An electron counting modification to potentials for

JOURNAL OF APPLIED PHYSICS 99, 064908 共2006兲

An electron counting modification to potentials for covalently bonded surfaces X. W. Zhou,a兲 D. A. Murdick, and H. N. G. Wadley School of Engineering and Applied Science, University of Virginia, Charlottesville, Virginia 22904

共Received 20 September 2005; accepted 26 January 2006; published online 24 March 2006兲 The surface structure of covalently bonded semiconductor materials undergoes reconstructions that are driven by electron redistribution between dangling and interatom bonds. Conventional interatomic potentials account for neither this electron redistribution nor its effects upon the atomic structure of surfaces. We have utilized an electron counting analysis to develop a surface interatomic potential that captures many of the effects of electron redistribution upon the surface structures of covalently bonded materials. The contributions from this potential decrease rapidly to zero beneath a surface. As a result, this surface potential can be added to many interatomic potentials for covalent materials without affecting its predictions of bulk properties such as cohesive energy, lattice parameters, and elastic constants. We demonstrate the approach by combining the surface potential with a recently proposed bond order potential and use it in a molecular statics simulation of the atomic reconstruction of a well studied 共001兲 GaAs surface. Many of the experimentally observed surface reconstructions are well predicted by the surface modified potential. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2180406兴 I. INTRODUCTION

Semiconductor materials such as GaAs are widely used in electronic devices. Examples include metal semiconductor field effect transistors 共MESFETs兲 and GaAs-based photonic heterostructures. GaAs MESFETs are an essential component of today’s cellular phones and many other wireless communication devices, direct broadcasting systems, global positioning systems, fiber optic drivers and receivers, collision avoidance systems, and phased array radars.1 GaAs-based photonic devices are used in light emitting diodes, lasers, infrared detectors, and solar cells.2 Each of these devices functions by manipulating the transport of electron charge. Recently, Kikkawa and Awschalom3 have discovered that the electron spin lifetime in GaAs films might be long enough to enable the design of electron spin based devices.4–6 The transport of selective spins can be controlled magnetically using ferromagnetic semiconductors such as 共GaMn兲As.7 This has restimulated an interest in the growth of GaAs thin films doped with magnetic metals.8 The GaAs thin films used by the devices above are grown by molecular beam epitaxy deposition.8 The convergence upon optimal vapor deposition processes for each device has often been prolonged because of a lack of understanding of the interrelationships among vapor deposition conditions, the atomic assembly mechanisms that occur on a growth surface, and the resulting film’s atomic scale structure 共including residual stresses and defect population兲. Molecular dynamics 共MD兲 simulations of vapor deposition provides a potentially useful means of investigating the dynamics of the atomic assembly processes and identifying the linkages between processing and defect structures.9 While the approach has been successfully used to improve a兲

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the growth of metallic multilayers,10,11 its extension to covalently bonded semiconductor systems has been problematic. The critical impediment has been the absence of a high fidelity interatomic potential that can accurately predict interatomic forces in covalently bonded materials. Such potentials are essential for molecular mechanics calculations of the critical surface transport phenomena controlling atomic structures. The issue is not a dearth of potentials for covalent materials. To the contrary, numerous angular dependent interatomic potentials have been proposed for semiconductor materials. They include the many variants of the Stillinger-Weber,12 Tersoff,13–16 and more recently the analytic bond order potential 共BOP兲.17–21 The problem is that these potentials are unable to predict the surface structures seen in covalent materials. Unlike simple metal systems, a variety of surface reconstructions can form on the surface of a semiconductor film. For example, nearly a dozen surface reconstructions have been experimentally observed on the 共001兲 GaAs surface.22,23 Density functional theory calculations indicate that these surface reconstructions have lower energies than other surface structures.24–26 It has been recently shown that none of the potentials identified above are able to predict the relative energies of these surface structures.27 During the growth of a thin film, atoms arriving at the surface rotate and migrate to minimize energy. This involves both making and breaking bonds with surface and underlying 共bulk兲 atoms. Some bonds on the surface can lead to the formation of surface dimers. When an atom is incorporated into a surface, the bonds formed often do not consume all the valence electrons of the surface atoms. The remaining electrons then occupy dangling bonds. In GaAs, dangling bonds on the arsenic atoms are of significantly lower energy than those on gallium atoms, and both are of higher energies than

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FIG. 1. 共Color online兲 Various 共001兲 GaAs surfaces: 共a兲 As-terminated 共1 ⫻ 1兲, 共b兲 共1 ⫻ 2兲, 共c兲 共2 ⫻ 1兲, 共d兲 ␤共2 ⫻ 4兲, 共e兲 ␤2共2 ⫻ 4兲, 共f兲 ␣2共2 ⫻ 4兲, 共g兲 ␨共4 ⫻ 2兲, and 共h兲 c共4 ⫻ 4兲-75%.

the normal Ga–As covalent bonds. The electron occupancies of the interatom and various dangling bonds then governs the energy of the surface. Under equilibrium conditions, a surface will reconstruct to minimize this energy. All conventional interatomic potentials ignore the reductions in surface energy that are facilitated by dynamic redistribution of electrons on the surface as these reconstructions occur. This, in part, is why they fail to correctly predict the surface structure. The incorrect surface structures then introduce inaccuracies into simulations of the surface atomic assembly mechanisms responsible for the growth, atomic scale structure, and defect populations of semiconductor thin films. The effect of surface reconstruction to modify the numbers of various interatom and dangling bonds so that all electrons can be accommodated into low energy levels is naturally addressed in the density functional theory calculations. By incorporating the self-consistent electron redistribution effect on the on-site energies, Chadi28 and Qian et al.29 showed that the tight-binding calculations can also predict well the 共001兲 GaAs surface reconstructions. However, both the density functional theory and the tight-binding calculations are computationally very expensive. Pashley proposed empirical criteria for identifying low energy surface reconstructions and concisely summarized them with an electron counting 共EC兲 rule.23,30 It identifies surface structures where the number of valence electrons matches the number of available low energy bonds on a surface. In accordance, we recently proposed an approximate method for introducing the effects of electron redistribution into energy calculations of GaAs surface structures. It essentially utilized the electron counting rule to calculate an energy penalty for the surface structures that result in electron

population of high energy level dangling bonds.27 This approach was able to successfully predict the surface energies of many of the low energy structures of 共001兲 GaAs surfaces.27 Here we extend this approach and propose a surface energy term that, when added to an interatomic potential, improves the molecular dynamics simulations of GaAs surfaces. While we focus our discussion on the 共001兲 GaAs surface, the method is general to other systems where dynamic electron redistribution drives surface structure transformations. II. GAAS „001… SURFACE STRUCTURES

Arsenic and gallium atoms have five and three valence electrons, respectively. When atoms are widely separated 共unbonded兲, the valence electrons in arsenic and gallium atoms occupy the three p and one s electron energy levels. In GaAs lattices, these energy levels are always hybridized. As a result, arsenic and gallium atoms can be equivalently viewed as having four degenerate hybrid energy levels, each with an energy equal to the average energy of the three p and one s electron energy levels. These atoms are tetrahedrally coordinated and form four identical Ga–As bonds in an ideal zinc-blende 共Z.B.兲 GaAs lattice structure. In this lattice, each bond is fully occupied by two electrons, with 5 / 4 of the electrons donated from the arsenic atom and 3 / 4 of the electrons donated from the gallium atom. During the vapor phase growth of an 共001兲 GaAs thin film, Ga atoms and As2 molecules condense on the surface and rearrange themselves into a surface structure. The surface reconstruction that forms will depend on the growth conditions and surface chemistry.23 Many atomic configura-

