- Email: [email protected]
Dislocation core properties in semiconductors
PERGAMON
Solid State Communications 118 (2001) 651±655
www.elsevier.com/locate/ssc
Dislocation core properties in semiconductors JoaÄo F. Justo a,*, A. Antonelli b, A. Fazzio a b
a Instituto de FõÂsica da Universidade de SaÄo Paulo, CP 66318, CEP 05315-970, Sao Paulo, Brazil Instituto de FõÂsica Gleb Wataghin, Universidade Estadual de Campinas, CEP 13083-970, Campinas, Sao Paulo, Brazil
Received 12 January 2001; accepted 30 January 2001 by C.E.T. GoncËalves da Silva; received in ®nal form by the Publisher 30 April 2001
Abstract Using ab initio calculations, we computed the core reconstruction energies of {111} 308 partial dislocations in zinc-blende semiconductors. Our results show a direct correlation between core reconstruction energies and the experimental activation energies for the velocity of 608 dislocations. The electronic structure of unreconstructed dislocation cores comprises a half-®lled band, which splits up in bonding and antibonding levels upon reconstruction. The levels in the electronic gap come from the core of b dislocations, while the levels related to a dislocations lie on the valence band. q 2001 Elsevier Science Ltd. All rights reserved. PACS: 61.72.Lk; 61.72.Bb; 62.20.Fe Keywords: A. Semiconductor; C. Dislocations and disclinations; D. Electronic band structure; D. Electronic states (localized)
Current requirements for device miniaturization have presented remarkable challenges to semiconductor scientists. At some point, electronic devices will reach dimensions of the extended defects (dislocations and grain boundaries), which are always present in the material. Thus, it is imperative to understand the physical properties of these extended defects. Dislocations in semiconducting materials affect both the electronic and mechanical properties of the material [1]. Learning about their properties of formation, motion and interaction with other defects [2] may bring important contributions to the development of new and more reliable devices. In silicon, dislocation mobility is considerably low at normal operating conditions T , 6008K: On the other hand, under equivalent conditions, dislocation mobility is considerably higher in III±V and II±VI compounds. Considering that all these crystalline materials have a diamond cubic or zinc-blende structure, and therefore, the same type of dislocation activity, it is relevant to identify the origin of such large scattering of dislocation velocities. In this paper we investigated the energetics involved in core reconstruction of partial dislocations in several zinc-blend
semiconductors and correlated those energies to the experimental activation energies for dislocation motion. In addition, we investigated the electronic properties of unreconstructed and reconstructed core structures, which allowed us to build a realistic model for the electronic band structure associated with dislocation core. Dislocations in zinc-blende semiconductors generally belong to the {111} glide sets [3,4]. Due to the high Peierls barriers, as a result of the strong covalent bonding, dislocations in these materials are aligned along k110l directions. It is energetically favorable for (screw and 608) dislocations to dissociate into (30 and 908) partial dislocations, forming a stacking fault between the partials. In a III±V compound, dissociation into partials can generate three types of dislocations: a, comprising a 308 partial and a 908 partial with the core formed by type V atoms, b, consisting of a 308 partial and a 908 partial with the core formed by type III atoms, and a screw dislocation comprising two 308 partials, one of each type. Obviously, a and b dislocations are identical in a type IV semiconductor. Dislocations glide conservatively in {111} glide planes. Their velocities are controlled by thermally activated mechanisms of kink formation and propagation, exhibiting the behavior [3]
* Corresponding author. Tel.: 155-11-3818-7018; fax: 155-113818-6831. E-mail address: [email protected] (J.F. Justo).
