthesis methodology. Itwould be a tremendous achievement to make a macroscopic sample ofcrystallites of this size, identical at the atomic scale. Molecular beamtechnology is central to the study of smaller, mass selected clusters, and isundergoing vigorous development at present. It has not yet been shown,however, that clusters of this large size can be made, characterized, and/oraccumulated in useful amounts by using beam technology.

Various forms of"arrested precipitation" are used to make crystallites,either at high temperature in hosts such as silicate glasses or NaC1, or atroom temperature in solution. In the solid hosts, the optical spectra oftenshow relatively sharp spectral features; this suggests that high qualitycrystallites have been made (40-49). Size can be varied through differentannealing procedures, but it is difficult physically to characterize and/orpurposefully modify the crystallites. Synthesis in solution creates a meta-stable colloid that can be modified by adsorption of dissolved species (50)..


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Yet in situ characterization of structure and/or surface species is difficult,and a broader distribution of sizes is often present. GaAs crystallites madeby molecular beam epitaxy on amorphous silica have been reported (51).Gas aggregation methods have also been used to make crystallites in thequantum size effect regime (52).

We synthesize crystallites that have organic moieties bound to thesurface. A key synthesis step occurs in an inverse micelle, in which, forexample, CdSe crystallites reside with surfactant molecules adsorbed ontheir otherwise reactive surfaces. As outlined in Figure 5, an organo-metallic reagent containing the Se-phenyl moiety displaces the surfactantand reacts with the Cd rich surface. When phenyl groups are chemicallybonded to the crystallite, it becomes hydrophobic. This surface deri-vatization prevents fusion of crystallites, and enables recovery of macro-scopic amounts of pure, size-selected, "capped" quantum crystallites as afree-flowing powder (1). Quantum crystallite powders stabilized with inorganic polymer adsorbed on the surface have also been reported (53).In a few cases, capped, layered particles, formed from two differentsemiconductors, have been grown in inverse micelle media (54). Suchcapped crystallites dissolve in organic liquids and can then be furthermodified and/or physically characterized. Certain yeasts are also knownto grow CdS particles of approximately 20 ~ capped with short peptides,also isolateable in pure form (55).

Capped CdSe crystallites in the 20 to 50 A range have been characterizedby powder X-ray scattering, transmission electron microscopy, elementalanalysis, 77Se NMR (56), IR, resonance Raman, fluorescence, and absorp-tion hole-burning spectroscopy. The internal crystallite structure is betterunderstood than the surface structure. Figure 6 shows powder X-raypatterns and theoretical fits, calculated numerically from the Debye equa-

~ PhSeTMS

@

~ = SURFACTANT MOLECULE

Figure 5 Schematic diagram of synthetic process that "caps" a CdSe crystallite with phenylgroups (Ref. 1).


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0

Figure 6

QUANTUM DOTS 489

(a)

INTERMEDIATE BETWEENZINC BLENDE

AND WUR]’ZITE

~]~ --EXPERIMENT

20 30 40 50 60

20

I-z::Do(J

(b)

NEAR WURTZITE

0 10 20 30 40 ,.50 60

2e

Bragg powder X-ray scattering spectra of CdSe crystallites (from Ref. 57).

tion for two different ca. 35 ~ CdSe syntheses (57). One reaction yieldsessentially wurtzite single crystallites, whereas the other yields particlesthat switch between wurtzite and zinc blende, apparently randomly, asgrowth occurs. Modeling shows that powder patterns measure averagelong range structure, and are relatively insensitive to deviations from theaverage. Such deviations can be either random (thermal motion and/ordefects) or systematic (position dependent strain and/or surface recon-struction). NMR and EXAFS measurements, however, do provide partialinformation about surface bonding and structure (P. R. Reynders and M.Marcus, unpublished results).


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Spectroscopic measurements are limited by the fact that real sampleshave a distribution of sizes and shapes, and most probably a distributionof surface compositions and structures for a given size and shape. Theseeffects create inhomogeneous spectroscopic broadening and complex aver-aging in photophysics. For example, a CdSe crystallite distribution, of40 /~ average diameter and standard deviation 10%, has a 1500 cm-t

inhomogeneous absorption width, as calculated from Eq. 7. There is apartial analogy here with low temperature protein spectroscopy, in whichnumerous conformational isomers are "frozen in" and create inhomo-geneous broadening.

