Optical Properties of Spherical Metal Nanoparticles Coated with an Oxide Layer

Abstract

The optical properties of an ensemble of metal nanoparticles coated with an oxide layer have been investigated within framework of the classical theory. The influence of the electron-scattering mechanisms on the polarizability of nanoparticles is analyzed. The limiting case of a thin oxide layer is considered, and analytical expressions for the real and imaginary parts of the polarizability are derived. The evolution of the frequency dependences of the polarizability and extinction coefficient upon variation in the particle size and oxide-layer thickness is investigated. It is shown that the consideration of the size dependence of the surface component of the relaxation time changes the character of the size dependence of the frequency of surface plasmons of two-layer nanoparticles.

APPENDIX. FORMULAS FOR THE REAL AND IMAGINARY PARTS OF POLARIZABILITY

Let us derive expressions for ${{\Xi }^{{ - 1}}}$ and $\Lambda$ in the first order of smallness with respect to parameter q:

$$\Xi = [2\epsilon _{{{\text{oxide}}}}^{2} + {{\epsilon }_{{{\text{oxide}}}}}{{\epsilon }_{1}} + 2{{\epsilon }_{1}}{{\epsilon }_{m}} + 4{{\epsilon }_{{{\text{oxide}}}}}{{\epsilon }_{m}}$$

$$ - \;(2\epsilon _{{{\text{oxide}}}}^{2} + 2{{\epsilon }_{1}}{{\epsilon }_{m}} + 2{{\epsilon }_{1}}{{\epsilon }_{{{\text{oxide}}}}} - 2{{\epsilon }_{{{\text{oxide}}}}}{{\epsilon }_{m}})$$

$$ \times \;(1 - 3q){{]}^{2}} + \epsilon _{2}^{2}{{[{{\epsilon }_{{{\text{oxide}}}}} + 2{{\epsilon }_{m}} + 2({{\epsilon }_{{{\text{oxide}}}}} - {{\epsilon }_{m}})(1 - 3q)]}^{2}}$$

$$ = 9\{ {{[\epsilon _{{{\text{oxide}}}}^{{}}({{\epsilon }_{1}} + 2{{\epsilon }_{m}}) + 2q(\epsilon _{{{\text{oxide}}}}^{{}} - {{\epsilon }_{m}})(\epsilon _{{{\text{oxide}}}}^{{}} - {{\epsilon }_{1}})]}^{2}}$$

$$ + \;\epsilon _{2}^{2}{{[{{\epsilon }_{{{\text{oxide}}}}} - 2q({{\epsilon }_{{{\text{oxide}}}}} - {{\epsilon }_{m}})]}^{2}}\} $$

$$ = 9\epsilon _{{{\text{oxide}}}}^{2}(\epsilon _{2}^{2} + {{({{\epsilon }_{1}} + 2{{\epsilon }_{m}})}^{2}})$$

$$ \times \;\left\{ {1 + 4q\frac{{({{\epsilon }_{{{\text{oxide}}}}} - {{\epsilon }_{m}})({{\epsilon }_{1}} + 2{{\epsilon }_{m}})({{\epsilon }_{{{\text{oxide}}}}} - {{\epsilon }_{1}}) - \epsilon _{2}^{2})}}{{{{\epsilon }_{{{\text{oxide}}}}}(\epsilon _{{{\text{oxide}}}}^{2} + {{{({{\epsilon }_{1}} + 2{{\epsilon }_{m}})}}^{2}})}}} \right\}.$$

Then,

$$\begin{gathered} {{\Xi }^{{ - 1}}} = \frac{1}{{9\epsilon _{{{\text{oxide}}}}^{2}(\epsilon _{2}^{2} + {{{({{\epsilon }_{1}} + 2{{\epsilon }_{m}})}}^{2}})}} \\ \times \;\left\{ {1 - 4q\frac{{({{\epsilon }_{{{\text{oxide}}}}} - {{\epsilon }_{m}})[({{\epsilon }_{1}} + 2{{\epsilon }_{m}})({{\epsilon }_{{{\text{oxide}}}}} - {{\epsilon }_{1}}) - \epsilon _{2}^{2}]}}{{{{\epsilon }_{{{\text{oxide}}}}}(\epsilon _{2}^{2} + {{{({{\epsilon }_{1}} + 2{{\epsilon }_{m}})}}^{2}})}}} \right\}. \\ \end{gathered} $$

(A.1)

