Transient optical properties as a signature of structural changes within FEL irradiated solids

a signature of stru tural hanges within

FEL irradiated solids

Dissertation zur Erlangung des Doktorgrades

des Fa hberei hes Physik

der Universität Hamburg

vorgelegt von

Vi tor Tka henko

aus

Dubna (RUS)

Prof. Dr. AlexanderLi htenstein

Guta hter der Disputation: Prof. Dr. NinaRohringer

Dr. Mark Prandolini

Dr. Martin Beye

Datum der Disputation: 31.07.2017

Vorsitzender des Fa h-Promotionsauss uhsses: Prof. Dr. Mi hael Rübhausen

Leiter des Fa hberei hes Physik: Prof. Dr. Mi hael Pottho

Dekan der Fakultät für Mathematik,

Moderne Freie-Elektronenlaser (FEL) im XUV- und Röntgenberei h liefern Energie, die

ausrei ht, um Festkörpersysteme auf einer ultrakurzen Zeitskala aus dem

Glei hgewi ht-szustand zu bringen. Die Änderungen in der komplexen Dielektrizitätsfunktion und die

ans hlieÿenden Modikationen der messbaren optis hen Koezienten spiegeln dann die

Strukturentwi klung des Materials auÿerhalb des Glei hgewi hts wider. Die optis he

Un-tersu hung des Materials erlaubt es, die Reektivität oder den Transmissionkoezienten

mitFemtosekundenauösung zu messen.

Aufgrund der Eigens haften ihrer Bandstruktursind Halbleitervon besonderem

Inter-esseinder FEL-basiertenWissens haft. PhotoneneinesRöntgen-FELs könnenAtome

ion-isierenundElektronenvomValenzbandoderausinnerenS halenindasLeitungsbandeines

Halbleiters anregen. Im Falle von groÿen Ladungsträgerdi hten im Leitungsband ändert

si h dieFlä he der potentiellen Energieder Atome erhebli h. Die ans hlieÿende Dynamik

der Atome führt zu strukturellen Transformationen und irreversiblen Phasenübergängen

auf einer Zeitskala von einigen

100

fs. Andererseits können Laserpulse au h thermis he Phasenübergänge über Elektronen-Phononenkopplung und infolgedessen Erwärmung des

Materials aufeiner Zeitskalavon

≈ 1

psoder längereinleiten.

Die vorliegende Arbeit untersu ht drei vers hiedene Materialien -Diamant, Silizium

undGalliumarsenid-dieRöntgen-FEL-Strahlungausgesetztwerden. Dieentwi kelten

the-oretis henModelle,mithilfedererdieoptis heAntwortderuntersu htenMaterialien

unter-su ht wird, basieren auf semi-empiris hen Ansätzen wie, z.B., dem Tight-Binding-Modell,

die einem die Mögli hkeit geben, die Zeitentwi klung eines Systems miteiner groÿen

An-zahl vonAtomenzu behandeln. Dieoptis hen Eigens haften, diedur h strukturelle

Mod-ikationenbeeinusst werden, stellendann dieVerbindung zwis hen den mikroskopis hen

Parametern und den experimentellen Observablen her. Entspre hende Experimente mit

diesenMaterialienwurden anFreien-Elektronen-LasernimXUV- und Röntgenberei hwie

FLASH,FERMIElettra,LCLSundSACLAdur hgeführt. Diegewonnenenoptis hen

Present-day XUV and X-rayfree-ele tron lasers deliver the energy su ient todrivesolid

systemsoutofequilibriumonanultrashorttimes ale. Stru turalevolutionofthematerial

in non-equilibrium is then ree ted in the modi ation of its omplex diele tri fun tion

and subsequent hanges of the observable opti al oe ients. The opti al probing of the

material allows to measure the ree tion or transmission oe ients with a femtose ond

time resolution.

Due to the features of their band stru ture, semi ondu tors are of parti ular interest

for the FEL-related resear h. X-ray FEL photons are apable to ionize atoms and ex ite

valen e band or ore hole ele trons to the ondu tion band in them. In ase of a large

density of arriers in the ondu tion band the potential energy surfa e of atoms

signi- antly hanges. The subsequent atomi dynami s leads tostru tural transformationsand

irreversible phase transitions on a time s ale of a few hundred fs. On the other hand,

laser pulses may also indu e thermal phase transitions via ele tron-phonon ouplingand,

onsequently, latti e heatingon atime s ale of

∼

1 psor longer.

The thesis studies three dierent materials - diamond, sili on and gallium arsenide

-exposed to X-ray FEL radiation. The developed theoreti al models evaluating the

op-ti al response of investigated materials are based on semi-empiri al approa hes, su h as

tight-binding s heme, whi h give an opportunity to treat the time evolution of the

sys-tem with a large number of atoms. The opti al properties, being ae ted by stru tural

modi ations, then set up the link between the mi ros opi parameters and

experimen-tal observables. Corresponding experiments with these materialswere performed at su h

XUV/X-ray FEL fa ilitiesasFLASH, FERMIElettra, LCLS and SACLA.The obtained

2015 Modelingof NonthermalSolid-to-SolidPhaseTransitioninDiamond Irradiatedwith

Femtose ond x-ray FEL Pulse

N. Medvedev,

V. T kachenko

,B Ziaja.

Contributions toplasma physi s,vol. 55, p. 12(2015)

Time-resolved observation of band-gap shrinking and ele tron-latti e thermalization

within X-ray ex ited gallium arsenide

B. Ziaja,N. Medvedev,

V. T kachenko

, T. Maltezopoulos, W. Wurth. S ienti Reports, vol. 5,p. 18068(2015)

2016 Transient opti alproperties of semi ondu tors under femtose ond x-ray irradiation

V. T kachenko

, N. Medvedev, Z. Li,P. Piekarz, B. Ziaja. Physi s Review B, vol. 93, p. 144101 (2016)

Transient hanges of opti al properties in semi ondu tors in response to

femtose -ond laser pulses

V. T kachenko

, N. Medvedev, B. Ziaja. Applied S ien es, vol. 6, no. 9,p. 238 (2016)

Soft x-rays indu e femtose ond solid-to-solid phase transition

S.Toleikis,B. Ziaja.

arXiv:1612.06698(2016); Nature Communi ations, submitted (2017)

2017 Ele tron-ion oupling in semi ondu tors beyond Fermi's goldenrule

N. Medvedev, Z.Li,

V. T kachenko

,B. Ziaja. Physi s Review B, vol. 95, p. 014309 (2017)

Pulse duration in seeded free-ele tron lasers

P. Finetti, H. H

¨o

ppner, E. Allaria, C. Callegari,F. Capotondi, P. Cinquegrana, M. Coreno, R. Cu ini, M.B. Danailov, A.Demidovi h, G. De Ninno, M. Di Fraia, R.

Feifel, E. Ferrari, L. Fr

¨o

hli h, D. Gauthier, T. Golz, C. Grazioli, Y. Kai, G. Kurdi, N. Mahne, M. Manfredda, N. Medvedev, I.P. Nikolov, E. Pedersoli, G. Pen o, O.

Plekan,M.J.Prandolini,K.C.Prin e,L.Raimondi,P.Rebernik,R.Riedel,E.

Rous-sel, P. Sigalotti, R. Squibb, N. Stojanovi ,S. Stranges, C. Svetina, T. Tanikawa, U.

