The combined optical and magnetic properties of magnetic-plasmonic core-shell nanoparticles (NPs) makes them ideal candidates for many applications in biomedical fields. Plasmonic properties of the shell gives rise to Surface Enhanced Raman Scattering (SERS) that can be utilized for sensitive detections, while magnetic properties are useful for magnetic separation and magnetic guided delivery. The plasmonic properties of the shell depends on both the size and shape of the core and shell, and this property, in principle, can be calculated using the Discrete Dipole Approximation (DDA) method. However, since the DDA is an approximation method, its accuracy to calculate the plasmonic properties of the shell, especially the near-field enhancement relevant to SERS, has not been examined carefully. We present a systematic test on the accuracy of the DDA to calculate the plasmonic properties in terms of both the extinction spectra and the near-field enhancement of the magnetic-plasmonic core-shell NPs. Accuracy of the DDA method was first investigated in comparison to Mie theory results for spherical core-shell NPs, since Mie theory gives the exact solution to spherical shaped particles. DDA calculations were further extended to core-shell nanoparticles with octahedral cores. We elucidate convergence of the DDA results by considering the effects of dipole distance and shell thickness in regard to the NP spectral properties. This work validates application of the DDA methods for calculating electrodynamic properties of core-shell NPs and highlights plasmonic properties of core-shell with non-spherical cores.

## INTRODUCTION

Magnetic-plasmonic core shell nanoparticles (NPs) possess dual magnetic and plasmonic properties and have widespread applications in biomedical fields.^{1–5} The magnetic cores such as iron oxide (IO) are greatly desired for applications such as magnetic separation, magnetic resonance imaging or magnetic guided drug delivery. The IO-cores can be chemically stabilized by coating them with noble metals, such as Au, that not only provides a chemically inert surface, but also introduces interesting plasmonic properties which can be utilized for sensing, imaging, and photothermal therapy. Bhana et al^{6–8} developed IO-Au core-shell NPs that are capable of dual enrichment and detection of circulating tumor cells (CTCs) resulting in an increased sensitivity to CTC detection. In that study, the IO-Au NPs were coated with Raman active reporter molecules on an Au-shell that gives rise to Surface Enhanced Raman Scattering (SERS). Simultaneously, the magnetic core allows the NPs to pull down CTC via antibody targeting, hence achieving dual enrichment and detection in one nanoconstruct. The SERS feature of the NPs provide spectral fingerprint-like signals and is far-superior than fluorescence-based detection methods. The SERS observed on plasmonic NPs is primarily a result of the enhanced electric field near the metal surface due to the collective oscillations of the free electrons in metallic NPs upon exposure to the oscillating electromagnetic fields of incident light.^{8,9} One may define a near-field enhancement factor, *R*(*ω*) = |** E**(

*ω*)|

^{2}/|

*E*_{0}(

*ω*)|

^{2}, where |

*E*_{0}(

*ω*)| is the incident electric field intensity, |

**(**

*E**ω*)| is the electric field near the plasmonic shell surface, and ω is the frequency of the electromagnetic radiation. The SERS enhancement factor is approximately equal to square of

*R*(ω) if the difference in frequencies between the incident light and emitted Stokes-shifted Raman signal is ignored. Depending on the experimental setup, the relevant data associated with the SERS intensity is either the enhancement

*R*(

*ω*) averaged over the particle surface <

*R*(ω)>, or the maximum

*R*

_{max}(ω) in the case of single molecule SERS. Both quantities are of interest to examine.

^{8}SERS detection using such magnetic-plasmonic core-shell NPs have many promising applications in nanomedicines.

