Energetics and Dynamics of Laser-Assisted Field Emission from Silicene Nanoribbons: Time-Dependent First-Principles Study

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We investigate laser-assisted field emission (LAFE) from a silicene nanoribbon (SiNR) using time-dependent density functional theory (TDDFT) simulation. The emission mechanism in the present study is considered to be over-barrier photoemission and is found to be governed by electronic dipole transitions, the characteristics of the excited states, and the energy levels of the excited states relative to the potential hump. The qualitative features of emission from the SiNR are similar to those from graphene nanoribbons (GNRs). The emission currents from SiNR, however, are found to be much larger than those from GNR for the same laser parameters in spite of a larger work function for SiNR. We reveal the emission currents in real time and space on an atomic scale, and observed current being driven back and forth in the early stage of emission. We further elucidate the dynamical correlation among the laser pulse, the Kohn-Sham potential and emission currents wiggling under the ponderomotive force. [DOI: 10.1380/ejssnt.2015.115]

본문

Energetics and Dynamics of Laser-Assisted Field Emission from Silicene Nanoribbons:

Time-Dependent First-Principles Study

Taishi Higuchi, Chunping Hu, and Kazuyuki Watanabe _∗_

_Department_ _of_ _Physics,_ _Tokyo_ _University_ _of_ _Science,_

_1-3_ _Kagurazaka,_ _Shinjuku,_ _Tokyo_ _162-8601,_ _Japan_
(Received 12 December 2014; Accepted 16 March 2015; Published 28 March 2015)

We investigate laser-assisted field emission (LAFE) from a silicene nanoribbon (SiNR) using time-dependent
density functional theory (TDDFT) simulation. The emission mechanism in the present study is considered to be
over-barrier photoemission and is found to be governed by electronic dipole transitions, the characteristics of the
excited states, and the energy levels of the excited states relative to the potential hump. The qualitative features
of emission from the SiNR are similar to those from graphene nanoribbons (GNRs). The emission currents from
SiNR, however, are found to be much larger than those from GNR for the same laser parameters in spite of a larger
work function for SiNR. We reveal the emission currents in real time and space on an atomic scale, and observed
current being driven back and forth in the early stage of emission. We further elucidate the dynamical correlation
among the laser pulse, the Kohn-Sham potential and emission currents wiggling under the ponderomotive force.

[DOI: 10.1380/ejssnt.2015.115]

Keywords: Laser-assisted field emission; Silicene; Time-dependent density functional theory

I. INTRODUCTION

Studies of ultrashort-pulse field emission (FE) from
nanoscale metal tips using femtosecond laser pulses [1, 2]
have attracted great interest because of the possibility of
spatiotemporal resolution down to an atomic scale with
an attosecond range utilizing a coherent electron source.
Recently, an ultrashort-pulse FE source with site selectivity on the scale of a few tens of nanometers was reported

[3, 4]. Emission mechanisms have been investigated and
categorized phenomenologically by several processes such
as photofield emission, multi-photon photoemission (or
above-threshold photoemission (ATP)) and optical field
emission. Which of these processes is dominant depends
on the value of the Keldysh parameter _γ_, which is determined by laser parameters and the work function of
the emitter [5]. Optical field emission begins to play an
important role in attosecond physics with an increasing
laser field [6–10]. The electron rescattering with surface
atoms [9, 11] that results in a plateau and a cutoff in
the photoemission spectrum [9, 10] and the carrier envelope phase (CEP) [10, 12, 13] of few-cycle near-infrared
laser pulses are typical phenomena in attosecond emission
dynamics. In order to further understand the emission
processes, different calculation methods have been presented in order to interpret the emission mechanisms. A
modified Fowler-Nordheim formula has been utilized for
optical field emission [1, 2], and the Boltzmann equation
has been solved for photofield emission [3, 4, 14]. More recently, the finite-difference time domain method has provided a field strength distribution around the tip apex in
100 nm dimensions with the presence of the CEP [9]. The
authors of Ref. 9 also applied a 1D time-dependent density functional theory (TDDFT) [15] simulation of jellium
substrate combined with 3D-quasiclassical Monte Carlo
simulations to the time-dependence analysis of emitted
electrons and their energy spectra in the ATP processes.

