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Surface electronic properties of Si(337)
Surface Science 437 (1999) 299–306 www.elsevier.nl/locate/susc
Surface electronic properties of Si(337) Yu Jia a,b, *, Xing Hu a, Bingxian Ma a, Jun Wan b, Ling Ye b a Department of Physics and Key Laboratory of Material Physics, Zhengzhou University, Zhengzhou 450052, People’s Republic of China b Surface Physics Laboratory (National Key Laboratory), Fudan University, Shanghai 200433, People’s Republic of China Received 29 January 1999; accepted for publication 27 May 1999
Abstract Using the scattering-theoretic method and employing the nearest-neighbor tight-binding formalism to describe the bulk electronic structure, we present the first calculations of electronic properties of the Si(337) surface. The surface band structure of Si(337) together with the projected bulk band, the wave-vector-resolved layer densities of states and the atom-resolved layer densities of states are presented. These results show that there are six surface bound states in the range from −12.0 to 2.0 eV. The localized surface features and orbital properties of these surface states along C–Y–S–X–C high symmetry lines of surface Brillouin zone are discussed. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Density of states; Resonance states; Scattering-theoretic method; Surface states
1. Introduction The electronic properties of low index silicon surfaces, such as Si(100) and Si(111), have been studied intensively, but the investigations on high index surfaces are lacking. One reason is that it is a generally believed that the high index silicon surfaces are unstable and easy to facet into low index planes upon annealing [1–6 ]. However, some high index surfaces are exceptions, such as the Si(112) and Si(113) surfaces. The experimental results of low-energy electron diffraction (LEED) show that these high index surface are stable; for example, the 3×1 and 3×2 reconstructions of the Si(113) surface have been observed in LEED experiments [7,8]. A stable high index silicon surface could be used as a substrate for heteroepitaxial growth of compound semiconductors and it can * Corresponding author. Fax: +86-0371-7973895. E-mail address: [email protected] ( Y. Jia)
hinder the formation of some defects, e.g. the antiphase domain; thus the quality of thin films could be improved. For this reason some theoretical investigations on the high index surfaces have also been carried out to study both the atomic and the electronic structure [9,10]. Theoretically, it is thought that the Si(112) surface is an ideal substrate for heteroepitaxial growth of compound semiconductors [11,12]. Xing and coworkers [4,13,14] have studied the geometrical structure of the Si(337) surface by the LEED method, and they showed that, in the area adjacent to Si(112) surface, the best and stable surface for epitaxial growth is not the Si(112) surface, but the Si(337) surface. For the Si(337) surface, Xing and coworkers [4,13,14] have found a 1×1 stable surface structure by LEED measurement. In addition to a strong Si(337)1×1 surface, they also observed for the first time a 2×1 reconstruction LEED pattern along the [11: 0] direction recently [14]. In their work, they also studied the relation
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between the change of the electronic affinity potential and crystallographic directions and found a minimum value of the potential on the Si(337) surface as well as on the three low-index surfaces and the Si(113) high-index surface. Since the electronic affinity potential reflects the relative amplitude of the surface dipole moment, the minimum of the potential indicates that the Si(337) surface is a stable high-index one. Studies by Baski and Whitman [2] on the Si(112) surface using a scanning tunneling microscope (STM ) also showed stable (337) facets on surface. To our knowledge, so far there has been no report on the electronic states of the Si(337) surface, so we will study the electronic states of Si(337) surface in the present work. In this paper, we will use the nearest-neighbor tight-binding model sp3s1 proposed by Vogl et al. [15] to describe the bulk electronic structure and the ideal Si(337) surface will be produced by the removing layers method. The electronic structure of the Si(337) surface is studied by using the Green function method in the frame of the scattering theory. The wavevector-resolved layer densities of states (LDOS) and the atom-resolved LDOS are calculated and some properties of their surface states are discussed.
erties of the perturbed solid are determined by solution of (H0+U )|y=E|y.
(2)
To obtain an efficient solution of Eq. (2), we employ the scattering theory method. The bound states and resonance states are entirely determined in terms of the Green function G, which can be obtained by solving Dyson’s equation: G=G0+G0UG.
(3)
The bulk Green function G0 in Eq. (3) is defined as G0=lim (E+ie−H )−1 e 0 and the bulk density of states is given by N0(E)=−
1 p
Tr Im G0(E).
