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

1. Problem Statement

The paper addresses the fundamental physics of ultrafast optically induced electron tunneling in narrow metallic gaps (sub-nanometer to nanometer scale). While Scanning Tunneling Microscopy (STM) and optical nanoantennas have enabled the study of electron dynamics with atomic spatial and femtosecond temporal resolution, a comprehensive, parameter-free theoretical understanding of the transition between different transport regimes remains challenging.

Key challenges include:

• Regime Transition: Distinguishing between Photon-Assisted Tunneling (PAT) (where electrons absorb discrete photons to tunnel through a barrier) and Optical Field Emission (where strong fields adiabatically suppress the barrier, allowing tunneling).
• Coherence vs. Incoherence: Determining whether electron transport is an incoherent sequence of excitation and tunneling or a coherent, single-step process driven by the optical field waveform.
• Parameter Dependence: Understanding how the transport mechanism evolves with optical field strength ($E_g$), gap size ($d_{gap}$), applied DC bias ($U$), and Carrier Envelope Phase (CEP).
• Theoretical Limitations: Existing perturbative theories (e.g., Tien-Gordon) fail for strong fields and short pulses, while many-body simulations often lack the ability to isolate specific transport channels without adjustable parameters.

2. Methodology

The authors employ a multi-faceted theoretical approach combining rigorous quantum many-body simulations with analytical and semi-classical models to ensure robustness and physical insight.

A. Time-Dependent Density Functional Theory (TDDFT)

• System: A model system of two parallel cylindrical nanowires (radius $R=5$ nm) separated by a narrow vacuum gap ($d_{gap} = 0.8 - 2$ nm).
• Model: Uses a free-electron (jellium) model for the metal to allow for larger system sizes relevant to experiments, avoiding the prohibitive cost of full atomistic descriptions.
• Implementation: Solves the time-dependent Kohn-Sham equations using the Adiabatic Local Density Approximation (ALDA).
• Key Feature: The work function is adjusted to account for the missing long-range image potential in standard ALDA, ensuring the tunneling barrier height matches classical expectations.
• Inputs: Single-cycle and multi-cycle optical pulses (femtosecond duration) and optional DC bias (simulated via a slow THz field).

B. One-Active-Electron (WPP) Model

• Approach: A 1D numerical solution of the time-dependent Schrödinger equation (TDSE) for a single electron active in the transition.
• Purpose: Provides intuitive access to single-particle dynamics, energy spectra, and symmetry constraints that are masked in many-body TDDFT. It uses model potentials including image charges.

C. Semi-Classical Strong-Field Theory (SFT)

• Approach: An analytical framework based on the saddle-point method (strong-field approximation).
• Purpose: Derives analytical expressions for electron transfer probabilities in different regimes (perturbative PAT, multiphoton, and field emission). It introduces parameters like the Keldysh parameter ($\gamma$) and a new parameter $\zeta$ to define regime boundaries.

3. Key Contributions

1. Parameter-Free Quantitative Analysis: The study provides a "parameter-free" TDDFT description of optically induced transport, directly comparing with experimental data without fitting adjustable parameters for excitation or relaxation.
2. Identification of Transport Regimes: The authors map the transition from perturbative PAT (dominated by low-order photon absorption) to multiphoton regimes and finally to optical field emission as a function of field strength and gap geometry.
3. Symmetry and CEP Control:
   • In symmetric gaps (zero bias), the net electron transfer driven by single-photon absorption ($\ell=1$) is zero due to symmetry cancellation. The lowest order contribution is two-photon absorption ($\ell=2$).
   • CEP control of net current is only effective in the multiphoton regime ($\ell \ge 2$). In the single-photon regime, CEP sensitivity vanishes.
   • Applying a DC bias breaks symmetry, enabling the observation of the $\ell=1$ channel.
4. Unified View of "Fast" Transport: The results confirm that optically induced transport is a coherent, one-step process. Electron excitation and tunneling cannot be separated in time; the electron is launched directly by the optical field waveform.

4. Key Results

A. Regime Transitions and Photon Order

• Perturbative Regime ($\zeta > 1$): At low field strengths, transport is dominated by the lowest symmetry-allowed photon order ($\ell_{min}$).
  • Symmetric gap: $\ell_{min} = 2$ ($N \propto E_g^4$).
  • Asymmetric gap (with bias): $\ell_{min} = 1$ ($N \propto E_g^2$).
• Multiphoton Regime ($\zeta < 1, \gamma \gg 1$): As field strength increases, higher-order channels contribute. The effective photon order $n$ (extracted from $N \propto E_g^{2n}$) converges to the threshold value determined by the barrier height: $n \approx W_{tb}/\hbar\omega$.
• Field Emission Regime ($\zeta < 1, \gamma \ll 1$): At very high fields, the barrier is adiabatically suppressed. Transport follows a Fowler-Nordheim-like scaling, and the concept of discrete photon absorption becomes less distinct.

B. Effect of DC Bias and Gap Size

• Bias: A DC bias lowers the tunneling barrier, increasing the net current and enabling the $\ell=1$ channel in symmetric geometries. It also shifts the transition point between perturbative and multiphoton regimes to lower field strengths.
• Gap Size: Increasing the gap width ($d_{gap}$) broadens the barrier, suppressing tunneling. For wider gaps (e.g., 2 nm), the system is dominated by over-the-barrier transitions (multiphoton photoemission) rather than tunneling, unless the bias is significantly increased.

C. Comparison with Experiments

The authors successfully reproduced trends from recent experiments on:

• Gold Bowtie Antennas (Luo et al.): The theory correctly predicts the transition from single-photon to multiphoton dominance and the effective photon order ($n \approx 2$) observed in experiments.
• Silver STM Junctions (Lin et al.): The model reproduces the dependence of photocurrent on bias and gap size, confirming that the "fast" coherent component dominates the signal in these ultrafast experiments.

D. Energy Spectra

Analysis of the electron energy spectra (via WPP) reveals distinct sidebands corresponding to $\ell$-photon absorption. As the field strength increases, the distribution shifts from tunneling-dominated (energies below the barrier) to over-the-barrier transitions (energies above the barrier), with the cutoff energy determined by the work function and field parameters.

5. Significance and Outlook

• Fundamental Physics: The work clarifies the nature of light-matter interaction in nanogaps, establishing that ultrafast electron transport is a coherent, field-driven phenomenon rather than a sequential excitation-tunneling process.
• Technological Impact: The findings are crucial for the development of Petahertz electronics and Ultrafast STM (USTM). Understanding the CEP dependence and the role of bias allows for the precise control of electron currents at the attosecond scale.
• Methodological Advance: The combination of parameter-free TDDFT with analytical SFT provides a robust framework for interpreting complex nanoscale experiments, bridging the gap between quantum many-body theory and observable macroscopic currents.
• Future Directions: The authors note that while the "fast" coherent component is captured, future work must address electron relaxation (thermalization) and band-structure effects (d-band electrons) to fully describe the "slow" thermionic components of emission observed in some experiments.

In summary, this paper provides a comprehensive, first-principles theoretical framework that deciphers the complex interplay of photon absorption, tunneling, and field emission in metallic nanojunctions, offering a predictive tool for next-generation ultrafast optoelectronic devices.