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Basal plane partial dislocations in silicon carbide
ARTICLE IN PRESS
Physica B 340–342 (2003) 160–164
Basal plane partial dislocations in silicon carbide c . A.T. Blumenaua,*, R. Jonesb, S. Oberg , P.R. Briddond, T. Frauenheima Department of Physics, Universitat . Paderborn, D - 33098 Paderborn, Germany b School of Physics, University of Exeter, Exeter EX4 4QL, UK c ( Sweden Department of Mathematics, Lulea( University of Technology, S-97187 Lulea, d Department of Physics, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, UK a
Abstract Under operating conditions (forward bias) bipolar 4H- and 6H-SiC devices are known to degrade rapidly through stacking fault formation and expansion in the basal plane. It has been suggested that a recombination-enhanced dislocation glide (REDG) mechanism allows the bordering Shockley partial dislocations to overcome their barrier to glide motion and thus results in the observed stacking fault growth. In this work, we investigate the structure and properties of the participating Shockley partials by means of density functional-based atomistic calculations. Their glide motion is modelled in a process involving the formation and subsequent migration of kinks. This in combination with an analysis of the electronic structure of the partials allows an identification of those types which will be affected by the REDG mechanism. r 2003 Published by Elsevier B.V. PACS: 61.72.Lk; 71.15.Nc; 71.55.i Keywords: Dislocations; Silicon carbide; Electronic structure; Glide motion
1. Introduction The wide band gap semiconductor silicon carbide (SiC) is a promising material for highpower, high-temperature and high-frequency applications. However, under forward bias, bipolar 4H- and 6H-SiC devices (high-power diodes) are known to degrade rapidly through stacking fault formation and expansion in the basal plane [1–5]. This results in a considerable drop in the forward voltage, rendering the device useless after few days of constant operation. The expanding stacking
*Corresponding author. E-mail address: [email protected] (A.T. Blumenau). 0921-4526/$ - see front matter r 2003 Published by Elsevier B.V. doi:10.1016/j.physb.2003.09.046
faults are of triangular and rhombic shape [3] and their edges were identified to be Shockley partial dislocations with Burgers vectors of 13/1 1% 0 0S [5,6]. It has been suggested that a recombinationenhanced dislocation glide mechanism (REDG) is responsible for the observed effect. This REDG mechanism requires non-radiative electron–hole recombination sites to lie along the dislocation line [7,8]. Part of the energy released following recombination has then to be directed into the formation and migration of kinks at the dislocation which are then able to move at or near room temperatures [9]. In this work, the atomic and electronic structure of the participating basal plane partial dislocations are investigated theoretically. The barriers to
ARTICLE IN PRESS A.T. Blumenau et al. / Physica B 340–342 (2003) 160–164
In this work, two computational methods with a different level of approximation to density functional theory (DFT) have been applied: Densityfunctional-based tight-binding method (DFTB) and ab initio modelling program (AIMPRO). The DFTB method is a tight-binding method using a minimal basis of atomic orbitals (linear combination of atomic orbitals, LCAO). The twocentre Hamiltonian and overlap matrix elements are obtained from atom-centred valence electron orbitals and the atomic potentials from single atom DFT calculations. Exchange and correlation contributions in the total energy as well as the core–core repulsion are taken into account by a repulsive pair-potential. The latter is obtained by comparison with DFT calculations for selected reference systems [11]. The AIMPRO method uses the pseudopotentials of Bachelet et al. [12] in combination with a Gaussian basis set and has been described in detail elsewhere [13]. This method requires greater computing resources and thus in this work, geometrical optimisation as well as the calculation of formation energies and barriers are carried out using DFTB with models containing around 600 atoms. Comparison for smaller models gave good agreement between both methods.
