Letter pubs.acs.org/JPCL
Time-Dependent, Optically Controlled Dielectric Function Maxim Artamonov and Tamar Seideman* Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States ABSTRACT: We suggest optical modulation of the dielectric function of a molecular monolayer adsorbed on a metal surface as a potential means of controlling plasmon resonance phenomena. The dielectric function is altered using a laser pulse of moderate intensity and linear polarization to align the constituent molecules. After the pulse, the monolayer returns to its initial state. Time-dependent, optically controlled dielectric function is illustrated by molecular dynamics calculations.
T
In what follows, we present the results of molecular dynamics simulations that illustrate the possibility of controlling the plasmon resonance by manipulating in a time-dependent fashion the dielectric function of a monolayer absorbed on the metal surface using a moderately intense laser pulse to transiently realign the constituent molecules. Laser alignment of molecules in the gas phase is a well-studied subject,58,59 and the concept of alignment has been extended to solvated60 molecules, torsional control,61−65 and, pertinently to the present study, molecular switches.66 The monolayer model consists of anthracenethiol (ANT) self-assembled67 on the Au(111) surface. For simplicity, we assume that every absorption site is on-top and correspondingly adopt the hexagonal (2 × 2) monolayer structure with the unit cell size a = 2aAu = 5.77 Å. The ANT moiety is held rigid while the Au−S bond is allowed to stretch. The body-fixed (BF) frame is defined with its origin at the sulfur atom, the Z-axis collinear with the S−C bond, the Y-axis in the plane, and the Xaxis out of plane of the molecule (see Figure 1a). The BF frame rotates with respect to the space-fixed (SF) frame. In defining the interactions of an ANT adsorbate with its environment, it is convenient to describe this rotation using the Euler angles68 because they represent three independent elementary rotations. Thus, the polar angle θ is the angle between the SF z- and BF Z-axes, and the azimuthal angle χ is the angle of the rotation about the BF Z-axis (see Figure 1b). The third Euler angle, ϕ, describing the rotation about the SF z-axis is not used explicitly in the formulation because the steric interaction between the ANT molecules is accounted for using Lennard-Jones (LJ) pair potential, and the surface is treated implicitly (see below). The interactions of an ANT molecule with its neighbors, the surface, and the laser field give rise to the total potential V =
he ability of metal nanoparticles or nanopatterned metal surfaces to support localized surface plasmons (a collective oscillation of the conduction electrons influenced by incident light) gives rise to important properties of such nanomaterials, among them, field enhancement and localization. These properties fuel growing interest in plasmonic nanomaterials for a diverse range of applications including spectroscopy,1 subwavelength imaging,2 photolithography,3 electro-optical devices,4,5 photovoltaics,6 and sensing.7 The localized surface plasmon resonance conditions depend on the composition, size, and shape of the nanoparticles or nanopatterns as well as the dielectric function of the surrounding medium.8 The latter dependence is utilized in developing bioand chemical sensors,7,9−15 where a change in the local dielectric environment caused by the presence of a stimulus results in a detectable change in the plasmon resonance. Whereas these conventional applications involve a static medium dielectric functions, a time-dependent dielectric function whose temporal modulation is controllable could serve to build active plasmonic devices, for example, tunable nanoantennas16−18 and filters,19,20 plasmonic switches,21−25 modulators, 26−28 splitters, 29,30 and tunable plasmonic lenses.31,32 A time-dependent dielectric function may find a variety of interesting applications other than nanoplasmonics. These include the generation of exotic forms of chirped light, as discussed, for example, in ref 33, capture and reemission of light proposed in ref 34, optical stopping of light,35 compression, broadening,36 and spectral manipulation36,37 of pulses, as well as designing liquid-crystal-38−40 and semiconductor-based41−43 tunable photonic band gap materials and electro-optically manipulated waveguides44−47 and switches.48−50 Several ways of generating a time-dependent dielectric function were demonstrated,33−57 but tuning the dielectric function of a surface layer remains desirable. © XXXX American Chemical Society
Received: November 3, 2014 Accepted: January 2, 2015
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ANT attached to a seven-atom gold cluster similar to the one used in ref 72 and are summarized in Table 1. The calculations Table 1. Constants of the Potential and Polarizability Functionsa kr r0 kθ θ0
7.631 2.412 4.532 1.304
α0 αr αθ αχ
−71.83 52.60 164.5 −22.30
a
The units of energy, length, charge, and angle are eV, Å, e, and rad, respectively.
were performed using the X3LYP73 functional in the NWCHEM package74 with the 6-311G75 basis set for the sulfur, carbon, and hydrogen atoms and the effective core potential CRENBL-ECP76 for gold. The polarizability was calculated77−79 at the same time as the energy, and the zzcomponent in the SF frame was fitted to a simple functional form 1 αzz(r , θ , χ ) = α0 + αrr + αθ cos θ + αχ cos2 χ 4πε0
Figure 1. (a) Definition of the BF frame. (b) Definition of the Euler angles θ and χ.
