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Lattice gas model for semiconductor (111) surface reconstruction
Surface Science Letters North-Holland
232 (1990) L201-L204
LA’ITICE GAS MODEL You Gong
FOR SEMICONDUCTOR
(111) SURFACE
RECONSTRUCTION
HA0
Physics Department,
West Virginia Uniuersity, Morgantown,
WV 26506, USA
and Laura
M. ROTH
Physics Department, Received
24 January
State University of New York, Albany, 1990; accepted
for publication
NY 12222, USA
8 March
1990
A simple lattice gas model is proposed for semiconductor (111) surface reconstruction. By signing energy parameters to the gas particles and the interaction among them we found that the local structure for the silicon (111) surface (7 x 7) reconstruction and the germanium (111) surface ~$2 X 8) reconstruction should be different, and that the formation of the dimers on the surfaces plays a very important rule in the surface reconstructions. We suggest that a dimer instead of a simple adatom should be the basic building unit for the germanium (111) surface c(2 X 8) reconstruction.
Silicon and germanium exhibit similar behavior in surface reconstruction [l]. On the (111) surface they both show the (2 X 1) reconstruction immediately after cleavage in a high vacuum. Both (2 x 1) structures are metastable and are converted to stable structures after annealing [2-61. The annealing produces a difference: for silicon the new structure is (7 x 7) and for germanium it is c(2 x 8). For a chemisorbed (111) surface the reconstruction of both crystals also shows similar behavior [7,8]. Scanning tunneling microscopy (STM) and other experiments [9,10] on germanium layers grown on silicon substrate by molecular epitaxy reveal a further relation of the surface reconstruction, in which the germanium surface shows (5 x 5), (7 x 7) (2 x 2) and c(4 x 2) reconstructed species as metastable states before it reaches the stable structure of c(2 X 8). Similarities in the (111) surface reconstructions for both crystals seem to suggest that there is some unique rule to govern the surface reconstructions. Regardless of the exact structure of the reconstructed surface one can introduce the surface gas 0039-6028/90/$03.50
0 1990
Elsevier Science Publishers
idea to investigate the general properties of the surface reconstruction [ll-131. Each atom on a cleaved silicon or germanium (111) surface has one dangling bond, which is unstable, and a surface reconstruction is expected to take place in such a
.
. 0 .
8
-0. .
b
. C
.
.
.
d
Fig. 1. (a) Reconstruction building particle where a dot stands for a surface atom and a circle for a second layer atom. (b) Dimer. (c) (fi x 6) reconstruction. (d) (2 x 2) reconstruction.
B.V. (North-Holland)
You Gong Hao,L.M. Rorh/ Lattrce gasmodelfor semiconductor (111) surface reconstruction
way as to reduce the total number of the dangling bonds. It does not matter if such a reduction of the dangling bonds is due to chemical adsorption of a foreign atom or an adatom from the bulk; a combination of three closest dangling bonds produces a surface particle as shown in fig. la (we do not discuss a “one to one” adsorption which gives a (1 X 1) reconstruction). These particles behave like a two-dimensional gas on the surface, with restriction of their position and interaction by the structure of the substrate. Now if the radius of this particle is about A&/2, where A is the lattice constant of the surface hexagonal mesh, a close packed structure of such particles produces a (6 X a) structure shown in fig. lc. For a radius of A the structure is (2 X 2) of fig. Id. If the radius of the particles is somewhere between the two radii, a further combination, or reaction between the particles, will occur. In the simplest case the combination provides dimers bonded by two nearest particles of radius slightly larger than A&/2 but less than A, shown in fig. lb, which then leads to a c(2 x 8) structure of fig. 2 by close packing of such dimers. The local two-fold symmetry of the dimers however, violates the nature of the three-fold symmetry of the substrate. If the interaction of the surface particles with the substrate is strong the dimer cannot be packed so closely. Then the surface will have a (n x n) structure given in fig. 3, which obeys the three-fold symmetry.
a
Fig. 3. (a) (3 X 3); (b) (7 X 7) reconstructions. where c shows the “corner” or the intersection of the dimer rows.
