#### Abstract

Gold nanoparticles have been widely used during the past few years in various technical and biomedical applications. In particular, the resonance optical properties of nanometer-sized particles have been employed to design biochips and biosensors used as analytical tools. The optical properties of nonfunctionalized gold nanoparticles and core-gold nanoshells play a crucial role for the design of biosensors where gold surface is used as a sensing component. Gold nanoparticles exhibit excellent optical tunability at visible and near-infrared frequencies leading to sharp peaks in their spectral extinction. In this paper, we study how the optical properties of gold nanoparticles and core-gold nanoshells are changed as a function of different sizes, shapes, composition, and biomolecular coating with characteristic shifts towards the near-infrared region. We show that the optical tenability can be carefully tailored for particle sizes falling in the range 100–150 nm. The results should improve the design of sensors working at the detection limit.

#### 1. Introduction

The development of biosensors devices requires sophisticated approaches to detect and to identify the analytes [1–3]. Because the sensor signal-to-noise ratio increases with decreasing size for many devices, many researchers are expending considerable effort for miniaturizing sensing devices down to nanoscale size [3–7]. In the past decade several authors have analyzed the effect of flow, size, and adsorption isotherms on biomolecular adsorption and in a limited way the effects at nanometer length scales. The role of nanoscale size can be shown examining a simple sensor geometry (e.g., hemisphere) that absorbs the analyte [6, 8, 9]. The maximum number of molecules, , that can accumulate on a sensor due to irreversible adsorption may be determined by
where is the total flux (molecule s^{−1}), is the flux at the sensors (molecules s^{−1}m^{−2}), is the unit area, is the sensor area, and is time. Time dependence in (1) involves a crucial question for clinical applications; that is, if any analyte molecule that contacts the sensor surface can be detected, what is the minimum detectable concentration for a given accumulation time? Sheehan and Whitman [6] showed that DNA microarrays with a 200 *μ*m diameter hemisphere can detect ~1 fM in ~1 min, which is actually very close to the actual DNA detection limit, ~20 fM. Analogously, from (1) is possible to deduce the accumulation events
where is Avogadro’s number, is the initial solution concentration, is the radius of the sensing surface, and is a diffusion constant. It is remarkable that (2) is linear in both radius and time. This implies that when reducing the size of the absorbing surface, the time for a critical accumulation of events increases consequently. The advent of nanotubes and nanowires as biosensors introduces new geometries into the sensing activity with some changes to accumulation events as in (2).

The use of nanotechnologies for diagnostic applications meets the rigorous demands of clinical standards sensitivity with cost effectiveness. Today, main nanodiagnostic tools include quantum dots (QDs), gold nanoparticles, and cantilevers [10]. The effectiveness of nanoparticles as biomedical imaging contrast and therapeutic agents depends on their optical properties. Biosensing applications based on surface plasmon resonance shifts need strong resonance in the wavelength sensitivity range of the instrument as well as narrow optical resonance line widths. For actual *in vivo* imaging and therapeutic applications, the optical resonance of the nanoparticles is strongly desired to be in the near-infrared (NIR) region of the biological water window, where the tissue transmissivity is the highest [11–15].

The optical properties of gold nanoparticles in the visible and near-infrared (vis-NIR) domains are governed by the collective response of conduction electrons. These form an electron gas that moves away from its equilibrium position when perturbed by an external light field, thus creating induced surface polarization changes that act as a restoring force on the electron gas. This results in a collective oscillatory motion of the electrons similar to the vibrations of a plasma and characterized by a dominant resonance band lying in the vis-NIR for gold and called plasmon excitations. Thus the surface of gold nanoparticles can be used as a sensing element because when biomolecules attach to such surface the binding event can be revealed by optical changes. In addition, it is possible to calculate the maximum number of possible binding events and the consequent optical response when the shape and size of the gold particles are known.

In this paper, we will calculate and simulate the optical tunability of gold nanoparticles and core-gold nanoshells (optical changes with nanoparticles sizes) using the classical Mie theory and discrete dipole approximation (DDA). We will show that when biomolecules bind to gold surface optical shifts toward the near-infrared region are produced, and wavelengths changes can be accurately quantified. The accurate knowledge of the optical tunability could allow for improving sensing performance, sensitivity, and figure of merit of biosensors operating at detection limit in short time as requested in clinical applications.