^{2,10–15}

Plasmonic properties of Au shell NPs can be tuned by changing the size and shape of both the core and the overall particle.^{16–18} Optical property studies through experimental techniques such as absorption spectroscopy (UV-Vis), atomic force microscopy (AFM), scanning tunneling microcopy (STM), and surface enhanced Raman scattering (SERS), combined with theoretical/computational calculations, are capable of providing great understanding on the structure/property relationships.^{19} Several numerical methods are available that calculate electrodynamic properties of arbitrarily-shaped core-shell nanostructures including Boundary Element Methods (BEM), Finite Element Methods (FEM), Finite Difference Time Domain methods (FDTD) and the Discrete Dipole Approximation (DDA).^{20–25} The DDA was initially developed to quantify light scattering properties of dust particles,^{26} and was applied to metallic NPs to calculate extinction spectra and Raman intensities of a variety of systems by Schatz and coworkers.^{27–31} They reported extinction spectra as a function of increased dipole number as well as Raman enhancement convergence for isolated, coupled spheroid, and tetrahedron particles on a flat surface.^{27} The DDA is also applicable in calculating surface plasmon resonance (SPR) in core-shell spherical nanoparticles revealing that shells in such systems greatly influence SPR properties.^{25} But applicability of the DDA to core-shell nanoparticles with arbitrary shapes has not been examined carefully. In the present work, we highlight the use of the DDA to calculate spectral properties of IO-Au core-shell NPs with non-spherical shapes. We have restricted studies to 35 nm iron oxide (Fe_{3}O_{4}) cores of different shapes, coated with varying gold shell thicknesses, since our goal is to test the applicability of the DDA to calculate the near-field enhancement. Our work tests applicability of the DDA methods for calculating electrodynamic properties of core-shell NPs and highlights impact of core shape on plasmonic properties of core-shell NPs.

## THEORY

### Extended Mie theory

Spherical geometric restriction of NPs allows for calculations of light scattering and absorption utilizing Mie theory, which, provides exact solution to the Maxwell equations.^{32} Extension of Mie theory to core-shell spheres was first introduced by Aden and Kerker^{33} and was later extended to multilayer spheres using a recursive algorithm.^{34,35} Briefly, the Maxwell equation is solved for a multilayer sphere, where each layer is characterized by a relative refractive index *m*_{l}*=N*_{l}*/N* where *N*_{l} and *N* are the refractive indices of the *l*th layer and of the medium outside the sphere, respectively, with *l*=1,2…*L*. The magnetic permeability is assumed to have the free space value μ=μ_{0} everywhere. The incident wave is polarized along the *x* direction propagating in the *z*-direction. Following the notation outlined in Bohren and Huffman’s book,^{36} the scattering electric field $ E \u2192 l $ and magnetic field $ H \u2192 l $ in the *l*th layer is expanded as the sum of incoming and outgoing parts:

where *E*_{n} = *i*^{n}*E*_{0}(2*n* + 1)/*n*(*n* + 1), ω is angular frequency of the light, and $ M \u2192 e 1 n j $, $ M \u2192 o 1 n j , N \u2192 e 1 n j , N \u2192 o 1 n j $ (*j*=1,3) are the vector harmonic functions with the radial dependence given as a spherical Bessel function *j*_{n}(*k*_{l}*r*) for *j=*1, and $ h n 1 k l r $ for *j*=3. The four coefficients in the above expansion, $ a n l , b n l , c n l $ and $ d n l $ can be determined by the boundary condition, namely, the tangential component of electric fields and magnetic fields have to be continuous at the boundary. In the first layer, 0 ≤ *r* ≤ *r*_{1}, there are no outgoing fields, so the coefficients, $ a n 1 = b n 1 =0$. In the region outside the sphere, the *L*+1 layer, the total external field is the sum of incident field and scattered, $ E \u2192 = E \u2192 i + E \u2192 s $, where the incident and scattered fields can also be expanded as:

Comparing Equation (1) with Equation (3) and (4), one can see that $ a n = a n L + 1 $, $ b n = b n L + 1 $, $ c n L + 1 = d n L + 1 =1$. The extinction coefficient, *Q*_{ext}, due to absorption and scattering of electromagnetic radiation by the particle, is given by *Q*_{ext}= *Q*_{a} + *Q*_{s} where *Q*_{s} and *Q*_{a} are defined as follows:

Here, the integral is taken over any hypothetical surface that enclose the entire particle but that is also far enough removed from the particle surface, and *R* is the radius of the particle. This scattering component is also referred to as the far-field, as only this component can reach a distance much further away from the particle. The above integral leads to the familiar equation for *Q*_{ext}:

The electric field near the particle surface contributes to the surface enhancement of Raman signals. One can calculate *Q*_{NF} according to the following integral.^{37}

The above integral is the average of the scattered electric field $ E \u2192 s 2 $ near the particle surface. Substituting the expansion of scattered electric fields in Eq. (4) into this integral, one gets

where $ h n 2 x $ is the spherical Hankel function of the second kind. This *Q*_{NF} is the surface-averaged near-field enhancement factor, EF= $R \omega = E 2 / E 0 2 $ where $ E 2 $ is the electric field averaged over the particle of surface (note the Q_{NF} calculated here differ by a factor 4 from that in Ref. 37).

