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First principles modeling of intermediate range order in amorphous SiSe2
Computational Materials Science 33 (2005) 106–111 www.elsevier.com/locate/commatsci
First principles modeling of intermediate range order in amorphous SiSe2 Massimo Celino
a,*
, Carlo Massobrio
b
a
b
Ente per le Nuove Tecnologie, l’Energia e l’Ambiente, UTS Materiali e Nuove Tecnologie, C.R. Casaccia, CP 2400, I-00100 Roma, Italy Institut de Physique et de Chimie des Mate´riaux de Strasbourg, 23 rue du Loess, F-67037 Strasbourg, France
Abstract The structural properties of amorphous SiSe2 are studied by performing extensive first-principle molecular dynamics simulations. Comparison with experiments in terms of total neutron scattering factor and nuclear magnetic resonance data confirms the reliability of our model. Partial pair correlation functions and ring statistics are used to identify the structural units whose spatial organization gives raise to the intermediate range order. An accurate analysis reveals that amorphous SiSe2 is largely a chemically ordered system consisting mainly of a network of silicon centered tetrahedra. Different types of connections among the tetrahedra can be identified and classified. In particular edge-sharing connections seem to play a fundamental role in the establishment of the intermediate range order and in the appearance of the first sharp diffraction peak in the total neutron scattering factor. Ó 2004 Elsevier B.V. All rights reserved. PACS: 61.43.Dq Keywords: Amorphous; Intermediate range order; Molecular dynamics
1. Introduction A renewed interest exists in amorphous semiconductors and their applications, from both fundamental and applied points of view. However, * Corresponding author. Tel.: +39 063048 3871; fax: +39 063048 4230. E-mail address: [email protected] (M. Celino).
identifying the microscopic features relevant at the macroscopic level is by far not straightforward for these systems. Typical amorphous materials are characterized by short range order (SRO), meaning that the order does not extend beyond nearest neighbour distances. On the contrary, structural organization involving larger distances is commonly termed intermediate range order (IRO) and is a characteristic feature of networkforming liquid and amorphous materials [1]. The
0927-0256/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.12.044
M. Celino, C. Massobrio / Computational Materials Science 33 (2005) 106–111
first sharp diffraction peak (FSDP) in the total neutron structure factor is an unambiguous fingerprint of IRO. Representative materials belonging to this class are AX2 oxides and chalcogenides (A = Ge, Si; X = O, S, Se). A large set of results have appeared in literature both to describe in detail the atomic structure and to propose models able to account for the IRO [2–7]. Amorphous SiSe2 (a-SiSe2) can be considered a prototype of this class of networks. Its crystalline structure is exclusively composed by infinite parallel chains of edge-sharing (ES) tetrahedra [8]. The extent of the similarities between crystalline and amorphous structure is one of the interesting points to be addressed by first-principles modeling. In the disordered phase, a-SiSe2 appears to be composed by a combination of edge-sharing and corner-sharing (CS) tetrahedra [9–12], as established via Raman and diffraction data. In particular spinning-nuclear magnetic resonance (MAS–NMR) experiments have identified and quantified three different Si sites from which it is possible to extract the percentage of ES and CS tetrahedra in the atomic structure [4]. Molecular dynamics simulations, based on a classical scheme, have confirmed the presence of both CS and ES tetrahedra, even if they are not able to give the right proportions [5]. Jackson has interpreted the Raman spectra of a-SiSe2 on the basis of a first-principles cluster model approach, by focusing on isolated molecular entities which are expected to exist in the real glass [6]. In this paper a full description of the atomic structure of a-SiSe2 is provided by means of extensive first-principle Molecular Dynamics simulations. The structure is described in terms of rings and sequences of tetrahedra enlighting the underlying network. The reliability of our model is confirmed by the good agreement with experimental results in terms of both total neutron structure factor and MAS–NMR data.
