Role of electron tunneling in the nonlinear response of plasmonic nanogaps

We report on the theoretical study of the second and third harmonic generation in plasmonic dimer nanoantennas with narrow gaps. Our study is based on the time dependent density functional theory. This allows us to address the nonlinear response of a tunneling junction in a subnanometric plasmonic gaps with a quantum calculation, which goes beyond conventional classical local and nonlocal treatments. We demonstrate that the nonlinear electron transport in a plasmonic junction, associated to the corresponding strong field enhancement in the narrow gap, allows to reach orders of magnitude enhancement in the efficiency of the second and third harmonic generation. Depending on the size of the junction and the frequency of the fundamental incident wave, we show that the frequency conversion in plasmonic dimer gaps can be determined by (i) the intrinsic nonlinearity of each individual nanoparticle, (ii) the nonlinear ac tunneling current across the gap, and (iii) the resonant excitations of the plasmon modes of the dimer. The study of the nonlinear response of plasmonic gaps within a full quantum treatment allows us to understand the fundamental mechanisms of nonlinearity in nanoplasmonics.

Figure 1

Sketch of the studied systems. (a) Spherical Al nanoparticle dimer. An individual nanosphere has a radius of $R_{\mathrm{sph}}=11\AA$ and comprises 1012 electrons. (b) Dimer of Na cylinders, infinite along $x$ axis. An individual cylinder has a radius $R_{\mathrm{cyl}}=30.7\AA$ and comprises 800 electrons per 1 nm length. The origin of coordinates is at the middle of the junction and the dimer $z$ axis is such that the $(x,y,z=0)$ plane is the symmetry plane of the system. The incident electromagnetic field is a plane wave polarized along the $z$ axis and propagating along the $y$ axis.

Figure 2

Spectral analysis of the $z$ component of the induced dipole for a dimer of cylindrical Na nanoparticles separated by the junction of the width $d_{\mathrm{gap}}=3.5\AA$. Results of the TDDFT calculations are shown as a function of the frequency $\mathrm{\Omega }$. The incident Gaussian electromagnetic pulse has carrier frequency $\omega =0.6$ eV and duration $2\tau =40$ fs.

Figure 3

[(a) and (b)] Color plots of the optical absorption cross section $\sigma$ of a dimer of cylindrical Na (a) and spherical Al nanoparticles (b). TDDFT results are presented as a function of the size of the gap between nanoparticles $d_{\mathrm{gap}}$ and frequency of the incident plane wave $\omega$. Labels indicate the plasmon modes responsible for the absorption resonances: bonding dipole plasmon (BDP), bonding quadrupole plasmon (BQP), the lowest (dipole) charge transfer plasmon (CTP), and the higher-energy charge transfer plasmon (CTP'). (c) Cylindrical Na dimer. The gap-size dependence of the enhancement of the incident field ${\text{R}}_{\text{E}}=\left|E_{\omega }^g/E_{\omega }\right|$ (solid lines) and of the amplitude of the normalized ac current $\left|I_{\omega }/E_{\omega }\right|$ between nanoparticles (dashed lines). The TDDFT results are obtained in the middle of the gap for different frequencies of the linearly polarized incident electromagnetic wave $\omega =0.6$ eV ($\lambda =2066$ nm), $\omega =0.8$ eV ($\lambda =1550$ nm), and $\omega =1.2$ eV ($\lambda =1033$ nm) as explained in the insert. Here, $E_{\omega }$ is the incident field, $E_{\omega }^g$ is the induced ac field in the middle of the gap, and $I_{\omega }$ is the electron current through the gap at optical frequency. (d) Hyperpolarizability ${\alpha }^{\left(3\right)}\left(3\omega \right)$ of an individual Na nanowire. The values are given in atomic units (a.u.). The relation $1\mathrm{esu}=0.1985\times {10}^{40}$ a.u. can be used for conversion of the third-order hyperpolarizability to electrostatic units (esu).