Q vd v0 exp 2 kB T
0038-1098/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(01)00197-1
1
652
J.F. Justo et al. / Solid State Communications 118 (2001) 651±655
Fig. 1. Atomic con®guration of a [111] plane with a 308 partial dislocation lying in the (110) direction. In the ®gure, (a) and (c) show unreconstructed a and b dislocation, respectively, while (b) shows a fully reconstructed dislocation. White and grey circles represent type III and type V atoms, respectively. Black circles represent the atoms in the dislocation core.
where v0 is a constant velocity, Q the activation energy, and kB is the Boltzmann constant. It has been pointed out that core effects play a major role on dislocation mobility, specially in semiconductors [5]. The relevance of such effects derive directly from the reconstruction of the dislocation cores, i.e. core atoms recover their fourfold coordination by forming bonds with other core atoms. This can be quanti®ed by the reconstruction energy, i.e. the energy required to break an intracore bond. Dislocation motion involves breaking and formation of core bonds. Therefore, the larger the reconstruction energy, the more dif®cult for a dislocation to move, and, consequently, the larger the activation energy for the dislocation velocity [6,7]. Although there is an intuitive connection between core reconstruction and dislocation mobility, this has not yet been shown quantitatively. Here we have unambiguously established a connection between core reconstruction energy of dislocations, a microscopic property, and the dislocation activation energy, a macroscopic property. This was achieved by computing the reconstruction energy
of partial dislocations in several diamond cubic and zincblend materials using total-energy ab initio calculations and comparing them to the respective experimental activation energies. We focused our attention on the reconstruction of 308 glide partial (both a and b) dislocations. This choice has been motivated by the experimental results on the velocity of 608 dislocations in zinc-blend materials, since it has been shown that it is the 308 partial which controls the dislocation mobility [8]. In our ab initio calculations we used a reference orthorhombic periodic cell with 96 atoms for all materials we investigated. A 308 partial dislocation dipole was introduced by displacing the atoms according to the known solution for the displacement ®eld of a dislocation [3]. The ®nal dislocation dipole contained an a and a b dislocation in a type III±V material, or two identical dislocations in a type IV material. The cell geometry is such that the dislocations of the dipole are four lattice parameters (along the k112l direction) apart in the glide plane, preventing core±core interactions. Fig. 1 shows a schematic description of (a and b) unreconstructed and reconstructed 308 partial dislocation cores on the {111} glide plane. Core reconstruction involves bond formation between atoms of the same species. Reconstruction energies of a and b partial dislocations were computed by taking the difference in total energies of the con®gurations 1(b) and 1(a) and the con®gurations 1(b) and 1(c), respectively. Full atomic relaxation of the con®gurations presented in Fig. 1 is a remarkable computational task using ab initio methods. To overcome this dif®culty, we initially relaxed the structures with a less computationally expensive model, using interatomic potentials. Then, those atomic relaxed con®gurations were used as input to the ab initio calculations, when full relaxation could then be performed. However, this initial relaxation requires a reliable interatomic potential, at least for the dislocation core properties. An interatomic potential for silicon, called environment dependent interatomic potential (EDIP), has recently been developed [9,10]. This model provides a superior description of dislocation core properties in Si [6,10]. We used this
Table 1 Calculated reconstruction energy (Erec, in eV/reconstructed bond) of 308 (a and b) partial dislocations and relative distance (d) between core atoms (in percentage of a crystalline bond) for several type III±V and IV semiconductors. The table also presents the experimental activation energies (Q) for 608 (a and b) dislocation velocities. For type IV materials, a and b results are identical, and the results are presented as a dislocations a Erec (eV)
da (%)
b Erec (eV)
db
Si Ge AlP AlAs GaP GaAs
0.92 0.64 0.47 0.44 0.47 0.43
5.9 6.2 1.2 8.1 5.2 8.7
0.87 0.70 0.83 0.56
9.2 5.3 2 0.4 1.4
GaSb
0.50
12.0
0.63
2 3.4
Qa (eV)
Qb (eV)
2.20 [17] 1.62 [18] 1.45 [19] 1.30 [20] 1.00 [21] 1.30 [22,23]
1.68 [19] 1.30 [20] 1.40 [21] 1.60 [22,23]
J.F. Justo et al. / Solid State Communications 118 (2001) 651±655
653
Fig. 2. Calculated core reconstruction energies for 308 partial dislocation versus experimental activation energies of 608 dislocations in type IV, Si (A) and Ge (S), and type III±V, GaP (K), GaAs (L), and GaSb (W), semiconductors (according to Table 1). Open and closed symbols represent a and b dislocations, respectively. For dislocations in type IV materials, symbols are half-®lled. Error bars represent the estimated error of the experimental data.
