P1.44 The Issue of Data Density and Frequency with EnKF Radar Data Assimilation in a Compre

High power pulsed magnetron sputtering:A review on scienti fic and engineering state of the artK.Sarakinos a ,⁎,1,J.Alami b ,1,S.Konstantinidis c ,1a Materials Chemistry,RWTH Aachen University,52074Aachen,Germanyb Sulzer Metaplas GmbH,Am Böttcherberg 30-38,51427,Bergisch Gladbach,GermanycLaboratoire de Chimie Inorganique et Analytique,Universitéde Mons,Avenue Copernic 1,7000Mons,Belgiuma b s t r a c ta r t i c l e i n f o Article history:Received 31March 2009Accepted in revised form 6November 2009Available online 19January 2010Keywords:HPPMS HiPIMSIonized PVDHigh power pulsed magnetron sputtering (HPPMS)is an emerging technology that has gained substantial interest among academics and industrials alike.HPPMS,also known as HIPIMS (high power impulse magnetron sputtering),is a physical vapor deposition technique in which the power is applied to the target in pulses of low duty cycle (b 10%)and frequency (b 10kHz)leading to pulse target power densities of several kW cm −2.This mode of operation results in generation of ultra-dense plasmas with unique properties,such as a high degree of ionization of the sputtered atoms and an off-normal transport of ionized species,with respect to the target.These features make possible the deposition of dense and smooth coatings on complex-shaped substrates,and provide new and added parameters to control the deposition process,tailor the properties and optimize the performance of elemental and compound films.©2009Elsevier B.V.All rights reserved.Contents 1.Introduction ..............................................................16622.HPPMS:power supplies and processes .................................................16623.Plasma dynamics in HPPMS ......................................................16643.1.On the HPPMS plasma .....................................................16643.2.Determination of plasma properties:a theoretical overview ....................................16653.3.The HPPMS plasma properties ..................................................16663.3.1.The effect of the pulse on/off time con figuration on the plasma properties .........................16683.3.2.Rarefaction in the HPPMS plasma ............................................16693.4.Instabilities in the HPPMS plasma ................................................16713.4.1.Plasma instabilities:an overview ............................................16713.4.2.Plasma initiation and the relationship between plasma instabilities and the chargetransport in the HPPMS plasma .....16714.Plasma-surface interactions and deposition rate .............................................16734.1.Non-reactive HPPMS ......................................................16734.1.1.Target-plasma interactions and target erosion rate ....................................16734.1.2.Transport of ionized sputtered species ..........................................16754.2.Reactive HPPMS ........................................................16755.Growth of thin films by HPPMS ....................................................16765.1.HPPMS use for deposition on complex-shaped substrates .....................................16765.2.Interface engineering by HPPMS .................................................16775.3.The effect of HPPMS on the film microstructure .........................................16785.4.Phase composition tailoring of films deposited using HPPMS ...................................16796.Towards the industrialization of HPPMS ................................................16816.1.Examples of industrially relevant coatings by HPPMS .................................. (1681)Surface &Coatings Technology 204(2010)1661–1684⁎Corresponding author.Tel.:+492418025974,+492204299390,+3265554956.E-mail address:sarakinos@mch.rwth-aachen.de (K.Sarakinos).1All authors have contributed equally to the preparation of themanuscript.0257-8972/$–see front matter ©2009Elsevier B.V.All rights reserved.doi:10.1016/j.surfcoat.2009.11.013Contents lists available at ScienceDirectSurface &Coatings Technologyj o u r n a l h om e p a g e :w w w.e l s ev i e r.c o m /l o c a t e /s u r fc o a t6.2.Up-scaling of the HPPMS process (1682)6.3.Positioning HPPMS in the coating and components market (1682)Acknowledgements (1682)References (1682)1.IntroductionThinfilms are used in diverse technological applications,such as surface protection and decoration,data storage,and optical and microelectronic devices.The increasing demand for new functional films has been a strong incentive for research towards not only understanding the fundamentals and technical aspects of thinfilm growth,but also developing new deposition techniques which allow for a better control of the deposition process.Among the various methods employed forfilm growth,Physical Vapor Deposition(PVD)techniques, such as magnetron sputtering,are widely used[1].Magnetron sputtering is a plasma-based technique,in which inert gas(commonly Ar)atoms are ionized and accelerated as a result of the potential difference between the negatively biased target(cathode)and anode.The interactions of the ions with the target surface cause ejection(sputtering)of atoms which condensate on a substrate and form