The inverse Faraday effect (IFE) is an opto-magnetic phenomenon that produces static magnetic fields in a wide range of materials during illumination with circularly polarized light. This study analyzes non-magnetic gold (Au) metal nanostructures, providing insight into plasmonic enhancement of the magnetic and optoelectronic phenomena associated with the IFE. We report a simple numerical approach in combination with full-wave optical simulations (finite-difference time-domain method) for tracking the optically-induced motion of electrons inside plasmonic nanostructures that gives rise to the IFE. In addition to static magnetic fields, a circulating drift current is observed, where the direction of current is the same as the chirality of the circularly polarized light. Our results indicate a significant enhancement of this drift current by ~100 times in Au nanoparticles due to larger optical field gradients in comparison with bulk Au films. We also report on the size, geometry, and spectral dependence of the induced drift currents and static magnetic fields, which we predict can exceed 110−3 T under 1015 W m−2 optical intensity for spherical Au nanoparticles. Our results inform the development of new classes of magneto-optic and optoelectronic behavior that can be obtained via direct manipulation of electron dynamics by the optical fields inside metals.
© 2017 Optical Society of America
The intense optical concentration provided by plasmon resonances has attracted significant scientific interest, motivating numerous studies of plasmonic nanostructures in applications that benefit from subwavelength field enhancement, such as photochemistry, sensing, optoelectronics, and non-linear optics [1–5]. A plasmon resonance occurs in materials with a high concentration of mobile carriers when optical fields couple to the collective displacement motion of electrons . Large fluctuations of charge density at ‘hot spots’ near the surface of resonant geometries intensify the polarization of the incident field. In contrast with linearly polarized illumination, which has zero magnitude electric field twice during an optical cycle, the electric field of circularly polarized radiation rotates with constant magnitude in the plane normal to light propagation. Consequently, if plasmonic structures are resonant with circularly polarized excitation, it is possible for them to exhibit regions of strongly modified carrier density for the duration of an optical cycle. We show here how this resonant, rotating charge displacement can provide significant enhancement of the direct conversion of circularly polarized radiation into static magnetic fields via the IFE, as well as provide a strong driving force for persistent circulating drift currents.
As depicted in Fig. 1, the IFE is an opto-magnetic phenomenon [7,8] manifest as a static magnetic field that is parallel or anti-parallel with the axis of circularly polarized excitation, based on the chirality of the radiation. First theoretically proposed as a consequence of Faraday rotation , the IFE has been observed in a variety of dielectric materials with large Verdet constants, such as rare-earth transition metal compounds [10–16]. Recently there has been renewed interest in the IFE as a means to provide optically modulated ultrafast control of magnetization for memory storage applications and advanced magnetic devices [10,13,15]. In this context plasmonic nanostructures have also been proposed as a possible route for enhancing optical field strength and confinement for application of the IFE [8,17–20]. Notably, Smolyanimov et al.  showed static magnetism in metallic nanohole arrays induced by circularly polarized illumination at wavelengths corresponding to surface plasmon excitations. Anomalously large signals in magnetic circular dichroism (MCD) experiments of Au colloids have also implicated plasmonic enhancement of IFE phenomena [21,22].