We now treat the lattice gas mode1 more specifically by assigning energy parameters to the particles and to the interactions among them. The energy parameters we used are: the formation energy of the particles, a; the bonding energy of dimers, b; and the formation energy of the intersection crossed by dimer rows, c, which we called “corner” previously [ll]. We have eliminated the energy parameters for interaction among the particles and the detailed interactions in the formation of these intersections in order to simplify the analysis. A mode1 involving more energy parameters essentially gives the same conclusion. Using these energy parameters we estimated the energy per surface atom for various structures: E(nxn) E(c(2
Fig. 2. c(2
x 8)
reconstruction
=(Y+j3/n+y/nz,
(1)
x 8)) = (Y+ p/12,
(2)
E(2 x 2) = (Y,
(3)
You Gong Hao, L.M. Roth / Lattice gas model for semiconductor
E-
(I I I) surface reconstruction
n
,
Fig. 4. Total energy versus n. Dashed line shows that when n c 6, the stable structure is (5 x 5). For n > 6 the structure is c(2 X 8) as the dash-dotted line shows. Dotted line shows that (7 X 7) could have lower energy than c(2 X 8) after the c(2 X 8) line is raised by AE.
where a=a/4, /3=6b/4 and 4c)/4. Since we are interested in a E( n X n) is minimum for n = n, . . . ), we found that y must be must be negative by considering second derivatives of E(n). This mum of
y=(-a-6b+ structure where (n = 3, 5, 7, 9, positive and p the first and gives the mini-
E(n,Xn,)=cX-/32/4y
(4)
at n0
=
-2y/p.
A simple immediately
(5) analysis of these equations that when n goes to infinity
gives us
E( n x Fr) = E(2 x 2) = CY and it has the highest structures, and that for -Y<3P
total
energy
of all these
(6)
the c(2 x 8) structure will have the lowest total energy. In fact by fitting eq. (6) to eq. (5) we found that E(c(2 X 8)) is the most stable structure except for n, less than 6.
The conclusion that the c(2 X 8) structure is the most stable structure is not consistent with the experimental discoveries on a silicon (111) surface. This controversy seems to suggest either that the simple model is not capable of explaining the silicon (111) surface reconstruction or that the silicon (7 x 7) is not a final stable structure. The widely accepted dimer-adatom-stacking fault (DAS) model for silicon (111) surface (7 X 7) reconstruction [14] has a junction of two stacking fault species under the dimers which increases the total energy drastically when used in a c(2 X 8) structure [15]. To account for this, we add a term AE in eq. (2) to represent this energy increase. In fig. 4 we plot the total energy for various structures against n with p fixed. A surface reconstruction produces the particles and dimers which determine the energy parameters (Y, /3 and y for the particular surface structure. If the relation of p and y gives n = 5, then along the dashed line a stable structure is (5 x 5). If however, n > 6, the most stable structure is c(2 x 8) unless the c(2 X 8) line is raised by AE. The total energy for (5 X 5) (7 x 7) and c(2 x 8) are very close and they are all
You Gong Hue, L. M. Roth / Lattice gm model for semiconductor
related to fi. This fact suggests that the formation of dimers plays a very important role in all these surface reconstructions. The most stable structure for silicon being (7 x 7) reveals that the local structure of the surface reconstructions for silicon and germanium must be different according to our model. On a germanium (111) surface a simple adatom model seems to be accepted widely. According to our model however, a simple adatom model should be either a (2 X 2) or (6 x 6) structure. The fact that the c(2 x 8) is the final stable structure implies that there should be bonded pairs of adatoms for an adatom model. In conclusion we have developed a lattice gas model to study the silicon and germanium (111) surface reconstruction. According to our model we found that the local structure of the reconstruction for the two crystals must be very different; otherwise only the c(2 x 8) or the (5 X 5) should be the final structure. We also found that the formation of dimers is crucial for these surface reconstructions. We suggest that a dimer should be the basic unit instead of a simple adatom for the germanium (111) surface c(2 X 8) reconstruction.
