Strong-field effects in the photo-emission spectrum of the C$_{60}$ fullerene

1.6 MB·

by  Cong-Zhang Gao

Strong-field effects in the photo-emission spectrum of the C fullerene
60
C.-Z. Gao,1,2 P. M. Dinh,1,2 P. Klüpfel,3 C. Meier,3 P.-G. Reinhard,4 and E. Suraud1,2
1Université de Toulouse; UPS; Laboratoire de Physique Théorique, IRSAMC; F-31062 Toulouse Cedex, France
2CNRS; UMR5152; F-31062 Toulouse Cedex, France
3Université de Toulouse; UPS; Laboratoire Collisions-Agrégats-Réactivité, IRSAMC; F-31062 Toulouse Cedex, France
4Institut für Theoretische Physik, Universität Erlangen, Staudtstraße 7, D-91058 Erlangen, Germany
(Dated: draft, January 15, 2016)
Considering C as a model system for describing field emission from the extremity of a carbon
60
nanotip, we explore electron emission from this fullerene excited by an intense, near-infrared, few-
cycle laser pulse (1013-1014 W/cm2, 912 nm, 8-cycle). To this end, we use time-dependent density
6
functional theory augmented by a self-interaction correction. The ionic background of C is de-
1 60
scribedbyasoftjelliummodel. Particularattentionispaidtothehighenergyelectrons. Comparing
0
thespectraatdifferentemissionangles,wefindthat,asamajorresultofthisstudy,thephotoelec-
2
tronsarestronglypeakedalongthelaserpolarizationaxisformingahighlycollimatedelectronbeam

n intheforwarddirection,especiallyforthehighenergyelectrons. Moreover,thehigh-energyplateau
a cut-off found in the simulations agrees well with estimates from the classical three-step model. We
J alsoinvestigatethebuild-upofthehigh-energypartofaphotoelectronspectrumbyatime-resolved

4 analysis. In particular, the modulation on the plateau can be interpreted as contributions from
1 intracycle and intercycle interferences.

]
s
u I. INTRODUCTION Asastartingpoint,weshallfocusouranalysisontheac-
l tualextremityofthetipwhichpracticallyconsistsinthe
c
cap of a nanotube (the tip is formed by rolling graphene
- Emission from nanometric tips subject to possibly in-
m sheetswithvariousradiiandhelicalstructures). Thecap
tense laser fields is of strong current interest as it con-
of the carbon nanotubes used in the above mentioned
at stitutes a promising source of a well collimated and co- photo-emissionexperimentsaresimilarinsizeandstruc-
. herent beam of ultrafast electrons [1], which can be used
s ture to a C cluster [13]. In this first step we shall thus
c in electron diffraction experiments [2] as well as electron 60
use C as a test model for the exploratory studies pre-
i microscopy[3]. Suchsystemscouldbeespeciallyinterest- 60
s sented here. We consider the response of C subject to
y ing in the recollision regime (in which emitted electrons 60
anintenselaserirradiationandcharacterizethisresponse
h recollide with the tip in the course of the laser pulse)
in terms of photo-electron spectra (PES).
p as this could allow to generate extremely short electron
[ The paper is outlined as follows: Section II briefly in-
pulses [4] and thus pave the way to sub-femtosecond troduces the theoretical framework and computational

1 and sub-nanometric probing of matter [5]. Most studies
details. Section III presents and discusses the results.
v have up to now focused on metallic nanotips. In par-
Section IV completes the paper with a conclusion.
6 ticular, laser-induced photo-emission of electrons in the
0 recollision regime, manifesting itself in the formation of
6
a plateau at high kinetic energies in the photo-electron
3 II. FORMAL FRAMEWORK
spectrum(PES),hasbeenveryrecentlyobservedintung-
0
. stentipsofnanometricsize[6–8]. Inthiscontext,carbon
1 A. Time-dependent local-density approximation
nanotubeshavealsobeenproposedaspromisingemitting
0
devices, since they possess an even smaller tip size than
6
the metal tips used so far [9]. These novel structures We describe electron dynamics by means of time-
1
: have mainly been characterized by field emission exper- dependent density-functional theory at the level of the
v iments [10, 11]. Due to this smaller tip size of carbon time-dependent local-density approximation (TDLDA).
i
X nanotubes, these systems might thus lead to even more The LDA is complemented by an average-density self-
promising sources for sub-femtosecond / sub-nanometric interaction correction (SIC), which has been shown to
r
a probing of dynamic processes in various areas of physics, provide a reliable theoretical framework of electron dy-
chemistry, biochemistry or material science. Only very namics in strong laser fields, in particular when the ex-
recently,laser-assistedphoto-electronstudies,whichhave citation leads to substantial ionization [14, 15]. The de-
been very successful in the case of metal nanotips, have tailed theoretical approach is given elsewhere, and shall
beenextendedtothesenoveltipsbasedoncarbonnanos- only briefly be summarized here. The time-dependent
tructures [12]. single-particle(s.p.) wavefunctionsϕj(r,t)areobtained
bysolvingthetime-dependentKohn-Shamequations[16]
In relation to this question of field emission from car-
(here and in the following we use atomic units) :
bonnanotips,thepresentworkaimsatstudyingtowhich
extent photo-emission by ultrafast laser pulses can reach ∂ 1
theintendedrecollisionregimeincarbonnanostructures. i ϕj(r,t)= 2+υC[ρ(r,t)]
∂t {−2∇
2
+ υ [ρ(r,t)]+υ (r,t) ϕ (r,t),(1a) of C are well reproduced, in particular the ionization
xc ext j 60
}
υext =υjel+υlas, (1b) potential (IP) at EIP = 0.56 Ry, that is identical to the
experimentalvalue[26],aHOMO-LUMOgapof0.14Ry,
where υ is the standard Coulomb potential and υ the whichiswellwithintherangeoftheexperimentalvalues
C xc
exchange-correlation potential from DFT [17] using the (0.12-0.15 Ry [27]), as well as a good description of the
exchange-functionalgivenin[18]. Thispotentialdepends photo-absorption spectrum (for details, see [23]).
