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Theoretical and numerical investigation of the size-dependent optical effects in metal nanoparticles

机译：金属纳米粒子中尺寸依赖的光学效应的理论和数值研究

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We further develop the theory of quantum finite-size effects in metallic nanoparticles, which was originally formulated by F. Hache, D. Ricard, and C. Flytzanis [J. Opt. Soc. Am. B 3, 1647 (1986)] and (in a somewhat corrected form) by S. G. Rautian [Sov. Phys. JETP 85,451 (1997)]. These references consider a metal nanoparticle as a degenerate Fermi gas of conduction electrons in an infinitely high spherical potential well. This model (referred to as the HRFR model below) yields mathematical expressions for the linear and the third-order nonlinear polarizabilities of a nanoparticle in terms of infinite nested series. These series have not been evaluated numerically so far and, in the case of nonlinear polarizability, they cannot be evaluated with the use of conventional computers due to the high computational complexity involved. Rautian has derived a set of remarkable analytical approximations to the series but direct numerical verification of Rautian's approximate formulas remained a formidable challenge. In this work, we derive an expression for the third-order nonlinear polarizability, which is exact within the HRFR model but amenable to numerical implementation. We then evaluate the expressions obtained by us numerically for both linear and nonlinear polarizabilities. We investigate the limits of applicability of Rautian's approximations and find that they are surprisingly accurate in a wide range of physical parameters. We also discuss the limits of small frequencies (comparable to or below the Drude relaxation constant) and of large particle sizes (the bulk limit) and show that these limits are problematic for the HRFR model, irrespective of any additional approximations used. Finally, we compare the HRFR model to the purely classical theory of nonlinear polarization of metal nanoparticles developed by us earlier [G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, Phys. Rev. Lett. 100, 47402 (2008)].
机译：我们进一步发展了金属纳米粒子中的量子有限尺寸效应的理论，该理论最初由F. Hache，D。Ricard和C. Flytzanis提出[J。选择。 Soc。上午。 B 3，1647（1986）]和（以某种程度更正的形式）由S. G. Rautian [Sov。物理JETP 85,451（1997）]。这些参考文献将金属纳米粒子视为在无限高的球形势阱中的导电电子的简并费米气体。该模型（以下称为HRFR模型）针对无限嵌套级数得出了纳米粒子的线性和三阶非线性极化率的数学表达式。到目前为止，尚未对这些序列进行数值评估，并且在非线性极化率的情况下，由于所涉及的计算复杂性高，因此无法使用常规计算机进行评估。 Rautian已对该系列导出了一组出色的分析近似值，但是直接对Rautian近似公式进行数值验证仍然是一个巨大的挑战。在这项工作中，我们导出了三阶非线性极化率的表达式，该表达式在HRFR模型中是精确的，但可以进行数值实现。然后，我们评估通过数值获得的线性和非线性极化率的表达式。我们研究了Rautian近似的适用范围，发现它们在各种物理参数中都具有惊人的准确性。我们还讨论了小频率（等于或低于Drude弛豫常数）和大颗粒尺寸（体积极限）的极限，并表明这些极限对于HRFR模型是有问题的，而与所使用的任何其他近似无关。最后，我们将HRFR模型与我们先前开发的金属纳米粒子非线性极化的纯经典理论[G. Y. Panasyuk，J。C. Schotland和V.A. Markel，物理学。牧师100，47402（2008）]。

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