2. Computational details The first-principles molecular dynamics framework [13] is applied to simulate liquid and amorphous SiSe2. The self consistent evolution of the
107
electronic structure during the motion is described within density functional theory [14]. We adopted a generalized gradient approximation (GGA) for the exchange and correlation part of the total energy [15] and norm conserving pseudopotentials are used for the core-valence interactions. A cutoff energy of 20 Ry yields accurate and converged properties for the dimer SiSe as well as for crystalline SiSe2. Amorphous structures have been obtained for two systems consisting of stoichiometric compositions of N = 120 and N = 144 atoms in ˚ periodically repeated cubic cells of size 15.61 A ˚ respectively. The density is equal to and 16.56 A the experimental value for a-SiSe2 at T = 300 K [10]. To ensure that our results are independent on the initial atomic configurations, they are chosen to be drastically different. For the N = 120 system, atoms are positioned randomly in the simulation box. We further disordered the configuration by equilibration at high temperature (T = 4000 K). Then we gradually lowered it to 1000 K in 15 ps before an additional equilibration for additional 9 ps. On the other hand, the initial configuration for N = 144 was obtained by adapting the crystal structure of SiSe2 to the cubic symmetry. This system is melted and thermalized at T = 1000 K for 15 ps. Both liquids SiSe2 (1-SiSe2) are then quenched from T = 1000 K to T = 300 K. Data collection takes place at thermal equilibrium over periods of 5 ps for each set of calculations. From these liquids we have generated, by quenching procedure, six amorphous atomic structures. In the following we report numerical results averaged on all the configurations.
3. Results The calculated neutron scattering structure factor is in very good agreement with experiment [10] (Fig. 1). Consideration of statistical errors allows to conclude that the position and the height of the peaks and minima are well reproduced over the entire k range. A first approach to the study of the atomic structure can be provided by the computation of the pair correlation functions gSiSi (r), gSiSe (r) and gSeSe (r) (see Fig. 2). The gSiSe (r) is characterized by a high
108
M. Celino, C. Massobrio / Computational Materials Science 33 (2005) 106–111
Fig. 1. Comparison between experimental and numerical total neutron scattering factor.
Fig. 2. Partial pair correlation functions gab of the amorphous system. Sticks in bold indicate the positions for the crystalline structure.
˚ , indicating the prepeak at the distance of 2.30 A dominance of nearest neighbor bonds between Si and Se, as in the crystalline structure. Integrating
under the area of this peak, the coordination number nSiSe is equal to 3.86. Heteropolar bonding is known as highly favored in a-SiSe2 [16]. In the pair correlation function gSeSe (r) the main peak is due to intratetrahedral second neighbors Se–Se distances. On the other hand, a small peak found at ˚ , is indicative of homopolar Se–Se bonding. 2.4 A This accounts for 2.5% of bonds involving Se atoms, the coordination number nSeSe being equal to 0.10. In the partial pair correlation gSiSi (r) the ˚ can be attributed to ES tetrahedra as peak at 3 A shown by the calculation of gSiSi (r) including only Si atoms in ES configurations as it will be shown in the following paragraphs. The feature dis˚ and 2.6 A ˚ is due to Si–Si cernible between 2.4 A homopolar bonds, each Si atom having on average nSiSi = 0.06 nearest neighbors of the same ˚. species, i.e. within a given shell of radius 2.6 A Remembering that the partial coordination of each atom is nSi = nSiSi + nSiSe and nSe = nSeSe + nSeSi, we obtain partial coordination numbers nSi = 3.95 and nSe = 2.04 very close to the values associated to a perfect chemically ordered tetrahedral network (CON) (nSi(CON) = 4, nSe(CON) = 2). This can be verified also by a closer inspection of the atomic structure by computing directly the coordination of each atom. The number of homopolar bonds Si–Si is less than 1% of the total number of bonds in the systems, in agreement with experimental findings. For what concern Se atoms, almost the totality of Se atoms are 2-fold coordinated. The bond angle distribution confirms (Fig. 3) the picture extracted by the pair correlation functions. In the case of Si–Se–Si only the peak at 80° is present in the crystal and it can be easily associated to angles due to ES connections between tetrahedra. The second peak at 109° is due to CS intertetrahedra connections that are not present in the crystal. The relative heights gives the percentage of ES and CS connections in the amorphous systems. The number of ES connections are about twice the number of CS connections. For what concern the Se–Si–Se the first peak is at 100°, remindful of the crystalline structure. The second peak at 109° can be viewed as the superposition of angles characteristic of the crystalline structure where tetrahedra structures are predominant features.