afilm[1,2].The condensation and film growth processes frequently occur far from the thermodynamic equilibrium,due to kinetic restrictions[2,3].Thus,a good control of both the thermodynamic and the kinetic conditions has implications on the growth dynamics and enables the tailoring of the structural,optical, electrical,and mechanical properties of thefilms[4].One way to control the growth dynamics is by heating the substrate during the deposition [2].The magnitude of the deposition temperature affects the energy transferred to thefilm forming species(adatoms)[2].This energy is,for instance,decisive for the activation of surface and bulk diffusion processes[5,6]and enables control over thefilm morphology[7–10]. Another source of energy are the plasma particles that impinge onto the growingfilm transferring energy and momentum to the adatoms[4,5]. The bombardingflux consists both of neutral and charged gas particles,as well as sputtered species.Numerous studies have shown that the energy, theflux,the angle of incidence,and the nature of the bombarding species are of importance for the properties of the depositedfilms[4,5,11,12].In general,the plasma particles exhibit a relatively broad energy distribu-tion with a mean value of several eV[13].The particle energy,andflux are affected by the target-to-substrate distance and the working pressure[1]. When the particles are charged the control of their energy can additionally be achieved by the use of electricfields,e.g.by applying a bias voltage to the substrate[4,14],while theirflux depends on a number of factors including the plasma density,the target power,and the magneticfield configuration at the target[4,14].It is therefore evident, that a high ion fraction in a depositingflux facilitates a more efficient and accurate control of the bombardment conditions providing,thus,added means for the tuning of thefilm properties.In magnetron sputtering processes the degree of ionization of the plasma particles is relatively low[13],resulting in a low total ionflux towards the growingfilm[13].As a consequence,in many cases bias voltage values of several tenths or even hundreds of V are required in order to increase the average energy provided to the deposited atoms and significantly affect thefilm properties[5].Moreover,the degree of ionization of sputtered species is typically less than1%[13,15,16].As a result,the majority of the charged bombarding particles is made of Ar+ions[13].This fact in combination with the relatively high bias voltages may cause subplantation of the Ar atoms in thefilm[17,18], leading to a generation of lattice defects,[18,19]high residual stresses [20–23],a deterioration of the quality of thefilm/substrate interface [21],and a poorfilm adhesion[24].Thus,the increase of the fraction of the ionized sputtered species has been an objective of many research works during the recent decades.Some of the approaches that have been demonstrated include the use of an inductively coupled plasma (ICP)superimposed on a magnetron plasma[25],the use of a hollow cathode magnetron[25],and the use of an external ion source[26]. Parallel to the above mentioned approaches Mozgrin et al.[27], Bugaev et al.[28]and Fetisov et al.[29]demonstrated in the mid90s that the operation of a conventional sputtering source in a pulsed mode,with a pulse duration ranging from1µs to1s and a frequency less than1kHz,allowed for pulsed target currents two orders of magnitude higher than the average target current in a conventional sputtering technique,such as direct current magnetron sputtering (dcMS)[27–29].These high pulse currents resulted,in turn,in ultra-dense plasmas with electron densities in the order of1018m−3[27–29],which are much higher than the values of1014–1016m−3 commonly obtained for dcMS[13,30].A few years later,Kouznetsov et al.[31]demonstrated in a publication,that received much acclaim,the high power pulsed operation of the sputtered target showing that in the case of Cu the high plasma densities obtained using this pulsed technique resulted in a total ionflux two orders of magnitude higher than that of a dcMS plasma,and a sputtered material ionization of ∼70%.The new deposition technique was called High Power Pulsed Magnetron Sputtering(HPPMS).In the years after Kouzsetov's report, HPPMS received extensive interest from researchers and led to a substantial increase of the HPPMS-related publications(Fig.1).Later, several research groups[25]adopted the alternative name High Power Impulse Magnetron Sputtering(HiPIMS)for this technique.The aim of this review paper is to describe the scientific and engineering state of the art of HPPMS.For this purpose,the principle and the basic structure of the existing HPPMS power supplies are described and the various existing approaches to produce high power pulses are demonstrated.The spatial and the temporal evolution of the HPPMS plasma and its interactions with the target and the growingfilm are discussed both from a theoretical and an experi-mental perspective.Furthermore,the effect of the HPPMS operation on the deposition rate and thefilm properties is presented.Finally,the use of HPPMS in industry and the challenges that the latter faces regarding the use of HPPMS are briefly addressed.2.HPPMS:power supplies and processesThe experimental