- Email: [email protected]
Simulation of the atomic arrangements in amorphous silicon and germanium
JOURNAL OF NoN-CRYSTALLINESOLIDS 8--10 (1972) 359--363 © North-Holland Publishing Co.
S I M U L A T I O N OF T H E A T O M I C A R R A N G E M E N T S I N A M O R P H O U S S I L I C O N AND G E R M A N I U M DOUGLAS HENDERSON and FRANK HERMAN IBM Research Laboratory, San Jose, California, U.S.A. The atomic arrangements in amorphous germanium and silicon are simulated by means of a computer program. In this treatment a disordered system of 64 atoms in a cubic box with periodic boundary conditions is taken as the initial configuration. Each atom is considered in turn and the four nearest and twelve second-nearest neighbors are moved radially towards the nearest and next nearest neighbor distances. The resulting radial distribution function is in good agreement with experiment. We are engaged in a theoretical study of the electronic structure of amorphous silicon and germanium. Our program consists of two stages; firstly, to simulate the atomic arrangements in these materials and, secondly, to use these atomic coordinates to calculate the resulting electronic structure. Experimental information regarding the structure of a material is obtained from scattering measurements of the radial distribution function (RDF). This function, which we denote by g (R), is the ratio of the local number density of atoms at a distance R from a given atom to the average number density of atoms, p = N / V , where N is the number of atoms in a volume V of the material. The experimental R D F , measured by Moss and Graczykl), of crystalline Si and of a vapor-deposited thin film of amorphous Si are shown in fig. 1. The quantity Ro=2.35 A is the nearest-neighbor distance in the crystal. A m o r p h o u s Si, although disordered, shows considerable structure. Just as in the crystal, there are four nearest neighbors and twelve second nearest neighbors around each a t o m at average distances of R o and x/(8/3)Ro, respectively. In addition, from low angle scattering data Moss and Graczyk conclude that the density of fully-annealed amorphous Si is very nearly equal to that of crystalline Si. The major difference between the two R D F ' s is the absence of a third neighbor peak. Similar conclusions have been made for amorphous Ge 2). Of the above conclusions, perhaps the most controversial is that concerning the density. Samples of amorphous Si and Ge are less dense than the crystal. The density of a sample of amorphous solid depends on its history and is thus not an unambiguous number. However, the model-building of Polk 3) as 359
360
D. HENDERSON AND F. HERMAN 80
0¢ ¢n
/1
40
amorphous
40
/J J
crystal
0
1
2
R/Ro Fig. I.
Radial distribution f u n c t i o n s o f crystalline a n d a m o r p h o u s Si (taken from ref. 1).
well as our own results support the view that fully-annealed amorphous Si and Ge have essentially the same density as their crystalline phases and that amorphous samples of lower density contain such imperfections as voids or unsatisfied bonds. A fully-annealed amorphous solid is, of course, an idealization never fully realized in practice. However, it is an appropriate starting point for theoretical studies just as the ideal crystal or the ideal gas of hard spheres are appropriate starting points in the theory of crystalline solids and of liquids, respectively. One interpretation, which we support, of the structure of amorphous Si and Ge, is that their structure is that of a random tetrahedral network in which each atom is linked to four nearest neighbors. The central atom and its four neighbors form a distorted tetrahedron. However, the tetrahedral bonds on neighbors are randomly oriented with respect to each other and any orientation including the extreme eclipsed and staggered configurations are possible. Thus, the first two peaks of the crystal R D F are retained in the amorphous R D F whereas the third peak in the crystal is absent. Amorphous Si and Ge although lacking a crystal structure have more structure than a liquid and cannot be considered merely as quenched liquids.