Teubner,

V. T kachenko

,S.Toleikis,M.Zangrando,B.Ziaja,F.Tavella,L.Giannessi. Physi s Review X, a epted (2017)

Pi ose ond Relaxation of X-ray Ex ited GaAs Bulk

V. T kachenko

, V. Lipp. N.Medvedev, B. Ziaja. HighEnergy Density Physi s,a epted (2017)

LIST OF PUBLICATIONS

1 Introdu tion 1

2 Theoreti al basis 5

2.1 Free-ele tronlaser . . . 5

2.2 Intera tion of X-ray photons with matter . . . 9

2.3 Ele tron-latti eintera tion . . . 11

2.4 Opti alproperties . . . 12

2.5 Ele troni band stru ture . . . 17

2.5.1 Drude model . . . 17

2.5.2 Ab initioapproa hes . . . 19

2.5.3 Tight-bindingmodel . . . 21

3 Constru tion of the model 23 3.1 Cal ulationof CDF inrandom phaseapproximation . . . 23

3.2 CDF formalism for inelasti s attering ross se tions . . . 27

3.3 XTANT model . . . 31

3.4 Opti al oe ients inequilibrium . . . 40

4 Non-equilibrium dynami s of FEL irradiated solids 44 4.1 Phase transitions . . . 44

4.2.1 Irradiationof diamondbelowand above graphitizationthreshold . . 47

4.2.2 Ee t of non-equilibrium ele tron kineti s onopti al properties . . 60

4.2.3 Opti allyindu ed radiationdamagein diamond . . . 61

4.2.4 Measured X-ray indu ed graphitizationof diamond . . . 64

4.2.5 Diusionpro esses . . . 72

4.3 Sili on . . . 76

4.3.1 Modelling of phase transitions insili on. . . 76

4.3.2 Opti allyindu ed radiationdamagein sili on . . . 81

4.3.3 Ee t of ele tron-ion ouplingon opti alproperties . . . 83

4.4 Galliumarsenide . . . 86

4.4.1 Ele tron-latti ethermalizationinGaAs . . . 86

4.4.2 Bandseparation in semi ondu tors . . . 94

5 Con lusions and outlook 102 5.1 Con lusions . . . 102

5.2 Outlook . . . 105

Appendi es 108

A Verlet algorithm 108

B Ehrenfest approximation in al ulation of potential energy surfa e 110

C Tight-binding parameters 112

D Cross se tions for inelasti ele tron s attering 116

Introdu tion

Ultrashort laser pulses(of femtose ond time duration,

τ = 10

−15

fs) generated by modern

free-ele tron lasers (FEL) in XUV and X-ray range (LCLS [1℄, SACLA [2℄, FERMI [3℄,

FLASH[4℄et .) are widelyused todayinphysi s,materials ien e, hemistry andbiology.

A broad range of systems from gases to lusters and strongly orrelated materials have

been already studied in this way. The pulse intera tion with matter ae ts its stru ture

onmole ularand atomi level. Dependingonthe pulse duration, intensity, oheren e and

in iden e angle, FEL radiation an trigger dierent pro esses inmatter, su h as hemi al

rea tions, radiation damage, phase transitions, and reates exoti states of matter et .

X-ray/XUV pulses an be used for various purposes. For example, hard X-rays (with

photonenergy

¯hω >

5keV) are of importan eforX-ray rystallography dueto theirlarge penetration depth in the material and the wavelengths omparable with the interatomi

distan e. On the other hand, soft X-rays (

¯hω < 5

keV) and XUV (10 eV

< ¯hω < 124

eV) are e ient for the pro esses where photoabsorption is strongly involved, as at su h

energies it dominates over the Compton s attering.

In ourworkweare fo usingonthe simulationof ele troni kineti sand atomi

dynam-i s in bulksemi ondu tors, whi h leads tostru tural transformations. The Fermi level in

semi ondu tors islo atedbetween the valen e bandand the ondu tion band. Thebands

in semi ondu tors may vary from

∼

0.1eV to a few eVsdepending on lassi ation. This implies the situation that in an external eld ele trons, whi h at the room temperature

o upyonlythevalen eband, anbeex ited,over omethebandgapandbetransferredto

the ondu tionband,thus reatingholesinthevalen ebandand for ingthesystem outof

equilibrium. If the number of ele tron-hole pairs is su ient, the onsequent interatomi

potential hanges,and atomi relo ationsthenleadtostru turalmodi ationsof

semi on-du tors. Thus, hanging stru tural properties of semi ondu tors are interesting not only

fromthe prospe tiveof theirappli ation inmaterials ien e, but alsoasaphenomenon, in

whi hinitialele tron-holeex itationtriggersa omplextransformationpro ess. In aseof

photoex itation it is often su ient to rea h the ne essary photon density with a single

FEL pulse in order to promoteenough ele trons tothe ondu tion band. It is alsoeasier

to deal with a single pulse from the simulation point of view as well as from the

experi-mentalperspe tive,asthemoderntimingtools andenepulsedurationwithfemtose ond

a ura y. Therefore, inour resear hwewill onsider simulationsand experimentswhere a

single FEL

pump

pulse was used.

The work ontains four main Chapters. In Chapter 2 we providethe theoreti al

ba k-groundwhi his vitalforour resear h. Anoverview of the prin ipleof FELsour e work is

given in the beginning of the Chapter. Then we subsequently dis uss pro esses that take

pla e within the material after the FEL shot, from absorption of photons by ele trons to

ele tron-latti e equilibration. Later we introdu e opti al properties, whi h an be

mea-suredduringtheevolutionofthesepro esses,anddis ussbandstru ture formalismswhi h

an be potentiallyapplied. In Chapter 3 we onstru t our theoreti almodelwhi h isable

toa ountforthepro esseshighlightedinChapter2andto al ulatetheopti alproperties

syn hronously with atomi dynami s. At the end of this Chapter we test the model on

the materialsin equilibrium. In Chapter 4 we ome to the a tual results of our work. In

this work, we dis uss the ultrafast (on a time s ale up to several pi ose onds,

t ∼ 10

−12

s) transient phenomena inthree materials - diamondas a rystal stru ture of arbon,

sil-i on and gallium arsenide. Parti ularly we study transient dynami s of the laser-indu ed

below damage threshold. The Chapter 5 is dedi ated to the on lusions of the work and

outlookof the future development of the modeland its appli ations.

Ultrafastphasetransitionsindiamondandsili on,whi hweobserve, analsoberesults

of a very high density of harge arriers in the ondu tion band and a orresponding

potential energy surfa e hange. Su h kind of phase transitions is alled non-thermal, as

it is not indu ed by ele tron-phonon oupling whi h is typi al for a onventional thermal

phasetransition. Note,thatallofthesephenomenahavebeenknownforsigni antamount

of time, but so far they have been investigated primarily as a onsequen e of irradiation

in opti al regime. In ontrast, here we on entrate on XUV/X-ray irradiation, proving

that su hee ts an o ur inthis photonenergy rangeaswell. Moreover, semi ondu ting

materials nowadays are used in X-ray opti s and then all possible X-ray damage ee ts

should be studiedto estimatetheir radiationtoleran e.

There are several ways to tra e the non-equilibrium dynami s in theory and

experi-ment. Here wetra e itby al ulationof the opti alresponse in irradiatedmaterials. This

methodis hosenasopti al onstantslikeree tivityortransmittan edistin tly

hara ter-izestru tural properties ofthe materials,and are a urately measured inthe experiments

onashorttimes ale. Thepump-probete hnique allowstoa quireatime-resolved pi ture

of the stru turalpro esses and tounveil theirme hanismsby measuring the opti alprobe

pulse signal intera ting with the material.

As the band stru ture of the semi ondu tors annot be des ribed a urately enough

withoutquantumtheoryandontheotherhandpure

ab initio

methodsare omputationally ine ient for the des riptions of the systems with many atoms under non-equilibrium in

time-resolvedmanner,wearriveatthesemi-empiri altransferabletight-bindingdes ription

of the band stru ture. In addition, we are using lassi al mole ulardynami s for treating

theatomi motion. Wealsoadopttheideaofseparatingallele tronsinthe solidsystemin

twodomainsandtreatingthem eitherwith MonteCarlomethodorwiththermodynami al

approa h,dependingontheirenergy. The al ulationofopti alpropertiesisbasedon

tight-binding approa h and omplex diele tri fun tion (CDF) formalism in the randomphase

(X-ray indu ed thermal and non-thermal phase transitions) [5℄ hybrid and transferable

between dierent materials. For predi tion of non-equilibrium temperature and

ele tron-phonon ouplinginGaAswealsoapplythetheoreti aldes riptionbasedonrateequations,

Theoreti al basis

2.1 Free-ele tron laser

A free-ele tron laser (FEL) is a light sour e whi h is able to generate oherent, tunable

high-powerradiation. Thegeneratedwavelengths bymodernFELs rangefrommi rowaves

toX-rays. ThekeyfeatureofFELisausageofanundulator(awiggler),whi hisessentially

a set of magnets, for ing the ele tron beam to wiggle transversely along the axis of the

undulator (wiggler). A eleration of ele trons results in the emission of photons asin the

syn hrotron. In anFEL the ele trons are united intomi robun hes and separated by one

opti al wavelength along the propagation axis, thus emitted radiation is mono hromati

and oherent. The feedba k of the emittedradiationontothe ele tron beam is alled

self-amplied spontaneous emission (SASE) [6, 7℄, whi h is employed on X-ray free ele tron

laser fa ilities (XFEL). The modern FELs in lude VUV and soft X-ray lasers su h as

FERMIElettra in Italy [3℄ and FLASH in Hamburg [4℄, and those designed for hard

X-rays the Lina Coherent Light Sour e (LCLS) in the USA [1℄, SACLA in Japan [2℄,

SwissFEL inSwitzerland[8℄andEuropeanXFELinHamburg [9℄,whi hisexpe ted tobe

in operationalregimein the se ond halfof 2017.