realization of HPPMS requires power supplies different than those used in conventional magnetron sputtering processes. Those power supplies must be able to provide the target with pulses of high power density(typically in the range of a few kW cm−2),while maintaining the time-averaged target power density in values similarto Fig.1.Number of HPPMS-related publications in the period1999–2008.1662K.Sarakinos et al./Surface&Coatings Technology204(2010)1661–1684those during dcMS (i.e.a few W cm −2).The low average target power density is necessary to prevent overheating of the cathode and damage of the magnets and the target.A number of research groups and companies have developed HPPMS arrangements for both laboratorial and industrial use.These devices exhibit similarities in their basic structure which is schematically depicted in Fig.2.A dc generator is used to load the capacitor bank of a pulsing unit,which is connected to the magnetron.The charging voltage of the capacitor bank ranges typically from several hundreds of V up to several kV.The stored energy is released in pulses of de fined width and frequency using transistors with a switching capability in the µs range,located between the capacitors and the cathode.The pulse width (also referred to as pulse on-time)ranges,typically,from 5to 5000µs,while the pulse repetition frequency spans from 10Hz to 10kHz.Under these conditions,the peak target current density may reach values of up to several Acm −2,which are up to 3orders of magnitude higher than the current densities in dcMS [25].The high target current densities during the pulse on-time are accompanied by differences in the electrical characteristics of the discharge,as manifested by the current –voltage (I –V )curves of a magnetron operating in a dcMS and an HPPMS mode (Fig.3).According to Thornton [32]the target and the voltage of the magnetron are linked by the power law I ∝Vnð1ÞIn dcMS processes the exponent n in Eq.(1)ranges from 5up to 15(Fig.3).In HPPMS the exponent n changes from values similar to dcMS at low target voltages to values close to the unity when high voltages are applied (Fig.3)[33,34].Despite their common basic architecture,the available HPPMS power supplies exhibit differences primarily in the width and the shape of the pulses they can deliver,and in whether they can sustain a constant voltage during the pulse on-time.Arrangements able to provide high power pulses were already developed in the mid 90s,in order to explore the feasibility of producing high-current quasi-stationary magnetron glow discharges for high rate deposition [27–29].However,it was in 1999when Kouznetsov et al.[31]emphasizedthat the high power pulsing could allow for a high ionization degree of copper vapor (70%)as well as a homogeneous filling of 1:2aspect ratio trenches (see Fig.4).The time-dependent target current and target voltage curves of the HPPMS power supply developed by Kouznetsov et al.[31]are plotted in Fig.5where it is shown that the ignition of the plasma is accompanied by a drop of the voltage from ∼1000V to ∼800V,as the current increases to its peak value ranging from 200to 230A at the highest applied working pressure [35,36].A characteristic of this power supply is that the loading voltage is not maintained constant during the pulse on-time,and its temporal evolution is rather determined by the size of the capacitor bank and the time-dependent plasma impedance.Other power generators were designed to deliver voltage pulses with a rectangular shape (see Fig.6),i.e.a constant voltage value during the pulse on-time [34,37–39].It is to be noted that pulses with widths larger than about 50µs allow for a saturation of the discharge current and establishment of a steady-state plasma [40]provided that enough energy is stored in the capacitor bank.A common problem encountered in sputtering processes is the occurrence of arcs.This phenomenon is particularly pronounced during the deposition from conducting targets covered by insulating layers and can be detrimental for the quality of the films,due to the ejection of µm sized droplets from the target [41,42].The high peak target currents during the HPPMS operation may enhance the frequency of the arc events [43].Therefore,sophisticated electronics can be used in conjunction with the HPPMS power supplies to limit their effect if formed [43].An alternative way to alleviate this problem is to operate HPPMS using short pulses with widths ranging from 5to 20µs [44].A waveform during this mode of operation is presented in Fig.7.Within this short period of time,the glow discharge remains in a transient regime [40](as manifested by the triangular form of the target current)so that an eventual glow-to-arc transition is prevented.These short pulses have been,for instance,shown to allow for a stable and an arc-free reactive deposition of metal oxide films [45,46].However,in order to eliminate the time lag for plasma ignition which would allow for the production of high-current pulses within this short period of time,a pre-ionization stage has to be implemented [44,47].The pre-ionization of the discharge ensures that a low density plasma is maintained between the pulses as the charge carriers are already present before the voltage is turned on.The plasma pre-ionization step may