RAPID COMMUNICATIONS PHYSICAL REVIEW B VOLUME 58, NUMBER 20 15 NOVEMBER 1998-II Short-range versus long-range electron-hole exchange interactions in semiconductor quantum dots A. Franceschetti, L. W. Wang, H. Fu, and A. Zunger National Renewable Energy Laboratory, Golden, Colorado 80401 ~Received 25 August 1998! Using a many-body approach based on atomistic pseudopotential wave functions we show that the electron- hole exchange interaction in semiconductor quantum dots is characterized by a large, previously neglected long-range component, originating from monopolar interactions of the transition density between different unit cells. The calculated electron-hole exchange splitting of CdSe and InP nanocrystals is in good agreement with recent experimental measurements. @S0163-1829~98!51144-7# One of the most intriguing features of the spectroscopy of clear, however, to what extent the exchange interaction in semiconductor quantum dots is the energy shift between the quantum dots is affected by dielectric screening. Recent di- zero-phonon absorption and emission peaks observed in size- rect calculations8,15 for InP and CdSe nanocrystals revealed selective spectroscopies of monodispersed samples.1–7 The that the unscreened exciton splitting is significantly larger redshift of the emission peak, of the order of 10 meV, has than the measured redshift, suggesting that the screening of been measured in Si,1 CdSe,2–5 InP,6 and InAs ~Ref. 7! the LR interaction may be important. nanocrystals grown with different techniques. Although sev- ~iv! How to calculate the exchange splitting. In standard eral models have been proposed to explain the redshifted approaches based on the effective-mass approximation emission, recent measurements and calculations for CdSe ~EMA! the e-h exchange interaction is described by a short- ~Refs. 2–5! and InP ~Refs. 6 and 8! nanocrystals indicate range phenomenological Hamiltonian acting on the exciton that, as first proposed by Calcott et al.,1 the redshift origi- envelope function:2–4 nates from the exciton splitting induced by the electron-hole (e-h) exchange interaction. According to this model, absorp- HEMA52j a3 DE d r 2r s J , ~1! tion takes place in a spin-allowed state, while emission oc- exch x x ~ h e! • curs from a lower energy spin-forbidden state. There are four open issues regarding the e-h exchange where j is a structure-dependent parameter (j5p/4 for cu- interaction in semiconductor quantum dots. bic lattice structure and j5p/3 for hexagonal lattice struc- ~i! The magnitude of the long-range exchange interaction. ture!, s and J are the electron and hole spin operators, ax is In general, the effective exchange interaction contains a the bulk exciton radius, and DEx is the bulk exchange split- short-range ~SR! component, which decays exponentially ting. This model relies heavily on the knowledge of ax and 2 with an effective length comparable with the bulk lattice DEx @;0.1 meV, 0.001 meV ~Ref. 7!#, which are often constant, and a long-range ~LR! component, which decays subject to large experimental uncertainties. The solutions of instead as a power law and extends over several lattice the model Hamiltonian ~1! fit well the observed redshift in constants.9 In bulk periodic semiconductors the LR interac- CdSe nanocrystals,2–4 but in the case of spherical zinc- tion is responsible for the nonanalytic behavior of the exciton blende quantum dots Eq. ~1! predicts a 1/R3 scaling of the dispersion as the exciton wave vector approaches zero,10 redshift with size,3 which is not observed in either InP ~Ref. which is observed spectroscopically as a longitudinal- 6! or InAs ~Ref. 7! nanocrystals where a ;R22 dependence transverse splitting. However, the magnitude of LR exchange is seen. While the prefactor of Eq. ~1! has been recently interactions in zero-dimensional quantum dots is still an questioned,16 the inability of the conventional EMA to repro- open question. duce the correct scaling law is particularly troublesome. ~ii! The physical origin of the LR exchange interaction. In this work we investigate the nature of the e-h exchange Conventional wisdom2–4,11 suggests that the LR exchange interaction in semiconductor quantum dots using a many- interaction in quantum dots originates, as in bulk semicon- body approach based on atomistic pseudopotential wave ductors, from dipole-dipole coupling of the transition density functions having a degree of accuracy comparable to ab ini- between unit cells. Under this assumption, the LR contribu- tio wave functions. We find the following. tion to the exchange splitting of s-like excitons in spherical ~i! For direct excitons the e-h exchange interaction has a quantum dots vanishes.11 It is not obvious, however, that sizable LR contribution, comparable in magnitude ~even