Terahertz radiation-induced sub-cycle field electron emission across a split-gap
dipole antenna
Jingdi Zhang, Xiaoguang Zhao, Kebin Fan, Xiaoning Wang, Gu-Feng Zhang, Kun Geng,
Xin Zhang, Richard D. Averitt
Supplemental material:
1. Kinetic energy emitted near the peak of in-gap THz E-field
To simulate the in-gap acceleration of electrons emitted at the peak amplitude of the THz
pulse, we use the non-relativistic equation of motion for a free electron in an AC field.
𝑚𝑒 𝑥̈ = −𝑒𝑓(𝑥)𝐸(𝑡)
𝑥
𝑥−𝑑
𝑓(𝑥) = 108 [exp (− ) + exp⁡(
)] + 60, 𝑥 ∈ [0, 2⁡𝜇𝑚]
𝑙𝑓
𝑙𝑓
𝐸(𝑡) = 𝐸0 𝑒𝑥𝑝 (−2𝑙𝑛2
𝑡2
) cos⁡(2𝜋𝜈𝑡)
𝜏2
Initial conditions: 𝑥𝑡=0 = 0; 𝑥̇ 𝑡=0 = 0,
where me is the electron rest mass, e elementary charge, f(x) is the spatial distribution of
field enhancement, lf is the decay length of in-gap THz field (~0.094 μm), τ is the pulse
duration of incident THz pulse (~1 ps), and ν is the central carrier frequency (~1 THz).
Figure 1S
Figure 1S (a) (b) shows the spatial distribution of field enhancement of f(x) and
time-domain THz pulse E(t), respectively. For the in-gap current simulations, we use a
THz pulse (E(t)) that peaks at t=0. That a free electron is located at the gold-vacuum
interface at t=0 that is subsequently accelerated by the in-gap E-field.
Figure 1S (c) (d) show the displacement and velocity of the free electron as a function of
time. The time scale for electron to complete its flight is 0.1 to 0.2 ps. This verifies the
sub-cycle nature of the emission process. One also notices that the calculated velocity is
over 3*107 m/s, which is a tenth of speed of light. Recall that the calculation is done at
incident field amplitude of 200 kV/cm. It is likely that the accelerated electron has entered
relativistic regime under THz field (200 kV/cm). This is because of the uniqueness of the
THz wave, i.e. high amplitude, low frequency, which is difficult to achieve in other
experiments using mid-IR and near-IR excitation. Figure 1S (e) (f) shows the velocity and
kinetic energy of emitted electron as a function of its displacement.
In general, electron emission from a metallic tip/gap is a two-step process: (1) Ionization
through MPI (Multi-photon ionization), ATI (Above threshold ionization) or TI (Tunneling
ionization); (2) Acceleration by the oscillating electric field. From the perspective of
maximum electron acceleration, it is advantageous to accelerate the electron in a strong
and slow-varying field, such that electrons can utilize the full energy of the
electro-magnetic field to gain maximum forward momentum while suppressing the quiver
motion. As such, THz pulses push electrons into high momentum regime more easily than
higher frequency fields.
2. Simulation of emitted electron energy spectrum
To calculate the energy spectrum of emitted electrons (Fig. 5, main text), we use a more
realistic in-gap THz field (in comparison to Fig. 1S b) obtained from time-domain
numerical simulations using CST microwave studio. We simulate the trajectory and
energy of the field emitted electrons initiated at different in-gap electric fields, instead of
only calculating the maximum kinetic energy of the electrons emitted at the peak of the
THz pulse as in Figure 1S. This is realized by shifting the time zero of the in-gap THz
E-field, and then using the same equations of motion as used for the for trajectory and
energy simulations in Figure 1S.
3. THz beam characteristics at the sample location
The size of our THz beam is ~1 mm FWHM. We used two different methods to
characterize the field strength of the THz electric field. (i). we used standard EO sampling
to characterize the peak field of THz pulse by measuring ∆I/I from the balanced
photodiode and calculating the field using standard expressions. (ii). another method
that gives a consistent electric field strength on a daily basis is to employ a doped (carrier
density 1*1016 cm3 ) GaAs metamaterial (GaAs MM) (for details see reference 11 from the
main text). The GaAs metamaterial resonance amplitude as a function of the incident THz
peak electric field shows a characteristic deflection point at incident field amplitude of
~150 kV/cm.