Adatom Interactions on GaN(0001) Surface I - ACS Publications

Article pubs.acs.org/JPCC

Adatom Interactions on GaN(0001) Surface I: Coverage-Dependent Adsorption Manjusha Chugh and Madhav Ranganathan* Department of Chemistry, Indian Institute of Technology Kanpur, Kanpur, India S Supporting Information *

ABSTRACT: The adsorption of an adatom on the surface of a crystal is affected by the surface coverage due to adatom− adatom and adatom−substrate interactions. We study the dependence of adsorption energy of Ga and N adatoms on wurtzite GaN(0001) as a function of decreasing surface coverage from 0.25 to 0.04 monolayers through first-principles calculations. The adsorption energies of Ga and N adatoms on the flat, clean GaN(0001) substrate do not converge with decreasing coverage. Further, it appears that the Ga and N adatoms significantly distort the substrate lattice from the flat configuration. We found that these distortions increase with increase in system size (or, equivalently, with reducing coverage). This observation is counterintuitive since it is expected that lattice distortions should decrease with decreasing surface coverage. We separate the different contributions to the adsorption energy and identify the part of adsorption energy that arises due to lattice distortions. This contribution appears to be much larger than the contribution due to dipolar interaction between adatoms and their periodic images. This also allows us to identify some surface reconstructions of clean GaN(0001) that have not been reported yet, which are lower in energy than the flat clean substrate. The adsorption energies calculated with respect to these reconstructed configurations converge with increasing system size. This size-dependent or coverage-dependent study suggests that the effect of lattice distortions should be carefully taken into account when calculating adsorption energy for periodic systems from first-principles methods. adatoms interactions, adatom-induced surface relaxations20 and reconstructions, etc.15,17 One example of significant adsorptioninduced surface reconstructions is the H/Pd(111) system.14 Adatom coverage (and, hence, adatom interactions) can also change the surface diffusion barriers. Properly identifying adsorption sites, adsorption geometries, and studying adatom−substrate and adatom−adatom interactions may also help in catalysis, where reaction sites and reaction paths play an important role.19,21−23 GaN is a wide and direct band gap semiconductor. The quality of GaN thin films is very important for their applications in blue and ultraviolet light-emitting diodes and lasers.24,25 The (0001) surface of GaN is technologically more important than other surfaces and is widely used in fabricating many optoelectronic devices.26,27 A large amount of research over the past 25 years has been carried out to understand the adsorption and diffusion of adatoms, surface reconstructions, and growing surface morphologies of GaN.28−35 Adsorption of heteroatoms and molecules on GaN has been investigated as a function of coverage of the surface.36−38 These studies on GaN have been done in the coverage range of 0.25−1 monolayer (ML). However, to our knowledge, no coverage-dependent

1. INTRODUCTION The adsorption energy of an atom on a surface changes due to the presence of another atom adsorbed on a nearby site.1−3 This effective interaction between two adatoms can be understood as a result of two different contributions: direct or “through space” and indirect or “through substrate”.4 Direct interaction includes contributions of electric dipoles, van der Waals interactions, and bonds between adatoms. The elastic effects from the perturbations in substrate atomic coordinates cause an indirect interaction. Both the dipolar and elastic interactions are expected to vary with lateral distance d as d−3 at large distances.5 Another indirect interaction is an electronic interaction mediated by the electronic band structure of the substrate. For metallic systems, this leads to an oscillatory interaction, analogous to Friedel oscillations, with an envelope function that goes either as d−2 or d−5 depending on nearby surface states.6−9 Several studies of the effect of adatom interactions on diffusion and growth morphology have been performed for metallic surfaces and for graphene.9−12 The process of adatom adsorption and its incorporation into the growing film is an important step in crystal growth. The surface coverage of adatoms affects the adsorption geometries, adsorption energies, dipole moments and work functions, and electronic and magnetic properties of a system.13−19 It is complicated to describe the adsorption trends with respect to adatom coverage because of the presence of direct and indirect © XXXX American Chemical Society

Received: December 6, 2015 Revised: March 25, 2016

A

DOI: 10.1021/acs.jpcc.5b11930 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

GaN A where EA/GaN slab , Eslab , and Eiso are the total energies of the adsorbate covered GaN slab, clean GaN slab, and free (isolated) A atom, respectively. To calculate the energy of the clean GaN slab, we used the usual method of cutting a plane perpendicular to the z-direction through bulk GaN to get a Ga-terminated surface and performing geometry optimization as described above. The surface after optimization remained flat (for all supercells), and we refer to this configuration as the flat clean slab (FCS). The energy per unit area was same for (2 × 2), (3 × 3), (4 × 4), and (5 × 5) FCS configurations. (An interested reader can refer to section S2 of the Supporting Information for a discussion on relative surface energy calculations.) We constructed another clean GaN slab configuration by removing the adatom (A) from the fully optimized adsorbate/substrate configuration (A/GaN) while retaining the substrate atoms at their relaxed positions. This configuration of the “clean” surface is referred to as the distorted clean slab (DCS) because in this configuration, the substrate atoms are distorted as compared to those in the flat clean slab (FCS) configuration, which was obtained directly from cutting the bulk GaN crystal. (The reader can refer to section S3 of the Supporting Information for additional details of FCS and DCS configurations.) We emphasize that both FCS and DCS are the “clean” GaN slab configurations. We have used the energies of these FCS and DCS configurations in place of energy of clean GaN slab (EGaN slab ) in eq 1 to calculate the adsorption energies of adatom A (Ga or N) with respect to FCS and DCS configurations, respectively. The clean slab energy was computed from the DCS using two methods. In the first method, only the electronic structure of the DCS configuration was optimized, whereas the ionic positions were kept fixed. This configuration is referred to as the DCS-scf to denote the self-consistent field calculations performed on the DCS configurations. In the other method, a full geometry optimization was performed starting from the DCS configuration, where the top three GaN bilayers were allowed to relax completely according to the force convergence criteria mentioned above. The structure obtained after geometry optimization of the DCS configuration is called DCS-rlx configuration. Thus, to calculate the EAad from eq 1, we have used total energies of FCS, DCS-scf, and DCS-rlx configurations in place of EGaN slab (as the energy of the clean GaN slab). The adsorption energy calculated according to eq 1 varies with system size and can be written as a sum of adsorption energy of an adatom on an infinite lattice, EA,∞ ad , and the interaction energy between the adatom and its periodic images, EAint(L). Thus, we can write