(4)
(5)
The perturbation potential U may introduce bound states that are localized near the surface. These bound states are determined by the discrete poles of the Green function G. The localization properties of the surface electronic states can be studied by calculating local, partial or total LDOS: N(E)=−
1 p
Tr Im G(E ).
(6)
2. Method 2.1. Basic equation In this section we briefly compile the basic equations of scattering theory method developed by Pollmann and coworkers [16,17] for treating locally perturbed surface solids. The unperturbed system is the perfect, three-dimensional periodic material and its electronic one-particle spectrum is given by the solution of Schro¨dinger’s equation H0|n, q=E (q)|n, q, (1) n where n is the index of the energy band and q is the usual three-dimensional wavevector. The surface is introduced as a two-dimensional periodic perturbation U localized in the direction perpendicular to the surface. The electronic prop-
2.2. Perturbation and the bulk Green function in a layer orbital representation In order to calculate the surface band structures from Eq. (6), we have to specify the perturbation U and represent the operators G0 and U in an appropriate basis set. As a basis set we use layer orbits |q, a, m, m, which are a linear combination of atomic orbits restricted to a particular layer. The two-dimensional wavevector is labeled as q and a, m, and m stand for the a atomic orbit of the basis atom m in the unit cell of the particular layer m considered in our calculations. To obtain an ideal Si(337) surface, we remove as many layers as necessary from the periodic bulk solid along the [337] direction to decouple the two resulting semi-infinite solids completely. In the
Y. Jia et al. / Surface Science 437 (1999) 299–306
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Cn∞ ∞ ∞ are the eigenvectors corresponding to the amm eigenvalues E (x, q) for a given wavevector q. n 3. Calculations
Fig. 1. Schematic diagram of the surface geometry for Si(337) surface: (a) side view; (b) top view.
layer orbital representation the perturbation U is diagonal, since the removal of layers is finished by shifting to infinity the on-site diagonal matrix elements on the layers to be removed. The resulting matrix elements of U are N
q, amm|U|q, a∞, m∞, m∞=lim ud ∞ d ∞ d ∞ ∑ d ◊ , mm aa mm mm ◊ u0 m =1 (7) where N is the number of layers to be removed. In Eq. (7), the sum of m◊ is restricted to the number of layer N. The matrix elements G0 are
q, amm|G0|q, a∞, m∞, m∞ 1
P
앀
67/2
lim ∑ eipx[g(m)−g(m∞)](2/앀67) 앀67/2 앀67 e 0 n − Cn1∞ ∞ ∞ (q, x)Cn (q, x) amm , ×dx a m m E+ie−E (x, q) (8) n with g(m)=m when the basis atom m in the unit cell of the m layer is in the anion position and g(m∞)=m∞+1 when m∞ in the unit cell of the m∞ 4 layer is in the cation position. The Cn and amm =
In our calculation, we take the lattice constant of Si to be a=0.540 nm. Fig. 1a and b respectively show the side view and the top view of the bulk truncated Si(337) ideal surface; the surface primitive cell of Si(337) is shown in the dashed line. Each line between two atoms represents one sp3 bond. There are two kinds of Si atom in each surface primitive cell; each is on the site of two different crystal lattices. One kind of Si atom is on the site of the cation crystal lattice and we label this Si1; the other is on the site of the anion crystal lattice and we label this Si2. For the bulk-truncated Si(337) surface, three atom-layers need to be cut off. Therefore, there will be seven dangling bonds in every primitive cell. Three of them are in the first layer, three in the second layer, and the other one is in the third layer. In comparison, the primitive cell of the Si(337) surface is larger than that of low index surfaces and some high index surfaces. There are 3.4 dangling bonds on every square lattice constant and this is larger than the density of dangling bonds of both the (112) and (113) surfaces, which have 3.3 and 1.8 dangling bonds respectively. We calculated the surface electronic structure of Si(337) by using the scattering theory method proposed by Pollmann and coworkers [16,17]. They used this method to calculate the electronic structure of low index surfaces, such as (100), (110) and (111) in Si, Ge and GaAs etc. Their results are in good agreement with the experimental data. In our calculation, the tight-binding sp3s1 model is used to describe the bulk electronic structure. This model is a semi-empirical method and its results are less accurate than those of the first-principles calculations; however, it can give more distinct properties in some aspects, such as orbital or local properties of electronic states. To produce infinite free Si(337) surfaces, three atomlayers should be removed so as to decouple two semi-infinite surfaces. Since the Si(337) surface has a 1×1 stable structure, the influence on the
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electronic states of the surface atoms relaxation is neglected in our calculation.
