- Email: [email protected]
Ab initio pseudopotential calculations for electronic and geometric structure of mixed Si–Ge dimers on the Si(001)-(1×2) and Si(001)-(2×4) surfaces
Surface Science 433–435 (1999) 909–914 www.elsevier.nl/locate/susc
Ab initio pseudopotential calculations for electronic and geometric structure of mixed Si–Ge dimers on the Si(001)-(1×2) and Si(001)-(2×4) surfaces S.C.A. Gay *, G.P. Srivastava Physics Department, University of Exeter, Stocker Road, Exeter EX4 4QL, UK
Abstract We report on ab initio pseudopotential calculations for electronic and geometric structure of mixed Si–Ge dimers on the Si(001)-(1×2) and Si(001)-(2×4) surfaces. The structural parameters obtained from our calculations provide some agreement with the recently proposed structure by Chen et al. based on adjacent substitutional Si–Ge dimers buckled in opposite directions. Dimer bonding and electronic states in the silicon band gap are also studied. It is noted that the c-(2×4) reconstruction provides a more efficient substrate relaxation mechanism than does the simple (1×2) reconstruction. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Silicon surface; Si–Ge dimers; Pseudopotentials
1. Introduction Due to their vast technological importance, Si(001)/Ge systems have been widely studied, both experimentally and theoretically. Like the clean Si(001) surface, a single monolayer (1 ML) Ge terminated Si(001) surface is thought to consist of buckled dimers [1,2], but at least one experimental work has reported symmetric dimers [3]. On the other hand, experimentalists appear to agree on a ˚ . Theory [1,4,5] dimer length of around 2.51–2.55 A predicts a shorter dimer bond length of about ˚ and always finds a tilted dimer. Strong 2.38 A experimental evidence exists which suggests that mixed Si–Ge dimers are formed [6,7] in the early stages of Ge growth on Si(001). Previous theoretical calculations by Jenkins and Srivastava [1] agree * Corresponding author. Fax: +44-0-1392-264111. E-mail address: [email protected] (S.C.A. Gay)
with experimental work [6,7] that the up atom in the mixed tilted dimer is the Ge atom. Recently, Chen et al [7] have put forward a reconstruction with adjacent dimers tilted in opposite directions, analogous to the low temperature reconstruction of the Si(001)c-(2×4) surface. They determine a dimer length and tilt angle for this reconstruction obtained by photoelectron diffraction experiments. However, their values disagree with those of Jenkins and Srivastava for the (1×2) reconstruction. In this paper we present more in depth calculations for the mixed Si–Ge dimers on Si(001)(1×2) and fresh calculations on Si(001)-(2×4). All calculations are performed using a plane wave pseudopotential and density functional method. 2. Method The results of all calculations presented in this paper are obtained by using the density functional
0039-6028/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 99 ) 0 05 2 8 -2
910
S.C.A. Gay, G.P. Srivastava / Surface Science 433–435 (1999) 909–914
Fig. 1. Diagram showing the relaxed geometry for the (a) (1×2) and (b) (2×4) unit cells. Our bond labelling convention is also shown.
