On the Calculation of Optical Characteristics and Dimensional Shifts of Surface Plasmons of Spherical Bimetallic Nanoparticles

Abstract

Formulas of efficient relaxation time are obtained for the cases when the free length of electrons is less or comparable with the characteristic dimensions of metallic regions. The frequency dispersion of optical characteristics of spherical bimetallic particles is calculated near plasma resonance in the absence of quantum-size effects. Retaining the style of common description of metallic particles based on the Mie–Drude theories, the frequency dependence of electric dipole polarizability of a two-layered metallic nanosphere was analyzed. The appearance of two polarizability maxima is the result of a difference in core and shell metals. The calculations are conducted for the following particles immersed into teflon: Au@Ag, Ag@Au, Au@Pt, Pt@Au, and Pt@Pd. It is demonstrated that the optical characteristics of bimetallic particles can be controlled by means of changing their composition and volumetric metal content. Cross-sections of absorption and scattering, as well as optical radiation efficiency of the particles in a wide spectrum range are calculated. Possible temperature of bimetallic particles at the absorption of an electromagnetic wave (for the photothermal therapy of malignant tumors) is estimated.

CALCULATION OF RELAXATION TIMES

Surface scattering. For noble metals at room temperature (${{\ell }^{{{\text{bulk}}}}}$ $\simeq$ 10–100 nm, de Broglie wavelength of electrons ~0.3 nm), surface irregularities of the atomic scale are noticeable. In this regard, we assume that the electron scattering on the surface of a bimetallic spherical particle is diffuse. The condition for applying the classical approach to the description of scattering is the $\hbar \omega$ $\ll$ ε_F inequality.

Expression for probability of electron scattering by all directions at angles that lie between (θ, θ + dθ) and (φ, φ + dφ) with respect to the tangent plane of the sphere (Fig. 1) is the following:

$$dW = {{W}_{0}}\cos \theta d\theta d\varphi ,$$

θ ∈ (0, π/2), φ ∈ (0, 2π).

In fact, this is a vector averaging (free length) $\vec {\ell }$ by all directions. For “subcritical” angles θ ≤ θ* where cosθ* = R_c/R, and for θ > θ*, the electron scatters in both metals. In this case, we write $\vec {\ell }$ as the sum of collinear vectors $\vec {\ell }$ = ${{\vec {\ell }}_{1}}$ + ${{\vec {\ell }}_{2}}$ + ${{\vec {\ell }}_{3}}$ and the average of the sum will be equal to the sum of the averages.

At the transformation of the vector into the following form ${{\vec {\ell }}_{1}}{\text{/}}{{{v}}_{{{\text{F}}{\text{,s}}}}}$ + ${{\vec {\ell }}_{2}}{\text{/}}{{{v}}_{{{\text{F}}{\text{,c}}}}}$ + ${{\vec {\ell }}_{3}}{\text{/}}{{{v}}_{{{\text{F}}{\text{,s}}}}}$, it direction remains the same (scattering on the metal boundary is neglected), and the only length is changed (now the dimensions of time), therefore averaging by directions is conducted in the same way.

The average relaxation time on the surface is determined as:

$${{\left\langle {\tau _{@}^{{{\text{surf}}}}} \right\rangle }_{R}} = \frac{{\int {\tau _{@}^{{{\text{surf}}}}dW} }}{{\int {dW} }}.$$

(A1)

Then,

$$\begin{gathered} {{\left\langle {\tau _{@}^{{{\text{surf}}}}} \right\rangle }_{R}} = \int\limits_0^{\theta^{*}} {\frac{{2R\sin \theta }}{{{{{v}}_{{{\text{F}}{\text{,s}}}}}}}\cos \theta d\theta } \\ + \;\int\limits_{\theta ^{*}}^{\pi /2} {\left( {\frac{{{{\ell }_{1}}}}{{{{{v}}_{{{\text{F}}{\text{,s}}}}}}} + \frac{{{{\ell }_{2}}}}{{{{{v}}_{{{\text{F}}{\text{,c}}}}}}} + \frac{{{{\ell }_{3}}}}{{{{{v}}_{{{\text{F}}{\text{,s}}}}}}}} \right)\cos \theta d\theta .} \\ \end{gathered} $$

(A2)

Solving the geometrical task, we have a quadratic equation of $\ell _{1}^{2}$ = $R_{{\text{c}}}^{2}$ – R² + $2{{\ell }_{1}}R\sin \theta$ with the following solution:

$${{\ell }_{1}} = R\sin \theta - {{(R_{{\text{c}}}^{2} - {{R}^{2}}{{\cos }^{2}}\theta )}^{{1/2}}}$$

and

$${{\ell }_{2}} = ({{R}^{2}} - R_{{\text{c}}}^{2} - \ell _{1}^{2}){\text{/}}{{\ell }_{1}},\quad {{\ell }_{3}} = 2R\sin \theta - {{\ell }_{2}} - {{\ell }_{1}}.$$

Extreme transitions transforming bimetallic particles into two different monometallic ones are easier to do at this stage.

