Surface modes in metal–insulator composites with strong interaction of metal particles

Abstract

We theoretically examine plasmonic resonance excited between two close metallic grains embedded into a dielectric matrix. The grains sizes are assumed to be much less than the wavelength of the electromagnetic wave in the dielectric medium and the grain’s separation is assumed to be much smaller than the grains sizes. A qualitative scheme is developed that enables one to estimate frequency of the plasmonic resonance and value of the field enhancement inside the gap. Our general arguments are confirmed by rigorous analytic solution of the problem for simplest geometry—two identical spherical grains.

Appendices

Appendix 1: Adiabatic approximation for quasi-plane gap

Consider dielectric gap between two bulk metal bodies. It is known that the surface plasmon wavenumber β in the gap is $2/d|\varepsilon|$ for ideally plane gap, where d is the thickness of the gap and $\varepsilon$ is the ratio the dielectric constants $\varepsilon_{{\rm m}}$ and $\varepsilon_{{\rm d}}$ in the metal and the dielectric, respectively. The expression is valid in the quasi-static limit, when $|\varepsilon|^{3/2}d\ll \lambda/\sqrt{\varepsilon_{{\rm d}}}.$

Let us derive wave equation for the surface plasmon, if the gap width slowly changes along the gap, so that adiabatic approximation for the wave propagation is applicable, when the wavelength is much less than the typical length where the gap width changes. Another derivation of the wave equation is given in Ref. [48].

Let us introduce Cartesian reference system and suppose for definiteness that the gap is symmetric in regards to OXY plane. Consider now a part of the gap, where the gap width d(x, y) is slightly nonuniform, and thus, has a gradient $\nabla^{\!\scriptscriptstyle \perp}d$ (we denote $\nabla^{\!\scriptscriptstyle \perp}$ to be gradient operator in OXY plane). We locally approximate the boundaries of the gap by two planes which have mutual inclination angle $\gamma = |\nabla^{\!\scriptscriptstyle \perp} d|.$ The adiabatic condition means that $\gamma\varepsilon \ll 1.$

The problem about surface plasmon wave propagation in a dielectric gap, whose boundaries are two intersecting planes, can be solved exactly. Let us direct OX axis along the gradient $\nabla^{\!\scriptscriptstyle \perp}d,$ and introduce polar reference system ${\rho,\varphi}$ in OXZ plane. The dielectric gap corresponds to angle $|\varphi|<\gamma/2$ and the gap width d = γ ρ (see Fig. 2). The solution for the quasi-electrostatic equation $\mathop{\hbox{div}}(\varepsilon({{\bf r}})\mathop{\hbox{grad}\Phi})=0$ can be written as $A(\rho,y)\sinh((2\varphi/\gamma)/\varepsilon)$ inside the gap (outside the gap at $\varphi>0$ the solution is $A(\rho,y)\exp(-(2\varphi/\gamma)/\varepsilon)/\varepsilon$). Wave equation on A(ρ, y) reads (ρ⁻¹∂_ρρ∂_ρ + ∂ ² _y  + β²)A = 0, where local wavenumber $\beta = -2/\varepsilon d.$ The equation can be rewritten in the form

$$ ((\nabla^{\!\scriptscriptstyle\perp})^2 + \beta^2)(\beta^{-3/2}E_z) = 0 . $$

(39)

In (39) we dropped high corrections in $\nabla^{\!\scriptscriptstyle\perp}d$ that is correct in adiabatic limit when the surface plasmon wavelength in the direction of the gap alternation $\nabla^{\!\scriptscriptstyle\perp}d$ is much less than the typical length were the gap width changes. It is worth to mention that WKB approximation for wave equation (39) is applicable at the same condition as the equation was obtained.

The dependence of the electric field enhancement factor G (12) on the resonance number n can be obtained from investigation of the electric field spatial structure based on the developed theory of surface plasmon mode in quasi-planar gap. Let us assume the number n of the mode to be large. It is applicable adiabatic approximation to describe the surface plasmon mode inside the gap in the case. It looks like

$$ \left(\frac{1}{\rho}\partial_\rho \rho\partial_\rho +\beta^2 \right) \frac{E_z}{\beta^{3/2}} =0, \quad \beta = -\frac{(\varepsilon \delta)^{-1}}{1+\rho^2/h^2}. $$