interatomic potential, combined with a conjugate gradient algorithm, to generate relaxed dislocation core structures (for reconstructed and unreconstructed cores). These relaxed structures were used for all the zinc-blend compounds, only renormalizing the lattice parameter for each compound. The reliability of this procedure could be tested by checking the atomic forces on the ab initio calculations for the input con®gurations. In all compounds, Hellmann±Feynman forces on all atoms were already Ê. below 0.2 eV/A All ab initio calculations [11] were performed within the density functional theory (DFT) and the local density approximation (LDA) framework. The Kohn±Sham equations were solved by the Car±Parrinello scheme [12] using norm-conserving pseudopotentials [13] in the Kleinmann±Bylander form [14]. The valence electron wave functions were expanded in a plane-wave basis set, with kinetic energy up to 16 Ry, which provides well converged results for all the compounds studied here. The sampling in the Brillouin zone was performed using a set of two k-points [15] in the dislocation direction. Geometric optimization of the atomic structure of all systems was performed by allowing atoms to relax until the Hellmann± Ê. Feynman forces were smaller than 0.02 eV/A Table 1 presents the reconstruction energies (in eV/reconstructed bond) of a and b partial dislocations for several type-IV and type III±V semiconductors. In all compounds, reconstruction energies of a dislocations are smaller than the ones of b dislocations. This is consistent with experimental results [17±23] which ®nd activation energies for
608 a dislocations smaller than the ones for 608 b dislocations. The ratio between reconstruction energies Ebrec =Earec is between 1.3 and 1.8. Experimental estimation for several compounds sets the ratio of activation energies Qb =Qa around 1.3 [22]. Table 1 also presents the relative distance between the reconstructed core atoms for all compounds, as compared to the respective crystalline bonds. Fig. 2 displays the results for reconstruction energies of 308 partial dislocation as a function of the available experimental activation energies for dislocation motion of 608 a and b dislocations in an undoped specimen, the error bar being the estimated experimental imprecision. According to the ®gure, there is an almost linear relation between reconstruction and activation energies. This is the ®rst direct evidence of the correlation between core reconstruction energy and dislocation activation energy. Our results also imply that we can predict the activation energy for a certain zinc-blend semiconductor only by computing the dislocation reconstruction energy. We should emphasize that this conclusion was reached by using only one supposition, that it is the 308 partial dislocation which controls the mobility of a 608 dissociated dislocation, which is a reasonable supposition considering all the available experimental data [8]. We have additionally investigated the electronic properties of reconstructed and unreconstructed dislocations. The current picture of the electronic structure associated to the dislocation core in semiconductors considers a band inside the materials gap. For an unreconstructed dislocation core, the electronic state at the core is a half-®lled
654
J.F. Justo et al. / Solid State Communications 118 (2001) 651±655
Fig. 3. Electronic band structure of unreconstructed a (a) and b (c), and reconstructed (b) dislocation cores of aluminum phosphide. The corresponding atomic con®gurations (a), (b), and (c) were computed according to Fig. 1(a)±(c) respectively. The calculated electronic gap of the material is 1.04 eV. Grey boxes represent the ®lled electronic bands.
