Ultrafast photoionization and excitation of surface-plasmon-polaritons on diamond surfaces

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by  Tzveta Apostolova

Ultrafast photoionization and excitation of surface-plasmon-polaritons on diamond
surfaces
Tzveta Apostolova1,2 and B. D. Obreshkov1
1 Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria and
2 Institute for Advanced Physical Studies, New Bulgarian University, 1618 Sofia, Bulgaria
A.A. Ionin3, S.I. Kudryashov3,4 S.V. Makarov3,4 N.N. Mel’nik3 A.A. Rudenko3
3Lebedev Physical Institute, 119991 Moscow, Russia and
7 4 ITMO University, 197101 St. Petersburg, Russia
1
0 Ultrafastplasmonicsofnovelmaterialshasemergedasapromisingfieldofnanophotonicsbringing
2 new concepts for advanced optical applications. Ultrafast electronic photoexcitation of a diamond
surfaceandsubsequentsurfaceplasmon-polaritons(SPPs)excitation arestudiedboththeoretically
b
e andexperimentally-forthefirsttime. Afterphotoexcitationontherisingedgeofthepulse,transient
F surface metallization was found to occur for laser intensity near 18 TW/cm2 due to enhancement
of the impact ionization rate; in this regime, the dielectric constant of the photoexcited diamond
2
becomesnegativeinthetrailingedgeofthepulsetherebyincreasingtheefficacywithwhichsurface

roughnessleadstoinhomogeneousenergyabsorptionattheSPPwave-vector. ThesetransientSPP
]
i wavesimprintpermanentfineandcoarsesurfaceripplesorientedperpendicularlytothelaserpolar-
c ization. Thetheoreticalmodelingissupportedbytheexperimentsonthegenerationoflaser-induced
s
periodic surface structure on diamond surface with normally incident 515-nm, 200-fs laser pulses.
-
l Sub-wavelength (Λ≈100 nm) and near wavelength (Λ≈450 nm) surface ripples oriented perpen-
r
t dicularly to the laser polarization emerged within the ablative craters with the increased number
m
of laser shots; the spatial periods of the surface ripples decreased moderately with the increasing
. exposure. Thecomparisonbetweenexperimentaldataandtheoreticalpredictionsdemonstratesthe
t
a roleoftransientchangesofthedielectricpermittivityofdiamondduringtheinitialstageofperiodic
m surface ripple formation upon irradiation with ultrashort laser pulses.
-
d
n I. INTRODUCTION pulses15, or even 50-100 nm on thin diamond films for
o 248-nmfs-laserpulses16 (downto30–40nmondiamond-
c like carbon after irradiation with 266-nm femtosecond
[ Diamond is a material, exhibiting unique mechanical, pulses)17,18 werereported,empiricallyscalingasthenor-
thermalandelectricalproperties,aswellashighelectron
2 malized laser wavelength λ/2n (n is the refractive index
and hole mobility1, promoting its high performance in
v ofdiamond), similarlyto otherdielectrics19,20. However,
microelectronic devices. At the same time, diamond is
0 despite some previous attempts15,21,22, the underlying
5 a basic ingredient in modern nanophotonics2,3. Due to photoexcitation of diamond surface and SPP waves still
6 its high refractive index in UV-VIS range, it is prospec-
remain unexplained.
4 tive material for all-dielectric4 and even hybrid metal-
0 dielectric nano-photonic devices and circuits5–7. More- Generally, spatial LIPSS periods Λ are known to
.
1 over, despite its dielectric character, similarly to silicon depend on the laser wavelength λ and the polariza-
0 it can be promptly turned by intense ultrashort laser tion of the laser electric field e and the number of
7 pulses into short-lived plasmonic state, becoming so- laser pulses23–27,29–31. The surface ripple period can be
1
called”virtualplasmonicmaterial”,supportingphotoex- slightlylessthanλ,succeedingthein-planeweakinterfer-
:
v citation and propagation of surface plasmon-polaritons ence of the incident transverse fs-laser wave and almost
i (SPPs)8–10, for potential applications in ultrafast opti- transverse surface polaritons10. These surface electro-
X
cal switching, spatial phase modulation and saturable magnetic modes, residing along the light cone line on
ar absorption8,11–14. Meanwhile, experimental ultrafast dispersion curves for the metallic or strongly photoex-
SPP photoexcitation on diamond surfaces was not real- cited dielectric surface with its dielectric permittivity
izedyet,eventhoughtheirpotentialimprintinginsurface ε and its intact dielectric with its dielectric permittiv-
m
reliefintheformofpolarization-dependentlaser-induced ity ε are photoexcited by the fs-laser pump pulse via
d
periodicalsurfacestructures(LIPSS,surfaceripples)was its scattering on permanent or laser-induced (e.g., phase
numerously evidenced15–17. Such experimental studies transition from diamond to glassy or diamond-like car-
weredevotedtothedesignandfabricationofbio-sensors, bon phase) cumulative surface relief roughness27,29–31,
employing the biocompatibility of the material, by ab- or prompt laser-induced ”optical roughness”32, if the
lative surface nanostructuring of its surface with high- condition e[ε ] e[ε ] is fulfilled26,33. Meanwhile,
m d
ℜ ≪ ℜ
intensity femtosecond (fs) laser pulses, assuring precise in the corresponding spectrally-narrow surface plasmon
deliveryofenergy,whileprecludingcollateralthermalef- resonance, occurring for the photoexcited surface at
fects. In the case of diamond, ultimate LIPSS periods of e[ε ] = e[ε ], the short-wavelength, longitudinal
m d
ℜ −ℜ
100–125 nm on diamond-like carbon for 800-nm fs-laser surface plasmons can similarly interfere with the inci-
2
dentwaveoramongthemselves(forcounter-propagating τ 1 fs, despite the neglect of collisional effects in the
e
∼
quasi-monochromatic surface plasmons), inducing sur- TDDFT simulation. The real part of dielectric function
face ripples with periods much lower than λ ( λ/2, λ/6, was well fitted by a Drude free-carrier response showing