(1 I I) surface reconstructmn
References
Ill N.P. Lieske, J. Phys. Chem. Solids 45 (1984) 821. PI M. Henzler, Surf. Sci. 36 (1973) 103. and V.I. Mashanov, Surf. Sci. 111 (1981) [31 B.Z. Olshanetsky 414. V.I. Mashanov and A.I. Nikiforov, Surf. [41 B.Z. Olshanetsky, Sci. 111 (1981) 429. [51 J.J. Lander, G.W. Gobell and Morrison, J. Appl. Phys. 34 (1963) 2298. [61 D. Haneman, Adv. Phys. 31 (1982) 165. S. Nakatani, T. Ishikawa and S. Kikuta, [71 T. Takahashi, Surf. Sci. 191 (1987) L825. J.S. Pedersen, M. Nielsen, F. Grey and PI R. Feidenhans’l, R.L. Johnson, Surf. Sci. 178 (1986) 927; J.S. Pedersen. R. Feidenhans’l, M. Nielsen, K. Kjazr, F. Grey and R.L. Johnson, Surf. Sci. 189/190 (1987) 1047. and B.S. Swartzentruber, [91 R.C. Becker, J.A. Golovchenko Phys. Rev. Lett. 54 (1985) L175. J.C. Bean, L.C. Feldman and W.M. UOI H.-J. Gossmann, Gibson, Surf. Sci. 138 (1984) L175. u11 You Gong Hao and L.M. Roth, in: Computer-Based Microscopic Description of the Structure and Properties of Materials, Proc. Mater. Res. Sot., Vol. 63. SUNY-Albany (1986). [I21 You Gong Hao, PhD Dissertation, Phys. Rev. B 36 (1987) 6209. [I31 D. Vanderbilt, Y. Tanoshiro, M. Takahashi, H. v41 K. Takayanagi, Motoyoshi, K. Yagi, in: Proc. 10th Int. Conf. on Electron Microscopy, 1982. B 4 [I51 G.X. Qian and D.J. Chadi, J. Vat. Sci. Technol. (1986) 1079.
232 (1990) L201-L204
LA’ITICE GAS MODEL You Gong
FOR SEMICONDUCTOR
(111) SURFACE
RECONSTRUCTION
HA0
Physics Department,
West Virginia Uniuersity, Morgantown,
WV 26506, USA
and Laura
M. ROTH
Physics Department, Received
24 January
State University of New York, Albany, 1990; accepted
for publication
NY 12222, USA
8 March
1990
A simple lattice gas model is proposed for semiconductor (111) surface reconstruction. By signing energy parameters to the gas particles and the interaction among them we found that the local structure for the silicon (111) surface (7 x 7) reconstruction and the germanium (111) surface ~$2 X 8) reconstruction should be different, and that the formation of the dimers on the surfaces plays a very important rule in the surface reconstructions. We suggest that a dimer instead of a simple adatom should be the basic building unit for the germanium (111) surface c(2 X 8) reconstruction.
Silicon and germanium exhibit similar behavior in surface reconstruction [l]. On the (111) surface they both show the (2 X 1) reconstruction immediately after cleavage in a high vacuum. Both (2 x 1) structures are metastable and are converted to stable structures after annealing [2-61. The annealing produces a difference: for silicon the new structure is (7 x 7) and for germanium it is c(2 x 8). For a chemisorbed (111) surface the reconstruction of both crystals also shows similar behavior [7,8]. Scanning tunneling microscopy (STM) and other experiments [9,10] on germanium layers grown on silicon substrate by molecular epitaxy reveal a further relation of the surface reconstruction, in which the germanium surface shows (5 x 5), (7 x 7) (2 x 2) and c(4 x 2) reconstructed species as metastable states before it reaches the stable structure of c(2 X 8). Similarities in the (111) surface reconstructions for both crystals seem to suggest that there is some unique rule to govern the surface reconstructions. Regardless of the exact structure of the reconstructed surface one can introduce the surface gas 0039-6028/90/$03.50
0 1990
Elsevier Science Publishers
idea to investigate the general properties of the surface reconstruction [ll-131. Each atom on a cleaved silicon or germanium (111) surface has one dangling bond, which is unstable, and a surface reconstruction is expected to take place in such a
.