TheXFELradiationhasawavelengthofanorderofAngstrom(

10

−10

m)whi his lose

femtose ond time duration), whi h isimmensely importantfor stru tural studies,be ause

the pulse outruns the radiationdamage. Together with brillian e(the number of photons

perse ondpropagating througha given ross se tionareaand withinagiven narrowsolid

angle and spe tral bandwidth) mu h higher than in syn hrotrons of any generation (see

Fig. 2.1)thismakesXFELauniquetoolforinvestigatingtime-resolveddynami sofmatter

onamole ularandatomi s ale. Therefore, XFELshavefound theirappli ationinphoton

and material s ien e; atomi , mole ular, and opti al physi s (AMO); stru tural biology;

hemistry;medi alphysi s. UltrafastX-rayimagingleadstoa reationofmole ularmovies

whi h may tell a lot about so far unexplored ultrafast pro esses within the materials.

High intensity of X-ray pulses allows to obtain single-shot dira tion images from single

mole ules with high spatial resolution.

SASE FEL pulses ontain independent, temporally oherent emission spikes [6℄. The

temporallengthof the spikesmay varyfrom hundreds of attose onds to several

femtose -onds. Re entinvestigationsshow[10℄that,forexample,atFLASHthegenerationofshort

FEL pulses with high temporal oheren e, lose to single spike is a hievable in the VUV

and soft X-ray range.

Fora uratemeasurements, ertainpulseparametershavetobea uratelydetermined.

One of the most important parameters is a pulse duration whi h is important in alltime

resolvedexperimentsandalsofortheproperdenitionofthepeakFELintensity. Thepulse

duration an be evaluatedby using auto orrelation measurements or via ross- orrelation

te hniques whi h in prin iple ompare the investigated pulse with a ertain model pulse.

Modern methodsare apableof providingtemporalmeasurementsof omplex pulseswith

multiplepeaksbothofspatialandtemporal hara ter[11℄. Theotherimportantproperties

of the pulse are the pulse energy and its shape. Withpulse shaping te hniques, the pulse

sour e an be modied interms of amplitude, phase and durationby spe ial devi es su h

as ampliersor ompressors.

Apart of the free-ele tron laser itself, another key point of X-ray spe tros opy is a

usage of the pump-probe te hniques. The pump-probeprin iplewas intensively employed

Figure 2.1: The order of peak brillian e (in

photons/s/mm

2

_/mr

2

_{/0.1%bandwidth}

) of X-ray

sour es ommissionedat dierenttimes. X-rayfreeele tron lasers anbe10 ordersofmagnitude

brighter than third generation syn hrotrons (e.g. Petra III, ESRF). The pi ture is taken from

two opti al pulses with dierent wavelength [12, 13℄. The invention of FELs su iently

extended thes opeofusingthe pump-probete hnique. Inapump-probes heme, theFEL

sour e anbeusedin ombinationwithanotherlaser,usuallyanopti alone. First,anFEL

shothits the sampleand indu esanex itationpro ess withinit. Then,withanadjustable

time delay aprobe pulse issent tothe sample(Fig. 2.2). Afterwards, the transmission or

ree tion signal of the probe pulse an be measured. By s anning the sample with probe

pulses with a denite delay, one an obtain the transmission or ree tion oe ient as a

fun tion of timedelay between pumpand probepulses, and insu h amanner re onstru t

the ex itation pro ess in the sample. So, briey speaking, the pump pulse laun hes the

pro ess and the probe pulse tra ks it. As the probe pulse signal ontains information

about the opti alproperties of the material,this impliesthat fromthe opti al properties,

measured in the experiment, we may a quire the information on the temporal pro esses

o uring withinthe material.

Nowadays,XUVorX-rayFEL pulsesmaybeusedasprobesaswell. Atthebeginning,

the FEL pump  FEL probe s hemes were based on time-delayed holography [14℄, where

the probe pulse was formed by the pump ree ted fromthe mirror, oron auto orrelation

prin iple, whereboth pumpand probepulses were generatedfromone ele tron bun hand

the in oming light was split in two parts by a fo using mirror [14, 15℄. Thus, only one

olour experiments were available. However in 2013 the LCLS group reported [14, 16℄

that by using a double undulator s heme, temporallyand spe trally separated pump and

probepulses anbegenerated. Thisopenedwideopportunitiesforstudiesofthe

radiation-matterintera tions. WithX-rayprobepulse,theX-raydira tionpatternfromthesample

is re onstru ted. Then, the evolution of Bragg peaks in dira tion images or proles is

Figure 2.2: S hemati pi ture of a pump-probe experiment. The time

τ

stands for the time

delaybetween the pumpand probe pulses.

2.2 Intera tion of X-ray photons with matter

Ingeneral,X-rayphotons areknown forthe apabilitytopenetratedeepintothematerial,

i.e., they have relatively high attenuation length in omparison to opti al photons or to

infraredradiation. AnotherimportantpropertyofX-rays,whi hwasalreadymentionedin

the previous Chapter, isthat anX-ray wavelength is omparable withthe size ofanatom

and with the interatomi distan e between atomsin a rystal stru tures of solids. Forthe

majority of materials, X-rays are also pra ti ally non-refra tive. All these hara teristi

features of X-rays make them an e ient tool for investigation of solid state stru tures,

along with ele tron dira tionand neutronography [17℄.

Irradiation of solids by X-ray photons triggers a number of pro esses within them.

Let us dis uss these pro esses in detail. Immediately after the start of X-ray photons

propagation within the material, they begin to photoionize atoms by removing bound

ele trons from the atomi shells. X-ray photons are able to ex ite the strongly bound

ele trons in the atoms from the ore shells, in su h a way reating ore holes. If the

photonenergyex eedsthe bindingenergyofele trons, aphotoionization rossse tionora

probabilityofemittinganele tronishigh. Ifasinglephotonenergy isbelowsu hbarriers,

multi-photon ionizationby several photons ombining their energies is still possible, and

intense oherent laser pulses raise the probability of multi-photon ionizationup.

ele tronsasphotons ex hangetheirkineti energywiththemintheeldofnu lei. Ex ited

ele trons after emission may also ionize atoms if they have enough energy: via inelasti

s attering. This pro ess is alled se ondary impa t ionization. The elasti s attering,

whi h does not lead to the signi ant ele tron energy loss during a s attering event, is

alsofeasible. Se ondaryele trons themselvesin rease ele trondensitywithinthematerial.

A highly harged system is reated as a result, and in nite-size samples, this an ause

Coulomb explosion after high-energy ele trons leave the system. Thermalizationof

ele -trons is a hieved through ele tron-ele tron ollisions ona time s ale of 10-100fs, whi h is

mu h faster than for an ion system be ause of a large mass dieren e between ions and

ele trons.

Re ombinationof ions with ele trons isessentially aninverse pro ess ofthe se ondary

ionization. In ase of three-body re ombination, a spe tator ele tron near anion re eives

thekineti energyreleasedbythere ombiningele tron. Theion hargede reasesduetothe

ele tron-ionre ombination,andtheion anexperien ephotoionizationagain. Thispro ess

highly depends on the ele tron harge and density. In ase of radiative re ombination, a

photon with the wavelength, orresponding tothe released energy, isemitted.

