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PHYSICAL REVIEW B 78, 193308 共2008兲 Relaxation models of the (110) zinc-blende III-V semiconductor surfaces: Density functional study Honggang Ye,1 Guangde Chen,1 Yelong Wu,1 Youzhang Zhu,1 and Su-Huai Wei2 1 Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China 2 National Renewable Energy Laboratory, Golden, Colorado 80401, USA 共Received 25 September 2008; published 17 November 2008兲 Clean III-V zinc-blende 共110兲 surfaces are the most extensively studied semiconductor surface. For conventional III-V compounds such as GaAs and InP, the surface relaxation follows a bond rotation relaxation model. However, for III-nitrides recent study indicates that they follow a bond-constricting relaxation model. Firstprinciples atom relaxation calculations are performed to explore the origin of the difference between the two groups of materials. By analyzing the individual shift trends and ionic properties of the top layer anions and cations, we attribute the difference between the conventional and nitride III-V compounds to the strong electronegativity of N, which leads to the s2 p3 pyramid bond angle to be larger than the ideal one in bulk 共109.5°兲. The general trends of the atomic relaxation at the III-nitrides 共110兲 surfaces are explained. DOI: 10.1103/PhysRevB.78.193308 PACS number共s兲: 68.35.bg, 68.47.Fg, 31.15.ae I. INTRODUCTION III-V semiconductor compounds such as GaAs and GaN are important materials for microelectronic and optoelectronic applications.1 Their physical properties have been extensively studied for several decades. Among them, surface reconstruction and relaxation have attracted much attention because they are closely related to the crystal growth and doping in these materials.2–4 As zinc-blende structure is favored by the majority of the III-V semiconductors, much of the attention has been focused on the low-energy closepacked nonpolar 共110兲 surface. The zinc-blende 共110兲 surface is characterized by a zigzag chain along the 关11̄0兴 direction with the 共1 ⫻ 1兲 translational symmetry. Earlier works had established a well accepted bond rotation 共BR兲 relaxation model for the 共110兲 surface of conventional III-V compounds.5–12 In this BR model, charge on the cation dangling bond is transferred to the anion dangling bond. To reduce the surface energy, the top layer anions move outward in favor of an s2 p3 bonding with three neighboring cations and the cations move inward in favor of an sp2 bonding with three neighboring anions, inducing a top layer buckling angle of about 30° and an almost conservation of the bulk bond length in the top layer. However, later studies for the IIInitrides found that they do not follow this established relaxation pattern.13–21 For the III-nitrides, the top layer bonds are constricted, the buckling angle is much smaller 共⬃10°兲, and the top layer N maintains nearly in the bulklike position after relaxation. To explain the different relaxation patterns between conventional III-V semiconductors and III-nitrides 共110兲 surface, it has been suggested that the strong ionic character in the III-nitrides causes less rehybridization at the surface to form a local anion-centered pyramidal geometry and the large charge transfer from cation to anion leads to a stronger Coulomb attraction between the surface anions and cations.14–16 However, this explanation based on ionicity is not very strict and convincing. According to the ionicity scales given by Pauling,22 Phillips23 and others,24,25 although the ionicity of III-nitrides is wholly strong, there is not an obvious jump 1098-0121/2008/78共19兲/193308共4兲 between the ionicity of III-nitrides and conventional III-V semiconductors. Especially, BN is shown to be less ionic than GaAs and InP,23,25 but BN behaves undoubtedly like other nitrides instead of conventional III-V semiconductors. So, interpretation beyond the bond ionicity picture is needed to explain the qualitatively different relaxation behavior between conventional III-V semiconductors and III-nitrides 共110兲 surfaces. In