These recent studies have prompted reassessment of the mechanism of the IFE in terms of the microscopic motion of charges, as opposed to the traditional phenomenological analysis [23,24], due to some debate about the appropriate description of momentum conservation [25,26]. In particular, Hertel et al. showed that a simple microscopic model of the IFE in metals can be obtained by separately considering slow time varying (or time-averaged) and oscillating contributions to electron motion in the continuity equation for a plasma subject to circularly polarized radiation . This result is briefly summarized here. Equation (1) describes the continuity equation with separate contributions from time-averaged drift velocity,, and electron density,, that is constant or slowly changing after several optical cycles, in addition to the oscillatory contributions at the optical frequency, and , where and .Eq. (2).Fig. 1(b), red traces] [8,26]. These solenoid-like currents gives rise to macroscopic static magnetization,, with
A 2nd-order perturbation analysis of the solenoid-like electron trajectories by Hertel et al. additionally predicts macroscopic drift current densities, , if optical field gradients are present inside the metal . The microscopic origin of this additional drift current is the slightly lop-sided trajectory of each electron in the solenoid-like path, i.e. not perfectly harmonic motion, if the electrons do not experience the same magnitude optical field driving their motion throughout the entirety of the optical cycle. This drift current flows along the isolines of the field amplitude, e.g. around the perimeter of a Gaussian beam spot in the plane normal to the optical excitation axis, and consequently circulates the optical spot at a rate much slower than the optical frequency. This macroscopic, persistent drift current density is given byEq. (3). Note that plane wave excitation will still produce field gradients inside a plasmonic nanoparticle, due to the stronger optical field enhancement near the particle surface. However, the gradient of the optical field moving from the center of the particle towards the exterior has the opposite sign compared with a Gaussian beam spot on a metal film, and hence the same chirality of optical excitation is expected to produce drift currents with an opposite sign along the azimuth for these different geometries.
Summarized schematically in Fig. 1, this analysis provides an intriguing insight into the photophysics of metals subject to circularly polarized excitation. In particular, the analysis that predicts persistent drift currents [Eq. (4)] builds from the linear solutions to Maxwell’s equations for simplified geometries, e.g. thin films, and considers the effect of small modifications of the electron paths due to field gradients, with the assumption that the perturbation is too small to modify the 1st-order solution of the field distribution in the metal. A challenge, however, is that these results cannot readily be derived using powerful computational methods, such as full-wave optical simulations, because such electrodynamics solvers are typically limited to linear solutions to Maxwell’s equations. Such 1st-order solutions define purely harmonic responses at the optical frequency, with no slow-varying or non-changing higher order contributions, which are crucial for analyzing the persistent electric drift current that results from the IFE.
As an alternative to the approach of Hertel et al. [7,25], we have therefore developed a simple numerical method in order to investigate the higher order effects of circularly polarized light in materials with plasmonic field enhancements, and in particular, to provide further insight into electron motion and the occurrence of drift currents due to the IFE in more complex materials and geometries. The linear solutions to Maxwell’s equations serve as a starting point for our calculations that track the time-dependent trajectory of electrons inside the structure subject to the optical forces. The advantage of our method is that it uses the full complex dielectric function of the structures being modeled, as opposed to assuming a Drude metal, and it is suitable for analysis of the motion of charges through arbitrary geometries in three spatial dimensions, as it is easily combined with commercial full-wave optical solvers. Crucially, our calculations are robust when intense field gradients are present near the inside surfaces of plasmonic nanostructures, where drastic changes in the optical field may entail that a perturbation method is inadequate. By explicitly tracking the motion of charges inside plasmonic structures induced by radiation, our report aims to identify conditions that maximize both the magnetic and optoelectronic phenomena associated with the IFE.
A commercial full-wave electrodynamics solver package (Lumerical Solutions) implementing the finite-difference time-domain method for computing Maxwell’s relations was used in conjunction with a homemade Matlab code (see the Code 1 Ref .). Our code numerically calculates the internal electron motion induced by a circularly polarized optical source, based on the time and spatial dependence of the instantaneous electric currents that are obtained from the linear solutions to Maxwell’s equations inside the metal. We analyzed 5 nm, 10 nm, and 100 nm diameter Au nanoparticles, as well as a 100 nm thick Au thin film, considering the electron motion within a 2-D plane that bisected the structure normal to the light source. In the case of thin film stimulations, light sources with a Gaussian intensity distribution (beam waist radius of 500 nm) were employed in order to define regions with optical field gradients.