one-dimensional band in the band gap. For a reconstructed dislocation core, this one-dimensional band splits up into a ®lled bonding and an empty antibonding state. As has been pointed by some authors [16], this model is only intuitive and certainly requires re®nement, especially for the case of III±V compounds. To investigate the electronic structure of the dislocation core, we focused our study for the case of aluminum phosphide (AlP). Still all the conclusions presented here may be extended to other zinc-blend materials. Our choice was motivated by the fact that for all the systems studied here, AlP was the material with the largest energy gap. This way, the identi®cation of the onedimensional band would become more evident than in the other cases. Fig. 3 shows the electronic band structure for the atomic structures displayed in Fig. 1, considering a large set of k-points in the ®rst Brillouin zone along the direction of the reciprocal lattice vector related to the dislocation line direction. First, we should emphasize that we found a very small dispersion in any direction normal to the dislocation line (lower than 40 meV), as compared to the dispersion of about 400 meV in the dislocation direction. The dispersion in the electronic energy levels results from a one-dimensional band related to the dislocation line direction. Let us now consider the reconstruction of the b dislocation (Fig. 3b and c). In the unreconstructed case (Fig. 3c), bonding and antibonding levels form a half-®lled energy band on the top of the energy gap. On the other hand, in the reconstructed case (Fig. 3b), bonding and antibonding levels split away, forming a gap of 0.15 eV. The last occupied level comes
from the Al±Al core interaction. The lowering of this energy band towards the middle of the energy gap, as a result of reconstruction, is consistent with the lowering in total energy of the system. In the reconstructed case, the electronic charge distribution of the last occupied level is highly localized at the interstitial region between Al core atoms. For the unreconstructed case, the charge distribution of the last occupied level is highly delocalized inside the dislocation (Al core atoms). For the case of a dislocation reconstruction, the situation is considerably different. The reconstructed bond between the P±P core atoms generates an energy level resonant in the valence band, 4 eV below the top of that band. In the a unreconstructed dislocations, the highest occupied level (Fig. 3a), which forms an energy band in the bottom of the gap, comes from the Al±Al reconstructed bond. The highest occupied level in Fig. 3b is also related to a reconstructed Al±Al bond in the dislocation core. However, although the total energy corresponding to Fig. 3b is lower than that of Fig. 3a, this level appears higher into the energy gap. This can be understood in the following manner: in the situation corresponding to Fig. 3b, the a dislocation is also reconstructed, giving rise to a level resonant in the valence band which has the same symmetry as that of the highest occupied level. Thus, the repulsion between levels of identical symmetry pushes the highest occupied level up. These results for the electronic band structure of dislocation core should be taken with caution. DFT±LDA calculations are known to give a reasonably poor description of electronic excited states, generally providing a smaller materials
J.F. Justo et al. / Solid State Communications 118 (2001) 651±655
gap as compared to the experimental data. Still, DFT±LDA calculations provide a reasonably reliable description of the relative dispersion of electronic bands. In summary, we have investigated the core properties of partial dislocations in several semiconductors. We have identi®ed for the ®rst time a direct connection between core reconstruction energy of partial dislocations and activation energy of dislocation velocity. Considering the results obtained by our calculations, as well as the available activation energies for dislocation motion of II±VI zincblende semiconductors [24], which are below 1 eV, we estimate the reconstruction energies for partial dislocations in II±VI materials to be around 0.25 eV/reconstruction bond. Furthermore, we have unveiled the band structure of the dislocation core. The b dislocations control the mobility of the dislocations, as a result of a larger core reconstruction energy. Moreover, they control the electronic activity of the dislocations, being responsible for the dislocation-related energy levels which appear in the materials gap. Acknowledgements The authors acknowledge partial support from Brazilian agencies FAPESP and CNPq. The calculations were performed at the LCCA-CCE of the Universidade de Sao Paulo. References [1] H. Alexander, in: F.R.N. Nabarro (Ed.), Dislocations in Solids, Vol. 7, North-Holland, Amsterdam, 1986, p. 115. [2] W. Cai, V.V. Bulatov, J.F. Justo, A.S. Argon, S. Yip, Phys. Rev. Lett. 84 (2000) 3346.
