Rate theory of acceleration of defect annealing driven by discrete breathers

18 Pages·2015·5.08 MB·English

Rate theory of acceleration of defect annealing
driven by discrete breathers
VladimirI.Dubinko,JuanF.R.Archilla,SergeyV.Dmitriev,and
VladimirHizhnyakov
1 Introduction............................................................. 2
2 Discretebreathersinmetalsandsemiconductors .............................. 2
3 DBexcitationunderthermalequilibriumandexternaldriving ................... 9
4 AmplificationofSb-vacancyannealingrateingermaniumbyDBs ............... 12
5 Summary ............................................................... 16
References ..................................................................... 17
AbstractNovelmechanismsofdefectannealinginsolidsarediscussed,whichare
basedonthelargeamplitudeanharmoniclatticevibrations,a.k.a.intrinsiclocalized
modes or discrete breathers (DBs). A model for amplification of defect annealing
rate in Ge by low energy plasma-generated DBs is proposed, in which, based on
recentatomisticmodelling,itisassumedthatDBscanexciteatomsarounddefects
rather strongly, giving them energy ≫k T for (cid:24)100 oscillation periods. This is
B
shown to result in the amplification of the annealing rates proportional to the DB
flux, i.e. to the flux of ions (or energetic atoms) impinging at the Ge surface from
inductivelycoupledplasma(ICP).
Key words: Anharmonic lattice vibrations, discrete breathers, intrinsic localized
modes,defectannealing
V.I.Dubinko
NSC Kharkov Institute of Physics and Technology, Akademicheskya Str. 1, Kharkov 61108,
Ukraine,e-mail:[email protected]
J.F.R.Archilla
GroupofNonlinearPhysics,UniversidaddeSevilla,ETSIInforma´tica,Avda.ReinaMercedess/n
41011,Seville,Spain,e-mail:[email protected]
S.V.Dmitriev
InstituteforMetalsSuperplasticityProblems,RAS,KhalturinStr.39,Ufa450001,Russia
National Research Tomsk State University, Lenin Prospekt 36, Tomsk 634050, Russia, e-mail:
[email protected]
V.Hizhnyakov
Institute of Physics, University of Tartu, Riia 142, EE-51014 Tartu, Estonia, e-mail:
hizh@fi.tartu.ee
1
2 V.I.Dubinko,J.F.R.Archilla,S.V.DmitrievandV.Hizhnyakov
1 Introduction
A defect lying in the band gap with energy > 0:1 eV from either band edge is
termeddeep.AsknownfromthestudiesofpropertiesofdefectsinGe[1,2,4,5,30],
Ar ions arriving at a semiconductor surface with very low energy (2 - 8 eV) are
annihilating defects deep inside the semiconductor. Several different defects were
removedormodifiedinSb-dopedgermanium,ofwhichtheE-centerhasthehigh-
estconcentration,asdescribedindetailsinRef.[1,2].Novelmechanismsofdefect
annealinginsolidsarediscussedinthiswork,whicharebasedonthelargeampli-
tude anharmonic lattice vibrations, a.k.a. intrinsic localized modes (ILMs) or dis-
cretebreathers(DBs).Thearticleisorganizedasfollows.InSect.2,ashortreview
onDBpropertiesinmetalsandsemiconductorsispresentedbasedontheresultsof
molecular dynamics (MD) simulations using realistic many-body interatomic po-
tentials.InSect.3,aratetheoryofDBexcitationunderthermalheatingandunder
non-equilibriumgasloadingconditionsisdeveloped.InSect.4,amodelforampli-
fication of defect annealing rate in Ge by plasma-generated DBs is proposed and
comparedwithexperimentaldata.TheresultsaresummarizedinSect.5.
2 Discretebreathersinmetalsandsemiconductors
DBsarespatiallylocalizedlarge-amplitudevibrationalmodesinlatticesthatexhibit
stronganharmonicity[14,20,33,35].Theyhavebeenidentifiedasexactsolutionsto
a number of model nonlinear systems possessing translational symmetry [14] and
successfullyobservedexperimentallyinvariousphysicalsystems[14,29].Presently
theinterestofresearchershasshiftedtothestudyoftheroleofDBsinsolidstate
physicsandtheirimpactonthephysicalpropertiesofmaterials[9,10,12,13,29,36].