study of the effect of individual Ga and N adatoms has been carried out in the lower coverage range, for example, from 0.04 to 0.25 ML. In this article, we investigate the effect of coverage on the adsorption of Ga and N adatoms on Ga-terminated wurtzite GaN(0001) surface using ab initio calculations. We choose to focus our study on the Ga-terminated GaN(0001) since it is more stable than the N-terminated GaN(0001) surface.39 We study how the coverage of the surface affects the adsorption energy, surface relaxations, and reconstructions. We also calculate the dipolar interactions between the adsorbed adatom and its periodic images. This is a fundamental study of adatom interactions on a polar surface. The rest of this article is organized as follows. In section 2, we describe the methodology used and the various types of calculations performed. In section 3.1, we show the key adsorption sites on GaN(0001) and their energetics. The main results of this article, namely the variation of adatom adsorption energy with system size, are shown in section 3.2. All the analysis and discussions are done in section 4. Section 4.1 deals with adatom induced lattice distortions, distorted clean slab (DCS) calculations, and calculation of dipolar interactions between the adatoms. Charge density analysis is done in section 4.2. Possible reconstructions of GaN(0001) surface are discussed in section 4.3. Concluding remarks are made in section 5.

2. COMPUTATIONAL DETAILS We have performed first-principles total energy calculations in the framework of density functional theory as implemented in PWscf code of Quantum Espresso distribution.40 The wave functions were expanded in plane waves up to a kinetic energy cutoff of 70 Ry. Ultrasoft pseudopotentials41 were employed for both Ga and N atoms, and Ga 3d electrons were treated as valence electrons. Exchange and correlation energy was treated in the generalized gradient approximation parametrized by the Perdew−Burke−Ernzerhof functional.42 The lattice vectors employed here were a⃗ = a(1,0,0), b⃗ = a(1/2, √3/2,0), and c ⃗ = a(0,0,c/a), corresponding to a wurtzite unit cell. The supercells used in our calculations were formed by multiples of a⃗ and b⃗ for lateral dimensions. These supercells had a slab geometry with vacuum gaps in the z-direction. The surfaces were modeled using supercells consisting of six GaN bilayers with a vacuum gap of 13 Å. We have restricted our calculations to the Ga-terminated surface. The dangling bonds of the bottom layer N atoms were passivated with pseudohydrogens of charge 0.75e in order to prevent any charge transfer between two sides of the slab. The upper three bilayers and adatoms were allowed to relax until the force exerted on each atom was less than 0.025 eV/Å, while the bottom three bilayers and pseudohydrogens were kept fixed to mimic bulk-like behavior. The Brillouin-zone integrations were performed using a Monkhorst−Pack grid.43 We have done calculations using (2 × 2), (3 × 3), (4 × 4), and (5 × 5) surface supercells. The corresponding k-points for the above supercells are (4 × 4 × 1), (3 × 3 × 1), (2 × 2 × 1), and (1 × 1 × 1), respectively. These k-points have been tested to give same accuracy for each system. (Tests for these k-mesh convergences have been shown in section S1 of the Supporting Information.) The adsorption energy for an adatom A (Ga or N) was calculated as A/GaN GaN A EadA = Eslab − Eslab − E iso

A EadA(L) = EadA, ∞ + E int (L )

(2)

where L represents the lateral size of the system. Thus, calculations of adsorption energy as a function of L can be used to analyze the interactions between adatoms. We performed two types of calculations for the adsorbate/ substrate configurations. The first type is referred to as “fullrelax” calculation, in which the coordinates of top three GaN bilayers and the adatom were allowed to relax completely. In the second one, we fixed the atomic coordinates of the slab at the FCS configuration and optimized the coordinates of the adatom only. This type of calculation is referred to as “fixedslab” calculation. The purpose of doing fixed-slab calculations is to see the effect of lattice relaxations on the total energy of a system.

(1) B


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The Journal of Physical Chemistry C To see the change in charge density upon adsorption, we used the formula Δρ( r ⃗) = ρ( r ⃗) − ρGaN ( r ⃗) − ρad ( r ⃗)

Table 1. Calculated Structural Parameters and Thermodynamic Properties of Wurtzite GaN and α-Ga Bulk Using PBE: Lattice Parameters a, c/a, b/a, and u and Cohesive Energy, Ecoh; Values from Experiments and Prior Calculations

(3)

where ρ(r)⃗ is the total charge density of the adsorbate/ substrate system and ρGaN(r)⃗ is the charge density of a system obtained by removing the adatom from the adsorbate/substrate system and performing self-consistent field calculations at fixed atomic coordinates (i.e., DCS-scf calculations). ρad(r)⃗ is the charge density of an isolated adatom located at the adsorbate site. Such charge density difference plots show the regions of charge accumulation and depletion when an adatom is adsorbed on a substrate. To estimate the dipolar contribution of the interaction between adatoms, we adopted the following procedure. We calculated the dipole moment (μ) in the z-direction for a given slab as44 μ=−

∫ ρ(z)z dz + ∑ Ziezi i

GaN bulk a (Å) c/a b/a u Ecoh (eV)

α (Δμ)2 4π ϵ0 r 3

expt

PBE26

this work

expt

3.221 1.626

3.19047 1.62647

3.252 1.629

4.574 1.719 1.010

4.51050 1.69550 1.00150

0.377 −8.61

0.37747 −9.0651

0.376 −8.54

−2.68

−2.8152

corresponding experimental and theoretical values.26,47 We obtained a heat of formation of GaN of −1.00 eV, which is in agreement with other studies.36,48 We got a spontaneous polarization as −0.033 C/m2, comparable with prior studies.49 We also performed some preliminary calculations for GaN(0001) surface using a (2 × 2) supercell. We calculated the adsorption energies and surface relaxations of different adsorption sites on GaN(0001) surface and compared these with reported results. These preliminary calculations are described in section 3.1. The main result of this article, the variation of adsorption energy with surface coverage, is described in section 3.2. 3.1. Adsorption Sites, Their Energies, and Surface Relaxations. Three different high-symmetry adsorption sites on a clean GaN(0001) surface are shown in Figure 1a. The