The objectives of our computational study are to understand the origin of the electronic states responsible

_∗_ Corresponding author: kazuyuki@rs.kagu.tus.ac.jp

for laser-assisted excitation and emission from nanostructures and demonstrate ultrafast electron dynamics in real
time. We first perform density-functional theory (DFT)
calculations of 3D-atomic nanostructures to determine the
electronic band structures. Next, time evolution of electron emission from the atomic surface is demonstrated
by TDDFT simulations to elucidate the dynamical correlations among the femtosecond laser pulse, Kohn-Sham
(KS) potential and emitted electrons on a femtosecond
time scale. The TDDFT method has become a standard tool for studying the electronic states of excitations
and non-equilibrium processes for many-electron systems.
Since the KS Hamiltonian includes electron-electron interaction (Hartree and exchange-correlation), electron-ion
interaction, and the interaction of electrons with external
fields (DC field and laser field) automatically, the time
evolution of electron wavefunctions obtained from the solution of the time-dependent KS equation gives detailed
information on excitation and emission dynamics.

In our previous study on this topic with TDDFT simulations, we clarified the microscopic mechanism of electron emission from graphene nanoribbons (GNRs) by an
analysis of the electronic dipole transition matrix element
and the electronic properties of the emitting states [16].
We observed photoemission for the laser parameters used
in the simulation and found that the emission properties
strongly depend on the electronic structures of GNRs.
In the present study, we perform a simulation of laserassisted field emission (LAFE) from silicene nanoribbons
(SiNRs) [17], which are similar to GNRs in electronic
state and atomic structure. Silicene, which is a one atom
thick silicon sheet arranged in a honeycomb lattice like
graphene, has been shown to have properties different
from graphene. Although the electrons in silicene were
considered to behave as Dirac fermions [18], many studies
have shown that silicene synthesized on Ag(111) surfaces
does not have Dirac fermions due to the strong coupling
with the substrate [19]. Further, even freestanding silicene
does not have a flat layer but a buckled one. (See a review
article 20 for the recent studies on silicene). Therefore, it
is of interest to compare the electron emission properties
of SiNRs with those of GNRs, as done in this study.

ISSN 1348 0391 _⃝_ c 2015 The Surface Science Society of Japan (http://www sssj org/ejssnt) 115

ELaser

EDC

2.7Å

A

II. METHOD AND MODEL

Throughout this section, we adopt atomic units in
which ℏ = 1, _m_ = 1 and _e_ 2 = 1, unless stated otherwise.
The time-dependent KS equation is given below as

_i_ _∂t_ _∂_ _ψi_ ( _r_ _, t_ ) = _H_ ( _r_ _, t_ ) _ψi_ ( _r_ _, t_ ) _,_ (1)

where _ψi_ ( _r_ _, t_ ) is the KS wavefunction of the valence electron, and _i_ denotes the level index that runs over all occupied valence states. The Hamiltonian is constructed by

(a)

(b)

0
(c)(c)

B C

−0.05

−0.1

0 10 20

Time[fs]

∑

1 2 _∇_ 2 +

_H_ ( _r_ _, t_ ) = _−_ 1 2

_V_ L( _r_ _−_ _R_ _α_ )

c urre nt

24.4Å

_α_

∑ _α_

_V_ NL _l,m_ ( _r_ _−_ _R_ _α_ ) ˆ _Pl_

_α,l,m_

∫

_d_ _r_ _′_ _ρ_ ( _r′_ _, t_ )

_|_ _r_ _−_ _r_ _′_ _|_

+ _µ_ xc[ _ρ_ ( _r_ _, t_ )] + _v_ ext( _r_ _, t_ ) _,_ (2)

where _R_ _α_ ( _t_ ) is the position of the _α_ th ion, _V_ L( _r_ _−_ _R_ _α_ ) is
the local pseudopotential and _V_ NL _l,m_ ( _r_ _−_ _R_ _α_ ) is the nonlo-

cal pseudopotential [21] for the _α_ th ion at _R_ _α_, formulated
by Troullier and Martins [22] based on the KleinmanBylander separable form in real space [23]. _P_ ˆ _l_ is the projection operator on the angular momentum ( _l_ ) and _m_ is
a magnetic quantum number that takes integral values
from _−l_ to _l_ . _µ_ xc is the exchange-correlation potential,
and _v_ ext( _r_ _, t_ ) is the external field potential.

