An analytic model for the optical properties of gold

P. G. Etchegoin, E. C. Le Ru, M. Meyer
2006 Journal of Chemical Physics  
The importance of gold ͑Au͒ as a substrate in biotechnology ͑and in the form of nanoparticles of different types 1 ͒ cannot be underestimated. Gold plays an important role in a myriad of plasmonics-type spectroscopies ͓such as surface enhanced raman scattering ͑SERS͔͒ where it is widely preferred for biological applications over silver ͑Ag͒ for its relatively easier surface chemistry, the possibility of attaching molecules via thiol groups, good biocompatibility, and chemical stability. The
more » ... cal properties of silver in the visible/near-uv range are fairly well represented by a simple Drude model, and several parametrizations of the dielectric function ⑀͑͒ have appeared in the literature 2 and have been used subsequently for different simulations. 3 One could argue that an analytic model for ⑀͑͒ is not really necessary, and that it is always possible to resort to interpolations of the experimental data. However, the advantages of having a realistic analytic representation of ⑀͑͒ with a small number of physically meaningful parameters are self-evident, not only for simulations but also to understand situations where the intrinsic parameters of the metal might be modified by external perturbations. Compared to Ag, the optical properties of Au are more difficult to represent in the visible/near-uv region with an analytic model. The reason for that is the more important role in the latter played by interband transitions in the violet/ near-uv region. Au has at least two interband transitions at ϳ 470 and ϳ330 nm that do play an important role and must be included explicitly if a realistic analytic model for ⑀͑͒ is sought. Their line shapes ͑coming from the joint density of states of the interband transitions 4 ͒ are not very well accounted for by a simple Lorentz oscillator ͑as in a simple molecular transition͒, and attempts to add simple Lorentz oscillators to a Drude term to account for the interband transitions very rapidly face limitations. 5 The reason for this is that the transitions have a somewhat asymmetric line shape than could only be accounted for with simple Lorentz oscillators by adding a series of additional "artificial" transitions, with the consequent increase in the number of ͑mostly un-physical͒ parameters. It is clear that with a sufficiently large number of Lorentz oscillators any line shape can be modeled, but the parameters of the model are then ill defined and have no real meaning; they do not provide more insight than a ͑equally unphysical͒ fit with high-degree polynomial, or a simple numerical interpolation of the experimental data. A different type of analytic model for the two interband transitions in gold in the violet/near-uv region has to be included to achieve a reasonable representation of ⑀͑͒ with a minimum set of parameters. It is possible to include a family of analytical models ͑called critical points͒ for transitions in solids, which satisfy a set of minimum requirements ͑like Kramers-Kronig consistency͒ and reproduce most of the line shapes in ⑀͑͒ observed experimentally. Critical point analysis of interband transitions in ⑀͑͒ is a subject with a longstanding tradition in semiconductors. 6 Following this approach, the frequency-dependent optical properties of Au in the visible/near-uv range can be very well represented by an analytic formula with three main contributions, to wit, where the first and second terms are the standard contribution of a Drude model 7 with a high-frequency limit dielectric constant ⑀ ϱ , a plasma frequency p , and a damping term ⌫, while G 1 ͑͒ and G 2 ͑͒ are the contributions from the aforementioned interband transitions ͑gaps͒. For the latter, we use the critical point transitions described in Ref. 8, namely, where the symbols have the following meaning: ͑i͒ C i , amplitude; ͑ii͒ i , phase; ͑iii͒ i , energy of the gap; ͑vi͒ ⌫ i , broadening; and ͑v͒ i , order of the pole. For Ϸ i the second term in ͑2͒ is less important than the first ͑it is the contribution from the pole in the negative frequency axis in the complex plane͒, but it needs to be added to the first term to ensure an expression which is fully Kramers-Kronig consistent. We assume throughout that the temporal variation of all fields goes as exp͑−it͒; the opposite sign implies a change in the sign of the imaginary part of ⑀͑͒ and is the convention mostly used in engineering. As explained in Ref. 8, critical points like ͑2͒ have the only drawback that they do not satisfy the "plasma sum rule" but this is not a limitation when they are used over a finite frequency range. The order of the poles ͑ i ͒ needs to be decided for each gap. In principle, the order is connected to the dimensionality of the Van Hove singularity in the joint density of states for the interband transitions. 4 In practice, however, many features in the dielectric function of solids involving several close transitions ͑and affected by broadening and limited experimental resolution͒ can be represented equally well by poles of more than one order. A pragmatic approach is to start with simplest order poles until a desired accuracy is obtained. The meaning
doi:10.1063/1.2360270 pmid:17092118 fatcat:wkd6vxe3xneanpfz6q7soz5kty