(4)

where ρ(z) is the x−y integrated valence charge density, Zi is the atomic number of an ion i, e is the charge of one electron, and zi is the z-coordinate of ion i. Wurtzite GaN has a net polarization (dipole moment per unit volume) along the z-direction. To calculate the dipole contribution due to the adatom only, we have subtracted the dipole moment of the clean GaN(0001) slab (DCS-scf) from the dipole moment of the adsorbate system: Δμ = μad/GaN − μDCS. This Δμ refers to the dipole moment change due to the adatom and is used to calculate the dipolar interaction energy between an adatom and its periodic image using the following formula: Edip(r ) =

Ga bulk

this work

(5)

where r is the distance between the adatom and its periodic image. α should be 2.0 if the dipoles are on a metal surface and 1.0 if they are in a vacuum.5 Since GaN is a semiconductor and density of free charge carriers is smaller than that of a metal surface, the screening effect of GaN(0001) surface would be very small. So we took α to be 1.0. We have explicitly summed up the dipolar interaction contributions due to the periodic images using special functions.45 We have also calculated the adsorption energy of a (2 × 2)Ga adatom configuration using dipole correction.46 We found that the difference in adsorption energy of this configuration with and without dipole correction is about 0.008 eV. The energy difference is small probably because we are using a pseudohydrogen termination and a sufficient vacuum gap which prevents the interaction between two sides of the slab. Hence, all our reported results are for calculations performed without dipole correction.

Figure 1. (a) Top view of GaN(0001) surface, showing (1 × 1) and (2 × 2) surface supercells along with three high-symmetry adsorption sites for adatoms (ontop, hcp, and fcc). Ga (N) atoms are represented as large (small) pink (blue) spheres. (b) Side view of Ga-hcp configuration showing various vertical distances characterizing surface relaxations. dad is the average distance of the adatom from the topmost substrate Ga layer. dGaN is the distance between Ga adatom and N atom of the second substrate layer for this configuration. dNN (not shown in figure) is the corresponding distance for N-hcp configuration. d12 and d23 are average interlayer spacings as shown.

adatom at the ontop site sits directly above the Ga atom of the substrate layer whereas the adatom at the hcp site sits directly above the N atom of the subsequent substrate layer. The adatom at fcc site sits at the hollow site above the substrate surface. Using a (2 × 2) supercell, we first make a comparison of the energies and surface relaxations of the adatom configurations at different adsorption sites. We define the structural parameters characterizing the surface relaxations in Figure 1b. The calculated surface relaxations are tabulated in Table 2. Some structural parameters from earlier theoretical results are also listed in Table 2, and we see a close agreement in our calculated values with earlier theoretical results. The adsorption energy values for the three adsorption sites are also

3. RESULTS To test the validity of our computational methods, we did some parametrization calculations for bulk GaN and saw whether our results are matching with the reported results. We compared several structural parameters and cohesive energies of bulkGaN, bulk-Ga, and N2 molecule with experimental and other ab initio calculations. Some of the computed bulk properties are shown in Table 1. The computed bulk lattice parameters a = 3.221 Å and c/a = 1.626 are in good agreement with the C


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surface Ga atom, which does not form a bond with the adatom, is called the rest atom. In the case of N-fcc configuration, the bond length between N adatom and the surface Ga atoms is about 2.00 Å. The surface Ga atoms shift laterally toward fcc site by an amount of 0.18 Å (not shown in Figure 2), and the vertical buckling of the surface Ga plane is about 0.63 Å. In the case of Ga-hcp configuration, the lateral shift of the Ga atoms toward hcp site is only about 0.02 Å (not shown in Figure 2), and the vertical buckling of the surface Ga plane is about 0.45 Å. Thus, it seems that N adatom causes more distortions of the substrate than the Ga adatom. We have also compared these distances with earlier theoretical results20,26 and found our results consistent with them. It is a common practice to construct maps showing displacement and force on each atom obtained after geometry optimization.11,55 Figure 3 shows the displacement maps for (2

Table 2. Calculated Surface Relaxations, Described by Various Distances Shown in Figure 1b, for Clean and (2 × 2) Adatom Configurationsa Ga adatom clean dGaN dNN db

on-top

fcc

on-top

hcp

fcc

2.00 2.0526 1.30 0.62

2.57 2.68

dad d12

0.66

2.96 0.66

d23

2.00

1.99

Ead

hcp

N adatom

−2.42

2.49 2.5026 1.80 0.67 0.6553 1.97 1.9253 −3.83

2.52 2.5326 1.81 0.67

1.82 2.04 0.63

2.51 2.09 2.0926 1.44 0.63

1.97

2.00

2.02

2.00

−3.62

−2.43

−4.90

−5.67

a

db is the bond length of the adatom with the surface Ga atoms. All the distances are in Å, and the adsorption energy is in eV. The corresponding values of d12 and d23 for bulk-GaN are 0.64 and 1.97 Å, respectively.

listed in Table 2. Note that the adsorption energy values are all negative and the adsorption site with the most negative value of adsorption energy corresponds to the preferred adsorption site. From the adsorption energy values, we see that the Ga adatom prefers to be adsorbed at the hcp site. This can be understood as due to attraction with N atom directly below and formation of bonds with the three Ga atoms of the surface. The Ga-hcp configuration is only slightly more stable (0.21 eV) than the Ga-fcc configuration. Ga-on-top position is unfavorable because of presence of unsaturated dangling bonds. The (2 × 2) N adatom structure at the fcc site is more stable than N-hcp and N-on-top configurations. In case of N-hcp configuration, there is a repulsion between N adatom and the N atom present in the subsurface layer. Hence, this configuration is less stable than N-fcc. The N adatom at the on-top position is highly unfavorable because it has only one bond with Ga atom lying directly below it and has three unsaturated dangling bonds. Our calculated relative stabilities of different adatom configurations are comparable with earlier results.20,26,54 We have also calculated surface formation energies of these adatom configurations as a function of N chemical potential. These have been shown in section S2 of the Supporting Infomation and are found to be in agreement with reported theoretical results.26 Specifically, for (2 × 2) N-fcc and Ga-hcp configurations, we have shown the details of surface relaxations in terms of bond lengths, vertical buckling of surface Ga plane, and lateral shifts of the surface Ga atoms in Figure 2. In both these configurations, the adatom forms bonds with three surface Ga atoms out of total four present on a (2 × 2) unit cell. The