theory in its local approximation. The Ceperley– Alder electron correlation scheme is used in the form parameterised by Perdew and Zunger [8]. Ion–electron interactions are treated by using the norm conserving pseudopotentials of Bachelet, Hamann and Schlu¨ter [9]. Relaxation of atomic and electronic degrees of freedom is achieved by solving the Kohn–Sham equations. Wave functions are expanded using a planewave basis set. We model the surface system in periodic slab geometry. Our unit cells have the natural periodicity of the surface as well as an artificial periodicity introduced in the surface normal direction. The (1×2) unit cell has a length equivalent to 12 atomic layers of Si, whereas the (2×4) unit cells have a length equivalent to 9 atomic layers. In both cases, mixed Si–Ge dimers are considered on one side of the slab with hydrogen passivation of
the opposite face of the slab. The (1×2) slab contains 7 layers of Si of which the back two are kept frozen, and the (2×4) slabs contain 5 layers of Si of which the back layer is kept frozen. All non-frozen atoms are allowed to relax into their minimum energy configuration by using a conjugate gradient technique [10]. Surface geometry and band structures were obtained using an 8 Ryd cut-off for the plane wave basis. Four special k-points were used for sampling the Brillouin zone for the (1×2) system, whereas the zone centre (C9 ) was used for the (2×4) systems. 3. Results Considering initially the pure single dimer (1×2) reconstruction, as in the previous paper
Table 1 ˚ ) and the Si–Ge dimer tilt angle on Si(001) surfaces Bond lengths and distances (in A
(1×2) (2×4) Chen et al. [7]
d 1
d 2
v
A
B 1
B 2
S
C
E
0.81 0.70 –
0.75 0.71 0.43
20.4 17.0 31.0
2.33 2.42 2.43
2.44 2.39 2.63
2.33 2.27 –
2.43 – –
2.33 2.31 –
2.38 2.36 –
S.C.A. Gay, G.P. Srivastava / Surface Science 433–435 (1999) 909–914
Fig. 2. Total charge density plots for (a) a vertical slice through the Si–Ge dimer, (b) an oblique slice through the Ge–Si backbond and (c) an oblique slice through the Si–Si backbond. The values are normalised to the total number of electrons inside the unit cell.
[1], we found a geometry in the dimer region significantly different from that reported by Chen et al. [7]. Our dimer tilt angle of 20.3° is much smaller than the reported value of 31°. Likewise, our dimer length was also in disagreement at ˚ . Fig. 1 and Table 1 show these measure2.33 A ments as well as other bond lengths in the tip few adsorbate layers. We obtain a shorter Ge–Si back˚ instead of 2.63 A ˚ bond (B in Fig. 1a at 2.44 A 1 found by Chen et al.). Furthermore Chen et al. found a much flatter Si–Si backbond than we did, ˚ ), d in Fig. 1a, with our vertical component (0.75 A 2 ˚ being almost twice as large as theirs (0.43 A ). The
911
dimer formation induces considerable strain in the substrate layers with bonds being distorted by up to 4%. The substrate relaxation results in all Si–Si bonds directly beneath the dimer row, marked with a C in Fig. 1a, being compressed, with the exception of the stretched bond labelled with an S. On the other hand, those outside the line of the dimer row, marked with an E in Fig. 1a, are elongated with respect to the bulk Si length of ˚ . Not shown in Fig. 1 is the vertical separa2.35 A tion between the two Si atoms in the top substrate ˚ . The vertical layer: this was found to be 0.08 A separation between the two Si atoms in the second ˚. substrate layers is somewhat larger at 0.31 A In going to a (2×4) reconstruction, we found that several differences in the dimer geometry came into play. The unit cell used was a true (2×4) unit cell with no enforced symmetry, so as not to bias a c-(2×4) reconstruction and yet not rule it out. With the four dimers all anti-buckling with respect to their adjacent dimers it was found that the unit cell remained essentially c-(2×4) and all four dimers were equivalent to each other except in orientation. Of the other five permutations of relative dimer orientations, only one other was investigated. This was the one with the four dimers pointing the same way, and was simply the (1×2) relaxed geometry augmented up to a (2×4) reconstruction. Energetically the anti-buckling dimers were slightly more favoured than the four dimers all tilting the same way. This difference in energy was about 0.25 eV per dimer. Structurally we found a longer dimer bond length for the (2×4) reconstruction, in good agreement with that of Chen et al. However, the tilt angle was still a long way off that of Chen et al. and, if anything, had gone down by about 3° from our value obtained for the (1×2) unit cell. The substrate distortions were, on the other hand, greatly reduced, with all the bonds between the two top substrate layers having the same length (see Fig. 1). All top layer substrate Si atoms were now at the same vertical height and the vertical separation between adjacent second layer substrate Si atom has been reduced ˚ to 0.20 A ˚ . This can be understood in from 0.31 A terms of a stress relief mechanism which has reduced both the asymmetry and the magnitude
912
S.C.A. Gay, G.P. Srivastava / Surface Science 433–435 (1999) 909–914
Fig. 3. Projected electronic band structure for mixed Si–Ge dimer within the (1×2) reconstruction. The hatched regions represent the bulk projected Si(001)-(1×2) band structure. Occupied surface states are shown as thick lines, whereas the lowest non-occupied surface state is shown as a thin line.