..)26,34–36. Importantly, in the former case, the surface that e[ε ] is sensitive to the total number density of
m
ℜ
polariton-mediated, near-wavelength ripples are always excited electrons and not to the detailed distribution of
oriented perpendicularly to e (their wavevector κ e), electron-hole pairs, while sensitivity to the nonequilib-
||
while the fine nanoripples can be oriented in both ways, rium distribution of the phototexcited carriersmanifests
depending which – red or blue – shoulder of the surface in the imaginary part of the dielectric constant. Subse-
plasmon resonance is involved37. Laser exposure (the quently, TDDFT was applied to study ablation of silica
number of incident pulses per spot, N) is known to in- subjected to ultrashort laser pulses54. The comparison
fluence LIPSS (both ripples and nanoripples36) to much betweentheestimatedsurfaceablationthresholdandthe
less extent, inducing about 30% reduction in their pe- experimental data suggests a non-thermal mechanism in
riods versus exposures, increasing to N 102-10329–31. the laserablationofsilica by fs-laserpulses, furthermore
Other effects – angle of incidence/laser∼polarization38, theoretical ablative crater depths agree with the mea-
intactdielectric39–41 indicate someemergingpossibilities sured ones. The drawback of this approach is its limi-
inreductionofLIPSSperiods, butshouldbe exploredin tation to very short laser-matter interaction timescales
details yet. Meanwhile, nanoscale hydrodynamics insta- (less than 10 fs).
bilitiesoflaser-inducedsurfacemeltwerealsoconsidered
andexploredasanalternativetothediverseelectromag-
netic approaches28,42,43.
Since the prompt dielectric permittivity of the pho- Inthepresentpaper,wepresenttheoreticalandexper-
toexcited surface appears to be crucial for excita- imental results for the laser ablation and LIPSS forma-
tion either near-wavelength surface polaritons, or sub- tion on diamond surfaces subjected to normally incident
wavelength surface plasmons, prompt photoexcitation 515-nm, 200-fs laser pulses. Our theoretical modeling of
(photoionization) of diamond, directly affecting its di- LIPSSformationondiamondsurfacesisbasedonnumer-
electric permittivity, should be explored in details. icalsolutionofthe time-dependentSchr¨odingerequation
Therearenumeroussemi-empiricalapproachestoexplain (TDSE) in bulk diamond subjected to a single intense
LIPSS formation e.g.24,44–47, corroborating the experi- laser pulse. The theory describes the electron dynam-
mental evidence, but no genuine microscopic approach ics quantum mechanically in the single-active-electron
is invoked so far. The basic physical processes involve approximation. Collisional de-excitation of the pho-
excitation of electron-hole pairs, often parameterized by toexcited carriers and subsequent impact ionization are
Keldysh approximate formulas. Photoionization may treated within rate equation approach and an optical
produce highly energetic electrons that collisionally ion- breakdownthresholdis derived. Due to the contribution
ize the valence band and produce more electrons in the of the impact ionization the real part of the bulk dielec-
conduction band. The multiplication of carriers may tric constant of the irradiated diamond becomes nega-
causeopticalbreakdownofbulkdiamond. Thecollective tive in the trailing edge of the pulse resulting in plasma
response of charge carriers screens out the laser electric that is opaque to the incident radiation. The inhomo-
field inside the bulk when the number density is suffi- geneous energy deposition in the surface was modeled
ciently large. At some instant of time the bulk dielec- with the Sipe-Drude efficacy factor theory19,47 in terms
tric function may become negative at the laser wave- oftime-dependentdielectricfunctionoffreecarriers. The
length, allowing excitation of SPP at the rough surface applicability of this efficacy factor theory for LIPSS for-
and LIPSS formation via the optical interference mech- mation in laser-irradiated dielectrics was confirmed by
anism. The dielectric properties of the laser-irradiated numerical solutions of the Maxwell’s equations at sta-
material in most cases are parameterized with Drude tistically rough surfaces55. The paper is organized as
model19,29,35,48–50, which combines the ground state re- follows. In Sec. II we present the theoretical approach
sponse with the laser-inducedfree-carrierresponse. This to describe LIPSS formation on diamond surfaces. Sec.
model usually requires three free parameters – the num- III presents results for the ablative craters that were ex-
berdensityofelectron-holepairs,thefree-carriereffective perimentally produced on the surface of monocrystalline
mass and the Drude damping time, which are adjusted diamond by multiple femtosecond laser pulses and the
to fit experimental data. Ref.51,52 proposed more elabo- subsequent emergence of fine and coarse surface ripples
rate model for the optical dielectric function, which im- with the increasing number of laser shots. The thresh-
plements state- and band-filling effects, renormalization oldsforsurfaceablationandnano-structuringofdiamond
of the band structure and free-carrier response. The di- andtheirdependenceonthesuperimposedpulsenumber
electric function oflaser-excitedsiliconwas studied from are obtained. The experimental data for the observed
first principles using the time-dependent density func- surface ripple periods is consistently interpreted within
tional theory (TDDFT)53. A distinguishing feature in the Sipe theory based on free-carrier Drude response of
the linear response of the photoexcited silicon is a plas- the laser-excited diamond. Sec. IV contains our main
mon peak with large Drude damping time as short as conclusions.