Figure 3. Atomic displacement maps of (a) Ga and (b) N adatom on a (2 × 2) unit cell. The Ga/N adatom position is shown by the green arrowhead near the top center of each figure. The displacements of substrate Ga (N) atoms are shown by pink (blue) arrows. Only the top two GaN bilayers of the slab are shown. The arrows are scaled for better visualization while ensuring that the arrows do not overlap.

× 2) Ga-hcp and N-fcc adatom configurations. From these maps and analysis of the above data, we see that Ga and N adatoms cause large distortions of the underlying lattice and N adatom causes more lattice distortions than the Ga adatom. 3.2. Size Dependence of Adsorption Energies. From the above calculations, we see that the hcp site for Ga and fcc site for N are the preferred adsorption sites for a (2 × 2) unit cell which corresponds to 0.25 monolayer (ML) coverage. The preference of these sites remains the same for both the adatoms at further low coverages. We consider the effect of increasing the system size (L) or, equivalently, reducing the coverage (1/ L2) on adsorption energy, keeping the adatoms at their preferred sites. We first show the results of adsorption energies calculated with respect to the flat clean slab (FCS) configuration (using FCS configuration as the clean slab, GaN which is done by replacing EGaN slab by EFCS in eq 1). So we can write eq 1 in a modified form: A/GaN GaN A EadA,FCS = Eslab − E FCS − E iso

(6)

EA,FCS ad

where is the adsorption energy of an adatom A, calculated using FCS as the clean slab configuration, EA/GaN is slab the energy of the fully relaxed adsorbate/substrate system, EGaN FCS is the energy of the clean GaN slab (FCS here), and EAiso is the energy of an isolated adatom. We performed both full-relax and fixed-slab calculations. We took (2 × 2), (3 × 3), (4 × 4), and (5 × 5) supercells with the corresponding coverages of 0.25, 0.111, 0.062, and 0.04 ML. Numbers of GaN bilayers and vacuum gap are all same for all the unit cells considered above. As described in section 2, we have chosen different number of k-points for different unit cells in order to maintain the same

Figure 2. (a) Ga-hcp and (b) N-fcc (2 × 2) configurations showing vertical buckling of the surface Ga atoms. All lengths are in Å. D


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In the following section, we show several techniques to analyze and understand the reasons for the nonconvergence of adsorption energy with decreasing coverage.

accuracy of different calculations. The calculated adsorption energies as a function of increasing system size L are shown in Figure 4. Adsorption energies for full-relax calculations are shown in Figure 4a, while for fixed-slab calculations are shown in Figure 4b.

4. ANALYSIS AND DISCUSSION 4.1. Lattice Distortions, Dipolar Interactions, and Convergence of Adsorption Energy. For the (2 × 2) Gahcp and N-fcc configurations, we have shown the buckling of the surface plane and lateral shifts of surface atoms (lattice distortions) in Figure 2. These values are in close agreement with the prior results.20,26 The buckling of surface plane, lateral shifts, and downward movement of some substrate atoms are also seen in larger unit cells. We show some of these lattice distortions in the charge density difference plots (calculated according to eq 3) in Figure 5. The plane of these plots passes

Figure 4. Adsorption energies plotted as a function of surface supercell size for Ga (black circles) and N (red squares) adatoms for (a) fullrelax and (b) fixed-slab calculations. Lines in (a) and (b) are guides to the eye.

From Figure 4a (full-relax calculations), we see that the adsorption energy becomes more negative with increasing system size (or alternatively, with decreasing coverage). This behavior is expected since the interaction between the adatom and its own periodic images is repulsive. But we notice some unexpected nonuniform behavior of adsorption energy with respect to decreasing coverage. It remains almost constant for (2 × 2) and (3 × 3) unit cells and decreases (becomes more negative) almost linearly for (4 × 4) and (5 × 5) unit cells. In general, the adsorption energy is expected to become more negative with decreasing coverage because of the reduced repulsive interaction between the adatom and its periodic images as the coverage is decreased (system size is increased). But here, we see that the decrease is unexpectedly large for (4 × 4) and (5 × 5) unit cells. This surprising observation may be attributed to some long-range interactions between the adatom and its periodic images. However, we expect that the interactions should decrease with increasing system size, and the adsorption energy should become almost constant for sufficiently large unit cells. We analyze this point further with the help of fixed-slab calculations, where the slab is kept fixed at the FCS configuration and only the adatom is allowed to relax. From Figure 4, we note that the adsorption energy is less negative for fixed-slab calculations than that for the full-relax calculations and becomes almost the same for Ga and N adatoms. Further, it seems to converge with increasing system size. The main difference between full-relax and fixed-slab calculations is that the latter do not have substrate−lattice distortions due to the adatom. This suggests that the reason for the nonconvergence of the adsorption energy in the full-relax calculations is related to the lattice distortions due to the adatom, which may stabilize the adatom configurations (leading to more negative adsorption energies as compared to fixed-slab calculations). It appears that the lattice distortions for (4 × 4) and (5 × 5) are much larger than those of (2 × 2) and (3 × 3) unit cells. This observation is unexpected since the adatom induced lattice distortions should decay with increasing system size. It may happen that the lattice distortions (relaxations) may get suppressed for smaller systems because of the periodicity of the unit cells. Thus, adsorption energy becomes more negative for larger unit cells.