VAA + VAG + Vstr + Vbnd + Vind, where VAA and VAG are the van der Waals (vdW) interactions among the ANT molecules and between each ANT and the gold surface, with the LJ parameters taken from refs 69 and 70, respectively. Because of the large number of surface atoms involved in the summation, VAG is precalculated prior to running the simulations on a grid composed of 100 equidistant points in each of the r, cos θ, and χ coordinates in the ranges [0.5,6] Å for r, [−1,1] for cos θ, and [0,2π] for χ. During the simulations, VAG is interpolated from the precalculated values using cubic splines.71 The much weaker dependence of VAG on ϕ due to the surface corrugation is disregarded. The stretching and bending motions are described with harmonic potentials,70 Vstr(r) = (1/2)kr(r − r0)2 and Vbnd(θ) = kθ(2 sin2 θ0)−1(cos θ − cos θ0) 2. These two potentials could, in principle, be incorporated into VAG; however, we treat them explicitly as they are easy to evaluate and serve as convenient means to quantify the stretching and bending dynamics of molecules in the monolayer. The last term in V is the laser field−matter interaction potential, Vind(r,θ,χ,t) = −(1/4)αzz(r,θ,χ)εz(t)2, where αzz(r,θ,χ) is the zz-component of the polarizability tensor in the SF frame. The field is linearly polarized at an oblique angle to the surface, as commonly taken in experiments, and εz(t) is the component of the field along the surface-normal (the z-axis). The most polarizable molecular axis (Z-axis in Figure 1a) aligns with the field component of the largest amplitude, while the second most polarizable molecular axis (Y-axis in Figure 1a) aligns with the second field component, resulting in three-dimensional alignment.58,59 Thus, the field component parallel to the surface will hinder the rotation in the Euler angle ϕ and align the planes of the ANT molecules to a single direction in space. For a judicial choice of the field incident angle, however, the parallel alignment is small with respect to the alignment produced by the surface-normal field component because of the combined effect of (i) the smaller amplitude of the parallel field component and (ii) lesser polarizability of the molecule along the Y-axis. Here, we neglect the smaller component as well as the (generally much smaller) off-diagonal terms of the polarizability tensor so as to simplify the presentation. The laser field frequency is far off transition frequencies; hence, as illustrated elsewhere,59 the interaction depends on the field envelope but not on its oscillations at the optical frequency, ϵ(t) = (1/2)[εz(t) exp(iωt) + c.c.] in Vind, where εz(t) = E0s(t)ẑ with the peak amplitude E0 and a smooth envelope s(t). The parameters in Vstr(r) and Vbnd(θ) were determined from a set of density functional theory (DFT) calculations of an
(1)
The simulations were performed on an ensemble of 672 ANT molecules. The equations of motion were integrated using the velocity Verlet80 for the stretching coordinate and the rotational velocity Verlet algorithm81 for the rotational coordinates with time step of 0.49 fs. The initial configuration was generated at 10 K using a Berendsen82 thermostat. The thermostat was not used during the main simulation runs. The laser pulse was centered at 196 ps from the beginning of the simulation with a peak intensity of 11 TW/cm2. For clarity, the data presented below were smoothed by computing a moving average with the Hamming window of 1.5 ps width. The ANT polarizability in the SF frame is orientationally dependent. As the laser pulse intensity increases, the tilt of the molecules decreases so as to maximize the zz-component of the polarizability tensor and thus minimize Vind. The change in polarizability of the monolayer results in the change in its dielectric function given by83 ε=1+
N ⟨α⟩ ε0A⟨L⟩
(2)
where ⟨α⟩ and ⟨L⟩ are the ensemble averages of αzz and the length of an ANT molecule in the z-direction, respectively, N is the number of ANT molecules, and A is the simulation area. The dielectric function of the unperturbed monolayer is 8.18 and increases to 8.94 at the peak of the pulse. This change corresponds to a 20−46 nm shift in the surface plasmon resonance, taking the values of 150−350 nm per refractive index unit reported for gold nanoparticles.7 The response of the dielectric function to the laser pulse is shown in Figure 2 together with the ensemble-averaged values of cos2 θ and cos2 χ, which measure the overall alignment of ANTs in the monolayer. With the molecular frame defined in Figure 1 and the Euler angle convention of ref 68, the increase in ⟨cos2 θ⟩ toward the pulse peak signifies a decrease in the angle between the S−C bond and the surface-normal. Similarly, the decrease in ⟨cos2 χ⟩ indicates the molecular plane becoming almost perpendicular to the gold surface. This general picture is supported by a more detailed view offered by the angular distributions in Figure 3. The initial bimodal θ distribution, 321
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their orientations on average are very close to their orientations before the pulse (see Figure 2). Moreover, a repeat simulation with the initial configuration at 200 K produced the same qualitative results. The effect of the laser pulse on the potential energy terms is shown in Figure 4. The vdW interaction energy between the
Figure 2. Time evolution of the average values of squared cosines of the angles θ (blue, triangles) and χ (red, filled circles) and the dielectric function ε (black, open circles).