. 0 .
8
-0. .
b
. C
.
.
.
d
Fig. 1. (a) Reconstruction building particle where a dot stands for a surface atom and a circle for a second layer atom. (b) Dimer. (c) (fi x 6) reconstruction. (d) (2 x 2) reconstruction.
B.V. (North-Holland)
You Gong Hao,L.M. Rorh/ Lattrce gasmodelfor semiconductor (111) surface reconstruction
way as to reduce the total number of the dangling bonds. It does not matter if such a reduction of the dangling bonds is due to chemical adsorption of a foreign atom or an adatom from the bulk; a combination of three closest dangling bonds produces a surface particle as shown in fig. la (we do not discuss a “one to one” adsorption which gives a (1 X 1) reconstruction). These particles behave like a two-dimensional gas on the surface, with restriction of their position and interaction by the structure of the substrate. Now if the radius of this particle is about A&/2, where A is the lattice constant of the surface hexagonal mesh, a close packed structure of such particles produces a (6 X a) structure shown in fig. lc. For a radius of A the structure is (2 X 2) of fig. Id. If the radius of the particles is somewhere between the two radii, a further combination, or reaction between the particles, will occur. In the simplest case the combination provides dimers bonded by two nearest particles of radius slightly larger than A&/2 but less than A, shown in fig. lb, which then leads to a c(2 x 8) structure of fig. 2 by close packing of such dimers. The local two-fold symmetry of the dimers however, violates the nature of the three-fold symmetry of the substrate. If the interaction of the surface particles with the substrate is strong the dimer cannot be packed so closely. Then the surface will have a (n x n) structure given in fig. 3, which obeys the three-fold symmetry.
a
Fig. 3. (a) (3 X 3); (b) (7 X 7) reconstructions. where c shows the “corner” or the intersection of the dimer rows.
We now treat the lattice gas mode1 more specifically by assigning energy parameters to the particles and to the interactions among them. The energy parameters we used are: the formation energy of the particles, a; the bonding energy of dimers, b; and the formation energy of the intersection crossed by dimer rows, c, which we called “corner” previously [ll]. We have eliminated the energy parameters for interaction among the particles and the detailed interactions in the formation of these intersections in order to simplify the analysis. A mode1 involving more energy parameters essentially gives the same conclusion. Using these energy parameters we estimated the energy per surface atom for various structures: E(nxn) E(c(2
Fig. 2. c(2
x 8)
reconstruction
=(Y+j3/n+y/nz,
(1)
x 8)) = (Y+ p/12,
(2)
E(2 x 2) = (Y,
(3)
You Gong Hao, L.M. Roth / Lattice gas model for semiconductor
E-
(I I I) surface reconstruction
n
,
Fig. 4. Total energy versus n. Dashed line shows that when n c 6, the stable structure is (5 x 5). For n > 6 the structure is c(2 X 8) as the dash-dotted line shows. Dotted line shows that (7 X 7) could have lower energy than c(2 X 8) after the c(2 X 8) line is raised by AE.
where a=a/4, /3=6b/4 and 4c)/4. Since we are interested in a E( n X n) is minimum for n = n, . . . ), we found that y must be must be negative by considering second derivatives of E(n). This mum of
y=(-a-6b+ structure where (n = 3, 5, 7, 9, positive and p the first and gives the mini-
E(n,Xn,)=cX-/32/4y
(4)
at n0
=
-2y/p.
A simple immediately
(5) analysis of these equations that when n goes to infinity
gives us
E( n x Fr) = E(2 x 2) = CY and it has the highest structures, and that for -Y<3P
total
energy
of all these
(6)
the c(2 x 8) structure will have the lowest total energy. In fact by fitting eq. (6) to eq. (5) we found that E(c(2 X 8)) is the most stable structure except for n, less than 6.