in the distortions of the top few substrate layers. This demonstrates that there is a considerable advantage in the adsorbate layer taking up a c-(2×4) reconstruction rather than the smaller (1×2). Looking at the total charge density plots for the (1×2) reconstruction, one can see that all the surface bonds are covalent. Fig. 2a shows a vertical slice through the dimer and shows the dimer bond to be perfectly covalent. Fig. 2b and c show the two backbond planes along side one another showing that both the Ge–Si and the Si–Si backbonds are covalent. Inspecting the region above the Ge atom in Fig. 2a also reveals a local maximum which is associated with the occupied lone pair on the Ge atom. It should be noted that Ge is slightly more electronegative than Si and, as such, one would expect the Ge atom to attract the two shared electrons in preference to the Si atom. Electrostatic considerations then explain that the Ge atom rises up as it feels a stronger repulsive force from the electrons in the substrate than does the Si atom of the dimer. Similar plots for the c-(2×4) reconstruction reveal the same characteristics. Fig. 3 shows the projected electronic band struc-
ture for the (1×2) reconstruction. The Si fundamental band gap is clear of any occupied surface states. There is one surface unoccupied state (D ) 2 that does penetrate this region, which is localised on the down component of the Si–Ge mixed dimer, namely the Si atom, and is due to the dangling bond orbital of the Si atom that has donated its electron to the Ge atom. However, this band does not intersect any of the valence bands and as such the surface remains semiconducting. Of the three occupied surface states in the band gap region, the highest in energy is associated with the fully occupied dangling orbital on the Ge atom (D ). The 1 second highest occupied state belongs to the Si–Si bonds between the top two substrate layers below the Si atom of the mixed dimer (Sub), corresponding to the ‘S’ bond in Fig. 1a. The third highest occupied surface band is localised around the actual Si–Ge dimer bond (Dm). This is only present over a short segment of k-space at the bottom of the valley of the Si projected band structure around the K 9 point. This state appears to be pps in nature. Partial charge density plots for these four surface states are shown in Fig. 4. The highest occupied and lowest unoccupied surface states were expected and are very similar to
S.C.A. Gay, G.P. Srivastava / Surface Science 433–435 (1999) 909–914
913
Fig. 4. Diagram showing four partial charge densities at the K 9 point: (a) lowest unoccupied state, (b) highest occupied state, (c) second highest occupied state and (d ) third highest occupied state. Vertical slices through the dimer are considered for (a), (b), and (d), whereas a vertical slice through the top layer Si atoms is considered for (c).
those that one would obtain for both the clean Si(001)-(1×2) and Ge(001)-(1×2) surfaces [11]. This is certainly not surprising as Si and Ge are both group IV elements. In going to the (2×4) reconstruction the picture remains very similar. The K 9 valley is shallower due to folding of the projected bulk states and leaves only the D and the D as clear surface 1 2
states. There are now, of course, four of each of these two types of bands. The four D bands are 1 degenerate at the K 9 point and the same is true of the D bands. Both states split off into two pairs 2 away from the K 9 point. This can be understood in terms of the folding of the D and D states 1 2 shown in Fig. 3 into the smaller Brillouin zone of the (2×4) unit cell.