M. Celino, C. Massobrio / Computational Materials Science 33 (2005) 106–111
109
Table 1 Percentages of Si atoms which do not belong to fourfold rings (Si(0)) which belong to one fourfold ring (Si(1)) and to two fourfold rings (Si(2))
Si(0) Si(1) Si(2)
CMD-a-SiSe2
ex-a-SiSe2
a-SiSe2
48 46 6
26 52 22
30 61 9
Our results (a-SiSe2) are listed together with results from experiments (ex-a-SiSe2) (Ref. [4]), and classical molecular dynamics (CMD-a-SiSe2) (Ref. [5]).
Fig. 3. Bond angle distribution of the amorphous systems. Bold sticks indicate crystalline values.
We have computed the distribution of rings focusing our attention on 4-fold rings from which we are able to understand how many CS and ES Si atoms are present in the a-SiSe2. The environment of Si atoms can be divided in three classes: 1. Si atoms that participate in only CS configurations. These Si atoms do not belong to any 4-fold ring. We can call them Si(0). 2. Si atoms being part of only one 4-fold ring. We can call them Si(1). 3. Si atoms belonging to two 4-fold rings. We can call them Si(2). We report the relative population of Si(0), Si(1) and Si(2) atoms and we compare them with experimental data [4] and with molecular dynamics results based on classical potential [5]. As shown in Table 1 our results reproduce better the experimental results than classical MD simulations. In particular the values for Si(0) and the total number of edge-sharing Si atoms, Si(1) + Si(2) improve significantly upon the results obtained with classical potentials. Since by using the ring statistics we can determine if a Si atom is of the type Si(0), Si(1) or Si(2), we are able to discriminate, in the computation of the pair correlation functions and bond angle distribution, the contributions of
the ES and CS Si atoms (Fig. 4). In the upper part we observe that ES Si atoms contribute to the appearance of the first and the second peak. On the contrary the third peak is due to the presence of CS Si absent in the crystal. In the lower part of Fig. 4 the ES and CS contributions to the bond angle distributions are reported. In particular, it is possible to assign, in the Si–Se–Si distribution, the peak at 80° to the presence of ES, while the second peak at 109° is related to contributions of CS atoms. For what concern the Se–Si–Se distribution, the double peak found in Fig. 3 is due to ES atoms and not to CS atoms. To get more insight in the contribution of the different subunits to the presence of the FSDP in
Fig. 4. Pair correlation distribution function Si–Si and bond angle distributions are decomposed in the contributions from edge-sharing and corner-sharing Si atoms.
110
M. Celino, C. Massobrio / Computational Materials Science 33 (2005) 106–111 Table 2 Number of triads centered on the Si(0), Si(1)) and Si(2) types of Si atoms x-Si(2)-x 1-2-1 2-2-2 0-2-2 1-2-2 0-2-0 0-2-1
x-Si(1)-x 5 0 0 1 0 1
2-1-1 2-1-0 2-1-2 1-1-0 1-1-1 0-1-0
x-Si(0)-x 6 7 1 24 21 6
0-0-0 1-0-0 1-0-1 2-0-2 2-0-1 2-0-0
9 31 14 0 1 0
Configurations occurring via a threefold Se atom are given in bol. Ô2Õ, Ô1Õ, Ô0Õ stand for Si(2), (Si(1)) and Si(0) respectively.
Fig. 5. Magnified view of the FSDP region of the total neutron structure factor Stot computed with different sets of Si atoms.