(40)

in cylindrical coordinates. Solution at ρ ≲ h is $E_z\propto {{\rm J}}_0(\rho/(|\varepsilon| \delta)),$ where J₀ is Bessel function of zero order. To continue the solution into region $\rho\gtrsim h,$ it is useful to rewrite Eq. (40) as

$$ (\partial_\rho^2+\beta_\rho^2)(\sqrt{\rho}E_z/\beta^{3/2})=0 $$

where β ²_ρ  = β² − 1/(4ρ²), and use the WKB approximation to solve the equation. The WKB approximation is applicable between the turning points, when ρ _s  < ρ < ρ _c . The turning points are determined by equating wavenumber β_ρ to zero, thus, $\rho_s = |\varepsilon|\delta/2\sim h/n$ and $\rho_c= 2h^2/(|\varepsilon|\delta) \sim{n}h.$ The solution for the electric field profile between the points is $E_z\propto\cos\left(\int^\rho\beta(\rho^\prime)\hbox{d}\rho^\prime\right)/(\beta\sqrt{\rho}),$ since β_ρ is close to β in the region. Note that the integral under the cosine converges at ρ ∼ h that validates the previous evaluation (8). At ρ ≲ h the wavenumber is constant and E _z decays as $1/\sqrt{\rho},$ that corresponds to previously obtained solution in terms of Bessel function. The main part of Ohmic dissipation occurs in the region, it can be estimated as $I_Q\sim\omega\varepsilon^{\prime\prime}(E_c/n\varepsilon)^2(\sqrt{a\delta})^3,$ that coincides with evaluation (11) for first mode. At h < ρ < nh, the wavenumber decreases as 1/ρ² and the electric field decays as 1/ρ^5/2. Although the filed decreases quite rapidly in the region, the main part of surface charge is accumulated on the scales. The value of the charge is determined by the last region with constant sing of E _z before the turning point ρ _c . The width of the region is quite large, $\Updelta\rho\sim{n}h,$, thus, the surface charge accumulated in the region is ∼E _c aδ/n. At larger distances ρ > ρ _c , the adiabatic approximation is not applicable. The region corresponds to static limit, where potential difference between the surfaces does not depend on distance from the axis of the system. Thus, evaluation (10) looks now as $E_c/n \sim (d/a^2\delta) \ln (a/\delta).$ This means that the enhancement factor G obtains additional factor 1/n for modes with n > 1.

Fig. 2

Dielectric gap with slightly varying width

Appendix 2: Recurrence equation and electric field calculation between two spherical grains

The basis of eigen functions for Laplace operator can be chosen as follows in bispherical reference system:

$$ \phi_{\alpha,m\pm}=\sqrt{\cosh\xi-\mu}{\textstyle \exp\!\left(\pm\!\left(\alpha+\frac{1}{2}\right)\xi\right)}\ \hbox{P}_{\!\alpha}^{|m|}(\mu)e^{im\varphi}, $$

(41)

where $\hbox{P}_{\!\alpha}^{|m|}$ is Legendre associated polynomial, α ≥ |m| and $\mu = \cos\eta.$ The recurrence equation (23) arises because of the boundary condition (4) and the normal derivative meshes the neighbouring harmonics. We used [Eq. 14.10.3, 42] when derivating the coefficients in the recurrence equation (23), which are

$$ \begin{aligned} X_{\alpha,m}^\pm &= -\left(\alpha+{\textstyle \frac{1}{2}}\pm\left(m+{\textstyle \frac{1}{2}}\right)\right)\\ & \quad\times \left\{\cosh\left[\!\left(\alpha+{\textstyle \frac{1}{2}}\pm1\right)\xi_0\right)]+\varepsilon \sinh\left[\!\left(\alpha+{\textstyle \frac{1}{2}}\pm1\right)\xi_0\right]\right\},\\ X_{\alpha,m} &= \varepsilon\left[e^{-\xi_0}+2\alpha\cosh\xi_0\right] \sinh\left[\!\left(\alpha+{\textstyle \frac{1}{2}}\right)\xi_0\right] \\ &\quad + \cosh\left[\!\left(\alpha+{\textstyle \frac{3}{2}}\right)\xi_0\right]+ \ 2\alpha \cosh\xi_0 \cosh\left[\!\left(\alpha+{\textstyle \frac{1}{2}}\right)\xi_0\right], \end{aligned} $$

