Ultrafast optically induced tunneling in narrow metallic gaps from the time dependent density functional perspective

TL;DR

The paper addresses ultrafast optically induced tunneling in narrow metal gaps by combining parameter-free TDDFT with one-electron and semiclassical strong-field theories. It identifies photon-assisted tunneling channels across a range of gap sizes and biases, showing how the transport switches from perturbative multiphoton absorption to optical-field–driven emission as field strength increases. The approach reproduces key experimental trends in bowtie antennas and STM-like junctions without fitting parameters, and provides a cohesive framework linking microscopic many-body dynamics to semiclassical transport pictures. The results have significant implications for ultrafast nanoelectronics and coherent control in scanning probe technologies, while clarifying the roles of gap size, bias, and waveform in dictating dominant transport channels.

Abstract

In this work, using the time-dependent density functional theory, we address the electron tunneling triggered by short (single-cycle and several-cycle) optical pulses in narrow metallic gaps under conditions relevant for actual experiments. We identify photon-assisted tunneling with one-photon, two-photon, and higher-order photon absorption, and we discuss the effect of the tunneling barrier, applied bias, and strength of the optical field on transition from photon-assisted tunneling (weak optical fields) to the optical field emission at strong optical fields. The numerical single-electron calculations and an analytical strong-field theory model are used to gain deeper insights into the results of the time-dependent density functional theory calculations. Additionally, our parameter-free calculations allow us to retrieve and explain recent experimental results on optically induced transport in narrow metallic gaps under an applied dc bias.

Figures (12)