4. Results and discussion 4.1. Surface band structure We present the surface band structure along the high symmetric lines C–Y–S–X–C of the surface Brillouin zone. The high symmetric points of the surface Brillouin zone are
A S B A A B
C(0.0, 0.0), Y 0.0,
and X
34앀2 67
2
67
,S
33앀2 67
,
S B 2
67
, 0.0
(the units are 2p/a). In Fig. 2, the points pattern gives the projected bulk bands into the (337) plane and the density of points gives a visual impression of the density of bulk states for a given q. The dashed line represents surface bound states in the gaps and the surface resonance states in the point pattern. Unlike the low-index surfaces of Si(100) and Si(110), the Si(337) surface has only one main pocket and a few small pockets in the region of the bulk valence bands. These pockets are the energy regions in the two-dimensional surface Brillouin zone without projected bulk bands. The main pocket is located in the region of −6 to −3 eV along Y–S–X lines, and the small pockets are located in the region −8 to −7 eV. We call this region the stomach gap. Since the stomach gap is small, most of the surface features are resonance in this energy region. All of these resonance states result from changes in the bonds lying parallel to the layers or from changes in the backbonds between the first and second, the second and third or the third and fourth layers. By the creation of the Si(337) surface, these bonds are more or less affected and thus give rise to changes within the bonding bands. The resonance states show mainly s orbital character below −7 eV and p character above −5 eV. In the region from −7 to −5 eV, it shows both s and p orbital characters.
Fig. 2. Surface band structure of Si(337) surface together with the projected bulk bands.
In the fundamental gaps and the stomach gaps, we find six localized surface states that are derived from the unoccupied and occupied seven dangling bonds. We labeled these surface states as S1, S2, S3, S4, S5 and S6 respectively. The surface states
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Fig. 3. Wavevector-resolved layer density of states of the X point on the first four layers of the Si(337) surface (arbitrary units).
Fig. 4. Wavevector-resolved layer density of states of the Y point on the first four layers of the Si(337) surface (arbitrary units).
S1, S2 and S3 show downward dispersion in the Y–S line and upward dispersion in the X–C symmetry direction and show flat dispersion along the C–Y and S–X lines. Near the top of the valence band, the surface states S4 and S5 turn into the resonance states in the vicinity of the C point and it can be noted that these two surface states show different dispersion behavior. In addition, the energy levels of S2 and S3, and of S4 and S5 are each degenerated along the high symmetric line S– X. To explain the local character of these surface states, the layer-resolved density of states of the X and Y points in the two-dimensional surface Brillouin zone are given in Figs. 3 and 4 respectively. By analyzing the amplitude of these surface states in each layer, we can determine in which layer these states are localized and then we can conclude the relation between these surface states
and chemical bonds between different layers. From Fig. 3, it can easily be found that the LDOS of the surface states S1 and S5 have a large occupation in the first layer. As the number of layers below the surface increases, the amplitude of the LDOS reduces quickly. This shows that they are mainly localized in the first layer and originate mainly from the three dangling bonds in the surface layer. Similarly, the LDOS of S2 and S4 have a large occupation in the second layer but a small occupation in other layers, so it can be concluded that they originate mainly from the three dangling bonds in the second layer. In the third layer, the surface state S5 has a large amplitude of layer density states, but has a small amplitude in other layers. This shows that S5 arises from the one dangling bond in the third layer. From Fig. 4, we can also obtain similar results
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Fig. 5. (a) Localized layer density of states of the Si(337) surface; (b) localized layer density of states of the Si1 atom of the Si(337) surface; (c) localized layer density of states of the Si2 atom of the Si(337) surface (arbitrary units).