655
[3] J.P. Hirth, J. Lothe, Theory of Dislocations, Wiley, New York, 1982. [4] J.F. Justo, M. de Koning, W. Cai, V.V. Bulatov, Phys. Rev. Lett. 84 (2000) 2172. [5] M.S. Duesbery, G.Y. Richardson, Crit. Rev. Solid State 17 (1991) 1. [6] J.F. Justo, V.V. Bulatov, S. Yip, J. Appl. Phys. 86 (1999) 4249. [7] V.V. Bulatov, J.F. Justo, W. Cai, S. Yip, Phys. Rev. Lett. 79 (1997) 5042. [8] H. Alexander, H. Gottschalk, Inst. Phys. Conf. Ser. 104 (1989) 281. [9] M.Z. Bazant, E. Kaxiras, J.F. Justo, Phys. Rev. B 56 (1997) 8542. [10] J.F. Justo, M.Z. Bazant, E. Kaxiras, V.V. Bulatov, S. Yip, Phys. Rev. B 58 (1998) 2539. [11] M. Bockstedte, A. Kley, J. Neugebauer, M. Schef¯er, Comput. Phys. Commun. 107 (1997) 187. [12] R. Car, M. Parrinello, Phys. Rev. Lett. 55 (1985) 2471. [13] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [14] L. Kleinman, D.M. Bylander, Phys. Rev. Lett. 48 (1982) 1425. [15] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [16] T. Suzuki, S. Takeuci, H. Yoshinaga, Dislocation Dynamics and Plasticity, Springer, Berlin, 1991. [17] M. Imai, K. Sumino, Philos. Mag. A 47 (1983) 599. [18] I. Yonenaga, K. Sumino, Appl. Phys. Lett. 69 (1996) 1264. [19] I. Yonenaga, K. Sumino, J. Appl. Phys. 73 (1993) 1681. [20] I. Yonenaga, K. Sumino, J. Appl. Phys. 65 (1989) 85. [21] S.K. Choi, M. Mihara, T. Ninomiya, Jpn. J. Appl. Phys. 16 (1977) 737. [22] S.K. Choi, M. Mihara, T. Ninomiya, Jpn. J. Appl. Phys. 17 (1978) 329. [23] M. Omri, J.P. Michel, A. George, Philos. Mag. A 62 (1990) 203. [24] I. Yonenaga, J. Appl. Phys. 84 (1998) 4209.
Solid State Communications 118 (2001) 651±655
www.elsevier.com/locate/ssc
Dislocation core properties in semiconductors JoaÄo F. Justo a,*, A. Antonelli b, A. Fazzio a b
a Instituto de FõÂsica da Universidade de SaÄo Paulo, CP 66318, CEP 05315-970, Sao Paulo, Brazil Instituto de FõÂsica Gleb Wataghin, Universidade Estadual de Campinas, CEP 13083-970, Campinas, Sao Paulo, Brazil
Received 12 January 2001; accepted 30 January 2001 by C.E.T. GoncËalves da Silva; received in ®nal form by the Publisher 30 April 2001
Abstract Using ab initio calculations, we computed the core reconstruction energies of {111} 308 partial dislocations in zinc-blende semiconductors. Our results show a direct correlation between core reconstruction energies and the experimental activation energies for the velocity of 608 dislocations. The electronic structure of unreconstructed dislocation cores comprises a half-®lled band, which splits up in bonding and antibonding levels upon reconstruction. The levels in the electronic gap come from the core of b dislocations, while the levels related to a dislocations lie on the valence band. q 2001 Elsevier Science Ltd. All rights reserved. PACS: 61.72.Lk; 61.72.Bb; 62.20.Fe Keywords: A. Semiconductor; C. Dislocations and disclinations; D. Electronic band structure; D. Electronic states (localized)
Current requirements for device miniaturization have presented remarkable challenges to semiconductor scientists. At some point, electronic devices will reach dimensions of the extended defects (dislocations and grain boundaries), which are always present in the material. Thus, it is imperative to understand the physical properties of these extended defects. Dislocations in semiconducting materials affect both the electronic and mechanical properties of the material [1]. Learning about their properties of formation, motion and interaction with other defects [2] may bring important contributions to the development of new and more reliable devices. In silicon, dislocation mobility is considerably low at normal operating conditions T , 6008K: On the other hand, under equivalent conditions, dislocation mobility is considerably higher in III±V and II±VI compounds. Considering that all these crystalline materials have a diamond cubic or zinc-blende structure, and therefore, the same type of dislocation activity, it is relevant to identify the origin of such large scattering of dislocation velocities. In this paper we investigated the energetics involved in core reconstruction of partial dislocations in several zinc-blend
semiconductors and correlated those energies to the experimental activation energies for dislocation motion. In addition, we investigated the electronic properties of unreconstructed and reconstructed core structures, which allowed us to build a realistic model for the electronic band structure associated with dislocation core. Dislocations in zinc-blende semiconductors generally belong to the {111} glide sets [3,4]. Due to the high Peierls barriers, as a result of the strong covalent bonding, dislocations in these materials are aligned along k110l directions. It is energetically favorable for (screw and 608) dislocations to dissociate into (30 and 908) partial dislocations, forming a stacking fault between the partials. In a III±V compound, dissociation into partials can generate three types of dislocations: a, comprising a 308 partial and a 908 partial with the core formed by type V atoms, b, consisting of a 308 partial and a 908 partial with the core formed by type III atoms, and a screw dislocation comprising two 308 partials, one of each type. Obviously, a and b dislocations are identical in a type IV semiconductor. Dislocations glide conservatively in {111} glide planes. Their velocities are controlled by thermally activated mechanisms of kink formation and propagation, exhibiting the behavior [3]
* Corresponding author. Tel.: 155-11-3818-7018; fax: 155-113818-6831. E-mail address: [email protected] (J.F. Justo).