on the actual electron density Theuseofthejelliumapproximation,nevertheless,re-
quires some words of caution. The standard procedure
(cid:88)Nel is to use a detailed ionic background coupled to the elec-
ρ(r,t)= ϕ (r,t)2. (2)
| j | trons through pseudo-potentials. There exist numerous
j=1 investigations for C [28, 29] and other carbon nanos-
60
where N is the number of electrons. The external one- tructures such as graphene [30] or nanotubes [31]. An
el
body potential υext is composed from the potential of elaborate description of C60 with an involved orienta-
theionicbackgroundpotential,heremodeledasajellium tion averaging is needed for detailed observables such as
(see section IIB), and the potential of the laser field (see photo-electron spectra (PES) and photo-angular distri-
section IIC). butions (PAD) [32–34]. A key issue in the present inves-
tigations is the ponderomotive motion of the electron in
thelaserfieldassociatedwithhugeexcursionsoftheelec-
B. Ionic background tron[35]. Thisrequiresextremelylargesimulationboxes
which become unaffordable for a grid representation in
The positively charged ionic background is approxi- full3D.Thejelliummodel,togetherwiththelinearlypo-
mated by a jellium model. Specifically for C consid- larized laser field, has cylindrical symmetry. This allows
60
ered here, all carbon ions are arranged into a shell-like ustouseacylindrical(2dimensional)boxwhichrenders
structure [19]. The model for the jellium potential υ the necessary huge boxes feasible. As an additional ben-
jel
for the ionic background reads : efit, we can also compute angular distributions without
the extra expense of orientation averaging [36, 37]. As
υ (r)= (cid:90) d3r(cid:48)ρjel(|r(cid:48)|) +υ (r), (3a) farasthejelliummodelisconcerned, itisapowerfulap-
jel − r r(cid:48) ps | | proximation as it provides an appropriate description of
| − |
ρ (r)=ρ g(r), (3b) many features of the electronic structure and dynamics
jel 0
insolids[38], clusterphysics[39,40], andC [21]. How-
υ (r)=υ g(r), (3c) 60
ps 0 ever, two aspects have been sacrificed. The first one is
1 1
g(r)= (3,d) that the returning electron collides with the jellium well
1+exp[(r R−)/σ] 1+exp[(R+ r)/σ] instead of with a carbon ion. Although the potential of
− −
∆R the ionic background is very steep, we are probably un-
R =R . (3e)
± ± 2 derestimating the actual yield of high-energy electrons.
The second aspect is that we ignore ionic motion. As a
Thejelliumdensityρ ismodeledbyasphereofpositive
jel consequence, we miss effects from phonon coupling [41]
charge with a void at the center [20–22]. The Woods-
aswellasfromelectronicdissipation[42]. However,since
Saxon profile g generates a transition from bulk shell
weareusingfemtosecondlaserpulses, theireffectshould
to the vacuum (inside and outside), providing soft sur-
not drastically affect the main findings presented here.
faces. Furthermore, we employ a pseudo-potential υ
ps
in addition to the potential created by the jellium den-
sity, as proposed in Ref. [23], which is tuned to ensure
C. Laser field
reasonable values of the single-particle energies. The
average radius R of the jellium cage is taken from ex-
The laser field is taken to be linearly polarized along
perimental data as R = 6.7 a [24]. The thickness of
0 the z direction, with a sin2-shaped envelope,
the jellium shell ∆R, the surface softness σ, and the
depth of the potential well υ0 are adjustable parame- Elas(t)=E0sin2(πt/Tpulse)sin(ωlast+φCEP)ez . (4)
ters for which we use here σ = 0.6 a , υ = 1.9 Ry, and
0 0
∆R = 2.57 a . The bulk density ρ is determined such Within the dipole approximation in length gauge, the
0 0
that (cid:82) d3rρjel(r) = Nel = 238. Note that this num- laser-electroninteractionisgivenbyVlas(r,t)=−Elas(t)·
ber of electrons is different from 240 for a real C60, but r. We use a laser frequency ωlas=1.36 eV (=912 nm)
no jellium model is capable to place the electronic shell and a total duration Tpulse = 24 fs. The laser strength
closure at this value so far (unless one uses a deliberate E0 is varied from 0.0113 V/a0 to 0.0453 V/a0, corre-
modification of the occupation numbers [25]). Most jel- sponding to laser intensities Ilas from 1013 W/cm2 to
lium models for C have the shell closure at N = 250 1.6 1014 W/cm2. It is instructive to characterize
60 el ×
[20–22]. The present model with soft surfaces comes to these laser conditions in terms of the Keldysh parame-
(cid:112)
N = 238 which is much closer to reality. By virtue of ter γ = 2E ω /E where E is the ionization po-
el IP las 0 IP
thechoiceofmodelparameters, theelectronicproperties tential [43]. The current combination of ω and E
las 0
3
spans the interval 0.55 γ 2.2, i.e., from multi- which characterizes the electronic response in time. It is
≤ ≤
photon ionization for γ > 1 to tunneling ionization for mostly used to compute photo-absorption spectra using
γ < 1. This transition has been experimentally studied spectral analysis [50]. Here we use it in the time do-
in PES of rare gases [44]. In the above expression, φ main to visualize the electron dynamics of the system.