be achieved by super-imposing a secondary plasma (such as a dc,a microwave,or an inductively coupled plasma)on the HPPMS plasma or by running the discharge at relatively high frequencies [44,47–49].The latter pre-ionization method is extensively implemented in mid-frequency pulsed plasmas [50–53].In contrast to the short pulses described above,long pulses with a width of several thousands of µs can also be used [54](Fig.8).In this case the pulse is composed of two (or more)stages.TheFig.2.Basic architecture of an HPPMS power supply.The dc generator charges the capacitor bank of a pulsing unit.The energy stored in the capacitors is dissipated into the plasma in pulses of well-de fined width and frequency using ultra fastswitches.Fig.3.Current –voltage curves of a magnetron during operation in dcMS and HPPMS mode.The change of the slope from 7to 1at a voltage value of 650V indicates loss of the electron con finement (data taken from [33]).Fig. 4.Cross-section SEM image of two via holes with an aspect ratio 1:2homogeneously filled by Cu using HPPMS (reprinted from [31]after permission,1999©Elsevier).1663K.Sarakinos et al./Surface &Coatings Technology 204(2010)1661–1684first step of the discharge is made of a weakly ionized regime,while the second part of the pulse brings the glow discharge to the high ionization state.The weakly ionized regime allows for the formation of a stable discharge prior to entering the highly ionized regime which helps to suppress the arc formation during the high power mode of operation [54].3.Plasma dynamics in HPPMS 3.1.On the HPPMS plasmaPlasmas are partially ionized gases characterized by,for example,the charge (electron)density or the electron energy distribution function [55].A classi fication of different plasmas could be based on the charge density as shown in Fig.9,where the magnetron sputtering plasmas are shown to exhibit densities ranging from ∼1014up to 1020m −3.It is known that high density plasmas such as arc plasmas exhibit high ionization fractions [25,31].This is not the case for conventional magnetron plasmas where the electron density does not exceed the value of 1016m −3,penning ionization is the main ionization mechanism [30]and therefore the ionization of the sputtered species is at the best of few percent [13].In order to achieve a high ion fraction in a discharge it is necessary to promote the electron impact ionization [55]which is best realized by increasing the electron density and temperature.Different approaches have been made to achieve this,for example by using an electron cyclotron resonance (ECR)plasma as a plasma electron source [56],or by employing an inductively coupled plasma (ICP)[57,58].A higher frequency of electron impact ionization collisions can alsobeFig.5.Target current and voltage waveforms produced by the power supply developed by Kouznetsov et al.at working pressures of (a)0.06Pa,(b)0.26Pa,(c)1.33Pa and (d)2.66Pa.The charging voltage of 1000V drops down to 800V and the plasma is ignited.Peak target currents of up to 230A are achieved.The time lag for the ignition of the plasma decreases when the pressure is increased (reprinted from [35]after permission,2002©Elsevier).Fig.6.Rectangular shape voltage waveforms generated by the HPPMS power supply developed by MELEC GmbH.The data were recorded from an Ar –Al HPPMS discharge operating at pulse on/off time con figuration 25/475µs (K.Sarakinos,unpublisheddata).Fig.7.Temporal evolution of target voltage and current during operation using a 10µs long pulse.The target current is interrupted before the plasma enters into the steady-state regime.The data were recorded from an Ar –Ti HPPMS discharge (S.Konstantinidis unpublished data).1664K.Sarakinos et al./Surface &Coatings Technology 204(2010)1661–1684achieved by increasing the power to the sputter source and therewith the electron density.This approach is,however,limited since a power excess can result in an overheating of the target or an exceeding of the Curie temperature of the magnetron's magnets.Thus,the work of Kouznetsov et al.[31]opened a new era of magnetron sputtering deposition and plasma studies,since it allowed for the generation of highly ionized plasmas using a conventional magnetron source.The early analyses of the HPPMS plasma showed that its temporal characteristics are complex and unique [35,59,60].Understanding the discharge evolution and the mechanisms that take place in the plasma is therefore of utmost importance as this sheds light on the growth conditions during the thin film deposition.To achieve this,modeling and a number of analytical studies have been carried out,using Langmuir probes as well as optical emission,mass,and absorption spectroscopy techniques.These studies are reviewed in the following chapters and the basic properties of the HPPMS plasma are thus outlined.3.2.Determination of plasma properties:a theoretical overview Plasmas are commonly modeled as fluids,and each species is described with its fluid equation [61].The fluid approximation is suf ficiently accurate to describe the majority of observed plasma phenomena.When this is not viable,the accumulated behavior of anensemble of charged particles is described using a statistical approach,in the form of the velocity distribution function which is given as f (r,u,t )d 3r d 3u ,i.e.the number of particles inside a six-dimensional phase space volume d 3r d 3u at (r ,u )and time t [55].Here,r is the position vector and u is the speed vector.Knowing the distribution function,fundamental features of a large ensemble can be quanti fied,i.e.by integrating over the velocity,the average (macroscopic)quantities associated with the plasma are de fined as:n ðr ;t Þ=∫f ðr ;u ;t Þd u plasmanumber ðion ;electron ;orneutral Þdensityð2Þu Àðr ;t Þ=1n∫u f ðr ;u ;t Þd u plasmabulkfluid velocity ð3ÞεÀðr ;t Þ=m 2n∫ðu À−u Þ2f ðr ;u ;t Þd u plasmameankineticenergy ð4Þwhere m is the particle (ion,electron,or neutral)mass.This procedure of integrating over velocity space is referred to as taking velocity moments,with each one yielding a physically signi ficant quantity.Knowledge of the electron energy (velocity)distribution function (EEDF)permits us to determine important parameters.The EEDF can be measured using a Langmuir probe,as was demonstrated by Druyvesteyn [55]:g e ðV Þ=2m e A pr 2eV m12d 2I edV ð5Þwhere A pr is the probe area,I e is the electron current,m is the electronmass,and V =V pl −V b is the potential difference between the plasma potential and the probe potential.With the change of variables ε=eV ,the electron density n e is determined as:n e =∫∞0g e ðεÞd εð6ÞPlasma parameters,such as the electron density,are easily obtained if the mathematical description of the EEDF is Maxwellian.The Maxwellian distribution is characterized by its single temperature and is obtained when plasma electrons undergo enough collisions and are in equilibrium with other plasma components and with the electron ensemble itself.At a low pressure,the EEDF is generally non-Maxwellian,and the electron temperature is thought of as aneffectiveFig.8.Target voltage and current waveforms during the long-pulse operation.In the first 500µs a weakly ionized plasma is generated which is followed by a highly ionized steady-state plasma (reprinted from [54]after permission,Society of Vacuum Coaters ©2007).Fig.9.Plasma density is used in order to classify plasmas.The HPPMS plasma spans over a large density range depending on the pulse on/off time con figuration used for a process and presents therefore a tool for better choosing the plasma dynamics needed for the deposition process.The abbreviation ECR stands for Electron Cyclotron Resonance (taken from [36]).1665K.Sarakinos et al./Surface &Coatings Technology 204(2010)1661–1684electron temperature,T eff∼23〈ε〉,representing the mean electron energy determined from the EEDF according to[55]:〈ε〉=1n e ∫∞εg eðεÞdεð7Þ3.3.The HPPMS plasma propertiesOne of the earliest works related to the study of the properties of the HPPMS plasma was carried out by Gudmundsson et al.[35,59]. Using a Langmuir probe[62]in an Ar–Ta plasma,they measured the temporal behavior of the EEDF by recording the time-dependent probe current curves.It was shown that the EEDF evolved from a Druyvesteyn-like distribution with a broad energy distribution(high average energy)during the pulse to a double Maxwellian distribution (i.e.a distribution that is composed of two-distinct energy-distribu-tions)toward the end of the pulse,andfinally a Maxwellian-like distribution hundreds ofµs after the pulse had been switched off.This is shown in Fig.10,where the temporal evolution of the EEDF is also seen to be affected by the working pressure[35].Similar results were obtained by Pajdarova et al.[63],although the double Maxwellian distribution was observed from the beginning of the pulse,which could be explained by the nature of the target material(Cu in this case)and the HPPMS parameters used in the experiments.Further-more,it was found that the effective electron temperature peaked a fewµs after the pulse start,and at the same time,independent of measurement position[35].This is indicative of the existence of high energy electrons accelerated by the target voltage,and thus escaping the confinement region in the vicinity of the cathode[35].InaFig.10.The EEDF(electron energy distribution function)for an HPPMS plasma with a100µs on-time and a50Hz frequency at three chamber pressures(a)0.25,(b)1.33,and(c) 2.66Pa.The distribution function changes during the pulse from a Druvesteyn distribution to a double-Maxwellian distribution(reprinted from[35]after permission,Elsevier©2002).1666K.Sarakinos et al./Surface&Coatings Technology204(2010)1661–1684different study by Seo et al.[64],Langmuir probes were utilized to determine the properties of a HPPMS plasma at a higher pulsing frequency of 10kHz and a duty cycle of 10%.By using such a high frequency,the electron density reached a value of 1.5×1017m −3,which could be seen as a relatively low density HPPMS plasma.The temporal evolution of the energy distribution in this plasma showed ef ficient electron heating,which took place within the first few µs from the pulse start.This work revealed that the plasma dynamics presented in the studies by Gudmundsson et al.