3
II. THEORETICAL APPROACH

15
A. Inhomogeneous energy deposition 10
V]
e 5
In order to model theoretically LIPSS formation in gy [ 0
femtosecond-laser-exciteddiamond, we apply the ab ini- er -5
n
tio theory developed by Sipe24. In this picture, the E -10
laser beam is incident on a rough surface, the (perma- -15
nent or laser-induced) roughness is assumed to be con- -20
fined within a surface region (selvedge) of thickness l 0.0 0.2 0.4 0.6 0.8 1.0
momentum [2 /a0]
muchsmallerthanthelaserwavelengthλ. Theoptically-
induced polarization in the selvedge generates surface- FIG. 1. Band structure of bulk diamond along the ∆-line.
scattered waves that interfere with the refracted laser The momentum is measured in units of 2π/a0, where a0 =
3.57˚A is thebulk lattice constant.
beam leading to inhomogeneous energy deposition into
the surface. The inhomogeneous energy absorption can
be described by the function
cle approximation based on the 3D TDSE. In a long-
A(κ) b(κ)η(κ;κ ), (1) wavelength approximation the light pulse is represented
i
∼| |
by a spatially uniform time-dependent electric field and
whereκ isthe componentofthelaserpropagationwave
i velocitygaugeisusedthroughoutthecalculations56. The
vectorparalleltothesurface,b(κ)isameasureofsurface
static bulk band structure is represented by the lowest 4
roughnessat wave-vectorκ=(κ ,κ ) and η(κ;κ ) is an
x y i valencebandsand16unoccupiedconductionbands. The
efficacy factor describing the contribution to the energy
BrillouinzonewassampledbyaMonteCarlomethodus-
absorption at the LIPSS wave vector κ. The prediction ing2000randomlygeneratedk-points. Thetimestepfor
ofEq.1isvalidiftheselvedgethicknessissmallcompared
integrationof the equations of motionwas δt=0.03a.u.
totheLIPSSperiod,i.e. κl 1shouldbesatisfied. The
≪ The static band structure along the ∆-line is shown
efficacyfactoressentiallyincorporatesthemodificationof
in Fig.1. Carrier excitation occurs through the direct
thesurfacemorphologyandthevariationofthedielectric
gap at the Γ point, however excitation into higher lying
constant ε of the photoexcited diamond. For normally
conduction bands is also a relevant process for the con-
incident s-polarized laser pulse with κ =0, the efficacy
i sidered laser intensity range I [1,50] TW/cm2. During
factor(asafunctionofthe normalizedwave-numberκ= ∈
the irradiation of the diamond surface with pulsed 200
λ/Λ) can be written as η(κ)=4π e[ν(κ)] with
|ℜ | fs-laser,the total number of electrons generatedinto the
κ 2 κ 2 conduction band is given by a Brillouin zone integral
ν(κ)= h (κ) y +h (κ) x γ t 2, (2)
ss κκ t s
(cid:20) (cid:16) κ (cid:17) (cid:16) κ (cid:17) (cid:21) | | d3k
where the response functions hss and hκκ ρe(t)= Z 4π3fnk(t), (4)
κκ κ κ ǫXn>0 BZ
m v m
h (κ)=2i , h (κ)=2 , (3)
ss κκ
κ +κ εκ +κ
v m v m where fnk is the occupation number of the n-th conduc-
are given in terms of the transient bulk dielectric func- tionbandandkisthecrystalmomentum. Theelectronic
tion ε(ω;t) (cf. Sec. Optical properties), κ = √κ2 1 excitation energy per unit cell is given by
v
−
and κ = √κ2 ε, the Fresnel transmission coef-
m
ficient t = 2/(1−+ √ε 1) in the absence of the d3k
selvedge,sthe effective tran−sverse susceptibility function Eex(t)= Z 4π3 hψnk(t)|i∂t|ψnk(t)i−E0, (5)
γ = (ε 1)/4π ε (1 f)(ε 1)[h(s) Rh (s)] , the ǫXn<0 BZ
t I
− { − − − − }
surface roughness characterized by shape s and filling f
where ψnk(t) are the time-evolved Bloch orbitals of va-
factors, R = (ε 1)/(ε + 1) and the shape functions
lence electrons and E is the ground-state energy. The
− 0
h(s) = √s2+1 s,hI(s) = (√s2+4+s)/2 √s2+1. timeevolutionofthefree-electrondensityisshowninFig.