914
S.C.A. Gay, G.P. Srivastava / Surface Science 433–435 (1999) 909–914
4. Conclusions
Acknowledgements
We have presented well-relaxed geometries for mixed Si–Ge dimers on the Si(001) surface within both a (1×2) and a (2×4) reconstruction. The (2×4) reconstruction actually took up a c-(2×4) reconstruction and differed considerably from the (1×2) reconstruction. Notably, the dimer length was found to be longer in the (2×4) reconstruction and agreed well with the values obtained by Chen et al. Furthermore, we found that the relaxation mechanism was more efficient in the (2×4) than the (1×2) and resulted in lowering the energy by about 0.25 eV per dimer. However, the tilt angle results obtained in our calculations differ greatly from that obtained by Chen et al. We have also presented electronic band structure information and found similar dispersions to those one obtains for the clean Si(001) or the clean Ge(001) surfaces. Both the dimer–substrate backbonds and the Si–Ge dimer bonds were found to be covalent.
S.C.A.G. is grateful to the EPSRC ( UK ) for financial support.
References [1] S.J. Jenkins, G.P. Srivastava, J. Phys.: Condens. Matter 8 (1996) 6641. [2] E. Fontes, Y.R. Patel, F. Comin, Phys. Rev. Lett. 70 (1993) 2790. [3] R. Shioda, J. van der Weide, Phys. Rev. B 57 (1998) R6832. [4] J.-H. Cho, S. Jeong, M.-H. Kang, Phys. Rev. B 50 (1994) 17139. [5] P. Kru¨ger, J. Pollmann, Phys. Rev. Lett. 72 (1994) 1130. [6 ] L. Patthey, E.L. Bullock, T. Abukawa, S. Kono, L.S.O. Johansson, Phys. Rev. Lett. 75 (1995) 2538. [7] X. Chen, D.K. Saldin, E.L. Bullock, L. Patthey, L.S.O. Johansson, J. Tani, T. Abukawa, S. Kono, Phys. Rev. B 55 (1997) R7319. [8] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [9] G.B. Bachelet, D.R. Hamann, M. Schlu¨ter, Phys. Rev. B 26 (1982) 4199. [10] A. Umerski, G.P. Srivastava, Phys. Rev. B 51 (1995) 2334. [11] P. Kru¨ger, J. Pollmann, Phys. Rev. Lett. 74 (1995) 1155.
Ab initio pseudopotential calculations for electronic and geometric structure of mixed Si–Ge dimers on the Si(001)-(1×2) and Si(001)-(2×4) surfaces S.C.A. Gay *, G.P. Srivastava Physics Department, University of Exeter, Stocker Road, Exeter EX4 4QL, UK
Abstract We report on ab initio pseudopotential calculations for electronic and geometric structure of mixed Si–Ge dimers on the Si(001)-(1×2) and Si(001)-(2×4) surfaces. The structural parameters obtained from our calculations provide some agreement with the recently proposed structure by Chen et al. based on adjacent substitutional Si–Ge dimers buckled in opposite directions. Dimer bonding and electronic states in the silicon band gap are also studied. It is noted that the c-(2×4) reconstruction provides a more efficient substrate relaxation mechanism than does the simple (1×2) reconstruction. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Silicon surface; Si–Ge dimers; Pseudopotentials
1. Introduction Due to their vast technological importance, Si(001)/Ge systems have been widely studied, both experimentally and theoretically. Like the clean Si(001) surface, a single monolayer (1 ML) Ge terminated Si(001) surface is thought to consist of buckled dimers [1,2], but at least one experimental work has reported symmetric dimers [3]. On the other hand, experimentalists appear to agree on a ˚ . Theory [1,4,5] dimer length of around 2.51–2.55 A predicts a shorter dimer bond length of about ˚ and always finds a tilted dimer. Strong 2.38 A experimental evidence exists which suggests that mixed Si–Ge dimers are formed [6,7] in the early stages of Ge growth on Si(001). Previous theoretical calculations by Jenkins and Srivastava [1] agree * Corresponding author. Fax: +44-0-1392-264111. E-mail address: [email protected] (S.C.A. Gay)
with experimental work [6,7] that the up atom in the mixed tilted dimer is the Ge atom. Recently, Chen et al [7] have put forward a reconstruction with adjacent dimers tilted in opposite directions, analogous to the low temperature reconstruction of the Si(001)c-(2×4) surface. They determine a dimer length and tilt angle for this reconstruction obtained by photoelectron diffraction experiments. However, their values disagree with those of Jenkins and Srivastava for the (1×2) reconstruction. In this paper we present more in depth calculations for the mixed Si–Ge dimers on Si(001)(1×2) and fresh calculations on Si(001)-(2×4). All calculations are performed using a plane wave pseudopotential and density functional method. 2. Method The results of all calculations presented in this paper are obtained by using the density functional
0039-6028/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 99 ) 0 05 2 8 -2
910
S.C.A. Gay, G.P. Srivastava / Surface Science 433–435 (1999) 909–914
Fig. 1. Diagram showing the relaxed geometry for the (a) (1×2) and (b) (2×4) unit cells. Our bond labelling convention is also shown.