The ore holes formed by removing ele trons from the ore shells typi ally relax via

Auger ee t in light elements. An ele tron from a higher energy level may ll a va an y

(e.g.,anele tron fromL-levelllstheK-shell ore hole). The released energy, equaltothe

dieren ebetween the bindingenergies of the levels, may thenbetransferred toa se ond,

higher-lying ele tron. Thus, this so- alled Auger-ele tron will be removed from the shell,

additionally ionizingthe atom. The kineti energy of an Auger-ele tronthus depends on

the type of the atom and its shell stru ture. Another way of lling the va an y in

inner-shellhole isaradiativepro ess whenthe energyis arried-oby thephoton. Heavy atoms

withalargeatomi numberandlargetransitionenergiesprimarilyde ayviasu hradiative

2.3 Ele tron-latti e intera tion

After ele trons absorb energy from the in ominglaser pulse, they start to ex hange their

energy with ea h other as we stated in the previous Se tion. At the same time they also

ex hange their energy with latti e, but in ea h s attering event they an transfer only a

small amount of energy. Therefore, this pro ess lasts on a longer time s ale (typi ally

within several pi ose onds). This stage of the system evolution is alled ele tron-latti e

equilibration. By this ex hange, the system evolves towards the equilibrium. After the

photoionizationeventsandele tron as ading,theele trontemperatureex eedsthelatti e

temperature. Theele tronsubsystemthenloosesthegainedenergyviaemissionofphonons

or the so- alledele tron-phonon oupling [18℄. In solid state physi s a phonon represents

a quant of the vibrational mode of the rystal latti e. Thus, the a umulated energy of

ele tronsinnon-equilibriumisbeingtransformedtothevibrationalex itationofthelatti e.

Atomsofthelatti egainkineti energyand starttomovefromtheirequilibriumpositions.

The temperature of ele trons starts de reasing, while atomi temperature, in ontrast, is

in reasing. Ultimate attening of the temperatures of the two subsystems results in the

ele tron-latti e thermalization.

Another pro ess in a bulk material that lowers ele tron temperature is the thermal

diusion of hot ele trons from the laser ex ited area to unex ited old regions within the

material[18℄. Ingeneral,thediusionmeansthatparti lesfromtheregionswiththeirhigh

on entration penetrate into the regions of their lower on entration to reate a uniform

arrierdistribution. Inphysi sofsemi ondu tors,theEinsteinapproa h,originallyapplied

to gases, is widely used to des ribe ele troni diusion. A ording to it, the diusion

oe ient, dening essentially the speed of the diusion pro ess, is obtained from the

2.4 Opti al properties

XTANTmodel(X-rayindu ed thermalandnon-thermalphasetransitions)proveditselfto

be an e ient tool for investigating a non-equilibrium evolution of ele troni and atomi

subsystems inXUVandsoftX-rayirradiatedsolids[5,20,21℄. Su himportantparameters

as band gap width, potential and kineti energies of ele troni and atomi subsystems,

ele tron and atomi temperatures, number of ele trons in the ondu tion band an be

estimated with XTANT. Also, atomi snapshots an be re orded. However, in order to

make dire t omparisons of the model predi tions with an experiment, we need some

ma ros opi experimentalobservables.

Opti al oe ientsof materialssu h asree tivity and transmittan eare widely

mea-sured in numerous experiments. The modern pump-probe te hniques, where the opti al

probefollows the pumppulsewith a ertaintime delay, allowtomeasure transientopti al

properties of irradiated materials. The time resolution of su h pump-probe experiments

an a hieve a fewfemtose onds [2224℄. Thus, the pro esses going oninside the materials

ona femtose ond time s ale an be dete ted, ifthese pro esses inuen e the opti al

prop-erties. For example, ele troni ex itation and espe ially phase transitions lead to abrupt

hangesof opti alpropertieswithinmaterials. By omparingtransientopti al oe ients,

obtained in the experiment, with the theoreti ally modelled ones, we an get

informa-tion on a presen e and a time s ale of the indu ed stru tural transformations and phase

transitions.

Opti alpropertiesofamaterial anbeexpressedthrough a omplexindex ofrefra tion

e

n

[25℄:

e

n(ω) = n(ω) + ik(ω).

(2.1)

Here

n(ω)

and

k(ω)

are orrespondinglyreal and imaginary parts ofthe omplexindex of refra tion,whi h are dependent onthe frequen y of the probepulse.

Theree tion oe ient(orree tivity)ofthematerialisafra tionofthein identlight

energythatwasree tedby thesurfa eofthematerial. Correspondingly,the transmission

transmit-Figure 2.3: S hemati pi tureofthe in identray

k

,ree tedray

k

r

andthetransmittedray

k

t

and the orrespondingangles

θ

,

θ

r

,

θ

t

on theborder between two media.

tedthrough thematerial. In ident,ree tedandtransmittedraysare s hemati allyshown

in Fig. 2.3. They are determined by the boundary onditions for ele tromagneti waves

onthe border ofthe materials. The ree tivity oe ientisdened by the Fresnel lawas:

R =









cos θ − en cos θ

_{cos θ + e}

_{n cos θ}

t

_t









2

,

(2.2)

where angle

θ

is an angle of the ray propagation in respe t to the normal in the va uum (

e

n

vac

= 1)and

θ

t

isanangle tothe normalinthe material. A ording tothe fundamental Snell's law

n

vac

sin θ = n sin θ

t

one an rewrite the expression Eq. (2.2) for the ree tivity oe ient as:

R =











cos θ −

p

e

n

2

_{− sin}

2

_θ

cos θ +

p

_e

n

2

_{− sin}

2

_θ











2

.

(2.3)

The transmission oe ient of the material also depends on the material thi kness

d

and the wavelength of the in ident probe pulse

λ

. As we usually deal with thi k bulk material and femtose ond probe pulses, in most ases we an approximate our model to

onlyrstray propagationwithnointerferen e ee tsin luded frommultipleree tionson

the material boundaries. In this ase, the expression for the transmission oe ient an

be writtenin the followingform [26, 27℄:

T =











4 cos θ

p

_e

n

2

_{− sin}

2

θ · e

−i

2πd

λ

√

e

n

2

_−sin

2

_θ

(cos θ +

p

_e

n

2

_{− sin}

2

_θ)

2











2

,

(2.4)

The absorption oe ient an be obtained from the normalization ondition:

A = 1 − T − R.

(2.5)

The omponentsof the omplex index of refra tion are onne ted with omponentsof

a omplexdiele tri fun tion (CDF)

ε(ω) = ε

′

_{+ i · ε}

′′

by relations:

n

2

₌

1

2

√

ε

′ 2

_{+ ε}

′′ 2

_{+ ε}

′



_,

k

2

=

1

2

√

ε

′ 2

_{+ ε}

′′ 2

_{− ε}

′



_.

(2.6)

The omplexdiele tri fun tionisanessentialproperty ofthematerialwhi h hara terizes

itsresponse to anexternal ele tri eld. CDF is tightlyboundwith the band stru ture of

solidsasitsimaginarypartisdire tlyrelatedtotheprobabilityofphotoabsorption. Ithas

ele tromagneti origin and an be dened through Maxwell'sequations:

∇ · D = ρ,

∇ × E +

∂B

∂t

= 0,

∇ · B = 0,

∇ × H −

∂D

∂t

= J,

(2.7)

where

E

denotes the ele tri eld,

H

denotes the magneti eld,

ρ

isafree hargedensity,

J

is afree urrent density,

D

is aele tri displa ement,

B

isa magneti uxdensity. The rstequationrepresentstheGauss'slawanddes ribeshowthe hargedensity

ρ

withinthe materialis onne ted to the ele tri displa ement

D

. In turn, the ele tri displa ement is onne ted with a ve tor of an ele tri eld and an ele tri dipolepolarizationve tor

P

:

D

= ǫ

0

E

+ P,

(2.8)

where

ǫ

0

isthe va uumele tri permittivity(whi hisafundamental onstantexpressed in SI units). Considering the linear proportionality of the polarizationto the magnitude of

the applied eld and isotropi ity of the material inspa e, the polarizationve tor and the

ele tri displa ement an bewritten as follows [28℄:

P

= ǫ

0

χ

e

E,

where

χ

e

is the linear ele tri sus eptibility. The diele tri onstant

ε

is therefore dened as

ε = 1 + χ

e

. In other words, this is the ratio of the ele tri permittivity in the material to the permittivity in va uum. In an isotropi materialthe diele tri fun tion is a s alar

quantity. Inageneral ase,this isatensorrelatingea h omponentof

P

toall omponents of

E

. It is dependent on many fa tors, hara terizingthe state of the material su h as its temperatureand pressure, the mole ularand atomi stru ture, ex itation level et . [28℄

The propagating ele tri eld is des ribed by the equation:

E

= E

0

e

i(kx−ωt)

,

(2.10)

where the frequen y

ω

and the magnitude of the wave ve tor

k

are relatedas [28℄:

ω(k) =

_√

1

εǫ

0

µµ

0

k.

