Surfaces are the cutting edge of materialsscience. Molecules from the environ-mental gas or liquid phase come in con-tact with a material at its surface, and

chemical bonds of these approaching moleculesare cut and new bonds formed. To understandhow materials function, and how to produce andenhance them, we must understand surfaces.

We focus here on advances in this field, specif-ically, the development of methodologies thatcombine density-functional theory with elasticitytheory, thermodynamics, or statistical mechanics.The resulting approaches, while computationallyelaborate, let us treat thousands of atoms to followtheir wanderings and interplay over time scalesfrom picoseconds up to seconds. Most important,the methods let us analyze the results and gainmuch insight into how surfaces function.

The examples we discuss relate to semicon-ductor growth and nanotechnology, but the ap-proaches have much wider application: Re-searchers are using analogous methods to modelthe growth of thin magnetic metal films,1 catal-ysis, and corrosion.2

Advancing materials scienceMaterials science seeks to design new materi-

als, improve the quality of existing materials, andmake the production process more efficient—that is, cheaper. Current experimental methodsto develop new materials still use trial and error.Theoretical modeling of technologically rele-vant chemical processes still employs mostlyphenomenological methods (such as rate equa-tions or hydrodynamic theories) together with“effective” parameters. These parameters typi-cally have limited physical meaning and applyonly within narrow temperature, pressure, ormaterial composition limits. Some researchersrefer to this as “modeling without microscopicunderstanding.” Despite the term’s negativeslant, this approach has proven the most valu-able to date.

Clearly, however, we need improved methods,and we posit that the next step in basic researchinvolves developing a theoretical approach “withmicroscopic understanding.” This article de-scribes our progress in and perception of devel-oping such a predictive theory of materials.

ChallengesUnderstanding material surface properties has

become critical to improving the technology onwhich our modern life is built. For example,semiconductor production involves depositing

16 COMPUTING IN SCIENCE & ENGINEERING

SURFACE KNOWLEDGE: TOWARD A PREDICTIVE THEORYOF MATERIALS

To understand how materials function, we must understand how molecules from theenvironment interact with their surfaces. This article describes new methodologies forstudying surface dynamics at the atomic level over a range of time scales.

M A T E R I A L SS C I E N C E

PETER KRATZER AND MATTHIAS SCHEFFLER

Fritz-Haber-Institut der Max-Planck-Gesellschaft

1521-9615/01/$10.00 © 2001 IEEE

NOVEMBER/DECEMBER 2001 17

thin layers of a semiconductor, insulator, ormetal, then structuring and processing thesefilms with various techniques. Molecular mod-eling can advance our atomistic understandingand hence implementation of these processes.

We stress general principles of molecularmodeling by focusing on two examples:

1.deposition of modern compound semicon-ductors, such as GaAs and InAs, and

2. spontaneous structure formation in thesesystems, the so-called self-assembly ofnanoscale islands.

If these islands are embedded in a matrix ofsemiconducting material, covering them with acapping layer, they can be used as quantum dots.These nanoscale structures confine charge car-riers (electrons or holes) due to quantum-me-chanical effects. Researchers already use themfor light-emitting diodes, lasers, and single-elec-tron transistors, and future applications mightinclude quantum computing.3

Modeling thin film deposition requires de-scribing a sample area of at least mesoscopic size,say 1000 nm2, involving several thousands ofatoms. The time scale the simulation coversshould correspond with the actual time neededto deposit one atomic layer; that is, seconds.

However, the atomistic processes that governthe physics and chemistry of deposition, adsorp-tion, and diffusion operate in the length and timedomains of 0.1 to 1 nm and femto- to picosec-onds. To model film growth and incorporate in-formation from the atomistic properties, there-fore, we must cover huge length scales and timescales: from 10–10 to 10–6 m and from 10–15 to100 seconds. Figure 1 shows the range of scalesinvolved.

Smaller length scales of a few hundred atomstypically suffice to discern, for example, theatomic structure of a step on a surface and itsrole for chemical reactions and atom diffusion.To produce trustworthy molecular modeling,however, we must address the gap between theatomic and the practically relevant time scales,and the crucial role of statistical mechanics.

Simulating crystal growth requires coveringlarge time scales because often rare events rulethe whole scenario. The events are rare becausesome atomistic effects are thermally activated inmaterials processing and occur with an exponen-tially small probability yet might ignite impor-tant, possibly crucial follow-up surface modifica-tions. The rare-event problem precludes the use

of straightforward (brute force) molecular-dy-namics (MD) techniques (both empirical and,even more so, ab initio MD) to simulate thermallyactivated processes at realistic temperatures.