CEP
is the carrier-envelope phase (CEP). A recent combined In the following, we will consider particularly the dipole
experimental/theoreticalstudyonstrong-fieldionization moment parallel to the laser polarization, that is D .
z
in C [45] reported a remarkable CEP effect. In these Most importantly, we will concentrate our analysis of
60
studies,apulsedurationof4fsandacentralfrequencyof electron emission on the angular-resolved photo-electron
ω = 1.72 eV have been used. However, in the present spectra (ARPES) yield (E ,θ), that is the yield of
las kin
Y
work, much longer pulses (of about 8 optical cycles) are asymptotic kinetic energies E of electrons emitted in
kin
used. We have performed a systematic analysis by vary- directionofangleθ. Toevaluate fromourTDDFTsim-
Y
ing the CEP and found no significant influence on the ulations, we employ the method initiated in [51, 52] and
PES. As a consequence, only results for φ = 0 are extended in [53] to the case of strong fields as used here.
CEP
shown below. In brief, the PES is computed by recording the single-
particle wave functions ψ (t,r ), j = 1,...,N , at se-
j M el
lected sampling points r near the absorbing bound-
M
D. Numerical representation ary. Once the simulation is completed, one computes
the Fourier transform from time to frequency domain
A detailed description of the numerical treatment can ψ(cid:102)j(Ekin,rM) augmented by a phase factor accounting
be found in [46, 47]. Here, we give a brief account for the external field [53]. The PES then reads
and specify the actual numerical parameters used. Wave
functions,densities,andfieldsarerepresentedonacylin- (cid:88)Nel
drical grid in coordinate space. As already mentioned, a (Ekin,θ) ψ(cid:102)j(Ekin,rM)2 . (7)
Y ∝ | |
major issue of the present investigation is the rescatter- j=1
ing of electrons which requires very large computational
The sampling points are chosen to cover a mesh of emis-
boxes for a complete description of the huge electron ex-
sion angles θ. This angle θ at detection point r is
M
cursions. We have thus made systematic investigations
defined with respect to the laser polarization e : for-
z
on the impact of box parameters. The final choice is a ward and backward emissions correspond to θ = 0◦ and
compromisebetweenacceptablenumericalcost,accuracy θ = 180◦ respectively. The energy and angle resolution
and robustness. The chosen dimensions of the numerical for the PES is 0.04 eV and 1◦ respectively.
box are 500 a along the laser polarization direction (z
0
axis) and 250 a in radial direction (r coordinate). The
0
gridspacingistakentobe0.5a0,whichallowsustorep- III. RESULTS AND DISCUSSIONS
resentkineticenergiesup 140eV.Theelectronicground
∼
stateisdeterminedbythedampedgradientmethod[48].
A. Angular dependence of the photoelectron
The Kohn-Sham wave functions are propagated in time
spectra
using the time-splitting technique [49], and absorbing
boundary conditions are used to remove all (emitted)
The upper panel of Fig. 1 shows the full ARPES at
electrons which have reached the boundaries of the box.
intensity I =1.6 1014 W/cm2 complemented by the
They consist of 70 grid points (=35 a0) at each of the angular dilsatsribution×(PAD) on the right. This PAD is
margins.
obtained from integration of the ARPES over the ki-
netic energy interval 50–160 eV, while the lower panel
shows the PES for selected angles, as indicated. The
E. Observables
ARPES clearly indicates that electron emission is more
pronounced for higher energies and is strongly focused
We have studied various observables related to the re-
in forward direction (θ = 0◦) : The ratio of the yield at
sponse of the system, in particular to ionization. The
0◦ and at 90◦ increases from 100 at low energies to
total ionization is calculated as the difference between 105 in the high energy range∼(100–150 eV). Analyzing
the initial number of electrons N and those left in the ∼
el the angular distributions in Fig. 1 (b) yields a focusing
box at a given time t : of 7◦ (full width at half maximum), similar to values
(cid:90) obs∼erved in gold nanotips [54].
N (t)=N d3rρ(r,t). (5)
esc el− In Fig. 1(c), we present the photoelectron spectra for
differentemissionangles,asindicated. AllfourPESstart
This quantity gives an indication on the charge state of
with a nearly exponential decrease and then, for higher
C afterirradiation[15]. Anotherobservableistheelec-
60 energies, develop into a broad plateau which extends up
tronic dipole moment
to a distinct cut-off. We will discuss this structure in
(cid:90) more details in the next section. Note that this typi-
D(t)= d3rρ(r,t)r, (6)
cal pattern is well known and can be qualitatively un-
4
dependence of the photoelectron spectra.