af- only dipolar LR interactions exist in quantum dots. ter screening! with the SR contribution. The LR and SR ~iii! The screening of the LR exchange interaction. The components have distinct dependencies on the quantum dot effects of dielectric screening on the e-h exchange interac- size. tion are quite controversial.12–14 It is generally understood ~ii! The LR component does not originate from dipole- that in bulk semiconductors the SR exchange interaction dipole interactions between unit cells, as in the case of bulk should be unscreened, while the LR exchange interaction excitons, but from monopole-monopole interactions that are should be screened by the bulk dielectric tensor.14 It is not peculiar to quantum-confined systems. 0163-1829/98/58~20!/13367~4!/$15.00PRB 58 R13 367 ©1998 The American Physical Society RAPID COMMUNICATIONS R13 368 A. FRANCESCHETTI, L. W. WANG, H. FU, AND A. ZUNGER PRB 58 ~iii! Using a phenomenological, distance-dependent di- electric constant, we show that the LR component of the exchange interaction in quantum dots is significantly reduced by screening effects. ~iv! We are able to calculate the exciton splitting of semi- conductor quantum dots without any a priori empirical in- formation on the bulk exchange splitting or exciton radius. Our results are in good agreement with the experimentally measured redshift in a number of systems. The low-lying excited states of a quantum dot are calcu- lated by a configuration-interaction expansion of the many- particle wave function in terms of single-substitution Slater determinants $Fvc%, obtained from the ground state F0 by exciting an electron from the occupied single-particle state cv(x) of energy ev to the unoccupied single-particle state cc(x) of energy ec @here x[(r,s), where s5 , is the spin variable#. By taking the ground-state energy↑ as↓ energy zero, the matrix elements of the many-particle Hamiltonian in the representation $F % are H vc ^Fvcu uFv8c8&5~ec2ev! dv,v8dc,c82Jvc,v8c81Kvc,v8c8 , H ~2! where J and K are the Coulomb and exchange integrals: c* x ! c* x ! c x ! c x ! v8~ 1 c ~ 2 v~ 1 c8~ 2 Jvc,v8c85 FIG. 1. ~a! The unscreened exchange integral K0 (S) ~nor- EE ¯e~r ,r ! ur 2r u vc,vc 1 2 1 2 malized by its value at S `) as a function of the cutoff radius S → 3dx dx , ~3! for GaAs (R522.5 Å), InP (R517.4 Å), and CdSe (R519.2 Å) 1 2 0 spherical quantum dots. The asymptotic values of Kvc,vc(`) are * * 18.1, 21.3, and 18.2 meV, respectively. The arrows denote the cv ~x1! cc ~x2! cc8~x1! cv~x2! K 5 8 Wigner-Seitz radii of the three materials. ~b! shows, for the same vc,v8c8 EE ¯ ¯ e~r1 ,r2! ur12r2u quantum dots, the distance-dependent dielectric constant e(S) used in Eqs. ~3! and ~4! to screen the e-h interaction. 3dx1 dx2 . ~4! states. Thus, we can selectively solve Eq. ~5! for the band- Note that J and K have both diagonal (v5v8,c5c8) and edge states using the folded spectrum method.19 off-diagonal components. The screening of the e-h interac- We consider spherical nanocrystals made of zinc-blende tion, caused by the polarization of the medium, is described GaAs and InP and wurtzite CdSe, with effective radii rang- phenomenologically by the microscopic, position-dependent ing from ;6Åto;23 Å. The effective radius is defined in dielectric constant ¯e, and will be discussed later. A diago- terms of the number of atoms in the dot (Ndot)asR 1/3 nalization of the Hamiltonian matrix ~2! yields the low- 5a0(gNdot) , where a0 is the bulk lattice constant, g energy excited states of the quantum dot. 53/32p for zinc-blende dots, and g53A3c0 /32pa0 for wurtzite dots. The surface dangling bonds are passivated us- The single-particle wave functions ci(r,s) and energies ing hydrogenlike potentials. Crystal-field effects, which were ei entering Eqs. ~2!–~4! are obtained here using an atomistic 2–4 pseudopotential approach that avoids effective-mass approxi- previously treated perturbatively, are naturally described here by specifying the atomic positions. mations. The total, microscopic pseudopotential V ps of the system consisting of the quantum dot and the surrounding The exchange integrals Kvc,v8c8 of Eq. ~4! include both barrier is written as superposition of atomic potentials, which the SR part and the LR part of the e-h exchange interaction. 8,17 The effective range of the SR interaction in bulk periodic are derived from measured bulk transition energies, effec- 9 tive masses, and deformation potentials, and from ab initio solids is comparable with the Wigner-Seitz radius Rws . To bulk wave functions calculated in the framework of density- establish the effective range of the exchange interaction in 21 functional theory. The single-particle states are the solutions quantum dots, we replace the Coulomb potential ur12r2u of the Schro¨dinger equation in Eq.