Researchers have devised special MD tech-niques to alleviate this difficulty,4 but even withthese techniques only certain problems in statis-tical mechanics are accessible to MD simulations.The interplay and interdependence between dif-ferent molecular processes presents an even moresevere—and not often apparent—problem. Aprocess’ importance, measured as how often it oc-curs during a given time interval, relates not justto its intrinsic rate but also to other processes re-quired to create, replenish, or prolong the life-time of an important precursor configuration.

Such interdependencies might emerge only afterfairly long simulations—perhaps milliseconds (asin the example of GaAs growth discussed later)—and after extensive sampling of the system’s phasespace, yet they significantly affect the simulation’soutcome. In our example of film growth, the mor-phology of the growing film would be affected. Wewill show here how combining density-functionaltheory (DFT) with kinetic Monte Carlo (kMC)simulations offers an efficient and accurate way toaccess sufficiently long time scales.

While many aspects of materials science andmolecular modeling require bridging length andtime scales, the systems we focus on here havean important similarity that facilitates the treat-ment and that we consequently exploit: Duringadsorption, diffusion, and epitaxial growth, thematerial retains its basic crystalline structure.This lets us assign a crystal lattice to the atoms’possible stable or metastable positions, while(necessarily) including atomic displacements

10–15

1

10–3

10–6

10–9

10–9

Leng

th (

m)

Macroscopicregime

Mesoscopicregime

Electronicregime

Time (s)10–3 1

Figure 1. Molecular modeling on the basis of first-principles electronic structure calculations requirescovering the length and time scales from the electronic to the mesoscopic or even macroscopic.

18 COMPUTING IN SCIENCE & ENGINEERING

away from the lattice sites. Such atomic dis-placements result from strain fields or the ab-sence of a full coordination shell of neighbors,such as near a step edge or defect, or occur as a(maybe local) surface reconstruction.

The simulation’s reliability and computationalcost depend on the quality level chosen to de-scribe the elementary processes—that is, how wetreat the electrons and the atom–atom interac-tions. Traditional approaches have been empiri-cal—for example, bond-strength–bond-orderpotentials or even simpler bonding descriptions.When chemical bonds between atoms break ornew bonds form, however, we need a descriptionthat accounts for the electrons’ quantum-me-chanical nature. This applies to surfaces in gen-eral (since bonds present in the bulk had to bebroken to create the surface), and even more soto chemical reactions occurring on them.

Recent work shows considerable progress inunderstanding surfaces and surface chemical re-actions using ab initio methods, particularly usingDFT for total-energy calculations.5 Analogousto the multiplicity of scales shown in Figure 1,we propose a multistep approach:

1.We perform DFT calculations for possiblycomplex system aspects that don’t requireknowledge of the full size of the system be-ing modeled. Examples include a particularfacet of a crystal, or a set of atomic configu-rations that occur during growth. Usually,before modeling can begin, we must collectdata from many such DFT calculations in adatabase.

2.Using this database, we run code that per-forms the simulation—that is, treats thethermodynamics or statistical mechanics.

3. We propose a learning approach for futurework, starting with an approximate descrip-tion of the atomic interactions by analytic po-tentials. This will let us explore a huge num-ber of molecular processes with modestcomputational effort, perhaps by using somespecial accelerated MD technique. As thesimulation proceeds, we identify the impor-tant processes to be calculated with improvedaccuracy using DFT, then use these DFT re-sults to improve the database obtained fromthe initial (semi-empirical) description.

Density-functional theory calculationsTo describe bonding in a solid or at its surface,

we must calculate its electronic structure, which

implies a quantum-mechanical description of themany-electron problem.5 In computationalterms, electronic degrees of freedom and theirassociated length scale come into play, which canmake simulations costly. DFT represents an ef-ficient approach to describe quantum effects inthe electronic structure of a polyatomic system.

This approach is exact in principle, but despiteDFT’s remarkable success in the past decade, wemust remember that in daily practice we areworking with approximate density functionals,whose performance we must assess for each sys-tem at hand. Once we’ve selected an appropri-ate density functional, a DFT calculation of theelectronic structure amounts to solving separateSchrödinger-like equations for all electrons inthe system, with an additional potential ac-counting for the electrons’ many-particle (ex-change and correlation) effects.

Although they substantially simplify the nu-merical task compared to many-particle meth-ods, DFT calculations still require much datahandling because during the calculation we muststore and access the information pertinent to thewave functions of all electrons in the problem.The computationally most straightforward wayof representing the (complex-valued) wave func-tions is to store them on a grid, either in realspace or by storing their Fourier coefficients.Many computers offer efficient fast Fouriertransform routines that permit switching be-tween wave function representations in realspace and in reciprocal (wave-vector) space.These algorithms form the heart of modernelectronic structure codes.