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tions have been observed and/or envisioned. Figure 1 shows eight possible surface structures for 共001兲 GaAs. In the figure, the arsenic and gallium atoms are represented by dark 共dark orange兲 and light 共gray兲 circles. The size of the atoms increases as the atoms approach the surface from the interior lattice. In Figs. 1共a兲–1共h兲, the upper part of each figure is a plan view while the lower part is a front view of the surface region of the crystal. Of all the surfaces shown in Fig. 1, only the ␤2共2 ⫻ 4兲, ␨共4 ⫻ 2兲, and c共4 ⫻ 4兲-75% surface reconstructions are observed and are thought to be the lowest energy structures for GaAs 共001兲 surfaces.25,31,32 The ␤共2 ⫻ 4兲 and ␣2共2 ⫻ 4兲 surface reconstructions have also been predicted to have low energies. All these low energy structures contain missing atoms in the top two or three atomic planes. Since the surface atomic configuration formed during a thin film growth is sensitive to the temperature and deposition conditions used to create the film, the relative stability of the various surface reconstructions must be evaluated using a surface phase diagram.29,33 In this approach, the relative surface energy is expressed as a function of arsenic vapor chemical potential. This allows a mapping of the surface stability against arsenic concentration above the surface. Density functional theory calculations24–26 have shown that at the lowest arsenic chemical potentials, a Ga-rich ␨共4 ⫻ 2兲 surface has the lowest energy. As the arsenic chemical potential is gradually increased, the lowest energy surface switches first to the ␣2共2 ⫻ 4兲 surface, then to the ␤2共2 ⫻ 4兲 surface, and finally 共at the highest arsenic chemical potential兲 to the As-rich c共4 ⫻ 4兲 − 75% surface. The calculations have also shown that the energy of the ␤共2 ⫻ 4兲 surface is very close to that of the ␤2共2 ⫻ 4兲 surface.34–36 These density functional theory predictions are generally in good agreement with the experimental observations of surface reconstructions.31 We have investigated the utility of a wide range of interatomic potentials for calculating the 共001兲 GaAs surface phase diagram.27,37 Neither the predicted energies nor the relative energy order of the various surface reconstructions agree with the density functional theory calculations. All the potentials predict a 共2 ⫻ 1兲 surface as the lowest energy surface at low arsenic chemical potentials and a 共1 ⫻ 2兲 surface as the lowest energy surface at high arsenic chemical potentials. One of the challenges confronting the development of improved potentials for covalent materials arises from the long range over which a surface cooperatively undergoes reconstruction. These characteristic distances are very large compared to the cutoff distance of the potentials. For instance, the ␤共2 ⫻ 4兲 surface reconstruction in Fig. 2 requires a missing dimer to occur for every four dimers in the 关110兴 dimer row direction. The missing dimer must “know” about other atom locations on a three dimer spacing length scale. This is a distance of about 12 Å on the 共001兲 GaAs surface. Current potentials use a nearest-neighbor cutoff distance 共艋3.7 Å兲 and always define high energies for underbonded 共less than tetrahedral coordination兲 configurations at a surface. As an example, consider forming a ␤共2 ⫻ 4兲 surface from a 共2 ⫻ 1兲 surface by removing some arsenic dimers. In

J. Appl. Phys. 99, 064908 共2006兲

FIG. 2. 共Color online兲 The ␤共2 ⫻ 4兲 surface. The black bars are the Ga–As bonds between the top arsenic and the underlying gallium atoms and the unfilled bar refers to the As–As bond between a top arsenic dimer.

this process, the underlying gallium atoms become underbonded and the ␤共2 ⫻ 4兲 structure is then predicted to be less stable by all of the conventional potentials. At first glance, the use of a longer-range cutoff distance 共in excess of 12 Å兲 might enable the long coordination distances to be assessed in a simulation. However, the computational cost for even the simplest angular dependent interatomic potentials rises as the fifth power of the cutoff distance. The use of a 12 Å cutoff distance would result in prohibitively expensive simulations. Fundamentally, the observed surface reconstructions are driven by electron redistribution between interatom and dangling bonds over length scales of an atom or two.30 Recall that the electron distribution of each atom is affected by the electron distributions of its neighboring atoms. In practice, therefore, information about the local electron distribution can be relayed from one atom to another over a long distance even though each atom only redistributes its electrons to the nearest-neighbor bonds. The long-range features of surface reconstructions might therefore be captured using short-range potentials if the effects of electron redistribution are incorporated. III. ELECTRON REDISTRIBUTION IN SURFACE ENERGY LEVELS

An arbitrary covalently bonded 共001兲 GaAs surface is shown in Fig. 3. Unlike a bulk GaAs crystal where each atom forms four bonds with four neighboring atoms, atoms on the surface have less neighbors and therefore form less bonds. Consequently, if a surface atom, i, forms iZ 共iZ ⬍ 4兲 bonds with its iZ neighbors, then it has the potential to form additional 4 − iZ bonds whenever new atoms are added to its vicinity. These 4 − iZ unrealized bonds are referred as dangling bonds. Figure 3 shows the five different types of bonds that can then form: 共i兲 As–As dimer bonds between the two arsenic atoms, 共ii兲 Ga–Ga dimer bonds between the two gallium atoms, 共iii兲 Ga–As bonds between adjacent gallium and arsenic atoms 共similar to those in the GaAs bulk lattice兲, 共iv兲 arsenic dangling bonds, and 共v兲 gallium dangling bonds. Harrison38 has estimated the energy of all but the Ga–Ga bond. Using the bond orbital approximation39 and a first

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gling bond. This global EC model has been combined with other classic potentials to better predict the GaAs surface phase diagram.27 The global EC rule model does not address the local electron population at the individual atom scale. As a result, it cannot penalize the energy of these local atomic configurations and fails to address the energy of a simulated system as a function of all its atom positions. It therefore cannot be used for atomistic simulations of atomic structures. Here we extend the global EC model to develop a local electron counting potential 共ECP兲 that relates a system’s energy to its atom positions.

FIG. 3. 共Color online兲 Plan view of a 共001兲 GaAs surface showing the various bond types.

principles calculation of tight-binding parameters, we have recalculated the energy levels of all five of the bonds shown in Fig. 3.27 The results are reproduced in Table I. We confirmed Harrison’s observation38 that the gallium dangling bond energy level is significantly higher than those of the other bonds. Obviously, the redistribution of valence electrons on a GaAs surface into the lowest available energy levels reduces the surface energy. The number of surface 共valence兲 electrons 共per unit of surface area兲 is determined by the arsenic and gallium fractions on the surface. For a given atom fraction, the surface energy can be minimized if the surface atoms move to create a surface structure where electrons can populate the low energy bonds. Pashley30 has summarized these observations in an electron counting rule. The EC rule can be used to identify the low energy surface configurations of GaAs. These configurations have the low energy Ga–As, Ga–Ga, and As–As bonds and the arsenic dangling bonds fully occupied 共by two electrons兲 while leave the high energy gallium dangling bonds empty. This simple rule has been successfully used by Pashley30 and others31 to explain many of the surface reconstructions of GaAs and other semiconductors. Conventional interatomic potentials do not account for the dynamic electron redistribution and the associated surface potential energy changes during atomic assembly. It has been recently shown that if the total number of surface valence electrons and the total number of low energy surface bonds can be compared, it is possible to penalize the energy of the surface structures that do not satisfy the EC rule.27 This approach computes a surface potential energy based on a “global” electron population at the surface and requires no knowledge of the electron populations of each bond and danTABLE I. Energy levels for As–As, Ga–Ga, and Ga–As interatom bonds and As and Ga dangling bonds 共DBs兲. Bond type