Figure 5. 2D contour plots of charge redistribution for Ga (N) adatom in left (right) column for (a) (2 × 2), (b) (3 × 3), (c) (4 × 4), and (d) (5 × 5) unit cells. The adatom is represented by “O” in these plots. “p” refers to the surface Ga atom directly bonded to the adatom. “q” (“r”) refers to the surface Ga atoms which show maximum upward (downward) displacements when an adatom is adsorbed. The patterns seen in each case are marked with dashed lines. The range of charge density (in e/Å3) is shown in the bottom panel.

through the plane formed by three surface Ga atoms which are nearest to the adatom (which form bonds with the adatom, labeled as “p” in Figure 5). These plots show the charge redistribution taking place due to the interaction between the adatom and the substrate. We focus on the lattice distortions using these charge density difference plots here and will do the charge analysis in the later part of the article. In Figure 5, we have connected the surface Ga atoms showing an upward displacement for (3 × 3), (4 × 4), and (5 × 5) adatom configurations using dashed lines. The atoms in a dashed pattern for a particular unit cell are at the same height and are E


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The Journal of Physical Chemistry C labeled as “q” in Figure 5. We see that as we go for larger unit cells, more and more atoms are displaced, and there seems to be more dashed patterns. In Table 3, we have listed the lattice

after removal of the effect of lattice relaxations on the adsorption energies, they converged as coverage is decreased. Thus, we can have identified that lattice distortions are the major reason for the observed nonconvergence of the adsorption energies in the case of full-relax calculations (calculated using FCS as a reference for clean slab in eq 6) as a function of decreasing coverage. The difference between adsorption energy values calculated using FCS and DCS-scf as a reference for clean slab (columns 3 and 4 of Table 4) is an estimate of contribution of lattice distortions toward the total adsorption energy. Surprisingly, by looking at the values of adsorption energies of Ga adatom from full-relax and scf calculations, we noticed that the adsorption energy calculated with respect to DCS-scf, EA,DCS‑scf is less negative than that ad . As can be seen from eqs calculated with respect to FCS, EA,FCS ad 6 and 7, this implies that the energy of the DCS-scf configuration, EGaN DCS‑scf, is more negative than the energy of the FCS configuration, EGaN FCS . Notice that both DCS-scf and FCS represent the clean GaN slab configurations and their total energies are negative. This implies that the DCS-scf configuration (for the Ga adatom) is more stable than the FCS configuration, for all four system sizes considered. Similarly, by comparing the adsorption energy values for N adatom for full-relax and scf method, we see that the FCS is more stable than the DCS-scf for (2 × 2) and (3 × 3) unit cells, but the DCS-scf is more stable than FCS for (4 × 4) and (5 × 5) unit cells. We have tabulated the total energy per unit area for all the FCS and DCS configurations in section S3 of the Supporting Information. To look into this matter more carefully, we did geometry optimization of distorted clean slab (DCS) of Ga and N configurations for all the unit cells considered. For this, we took DCS configurations and allowed the top three GaN bilayers to relax according to the procedure described in the Computational Details section. The configurations obtained after the geometry optimization of DCS are called DCS-rlx configurations. We saw that atoms of the DCS configurations did not relax to the flat clean slab (FCS) configurations after geometry optimization. For the Ga DCS case, the atomic positions after geometry optimization were close to their positions at which they were when the Ga adatom was present. For a better understanding, the reader can look at section S3 of the Supporting Information. The N DCS configurations were significantly altered during geometry optimization. The DCSrlx configurations of N became identical to the DCS-rlx configurations of the Ga for the corresponding unit cells. We also used DCS-rlx energy as the clean slab energy in eq 1 and calculated the corresponding adsorption energy, EA,DCS‑rlx , ad for all the unit cells considered here (using an equation similar to eq 6 or 7). These adsorption energy values are tabulated in Table 4, column 5. We again observed the convergent behavior of adsorption energy with respect to system size, as was with DCS-scf. We found that the adsorption energies of Ga and N adatoms calculated with respect to the DCS-rlx configurations, EA,DCS‑rlx , were less negative than those calculated with respect ad to the FCS configurations, EA,FCS (again, both FCS and DCS-rlx ad refer to the clean GaN slab configurations), and this difference is larger for larger systems (columns 3 and 5 of Table 4). This implies that the energy of the DCS-rlx configuration, EGaN DCS‑rlx, is more negative than the energy of the FCS configuration, EGaN FCS . Thus, we conclude that the DCS-rlx configurations are more stable than the FCS configurations for all the system sizes considered for both the adatoms, and the difference in their

Table 3. Displacements of Labeled Atoms (p, q, and r) along the z-Direction in Figure 5 (All Displacements in Å) Ga adatom Δzp (2 (3 (4 (5

× × × ×

2) 3) 4) 5)

0.11 0.18 0.30 0.24

N adatom

Δzq

Δzr

Δzp

0.20 0.47 0.37

−0.35 −0.29 −0.30 −0.36

0.14 0.16 0.15 0.11

Δzq

Δzr

0.13 0.62 0.39

−0.49 −0.22 −0.36 −0.37

distortions of few top layer substrate atoms (labeled in Figure 5) caused because of the adatom adsorption. It is clear from Table 3 that the magnitude of upward and downward displacement of the substrate atoms is large for (4 × 4) and (5 × 5) unit cells. These distortions may stabilize the (4 × 4) and (5 × 5) adatom configurations more and hence make the adsorption energy more negative. It appears that the large adatom-induced lattice distortions with respect to the flat clean slab (FCS) can lead to significant contribution to the adsorption energy (see the adsorption energies for full-relax calculations in Figure 4a). To separate the contribution of lattice distortions to the adsorption energy, we performed calculations as described below. We considered DCS-scf calculation, wherein we took fully relaxed adsorbate/ substrate configuration, removed the adatom, and performed self-consistent field calculations (see section S3 of Supporting Information for details). This DCS-scf energy is used as the GaN clean slab energy (replace EGaN slab by EDCS−scf in eq 1) to calculate the adsorption energy of an adatom. So we can write eq 1 in the following modified form: A/GaN GaN A EadA,DCS‐scf = Eslab − E DCS ‐scf − E iso

(7)