Our computational analysis first determines the instantaneous current densities at a given time, , and location, , in the metal, assuming that all material properties within the simulations are expressed via an effective material permittivity,  The instantaneous displacement field, , is a function of the optical field, , that is output from the full-wave simulations.Eq. (5) as:Eq. (7) will always indicate a net current density of zero moving through a specific position in the metal after one full optical cycle. While this is sufficient to describe the microscopic harmonic electron motion that results in the static magnetic fields of the IFE, the calculations do not indicate the persistent macroscopic drift currents due to the IFE that are of great interest in this study.
To overcome this limitation, our analysis tracks the motion of a particular differential volume element of charges inside the structure subject to the time-dependent optical forces, in order to determine how much that charge packet changes position after one full optical cycle. Packets of charge will only start and finish in different positions after an optical cycle if optical field gradients are present inside the metal. We assume that the instantaneous field distribution reported by the full wave simulation is accurate. Therefore, we also implicitly assume a uniform charge distribution is preserved in the metal over many optical cycles, such that we neglect any redistribution of the electric field due to other non-linear phenomena, like plasmon drag effects . Figure 2(a) provides a schematic of the computational approach.
In order to trace the trajectory of a packet of charges, we first discretize the time axis as, whereis the number of time steps within one full optical cycle, with the assumption that the forces do not change in between time steps. We next calculate how the packet of charges move from an initial position, , to a new position, , in a time of duration, , with drift velocity, .Fig. 2(b)] this method approximates the continuous motion of a charge packet.
Electron trajectories corresponding to solenoid-like displacement motion and net drift velocities were obtained [Fig. 1], in the limit that the resulting answer converged to a value that did not change significantly with more total time steps,. As summarized in Fig. 2(b), answers converged more quickly with a lower value of (~2 106) and at a higher optical intensity of 1015 W m−2, which is employed throughout this study unless otherwise stated. The radial symmetry of the structures allowed all calculations to be performed for charge packets with initial positions along the particle radius on the positive x-axis, as long as an optical phase offset was included to account for the optical fields experienced at different angular positions in the particle during the optical cycle (see Code 1 Ref .). The resultant net drift current for a given distance from the center of the particle was calculated by averaging the value associated with all phase offsets, in order to better compare our results with the behavior expected from an ensemble sample, e.g. a colloid suspension. The calculated drift current corresponds to electrons circulating throughout the entire structure in rings that are a fixed radius from the center, producing an induced magnetic field at the particle center. The induced magnetic field due only to drift currents was calculated using the Biot-Savart law by summing the results from all of the current-carrying rings. The local magnetic field due only to the near-harmonic motion at the optical frequency was calculated by determining the magnitude and radius of the small solenoid-like displacement current at a given position. The total induced magnetic moment, due to both the motion at optical frequencies and drift currents, was also estimated for the 100 nm particle and the thin film. In order to better compare with other reports , the thin film calculation analyzed the electron motion in a slab of metal with a thickness of 10 nm localized in the center of the simulation region.
3. Results and discussion
The rotating dipole of the radiation causes two distinct types of electron motion, summarized in Fig. 1(b). First, the electric field induces sub-angstrom displacement motion of each electron through a circular path that produces small solenoid-like circular currents at the optical frequency. As outlined by others [7,26], the sum of all these circulating currents can primarily account for the static magnetic field observed during the IFE. As discussed below, we find a significant enhancement of the IFE due to the optical field enhancement provided by plasmonic resonances. However, especially because of large optical field gradients near ‘hot spots’ on the interior side of the surface of resonant geometries, electrons do not experience the same magnitude of electric field throughout their path, and hence do not undergo perfectly harmonic motion. Rather, electrons start and finish in slightly different positions after one optical cycle, thus resulting in a net drift current that circulates through regions with a field gradient with the same chirality as the circularly polarized excitation. We also obtain an even larger enhancement of circulating drift currents in plasmonic geometries, by two orders of magnitude in comparison with bulk metals under similar optical powers (Fig. 3 and Fig. 5), as a result of the greatly amplified optical field gradients.