[35,59]are to a large extent general features of the low duty cycle pulsed plasmas regardless of the frequency.The early studies on the HPPMS plasma characteristics also showed the potentially high ionization fraction of the sputtered material that could be achieved by using this technique [16,25,33,60,65,66].Spectroscopic analyses,such as optical emission spectroscopy [67–69],mass spectroscopy [69],and absorption spectroscopy [68,69]were,therefore,employed in order to better quantify and understand the average and time-dependent evolution of the ion population in the HPPMS plasma.The optical emission studies [33,65,70–72],were performed both for the HPPMS and the dcMS discharges and showed the much higher metal ion emission intensity in HPPMS (Fig.11).However,in order to better quantify these observations,absorption spectroscopy [70,71,73,74]and mass spectroscopy [75,76]studies were performed,con firming the much higher ionization of the sputtered material and of the sputtering gas achieved in HPPMS.One well studied material in this regard is Ti.Mass spectroscopy measurements of the peak ion compositions in an Ar –Ti discharge for the pulse on-time of 100µs and a frequency of 50Hz [75]showed that the ionized part of the plasma contained 50%of Ti 1+,24%of Ti 2+,23%of Ar 1+,and 3%of Ar 2+ions.When longer pulse on-times of several hundreds of µs were used even Ti 3+and Ti 4+could also be detected [77].It is deduced that for the formation of multiply charged ions certain conditions should be ful filled [77];(i)the discharge should contain a high enough number of metal ions,(ii)the pulse on-time should be long enough (and the capacitance of the used power supply should be large enough to store enough energy for the whole pulse),(iii)the self-sputtering yield (i.e.the sputtering yield of the target material from ionized target species)should be low,and (iv)the ionization energy of each ionization step should not be too high.These two examples show clearly that the HPPMS plasma properties depend not only on the plasma density and energy distribution but also,to a large degree,on the pulse on/off time con figuration.Since the introduction of the HPPMS technique,most of the discharge studies have been performed for pulses with pulses on-time longer than 50µs.Exceptions to this can be found in the works by Konstantinidis et al.[70,73],Ganciu et al.[44],and Vasina et al.[48],where shorter pulses ranging between 5and 30µs have been used,an example of which is shown in ing time-resolved optical emission spectroscopy in a Ti –Ar discharge,an increase in the line intensity ratio of the ion-to-neutral Ti and Ar species with increasing the pulse duration was observed [70].This means that the ionization degree increases with increasing pulse length,indicating the necessity for the sputtered atoms to reside a suf ficient time in the target vicinity for the vapor to become ionized.The increase of the AremissionFig.11.Optical emission spectroscopy from a typical HPPMS plasma.The emission from the Ti and Ar species increases greatly in HPPMS indicating the high ionization of the metal and gas species (data taken from [72]).Fig.10(continued ).1667K.Sarakinos et al./Surface &Coatings Technology 204(2010)1661–1684。

The first theorem of Castigliano is a powerful tool for finite element formulation, as is now illustrated for the bar element.
8
x
u2
L
u1
[1.30]
x
E x
E
u2
L
L
which is exactly the relation shown by strength of materials theory
W
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o k d
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1 2
k
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(1.40)
Mechanical work
Strain energy
3
we observe that the work and resulting elastic potential energy are quadratic functions of displacement
u1
[1.31]
ue d [1.43] 0
Combining Equations 1.30, 1.31, and 1.43, the total strain energy for the bar element is given by
Ue
1 2
x xV
1 2
E
u2
L
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(1.48)
U e
i
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7
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U

a r X i v :m t r l -t h /9609008v 1 25 S e p 1996Strained tetragonal states and Bain paths in metalsP.Alippi 1,P.M.Marcus 1,2,and M.Scheffler 11Fritz-Haber-Institut der Max-Planck-Gesellschaft,Faradayweg 4-6,D-14195Berlin-Dahlem,Germany2IBM Research Center,Yorktown Heights,N.Y.10598,USA(February 6,2008)Paths of tetragonal states between two phases of a material,such as bcc and fcc,are called Bain paths.Two simple Bain paths can be defined in terms of special imposed stresses,one of which applies directly to strained epitaxial films.Each path goes far into the range of nonlinear elasticity and reaches a range of structural parameters in which the structure is inherently unstable.In this paper we identify and analyze the general properties of these paths by density functional theory.Special examples include vanadium,cobalt and copper,and the epitaxial path is used to identify an epitaxial film as related uniquely to a bulk phase.PACS numbers:64.70.Kb,61.50.Ks,68.56.Eq,62.20.DcPseudomorphic epitaxy of a cubic or tetragonal (001)film typically results in a strained tetragonal structure.If the stresses on the tetragonal state vanish and also the state corresponds to a local minimum of energy with respect to tetragonal deformations,the structure will be called a tetragonal phase .Such a phase will be stable or metastable depending on whether it has the lowest energy compared to other minima.Frequently metals have two tetragonal phases;sometimes both are cubic,e.g.,bcc and fcc Na and Rb [1,2];sometimes one is cubic