the present work, we calculated the 共110兲 surface relaxation of all III-nitrides and GaAs, a prototype of conventional III-V compound, and obtained consistent results with previous reports. Different from previous reports, our analysis indicates that the intrinsic atomic Pauling electronegativity difference between the anions in the conventional III-V compounds and III-nitrides is the main reason that leads to different relaxation behavior at the 共110兲 surface of these materials. This model can successfully explain the individual displacements of the anions and cations as well as the small buckling angle variation at the III-nitrides 共110兲 surfaces. II. CALCULATION METHOD Density functional theory based calculations are performed within the generalized gradient approximation 共GGA兲 共Ref. 26兲 framework as implemented by the Vienna Ab initio Simulation Package 共VASP兲 code.27,28 The Ga 3d and In 4d electrons are treated as valance electrons. The interaction between core and valence electrons are treated with the projector augmented wave method.29 The energy cutoff for the basis function is 500 eV for III-nitrides and 350 eV for GaAs. We employ Monkhorst-Pack sampling scheme with k-point mesh of 7 ⫻ 10⫻ 1 for BN, 6 ⫻ 8 ⫻ 1 for AlN and GaN, 5 ⫻ 7 ⫻ 1 for InN, and 4 ⫻ 6 ⫻ 1 for GaAs.30 The slab models are built containing eleven atomic layers with 12 Å vacuum spaces separating the slabs. The top three layers at both sides of the slab are allowed to relax by minimizing the quantum-mechanical force on each ion site to be less than 0.01 eV/ Å. The other layers are fixed in the optimized bulk configuration. Test calculations show that the cell size is converged. Side and top views of the ideal and relaxed slab 193308-1 ©2008 The American Physical Society PHYSICAL REVIEW B 78, 193308 共2008兲 BRIEF REPORTS models for III-nitrides and conventional III-V compounds are shown in Fig. 1. The lattice parameters used in building the slab models are 3.625 Å for BN, 4.399 Å for AlN, 4.543 Å for GaN, 5.048 Å for InN, and 5.742 Å for GaAs, which are obtained by optimizing the corresponding bulk primitive cell and agree well with the experimental values.31 III. RESULTS AND DISCUSSION FIG. 1. 共Color online兲 Schematic representation of the relaxed 共filled circles兲 and unrelaxed 共empty circles兲 atomic positions for GaAs and GaN. 共a兲 side view of GaAs; 共b兲 side view of GaN; 共c兲 top view of GaAs; 共d兲 top view of GaN. The numbers 1 and 2 in 共c兲 and 共d兲 denote the atom layers. When the 共110兲 surface is cleaved from a zinc-blende crystal, the top layer anion and cation become threefold coordinated with one dangling-bond point away from the surface. To achieve lower energy, charges are transferred from the high-energy cation dangling bond to the low-energy anion dangling bond, thus satisfying the electron counting rule. Accompanied with the charge transfer, the anion tends to have the local pyramidal s2 p3 configuration, whereas the cation tends to have the sp2 local planar configuration, resulting in a buckled top layer. Following the custom, the main features of the 共110兲 surface relaxation are described by two parameters:11 the top layer rotation angle ␻ and the top layer bond constriction ⌬b. Additionally, the horizontal and vertical shifts of each anion and cation are represented by ⌬Ai, ⌬Ai,⬜, ⌬Ci, and ⌬Ci,⬜, respectively, where the index i denotes the atom layer. By symmetry, the horizontal shift here is along the 关001兴 direction, and the vertical shift is along the 关110兴 direction. The calculated data for each material are shown in Table I and compared with previous reports.13–15 The relative displacement between anion and cation can be simply derived from the position shift of each atom. It can be seen from Table I that the results of the presentwork are consistent with previous reports. The top layer TABLE I. Calculated structure parameters for the relaxed zinc-blende group-III nitrides and GaAs 共110兲 surfaces in comparison with previous reports. BN AlN GaN InN GaAs Present Ref. 15 Ref. 13 Present Ref. 15 Ref. 13 