Comment on “Impact Ionization and Auger Recombination Rates in

J. Phys. Chem. C 2010, 114, 16859

Comment on “Impact Ionization and Auger Recombination Rates in Semiconductor Quantum Dots”

WLR 0 ) K0U0

K0 )

1 f (R ) f (R ) A dR1 dR2 f *(R 2′ 2) f *(R 1′ 1) |R1 - R2 | 1 1 2 2

U0 )

1 Ω2

∫Ω ∫Ω dr1 dr2 u*(r 2′ 2) u*(r 1′ 1) u1(r1) u2(r2) (4)

ReceiVed: April 23, 2010; ReVised Manuscript ReceiVed: August 27, 2010 Recently, a paper was published by Fu et al.1 on an envelope function approach to estimate the impact ionization and Auger recombination rates in CdSe quantum dots. The accurate evaluation of the Coulomb matrix elements is crucial for correct determination of these quantities. However, we believe that the evaluation of the Coulomb scattering amplitudes for Augertype processes, i.e., for impact ionization or Auger recombination, is erroneous in ref 1. Specifically, the effect of Bloch functions on the long-range Coulomb scattering amplitude is not treated properly. To demonstrate this let us consider Coulomb scattering matrix element, whose general form can be written as

W)

1 ψ (r ) ψ (r ) A dr1 dr2 ψ*(r 2′ 2) ψ*(r 1′ 1) |r1 - r2 | 1 1 2 2

(1) Here ψi(r), i ) 1, 1′, 2, 2′, is the wave function of singleparticle states in both conduction (electrons) and valence (holes) bands. Within the envelope function approximation, the total wave function can be partitioned as2 ψi(r) ) fi(r) ui(r), where ui(r) ){ue(r), uh(r)} is the band-edge Bloch function for electrons and holes, respectively, and fi(r) is the envelope function. Here, for simplicity we neglect the spin projections and the multiple valence bands in CdSe. The inclusion of these factors is straightforward and does not affect the conclusions. The standard way of dealing with eq 1 is to assume that envelope functions vary slowly over unit cell.3,4 This leads to separation of the Coulomb matrix element into a sum containing long-range, WLR, and short-range contributions. The often dominating long-range component can be further approximated by using the multipole expansion

1 1 (r · R) + ... ≈ |R + r| R R3

(2)

The first term in this expansion leads to the monopole contribution to the long-range Coulomb scattering amplitude * Corresponding author. E-mail: [email protected].

(3)

where

Kirill A. Velizhanin* Center for Nonlinear Studies (CNLS)/T-4, Theoretical DiVision, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

16859

with Ω being the unit cell volume. The integrals in U0 are taken over unit cell. K0 resembles the initial Coulomb matrix element, eq 1, except the former is evaluated over envelope functions instead of total wave functions. U0 turns out to be an orthogonality relation for Bloch functions. For example, if direct Coulomb matrix element is evaluated between an electron and a hole, the contribution of Bloch functions becomes

U0 )

1 Ω2

∫Ω ∫Ω dr1 dr2 u*(r h 2) u*(r e 1) ue(r1) uh(r2) ) 1 (5)

which leads to W0LR ≡ K0.3,5 However, in the case of Augertype processes U0 contains odd number of Bloch functions of either type (electron or hole) resulting in U0 ) 0 because, e.g., uh(r) ) 0. Therefore, in Auger-type processes the ∫Ω dr u*(r) e monopole component of the long-range Coulomb scattering amplitude vanishes exactly. Nevertheless, eq 8 in ref 1 implies W0LR ) K0 * 0 for Auger-type processes, which we consider erroneous. Instead, the dipole contribution to the long-range Coulomb interaction, originating from the second rhs term in eq 2, has to be used, since Kane momentum-matrix element ∫Ω dr u*(r) r uh(r) is not necessarily zero. e Finally, we note that the considerations above are materialspecific. Indeed, the monopole contribution to the long-range Coulomb does not vanish for Auger-type processes in narrow band gap semiconductors (e.g., PbSe and PbS), since significant coupling exists between conduction and valence bands.3 Acknowledgment. This work was supported by Center for Nonlinear Studies (CNLS), LANL. The author thanks A. Piryatinski for discussions. References and Notes (1) Fu, Y.; Zhou, Y.-H.; Su, H.; Boey, F. Y. C.; Ågren, H. J. Phys. Chem. C 2010, 114, 3743. (2) Ihn, T. Semiconductor Nanostructures: Quantum states and electronic transport; Oxford University Press: Oxford, 2010. (3) Kang, I.; Wise, F. W. J. Opt. Soc. Am. B 1997, 14, 1632. (4) Nomura, S.; Segawa, Y.; Kobayashi, T. Phys. ReV. B 1994, 49, 13571. (5) Gupalov, S. V.; Ivchenko, E. L. Phys. Solid State 2000, 42, 2030.

JP103681Y

10.1021/jp103681y  2010 American Chemical Society Published on Web 09/10/2010