− −
When e[ε]<0,theresponsefunctionhss exhibitssmall 2a, for linearly polarized electric field along the (1,1,1)
ℜ
kinks near the light line κ 1, in contrast hκκ exhibits direction with intensity 30 TW/cm2. Carrier generation
≈
sharp resonance structure due to the excitation of sur- occursefficientlypriortothepeakofthepulse. Transient
face plasmons and diverges at the (complex) SPP wave- charge density oscillations following the laser period are
number κSPP = ε/(1+ε). duetoquivermotionoffreeelectronsintheelectricfield.
p Anelectron-holeplasma(EHP)withnumber density ex-
ceeding 1021cm−3 is established shortly after the peak
B. Photoexcitation intensity. Thecycle-averagedphotoelectronyield,shown
in Fig. 2b, is a slowly varying function of time. Car-
Photoexcitation and the dielectric response of laser- rier generation on the rising edge of the pulse competes
irradiated diamond are treated in independent parti- with recombination on the trailing edge of the pulse to
4
(a) (b) (с)
-3log10[cm]r 2212 -3log10[cm]r 2212 -3log10[cm]r 122901 log (arb. units)----5432 (a) log (arb. units)----5432 (b) log (arb.units)---432 (c)
20200 300 400t i m 5e00 [ f s 6]00 700 800 20200 400 t i m e [6f0s0 ] 800 18200 400 t i m e 6[0f0s ] 800 -60 Kinetic5 energy [e1V0] -60 K5inetic10 ener1g5y [e2V0] 25 -50 K5inetic10 Ene1r5gy [e2V0] 25
FIG. 2. Time evolution of the free-electron density in di- FIG. 4. Density of conduction states after the irradiation of
amond irradiated by 200fs laser pulse with intensity 30 bulkdiamondwith200fs laserpulse. Theenergyismeasured
TW/cm2, linearly polarized along the (1,1,1) direction. The relativetotheconductionbandminimum. Thelaserintensity
red curve shows the cycle-averaged electron density and the is I =10, 20 and 30 TW/cm2in Fig. (a-c), respectively. The
blue curve is the electron density. (b) The cycle-averaged verticaldashedlineinFig.aindicatesthethresholdforimpact
carrier densities for intensity I=30, 40 and 50 TW/cm2 are ionization
shown by the dashed-dotted, dashed and solid lines, respec-
tively. The position of the pulse peak is indicated by the
vertical dashed line and Fig. (c) presents the number den- ofthecycle-averagedenergygainfollowscloselytheenve-
sity of non-adiabatically excited carriers for intensity I=30, lope of the laser pulse during the first half of the driving
40 and 50 TW/cm2 pulse and reaches 1.5 eV/atom at the peak of the pulse
that is small as compared to the cohesive energy of dia-
2.5 4 mond 7.37 eV/atom. After the pulse peak, electron-hole
(a) (b)
m]2.0 pairsrecombinebytransferringpartoftheirenergyback
o 3
eV/at1.5 2 troevtehresirbaledisaitnicoentfiheeldt.imEenedreglyayexicnharensgteorisatnioontcoofmepqlueitleibly-
y [1.0
g rium gives rise to a net energy gainof 0.5 eV per carbon
Ener0 .5 1 atom after the end of the pulse. The deposited energy
0.0 0 increases steadily with the increase of the intensity, i.e.
200 300 400 500 600 700 800 200 300 400 500 600 700 800
time [fs] time [fs] for I=50 TW/cm2, it reaches 2 eV/atom. Since this ex-
citation energy is still lower than the diamond cohesive
FIG. 3. (a) Instantaneous excitation energy of electrons
energy, Fig. 3(b) shows that ablation threshold is not
interacting with 200fs laser pulse with intensity 30 TW/cm2 reached up to I=50 TW/cm2.
(linearlypolarizedalongthe(1,1,1)direction),thegreencurve
showsthecycle-averagedelectronicexcitationenergy. (b)The
cycle-averaged excitation energy for laser intensity I=30, 40
and 50 TW/cm2 is shown by the dashed-dotted, dashed and C. Impact ionization and optical breakdown
threshold
solid lines, respectively. The position of the pulse peak is
indicated bythe vertical dashed line.
Forthe 200fspulsedurationandintensities lowerthan
50 TW/cm2 the electron density produced by photoion-
determine the final photoionization yield. Recombina- ization is below the critical one. That suggests that im-
tionofcarriersbecomes unlikely withthe increasedlaser pactionizationistherelevantprocessthatdeterminesthe
intensity, cf. also Fig. 2b. The cycle averaged electron optical breakdown threshold. In Fig.4 (a-c) we plot the
yieldincludescontributionsduetocreationofrealaswell densityofconductionstatesaftertheendofthepulse. It
as virtual electron-hole pairs. Since adiabatic evolution isseenthatthelaserhascreatedelectron-holepairswith
does not produce any real excitation of the crystal, the well-defined energies. This non-thermal distribution re-
carrierdensityshouldbecalculatedwithrespecttoadia- laxes towards the equilibrium Fermi-Dirac distribution
batically evolvedground state orbitals that are obtained on a time scale ranging over few tens of a femtoseconds
from the static Bloch orbitals with shifted crystal mo- to a picosecond57,58 without changing the electron num-
mentumk(t)=k+A(t),i.e. ρ(t)=ρe(t) ρad(t),where ber density. Photoelectrons are excited into the lowest
−
the adiabatic density is conduction band across the direct gap (with energies 2
d3k eV above the conduction band minimum) and substan-
ρad(t)= Z 4π3|hunk|unk(t)i|2, (6) tial fraction of carriers occupy higher lying conduction
ǫXn<0 BZ bands with energy abovethreshold for impact ionization
(specifiedbytheindirectgap5.4eV).Thesehighlyener-
and {|unki} are the lattice-periodic Bloch states. The geticelectronsmaycollisionallyde-excitetolowerenergy
number density of photoexcited carriers is shown in Fig.
statesandtheirexcessenergyisspenttopromotevalence
2c. It canbe seenthat discarding contributions of virtu-
electrons into the conduction band.
ally excited electron-hole pairs leads to reduction in the
We further assumethat the time evolutionofthe elec-
number density by an order magnitude near the peak of
tron density is governedby a rate equation59,60
the pulse.