From the time-dependent KS wavefunctions, the electronic density _ρ_ and emission current density J are respectively calculated as

FIG. 1. (a) Side view of hydrogen (H)-terminated SiNR. (b)
Top view of H-SiNR under an electrostatic field (EDC) and
a laser pulse (ELaser). The Si and H atoms are indicated by
large and small balls, respectively. The currents are emitted
toward the right vacuum region. The unit cell used in the
calculations is shown by the dashed box. (c) Time evolution
of sum of electrostatic field and laser field.

points were chosen along the ribbon axis in the first Brillouin zone. A complex absorbing potential (CAP) [27]
is introduced in the vacuum to absorb emitted electrons,
preventing them from being reflected at the boundary of
the unit cell.

The emission current is measured in an observation
plane placed at 10 ˚ A above the H atoms, while three
positions labeled by A, B, and C are chosen for monitoring the spatiotemporal evolution of the current densities.
The distance from the H atom to position A is 2.7 ˚ A,
while the A-B and B-C distances are both 1.2 ˚ A. Figure 1(c) shows the time evolution of the electric field applied to the SiNR, which consists of an electrostatic field
of _−_ 0 _._ 075 V/ ˚ A and a laser field with a strength of 0.05
V/ ˚ A (power of 3 _._ 3 _×_ 10 10 W/cm 2 ) and an energy of 6 eV.
The laser field is spatially uniform and has a Gaussian
envelope with a FWHM of 6 fs.

III. RESULTS

We first calculated the SiNR band structures in the
absence of external fields together with the averaged KS
potentials along the _x_ axis under an electrostatic field of
_E_ DC = _−_ 0 _._ 075V/ ˚ A, as shown in Fig. 2(a) and (b), respectively. To compare, we also made the same calculations for the counterpart GNR with hydrogen-terminated
zigzag edges and the results are shown in (c) and (d),
which are essentially the same as Fig. 2(a) and (b) in
Ref. 16, respectively except that the Kohn-Sham potential in Fig. 2(b) in Ref. 16 was put downward slightly due
to a different energy origin. In all figures, the energies
are measured from the Fermi energies. In both (a) and
(c), flat bands near the X point, which correspond to the
edge states, and quasi-continuum states above an energy
of about 4 eV, which correspond to the vacuum levels,
can be observed. Quantitatively, the energy-level spacings are smaller for the SiNR than for the GNR, which
means that there might be more contributing channels to

_ρ_ ( _r_ _, t_ ) =

∑

_i_

_fi|ψi_ ( _r_ _, t_ ) _|_ 2 _,_ (3)

and

J ( r _, t_ ) =

∑

_i_

1
2 _i_ [ _ψ_ _i_ _∗_ ( r _,_ t ) _∇ψi_ ( r _, t_ ) _−_ _ψi_ ( r _, t_ ) _∇ψ_ _i_ _∗_ ( r _, t_ )] _,_

(4)
where _fi_ is the occupation number of the _i_ -th KS level.
In the present study, the time evolution operator is approximated by a fourth-order Taylor expansion method

[24]. The Ceperley-Alder functional [25] for local-density
approximation with Perdew-Zunger parameterization [26]
is used for the exchange-correlation interaction. The KS
Hamiltonian includes the potentials of an electrostatic
field and a laser field, which are simultaneously applied
to the target nanostructure.

