Thin Films of Nanostructured Noble Metals

Abstract

The optical characterisation of nanostructured noble metal films is discussed. A good description of the optical response of a 2D layer of identical metal particles on a support is obtained with the so-called Thin Island Film theory, which is based on the polarisability of an individual nanoparticle. The influence of particle shape, density, and the optical properties of both the support and the surrounding ambient is discussed as well as the ability of the commonly used Maxwell Garnett approximation to model these films. The possibilities and limits of this approach is illustrated with several examples. A short outlook towards the ellipsometric analysis of metamaterials is given.

Appendices

Appendix

Lorentzian Line Shape for the Polarizability in the Dipole Approximation

The polarizability of a particle with radius a and screening parameter k to incorporate shape effects is:

$$\begin{aligned} \upalpha =4{\uppi }\mathrm{{a}}^{3}\frac{\upvarepsilon -\upvarepsilon _{\mathrm{{a}}} }{\upvarepsilon +\;\mathrm{{k}}{\upvarepsilon _{\mathrm{{a}}}} } \end{aligned}$$

(A.1)

Inserting the Drude dielectric function:

$$\begin{aligned} \upalpha&= 4{\uppi }\mathrm{{a}}^{3}\frac{\upvarepsilon _\infty -\upvarepsilon _{\mathrm{{a}}}\;-\;\frac{\mathrm{{E}}_{\mathrm{{p}}}^2 }{\mathrm{{E}}^{2}+\;\mathrm{{i}}\Gamma \mathrm{{E}}}}{\upvarepsilon _\infty +\;\mathrm{{k}}{\upvarepsilon _{\mathrm{{a}}}} -\frac{\mathrm{{E}}_{\mathrm{{p}}}^2 }{\mathrm{{E}}^{2}+\;\mathrm{{i}}\Gamma \mathrm{{E}}}}\nonumber \\&=4{\uppi }\mathrm{{a}}^{3}\frac{\left( {\frac{\upvarepsilon _\infty -\upvarepsilon _{\mathrm{{a}}} }{\upvarepsilon _\infty +\;\mathrm{{k}}{\upvarepsilon _{\mathrm{{a}}}} }} \right)\left( {\mathrm{{E}}^{2}+\;\mathrm{{i}}\Gamma \mathrm{{E}}} \right)-\frac{\mathrm{{E}}_{\mathrm{{p}}}^2 }{\upvarepsilon _\infty +\;\mathrm{{k}}{\upvarepsilon _\mathrm{{a}}} }}{\mathrm{{E}}^{2}+\;\mathrm{{i}}\Gamma \mathrm{{E}}-\frac{\mathrm{{E}}_{\mathrm{{p}}}^2 }{\upvarepsilon _\infty +\;\mathrm{{k}}{\upvarepsilon _{\mathrm{{a}}}}}}\nonumber \\&=4{\uppi }\mathrm{{a}}^{3}\frac{\chi \left( {\mathrm{{E}}^{2}+\;\mathrm{{i}}\Gamma \mathrm{{E}}} \right)-\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 }{\mathrm{{E}}^{2}-\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 +\;\mathrm{{i}}\Gamma \mathrm{{E}}}\nonumber \\&=4{\uppi }\mathrm{{a}}^{3}\frac{\chi \left( {\mathrm{{E}}^{2}-\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 +\;\mathrm{{i}}\Gamma \mathrm{{E}}} \right)+\;\left( {\chi -1} \right)\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 }{\mathrm{{E}}^{2}-\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 +\;\mathrm{{i}}\Gamma \mathrm{{E}}}\nonumber \\&=4{\uppi }\mathrm{{a}}^{3}\left( {\chi +\;\left( {1-\chi } \right)\frac{\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 }{\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 -\mathrm{{E}}^{2}-\mathrm{{i}}\Gamma \mathrm{{E}}}} \right). \end{aligned}$$

(A.2)