In order to provide deeper insight into the accuracy and compatibility of our numerical method for a wide range of structures, we performed stimulations of Au thin films and compare our results with those obtained by the 2nd-order perturbation analysis outlined by Hertel et al. for the same geometry . Figure 3 shows the electrical field from a Gaussian optical spot (black trace) and the current density that circulates the beam waist as a function of distance from the center for a wavelength of 450 nm. The current density results from the gradient of the optical field, leading to the highest value around 250 nm from the center of the spot and then decreasing with increasing distance. Our calculated current density results agree well with that obtained from the 2nd-order perturbation method of Hertel et al. when the particle is treated as a Drude metal. However, there is a systematically lower magnitude current density (see red trace) determined by our method when a more empirically-based dielectric function for Au is implemented (Johnson and Christy ), reflecting the more accurate measure of damping losses. We also estimate the contribution to the magnetic moment of the film at the center due to solenoid-like electron motion at the optical frequency as well as due to drift current to be 1.6 10−18 J T−1 and 2.7 10−19 J T−1, respectively, again in good agreement of the prediction form Hertel et al. as well as ultrafast measurements of Au thin films . The radial frequency of the drift current is approximately nine orders of magnitude slower than the optical frequency. Importantly, both our calculations and the 2nd-order perturbation method show drift current that circulates the opposite direction as the chirality of the circularly polarized excitation in this thin film geometry, due to the opposite sign of the field gradient, i.e. decreasing field intensity moving out from the center, compared with the opposite trend inside metal nanoparticles. Here the drift current provides magnetization that counteracts the magnetization due to electron motion at the optical frequency, highlighting the importance for considering the interaction of phenomena due to the IFE for a desired application.
We applied our method to calculate both the drift current and magnetic fields induced by circularly polarized light in 100 nm diameter spherical nanoparticles, assuming a dielectric environment of air. First, the absorption and scattering cross-section spectra were calculated. The absorption cross-section displays a peak at ~517 nm whereas the scattering cross-section exhibits a peak at ~538 nm, in good agreement with other reports [31,32]. Figure 4(a) displays a current density map and vector plot of the optically induced drift currents under an incident light intensity of 1015 W m−2 on resonance with the absorption maximum. These data show that there are circulating drift currents in the entire nanostructure with the same chirality as the circularly polarized excitation. However, the magnitude of drift current near the inside of the particle surface is ~100 times greater than that near the particle center. This enhancement is attributed to the greatly increased optical field gradient near the exterior of the particle [Fig. 6(a)]. In addition, the magnetic field due only to this drift current was determined using the Biot-Savart formula for a current-carrying circular loop. A magnetic field of ~1.1 10−5 T is generated at the center of the particle, corresponding to a drift current-induced magnetic moment of ~1.96 10−21 J T−1. In comparison, Fig. 4(b) maps the local static magnetic field resulting only from the solenoid-like near-harmonic motion associated with the IFE, i.e. neglecting the contribution from the drift current. This magnetization is greater near the exterior of the particle, enhanced by the same amount as the optical field (~4.8 times), as compared with bulk films, due to the plasmonic optical field concentration near the particle surface. By spatially integrating the local magnetization, we calculate an induced magnetic moment of ~1.7 10−19 J T−1. Thus the contribution to the total magnetic moment due to the slower moving drift current is much smaller than that generated by the solenoid-like motion at the optical frequency. These results indicate that non-magnetic plasmonic materials, such as Au, can behave as optically switched static magnetic materials in addition to providing new strategies for generating optically induced electrical currents.