The upper panel of Figure 7 shows the 10 K absorption spectrum of

ENERGY (eV)2.5 2.0

0.6 1I , I I I I i I

0.3

0.0

0.:3

0.0 --

450 550 650WAVELENGTH (nm)

b’igure 7 Hole-burning ~pectra of capped CdSe crystallites at low temperature (from Ref.(59a,b). Upper panel: optical absorption. Lower panel: change in optical density observed 10nsec after pumping with an intense, spectrally narrow laser.


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QUANTUM DOTS 491

capped CdSe clusters in polystyrene. The broad feature at 530 nm isassigned as the ls-ls excited state. Transient photophysical hole burningexperiments show that this width is largely inhomogeneous (59a,b). photoexcited crystallite luminesces on a nanosecond to microsecond timescale, and during this period a narrow hole can be observed in the remain-ing ground state absorption. This sample shows a 940 cm-1 inhomo-geneous width, and a ls-ls homogeneous lineshape exhibiting weakFranck-Condon coupling to an optical vibration. In this 205 cm- 1 normalmode, Cd ions and Se ions vibrate out of phase with each other along thecrystallite radius. This coupling of the excited state to this specific opticalvibration is also seen in CdSe resonance Raman (RR) spectra (60). [Opticvibration RR experiments on larger ZnTe crystallites have been recentlyreported (52).]

If the excited state potential for this mode is modeled as a displacedharmonic oscillator, then the normalized displacement is about four timessmaller in the crystallite than in the bulk (60). That is, the excited stateFranck-Condon coupling is far weaker in the quantum crystallite than inthe bulk. This result is unambiguously demonstrated in a series of cappedCdS cluster powders, in which a smooth variation with size occurs (61). was earlier predicted, from analysis of Frohlich vibronic coupling, that thecoupling to optic vibrations should decrease in small particles as electronhole overlap increases (62).

The homogeneous ls-ls lifetime in small CdS crystallites, derived fromthe RR fit, is 50 fs (61). CdSe crystallite ls-ls optic vibronic transitionsalso show a homogeneous width (59a,b) on the order of 100 em-1; thismay be "lifetime" broadening and/or coupling to low frequency motionsin the excited state. Bulk hydrogenic excitons, with no surface present,typically show far sharper optic vibronic spectra.

Luminescence excitation experiments can also reveai the homogeneousabsorption spectra of size selected crystallites. The 25 K absorption spectraof the wurtzite crystallites whose X-ray scattering spectra are displayed inFigure 5 appear in the top of Figure 8. Also shown is the luminescencespectrum observed for excitation in the ultraviolet, where homogeneouswidths are large and all crystallites are excited equally. The width ofthis structureless luminescence spectrum is almost entirely inhomogeneous(63). The excitation spectrum of a narrow band of luminescence at 530 is shown in the lower panel. This spectrum shows more structure than theabsorption spectrum. It preferentially records the luminescence of a fewof the smaller crystallites whose spectra are shifted to higher energy. Thels-ls peak observed is actually the (1,0) optic vibronic transition; the(2, 0) is visible as a shoulder.

At least three additional electronic states can be resolved in the excitation


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=32~ WURTZITE CdSe CRYSTALLITES

~25K

LUMINESCENCE

LUMINESCENCE EXCITATION /~

ls-Is(A) /ls-ls

~ ~s-lp /

EMISSION I

350 400 450 500 550WAVELENGTH (nrn)

Figure 8 Steady state fluorescence excitation spectrum of CdSe crystallites at low temperaturein an organic glass (Ref. 63). Upper panel: optical absorption and fluorescence spectra. Lowerpanel: excitation spectrum of 1 nm wide band of luminescence centered at 530 nm.

spectrum. The bump at ca. 500 nm may be the ls-lp transition, and theweak feature at 460 nm may be the ls-2s transition. The stronger andbroader feature at 440 nm, also seen directly in absorption, is probablythe higher spin orbit component of ls-ls. The broad continuum in theultraviolet corresponds to higher excited states including lp~lp.