The electronic excitation energy is shown in Fig. 3(a) dρ
for laser intensity 30 TW/cm2. The temporal variation =G(t) R(t)+wimp(I)ρ (7)
dt −
5
1]1.2 2 22
-s (a) (b)
19-3n/dt [10cmf00..06 -1 logw [ps]10imp01 -3og [cm] 2201
d l 19
-0.6 -1
200 300 400 500 600 700 800 10 12 14 162 18 18
time[fs] Intensity [TW/cm] 200 300 400 500 600 700 800
time[fs]
FIG. 5. Fig. (a) Time-dependentrates includingcarrier gen-
eration(positivepart)andlaser-inducedrecombination(neg- FIG. 6. Conduction electron density due to photoionization
ative part). The laser intensity is 10 TW/cm2(dotted), 20 only (dashed line) and including the impact ionization (solid
TW/cm2(dashed)and30TW/cm2(solidline). Fig.(b)shows line). The laser intensity is 18TW/cm2. The vertical dotted
theintensity-dependentimpact ionization rate. line indicates the position of thepulse peak.
D. Optical properties
includingcarriergenerationG(t)andrecombinationR(t)
rates supplemented by an intensity-dependent impact
ionization rate w obtained as a weighted-average of Sincetheabsorptionofthefemtosecondlaserpulsesin
imp
the field-free ionization rate diamondresultsinthegenerationofnearlyfreeelectrons
in the conduction band on timescales smaller than the
∞dǫρ(ǫ;I)P (ǫ) electron-phononrelaxationtime62,wedescribethelinear
wimp(I)= Rǫi ∞dǫρ(ǫ;iIm)p (8) response of the photoexcited diamond by a free-carrier
0 Druderesponse19 usingoftime-dependentplasmon-pole-
R
approximation for the density-density correlation func-
here ρ(ǫ;I) is the density of conduction states after the
tion of the Coulombically interacting electron gas63
end of the pulse (cf. Fig.4a-c),
t
ρ(ǫ;I)= Z d4π3k3fnk(I)δ(ǫ−ǫnk), (9) S(t,t′)=−θ(t−t′)ωp3/2(t′)ωp−1/2(t)sinZt′ dτωp(τ)(,10)
ǫXn>0 BZ
where θ(t) is the Heaviside step function, ω (t) =
p
1/2
iPoinmizpa(ǫt)io=nrPat0e(ǫfo−rdǫii)a4m.5onisdt,hǫeiesntehregtyh-dreespheonlddefnotriimmppaacctt (cid:16)ερ0(mt)e(cid:17) is the bulk plasma frequency, ε0 and me are
i the vacuum permittivity constant and the free-electron
ionization (5.42 eV) and P =3.8 1010s−1eV−4.561.
0 × mass,respectively. Inlongwavelengthapproximationthe
In contrast to the standard perturbative result based
spatial dispersion of the bulk plasmon is neglected. The
on Keldysh theory valid for monochromatic laser radi-
Fourier transformation of the correlation function is the
ation the calculated carrier generation and recombina-
transientfrequency dependent inversedielectric function
tion rates shown in Fig.5a do not follow the temporal
of the free-electron plasma
profile of the laser pulse. This result suggests that the
pulse shape and pulse duration are relevant control pa- t
rametersfornon-adiabaticelectrondynamicsinthelaser ε−1(ω;t)=1+ dt′ei(ω+iδ)(t−t′)S(t,t′), (11)
irradiated diamond. The key features are generation of Z
−∞
denseplasma50fspriortothepulsepeakandsubsequent
laser-induced recombination of electron-hole pairs in the where δ = 1/τ is a free-carrier polarization dephasing
e
trailing edge of the pulse. rate, which we shall treat as a free parameter. If the
Fig.5b shows the impact ionization rate that depends time delay in the build up of screening in the optically
inhighlynon-linearwayonthelaserintensity. Thisnon- excited plasma can be neglected, the classical Drude di-
linear and non-monotonic intensity-dependence reflects electricfunction isrecoveredε−1(ω;t)=ω2/(ω2 ω2(t))
− p
thepopulationofhigher-lyingconductionbands(cf. also withparametrictimedependenceofthebulk plasmafre-
Fig.4). It is seen that the impact ionization rate reaches quency.
few tens of inverse picosecond for I > 15 TW/cm2. In InFig.7a-c,weplottherealandimaginarypartsofthe
Fig.6 we plot the EHP density with and without the im- dielectric function for laser intensity 18 TW/cm2. The
pactionizationterm. Thiscomparisondemonstratesthat screeningchargedensityaccumulatesduringthefirsthalf
photoionization produces the seed electrons needed for of the pulse (t < 0). Over that time interval the laser
the impact ionizationon the rising edge of the pulse and frequency is above the plasma frequency and the dia-
thentheconductionelectrondensitygrowsexponentially mond surface remains transparent to the incident radi-
afterthepulsepeakresultingindenseplasma(withden- ation. The frequency dependent dielectric function dis-
sity 1022 cm−3) 50fs after the peak of the pulse. plays oscillations in the spectral range below the laser
6

6 15
(a) 5 (b) 10 (c)
3 5
0 0
0 -5
-3 -5 -10
L -15
-6 -10 -20
-25
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 5 6
[eV] [eV] [eV]
FIG. 7. Frequency dependence of the real (solid line) and
imaginarypart(dashedline)oftheinversedielectricfunction
of photoexcited carriers subjected to 200 fs laser pulse with
intensity18TW/cm2. Thetimeintervalismeasured relative
to the peak of the pulse (t = 0). In Fig. (b) t = 50 fs, and
in Fig. (c) t=250 fs. The photon energy is indicated by the
vertical dotted line.