So far, we’ve discussed computer codes usingthe pseudopotential–plane-wave method, which ex-plicitly treats only the atoms’ valence shell elec-trons. This method creates efficient codes thatpermit calculations on several hundred atoms onparallel architectures.6 Methods for solving thefull problem for all electrons numerically, suchas the linearized augmented plane-wave(LAPW) method, are also available and have alsobeen ported to parallel computers,7,8 but thehigher demands they place on computer re-sources to describe the entire electronic systemrestricts them to problems involving feweratoms, presently up to about 100.

Figure 2 illustrates how we use DFT calcula-tions relevant to the modeling of island nucleationand crystal growth by molecular beam epitaxy(MBE). We model a crystal surface by repeatingthe group of atoms shown in Figure 2 periodi-cally. Figure 2a shows a cut perpendicular to the

NOVEMBER/DECEMBER 2001 19

surface in side view, with As atoms displayed inblue and Ga atoms in green. The yellow/orangecontour surface indicates the valence electrondensity. Closed shells of electron density surroundthe As anions, while the Ga cations are depletedof electronic charge. Upon adsorption (Figure2b), the As2 molecule’s electron density agglom-erates with the substrate’s electron density. We offer a more detailed animation of this process at www.fhi-berlin.mpg.de/th/publications/img/ie-as2-alpha.gif.

Figure 2 shows an As2 molecule adsorbing onthe GaAs(001) surface where some Ga atomswere deposited before. Surface scientists refer tothis surface as being locally in the α-reconstruc-tion, a state characterized by bonds between twosurface Ga atoms (a feature absent in the bulk).Adsorption of the As2 molecule breaks theGa–Ga bonds while As–Ga bonds form simul-taneously between the molecule and the surface.The DFT calculation lets us monitor thisprocess by inspecting the electron density ateach stage of the approaching As2 molecule. Inthis way we gain insight into the nature of chem-ical reactions occurring at the surface. Thisknowledge, together with the calculated total en-ergies, forms part of the input for using statisti-cal mechanics to understand growth.

Modeling island nucleationWhile the DFT calculations just described

consider zero temperature and pressure, model-

ing of thin film deposition must treat finite tem-perature and pressure, and we must even go be-yond thermodynamic equilibrium theory: A fluxof deposited atoms or molecules implies we aredealing with an open system; we therefore needa kinetic description of molecular processes suchas adsorption, desorption, diffusion, and even-tually island nucleation and growth.

Our approach exploits the fact that we canrepresent the atomic configurations that occurduring growth on a crystal lattice. We thus re-place the physical (Newtonian) surface atom dy-namics with a “discrete dynamics”: The atomscan move from one lattice site to another or ap-pear on and disappear from the surface (adsorp-tion and desorption) by discrete jumps; aftereach jump, the system equilibrates in the newgeometry before a transition to another state oc-curs as a thermal fluctuation. Thus, at the tem-peratures used for the film deposition, thesejumps constitute rare events in the sense de-scribed earlier.

By concentrating on the jumps, we eliminatethe (mostly) superficial information about atomdynamics between rare events—that is, the dis-crete states’ vibrations and the associated timescale. We don’t neglect this information butrather describe it in terms of a probability pref-actor, discussed later. Each discrete event entersin the simulation with a rate we can determinerigorously to account for the dynamics on theeliminated time scale.

We can demonstrate this for a simple exam-

Figure 2. DFT electronic-structure calculation for the adsorption of an As2 molecule on the GaAs(001)surface. In (a), the As 2 molecule hovers above the surface (yellow balls in the upper part of thepicture). In (b), the molecule has adsorbed onto the surface and established chemical bonds with thesurface Ga atoms. The closed yellow surfaces represent contours of constant electron density. The bluedots represent the As nuclei; the green dots, the Ga atoms.

(a) (b)

20 COMPUTING IN SCIENCE & ENGINEERING

ple: a single particle’s surface wanderings.9 Onthe atomistic level, the particle’s motion is gov-erned by the potential-energy surface (PES),which is the potential energy experienced by thediffusing adatom

, (1)

where Etot (Xad, Yad, Zad, {RI}) is the ground-stateenergy of the many-electron system (also re-ferred to as the total energy) at the atomic con-figuration (Xad, Yad, Zad, {RI}). The PES repre-sents the minimum total energy with respect tothe z-coordinate of the adatom Zad and all coor-dinates of the substrate atoms {RI}. If we disre-gard vibrational contributions to the free energyfor a moment, the minima of the potential-en-ergy surface represent the adatom’s stable andmetastable sites.