Q vd v0 exp 2 kB T
0038-1098/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(01)00197-1
1
652
J.F. Justo et al. / Solid State Communications 118 (2001) 651±655
Fig. 1. Atomic con®guration of a [111] plane with a 308 partial dislocation lying in the (110) direction. In the ®gure, (a) and (c) show unreconstructed a and b dislocation, respectively, while (b) shows a fully reconstructed dislocation. White and grey circles represent type III and type V atoms, respectively. Black circles represent the atoms in the dislocation core.
where v0 is a constant velocity, Q the activation energy, and kB is the Boltzmann constant. It has been pointed out that core effects play a major role on dislocation mobility, specially in semiconductors [5]. The relevance of such effects derive directly from the reconstruction of the dislocation cores, i.e. core atoms recover their fourfold coordination by forming bonds with other core atoms. This can be quanti®ed by the reconstruction energy, i.e. the energy required to break an intracore bond. Dislocation motion involves breaking and formation of core bonds. Therefore, the larger the reconstruction energy, the more dif®cult for a dislocation to move, and, consequently, the larger the activation energy for the dislocation velocity [6,7]. Although there is an intuitive connection between core reconstruction and dislocation mobility, this has not yet been shown quantitatively. Here we have unambiguously established a connection between core reconstruction energy of dislocations, a microscopic property, and the dislocation activation energy, a macroscopic property. This was achieved by computing the reconstruction energy
of partial dislocations in several diamond cubic and zincblend materials using total-energy ab initio calculations and comparing them to the respective experimental activation energies. We focused our attention on the reconstruction of 308 glide partial (both a and b) dislocations. This choice has been motivated by the experimental results on the velocity of 608 dislocations in zinc-blend materials, since it has been shown that it is the 308 partial which controls the dislocation mobility [8]. In our ab initio calculations we used a reference orthorhombic periodic cell with 96 atoms for all materials we investigated. A 308 partial dislocation dipole was introduced by displacing the atoms according to the known solution for the displacement ®eld of a dislocation [3]. The ®nal dislocation dipole contained an a and a b dislocation in a type III±V material, or two identical dislocations in a type IV material. The cell geometry is such that the dislocations of the dipole are four lattice parameters (along the k112l direction) apart in the glide plane, preventing core±core interactions. Fig. 1 shows a schematic description of (a and b) unreconstructed and reconstructed 308 partial dislocation cores on the {111} glide plane. Core reconstruction involves bond formation between atoms of the same species. Reconstruction energies of a and b partial dislocations were computed by taking the difference in total energies of the con®gurations 1(b) and 1(a) and the con®gurations 1(b) and 1(c), respectively. Full atomic relaxation of the con®gurations presented in Fig. 1 is a remarkable computational task using ab initio methods. To overcome this dif®culty, we initially relaxed the structures with a less computationally expensive model, using interatomic potentials. Then, those atomic relaxed con®gurations were used as input to the ab initio calculations, when full relaxation could then be performed. However, this initial relaxation requires a reliable interatomic potential, at least for the dislocation core properties. An interatomic potential for silicon, called environment dependent interatomic potential (EDIP), has recently been developed [9,10]. This model provides a superior description of dislocation core properties in Si [6,10]. We used this
Table 1 Calculated reconstruction energy (Erec, in eV/reconstructed bond) of 308 (a and b) partial dislocations and relative distance (d) between core atoms (in percentage of a crystalline bond) for several type III±V and IV semiconductors. The table also presents the experimental activation energies (Q) for 608 (a and b) dislocation velocities. For type IV materials, a and b results are identical, and the results are presented as a dislocations a Erec (eV)
da (%)
b Erec (eV)
db
Si Ge AlP AlAs GaP GaAs
0.92 0.64 0.47 0.44 0.47 0.43
5.9 6.2 1.2 8.1 5.2 8.7
0.87 0.70 0.83 0.56
9.2 5.3 2 0.4 1.4
GaSb
0.50
12.0
0.63
2 3.4
Qa (eV)
Qb (eV)
2.20 [17] 1.62 [18] 1.45 [19] 1.30 [20] 1.00 [21] 1.30 [22,23]
1.68 [19] 1.30 [20] 1.40 [21] 1.60 [22,23]
J.F. Justo et al. / Solid State Communications 118 (2001) 651±655
653
Fig. 2. Calculated core reconstruction energies for 308 partial dislocation versus experimental activation energies of 608 dislocations in type IV, Si (A) and Ge (S), and type III±V, GaP (K), GaAs (L), and GaSb (W), semiconductors (according to Table 1). Open and closed symbols represent a and b dislocations, respectively. For dislocations in type IV materials, symbols are half-®lled. Error bars represent the estimated error of the experimental data.
