a r X i v :c o n d -m a t /980
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404v 1 [c o n d -m a t .m t r l -s c i ] 30 S e p 1998
Atomistic modelling of large-scale metal film growth fronts
U.Hann,1P.Vogl,1and Vincenzo Fiorentini 1,2
1)Physik-Department and Walter Schottky Institut,Technische Universit¨a t M¨u nchen,D-85748Garching,Germany
2)Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica,Universit`a di Cagliari,Italy
(February 7,2008)
We prent simulations of metallization morphologies under ionized sputter deposition condi-tions,obtained by a new theoretical approach.By means of molecular dynamics simulations using a carefully designed interaction potential,we analyze the surface adsorption,reflection,and etching reactions taking place during Al physical vapor deposition,and calculate their relative probability.The probabilities are then employed in a feature-scale cellular-automaton simulator,which pro-duces calculated film morphologies in excellent agreement with scanning-electron-microscopy data on ionized sputter deposition.PACS:81.10.Aj,81.15.-z,68.55.-a
In this Letter we prent results of a hierachy of theo-retical models developed to describe the growth of metal thin films.Atomic-scale molecular dynamics (MD)and a feature-scale cellular-automaton simulator are combined to yield realistic simulations of film growth during physi-cal vapor metallization of contact vias typical of micon-ductor device technology.The MD simulations,account-ing in full for the microscopic many-atom dynamics,are ud to predict the reaction rates of the process rele-vant in physical vapor deposition;the cellular automaton simulator incorporates the reaction rates thus obtained,enabling us to predict and understand the topography of µm-scale film fronts,and their relationship to substrate geometry,incident beam energy,and angular beam dis-tribution.Our approach yields a consistent and compu-tationally efficien
t scheme to predict the topography of metal films on arbitrarily-shaped substrates.A compar-ison with scanning-electron-microscopy data on sputter-deposited Al-covered trenches demonstrates an excellent level of agreement between theory and experiment.Our theoretical approach proceeds in three steps,namely (a)a classical interatomic interaction potential for Al is developed,(b )reaction rates for Al atoms in-cident on Al surfaces are calculated therewith in a MD simulation,and (c )a cellular automaton is developed and employed to simulate µm-scale film fronts,using the re-action rates extracted from MD.In the following,we an-alyze this procedure and prent the relevant results for each step.
Interatomic potential –Previously developed classi-cal interatomic potentials 1–6for Al-Al interactions have mostly focud on bulk and molecular properties.The Ercolessi-Adams potential 6is a partial exception since,besides reproducing quantitatively the bulk and elastic properties of Al,it also yields interlayer relaxations 6in good agreement with experiment.7Using this potential,we obtain surface energies in excellent agreement with ab initio calculations.8We thus cho this model as a start-ing point to develop a new classical many body potential for Al.Our own potential is carefully designed to repro-duce the properties of Al aggregates in a wide range of
bonding configurations (from bulk Al to Al surfaces and Al molecules),with special regard to surface p
roperties.Its functionalities significantly extend tho of previous embedded atom models,while maintaining a high level of accuracy in reproducing Al bulk properties and surface formation energies.6First ,we introduced an exponential repulsive pair potential 9,10to account for the short-range interaction of Al atoms with kinetic energies exceeding 10eV;this is a key requirement,as the kinetic energies of de-posited atoms reach over 150eV during ionized physical vapor deposition.Second ,the embedding function has been readjusted to reproduce obrved properties of low-density Al structures;after the changes,we obtained a significantly improved agreement of veral reference quantities 11in comparison to ab inito results 12,13and/or experiment.14Third ,we introduced a 5th-order polyno-mial cutofffunction,smoothly cutting offthe interaction range of our potential at an interatomic distance of 5.56˚A ,slightly larger than third-nearest neighbor distance in bulk Al.
As a further test of our potential,we calculated the barriers for homodiffusion on the low-index faces of fcc Al.Specifically,we considered hopping on Al (110)along and orthogonal to the [1
Al (111)hopping 0.040.04a Al (100)hopping 0.600.68a ,0.65b
Al (100)exchange 0.500.35a Al (110)⊥hopping 1.13 1.06a Al
(110) hopping 0.30
0.60a
a
Reference 12;b Reference 13
Reaction rates from molecular dynamics–In the c-ond step of our approach,reaction probabilities were cal-culated in classical molecular dynamics simulations us-ing our Al interaction potential.In particular we deter-mined,as a function of the energy and off-normal angle of incident Al atoms,the probability of three process: adsorption,reflection,and etching(in the latter,the in-coming atom’s impact on the surface caus the kick-out of one or more substrate atoms).Supercells containing 1320atoms arranged in10atomic layers are employed; cell dimensions are chon so as to avoid artifacts of the in-plane periodicity.The starting configuration is chon to be a(111)surface,the one Al surface with the lowest formation energy.All atomic coordinates are allowed to evolve dynamically,except tho of the two bottom lay-ers of the supercell.The surface temperature is t at
450K(i.e about1/2of the melting temperature,and20 %larger than the bulk Debye temperature).
