Infrared and Raman selection rules for elastic ... - APS Link Manager

PHYSICAL REVIEW B 74, 197401 共2006兲

Comment on “Infrared and Raman selection rules for elastic vibrations of spherical nanoparticles” S. V. Goupalov A. F. Ioffe Physico-Technical Institute, 26 Polytechnicheskaya, 194021 St. Petersburg, Russia

Lucien Saviot Laboratoire de Recherche sur la Réactivité des Solides, UMR 5613 CNRS – Université de Bourgogne, 9 Avenue A. Savary, BP 47870, 21078 Dijon, France

Eugène Duval Laboratoire de Physico-Chimie des Matériaux Luminescents, UMR 5620 CNRS – Université Lyon I, 43 Boulevard du 11 Novembre 69622, Villeurbanne Cedex, France 共Received 23 December 2005; revised manuscript received 11 May 2006; published 13 November 2006兲 This Comment demonstrates again that selection rules established several years ago for the infrared absorption and the Raman scattering by vibrations of a spherical nanoparticle are correct and that the claimed errors about them are wrong. DOI: 10.1103/PhysRevB.74.197401

PACS number共s兲: 78.30.⫺j, 63.22.⫹m, 62.30.⫹d

Fourteen years ago, it was demonstrated,1 by using group theory, that the vibration modes of a spherical nanoparticle, which are visible by Raman scattering are the spheroidal modes with the total angular momentum ᐉ = 0 and 2. By infrared, only the spheroidal modes with ᐉ = 1 can be observed.1 The Raman and infrared transitions are forbidden for the torsional modes. In a recent paper, Kanehisa claimed that these selection rules are erroneous.2 Starting from the original paper of Lamb3 and like in the book of Eringen and Suhubi,4 the author deduced the displacement field corresponding to the different modes and determined the parity of these displacement fields.2 However, it is important to realize that, unlike the parity of a scalar function, the parity of a polar vector function, such as the displacement field, does not coincide with the parity of the orbital angular momenta associated with the function. This is in agreement with the paper of Montagna and Dusi.5 Failing to notice this difference, Kanehisa wrongly concluded that “the assignment of parity in previous works neglected the vector nature of the elastic field.” Doing the parity assignment the right way, the spheroidal and torsional modes characterized by an even ᐉ are, respectively, even and odd, as clearly shown before.1 This confirms that the breathing mode 共ᐉ = 0兲 and the spheroidal mode with ᐉ = 2 are even. It is emphasized, that, like in molecules, for example, in benzene, the breathing mode is always even. Therefore, by these very simple considerations, the selection rules established in Ref. 1 are demonstrated to be perfectly correct. In particular, they apply to the Raman scattering resonant with the electronic dipolar plasmon in metallic 共Ag, Au, Cu兲 nanoparticles, but the effect of resonance is stronger for spheroidal modes with ᐉ = 2 than for the breathing modes 共ᐉ = 0兲.6–10 Goupalov and Merkulov showed11 that, in spherical semiconductor nanocrystals, the breathing mode always participates in the Raman scattering 共though the corresponding Raman feature can be suppressed due to a partial cancellation of the conduction band electron and the valence band hole contributions to the exciton-phonon coupling兲 while participation of the speroi1098-0121/2006/74共19兲/197401共2兲

dal modes with ᐉ = 2 becomes possible for semiconductors with the complex valence band structure. They used the quasiparticle description12–15 and considered a particular mechanism of exciton-phonon interaction for semiconductor nanocrystals. In order to avoid further confusion, below we present our group-theoretical arguments in more details. It is convenient to construct the solutions of the equation of motion for continuum displacements of an elastic sphere as mutual eigenfunctions of the total angular momentum opˆ + Iˆ 共where L ˆ is the operator of the orbital angular erator 艎ˆ = L ˆ momentum, and I is the spin operator corresponding to I ˆ , and the = 1兲, operator of its projection to an arbitrary axis m parity operator Pˆ.11 For the spheroidal modes, displacements are linear combinations 共with angular-independent coeffiL 共␪ , ␾兲 共Ref. 16兲 cients兲 of the vector spherical harmonics Yᐉm with L = ᐉ ± 1 共or L = 1 for ᐉ = 0兲 while, for the torsional modes, they are proportional to the vector spherical harmonics with L = ᐉ. The idea behind the symmetry analysis of Duval1 is best explained in the Tinkham’s book.17 The matrix element of the Kramers-Heisenberg formula12,18 is calculated on the vibrational wave functions 共which for each phonon mode are those of one-dimensional linear oscillator兲, and the usual selection-rule theorem is applied.17 As a result, a phonon can participate in a first-order Raman process if and only if its irreducible representation is the same as one of the irreducible representations which occur in the reduction of the representation of the polarizability tensor.1,14,17 Macroscopically, the same selection rules follow from the analysis of the number of independent components of the third-rank tensor of susceptibility 共or polarizability兲 derivatives.13,15,19 Thus, it remains to determine according to which representation of L 共␪ , ␾兲 transforms. By the rotation-inversion group Yᐉm definition,16 the corresponding irreducible representation is characterized by the angular momentum ᐉ and what remains to determine is the parity. When the parity operator is applied to a 共polar兲 vector function, we must take into account the