The kinetic description of growth defines theunderlying atomistic processes, such as diffusionor desorption, in terms of the probabilities withwhich they occur. Under rather mild assump-tions usually fulfilled for processes at a surface,for the rate ΓJ of a molecular process J (diffu-sion, for example), the rate law reads

, (2)

where ∆FJ is the difference in the Helmholtzfree energy between the maximum (saddle point)and the minimum (initial geometry) of the PESalong the reaction path of the process J. T is thetemperature, kB the Boltzmann constant, and hthe Planck constant. The following equationgives the free energy of activation ∆FJ that thesystem needs to move from the initial positionto the saddle point:

. (3)

The internal energy U in this expression con-sists of the (static) total energy EPES and the vi-brational energy Uvib. ∆UJ is the difference ofthe system’s internal energy with the particle atthe saddle point and at the minimum belongingto process J, and ∆Svib

J is the analogous differ-ence in the vibrational entropy. We cast the rateof the process J as follows:

, (4)

where

,

represents the attempt frequency.We obtain both basic quantities in Equation

4, Γ0J and ∆EJ, from DFT calculations. For sin-

gle-particle diffusion, we can read ∆EJ directlyfrom the PES. In this case, we find that Γ0

J is ofthe same order, roughly 1013 s-1, for variousprocesses. As a first approximation, we can thusoften avoid recalculating Γ0

J for each process, be-cause variations in Γ0

J by a factor of 5 are typi-cally irrelevant compared to variations of ∆EJ inthe exponent. However, when many atoms aremoving in a collective manner, the prefactormight be significantly different. For processesinvolving several particles, such as nucleation,attachment to islands, or motion along theatomic step that marks an island edge, we mustaccount for the interactions between particles.We can also access these quantities with DFTcalculations by calculating the PES, or partsthereof, for all microscopic situations that occurduring growth.

However, this involves performing calcula-tions for many different atomic arrangements.The near-sightedness of nature substantially re-duces the workload: We can usually describe asurface atom’s local chemical bonding by con-sidering a local environment of its nearest andnext-nearest neighbors, provided we properlyconsider the boundary conditions. This near-sightedness reduces a simulation to a recurrenceof a possibly large but finite number of local con-figurations.

To obtain these configurations’ binding ener-gies and barriers, we perform DFT calculationsand collect the results. Next, a computer pro-gram performing a kinetic Monte Carlo growthsimulation reads the barriers from a file and cal-culates a list of rates according to Equation 4.Each simulation step selects a particular processrandomly with a probability proportional to itscontribution to the total rate. The simulationtime advances by an increment given by the in-verse total rate multiplied by a random numberdrawn from an exponentially decaying probabil-ity distribution.

This establishes a physically meaningful map-ping between simulation time and physical time,thereby exploiting the fact that the microscopicprocesses eligible within an atomistic time in-terval can be treated as independent Poissonianprocesses.10 A growth simulation consists of bil-lions of such simulation steps.

Γ J B J Jk T h S k U k T0 = ( ) ∆ − ∆( )/ exp / /vibB

vibB

Γ Γ ∆J J JE k T= −( )0 exp B

∆ ∆ ∆F U T SJ J J= − vib

Γ ∆J Jk T

hF k T= −B

Bexp( / )

E X Y

E X Y ZZ R

II

PES

totad ad ad

minad

ad ad,

, , ,,

( ) =

{ }( ){ }R

NOVEMBER/DECEMBER 2001 21

Introducing a lattice-gas Hamiltonian11,12 letsus conveniently represent the atomic interactiondata obtained from DFT calculations. We dis-cuss this approach schematically for a specificsystem, homoepitaxy of GaAs by MBE fromatomic Ga and molecular As2 sources on theGaAs(001) substrate. Under usual MBE growthconditions, the GaAs(001) surface displays a sur-face reconstruction terminated by pairs of Asdimers, which alternate with “trenches” runningin the [

–110] direction. These trenches stem from

a missing pair of As dimers in the surface layerand two missing Ga atoms per unit cell in thelayer below.

MBE experiments on GaAs(001) and DFT cal-culations13,14 show that arsenic molecules and gal-lium atoms interact quite differently with this sur-face. While Ga atoms adsorb with unit stickingprobability, As2 molecules stick to the surface onlyafter Ga deposition. Surface diffusion distributesthe deposited Ga atoms. Growth depends on Gamobility because strong As2 chemisorption sitesonly arise after a suitable local arrangement of Gaadatoms. Once such a site forms, the As2 mole-cules can adsorb without dissociation and becomepart of the surface in the form of As dimers. Con-versely, As dimers can leave the surface by ther-mally activated desorption.

