Density functional theory meets statistical physics: from the atomistic to the mesoscopic properties...

SURFACE AND INTERFACE ANALYSISSurf. Interface Anal. 2006; 38: 1158–1163Published online 22 May 2006 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/sia.2373

Density functional theory meets statistical physics:from the atomistic to the mesoscopic properties ofalloys†

Stefan Muller∗

Lehrstuhl fur Festkorperphysik, Universitat Erlangen-Nurnberg, Staudtstr. 7, D-91058 Erlangen, Germany

Received 3 March 2006; Revised January 2005; Accepted 6 March 2006

Today, calculations based on density functional theory (DFT) allow us to study a number of metal alloyproperties, as e.g. formation enthalpies, or electronic and elastic properties of intermetallic compounds.However, such so-called (ab initio) calculations can only be applied as long as the alloy structure requiresonly small unit cells for the crystallographic description. This limitation can be overcome by combiningDFT calculations with the so-called Cluster Expansion (CE) methods and Monte-Carlo (MC) simulations.The concept will be applied to study two characteristic type of interfaces, namely, the boundary betweensolid solution and precipitate as well as the alloy’s surface. Copyright 2006 John Wiley & Sons, Ltd.

KEYWORDS: computational materials science; ab initio methods; metal alloys; segregation

INTRODUCTION

Modelling surface and interface properties of binary metalalloys often demands the treatment of phenomena on amesoscopic scale, like e.g. short-range order. Consequently,such studies cannot be done directly by the so-called first-principles methods based on density functional theory (DFT)because then they would be limited to model systemscontaining about 100–500 atoms. Possible solutions tothis dilemma are to simplify the electronic Hamiltonian,1,2

or to forbid symmetry-lowering geometric relaxations.3 – 6

However, such approximations only reach a quantitativeagreement with experimental data for certain systems, e.g.if the atomic size difference between two elements is smallenough so that the atomic relaxation energies do not playa crucial role. An alternative access that does not demandsuch approximations is the combination of DFT calculationswith methods from statistical physics. The latter can provideus with both, a lever to the mesoscopic world and theconsideration of temperature-dependent phenomena. Themethod used for the studies presented in this paper isCluster Expansion (CE)7 and is based on the idea that if onedivides a well-defined underlying lattice (e.g. fcc-based) intocharacteristic figures such as pairs, triangles, etc., then, it canbe shown analytically7 that the energy of any configuration� on this lattice can be written as a linear combination of thecharacteristic energies Jf of the figures. In practice, the onlyerror we make is that the sum must be truncated at some

ŁCorrespondence to: Stefan Muller, Lehrstuhl fur Festkorperphysik,Universitat Erlangen-Nurnberg, Staudtstr. 7, D-91058 Erlangen,Germany. E-mail: [email protected]†Paper presented at ECASIA’05; The 11th European Conference onApplications of Surface and Interface Analysis, 25–30 September2005, Vienna, Austria.

point. The ‘art’ is now to develop efficient tools in order tocontrol this error, i.e. to ensure that it is smaller than theaccuracy needed for the problem of interest. This conceptwill be discussed in the following section. The ansatz can beextended to finite temperatures by using the characteristicinteractions Jf in Monte-Carlo (MC) simulations. We even canextend the field of applications to non-equilibrium processesby switching from thermodynamic MC to kinetic MC.8

As an example, the approach will be applied to twocharacteristic types of interfaces, both shown in Fig. 1.Figure 1(a) presents the result of a kinetic MC simulation8

in order to understand the formation and evolution of Zn-precipitates in an Al-rich Al-Zn solid solution. Precipitatesare formed by quenching a solid solution deep into a two-phase region of a phase diagram followed by sample aging,and play an important role in heat-treatable alloys like Al-rich Al-Zn alloys as they act as obstacles for the dislocationmotion through the crystal. Hereby, the orientation of theinterface between solid solution and precipitate is essentialfor the technical application of such materials. Figure 1(b)displays the predicted segregation profile of a Pt25Rh75(111)surface. Indeed, this alloy surface represents an interestingcandidate with respect to the replacement of element crystalsby alloy crystals for catalytic reactions. Although the bulkmaterial of this disordered alloy on the average contains 25at% of Pt only, the top layer is tremendously enriched byPt (black atoms). Both interfaces will be briefly discussed inthe Section on Application to Interfaces. First of all, we willanswer the question of how to do such calculations.

THE THEORETICAL APPROACH

Calculations of the first-principles on the basis of the densityfunctional approach have been very useful to understand

Copyright 2006 John Wiley & Sons, Ltd.