Assuming R_c, β_c → 0, we have in (A2) θ* → π/2, ${{\ell }_{2}}{\text{/}}{{{v}}_{{{\text{F}}{\text{,c}}}}}$ → 0; result is the following:

$${{\left\langle {\tau _{@}^{{{\text{surf}}}}} \right\rangle }_{R}} \to R/{{{v}}_{{{\text{F}}{\text{,s}}}}}.$$

Assuming R_c → R, β_c → 1, we have in (A2) θ* → 0, (${{\ell }_{1}}$ + ${{\ell }_{3}}$)/${{{v}}_{{{\text{F}}{\text{,s}}}}}$ → 0; result is the following:

$${{\left\langle {\tau _{@}^{{{\text{surf}}}}} \right\rangle }_{R}} \to R/{{{v}}_{{{\text{F}}{\text{,c}}}}}.$$

Calculating integrals in (A2) considering expressions for ${{\ell }_{1}}$, ${{\ell }_{2}}$, and ${{\ell }_{3}}$, we finally obtain:

$${{\left\langle {\tau _{@}^{{{\text{surf}}}}} \right\rangle }_{R}} \to \frac{R}{{\mathcal{A}{{{v}}_{{{\text{F}}{\text{,s}}}}}}},$$

$$\frac{1}{\mathcal{A}} = 1 + \left( {\frac{{{{{v}}_{{{\text{F}}{\text{,s}}}}}}}{{{{{v}}_{{{\text{F}}{\text{,c}}}}}}} - 1} \right)$$

(A3)

$$ \times \;\left[ {\beta _{{\text{c}}}^{{1/3}} + \frac{1}{2}(1 - \beta _{{\text{c}}}^{{2/3}})\ln \left( {\frac{{1 - \beta _{{\text{c}}}^{{1/3}}}}{{1 + \beta _{{\text{c}}}^{{1/3}}}}} \right)} \right].$$

Radiation damping. Using in formula (8)

$$\operatorname{Re} \sigma (\omega ) = \frac{{{{\sigma }_{0}}}}{{1 + {{{(\omega {{\tau }^{{\operatorname{eff} }}})}}^{2}}}},\quad {{\sigma }_{0}} = \frac{{{{e}^{2}}\bar {n}{{\tau }^{{\operatorname{eff} }}}}}{{{{m}^{{\operatorname{eff} }}}}}$$

and inequality in the area of plasma frequencies ωτ^eff $\gg$ 1 and, consequently, replacement of ω → ω_bp, we obtain:

$${{\left\langle {\tau _{@}^{{{\text{rad}}}}} \right\rangle }_{R}} = \frac{9}{2}{{c}^{3}}\epsilon _{{\text{m}}}^{{1/2}}{{\left\langle {\mathcal{B}{{\tau }^{{{\text{rad}}}}}} \right\rangle }_{R}},\quad \mathcal{B} = \frac{1}{{\omega _{{{\text{bp}}}}^{3}V}}.$$

(A4)

1. Case (9). Similarly to (A2), we write

$$\begin{gathered} {{\left\langle {\tau _{@}^{{{\text{rad}}}}} \right\rangle }_{R}} = \frac{9}{2}{{c}^{3}}\epsilon _{{\text{m}}}^{{1/2}}\left[ {\int\limits_0^{\theta^{*}} {{{\mathcal{B}}_{{\text{s}}}}\frac{{2R\sin \theta }}{{{{{v}}_{{{\text{F}}{\text{,s}}}}}}}\cos \theta d\theta } } \right. \\ \left. { + \;\int\limits_{\theta^{*}}^{\pi /2} {\left( {{{\mathcal{B}}_{{\text{s}}}}\frac{{{{l}_{1}}}}{{{{{v}}_{{{\text{F}}{\text{,s}}}}}}} + {{\mathcal{B}}_{{\text{c}}}}\frac{{{{l}_{2}}}}{{{{{v}}_{{{\text{F}}{\text{,c}}}}}}} + {{\mathcal{B}}_{{\text{s}}}}\frac{{{{l}_{3}}}}{{{{{v}}_{{{\text{F}}{\text{,s}}}}}}}} \right)\cos \theta d\theta } } \right]. \\ \end{gathered} $$

(A5)

Coefficient of ${{\mathcal{B}}_{{\text{c}}}}$ contains the core volume V_c = 4π$R_{{\text{c}}}^{3}$/3 = β_cV₀, and ${{\mathcal{B}}_{{\text{s}}}}$ coefficient contains the shell volume V_s = 4π(R³ – $R_{{\text{c}}}^{3}$)/3 = (1 – β_c)V₀, where V₀ = 4πR³/3 is the total volume.