3. Straight dislocations To model the straight dislocation core, a unit cell is constructed by inserting a single dislocation into an infinite cylinder whose surface dangling bonds are hydrogenated. The dislocation is aligned with the axis of the cylinder and is periodic along ( of vacuum the dislocation line. More than 100 A
30° Si [1120]
2. Computational methods
separates cylinders stacked on a square lattice. Hence, this periodic cluster (or supercell–cluster hybrid) approach respects the periodicity of the dislocation along the line and avoids artificial dislocation–dislocation interactions which occur when a dipole is inserted into a supercell. The cylinders have a period twice that of the cubic or hexagonal lattice and the unit cell contains B600 atoms. Structural relaxations are performed at the G-point only. More details on this approach are given in Ref. [14]. Fig. 1 shows the relaxed core structure of a Si terminated 30 partial in 2H-SiC. On the terminating line of Si atoms, Si–Si reconstruction bonds are formed resulting in a double-period structure. These reconstruction bonds are about 0.7% stretched compared to bulk Si. The situation is similar at the carbon terminated 30 partial (not shown). However compared to diamond, here the C–C reconstruction bonds are stretched considerably by 17%. The structure of the single-period (SP) and the double-period (DP) core reconstructions of the 90 partial have been discussed in Ref. [10]. Also for the 90 partial, the C–C bonds were found considerably stretched (10–15%) and the Si–Si bonds were of almost bulk Si bondlength (0.4– 1.4% compressed). Identical calculations were carried out in 3C-SiC giving almost identical results. Therefore, the stacking seems to have no major influence and we are confident that our results are representative for 4H and 6H as well. From linear elasticity theory, it is known that the elastic energy contained in a cylinder of radius R around the dislocation depends logarithmically
[0001]
partial glide motion are obtained by modelling the formation and migration of kinks along the dislocation line. The main emphasis lies on the 30 Shockley partial, as the 90 partial has been investigated earlier [10]. However, to facilitate comparison, selected results for the 90 partials are reproduced.
161
: faulted region
Fig. 1. The relaxed core structure of the 30 silicon terminated partial. The structure is shown both looking along the dislocation line ½1 1 2% 0 and in the basal plane (0001). The region of the intrinsic stacking fault is shaded.
ARTICLE IN PRESS A.T. Blumenau et al. / Physica B 340–342 (2003) 160–164
on the radius [15]: EðRÞ ¼
kðbÞjb~j2 R ln þ Ec ; Rc 4p
RXRc :
ð1Þ
The energy factor kðbÞ and hence the gradient of ( can be obtained from the energy vs. lnðR=AÞ elastic constants, the line direction and the Burgers vector b~ [15]. Anisotropic linear elasticity theory using the DFTB elastic constants yields kðbÞ ¼ 195 and 249 GPa for the 30 and 90 partials in 3C. For 2H, we find 191 and 246 GPa; respectively, indicating the errors within our calculations. The core energy Ec can be obtained in comparison with the atomistic calculations: The DFTB method allows an easy definition of the formation energy projected onto atom sites. This is used to define a radial formation energy Ef ðRÞ which can be compared directly with EðRÞ in Eq. (1). Table 1 gives the resulting energy factors and core energies in 3C-SiC and 2H-SiC, the latter given as formation energies with respect to stoichiometric bulk SiC. To reduce the computational effort, periodic cluster models smaller than those used to obtain the core energies, containing about 120 atoms, are used to calculate the electronic band structures with the DFT-pseudopotential code AIMPRO. The structures are relaxed, using a MonkhorstPack 2 1 1 set of k~-points [16]. The projected electronic band structure of the relaxed structure is then calculated at 21 different k~-points along the dislocation axis. Fig. 2 shows the projected band structures of the 30 partials in 2H-SiC. Both for
the C and the Si terminated partial, the localised gap states below the conduction band minimum (CBM) are related to the stacking fault. For 2H, we find these states as deep as B0:5–0:6 eV below the CBM. Similar bands have been found in 4H and 6H (B0:2 eV below the CBM) [17,18] and for intrinsic faults in 4H (B0:3 eV below the CBM) [19]. The striking difference between the C and the Si terminated partial is the presence of a localised defect band (reaching as far as 0:4 eV above the valence band maximum) related to the Si–Si reconstruction bonds. No such band is found at the C partial. Hence, results are very similar to those for the 90 partials [10].
(eV)
162
4
2
0
-2 30° Si
30° C
Fig. 2. The projected bandstructures of the silicon and the carbon terminated 30 partial (aligned at the valence band maximum).
Table 1 30 Si
kðbÞ ( Ec ð6 AÞ
(GPa) ( ðeV=AÞ
90 Si (SP)
3C
2H
3C
2H
3C
2H
203 0.48
194 0.47
249 0.68
251 0.69
250 0.59
242 0.60
30 C
kðbÞ ( Ec ð6 AÞ
(GPa) ( ðeV=AÞ
90 Si (DP)
90 C (SP)
90 C (DP)
3C
2H
3C
2H
3C
2H
194 0.86
197 0.85
242 1.27
237 1.27
244 1.17
244 1.17
( of the investigated Shockley partials in stoichiometric 3C Energy factors kðbÞ; and core energies Ec (corresponding to a radius of 6 A) and 2H SiC. kðbÞ is obtained as fit to Er ðRÞ vs. InðR=Rc Þ plot following Eq. (1).