This is a Lorentzian line shape with the same broadening $\Gamma$ as the Drude lineshape and with

$$\begin{aligned} \chi =\frac{\varepsilon _\infty -\varepsilon _{\mathrm{{a}}}}{\varepsilon _\infty +k{\varepsilon _\mathrm{{a}}}}\;\;\quad {\text{ and}}\;\quad E_{{0},dip}^2 =\frac{E_{\mathrm{{p}}}^2 }{\varepsilon _\infty +k{\varepsilon _{\mathrm{{a}}}}}. \end{aligned}$$

(A.3)

Lorentzian Line Shape in the Maxwell Garnett Approach

The Maxwell Garnett approach provides an effective dielectric function according to:

$$\begin{aligned} \frac{\upvarepsilon _{\mathrm{{eff}}}}{\upvarepsilon _{\mathrm{{a}}}}=\frac{\upvarepsilon (1+2\mathrm{{f}})+\upvarepsilon _{\mathrm{{a}}}(\mathrm{{k}}_{\mathrm{{s}}}-2\mathrm{{f}})}{\upvarepsilon (1-\mathrm{{f}})+\upvarepsilon _{\mathrm{{a}}}(\mathrm{{k}}_{\mathrm{{s}}} +\,\mathrm{{f}})}. \end{aligned}$$

(A.4)

For a Drude metal particle this leads to:

$$\begin{aligned} \frac{\upvarepsilon _{\mathrm{{eff}}} }{\upvarepsilon _{\mathrm{{a}}}}&=\frac{\left( {\upvarepsilon _\infty -\frac{\mathrm{{E}}_{\mathrm{{p}}}^2 }{\mathrm{{E}}^{2}+\;\mathrm{{i}}\Gamma \mathrm{{E}}}} \right)(1+2\mathrm{{f}})+\upvarepsilon _{\mathrm{{a}}}(\mathrm{{k}_{s}} -2\mathrm{{f}})}{\left( {\upvarepsilon _\infty -\frac{\mathrm{{E}}_{\mathrm{{p}}}^2 }{\mathrm{{E}}^{2}+\;\mathrm{{i}}\Gamma \mathrm{{E}}}} \right)(1-\mathrm{{f}})+\upvarepsilon _{\mathrm{{a}}}(\mathrm{{k}}_{\mathrm{{s}}} +\,\mathrm{{f}})} \nonumber \\&=\frac{\upvarepsilon _{\infty ,\mathrm{{MG}}} \left( {\mathrm{{E}}^{2}+\,\mathrm{{i}}\Gamma \mathrm{{E}}} \right)-\mathrm{{E}}_{{0},\mathrm{{MG}}}^2 \frac{1+2\mathrm{{f}}}{1-\mathrm{{f}}}}{\mathrm{{E}}^{2}-\mathrm{{E}}_{{0},\mathrm{{MG}}}^2 +\,\mathrm{{i}}\Gamma \mathrm{{E}}} \nonumber \\&=\mathop {\upvarepsilon }\nolimits _{\infty ,\mathrm{{MG}}} +\frac{\mathrm{{E}}_{\mathrm{{p}},\mathrm{{MG}}}^2 }{\mathrm{{E}}_{{0},\mathrm{{MG}}}^2 -\mathrm{{E}}^{2}-\mathrm{{i}}\Gamma \mathrm{{E}}} \end{aligned}$$

(A.5)

which is a Lorentzian line shape with

$$\begin{aligned} \mathop {\upvarepsilon }\nolimits _{\infty ,\mathrm{{MG}}}&=1+\frac{3\mathrm{{f}}(\upvarepsilon _\infty -\upvarepsilon _{\mathrm{{a}}})}{\upvarepsilon _\infty (1-\mathrm{{f}})+\upvarepsilon _{\mathrm{{a}}}(\mathrm{{k}}_{\mathrm{{s}}} +\,\mathrm{{f}})}\\ \nonumber \mathrm{{E}}_{{0},\mathrm{{MG}}}^2&=\frac{(1-\mathrm{{f}})}{\upvarepsilon _\infty (1-\mathrm{{f}})+\upvarepsilon _{\mathrm{{a}}}(\mathrm{{k}}_{\mathrm{{s}}} +\,\mathrm{{f}})}\mathrm{{E}}_{\mathrm{{p}}}^2 . \end{aligned}$$