As–As

Ga–Ga

Ga–As

As DB

Ga DB

Energy level 共eV兲

−8.26

−3.84

−5.96

−3.69

−0.23

IV. A SURFACE INTERATOMIC POTENTIAL

In conventional potentials, all energies are calculated relative to that of the isolated atoms 共i.e., the potential energy of an isolated atom is assumed to be zero兲. A well parametrized conventional potential is assumed to have correctly predicted the potential energy of bulk crystals and isolated dimers. These conventional potentials do not consider the dynamic electron redistribution amongst interatom and dangling bonds at a surface. Here we introduce an additive EC energy term that only becomes nonzero when a local region of a surface structure deviates from those satisfying the EC rule conditions. We have designed this EC potential so that it can be fitted to the system of interest and superimposed upon a conventional potential to provide a more complete description of interatomic interactions. In order to retain the integrity of the conventional potential, it is necessary for the EC term to predict zero energies for isolated atoms, dimers, and bulk crystals since the conventional potential is fitted to predict these quantities correctly. Unlike the global EC rule model which requires predefined bonds, the bonds encountered during atomistic simulations dynamically form and their atom separations can be widely distributed. The interatomic potential must therefore be a continuous function of atom positions and a bond between neighboring atoms must gradually disappear when the atoms are pulled apart. An EC energy penalty term for a potential can be directly derived from Pashley’s EC rule.23 The first step is to determine the number of electrons partitioned into bonds and dangling bonds. We can assume that no dangling bonds are present in equilibrium lattice structures. Since all valence electrons must then be distributed in interatom bonds and these interatom bonds are symmetric around each atom in a bulk lattice, the number of electrons in the bonds of the equilibrium bulk lattice can be predetermined and does not need to be solved dynamically during a simulation. The electron distribution in the equilibrium bulk lattice bonds is therefore referred as a static electron distribution. As an equilibrium bulk crystal is pulled apart, interatomic bonds are gradually broken and dangling bonds gradually form on a slowly created free surface. The static electron distribution is then required to decay to zero as the electrons previously in the interatom bonds retreat to the

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dangling bonds. Such a static electron distribution would naturally capture the continuous formation/destruction of bonds. While a continuous electron transfer between bonds and dangling bonds on a surface is essential to be incorporated in an ECP, the number of electrons that can be transferred must be bounded. For instance, it is physically meaningless for any bond to have a negative number of electrons and no bond should contain more than two electrons due to Pauli principle. These electron bounds can be implemented by applying a simple constraining energy.40 The total EC contribution can therefore be divided into 共i兲 an EC rule bond energy arising from violations of the EC rule for interatom bonds, 共ii兲 an EC rule dangling bond energy arising from violations of the EC rule for dangling bonds, and 共iii兲 an electron constraining energy arising from violation of electron bounds. Finally, an electron counting approach that distinguishes surfaces from bulk crystals, isolated atoms, and dimers must be developed. A. Static electron distribution in bulk lattice bonds

Consider first the static electron distribution. Suppose that within an equilibrium bulk crystal, bond ij receives qij electrons from atom i and q ji electrons from atom j. The matrices qij and q ji then quantify the static electron distribution within the lattice. In general, different values of qij can be expected for different lattice structures since they have different atomic coordinations, bond lengths, and bond angles. Note that a change in the bond angle or atom coordination is always associated with a corresponding change in the equilibrium bond length. This means that to a good approximation, the static electron distribution can be uniquely defined by a radial function qij共rij兲. qij for various atom pair types 共i = Ga, As, j = Ga, As兲 can be easily determined for equilibrium bulk lattices where no dangling bonds are present. As an example, consider the equilibrium GaAs zinc-blende crystal. Each arsenic atom forms four bonds with four gallium neighbors. Due to symmetry, each arsenic atom must provide 41 of its five valence electrons to each bond. As a result, qAsGa共r1兲 = 1.25, where r1 is the nearest-neighbor distance in an equilibrium zincblende GaAs lattice structure. Similarly, qGaAs共r1兲 = 0.75 for a gallium centered tetrahedral bond in the same structure. Likewise, in an equilibrium fcc arsenic crystal, each arsenic 5 , atom has 12 nearest arsenic neighbors, and so qAsAs共r2兲 = 12 where r2 is the nearest-neighbor distance in equilibrium bulk fcc arsenic. A continuous, differentiable form for qij共rij兲 can be deduced by analyzing many lattice structures. Twenty seven lattice structures of the GaAs system 关共␣As兲As, 共␣Ga兲As, 共B12兲As, 共␤Ga兲As, 共␤Sn兲As, 共bcc兲As, 共dc兲As, 共fcc兲As, 共fct兲As, 共hcp兲As, 共sc兲As, 共sh兲As, 共␣As兲Ga, 共␣Ga兲Ga, 共b12兲Ga, 共␤Ga兲Ga, 共␤Sn兲Ga, 共bcc兲Ga, 共dc兲Ga, 共fcc兲Ga, 共fct兲Ga, 共hcp兲Ga, 共sc兲Ga, 共sh兲Ga, 共B1兲GaAs, 共B2兲GaAs, and 共Z.B.兲 GaAs兴, were used to determine the static electron distribution functions for the four bond types present: qAsAs共r兲, qGaGa共r兲, qAsGa共r兲, and qGaAs共r兲. We found that the results can be summarized using a spline function,

TABLE II. Static electron distribution function parameters qij0 , ra,ij, and rc,ij. Bond type

qij0

ra,ij

rc,ij

As–As Ga–Ga As–Ga Ga–As

1.25 0.75 1.25 0.75

2.800 2.600 2.700 2.700

3.500 2.918 3.220 3.220

冦

q0ij ,

qij共r兲 = q0ij 0,

r 艋 ra,ij

共3ra,ij − rc,ij − 2r兲共r − rc,ij兲2 , 共ra,ij − rc,ij兲3 rc,ij 艋 r,

ra,ij ⬍ r ⬍ rc,ij

冧

共1兲

where the parameters q0ij, ra,ij, and rc,ij obtained from these 27 lattices are listed in Table II. Equation 共1兲 is plotted in Fig. 4. It can be seen that qij共r兲 has a constant value of q0ij = Vi / 4 共Vi is valence electrons for atom i兲 at small r. This value corresponds to that of the zinc-blende 共diamond cubic兲 structure, which has a smaller nearest-neighbor spacing than all other structures. As r is increased, qij共r兲 decreases to zero. The extent of formation of a bond ij can be characterized by a normalized bond formation function qij共r兲 / q0ij. We can assume that atoms such as arsenic and gallium have a capacity of forming four 共tetrahedral兲 complete bonds such as those in zinc-blende structure. The bond formation function qij共r兲 / q0ij for these complete bonds equals unity. It should be pointed out that atoms may form more bonds in other structures. For instance, an atom forms 12 bonds in a fcc structure. However, structures with more bonds are associated with longer bond lengths. The bond formation function qij共r兲 / q0ij is then less than unity for these bonds. This means that fcc bonds can be considered as partial bonds. The assumption that atoms have a capacity to form four complete bonds then is still valid even when the structure forms more partial bonds. The bond formation function qij共r兲 / q0ij approaches unity as the bond length r decreases. It captures the continuous formation of “one” bond when atoms i and j are brought together. The bond formation function decays to zero as the

FIG. 4. Static electron distribution function qij共r兲 for As–As, Ga–Ga, As– Ga, and Ga–As bonds.