2. Case (10). Then,

$$\begin{gathered} {{\left\langle {\tau _{@}^{{{\text{rad}}}}} \right\rangle }_{R}} = \frac{9}{2}{{c}^{3}}\epsilon _{{\text{m}}}^{{1/2}}\left[ {\int\limits_0^{\theta ^{*}} {{{\mathcal{B}}_{{\text{s}}}}\tau _{{\text{s}}}^{{{\text{bulk}}}}\cos \theta d\theta } } \right. \\ \left. { + \;\int\limits_{\theta^{*}}^{\pi /2} {(2{{\mathcal{B}}_{{\text{s}}}}\tau _{{\text{s}}}^{{{\text{bulk}}}} + {{\mathcal{B}}_{{\text{c}}}}\tau _{{\text{c}}}^{{{\text{bulk}}}})\cos \theta d\theta } } \right]. \\ \end{gathered} $$

(A6)

As a result of integration in (A5) and (A6), we obtain, respectively, the expressions:

$${{\left\langle {\tau _{@}^{{{\text{rad}}}}} \right\rangle }_{R}} = \frac{9}{2}\frac{{\epsilon _{{\text{m}}}^{{1/2}}}}{{{{V}_{0}}}}\frac{R}{{{{{v}}_{{{\text{F}}{\text{,s}}}}}}}\frac{{{{c}^{3}}}}{{\omega _{{{\text{bp}},{\text{s}}}}^{3}}}$$

$$ \times \;\left[ {\frac{{1 - \beta _{{\text{c}}}^{{1/3}} - \frac{1}{2}(1 - \beta _{{\text{c}}}^{{2/3}})\ln \frac{{1 - \beta _{{\text{c}}}^{{1/3}}}}{{1 + \beta _{{\text{c}}}^{{1/3}}}}}}{{1 - {{\beta }_{{\text{c}}}}}}} \right.$$

(A7)

$$\left. { + \;\frac{{{{{v}}_{{{\text{F}}{\text{,s}}}}}}}{{{{{v}}_{{{\text{F}}{\text{,c}}}}}}}\frac{{\omega _{{{\text{bp}},{\text{s}}}}^{3}}}{{\omega _{{{\text{bp}},{\text{c}}}}^{3}}}\frac{{\beta _{{\text{c}}}^{{1/3}} + \frac{1}{2}(1 - \beta _{{\text{c}}}^{{2/3}})\ln \frac{{1 - \beta _{{\text{c}}}^{{1/3}}}}{{1 + \beta _{{\text{c}}}^{{1/3}}}}}}{{{{\beta }_{{\text{c}}}}}}} \right],$$

and

$$\begin{gathered} {{\left\langle {\tau _{@}^{{{\text{rad}}}}} \right\rangle }_{R}} = \frac{9}{2}\frac{{\epsilon _{{\text{m}}}^{{1/2}}}}{{{{V}_{0}}}}\frac{{{{c}^{3}}}}{{\omega _{{bp,{\text{s}}}}^{3}}}\left[ {\frac{{[2 - {{{(1 - \beta _{{\text{c}}}^{{2/3}})}}^{{1/2}}}]\tau _{{\text{s}}}^{{{\text{bulk}}}}}}{{1 - {{\beta }_{{\text{c}}}}}}} \right. \\ \left. { + \;\frac{{\omega _{{{\text{bp}},{\text{s}}}}^{3}}}{{\omega _{{{\text{bp}},{\text{c}}}}^{3}}}\frac{{[1 - {{{(1 - \beta _{{\text{c}}}^{{2/3}})}}^{{1/2}}}]\tau _{{\text{c}}}^{{{\text{bulk}}}}}}{{{{\beta }_{{\text{c}}}}}}} \right]. \\ \end{gathered} $$

(A8)

Bulk scattering. Keeping the style of calculations in the diffuse approximation, we write:

$$\begin{gathered} {{\left\langle {\tau _{@}^{{{\text{bulk}}}}} \right\rangle }_{R}} = \int\limits_0^{\theta^{*}} {\tau _{{\text{s}}}^{{{\text{bulk}}}}\cos \theta d\theta } \\ + \;\int\limits_{\theta ^{*}}^{\pi /2} {(2\tau _{{\text{s}}}^{{{\text{bulk}}}} + \tau _{{\text{c}}}^{{{\text{bulk}}}})\cos \theta d\theta } \\ = [2 - {{(1 - \beta _{{\text{c}}}^{{2/3}})}^{{1/2}}}]\tau _{{\text{s}}}^{{{\text{bulk}}}} \\ + \;[1 - {{(1 - \beta _{{\text{c}}}^{{2/3}})}^{{1/2}}}]\tau _{{\text{c}}}^{{{\text{bulk}}}}. \\ \end{gathered} $$

(A9)

Assuming R_c, β_c → 0, we should set $\tau _{{\text{c}}}^{{{\text{bulk}}}}$ → 0. Then, ${{\left\langle {\tau _{{\text{@}}}^{{{\text{bulk}}}}} \right\rangle }_{R}}$ = $\tau _{{\text{s}}}^{{{\text{bulk}}}}$.

Assuming R_c, β_c → 1, we should set $\tau _{{\text{s}}}^{{{\text{bulk}}}}$ → 0. Then, ${{\left\langle {\tau _{{\text{@}}}^{{{\text{bulk}}}}} \right\rangle }_{R}}$ = $\tau _{{\text{c}}}^{{{\text{bulk}}}}$.

When the article has been written, we read the work [43] which considers classic electron scattering in a two-layer spherical particle and a critical angle is introduced in formula (37) (in our work, this is θ* in Fig. 1 and in the Appendix).