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THEORETICAL STUDIES OF THE ATOMIC AND ELECTRONIC STRUCTURES OF GRAIN
BOUNDARIES IN SILICON AND SILICON-CARBIDE
M. Kohyama, Y. Ebata, S. Rose, M. Kinoshita, R. Yamamoto
To cite this version:
M. Kohyama, Y. Ebata, S. Rose, M. Kinoshita, R. Yamamoto. THEORETICAL STUDIES OF THE ATOMIC AND ELECTRONIC STRUCTURES OF GRAIN BOUNDARIES IN SILI- CON AND SILICON-CARBIDE. Journal de Physique Colloques, 1990, 51 (C1), pp.C1-209-C1-214.
�10.1051/jphyscol:1990132�. �jpa-00230290�
THEORETICAL STUDIES OF THE ATOMIC AND ELECTRONIC STRUCTURES OF GRAIN BOUNDARIES IN SILICON AND SILICON-CARBIDE
M. KOHYAMA, Y. EBATA, S. KOSE, M. KINOSHITA and R. YAMAMoTo*
Glass and Ceramic Material Department, Government Industrial Research :nstitute, Osaka, 1-8-31, Midorigaoka, Ikeda, Osaka 563, Japan
Research Center of Advanced Science and rechnofogy, University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153, Japan
Abstract - The atomic and electronic structures of a facetted twin boundary in Si and a twin boundary in 6-Sic have been studied theoretically for the first time. The atomic structure of the {211}/{111} facets in Si based on the model by Bourret and Bacmann has been optimised by the Keating model, and the electronic structure has been calculated using a supercell technique and a tight-binding model. It has been found that the distorted region at one of the two types of the facet junctions much influences the electronic structure as compared with the other regions. However, there are no states introduced inside the minimum gap. The atomic and electronic structure of the {l221 1=9 twin boundary in 6-Sic has been calculated using the self-consistent tight-binding method. The calculated boundary energy has shown that the atomic model slmilar to that in Si or Ge can exist stably in &Sic, although the Coulomb interaction energy caused by the presence of the wrong bonds is a large part of the boundary energy. The boundary band structure has no deep states in the gap, although the localised states at the wrong bonds have been found at the band edges.
I. INTRODUCTION
Significant advances have been made in the understanding of atomic and electronic structures of grain boundaries in covalent semiconductors such as Si and Ge. From various experiments /l-3/, observations / 4 / and theoretical calculations / 5 - g / . it can be considered that the frequently observed coincident-site-lattice (CSL) boundaries in Si or Ge are electrically non- active because of atomic reconstruction at the interfaces. We have shown by calculations using the semi-empirical tight-binding method /10/ that there are no deep states in the fundamental gap in the {211} 2=3 boundary and in the {l301 1=5 boundary in Si /g/. In these calculations, we have found some boundary localised states at the band edges, but these states are buried under the bulk band edges of the density of states. It seems that the electrical activity of grain boundaries is associated with the irregularity such as defects induced in the CSL boundaries, irregular boundaries, or segregated impurities. The next step in the study of grain boundaries in semiconductors is to confirm the origins of the observed activity.
Recently, several types of defects in the CSL boundaries .in semiconductors such as dislocations, steps, facets and dissociations have been observed using HREM /4,11,12/.
Nowadays, it is essential to study the structural and electronic properties of these defects in the CSL boundaries as well as the effects of segregation or precipitation of impurities.
In this paper, we have calculated the atomic and electronic structure of the {211}/{111) facets for the first time as the defects in the CSL boundaries in Si.
On the other hand, various properties of ceramics such as sintering, mechanical and electronic properties depend on grain boundaries. Nowadays, it is possible to obtain experimentally valuable information about atomic structures of grain boundaries also in ceramics /13,14/.
However, on the theoretical side, only grain boundaries in ionic ceramics such as MgO and NiO /15/ have been dealt with from a microscopic viewpoint. It is necessary to deal with grain boundaries in covalent ceramics such as Sic and Si3N4 theoretically. As a first step, it is necessary to investigate the CSL boundaries. It is interesting to examine whether the above mentioned features of grain boundaries in covalent semiconductors are applicable to grain boundaries in covalent ceramics. In this paper, we have constructed an atomic model of the (1221 1=9 grain boundary (second-order twin boundary) in 6-Sic and calculated theoretically the stable atomic configuration, boundary energy and electronic structure for the first time.