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bond length r increases. It hence also captures the continuous breaking of the bond as atom i is pulled away from atom j and the electrons originally in the ij bond continuously retreat into dangling bonds. As a result, while Eq. 共1兲 and Fig. 4 are derived from equilibrium bulk structures, they can also describe the continuous transfer of electrons from bonds to dangling bonds as the equilibrium lattice structure is continuously stretched apart. In principle, a covalently bonded atom such as an arsenic or gallium atom can form at most four complete bonds. The number of bonds formed by an atom in a given structure can therefore be measured as a fraction of four. For an atom i, this atom bond fraction, Fi, can be simply defined as Fi =

Z q 共r 兲/q 0 兺ij=i ij ij ij 1

4

− ⌬Fi =

Z q 共r 兲 兺ij=i ij ij 1

Vi

− ⌬Fi ,

共2兲

where i1, i2 , . . ., iZ is a list of the iZ neighbors of atom i. The ⌬Fi term is an overbonding function defined as

⌬Fi =

冦

Z q 共r 兲 兺ij=i ij ij 1

0,

Vi

Z q 共r 兲 兺ij=i ij ij 1

Vi

− 1,

艋1 Z q 共r 兲 兺ij=i ij ij 1

Vi

⬎ 1.

冧

共3兲

The overbonding function is introduced to ensure that the atom bond fraction, Fi, does not exceed unity when an atomic configuration is compressed. With this modification, the value of Fi then always lies between zero and unity. With the atom bond fraction, Fi, known, the number of dangling bonds of atom i, ni, can then be calculated as ni = ␯i共1 − Fi兲,

共4兲

where ␯i 共=4兲 is the characteristic tetrahedral coordination of arsenic and gallium atoms. As discussed above, qij共rij兲 represents the total number of electrons that atom i donates to the ij bond in an equilibrium bulk lattice. This also holds for a stretched bulk lattice as qij共rij兲 also captures the breaking of the bond. However, qij共rij兲 cannot be used to represent the total number of electrons that atom i donates to the ij bond in a compressed bulk lattice. This is because if an atom i donates qij共rij兲 electrons to the ij bond with its jth neighbor, then it will donate a total Z qij共rij兲, to all the iZ bonds it forms number of electrons, 兺ij=i 1 with the iZ neighbors. When a lattice is compressed, the number of neighbors around an atom increases. The total number of electrons required for atom i to fill all its bonds may then exceed its valence electrons. Suppose that in a generalized 共including equilibrium, stretched, and compressed兲 bulk lattice, an ij bond receives sij electrons from atom i and s ji electrons from atom j. The bulk lattice bond electron distributions, sij and s ji, can then be defined as sij = and

qij共rij兲 1 + ⌬Fi

共5兲

s ji =

q ji共rij兲 . 1 + ⌬F j

共6兲

It can be seen that for equilibrium and stretched bulk lattices, the overbonding ⌬Fi and ⌬F j terms equal zero, and so sij and s ji are equivalent to qij and q ji. Unlike qij and q ji, the bulk lattice bond electron distributions defined by sij and s ji can always match the available atom valence electrons even when the lattice is highly compressed. B. Dynamic electron distribution in near-surface bonds

Unlike the bulk lattice bond electron distribution sij, which can be calculated using the predetermined 共static兲 function qij共r兲, Eq. 共5兲, the electrons in bonds at a surface redistribute according to the local structures and the resulting surface energy is minimized when the EC rule criterion is achieved. The electrons populate a set of bonds and dangling bonds that evolve as a surface reconstructs and so they must be deduced dynamically. We assume that a surface bond between atoms i and j receives aij electrons from atom i and a ji electrons from atom j; aij and a ji vary as the local environment changes and define the dynamic electron distribution in the surface bonds. During molecular dynamics simulations, a bulk bond must smoothly evolve into a surface bond if the surrounding atoms gradually move away. A surface order function, Pij, can be used to characterize the nature of the bond between atoms i and j. We can make Pij lie between zero and one and ensure that as the ij bond changes from a bulk bond to a surface bond, Pij increases linearly from zero to one. We can then imagine that a general bond has a 共1 − Pij兲 bulk lattice component and a Pij surface component. The number of electrons such a bond receives from atoms i and j is then given, respectively, by the general bond electron distributions gij and g ji. gij and g ji, can be calculated as gij = Pijaij + 共1 − Pij兲sij

共7兲

g ji = Pija ji + 共1 − Pij兲s ji .

共8兲

and In Eq. 共8兲, we have assumed Pij = P ji. The total number of electrons in a general bond formed between atoms i and j, ␰ij, is then the sum of the two contributions as follows:

␰ij = Pij共aij + a ji兲 + 共1 − Pij兲共sij + s ji兲.

共9兲

C. EC rule bond energy

The EC rule criterion for interatom bonds discussed above requires that the total number of electrons in a complete 共i.e., short bond length兲 bond at a surface equals two. Deviation from this electron number is taken to result in an additional EC rule bond energy penalty. This criterion, however, is applicable only to a short length bond at a surface. To develop an ECP, the concept needs to be extended to bonds with arbitrary bond lengths in arbitrary configurations. Assuming that when the EC rule is satisfied the number of electrons in a general bond is ␰0ij. At a surface where the

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surface order function Pij = 1, ␰0ij should approach two when the bond length rij is small and then continuously decrease to zero as rij is increased. This will ensure that the original EC rule applies at short bond lengths and that the bond is continuously broken as the bond length increases. Inside the lattice where the surface order function Pij = 0, ␰0ij should approach sij + s ji. This ␰0ij precisely gives the sum of Eqs. 共5兲 and 共6兲 required for the total number of electrons in a bulk bond. To a good approximation, we can therefore rewrite the EC rule criterion for interatom bonds as

␰0ij = Pij

冋

册

qij共rij兲 q ji共rij兲 + + 共1 − Pij兲共sij + s ji兲. q0ij q0ji

共10兲

It can be seen that Eq. 共10兲 generalizes the EC rule to interatom bonds with arbitrary bond lengths in arbitrary configurations. Clearly, the EC rule bond energy penalty is minimized 共to zero兲 when the total number of electrons in the ij bond, ␰ij, reaches the value defined by the modified EC rule criterion for interatom bonds, ␰0ij. By taking a Taylor expansion of the EC rule bond energy penalty in the vicinity of ␰0ij and retaining the first order term, we can write an EC rule bond energy penalty, Eb, for deviation from the EC rule criterion for interatom bonds N

iZ

N

iZ

bij共␰ij − ␰0ij兲2 兺 i=1 j=i

Eb = 兺

1

兺 bijP2ij i=1 j=i

=兺

1

冋

aij + a ji −

qij共rij兲 共q jirij兲 − q0ij q0ji

册

2

,

共11兲

where the second derivative coefficient, bij, can be viewed as a weight that determines the strength of the energy penalty due to the violation of the EC rule criterion for interatom bonds. D. EC rule dangling bond energy

