16860
J. Phys. Chem. C 2010, 114, 16860
Reply to “Comment on ’Impact Ionization and Auger Recombination Rates in Semiconductor Quantum Dots’” Y. Fu,*,† Y.-H. Zhou,‡ Haibin Su,‡ F. Y. C. Boey,‡ and H. Ågren†,‡ Department of Theoretical Chemistry, School of Biotechnology, Royal Institute of Technology, S-106 91 Stockholm, Sweden, and DiVision of Materials Science, Nanyang Technological UniVersity, 50 Nanyang AVenue, Singapore 639798
uck(r) ) uc(r) + k∇kuc(r) + ... uvk(r) ) uv(r) + k∇kuv(r) + ...
ReceiVed: June 25, 2010; ReVised Manuscript ReceiVed: August 30, 2010 We have read the comment by K. A. Velizhanin, and our reply is as follows. Consider an Auger-type process where an incident high-energy electron in the conduction band with a wave vector k1 collides with a second electron that occupies a valence band state k′, 2 resulting in two conduction-band electrons k1′ and k2′. The general expression for the scattering matrix element of this process is
A Ψ*ck (r1) Φ*vk (r2) |r1 -1 r2| Ψck (r1) Ψck (r2) dr1 dr2 1
' 1
2
' 2
(1) see, for example, the textbook of Landau and Lifshitz.2 Here Ψck(r) [Φvk(r)] denotes the total wave function of conductionband electron (valence-band hole) state k. For semiconductor systems and within the effective mass approximation, Ψck(r) ) ψck(r) uck(r) and Φvk(r) ) φvk(r) uvk(r), where ψ and φ are envelop functions of conduction-band electron and valence-band hole, respectively, and u’s are periodic Block functions, the above expression becomes
A ψ*ck (r1) u*ck (r1) φ*vk (r2) u*Vk (r2) |r1 -1 r2| 1
1
2
2
×
ψck'1(r1) uck'1(r1) ψck'2(r2) uck'2(r2) dr1 dr2 (2) See, for example, eq 6.128 in ref 3. Following overlap integrals are thus involved
I1 ) I2 )
∫cell u*ck (r1)uck (r1) dr1 ∫cell u*vk (r2) uck (r2) dr2 1
' 1
2
' 2
(4)
where uc(r) and uv(r) are periodic Bloch functions at the conduction- and valence-band edges, respectively. ∫cellu*(r v 2) uc(r2) dr2 ) 0. Substituting these expressions into the overlap integrals we obtain the squared overlap integral in terms of the heavy-hole mass mv3,4
|I2 | 2 )
(
)
p2 1 1 + |k - k2′ | 2 2Eg m0 mv 2
(5)
where Eg is the energy band gap of the bulk material. By using the inverse Bohr radius of shallow impurities as a measure about the k values, Landsberg and Adams obtained that |I2| ) 0.223 for shallow-impurity-assisted Auger-type processes in bulk CdS and |I2| ) 0.265 in GaAs.4 Note that the inverse Bohr radius of shallow impurities in bulk semiconductors is small. For quantum dots (QDs) under investigation, the effective Bohr radius of the electron and hole distribution in the QD is largely determined by the QD size, which is about 5 nm, i.e., very small compared with the Bohr radius of shallow impurities in bulk semiconductors (about 100 nm in CdS and GaAs4). This results in an abnormal large value of |I2| that exceeds unity. The abnormal large value of |I2| is the result of eqs 4 which is valid only for small k. Under this specific circumstance, we approximate |I2| ) 1. In other words, the electrons and holes in QDs, described by effective masses with the presence of the QD confinement potentials, interact with each other via the Coulomb force of eq 8 in ref 1; see ref 5. The approach has been adopted for describing carrier interactions in many electronic devices such as tunnel junctions where the kinetic energies of relevant carriers are large; see for examples refs 6 and 7. References and Notes
(3)
in the evaluation of the Auger-type scattering processes. The first overlap integral can be approximated to unity. Because of the orthogonality of u functions for different bands but the same k, the second overlap integral I2 is, in crudest approximation, * To whom correspondence should be addressed. † Royal Institute of Technology. ‡ Nanyang Technological University.
zero, as stated in the comment. This however is not correct since the periodic Bloch functions are functions of wave vectors; see eq 6.149 on p 274 in Ridley.3 We now estimate the second overlap integral for the Augertype processes under investigation of ref 1. For small k we can write
(1) Fu, Y.; Zhou, Y.-H.; Su, H.; Boey, F. Y. C.; Ågren, H. J. Phys. Chem. C 2010, 114, 3743–7. (2) Landau, L. D.; Lifshitz, E. M. Quantum Mechanics, 3rd ed.; Pergamon Press: New York, 1962; p 278. (3) Ridley, B. K. Quantum Processes in Semiconductors; Clarendon Press: Oxford, 1988; p 269-78. (4) Landsberg, P. T.; Adams, M. J. Proc. R. Soc. London A 1973, 334, 523–39. (5) Abrahams, E. Phys. ReV. 1954, 95, 839–10. (6) Takenaka, N.; Inoue, M.; Inuishi, Y. J. Phys. Soc. Jpn. 1979, 47, 861–8. (7) Rodina, P.; Ebert, U.; Hundsdorfer, W.; Grekhov, I. J. Appl. Phys. 2002, 92, 958–64.
JP1058676
10.1021/jp1058676 2010 American Chemical Society Published on Web 09/10/2010