ARTICLE IN PRESS A.T. Blumenau et al. / Physica B 340–342 (2003) 160–164
and angles due to the different character of the Si–Si and C–C bonds. The kink formation energies are obtained by introducing kink pairs on a dislocation embedded in a cluster as described in Ref. [10]. The elastic kink–kink interaction energy within the cluster is subtracted following Ref. [15]. For the smallest possible double kink, the interaction energy is found as 0:49 eV: Since the structures of left and right kinks are very different, one cannot expect similar formation energies. Hence, the single-kink energies are not individually found, but only the sum of two kink formation energies. We calculated the sum of formation energies for all possible four combinations of left with right kinks. LK1–RK1 occurs as the lowest energy pair. The energies of other combinations can be found by adding the appropriate D terms given in Table 2. To find the intermediate saddlepoint structure between two different kinks—and hence the migration barrier between them—the elementary kink migration step was parameterised by the coordinates of each of two core (primary) atoms at the kink projected onto the connecting line between their initial and final positions. Varying these two parameters independently, yields a twodimensional energy surface. Within the vicinity of the saddle point, the parameter mesh was chosen 1 as 100 : For details, see Ref. [10] where the method is explained. Table 2 displays the resulting barriers Wm found for right and left kinks. In the case when obstacles (point defects and impurities) are not present, and for short dislocation segments, the glide velocity is controlled by the sum of the double-kink formation energy 2Ef and the kink migration barrier Wm ; whereas for
4. Thermally activated glide Dislocation glide arises from an external stress acting on the dislocation. When the stress is insufficient to overcome the Peierls barrier, kinks must be generated at the dislocation by a thermal process and motion occurs by their migration along the dislocation line. Due to the doubleperiodicity present in the reconstructed 30 dislocation shown in Fig. 1, two varieties of left (LK1 and LK2) and right kinks (RK1 and RK2) occur. Fig. 3 shows the four different kink structures at the silicon-terminated 30 partial. All atoms in these structures are four-fold coordinated suggesting relatively low kink formation energies. Qualitatively, the structures at the carbon partial are very similar, with only differences in bond lengths
LK1
LK2
RK1
163
RK2
Fig. 3. The two different left kinks (LK) and the two right kinks (RK) of the 30 silicon terminated partial. All structures are shown in the basal plane and the region of the intrinsic stacking fault is shaded.
Table 2
Si C
2Ef ð1Þ
DðLKÞ
Wm ðLKÞ
DðRKÞ
Wm ðRKÞ
Q% 30
Q90
1.62 2.21
2.87 2.11
3.79 3.00
0.28 0.33
2.87 1.78
4.95 (4.14) 4.60 (3.50)
4.09 (3.58) 2.74 (2.29)
Kink formation energies Ef and migration barriers Wm for the 30 Shockley partials. 2Ef ð1Þ ¼ Ef ðLK1Þ þ Ef ðRK1Þ denotes the sum of the formation energies of the low-energy left and the low-energy right kink. DðLKÞ gives the energy difference between the low-energy (LK1) and the high-energy left kinks (LK2) and DðRKÞ denotes the same quantity for right kinks. To facilitate comparison with the 90 partials, an average activation energy Q% 30 ¼ ð2Ef ðLK1Þ þ Wm ðLKÞ þ 2Ef ðRK1Þ þ Wm ðRKÞÞ=2 is given. The glide activation energy assumes short dislocation segments and the values in brackets give the corresponding values for long segments. For comparison the corresponding values for the 90 partials are given in the last column [10].
ARTICLE IN PRESS A.T. Blumenau et al. / Physica B 340–342 (2003) 160–164
164
long segments the glide activation energy is given as Q ¼ Ef þ Wm [15]. Having two different barriers for left and right kink migration, respectively, results in a modified expression for the partial velocity [20]
experiment [4]. Such a mechanism would further explain the experimental observation that only one type of partial—the Si partial—is highly mobile under forward bias [21].
vdisl p eðEf ðLK1ÞþEf ðRK1ÞÞ=kT
½eWm ðLKÞ=kT þ eWm ðRKÞ=kT :
ð2Þ
The kinks with the lower migration barrier dominate the glide process. However, to allow an easy comparison, as a crude approximation an average activation barrier Q% 30 ¼ ð2Ef ðLK1Þ þ Wm ðLKÞ þ 2Ef ðRK1Þ þ Wm ðRKÞÞ=2 can be defined for short segments. Values for short and long segments are given in Table 2. At both the 30 and the 90 partials, the carbon cores are found with lower activation energies than the silicon cores. This might appear to be counter intuitive as the bonds in diamond are undoubtedly stronger than those in silicon. However, one has to keep in mind that the C–C reconstruction bonds at the partial are stretched considerably when compared with diamond, and it is this effect which lowers the energy required to break the reconstructed bond necessary for kink migration.