frequency due to the time lag in the build up of screen-
ing. Becauseoftheimpactionization,thelaserfrequency
falls off below ωp after the pulse peak (t = 25 fs) when FIG. 8. SEM images of ablative craters on the diamond sur-
the plasma is reflective for the incident radiation and an faceforN=1(a),2(b),30(c),100(d),300(e)and1000(f)
optical breakdown threshold is reached. In this regime, pulses. Thescalebarsaresomewhat differentoneach image.
thedielectricfunctionessentiallyexhibitstheDrudeform
with time-dependent bulk plasma frequency ω (t). For
p
the transiently increasing carrier density, e[ε] passes threshold I (4) 14.4 TW/cm2 (Fig.8), but for longer
abl
ℜ ≈
the narrow surface plasmon resonance at e[ε] = 1, exposuresN 1 the thresholdintensity decreasesdown
with κ 1 and Λ/λ 1, and becomℜes large a−nd to I (1000)≫ 2.1 TW/cm2, following the well-known
SPP abl
negative in≫the trailing ed≪ge of the pulse (t > 100 fs) accumulation≈relationship I(N) = I(4)N−α, where α =
with e[ε] > 0, with corresponding κ 1. During 0.16 0.03(Fig.9). Thespallativeoriginoftheexternal
SPP
ℑ ≥ − ±
thisplasmonically-activephaseofthelaser-irradiateddi- crater is clearly seen as the sharp crater edge in Fig.8f,
amondthe SPP-laserinterferencemechanismofinhomo- however, at higher exposures another ablation mecha-
geneous energy deposition is effective and leaves perma- nism – apparently,phase explosion– comes into play for
nent imprints on the surface morphology after the con- I I (N),formingthedeepcentraldipsanddestroy-
0 abl
≫
clusion of the pulse. ing the intermediate LIPSS (Figs.8d-f).
The observed cumulative decrease of the surface abla-
tion threshold can be related, e.g., to the increasing col-
III. COMPARISON OF THEORY AND oration shown by SEM as darker ablated spots in Fig.8,
EXPERIMENT aswellastostress,structuraldamageandablativemod-
ification of the crater surface (Fig.8a-c). In particular,
A. SPP-mediated surface ripples in diamond: micro-Raman characterization of the craters, exhibiting
generation and characterization only slightly displaced D-band with low-intensity back-
ground (Fig.10), is in agreement with some previous fs-
SPP-mediated surface ripples were produced on laser nanostructuring studies on diamond surfaces16,23,
a 0.5-mm thick plate of monocrystalline A-type showing rather clean nanostructured surfaces. The
diamond nanostructured with the help of laser low-intensity ultrabroad (1100-11400 cm−1) difference
nano/microfabrication workstation64. The sample was spectral band is known to yield from luminescence of
arranged on a three-dimensional motorized translation nanoscale clusters65,rather than from the pump radia-
micro-stage under PC control and moved from spot to tion, since both these spectra exhibit similar D-band in-
spot to make possible ablation of its fresh spots at vari- tensities and the pump radiation was cut in the exper-
able number ofpulses N. Single- and multi-shotablation iments in the same way. Moreover, the displaced (∆
of the sample was produced by 515-nm, 220-fs TEM - 0.1 cm−1) D-band shown by the corresponding bipola≈r
00
modelaserpulsesweakly(NA 0.1)focusedintoafocal band in the difference Raman spectrum (Fig.10), indi-
≈
spot with a 1/e-radius about 5.5 µm at the energy E = cates the internal residual stresses 0.3 kbar, according
∼
3.4 µJ (the peak intensity I 10 TW/cm2). The re- totheknowncalibrationcoefficientforthisband 0.336
sulting single- and multi-shot0 c≈raters were characterized cm−1 / kbar66. ≈
by means of a scanning electronmicroscope (JSM JEOL Fine and coarse surface ripples appear within the ab-
7001F)) and a Raman microscope U-1000 (Jobin Yvon) lative craters, starting from N > 3, inside the sur-
at the 488-nm pump laser wavelength. face regions limited by I (N) I I (N) and
FR CR
≤ ≤
Surface ablation of the crystalline diamond occur for I (N) I I (Fig.11), respectively. These thresh-
CR 0
≤ ≤
fs-laser intensities, exceeding the single-shot ablation olds exhibit two different trends with the increasing ex-
7

)
2
m
c
W/ 10 -0.02–0.01
T
( CR
d
ol -0.12–0.04
h
s
e FR
r
h
t
n -0.16–0.03 ABL
o
ti
a
c
fi
di
o
m 1
1 10 100 1000
exposure N (pulses)
FIG. 11. SEM images of ablation crater edge (ABL), fine
FIG.9. N-dependentvariationofablation (ABL)andnanos-
(FR) and coarse (CR) rippled regions within the craters on
tructuring (coarse and fine ripples, CR and FR) thresholds
thediamond surface for N = 10 (a), 30 (b), 100 (c), and 300
with thecorresponding linear fitting lines and slopes.