and the other phase is tetragonal,e.g.,Ti and V [2,3].They can,of course,also have phases with other structures,e.g.,Ti also has a hcp phase.Many paths can go from one tetragonal phase to the other.If the geometries along such a path have tetrag-onal symmetry and if they connect bcc and fcc phases,the paths have been called Bain paths [4].A purpose of the present work is to define and discuss a partic-ular Bain path which will be called the epitaxial Bain path (EBP).The EBP is produced by isotropic stress or strain in the (001)plane of tetragonal phases accompa-nied by vanishing stress perpendicular to the plane,such as pseudomorphic epitaxy produces on an (001)cubic or tetragonal film.Epitaxy provides a valuable means of stabilizing metastable phases and of putting phases under very large strains,both tensile and compressive in the plane of the epitaxy.The EBP of a material identifies the phase that has been strained,checks quantitatively the elastic behavior,which can be highly nonlinear,and predicts which phase of the material will form on a given substrate.Thus,in order to understand the properties of epitaxial films and new materials,the knowledge of the EBP is indispensable.A different Bain path has long been discussed,partic-ularly by Milstein [1],in which uniaxial stress is applied to a tetragonal state along the [001]axis accompanied by zero stress in the (001)plane;this path is conveniently called the uniaxial Bain path (UBP).We compare the two paths,EBP and UBP,which are both physically realizable.We show that both have the same lowestpossible maximum energy or barrier energy of all Bain paths between the two tetragonal phases.However the EBP has a special value in relating strained tetragonal structures produced by epitaxy to particular tetragonal phases.Depending on where the tetragonal structure lies on the EBP,we shall show that a clear choice can be made which classifies the structure as a strained form of a particular phase.Thus for materials with both a bcc and a fcc phase,the bulk (i.e.,the interior)of an epitax-ial film can be uniquely classified as a strained bcc phase or a strained fcc phase.Experimental geometry analyses of the bulk structure are frequently made by low-energy electron diffraction of ultrathin epitaxial films,which can then be placed on the EBP in the tetragonal plane to identify the phase of which it is the strained form.We employ density functional theory together with the full-potential linearized augmented plane wave method [5].An important advantage of a first-principles calculation,i.e.,a calculation with no adjustable parame-ters,is that it applies just as reliably to highly strained or even unstable geometries,as it does to the unconstrained ground state,i.e.,to the stable phase.The same relia-bility cannot be asserted for empirically adjusted poten-tials,such as the empirical pseudopotential model used in Ref.[1],which is adjusted to reproduce ground-state properties of only certain structures.The first-principles calculation provides a check on the validity of empirical potentials far from the structural ground state and of linear elasticity theory.The calculation proceeds by evaluating the total energy E (a,c )in eV/atom as a function of the tetragonal lat-tice constants a and c .Contour lines of constant E are plotted in Fig.1for vanadium on the (a/a bcc -V/V bcc )plane,where V =ca 2/2is the volume per atom.The contour lines show clearly two minima,one at the bcc structure and one at a tetragonal structure,and between them a saddle point at the fcc structure.Thus vanadium has two fairly deep and well-separated minima and so is a good choice to illustrate the tetragonal paths between minima [6].Since E has an extremum at the cubic struc-tures [3,7],the saddle point will occur exactly at the fcc structure.AlsoplottedinFig.1aretheEBP and the UBP.The EBP is calculated from the minima of E (c )at constant a ,which corresponds to the epitaxial situation,i.e.,a is held fixed and c adjusts to minimize E (c )and thus makes the out-of-plane stress,i.e.,the stress along [001],vanish.Conversely,the UBP calculation fixes c and lets a adjust to minimize E (a )making the in-plane stress vanish.Both paths go through the minima and the saddle point,hence the maximum energy on each path is the same.The saddle point is called in Ref.[1]“a special un-stressed tetragonal state ...at a local energy maxi-mum”[8].Obviously any path on the tetragonal plane that does not go through the saddle point must go through states of higher energy than the saddle point,since it will cross contours in either the upper or lower central sectors formed by the saddle-point contours at E =0.29eV/atom.Two common paths that do not go through the saddle point,hence have higher maximum energies than the EBP or UBP,are : 1.the constant volume paths at the volumes of the minima,which differ from the saddle-point volume.These paths have been called volume conserving Bain paths [2,3];2.the hy-drostatic pressure path,which has equal in-plane and out-of-plane stress components.Figure 2plots E vs.a along the EBP and UBP.Also indicated is the range from a =2.58˚A (c/a =1.6)to a =2.66˚A (c/a =1.2)for which we find along the EPB the tetragonal structure to be inherently unstable [9].In this range the slope of the EBP changes sign,and the stability condition˜c 11˜c 33−˜c 213≥0(1)is violated.Here ˜c ij are calculated from ˜c 11=(a 2/V )∂2E (a,c )/∂a 2,˜c 13=(ac/V )∂2E (a,c )/∂a∂c ,˜c 33=(c 2/V )∂2E (a,c )/∂c 2for the strained lattice [10].Tetragonal states outside the range of Eq.