the total neutron structure factor, we have calculated the neutron structure factor considering only selected sets of Si atoms. Since we can distinguish Si atoms of the type Si(0), Si(1) and Si(2), we have computed the neutron structure factors taking into account all the Se atoms and only predefined sets of Si atoms. We have defined 5 sets of Si atoms: (1) Si(1) + Si(2), (2) Si(0) + Si(1), (3) Si(0) + Si(2), (4) Si(1) and (5) Si(0). The results are reported in Fig. 5 where a magnified view in the FSDP region is presented for the total neutron structure factor. The most important contributions come from structure factors computed by using both Si(1) and Si(2) silicon atoms. However, also the structure factors computed only on Si(1) atoms give a good estimate of the FSDP height. This suggests an interesting correlation between the establishment of intermediate range order and the number of edge-sharing connections, at least in the present system. Having analyzed the amorphous system in terms of rings, let us now examine if there are chainlike molecular fragments. In our calculation, first we determine the Se atoms which are common nearest neighbors of Si atoms and then we account for all possible paths connecting three Si atoms
through shared Se atoms. As shown in Table 2, the highest value for Si triads is found in the case of Si(1)–Si(0)–Si(0) and Si(1)–Si(1)–Si(0), followed by Si(1)–Si(1)–Si(1) and Si(1)–Si(0)–Si(1). These configurations arise from CS and ES connections and do not necessarily involve any miscoordination. Focusing on the Si(2)-centered triads, Si(1)– Si(2)–Si(1) is by far the most frequent, while Si(2)–Si(2)–Si(2) chains are absent. Among the 18 sequences of three neighboring Si atoms, 6 are made possible by the presence of threefold Se atoms, corresponding to two adjacent fourfold rings sharing one Si–Se bond. However, the number of these configurations does not exceed 1. Overall, the amount of the various Si(0)- and Si(1)-centered triads indicates that the dominant structural motifs in a-SiSe2 contain both CS and ES tetrahedra. Therefore, the count of sequences expected to characterize the network cannot a priori be restricted to only one kind (CS or ES) of intertetrahedral connection but it has to include combinations of both. In the end, we conclude this section studying chains longer than triads in a-SiSe2. In Table 3 are reported the number of chains composed by four interconnected tetrahedra by focusing on those featuring Si(2) atoms and a series of Si(1) atoms. The only chains that can be found are 1-1-2-1 and 1-1-1-1. These results are consistent with the numbers reported in Table 2 where 1-21 and 1-1-1 are among the most abundant structures for triads not involving Si(0) atoms. We notice that the 2-2-2-2 chain, remindful of the crystalline structure is absent, as expected since no 2-2-2 triads were formed (see Table 2).
M. Celino, C. Massobrio / Computational Materials Science 33 (2005) 106–111 Table 3 Number of chains in the amorphous SiSe2 composed by four Si tetrahedra centered on the Si(0), (Si(1)) and Si(2) types of Si atoms Chains
a-SiSe2
1-2-2-1 1-2-2-0 0-2-2-1 1-1-2-1 1-1-1-1 1-2-2-2 2-2-2-2
0 0 0 5 12 0 0
4. Conclusions Disordered network-forming materials are challenging system for atomic-scale simulations, since the delicate balance of ionic and covalent bonding requires appropriate first-principles treatment of the electronic structure. The case of amorphous SiSe2 is a revealing example of the predictive power of ab-initio based approaches. We obtained correct proportions of the basic structural units composing the network, by improving substantially over the classical molecular dynamics methods. Concerning intermediate range order, we performed an analysis of the contributions associated with the different kinds of tetrahedral connections. This shows an interesting correlation between the appearance of the first sharp diffraction peak and the edge-sharing connections.