11. THE {211}/{111) FACETS IN SILICON 11-1. ATOMIC MODEL
The 1=3 first-order twin boundaries often show facetting on {Ill} and {211} planes in polycrystalline Si and Ge /1,4,11,16/. There have been several experiments indicating that the electrical activity is associated with these types of facets or steps /1,16/. We have
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990132
Cl-210 COLLOQUE DE PHYSIQUE
constructed an atomic model of the {211}/{111} microfacets in Si based on the model by Bourret and Bacmann /4/ from the HREM image of a coherent {Ill} step in the {211} C=3 twin boundary in Ge as shown in Fig. 1. It is interesting that complete reconstruction is possible also at the junctions of the {l111 and {211] planes by introducing recostruction schemes along the to111 axis. There are two types of the facet junctions. One of these two contains at A in Fig. 1 a particular reconstruction scheme, the D1+S reconstruction /4/, which introduce a large shear around itself. The other junction contains at F in Fig.1 an ordinary reconstruction scheme, the D1 reconstruction, as well as in the {211) segment. It is important to examine the structural and electronic properties of the regions of the facet-junctions because the properties of the {211) and the {Ill) segments themselves have already been examined /7-9/.
Fig. 1. t0111 projection of the atomic model of the {211)/{111}
facets in Si. Atoms represented by open and closed circles are displaced from each other by about /%i0/4 along the [0111 direction. The structural units at the interface are indicated by the symbols proposed by Papon and Petit. Atoms indicated by A, B, C, D, E and F constitute the reconstructed bonds along the toll! direction.
11-2. METHOD OF CALCULATION
We have constructed a supercell geometry for calculation of the electronic structure as shown in Fig. 1. Periodicity from 0 to 0' is imposed along this direction. The {211} segment contains one and a half periods of the cm(2x2) strucrture /4/. An additional periodicity is imposed by arranging other facetted twin boundaries with inversion symmetry alternately. The symmetric point is indicated by S in Fig. 1. Each unit cell contains 704 atoms. Of course, the present supercell geometry has been selected by' the restriction of computing power. The effects of the sizes of the respective segments and the distance between the neighboring symmetric facetted twins are important in connection with the elastic fields caused by the different rigid body translations of the {211} and {l111 interfaces, and this problem should be investigated in future.
Because of the large number of the atoms, the optimum atomic structure in the supercell geometry was determined by energy-minimisation calculations in the Keating model /17/. The optimum rigid body translation between the two crystals constitutinq the facetted twin was also determined. The electronic structure has been calculated using a first-neighbor tight- binding model with parameters given by Chadi /10/, which provide a reasonable description of the electronic structure of bulk Si. These parameters were assumed to have a d-2 dependence on the nearest-neighbor distance d. The two-dimensional periodicity parallel to the (011) atomic plane in the present supercell geometry, that is the X-z plane in the present notation, is fairly large as compared with the period along the [Oll) axis. Thus, the band structure along the [0111 direction has been calculated for several k points with k,=k,=O.
11-3. RESULTS AND DISCUSSION
The rigid body translation along the [ilil direction inherent in the {211f C=3 twin boundary moves the upper crystal toward the right-hand side against the lower crystal in Fig. 1, but the presence of the {l111 steps decreases this translation. Thus the compressive stress is induced for the {Ill} interfaces. The atomic structure of the boundary in the central region of the I2111 facet is not so different from that qf the straight {211} C=3 boundary in our previous calculation / g / . Local distortion energies around the D1 reconstruction at F in Fig.
1 are in the same range as those in the region of the 12111 segment. However, relatively large distortion energies exist around the D1+S reconstruction at A in Fig. 1 because this type of reconstruction requires a shear along the [Oil] axis. The deviations in bond lengths in this region are in the same range as those in the other regions, but the deviations in bond angles are relatively large in this region. Particularly, the bond-angle deviations around the atom indicated by P in Fig. 1 range from -33.1° to +25.4O and the energy of this atom is
observation is probably connected with the present atomic model where high distortion energy must be introduced at only one side of the facet junctions.
WAVE VECTOR k y WAVE VECTOR kv
Fig. 2. (a) Energy band structure along the to111 direction of the atomic model of the {211)/
{Ill) facets in Si shown in Fig. 1. Calculated eigen states of the 704-atom unit cell are represented. c is J2a0/2. (b) Energy band structure of the atomic model with the
reconstruction at A in Fig. 1 replaced by dangling bonds.
The calculated electronic structure is shown in Fig. 2(a). There are no deep states in accordance with the absence of dangling bonds. However, there exists a doubly degenerated band inside the gap indicated by an arrow. This shallow band is different from the boundary localised states at the band edge in the I2111 x=3 twin boundary found in our previous calculation /g/. By the analysis of the eigen vectors, some states of this band are found to be localised in the vicinity of the facet junction including the D1+S reconstruction. In order to elucidate the origin of this shallow band more clearly, we have calculated the electronic structure of the atomic model where the D1+S reconstruction at A in Fig. 1 is replaced by dangling bonds. This structure has been relaxed and large distortions around the Dl+S reconstruction are removed. The results are shown in Fig. 2(b), Two doubly degenerated dangling bond bands appear in the fundamental gap. But the band indicated by an arrow in Fig.