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©2006 The American Physical Society

PHYSICAL REVIEW B 74, 197401 共2006兲

COMMENTS

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20 ible representation D共ᐉ兲 g of the rotation-inversion group, and 1 the parity assignment of Duval is correct. The last argument proves that Kanehisa’s paper2 is based on the wrong prerequisites and is misleading.

Therefore, the spheroidal modes with the total angular momentum ᐉ = 0 , 2 transform under the even 共gerade兲 irreduc-

The authors thank M. Kibler for approving the group theory arguments.

Duval, Phys. Rev. B 46, 5795 共1992兲. Kanehisa, Phys. Rev. B 72, 241405共R兲 共2005兲. 3 H. Lamb, Proc. London Math. Soc. 13, 189 共1882兲. 4 A. C. Eringen and E. S. Suhubi, Elastodynamics 共Academic, New York, 1975兲, Vol. II, pp. 804–833. 5 M. Montagna and R. Dusi, Phys. Rev. B 52, 10080 共1995兲. 6 H. Portales, L. Saviot, E. Duval, M. Fujii, S. Hayashi, N. D. Fatti, and F. Vallée, J. Chem. Phys. 115, 3444 共2001兲. 7 H. Portalès, L. Saviot, E. Duval, M. Gaudry, E. Cottancin, M. Pellarin, J. Lermé, and M. Broyer, Phys. Rev. B 65, 165422 共2002兲. 8 A. Courty, I. Lisiecki, and M. P. Pileni, J. Chem. Phys. 116, 8074 共2002兲. 9 A. Courty, A. Mermet, P. A. Albouy, E. Duval, and M. P. Pileni, Nat. Mater. 4, 395 共2005兲. 10 G. Bachelier and A. Mlayah, Phys. Rev. B 69, 205408 共2004兲. 11 S. V. Gupalov and I. A. Merkulov, Fiz. Tverd. Tela 共S.-Peterburg兲 41, 1473 共1999兲 关Phys. Solid State 41, 1349 共1999兲兴. 12 R. Martin and L. Falicov, Light Scattering in Solids, edited by M. Cardona 共Springer, New York, 1975兲.

13 P.

fact that the reversal of the directions of the axes changes the sign of all the components of the vector.18 Hence,16,18 L L L PˆYᐉm 共␪, ␾兲 ⬅ − Yᐉm 共␲ − ␪, ␲ + ␾兲 = 共− 1兲L+1Yᐉm 共␪, ␾兲.

1 E.

2 M.

Yu and M. Cardona, Fundamentals of Semiconductors: Physics and Materials Properties 共Springer, New York, 2002兲. 14 R. Loudon, Adv. Phys. 13, 423 共1964兲. 15 W. Hayes and R. Loudon, Scattering of Light by Crystals 共Wiley, New York, 1978兲. 16 D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum 共World Scientific, Singapore, 1988兲. 17 M. Tinkham, Group Theory and Quantum Mechanics 共McGrawHill, New York, 1964兲. 18 V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics 共Pergamon, Oxford, 1982兲. 19 M. Lax, Symmetry Principles in Solid State and Molecular Physics 共Wiley, New York, 1974兲. 20 A more rigorous proof of this statement can be obtained by a successive application of the following sections of Ref. 16: 7.3.1; 7.1.1; 3.1.1; 3.1.4; 7.3.4 along with the fact that displacements in the spheroidal modes with the total angular momentum ᐉ±1 共␪ , ␾兲. ᐉ are linear combinations of Yᐉm

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