111
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13] [14]
[15]
[16]
S.R. Elliott, Nature 354 (1991) 445. P. Boolchand, W.J. Bresser, Nature 410 (2001) 1070. L.F. Gladden, S.R. Elliott, Phys. Rev. Lett. 59 (1987) 908. M. Tenhover, R.D. Boyer, R.S. Henderson, T.E. Hammond, G.A. Shave, Solid State Commun. 65 (1988) 1517. G.A. Antonio, R.K. Kalia, A. Nakano, P. Vashishta, Phys. Rev. B 45 (1992) 7455. K. Jackson, S. Grossman, Phys. Rev. B 65 (2001) 012206. M. Celino, C. Massobrio, Phys. Rev. Lett. 90 (2003) 125502. J. Peters, B. Krebs, Acta Crystallogr. B 38 (1982) 1270. J.E. Griffiths, M. Malyj, G.P. Espinosa, J.P. Remeika, Solid State Commun. 53 (1985) 587. R.W. Johnson, D.L. Price, S. Susman, M. Arai, T.I. Morrison, G.K. Shenoy, J. Non-Cryst. Solids 83 (1986) 251. M. Arai, D.L. Price, S. Susman, K.J. Volin, U. Walter, Phys. Rev. B 37 (1988) 4240. D. Selvanathan, W.J. Bresser, P. Boolchand, Phys. Rev. B 61 (2000) 15061. R. Car, M. Parrinello, Phys. Rev. Lett. 55 (1985) 2471. The simulations were performed using for norm-conserving pseudopotentials the computer program described in A. Pasquarello, K. Laasonen, R. Car, C. Lee, D. Vanderbilt, Phys. Rev. Lett. 69 (1992) 1982, and in K. Laasonen, A. Pasquarello, R. Car, C. Lee, D. Vanderbilt, Phys. Rev. B 47 (1993) 10142. J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. R.W. Johnson, D.L. Price, S. Susman, M. Arai, T.I. Morrison, G.K. Shenoy, The structure of silicon selenium glasses, J. Non-Cryst. Solids 83 (1986) 251.
First principles modeling of intermediate range order in amorphous SiSe2 Massimo Celino
a,*
, Carlo Massobrio
b
a
b
Ente per le Nuove Tecnologie, l’Energia e l’Ambiente, UTS Materiali e Nuove Tecnologie, C.R. Casaccia, CP 2400, I-00100 Roma, Italy Institut de Physique et de Chimie des Mate´riaux de Strasbourg, 23 rue du Loess, F-67037 Strasbourg, France
Abstract The structural properties of amorphous SiSe2 are studied by performing extensive first-principle molecular dynamics simulations. Comparison with experiments in terms of total neutron scattering factor and nuclear magnetic resonance data confirms the reliability of our model. Partial pair correlation functions and ring statistics are used to identify the structural units whose spatial organization gives raise to the intermediate range order. An accurate analysis reveals that amorphous SiSe2 is largely a chemically ordered system consisting mainly of a network of silicon centered tetrahedra. Different types of connections among the tetrahedra can be identified and classified. In particular edge-sharing connections seem to play a fundamental role in the establishment of the intermediate range order and in the appearance of the first sharp diffraction peak in the total neutron scattering factor. Ó 2004 Elsevier B.V. All rights reserved. PACS: 61.43.Dq Keywords: Amorphous; Intermediate range order; Molecular dynamics
1. Introduction A renewed interest exists in amorphous semiconductors and their applications, from both fundamental and applied points of view. However, * Corresponding author. Tel.: +39 063048 3871; fax: +39 063048 4230. E-mail address: [email protected] (M. Celino).
identifying the microscopic features relevant at the macroscopic level is by far not straightforward for these systems. Typical amorphous materials are characterized by short range order (SRO), meaning that the order does not extend beyond nearest neighbour distances. On the contrary, structural organization involving larger distances is commonly termed intermediate range order (IRO) and is a characteristic feature of networkforming liquid and amorphous materials [1]. The
0927-0256/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.12.044
M. Celino, C. Massobrio / Computational Materials Science 33 (2005) 106–111
first sharp diffraction peak (FSDP) in the total neutron structure factor is an unambiguous fingerprint of IRO. Representative materials belonging to this class are AX2 oxides and chalcogenides (A = Ge, Si; X = O, S, Se). A large set of results have appeared in literature both to describe in detail the atomic structure and to propose models able to account for the IRO [2–7]. Amorphous SiSe2 (a-SiSe2) can be considered a prototype of this class of networks. Its crystalline structure is exclusively composed by infinite parallel chains of edge-sharing (ES) tetrahedra [8]. The extent of the similarities between crystalline and amorphous structure is one of the interesting points to be addressed by first-principles modeling. In the disordered phase, a-SiSe2 appears to be composed by a combination of edge-sharing and corner-sharing (CS) tetrahedra [9–12], as established via Raman and diffraction data. In particular spinning-nuclear magnetic resonance (MAS–NMR) experiments have identified and quantified three different Si sites from which it is possible to extract the percentage of ES and CS tetrahedra in the atomic structure [4]. Molecular dynamics simulations, based on a classical scheme, have confirmed the presence of both CS and ES tetrahedra, even if they are not able to give the right proportions [5]. Jackson has interpreted the Raman spectra of a-SiSe2 on the basis of a first-principles cluster model approach, by focusing on isolated molecular entities which are expected to exist in the real glass [6]. In this paper a full description of the atomic structure of a-SiSe2 is provided by means of extensive first-principle Molecular Dynamics simulations. The structure is described in terms of rings and sequences of tetrahedra enlighting the underlying network. The reliability of our model is confirmed by the good agreement with experimental results in terms of both total neutron structure factor and MAS–NMR data.
