Optical and transport properties of spheroidal metal nanoparticles with account for the surface effect

The kinetic approach is applied to develop the Drude-Sommerfeld model for studying the optical and electrical transport properties of spheroidal metallic nanoparticles when the free electron path is much greater than the particle size. For the nanoparticles of an oblate or a prolate spheroidal shape a dependence of the dielectric function and the electric conductivity on a number of factors, including the frequency, the particle radius, the spheroidal aspect ratio, and the orientation of the electric field with respect to the particle axes, has been been found. The oscillations of the real and imaginary parts of the dielectric permeability have been found with increasing particle size at some fixed frequencies or with frequency increasing at some fixed radius of a nanoparticle. The results obtained in the kinetic approach are compared with those known from the classical model.

Figure 1

The dependence of the ratio ${\epsilon }_{\perp }^{\prime \prime }/{\epsilon }_{\parallel }^{\prime \prime }$ for prolate ($R_{\perp }/R_{\parallel }=0.1$, upper curve) and oblate ($R_{\perp }/R_{\parallel }=10$, lower curve) Au particles (with $R=50$ Å) vs the frequency ratio $\omega /{\nu }_s$, where ${\nu }_s={\upsilon }_F/\left(2R\right)$.

Figure 2

The dependence of ${\epsilon }_{\perp }^{\prime },{\epsilon }_{\parallel }^{\prime }$ for both the prolate ($R_{\perp }/R_{\parallel }=0.1$, solid lines) and the oblate ($R_{\perp }/R_{\parallel }=10$, dashed lines) Au particles (with $R=50$ Å) vs the frequency ratio $\omega /{\nu }_s$. Thick curves are for parallel components, and thin curves are for perpendicular components of ${\epsilon }^{\prime }$.

Figure 3

The real ${\sigma }^{\prime }$ (solid lines) and imaginary ${\sigma }^{\prime \prime }$ (dashed lines) parts of the ratio of the electric conductivity to the statical conductivity ${\sigma }_0$ vs the frequency ratio for a spherical Au particle with $R=200$ Å. The thin lines correspond to the results obtained using the kinetic method, and the thick lines correspond to the Drude-Sommerfeld formulas.

Figure 4

The dependence of ${\epsilon }^{\prime }$ for a spherical Au particle vs the radius $R$ at the frequencies of $\omega =$ $5.7\times {10}^{14}$ s${}^{-1}$ (long-dashed line), $6\times {10}^{14}$ $s^{-1}$ (solid line), and $6.3\times {10}^{14}$ s${}^{-1}$ (short-dashed line).

Figure 5

The dependence of ${\epsilon }^{\prime \prime }$ for a spherical Au particle across the radius $R$ at the frequencies of $\omega =$ $5\times {10}^{14}$ s${}^{-1}$ (long-dashed line), $6\times {10}^{14}{s}^{-1}$ (solid line), and $7\times {10}^{14}$ s${}^{-1}$ (short-dashed line).

Figure 6

The frequency dependence of the reduced dielectric constant of Au particles with different $R$ (Å): 50 (thick lines) and 150 (thin lines). Solid lines are for a prolate particle with $R_{\perp }/R_{\parallel }=0.1$, and dashed lines are for an oblate one with $R_{\perp }/R_{\parallel }=10$.

Figure 7

The dependence of the real part of the dielectric constant of Au particles with different $R$ (Å), 50 (thick lines) and 150 (thin lines), on the ellipsoid aspect ratio at a frequency of plasmon resonance $\omega ={\omega }_{pl}/\sqrt{3}\simeq 7.91\times {10}^{15}$ s${}^{-1}$. Solid lines are for the parallel component, and dashed lines are for the perpendicular component.

Figure 8

The dependence of the imaginary part of the dielectric constant of Au particles with different $R$ (Å), 50 (thick lines) and 150 (thin lines), on the ellipsoid aspect ratio at the frequency of a plasmon resonance $\omega \simeq 7.91\times {10}^{15}$ s${}^{-1}$. Solid lines are for the parallel component, and dashed lines are for the perpendicular component.

Figure 9

Linewidth $\Gamma \left(R\right)$ of the surface plasmon resonance as a function of radius for Na nanoparticles in units of the Bohr radius $a_B\simeq 0.53$ Å. The smooth term ${\Gamma }_0\left(R\right)$ is given by Eq. (89) (dashed line), and the solid line corresponds to the sum of Eqs. (89) and (90).