2(a) does not exist in Fig. 2(b). The other states are not so different from each other in Fig. 2(a) and Fig. 2(b). Thus, it is clear that the shallow band in Fig. 2(a) is connected with localised distortions caused by the D1+S reconstruction at the facet junction.
In this way, it is clear that the disordered region including the Dl+S reconstruction at the facet junction much more perturbs the electronic structure than the {211) and {Ill)
interfaces. However, it should be noted that the shallow band caused by this region exists above the bulk conduction-band minimum and does not lie inside the minimum gap in Fig. 2(a).
Of course, we do not deny a possibility that the energy levels of this band might be changed in calculations by more reliable theoretical schemes than the present tight-binding model, and this band might be connected with band tails observed in polycrystalline semiconductors.
However, the present results agree with the prediction that only very strongly distorted bonds can induce localised states inside the fundamental gap in Si /18/.
111. THE {l221 1=9 GRAIN BOUNDARY IN SILICON-CARBIDE 111-1. ATOMIC MODEL
Recently, Hiraga has obtained a HREM lmage of the 11221 1=9 grain boundary in CVD @-sic /13/.
He has found that there exist coherent matching and periodic structure at the interface. By comparing this image with a HREM image of the same type of boundary in Ge /19/, we have found that there is a great similarity between the two images. It has been already shown that the {l221 1=9 grain boundary in Ge and Si has an atomic structure consisting of zigzag arrangement of 5-membered and 7-membered rings /5,6,19,20/. Thus, a similar atomic model can be
constructed for this type of boundary in &sic. In this model, the interface consists of zigzag arrangement of 5-membered and 7-membered rings and respective atoms constitute covalent
Cl-212 COLLOQUE DE PHYSIQUE
bonds between four neighboring atoms. However, wrong bonds between Si atoms and between C atoms exist alternately at the interface.
111-2. METHOD OF CALCULATION
We have used the self-consistent tight-binding (SCTB) method /21,22/. This method can deal with covalency and ionicity on an equal footing, and is capable of calculating electronic structure, total energy and atomic forces sufficiently rapidly for the lattice relaxation of extended defects. The essence of this method can be described as follows. The tight-binding Hamiltonian is expressed using a local orbital basis set. Charge transfer effects are included self-consistently in the on-site elements as
where i and j denote atoms in the unit cell.
3
is the lattice vector in the periodic system.The first term is the orbital energy of the neutral free atom. Ui is an average of the intra- atomic ~oulomb'integrals of the valence electrons. Zi and Qi are the charge of the ion core and the self-consistent valence electron occupancy. The third term is the electro-static inter-atomic potential at site
Zi.
v(fj+3,Xi) is a effective inter-atomic function, which is given so as to express a simple Coulomb potential for large distances and to include the effects of charge overlap for short distances. The fourth term is the non-orthogonality correction and given by the overlap matrix elements between atomic orbitals. The total energy is given by subtracting the doubly counted electron-electron electrostatic energy from the band structure energy and adding the ion-ion electrostatic energy. The binding energy EB per unit cell, which is the difference between the total energies of the system and the free atoms, can be written as a sum of four terms, Epr,, EMad: Ecov and Eov /22/. Epro is the promotion and intra-atomic Coulomb energy. EMad is the ~nter-atomic Coulomb energy. Ecov is the covalent energy and E,, is the overlap interaction energy, which expresses the increase in the kinetic energy of the electrons upon compression, Atomic forces are also given very easily /22/ via the Hellman-Feynman theorem as well as other tight-binding theories /10/. OneS and three p valence orbitals per atom are included in the basis set. We have determined the parameter values and functional forms in this method and have examined that various structural and electronic properties of B-Sic, Si and diamond can be reasonably reproduce. A weak point is the only qualitative reproduction of the conduction band as well as other tight-binding theories. This problem will be overcome by including excited orbitals in the basis set.
All the calculations are carried out with use of a supercell technique, where periodicity normal to the boundary plane is imposed by stacking symmetric boundary planes alternately.
The band-structure calculation is carried out with respect to a large unit cell containing two symmetric interfaces. Both 80-atom and 144-atom supercells have been used in order to examine the dependence on the size of the cell. The optimum rigid-body translation between the two grains was determined by iterating lattice relaxation. In addition, we have also carried out similar calculations for the similar atomic model of the same type of grain boundary in Si using the same theoretical method for comparison. This structure in Si has already been examined by other theoretical schemes /5.6/.
111-3. RESULTS AND DISCUSSION
o i i
Fig. 3. 10111 projection of relaxed atomic structure of the {l221 Z = 9 grain boundary in B-Sic. The circles and the double circles indicate C atoms and Si atoms, respzctively.