theory in its local approximation. The Ceperley– Alder electron correlation scheme is used in the form parameterised by Perdew and Zunger [8]. Ion–electron interactions are treated by using the norm conserving pseudopotentials of Bachelet, Hamann and Schlu¨ter [9]. Relaxation of atomic and electronic degrees of freedom is achieved by solving the Kohn–Sham equations. Wave functions are expanded using a planewave basis set. We model the surface system in periodic slab geometry. Our unit cells have the natural periodicity of the surface as well as an artificial periodicity introduced in the surface normal direction. The (1×2) unit cell has a length equivalent to 12 atomic layers of Si, whereas the (2×4) unit cells have a length equivalent to 9 atomic layers. In both cases, mixed Si–Ge dimers are considered on one side of the slab with hydrogen passivation of
the opposite face of the slab. The (1×2) slab contains 7 layers of Si of which the back two are kept frozen, and the (2×4) slabs contain 5 layers of Si of which the back layer is kept frozen. All non-frozen atoms are allowed to relax into their minimum energy configuration by using a conjugate gradient technique [10]. Surface geometry and band structures were obtained using an 8 Ryd cut-off for the plane wave basis. Four special k-points were used for sampling the Brillouin zone for the (1×2) system, whereas the zone centre (C9 ) was used for the (2×4) systems. 3. Results Considering initially the pure single dimer (1×2) reconstruction, as in the previous paper
Table 1 ˚ ) and the Si–Ge dimer tilt angle on Si(001) surfaces Bond lengths and distances (in A
(1×2) (2×4) Chen et al. [7]
d 1
d 2
v
A
B 1
B 2
S
C
E
0.81 0.70 –
0.75 0.71 0.43
20.4 17.0 31.0
2.33 2.42 2.43
2.44 2.39 2.63
2.33 2.27 –
2.43 – –
2.33 2.31 –
2.38 2.36 –
S.C.A. Gay, G.P. Srivastava / Surface Science 433–435 (1999) 909–914
Fig. 2. Total charge density plots for (a) a vertical slice through the Si–Ge dimer, (b) an oblique slice through the Ge–Si backbond and (c) an oblique slice through the Si–Si backbond. The values are normalised to the total number of electrons inside the unit cell.