Ref. 14 Present Ref. 15 Ref. 13 Ref. 14 Present Ref. 15 Ref. 13 Ref. 14 Present Ref. 14 ␻ 共deg兲 ⌬b 共%兲 ⌬A1 共Å兲 ⌬A1,⬜ 共Å兲 ⌬C1 共Å兲 ⌬C1,⬜ 共Å兲 ⌬A2 共Å兲 ⌬A2,⬜ 共Å兲 ⌬C2 共Å兲 ⌬C2,⬜ 共Å兲 17.7 15.74 16.6 12.0 11.61 11.9 11.7 14.3 14.29 17.5 14.3 13.4 13.13 10.6 14.4 30.0 30.1 7.06 7.8 6.6 5.92 3.6 2.9 5.9 5.59 5.3 2.8 4.9 4.75 4.9 3.3 4.3 1.14 0.9 0.06 −0.01 −0.18 −0.22 0.02 0.03 −0.01 0.06 0.05 −0.02 −0.18 −0.20 0.02 0.01 0.00 0.05 0.02 0.02 −0.22 −0.21 0.02 0.01 −0.01 0.05 0.04 0.00 0.05 0.05 −0.17 −0.23 −0.18 −0.20 0.05 0.01 0.02 0.01 0.03 −0.01 0.07 0.04 −0.15 −0.15 0.24 0.42 −0.38 −0.37 −0.45 −0.27 0.04 −0.06 0.02 0.13 0.05 −0.08 0.09 0.23 193308-2 PHYSICAL REVIEW B 78, 193308 共2008兲 BRIEF REPORTS FIG. 2. 共Color online兲 The model used to predict the bond angles of AC3 pyramid with group-V anion A at the center and coordinated with three group-III cations. The dangling bonds of cations are saturated by 5/4 electrons charged hydrogen atoms. buckling angles for III-nitrides vary from 13.4° for AlN to 17.7° for BN, much smaller than the values of 30.0° for GaAs. The top layer bond-length constriction for III-nitrides are between 4.75⬃ 7.06%, qualitatively different from the 1.14% for GaAs. The vertical displacement of top layer N is between −0.02⬃ 0.05 Å, i.e., it is close to the bulk position, however, the 0.24 Å displacement for top layer As indicates a large outward movement of top layer As in GaAs. The calculated results, thus, show the main features of 共110兲 surface relaxation, i.e., the III-nitrides follow a bondconstricting rotation model with the top layer N atoms maintaining in the bulklike position; whereas for conventional III-V compounds, such as GaAs, they follow a bondconserving rotation model with the top layer anions moving outward. To understand the relaxation of the anions at the 共110兲 surface, we first analyze the bond angles of the AC3 in the pyramidal configuration. This is because when the anion A at the 共110兲 surface forms the pyramid, the pyramid bond angle is directly correlated with the top layer rotation angle ␻ and the top layer atomic displacement. When anion moves outward, the rotation angle ␻ increases but the bond angle decreases, whereas when anion moves inward, the rotation angle ␻ decreases but the bond angle increases. For C = H and A = N, P, As, and Sb, the computed bond angles of NH3, PH3, AsH3, and SbH3 molecules are 106.6°, 92.4°, 90.8°, and 90.6°, respectively. For C = Ga, the computed bond angles, using the geometry described in Fig. 2, are 115.9°, 106.6°, 104.9°, and 104.7°, respectively, when the cation layer is fixed in bulk size 共the bond angle and bond length are dependent兲 and 116.3°, 95.0°, 93.2°, and 92.9°, respectively, when the cation layer is allowed to relax freely 共the bond angle and bond length are independent兲. The real bond angles at surfaces should be between them. The above data indicate that large difference exists between NC3 and AC3 共A = P , As, Sb兲 and that with the same cation C atom, the bond angles of AC3 pyramid increases with the decreasing atomic number of anion A. This can be understood by valence shell electronic pair repulsion 共VSEPR兲 model.32 When the atomic number of A decreases, the Pauling electronegativity of A increases 共the Pauling electronegativity of N, P, As, and Sb are 3.04, 2.19, 2.18, and 2.05, respectively兲,33 thus the charges on anion-cation bonds become closer to each other. Therefore, to avoid the electronic repulsion between bond charges, the bond angle increases. Although the bond angles of AC3 pyramid configuration are mainly determined by the center atom A, the C atoms also have effect on them. The VSEPR model tells us that the bond angle should increase when the Pauling electronegativity of C decreases. This is exactly what we observed when H is replaced by Ga. It is interesting to notice that the bond angle of about 116° for NGa3 is larger than ideal tetrahedral angle of 109.5°, but the bond angle for PGa3, AsGa3, and SbGa3 is smaller than 109.5°. It indicates that, as the 共110兲 surface is cleaved, the top layer