5. Conclusions For all partials, we found stretched C–C reconstruction bonds or, respectively, bulk like Si–Si reconstruction bonds, depending on the type of partial (C or Si terminated). This results in a higher thermal mobility of the C-terminated partials. Electrically, however, the C-terminated partials appear to be inert—only the Si partials induce filled bands low in the gap. This points towards an REDG mechanism based on a nonradiative recombination between the stacking fault-related states and those induced by the Si partial. Assuming the latter states in 4H to be around 0:3 eV below the CBM together with a gap of 3:3 eV; and the states of the Si partials to be 0:4 eV above the VBM, gives a rough estimate of 2:6 eV for the non-radiative recombination. Considering the band gap error in DFT, this is in reasonable agreement with the 2:2 eV suggested by
References [1] H. Lendenmann, F. Dahlquist, N. Johansson, R. ( Nilsson, J.P. Bergman, P. Skytt, Mater. . Soderholm, P.A. Sci. Forum 353–356 (2000) 727. [2] A. Galeckas, J. Linnros, B. Breitholtz, H. Bleichner, Mater. Sci. Forum 353–356 (2000) 389. ( Nilsson, U. Lindefelt, [3] J.P. Bergman, H. Lendenmann, P.A. P. Skytt, Mater. Sci. Forum 353–356 (2000) 299. [4] A. Galeckas, J. Linnros, P. Pirouz, Appl. Phys. Lett. 81 (2002) 883. . [5] J.Q. Liu, M. Skowronski, C. Hallin, R. Soderholm, H. Lendenmann, Appl. Phys. Lett. 80 (2002) 749. ( Persson, L. Hultman, H. Jacobson, J.P. Bergman, [6] P.O.A. E. Janz!en, J.M. Molina-Aldareguia, W.J. Clegg, T. Tuomi, Appl. Phys. Lett. 80 (2002) 4852. [7] J.D. Weeks, J.C. Tully, L.C. Kimerling, Phys. Rev. B 12 (1975) 3286. [8] H. Sumi, Phys. Rev. B 29 (1984) 4616. [9] K. Maeda, S. Takeuchi, Enhancement of dislocation mobility in semiconducting crystals by electronic excitation, in: F.R.N. Nabarro, M.S. Duesbery (Eds.), Dislocations in Solids, Vol. 10, North-Holland, Amsterdam, 1996, p. 444. [10] A.T. Blumenau, C.J. Fall, R. Jones, M.I. Heggie, P.R. . Briddon, T. Frauenheim, S. Oberg, J. Phys.: Condens. Matter 14 (2002) 12 741. [11] T. Frauenheim, G. Seifert, M. Elstner, Z. Hajnal, G. Jungnickel, D. Porezag, S. Suhai, R. Scholz, Phys. Stat. Sol. (B) 217 (2000) 41. [12] G.B. Bachelet, D.R. Hamann, M. Schluter, . Phys. Rev. B 26 (1982) 4199. [13] P. Briddon, R. Jones, Phys. Stat. Sol. (B) 217 (2000) 131. [14] A.T. Blumenau, M.I. Heggie, C.J. Fall, R. Jones, T. Frauenheim, Phys. Rev. B 65 (2002) 205205. [15] J.P. Hirth, J. Lothe, Theory of Dislocations, 2nd Edition, Wiley, New York, 1982. [16] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. . [17] H. Iwata, U. Lindefelt, S. Oberg, P.R. Briddon, Mater. Sci. Forum 529 (2002) 389. . [18] H. Iwata, U. Lindefelt, S. Oberg, P.R. Briddon, Phys. Rev. B 65 (2002) 033203. [19] M.S. Miao, S. Limpijumnong, W.R.L. Lambrecht, Appl. Phys. Lett. 79 (2001) 4360. [20] R.W. Nunes, J. Bennetto, D. Vanderbilt, Phys. Rev. B 57 (1998) 10388. [21] M. Skowronski, J.Q. Liu, W.M. Vetter, M. Dudley, C. Hallin, H. Lendenmann, J. Appl. Phys. 92 (2002) 4699.