B. Impact of laser intensity
1. PES as a function of laser intensity
To study the influence of the laser intensity on PES
of irradiated C , four intensities have been explored,
60
namely I , 2I , 4.6I , and 8I where I = 2
0 0 0 0 0
1013 W/cm2, corresponding to values of the Keldys×h
parameter γ = 1.5, 1.1, 0.7, and 0.5 respectively. Fig-
ure2collectstheresultsforthePESinforwarddirection
(θ = 0◦). They exhibit similar pattern at all four con-
10 4
−
10−2 10−6
10 8
−
nits) 10−4 10−10 ωlas
u 0 2 4 6 8 10 12
arb. 10−6 Ekin(eV)
(
0) 8I0
= 10 8
θ − 4.6I0
,n
ki
E(Y10−10 2I0
Figure 1. (a): Color map of angle-resolved photo-electron
spectra(ARPES)asafunctionofkineticenergyandemission
10 12
angle, in logarithmic scale for the yield, from C excited by −
60
I
a laser pulse with following parameters : ωlas = 1.36 eV, 0
T = 24 fs, I = 1.6×1014 W/cm2. (b): PAD obtained
pulse las
fromintegrationoftheARPESoverthehigh-energyrangeof 0 25 50 75 100 125 150
[50–160 eV], normalized to 1 at 0◦. The dashed line marks E (eV)
kin
the cone angle at full width at half maximum. (c): PES for
emission angles at θ = 0◦,30◦,60◦,and 90◦ (in logarithmic Figure 2. Samples of photo-electron spectra in forward di-
scale), with an arrow indicating the 10Up cutoff energy. rection in C60 irradiated by laser pulses with ωlas = 1.36 eV
(912 nm), T = 24 fs, and four intensities as indicated
pulse
(I =2×1013 W/cm2). The curves are vertically shifted for
0
a better graphical discrimination. Inset : zoom in the region
derstood by the so called “three-step model’’ [55]: the 0<E <13 eV contributed by direct emissions.
kin
electron is ionized during the laser field through tunnel
ionization, then accelerated in the electric field, driven sidered intensities : on the low-energy side, they show
back to the ion core and gains a large amount of energy a more or less extended plateau (see inset), then turn
through recollision with the latter. Within this simple to a rapid exponential decrease, and finally develop a
picture, the cut-off appears at ∼ 10Up with Up = 4Iωlal2sas broad second plateau which extends to high energies.
being the ponderomotive energy. Here, the angular de- Both plateaus are shifted towards higher kinetic energy
pendence of the cut-off roughly follows a cosθ depen- with increasing laser intensity, whereby the width of the
dence, similar to the angular tendency of cut-off shown plateau increases as well and thus the upper energy cut-
in [56, 57]. A further interesting finding is the os- off of the second plateau increases accordingly.
cillatory structure of the PES within the high energy We first concentrate on the PES at low energies (<
plateau. Similar structures have been observed in other 13 eV) magnified in the inset of Fig. 2. For the two low-
systems [58, 59], where they were interpreted as inter- est intensities (I and 2I ), they show a dense sequence
0 0
ferenceeffectsofdifferentelectrontrajectoriesleadingto ofpeakswhichareseparatedbythephotonenergyω =
las
the same final kinetic energy. In Sec. C, we will ana- 1.36eV. ThisisthetypicalpatternforAbove-Threshold
lyze the time evolution of the spectra, and we will show Ionization(ATI),whichhasbeenwellstudiedforC ex-
60
that this interpretation is consistent with our results. In perimentally[60,61]andtheoretically[32,33,62]. These
the following section, we will first address the intensity equi-spaced patterns are washed out when the laser in-
5
tensity increases to the two highest values (4.6I , 8I ).
0 0
This can have several reasons: first, due to the signifi- 150
cant ionization at these intensities, the cluster progres-
)
sively becomes charged during the laser pulse, leading to V120
e
changes in the electronic structure. This effect has been (
ff
observedinsodiumclusters,seee.g.[51,62],andprelim- to 90
inary investigations have clearly shown similar effects in Cu
thepresentcasetoo. However,additionalblurringdueto u
a 60
other effects, like the space charge, may also contribute, e
t
a
and will be analyzed in future studies. Pl
30
SFA cutoff law
TDDFT estimate
0
2. Analysis of the high energy plateau
1 4 7 10 13 16
Laser intensity (1013 W/cm2)
As mentioned above, the high-energy plateau is gen-
erated by strong-field ionization (SFI) mechanism, and
Figure 3. Cutoff energies of the high-energy plateau in the
thecharacteristiccut-offscanbefoundapproximatelyat
PESasfunctionsofthelaserintensity,fromTDLDAcalcula-
10.007U [56]. Moreover,thederivationofasemiclassical
p tionscomparedwiththeestimate(8)deducedfromthestrong
cut-off law [63], based on the Strong Field Approxima-
field approximation (SFA).
tion (SFA), reveals that the IP also plays a role in the
SFI regime, and can be estimated by [63]:
2 1013 W/cm2 (left column) and another one at higher
Ec(uStFA) =10.007Up+0.538EIP , Up = 4Iωla2s , (8) iunm×tenn)s.itTyhoefcIularsre=nt4d.6enIs0it=yi9s.2pl×ott1e0d13inWa/2cDmd2e(nrsigithytmcoalp-
las
as a function of time (horizontal axis) and space coordi-
Because of the inverse quadratic dependence of U on
p natez(verticalaxis),withpositivej inred(orgray)and
z
the laser frequency ω , the low value of 1.36 eV used
las negative j in green (or light gray). It is also instructive
z
heredeliversalargeponderomotiveenergyandlargecut-
to compare these maps with the time evolution of the
offs in the PES. For instance, at 4.6I , we have U =7.1
0 p dipole moment defined in Eq. (6) which is plotted in the
eV, largely exceeding the photon energy and being even
toppanelofthefigure. Notethatthedipolemomentand
comparable to EIP, and Ec(uStFA) =75 eV in Fig. 2. thecurrentdensityarerelatedbythecontinuityequation
We have extracted the energy cut-off from our calcu- which reads :
lated PES, denoted by E(TDLDA) to compare it to the (cid:90)
simple estimate E(SFA) gciuvten by Eq. (8). More pre- ∂Dz = dz jz . (9)
cut ∂t
cisely, for a given PES, E(TDLDA) is obtained as the in- z
cut ThesignofthetimederivativeofD isthusequaltothe
tersect of two fitting exponential curves at each side of z
one of the dominant part of j .