To describe As dimers’ binding energy at thereconstructed surface and Ga adatom binding atvarious sites, we propose a lattice-gas Hamil-tonian that consists of a single-site contributionH0 plus an interaction term:

H = H0 + Hint . (5)

Let k be a shorthand notation for a lattice siteand nk a discrete variable that takes on the value 1if an As dimer occupies the site or 0 if it’s empty.Similarly, mk is an occupation variable describingthe Ga adatoms. Then H0 takes on the form

. (6)

The reconstructed GaAs(001) surface actuallypresents some complexity because of the need todistinguish several inequivalent lattice sites dueto the reconstruction. To clarify the basic idea,we leave these complications aside for the mo-ment. We describe deviations of the energeticsfrom the single-site values due to different re-constructions and different possible arrange-ments of Ga adatoms in the interaction part of

the lattice-gas Hamiltonian,

.(7)

Here, the indices k + a and k + b denote nearestand next-nearest neighbor sites relative to k. DFTcalculations performed for a set of atomic config-urations determine the interaction parameters

.

Adding more interaction types further refinesthe description by the lattice-gas Hamiltonian,if required.

To set up a kMC simulation, we need thebinding energies as well as the energy barriersto calculate the rates. These barriers show con-siderable diversity because each specific process,from an initial state with a given local environ-ment through a transition state to a final statewith a different local environment, requires aspecific barrier. We must be careful not to vio-late the principle of detailed balance. This fun-damental principle of statistical mechanics re-quires that the quotient of any forward andbackward process rates, obtained by interchang-ing initial and final states, equals

, (8)

where ∆F shows how the Helmholtz free energydiffers in the initial and final state (or the Gibbsfree energy, if desorption into the gas phase isinvolved).

Figure 3 shows what we can learn from kMCsimulations, specifically, a DFT+kMC simulationof GaAs homoepitaxy on GaAs(001) at T= 700 K,Ga flux 0.1 monolayers/s, and As2 pressure p =0.85 × 10–5 mbar. It shows a small part of the totalsimulation area, with a “trench” typical for theGaAs surface reconstruction in the center. Ga andAs substrate atoms appear in green and dark blue,Ga adatoms in yellow, and freshly adsorbed Asdimers in light blue. For the full movie, go towww.fhi-berlin.mpg.de/th/publications/img/isl-gaas-front.mpg.13

At the chosen temperature and pressure con-ditions we observe island nucleation in the re-constructed surface’s trenches. Island growth

Γ Γ ∆fi if F k T( ) ( ) = −( )exp B

V V V V Vn n n n1 2 1 2As As Ga Ga

trioGa and , , , ,

H V n n V n n

V m m V m m

V m m m

n k k ak a

n k k bk b

n k k a n k k bk bk a

k k a k ak a a

int, ,

,,

, ,

...

= +

+ +

+ +

+ +

+ +

+ + ′′

∑ ∑

∑∑

∑

12

12

12

12

13

1 2

1 2

As As

Ga Ga

trioGa

H E n E m

V n m

b kk

b kk

k kk

0 = +

+

∑ ∑∑−

As Ga

Ga As

22 COMPUTING IN SCIENCE & ENGINEERING

proceeds along the trench, thereby extendinginto a new layer.

Unlike MBE experiments, these simulationslet us study how temperature affects thegrowth morphology in a wide range and at theatomic level. By varying the temperature usedin the simulations, we find that Ga surface dif-fusion alone causes the island morphology fortemperatures up to about 700 K. At highertemperatures, the adsorption of As2 moleculesat reactive surface sites becomes reversible,and arsenic losses due to desorption becomeapparent. The temperature window MBEcrystal growers frequently use permits a com-promise between high Ga adatom mobilityand stability of As complexes that leads to lowisland density.13,14

Modeling strained heterostructureswith thousands of atoms

For growing GaAs films, crystal growers use,for example, molecular beam epitaxy to depositgallium (Ga) and As2. With this technique, it iseasy to grow heteroepitaxial films by switchingthe Ga flux at some point to another material,such as indium (In). The mechanical strain re-

sulting from the misfit between the substrate andfilm crystal lattices is often critical to the result-ing film morphology, and under certain condi-tions the spontaneous formation (self-assembly)of quantum dots, consisting of several thousandatoms each, will occur. Here we focus on themethodology to calculate the size and shape ofsuch mesoscopic structures, assuming they arein thermodynamic equilibrium.

A simplified picture of heteroepitaxial growthshows two limiting cases. In the first, the de-posited material forms a smooth homogeneouslystrained film; in the second, the deposited mate-rial evolves into 3D islands. Frequently, these is-lands are pseudomorphic with the substrate—that is, the deposited material lattice continuesthe substrate host lattice without perturbationssuch as dislocations or grain boundaries being in-troduced. As a consequence, the islands are elas-tically strained at their base, where the depositedmaterial is forced to take on the substrate’s lat-tice constant, but the strain is relaxed near the is-land top, where the atomic layers can take on thegeneric lattice constant of deposited material, asFigure 4 shows.15 This strain relief constitutesthe thermodynamic driving force that favors theformation of 3D islands. On the other hand, the

Figure 3. Snapshots from aDFT+kMC simulation of GaAshomoepitaxy. Ga and As substrate atoms appear ingreen and dark blue, Gaadatoms in yellow, and freshly adsorbed As dimers in light blue. (a) Ga adatoms preferentially wander around in the trenches. (b) Under thegrowth conditions used here,an As2 molecule adsorbing on a Ga adatom in the trench initiates island formation. (c)Growth proceeds into a newatomic layer via Ga adatomsforming Ga dimers. (d) Eventu-ally, a new layer of arsenic startsto grow, and the island extendsitself towards the foreground,while more material attachesalong the trench.