(2.11)

The oe ients

µ

0

and

µ

are magneti permeability of va uum and of the material orre-spondingly. The phase velo ity of the wave is then determined from the known formula

as:

v =

ω

k

=

1

√

_εǫ

0

µµ

0

=

c

e

n

,

(2.12) where

c = (ǫ

0

µ

0

)

−1/2

isthephasevelo ityinva uum,and

e

n = √εµ

istherefra tiveindexof themedium. Forfrequen iesintheopti alpartofthespe trumthemagneti permeability

µ

= 1. Thus, we ometo the relation that wasrst mentioned inEq. (2.6).

Apart fromwell-known ree tivity and transmissivity, other opti al parameters of the

ex ited materials an be measured during the experiment. Let us briey highlight these

opportunities, althoughwe donot investigate them laterin this study.

An exampleof another opti alproperty whi h an be measuredin the regimeof linear

responseisthelightemittan eorlumines en e[28℄. Thisee tisbasedonthespontaneous

emissionof radiationand anbeexplainedonlyonthequantumlevel. Emissiondue tothe

lumines en e has a non-thermal nature and o urs after light absorption. Lumines en e

is possible if the spe trum of the matter is dis rete and its energy levels are separated

-that is the reason why metals do not produ e lumines en e. After being ex ited to the

energy

E

1

, releasingthe photonwiththeenergy

E

2

− E

1

= ¯hω

. Thelumines en erate an be estimated as a ratio of the number of emitted photons

N

e

to the number of absorbed photons

N

p

. The frequen y of the emitted light is usually dierent from the one of the absorbed light.

The experimental values of lumines en e an be measured in a time-resolved way, or

as a fun tionof arrier density [28℄. For sub-pi ose ond measurements, the up- onversion

methoddes ribedbyShahetal.[29℄iswidelyused. Themethodhasmu hin ommonwith

pump-probe te hniques. The pump pulse ex ites the system, and the probe pulse follows

ituntilsomelumines entlighthas alreadybeen emitted. Then, aspe ial interferen e lter

sele ts adenite frequen y whi h isin afo us of the experiment.

An example of the non-linear opti al ee t is the se ond-harmoni generation (SHG),

whenthesample onvertsthein identradiationoffrequen y

ω

totheradiationoffrequen y

2ω

[28℄. This is a parti ular ase of a sum frequen y generation, and it is feasible only in media without inversion symmetry, i.e., if its rystal stru ture does not belong to the

point group with the inversion enter of symmetry. A non-linear bulk material gives a

ontribution to the se ond order of sus eptibility

χ

(2)

. This ontribution depends on the

orientationofmole uleswithinthematerial,i.e.,itsstru turalorder. Thebehaviourof

χ

(2)

during a laser-indu ed phase transition in GaAs was onsidered, for instan e, in [30℄. In

this ase,the se ond-harmoni signaldependsonthenon-linearse ond-order sus eptibility

aswellasonthe lineardiele tri fun tions

ǫ(ω)

and

ǫ(2ω)

. Thesedependen ieshaveto be sorted out. By measuring the se ond-harmoni signal as a probe, the disordering of the

rystallinelatti e of GaAs an be dete ted, when the

χ

(2)

goesto zero.

The more general ase is a sum-frequen y generation (SFG), when due to the

annihi-lation of two in oming photons at dierent frequen ies

ω

1

and

ω

2

, a photon at frequen y

ω

=

ω

1

+

ω

2

is emitted. The signals at visiblefrequen ies are very oftengenerated from near-infraredandvisiblebeams. Therearealsoevenmore ompli atedte hniquesavailable

2.5 Ele troni band stru ture

The basis for the models des ribing stru tural evolution of the atomi and mole ular

sys-temsisana uratedeterminationoftheirele troni bandstru ture. Here,wewillintrodu e

the variousmodels used forthis purpose and dis usstheir advantages and drawba ks.

2.5.1 Drude model

The Drude model for ele trons was proposed by Paul Drude in 1900, and it was one of

the rst attemptstoexplain ele tro-and thermal ondu tivity inmetals afterdis overyof

an ele tron by Thomson. The model isbased onthe kineti theory and assumes lassi al

behaviour of free-ele tron gas in an external eld of positively harged heavy stationary

ions.

The ele trons in the model are represented as ideal elasti spheres whi h oat along

straight traje tories and ollide with ions, hanging the dire tion of their oating after a

ollision. The duration of the ollision

τ

c

tends to zero, i.e. it is supposed to be instanta-neous [33℄. Thelong-range intera tions between ele trons or between anele tron and ions

are negle ted ex ept at the instant of the ollision. The hara teristi parameters of the

ele tron behaviour are the relaxation time

τ

, i.e., average time between two ollisions for one propagating parti le, and a mean free path of an ele tron,

λ = v

0

τ

, where

v

0

is the average speed of the ele tron.

Despite the simpli ityof the Drude model and a number of approximations made (for

ele tron ollisions and s attering), for a long time it has been used for the estimation of

thermal and ele tri properties in metals,and is stillapplied fortheir qualitative

des rip-tion. However, ithasbeenknownthatthedrawba ksoftheDrudetheorymayleadtofalse

predi tionsofopti alpropertiesinmetals,rstofall,be auseofthe ompli atedfrequen y

dependen ewhi h an notbetaken intoa ountbythe Drudemodel. Even sodium,whi h

isametaloftherstgroupintheperiodi table,reveals thefrequen y dependen ewhi his

notreprodu edbyDrude-modelpredi tions[33℄. Moreover,thebandstru tureof

are boundthere and the free-ele tron approximation isnot appli able for them.

A omplexdiele tri fun tionintheDrude model, whi hdoesnot take interband

tran-sitions inthe semi ondu tor intoa ount, is dened as [34℄:

ε(ω) = ε

0

+

i

ω

ω

p

2

τ

1 − iωτ

,

(2.13)

where

ε

0

is the unex ited material diele tri onstant;

ω

p

=

p

n

e

e

2

/m

∗

ε

0

is a plasma frequen y, with

n

e

being the free-ele tron density (of ele trons ex ited to the ondu tion band), and

m

∗

is the ee tive ele tron mass;

τ

is the relaxation time. The plasma fre-quen y hara terizes eigenfrequen y of the free-ele tron gas in response to a small harge

separation. So, it orresponds tothe ele tron density os illationin the ondu ting media.

The photoabsorption ina semi ondu tor an be interpreted asa promotionof an

ele -tron fromthe valen e to the ondu tion band. An inverse pro ess of anele tron transfer

fromthe ondu tion tothe valen e band isalsopossible. These pro essesare alled

inter-bandtransitions. Generally,thediele tri fun tion ontainsa ontributionfromtheDrude

response of free arriersand a ontributionfrominterband transitions [28℄:

ε(ω, t) = 1 + 4π(ξ

interband

(ω) + ξ

Drude

(ω)).

(2.14)

Approa hing to the non-equilibrium ex itation regime, the appli ability of the Drude

model is getting even more problemati . At high pump uen es (e.g., above

0.5 kJ/m

2

at

¯hω

= 1.9 eV for GaAs) the Drude model does not des ribe hanges in the diele tri onstantadequately[30℄astheex itation ausessigni ant hanges intheele troni band

stru ture. Thus, theterm

ξ

interband

(ω)

beginstodominateinthe ontributiontotheopti al sus eptibilityovertheterm

ξ

Drude

(ω)

andree tstheresponseoftheCDFtotheex itation. It shows the ne essity of estimating the interband ontribution beyond the Drude model

that we perform in Se tion 4.4. Whileon longer time s ales (belowthe damagethreshold

and thermal melting), the Drude model is still apable of produ ing results lose to the

experimental ones, on short time s ales after the ex itation by ultrafast XUV or X-ray

pulses, non-thermalee ts mayplay asigni ant role,and some othermodelingapproa h

2.5.2 Ab initio approa hes

In physi s

ab initio

methods imply al ulations from the rst prin iples by employing established laws of nature without any approximations or tting parameters. However,

depending onthe ommunity,a term 'ab initio' issometimes also applied tothe methods

thatpartiallyrelyonapproximate s hemes,forexample,tosu hasthe well-knowndensity

fun tionaltheory (DFT).