The coefficients in the expansions in Eq. (1)–(4) are obtained through the recursive algorithm suggested by Yang.^{34} The readers should refer to the original article by Yang et al. for the details of how a recursive algorithm allows for the determination of these coefficients for each dielectric layer of the NP of interest. We implemented this algorithm and have examined the impact of the core dielectric properties on the plasmonic properties in previous publications.^{38,39}

### Discrete dipole approximation

Extinction spectra and near-field enhancement factors of Au and IO-Au core-shell NPs were also calculated using the DDA method, implemented in the DDSCAT 7.3 software package.^{40–42} The DDA is an approximation method used to solve Maxwell’s equations where exact solutions, such as Mie theory, are not available due to the shape of the target material. The DDA method has been widely used and is discussed in detail elsewhere.^{10,38,39} Computed electric fields, *|**E*_{j}*|/|*E_{0}|, were obtained for all dipole locations within simulated particle volumes and in an extended volume surrounding the particle and plotted with Paraview to visualize E-field enhancements. Dipole distance of simulated particle systems, as a function of convergence to Mie theory, is of interest in this work as this variable is directly linked to computational expense and accuracy. The DDA calculations were performed using incremental dipole distances to check the effect of dipole density for Au and IO-Au NPs. The average near-field enhancement, EF= $<R \omega >=< E \omega 2 >/ E 0 \omega 2 $ is determined by averaging over all dipoles one lattice point away from the simulated particle surface.

We simulated the extinction and near-field spectra for spherical gold nanoparticles (50 nm) and extended the calculations for core-shell IO-Au nanoparticles, which, were compared to Mie theory. An octahedral IO-core was also used to compare against the spherical IO-core to elucidate the impact of core shape on the spectra. Gold shells and IO cores were modeled using optical constants determined by Johnson and Christy^{43} and Goossens et al.,^{44} respectively. A house developed Fortran 90 code was used to generate particle representations as material-specific point dipoles on a cubic lattice for the subsequent DDA calculations.

## RESULTS & DISCUSSIONS

### Spherical solid Au nanoparticles

Figure 1a presents the DDA computed extinction spectra of 50 nm Au NPs as a function of dipole distance (0.25nm, 0.5nm, 0.75nm and 1 nm), while Figure 1b shows corresponding results for average near-field enhancement factor spectra. The near-field EF is defined as the averaged E-field intensity, <|*E*^{2}|>/|*E*_{0}|^{2}, for all surface sites, where *E*_{0} is the incident E-field intensity used in the calculation. Surface sites in the DDA calculations are defined as any medium lattice site where one of its nearest neighbors is occupied by the particle (or one lattice point removed from particle surface). A smaller dipole distance corresponds to a finer grid spacing, and an overall higher number of dipoles used in the DDA calculations, which, usually leads to a more accurate result. Convergence of the DDA results toward expected Mie results, in terms of extinction spectra, is quickly obtained without the need of increased dipole density as there is essentially no difference in computed DDA spectra (Figure 1a). The DDSCAT manual suggests using a criterion of |*m*|*kd*<1, or a more conservative criterion of |m|k*d*<0.5, to provide a resulting dipole distance (*d*) that should generate accurate results. According to this criterion, the desired *d* for a 50 nm Au NP is approximately 17 nm or 8 nm, respectively. The dipole distance used in this study is much smaller than the suggested value in the DDSCAT manual. Not surprisingly, the extinction spectra calculated is essentially indistinguishable from Mie theory. The near-field enhancement spectra, however, does not converge to anticipated Mie theory results as quickly as the extinction spectra (Figure 1b). The most noticeable difference is the presence of a shoulder peak at approximately 680 nm in the DDA spectra. This peak diminishes slightly when a smaller dipole distance is used, but does not go away entirely.