UntilrecentlytheevidencefortheDBexistenceprovidedbydirectatomisticsim-
ulations, e.g. MD, was restricted mainly to one and two-dimensional networks of
coupled nonlinear oscillators employing oversimplified pairwise inter-particle po-
tentials [14,20,33]. Studies of the DBs in three-dimensional systems by means of
MD simulations using realistic interatomic potentials include ionic crystals with
NaCl structure [21,25], graphene [6,23,27], graphane [28], semiconductors [37],
pure metals [15,18,32,36], and ordered alloys [31]. For the first time the density
functional theory (DFT) was applied to the study of DBs, using graphane as an
example[7].
DBs have very long lifetime because their frequencies lie outside the phonon
band. Monatomic crystals like pure metals and semiconductors such as Si and Ge
donotpossessgapsinthephononspectrum,whilecrystalswithcomplexstructure
oftenhavesuchgaps,forexample,diatomicalkalihalidecrystalsandorderedalloys
withalargedifferenceintheatomicmassofthecomponents.Forthecrystalspos-
sessingagapinthephononspectrumtheso-calledgapDBswithfrequencieswithin
thegapcanbeexcited.Thiscasewillnotbediscussedhereandinthefollowingwe
focusontheDBshavingfrequenciesabovethephononband.
Ratetheoryofaccelerationofdefectannealingdrivenbydiscretebreathers 3
2.1 Metals
IntheworkbyKiselevetal.[24]ithasbeendemonstratedthat1Dchainofparticles
interactingwiththenearestneighborsviaclassicalpairwisepotentialssuchasToda,
Lennadrd-Jones or Morse cannot support DBs with frequencies above the phonon
band.TheywereabletoexciteonlygapDBswithfrequencieslyingwithinthegap
ofthephononspectrumbyconsideringdiatomicchains.Inlinewiththeresultsof
this work, it was accepted for a long time that the softening of atomic bonds with
increasingvibrationalamplitudeisageneralpropertyofcrystals,whichmeansthat
theoscillationfrequencydecreaseswithincreasingamplitude.ThereforeDBswith
frequenciesabovethetopphononfrequencywereunexpected.
However, in 2011, Haas et al. [15] have demonstrated by MD simulations us-
ingrealisticmany-bodyinteratomicpotentialsthatDBswithfrequenciesabovethe
phonon spectrum can be excited in fcc Ni as well as in bcc Nb and Fe [15,18].
SimilarresultswereobtainedforbccFe,V,andW[32].
The point is that the realistic interatomic potentials, including Lennard-Jones
and Morse, have an inflection point meaning that they are composed of the hard
core and the soft tail. This is typical for interatomic bonds of any complexity, in-
cluding many-body potentials. Physically the soft tail is due to the interaction of
theouterelectronshellsoftheatoms,whilethehardcoreoriginatesfromthestrong
repulsiveforcesbetweennucleiandalsofromthePauliexclusionprincipleforinner
electrons(fermions)thatcannotoccupythesamequantumstatesimultaneously.It
is thus important which part of the interatomic potential (hard or soft) contributes
moretothedynamicsofthesystem.Asitwasshownin[24],theasymmetryofthe
interatomicpotentialsresultsinthethermalexpansioneffectwhenlargervibrational
amplitudes,atzeropressure,causethelargerequilibriuminteratomicdistanceand
hence, a larger contribution from the soft tail. If thermal expansion is suppressed
somehow,thenthehardcoremanifestsitself.Todemonstratethisletusconsiderthe
Morse chain of unit mass particles whose dynamics is described by the following
equationsofmotion
u¨n=U′(h+un+1(cid:0)un)(cid:0)U′(h+un(cid:0)un(cid:0)1); (1)
whereu (t)isthedisplacementofthenthparticlefromthelatticeposition,histhe
n
latticespacing,
U(r)=D(e(cid:0)2a(r(cid:0)rm)(cid:0)2e(cid:0)a(r(cid:0)rm)); (2)
istheMorsepotential,whereristhedistancebetweentwoatoms,D,a,r arethe
m
potentialparameters.ThefunctionU(r)hasaminimumatr=r ,thedepthofthe
m
potential(thebindingenergy)isequaltoDandadefinesthestiffnessofthebond.