Y. Jia et al. / Surface Science 437 (1999) 299–306
as from Fig. 3. However, there are two main features that need to mentioned at the Y point of the surface Brillouin zone. Firstly, the difference in the energy levels between the surface states S2 and S4 is smaller than that between the surface states S1 and S5, since the atoms in the second layer are closer to the bulk environment than those in surface layer. Secondly, the occupation of the surface state S1 is mainly localized in the surface and the third layer, but there is almost no occupation in the second layer. Similarly, the surface state S3 is also mainly localized in the third layer and the surface layer. In the stomach gap of the bulk bands that appear in the vicinity of S–X direction in the energy region from −3.0 to −7.0 eV, a localized surface state S6 appears in the lower part of the stomach gap and this state almost has no dispersion along the high symmetric line S–X, but it has strong downward dispersion along Y–S and strong upward dispersion along X–C. From the layerresolved density of states, we find it has a large occupation of electronic states in the first four layers below the surface. So S6 is mainly related to the change of bonds formed by the atoms between the surface layer and the second layer, and the second layer and the third layer, which we call backbond states. In the small pockets, there are also some surface states that show the same orbital character as the S6 surface states. It should be mentioned that there is an unoccupied antiresonance surface state that lies above the conduction band minimum. It originates mainly from the mixed sp3 hybrid states of the Si2 atom. This surface has the same dispersion relation as the surface states S1 along the C–Y–S–X–C.
305
bond of the Si1 atom. The dominant contributors are the s-like and p-like orbits of Si1; the occupied dangling bond states (S4 and S5) lie close to the top of valence band maximum and they come from the sp3 hybridized bond of Si2 atom. (2) In the stomach gap, the localized surface state S6 is composed of s-like orbits of the Si2 atom and p-like orbits of the Si1 atom.
5. Conclusions In conclusion, we have studied the electronic states of the high-index Si(337) 1×1 surface by using the tight-binding method. The results show that there exist six surface bound states in the range from −12.0 to +2.0 eV. In the fundamental gap, three of these surface states lying close to the bottom of conduction band maximum come from the Si atom in the site of the cation crystal lattice and the others from the Si atom in the site of the anion crystal lattice. The local and the orbital properties of these surface states are discussed. The results are useful for further study of the electronic properties of the Si(337) surface from both theoretical and experimental aspects.
Acknowledgements We are grateful to Professor Yu-ping Huo of Zhengzhou University and Professor Y.R. Xing of the Institute of Semiconductors of Academia Sinica for their helpful discussions. This work was supported by the Natural Science Foundation of Henan, China, grant: 984040500.
4.2. Localized density of states and the orbital characters References To explain the orbital character of these surface states, The localized LDOS and atom-resolved localized LDOS on different layers for the Si(337) surface are shown in Fig. 5a–c. From Fig. 5, we can see the following. (1) In the fundamental gap, the occupied dangling bond states (S1, S2 and S3) lie close to the bottom of the conduction band maximum and they come from the sp3 hybridized
[1] [2] [3] [4]
X.S. Wang, W.H. Weinberg, Surf. Sci. 314 (1994) 71. A.A. Baski, L.J. Whitman, Phys. Rev. Lett. 74 (1995) 956. W. Ranke, Y.R. Xing, Phys. Rev. B 31 (1985) 2246. X.R. Xing, J.P. Zhang, J.A. Wu, Surf. Sci. 232 (1990) L215. [5] W. Ranke, Phys. Rev. B 41 (1990) 5243. [6 ] E.D. Williams, N.C. Bartelt, Science 251 (1991) 393. [7] B.Z. Olshanetsky, V.I. Mashano, Surf. Sci. 111 (1981) 414.
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[8] U. Myler, K. Jacobi, Surf. Sci. 220 (1989) 353. [9] K.A. Feng, X.M. Hu, Z. Lin, X.R. Xing, Appl. Surf. Sci. 120 (1997) 94. [10] Y.P. Feng, T.H. Wee, C.K. Ong, H.C. Poon, Phys. Rev. B 54 (1996) 4766. [11] H. Kroemer, J. Crystal Growth 81 (1987) 193. [12] S.L. Wright, M. Jnada, H. Kroemer, J. Vac. Sci. Technol. 21 (1982) 543.
[13] Y.R. Xing, W. Ranke, Chin. J. Semicond. 15 (1994) 136 in Chinese. [14] X. Hu, E.G. Wang, Y.R. Xing, Appl. Surf. Sci. 103 (1996) 317. [15] P. Vogl, H.P. Hjalmarson, J.D. Dow, J. Chem. Solids 44 (1983) 365. [16 ] J. Pollmann, S.T. Pantelides, Phys. Rev. B 18 (1978) 5524. [17] I. Ivanov, A. Mazur, J. Pollmann, Surf. Sci. 92 (1980) 365.