After donating gij electrons 关see Eq. 共7兲兴 to form bonds with neighboring atoms j 共j = i1 , i2 , . . . , iZ兲, an atom i may still retain some valence electrons which are then available to populate dangling bonds. The number of these valence electrons is given by iZ

dini = Vi − 兺 gij ,

共12兲

j=i1

where di is the number of electrons per dangling bond and ni is number of dangling bonds 关Eq. 共4兲兴. The EC rule criterion for surface dangling bonds discussed above requires that the total number of electrons in the dangling bonds of a surface atom i be ␣ini, where the parameter ␣i equals two for arsenic atoms and zero for gallium atoms. Any deviation from this number of electrons for a surface atom results in an EC rule dangling bond energy penalty. This criterion, however, is only applicable to surface atoms. It needs to be extended to general atoms in order to be used by an ECP. An EC energy penalty modification to an interatomic potential must predict a zero EC rule energy penalty for

structures such as bulk crystals, isolated atoms, and dimers where the classic potentials already produce correct results. This can be achieved by making the EC rule dangling bond energy penalty zero for any atom i in these three structures regardless of its number of dangling bond electrons, dini. This enables the number of electrons in the dangling bonds to be freely adjusted to satisfy the EC rule criterion in the interatom bonds. An EC rule criterion for dangling bonds therefore needs to be developed which can be applied to every atom, but imposes a dangling bond energy penalty only for surface atoms. To achieve this, we need to distinguish atoms on surfaces from isolated atoms as well as those in bulk crystals and dimers. Lattice atoms, surface atoms, dimer atoms, and isolated atoms have increasingly reduced coordination numbers. A low atom coordination function, Si, can be created to distinguish if an atom i is in a dimer or an isolated configuration, as opposed to the surface or interior of a lattice. To achieve this, Si can be made a continuous function that increases from zero to unity as an atom i is transformed within the bulk lattice to a surface location. It then remains at unity as the atom is further moved into a dimer or an isolated environment. Similarly, a high atom coordination function, Zi, can be created to quantify the extent to which an atom i is located in the bulk or on a surface, as opposed to a dimer or an isolated configuration. To achieve this, Zi can be made a continuous function that increases from zero to unity as atom i is transformed from an isolated atom, a dimer atom, to a surface atom. It then remains at unity as the atom is further moved to the interior of a lattice. The quantity SiZi then only equals unity when atom i is on a surface. SiZi, therefore, captures the gradually increased surface component of an atom as it moves to the surface from any other configurations. The combined parameter, SiZi, enables the EC energy penalty for dangling bonds to be applied to any atoms. Suppose that when the EC rule is satisfied, the total number of electrons in the dangling bonds of atom i is do,ini, where do,i represents the number of electrons per dangling bond. For surface atoms where SiZi = 1, di,oni should equal ␣ini to comply with the original EC rule for surface atoms. For isolated atoms, dimer atoms, and bulk atoms where SiZi = 0, di,oni should equal dini so that the extended EC rule for dangling bonds is always satisfied regardless of the actual number of dangling bond electrons, dini. To a good approximation, we have

do,ini = SiZi␣ini + 共1 − SiZi兲dini .

共13兲

It can be seen that Eq. 共13兲 generalizes the extended EC rule for dangling bonds of atoms in any configuration. Clearly, the EC rule dangling bond energy penalty is minimized 共to zero兲 when dini = do,ini. Again by taking a Taylor expansion of the EC rule dangling bond energy in the vicinity of do,ini and retaining only the first order term, we can write a dangling bond energy, Ed, for deviation from the EC rule for the dangling bonds,

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064908-8

J. Appl. Phys. 99, 064908 共2006兲

Zhou, Murdick, and Wadley N

N

Ed = 兺 wdi共dini − do,ini兲 = 兺 wdi共SiZi兲2共dini − ␣ini兲2 , 2

i=1

i=1

共14兲 where the second derivative coefficient, wdi, can be viewed as a weight controlling the strength of the energy penalty. E. Electron constraining energy

The ECP proposed above enables atoms to redistribute their valence electrons among interatom bonds and dangling bonds at a surface to reduce the potential energy. However, this redistribution must be bounded between zero and two since the number of electrons in a bond or a dangling bond cannot become negative and cannot violate the Pauli electron principle. The dynamic electron redistribution is therefore constrained. For dangling bonds, 0 艋 dini 艋 2ni, and for interatom bonds, 0 艋 aij + a ji 艋 2. It should be noted that while 0 艋 aij + a ji 艋 2, either aij or a ji can be negative or larger than two. This means that atoms can receive electrons from the bond 共rather than give electrons to the bond兲 or give more than two electrons to the bond. In a Ga–As dimer, for instance, the gallium atom can give all three of its valence electrons to the Ga–As bond. Since a stable Ga–As bond only needs two electrons, the arsenic atom receives, not provides, an electron from the Ga–As bond. The arsenic atom then uses this acquired electron and its five valence electrons to form three stable dangling bonds. Like all potentials used for atomistic simulations, the ECP must be a continuous function of atom position. This requires that aij and a ji are continuous as atoms i and j are moved. Obviously, both aij and a ji must continuously decay to zero as the separation distance between i and j is increased to the cutoff distance of the potential since at that separation distance the ij bond, by definition, no longer exists. However, it can be seen from Eq. 共11兲 that for an isolated bond, minimization of the EC rule bond energy would only enable the sum of aij and a ji 共instead of aij and a ji separately兲 to become zero at the cutoff bond length. Even this is not guaranteed when multiple bonds are involved. As a result, the aij and a ji calculated by minimizing the EC rule bond energy alone are not continuous at the cutoff bond length between atoms i and j. For example, consider a high EC rule energy surface. Let us assume that the high EC rule energy originates from a local region near an atom j where three extra valence electrons cannot find low energy levels. Imagine then that an arsenic vapor atom i just outside the cutoff distance of atom j is moved to just within the cutoff distance. Almost no energy change should occur during this process if the energy is a continuous function of relative atom position. Similarly, both aij and a ji should remain almost zero as atom i is moved. Nonetheless, Eqs. 共11兲 and 共14兲 predict an abrupt energy reduction and changes in both aij and a ji at the sudden appearance of the 共highly stretched兲 ij bond. This can be seen by assuming aij = −3 and a ji = 3. While the energy minimization condition of aij + a ji = 0 required by Eq. 共11兲 is still satisfied, the three extra surface electrons, which cause the original surface to have a high EC rule energy, are trans-

ferred to the newly added arsenic atom. This enables a filling of its four low energy dangling bonds with its five valence electrons and the reduction of the total EC rule energy for the system. Physically, the capability of an atom to give or receive electrons from a bond decays to zero as the bond length is increased to the cutoff distance. Continuous energy and electron distribution functions can both be enforced if we implement a bond length dependent redistribution capability. This can be achieved by introducing a lower electron bounding function ␭ij共rij兲 and an upper electron bounding function ␮ij共rij兲 in such a way that an atom i is only capable of giving a total number of electrons gij to the ij bond when ␭ij共rij兲 艋 gij 艋 ␮ij共rij兲. Both ␭ij共rij兲 and ␮ij共rij兲 are set to decay to zero at the cutoff distance, while ␭ij共rij兲 becomes very small and ␮ij共rij兲 becomes very large as the bond length rij is decreased. As a result, enforcing the condition ␭ij共rij兲 艋 gij 艋 ␮ij共rij兲 has no effects on the EC rule energy calculations at a small bond length, but results in zero gij 关and consequently aij, Eq. 共7兲兴 at the cutoff bond length. In order for ␭ij共rij兲 and ␮ij共rij兲 to have no effect on the energies calculated at small bond lengths, we assume that they take the valence limiting values at small bond lengths. The maximum number of electrons atom i can attempt to give to the ij bond is bounded by its valence Vi. On the other hand, the maximum number of electrons atom j can receive is 2␯ j − V j 共the tetrahedral coordination number ␯ j = 4兲 in order to avoid overfilling its outer shell. If the ij bond takes two electrons, the maximum number of electrons atom i can give to the ij bond that is acceptable to atom j is 2 + 2␯ j − V j. In other words, the maximum number of electrons atom i can give to the ij bond is min共Vi , 2 + 2␯ j − V j兲. The minimum number of electrons atom i can give to the ij bond equals the negative value of the maximum number of electrons atom i can receive from the ij bond. The maximum number of electrons atom i can attempt to receive is a number that completely fills its outer shell. But atom i cannot receive more electrons once atom j gives all its valence electrons to the ij bond. Thus, the minimum number of electrons atom i can give to the ij bond is max共Vi − 2␯i , 2 − V j兲. Multiplying these limits by the radially dependent bond formation function, qij共rij兲 / q0ij, gives a good choice for ␭ij共rij兲 and ␮ij共rij兲, ␭ij共rij兲 = max共Vi − 2␯i,2 − V j兲

qij共rij兲 q0ij

共15兲

␮ij共rij兲 = min共Vi,2 + 2␯ j − V j兲

qij共rij兲 . q0ij

共16兲

and

The electron bounds identified above can be effectively implemented by adding six electron constraining energy terms to the total EC rule energy,40 N