EA,DCS‑scf ad

where is the adsorption energy of adatom A, calculated using the DCS-scf energy as the clean slab energy. EA/GaN is the slab energy of fully relaxed adsorbate/substrate system, EGaN DCS‑scf is the energy of clean GaN slab (DCS-scf configuration here), and EAiso is the energy of an isolated adatom. The adsorption energy calculated using the DCS-scf as a reference for the clean slab energy excludes the effect of lattice relaxations on the adsorption energy. These adsorption energy values obtained for each unit cell are shown in Table 4 (column 4). From column 4 of Table 4, we can see that adsorption energies calculated in the way described above (eq 7) are almost constant for the coverages considered. It shows that Table 4. Adsorption Energies for Ga and N Adatom Configurations Calculated with Respect to Different References for Clean Slab system

adatom

EA,FCS ad

EA,DCS‑scf ad

EA,DCS‑rlx ad

EA,fixS ad

× × × × × × × ×

Ga Ga Ga Ga N N N N

−3.83 −3.93 −4.84 −5.47 −5.67 −5.63 −6.45 −7.20

−3.76 −3.74 −3.63 −3.70 −5.97 −5.78 −5.92 −6.08

−3.70 −3.67 −3.56 −3.62 −5.53 −5.41 −5.18 −5.35

−3.34 −3.37 −3.37 −3.36 −3.56 −3.62 −3.61 −3.64

(2 (3 (4 (5 (2 (3 (4 (5

2) 3) 4) 5) 2) 3) 4) 5)

F


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The Journal of Physical Chemistry C stability is greater for larger systems. We will analyze the relative stability of DCS-rlx and FCS through charge analysis in the later part of the article. For completeness, we have also tabulated the adsorption energy values obtained from the fixedslab calculations in Table 4, column 6. The nonconvergence of adsorption energy may also be related to dipolar and electronic interactions and their coupling to lattice distortions. Here, we try to evaluate the role of these interactions. The separation of all these interactions is a little more difficult because of the large lattice distortions due to the adatom. We calculated dipole moment for each adatom configuration and for the corresponding distorted clean slab (DCS-scf) configuration using eq 4. From these, we calculated the change in dipole moment of a configuration due to adatom adsorption (Δμ). The dipole moments and the change in dipole moments for different configurations are shown in Table 5. We found Table 5. Dipole Moment Values for Ga and N Adatom Configurations for Different Unit Cells Ga adatom

N adatom

system

μGa/GaN

μDCS‑scf

ΔμGa

μN/GaN

μDCS‑scf

ΔμN

× × × ×

−4.37 −9.81 −20.22 −28.92

−4.81 −11.04 −20.34 −29.60

0.44 1.23 0.12 0.68

−5.95 −11.66 −19.75 −29.08

−3.85 −10.40 −19.05 −27.85

−2.10 −1.26 −0.70 −1.23

(2 (3 (4 (5

2) 3) 4) 5)

Figure 6. Adsorption energies for (a) Ga and (b) N adatoms with respect to unit cell size. “a” and “b” labels in the figure refer to the adsorption energies calculated using FCS (in full-relax calculations) and DCS-rlx as the clean slab energies. “c” refers to the adsorption energies calculated from the fixed-slab calculations. The dipolar interaction between the adatom and its periodic images is shown by label “d”.

that the change in dipole moment due to adatom adsorption is very small. The reasons for the small change in dipole moment for a polar material like GaN are not obvious. It may be due to the fact that we are studying Ga and N adatoms as opposed to heteroatoms on GaN surface. It is also possible that our method underestimates the dipole moment because our method involves the subtraction of dipole moment of the clean slab from that of the slab with adatom. We also calulated the dipole moment using charge transfer and using work function method but did not see any increase in its value (see section S4 of the Supporting Information). Nevertheless, we used Δμ in eq 5 and summed over all images to calculate the dipole−dipole interaction energy between the adatom and its periodic images. We plotted this dipolar interaction energy with respect to increasing system size in Figure 6. In Figure 6, we also plotted adsorption energies of Ga and N adatom configurations. The adsorption energies calculated using FCS (full-relax) and DCS-rlx configurations as a reference for clean slab are shown along with those calculated from the fixed-slab calculations to compare their magnitudes. The adsorption energies calculated using DCS-rlx and from fixed-slab calculations are almost constant with respect to increasing system size, indicating convergent behavior. The difference between adsorption energies from DCS-rlx and from fixed-slab calculations is mainly due to the electronic interaction between the adatoms. This difference is larger for the N adatom as compared to that for the Ga adatom. It may be due to higher electronegativity of N as compared to that of Ga. In Figure 6, the difference between adsorption energy values of FCS and DCS-rlx is a measure of amount of elastic interactions between the adatom and its periodic images (which arises because of the lattice relaxations). If we compare the magnitude of dipolar and elastic interactions, we see that the dipolar interactions are very small as compared to the magnitude of elastic interactions. The reason for small magnitude of dipolar interaction between

adatoms is again due to the observed small change in dipole moment upon adatom adsorption. Thus, we can say that the main contribution to the observed nonconvergence of adsorption energies comes from the elastic interactions (lattice distortions) between the adatoms. The calculation of substratemediated electronic interaction between the adatoms is not straightforward for these systems, but since the semiconductor GaN has smaller amount of free charge carriers as compared to a metal, these electronic interactions are expected to be very small. In ongoing work, we are looking more closely at adatom interactions by placing multiple adatoms on the surface. These results will be presented elsewhere. Thus, the above analysis suggests that, for Ga-terminated GaN(0001) surface, the adatoms cause large lattice distortions, and these are larger for larger system sizes. These large lattice distortions may stabilize the particular adatom configuration and lead to more negative adsorption energy as the system size is increased. It is also observed that the distorted clean slab (DCS-rlx) of the above configurations is more stable than the flat clean slab (FCS). The configuration obtained after removal of adatom from the adsorbate/substrate system does not go back to the flat clean slab (FCS) configuration after geometry optimization. In the next part of the article, we will try to understand the reasons for how the large lattice distortions are stabilizing the larger systems and also the origin of higher stability of DCS-rlx as compared to the FCS through the charge density analysis. 4.2. Charge Density Analysis. In Figure 5, the change in charge distribution caused due to the Ga and N adatoms for all the unit cells considered is shown in the chosen plane. Because of the bonding of adatom with the GaN substrate, there appears a triangular region of electron density variation around the adatom location in the chosen plane. The regions where the G