(d)pulses. Thescalebarsaredifferentoneachimageandthe
bi-lateral arrow in a) shows the laser polarization.

) Fig. 11 shows that considerable CR erosion is present
s
nit4000 for N = 30 and 100.
u Mostimportantly,thesmalldifferencebetweentheCR
rb. periods ( 0.45 µm, wavenumber 2.2µm−1) and the
a
y (3000 laserwave≤lengthλ(0.515µm, wave≥number 1.9µm−1)
sit points out that long-wavelength micron-sca≈le ( 3µm)
n ∼
e2000 perturbations of surface relief (permanent or cumula-
t
n tive ones – e.g., the spallative crater edge for N 1) or
n i optical characteristics (prompt or cumulative one≥s)32,33
a
m1000 are responsible for excitation of the underlying near-
a
R wavelength plasmon-polaritons. The corresponding FR
andCRperiods decreaseversusN – from0.13 0.03till
0
±
1200 1400 1600 1800 0.09 0.03µm and from 0.45 0.04 till 0.38 0.04µm
-1 ± ± ±
wavenumber (cm ) (Fig.12), respectively, in agreement with cumulative
trends known for FR and CR29–32
FIG. 10. Raman spectra of theD-bandfor thereference dia-
mond spot (bottom black curve) and the 10-shot crater (top
purplecurve). B. Interpretation of LIPSS as imprints of transient
SPP modes
posure N – monotonous decrease for IFR(N) scaling as To make possible identification and interpretation of
β = 0.12 0.04forintensitiesfrom8.8 0.5to3.14 0.5 experimentally obtained SPP modes we plot the efficacy
TW/−cm2 (±Fig.9) and almost no varia±tion for ICR±(N) factorasafunctionofthewavevectorκinanarrowlaser
(scaling as γ = 0.02 0.01) for intensities in the range intensityrangeabovetheopticalbreakdownthresholdin
from14.8 0.5t−o14.1±0.5TW/cm2 (Fig.9). Theminor Fig.13 a-b. The transient bulk dielectric function was
± ±
variationofICR(N)potentiallyindicatesthattheCRare evaluated at the laser wavelength, i.e. ε(t) = ε(ωL;t).
formedduetoscatteringmechanism,i.e. independenton The surface roughness was modeled as a collection of
the surface absorption, while surface absorption is more spherically-shapedislandscorrespondingtostandardval-
crucial for the formation of fine ripples. ues s = 0.4 and f = 0.1 for the shape and filling fac-
Moreover, in comparison to fine ripples with thresh- tors respectively. For normally incident light pulse, the
old I (N) I (N), coarse ripples, having consider- numerical results are weakly dependent on the specific
FR abl
≥
ably higher threshold I (N) I (N), disappear in parameters describing surface morphology and therefore
CR abl
≫
the central crater part for N > 100 because of the pro- the transient dielectric constant is the most significant
nounced ablation in this region (cf. Fig.8 and Fig.11). in determining the efficacy factor. Here we demonstrate
8
✂
✄
✝✞
☎
✆
) ✝ ✂
m 0.4 CR (cid:0) ✞
d ( ✁✝ ✞
o ✁✆
ri ✁☎
e ✁✂
p ✁✄
e ✁✂
pl 0.2 ✁✂ ✁✄ ✁☎ ✁✆ ✁✝ ✞ ✝ ✆ ☎ ✄ ✂ ✁✂ ✁✄ ✁☎ ✁✆ ✁✝ ✞ ✝ ✆ ☎ ✄ ✂
p ✟ ✟
ri
FR
FIG. 13. 2D intensity map of the logarithm of the transient
efficacyfactoroflaser-irradiateddiamond,asafunctionofthe
0.0
1 10 100 1000 normalized(tothelaserwavelength)LIPSSwavevectorcom-
ponents(κx,κy),forInstantaneousbulkplasmafrequency(a)
exposure N (pulses)
ωp(t)=1.5ωL and(b)ωp(t)=1.43ωL. Thelaserbeamislin-
early polarized along the y-axis and is normally incident to
FIG. 12. N-dependentvariation of CR and FR periods. thesurface.
that the main features in the inhomogeneous energy de-
Thesurfaceplasmonpeakintheefficacyfactorisalsoaf-
position in the surface as represented by the position of
fected by the relaxation time parameter τ as shown in
thepeaksofthetransientefficacyfactorareincorrespon- e
Fig. 14b. If τ is decreased to 10 fs, the surface plas-
dence with the experimentally observed LIPSS periods. e
moncuspturnsintoadip,whichhinderstheefficienten-
In a very narrow laser intensity range, when the laser
ergyabsorptionatthe surfaceplasmonwavevector. This
frequency nearly matches the surface plasma frequency
dependence suggests that the Drude carrier relaxation
(Fig. 13a), the efficacy factor has large contribution due
time parameter influences prompt feedback mechanisms
to excitation of the surface plasmon resonance (SPR).
involved in the formation of surface ripples. Indeed in
In this early stage, the spatial extension of the elec-
the high-frequency limit with ω τ 1, the metalized
tromagnetic field inside the bulk associated with SPP L e ≫
surface behaves as a nearly ideal inductor, while in the
is determined by the SPR decay constant κ , which
m low-frequency limit ω τ 1, the resistive Ohmic losses
at short wavelengths κm → 1/l defines a skin-depth result in electron heatLineg≪in the skin layer.