WetakeD=1,r =1anda=5.Fortheconsideredcaseofthenearest-neighbor
m
interactionstheequilibriuminteratomicdistanceish=r =1.
m
InframeofthemodelgiveninEqs.(1),(2)westudythedynamicsofthestag-
geredmodeexcitedwiththeuseofthefollowinginitialconditions
u (0)=Acos(pn)=((cid:0)1)nA; u˙ (0)=0; (3)
n n
4 V.I.Dubinko,J.F.R.Archilla,S.V.DmitrievandV.Hizhnyakov
inthechainofNparticles(Nisanevennumber)subjectedtotheperiodicboundary
conditions, u (t)=u (t). Our aim is to find the frequency of the mode as the
n n+N
functionofthemodeamplitudeAforthetwocases.Firstlythechainisallowedto
expand,andforgivenA>0theinteratomicdistanceh>1issuchthatthepressure
p=0.Inthesecondcasethethermalexpansionissuppressedbyfixingh=1forany
A.Inthiscase,ofcourse,forA>0onehas p>0.Theresultsforthetwocasesare
showninFig.1(a)and(b),respectively.In(a)thefrequencyofthemodedecreases
withA,whilein(b)theoppositetakesplace.
In the numerical experiments by Haas et al. [15] is was found that the DBs in
puremetalsareextendedalongaclose-packedatomicrow.Theatomssurrounding
the atomic row where DB is excited create the effective periodic on-site potential
thatsuppressesthethermalexpansionoftherowandthatiswhytheDBfrequency
increaseswithincreasingamplitude.Theon-sitepotentialwasnotintroducedinthe
1DmodelbyKiselevetal.[24]and,naturally,thermalexpansiondidnotallowfor
theexistenceofDBswithfrequenciesabovethephononband.
Notably,theexcitationenergyofDBsinmetalscanberelativelysmall(fractions
of eV) as compared to the formation energy of a stable Frenkel pair (several eV).
Moreover,ithasbeenshownthatDBsinpuremetalsarehighlymobileandhence
theycanefficientlytransferenergyandmomentumoverlargedistancesalongclose-
packed crystallographic directions [18,32,36]. Recently, a theoretical background
hasbeenproposedtoascribetheinteractionofmovingDBs(a.k.a’quodons’-quasi-
Fig.1 Solidlinesshowfrequencyofthestaggeredmode(leftordinate)asthefunctionofamplitude
forthecaseof(a) p=0and(b)h=1.Dashedlinesshow(a)hand(b) p(rightordinate)asthe
functionsofA.Theresultsforthe1DMorselattice(1),(2)withtheinitialconditions(3).
Ratetheoryofaccelerationofdefectannealingdrivenbydiscretebreathers 5
particles propagating along close-packed crystallographic directions) with defects
inmetalstoexplaintheanomalouslyacceleratedchemicalreactionsinmetalssub-
jectedtoirradiation.RussellandEilbeck[34]havepresentedexperimentalevidence
for the existence of quodons that propagate great distances in atomic-chain direc-
tions in crystals of muscovite, an insulating solid with a layered crystal structure.
Specifically,whenacrystalofmuscovitewasbombardedwithalpha-particlesata
givenpointat300K,atomswereejectedfromremotepointsonanotherfaceofthe
crystal, lying in atomic chain directions at more than 107 unit cells distance from
the site of bombardment. Irradiation may cause continuous generation of DBs in-
sidematerialsduetoexternallatticeexcitation,thus’pumping’amaterialwithDB
gas[10,12].
Inordertounderstandbetterthestructureandpropertiesofstandingandmoving
DBs,considerthewaysoftheirexternalexcitationinFebyMDsimulations[36].
AstandingDBcanbeexcitedbyapplyingtheinitialdisplacementstothetwoadja-
centatomsalongtheclose-packed[111]directionwiththeoppositesignstoinitiate
their anti-phase oscillations, as shown in Fig. 2(a). The initial displacements (cid:6)d
0
determine the DB amplitude, frequency and, ultimately, its lifetime. DBs can be
excitedinafrequencyband(1.0-1.4)(cid:2)1013 HzjustabovetheDebyefrequencyof
bccFe,andDBfrequencygrowswithincreasingamplitudeasexpectedforthehard
type anharmonicity due to the major contribution from the hard core of the inter-
atomicpotential.Initialdisplacementslargerthanjd j=0:45A˚ generateachainof
0
focusons, while displacements smaller than jd j=0:27 A˚ do not provide enough
0
potential energy for the system to initiate a stable DB and the atomic oscillations
decayquicklybylosingitsenergytophonons.ThemoststableDBscansurviveup
to400oscillations,asshowninFig.2(b),andultimatelydecayinastepwisequan-
tumnaturebygeneratingburstsofphonons,ashasbeenpredictedbyHizhnyakov
asearlyasin1996[17].