(a) (b)

(c) (d)

NOVEMBER/DECEMBER 2001 23

tendency to minimize the surface free energy fa-vors the formation of a smooth film.

In fact, real system behavior often lies some-where between the two limiting cases sketchedearlier: A smooth film may form after depositiononly up to a certain thickness, and 3D islandgrowth takes over if more material is deposited.This results from an energetic trade-off: whileelastic strain relief would favor formation of is-lands rather than a smooth film, additional sur-faces forming at these islands’ side facets cost energy.

In a real system, whether and how large islandsform depends on the energetic balance betweenthe surface contribution and an elastic contribu-tion from both the islands’ bulk and the sub-strate. Under such conditions, island formationwould follow what’s known as Stranski-Kras-tanov growth mode. We’ve developed an ap-proach to understanding this delicate balance bycombining results from the continuum elastic-ity theory of solids with calculated surface for-mation energies obtained from density-func-tional theory.

To illustrate, we discuss a specific example, theformation of small InAs islands after depositionof an InAs film on a GaAs substrate. This com-bination typically yields islands with a baselength between 10 and 20 nm in the experiment,as Figure 5 shows, and such islands consist of10,000 to 50,000 atoms each. Most atoms inthese islands have a full coordination shell ofneighbors and undergo only small relaxationsfrom their bulk positions. Only the surfaceatoms have a substantially different atomic en-vironment and experience larger displacementsdue to the surface reconstruction.

Although the elastic relaxations release littleenergy per bulk atom, bulk elastic effects can besignificant—particularly for the largest islandsbecause the elastic energy scales linearly with thenumber of atoms in the island. Other energeticcontributions coming from surface energychanges during island formation counteract theelastic contribution. We compare the total en-ergy of an InAs film (thickness Θ0; see Figure 5,right) with the islands’ energy at a given densityand a thinner film (thickness Θ; see Figure 5,left). Obviously, both situations must involveequal amounts of material.

To understand the energetics driving islandformation, we find it neither necessary nor fea-sible to employ a full DFT description of allatoms involved but have instead developed a hy-brid approach. We write the key quantity, the

energy gain per unit volume (or per atom), as asum of three contributions,

Etot/V = Eelast/V + Esurf/V + Ew1/V, (9)

where Etot stands for the total energy gain of anisland with volume V. Eelast shows the differencein elastic strain energy between a situation withislands and a homogeneously strained film. Weevaluate this quantity, involving small displace-ments, by numerically solving the elastic contin-uum equations, using, for instance, the finite-el-ement method. Mechanical engineers regularlyuse commercially available software for a similartask, calculating strain in structures such as

Figure 4. Strain distribution, as obtained from finite-element calculations15 for (a) a pyramidal and (b) a truncated InAs island coherently grown on a GaAs(001) substrate. The color indicates thetrace of the strain tensor on (010) cross sections through the islands.

(a)

(b)

0.1500

0.1125

0.0750

0.0375

0.0000

–0.0375

–0.0750

–0.1125

–0.1500

InAs island

Substrate

(101)

(101)(001)

a

b

InAs Θ0 MLΘGaAs

Figure 5. Schematic illustration of the formation of coherent islands on the substrate surface. Θ0represents the nominal coverage (the total amount of deposited material) and Θ is the wetting-layerthickness, measured in monolayers (ML).

24 COMPUTING IN SCIENCE & ENGINEERING

bridges and parts of machinery. The extrapola-tion of this approach to nano-sized structures isjustified as long as they consist of several thou-sands of atoms.

The Esurf and Ewl contributions come from theextra surfaces that islands create and from ener-getic changes due to thinning of the wettinglayer, respectively. DFT calculations, using slabgeometry to model one unit mesh of the surfacereconstruction, work well for these quantities,which originate from surface bond breaking anddistortions. Such calculations require only mod-est numbers of atoms and are feasible on a mod-ern workstation.