the plateau, similarly to the procedure applied on ex- z
perimental data [64]. The comparison of E(TDLDA) and One observes successive fringes in jz connected to the
cut change of sign of the derivative of D or, in other words,
Ec(uStFA) is presented in Fig. 3. We first note that both to the oscillations of Dz in time, aszexemplified by ver-
energy cut-offs grow linearly with laser intensity, point-
tical dashes in a case when the sign is negative. For the
ing towards the validity of the simple classical scaling
highest laser intensity (right column of Fig. 4), a sizable
law. However, our numerical simulations show a slightly
backflow of the current density occurs during pulse du-
steeper slope. If one interprets the numerical results ob-
ration, especially between 5 and 15 fs, where the field
tainedbytheTDLDAapproachinthelightofthesimple
amplitude is maximal. For instance, in the time interval
classical scaling law (8), the difference may be explained
indicatedbythetwoverticaldashedlines,themajorityof
byafieldenhancementofabout5to10%. Furtherinves- electronspossessanegativej (theyarethuspulledaway
z
tigations are needed to confirm this interpretation. Note
from the C ), while a non-negligible amount exhibits a
60
howeverthatasimilareffecthasbeenobservedinmetal-
positive j , which means that they are pulled back to-
z
lic nanotips [7, 54, 65].
wards C and that recollision is possible. At the lowest
60
I , this backflow still exists but is much weaker, see in-
las
setinthebottomleftpanelforwhichthecurrentdensity
3. Ponderomotive oscillations scale has been divided by 100 to allow the visualization
of this weak counterflow. The amplitude of this quiver
Tobettervisualizetheponderomotivemotion,weshow motion can be estimated from a purely classical model
in the lower panels of Fig. 4 the current density j along as l = αE /ω2 where α is the field enhancement factor
z q 0
the laser polarization axis z. Two intensities are con- (here about 1.05). This yields l 12 a and 25 a
q 0 0
∼ ∼
sidered, one at the moderate side with I = I = at the low and the high I respectively. These classical
las 0 las
6
)00.12 I = 2.0 1013 W/cm2 I = 9.2 1013 W/cm2
a las las
× ×
f0.06
o
ts 0
i
n
u-0.06
(
Dz-0.12
40 a)0 200 4 2)
( −0
z 20 a
) 20 − 2 1−
a0 12Ti1m4e1(6fs)18 fs
f 4
o −
s 0 0 0
t 1
i
n f
u o
(-20 -2 ts
z i
n
u
(
-40 -4 z
j
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 40
Time (fs) Time (fs)
Figure 4. Electronic dipole moment D as a function of time (top row), and current density j as a function of time and
z z
z coordinate (bottom row, 2D density map) of C along laser polarization axis (z direction), after irradiation by a laser
60
pulse with ω = 1.36 eV, T = 24 fs, and with two different intensities I = 2.0×1013 W/cm2 (left column) and
las pulse las
I =9.2×1013 W/cm2 (right column). Positive j are depicted in red (or gray), and negative j are in green (or light gray).
las z z
For a given color (or type of gray), the brighter the shade, the smaller the absolute value of j . The inset in the left bottom
z
panel magnifies the map between 10 and 20 fs with a scale in current density divided by 100.
l agree well with the amplitudes one can read off from the tunnel ionization probability is maximal.
q
the current density maps. The large ponderomotive os- Since we focus on electron emission in the forward di-
cillationsterminateassoonastheexternalfielddiesout. rection, we have chosen, for the time-resolved analysis,
The further evolution still shows a succession of positive instantsbasedonaclassicalpictureoftheelectronemis-
and negative fringes of j , but electrons in the z >0 and sion. More precisely, the “birth” times of the electrons,
z
thez <0regionsdonotoscillateinphaseanymore. This denotedbyt ,aretakenatthethreelargestmaxima
b1/2/3
is particularly visible for the highest I above 20 fs. oftheelectricforce,thatisatrespectively2.75,3.75,and
las
4.75 t , where t = 3 fs is the single optical cycle. We
oc oc
have indicated these birth times in Fig. 5(a). We then
C. Time-resolved analysis correlate tb1/2/3 with the detection time tf1/2/3. Since
the(fastest)electronsgeneratedatt needtimetoreach
b
the boundary where they are detected, we take into ac-
As mentioned above, in addition to the high-energy
count a time delay ∆t which consists of two terms. The
cut-off, we see modulations of the PES within the broad
first one is the “return” time for the electron to recollide
plateau for the two cases with the higher intensities in
with the target and is about 0.75t . The second one is
Fig. 2. Similar structures have been experimentally ob- oc
the time for electrons emitted from the rescattering site
served in the photoemission spectra of rare gases (i.e.,
to travel to the detection points R = 90 a near the
argon atom [66]) ionized by strong infrared laser pulses. b 0
boundary of the simulation box. This “traveling” time
Thesestructureshavebeeninterpretedasinterferenceef-
is (R R)/√2E for electrons recorded with kinetic
fectsfromseveralelectrontrajectoriesgeneratedeitherin b kin
−
energy E . All in all, we have used :
thesameopticalcycleorinthesubsequentopticalcycle, kin
leading to the same final states [67]. In particular, in t t +∆t, (10a)
f b
this work, two types of trajectories have been identified, ≈ (cid:112)
∆t=0.75t +(R R)/ 2E . (10b)
labeled as “short’’ and “long’’ trajectories, due to their oc b− kin
different excursion times. Within the three-step model, For t , we used E = 70 eV leading to ∆t = 1.04 t ,
b1 kin oc
the electrons are born close to the field maxima, where while for t and t , we took E = 135 eV and then
b2 b3 kin
7
obtained ∆t = 0.96 t . Plugging these numbers in tra at later times (also see the corresponding ionization
oc
