Ab initio study of atomic disorder on As-rich GaAs(111)A surface

Surface Science 641 (2015) 330–335

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Ab initio study of atomic disorder on As-rich GaAs(111)A surface O. Romanyuk a,⁎, P. Mutombo a, F. Grosse b a b

Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, CZ-162 53 Prague 6, Czech Republic Paul-Drude-Institut für Festkörperelektronik, Hausvogteiplatz 5–7, D-10117 Berlin, Germany

a r t i c l e

i n f o

Available online 29 January 2015 Keywords: GaAs(111) Surface reconstructions Surface kinetics Density functional theory

a b s t r a c t Mechanisms for the appearance of disorder on the As-rich GaAs(111)A surface were investigated employing density functional theory (DFT). Focus was given to the As trimer interactions by considering different surface symmetries and rest site occupations. The (2 × 2) and the c(4 × 2) structure models with As trimer and an As rest site were found the most energetically stable under the As-rich experimental conditions at T = 0 K. Low interactions between neighboring As trimers causes disorder in thermodynamic equilibrium at finite temperatures. A careful analysis of the configurational entropy contributions including the different statistics was carried out. The experimentally observed As-rich (2 × 2) structure was confirmed to be kinetically stabilized. The stabilization mechanism is discussed with respect to the As trimer migration on the surface, which is limited by a large diffusion barrier through the As rest sites. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Low index group III–V semiconductor surfaces were studied in numerous publications over the last decades. Renewed interest appeared with epitaxial growth of nanostructures, including quantum dots and more recently nanowires. The knowledge of the surface structures and their energetics is essential e.g. for the determination of equilibrium shapes [1] or the basis for studying the growth dynamics [2,3]. GaAs is the protypical group III–V compound. Its clean polar surfaces exhibit a large number of surface reconstructions depending on the preparation conditions including temporal evolution of substrate temperature as well as chemical potentials of the constituents. The atomic surface structure of the GaAs(001) reconstructions was well studied [4]. The GaAs(111) surfaces, both cation terminated (111)A and anion terminated (111)B have attracted less attention, however. The GaAs(111)B surface was found to exhibit many different reconstructions depending on substrate temperature and As/Ga flux concentration ratio. A phase diagram of the GaAs(111)B surface includes the (2 × 2), pffiffiffiffiffiffi pffiffiffiffiffiffi (1 × 1)LT, 19  19 , and (1 × 1)HT reconstructions [5]. In contrast to the GaAs(111)B surface, only a (2 × 2) diffraction symmetry was reported on the cation terminated GaAs(111)A surface by reflection highenergy electron diffraction (RHEED) [5,6]. Two phases of the (2 × 2) structure were suggested for the GaAs(111)A surface. The Ga-rich (2 × 2) phase is Ga terminated and contains one Ga vacancy per surface unit cell (the vacancy buckling model, VB). This structure was confirmed by many techniques including low-energy electron diffraction ⁎ Corresponding author. E-mail address: [email protected] (O. Romanyuk).

http://dx.doi.org/10.1016/j.susc.2015.01.015 0039-6028/© 2015 Elsevier B.V. All rights reserved.

(LEED) [7], grazing incidence X-ray diffraction [8], scanning tunneling microscopy (STM) [6], and ab initio calculations using density functional theory (DFT) [1,9–11]. The VB model was also confirmed for the InAs [12], GaSb [13], InP [14] and InSb [15,16] surfaces. The second (2 × 2) phase on the GaAs(111)A phase diagram is an As-rich (2 × 2) phase. It was suggested that this structure contains one As trimer per surface unit cell on the T4 atomic site [1]. A similar model was suggested for the GaAs(111)B (2 × 2) surface [6]. Recently, the (2 × 2) structure of the GaSb(111)A surface was investigated by ab initio calculations with respect to stabilization by configurational entropy [17]. The stability of the (2 × 2)-VB model was confirmed for the Ga-rich conditions similar to the GaAs(111)A surface. Under more Sb-rich conditions the former (2 × 2) trimer model was improved by replacing the Ga atom in the second atomic layer at the rest site by Sb. A similar result was confirmed for the As-rich GaAs(111)A (2 × 2) surface [17]. Later, the formation process of the As-rich GaAs(111)A (2 × 2) phase was investigated by X-ray photoelectron spectroscopy (XPS), rocking curve RHEED, and STM measurements [18]. The As trimer with the As rest site, T(II) model in the following, was confirmed experimentally. The structure was found to be kinetically stabilized. An amorphous As film deposition and sequential substrate annealing are required to produce the As-rich (2 × 2) structure, whereas the As-rich structure was not obtained by cooling of the sample under constant As2 flux. In addition, multiple phase coexistence including As trimers on top, T4, and hollow, H3, site positions on the surface was suggested. The agreement between the experimental and theoretical RHEED rocking curves could be achieved assuming a multiple domain structure. If the observed behavior can be understood by thermodynamic equilibrium or if kinetics is responsible can only be answered by calculating the interactions between the reconstruction domains.