interatomic potential, combined with a conjugate gradient algorithm, to generate relaxed dislocation core structures (for reconstructed and unreconstructed cores). These relaxed structures were used for all the zinc-blend compounds, only renormalizing the lattice parameter for each compound. The reliability of this procedure could be tested by checking the atomic forces on the ab initio calculations for the input con®gurations. In all compounds, Hellmann±Feynman forces on all atoms were already Ê. below 0.2 eV/A All ab initio calculations [11] were performed within the density functional theory (DFT) and the local density approximation (LDA) framework. The Kohn±Sham equations were solved by the Car±Parrinello scheme [12] using norm-conserving pseudopotentials [13] in the Kleinmann±Bylander form [14]. The valence electron wave functions were expanded in a plane-wave basis set, with kinetic energy up to 16 Ry, which provides well converged results for all the compounds studied here. The sampling in the Brillouin zone was performed using a set of two k-points [15] in the dislocation direction. Geometric optimization of the atomic structure of all systems was performed by allowing atoms to relax until the Hellmann± Ê. Feynman forces were smaller than 0.02 eV/A Table 1 presents the reconstruction energies (in eV/reconstructed bond) of a and b partial dislocations for several type-IV and type III±V semiconductors. In all compounds, reconstruction energies of a dislocations are smaller than the ones of b dislocations. This is consistent with experimental results [17±23] which ®nd activation energies for
608 a dislocations smaller than the ones for 608 b dislocations. The ratio between reconstruction energies Ebrec =Earec is between 1.3 and 1.8. Experimental estimation for several compounds sets the ratio of activation energies Qb =Qa around 1.3 [22]. Table 1 also presents the relative distance between the reconstructed core atoms for all compounds, as compared to the respective crystalline bonds. Fig. 2 displays the results for reconstruction energies of 308 partial dislocation as a function of the available experimental activation energies for dislocation motion of 608 a and b dislocations in an undoped specimen, the error bar being the estimated experimental imprecision. According to the ®gure, there is an almost linear relation between reconstruction and activation energies. This is the ®rst direct evidence of the correlation between core reconstruction energy and dislocation activation energy. Our results also imply that we can predict the activation energy for a certain zinc-blend semiconductor only by computing the dislocation reconstruction energy. We should emphasize that this conclusion was reached by using only one supposition, that it is the 308 partial dislocation which controls the mobility of a 608 dissociated dislocation, which is a reasonable supposition considering all the available experimental data [8]. We have additionally investigated the electronic properties of reconstructed and unreconstructed dislocations. The current picture of the electronic structure associated to the dislocation core in semiconductors considers a band inside the materials gap. For an unreconstructed dislocation core, the electronic state at the core is a half-®lled
654
J.F. Justo et al. / Solid State Communications 118 (2001) 651±655
Fig. 3. Electronic band structure of unreconstructed a (a) and b (c), and reconstructed (b) dislocation cores of aluminum phosphide. The corresponding atomic con®gurations (a), (b), and (c) were computed according to Fig. 1(a)±(c) respectively. The calculated electronic gap of the material is 1.04 eV. Grey boxes represent the ®lled electronic bands.