The near-field EF determines the surface enhancement of Raman molecules adsorbed on the nanoparticles. When comparing experimentally measured and computed EFs, there are some complications. Experimentally observed EFs could be a result of a single surface molecule located at a hot spot where EF is maximized, or it could be based on the average of EF over all molecules adsorbed on the particle surface. Assuming Raman molecules are adsorbed on the Au NP with uniform monolayer coverage, the relevant EF should be equated to the <|E^{2}|> averaged over all surface points. We see from the above data, that the average <|E^{2}|> value is approximately 15 at 550 nm, but if the EF is coming from the hot spot, then the highest |E^{2}| value is a better representation of experimental conditions. Figure 2A presents a plot of all EF values for all surface points obtained with two different dipole distance values, *d* = 0.4 nm and *d* = 1.0 nm, at λ=550 nm. The maximum |E^{2}| value increases from 70 to 90 when the lattice size is adjusted. Hao and Schatz have also shown that using a dipole distance of 0.25 nm, in comparison to 1.0 nm, leads to a 2 to 3-fold increase in E-field intensity near the surface.^{29} This indicates some challenges to when trying to compute the relevant SERS EF using computational approaches. The precise magnitude of the EF due to hot spots is dependent on the computational details such as grid space used in DDA. Figure 2B compares the histogram distribution of all |E^{2}| values over the surface points with different dipole distances. As the dipole distance decreases, the histogram did not change much, except it has more high E-field tail. The EF values averaged over the particle surface however did not change significantly as seen in Figure 1.

### Spherical core-shell IO-Au nanoparticles

Next, we compare the accuracy of the DDA calculations to Mie theory for spherical magnetic plasmonic core-shell NPs. The first testing system is a spherical IO-core with a 35 nm diameter and 10 nm Au shell. Figure 3A compares computed extinction spectra for spherical IO-Au core-shell NPs based on the DDA calculations with dipole distances varied between 0.5 nm and 1.0 nm, respectively. Figure 3B presents corresponding results for the average near field enhancement factor *R*(ω) of the same systems. The DDA results converge with Mie theory results regardless if dipole distance is at 1 nm or 0.5 nm. Comparison of extinction spectra and near-field results of these IO-core NPs to 50 nm Au NP results (Figure 1) reveals an approximate 100 nm red-shift with the introduction of the IO core. The dielectric properties of the core material significantly impact both the peak position and intensity of the extinction spectra.^{39} The DDA results in Figure 3 are in excellent agreement with Mie theory, and are independent of the two dipole distances studied, thus demonstrating that the DDA method can accurately describe the electrodynamics of such core-shell nanoparticles.

Similarly, we computed optical properties of spherical IO-Au core-shell NPs with a shell thickness of 5 nm using the DDA method and compared these results to Mie theory. As the shell thickness decreases, the extinction spectra and near-field enhancement spectra are further red shifted for another 100 nm (Figure 4A and 4B). An earlier study has suggested that for IO core, the plasmon peak red shift as the shell thickness increases, in contrast to silica core.^{45} This suggestion however was not correct. Chaffin et al have investigated impact of core dielectric properties on the plasmonic behavior of core-shell nanoparticles and have shown that the shift in plasmonic peak with shell thickness need to consider the overall size of the particle.^{39} The average near-field enhancement factor shows an increase (Figure 4B) in comparison to solid Au NPs (Figure 1) and with 10nm Au shell (Figure 3B). Unlike in the case of 10nm Au shell, the DDA calculation with dipole distance 1.0nm clearly is not agreeing with Mie theory prediction. Reducing the dipole distance to 0.5 nm gives a better agreement with the Mie theory. This demonstrates that for thin shell, a smaller dipole distance is needed to obtain accurate results from DDA calculation.

### IO-Au core shell nanoparticles with octahedral core

DDA calculations are extended to predict extinction and near-field properties of core-shell NPs with an octahedral core for which Mie theory cannot be used. Figures 5 presents the extinction and near-field enhancement factor with an octahedral core with edge length of 35 nm compared with a spherical core of 35 nm. The volume of the two cores in this comparison are not equal as the volume of an octahedron is given by *V*∼0.471*a*^{3}, with *a* being the edge length, and the volume of spherical core is given by *V* ∼ 0.523*d*^{3}, with *d* being the diameter of the sphere. The Mie theory results for spherical core are included in Figure 5 merely as a reference. The shell thickness for these systems is approximately 10 nm (the overall particle diameter is fixed at 55nm in both cases). The two dipole distances used in DDA calculations did not influence the results. The extinction spectra are broader for octahedral core-shell NPs than for the spherical core-shell NPs. The near-field EF spectra are also broader for octahedra cores than for the spherical cores.