(A.6)

The width $\Gamma$ of the Lorentzian is the same as the original Drude broadening. The enumerator $\mathrm{{E}}_{\text{ p,MG}}^2$ in the last term of Eq. 6.14 can be rewritten

$$\begin{aligned} \mathrm{{E}}_{\mathrm{{p}},\mathrm{{MG}}}^2&=\mathrm{{E}}_{{0},\mathrm{{MG}}}^2 \left( {\frac{1+2\mathrm{{f}}}{1-\mathrm{{f}}}-\mathop {\upvarepsilon }\nolimits _{\infty ,\mathrm{{MG}}} } \right)\nonumber \\&= 3\mathrm{{f}}\frac{\upvarepsilon _{\mathrm{{a}}}(1+\,\mathrm{{k_s}} )}{\left[ {\upvarepsilon _\infty (1-\mathrm{{f}})+\upvarepsilon _{\mathrm{{a}}}(\mathrm{{k}}_{\mathrm{{s}}} +\,\mathrm{{f}})} \right]^{2}}\mathrm{{E}}_{\mathrm{{p}}}^2 . \end{aligned}$$

(A.7)

Lorentzian Line Shape in the TIF Approach

The derivation of the dimensionless polarizability lineshape for a Drude metal is very similar to A1:

$$\begin{aligned}&{\upalpha }^{\prime }=(1+\,\mathrm{{k}})\frac{\upvarepsilon _\infty -\upvarepsilon _{\mathrm{{a}}}-\frac{\mathrm{{E}}_{\mathrm{{p}}}^2 }{\mathrm{{E}}^{2}+\;\mathrm{{i}}\Gamma \mathrm{{E}}}}{\upvarepsilon _\infty +\;\mathrm{{k}}{\upvarepsilon _{\mathrm{{a}}}}-\frac{\mathrm{{E}}_{\mathrm{{p}}}^2 }{\mathrm{{E}}^{2}+\;\mathrm{{i}}\Gamma \mathrm{{E}}}} \nonumber \\&=(1+\,\mathrm{{k}})\left( {\chi +\left({1-\chi } \right)\frac{\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 }{\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 -\mathrm{{E}}^{2}-\mathrm{{i}}\Gamma \mathrm{{E}}}} \right). \end{aligned}$$

(A.8)

This is again a Lorentzian line shape with the same broadening $\Gamma$ as the Drude lineshape and with as for the polarizability

$$\begin{aligned} \chi =\frac{\upvarepsilon _\infty -{\upvarepsilon _{\mathrm{{a}}}}}{\upvarepsilon _\infty +\;\mathrm{{k}}{\upvarepsilon _{\mathrm{{a}}}}}\;\;{\text{ and}}\;\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 =\frac{\mathrm{{E}}_{\mathrm{{p}}}^2 }{\upvarepsilon _\infty +\;\mathrm{{k}}{\upvarepsilon _{\mathrm{{a}}}}}. \end{aligned}$$

(A.9)

The parallel component of the dielectric function is thus (Eq. 6.6):

$$\begin{aligned} \frac{\upvarepsilon _{\mathrm{{effP}}} }{\upvarepsilon _{\mathrm{{a}}}}=1+\,\mathrm{{f}}(1+\,\mathrm{{k}}_{\mathrm{{P}}} )\chi +\frac{\mathrm{{f}}(1+\,\mathrm{{k}}_{\mathrm{{P}}} )(1-\chi )\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 }{\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 -\mathrm{{E}}^{2}-\mathrm{{i}}\Gamma \mathrm{{E}}} \end{aligned}$$