2. Computational details The first-principles molecular dynamics framework [13] is applied to simulate liquid and amorphous SiSe2. The self consistent evolution of the
107
electronic structure during the motion is described within density functional theory [14]. We adopted a generalized gradient approximation (GGA) for the exchange and correlation part of the total energy [15] and norm conserving pseudopotentials are used for the core-valence interactions. A cutoff energy of 20 Ry yields accurate and converged properties for the dimer SiSe as well as for crystalline SiSe2. Amorphous structures have been obtained for two systems consisting of stoichiometric compositions of N = 120 and N = 144 atoms in ˚ periodically repeated cubic cells of size 15.61 A ˚ respectively. The density is equal to and 16.56 A the experimental value for a-SiSe2 at T = 300 K [10]. To ensure that our results are independent on the initial atomic configurations, they are chosen to be drastically different. For the N = 120 system, atoms are positioned randomly in the simulation box. We further disordered the configuration by equilibration at high temperature (T = 4000 K). Then we gradually lowered it to 1000 K in 15 ps before an additional equilibration for additional 9 ps. On the other hand, the initial configuration for N = 144 was obtained by adapting the crystal structure of SiSe2 to the cubic symmetry. This system is melted and thermalized at T = 1000 K for 15 ps. Both liquids SiSe2 (1-SiSe2) are then quenched from T = 1000 K to T = 300 K. Data collection takes place at thermal equilibrium over periods of 5 ps for each set of calculations. From these liquids we have generated, by quenching procedure, six amorphous atomic structures. In the following we report numerical results averaged on all the configurations.
3. Results The calculated neutron scattering structure factor is in very good agreement with experiment [10] (Fig. 1). Consideration of statistical errors allows to conclude that the position and the height of the peaks and minima are well reproduced over the entire k range. A first approach to the study of the atomic structure can be provided by the computation of the pair correlation functions gSiSi (r), gSiSe (r) and gSeSe (r) (see Fig. 2). The gSiSe (r) is characterized by a high
108
M. Celino, C. Massobrio / Computational Materials Science 33 (2005) 106–111
Fig. 1. Comparison between experimental and numerical total neutron scattering factor.
Fig. 2. Partial pair correlation functions gab of the amorphous system. Sticks in bold indicate the positions for the crystalline structure.
˚ , indicating the prepeak at the distance of 2.30 A dominance of nearest neighbor bonds between Si and Se, as in the crystalline structure. Integrating
under the area of this peak, the coordination number nSiSe is equal to 3.86. Heteropolar bonding is known as highly favored in a-SiSe2 [16]. In the pair correlation function gSeSe (r) the main peak is due to intratetrahedral second neighbors Se–Se distances. On the other hand, a small peak found at ˚ , is indicative of homopolar Se–Se bonding. 2.4 A This accounts for 2.5% of bonds involving Se atoms, the coordination number nSeSe being equal to 0.10. In the partial pair correlation gSiSi (r) the ˚ can be attributed to ES tetrahedra as peak at 3 A shown by the calculation of gSiSi (r) including only Si atoms in ES configurations as it will be shown in the following paragraphs. The feature dis˚ and 2.6 A ˚ is due to Si–Si cernible between 2.4 A homopolar bonds, each Si atom having on average nSiSi = 0.06 nearest neighbors of the same ˚. species, i.e. within a given shell of radius 2.6 A Remembering that the partial coordination of each atom is nSi = nSiSi + nSiSe and nSe = nSeSe + nSeSi, we obtain partial coordination numbers nSi = 3.95 and nSe = 2.04 very close to the values associated to a perfect chemically ordered tetrahedral network (CON) (nSi(CON) = 4, nSe(CON) = 2). This can be verified also by a closer inspection of the atomic structure by computing directly the coordination of each atom. The number of homopolar bonds Si–Si is less than 1% of the total number of bonds in the systems, in agreement with experimental findings. For what concern Se atoms, almost the totality of Se atoms are 2-fold coordinated. The bond angle distribution confirms (Fig. 3) the picture extracted by the pair correlation functions. In the case of Si–Se–Si only the peak at 80° is present in the crystal and it can be easily associated to angles due to ES connections between tetrahedra. The second peak at 109° is due to CS intertetrahedra connections that are not present in the crystal. The relative heights gives the percentage of ES and CS connections in the amorphous systems. The number of ES connections are about twice the number of CS connections. For what concern the Se–Si–Se the first peak is at 100°, remindful of the crystalline structure. The second peak at 109° can be viewed as the superposition of angles characteristic of the crystalline structure where tetrahedra structures are predominant features.