• Figure 1: Sketch of the processes behind the electron transport in the gap under different conditions. The potential of the junction between two metallic leads is shown as function of the $x$-coordinate perpendicular to the surface. $\mathcal{E}_F$ stands for the Fermi energy, $\hbar \omega$ is the energy of an absorbed photon. Panel a: The multiphoton regime of electron transport for narrow junction (weak fields). No bias is applied. The $\ell$-photon absorption (vertical solid arrows, $\hbar \omega$) can lead to tunneling through the potential barrier reduced by $\ell \hbar \omega$ (here $\ell=1$ or $\ell=2$), or to a classically allowed over-the-barrier transition when $\ell \hbar \omega$ is larger than the height of the tunneling barrier (here $\ell=3)$. The different $\ell$ channels of electron transfer are shown with dashed blue arrows. Panel b: The same as panel a, but the width of the junction is large. The electron tunneling contribution is negligible. Instead, electron transport is dominated by classically allowed over-the-barrier transitions. The process can be seen as electron emission followed by electron propagation in the vacuum gap. Panel c: The same as panel b, but a dc bias $U$ is applied. This reduces the potential barrier and permits tunneling. Along with over-the-barrier transitions assisted by multiphoton absorption, the dc tunneling (solid grey arrow) or tunneling induced by $\ell$-photon absorption is possible. Panel d: The optical field emission regime (strong fields) where an electron tunnels through the potential barrier reduced by the optical field (solid blue arrow).
• Figure 2: Sketch of the studied system. Cross section $(x,y)$ of a dimer of two identical parallel cylindrical nanowires infinite along the $z$-axis. The nanowires are separated by a narrow gap of width $d_{\rm{gap}}$ in the (sub-)nm range. The middle of the gap is located at $(x=0,y=0)$. The nanowire radii are $R=5$ nm so that the system represents, e.g., the gap between the tips of a bowtie nanoantenna (represented by lines) as used in experiments rybka2016Ludwig2019Luo2023Luo2024. An $x$-polarized optical pulse is incident on the nanowires along the $y$-axis leading to optically induced electron transport across the gap. The inset shows the $x$-component of the time-dependent electric field of the single-cycle pulse with CEP$=0$.
• Figure 3: Sketch of the electron transport described by the SFT. Panel a: When the parameter $\zeta > 1$, the electron is excited by absorbing a few photons and then tunnels through the junction. In this perturbative regime of PAT, the contribution of the $\ell$th channel to the transport decreases with $\ell$ as $E_{\rm{gap}}^{2\ell}$ (see Eq. \ref{['eq:Pedersen']}). The photon-assisted channels do not exceed the barrier threshold so that we observe solely the PAT. Panel b: When $\zeta < 1$ but $\gamma \gg 1$, electron transport occurs in the multiphoton regime. The electron can absorb sufficient energy to be excited over the barrier and into a Volkov state (above-threshold ionization, ATI). The effective photon absorption reaches the barrier threshold. Although under-the-barrier PAT still occurs, it is not the dominant transport mechanism. Panel c: When $\zeta < 1$ and $\gamma \ll 1$, the potential barrier is adiabatically suppressed by the field, allowing the electron to efficiently tunnel through the junction.
• Figure 4: Electron transfer induced by a $x$-polarized single-cycle optical pulse ($\omega=0.95$ eV, CEP$=\pi$) across the gap of a nanowire dimer with work function $\Phi_{\rm{M}}=5$ eV. The electric field of the pulse is shown in the inset of panel a. Panel a: Electron transfer $\mathcal{N}$ defined as the net number of electrons transferred per pulse and per nm length of the dimer. TDDFT results (dots) calculated for a width of the gap of 0.8 nm (red), 1 nm (blue), and 2 nm (black) are shown as a function of the electric field in the gap $E_{\rm{g}}$ and of the Keldysh parameter $\gamma$. The lines of the corresponding color display the fit by the $\mathcal{N} \propto E_{\rm{g}}^{2n}$ dependence characteristic for the multiphoton regime. The effective number of photons $n$ is given in the legend. The vertical red bar marks the regime transition at $\zeta=1$ for the 0.8 nm gap. Panel b: Self-consistent ground-state potential $V_{\rm{gs}}(x,y=0)$ in the gap region as a function of the $x$-coordinate along the dimer axis. Results are shown for a gap of the width of 0.8 nm (red), and 1 nm (blue). The energy is measured with respect to the vacuum level. $\mathcal{E}_F=-\Phi$ is the Fermi level energy corresponding to the effective work function $\Phi=4.2$ eV used in the TDDFT (see Table \ref{['table1']}). Dashed arrows indicate electron tunneling and over-the-barrier transitions induced by $\ell$-photon absorption. The latter is indicated with vertical arrows. The $\ell=1$ tunneling channel is not observed in panel a.
• Figure 5: Model 1D WPP study of electron transfer induced by the $x$-polarized single-cycle optical pulse ($\omega=0.95$ eV, CEP$=0$ and CEP$=\pi$) across metal leads separated by a 0.8 nm gap. The metal work function is $\Phi_{\rm{M}}=5.0$ eV. Initially, an electron occupies an orbital at the Fermi energy $\mathcal{E}_F$ in the left lead. The electric field of the single-cycle pulse is given by Eq. \ref{['eq:Egap']}. The probability $\mathcal{P}_{L \rightarrow R}$ of electron transfer from the left to the right electrode across the junction is shown with solid dots as a function of the field amplitude in the gap $E_{\rm{g}}$. Blue dots: $\mathcal{P}_{L \rightarrow R}(0)$ obtained with CEP$=0$. Green dots: $\mathcal{P}_{L \rightarrow R}(\pi)$ obtained with CEP$=\pi$. Red dots show the difference $\mathcal{N}=\mathcal{P}_{L \rightarrow R}(\pi)-\mathcal{P}_{L \rightarrow R}(0)$ representing the net electron transfer induced by the CEP$=\pi$ single-cycle optical pulse between the two leads. Lines: fit by the $E_{\rm{g}}^{2n}$ dependence. The effective number of photons $n$ is given in the legend. Clear changes of $n$ occur around $E_{\rm{g}}=3$ V/nm ($\zeta=1$) marked by a vertical grey bar. The inset shows the dependence of the model potential in the gap region on the $x$-coordinate. Energy zero corresponds to the vacuum level. For further details see the text.