#### 2. Nanoscale-Sized Systems for Detection Limit

The increased demand for sensitivity requires that a diagnostically significant interaction occurs between analyte molecules and signal-generating particles, thus enabling detection of a single analyte molecule. Nanotechnology has enabled one-to-one interaction between analytes and signal-generating particles such as QDs and gold nanoparticles. Here, we review briefly the most important nanoscale tools for detection limit, such as QDs, cantilevers, and gold nanoparticles [10, 16].

QDs are semiconductor nanocrystals, characterized by strong light absorbance, that can be used as fluorescent labels for biomolecules. A typical QD has a diameter of 2–10 nm and is usually composed of a core consisting of a semiconductor material enclosed in a shell of another semiconductor material with a larger spectral bandgap. When a QD absorbs a photon with energy higher than the bandgap energy of the composing semiconductor, an exciton (electron-hole pair) is created. As a result, a broadband absorption spectrum occurs because of the increased probability of absorption at shorter wavelengths. The recomposition of the exciton, generally characterized by a long lifetime >10 ns, to a lower energy state leads to emission of a photon with a narrow symmetric band [17–19]. QDs are reported to emit with lifetimes of 5–40 ns, whereas conventional dyes emit in a time less than 2 ns. This characteristic produces a strong and stable fluorescence signal. In addition, due to quantum confinement effect a direct relationship exists between the QD size and the values of the quantized energy levels. Methods for tracking and detecting QDs are numerous and include fluorometry and several types of microscopy, such as fluorescence, confocal, total internal reflection, wide-field epifluorescence, near-field optical microscopy, and multiphoton microscopy. The choice of detection techniques depends on the emission of wavelengths; for example, QDs composed by CdSe/NzS emit in the 530–630 nm range and InP and InAs QDs emit in the NIR range, while PbS emits in the 850–950 nm range and PbSe QD in the mid-infrared range.

Another nanoscale detection technique is based on cantilevers [20–22]. Cantilevers are small beams similar to those used in atomic force microscopy and they operate detection by use of nanomechanical deflections. An instructive example of operating cantilevers is the detection of DNA hybridization. The cantilever surface holds a particular DNA sequence capable of binding to a specific target. When the hybridization occurs with the cantilever single-stranded DNA, mechanical stress produced by the rebinding process underlying the hybridization deflects the cantilever, with a measurable deflection by the optical lever method proportional to the amount of DNA hybridized. This technique can be used as a microarray, allowing for multiple analyses. However, this method currently needs further development to solve the problem of nonspecific binding [23].

Gold nanoparticles and gold nanoshells provide great sensitivity for the detection of DNA antibodies and proteins [24–28]. The instrumental platform for the detection of biomolecules including gold particles is based on surface plasmon resonance (SPR). SPR is an optical technique that measures the refractive index of very thin layers of material adsorbed on a metal. It offers real-time in situ analysis of dynamic surface events and is capable of defining rates of adsorption and desorption for surface interactions. Plasmon-plasmon resonance, resulting from the interaction of locally adjacent gold nanoparticle labels that have bound to a target, produces changes in optical properties that can be used for detection. It is known that the characteristic red color of gold colloid changes to a bluish-purple color on colloid aggregation because of this effect. Analogously, the reduction of gold particle sizes involves shifts toward the NIR region [29]. Raman spectroscopy is a favored detection method using silver in the visualization process. In this case, gold nanoparticles can be coated with silver shells; silver-coated gold particles less than 100 nm in size have strong light-scattering properties and can easily be detected by optical microscopies operating at the detection limit for oligonucleotides down to ~10 fM, that is, nearly 50-fold lower than conventional fluorophore-based methods. Gold nanoshells could allow direct, rapid, and economically feasible analysis of whole blood samples. Generally, a nanoshell consists of concentric spherical nanoparticles with a dielectric core, meanly consisting of gold sulfide or silica, surrounded by a thin gold shell. Variations in the relative thickness of the core and outer shell allow the optical resonance of gold to go into the midinfrared region. Other relative surface properties can be used to focus the absorption wavelength range into the near-infrared, just above the absorption of hemoglobin and below that of water [30]. This characteristic aids in avoiding interference from hemoglobin giving the possibility to do a direct analysis of whole blood. An additional advantage of gold nanoshells is the excellent biocompatibility.