(A.10)

$$\begin{aligned}&={\mathop {\upvarepsilon }\nolimits _{\infty ,\mathrm{{dip}}}} +\,\mathrm{{f}}\frac{\upvarepsilon _\mathrm{{a}}(1+\,\mathrm{{k}_{P}} )^{2}}{\upvarepsilon _\infty +\,\mathrm{{k}}_{\mathrm{{P}}}{\upvarepsilon _\mathrm{{a}}}}\frac{\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 }{\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 -\mathrm{{E}}^{2}-\mathrm{{i}}\Gamma \mathrm{{E}}} \nonumber \\&={\mathop {\upvarepsilon }\nolimits _{\infty ,\mathrm{{dip}}}} +\frac{\mathrm{{E}}_{\mathrm{{p}},\mathrm{{dip}}}^2 }{\mathrm{{E}}_{{0},\mathrm{{dip}}}^2 -\mathrm{{E}}^{2}-\mathrm{{i}}\Gamma \mathrm{{E}}} \end{aligned}$$

(A.11)

with

$$\begin{aligned} {\mathop {\upvarepsilon }\nolimits _{\infty ,\mathrm{{dip}}}}&=1+\,\mathrm{{f}}(1+\,\mathrm{{k}}_{\mathrm{{P}}} )\frac{\upvarepsilon _\infty -\upvarepsilon _{\mathrm{{a}}}}{\upvarepsilon _\infty +\;\mathrm{{k}}_{\mathrm{{P}}} {\upvarepsilon _{\mathrm{{a}}}}}\\ \nonumber \mathrm{{E}}_{\mathrm{{p}},\mathrm{{dip}}}^2&=\mathrm{{f}}\frac{\upvarepsilon _{\mathrm{{a}}}(1+\,\mathrm{{k}}_{\mathrm{{P}}} )^{2}}{\left( {\upvarepsilon _\infty +\;\mathrm{{k}}_{\mathrm{{P}}}{\upvarepsilon _{\mathrm{{a}}}}} \right)^{2}}\mathrm{{E}}_{\mathrm{{p}}}^2 . \end{aligned}$$

(A.12)

The resonance energy $\mathrm{{E}}_{0,{\text{ dip}}}$ in the dipolar approximation of the TIF model looks at first the same as that of the isolated particles. However, do note that the screening factor k not only includes shape effects, but also image and neighbour interaction effects.

Quadrupole Expression for the Excess Polarizability

Haarmans and Bedeaux [18] derived an explicit form for the expression up to quadrupole order for excess surface polarizability $\upgamma$ and $\upbeta$:

$$\begin{aligned}&{\mathop {\upgamma }\nolimits _{\mathrm{{qu}}}} ={\upphi } \frac{4\mathrm{{a}}{\upvarepsilon _{\mathrm{{a}}}}}{3}\frac{{\updelta }{\upvarepsilon } \left( {1+\,\mathrm{{L}}1_{\mathrm{{p}}} {\updelta }{\upvarepsilon } } \right)}{\left( {1+\,\mathrm{{L}}_{\mathrm{{p}}}{{\updelta }{\upvarepsilon }} } \right)\left( {1+\,\mathrm{{L}}1_{\mathrm{{p}}} {\updelta }{\upvarepsilon } } \right)+\Lambda _{\mathrm{{p}}}{{\updelta }{\upvarepsilon }} ^{2}}\end{aligned}$$

(A.13)

$$\begin{aligned}&{\mathop {\upbeta }\nolimits _{\mathrm{{qu}}}} ={\upphi } \frac{4\mathrm{{a}}}{3{\upvarepsilon _{\mathrm{{a}}}}}\frac{{\updelta }{\upvarepsilon } \left( {1+\,\mathrm{{L}}1_{\mathrm{{z}}} {\updelta }{\upvarepsilon } } \right)}{\left( {1+\,\mathrm{{L}}_{\mathrm{{z}}}{\updelta }{\upvarepsilon } } \right)\left( {1+\,\mathrm{{L}}1_{\mathrm{{z}}}{\updelta }{\upvarepsilon }} \right)+\Lambda _{\mathrm{{z}}}{{\updelta }{\upvarepsilon }} ^{2}} \end{aligned}$$

(A.14)

Here, $L_\mathrm{{p}}$ and $L_\text{ z}$ represent the dipolar correction terms and $L1_{\mathrm{{p}}}$ and $L1_{\mathrm{{z}}}$, $\Lambda _{\mathrm{{p}}}$ and $\Lambda _{\mathrm{{z}}}$ are the quadrupole depolarization factors.