From the atomistic to the mesoscopic properties of alloys 1159

(a) (b)

Figure 1. Two characteristic types of interfaces in binary metal alloys modelled via CE and MC: (a) Zn-precipitates in Al-rich Al-Znalloys8 (only Zn atoms are shown). Hereby the orientation of the precipitate-solid solution interface plays a crucial role for themechanical properties of the alloy. (b) Four top layers of an Pt25Rh75(111) surface. The top layer is tremendously enriched by Pt(68%) compared to the bulk.

the stability of simple single-phased materials. The accessis based on the Hohenberg-Kohn-theorem,9 which statesthat the energy of a system of interacting electrons inan external potential depends only on the ground-stateelectronic density. In our case, namely, the investigationof solid structures, the external potential is the Coulombpotential caused by the nuclei in a solid. In practice, thecalculation of the ground-state density is realized by theself-consistent solution of a set of single particle equations,called Kohn-Sham equations:10[

� h2

2mr2 C Ve�nuc�r� C VH�r� C VXC�r�

]i�r� D εii�r� �1�

In this Schrodinger-like equation, the first term on the left siderepresents the kinetic energy operator, Ve�nuc the Coulombpotential due to the nuclei, VH the Hartree potential, and VXC

is the exchange correlation potential. In this work, for VXC

the local density approximation (LDA)11,12 is applied to theAl-Zn system, and the generalized gradient approximation(GGA)13 to the Pt-Rh system. While the former system wasmodelled via PEtot (‘Parallel total energy’; this code wasdeveloped at the National Renewable Energy Laboratory andlater at NERSC Computer Centre, Berkeley. Some details canbe found in the Appendix of Ref. 30.), for the latter systemthe VASP code (‘Vienna ab initio simulation package’) wasapplied.14,15 For the following, it is important to mention thattotal energies of individual compounds calculated via DFTmust correspond to the geometrically fully relaxed configuration,i.e. the structure has to be optimized (consistent with thesymmetry of the structure) with respect to the aspect ratio ofunit cell vectors, cell-internal atomic displacements, and thevolume of the unit cell.

Next, we use these energies in a so-called CE in order toconstruct effective pair and multibody interactions. This canbe realized by transforming the ‘alloy problem’ to an Isingmodel. Each atom i of an A1�xBx alloy with concentrationx of B atoms is assigned to a spin-value Si D �1, if i is anA-atom, and to Si D C1, if i is a B-atom. Then, the energy ofeach configuration can be expressed by an Ising-expansion:

E�� D J0 C∑

i

JiSi�� C∑j<i

JijSi��Sj��

C∑k<j<i

JijkSi��Sj��Sk�� C Ð Ð Ð �2�

The first two terms on the right define the energy of therandom alloy (with zero mutual interactions), the third termcontains all the pair interactions, the fourth all the three-bodyinteractions, etc. This equation can be brought to a compactform by introducing a correlation function NF for each classof symmetry-equivalent figures F:

NF�� D 1NDF

∑f

Si1��Si2 �� . . . Sim �� 3�

Here, DF gives the number of figures of class F per site. Theindex f runs over the NDF figures in class F, and m denotesthe number of sites of figure f . Then, Eqn (2) becomes16

E�� D N∑

F

DF NF��JF �4�

This ‘rewriting’ of Eqn (2), however, does not help answerthe question of how the interactions JF of the system canbe determined. A possible solution is given by the directinversion method.17 For a set (about 15–150 depending on thecomplexity of the problem) of N� mostly ordered structureswith typically no more than 2–100 basis atoms, the energiesEDFT of the geometrically relaxed structures are calculatedvia DFT. Then, the N� energies are used to construct NF

interactions JF by solving

N�∑ω� jEDFT�� N

NF∑DFJF NF��j2 D Minimum �5�

with respect to the NF values of JF. Here, ω� are the weightfactors that e.g. allow us to give stable configurations(ground-states) a higher weight in order to describe themas accurately as possible. Naturally, NF � N� in order toprohibit an overestimation of the interactions. As recentlyshown by Hart et al. the efficiency of the method can betremendously improved by applying genetic algorithms forthe selection of the figure classes F18, especially if the pool of

Copyright 2006 John Wiley & Sons, Ltd. Surf. Interface Anal. 2006; 38: 1158–1163DOI: 10.1002/sia

1160 S. Muller

all possible figures becomes large, e.g. in systems with lowsymmetry as CE for hexagonal systems19 or surfaces.20