The optical tunability of gold nanoparticles and core-gold nanoshells resides in plasmon resonance properties. It is well known that the plasmon resonance of metal nanoparticles is strongly sensitive to the nanoparticles’ size and shape and the dielectric properties of the surrounding environments. We will simulate the optical tunability of gold nanoparticles and gold nanoshells (optical changes with nanoparticles’ sizes and surface biomolecular coating) using the classical Mie theory and DDA-modified approach. Finally, we will obtain the wavelength shift as a function of different size dimension, environments, and biomolecular coating layer of the gold nanoparticles. The accurate knowledge of the optical tunability and wavelength shifts should allow for development of biosensors operating at detection limit in short time as requested in clinical applications.

The design of biosensors working at the detection limit must attempt to compare sensing performance, sensitivity, and figure of merit (FOM) [31]. The sensitivity is defined as the ratio of the resonant wavelength shift to the variation of the surrounding refractive index , while the FOM is defined as the ratio of the refractive index sensitivity to the resonance width : Resolution is typically defined as the minimum detection limit. In addition, sensitivity and thence resolution can be modified by size and geometry of the gold nanoparticles and nanostructure environment. In general, the adsorbed mass on the sensor surface can be approximated by the De Freijter’s formula, which is based on the refractive index change [32] where is the dimension of the adsorbed species (biomolecule) falling in the range of nanometers, is the refractive index difference between the medium and the species, and is the correspondent biomolecular refractive increment. Analogously, the spectral response of the refractometric nanoplasmonic sensors can be described by where is the refractive index sensitivity, expressed in plasmon resonance peak shift per refractive index unit (RIU), and is the effective refractive index of the adsorbate layer. In the simplest approximation, the surface plasmon resonance-induced evanescent field decays exponentially from the surface as with a decay length . The effective refractive index of the adsorbate layer is described as Using (5)-(6) into (4) we obtain a relationship of the surface plasmon resonance peak shift with the surface coverage of the adsorbate: In the next section, we will provide the accurate calculation of wavelength shifts due to increased size of gold nanoparticles and coating of gold surface in core-gold nanoshells. Equation (7) establishes that the accurate knowledge of such shifts gives the possibility to design gold nanoparticles based sensors working at the biomolecular detection limit.

#### 3. Optical-Infrared Response of Gold Nanoparticles and Core-Gold Nanoshells: Calculation and Simulation Methods

The complexity of the electromagnetic field in the presence of arbitrarily shaped nanoparticles is such that Maxwell’s equations must be solved using numerical methods [33]. The far-field performance of a nanoparticle is summarized in the wavelength-dependent absorption and scattering cross sections, and as a consequence, the optical properties of gold nanospheres and silica-gold nanoshells will be quantified in terms of their calculated absorption and scattering efficiency. The light intensity transmitted through a dilute dispersion (assuming no more than one photon-particle collision per photon) is , where is the incident light intensity, is the number of particles per unit volume, is the path length inside the dispersion, and and are the wavelength-dependent absorption and scattering cross sections, respectively. The optical modeling of nanoparticles thus relies on the solution of Maxwell’s equations for each specific geometry and set of illumination conditions, assuming a local dielectric function, , and description of the materials involved. When dipolar contribution in plasmon excitation is prominent, then the far-field cross sections can be obtained from the polarizability using the following expressions [34]: where is the environmental permittivity of the medium outside the particle and is the light wavelength.