M. Celino, C. Massobrio / Computational Materials Science 33 (2005) 106–111
109
Table 1 Percentages of Si atoms which do not belong to fourfold rings (Si(0)) which belong to one fourfold ring (Si(1)) and to two fourfold rings (Si(2))
Si(0) Si(1) Si(2)
CMD-a-SiSe2
ex-a-SiSe2
a-SiSe2
48 46 6
26 52 22
30 61 9
Our results (a-SiSe2) are listed together with results from experiments (ex-a-SiSe2) (Ref. [4]), and classical molecular dynamics (CMD-a-SiSe2) (Ref. [5]).
Fig. 3. Bond angle distribution of the amorphous systems. Bold sticks indicate crystalline values.
We have computed the distribution of rings focusing our attention on 4-fold rings from which we are able to understand how many CS and ES Si atoms are present in the a-SiSe2. The environment of Si atoms can be divided in three classes: 1. Si atoms that participate in only CS configurations. These Si atoms do not belong to any 4-fold ring. We can call them Si(0). 2. Si atoms being part of only one 4-fold ring. We can call them Si(1). 3. Si atoms belonging to two 4-fold rings. We can call them Si(2). We report the relative population of Si(0), Si(1) and Si(2) atoms and we compare them with experimental data [4] and with molecular dynamics results based on classical potential [5]. As shown in Table 1 our results reproduce better the experimental results than classical MD simulations. In particular the values for Si(0) and the total number of edge-sharing Si atoms, Si(1) + Si(2) improve significantly upon the results obtained with classical potentials. Since by using the ring statistics we can determine if a Si atom is of the type Si(0), Si(1) or Si(2), we are able to discriminate, in the computation of the pair correlation functions and bond angle distribution, the contributions of
the ES and CS Si atoms (Fig. 4). In the upper part we observe that ES Si atoms contribute to the appearance of the first and the second peak. On the contrary the third peak is due to the presence of CS Si absent in the crystal. In the lower part of Fig. 4 the ES and CS contributions to the bond angle distributions are reported. In particular, it is possible to assign, in the Si–Se–Si distribution, the peak at 80° to the presence of ES, while the second peak at 109° is related to contributions of CS atoms. For what concern the Se–Si–Se distribution, the double peak found in Fig. 3 is due to ES atoms and not to CS atoms. To get more insight in the contribution of the different subunits to the presence of the FSDP in
Fig. 4. Pair correlation distribution function Si–Si and bond angle distributions are decomposed in the contributions from edge-sharing and corner-sharing Si atoms.
110
M. Celino, C. Massobrio / Computational Materials Science 33 (2005) 106–111 Table 2 Number of triads centered on the Si(0), Si(1)) and Si(2) types of Si atoms x-Si(2)-x 1-2-1 2-2-2 0-2-2 1-2-2 0-2-0 0-2-1
x-Si(1)-x 5 0 0 1 0 1
2-1-1 2-1-0 2-1-2 1-1-0 1-1-1 0-1-0
x-Si(0)-x 6 7 1 24 21 6
0-0-0 1-0-0 1-0-1 2-0-2 2-0-1 2-0-0
9 31 14 0 1 0
Configurations occurring via a threefold Se atom are given in bol. Ô2Õ, Ô1Õ, Ô0Õ stand for Si(2), (Si(1)) and Si(0) respectively.
