3 January 2013
Both metallic and semiconductor nanoparticles exhibit surface plasmon resonance (SPR) when excited by an EM field at certain frequencies. SPR, the collective oscillation of the conduction band electrons, can be utilised to confine EM energy near a conductor-dielectric interface in sub-wavelength dimensions. At optical frequencies, SPR in metal particles has allowed construction of sub-wavelength waveguides, superlenses and optical circuit components. Because of the lower carrier density in semiconductors, SPR occurs around the terahertz, and far-infrared bands, instead of the visible and UV range as in metals. With recent advancements in terahertz technology there is a need for efficient signal interconnect, active and passive circuit components in this range.
Traditional models of conductive nanoparticles are based on the classical solution obtained by Mie in 1908, solving Maxwell’s equations for scattering and absorption of EM waves by a sphere. For dielectric materials like solids and liquids containing polar molecules, Mie theory has been employed with immense success in understanding the interaction of light with the molecular species, providing insight into characteristics including their dynamics and hence the chemical bonds. When mobile charges are present, their effects on the absorption and scattering of waves can be accounted for by using a term in the dielectric function to form an effective complex permittivity. However, in original form, the Mie resonance condition cannot explain the shift in the absorption peak with change in particle size.
Another limitation of the theory, although not an intrinsic one, is reliance on a dielectric function to account for the motion of mobile charges, often represented by a bulk conductivity assumed to be constant over the sphere’s interior. But unlike bound charges as in molecules, electrons can move from one side of the sphere to the other; screening out the electric field, given enough time to accumulate. Revealed at low frequencies, in such regimes the sphere’s conductivity is not uniform and the fixed conductivity assumption used in forming the complex permittivity breaks down; surface effects dominate the bulk properties of the material.
To properly account for charge carrier motion in a conductive particle, transport equations must be employed in conjunction with Maxwell’s equations. This allows the velocity and concentration of the charge carriers to be functions of position and time, relieving the constraint of a uniform conductivity across the particle.
Based on such a transport formulation we have obtained analytic and numerical results for the polarisation induced on the particle by a terahertz electric field. The dipole moment derived from the response enables systematic construction of an equivalent circuit capturing the field-carrier interaction in the semiconductor particle. The physical origin of each circuit element can be traced to the charge species in the particle and their role in the polarisation process. They can be expressed in terms of material parameters and particle radius. The equivalent circuit enables the analysis of composite materials containing multiple semiconductor nanoparticles using network theory and effective medium theory.
With circuit elements expressed in terms of material parameters one can readily obtain the induced dipole moment on the nanoparticle without having to directly solve the computationally intensive EM problem. The saving in computational effort is even more substantial when aggregates of nanoparticles are analysed. In the same way as the scattering parameters facilitate the analysis of a complex microwave network, the generalised impedance of the nanoparticle simplifies calculation of the induced dipole moment on an individual nanoparticle as well as an aggregation of it.
This enables one to determine the effects of the nanoparticle on its surroundings, which can range from a cluster of similar nanoparticles to waveguiding structures and signal processing components in the terahertz frequency range. The equivalent circuit can also be employed to guide the synthesis of new nanoparicles with heterogeneous internal structures to achieve novel polarisation properties for sensing and terahertz circuitry applications.
The work was prompted by the observation that complex permittivity functions encompassing bulk conductivity (or equivalently complex conductivity functions) lead to an unrealistic induced dipole moment at low frequency for a nanoparticle containing mobile charges. We also knew from our earlier work on planar waveguides with semiconductor substrates that field screening effects of the charge carriers can have significant influence on the guided wave and the field distribution. A formulation based on simultaneously solving the transport equations of the charge carriers and the Maxwell’s equations is the key to resolving these issues. The framework for the analysis and numerical simulation was built by a former team member, M. Yan, the current team focuses on computation efforts and the quest for closed form solutions that provide physical insights. Because of the rapid variation in the charge density in the region close to the surface, judicial choice of gridding at the surface is vital for numerical stability and accuracy.
In constructing the equivalent circuit, many configurations were proposed and most abandoned for failing to meet all the requirements. For example, some had correct frequency response but contained physically unrealisable elements, such as frequency dependent resistors. We begin with choosing circuit topology by considering the physical process, then assessing their asymptotic response at the static and high frequency limits. Algebraic forms of expressions for the elements in terms of physical parameters and frequency are proposed and refined by numerical fitting to response functions given by analytic solutions and EM simulations. Once the functional forms meet the response requirements, they are further checked for physical relevance and sensitivity to parameter variations.
We are studying equivalent circuit development for core-shell structured nanoparticles and the influence of a static field superimposed on the dynamic field. An important component in applying network theory in treating aggregates of nanoparticles is an accurate circuit model for the coupling of the field between two or more nanoparticles with controlled spacing between them. This is a formidable task, but with the rapid development in EM simulation tools and many excellent studies on related configurations by analytic methods in the literature, we are optimistic that our efforts will soon bear fruit.
We have long term collaborations with Argonne National Laboratory and Fermilab on new concepts in particle accelerators and microwave instrumentation, including development of a micro-machined terahertz spectrometer on planar structure to study molecular dynamics in aqueous solutions (image above). This is expected to substantially reduce required sample size compared to free-space measurement methods.
This article is based on the Letter: Transport Based Equivalent Circuit for a Semiconductor Nanoparticle in the Terahertz Frequency Range [new window]
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