[1], we found a geometry in the dimer region significantly different from that reported by Chen et al. [7]. Our dimer tilt angle of 20.3° is much smaller than the reported value of 31°. Likewise, our dimer length was also in disagreement at ˚ . Fig. 1 and Table 1 show these measure2.33 A ments as well as other bond lengths in the tip few adsorbate layers. We obtain a shorter Ge–Si back˚ instead of 2.63 A ˚ bond (B in Fig. 1a at 2.44 A 1 found by Chen et al.). Furthermore Chen et al. found a much flatter Si–Si backbond than we did, ˚ ), d in Fig. 1a, with our vertical component (0.75 A 2 ˚ being almost twice as large as theirs (0.43 A ). The
911
dimer formation induces considerable strain in the substrate layers with bonds being distorted by up to 4%. The substrate relaxation results in all Si–Si bonds directly beneath the dimer row, marked with a C in Fig. 1a, being compressed, with the exception of the stretched bond labelled with an S. On the other hand, those outside the line of the dimer row, marked with an E in Fig. 1a, are elongated with respect to the bulk Si length of ˚ . Not shown in Fig. 1 is the vertical separa2.35 A tion between the two Si atoms in the top substrate ˚ . The vertical layer: this was found to be 0.08 A separation between the two Si atoms in the second ˚. substrate layers is somewhat larger at 0.31 A In going to a (2×4) reconstruction, we found that several differences in the dimer geometry came into play. The unit cell used was a true (2×4) unit cell with no enforced symmetry, so as not to bias a c-(2×4) reconstruction and yet not rule it out. With the four dimers all anti-buckling with respect to their adjacent dimers it was found that the unit cell remained essentially c-(2×4) and all four dimers were equivalent to each other except in orientation. Of the other five permutations of relative dimer orientations, only one other was investigated. This was the one with the four dimers pointing the same way, and was simply the (1×2) relaxed geometry augmented up to a (2×4) reconstruction. Energetically the anti-buckling dimers were slightly more favoured than the four dimers all tilting the same way. This difference in energy was about 0.25 eV per dimer. Structurally we found a longer dimer bond length for the (2×4) reconstruction, in good agreement with that of Chen et al. However, the tilt angle was still a long way off that of Chen et al. and, if anything, had gone down by about 3° from our value obtained for the (1×2) unit cell. The substrate distortions were, on the other hand, greatly reduced, with all the bonds between the two top substrate layers having the same length (see Fig. 1). All top layer substrate Si atoms were now at the same vertical height and the vertical separation between adjacent second layer substrate Si atom has been reduced ˚ to 0.20 A ˚ . This can be understood in from 0.31 A terms of a stress relief mechanism which has reduced both the asymmetry and the magnitude
912
S.C.A. Gay, G.P. Srivastava / Surface Science 433–435 (1999) 909–914
Fig. 3. Projected electronic band structure for mixed Si–Ge dimer within the (1×2) reconstruction. The hatched regions represent the bulk projected Si(001)-(1×2) band structure. Occupied surface states are shown as thick lines, whereas the lowest non-occupied surface state is shown as a thin line.
in the distortions of the top few substrate layers. This demonstrates that there is a considerable advantage in the adsorbate layer taking up a c-(2×4) reconstruction rather than the smaller (1×2). Looking at the total charge density plots for the (1×2) reconstruction, one can see that all the surface bonds are covalent. Fig. 2a shows a vertical slice through the dimer and shows the dimer bond to be perfectly covalent. Fig. 2b and c show the two backbond planes along side one another showing that both the Ge–Si and the Si–Si backbonds are covalent. Inspecting the region above the Ge atom in Fig. 2a also reveals a local maximum which is associated with the occupied lone pair on the Ge atom. It should be noted that Ge is slightly more electronegative than Si and, as such, one would expect the Ge atom to attract the two shared electrons in preference to the Si atom. Electrostatic considerations then explain that the Ge atom rises up as it feels a stronger repulsive force from the electrons in the substrate than does the Si atom of the dimer. Similar plots for the c-(2×4) reconstruction reveal the same characteristics. Fig. 3 shows the projected electronic band struc-
ture for the (1×2) reconstruction. The Si fundamental band gap is clear of any occupied surface states. There is one surface unoccupied state (D ) 2 that does penetrate this region, which is localised on the down component of the Si–Ge mixed dimer, namely the Si atom, and is due to the dangling bond orbital of the Si atom that has donated its electron to the Ge atom. However, this band does not intersect any of the valence bands and as such the surface remains semiconducting. Of the three occupied surface states in the band gap region, the highest in energy is associated with the fully occupied dangling orbital on the Ge atom (D ). The 1 second highest occupied state belongs to the Si–Si bonds between the top two substrate layers below the Si atom of the mixed dimer (Sub), corresponding to the ‘S’ bond in Fig. 1a. The third highest occupied surface band is localised around the actual Si–Ge dimer bond (Dm). This is only present over a short segment of k-space at the bottom of the valley of the Si projected band structure around the K 9 point. This state appears to be pps in nature. Partial charge density plots for these four surface states are shown in Fig. 4. The highest occupied and lowest unoccupied surface states were expected and are very similar to
S.C.A. Gay, G.P. Srivastava / Surface Science 433–435 (1999) 909–914
913
Fig. 4. Diagram showing four partial charge densities at the K 9 point: (a) lowest unoccupied state, (b) highest occupied state, (c) second highest occupied state and (d ) third highest occupied state. Vertical slices through the dimer are considered for (a), (b), and (d), whereas a vertical slice through the top layer Si atoms is considered for (c).
