# Brus Jcp 1984 Colloidal Quantum Dots Theory

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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