Ec1 = 兺 wc1H关− dini兴关dini兴2 ,

共17兲

i=1

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064908-9

J. Appl. Phys. 99, 064908 共2006兲

Zhou, Murdick, and Wadley N

Ec2 = 兺 wc2H关dini − 2ni兴关dini − 2ni兴2 ,

共18兲

i=1 N

E c3 = 兺

iZ

兺 wc H关− 共aij + a ji兲兴共aij + a ji兲2 ,

i=1 j=i1 N

共19兲

3

iZ

wc H共aij + a ji − 2兲共aij + a ji − 2兲2 , 兺 i=1 j=i

E c4 = 兺

4

共20兲

1

N

iZ

wc H兵− 关gij − ␭ij共rij兲兴其关gij − ␭ij共rij兲兴2 , 兺 i=1 j=i

E c5 = 兺

5

共21兲

1

N

E c6 = 兺

iZ

兺 wc H关gij − ␮ij共rij兲兴关gij − ␮ij共rij兲兴2 ,

i=1 j=i1

6

共22兲

where H共x兲 is a Heaviside step function such that H共x兲 = 0 when x ⬍ 0 and H共x兲 = 1 when x 艌 0, and wc1-wc6 are constants. It can be seen that Eqs. 共17兲 and 共18兲 are both zero when 0 艋 dini 艋 2ni, Eqs. 共19兲 and 共20兲 are both zero when 0 艋 aij + a ji 艋 2, and Eqs. 共21兲 and 共22兲 are both zero when ␭ij共rij兲 艋 gij 艋 ␮ij共rij兲. As a result, the addition of Eqs. 共17兲–共22兲 has no effect on the EC rule calculations when the dynamic electron distributions are within the physically meaningful ranges. However, when electrons fall out of these bounds, Eqs. 共17兲–共22兲 impose energy penalties. The parameters wc1-wc6 are then chosen to be relatively large numbers so that these electron constraining energy terms overpower Eqs. 共11兲 and 共14兲 to enforce the electron bounds. F. Local environment defining functions

The formalism discussed above requires three functions, the low coordination Si, the high coordination Zi, and the surface order Pij, to calculate the local environment of an atom and a bond. We have discussed the need for these functions above and give here their proposed forms. Functions Si, Zi, and Pij all change from zero to unity as the environment changes. They can hence be described by a general equation

X共x兲 =

冦

0, 1 2

x 艋 xs

冉

− 21 cos ␲

1,

x f 艋 x,

冊

x − xs , x f − xs

xs ⬍ x ⬍ x f

冧

共23兲

where X = Si, Zi, and Pij, x is the corresponding environment dependent parameter for the three functions, and xs and x f define the range of x within which the function X changes from zero to unity. The low coordination function Si has a high value for relatively low coordinated atoms such as isolated, dimer, or surface atoms. The number of dangling bonds of atom i, ni, can distinguish atom in these low coordinated environments from those in equilibrium or compressed bulk lattices. The value of ni in a stretched bulk lattice, however, can become close to that in the low coordinated environments. Imagine

that the stretched environment of an atom i can be unstretched by rescaling the distances between i and its neighbors so that the nearest distance between any atom pair of i and its neighbors equals a predefined equilibrium bulk 共say, zinc-blende兲 nearest-neighbor distance. The number of dangling bonds for the rescaled configuration, noi , is then normalized to that of the equilibrium 共zinc-blende兲 structure and becomes independent of the lattice strain. noi can therefore quantify the smooth transition when an atom moves from an 共equilibrium or strained兲 bulk to other lower coordinated environments. Clearly, for the Si function X = Si, the environment variable can be chosen as x = noi , with the parameters xs = ns = 0.25 and x f = n f = 0.50. The high coordination function Zi has a high value for relatively highly coordinated atoms such as bulk or surface atoms. The bond fraction parameter of atom i, Fi, can distinguish the bulk and surface atoms from atoms in other lower coordinated environments. For the Zi function X = Zi, the environment variable can be chosen as x = Fi, with xs = Fs = 0.26 and x f = F f = 0.40. The surface order function Pij characterizes the environment of a bond. It can be seen from Eq. 共7兲 that when Pij = 0 共bulk bonds兲, the number of electrons that atom i gives to the ij bond exactly equals the predetermined bulk electron distribution sij 关or qij, see Eq. 共5兲兴. In such cases, the dynamically distributed electrons, aij, make no contribution to the total number of electrons in the ij bond. To the contrary, when Pij = 1 共surface bonds兲, the electrons in the ij bond completely come from the dynamically distributed electrons, aij and a ji. Furthermore, the EC rule violating bond energy is minimized when aij + a ji = qij共rij兲 / q0ij + q ji共rij兲 / q0ji 关Eq. 共11兲兴, as opposed to aij and a ji independently reaching their separate optimum values. Obviously, an increase in aij and a corresponding decrease in a ji would not change aij + a ji, but would increase the electron transfer from atom i to atom j. It is this formalism that enables electron redistribution at the surface. The correct choice of the functional form of Pij is essential to the utility of the potential. This can be illustrated by examining two extreme cases. First, if we incorrectly assume that all bonds in the system were bulk lattice bonds with Pij = 0, then no electron redistribution would occur and the surface potential would incorrectly predict the energy of surface reconstructions. If instead we incorrectly assume that all bonds in the system were surface bonds with Pij = 1, then electron redistribution would occur in all bonds and this also causes problems. For instance, suppose a surface has a local region A where excess valence electrons cannot find low energy levels, and another local region B where there are extra low energy levels. Such a surface should have a high EC rule violating energy because in covalently bonded materials, the excess electrons at region A cannot by themselves transfer to region B. However, if every bond between A and B were allowed to incorrectly transfer electrons, then the electrons between A and B can be shifted bond by bond, resulting in the disappearance of the excess electrons at A and the filling of the low energy levels at B. The ECP would then incorrectly predict a low energy surface.