Surface electronic properties of Si(337) Yu Jia a,b, *, Xing Hu a, Bingxian Ma a, Jun Wan b, Ling Ye b a Department of Physics and Key Laboratory of Material Physics, Zhengzhou University, Zhengzhou 450052, People’s Republic of China b Surface Physics Laboratory (National Key Laboratory), Fudan University, Shanghai 200433, People’s Republic of China Received 29 January 1999; accepted for publication 27 May 1999
Abstract Using the scattering-theoretic method and employing the nearest-neighbor tight-binding formalism to describe the bulk electronic structure, we present the first calculations of electronic properties of the Si(337) surface. The surface band structure of Si(337) together with the projected bulk band, the wave-vector-resolved layer densities of states and the atom-resolved layer densities of states are presented. These results show that there are six surface bound states in the range from −12.0 to 2.0 eV. The localized surface features and orbital properties of these surface states along C–Y–S–X–C high symmetry lines of surface Brillouin zone are discussed. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Density of states; Resonance states; Scattering-theoretic method; Surface states
1. Introduction The electronic properties of low index silicon surfaces, such as Si(100) and Si(111), have been studied intensively, but the investigations on high index surfaces are lacking. One reason is that it is a generally believed that the high index silicon surfaces are unstable and easy to facet into low index planes upon annealing [1–6 ]. However, some high index surfaces are exceptions, such as the Si(112) and Si(113) surfaces. The experimental results of low-energy electron diffraction (LEED) show that these high index surface are stable; for example, the 3×1 and 3×2 reconstructions of the Si(113) surface have been observed in LEED experiments [7,8]. A stable high index silicon surface could be used as a substrate for heteroepitaxial growth of compound semiconductors and it can * Corresponding author. Fax: +86-0371-7973895. E-mail address: [email protected] ( Y. Jia)
hinder the formation of some defects, e.g. the antiphase domain; thus the quality of thin films could be improved. For this reason some theoretical investigations on the high index surfaces have also been carried out to study both the atomic and the electronic structure [9,10]. Theoretically, it is thought that the Si(112) surface is an ideal substrate for heteroepitaxial growth of compound semiconductors [11,12]. Xing and coworkers [4,13,14] have studied the geometrical structure of the Si(337) surface by the LEED method, and they showed that, in the area adjacent to Si(112) surface, the best and stable surface for epitaxial growth is not the Si(112) surface, but the Si(337) surface. For the Si(337) surface, Xing and coworkers [4,13,14] have found a 1×1 stable surface structure by LEED measurement. In addition to a strong Si(337)1×1 surface, they also observed for the first time a 2×1 reconstruction LEED pattern along the [11: 0] direction recently [14]. In their work, they also studied the relation
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between the change of the electronic affinity potential and crystallographic directions and found a minimum value of the potential on the Si(337) surface as well as on the three low-index surfaces and the Si(113) high-index surface. Since the electronic affinity potential reflects the relative amplitude of the surface dipole moment, the minimum of the potential indicates that the Si(337) surface is a stable high-index one. Studies by Baski and Whitman [2] on the Si(112) surface using a scanning tunneling microscope (STM ) also showed stable (337) facets on surface. To our knowledge, so far there has been no report on the electronic states of the Si(337) surface, so we will study the electronic states of Si(337) surface in the present work. In this paper, we will use the nearest-neighbor tight-binding model sp3s1 proposed by Vogl et al. [15] to describe the bulk electronic structure and the ideal Si(337) surface will be produced by the removing layers method. The electronic structure of the Si(337) surface is studied by using the Green function method in the frame of the scattering theory. The wavevector-resolved layer densities of states (LDOS) and the atom-resolved LDOS are calculated and some properties of their surface states are discussed.
erties of the perturbed solid are determined by solution of (H0+U )|y=E|y.
(2)
To obtain an efficient solution of Eq. (2), we employ the scattering theory method. The bound states and resonance states are entirely determined in terms of the Green function G, which can be obtained by solving Dyson’s equation: G=G0+G0UG.
(3)
The bulk Green function G0 in Eq. (3) is defined as G0=lim (E+ie−H )−1 e 0 and the bulk density of states is given by N0(E)=−
1 p
Tr Im G0(E).