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The Journal of Physical Chemistry C electron density is red correspond to charge depletion and where it is blue/purple correspond to charge accumulation due to the adatom. From the figure, we see that N adatom causes greater charge redistribution than the Ga adatom. It is apparent by the presence of alternate positive (charge accumulation) and negative (charge depletion) charge regions in the N adatom charge density difference plots. This is expected since the surface of the substrate is Ga-terminated. In the (2 × 2) N adatom charge density difference plot in Figure 5a, the charge transfer from the Ga rest atom toward the N adatom is visible by the small red circular region around the Ga rest atom position in the plot. To study the complete details of charge transfer and bonding characteristics of the adatoms, we need to see the charge densities in planes other than the ones we have shown. This is not the main interest of this article, so we restrict our discussion on charge densities to the single chosen plane. The charge density difference in Figure 5 (calculated according to eq 3) captures the part of the charge redistribution that may be ascribed exclusively as due to the adatom. In that sense, it fails to capture the part of the charge redistribution due to the lattice distortions. In order to see the effect of lattice distortions on the charge redistribution for the Ga and N adatom configurations, we define the following charge density difference: GaN ( r ⃗) − ρad ( r ⃗) Δρ1( r ⃗) = ρ( r ⃗) − ρFCS

Figure 7. Charge density difference maps for Ga (N) adatom configurations in left (right) column calculated according to eq 8 for (a) (2 × 2), (b) (3 × 3), (c) (4 × 4), and (d) (5 × 5) unit cells. The adatom is labeled by “O” in all the unit cells. The range of charge density (e/Å3) is shown in the bottom panel.

(8)

where ρ(r)⃗ and ρad(r)⃗ have the same meanings as defined in eq 3. ρGaN FCS (r)⃗ is the charge density of the flat clean slab (FCS). Δρ1(r)⃗ captures the electron density redistribution due to the adatom induced lattice distortions. The charge density difference plots for Ga and N adatom configurations (calculated according to eq 8) are shown in Figure 7. The planes of the charge density difference maps in Figure 7 pass through the plane formed by three surface Ga atoms which are nearest to the adsorbed adatom (form bonds with the adatom). From Figure 7, we can see some positive charge (blue or purple) regions in/around some of the surface Ga atoms, away from the adatom locations. These Ga atoms are exactly the atoms that were substantially displaced from their FCS positions and marked by the dashed lines patterns in Figure 5. It appears that due to the adatom-induced lattice distortions, the electron densities of the surface Ga atoms overlap. This overlapping of the electron densities of the surface Ga atoms reduces the energy of the corresponding configuration. Since there are more lattice distortions in larger unit cells, the size of the positive charge density regions is greater. These observations indicate that the larger unit cells adatom configurations are more stable. This is the reason that adatom adsorption energy calculated on a larger unit cell is more negative. Because of computational limitations, we could not perform calculations for still larger unit cells and see the length scale up to which the adatom affects the movement of substrate atoms. Next, we try to understand the reason for higher stability of DCS-rlx configurations as compared to the FCS, corresponding to each unit cell. As we have said before, the DCS-rlx is also a clean slab configuration like FCS but with the atoms displaced (distorted) with respect to the FCS. To see the effect of these lattice distortions on the charge redistribution for a clean surface, we define another quantity: GaN GaN Δρ2 ( r ⃗) = ρDCS ( r ) − ρFCS ( r ⃗) − rlx ⃗

GaN where ρDCS‑rlx (r)⃗ is the charge density of the DCS-rlx configuration and ρGaN FCS (r)⃗ is the charge density of the FCS configuration. Figure 8 represents the charge density difference plots for Ga and N configurations calculated according to eq 9. The planes of these charge density difference plots pass through the plane formed by three surface Ga atoms which were nearest to the adatoms (these three surface Ga atoms had formed bonds with the adatoms in the respective adatom configurations). From Figure 8, we can see the positive charge density regions around some surface Ga atoms. This indicates that the orbitals of the surface Ga atoms of the distorted clean slab (DCS-rlx) configurations undergo some partial overlapping. This overlapping of the electron density of the atoms reduces the energy of DCS-rlx configurations. Since no such orbitals overlap is present in the flat clean slab (FCS) configurations, the DCS-rlx configurations are more stable than the FCS configurations for all the unit cells considered in our work. Further, if we compare the charge density difference plots of Ga and N configurations in Figure 8, we can see that Ga DCSrlx charge density plots are identical to those of N DCS-rlx plots with a translated unit cell. We have compared the energies of DCS-rlx configurations of both Ga and N for particular unit cells and saw that the energies are same (these energy values are reported in Table S1 of the Supporting Information). This means that there is a single stable DCS-rlx configuration for a particular system size. It is to be noted that there occurs a significant change in the atomic coordinates of N DCS configurations after geometry optimization. It can be seen from the charge density maps of Figures 7 and 8. While the maps for Ga configurations are almost identical (left column of both figures), there seems a large change in the charge density maps for N configurations (right columns).