O. Romanyuk et al. / Surface Science 641 (2015) 330–335

The absorption of As on the GaAs(111)A surface was studied by ab initio DFT calculation in the past [19,20]. The VB and As trimer models were confirmed to be stable. The absorption of As at the Ga vacancy site was found unfavorable without the presence of an As adatom or an As trimer on top of the Ga layer. The surface energy of the As adatom structure, however, was found much higher than the one of the As trimer model [10]. Atomic structure with As substitution site in the Ga layer was not considered in these studies. In the current paper, the basic structural motifs for GaAs(111)A reconstructions are identified and their interactions are studied. The reconstruction phase diagram is extended including Bravais lattice diversity [21]. Total energy calculations are carried out on (2 × 2),  pffiffiffi pffiffiffi pffiffiffi c(4 × 2), (4 × 2), 2 3  2 3 R30° [2 3 in short], and c(4 × 4) unit cells. The structural motifs include Ga vacancies, As trimers, and As or Ga rest sites within these surface unit cells. Finite temperature phase concentrations at thermodynamic equilibrium are derived by means of partition function calculations.

applied for unit cells with different periodicity and symmetry [17]. For larger unit cells, the product (m × n) increases, whereas the symmetry related term g changes according to the additional symmetry operations (translations, glide operation, rotations). Adsorption–desorption behavior of As trimers on As-rich surfaces is treated by comparing the chemical potential of the solid μAs with that of the molecular counterpart in the vapor phase μ As2 [28–30]. μAs per atom is obtained by computing desorption energy ΔE of n As atoms from the surface: nΔEAs ¼ Etot −Ere f −n

" 2μ As2 ¼ −kB Tln

bulk

ΔγA ¼ Esurf −ðnAs −nGa Þμ As −nGa μ GaAs ;

ð1Þ

where Esurf represents the total energy of the system, μi the chemical potential of species i, ni the number of atoms of the species i, and A is the unit cell area, respectively. The Ga-rich and As-rich experimental conditions correspond to the chemical potential range [1]: bulk

bulk

μ As −ΔH f b μ As b μ As ;

ð2Þ

where ΔHf = −0.822 eV is the computed heat of formation of GaAs. The bulk μ bulk Ga and μAs chemical potentials are computed for the orthorhombic α-Ga [27] and rhombohedral As phases, respectively. The converged GaAs bulk lattice constant of 5.536 Å is used. Phase concentration at an elevated temperature is estimated by the partition function calculations including the configuration entropy contributions [17]. The concentration of a phase i in thermodynamic equilibrium at finite temperature is ci = Zi/Z, i ∈ S, where Z is the partition function and S covers all possible system states, i.e. phases. In pffiffiffi our calculations, As-rich phases with (2 × 2), (4 × 2), c(4 × 2), 2 3 and c(4 × 4) unit cells are used. The partition function is expressed as follows [17]: Z¼

X i

Zi ¼

X i

  Δγi A ; g i exp − kB T

ð3Þ

where Δγi is the surface energy of the ith phase with area A, kB is the Boltzmann constant, T is the temperature, and g is a degeneracy factor, which is related to the unit cell size (m × n) cells and cell symmetry. The summation is over allowed inequivalent structures. Eq. (3) is

EAs2

ð4Þ

2

where Etot is the total energy of the relaxed As-rich surface with adsorbate atoms, Eref is the total energy of the relaxed structure without adsorbate atoms, and EAs2 is the total energy of the As2 molecule. Gas chemical potential (per atom) is obtained by partition functions calculation [20,30]:

2. Computational details The total energy calculations are carried out using the ABINIT computer code [22,23]. The local-density approximation for the exchange–correlation energy functional is applied. Norm-conserving pseudopotentials [24] of the Troullier–Martins type [25] are used to describe the atomic species. The electronic wave functions are expanded in a plane wave basis. A kinetic energy cutoff of 12 Hartree (Ha) and a k point set corresponding to 12 × 12 per (1 × 1) surface Brillouin zone [26] are used. Surface structures are constructed using the repeated supercell approach with slab thickness of four GaAs bilayers and trimer layer. The vacuum gap thickness of 10 Å is used. The bottom As layer of the slab is passivated by pseudohydrogens with 0.75 electronic charges. Atomic coordinates are adjusted until the interatomic forces become smaller than 10− 4 Ha/Bohr, whereby only the three bottom layers (Ga–As–pseudo-H) are kept fixed. The surface free energy density is defined by the following expression [1]:

331

kB T  d ξtrans  ξrot  ξυib pAs2

# ð5Þ

where kB is Boltzmann constant, T is the gas temperature, pAs2 is the As2 beam equivalent pressure (BEP), d = 1 is the degree of degeneracy of the diatomic As2 electron energy levels [valence electrons ground state is (σ4s)2 (σ4s⁎)2 (π4p)4 (σ4p)2], and ξtrans, ξrot, and ξυib are the molecular partition functions for the translation, vibration, and rotation motions, respectively. Partition functions are defined as:   2 3=2 ξtrans ¼ 2πmkB T=h

ð6Þ

ξrot ¼ kB T=σ B

ð7Þ

  −hν=kB T −1 ξυib ¼ 1−e

ð8Þ

where m is the mass of the As2 molecule, h is Planck constant, σ = 2 is the symmetric number of As2 molecule [29],B ¼ 0:1 cm–1 is the computed rotational constant of the As2 molecule with As\As bond length of r = 2.13 Å, and ν is the vibration frequency. As trimer adsorption–desorption diagram is derived by comparing μAs with μ As2 : As desorption or adsorption correspond to the μ As Nμ As2 or μ As b μ As2 conditions, respectively. 3. Results and discussion 3.1. Ground state surface stability diagram All structure models considered in the paper fulfill the electron counting model (ECM) [31], which is a guiding principle to evaluate stable surface reconstructions with a specific surface stoichiometry. The ECM requires that the energetically high-lying dangling bond states of the group III element (Ga) to be empty, and those dangling bond states of the group V element (As), which are lower in energy, to be filled. Hence, surfaces fulfilling ECM are likely semiconducting. In particular, a large variety of structure configurations are possible for (111)A surfaces [32]. The As-rich GaAs(111)A (2 × 2) models are built up from the As trimers and Ga or As rest sites. The partial electronic charges for this surface structure motifs are − 3/4 and 3/4 electrons per (1 × 1) cell. A total electronic charge is zero for these models and, thus, these structures obey the ECM. In the past, As trimer stacking faults on the As-rich GaAs(111)A [33,34] and GaAs(111)B [6,35] (2 × 2) surfaces were observed by STM. Domains with local (2 × 2) and c(4 × 2) periodicity were observed. A c(4 × 2) structure can be considered as a disordered (2 × 2) structure: two (2 × 2) atomic rows are shifted by one surface lattice