DFTiswidelyusedbothinsolidstatephysi sandquantum hemistry. Ithas onrmed

its potential in dening the ele troni band stru ture in many-body systems. DFT is

based on the prin iple that the ground state of the many-parti le system is determined

uniquely by the ele tron density (Hohenberg-Kohn theorem), whi h an be obtained from

a S hr

¨o

dinger equation [35℄. The parti le orbitals are dened by Kohn-Sham equations. Forthe solutionof Kohn-Sham equations dierent basis sets are used [36, 37℄.

The most simple on ept is to use plane waves whi h are not suitable for qui kly

varying potentials, unless avery large number of the waves is onsidered. More advan ed

approa hes use augmented fun tions. They in lude, e.g., augmented plane wave (PAW),

mun tin orbital (MTO). These methods are designed totreat the energy dependen e of

thebasisfun tions. Thethirdtypeofmethodsuseslo alizedorbitals,su has,forexample,

a linear ombination of atomi orbitals (LCAO), whi h is a quantum superposition of

atomi orbitals[38℄.

For studying the properties of many-body systems in non-equilibrium,evolvingin the

presen e of time-dependent potentials, an extension of the onventional DFT, whi h is

alled time-dependentdensity fun tionaltheory (TDDFT) wasperformed. TDDFT relies

on Runge-Gross (RG) theorem, whi h is basi ally an analogue of the Hohenberg-Kohn

theorem for time-dependent systems. RG-theorem governs the relationbetween the

time-dependentmanyparti lestateandthe orrespondingtime-dependentdensity. Analogously

tothe stationary ase,the hara teristi Hamiltonian ontains thekineti energy operator

ˆ

operator

V (t)

ˆ

of the ele trons in the time-dependent potential

V (r, t)

[35, 39℄:

ˆ

H = ˆ

T + ˆ

V (t) + ˆ

W.

(2.15)

OneofthewidelyusedDFTmethodsinsemi ondu torsstudiesislo aldensity

approx-imation(LDA) [35℄. Themain on ept of LDA istobuildthe ex hange- orrelationenergy

E

xc

from the ex hange- orrelation energy per parti le

ǫ

xc

of the homogeneous ele tron gas [35,40℄. The dependen e on

ǫ

xc

(n)

anbe omputedby quantum MonteCarlos heme. Generally,LDA des ribes well ovalent, metalli and ioni bonds, but works inadequately

for hydrogen and Vander Waalsbonds.

The ex hange- orrelation hole

P

xc

(r

1

, r

2

)

isthe probability ofnding anele tron at

r

2

giventhatthereisanele tronat

r

1

. Itmust fulllthe normalization onditiontoyield ex-a tlyoneele tron. ThereasonwhyLDAworksisthatitsa uratepredi tionofthex -hole

whi hisreasonablywellreprodu edinLDA[40℄. Inturn, theele tron-ele tronintera tion

depends ex lusively onthe spheri al average of the x -hole. However, this approa h does

not work for the ex ited states and pro esses beyond Born-Oppenheimer approximation.

Also, the al ulation of the band gap width for various semi ondu tors [41℄ and bonding

energies between atoms as a fun tion of distan e even for a simple

H

2

mole ule [42℄ by LDA disagrees with the experiment.

Ingeneral,DFTmethodsreprodu ebandstru turewithahigha ura y. However, due

tothehigh omputational osts,they areinappropriateforinvestigationsofhighly-ex ited

systems with high time resolution.

Another ab initio approa h that an be implemented is based on the

Hartree-Fo k-Slater (HFS) method and realized, for example, in the XMOLECULE ode [43℄. It an

al ulate an ele troni stru ture for mole ules irradiated with high intensity X-rays. In

this s heme, mole ular orbitals are essentially linear ombinations of atomi orbitals for

mole ular ore-hole states (the LCAO s heme is employed). The atomi orbitals are

al- ulatedwith the helpof the numeri algrid-based method. Insu h away, grid parameters

and trun ations hemes an beiterativelyadjusted. The s hemeis apableofreprodu ing

ground state,therefore, it an be appliedfor non-equilibriumdynami s.

In the HFS method mole ular orbitals (MO) and orbital energies

ǫ

i

are the result of solving a single-ele tronS hr

¨o

edinger equation:



−

1

₂

∇

2

+ V

ext

(r) + V

H

(r) + V

X

(r)



ψ

i

(r) = ǫ

i

ψ

i

(r).

(2.16)

Here,

V

ext

(r)

is the external nu lei potential,

V

H

(r)

is the Hartree potential whi h de-s ribesthe Coulombintera tion between the ele trons, and

V

X

(r)

represents theex hange intera tion, approximated by the Slater ex hange potentialfor zero temperature [43, 44℄:

V

X

(r) = −

3

2



3

π

ρ(r)



1

3

,

(2.17)

where

ρ(r)

is the ele troni density. The presen e of the Slater ex hange potential distin-guishestheHartree-Fo k-Slatermethodfromthe Hartree-Fo kmethod. Regardingthe

ex- hangepotentialatanitetemperature,dierentimplementationshavebeenproposed[44℄,

e.g., in [4547℄

2.5.3 Tight-binding model

Tight-binding model is a well established semi-empiri al method for the band stru ture

omputationinsolids. The name'tight-binding' impliesthat theele trons inthe modeled

solids are tightly bound to atoms and intera t only with the nearest neighbours among

all surrounding atoms. This denition already emphasizes the prin ipal dieren e of the

tight-bindingmodelfromthe Drude modelwith its free-ele tron approximation.

Theperiodi ityofatomsinthesolid( rystal) anbetakenintoa ountinseveralways,

e.g. byusingthededi atedBlo hele tronwavefun tions,whi hareagoodapproximation

for metals [38℄. Otherwise, ele trons an be des ribed as slowly moving parti les whi h

are 'tightlybound'toa deniteatom. Thus, the band stru ture of the rystal isbased on

the superposition of the lo alized wave fun tions for isolated atoms. This des ription is

The formal mathemati al basis of the method is a tight-binding Hamiltonian, whi h an be written asfollows [5,18℄:

H

TB

=

X

ijην

H

ijην

,

H

ijην

= ǫ

iη

δ

ij

δ

ην

+ t

ην

ij

(1 − δ

ij

).

(2.18)

Here

i

and

j

stand forthe atoms,

η

and

ν

stand forthe atomi orbitals,

t

ην

ij

isthe hopping matrix element, the oe ients

ǫ

iη

in lude the on-site energy of atoms. Theoreti ally, any number of orbitals an be in luded in the se ond term of the Hamiltonian. However,

in the approa h proposed by Slater and Koster

t

ην

ij

plays rather a role of a set of tting parameters for the band stru ture [48℄. It is then not a sum or integral over all orbitals.

Thus, oe ients

t

ην

ij

an be used for the interpolation between the energies obtained for denite

k

-points from the experiment or ab initio al ulations [18℄. For many materials, itis su ientto hoose justseveral orbitals fora urate des ription of theband stru ture

and other material properties, and an

ansatz

1

for the form of hoppingparameters, whi h

analyti al expression may vary. The a ura y of a TB model depends on the parti ular

hoi e of the set of the basis fun tions and the a ura y of their tting.

Usually, tight-binding parameters are adjusted to the ground state of the material.

TB is then e ient for band stru ture al ulations in the stati regime. However, TB

an be also extended for the ex ited system, whi h an evolve with hanging atomi

ge-ometry. This means that hopping parameters must be valid for dierent stru tures and

su h modi ation of TB is then named

transf erable

tight-binding model. The transfer-able tight-binding approa h is e ient in predi ting repulsion between the bands whi h

inuen es the size of the band gap [5℄. Apart of this, tight-bindingapproa h an des ribe

a urately enoughthe hemi albonding,densityof states,Fermienergy,and spe i

ele -tron properties.