(9) H


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configuration does not go back to the flat clean slab (FCS) after geometry optimization, and this relaxed DCS configuration is energetically more stable than the corresponding FCS configuration for each system size. Both FCS and DCS-rlx represent the clean GaN(0001) surface configurations, and both are stable to small displacements. The higher stability of DCS-rlx configurations indicates that these DCS-rlx configurations represent some possible reconstructions of the clean Ga-terminated GaN(0001) surface, which have lower energies than the unreconstructed clean surface (FCS). There are several discrepancies between the theoretical and experimental surface reconstructions and electronic surface states for the clean GaN(0001) surface.57,58 Almost all the previously reported reconstructions on GaN(0001) surface have adatoms on the Ga-terminated GaN(0001) surface. It has been said that clean GaN(0001) surface does not reconstruct like the traditional semiconductors like GaAs, Si, etc.20 Recently, Kempisty et al. have theoretically predicted a stable reconstruction of clean GaN(0001) surface, which is found to be more stable than the flat (1 × 1) GaN surface.59 They have called it as (2 × 1) reconstruction in which there are alternate rows of Ga atoms having a height difference of almost 0.6 Å. They stated that the (2 × 1) reconstructed surface could explain the surface states obtained in the experiments.58 In a more recent work, Chen et al. have used the (2 × 1) reconstructed configuration instead of the flat (1 × 1) surface configuration to calculate the adsorption energy of water on GaN(0001).60 In our work, we found DCS-rlx configurations more stable than the FCS configurations for all the system sizes considered in this work. It is possible that there are other reconstructions of GaN(0001) which might be more stable than the DCS-rlx, but we have not found any in our work. It was purely coincidental that we were able to notice these reconstructions during our analysis of the nonconvergence of adsorption energy. We now compare the energies of these reconstructed surfaces with those of unreconstructed surfaces (FCS) for different unit cells considered. The relative energies of these DCS-rlx configurations as compared to FCS configurations are −3.7, −2.9, −8.9, and −8.2 meV/Å2 for (2 × 2), (3 × 3), (4 × 4), and (5 × 5) configurations, respectively. Thus, we see that the larger unit cells are more stable than than the smaller unit cells, and (4 × 4) and (5 × 5) configurations have almost same stability. We emphasize that due to computational restrictions, we are not able to carry out simulations using larger surface supercells.

Figure 8. Charge density difference maps for Ga (N) configurations in left (right) column calculated according to eq 9 for (a) (2 × 2), (b) (3 × 3), (c) (4 × 4), and (d) (5 × 5) unit cells. The range of charge density (e/Å3) is shown in the bottom panel.

We show another way to understand the stability of the DCS-rlx in terms of stability of the corresponding adsorbate system. The (2 × 2) Ga adatom is one of the stable reconstructions on GaN(0001) surface. In this configuration, there are six electrons in the dangling bond states of Ga atoms. It obeys electron counting rule (ECR)56 and has a reduced dangling bond density. It has fully occupied states above the valence band maximum (VBM) and empty states below the conduction band minimum (CBM)57 and is semiconducting. Similarly, a (2 × 2) N adatom configuration is semiconducting. It has eight electrons in the dangling bond states of the surface layer and obeys ECR. It has fully occupied states within the bandgap. The corresponding number of electrons in surface dangling bonds for (3 × 3), (4 × 4), and (5 × 5) supercells are 39/4, 15, and 87/4 for Ga adatom configurations and 47/4, 17, and 95/4 for N adatom configurations, respectively. Thus, the adatom configurations for these supercells do not follow ECR. However, their surface relaxations and the positive charge regions in the charge density difference plots (Figure 7) suggest that the surface away from the adatom undergoes some reconstruction in a way that the system lowers its energy by partially reducing the number of dangling bonds. There is significant overlapping between the surface Ga atoms orbitals for the DCS-rlx configurations of the chosen unit cells (Figure 8). Thus, there may be a reduction in the dangling bond density for these DCS-rlx configurations also. The reduction in the dangling bond density for these DCS-rlx configurations might stabilize these configurations relative to the FCS configurations. 4.3. Possible Reconstructions of GaN(0001). The above analysis indicates that the distorted clean slab (DCS)

5. CONCLUDING REMARKS In this article, we have seen the effect of adatom interactions on the adsorption energy of Ga/N adatom on the Ga-terminated GaN(0001) surface through DFT calculations. The adsorption energies of Ga and N adatoms calculated with respect to the flat Ga-terminated clean surface (FCS) become more negative with increasing system size or with decreasing coverage. The adsorption energies do not converge even for very small coverage. The adatoms cause significant distortions to the lattice, and this is manifested by displacements of the substrate atoms from their positions. In particular, we see patterns of surface atoms, away from the adatom, that are raised by about 0.2−0.6 Å over the surface. Through the charge density difference maps, we see that the charge densities on these surface atoms are able to overlap, and this leads to a stabilization of the surface. Because of the greater stabilization of adatom configurations of larger unit cells, the adsorption I


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energies for the larger unit cells are more negative. Similar patterns of the surface Ga atoms on the distorted clean surfaces (DCS) are also observed even after geometry optimization. The overlap of the electron densities on the atoms of these patterns lowers the energy of these DCS-rlx configurations relative to the flat clean surfaces (FCS). The stabilization energy due to these lattice distortions and the corresponding electron density overlap are able to explain the observed trends in the adsorption energy. The adsorption energies calculated by considering the DCS-rlx configurations as reference for clean slabs show convergent behavior with respect to increasing system size. Thus, our size-dependent or coverage-dependent study indicates that, to calculate the adsorption energy for a periodic system, one must consider the effect of lattice distortions. We find that the calculated change in dipole moment due to adatom adsorption is very small for this polar material. Hence, the dipolar interaction energy between the adatom and its periodic images is much smaller than the elastic interaction (due to lattice distortions). Thus, the major contribution to the observed trend in adsorption energy is due to the lattice distortions (elastic interactions). We also see that the charge density difference plots of DCS-rlx configurations of N adatom are same as those of DCS-rlx configurations of Ga adatom. Through our calculations, we notice that the distorted clean surface (DCS-rlx) has a lower energy than the flat clean surface (FCS) for all the unit cells considered. These DCS-rlx configurations represent possible reconstructions of the clean GaN(0001) surface. These reconstructions have not been directly detected in experiments as far as we know. The STMbased methods should be able to detect the distorted patterns since the Z-displacements (as shown in Table 3) are generally substantial (0.2−0.6 Å). This is an aspect that could be clarified by experiments.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b11930. k-mesh convergences; relative surface energy calculations; FCS and DCS configurations; dipole moment calculations; sample input and output files (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Ph +91 512 259 6037; Fax +91512-259 6806 (M.R.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge Defence Research and Development Organization, India, for their support through grant ER & IPR. Manjusha acknowledges CSIR, India, for her research fellowship and the High Performance Computing facility at IIT Kanpur.

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Adatom Interactions on GaN(0001) Surface I - ACS Publications

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