Although genetic algorithms are efficient in selecting thebest sets of figures for a given number of DFT-based inputenergies, they do not tell us which structures are importantas the input for construction of the effective interactions JF.Indeed, the weakness of the procedure described above isthe dependence of the interactions on the set of chosen inputstructures. Consequently, a way must be found to ensure thatselection of the number and type of relevant input structureswill be done by the system and not by the scientist. This canbe done by constructing a ground-state diagram in the veryearly stage of the construction of interactions. For a small setof input-structures (typically 10–20) a CE is constructed. TheCE is then used to calculate a ground-state line of the system,i.e. the dependence of the energy on the alloy composition.In the next step, for all structures on or near the ground-stateline, DFT calculations will be done and they will be addedto the set of input-structures used for the construction ofthe input-structures. This procedure has to be repeated untilthe ground-state line becomes stable, i.e. does not changeany longer, if input-structures are removed or added (detailscan be found e.g. in Ref.21). For the construction of surfaceCE for the Pt-Rh system, the linear-programming methodfor obtaining effective interactions by Garbulsky and Cederwas applied.22 In addition to the Conolly-Williams inverseapproach,17 Eqn (5), this method not only tries to reproducethe formation energies of individual structures in the CE fitbut also keeps the hierarchy between the formation energiesof individual structures, i.e. the relative energies betweenindividual structures.

In order to apply the CE ansatz to surfaces, our modelmust account for the broken symmetry at the surface. Thisreduces the conserved point symmetry operations of thebulk (e.g. the 48 operations for cubic fcc or bcc lattice) to amuch smaller orientation-dependent number. Consequently,the symmetric degeneration of CE figures decreases and,therefore, the number of geometrically different figures in theCE increases. Therefore, a larger number of input-structuresis necessary in order to stabilize the surface interactionscompared to the bulk. It is, however, possible to circumventa part of this problem by treating the surface interactionsas corrections of the bulk interactions. Since energies areadditive, we may write

HCEf D HVol

f C HSurff �6�

This ansatz was applied earlier by Drautz et al. to studythe energetics of Ni-rich Ni-Al surfaces.23 The advantage oftreating the surface interactions as ‘perturbation’ of the bulkinteractions comes from the fact that the DFT calculationsfor different surface terminations and segregation profileshave no longer to account for an infinite bulk reservoir. Weonly have to make sure that the DFT slab model is thickenough that the centre layer of the slab becomes bulk-like.The energy of a structure � can then be written as

E�� DN∑

iD1

NF∑ NF��DFJF CN0

F∑ N0F�Ri�D0

F�Ri�υJF�Ri�

�7�

We see that for the surface part the interactions becomesite dependent. Here, Ri defines the position of the atom iwith respect to the alloy surface. So, for an atom i withinthe segregation profile, every individual interaction Jf toneighbour atoms will be corrected to Jf C υJf �Ri�. Naturally,with increasing distance from the alloy surface, υJf ! 0and consequently the surface term (second term) in Eqn (7)become zero. In the case of the Pt25Rh75(111) surface it turnsout that already for the fourth layer from the top, υJf ! 0.The extended Hamiltonian, Eqn (7), can now be used inMC simulations. Starting from a random configuration,the segregation profile is determined as a function oftemperature by simulated annealing. The concept of CEallows us to choose a large enough number of atoms per layer(¾103 � 105 atoms per layer) so that the layer-dependentcorrelations between ordering and segregation can bestudied. In summary, the combination of surface CE and MCpermits us to study surface segregation for geometrically fullyrelaxed surfaces by purely quantum mechanical and statisticalprinciples, i.e. without any continuum approaches, empiricalparameters, or neglecting correlations between structure andsegregation.

APPLICATION TO INTERFACES

Boundary between precipitate and solid solutionProbably, the best-known example of heterogeneous phase-coexistence is phase separation of an alloy into itsconstituents24 A1�xBx ⇀↽ �1 � x�A C xB. Depending on thealloy’s concentration (A-rich or B-rich), decomposition reac-tions create so-called precipitates, which define an importantpart of the microstructure of many alloy systems. The earlystage of these reactions typically involves the formation ofcoherent precipitates that adopt the crystallographic lattice ofthe alloy from which they emerge.25 Coherent precipitateshave practical importance, as they act to impede dislocationmotion, and thus lead to ‘precipitation-hardening’ in manyalloys.25,26