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064908-10

J. Appl. Phys. 99, 064908 共2006兲

Zhou, Murdick, and Wadley

In light of this, we carefully analyzed a variety of surface reconstructions including the ␣共2 ⫻ 4兲, ␣共4 ⫻ 2兲, ␣2共2 ⫻ 4兲, ␣2共4 ⫻ 2兲, ␤共2 ⫻ 4兲, ␤共4 ⫻ 2兲, ␤2共2 ⫻ 4兲, ␤2共4 ⫻ 2兲, c共4 ⫻ 4兲-75%, ␥共2 ⫻ 4兲, and the ␨共4 ⫻ 2兲 surfaces.24,31 We discovered two criteria for defining Pij that are valid for all these surface reconstructions. First, we found that if either atom i or atom j is a surface atom, then the ij bond can transfer electrons. The electron transfer capability of the ij bond can be simply set to scale with the maximum surface component of atoms i and j. Second, even when both atoms i and j are bulk atoms, the ij bond may still transfer electrons, provided both atoms i and j have a surface neighbor. In this case, the electron transfer capability of the ij bond increases when the surface components of the two affecting neighbors of atoms i and j both increase while their distances from atoms i and j both decrease. Physically, the first condition is equivalent to requiring that if any surface atom seeks to optimize its dangling bond energy, it must ask its bonds to transfer electrons. The second condition is essentially a nearest-neighbor mechanism for two surface atoms to exchange electrons through their common neighboring bond. The parameter max共noi , noj 兲 captures the first condition using the maximum surface component between atoms i and j. The parameter nm i = max关nok qik共rik兲 / q0ik , k = i1 , i2 , . . . , iZ兴 gives the effect of the most influential neighbor of atom i. The parameter m min共nm i , n j 兲 thus captures the second condition because it has a large value only when both atoms i and j are strongly affected by at least one of their surface neighbors. Obviously, both first and second conditions can be incorporated in a m m o o bond environment parameter nm ij = max关min共ni , n j 兲 , ni , n j 兴. For the Pij function X = Pij, the environment variable can m m therefore be chosen as x = nm ij , with x f = ns = 0.5 and x f = n f = 0.75. G. The EC surface potential

The total EC potential energy, E, can be obtained as a sum of three terms, the EC rule bond energy, the EC rule dangling bond energy, and the electron constraining energy. Combining Eqs. 共11兲, 共14兲, and 共17兲–共22兲 the total EC rule surface potential is expressed as 6

E = E b + E d + 兺 E ci .

共24兲

i=1

The dangling bond electrons, dini, can be expressed in terms of dynamic electrons, aij, using the electron conservation equation, Eq. 共12兲. Equation 共24兲 can then be viewed as a function of only atom positions and the dynamic electron population, aij and a ji. Assuming that electrons always achieve equilibrium, aij and a ji can be solved from energy minimization considerations at any time step during an atomistic simulation. A conjugate gradient method coupled with a Newton-Raphson algorithm can efficiently minimize Eq. 共24兲 and solve for the aij variables.40 It should be noted that the model results in a continuous variation of aij with respect to atom motion. As a result, the efficiency of the calculations is significantly improved when the aij’s solved from the previous time step are memorized and started as the initial val-

TABLE III. Surface potential parameters bij, wc3, wc4, wc5, and wc6. Bond type

bij

w c3

w c4

w c5

w c6

As–As Ga–Ga As–Ga

6.00 6.00 6.00

100.00 100.00 100.00

100.00 100.00 100.00

100.00 100.00 100.00

100.00 100.00 100.00

ues for the numerical iteration in the next time step. Once aij 共and a ji兲 is known, Eq. 共24兲 becomes a function of atom positions only. The fact that all aij’s are solved from a ⳵E / ⳵aij = 0 condition means that the derivatives of aij’s with respect to atom positions and crystal sizes have no impact on the forces and stresses. Equation 共24兲 can therefore be used as a conventional potential. V. CHARACTERISTICS OF THE SURFACE POTENTIAL

The surface electron counting potential described above has been integrated with a previously parametrized BOP,41 implemented in a molecular dynamics code, and used to investigate the stability of 共001兲 GaAs surfaces. The EC potential discussed above can arbitrarily raise the surface energies of the 共1 ⫻ 2兲 and 共2 ⫻ 1兲 surfaces without affecting any EC rule satisfying structures. This provides a means to accurately model surface energies. The parametrization of the potential is rather simple. As described above, the function qij共r兲 can be determined from the lattice constants of a variety of bulk lattices. The values of bij and wdAs can be directly given using the analytical expressions given in Appendix A. The constraining parameters can be simply chosen to be relatively big numbers. Finally, it may require some fine adjustment of the parameters used in the qij共r兲 function and the xs and x f parameters used in the Si, Zi, and Pij functions to ensure that the EC rule satisfying surface structures are correctly reflected. The robustness of the ECP can be tested by examining the destabilization of the EC rule violating surfaces. For this purpose, we have chosen a set of parameters for bij, wdi, and wc1 − wc6, as listed in Tables III and IV. We found that a value of 100 for parameters wc1 − wc6 effectively established the electron transfer bounds. A value of 6.0 for parameters bij and wdAs and a value of 2.67 for parameter wdGa would raise the 共1 ⫻ 2兲 and 共2 ⫻ 1兲 surface energies by about 40 meV/ Å2. This energy causes the 共1 ⫻ 2兲 and 共2 ⫻ 1兲 surfaces to be less stable than the ␤2共2 ⫻ 4兲 surface for all the four potentials previously explored.27,37 To study surfaces, crystals with periodic boundary conditions in the horizontal axes and free boundary condition in the vertical axis can be used. For static energy calculations on fixed surfaces, double symmetric surfaces can be constructed on both free sides of the crystal. In this scenario, the TABLE IV. Surface potential parameters wdi, wc1, and wc2. Atom type

w di

w c1

As Ga

6.00 2.67

100.00 100.00

w c2 100.00 100.00

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064908-11

J. Appl. Phys. 99, 064908 共2006兲

Zhou, Murdick, and Wadley

TABLE V. Unit area energy of a ␤共2 ⫻ 4兲 surface as a function of crystal thickness. Crystal thickness 共planes兲 Unit area energy 共meV/ Å2兲

4

5

6

7

8

9

10

a

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

a

From crystal with double symmetric surfaces.

surface potential can be applied to the entire system. For simulations of vapor deposition where only growth on one 共top兲 surface is considered, a bottom plane needs to be chosen to terminate the crystal. By reducing Vi and ␯i by half for all the atoms in that plane, atoms below that plane are effectively treated as in the bulk and do not need to be considered in the calculations. It is important that the surface potential produces the same results regardless of the position of the bottom plane. To validate this, we calculated the EC rule energy per unit area of a ␤共2 ⫻ 4兲 surface at various crystal thicknesses. The results of this calculation are shown in Table V, where the value obtained from a crystal with double symmetric surfaces is also included. It can be seen that the results are indeed independent of the depth of the bottom plane and the same results were obtained from both double and single surface crystals. This is because the potential correctly incorporates the effect of surface and bulk environments. Calculations were carried out to determine the unit area surface EC rule energies for a variety of surfaces. The results are shown in Table VI. It verifies that the ␣共2 ⫻ 4兲, ␣共4 ⫻ 2兲, ␣2共2 ⫻ 4兲, ␣2共4 ⫻ 2兲, ␤共2 ⫻ 4兲, ␤共4 ⫻ 2兲, ␤2共2 ⫻ 4兲, ␤2共4 ⫻ 2兲, c共4 ⫻ 4兲-75%, ␥共2 ⫻ 4兲, and ␨共4 ⫻ 2兲 surfaces all have zero EC rule energies, consistent with the notion that these surfaces satisfy the EC rule. The calculations yielded electron populations. The electrons in interatom bonds and dangling bonds on the ␤共2 ⫻ 4兲 surface obtained from calculations are mapped in Fig. 5, where the number beside the arrow from atom i to atom j represents the total electrons that atom i gives to the ij bond, and the number beside the dangling bond arrow is the total electrons in that dangling bond. It can be seen that all the As–Ga bonds and As dangling bonds have two electrons, and the Ga dangling bonds are empty. This is fully in agreement with the empirical EC rule proposed by Pashley.23,30 In the ␤共2 ⫻ 4兲 surface, the number of top plane arsenic dimers, M As, and the number of second plane gallium dimers, M Ga, satisfy the relation M As = 0.75M Ga. The global EC rule model successfully predicted a zero EC rule energy for any As-terminated 共001兲 surfaces satisfying M As = 0.75M Ga.27 The global model, however, did not specify the local population of arsenic atoms on the surface. One issue