O. Romanyuk et al. / Surface Science 641 (2015) 330–335

constant relative to each other. ECM does fulfill for the c(4 × 2) structure. Despite the appearance of the shifted cells in STM observations, the c(4 × 2) diffraction patterns have never been reported. In Fig. 1(a), GaAs(111)A structure models are presented. The VB model consists of one Ga vacancy per (2 × 2) surface unit cell [1,7]. The T(I) is the As trimer on T4 site with Ga rest site model [1,12]. This structure was considered as a stable As-rich structure for many years but modified recently [17,18]: the T(II) structure model with the As rest site in the second layer and As trimer on the T 4 sites was confirmed for the As-rich experimental conditions. pffiffiffi In addition to the (2 × 2) structures, the GaAs(111)A 2 3 and pffiffiffi c(4 × 4) structures are considered. The 2 3 structure models cover all possible As-rich configurations with the As trimers and As substitutional sites within a surface unit cell (not shown here) similar to the pffiffiffi pffiffiffi GaSb(111)A 2 3 structure models [17]. In Fig. 1(a), the 2 3 -T(II) strucpffiffiffi ture, which has the lowest surface energy among the 2 3 structures, is only shown. The structure consists of three As trimers and three As rest sites per surface unit cell. The c(4 × 4)-T(II) [T(I)] structure consists of four As trimers and four As [Ga] rest sites per surface unit cell. Surface unit cells are marked by dashed lines in Fig. 1(a). Local atomic disorder on the surface is considered by shifting (2 × 2) unit cells with respect to each other by one surface lattice constant as h i h i along the 101 direction. A shift along the 110 direction produces symmetrically equivalent domains. In Fig. 1(b), different Bravais lattice types are presented. The shifted cell with the same (2 × 2) motif, i.e. T(II), T(I), or VB, forms the c(4 × 2) Bravais lattice. In the case of different occupancy of two neighboring (2 × 2) cells, i.e. combination of T(II) and T(I) for instance, the (4 × 2) Bravais lattices of type a or b are possible [Fig. 1(b)]. Bravais lattices with T(I), T(II), VB, and H(II) motifs are considered in the present paper, whereas the H(I) motif is excluded due to its high energy. In Fig. 2, the GaAs(111)A surface energy phase diagram for T = 0 K is shown. There are three stable phases. The (2 × 2)-VB structure is stable along a wide As chemical potential range (ΔHf b μAs − μ bulk As b −0.178). The (2 × 2)-T(II) and the c(4 × 2)-T(II) structures have almost the same energy with only an energy difference of 2 meV/(1 × 1), which is lower than the present calculation precision. These two configurations are represented by a single curve in Fig. 2. The (2 × 2)-H(II) and c(4 × 2)-H(II) structures have the same energy [a single curve in Fig. 2]. Thus, weak interaction between the As trimers occurs for the T(II) or H(II) trimer sites. The H(II) structure is less stable than the corresponding T(II) structures. A smaller energy difference between GaAs(111)A (2 × 2) T(II) and H(II) structures was reported recently. However, the p3m1 surface unit cell symmetry was not strictly preserved [18]. The larger number of relaxed parameters might affect

0.10 c(4x2)-VB

0.05 0.00 (2x2)-VB

-0.05

,

332

-0.10 -0.15

(2x2)-T(II) c(4x2)-T(II)

-0.20 -0.3 Ga-rich

-0.2

-0.1 As

(bulk) As

c(4x4)-T(I) T(I) H(II)T(I) T(II)T(I) H(II) T(II)H(II) c(4x4)-T(II) 0.0 As-rich

Fig. 2. (Color online) Energy phase diagram of the GaAs(111)A surface for T = 0 K. The thermodynamically stable phases are the vacancy buckling model (2 × 2)-VB, (2 × 2)pffiffiffi T(II), and the c(4 × 2)-T(II) structures. Each curve except the VB, 2 3 and c(4 × 4) models represents two phases: the (2 × 2) and the c(4 × 2) [or (4 × 2)] structures. The energy difference between these two configurations is negligible (few meV/(1 × 1)). The red curve represents the T(II)T(I) structure with (4 × 2)a and (4 × 2)b cells. The c(4 × 4)-T(II) is only 10 meV/(1 × 1) higher in energy than (2 × 2)-T(II).

the total energy. In the present paper, the surface unit cell symmetry is preserved in all cases. pffiffiffi The energy difference between the 2 3 -T(II) structure and the (2 × 2)-T(II) structure is 20 meV/(1 × 1). This difference is larger compared to the GaSb(111)A surface [17]. The total energies of T(II)T(I) structures with unshifted (4 × 2)a and shifted (4 × 2)b configurations are low at μAs − μbulk As = −0.178. Here a coexistence of VB, T(II), and T(I) structures is possible at finite temperature. The total energy of the c(4 × 4)-T(II) structure is higher by only 10 meV/(1 × 1) than the energy of the (2 × 2)-T(II) structure. A larger difference is found for the corresponding structures with Ga rest sites [c(4 × 4)-T(I)]. The interaction energy between Ga vacancies is also estimated by considering unshifted (2 × 2) and shifted c(4 × 2) cells with VB motifs. A large energy difference [28 meV/(1 × 1)] between this two phases is pffiffiffi found. The energy difference is even much larger for the 2 3 and c(4 × 4) cells with VB structural motifs (not shown here). In contrast to the As trimers, Ga vacancies interact strongly on the GaAs(111)A surface. 3.2. As trimer adsorption–desorption diagram As trimer desorption from different As-rich GaAs(111)A surface reconstructions was studied in the past [20]. An As trimer desorption temperature from the (2 × 2)-T(I) structure was derived (T ≃ 470∘C,

b)

a) VB

T(I)