(4)
(5)
The perturbation potential U may introduce bound states that are localized near the surface. These bound states are determined by the discrete poles of the Green function G. The localization properties of the surface electronic states can be studied by calculating local, partial or total LDOS: N(E)=−
1 p
Tr Im G(E ).
(6)
2. Method 2.1. Basic equation In this section we briefly compile the basic equations of scattering theory method developed by Pollmann and coworkers [16,17] for treating locally perturbed surface solids. The unperturbed system is the perfect, three-dimensional periodic material and its electronic one-particle spectrum is given by the solution of Schro¨dinger’s equation H0|n, q=E (q)|n, q, (1) n where n is the index of the energy band and q is the usual three-dimensional wavevector. The surface is introduced as a two-dimensional periodic perturbation U localized in the direction perpendicular to the surface. The electronic prop-
2.2. Perturbation and the bulk Green function in a layer orbital representation In order to calculate the surface band structures from Eq. (6), we have to specify the perturbation U and represent the operators G0 and U in an appropriate basis set. As a basis set we use layer orbits |q, a, m, m, which are a linear combination of atomic orbits restricted to a particular layer. The two-dimensional wavevector is labeled as q and a, m, and m stand for the a atomic orbit of the basis atom m in the unit cell of the particular layer m considered in our calculations. To obtain an ideal Si(337) surface, we remove as many layers as necessary from the periodic bulk solid along the [337] direction to decouple the two resulting semi-infinite solids completely. In the
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301
Cn∞ ∞ ∞ are the eigenvectors corresponding to the amm eigenvalues E (x, q) for a given wavevector q. n 3. Calculations
Fig. 1. Schematic diagram of the surface geometry for Si(337) surface: (a) side view; (b) top view.
layer orbital representation the perturbation U is diagonal, since the removal of layers is finished by shifting to infinity the on-site diagonal matrix elements on the layers to be removed. The resulting matrix elements of U are N
q, amm|U|q, a∞, m∞, m∞=lim ud ∞ d ∞ d ∞ ∑ d ◊ , mm aa mm mm ◊ u0 m =1 (7) where N is the number of layers to be removed. In Eq. (7), the sum of m◊ is restricted to the number of layer N. The matrix elements G0 are
q, amm|G0|q, a∞, m∞, m∞ 1
P
앀
67/2
lim ∑ eipx[g(m)−g(m∞)](2/앀67) 앀67/2 앀67 e 0 n − Cn1∞ ∞ ∞ (q, x)Cn (q, x) amm , ×dx a m m E+ie−E (x, q) (8) n with g(m)=m when the basis atom m in the unit cell of the m layer is in the anion position and g(m∞)=m∞+1 when m∞ in the unit cell of the m∞ 4 layer is in the cation position. The Cn and amm =
In our calculation, we take the lattice constant of Si to be a=0.540 nm. Fig. 1a and b respectively show the side view and the top view of the bulk truncated Si(337) ideal surface; the surface primitive cell of Si(337) is shown in the dashed line. Each line between two atoms represents one sp3 bond. There are two kinds of Si atom in each surface primitive cell; each is on the site of two different crystal lattices. One kind of Si atom is on the site of the cation crystal lattice and we label this Si1; the other is on the site of the anion crystal lattice and we label this Si2. For the bulk-truncated Si(337) surface, three atom-layers need to be cut off. Therefore, there will be seven dangling bonds in every primitive cell. Three of them are in the first layer, three in the second layer, and the other one is in the third layer. In comparison, the primitive cell of the Si(337) surface is larger than that of low index surfaces and some high index surfaces. There are 3.4 dangling bonds on every square lattice constant and this is larger than the density of dangling bonds of both the (112) and (113) surfaces, which have 3.3 and 1.8 dangling bonds respectively. We calculated the surface electronic structure of Si(337) by using the scattering theory method proposed by Pollmann and coworkers [16,17]. They used this method to calculate the electronic structure of low index surfaces, such as (100), (110) and (111) in Si, Ge and GaAs etc. Their results are in good agreement with the experimental data. In our calculation, the tight-binding sp3s1 model is used to describe the bulk electronic structure. This model is a semi-empirical method and its results are less accurate than those of the first-principles calculations; however, it can give more distinct properties in some aspects, such as orbital or local properties of electronic states. To produce infinite free Si(337) surfaces, three atomlayers should be removed so as to decouple two semi-infinite surfaces. Since the Si(337) surface has a 1×1 stable structure, the influence on the
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Y. Jia et al. / Surface Science 437 (1999) 299–306
electronic states of the surface atoms relaxation is neglected in our calculation.