We start our simulations with the incident Al atom placed outside the interaction range of the surface.Its initial kinetic energy is t in the range of25to125eV, and its starting angle offthe surface normal in the range 0◦to60◦,which corresponds to typical ionized physical vapor deposition conditions.The trajectories of the inci-dent atom,and of any other atom which may be etched away from the surface upon impact,are then monitored until either a certain time span has elapd,or the out-coming atoms(in the ca of reflection or etching)have traveled a distance of10˚A away from the surface.Ana-lyzing200trajectories per incident energy and angle,we collected a statistically significant sample of well-defined adsorption,reflection,and etching events.The relative probability of the corresponding process is calculated as the ratio of the number of events of each kind to the to-tal number.The typical statistical error in the reaction probabilities thus determined is below5%.
The behavior of reflection,adsorption,and etch rates
as a function of off-normal angle are summarized in Fig. 1for two reprentative incident kinetic energies,namely 25eV in panel(a),and100eV in panel(b).At low en-ergies(panel(a))the adsorption probability(thick solid line,solid squares)decreas from near unity for angles below20◦to approximately1/2for angles in the range of60◦.This decrea is compensated by a corresponding increa in the reflection probability(dashed line,solid triangles).At low energy,etch process(thin solid line, solid circles)are negligible at all angles.Increasing the incident kinetic energy to100eV,wefind considerable changes in the relative reaction probabilities.Even at small angles,the reflection probability is non-vanishing; the adsorption probability is correspondingly reduced, and is now below0.7at all angles.More interestingly, the etching probability is always non-zero,it reaches a maximum of0.4at50◦,then decreas as near-grazing angles are approached.For small deviations from normal incidence,the etch rate initially rais,since the proba-bility of a surface atom to gain mo
mentum directed away from the surface increas when the incoming atom ar-rives at an oblique angle at the surface.At large angles of incidence the etch rate drops becau of the competing specular reflection events.
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FIG.1.Reaction probabilities for Al atoms impinging on Al(111)with a kinetic energy of25eV(panel(a))and100 eV(panel(b))as a function of the off-normal angle.The process considered are adsorption(thick solid line,solid squares),reflection(dashed line,solid triangles),and etching (thin solid line,solid circles).
By adjusting the bias voltage between sputter source and deposition target during ionized physical vapor de-position,both the energy regimes just discusd are ex-perimentally accessible.It is expected that they lead to rather different trench topographies,which we simulate with a cellular-automaton technique.
Feature-scale cellular-automaton simulator–In the final step of our approach,we have developed a two-dimensional cellular automaton to model the growing film front on aµm scale.The automaton accounts for the effects offlux shadowing,adsorption,reflection,etching, and surface diffusion.The sim
ulated structure is repre-nted in cross ction by a two-dimensional grid.Each grid cell reprents an Al atom,and is assumed to have a physical length of2.5˚A(the effective atomic diameter in Al bulk):thus,for example,a1-µ–wide structure will be described by4000grid cells across.
The atoms are rially and independently emitted from the sputter source far above the surface according to a pre-determined angular and energetic distribution,and move on a straight trajectory,determined also by the
applied source-target bias,until they strike the growing film front.Interactions in the gas pha are neglected in view of the the low pressure (typically a few tenths of mTorr)and the resulting long mean free path typi-cal of sputter deposition.Spontaneous desorption is also negligible in all the conditions considered.
The impact angle and energy of the atoms hitting the
surface determine which process is activated upon impact.15This may be any one of the three (adsorption,reflection,etching)who rates have been previously cal-culated via MD simulations.If the atoms are reflected,or an etch process takes place,the path of the corresponding atoms is further traced until they hit the film surface for a cond time.The three basic process can then take place
again,and so forth.Finally,the atoms get either adsorbed,or escape back into the gas pha.
Adsorption prent some additional complications.The local diffusion and accomodation mechanism upon adsorption is a key ingredient in deposition models.A reasonable assumption is that incoming atoms are ac-comodated at a (local)minimum energy site within one diffusion length from the landing site,the diffusion length being determined by the surface temperature and mor-phology,and the deposition rate.In the cas of inter-est here,the problem of determining an effective diffu-sion length for the adsorbed atoms is quite formidable for veral reasons.On the one hand,one expects low effective diffusivity due to the very high experimental growth rate (not well controlled,but in the order of 0.5µm/minute,or roughly 40ML/c 16,17),and also becau the growing surface rapidly loos its low-index charac-ter becoming esntially disordered.On the other hand,collision energies are large,and (although energy transfer mechanisms on rough surfaces are largely unknown)one may expect multi atom events and transient mobility ef-fects to increa the effective diffusivity.We aim at using parameters that maximize the coordination of the new particle,with the constraint that the impact-to-final site distance is minimized (so that the local film curvature is minimized 18).