Eqs. (10), we got t = 3.79, 4.71, and 5.71 t re- in Fig.5(b)), showing that these contributions are negli-
f1/2/3 oc
spectively, as symbolized in Fig. 5(b). gible. Indeed, at the field maximum indicated by t , on
b1
The bottom panel of Fig. 5 presents three different the rising part of the pulse envelop, the field strength is
PES,notethateachonehasbeenanalyzedfromt=0up not yet sufficient for an efficient tunnel ionization. The
tooneofthe“final” timesintroducedabove. Inaddition, major part of the spectrum, containing the main struc-
tures, is obtained for the spectrum at t , which corre-
f2
Optical cycles sponds to a birth time tb2 at the highest field value of
the pulse. It shows a clear plateau, with distinct mod-
0 2 4 6 8 10
ulations. If one interprets these structures as interfer-
) 12 (a) tb1 tb2 tb3
0 encesfromtwoelectrontrajectories, itwouldcorrespond
a 8
/eV 4 twoelal breellaotwivethpehaospetiocaflδSpe=rioEdktinδ=t [638]fsw. itThhδist=v0a.l4u1efosf,
of 0 oc
δt, about one seventh of t , is in the same order as de-
s oc
t -4
ni duced by an extended schematic model for recent laser-
(u -8 nanotipinteractions,estimatedtoaboutonesixthinthis
F -12 case [65]. Within this interpretation of interference of
(b) different electron trajectories, the findings of the numer-
1
ically calculated spectra at t would correspond to “in-
f2
t tracycle” interferences, i.e., interference that stems from
10 2 f1
esc − electron trajectories originating during the same optical
N
tf2 tf3 period. Whencomparingtothespectrumattf3,onesees
10 4
− that the additional changes with respect to that at t
f2
are minor at the side of the extension of the plateau,
10−6 however, onestrikingdifferencecanbeobserved: theap-
1 (c) pearance of high-frequency modulations in the spectra.
full PES
Thesemodulationscorrespondtopeaksseparatedbythe
t
nits) 10−2 tff12 cthenesterasltfrruecqtuuernescyarωelacsr.eaFtreodmbya itnetmerpfoerraenlcpeosinotf oelfecvtireown,
u t
f3 trajectories that are born at times separated by the op-
b.
r tical period, and can thus be identified as “intercycle”
a 10 4
( − interferences. These are commonly observed for longer
)
0
pulses, where several field peaks have comparable max-
,n
Eki 10−6 ima, as it is in the present case.
( When compared to the full spectra, one sees slow con-
Y
vergence for the low-energy electrons. This can be un-
10 8
− derstood by the fact that these electrons, due to their
0 25 50 75 100 125 150 slower speed, only arrive at delays much larger than ∆t.
E (eV) Thissystematiceffectisclearlyvisibleinalltime-resolved
kin
spectra.
Figure5. Time-resolvedanalysisofphoto-emissionfromC To summarize, we have analyzed the time evolution of
60
irradiated by a laser pulse polarized in the z direction, with the spectra obtained by the fully numerical TDDFT cal-
ω = 1.36 eV, T = 24 fs and I = 1.6×1014 W/cm2. culations by choosing particular detection times to sepa-
las pulse las
(a) Time evolution of the force F acting on electrons, with rateasfullyaspossiblethecontributionsfromthediffer-
3 “birth” times indicated by arrows (see text for details). entfieldmaxima. Whilenotclaimingtohaveunambigu-
(b) Time evolution of the total ionization N , with 3 “fi-
esc ously identified the observed structures, we have shown
nal” times indicated by arrows and defined in Eqs. (10). (c)
that their characteristics, both in their time evolution
Full PES (black) and PES from data accumulated up to the
as well as with respect to their frequency fingerprints,
3 different final times.
are consistent with the picture of interfering intra- and
intercycle trajectories.
thefullPES(blackcurve)obtainedattheendofthelaser
pulse is shown. We can therefore progressively observe
how the full PES builds up in time. It should be noted
thatthisanalysisisstrictlyvalidonlyforthehigh-energy IV. CONCLUSIONS
electrons, since low-energy electrons arrive later, and are
thus not accounted for at the corresponding final times In this paper, we have analyzed the electron dynamics
t . of laser-excited C as a model case for the generation
f1/2/3 60
Asafirstresult, weseethatthespectraatt isabout of high-energy electrons from a carbon tip. We have ex-
f1
two to three orders of magnitude smaller than the spec- ploredtheresponseofthesystemtolaserfieldsofvarious
8
intensities for a frequency in the infrared domain which demonstrates how the high-energy plateau is generated
leads,athighintensities,tosignificantponderomotiveef- successivelyduringthelaserpulse. Inparticular,wehave
fects. Toanalyzesuchacomplexdynamics,wegobeyond shownthattheobservedstructuresandtheirtimeevolu-
single-active-electronapproachesandusetime-dependent tion stemming from different peaks of the field are con-
density-functional theory propagated in real time and sistent with the picture of intra- and intercycle interfer-
computed on a spatial grid. This approach allows, in ences. As far as the angular distribution is concerned,
particular, a proper description of collective electronic one of the most promising results of the presented study
motion as well as a detailed analysis of photo-electron is the strong focusing of the electrons in the forward di-
spectra (PES) and photo-angular distributions (PAD). rection, especially for the high energy electrons provided
One of the main numerical challenges in the present in- bytherecollisionprocess. Thisisanimportantfeaturein
vestigationwasthelargeboxsizerequiredtoaccountfor the context of using carbon nanotubes as future sources
the huge pathway of the ponderomotive motion of the ofcollimatedelectronbeamsfortime-resolveddiffraction
electrons. To make that feasible in a quantum mechani- experiments.
cal framework, we use a spherical jellium approximation
for the ionic background and handle the dynamics on a ACKNOWLEDGMENTS:
cylindrical grid.