To simulate the optical properties of gold nanoparticles, many methods can be found. Mie theory is one of such methods [35]. Mie scattering theory begins with Maxwell’s equations and the necessary boundary conditions, which are then transformed into spherical polar coordinates with a solution of the wave vector equation emerging. The coefficients for scattering arise and the theory is extended to the far-field solution based on a plane constructed from the incident and scattered waves. At this point, the Stokes parameters that are the measurable quantities and the formulation of matrix plane can be quantified. From Mie theory, once scattering matrices have been derived, information about the direction and polarization dependence of the scattered light can be extracted, so that absorption, scattering, and extinction cross sections for any arbitrary spherical particle with dielectric function can be calculated. Since extincted power is the sum of the scattered and absorbed power, the absorption cross section is simply , while the scattering and extinction cross sections can be calculated from the total cross section of a spherical particle, that can be written as where are the electric and magnetic scattering coefficients, respectively, and is the orbital momentum number (e.g., for a dipole). These coefficients admit analytical expressions in terms of spherical Bessel and Hankel functions for a homogeneous sphere [33] where , , and the prime represents the first differentiation with respect to the argument in parentheses. It should be noted that plasmon modes are related to the poles of . In fact, the magnetic contribution to scattering produces only resonances for particle sizes comparable to the wavelength. The multipolar polarizability is proportional to , and in particular, the dipolar polarizability reads , where the dipolar electric scattering coefficient can be readily calculated using and , so that (8) can be used instead of (9). The value of obtained is really accurate for describing gold spheres with diameters up to 200 nm and simplified expression can be found as, for example, where is the particle volume, and are the permittivities of the particle and the surrounding medium, and is the size parameters that recover the electrostatic limit for . Analogously, the dipolar polarizability of a coated gold nanosphere is given by where is the internal radius of the core material described by , the coating of permittivity extends up to a radius , and the medium outside the particle has permittivity . An analogous formula can be derived for a barium titanate core particle with a gold nanoshell. Calculations of the optical absorption and scattering efficiency of gold nanospheres and barium titanate-gold nanoshells will be presented below. The required parameters for the code were the value of the core and shell radii and , the complex refractive indices for the core, shell, and biomolecular coating, and the surrounding medium , , , and , respectively, Figure 1.

**(a)**

**(b)**

The discrete-dipole approximation (DDA) has been used as a complimentary method to Mie theory [36]. DDA is a flexible and powerful technique for computing scattering and absorption by targets of arbitrary geometry. DDA calculations require choices for the locations and the polarizabilities of the point dipoles that represent the targets. To approximate a certain geometry (e.g., a sphere or a core-shell-environment system, where the shell is made by gold) with a finite number of dipoles, we might consider using some number of closely spaced, weaker dipoles in regions near the target boundaries to do a better job of approximating the boundary geometry. For a core-shell system, we use the following algorithm to generate the dipole array. Given a coordinate reference system, we generate a trial lattice defined by a lattice spacing and coordinates of the lattice point nearest the origin. All lattice sites are located within the volume of the core-shell-surrounding system (where the surrounding is considered as a sphere enveloping the core-shell system). Try different values of and dipoles coordinates and maximize some goodness-of-fit criterion for a list of occupied sites . Each of these occupied sites represents a cubic subvolume of material centered on the site. Particular efforts have been addressed to distinguish between lattice sites near the surface and those in the interior. We therefore rescale the array by requiring that , so that the volume of the occupied lattice sites is equal to the volume of the original target. For each occupied sites, assign a dipole polarizability .

The assignation of dipole polarizabilities is a crucial question in a DDA method. Following the seminal paper of Draine and Flatau [37], the polarizability can be found analytically in the long-wavelength limit as a series expansion in the powers of , with the criterion that where is the refractive index and is the effective radius of the nanoparticles of volume . As a consequence, for materials with higher refractive indexes the DDA method can overestimate absorption cross sections. A customized code for the DDA calculations of gold nanoshells has been written adapting DDSCAT program that is a freely available code [38, 39].

The finiteness of the speed of light has important consequences that affect the response of gold nanoparticles. First of all, the electromagnetic field cannot penetrate beyond a certain depth inside the metal, the so-called skin depth, which is of the order of 15 nm in the vis-NIR. But more importantly, redshifts take place as the particles size increases, and retardation effects play a significant role when the diameter is a consistent fraction of the mode wavelength in the surrounding medium, which is related to the free-space wavelength through . In particular, opposite charges are separated by roughly one particle diameter in a dipole mode, so that the reaction of one end of the particle to changes produced in the other end takes place with a phase delay of the order of , and consequently, the period of one mode oscillation increases to accommodate this delay. Analogously, quadrupoles and higher-order modes produce more nodes in the distribution of the polarization charges induced on the surface of the particle, which reduce the effective interaction distance.