1

The word

ansatz

an be translated from German as an assumption about the form of unknown fun tion, whi h is made in order to fa ilitate solution of anequation orother problem (OxfordEnglish

Constru tion of the model

After sorting out the basi physi al prin iples whi h are ru ial to des ribe FEL

ex ita-tion of solids and opti al photons propagation, here we present the onstru tion of the

model based on these prin iples. As the main goal of the work is the determination of

transientopti alproperties,rst, wewilldis ussthe methodologyfor al ulatingthe

om-plexdiele tri fun tion. WhileCDF is onne ted withthe materialband stru ture, atthe

se ond step we will review the XTANT model whi h allows to al ulate it in onne tion

withmole ulardynami s al ulations. Finally,wewillshowthepredi tions ofXTANT for

opti al oe ientsin systems under equilibrium.

3.1 Cal ulation of CDF in random phase approximation

Inthepump-probes hemeweuseopti alpulsesasprobes,andthisgivesusanopportunity

touseanumberofapproximations,be ausetheresponseofthematerialinthis ase anbe

redu edwithagooda ura ytothe ontributionofvalen eele tronsonly. Fortheanalysis

ofthe diele tri fun tionfromthe mi ros opi pointofview,wemay applyasemi- lassi al

pi ture,treatingele tronsquantum me hani allyandbytreatingthe externalele tri eld

lassi ally. The Hamiltonianfor asingle ele tron reads as[28℄:

H

0

=

p

2

2m

e

The Lorentz for e whi h governs the motion of the parti le in lassi al ele tromagneti

eld is:

F = (−e)[E + v × B],

(3.2)

where

v

isavelo ityoftheele tronand

B

isthemagneti eldve tor,andthe bothvalues are dependent on the ele tri eld

E

. It means that we an negle t the quadrati se ond term. The ele tron-radiationHamiltonianthen reads as [28℄:

H

1

(t) = +ex · E(t) = −Υ · E(t).

(3.3)

Here

Υ

= −ex

is a dipolemomentoperator. Su h an approximation is alled the ele tri dipole approximation as the intera tion Hamiltonian

H

1

orresponds to the Hamiltonian foradipoleinastati eld

E

0

. As theele tri eldisalsospa e-dependent, itisexpressed as inEq. (2.10):

E

= E

0

(ω)e

i(kx−ωt)

.

(3.4)

TheFermiwaveve torofthe rystallatti eofismu hlargerthanthewaveve torofopti al

photons. This implies that opti al irradiation has a negligible ee t on the momentum

transfer tothe latti e. Then we an expand the

e

ikx

fromEq. (2.10) as

e

ikx

_{≈ 1 + ik · x + ...}

(3.5)

and negle t all the

k

-dependent terms [28℄. Su h an approximation in ondensed matter physi s is alled the random phase approximation (RPA). For the pump-probe s heme,

we an restri t ourselves to the RPA whi h implies

q

= |k − k

′

_{| → 0}

, where

k

and

k

′

orrespond to the rystal momentum in the initial and the nal state after the opti al

transition[27℄.

The frequen y-dependent CDF an be al ulated by various DFT s hemes. Among

them, the methodologies based on, for example, the full-potential linearized augmented

plane-wave[49℄ andthe proje tor-augmentedwave methodology [50℄,were presented. The

a ura y of the DFT predi tions for the frequen y-dependent CDF is usually high.

How-ever, the high omputational osts make them inappropriate for tra ing the materials

The semi-empiri al methods su h as tight-binding model, based on a set of lo alized

wave fun tions, are also able to provide reliable results for non-equilibrium state. As we

onsider a single frequen y of the external eld, the random phase approximation an

be applied. Within the RPA [5154℄ diele tri fun tion an be expressed by a Lindhard

formula, based on the rst-order perturbation theory [27, 55,56℄:

ε

αβ

(ω) = δ

α,β

+

e

2

_¯h

2

m

e

2

Ω ǫ

0

X

ην

F

ην

E

2

ην

f

ν

− f

η

¯hw − E

ην

+ i γ

.

(3.6)

In this equation

Ω

is the volume of the simulation box;

E

ην

=

E

ν

− E

η

is the transition energy between two eigenstates

|ηi

and

|νi

,

f

η

and

f

ν

are the transient o upation num-bers of the orresponding states, whi h are normalized to 2 (taking intoa ount the spin

degenera y);

m

e

is the mass of a free ele tron;

e

is the ele tron harge;

¯h

is the Plan k onstant;and

ǫ

0

isthe va uumpermittivityinSI units. Parameter

γ

isaninverse ele tron relaxationtime. Inour al ulations,weuseavalue

γ = 1.5×10

13

s

−1

. The hoi eof

γ

does not ae t the results beyond the broadening of peaks in the CDF [57℄. Values of

γ

below

10

14

s

−1

lead to almost identi al peaks in the opti al oe ients.

F

ην

are the diagonal elements of the os illatorstrength matrix and are dened as:

F

ην

= | hη|ˆp|νi |

2

,

(3.7)

where

hη|ˆp|νi

is a momentum matrix element between the two eigenstates:

hη|ˆp|νi =

X

R

a

R

′

a

,σ,σ

′

B

σ η

(R

a

)P (R

a

, R

′

a

)B

σ

′

_ν

(R

′

_a

).

(3.8)

In Eq. (3.8)

R

a

denotes the oordinates of atoms, and

σ

labels the atomi orbitals.

B

σ η

and

B

σ

′

ν

are the orrespondingeigenve tors of the Hamiltonian.

Energylevelsando upationnumbersare obtainedfromthe al ulationofthematerial

band stru ture atea h time step and from the temperatureequation in a framework of a

semi-empiri al self- onsistent model, whi h we will dis uss in Se tion 3.3. On the other

hand, the momentum matrix elements an not be obtained if the modeldoes not ontain

ˆ

p =

m

e

i¯

h

[ˆ

r, ˆ

H]

. Thus, the momentum matrix elements an be dened, as in work by Trani et al.[56℄:

P (R

a

, R

′

a

) =

m

e

i¯h

[R

a

− R

′

a

]H(R

a

, R

′

a

),

(3.9) where

H(R

a

, R

′

a

)

is the Hamiltonianmatrix.

Averagevalueofthe CDF anbe al ulatedfromthediagonalelementsofthediele tri

fun tion tensor:

hεi =

1

₃

(ε

xx

+ ε

yy

+ ε

zz

).

(3.10)

The nal expressions for the real and imaginary part of CDF, whi h enter Eq. (2.6), an

be writtenas follows [27℄:

Re(ε) = 1 +

e

2

_¯h

2

m

e

2

Ω e

0

X

ην

(f

ν

− f

η

) F

ην

(¯hω − E

ην

)

E

2

ην

((¯hω − E

ην

)

2

+ γ

2

)

,

Im(ε) = −

γ e

2

_¯h

2

m

e

2

Ω e

0

X

ην

(f

ν

− f

η

) F

ην

E

2

ην

((¯hω − E

ην

)

2

+ γ

2

)

.

(3.11)

Ingeneral ase,the al ulationoftheele troni stru turerequiresanumberofdierent

k

-spa eintegralsoverthewholeBrillouinzoneoritsirredu iblepart,obtainedbyaveraging ontributionsfromall

k

-pointsonameshinthere ipro allatti e. HamiltonianinEq. (3.9) an be repla ed by aFourier transform:

H(k) =

X

R

e

i k R

H(R).

(3.12)

Therearedierents hemesexistingofsolvingsu hintegralswithdierent hoi eof'spe ial

points'. Oneof themostprominentand a urateisMonkhortst-Pa ks heme[58℄for ubi

latti es. Therein, the latti eisdened by threeprimitivetranslation ve tors

t

1

,

t

2

and

t

3

, while the asso iated re ipro al-spa e ve tors an be dened as:

b

1

=

2π

Ω

t

2

× t

3

,

b

2

=

2π

Ω

t

3

× t

1

,

b

3

=

2π

Ω

t

1

× t

2

.

(3.13)

Here

Ω

is the unit ell volume. For a parti ular hoi e of the grid points we dene a sequen e of numbers [58℄:

u

r

= (2r − q − 1)/2q, at r = 1, 2, ..., q,

(3.14)

where

q

denes the number of spe ial points in the set in one dimension. Thus

k

-ve tor an be dened as:

k

prs

= u

p

b

1

+ u

r

b

2

+ u

s

b

3

.