T(II)

H(II)

(2x2)

[101]

as -T(II)

T(II)

c(4x2) T(II)

T(II) as

[110]

c(4x4)-T(II)

a

(4x2) nd

2

- As, 3

rd

T(II) T(I)

(4x2)b T(I) T(II)

ML ML

Fig. 1. (Color online) a) Ball-and-stick models of the GaAs(111)A structures. The VB model consists of one Ga vacancy per (2 × 2) unit cell. The As-rich models involve As trimers on top T4 pffiffiffi and hollow H3 sites. The rest atoms are occupied by Ga (I) or As (II) atoms. The 2 3 and c(4 × 4) models contain three or four trimers and As rest sites per surface unit cell, respectively. h i h i b) Possible translation periodicity on a surface. A c(4 × 2) structure is formed by the similar (2 × 2) cells shifted on one surface lattice constant as along 101 or 110 directions. A (4 × 2) structure is formed by dissimilar (2 × 2) cells [T(II)T(I) or T(II)H(II)].

O. Romanyuk et al. / Surface Science 641 (2015) 330–335

p = 2 × 10−7 Torr). Recently, phase transition temperature was measured close to T = 390 °C during heating and T = 290 °C during cooling under As2 flux [18]. The difference in the desorption temperatures was related to the difference in the calculated As adsorption energy of the As-rich surface models. In particular, an energetically unfavorable As ad-atom on the a-top site structure model was considered as a transition structure [20]. In the current paper, the As trimer desorption from energetically more stable T(I) and T(II) structures is considered. In Fig. 3 a), As2 chemical potential dependence on temperature is shown. μ As2 decreases with temperature [29]. The As trimer desorption from the (2 × 2)-T(I) surface occurs when the As chemical potential μAs (red line in the figure) is larger than μ As2 . Note, a one step As trimer desorption process from T(I) substrate is assumed here, i.e. the (2 × 2) to (1 × 1) transition occurs in the particular case. Moreover, a complete As trimer desorption from T(I) and T(II) surfaces is considered for all As-rich structures. In Fig. 3 b), the As trimer adsorption–desorption diagram is shown. The As trimer desorption temperatures for (2 × 2)-T(I), T(II), and c(4 × 2)-T(II), c(4 × 4)-T(II) domains are slightly different (within 60 °C range). In contrast to previous works, the computed As trimer adsorption/desorption temperatures are close to the experimental phase transition temperatures (290°–390°, p = 2 × 10−7 Torr) [18]. It should be mentioned, however, that the free energy of vibration is included in μ As2 calculations, whereas it is not for the bulk and surface atoms. The assumption that the latter contributes identical to all surface structures could cause an inaccuracy in the theoretical phase transition temperature. The As coverage is identical for all T(II) models but the local As trimer arrangement is different. The latter influences the As trimer kinetics. Therefore, the smearing of the phase transition temperature could be caused by atomic disorder and multiple phase co-existence.

3.3. Diversity of As trimer arrangement at elevated temperature In order to estimate the phase concentration at elevated temperature, the partition function is computed for T = 300 °C. The configurational entropy term was included into partition function calculation by means of degeneracy factors g, which depends on unit cell size and surface unit cell symmetry [17]. g gives the total number of all possible nonequivalent cell configurations. For instance, the (2 × 2)-T(II) structure has four possible nonequivalent configurations. These configurations can be generated by simple translation of the surface unit cell origin by 2 × 2 times. Rotation operations are part of the surface cell symmetry (p3m1) which have to be excluded (degenerated). Symmetry of the c(4 × 2)-T(II) structure is lower (p1m1). It gives additional three rotation domains, each of them can be translated four times. The g factor of the c(4 × 2)-T(II) structure is therefore 3 × 4 = 12. In Table 1, the surface unit cell size, the number of

As2

-1.1 -1.2 -1.3 -1.4 -1.5 -1.6 -1.7 -1.8

Table 1 Symmetry-determined degeneracy factor g for the cells with different space group symmetry. f is the ratio of the number of symmetry operations of the unreconstructed (1 × 1) and the reconstructed surface unit cell [17]. Structure