4. Results and discussion 4.1. Surface band structure We present the surface band structure along the high symmetric lines C–Y–S–X–C of the surface Brillouin zone. The high symmetric points of the surface Brillouin zone are
A S B A A B
C(0.0, 0.0), Y 0.0,
and X
34앀2 67
2
67
,S
33앀2 67
,
S B 2
67
, 0.0
(the units are 2p/a). In Fig. 2, the points pattern gives the projected bulk bands into the (337) plane and the density of points gives a visual impression of the density of bulk states for a given q. The dashed line represents surface bound states in the gaps and the surface resonance states in the point pattern. Unlike the low-index surfaces of Si(100) and Si(110), the Si(337) surface has only one main pocket and a few small pockets in the region of the bulk valence bands. These pockets are the energy regions in the two-dimensional surface Brillouin zone without projected bulk bands. The main pocket is located in the region of −6 to −3 eV along Y–S–X lines, and the small pockets are located in the region −8 to −7 eV. We call this region the stomach gap. Since the stomach gap is small, most of the surface features are resonance in this energy region. All of these resonance states result from changes in the bonds lying parallel to the layers or from changes in the backbonds between the first and second, the second and third or the third and fourth layers. By the creation of the Si(337) surface, these bonds are more or less affected and thus give rise to changes within the bonding bands. The resonance states show mainly s orbital character below −7 eV and p character above −5 eV. In the region from −7 to −5 eV, it shows both s and p orbital characters.
Fig. 2. Surface band structure of Si(337) surface together with the projected bulk bands.
In the fundamental gaps and the stomach gaps, we find six localized surface states that are derived from the unoccupied and occupied seven dangling bonds. We labeled these surface states as S1, S2, S3, S4, S5 and S6 respectively. The surface states
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303
Fig. 3. Wavevector-resolved layer density of states of the X point on the first four layers of the Si(337) surface (arbitrary units).
Fig. 4. Wavevector-resolved layer density of states of the Y point on the first four layers of the Si(337) surface (arbitrary units).
S1, S2 and S3 show downward dispersion in the Y–S line and upward dispersion in the X–C symmetry direction and show flat dispersion along the C–Y and S–X lines. Near the top of the valence band, the surface states S4 and S5 turn into the resonance states in the vicinity of the C point and it can be noted that these two surface states show different dispersion behavior. In addition, the energy levels of S2 and S3, and of S4 and S5 are each degenerated along the high symmetric line S– X. To explain the local character of these surface states, the layer-resolved density of states of the X and Y points in the two-dimensional surface Brillouin zone are given in Figs. 3 and 4 respectively. By analyzing the amplitude of these surface states in each layer, we can determine in which layer these states are localized and then we can conclude the relation between these surface states
and chemical bonds between different layers. From Fig. 3, it can easily be found that the LDOS of the surface states S1 and S5 have a large occupation in the first layer. As the number of layers below the surface increases, the amplitude of the LDOS reduces quickly. This shows that they are mainly localized in the first layer and originate mainly from the three dangling bonds in the surface layer. Similarly, the LDOS of S2 and S4 have a large occupation in the second layer but a small occupation in other layers, so it can be concluded that they originate mainly from the three dangling bonds in the second layer. In the third layer, the surface state S5 has a large amplitude of layer density states, but has a small amplitude in other layers. This shows that S5 arises from the one dangling bond in the third layer. From Fig. 4, we can also obtain similar results
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Fig. 5. (a) Localized layer density of states of the Si(337) surface; (b) localized layer density of states of the Si1 atom of the Si(337) surface; (c) localized layer density of states of the Si2 atom of the Si(337) surface (arbitrary units).