(3.15) Therefore,

q

3

points intotal an be found in the Brillouinzone.

3.2 CDF formalism for inelasti s attering ross se tions

The essential part of any numeri al treatment of XUV or X-ray irradiated system is an

a urate al ulationofthe total rossse tionsfor theele tron impa tionizationor

ex ita-tionof atoms. Thereare severalmethodsallowingforsu h al ulationsand usingdierent

ollision theories. The quality of the used wave fun tions is espe ially important for the

a ura yofthemodel. However, the abinitio al ulationsofthe wavefun tions,espe ially

forpolyatomi mole ules,ne essarytoobtaintheionization rossse tion,presentdi ulty.

Thus, experimentaldataandsemi-empiri altheoriesarewidelyusedforthe purpose. Very

often su h methods have a good a ura yfor high ex itation energies but fail todes ribe

the ross se tion properlyat lowerenergies.

One of the most known models isbased on atomi ross se tionsand is alled

Binary-En ounter-Bethe(BEB) model. It usesthe Mott rossse tion[59, 60℄. This modelisused

both for ross se tions in atoms and in mole ules, and it does not ontain any empiri al

parameters. The analyti formulafor the total impa tionization ross se tion per atomi

ormole ularorbital reads as follows:

σ

BEB

=

S

t + (u + 1)



Q ln t

2



1 − 1

t

2



+ (2 − Q)



1 −

1

_t

−

_{t + 1}

ln t



,

(3.16)

where

t = T /b

,

u = U/B

,

S = 4πa

2

0

N(R/B)

2

, a

0

=0.52918Å,and

R

=13.6057eV.Here

B

isthebindingenergy,

U

istheorbitalkineti energy,

N

isthe ele trono upation number,

AlthoughtheBEBmodelagreeswithmanyexperimentalionization rossse tionsfora

varietyofatomsandmole ulesevenatnearthresholdenergies,itmaygiveunreliableresults

for arbonbasedmaterialsasshown in[61℄. Insemi ondu tors theimpa tionizationrates

are strongly dependent on the band stru ture, and the ele tron s attering ross se tions

are al ulated with the usage of time-dependent perturbation expansions.

Aquitea uratemodelforboth highandlowimpa tenergies wasproposedin[62℄. An

empiri alformulaproposedthereforthemeanfreepathtakesintoa ounttheasymptoti s

atlowand high energies [62℄:

λ

ii

(E) =

√

E/



_{a(E − E}

th

)

b



+ [E − E

0

exp(−B/A)]/[A ln(E/E

0

) + B].

(3.17)

Here

E

isthe ele tron energy,

E

0

isthe tteddimensional oe ient,

E

th

is theionization threshold energy;

a

,

A

and

B

are material dependent onstants. The Eq.(3.17) was then tted to the known data for the impa t ionization mean free path for several elements

and ompounds. Out of that, the total ele tron impa t ionization ross se tions an be

obtained.

In our work,weimplementthe omplex diele tri fun tionformalism,whi h al ulates

the ross se tionfor impa t ionizationby an ele tron oranother hargedparti le,and the

mean free path of the parti le by using the known CDF spe tra. It is important to note

that the CDF we use here, is an experiment-based CDF t to the ground-state of the

material, and its role is dierent from the role of the transient CDF, that we dened in

Se tion3.1. Wearenot abletousethe transientCDFatea htimestepforthe al ulation

of the ross se tion be ause of the omputational and fun tional omplexities, as we will

demonstrate further. The CDF formalism for inelasti s attering ona system of strongly

bound ele trons is ombined with the Rit hie and Howie method [63℄. Within the Born

approximation, the ross se tion for the s attering ona su h system reads asin [64, 65℄:

d

2

_σ

dωdq

= AS(ω, q), A =

m

2

e

4π

2

_¯h

5

k

k

0

W (q),

S(ω, q) =

X

n

0

p

n

0

X

n













"

_N

X

j=1

exp(iqr

j

)

#

n

n

0













2

δ(ω + (E

n

0

− E

n

)/¯h).

(3.18)

In this equation

σ

is the total ross se tion for ele tron s attering,

A

is the probabilityof s atteringonanele tron,

S(ω, q)

isthedynami stru turefa torofele trons. Free-ele tron mass is denoted

m

e

,

k

0

and

k

are the initialand nal momentum ve tors of the ele tron,

r

j

isthepositionve tor ofea hof

N

parti leswithinthes attering system, theinitialand nalstatesofthissystem aresignedas

n

0

and

n

orrespondingly,and

E

n

0

and

E

n

are their respe tive energies. Finally,

p

n

0

is the statisti alweight of the initialstate.

The link between the stru ture fa tor

S(ω, q)

and the ma ros opi diele tri fun tion

ε(ω, q)

isprovided by the u tuation-dissipation theorem [64,66℄:

S(ω, q) =

q

2

4π

2

_n

e

Im



1

ε(ω, q)



,

(3.19)

where

n

e

is the ele tron density. The ross se tionfor s attering ona system of ele trons within the Born approximation an be writtenas [64, 66℄:

d

2

_σ

dωdq

=

2(Z

e

e)

2

π¯h

2

ν

2

1

q

Im



−1

ε(ω, q)



.

(3.20)

Z

e

is the ee tive harge of a parti le in the ele troni system, while

e

is the ele tron harge,

ν

isthevelo ity ofthe in identparti le. The meanfreepathis onne ted withthe ross se tion as

λ

=

1

n

e

σ

.

The algorithm,invented by Rit hieand Howie, onstru ts the approximateddiele tri

fun tion from the experimentallyavailable data onphoton s attering insolids. The CDF

parametrizationinthis methodisthe Drude-typetfor anensembleofatomi os illators,

whi h ontain three sets of adjustable oe ients [63, 64℄:

Im



−1

ε(ω, q = 0)



=

X

i

A

i

γ

i

¯hω

(¯h

2

ω

2

_{− E}

2

0i

)

2

+ (γ

i

¯hω)

2

.

(3.21)

The oe ient

E

0i

is the hara teristi energy of the os illator

i

, the oe ient

A

i

is the fra tion of ele trons that have the spe i energy

E

0i

, and

γ

i

is the

i

th energy damping oe ient. The summation isperformedover allos illators

i

.

The dipole approximation,

q = 0

, is assumed in the model, implying the absen e of the momentum transferfor masslessphotons. Bytting themodelfun tionsfrom theEq.

opti alproperties anberepresented eitherby

n

and

k

refra tionindi es(typi allyfor low photon energies below 30 eV) or by the attenuation oe ients for higher energies. The

imaginarypart of inverse CDF looks like[64, 67℄:

Im



−1

ε(ω, q = 0)



=

c

λ

ph

ω

,

(3.22)

where

λ

ph

is the mean free path till absorption, and

c

is the speed of light in va uum. During the implementation,the quality of the tting should be he ked with the spe i

sum rules. First,

ps

-sum-rule(perfe t s reening sum rule)[63, 64,67℄:

P

eff

=

2

π

¯

hω

Z

max

0

Im



−1

ε(ω, q = 0)



d(¯hω)

¯hω

.

(3.23)

If

¯hω

goes to innity, the value of

P

eff

must tend to 1. The se ond rule is alled the

f

-sum-rule[63, 64, 67℄:

Z

eff

=

2

πΩ

2

p

ω

Z

max

0

Im



−1

ε(ω, q = 0)



ωdω,

(3.24) where

Ω

p

2

_{= (4πn}

m

e

2

/m

e

)

1/2

is the plasma frequen y and

n

m

is the density of mole ules inthe solid under onsideration. If

¯hω

max

tends to innity,the

Z

eff

must tend tothe total number of ele trons per mole ule of a target. Finally, in order to extend the

approxima-tion from the dipole limit

q = 0

to massive parti les (ele trons), one should repla e the oe ients

E

0i

by the expressions

(E

0i

+ (¯

hq)

2

_/(2m

e

))

[63,64℄.

The ross se tion al ulations with Rit hie and Howie method are based on the rst

Born approximation, so it should be implemented only for high-energy ele trons. The

es-timations made in [68℄ show that the lower limitfor ele tron energies, whi h an provide

reliableresults,is

∼

10eV.Forlowerele tron energies, quantum ee ts ofthe band stru -turemustbetaken intoa ount. Thespe i parametersenteringintothemodelfun tions