Fig. 5. Magnified view of the FSDP region of the total neutron structure factor Stot computed with different sets of Si atoms.
the total neutron structure factor, we have calculated the neutron structure factor considering only selected sets of Si atoms. Since we can distinguish Si atoms of the type Si(0), Si(1) and Si(2), we have computed the neutron structure factors taking into account all the Se atoms and only predefined sets of Si atoms. We have defined 5 sets of Si atoms: (1) Si(1) + Si(2), (2) Si(0) + Si(1), (3) Si(0) + Si(2), (4) Si(1) and (5) Si(0). The results are reported in Fig. 5 where a magnified view in the FSDP region is presented for the total neutron structure factor. The most important contributions come from structure factors computed by using both Si(1) and Si(2) silicon atoms. However, also the structure factors computed only on Si(1) atoms give a good estimate of the FSDP height. This suggests an interesting correlation between the establishment of intermediate range order and the number of edge-sharing connections, at least in the present system. Having analyzed the amorphous system in terms of rings, let us now examine if there are chainlike molecular fragments. In our calculation, first we determine the Se atoms which are common nearest neighbors of Si atoms and then we account for all possible paths connecting three Si atoms
through shared Se atoms. As shown in Table 2, the highest value for Si triads is found in the case of Si(1)–Si(0)–Si(0) and Si(1)–Si(1)–Si(0), followed by Si(1)–Si(1)–Si(1) and Si(1)–Si(0)–Si(1). These configurations arise from CS and ES connections and do not necessarily involve any miscoordination. Focusing on the Si(2)-centered triads, Si(1)– Si(2)–Si(1) is by far the most frequent, while Si(2)–Si(2)–Si(2) chains are absent. Among the 18 sequences of three neighboring Si atoms, 6 are made possible by the presence of threefold Se atoms, corresponding to two adjacent fourfold rings sharing one Si–Se bond. However, the number of these configurations does not exceed 1. Overall, the amount of the various Si(0)- and Si(1)-centered triads indicates that the dominant structural motifs in a-SiSe2 contain both CS and ES tetrahedra. Therefore, the count of sequences expected to characterize the network cannot a priori be restricted to only one kind (CS or ES) of intertetrahedral connection but it has to include combinations of both. In the end, we conclude this section studying chains longer than triads in a-SiSe2. In Table 3 are reported the number of chains composed by four interconnected tetrahedra by focusing on those featuring Si(2) atoms and a series of Si(1) atoms. The only chains that can be found are 1-1-2-1 and 1-1-1-1. These results are consistent with the numbers reported in Table 2 where 1-21 and 1-1-1 are among the most abundant structures for triads not involving Si(0) atoms. We notice that the 2-2-2-2 chain, remindful of the crystalline structure is absent, as expected since no 2-2-2 triads were formed (see Table 2).
M. Celino, C. Massobrio / Computational Materials Science 33 (2005) 106–111 Table 3 Number of chains in the amorphous SiSe2 composed by four Si tetrahedra centered on the Si(0), (Si(1)) and Si(2) types of Si atoms Chains
a-SiSe2
1-2-2-1 1-2-2-0 0-2-2-1 1-1-2-1 1-1-1-1 1-2-2-2 2-2-2-2
0 0 0 5 12 0 0
4. Conclusions Disordered network-forming materials are challenging system for atomic-scale simulations, since the delicate balance of ionic and covalent bonding requires appropriate first-principles treatment of the electronic structure. The case of amorphous SiSe2 is a revealing example of the predictive power of ab-initio based approaches. We obtained correct proportions of the basic structural units composing the network, by improving substantially over the classical molecular dynamics methods. Concerning intermediate range order, we performed an analysis of the contributions associated with the different kinds of tetrahedral connections. This shows an interesting correlation between the appearance of the first sharp diffraction peak and the edge-sharing connections.
111
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