FIG. 5. 共Color online兲 Electron populations on the ␤共2 ⫻ 4兲 surface.

for ECP is if it can reveal the energy as a function of atomic population on the surface 共given the same chemical composition兲. In order to test this, we modified a ␤共2 ⫻ 4兲 surface by combining its surface arsenic dimers so that instead of missing one dimer for every four dimer positions, it misses three adjacent arsenic dimers for every 12 dimer positions. Obviously, the structure is still predicted to have a zero EC rule energy by the global EC rule model, while it should have a higher energy than the ␤共2 ⫻ 4兲 surface. Calculations have been carried out for this surface. Indeed, we found that the modified structure has an EC rule energy of about 70 meV/ Å2, as compared to a zero energy for the ␤共2 ⫻ 4兲 surface. It should be pointed out that the surface potential proposed above precisely implements the EC rule for the geometry of the bulk and surface crystals predicted by the BOP.41 The ECP can be superimposed to any existing bonding potential following the simple parametrization procedures described above. VI. CONCLUSIONS

A surface electron counting potential has been developed for correcting conventional interatomic potentials used for covalent semiconductor systems. This potential incorporates the effects of dynamic electron transfer among dangling, surface, and bulk lattice bonds, as approximated by the electron counting rule. The electron counting potential has a zero effect on the potential energy of bulk crystal structures, isolated atoms, or dimers. As a result, it can be superimposed on

TABLE VI. Unit area energy as a function of surface reconstructions. Surface structures Unit area energy 共eV/ Å2兲 Surface structures Unit area energy 共eV/ Å2兲

␣共2 ⫻ 4兲

␣共4 ⫻ 2兲

0.0

0.0

␤2共4 ⫻ 2兲 c共4 ⫻ 4兲 − 75% 0.0

0.0

␣2共2 ⫻ 4兲 ␣2共4 ⫻ 2兲 ␤共2 ⫻ 4兲 ␤共4 ⫻ 2兲 ␤2共2 ⫻ 4兲 0.0

0.0

0.0

0.0

0.0

␥共2 ⫻ 4兲

␨共4 ⫻ 2兲

共1 ⫻ 2兲

共2 ⫻ 1兲

¯

0.0

0.0

0.040

0.040

¯

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064908-12

J. Appl. Phys. 99, 064908 共2006兲

Zhou, Murdick, and Wadley

any existing potentials without affecting the predicted properties, such as cohesive energy, lattice constants, and elastic constants of bulk structures. Preliminary energy calculations using the potential correctly predicted the various surface reconstructions observed in experiments. It also reproduce the electron population in the bonds and dangling bonds that are predicted from the empirical EC rule.

ACKNOWLEDGMENTS

We are grateful to the Defense Advanced Research Projects Agency and Office of Naval Research 共C. Schwartz and J. Christodoulou, Program Managers兲 for support of this work through Grant No. N00014-03-C-0288. We also thank S. A. Wolf for numerous helpful discussions.

E共1⫻2兲 =

冋 冉

1 3 + aAsGa − 2 2 8bGaAs 4 a

冊

2

+ 2bAsAs共2aAsAs − 2兲2

册

+ 2wdAs共5 − 2aAsGa − aAsAs − 2兲2 ,

共A1兲

where a is the lattice constant of the zinc-blende GaAs crystal and a2 represents the area of the 共1 ⫻ 2兲 periodic cell. Note here that the electron constraining terms, Eqs. 共17兲–共22兲, are not included in Eq. 共A1兲 because electrons do not seek to exceed their bounds under the equilibrium condition. Energy minimization with respect to aAsGa and aAsAs results in an equilibrium EC rule surface energy of the 共1 ⫻ 2兲 surface, E共1⫻2兲 =

2bGaAsbAsAswdAs 2w共1⫻2兲 1 . = 2 a 4bAsAswdAs + bGaAs共4bAsAs + wdAs兲 9a2 共A2兲

APPENDIX: ANALYTICAL EXPRESSION FOR EC ENERGY OF „1 Ã 2… AND „2 Ã 1… SURFACES

The ECP parameters can be chosen so that the surface energies of the 共1 ⫻ 2兲 and 共2 ⫻ 1兲 surfaces match their desired values. For this purpose, it is very convenient to assume bGaGa = bAsAs = bGaAs = wdAs = w共1⫻2兲, and wdGa = w共2⫻1兲. Here we derive analytical expressions for w共2⫻1兲 and w共1⫻2兲 in the limit of infinitely large coefficients wc1 − wc6 共i.e., the electron bounds are exact兲. For the As-terminated 共1 ⫻ 2兲 surface shown in Fig. 1共b兲, consider the top arsenic and the second gallium planes. Within the 共1 ⫻ 2兲 periodic cell, there are four identical Ga–As bonds, one As–As dimer bond, and two arsenic dangling bonds 共one per arsenic atom兲. Due to symmetry, the four Ga–As bonds have an equivalent electron distribution. Similarly, the two arsenic dangling bonds contain an equal number of electrons. Each of the Ga–As bond receives 3 / 4 static electrons from the underlying gallium atom and aAsGa dynamic electrons from the top arsenic atom. The surface As–As dimer bond receives aAsAs dynamic electrons from each of the two arsenic atoms. The remaining arsenic valence electrons, which equals 5 − 2AsGa − aAsAs, then populate the arsenic dangling bond. According to Eq. 共24兲, the EC rule energy per unit surface area can be written as

E共2⫻1兲 =

For the Ga-terminated 共2 ⫻ 1兲 surface shown in Fig. 1共c兲, consider the top gallium and the second arsenic planes. Within the 共2 ⫻ 1兲 periodic cell, there are four identical Ga–As bonds, one Ga–Ga dimer bond, and two gallium dangling bonds. Each of the Ga–As bond receives 5 / 4 static electrons from the underlying arsenic atom and aGaAs dynamic electrons from the top gallium atom. The surface Ga–Ga dimer bond receives aGaGa dynamic electrons from each of the two gallium atoms. The remaining 3 − 2aGaAs − aGaGa gallium valence electrons then populate the gallium dangling bond. According to Eq. 共24兲, the EC rule surface energy can be written as E共2⫻1兲 =

冋 冉

1 5 + aGaAs − 2 2 8bGaAs 4 a

冊

2

+ 2bGaGa共2aGaGa − 2兲2

+ 2wdGa共3 − 2aGaAs − aGaGa兲2 + 8wc4

冉

5 + aGaAs − 2 4

冊

2

册

+ 2wc4共2aGaGa − 2兲2 . 共A3兲

Note here that the electron constraining terms cannot be eliminated because electrons in bonds seek to exceed their bounds under the equilibrium condition. Energy minimization with respect to aGaAs and aGaGa results in an equilibrium EC rule surface energy of the 共2 ⫻ 1兲 surface,

wd 2共bGaGa + wc4兲共bGaAs + wc4兲wdGa w共2⫻1兲 1 = Ga . 2 2 = a 4共bGaGa + wc4兲共bGaAs + wc4兲 + 共4bGaGa + bGaAs + 5wdGa兲wdGa 2a 2a2

共A4兲

Equation 共A4兲 assumes an infinity value of wc4. According to Eqs. 共A2兲 and 共A4兲, w共1⫻2兲 = 9a2E共1⫻2兲 / 2 and w共2⫻1兲 = 2a2E共2⫻1兲, where E共1⫻2兲 and E共2⫻1兲 can be viewed as the energy shifts required for a 共conventional兲 potential to predict the correct 共1 ⫻ 2兲 and 共1 ⫻ 2兲 surface energies.

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