A moving DB can be excited by introducing certain asymmetry into the initial
conditions.Particularly,thetranslationalkineticenergyE canbegiventothetwo
tr
central atoms of DB in the same direction along [111] atomic row. DB velocity
ranges from 0.1 to 0.5 of the velocity of sound, while travel distances range from
several dozens to several hundreds of the atomic spaces, depending on d and E
0 tr
[32,36]. Figure 3(a) shows a DB passing the two neighboring atoms with indices
3415and3416.InthemovingDBthetwocentralatomspulsatenotexactlyinanti-
phasebutwithaphaseshift.Inabout1ps((cid:24)10oscillations)theoscillationsofthese
twoatomsceasebuttheyareresumedatthesubsequentatomsalong[111]atomic
row.Inthisway,theDBmovesataspeedof2.14km/s,i.e.aboutthehalfspeedof
soundinbccFe.ThetranslationalkineticenergyoftheDBisabout0.54eV,which
is shared mainly among two core atoms, giving 0.27 eV per atom, which is close
to the initial kinetic energy of E =0:3 eV given to the atoms to initiate the DB
tr
translational motion. The deviation of the potential energy of the atoms from the
ground state during the passage of the DB is presented in Fig. 3(b). The maximal
deviation of energy is of the order of 1 eV. Thus, a moving DB can be viewed as
anatom-sizelocalisedexcitationwithlocaltemperatureabove1000Kpropagating
alongthecrystalatasubsonicspeed.
6 V.I.Dubinko,J.F.R.Archilla,S.V.DmitrievandV.Hizhnyakov
Fig.2 Oscillationofxcoordinateoftwoneighbouringatoms,2480and2479,ina[111]rowin
FeinastandingDBexcitedwithd =0:325A˚ [36].(a)InitialstageofDBevolution;(b)total
0
lifespanofDBshowingastepwisequantumnatureofitsdecay
2.2 Semiconductors
Similartometals,semiconductorspossessnogapinphononspectrumandthusDBs
mayexistonlyiftheirfrequencyispositionedabovethephononspectrum[15,37].
Such high-frequency DBs may be realized in semiconductors due to the screen-
ing of the short-range covalent interaction by the conducting electrons. Voulgar-
akis et al. [37] investigated numerically existence and dynamical properties of
DBs in crystalline silicon through the use of the Tersoff interatomic potential.
They found a band of DBs with lifetime of at least 60 ps in the spectral region
Ratetheoryofaccelerationofdefectannealingdrivenbydiscretebreathers 7
Fig.3 (a)Oscillationofxcoordinateoftwoneighbouringatoms,3415and3416ina[111]rowin
FeduringthepassageofamovingDB(d0=0:4A˚,Etr=0:3eV);(b)deviationofthepotential
energyoftheatomsfromthegroundstateduringthepassageofDB
(1:643(cid:0)1:733)(cid:2)1013 Hz, located just above the upper edge of the phonon band
calculated at 1:607(cid:2)1013 Hz. The localized modes extend to more than second
neighbors and involve pair central-atom compressions in the range from 6.1% to
8.6%ofthecovalentbondlengthperatom.Finitetemperaturesimulationsshowed
thattheyremainrobusttoroomtemperaturesorhigherwithatypicallifetimeequal
to 6 ps. Figure 4 shows DB generated in silicon modeled by the Tersoff poten-
tial[37].ItcanbeseenthattheDBisverypersistentandlocalized:itsvibrational
energyismainlyconcentratedinthebondbetweentwoneighboringatomsoscillat-
inginanti-phasemode.
Similar to silicon, germanium has a diamond crystal structure and readily pro-
ducesDBs[19],asdemonstratedinFig.5.AsinSi,theDB’senergyinGeiscon-
centrated in the central bond between two atoms oscillating in anti-phase mode.
8 V.I.Dubinko,J.F.R.Archilla,S.V.DmitrievandV.Hizhnyakov
Fig.4 (a)DBgenerationinsiliconmodeledbyTersoffpotentials.TheDBfrequencyis1:733(cid:2)
1013 Hz,whilevectors(magnifiedforvisualizationpurposes)denoteatomicdisplacementsfrom
equilibrium;onlyfirst(gray,redonline)andsecond(white)neighborstothecentral(black,blue
online)twobreatheratomsareincluded.Thedisplacementofthetwocentralbreatheratomsis
0.18 A˚. (b) Time evolutionof the silicon DB after 998 breather periods. The absolute value of
the displacements from equilibrium along the direction of motion of each atom is plotted. The
coordinatedoscillationsofcentral(solid),first(dotted),andsecond(dashed)neighboratomsare
indicated.ReproducedwithpermissionfromVoulgarakis,N.,Hadjisavvas,G.,Kelires,P.,Tsironis,
G.:ComputationalinvestigationofintrinsiclocalizationincrystallineSi.Phys.Rev.B69,113,201
(2004).Copyright(2004)AmericanPhysicalSociety
This means that potential barriers for chemical reactions in the vicinity of an DB
maybesubjectedtopersistentperiodicoscillations,whichhasbeenshowntoresult
inastrongamplificationofthereactionrates[13].Inthenextsectionweconsider
thewaysofDBexcitationinthermalequilibriumandunderexternaldriving.