SIMULATION OF THE ATOMIC ARRANGEMENTS
36l
Indeed liquid Si and Ge are metallic and demonstrate a coordination greater than four. The R D F of a liquid shows only a well-defined peak for nearest neighbors and little structure for larger R. The concept of a network is not useful in liquid studies. Polk 3) has built a model random tetrahedral network. We have attempted a similar study using a computer simulation. Our simulation is similar to that of Shevchik4). However, there are slight differences. He has used a finite static cluster of atoms and has built his cluster in a manner closely analogous to the actual deposition of an amorphous film. That is, his cluster is less dense than the crystal and contains a small number of unsatisfied bonds. On the other hand, our model simulates the combined effects of structural and thermal disorder. In addition, we have used a relatively small cluster of atoms in a box with periodic boundary conditions and have required that all bonds be satisfied. At first sight the use of periodic boundary might appear to be incompatible with a realistic structure of an amorphous solid unless the unit cell is extremely large. However, computer simulations of the structure of liquids have shown that unit cells containing as few as 32 molecules give realistic resultsS). To be sure, it may turn out that much larger unit cells are required to give reasonable results for the electronic structure amorphous Si and Ge. However, our present results show that the R D F of these materials can be simulated with a small number of atoms. Our scheme is very simple. Firstly, we locate N atoms in the diamond structure in a unit cell of dimension:
L = x/3
Ro,
(1)
where p and Pc are the number densities of the amorphous material and its crystal. The value of p is part of the imput data. Usually, we have used P=Pc. However, we have considered other values. In the present case we used N = 64. It may turn out that smaller values of N may be used but we have not yet attempted this. The purpose of this initial step is only to label the atoms and define L and certain other parameters. The second step is to disrupt the crystal structure by moving the atoms at random. The third and key part of the program now follows. Each atom is considered in turn and moved ten percent of the distance to the centroid of its four nearest neighbors. Then the four nearest neighbors and the twelve next nearest neighbors are moved radially ten percent of the distance to the nearest and next nearest neighbor distances of the crystal. Going once around the box means that each atom has been moved back
D. HENDERSON AND F. HERMAN
362
and forth sixteen times. Thus, the positions are only slightly more realistic than the starting arrangement. However, the process can be repeated. A satisfactory structure can be found after a few seconds of computing with the IBM 360/195 computer. No appreciable change is observed after 15 min of additional computing. As may be seen in fig. 2, the resulting R D F is in excellent agreement with that of amorphous Si. We have made runs for other densities and have found that very similar RDF's can be obtained even if p differs from Pc by as much as four percent. This is not surprising in view of the fact that the densities of amorphous Si and Ge are not well-defined. Strictly speaking, one must be careful in using our R D F for R>½L because of the periodic boundary conditions which we have employed. In view of this we have not plotted values of g (R) in fig. 2 for R > ½ L. However, we have looked at our g (R) for larger values of R and have found peaks at about 2.45-2.55 R o, 3.20-3.40 Ro, and 4.054.15 R o. This is in reasonable agreement with experimental results 2). On the basis of these studies we conclude that the structure amorphous Si
80
I
I
I /
c~ o et-
,/./
//
/
///
40
%
-
//
//
/ ///////~ ,0
.s / /
......
,t
f 1
l, 2
R/No Fig. 2. Radial distribution functions. The lower curve gives the experimental results of Moss and Graczyk z) and the upper histogram gives the results of our simulation.
SIMULATION OF THE ATOMIC ARRANGEMENTS
363
and Ge is that of a random tetrahedral network. Similar conclusions have been reached in the complementary studies of Shevchik4). Acknowledgement The authors are grateful to J. A. Barker and N. J. Shevchik for stimulating discussions. References 1) S.C. Moss and J. F. Graczyk, in Proc. Tenth Intern. Conf. Physics of Semiconductors, USAEC (1970) p. 658. 2) V. H. Richter and G. Breitling, Z. Naturforsch. 13a (1958) 988. 3) D. E. Polk, J. Non-Crystalline Solids 5 (1971) 365. 4) N. J. Shevchik and W. Paul, J. Non-Crystalline Solids 8--10 (1972) 381. 5) D. Henderson, J. A. Barker and S. Kim, Intern. J. Quantum Chem. 3S (1969) 265.