The calculation model is shown in Fig. 1. The target
is chosen to be a hydrogen (H)-terminated zigzag-edged
SiNR. Its atomic structure is similar to GNRs, as used in
the previous simulation in Ref. 16, except for the buckled
structure in Fig. 1(a). The optimized buckling distance
of the Si atoms in the _z_ -direction is 0.44 ˚ A, which agrees
well with previous calculations [18]. The directions of the
external fields, as well as the unit cell with the periodic
boundary condition, are given in (b). Both of the electrostatic and laser fields are applied on the _x_ axis. The
unit cell size is set at 65.5 _×_ 3.97 _×_ 15.0 ˚ A 3 . Twelve _ky_

116 http://www sssj org/ejssnt (J Stage: http://www jstage jst go jp/browse/ejssnt/)

(a) (b)

6
Φ =4.6eV

# (a) (b)

28th

30th

31
~~30~~

28 4 hump

−2

−4
0 10 20 30 40 50 60

x[Å]

-2

-4

-6

Γ X
ky

-2

Initial state

Φ =4.0eV

−2

−4
0 10 20 30 40 50

x[Å]

-4

-6

Γ ky X

FIG. 2. Energy band structures of (a) SiNR and (c) GNR. Integer numbers indicate the band indices. KS potentials under
the electrostatic field for the SiNR and GNR are shown in (b)
and (d), respectively. The dotted lines in figures (b) and (d)
indicate the work function (upper line) in the absence of an
electric field and the Fermi energy (lower line), respectively.
Figures (c) and (d) are essentially the same as Fig. 2(a) and
(b) in Ref. 16, respectively except for the details.

the LAFE from the SiNR. The KS potentials in (b) and
(d) are similar to each other, showing a linear decrease
from greater than _x ∼_ 25 ˚ A with a slope corresponding to
the applied field _E_ DC. The work function is larger for the
SiNR (4.6 eV) than for the GNR (4.0 eV), which means
that the FE should be more difficult from the SiNR than
that from the GNR in the absence of laser fields.

To further clarify the band contributions to the FE currents from the SiNR, we calculated the dipole transition
matrix elements, _V_ d = _⟨i|x_ ˆ _|f_ _⟩_ . These are the off-diagonal
matrix elements of the dipole operator _x_ ˆ between the initial states _|i⟩_ and the final states _|f_ _⟩_ corresponding to
the energy bands in Fig. 2(a). The calculated matrix elements at the Γ point are shown in Fig. 3(a), since the
elements at Γ are significantly larger than those at other
k points. The red color shows the largest _V_ d, and blue
shows the smallest. The levels marked red do not make
an important contribution to emission because the 26th
level is below and 20th level is far below the potential

FIG. 3. (a)Map of dipole matrix elements for SiNR at Γ point.
The red color shows the largest _V_ d, and blue shows the smallest. (b) Side view of the SiNR KS wavefunctions for the 28th
and 30th levels at Γ are shown.

hump, as seen in Fig. 2(b). In contrast, the 28th and
30th states are found to be emission states, although a
_V_ d(10 _→_ 28) (solid circle, the energy difference between
the two states is 6.8 eV) and _V_ d(14 _→_ 30) (dashed circle,
the energy difference between the two states is 6.0 eV) do
not seem large. The energy levels of the final states, _i.e._,
the 28th and 30th levels at the Γ point, are sufficiently
higher than the potential hump.

The corresponding KS wavefunctions for the final states
are shown in Fig. 3(b). It can be seen that these levels
have electronic structures with a _σ_ -like character having
electronic distribution along the direction parallel to the
external fields and protruding toward the vacuum. According to our previous study, the relative angle between
the direction of electronic distribution and that of applied
fields was found to be a critical factor for determining
the emission probability; electrons can be emitted efficiently from _σ_ -like states (parallel to the electric field)
but not from _π_ -like states (perpendicular to the electric
field). This has been verified both in conventional FE
from graphitic nanostructures and LAFE from a GNR

[16, 28]. Therefore, the orbital characteristics in Fig. 3(b)
are important for the discussion below. In determination
of the electronic states responsible for electron emission,
we took into account the value of _V_ d and the orbital characteristics of the emitting states. It would be necessary to
evaluate the the time-dependent decomposition of KohnSham orbitals that contribute electron emission for con

http://www sssj org/ejssnt (J Stage: http://www jstage jst go jp/browse/ejssnt/) 117

firmation of the emitting states. This will be the next
study.