Figure 2 shows the shift towards the NIR region for the efficiency of the absorption calculated using the Mie theory and DDA approach for a gold nanoparticle (radius 120 nm, black line) and a barium titanate (BaTiO_{3})-gold nanoshell (core 100 nm, gold 20 nm, red line). The barium titanate refractive index was taken to follow the dispersion formula as . The refractive index of the surrounding medium was considered to be at all wavelengths, close to the water.

Optical tunability plays a fundamental role for the identification of size, shape, and core-shell composition of the nanoparticles for biomedical sensors and detection limit definition. In order to define the optical tunability, the red shifts must be carefully identified. In Figure 3, the tunability of the extinction cross section of a gold nanoparticle as a function of the diameter size is reported, while in Figure 4, the same quantity for a core-gold nanoshell (core = BaTiO_{3}, gold shell = 20 nm) is reported as a function of different core/gold ratios. In the case of a gold nanosphere, the total extinction cross section increases linearly as the nanosphere size increases, with an initial value of for a gold nanosphere diameter of 40 nm. Approximately, linear behavior of total extinction cross section is waited for core/shells systems, Figure 4.

The increase in the extinction scattering with the nanoparticle volume has been related to increased radiative damping in larger nanoparticles based on experimental scattering spectra of gold nanospheres and core-gold systems [29–33]. This behavior suggests that larger nanoparticles of high core/shell ratios would be more suitable for biological sensors applications based on light scattering. In the case of nanoshells, the relative core-shell dimensions vary linearly the magnitude of light extinction. This implies that a decrease in the core/shell ratio can be seen to be an effective handle in increasing the scattering contribution to the total extinction. The next calculated quantity is the effect of a biomolecular adsorbed layer on the gold surface for a core/shell system with fixed ratio (80 nm for BaTiO_{3} and 20 nm for gold surface) on wavelength shifts for different layer dimensions. This quantity is particularly important for the definition of sensitivity response and FOM of a biosensor, as described by (3). In Figure 5, the progressive plasmon resonance maximum wavelength red-shift is shown as a function of increased homogeneous layers of adsorbed bovine serum albumin (BSA, 170 mg/mol). The calculation was made, as in the previous case, using (8) and (12), but including in the DDA approach progressive 1-to-3 supplemental layers with a known refractive index [40]. It is interesting to note that when increasing the size of the nanoparticle (from left black line to cyan line on right) an increasing wavelength red-shift is observed. The reduction of the peaks is an artifact effect and the absorption quantity should be normalized including the increasing nanoparticle volume.

Figure 5 represents the most important result of the present paper. In fact, once the red-shift, , is accurately known, it should be possible to quantify the refractive index changes, , (7), to design high sensitive biosensors operating at detection limit with effective optimal factor. The absolute magnitude of the optical cross section provides a limited reliable measure of the optical properties of a population of nanoparticles employed in the biomedical sensing activity or other real-life biomedical applications because broad size nanoparticle population can have a different optical response with respect to few nanoparticles but with higher sizes. The red-shifts as a function of nanoparticles diameter size, shown in Figure 5, are one quantity that can be measured in scanning near optical microscopy (SNOM) technique, measurements that will provide the next development of the present paper.

#### 4. Conclusions

The development of biosensors devices requires sophisticated approaches to detect and to identify the analytes. Because the sensor signal-to-noise ratio increases with decreasing size for many devices, many researchers are expending considerable effort for miniaturizing sensing devices down to nanoscale size. Gold nanoparticles have been widely used during the past few years in various technical and biomedical applications. In particular, the resonance optical properties of nanometer-sized particles have been employed to design biochips and biosensors used as analytical tools. The optical properties of nonfunctionalized gold nanoparticles and core-gold nanoshells play a crucial role for the design of biosensors where gold surface is used as a sensing component. Gold nanoparticles exhibit excellent optical tunability at visible and near-infrared frequencies leading to sharp peaks in their spectral extinction. In this paper, we have simulated how the optical properties of gold nanoparticles are changed as a function of different sizes, shapes, and composition. We have shown that the optical tunability can be carefully tailored for particle sizes falling in the range 100–150 nm. Nanoshells made by a barium titanate core have been considered and the optical response of gold surfaces coated by 1-to-3 layer of BSA was taken in consideration. The simulated results will help experimental optical tests that will be performed using nanoscale optical scanning technology.

#### Acknowledgment

M. D’Acunto wishes to acknowledge the NanoICT Project for useful support.