S I M U L A T I O N OF T H E A T O M I C A R R A N G E M E N T S I N A M O R P H O U S S I L I C O N AND G E R M A N I U M DOUGLAS HENDERSON and FRANK HERMAN IBM Research Laboratory, San Jose, California, U.S.A. The atomic arrangements in amorphous germanium and silicon are simulated by means of a computer program. In this treatment a disordered system of 64 atoms in a cubic box with periodic boundary conditions is taken as the initial configuration. Each atom is considered in turn and the four nearest and twelve second-nearest neighbors are moved radially towards the nearest and next nearest neighbor distances. The resulting radial distribution function is in good agreement with experiment. We are engaged in a theoretical study of the electronic structure of amorphous silicon and germanium. Our program consists of two stages; firstly, to simulate the atomic arrangements in these materials and, secondly, to use these atomic coordinates to calculate the resulting electronic structure. Experimental information regarding the structure of a material is obtained from scattering measurements of the radial distribution function (RDF). This function, which we denote by g (R), is the ratio of the local number density of atoms at a distance R from a given atom to the average number density of atoms, p = N / V , where N is the number of atoms in a volume V of the material. The experimental R D F , measured by Moss and Graczykl), of crystalline Si and of a vapor-deposited thin film of amorphous Si are shown in fig. 1. The quantity Ro=2.35 A is the nearest-neighbor distance in the crystal. A m o r p h o u s Si, although disordered, shows considerable structure. Just as in the crystal, there are four nearest neighbors and twelve second nearest neighbors around each a t o m at average distances of R o and x/(8/3)Ro, respectively. In addition, from low angle scattering data Moss and Graczyk conclude that the density of fully-annealed amorphous Si is very nearly equal to that of crystalline Si. The major difference between the two R D F ' s is the absence of a third neighbor peak. Similar conclusions have been made for amorphous Ge 2). Of the above conclusions, perhaps the most controversial is that concerning the density. Samples of amorphous Si and Ge are less dense than the crystal. The density of a sample of amorphous solid depends on its history and is thus not an unambiguous number. However, the model-building of Polk 3) as 359
360
D. HENDERSON AND F. HERMAN 80
0¢ ¢n
/1
40
amorphous
40
/J J
crystal
0
1
2
R/Ro Fig. I.
Radial distribution f u n c t i o n s o f crystalline a n d a m o r p h o u s Si (taken from ref. 1).
well as our own results support the view that fully-annealed amorphous Si and Ge have essentially the same density as their crystalline phases and that amorphous samples of lower density contain such imperfections as voids or unsatisfied bonds. A fully-annealed amorphous solid is, of course, an idealization never fully realized in practice. However, it is an appropriate starting point for theoretical studies just as the ideal crystal or the ideal gas of hard spheres are appropriate starting points in the theory of crystalline solids and of liquids, respectively. One interpretation, which we support, of the structure of amorphous Si and Ge, is that their structure is that of a random tetrahedral network in which each atom is linked to four nearest neighbors. The central atom and its four neighbors form a distorted tetrahedron. However, the tetrahedral bonds on neighbors are randomly oriented with respect to each other and any orientation including the extreme eclipsed and staggered configurations are possible. Thus, the first two peaks of the crystal R D F are retained in the amorphous R D F whereas the third peak in the crystal is absent. Amorphous Si and Ge although lacking a crystal structure have more structure than a liquid and cannot be considered merely as quenched liquids.