Having discussed the origin of the electronic states responsible for emission from the SiNR, we present the results of the emission currents, compared with those from
the counterpart GNR in Fig. 4. The results for laser energies of 6.0 and 6.8 eV are shown; for energies of 5.0 and
5.5 eV, we observed emission currents two orders of magnitude smaller (not shown here). In spite of the larger
work function of the SiNR (4.6 eV) compared with that
of the GNR (4.0 eV), the emission current from the SiNR
under the 6.8 eV laser is found to be about six times as
large as that of the GNR. The present results can be interpreted by a combination of the factors discussed above:
the _V_ d, the energy level of the emitting states relative to
the potential hump and the orbital characteristics of the
emitting states. In addition, we observed that there are
more emission channels (higher density of states) in the
SiNR.

For the 6.8 eV laser, _V_ d(10 _→_ 28) makes a dominant contribution to the FE current for the SiNR (see Fig. 3(a),
solid circle), while the transition from the 16th to the
20th level dominates the current from the GNR [16]. The
orbital characteristics for the 28th level of the SiNR enable the electrons to emit efficiently due to its protruding
structure near the H atom. As shown in Fig. 2(a), the
energy of the 28th level is sufficiently high such that the
emitted electrons can efficiently go beyond the potential
hump. Therefore, excitation from the 10th to the 28th
level is a favored channel for SiNR electron emission. The
number of emission channels also makes important contributions to the larger current of the SiNR. Under the 6.8
eV laser, electrons in the 13th to 17th levels can be excited
to the quasi-continuum above the 31st level, where the
transition probabilities are greatened by the high-density
of continuum states, even with a small _V_ d, and then the
excited electrons can be emitted efficiently towards the
vacuum. These are the reasons for the large current from
the SiNR under the 6.8 eV laser with an emission mechanism considered to be over-barrier photoemission [10].

For the 6.0 eV laser, _V_ d(14 _→_ 30) (dashed circle in
Fig. 3(a)), which looks small, is considered to be a possible channel of emission from the SiNR because the final
state is sufficiently high relative to the potential hump.
Compared with emission from the 28th level, that from
the 30th level could be less efficient because KS wavefunction projection of the level near the H atom is smaller than
that for the 28th level (see Fig. 3(b)), although the energy
of the state is above the potential hump. In addition to
the 30th level emission, excitation from the 17th level to
the quasi-continuum also contributes to the current under
the 6.0 eV laser. This explains the finite FE current from
the SiNR under the 6.0 eV laser. In contrast, the GNR
has no channels for emission under the 6.0 eV laser and
thus the FE current is negligible.

To understand the spatiotemporal FE dynamics in
more detail, we examine the time evolution of emitted
current density at the points labeled A, B and C under
the 6 eV laser as in Fig. 1. The corresponding results
are shown in Fig. 5. At point B, the KS potential under the electrostatic field without the laser has a spatially
maximum value (hump); this corresponds to _x_ _∼_ 28 ˚ A in
Fig. 2(b). The emission current density in Fig. 5(a) os

0.5

0.4

0.3

0.2

0.1

6.8eV

6eV

6.8eV

6eV

0 10 20

Time[fs]

FIG. 4. Time-dependent emission currents per unit length
from SiNR (red) and GNR (blue) under electrostatic field of
EDC = 0.075 V/ ˚ A and laser pulse with strength of ELaser =
0.05 V/ ˚ A at energy of 6.0 and 6.8 eV, respectively.

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

(a)(a) (b)(b) (c)(c)

(a)

(b)

(c)

AA0

CA'

0 10 20

Time[fs]Time [fs]

0 10 20

Time[fs]Time [fs]

Time[fs]Time [fs] Time[fs]Time [fs] Time[fs]Time [fs]

FIG. 5. Time evolution of the emission currents from SiNR at
the points labeled as A, B and C under conditions in Fig. 1
are shown as (a), (b) and (c), respectively.

cillates and increases in amplitude initially, it reaches a
maximum at _∼_ 12.5 fs, and then decreases. The oscillation
period corresponds to that of the laser. Similarly, the current density in (b) also shows an oscillating feature and
reaches a maximum at _∼_ 14 fs; however, the oscillation
amplitudes in (c) are greatly reduced with a maximum at
_∼_ 17 fs. The oscillations are the strongest in (a) because
position A is located closest to the ribbon edge, while the
other two points are located farther away, where FE currents from different channels can overlap and cancel out
the oscillations. This can also explain the delay in time
at which the maximum is reached in (b) and (c).