We must, however, perform many calculationsto study various surface compositions and recon-structions for each surface orientation. Compo-sition variations make the surface energies, andthus the islands’ shape, dependent on the gas-phase environment’s temperature and pressure.We can retain the elastic effects near the surfacethat differ from bulk elasticity in Equation 9 ifwe include contributions to the surface energylinear in the strain εjk. From DFT slab calcula-tions we obtain surface energies γ(i) of the ith facetand the coefficient σ(i)

jk preceding the linear term,which is unique to surface elastic properties.Likewise, we calculate the formation energy ofthe wetting layer γwl(Θ) within DFT. Thus wedesignate the surface energy contribution

, (10)

where A(i) denotes the area of the island’s ith sidefacet, and A0 its base area.

Equation 9 also describes the thinning of thewetting layer, defined as

,(11)

where γwl(Θ) is the combined surface and interfaceenergy of the wetting layer, and n is the area densityof islands. DFT calculations for thin films showthat γwl(Θ) is a function of the wetting layer thick-ness Θ. Only for films thicker than two atomic lay-ers of In does γwl(Θ) approach a constant. This hasimportant consequences for the distribution of de-posited material between islands and film.

The total-energy expression in Equation 9 letsus quantitatively study various island properties.Once we understand the shape-dependence ofthe elastic energy and the surface energies, we

can systematically study quantum dots’ stabilityas a function of their shape. To this end, we’veevaluated the energy gain Erelax + Esurf for a singleisland, for many trial shapes. Optimizing theshape results in an island with a flat top that dis-plays all low-index side facets. The exact shapedepends on the island’s size; in particular, the rel-ative importance of the island top’s horizontalfacet decreases and the island’s aspect ratio in-creases as the islands grow bigger.14

The above treatment relied on a thermody-namic argument, following the principle that agiven amount of material actually takes on theshape that corresponds to its lowest possible(free) energy. We suggest using the argumentwith some care, however, as the quantity Etot/Vfrom Equation 9 is a monotonically decreasingfunction of V. Thus, full thermodynamic equi-librium would correspond to an unrealistic sit-uation where all deposited material has collapsedinto a large single island. We can easily remedythis flaw by restricting ourselves to a constrainedequilibrium, the constraint being a fixed islanddensity n physically determined by growth con-ditions during early stages of deposition. Apartfrom being more physical, this theory lets us ob-tain information about not only the islands’shape but also their optimum size, once we knowthe island density and the total amount of de-posited InAs. The optimum island size corre-sponds to a minimum of Etot/V in Equation 9when evaluated for fixed values of n and Θ0.

Closely linked to this optimum island size isthe remaining thickness Θ of the wetting layerafter the islands develop fully. Only 10 to 30 per-cent of the total deposited material assemblesinto islands, depending on the nominal cover-age Θ0 and the island density n. The results alsoshow that a wetting layer thicker than onemonolayer always exists even after three-dimen-sional island formation has set in. This observa-tion, and the calculated island sizes and aspectratios, correspond with recent results of scan-ning-tunneling microscopy investigations.

Our modeling points to a possible method of at-taining a desired island size by properly choosingthe growth conditions and nominal coverage.16,17

Nearly everything in materials sci-ence proceeds from chemical bondformation and dissolution. At sur-faces, this typically involves the col-

lective behavior of many atoms, between 10 and200, which we can effectively calculate usingDFT. DFT gives us the total energy or PES on

E n Aw w w1 0 1 101= −( ) ( ) − ( )[ ]γ γΘ Θ

E

A Aijki

jki

jk

i w

i

surf =

+

− ( )( ) ( ) ( ) ( )∑∑ γ σ ε γ 1

00Θ

NOVEMBER/DECEMBER 2001 25

which the various atoms move.Sometimes the minima of this potential-en-

ergy surface suffice for calculations, as theseminima identify the stable and metastablegeometries—that is, the thermal-equilibriumstructures. Thermodynamic equilibrium theorydoes not always apply, however, because manysystems are in a local energy minimum. Under-standing which of the many PES local minimadetermine the system’s properties requires ana-lyzing the atomic motion along the potential en-ergy surface.

In other words, PES computation is not thefinal step but forms the basis for description ofthe atom dynamics. This, in turn, determines theprobability of molecular processes (such aschemical reactions) occurring. For the full pic-ture, then, in addition to the knowledge of thePES, we need a description of atomic motion,specifically, how atoms diffuse and what happenswhen they bump into each other.

Predictive modeling requires distinguishingbetween significantly more process types thansemi-empirical simulations, but we’ve shownsuch studies to be feasible. For example, theGaAs growth simulation presented here includesmore than 30 process types, each with a clearatomistic, physical meaning. The techniqueneeds further refinement, and we will learn toavoid pitfalls with each new case study. Already,however, first-principles kMC simulations haveprovided significant new insight into surfaceprocesses and their interplay.

References1. R. Pentcheva et al., “Initial Stages of Heteroepitaxy with Inter-

mixing: Co on Cu(001),” to be published.

2. C. Stampfl et al., “Catalysis and Corrosion: The Theoretical Sur-face-Science Context,” to be published in Surface Science, vol.500, 2001.