Cell size

Symmetry

f

g

Unreconstructed T(II), VB T(II) T(II)T(I) T(II)H(II) T(II)

(1 × 1) (2 × 2) c(4 × 2) (4 × 2)a,b (4 × 2)a,b pffiffiffi 2 3 c(4 × 4)

p3m1 p3m1 p1m1 p1m1 p1 p3m1

1 1 3 3 6 1

1 4 12 24 48 12

p1g1

3

48

T(II)

symmetry operations f and the g values are summarized for the different reconstructions. In Fig. 4, the dependence of the phase concentration ci on the As chemical potential at T = 300 °C is shown. The concentration of the VB phase is close to 100% up to μAs − μbulk As b −0.178. At a narrow chemical potential range close to the phase transition point, T(II) and T(I) structural motifs coexist on a surface: 20% of (4 × 2)a and 20% of (4 × 2)b phase is predicted. Under As-rich conditions, the concentrations of the c(4 × 2)-T(II), c(4 × 4)-T(II), and (2 × 2)-T(II) phases are equal to 57%, 21%, and 19%, respectively. Phases with concentration less than 5% were omitted in Fig. 4. The concentrations of the c(4 × 2)-T(II) and c(4 × 4)-T(II) phases are larger than the concentration of the (2 × 2)-T(II) phase due to configurational entropy. The latter has a higher surface unit cell symmetry and smaller unit cell size, i.e. a smaller g factor (Table 1). Thus, a phase coexistence and correspondingly atomic disorder on a surface are predicted for the GaAs(111)A surface under As-rich experimental conditions in thermodynamic equilibrium. The driving force of the disorder is the low interaction among the As trimers on a surface. Recently, rocking curve RHEED analysis was carried out on the Asrich GaAs(111)A surface [18]. The agreement between experimental and theoretical RHEED rocking curves was improved when two coherently scattered domains with T(II) and H(II) structures were considered. In other words, local atomic disorder or small domain coexistence on a surface was assumed. The incoherent combination of RHEED intensities did not improve the agreement. The rocking curve RHEED analysis of the T(II) and H(II) coexisting phases did not reach the level of satisfactory agreement, however [18]. This was explained by the presence of the atomic disorder on the surface. Our results support atomic disorder on a As-rich GaAs(111)A surface due to low interaction energy between As trimers and predict the coexistence of the (2 × 2)-T(II), c(4 × 2)-T(II), and c(4 × 4)-T(II) domain fractions at elevated temperature in thermodynamic equilibrium. It

b)

2

500 Ga-rich

450

T, oC

a)

333

[T(I)]

As

p(As2), Torr 10-3 10-4 10-5 10-6 -7 10

340

380

400

desorption

350 As-rich

300 adsorption

420 o T, C

460

500

250 1E-7

1E-6

(2x2)-T(I) (2x2)-T(II) c(4x4)-T(II) c(4x2)-T(II)

1E-5 1E-4 p(As2), Torr

1E-3

Fig. 3. (Color online) a) As2 chemical potential as a function of temperature for different BEP of As2 flux. Chemical potential of the As atom on (2 × 2)-T(I) structure is indicated. Adsorption– desorption behavior of As trimers on/from As-rich surfaces is shown in b). Experimentally observed transition temperatures during heating and cooling [18] are indicated by the ticks.

334

O. Romanyuk et al. / Surface Science 641 (2015) 330–335

Fig. 4. (Color online) The phase concentration diagram for substrate temperature T = 300 °C. At As-rich conditions, the c(4 × 4)-T(II) structure should dominate due to configurational entropy in thermodynamic equilibrium. Experimentally, however, the kinetically stabilized As-rich (2 × 2) phase is observed.