The conclusion that the c(2 X 8) structure is the most stable structure is not consistent with the experimental discoveries on a silicon (111) surface. This controversy seems to suggest either that the simple model is not capable of explaining the silicon (111) surface reconstruction or that the silicon (7 x 7) is not a final stable structure. The widely accepted dimer-adatom-stacking fault (DAS) model for silicon (111) surface (7 X 7) reconstruction [14] has a junction of two stacking fault species under the dimers which increases the total energy drastically when used in a c(2 X 8) structure [15]. To account for this, we add a term AE in eq. (2) to represent this energy increase. In fig. 4 we plot the total energy for various structures against n with p fixed. A surface reconstruction produces the particles and dimers which determine the energy parameters (Y, /3 and y for the particular surface structure. If the relation of p and y gives n = 5, then along the dashed line a stable structure is (5 x 5). If however, n > 6, the most stable structure is c(2 x 8) unless the c(2 X 8) line is raised by AE. The total energy for (5 X 5) (7 x 7) and c(2 x 8) are very close and they are all
You Gong Hue, L. M. Roth / Lattice gm model for semiconductor
related to fi. This fact suggests that the formation of dimers plays a very important role in all these surface reconstructions. The most stable structure for silicon being (7 x 7) reveals that the local structure of the surface reconstructions for silicon and germanium must be different according to our model. On a germanium (111) surface a simple adatom model seems to be accepted widely. According to our model however, a simple adatom model should be either a (2 X 2) or (6 x 6) structure. The fact that the c(2 x 8) is the final stable structure implies that there should be bonded pairs of adatoms for an adatom model. In conclusion we have developed a lattice gas model to study the silicon and germanium (111) surface reconstruction. According to our model we found that the local structure of the reconstruction for the two crystals must be very different; otherwise only the c(2 x 8) or the (5 X 5) should be the final structure. We also found that the formation of dimers is crucial for these surface reconstructions. We suggest that a dimer should be the basic unit instead of a simple adatom for the germanium (111) surface c(2 X 8) reconstruction.
(1 I I) surface reconstructmn
References
Ill N.P. Lieske, J. Phys. Chem. Solids 45 (1984) 821. PI M. Henzler, Surf. Sci. 36 (1973) 103. and V.I. Mashanov, Surf. Sci. 111 (1981) [31 B.Z. Olshanetsky 414. V.I. Mashanov and A.I. Nikiforov, Surf. [41 B.Z. Olshanetsky, Sci. 111 (1981) 429. [51 J.J. Lander, G.W. Gobell and Morrison, J. Appl. Phys. 34 (1963) 2298. [61 D. Haneman, Adv. Phys. 31 (1982) 165. S. Nakatani, T. Ishikawa and S. Kikuta, [71 T. Takahashi, Surf. Sci. 191 (1987) L825. J.S. Pedersen, M. Nielsen, F. Grey and PI R. Feidenhans’l, R.L. Johnson, Surf. Sci. 178 (1986) 927; J.S. Pedersen. R. Feidenhans’l, M. Nielsen, K. Kjazr, F. Grey and R.L. Johnson, Surf. Sci. 189/190 (1987) 1047. and B.S. Swartzentruber, [91 R.C. Becker, J.A. Golovchenko Phys. Rev. Lett. 54 (1985) L175. J.C. Bean, L.C. Feldman and W.M. UOI H.-J. Gossmann, Gibson, Surf. Sci. 138 (1984) L175. u11 You Gong Hao and L.M. Roth, in: Computer-Based Microscopic Description of the Structure and Properties of Materials, Proc. Mater. Res. Sot., Vol. 63. SUNY-Albany (1986). [I21 You Gong Hao, PhD Dissertation, Phys. Rev. B 36 (1987) 6209. [I31 D. Vanderbilt, Y. Tanoshiro, M. Takahashi, H. v41 K. Takayanagi, Motoyoshi, K. Yagi, in: Proc. 10th Int. Conf. on Electron Microscopy, 1982. B 4 [I51 G.X. Qian and D.J. Chadi, J. Vat. Sci. Technol. (1986) 1079.