(42)

$$ \begin{aligned} D_\alpha &= -\sqrt{2}aE^{\rm ext}_z\sinh\xi_0(\varepsilon-1) \\ &\quad\times \exp\left[-\!\left(\alpha+{\textstyle \frac{1}{2}}\right)\xi_0\right] \left[\alpha e^{\xi_0} - (\alpha+1)e^{-{\xi_0}}\right]. \end{aligned} $$

(43)

The condition ξ₀ ≪ 1 allows us to pass in recurrence equations (23) to continues limit far from special points. On the way, one obtains a linear differential equation of the second-order valid at $|\alpha-\alpha_\varepsilon|\gg1.$ The coefficients in the differential equation

$$ \left({\hbox{C}}_2^{|m|}\partial_\alpha^2 + {\hbox{C}}_1^{|m|}\partial_\alpha + {\hbox{C}}_0^{|m|} \right)B^m(\alpha) = D(\alpha) \delta^{m0} $$

(44)

are set by equations ${{\hbox{C}}_2^{|m|}}(\alpha) = (X_{\alpha,m}^++X_{\alpha,m}^-)/2,{{\hbox{C}}_1^{|m|}}(\alpha) = X_{\alpha,m}^+-X_{\alpha,m}^-, {{\hbox{C}}_0^{|m|}}(\alpha) = X_{\alpha,m} - (X_{\alpha,m}^++X_{\alpha,m}^-).$ We consider mainly the axial symmetric modes, which correspond to m = 0. Equation (44) looks like

$$ \Big(\alpha(\alpha-\alpha_\varepsilon)\partial^2_\alpha + (3\alpha-\alpha_\varepsilon)\partial_\alpha + 1 \Big)B = \sqrt{2}E^{\rm ext}_za $$

(45)

in the region α≪1/ξ₀. Two linear independent solution of homogeneous version of Eq. 45) are

$$ 1/(\alpha-\alpha_\varepsilon),\quad \ln(\alpha/\alpha_\varepsilon)/(\alpha_\varepsilon-\alpha), $$

(46)

where the first one is B ^hom_α introduced in Sect. 3.2. Partial solution of full equation (45) is $B = \sqrt{2}E^{\rm ext}_za$ [compare with (27)]. It is convenient to pass to variable u = αξ₀ in the limit $(\alpha-\alpha_\varepsilon)\gg 1,$ after what the differential equation on B takes the form

$$ \begin{aligned} \left(u\sinh u\ \partial^2_u + (2u\cosh u+\sinh u)\partial_u + e^u\right)B \\ = \sqrt{2}E^{\rm ext}_za\ e^{-u}(1-2u). \end{aligned} $$

(47)

Two linear independent solutions of homogeneous version of (47) are

$$ \frac{2/\xi_0}{e^{2u}-1}, \quad \frac{1}{e^{2u}-1} \int\limits_{1}^u\frac{e^{2u^\prime}}{u^\prime} \hbox{d}u^\prime. $$

(48)

These solutions are continuation of solutions (46) correspondingly. The first one decreases exponentially, whereas the second one decreases only as 1/α at large α ≫ 1/ξ₀. Thus, one should demand that the solution for expansion coefficient B _α should contain only the first solution in (46, 48) such that expression (24) should be satisfied. Partial solution of (47) is $2\sqrt{2}E^{\rm ext}_za u/(e^{2u}-1)$ [compare with (27)]. Let us implement matching of solutions of (45) and (47) using condition $\int_1^\infty B \hbox{d}\alpha=0$ which is the continuous analog of (24). As a result, we obtain

$$ \begin{aligned} B = -\sqrt{2}E^{\rm ext}_za \left\{\begin{array}{ll} \displaystyle 1 - b/(\xi_0(\alpha-\alpha_\varepsilon)), &\alpha\ll1/\xi_0 \\ \displaystyle 2(u -b)/(e^{2u}-1), &\alpha\gg \alpha_\varepsilon \end{array} \right. \end{aligned} $$

(49)

where $b = \pi^2/(12\ln(1/\xi_0)).$

Finally, the electric field z-component in the plane z = 0 is expressed through expansion coefficients B _α as follows

$$ E_z(\rho) = E^{\rm ext}_z \left[1+\frac{\sqrt{a}}{\sqrt{\delta}}(1-\mu)^{3/2} \sum\limits_{\alpha=0}^\infty (2\alpha+1) B_\alpha \hbox{P}_{\!\alpha}(\mu)\right]. $$

(50)