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as from Fig. 3. However, there are two main features that need to mentioned at the Y point of the surface Brillouin zone. Firstly, the difference in the energy levels between the surface states S2 and S4 is smaller than that between the surface states S1 and S5, since the atoms in the second layer are closer to the bulk environment than those in surface layer. Secondly, the occupation of the surface state S1 is mainly localized in the surface and the third layer, but there is almost no occupation in the second layer. Similarly, the surface state S3 is also mainly localized in the third layer and the surface layer. In the stomach gap of the bulk bands that appear in the vicinity of S–X direction in the energy region from −3.0 to −7.0 eV, a localized surface state S6 appears in the lower part of the stomach gap and this state almost has no dispersion along the high symmetric line S–X, but it has strong downward dispersion along Y–S and strong upward dispersion along X–C. From the layerresolved density of states, we find it has a large occupation of electronic states in the first four layers below the surface. So S6 is mainly related to the change of bonds formed by the atoms between the surface layer and the second layer, and the second layer and the third layer, which we call backbond states. In the small pockets, there are also some surface states that show the same orbital character as the S6 surface states. It should be mentioned that there is an unoccupied antiresonance surface state that lies above the conduction band minimum. It originates mainly from the mixed sp3 hybrid states of the Si2 atom. This surface has the same dispersion relation as the surface states S1 along the C–Y–S–X–C.
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bond of the Si1 atom. The dominant contributors are the s-like and p-like orbits of Si1; the occupied dangling bond states (S4 and S5) lie close to the top of valence band maximum and they come from the sp3 hybridized bond of Si2 atom. (2) In the stomach gap, the localized surface state S6 is composed of s-like orbits of the Si2 atom and p-like orbits of the Si1 atom.
5. Conclusions In conclusion, we have studied the electronic states of the high-index Si(337) 1×1 surface by using the tight-binding method. The results show that there exist six surface bound states in the range from −12.0 to +2.0 eV. In the fundamental gap, three of these surface states lying close to the bottom of conduction band maximum come from the Si atom in the site of the cation crystal lattice and the others from the Si atom in the site of the anion crystal lattice. The local and the orbital properties of these surface states are discussed. The results are useful for further study of the electronic properties of the Si(337) surface from both theoretical and experimental aspects.
Acknowledgements We are grateful to Professor Yu-ping Huo of Zhengzhou University and Professor Y.R. Xing of the Institute of Semiconductors of Academia Sinica for their helpful discussions. This work was supported by the Natural Science Foundation of Henan, China, grant: 984040500.
4.2. Localized density of states and the orbital characters References To explain the orbital character of these surface states, The localized LDOS and atom-resolved localized LDOS on different layers for the Si(337) surface are shown in Fig. 5a–c. From Fig. 5, we can see the following. (1) In the fundamental gap, the occupied dangling bond states (S1, S2 and S3) lie close to the bottom of the conduction band maximum and they come from the sp3 hybridized
[1] [2] [3] [4]
X.S. Wang, W.H. Weinberg, Surf. Sci. 314 (1994) 71. A.A. Baski, L.J. Whitman, Phys. Rev. Lett. 74 (1995) 956. W. Ranke, Y.R. Xing, Phys. Rev. B 31 (1985) 2246. X.R. Xing, J.P. Zhang, J.A. Wu, Surf. Sci. 232 (1990) L215. [5] W. Ranke, Phys. Rev. B 41 (1990) 5243. [6 ] E.D. Williams, N.C. Bartelt, Science 251 (1991) 393. [7] B.Z. Olshanetsky, V.I. Mashano, Surf. Sci. 111 (1981) 414.
306
Y. Jia et al. / Surface Science 437 (1999) 299–306
[8] U. Myler, K. Jacobi, Surf. Sci. 220 (1989) 353. [9] K.A. Feng, X.M. Hu, Z. Lin, X.R. Xing, Appl. Surf. Sci. 120 (1997) 94. [10] Y.P. Feng, T.H. Wee, C.K. Ong, H.C. Poon, Phys. Rev. B 54 (1996) 4766. [11] H. Kroemer, J. Crystal Growth 81 (1987) 193. [12] S.L. Wright, M. Jnada, H. Kroemer, J. Vac. Sci. Technol. 21 (1982) 543.
[13] Y.R. Xing, W. Ranke, Chin. J. Semicond. 15 (1994) 136 in Chinese. [14] X. Hu, E.G. Wang, Y.R. Xing, Appl. Surf. Sci. 103 (1996) 317. [15] P. Vogl, H.P. Hjalmarson, J.D. Dow, J. Chem. Solids 44 (1983) 365. [16 ] J. Pollmann, S.T. Pantelides, Phys. Rev. B 18 (1978) 5524. [17] I. Ivanov, A. Mazur, J. Pollmann, Surf. Sci. 92 (1980) 365.