Figure 6 compares the DDA results for IO-Au core-shell nanoparticle with 35 nm octahedra core to spherical IO-Au nanoparticles with equal core volumes. The spherical core now has a diameter of 33.8 nm, instead of 35nm, to allow for better comparison. These two figures reveal similar patterns as seen in Figure 5, the extinction spectra for octahedra core is broader in comparison to the spherical core, but the two peak positions are now aligned together at a similar wavelength. The near-field EF spectra for the octahedra core is slightly red shifted than the spherical core. The near-field spectra obtained from the DDA calculations for the spherical core has a shoulder peak and is not in full agreement with Mie theory, a similar problem encountered for single spherical 50nm Au NP (Figure 1B).

Although the average near field enhancement spectra obtained from octahedra core-shell NPs is similar to the spherical core-shell NPs, a closer look at the electric field around the particle reveals some difference. Figure 7 shows plots of the electric field intensity |E|^{2}/|E_{0}|^{2} maps for the systems highlighted in Figure 6. We compare the electric field maps between octahedra core with 35nm edge and spherical cores with 35nm diameter (Figure 7A and 7B) and with spherical core with equal volume (diameter 33.8nm) (Figure 7C). For the spherical IO-Au NP, the maximum E-field is located along the E-field polarization direction (the y-axis). The octahedra core, however, perturbs the electric field and the maximum E-field is enhanced on the edge of octahedra core. Note Figure 7B and 7D present the E-field maps in two planes perpendicular to each other. This localization of strong electric field intensity along the edges of metal nanoparticle (edge effect) is due to significantly enhanced charge density. The near-field EF plots for all surface points clearly demonstrates that there are more hot spots for an octahedral core than for a spherical core at all wavelengths (Figure 8). The EF value for an individual hot spot reaches approximately 170 at 660 nm for octahedron core (Figure 8B), while the EF value for hot spots with spherical core reaches about 70. Taking these results into consideration, one can conclude that having a non-spherical core can enhance SERS properties of Raman molecules adsorbed on IO-Au nanoparticles.

Lastly, we present the E-field maps for the same octahedra core-shell NPs at 660 nm when the polarization angle of the light is changed (see Figure 9). The E-field map shows a change in intensity in conjunction with a change in the direction of electric field. However, the averaged EF spectra with different polarization angles and the extinction spectra remains the same for this particle (data not shown).

## CONCLUSION

We present a test study on the accuracy of the DDA calculations of both extinction spectra and near-field EF spectra for IO-Au core-shell nanoparticles. By making use of a recursive algorithm to solve the extended Mie theory for multi-layer spherical particles, we compared DDA calculations for spherical particles against Mie theory for both the EF spectra and near-field enhancement spectra. We see that a convergence of the DDA results, as a function of dipole distance, is reached much more quickly than the near-field EF spectra. Also, when the Au-shell thickness is decreased, a smaller dipole distance is needed to reach the convergence.

After establishing the accuracy of the DDA calculations for spherical IO-Au NPs, we investigated the impact of a non-spherical core on IO-Au NPs. We observe that with an octahedral core, both the extinction spectra and EF spectra become broader than for a spherical core. In general, a broader EF spectrum provides a better SERS signal as one can gain better enhancement over a wider range of wavelength. In addition, we also see that although the averaged EF over all surface points with an octahedral core is close to that obtained with spherical-core shaped NPs, there are nevertheless more hot spots for the non-spherical core due to the edge effect on the core. Thus, a non-spherical core, in principle, can provide better SERS enhancement than can a spherical core.

## ACKNOWLEDGMENTS

We acknowledge the partial financial support for this work from NIH/NCI 1R15CA195509-01.

## REFERENCES

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_{3}/Au core/shell nanoparticles for bioseparation and immunoassay based on surface-enhanced Raman spectroscopy