one-dimensional band in the band gap. For a reconstructed dislocation core, this one-dimensional band splits up into a ®lled bonding and an empty antibonding state. As has been pointed by some authors [16], this model is only intuitive and certainly requires re®nement, especially for the case of III±V compounds. To investigate the electronic structure of the dislocation core, we focused our study for the case of aluminum phosphide (AlP). Still all the conclusions presented here may be extended to other zinc-blend materials. Our choice was motivated by the fact that for all the systems studied here, AlP was the material with the largest energy gap. This way, the identi®cation of the onedimensional band would become more evident than in the other cases. Fig. 3 shows the electronic band structure for the atomic structures displayed in Fig. 1, considering a large set of k-points in the ®rst Brillouin zone along the direction of the reciprocal lattice vector related to the dislocation line direction. First, we should emphasize that we found a very small dispersion in any direction normal to the dislocation line (lower than 40 meV), as compared to the dispersion of about 400 meV in the dislocation direction. The dispersion in the electronic energy levels results from a one-dimensional band related to the dislocation line direction. Let us now consider the reconstruction of the b dislocation (Fig. 3b and c). In the unreconstructed case (Fig. 3c), bonding and antibonding levels form a half-®lled energy band on the top of the energy gap. On the other hand, in the reconstructed case (Fig. 3b), bonding and antibonding levels split away, forming a gap of 0.15 eV. The last occupied level comes
from the Al±Al core interaction. The lowering of this energy band towards the middle of the energy gap, as a result of reconstruction, is consistent with the lowering in total energy of the system. In the reconstructed case, the electronic charge distribution of the last occupied level is highly localized at the interstitial region between Al core atoms. For the unreconstructed case, the charge distribution of the last occupied level is highly delocalized inside the dislocation (Al core atoms). For the case of a dislocation reconstruction, the situation is considerably different. The reconstructed bond between the P±P core atoms generates an energy level resonant in the valence band, 4 eV below the top of that band. In the a unreconstructed dislocations, the highest occupied level (Fig. 3a), which forms an energy band in the bottom of the gap, comes from the Al±Al reconstructed bond. The highest occupied level in Fig. 3b is also related to a reconstructed Al±Al bond in the dislocation core. However, although the total energy corresponding to Fig. 3b is lower than that of Fig. 3a, this level appears higher into the energy gap. This can be understood in the following manner: in the situation corresponding to Fig. 3b, the a dislocation is also reconstructed, giving rise to a level resonant in the valence band which has the same symmetry as that of the highest occupied level. Thus, the repulsion between levels of identical symmetry pushes the highest occupied level up. These results for the electronic band structure of dislocation core should be taken with caution. DFT±LDA calculations are known to give a reasonably poor description of electronic excited states, generally providing a smaller materials
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gap as compared to the experimental data. Still, DFT±LDA calculations provide a reasonably reliable description of the relative dispersion of electronic bands. In summary, we have investigated the core properties of partial dislocations in several semiconductors. We have identi®ed for the ®rst time a direct connection between core reconstruction energy of partial dislocations and activation energy of dislocation velocity. Considering the results obtained by our calculations, as well as the available activation energies for dislocation motion of II±VI zincblende semiconductors [24], which are below 1 eV, we estimate the reconstruction energies for partial dislocations in II±VI materials to be around 0.25 eV/reconstruction bond. Furthermore, we have unveiled the band structure of the dislocation core. The b dislocations control the mobility of the dislocations, as a result of a larger core reconstruction energy. Moreover, they control the electronic activity of the dislocations, being responsible for the dislocation-related energy levels which appear in the materials gap. Acknowledgements The authors acknowledge partial support from Brazilian agencies FAPESP and CNPq. The calculations were performed at the LCCA-CCE of the Universidade de Sao Paulo. References [1] H. Alexander, in: F.R.N. Nabarro (Ed.), Dislocations in Solids, Vol. 7, North-Holland, Amsterdam, 1986, p. 115. [2] W. Cai, V.V. Bulatov, J.F. Justo, A.S. Argon, S. Yip, Phys. Rev. Lett. 84 (2000) 3346.
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