The image effect is modulated by the factor $\mathrm{{B}}_{\mathrm{{sa}}}$ that describes the contrast between ambient and substrate:

$$\begin{aligned} \mathrm{{B}}_{\mathrm{{sa}}} =\left({\mathop {\upvarepsilon }\nolimits _{\mathrm{{a}}}}- {\mathop {\upvarepsilon }\nolimits _{\mathrm{{s}}}} \right)/ \left({\mathop {\upvarepsilon }\nolimits _{\mathrm{{a}}}}+\,{\mathop {\upvarepsilon }\nolimits _{\mathrm{{s}}}}\right) \end{aligned}$$

(A.15)

This contrast is quite considerable for semiconductors and metals $(\mathrm{{B}}_{\mathrm{{sa}}} {\approx }-1)$, while it is quite reduced for dielectrics. For instance for a glass substrate in a water ambient, $\mathrm{{B}}_{\mathrm{{sa}}}$ and thus the image effect, is reduced by an order of magnitude.

The dipolar and quadrupole depolarization factors as a function of surface coverage are:

$$\begin{aligned}&\mathrm{{L}}_{\mathrm{{p}}} =\frac{1}{3}\left[ {1-\frac{\upphi }{2}+\,\mathrm{{B}}_{\mathrm{{sa}}} \left( {\frac{1}{8}-\frac{\upphi }{4\sqrt{2}}} \right)} \right] \\&\mathrm{{L}}_{\mathrm{{z}}} =\frac{1}{3}\left[ {1+\upphi +\,\mathrm{{B}}_{\mathrm{{sa}}} \left( {\frac{1}{4}-\frac{\upphi }{2\sqrt{2}}} \right)} \right] \nonumber \\&\mathrm{{L}}1_{\mathrm{{p}}} =\frac{2}{5}\left[ {1-\frac{\upphi }{8}+\frac{\mathrm{{B}}_{\mathrm{{sa}}} }{8}\left(1-\upphi \left( {\frac{1}{\sqrt{2}}\frac{3}{8}} \right)\right)} \right] \nonumber \\&\mathrm{{L}}1_{\mathrm{{z}}} =\frac{2}{5}\left[ {1+\frac{3\upphi }{16}+\frac{3\mathrm{{B}}_{\mathrm{{sa}}} }{16}\left(1-\upphi \left( {\frac{1}{\sqrt{2}}\frac{3}{8}} \right)\right)} \right] \nonumber \\&\Lambda _{\mathrm{{p}}} =-\frac{\mathrm{{B}}_{\mathrm{{sa}}}^2 }{640}\left(-1+\frac{\upphi }{\sqrt{2}}\right)^{2} \nonumber \\&\Lambda _{\mathrm{{z}}} =-\frac{3\mathrm{{B}}_{\mathrm{{sa}}}^2 }{640}\left(-1+\frac{\upphi }{\sqrt{2}}\right)^{2}\nonumber \end{aligned}$$

(A.16)

The factors $\Lambda _{\mathrm{{p}}}$ and $\Lambda _{\mathrm{{z}}}$ are quite small and are only present if image effects play a role. Very often, $\Lambda _{\mathrm{{p}}}$ and $\Lambda _{\mathrm{{z}}} << 1$ in a coverage range up to 50 %. In this case the quadrupole contribution vanishes, and only the dipole contribution remains. Note that as a results of the image effect the sum of the depolarization factors

$$\begin{aligned} 2\mathrm{{L}}_{\mathrm{{p}}} +\,\mathrm{{L}}_{\mathrm{{z}}} =1+\frac{\mathrm{{B}}_{\mathrm{{sa}}} }{3}\left( {\frac{1}{2}-\frac{\upphi }{\sqrt{2}}} \right) \end{aligned}$$

(A.17)

no longer equals 1. This rule is broken as a result of the image effect.