Figure 5(a) illustrates the current density versus light intensity for a 100 nm particle, computed at three different wavelengths of 485 nm, 517 nm, and 800 nm. We observe a linear dependence on the optical intensity, in agreement with Hertel et al. . Figure 5(b) illustrates the circulating current density near the particle exterior as a function of wavelength for an optical intensity of 1015 W m−2. The maximum drift current density of ~1.3 1010 A m−2 corresponds to a wavelength of 575 nm. The current density decreases with decreasing wavelength below 575 nm, reaching a minimum value of ~5.9 108 A m−2 at ~400 nm. The current density also decreases slightly when probed above 575 nm. The maximum current density peak is red-shifted by ~35 nm in comparison to the scattering cross section peak. A cubic dependence of the drift current on wavelength [Eq. (4)] can account for this red-shift . As can be seen in Fig. 5(b), the magnetic moment exhibits a maximum value of ~1.7 10−19 J T−1 at 517 nm. This maximum at the peak absorption wavelength can be attributed to an enhanced contribution from electron motion at the resonance frequency. Figure 5(c) shows the current density as a function of distance from center of the 100 nm-diameter nanoparticle. For comparison, the current densities were also computed using the 2nd-order perturbation method of Hertel et al. (blue trace) where the particle is treated as a Drude metal . The current density increases with increasing distance from center, leading to the highest current density closest to the perimeter of the particle. This can be attributed to the greater field gradient near the inside edge of the particle. Our calculation shows good agreement with that obtained from the 2nd-order perturbation method. Note that the dip in current density a few nanometers inside the particle is due to free carriers screening the surface electric field, similar to an electric double layer in ionic solutions. A similar trend is also observed for the electric field inside the particle [Fig. 6(a)].
Particle size also plays a role in enhancing the drift current due to the IFE, due to the size dependence of the optical field profile. Figures 6(a-c) shows the electrical field as a function of absolute distance from the center for 100 nm, 10 nm, and 5 nm-diameter Au nanoparticles. The field increases with the distance, showing greater field enhancement at exterior for all three cases. It appears that there is a decrease in electric field a few nanometers inside the exterior for all three structures. This can be attributed to screening effects due to free carriers in the metal. Figures 6(d-f) displays current density as a function of absolute distance from the center for the same particles. The drift current densities were computed at an incident wavelength of 517 nm, 507 nm, and 499 nm for 100 nm, 10 nm, and 5 nm-diameter Au nanoparticles, respectively. The current density from a 5 nm-diameter particle shows higher values on average than those obtained from larger particles. Figure 6(g) plots the current density near the interior of the particle surface as a function of particle diameter. Somewhat counter intuitively, the current density increases with decreasing particle diameter, revealing a maximum current density of ~2.7 1010 A m−2, whereas the surface field enhancement increases with increasing particle diameter. We conclude that structures with sharper edges would be better targets for harvesting electrical currents, due to the even greater optical field gradients in smaller geometries. As expected, the current density in all nanoparticle structures is two orders of magnitude greater than that observed in bulk films at similar light intensity [Fig. 3].
We have numerically explored the use of Au nanostructures to enhance phenomena associated with the IFE. These phenomena include static magnetization in non-magnetic plasmonic metals during illumination, as well as drift currents that circulate inside the nanostructure normal to optical field gradients. Nanoparticles exhibit the largest circulating drift current nearest to the exterior due to greater field gradients at that location. The direction of the induced magnetic field due to electron motion at the optical frequency depends on the chirality of the circularly polarized light, while the direction that the drift current circulates, and consequently the magnetic field it produces, depends on the sign of the optical field gradient. We anticipate that the results and numerical techniques from this study can inform the development of new types of advanced magneto-optic materials, as well as help outline new methods for optical-to-electrical detection and energy conversion.
Welch Foundation (A-1886).
We thank Mr. Jarret C. Martin and Mr. Easwara Moorthy Essaki Arumugam for assistance with Matlab codes. This research made use of the High Performance Research Computing (HPRC) facility at Texas A&M University. We also thank the technical support of Lumerical Solutions Inc. for assistance with the Lumerical scripting language.
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