Figure 4. Time evolution of the mean potential energy of the vdW interaction among ANTs (blue, triangles), the Au−S bond stretch (red, filled circles), bend (cyan, squares), and the field−matter interaction (black, open circles).
ANT and the surface, VAG, is of the smallest magnitude with the least change and is omitted from the figure. Prior to the application of the laser field, the strongest interaction is among the ANT molecules, VAA. At the peak of the pulse, VAA decreases, indicating a more stable configuration, but the change is relatively small. Likewise, the Au−S stretching energy Vstr increases slightly at the peak of the pulse. This is consistent with the definition of αzz as the polarizability increases with an increase in the Au−S bond length. Aside from the field−matter interaction energy, the bending energy term, Vbnd, experiences the largest change with its average increasing from 0.21 eV for the unperturbed monolayer to the peak value of 0.57 eV. Figure 4 also demonstrates that the monolayer returns to its initial state after the pulse. Several features of the approach are interesting to point out before concluding. First, the method is general as all molecules are polarizable to a certain extent, and once adsorbed, they acquire anisotropy even if isotropic in the isolated form. Furthermore, the polarizability of the adsorbed molecules is expected to be larger than that of the isolated molecules due to delocalization of the metal electrons. Additionally, enhancement of the incident electromagnetic field by the surface corrugation is expected. The choice of the monolayer-forming molecules is important, however, as eq 2 notes that for an adequate change in the dielectric function, the molecule should possess sufficient polarizability anisotropy in the aligned and random configurations without a commensurate increase in the monolayer thickness. Additionally, because the field−matter interaction energy is proportional to the product of the polarizability anisotropy and the laser field intensity while the intensity is upper bounded by the onset of damage, a large polarizability anisotropy is advantageous. Clearly, chemical
Figure 3. Time evolution of the normalized θ and χ angular distributions. Color represents the fraction of molecules with a particular value of the angle.
peaked at ∼52 and 66°, narrows and shifts to ∼39 and 46°. Note that the equilibrium bending angle in the monolayer indicated by the θ distribution is smaller than the minimum of the bending potential, Vbnd, of ∼75° (θ0 in Table 1). This is because Vbnd was calculated using a single ANT molecule attached to a small cluster of gold atoms, while steric interactions between molecules in the monolayer constrain the molecules into a less inclined configuration. The initial χ distribution, peaked at ∼73 and 107°, shifts to a single peak at around 90°. The shifts in both distributions are consistent with maximizing the value of αzz, eq 1. After the pulse, the two distributions return to their initial states, albeit more diffused. The somewhat broader angular distribution after the pulse is the result of increased rotational temperature, but the slightly elevated temperature does not affect the conclusions of this study. While the ANT molecules move faster after the pulse, 322
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substitutions can assist in both increasing the polarizability anisotropy and reducing the well depth of Vbnd. For the system considered above, inserting methylene bridges between the S atom and the polarizable moiety,84 for example, would reduce the intensity required to induce orientation. Finally, depending on the application in mind, one can envision different temporal modulations of the dielectric function. Above, we considered the adiabatic case, where orientation follows the pulse envelope and the monolayer is re-established in the original configuration after the pulse. A possible way to maintain the alignment in the absence of the field in the adiabatic case is to induce a chemical reaction to produce cross-linking.85 In the nonadiabatic case, where the field envelope varies more rapidly than the bending motion, the molecules gain momentum as a consequence of the restoring force of the bending potential not being balanced by the field−matter interaction forces. Simulations with shorter pulses, down to 25 ps, produced results similar to the ones presented above except for the greater postpulse increase in the temperature. In the limit of very short (subpicosecond) pulses, the peak alignment occurs after the pulse. In this case, however, the temperature rise is too severe, calling for a more advanced model, one that would allow for dissipative channels. In summary, we introduced an approach to make materials with a controlled, time-modulated dielectric function by an extension of the concept of strong field coherent control from the domain of small gas-phase molecules to address a material science problem with broad applicability. Sharing a common feature of several strong field coherent control methods, the approach is very general as it relies on the interaction of the polarizability tensor of the molecule with a nonresonant laser field. Uniquely, it offers a wide variety of potential applications in nanoplasmonics as it allows introduction of a large change in the dielectric function of the medium of a given plasmonic construct on a rapid and largely controllable time scale. These potential applications include, for instance, switching on and off as well as enhancement and spectral tuning of the chemical sensitivity of nanoplasmonic constructs, in addition to modulating or switching the electron transport, energy transfer, and optical properties of the molecular layer.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We are grateful to the National Science Foundation’s MRSEC program (DMR-1121262) at the Materials Research Center of Northwestern University and to the Department of Energy (DE-FG02-04ER15612) for support.
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