As a major result, we have shown that the recollision C.-Z.G. thanks the financial support from China
regime can be reached for strong, but realistic, laser in- Scholarship Council (CSC) (No. [2013]3009). We
tensities. Wefindtheestablishmentofaplateaustretch- thank Institut Universitaire de France, European ITN
ing out to very high kinetic energies, e.g. to about 125 network CORINF and French ANR contract LASCAR
eV for I =1.6 1014 W/cm2, which is interpreted using (ANR-13-BS04-0007) for support during the realization
×
the well-known recollision mechanism and illustrated by of this work. It was also granted access to the HPC
using a map of the time dependence of the current dis- resources of CalMiP (Calcul en Midi-Pyrénées) under
tribution. The cut-off of the plateau is shown to follow the allocation P1238, and of RRZE (Regionales Rechen-
the semiclassical model based on the three-step model, zentrum Erlangen).
with a field which is enhanced by about 10% in our sim-
ulations. A detailed time-resolved analysis of the PES
[1] S. Hilbert, A. Neukirch, C. Uiterwaal, and H. Batelaan, [14] Y. Kawashita, T. Nakatsukasa, and K. Yabana, J. Phys.
J. Phys. B: At., Mol. Opt. Phys. 42, 141001 (2009). Condens. Matter 21, 064222 (2009).
[2] J. C. Williamson, J. Cao, H. Ihee, H. Frey, and A. H. [15] T.Fennel,K.-H.Meiwes-Broer,J.Tiggesbäumker,P.M.
Zewail, Nature 386, 159 (1997). Dinh, P.-G. Reinhard, and E. Suraud, Rev. Mod. Phys.
[3] V. A. Lobastov, R. Srinivasan, and A. H. Zewail, Proc. 82, 1793 (2010).
Natl. Acad. Sci. U. S. A. 102, 7069 (2005). [16] E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997
[4] P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, (1984).
P. Balcou, H. G. Muller, and P. Agostini, Science 292, [17] R. M. Dreizler and E. K. U. Gross, Density Functional
1689 (2001). Theory: AnApproachtotheQuantumMany-BodyProb-
[5] P.AgostiniandL.F.DiMauro,Rep.Prog.Phys.67,813 lem (Springer-Verlag, Berlin, 1990).
(2004). [18] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244
[6] R. Bormann, M. Gulde, A. Weismann, S. V. Yalunin, (1992).
and C. Ropers, Phys. Rev. Lett. 105, 147601 (2010). [19] H.W.Kroto,J.R.Heath,S.C.O’Brien,R.F.Curl,and
[7] M. Schenk, M. Krüger, and P. Hommelhoff, Phys. Rev. R. E. Smalley, Nature 318, 162 (1985).
Lett. 105, 257601 (2010). [20] M.J.PuskaandR.M.Nieminen,Phys.Rev.A47,1181
[8] M. Krüger, M. Schenk, P. Hommelhoff, G. Wachter, (1993).
C. Lemell, and J. Burgdörfer, New J. Phys. 14, 085019 [21] D. Bauer, F. Ceccherini, A. Macchi, and F. Cornolti,
(2012). Phys. Rev. A 64, 063203 (2001).
[9] D. L. Carroll, P. Redlich, P. M. Ajayan, J. C. Charlier, [22] E. Cormier, P.-A. Hervieux, R. Wiehle, B. Witzel, and
X. Blase, A. De Vita, and R. Car, Phys. Rev. Lett. 78, H. Helm, Eur. Phys. J. D 26, 83 (2003).
2811 (1997). [23] P.-G.Reinhard,P.Wopperer,P.M.Dinh,andE.Suraud,
[10] Y.Saito,K.Hamaguchi,K.Hata,K.Uchida,Y.Tasaka, in ICQNM 2013, The Seventh International Conference
F. Ikazaki, M. Yumura, A. Kasuya, and Y. Nishina, Na- on Quantum, Nano and Micro Technologies (2013), pp.
ture 389, 554 (1997). 13–17.
[11] N. Nilius, N. Ernst, and H.-J. Freund, Phys. Rev. Lett. [24] K. Hedberg, L. Hedberg, D. S. Bethune, C. Brown,
84, 3994 (2000). H. Dorn, R. D. Johnson, and M. De Vries, Science 254,
[12] M. Bionta, B. Chalopin, A. Masseboeuf, and B. Chatel, 410 (1991).
Ultramicroscopy (2014), in press. [25] M. E. Madjet, H. S. Chakraborty, J. M. Rost, and S. T.
[13] R. Saito, M. Fujita, G. Dresselhaus, and M. S. Dressel- Manson, J. Phys. B: At., Mol. Opt. Phys. 41, 105101
haus, Phys. Rev. B 46, 1804 (1992). (2008).
9
[26] D.L.Lichtenberger,M.E.Jatcko,K.W.Nebesny,C.D. [47] P.-G. Reinhard and E. Suraud, Introduction to Cluster
Ray, D. R. Huffman, and L. D. Lamb, Mater. Res. Soc. Dynamics (Wiley, New York, 2003).
Symp. Proc. 206, 673 (1990). [48] P.-G. Reinhard and R. Y. Cusson, Nucl. Phys. A 378,
[27] K. Sattler, Handbook of Nanophysics: Clusters and 418 (1982).
Fullerenes,HandbookofNanophysics(CRCPress,2010). [49] M. D. Feit, J. A. Fleck, and A. Steiger, J. Comp. Phys.