those that one would obtain for both the clean Si(001)-(1×2) and Ge(001)-(1×2) surfaces [11]. This is certainly not surprising as Si and Ge are both group IV elements. In going to the (2×4) reconstruction the picture remains very similar. The K 9 valley is shallower due to folding of the projected bulk states and leaves only the D and the D as clear surface 1 2
states. There are now, of course, four of each of these two types of bands. The four D bands are 1 degenerate at the K 9 point and the same is true of the D bands. Both states split off into two pairs 2 away from the K 9 point. This can be understood in terms of the folding of the D and D states 1 2 shown in Fig. 3 into the smaller Brillouin zone of the (2×4) unit cell.
914
S.C.A. Gay, G.P. Srivastava / Surface Science 433–435 (1999) 909–914
4. Conclusions
Acknowledgements
We have presented well-relaxed geometries for mixed Si–Ge dimers on the Si(001) surface within both a (1×2) and a (2×4) reconstruction. The (2×4) reconstruction actually took up a c-(2×4) reconstruction and differed considerably from the (1×2) reconstruction. Notably, the dimer length was found to be longer in the (2×4) reconstruction and agreed well with the values obtained by Chen et al. Furthermore, we found that the relaxation mechanism was more efficient in the (2×4) than the (1×2) and resulted in lowering the energy by about 0.25 eV per dimer. However, the tilt angle results obtained in our calculations differ greatly from that obtained by Chen et al. We have also presented electronic band structure information and found similar dispersions to those one obtains for the clean Si(001) or the clean Ge(001) surfaces. Both the dimer–substrate backbonds and the Si–Ge dimer bonds were found to be covalent.
S.C.A.G. is grateful to the EPSRC ( UK ) for financial support.
References [1] S.J. Jenkins, G.P. Srivastava, J. Phys.: Condens. Matter 8 (1996) 6641. [2] E. Fontes, Y.R. Patel, F. Comin, Phys. Rev. Lett. 70 (1993) 2790. [3] R. Shioda, J. van der Weide, Phys. Rev. B 57 (1998) R6832. [4] J.-H. Cho, S. Jeong, M.-H. Kang, Phys. Rev. B 50 (1994) 17139. [5] P. Kru¨ger, J. Pollmann, Phys. Rev. Lett. 72 (1994) 1130. [6 ] L. Patthey, E.L. Bullock, T. Abukawa, S. Kono, L.S.O. Johansson, Phys. Rev. Lett. 75 (1995) 2538. [7] X. Chen, D.K. Saldin, E.L. Bullock, L. Patthey, L.S.O. Johansson, J. Tani, T. Abukawa, S. Kono, Phys. Rev. B 55 (1997) R7319. [8] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [9] G.B. Bachelet, D.R. Hamann, M. Schlu¨ter, Phys. Rev. B 26 (1982) 4199. [10] A. Umerski, G.P. Srivastava, Phys. Rev. B 51 (1995) 2334. [11] P. Kru¨ger, J. Pollmann, Phys. Rev. Lett. 74 (1995) 1155.