3. R. Hughes, “Quantum Computation,” Computing in Science &Eng., vol. 3, no. 2, Mar./Apr. 2001, p. 26.

4. M.R. Sørensen and A.F. Voter, “Temperature-Accelerated Dy-namics for Simulation of Infrequent Events,” J. Chemical Physics,vol. 112, no. 21, June 2000, pp. 9599–9606.

5. K. Horn and M. Scheffler, eds., Handbook of Surface Science, Vol.2: Electronic Structure, Elsevier, Amsterdam, 2000.

6. M. Bockstedte et al., “Density-Functional Theory Calculations forPoly-Atomic Systems: Electronic Structure, Static and Elastic Prop-erties and Ab Initio Molecular Dynamics,” Computer PhysicsComm., vol. 107, nos. 1–3, Dec. 1997, pp. 187–222; www.fhi-berlin.mpg.de/th/fhimd (current 21 Sept. 2001).

7. A. Canning, W. Mannstadt, and A.J. Freeman, “Parallelization ofthe FLAPW method,” Computer Physics Comm., vol. 130, no. 3,Aug. 2000, pp. 233–243.

8. R. Dohmen et al., “A Parallel Implementation of the FP-LAPWMethod for Distributed-Memory Machines,” Computing in Sci-ence & Eng., vol. 3, no. 4, July/Aug. 2001, pp. 18–29.

9. C. Ratsch, P. Ruggerone, and M. Scheffler, “Density FunctionalTheory of Surface Diffusion and Epitaxial Growth of Metals,” Sur-face Diffusion: Atomistic and Collective Processes, NATO ASI Series B,vol. 360, 1997, pp. 83–101.

10. K.A. Fichthorn and W.H. Weinberg, “Theoretical Foundations ofDynamical Monte Carlo Simulations,” J. Chemical Physics, vol.95, no. 2, July 1991, pp. 1090–1096.

11. K.A. Fichthorn and M. Scheffler, “Island Nucleation in Thin-FilmEpitaxy: A First-Principles Investigation,” Physical Rev. Letters, vol.84, no. 23, June 2000, pp. 5371–5374.

12. K.A. Fichthorn, M.L. Merrick, and M. Scheffler, “A Kinetic MonteCarlo Investigation of Island Nucleation and Growth in Thin-FilmEpitaxy in the Presence of Substrate-Mediated Interactions,” tobe published in Applied Physics A.

13. P. Kratzer and M. Scheffler, “Reaction-Limited Island Nucleationin Molecular Beam Epitaxy of Compound Semiconductors,” tobe published.

14. P. Kratzer, E. Penev, and M. Scheffler, “First-Principles Studies ofKinetics in Epitaxial Growth of III–V Semiconductors,” to be pub-lished in Applied Physics A.

15. N. Moll, M. Scheffler, and E. Pehlke, “Influence of Surface Stresson the Equilibrium Shape of Strained Quantum Dots,” PhysicalRev. B, vol. 58, no. 7, Aug. 1998, pp. 4566–4571.

16. L.G. Wang et al., “Formation and Stability of Self-Assembled Co-herent Islands in Highly Mismatched Heteroepitaxy,” PhysicalRev. Letters, vol. 82, no. 20, May 1999, pp. 4042– 4045.

17. L.G. Wang et al., “Size, Shape, and Stability of InAs QuantumDots on the GaAs(001) Substrate,” Physical Rev. B, vol. 62, no.3, July 2000, pp. 1897–1904.

For more information on this or any other computingtopic, please visit our Digital Library at http://computer.org/publications/dlib.

Peter Kratzer is a computational physicist at the Fritz-Haber-Institut der Max-Planck-Gesellschaft. He re-ceived a PhD in theoretical physics from Technical Uni-versity Munich in 1993 with a PhD thesis aboutmodeling the dynamics of surface chemical reactions.He is conducting research in semiconductor materialsand nanostructures, and his main interests concern sur-face chemical reactions and epitaxy of compound ma-terials. Contact him at Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin-Dahlem, Germany; kratzer@fhi-berlin.mpg.de; www.fhi-berlin.mpg.de/th/th.html.

Matthias Scheffler is director of the Theory Departmentof the Fritz-Haber-Institut der Max-Planck-Gesellschaftand a professor at Technical University Berlin. His re-search concerns condensed-matter theory, materials,and the chemical physics of surfaces. His current inter-ests include developing first-principles methods (usingdensity-functional theory) for molecular simulations thatbridge the time and length scales from those of theatomistic processes to those that determine the prop-erties of realistic systems. Contact him at Fritz-Haber-In-stitut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin-Dahlem, Germany; scheffler@fhi-berlin.mpg.de; www.fhi-berlin.mpg.de/th/th.html.