The numbers inside the circles indicate the effecive charges of respective atoms expressed in units of e. The effective charges in the perfect crystal are '0.45e.
types of cells is small. The bond-angle deviations range from -23.1' to 24.1'. The bond- length deviations of normal Si-C bonds range from -2.5% to 2.2%. The length of the Si-Si wrong bond is shorter by 4.3% than that in bulk Si and the lenght of the C-C wrong bond is longer by 3.8% than that in bulk diamond. These values of the deviations of the bond angles and the Si-C bond lengths are in the same range as those in grain boundaries in Si in the present and previous calculations /5,6,8,9/. The static charge fluctuations exist especially around the wrong bonds and are localised within several atomic layers at the interface as shown in Fig. 3. On the other hand, the values of the charge fluctuations in the same boundary in Si are within '0.02e in the present calculation.
Table I shows the calculated total boundary energies per unit cell. By comparing the values of the two types of cells, it is clear that all the values are satisfactorily converged. The second line in Table I shows the energy increases mainly caused by the bond distortions. The third line shows the energy increases mainly caused by the Coulomb interaction. It is clear that the Coulomb energy is a large part of the boundary energy in B-Sic as compared with that in Si. This is caused by the presence of the wrong bonds at the interface. By comparing the interfacial energy with the estimated surface energies of Sic /23/, we can conclude that the present model of the {l221 E 9 boundary in B-Sic possibly exist stably. The present
calculated boundary energy in Si agrees well with the previously calculated values /5,6/.
Table I. Calculated energy values of the l1221 C=9 grain boundaries in B-Sic and Si. Energy increases per unit cell against the values of the perfect crystals are shown. Each unit cell contains two symmetric interfaces in the present supercell calculations. Egb is the
interfacial energy for one interface.
&sic Si
80-atom cell 144-atom cell 80-atom cell 144-atom cell
. -
0 1 2 3 0 0 1 2 3 0
Fig. 4. (a) The energy band structure of the {l221 ,?=9 grain boundary in 6-Sic. Points 0, 1, 2 and 3 are (0, 0, O), (n/R1, 0, 0). (n/R1, n/R2, 02, and (0, ' T / R ~ , 0) in the orthorhombic Brillouin zone, respectively, where ~ ~ = 3 / 2 / 2 a ~ and ~ ~ = / 2 / 2 a ~ . ( b ) Localised boundary states.
The states plotted are those for which the probability that an electron is located on the atoms constituting the wrong bonds at the interface exceeds 25%.
Figure 4 shows the calculated band structure along the lines in the plane with kZ=O in the orthorhombic Brillouin zone of the present supercell geometry. There are no deep states in the fundamental gap in accordance with complete reconstruction at the interface as well as in the case of Si. As compared with the projected band structure of the perfect crystal (the dotted lines in Fig. 4(b)), a great difference is the presence of the boundary localised
Cl-214 COLLOQUE DE PHYSIQUE
states at the band edges. These states are localised at the wrong bonds at the interface.
Sharply localised states at the C-C bonds appear at the bottom of and within the valence band, and above the conduction band. And sharply localised states at' the Si-Si bonds appear at the top of the valence band and at the bottom of the conduction band, and these states exist inside the gap. However, these states are shallow states and not deep states. Thus we can conclude that the present boundary in @-Sic is electrically non-active intrinsically. The reason why the wrong bonds do not cause deep states in the present case differently from the case of amorpous compound semiconductors /24/ is that the differences between the atomic levels of C and Si are not so large as compared with compound semiconductors. The present results are consistent with the calculation of antisite defects in @-Sic /25/.
IV. SUMMARY
We have studied theoretically a facetted twin boundary in Si and a twin boundary in B-Sic for the first time. There exist relatively large distortions at one of the two types of the facet junctions in the {211)/{111) facets in Si. This region much more perturbs the electronic structure than the {211) and {l111 interfaces, but introduces no states inside the minimum gap. Therefore, concerning the present atomic structures of the facet junctions, the origin of the observed electrical activity of the microfacets or steps /1,16/ should be attributed to extrinsic effects such as antiphase defects or impurities. It can be said that the present atomic model of the {l221 C=9 boundary in @-Sic exists stably and has no deep states in the fundamental gap. These results indicate a possibility that electrically non-active and
atomistically reconstructed boundaries can be constructed by arranging structural units in Sic as well as in Si or Ge. However, the Si-Si wrong bonds at the interface cause the shallow states in the gap and the Coulomb energy caused by the wrong bonds is a large part of the boundary energy. It is possible that boundary energies of Sic are much influenced by the number of wrong bonds at interfaces. This seems to be important in the case of general boundaries in Sic. And the effects of impurities in grain boundaries in Sic can be considered in connection with the wrong bonds. For example, there may be a possibility that impurity atoms such as oxygen are easy to intervene between Si atoms at the Si-Si wrong bonds.
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