[28] S. Korica, A. Reinköster, M. Braune, J. Viefhaus, 47, 412 (1982).
D.Rolles,B.Langer,G.Fronzoni,D.Toffoli,M.Stener, [50] F.Calvayrac,E.Suraud,andP.-G.Reinhard,Ann.Phys.
P. Decleva, et al., Surf. Sci. 604, 1940 (2010). (N.Y.) 255, 125 (1997).
[29] D. Toffoli and P. Decleva, Phys. Rev. A 81, 061201(R) [51] A. Pohl, P.-G. Reinhard, and E. Suraud, Phys. Rev.
(2010). Lett. 84, 5090 (2000), URLhttp://link.aps.org/doi/
[30] M.Araidai,Y.Nakamura,andK.Watanabe,Phys.Rev. 10.1103/PhysRevLett.84.5090.
B 70, 245410 (2004). [52] A. Pohl, P.-G. Reinhard, and E. Suraud, J. Phys. B 34,
[31] J.A.Driscoll,B.Cook,S.Bubin,andK.Varga,J.Appl. 4969 (2001).
Phys. 110, 024304 (2011). [53] P. M. Dinh, P. Romaniello, P.-G. Reinhard, and
[32] P.Wopperer,P.M.Dinh,P.-G.Reinhard,andE.Suraud, E. Suraud, Phys. Rev. A 87, 032514 (2013).
Phys. Rep. 562, 1 (2015). [54] D. J. Park, B. Piglosiewicz, S. Schmidt, H. Kollmann,
[33] C.-Z.Gao,P.Wopperer,P.M.Dinh,P.-G.Reinhard,and M. Mascheck, and C. Lienau, Phys. Rev. Lett. 109,
E. Suraud, J. Phys. B: At., Mol. Opt. Phys. 48, 105102 244803 (2012).
(2015). [55] P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).
[34] P. Wopperer, C. Z. Gao, T. Barillot, C. Cauchy, [56] G. G. Paulus, W. Becker, W. Nicklich, and H. Walther,
A. Marciniak, V. Despré, V. Loriot, G. Celep, C. Bor- J. Phys. B: At., Mol. Opt. Phys. 27, L703 (1994).
das, F. Lépine, et al., Phys. Rev. A 91, 042514 (2015). [57] C. Cornaggia, Phys. Rev. A 78, 041401 (2008).
[35] G. Herink, D. Solli, M. Gulde, and C. Ropers, Nature [58] R. Kopold and W. Becker, J. Phys. B: At., Mol. Opt.
483, 190 (2012). Phys. 32, L419 (1999).
[36] P.Wopperer,B.Faber,P.M.Dinh,P.-G.Reinhard,and [59] M. V. Frolov, N. L. Manakov, and A. F. Starace, Phys.
E. Suraud, Phys. Lett. A 375, 39 (2010). Rev. A 79, 033406 (2009).
[37] P.Wopperer,B.Faber,P.M.Dinh,P.-G.Reinhard,and [60] E. E. B. Campbell, K. Hansen, K. Hoffmann, G. Korn,
E. Suraud, Phys. Rev. A 82, 063416 (2010). M. Tchaplyguine, M. Wittmann, and I. V. Hertel, Phys.
[38] N. W. Ashcroft and N. D. Mermin, Solid State Physics Rev. Lett. 84, 2128 (2000).
(Saunders College, Philadelphia, 1976). [61] M.Kjellberg,O.Johansson,F.Jonsson,A.V.Bulgakov,
[39] W. Ekardt, Phys. Rev. Lett. 52, 1925 (1984). C. Bordas, E. E. B. Campbell, and K. Hansen, Phys.
[40] M. Brack, Rev. Mod. Phys. 65, 677 (1993). Rev. A 81, 023202 (2010).
[41] O. Gunnarsson, H. Handschuh, P. S. Bechthold, [62] A. Pohl, P.-G. Reinhard, and E. Suraud, J. Phys. B 37,
B. Kessler, G. Ganteför, and W. Eberhardt, Phys. Rev. 3301 (2004).
Lett. 74, 1875 (1995). [63] M. Busuladžić, A. Gazibegović-Busuladžić, and
[42] P.-G. Reinhard and E. Suraud, Ann. Phys. (N.Y.) 354, D. Milošević, Laser Phys. 16, 289 (2006).
183 (2015). [64] M.Krüger,M.Schenk,andP.Hommelhoff,Nature475,
[43] L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965). 78 (2011).
[44] E.Mevel,P.Breger,R.Trainham,G.Petite,P.Agostini, [65] M. Krüger, M. Schenk, M. Förster, and P. Hommelhoff,
A.Migus,J.-P.Chambaret,andA.Antonetti,Phys.Rev. J. Phys. B: At., Mol. Opt. Phys. 45, 074006 (2012).
Lett. 70, 406 (1993). [66] G. G. Paulus, F. Grasbon, H. Walther, R. Kopold, and
[45] H. Li, B. Mignolet, G. Wachter, S. Skruszewicz, W. Becker, Phys. Rev. A 64, 021401 (2001).
S. Zherebtsov, F. Süßmann, A. Kessel, S. A. Trushin, [67] M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier,
N. G. Kling, M. Kübel, et al., Phys. Rev. Lett. 114, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
123004 (2015). [68] D. Milošević, G. Paulus, D. Bauer, and W. Becker, J.
[46] F.Calvayrac,P.-G.Reinhard,E.Suraud,andC.A.Ull- Phys. B: At., Mol. Opt. Phys. 39, R203 (2006).
rich, Phys. Rep. 337, 493 (2000).