l = 1/κ l leading to strong concentration of the
s m
∼
electromagnetic field in the thin selvedge region. SPPs Because the efficacy factor theory does not fully ac-
need finite time to build up to incorporate the details of countforinterpulsefeedbackprocessesthatareundoubt-
the surface relief and interfere with the laser to modify edly important in the detailed development of morpho-
theFouriercomponentsofthesurfaceroughnessfunction logical features on the diamond surface our theoreti-
b(κ) viaperiodic laserablation. Inthis regimethe depo- cal results are not directly applicable to the multipulse
sition of laser energy into the surface plasmon wavevec- phase of LIPSS formation. However the experimental
tor causes formation of fine ripples with spatial periods data shows that once surface ripples are formed, expo-
around 100 nm, as observed in the periphery of the ab- sure by subsequent pulses has little effect on their spa-
lative craters, cf. Fig.11. The transverse-magnetic char- tial period and location, thus LIPSS formation should
acteristic of the SPP determines the orientation of the be possible already for a single-pulse irradiation, pro-
surface ripples. At a later time, the transiently increas- videdthatSPPcanbeexcited,e.g.,bysurfacedefects67.
ing number of conduction electrons makes the dielectric Once LIPSS are formed, the spectrum of the surface
constant large and negative at the laser wavelength, the roughness, b(κ) contains peaks at the SPP wavenum-
intensity map of η(κ;0) shrinks and concentrates on the ber causing enhanced inhomogeneous energy deposition
outer part of the circle κ=1 (cf. Fig 13b) which clearly and further growth of LIPSS as is also evidenced from
can be associatedwith the formationof near-wavelength the SEM images in Fig.11a-b. Furthermore the subse-
surface ripples oriented perpendicularly to the laser po- quentpulsesinteractwithperiodicallystructuredsurface
larization. Atthelongerwavelengthswithκ 0theskin hence a grating-assisted laser-surface coupling becomes
depth l =c/ω 80 nm is much smaller th→an the laser effective26 causing a decrease of the ripple wavelength.
s p
≈
wavelength. Therefore,abovetheSPRexcitationthresh- The experimental data shows only minor modification
old, the transiently increasingcarrierdensity resultsin a of LIPSS periods that is consistent with the hypothe-
shift of the SPP wave number from the high spatial fre- sis in Ref.26 that because of strong thermal effect at the
quencyregiontowardsthelightline(alsocausingexpan- crater center, the grating-assisted coupling is weak and
sionof the skindepth), andthis redshift is highly sensi- the ripple wavelength is unaffected by higher exposure,
tiveonthecarrierdensity(orlaserintensity),cf. Fig.14a. i.e. depends weakly on the superimposed pulse number.
9
laser energy deposition on photo-excited diamond sur-

12 face based on Sipe-Drude theory provided realistic and
10 (a) 10 (b) detailedinsightintomicroscopicmechanisms. Themodel
8 8 identifiestheimpactionizationasarelevantprocesscaus-
log10 46 log10 46 ionfgthoeptpicuallsebrreeasukldtoinwgninofpdlaiasmmoonndicainllyt-hacetitvreaisliunbgsterdagtee
withnegativebulkdielectricconstanttriggeringtheSPP-
2 2
laserinterferencemechanismforsurfacerippleformation.
0 0
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Fine ripples oriented perpendicularly to the laser polar-
/
ization emerge for intensities in a narrow range above
FIG.14. Transientvariationoftheefficacyfactorinthedirec- the optical breakdown threshold and the transient in-
tion perpendicular to the laser polarization. In Fig. (a), the crease of the carrier density above this threshold results
dashed-dotted line the EHP frequency ωp(t) = 1.4ωL is just in the formationof near-wavelengthsurface ripples. The
below the SPR excitation threshold, the solid line represents
interpulse feedback mechanisms involved in LIPSS for-
resonant excitation of surface plasmons corresponding to in-
mationarenotconsideredbythepresenttheoryandfur-
stantaneousbulkplasmafrequencyωp(t)=1.43ωL,thetran- ther work will be carried out in this direction. The ob-
sient increase of the free-carrier density with ωp(t)=1.45ωL
tained results lay the groundwork for utilizing diamond
(dashedline)andωp(t)=1.47ωL (dottedline)resultsinred-
as a plasmonic material supporting subwavelength and
shift of the surface plasmon peak towards the light line and
formation of near wavelength ripples. Fig.(b) demonstrates intense SPPsthatis verypromisingfor advancedoptical
the dependence of the efficacy factor on the Drude damping applications.
time τe = 100 fs (solid line) and τe = 10 fs (dashed line).
The laser beam is linearly polarized along the y-axis and is
normally incident to thesurface, λ and Λ designate thelaser ACKNOWLEDGEMENTS
wavelength and theLIPSS period, respectively.
This work was partially supported by the Russian
Foundation for Basic Research (projects nos. 17-52-
53003,17-02-00293and17-52-18023)andthegrantofthe
IV. CONCLUSION
RASPresidiumprogram,aswellasbytheGovernmentof
the Russian Federation (Grant 074-U01) through ITMO
The influence of prompt and cumulative optical feed- Visiting Professorship Program for S.I.K. This material
backcontributionsinmulti-shotfs-laserinduceddynam- is based upon work supported by the Air Force Office of
ics of surface ripples was investigated theoretically and ScientificResearch,AirForceMaterialCommand,USAF
experimentally. The numerical simulations of periodic under Award No. FA9550-15-1-0197(T.A.).
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