N atom in the nitrides tend to move inward to increase the bond angle. On the other hand, the top layer P, As, and Sb atoms in conventional III-V semiconductors tend to move outward in order to reduce the bond angle from the ideal tetrahedral values. For the top layer cations, they prefer to form a sp2 configuration so they must move inward to form a plane and push the surface anion outwards 共Fig. 1兲. For III-nitrides, the top layer N atoms try to move inward but the surface cations try to drive them outward so the top layer N atoms remain nearly in the bulklike positions, and the top layer cations only can partly realize sp2 configuration by bond constriction, which result in a small top layer buckling angle. For other III-V compounds, the top layer anions tend to move outward and the cations also try to drive them outward, the two effects add to each other, making the anions to move outward in a large scale and the cations almost fully realize sp2 configuration, which result in a large top layer buckling angle of about 30°. Based on the above relaxation mechanism, the other parameters in Table I can also be explained. The top layer anion parallel shift ⌬A1 is small and positive for III-nitrides 共0.00– 0.06 Å兲 but relative large and negative for GaAs 共−0.15 Å兲. This is also because the positive shift is helpful to increase the pyramid bond angle whereas the negative shift is helpful to decrease the bond angle. The two parameters ⌬A2, ⌬A2,⬜ in Table I indicate that the second layer anions shift to top-right direction 关relative to Figs. 1共a兲 and 1共b兲兴 for both III-nitrides and GaAs. This is because this movement is driven by the top layer cations tending to form planar sp2 configuration. The parallel shifts ⌬C2 of second layer cations for III-nitrides are negligible. Their vertical shifts ⌬C2,⬜ are also helpful to the bond angle increase of top layer N. For GaAs, the second layer Ga moves upward along the bond with the top layer As because it tends to conserve the Ga-As bond length. After understanding the relaxation pattern difference between III-nitrides and conventional III-V compounds, the variation in the top layer buckling angle between III-nitrides can also be understood conveniently based on the relaxation mechanism discussed above. As shown in Table I, the top layer buckling angles of III-nitrides decrease in the sequence of BN共17.7°兲 ⬎ GaN共14.3°兲 ⬎ In N共13.4°兲 ⬎ AlN共12.0°兲, which indicate that the bond angles of the NC3 pyramid increases in the same order. According to the VSEPR model,32 this trend can be easily understood by noticing that the electronegativity of cation decreases in the sequence of B共2.04兲 ⬎ Ga共1.81兲 ⬎ In共1.78兲 ⬎ Al共1.61兲 共Ref. 33兲 to avoid overlapping of valence electrons, the bond angle of the pyramid will increase. IV. CONCLUSION By analyzing the bond angle of 共110兲 surface pyramidal configuration, the shift trends of top layer anions and cations 193308-3 PHYSICAL REVIEW B 78, 193308 共2008兲 BRIEF REPORTS are predicted individually. The top layer N tends to move inward to increase the pyramid bond angles, but top layer P, As, and Sb tend to move outward to decrease the pyramid bond angles. The opposite trends induce the final surfaceatom configurations which are obviously different between III-N and conventional III-V semiconductors. Besides the main features, each displacement of the first and second layer atoms also is explained convincingly. Furthermore, based on this relaxation mechanism, the top layer buckling angle variation between III-nitrides is explained expediently by the Pauling electronegativity trend of group-III elements. It is 1 H. Morkoç, Nitride Semiconductors and Devices 共Springer, Ber- lin, 1999兲. 2 B. A. Haskell, S. Nakamura, S. P. DenBaars, and J. S. Speck, Phys. Status Solidi B 244, 2847 共2007兲. 3 K. Xu, J. Xu, P. Z. Deng, R. S. Qiu, and Z. J. Fang, Phys. Status Solidi A 176, 589 共1999兲. 4 T. Akasaka, Y. Kobayashi, and T. Makimoto, Appl. Phys. Lett. 90, 121919 共2007兲. 5 A. Kahn, E. So, P. 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