Another interesting feature observed in Fig. 5 is that
the oscillating current density in (a) and (b) changes polarity at the initial time; the emitted electrons are driven
back and forth every 0.69 fs. The average of the current
density is positive, which indicates emission of electrons
toward the vacuum. This shows the wiggle motion of the
emitted electrons is caused by the laser field [29], and the
polarity change in (b) implies that the electrons go back
into the ribbon.

Figure 6(a) shows the averaged KS potentials for the
SiNR along the _x_ axis at _t_ = 14.15 and 14.51 fs, denoted

118 http://www sssj org/ejssnt (J Stage: http://www jstage jst go jp/browse/ejssnt/)

(a)

(b)

(c)

−2

−4

t=14.51fs

t=14.15fs

10 20 30 40 50 60

x[Å]

14.15 fs

14.51 fs

9.20 fs

14.03 fs

18.87 fs

FIG. 6. (a) Averaged KS potentials for SiNR along _x_ axis at
_t_ = 14.15 and 14.51 fs, denoted by solid and broken lines, respectively. (b) Corresponding FE current density distribution
snapshots along _x_ -axis for (b) short and (c) long time scales.

by the solid and broken lines, respectively. The interval
between the two times is close to a half period of the laser
oscillation (0.34 fs); the slopes of the broken and solid
lines beyond the region _x_ = 30 ˚ A are -0.039 and -0.124
V/ ˚ A respectively. Here, it is noted in Fig. 6(a) that
the KS potential hump does not decrease noticeably with
increasing the time-dependent field strength. The feature
is understood by the fact that the charge polarizations at
both ends of the ribbon that is induced by the laser field
hinder the electron emission by increasing the amount of
the electric-dipole layers at ribbon edges.

Snapshots of the FE current density distribution at the
two times are shown in Fig. 6(b). In this short timescale,

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we cannot clearly see current propagation toward the vacuum. However, in Fig. 6(c), which are the snapshots over
a longer time scale, propagation of the emission current
toward the vacuum region is clearly seen, especially for
_t_ = 14.03 and 18.87 fs. As the laser has its peak at _t_
= 12 fs (Fig. 1(c)), the electrons begin to emit after the
laser peak (see the center panel of Fig. 6(c)) and are then
driven by the electrostatic field to the vacuum.

IV. CONCLUSION

In conclusion, the energetics and dynamics of LAFE
from nanoribbons were clarified by TDDFT simulations.
The emission mechanism, which is common to the SiNR
and GNR, is found to be governed by three factors: the
electronic dipole matrix element, the orbital characteristics of the excited states, and the energy levels of the
excited states relative to the potential hump. However,
the emission currents from the SiNR are found to be much
larger than from the GNR for the same laser parameters in
spite of the larger work function of the SiNR. This can be
interpreted by the higher density of states, or more channels for dipole transition in SiNRs than that of GNRs.
We demonstrated the dynamical correlation among the
laser pulse, the Kohn-Sham potential and emission currents wiggling under the laser field in real time and space
on an atomic scale, and observed emission current being
driven back and forth in the early stage of emission.

ACKNOWLEDGMENTS

This work was supported in part by the Project of
Materials Design through Computics: Complex Correlation and Non-Equilibrium Dynamics, a Grant-in-Aid for
Scientific Research on Innovative Areas, and the Next
Generation Super Computing Project, Nanoscience Program, MEXT, Japan. K.W. acknowledges partial financial support from MEXT through a Grant-in-Aid
(No. 22104007). Parallelized calculations were performed
on the supercomputers of the Institute for Solid State
Physics, University of Tokyo.

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