In practice,we t the following criteria for (a )the maximum diffusion distance,and (b )the nature of a minimum-energy site:(a )The maximum diffusion dis-tance is taken to be d max =d thr max =5grid spacings =12˚A .All sites within this distance from the landing site are analyzed;the chon final site is the one clost to the landing site,among tho with the highest local coordi-nation (to be defined below,point (b)).Becau of this,the effective diffusion length is actually rather smaller than the maximum value d max .Our specific choice of d max may be simply be regarded as a calibration factor designed to avoid dendritic structures or very flat sur-faces which are not obrved in the expe
rimental regime of interest here.However,we find a rather sharp change in the dependence of the surface roughness on d max ,from
rather weak for d max >d thr max to strong for d max <d thr max .19
Thus our value d thr max is effectively a crossover threshold between the two regimes mentioned.
(b )–A minimum-energy site is defined in terms of its local coordination 20as follows:for each candidate site within the pret maximum distance from the landing site (point (a ))we calculate the number of atoms con-tained in a circle center at the candidate site and with a radius of 7grid spacings.This criterion amounts to lecting the site with maximum average coordination;while similar to that of highest first neighbor coordina-tion,contrary to the latter it avoids pathological choices such as high coordination sites on highly ramified struc-tures.
Results of topography simulations –Figure 2depicts trench topographies predicted by our model for different deposition conditions,compared to scanning-electron-microscope pictures taken in similar conditions.16The structure size is 1.2µm across.In the cellular automa-ton simulation,the emitted atom energies are picked from a Thompson distribution centered at 3eV,as suggested experimentally.17,21The initial angles of the non-ionized atoms are chon from a collimated cosine
风湿的症状distribution with a maximum off-normal angle of 40◦.To mimic the experimental conditions,we assume that 80%of the emit-ted Al atoms get one-fold ionized;for the atoms,the trajectory and impact energy change according to the applied source–target bias.
(a)(b)
FIG.2.Film morphologies on trench structures predicted for different ionized magnetron sputtering conditions com-pared to experiments.16In panel (a)we t 80%ionization,and 10V bias,in panel (b)80%ionization,and 80V bias.Below,a SEM micrograph with experimental results.
We now describe our results for different values of the
the sputter source-to-target bias corresponding to the low
and high energy regimes identified in our MD calcula-
tions.The results in panel(a)of Fig.2were obtained
tting the bias to10eV.In agreement with experiment
(also displayed in Fig.2),we predict afilm growth front
of rounded shape on top of the feature and,due to geo-
metric shadowing,a reducedfilm thickness at the bottom
of the trench.The pile-up at the center of the trench is
not only of geometric origin,but also partly due to re-
介绍信模板
flections of atoms impinging on the trench sidewalls.In
panel(b)of Fig.2we report results obtained with a bias
of80eV.Our calculations predict,in accord with exper-
iment,a roof-like structure on top of the feature.This
structure is due to the preferential etching at angles of
50◦(e Fig.1(a))which leads to a lower deposition
rate on the roof-like structure.Simulations of structures
scaled down in size by a similarity transformation,
with the same geometry,relative sizes,and aspect ratios)
浪漫七夕produced very clo results,suggesting that the profiles
obtained are largely lf-similar at the length scales.22孟云卿
In conclusion,we have demonstrated the viability of
accurate simulations of mesoscopic thinfilm morphol-
ogy bad on atomic-scale simulations.We performed
detailed theoretical calculations of the probabilities for
surface reactions taking place during ionized physical va-
por deposition conditions,and combined the predic-
tions withµm-scale cellular-automaton simulations.Our
molecular dynamics calculations revealed strongly energy
猪肉豆腐丸子dependent adsorption,reflection,and etch rates(the lat-
ter exhibiting a distinct maximum for high incident ki-
netic energies at∼50◦).We were able to predict to-
pographies of metalfilms deposited on trench structures
under different ionized physical vapor deposition condi-
tions in remarkable agreement with experiment.Our re-
sults reprent a major step ahead over earlier thinfilm
growth models bad on rate equations.23which did not
incorporate beam-energy–dependent surface reactions on
an atomistic scale.
We gratefully acknowledgefinancial support by
Siemens AG.We thank Dr.A.Spitzer and Dr.A.Kersch
for valuable guidance throughout the project,and Dr.
Paolo Ruggerone for a critical reading.V.F.was sup-
ported by the Alexander von Humboldt-Stiftung during
his stay at the Walter Schottky Institut.
C,with A and C constants.
21E.Dullini,Nucl.Instr.and Meth.B2,610(1984).
22A.Barabasi and H.E.Stanley,Fractal Concepts in Surface
小投资项目Growth,(Cambridge UP,Cambridge1995).
23L.J.Friedrich,S.K.Dew,M.Brett,and T.Smy,Thin
Solid Films226,83(1995);S.S.Winterton,T.Smy,S.K.
Dew,and M.J.Brett,J.Appl.Phys.78,3572(1995);L.
J.Friedrich,D.S.Gardner,S.K.Dew,M.J.Brett,and
T.Smy,J.Vac.Sci.Technol.B15,1780(1997).