In general, quantum crystallite luminescence is quite sensitive to surfacecomposition and structure. Figure 9 schematically illustrates the typesof processes that may occur. The figure compares the discrete MOs ofnanometer crystallites with the standard band diagram of a bulk semicon-ductor. The CdSe luminescence observed in Figure 8 appears to be the ls-ls state in fluorescence, although a complete understanding of vibrationaland electronic reorganization between absorption and fluorescence hasnot yet been worked out. In this state, the electron and hole are inside thecrystailite; recall that the ls wavefunction has a node on the surface (inthe limit of an infinite barrier).


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QUANTUM DOTS

SPATIAL ELECTRONIC STATE CORRELATION DIAGRAM

493

CLUSTERBULK SEMICONDUCTOR

CONDUCTION BAND

/////////%)---;/’l i SHALLOW TRAP .....

BULK ig

}"////////// .... ,,,

Figure 9crystalline states (adapted from Ref. 68). Energies are not to scale.

LUMO

ADSORBEDMOLECULE

~’--SURFACESTATE

VALENCE BAND~’ ~ HOMO

~" DISTANCE "-=" ~CLUSTER DIAMETER

Schematic correlation diagram relating bulk semiconductor states to quantum

Even before quantum size effects were recognized, it was discovered thata single atom of a strongly electropositive metal on the surface of anaqueous colloidal CdS particle quenches crystallite luminescence (64, 65).This atom appears to act as a localized surface state, trapping the lselectron. Adsorbed organic species that have redox potentials inside thecluster "band gap" also act as surface traps, and charge transfer to thesespecies from crystallites has been observed via transient optical absorptionand RR spectroscopies (66). A novel quantum theory of the reverseprocess, electron transfer from an excited adsorbed molecule into asemiconductor particle, has recently been proposed and compared withexperiment (67a,b).

In some cases, a low energy, broad luminescence is observed that isthought to occur from electrons and/or holes individually trapped inuncharacterized, naturally present surface states (50). The lifetime andtemperature dependence of this luminescence are characteristic of sep-arately localized carriers strongly coupled to lattice phonons (68). At roomtemperature the recombination is principally nonradiative.

In the case of CdS colloids, adsorption of organic amines from solutioncan shift crystallite luminescence to higher energy, and increase the quan-


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turn yield (69). It appears thai: amine complexation to the surface eliminatessurface states that otherwise would trap the electron. In a similar fashion,an apparent monolayer of Cd(OH)2 on CdS (70), or ZnS on CdSe (54) convert surface trapped luminescence into apparent Is-Is luminescence.

These examples illustrate a remarkable and potentially useful propertyof quantum crystallites: The: excited state photophysics can be tailored,via surface modification, while the optical absorption can be left largelyunchanged. At the same time, the optical absorption can be independentlyvaried via the size and shape of the crystallite. In order to take fulladvantage of these ideas, advances in the methods of synthesis, surfacemodification, and characterization are necessary.

SUMMARY

We have reviewed the electronic quantum size effect in nanometer-scalefragments of inorganic tetrahedral semiconductors. The effect is a conse-quence of strong chemical bonding. We have described the nodal patternsand energies of discrete, size dependent crystallite molecular orbitals.Molecular orbital effects, along with Coulombic attraction between theelectron and hole, can be incorporated into an effective mass Hamiltonian.The resulting discrete eigenspectrum of crystallite excited states shows arich structure. Electron-hole correlation, and a continuous band structure,develop gradually with increasing size. Large crystallites, of diameter sev-eral times the bulk exciton Bohr radius, are predicted to have giant oscil-lator strengths and large resonant optical nonlinearities.

The calculated discrete electronic spectra may need modification toincorporate ultrafast excited state internal conversion processes. Quan-titative calculations will require improvement beyond the effective massapproximation. The atomic nature and structure of intrinsic surface states,and the general question of possible surface electronic bands and recon-struction, remain largely unexplored. The dephasing of physically large,yet correlated electron-hole bound states needs to be addressed. Experi-ments on CdSe crystallites clearly demonstrate the reality of the electronicquantum size effect. Detailed interpretations are partially hindercd byaveraging over inhomogeneous distributions of size, shape, and surfacecomposition.

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the quantum mechanics of larger semiconductor clusters...

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