SIMULATION OF THE ATOMIC ARRANGEMENTS
36l
Indeed liquid Si and Ge are metallic and demonstrate a coordination greater than four. The R D F of a liquid shows only a well-defined peak for nearest neighbors and little structure for larger R. The concept of a network is not useful in liquid studies. Polk 3) has built a model random tetrahedral network. We have attempted a similar study using a computer simulation. Our simulation is similar to that of Shevchik4). However, there are slight differences. He has used a finite static cluster of atoms and has built his cluster in a manner closely analogous to the actual deposition of an amorphous film. That is, his cluster is less dense than the crystal and contains a small number of unsatisfied bonds. On the other hand, our model simulates the combined effects of structural and thermal disorder. In addition, we have used a relatively small cluster of atoms in a box with periodic boundary conditions and have required that all bonds be satisfied. At first sight the use of periodic boundary might appear to be incompatible with a realistic structure of an amorphous solid unless the unit cell is extremely large. However, computer simulations of the structure of liquids have shown that unit cells containing as few as 32 molecules give realistic resultsS). To be sure, it may turn out that much larger unit cells are required to give reasonable results for the electronic structure amorphous Si and Ge. However, our present results show that the R D F of these materials can be simulated with a small number of atoms. Our scheme is very simple. Firstly, we locate N atoms in the diamond structure in a unit cell of dimension:
L = x/3
Ro,
(1)
where p and Pc are the number densities of the amorphous material and its crystal. The value of p is part of the imput data. Usually, we have used P=Pc. However, we have considered other values. In the present case we used N = 64. It may turn out that smaller values of N may be used but we have not yet attempted this. The purpose of this initial step is only to label the atoms and define L and certain other parameters. The second step is to disrupt the crystal structure by moving the atoms at random. The third and key part of the program now follows. Each atom is considered in turn and moved ten percent of the distance to the centroid of its four nearest neighbors. Then the four nearest neighbors and the twelve next nearest neighbors are moved radially ten percent of the distance to the nearest and next nearest neighbor distances of the crystal. Going once around the box means that each atom has been moved back
D. HENDERSON AND F. HERMAN
362
and forth sixteen times. Thus, the positions are only slightly more realistic than the starting arrangement. However, the process can be repeated. A satisfactory structure can be found after a few seconds of computing with the IBM 360/195 computer. No appreciable change is observed after 15 min of additional computing. As may be seen in fig. 2, the resulting R D F is in excellent agreement with that of amorphous Si. We have made runs for other densities and have found that very similar RDF's can be obtained even if p differs from Pc by as much as four percent. This is not surprising in view of the fact that the densities of amorphous Si and Ge are not well-defined. Strictly speaking, one must be careful in using our R D F for R>½L because of the periodic boundary conditions which we have employed. In view of this we have not plotted values of g (R) in fig. 2 for R > ½ L. However, we have looked at our g (R) for larger values of R and have found peaks at about 2.45-2.55 R o, 3.20-3.40 Ro, and 4.054.15 R o. This is in reasonable agreement with experimental results 2). On the basis of these studies we conclude that the structure amorphous Si
80
I
I
I /
c~ o et-
,/./
//
/
///
40
%
-
//
//
/ ///////~ ,0
.s / /
......
,t
f 1
l, 2
R/No Fig. 2. Radial distribution functions. The lower curve gives the experimental results of Moss and Graczyk z) and the upper histogram gives the results of our simulation.
SIMULATION OF THE ATOMIC ARRANGEMENTS
363
and Ge is that of a random tetrahedral network. Similar conclusions have been reached in the complementary studies of Shevchik4). Acknowledgement The authors are grateful to J. A. Barker and N. J. Shevchik for stimulating discussions. References 1) S.C. Moss and J. F. Graczyk, in Proc. Tenth Intern. Conf. Physics of Semiconductors, USAEC (1970) p. 658. 2) V. H. Richter and G. Breitling, Z. Naturforsch. 13a (1958) 988. 3) D. E. Polk, J. Non-Crystalline Solids 5 (1971) 365. 4) N. J. Shevchik and W. Paul, J. Non-Crystalline Solids 8--10 (1972) 381. 5) D. Henderson, J. A. Barker and S. Kim, Intern. J. Quantum Chem. 3S (1969) 265.