Ratetheoryofaccelerationofdefectannealingdrivenbydiscretebreathers 9
Fig.5 DBgeneratedingermaniummodeledbytheTersoffpotential.Displacementofoneofthe
twocentralatomsisshownwithasolidlineandofthefirstneighborbydashed(along[111]axis)
anddotted(perpendicularto[111]axis)lines.SeeRef.[19]
(cid:1) (cid:1)(cid:2)(cid:3)(cid:4)
(cid:2)
(cid:3)
(cid:10) (cid:5)
(cid:9)(cid:3)(cid:8) (cid:3)(cid:8) (cid:6)(cid:7)
Fig.6 (a)Sketchofthedouble-wellpotentiallandscapewithminimalocatedat(cid:6)xm.Theseare
stablestatesbeforeandafterreaction,separatedbyapotential”barrier”withtheheightchang-
ingperiodicallyorstochasticallywithintheV band.(b)Amplificationfactor,I0(V=kBT),forthe
averageescaperateofathermalizedBrownianparticlefromaperiodicallymodulatedpotential
barrieratdifferenttemperaturesandmodulationamplitudesV.Reproducedwithpermissionfrom
V.I.Dubinko,P.A.SelyshchevandJ.F.R.Archilla:Reaction-ratetheorywithaccountofthecrystal
anharmonicity Phys.Rev.E83041124(2011).Copyright(2011)AmericanPhysicalSociety.
3 DBexcitationunderthermalequilibriumandexternaldriving
Inthissection,fortheconvenienceofthereader,werepeatthemainpointsofthe
chemical reaction rate theory that takes into account the effect of DBs, following
theearlierworks[3,11,13].
TherateequationfortheconcentrationofDBswithenergyE,C (E;t)canbe
DB
writtenasfollows[13]
10 V.I.Dubinko,J.F.R.Archilla,S.V.DmitrievandV.Hizhnyakov
¶C (E;t) C (E;t)
DB =K (E)(cid:0) DB ; (4)
¶t DB t (E)
DB
where K (E) is the rate of creation of DBs with energy E >E and t (E) is
DB min DB
theDBlifetime.Ithasanobvioussteady-statesolution(¶C (E;t)=¶t=0):
DB
C (E)=K (E)t (E): (5)
DB DB DB
Inthefollowingsectionswewillconsiderthebreatherformationbythermalactiva-
tionandthenextendthemodeltonon-equilibriumsystemswithexternaldriving.
3.1 Thermalactivation
The exponential dependence of the concentration of high-energy light atoms on
temperature in the MD simulations [22] gives evidence in favor of their thermal
activationatarategivenbyatypicalArrheniuslaw[33]
( )
E
K (E;T)=w exp (cid:0) ; (6)
DB DB
k T
B
wherew istheattemptfrequencythatshouldbeclosetotheDBfrequency.The
DB
breatherlifetimehasbeenproposedin[33]tobedeterminedbyaphenomenological
law based on fairly general principles: (i) DBs in two and three dimensions have
a minimum energy E , (ii) The lifetime of a breather grows with its energy as
min
t =t0 (E=E (cid:0)1)z, with z and t0 being constants, whence it follows that
DB DB min DB
underthermalequilibrium,theDBenergydistributionfunctionC (E;T)andthe
DB
meannumberofbreatherspersiten (T)aregivenby
DB
( )
E
C (E;T)=w t exp (cid:0) ; (7)
DB DB DB
k T
B
E∫max exp((cid:0)E =k T) y∫max
n (T)= C (E;T)dE=w t0 min B yzexp((cid:0)y)dy; (8)
DB DB DB DB (E =k T)z+1
min B
Emin 0
∫
with y =(E (cid:0)E )=k T. Noting that G(z+1;x)= xyzexp((cid:0)y)dy is the
max max min B 0
secondincompletegammafunction,Eq.(8)canbewrittenas[13]:
( )
exp((cid:0)E =k T) E (cid:0)E
n (T)=w t0 min B G z+1; max min : (9)
DB DB DB (E =k T)z+1 k T
min B B
ItcanbeseenthatthemeanDBenergyishigherthantheaveragedenergydensity
(ortemperature):