Figure 4

Nonlinear response of a dimer of (a) cylindrical Na and (b) spherical Al nanoparticles. Results of the TDDFT calculations are shown as a function of the size of the gap between nanoparticles, $d_{\mathrm{gap}}$. Solid lines represent the enhancement of the intensity of the nonlinear dipole ${\mathfrak{R}}_n={\left|p_{n\omega }\right|}^2/\left(4{\left|p_{n\omega }^i\right|}^2\right)$ at the second ($n=2$) and third ($n=3$) harmonic frequency. Dashed lines represent the scaled ac current through the middle of the junction $\beta {\left|I_{n\omega }\right|}^2/\left(4{\left|p_{n\omega }^i\right|}^2\right)$ calculated at the second and third harmonic frequency. Different colors correspond to results obtained with different frequencies $\omega$ of the incident fundamental wave, as detailed in the inserts. For further details see the main text.

Figure 5

Analysis of the nonlinear response of the Na cylindrical dimer for the frequency of the fundamental wave $\omega =0.6$ eV. (a) Sketch of the geometry of the system and of the polarization of the fundamental wave. (b) Color plots of the current density ${\vec{j}}_{3\omega }$ induced at the third harmonic frequency calculated for several sizes of the gap $d_{\mathrm{gap}}$. The absolute value of the $z$-component of ${\mathbf{j}}_{3\omega }$, $\left|j_{z,3\omega }\right|\equiv \left|{\widehat{e}}_z{\mathbf{j}}_{3\omega }\right|$ is shown in the $(y,z)$ plane perpendicular to the cylinder axis. The color codes are explained in the insert. Blue frames indicate a regime of capacitive coupling between nanoparticles, and red frames indicate a conductive coupling regime. A violet frame is used for an intermediate situation at $d_{\mathrm{gap}}=6\AA$. (c) Time evolution of the nonlinear dipole $\text{Re}\left[e^{-i3\omega t}{p}_{3\omega }\right]$, charge transferred between nanoparticles $\text{Re}\left[e^{-i3\omega t}{Q}_{3\omega }\right]$, and current through the junction $\text{Re}\left[e^{-i3\omega t}{I}_{3\omega }\right]$ at the third harmonic frequency. Results are shown for $d_{\mathrm{gap}}=3\AA$, which corresponds to the maximum enhancement of the third harmonic intensity. Here, $\text{Re}\left[x\right]$ stands for the real part of $x$. All the quantities are scaled to 1 at the maximum. The color code is explained in the insert.

Figure 6

Analysis of the nonlinear response of an Al spherical dimer for $\omega =1$ eV. The dimer geometry is characterized by a gap of width $d_{\mathrm{gap}}=2\AA$, corresponding to the maximum enhancement of the third harmonic intensity. (a) Color plot of the current density ${\vec{j}}_{3\omega }$ induced at the third harmonic frequency. The real part of the $z$ component of the current density, $\text{Re}\left(j_{z,3\omega }\right)\equiv \left({\widehat{e}}_z{\vec{j}}_{3\omega }\right)$, is shown at the instant of time corresponding to the maximum third harmonic current through the junction. Results are shown in the $(x,z)$ plane with $z$ being the axis of the dimer (see Fig. 1). (b) Time evolution of the nonlinear dipole $\text{Re}\left[e^{-i3\omega t}{p}_{3\omega }\right]$, charge transferred between nanoparticles $\text{Re}\left[e^{-i3\omega t}{Q}_{3\omega }\right]$, and current through the junction $\text{Re}\left[e^{-i3\omega t}{I}_{3\omega }\right]$ at the third harmonic frequency. All the quantities are scaled to 1 at the maximum. The color code is displayed in the insert.

Figure 7

Third harmonic generation in a Na cylinder dimer. The polarization of the incident plane wave is sketched in the insert of (a). TDDFT results are presented as a function of the size of the gap between nanoparticles $d_{\mathrm{gap}}$. (a) Enhancement of the intensity of the emitted third harmonic, $\mathfrak{R}$, as compared to the case of the individual noninteracting nanoparticles. For the exact definition of $\mathfrak{R}$ see Eq. (9). (b) Hyperpolarizability, ${\alpha }^{\left(3\right)}\left(3\omega \right)$, of the same dimer. The values are given in atomic units (a.u.). The relation $1\mathrm{esu}=0.1985\times {10}^{40}$ a.u. can be used for conversion to electrostatic units (esu). Different colors correspond to results obtained with different frequencies $\omega$ of the incident fundamental wave, as detailed in the insert of (a).