PHYSICAL REVIEW B VOLUME 51, NUMBER 4 15 JANUARY 1995-II Quantum size effect on optical second-harmonic generation in small metallic particles Kuang Yao Lo and Juh Tzeng Lue Department of Physics and Department of Electrical Engineering, National Tsing Hua University, Hsin Chu, Taiwan, Republic of China (Received 16 May 1994; revised manuscript received 25 July 1994) Second-order susceptibilities of small metallic particles are calculated by the quantum size effect, in which the wave functions of conduction electrons are constrained in a small sphere. The electric- quadrupole term, which plays an essential role in the second-harmonic generation in metals, varies in- versely with different powers of the radius. The magnitude of the quadrupole susceptibility is enhanced by the quantum size effect as the particle size diminishes below 10 nm, and approaches the results evalu- ated from the free-electron-gas model as the particle radius goes to infinity. The enhancement of the second-harmonic reflectivity generated from small metallic particles, as calculated in this work, is in gen- eral agreement with experimental results. I. INTRODUCTION The linear and nonlinear optical properties of small metallic particles have stimulated numerous investiga- tions of the wave-length change of visible fluorescence and the enhancem. ent of second-harmonic generation by the reduction of particle size. The physics of ultradisper- sion media (UDM), consisting of microscopic ensembles of small particles with sizes from 1 to 100 nm, is recog- nized to be incoherent for bulk materials, especially the response to the electromagnetic field. Recently, a significant growth of the intensity of the second- harmonic generation (SHG) reflected from metallic-island films' and an enhancement of the Raman scattering of light by the molecules adsorbed on the surface of silver- island films have been detected. The phenomena of enhanced SHG due to the surface-plasmon resonance have been studied. Until now, the origin of the enhance- ment of SHG of small particles has not been well studied. For centrosymmetric solids, the electric-dipole field is prohibited from generalizing the second-harmonic wave, and the SHG is manifestly generated by the higher-order terms of electric-quadrupole and magnetic-dipole fields. The induced second-order polarization in metal sur- faces ' is the normal component of the electric field at the bound- ary, concluding that the electric quadrupole dominates the SHG of the metal surface. Other works by Liebsch and Govorkov et al. also proved that the dominant SHG source arises from the surface electric quadrupole term. Therefore, we do not intend to include the bulk magnetic dipole term in this calculation. In ultrahigh vacuum, the SHG intensity of the metal surface is stronger than that measured in air. The high nonlinear intensity is due to the local-electron-density fluctuation near the unbounded metal surface which is nearly free and isotropic. But in the UDM case, metallic particles are embedded in a dielectric material, such as glass, in which the quantum confinement is the more dominant effect than the local-electron-density fluctua- tion. In this work, we attempt to elucidate the quantum size effect on the second-order susceptibility for small metallic particles that confine the conduction electrons to behav- ing like a rigid sphere. The variation of the quantum ex- pectation value of the electric quadrupole term with respect to the particle size is calculated and compared with the experimental data. II. THEORY P (2co) =a[E(co) X H(co)]+PE(co) [V E(co) ], where the first term is the magnetic-dipole term arising from the bulk and the second term is the electric- quadrupole term relating to the surface current. Since the magnetic-dipole term depends on the field inside the bulk, which is much weaker than the field near the surface be- cause of the high attenuation rate of metals and due to the vast enhancement of surface-to-volume ratio for nanocrystalline particles, the contribution to the SHG by the magnetic dipole is much smaller than from the elec- tric quadrupole. Originally, Jha and Rudnick derived the nonlinear polarization source of metal surfaces due to conduction electrons and called attention to the impor- tance of surface terms that arise from the discontinuity of Inside a metal, due to the short screening length of charge particles, the quantum-sphere model (QSM) (Ref. 10) implying N independent electrons confined in a sphere of radius R is applicable. The normalized one- electron wave function in such a case is given by 1/2 2 a =R„t(„) Yt (O, y), where Yt is the spherical harmonics ( — l ~ m ~1), and j& is the spherical Bessel function of order l, with nth root a„t which relates to the eigenenergies of P„& by 01. 63-1829/95/51(4)/2467(6)/$06. 00 51 2467 1995 The American Physical Society