should be emphasized, however, that configurational entropy calculation approach gives a phase concentration probability without taking into account defects on the surface. Defects can form for example due to simultaneous cell shifts along two sides of the (2 × 2) cell. Surface formation kinetics including defect formation on a surface could be studied by kinetic Monte Carlo simulations [3] which is however out of the scope of the present paper. In Fig. 5, in-plane diffraction pattern symmetry is shown schematically for (a) (2 × 2)-T(II), (b) c(4 × 2)-T(II), and (c) c(4 × 4)-T(II) structures. Patterns are simulated for three rotation domains. All the patterns consist of (2 × 2) fractional order reflections, i.e. (2 × 2) periodicity would be enhanced, whereas intensity of the rest fractional order reflections could be suppressed due to atomic disorder. In case of phase coexistence, the (2 × 2) fractional order reflections involve scattering contributions from all these phases. Unit cell stacking faults or one-dimensional disorder on surfaces was investigated for GaAs(001) (2 × 4)/c(2 × 8) [36–40] and GaSb(001) (4 × 3) surfaces [21,36,41]. For these surfaces, weak interaction between neighboring surface unit cells along the specific crystallographic directions on a surface was found. Atomic disorder alters the diffraction intensities: diffuse streaks can appear or even diffraction spot extinction can occur. Only (2 × 2) diffraction patterns were experimentally observed for the As-rich GaAs(111)A and GaAs(111)B surfaces by RHEED along the high symmetrical crystallographic directions [6,18]. Diffuse streaks on diffraction patterns due to disorder have never been reported for Asrich phases even the c(4 × 2) periodicity was observed locally by STM

a)

b)

c)

[33–35]. On the other hand, the azimuthal scan RHEED patterns have revealed atomic disorder and diffuse streak intensities for other III–V(111) semiconductor surfaces [32]. The phase concentration diagram in Fig. 4 shows statistical distribution in thermodynamic equilibrium. Experimentally, the thermodynamic equilibrium might not be achieved: the As-rich GaAs(111)A (2 × 2) phase was found to be kinetically stabilized [18]. The As-rich phase cannot be stabilized from the VB structure by As2 flux only. An As2 deposition at low temperature and subsequent anneal are necessary to stabilize the As-rich (2 × 2) phase. We speculate, that vacancies of the well-ordered and stable (2 × 2)-VB structure are occupied by As atoms first during As2 deposition. One As rest atom and the As trimer can be formed by arsenic molecule dissociation (2 As2 → 1 As trimer + 1 As rest site). As trimer diffusion and As rest site re-arrangement are required to form c(4 × 2) or c(4 × 4) structures. A trimer flip through the As rest site, however, violates the ECM. We found, that the total energy of the (2 × 2)-T(II) structure with As trimer bond to As rest site and one Ga dangling bond has a 0.35 eV/(1 × 1) higher energy than the corresponding T(II) structure with an As rest site dangling bond. Thus, As rest sites on positions of the Ga vacancies of the VB structure limit the trimer diffusion on a surface. High substrate temperature would require to overcome the energy barrier and to re-arrange the As rest sites and trimers. Temperature increase, however, could cause As desorption from surface and Ga-rich VB phase could form. Thus, our calculations confirm that the experimentally observed As-rich GaAs(111)A (2 × 2) structure is kinetically stabilized. 4. Conclusions Stability of the As-rich GaAs(111)A surface was investigated by ab initio DFT calculations. Two structures were found to be stable in thermodynamic equilibrium: the (2 × 2)-T(II) and the c(4 × 2)-T(II) structures with As trimers and As rest site atoms. Two structures have almost the same surface energy. Therefore, there is a weak interaction among As trimers on the surface. Such low interaction leads to atomic disorder on the surface in thermodynamic equilibrium. At elevated temperatures, a few fractions of As-rich phases can coexist on a surface. Experimentally, only the (2 × 2) structure was observed. It is suggested that the (2 × 2) structure is kinetically limited due to a high diffusion barrier of the As trimer through the As rest site atom. Acknowledgments Support by the Grant Agency of the Czech Republic (Grant No. P204/10/P028) and Academy of Sciences of the Czech Republic (project No. M100101201) is gratefully acknowledged. The access to the MetaCentrum computing facilities provided under the program “Projects of Large Infrastructure for Research, Development, and Innovations” LM2010005 funded by the Ministry of Education, Youth, and Sports of the Czech Republic is highly appreciated. References

Fig. 5. (Color online) Schematic diffraction pattern symmetry for the (a) (2 × 2), (b) c(4 × 2), and (c) c(4 × 4) structures in T(II) configuration. Large and small sphere represent integer- and fractional-order beams, respectively. Green spheres mark the (2 × 2) fractional-order beams. All the patterns consist of the (2 × 2) fractional-order beams.

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