The question arises as to what controls the shapes and sizesof precipitates, which are found to be strongly correlated. Forexample, in Al-rich Al1�xZnx, in which many experimentalstudies have been performed to examine the shapes andstructure of precipitates (see e.g. Ref. 27 and referencestherein), the Zn-precipitates exhibit a remarkable series ofshapes and sizes.28 Small precipitates tend to be spherical,until they reach a critical size RC of about 15–25 A(dependenton the aging temperature) at which point they becomeellipsoidal/plate-like with the short axis parallel to the[111] direction. As an example, Fig. 2 shows the size andtemperature dependence of the precipitate shape resultingfrom our MC simulations. It can be seen that the experimentalfeature mentioned can also be observed in our simulations.Indeed, as shown earlier, there is even a quantitativeagreement between our prediction and experimental data.29

The flattening along the [111] direction is a quite unexpectedresult because most fcc-based metal alloys are hard alongthis direction. The answer is given by the nature of Znitself. Although Zn is naturally an hcp element, in Al-richAl1�xZnx alloys, it is forced to have the fcc structure of



Figure 2. Dependence of the calculated coherent fcc-Zn precipitate shape on the number of Zn atoms and temperature in Al-Znalloys.21

the alloy. It thus develops instability30 in the form of ananomalously low energy for the [111]-deformed unit cell.Interestingly, this instability can be described in terms ofelastic constants. Probably, while the most known fcc-bccinstability (Bain path) is characterized by a negative valueof C11 � C12, the Zn instability is due to a negative value ofC44. As a consequence of this unusual behaviour, [111] is theelastically softest direction of fcc-Zn, resulting in extremelysmall (<1 meV/atom) coherency strain energies along thisdirection. This [111]-soft precipitates are embedded in amatrix of Al that has an elastically soft [100] direction. Whilethe [100] softness of Al has been shown to yield spinodaldecomposition fluctuations along this direction,31 it doesnot control the plate orientation of (relatively) large Zn-precipitates due to the well-known result of Khachaturyan,32

which states that the habit plane of a precipitate is determinedby the elastic constants of the precipitate phase, and not bythose of the matrix. So, for Zn-rich alloys the strain wouldbe dictated by the elastic response of Al, and hence wouldbecome more and more isotropic.

While the flattening of the Zn-precipitates can beexplained by the discussed instability, the facetting ofthe precipitates for T ! 0 K is caused by the directiondependence of the interfacial energy. In order to demonstratethis dependence, Fig. (3) shows via CE calculated interfacialenergies of Aln Znn superlattices (SL) as function of periodlength n for five different crystal orientations. It can be seenthat for small n the interfacial energies are not only direction-dependent but also depend on n. In other words, the chemicalinteractions at the interfaces dominate the interfacial energy.This effect can e.g. be of importance for ultrathin sandwichfilms in multi-layer systems. With increase in n the influenceof the interfaces on the interfacial energy disappears.

Figure 3. Via CE calculated interfacial energies for Aln Znn

superlattices as function of direction and period length n.

I(n, OG) is now determined by the direction-dependentstrain energy necessary to maintain coherency betweenthe blocks of Al- and Zn-layers. Hereby, the formation of(100) and (111) orientated interfaces are clearly the lowestin energy. Consequently, it is not surprising that for lowtemperatures only those precipitate-solid solution interfacescan be observed. For higher temperatures this facetting iswiped out by entropy.

Segregation at surfaces: the Pt25Rh75(111) surfaceBesides the technical importance of the Pt25Rh75(111) alloysurface for catalytic reactions,33 there is a more fundamentalargument as to why this surface was chosen for a predic-tion. Although the phenomenon of segregation is controlled


1162 S. Muller

by thermodynamics, it is by no means clear that this equi-librium can always be reached in experiment. In the caseof Pt25Rh75, however, the existence of an equilibrium seg-regation profile is manifested by chemically resolved STMimages,34 low energy ion scattering (LEIS), and quantita-tive low energy electron diffraction (LEED) analyses.35 Thesestudies unambiguously show that for annealing tempera-tures above ¾1000 K a segregation profile can be observed,which no longer depends on the experimentally chosenannealing temperature of the sample.

The bulk phase diagram of the binary system Pt-Rh doesnot show any long-range ordered structures. Instead, an fcc-based solid solution is stable over the whole concentrationregime. These observations are manifested by our DFTcalculations for different intermetallic compounds and quasi-random structures speaking for a weak preference ofordering with negative formation enthalpies between 0and about �20 meV/atom, i.e. smaller than kT at roomtemperature. Regarding the Pt25Rh75(111) surface, chemicallyresolved STM pictures34 find short-range substitutionalordering in the top layer in the form of a preference forchemically unlike nearest neighbours. In agreement withquantitative LEED structure determinations35 there is asegregation of Pt atoms to the top layer resulting in a Ptconcentration of about 70%. The opposite is true for thelayer underneath. A depletion of Pt is found, while thethird and fourth layers from the top show nearly bulk-likeconcentrations. These findings will now be compared withour predictions resulting from a CE for this alloy surface incombination with the MC simulations.