(1)cannot be stabilized with any applied stresses.When a path on the tetragonal plane reaches the boundaries of this unstable region,the tetragonal crystal becomes unstable and an epitaxial film will not be able to form many layers at the bulk spacing.Epitaxial films strained to a state near the unstable region should show large changes in the phonon spectra as elastic constants weaken,which would be in-teresting to study.One consequence of the existence of an unstable region is that a clear conceptual distinction can be made be-tween strained states of the phase at lower c/a (for vana-dium the bcc phase)and strained states of the larger c/a phase (for vanadium at c/a =1.83).If there was a con-tinuous path of allowed states between the two phases,there would not be a clear criterion for regarding an ob-served strained state as strained from a particular one of the two phases.A second consequence of the unstable region is that thestrained system never gets to the saddle point,but trans-forms to another strained phase at an energy at or below the energy of the state where the path strikes the bound-ary of the unstable region.Thus for the EBP and UBP the breakdown energy is at least 0.05eV/atom lower than the saddle-point energy,and it is lower on the UBP.The EBP and UBP can be found for other metals by similar calculations;for Cu and Co our results are shown in Fig.3and Fig.4.The topology of the E (a,c )sur-face of cobalt is similar to the one of vanadium,with the two minima and the saddle point corresponding to the structures where the EBP and the UBP intersect.In this case,however,the stable cubic phase is face-centered (c/a =√2(and a >a fcc );we also show that the film is too far from the bcc region of the EBP to be regarded as strained bcc.Thus,we conclude that the properties of such Cu/Pd(001)film are not bulk like,but instead stabilizedby the(still ultrathin)film properties.In Fig.4(as in Fig.3)the epitaxial paths from linear elasticity theory are also drawn for fcc Cu(and Co). Epitaxial strain on a cubic(001)surface in the linear elastic approximation says normal stress vanishes,hence δc/c=−(2ν/(1−ν))δa/a,whereνis the Poisson ra-tio.A comparison with the calculated EBP in bothfig-ures shows how the linear approximation fails to be valid forδa/a equals to few percent.In the analysis of the Cu/Pd(100)film,it is evident that thefilm is already far from the linear elastic regime.Generalizations of this calculation of physically real-izable paths between phases are possible with existing stein[19]has considered the instability of tetragonal structures with respect to orthorhombic defor-mations and found such instabilities on the UBP outside the range of c/a between the two minima;these insta-bilities probably exist on the EBP also,but have not as yet been studied.It looks practicable to explore with present codes even more general paths of homogeneous strain between tetragonal phases that make more gen-eral deformations of the unit mesh than orthorhombic deformations.Possibly a path of deformed lattices could be found with lower maximum energies,as the path of trigonal lattices of Ref.[7].It would also be possible to find epitaxial paths of tetragonal states for slabs of afi-nite number of layers,which would include relaxations of surface layers.In summary,paths of tetragonal states between two tetragonal phases include two particularly simple cases, namely,those produced by epitaxial(isotropic biaxial) stress and by uniaxial stress.Both cases correspond to maintaining certain surfaces stress-free and both have minimal maximum energies at a saddle point of energy that both pass through.Along these paths a range of in-stability exists,which separates states that can be clearly regarded as strained from one or the other of the two phases.The epitaxial path permits definite identifica-tion of the equilibrium phases of strained epitaxialfilms and could usefully be found for all metals with tetragonal phases.0.76 0.82 0.88 0.941.00 1.06a/a bccV /V b c cFIG.1.Contour lines of constant E in eV/atom for vana-dium on the tetragonal plane showing a bcc minimum (thezero of E )with a bcc =2.93˚A and V bcc =12.6˚A 3(c/a =1),a minimum at a =2.41˚A (c/a =1.83)and V =12.8˚A 3,anda saddle point at a =2.65˚A (c/a =(2)1。

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I. INTRODUCTION
The microscopic foundations of thermodynamics are well established using the Gibbs hypothesis of statistical ensembles maximizing the Shannon entropy[1].When the thermodynamic limit can be taken, the various Gibbs ensembles for infinite systems converge to a unique thermodynamic equilibrium [28]. However, many systems studied in physics do not correspond to this mathematical limit of infinite systems[13] and, in fact, finite systems are now, per se, a subject of a very intense research activity, from metallic clusters[3, 4] to Bose condensates[5, 6], from nanoscopic systems[7] to atomic nuclei [8, 9] and elementary particles[10]. The question thus arises: can the equilibrium of a finite systems be