Similarly,

$$\operatorname{Re} \Lambda = 3{{\epsilon }_{{{\text{oxide}}}}}(3{{\epsilon }_{{{\text{oxide}}}}} + {{\epsilon }_{m}})\epsilon _{2}^{2} + 9\epsilon _{{{\text{oxide}}}}^{2}\epsilon _{1}^{2} - 18\epsilon _{{{\text{oxide}}}}^{2}\epsilon _{m}^{2}$$

$$ - \;4\epsilon _{{{\text{oxide}}}}^{3}{{\epsilon }_{m}} + 2{{\epsilon }_{{{\text{oxide}}}}}{{\epsilon }_{m}}\epsilon _{1}^{2} + 11\epsilon _{{{\text{oxide}}}}^{2}{{\epsilon }_{m}}{{\epsilon }_{1}}$$

$$ - \;3q[2\epsilon _{2}^{2}(6\epsilon _{{{\text{oxide}}}}^{2} - \epsilon _{m}^{2} + {{\epsilon }_{{{\text{oxide}}}}}{{\epsilon }_{m}}) - 16\epsilon _{{{\text{oxide}}}}^{2}\epsilon _{m}^{2}$$

$$ + \;12\epsilon _{{{\text{oxide}}}}^{2}\epsilon _{1}^{2} - 12\epsilon _{{{\text{oxide}}}}^{3}{{\epsilon }_{m}} - 2\epsilon _{{{\text{oxide}}}}^{3}{{\epsilon }_{m}} - {{\epsilon }_{{{\text{oxide}}}}}{{\epsilon }_{m}}\epsilon _{1}^{2}$$

$$ + \;11\epsilon _{{{\text{oxide}}}}^{2}{{\epsilon }_{m}}{{\epsilon }_{1}} + 12{{\epsilon }_{{{\text{oxide}}}}}\epsilon _{m}^{2}{{\epsilon }_{1}}] = (3{{\epsilon }_{{{\text{oxide}}}}}(3{{\epsilon }_{{{\text{oxide}}}}} + {{\epsilon }_{m}})\epsilon _{2}^{2}$$

$$ + \;9\epsilon _{{{\text{oxide}}}}^{2}\epsilon _{1}^{2} - 18\epsilon _{{{\text{oxide}}}}^{2}\epsilon _{m}^{2} - 4\epsilon _{{{\text{oxide}}}}^{3}{{\epsilon }_{m}} + 2{{\epsilon }_{{{\text{oxide}}}}}{{\epsilon }_{m}}\epsilon _{1}^{2}$$

(A.2)

$$ + \;11\epsilon _{{{\text{oxide}}}}^{2}{{\epsilon }_{m}}{{\epsilon }_{1}})\{ 1 - 3q[2\epsilon _{2}^{2}(6\epsilon _{{{\text{oxide}}}}^{2} - \epsilon _{m}^{2} + {{\epsilon }_{{{\text{oxide}}}}}{{\epsilon }_{m}})$$

$$ - \;16\epsilon _{{{\text{oxide}}}}^{2}\epsilon _{m}^{2} + 12\epsilon _{{{\text{oxide}}}}^{3}{{\epsilon }_{1}} - 2\epsilon _{{{\text{oxide}}}}^{3}{{\epsilon }_{m}} - {{\epsilon }_{{{\text{oxide}}}}}{{\epsilon }_{m}}\epsilon _{1}^{2}$$

$$ + \;11\epsilon _{{{\text{oxide}}}}^{2}{{\epsilon }_{m}}{{\epsilon }_{1}} + 12{{\epsilon }_{{{\text{oxide}}}}}\epsilon _{m}^{2}{{\epsilon }_{1}}][3{{\epsilon }_{{{\text{oxide}}}}}(3{{\epsilon }_{{{\text{oxide}}}}} + {{\epsilon }_{m}})\epsilon _{2}^{2}$$

$$ + \;9\epsilon _{{{\text{oxide}}}}^{2}\epsilon _{1}^{2} - 18\epsilon _{{{\text{oxide}}}}^{2}\epsilon _{m}^{2} - 4\epsilon _{{{\text{oxide}}}}^{3}{{\epsilon }_{m}} + 2{{\epsilon }_{{{\text{oxide}}}}}{{\epsilon }_{m}}\epsilon _{1}^{2}$$

$$ + \;11\epsilon _{{{\text{oxide}}}}^{2}{{\epsilon }_{m}}{{\epsilon }_{1}}{{]}^{{ - 1}}}\} ;$$

$$\operatorname{Im} \Lambda = 27{{\epsilon }_{m}}\epsilon _{{{\text{oxide}}}}^{2}{{\epsilon }_{2}}(1 - 3q).$$

(A.3)

Substituting (A.1)–(A.3) into (3), we obtain formulas (7) and (8) for the real and imaginary parts of the polarizability.