Owing to the small formation enthalpies for the orderedcompounds mentioned above, all the constructed effectivecluster interactions JF are also unusually small, possessingenergy values smaller than 25 meV per atom. These smallinteractions cannot explain the characteristic segregationprofile found for this surface, namely, the Pt enrichmentin the top layer, and the Pt depletion in the second layer.Our calculation found that the only relevant deviation of

the surface’s energetic properties from those of the bulkcomes by the surface itself. Owing to the symmetry break,the on-site energies of the individual atomic sites, whichare defined by J0 and J1 in Eqn (2), are different for thenear-surface layers. For only weakly ordering systems as thePt25Rh75(111) surface, these on-site energies represent a goodmeasure for the segregation behaviour. Actually, it turnsout that the top layer shows a tremendous tendency for anenrichment with Pt atoms reflected by an energy gain ofabout 0.2 eV per atom! Interestingly the opposite is true forthe layer underneath. Here, the on-site energy speaks for Ptdepletion and clustering of Rh atoms.

For an extension of our simulation to finite tempera-ture, the surface CE Hamitonian was installed in a classicalMetropolis MC algorithm. The MC simulations were per-formed for temperatures between 100 and 5000 K. A 40 ð 40atom cell per layer was used in the simulations, whichappeared to be sufficient for a quantitative description of thesegregation profile as well as for the substitutional ordering(see below). For four-surface layers this leads to 6400 atomicsites differing in their energetics from those of the bulk. Itturns out that above 1000 K the segregation profile is nearlytemperature-independent. For comparison we used experi-mental data35 corresponding to an annealing temperature of1400K. As mentioned above, for the Pt25Rh75(111) system thistemperature ensures that the equilibrium segregation profileis reached. Our MC simulations find that the Pt concentra-tion in the top layer decreases from 100% as expected forT D 0 K to about 70%, in excellent agreement with STM andLEED. Therefore, this depletion of Pt atoms must be drivenby entropy. Figure 4 compares an STM image with atomicand chemical resolution34 with our predicted one. It can beseen that there is excellent agreement between experimentand theory.

In order to determine the detailed atomic structuralproperties as multi-layer relaxation or buckling amplitudesof individual atoms, the configuration that is lowest information enthalpy and, additionally, close to the determined

Figure 4. Pt25Rh75(111) surface: Comparison between chemically resolved STM picture and our prediction (white atoms are Rhatoms; top view).



Figure 5. Comparison between experimentally (LEED)determined and predicted atomic structure. The interlayerdistances dij are described as deviation from the ideal bulklayer spacing. The dependence of the first interlayer spacingd12 from the chemical species makes clear that statisticalbuckling of top layer atoms takes place.

equilibrium segregation profile must be found. Since the CEallows for a grid search of huge configuration spaces withina few hours, millions of configurations can be considered insuch a ground-state search. Here, it turns out that alreadya simple (2 ð 2) supercell with four basis atoms per layeris sufficient to study the detailed atomic structure of thealloy’s surface. Figure 5 compares the structural parametersresulting from the DFT calculation for a long-range ordered(2 ð 2) supercell and from the quantitative LEED analysis35

We see that the geometrically fully relaxed structure showsa statistical buckling in the top layer, with Pt atoms beingon average clearly above the average layer plane as wasalready found before in the LEED analysis. In total, thereis an excellent agreement between the structural parametersdetermined by DFT and LEED.

SUMMARY

The combination of DFT calculations with CE and MCsimulations represents a powerful tool for studying interfaceproperties in binary systems from first-principles because itallows us to solve two fundamental problems in modellingalloy surfaces – treating huge model systems and hugeconfiguration spaces. The use of CE Hamitonian in MCsimulations permits us to extend our theoretical studiesto finite temperature. Since the construction of effectiveinteractions is based on an electronic structure theory,the access allows us to investigate the reasons for theobserved behaviour. While in the case of Zn-precipitatesin an Al-Zn solid solution the preference for (111) orientatedinterfaces is stabilized by the interfacial energies, in the

case of a Pt25Rh75(111) surface, the experimentally observedsegregation profile can be simply explained by the on-siteenergies of the near-surface layers. All results are in excellentagreement with experimental data.

AcknowledgementsThe author is indebted to Deutsche Forschungsgemeinschaft (DFG)for financial support. Special thanks to my group members OleWieckhorst and Markus Stohr.

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