Physica B 340–342 (2003) 160–164
Basal plane partial dislocations in silicon carbide c . A.T. Blumenaua,*, R. Jonesb, S. Oberg , P.R. Briddond, T. Frauenheima Department of Physics, Universitat . Paderborn, D - 33098 Paderborn, Germany b School of Physics, University of Exeter, Exeter EX4 4QL, UK c ( Sweden Department of Mathematics, Lulea( University of Technology, S-97187 Lulea, d Department of Physics, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, UK a
Abstract Under operating conditions (forward bias) bipolar 4H- and 6H-SiC devices are known to degrade rapidly through stacking fault formation and expansion in the basal plane. It has been suggested that a recombination-enhanced dislocation glide (REDG) mechanism allows the bordering Shockley partial dislocations to overcome their barrier to glide motion and thus results in the observed stacking fault growth. In this work, we investigate the structure and properties of the participating Shockley partials by means of density functional-based atomistic calculations. Their glide motion is modelled in a process involving the formation and subsequent migration of kinks. This in combination with an analysis of the electronic structure of the partials allows an identification of those types which will be affected by the REDG mechanism. r 2003 Published by Elsevier B.V. PACS: 61.72.Lk; 71.15.Nc; 71.55.i Keywords: Dislocations; Silicon carbide; Electronic structure; Glide motion
1. Introduction The wide band gap semiconductor silicon carbide (SiC) is a promising material for highpower, high-temperature and high-frequency applications. However, under forward bias, bipolar 4H- and 6H-SiC devices (high-power diodes) are known to degrade rapidly through stacking fault formation and expansion in the basal plane [1–5]. This results in a considerable drop in the forward voltage, rendering the device useless after few days of constant operation. The expanding stacking
*Corresponding author. E-mail address: [email protected] (A.T. Blumenau). 0921-4526/$ - see front matter r 2003 Published by Elsevier B.V. doi:10.1016/j.physb.2003.09.046
faults are of triangular and rhombic shape [3] and their edges were identified to be Shockley partial dislocations with Burgers vectors of 13/1 1% 0 0S [5,6]. It has been suggested that a recombinationenhanced dislocation glide mechanism (REDG) is responsible for the observed effect. This REDG mechanism requires non-radiative electron–hole recombination sites to lie along the dislocation line [7,8]. Part of the energy released following recombination has then to be directed into the formation and migration of kinks at the dislocation which are then able to move at or near room temperatures [9]. In this work, the atomic and electronic structure of the participating basal plane partial dislocations are investigated theoretically. The barriers to
ARTICLE IN PRESS A.T. Blumenau et al. / Physica B 340–342 (2003) 160–164
In this work, two computational methods with a different level of approximation to density functional theory (DFT) have been applied: Densityfunctional-based tight-binding method (DFTB) and ab initio modelling program (AIMPRO). The DFTB method is a tight-binding method using a minimal basis of atomic orbitals (linear combination of atomic orbitals, LCAO). The twocentre Hamiltonian and overlap matrix elements are obtained from atom-centred valence electron orbitals and the atomic potentials from single atom DFT calculations. Exchange and correlation contributions in the total energy as well as the core–core repulsion are taken into account by a repulsive pair-potential. The latter is obtained by comparison with DFT calculations for selected reference systems [11]. The AIMPRO method uses the pseudopotentials of Bachelet et al. [12] in combination with a Gaussian basis set and has been described in detail elsewhere [13]. This method requires greater computing resources and thus in this work, geometrical optimisation as well as the calculation of formation energies and barriers are carried out using DFTB with models containing around 600 atoms. Comparison for smaller models gave good agreement between both methods.
3. Straight dislocations To model the straight dislocation core, a unit cell is constructed by inserting a single dislocation into an infinite cylinder whose surface dangling bonds are hydrogenated. The dislocation is aligned with the axis of the cylinder and is periodic along ( of vacuum the dislocation line. More than 100 A
30° Si [1120]
2. Computational methods
separates cylinders stacked on a square lattice. Hence, this periodic cluster (or supercell–cluster hybrid) approach respects the periodicity of the dislocation along the line and avoids artificial dislocation–dislocation interactions which occur when a dipole is inserted into a supercell. The cylinders have a period twice that of the cubic or hexagonal lattice and the unit cell contains B600 atoms. Structural relaxations are performed at the G-point only. More details on this approach are given in Ref. [14]. Fig. 1 shows the relaxed core structure of a Si terminated 30 partial in 2H-SiC. On the terminating line of Si atoms, Si–Si reconstruction bonds are formed resulting in a double-period structure. These reconstruction bonds are about 0.7% stretched compared to bulk Si. The situation is similar at the carbon terminated 30 partial (not shown). However compared to diamond, here the C–C reconstruction bonds are stretched considerably by 17%. The structure of the single-period (SP) and the double-period (DP) core reconstructions of the 90 partial have been discussed in Ref. [10]. Also for the 90 partial, the C–C bonds were found considerably stretched (10–15%) and the Si–Si bonds were of almost bulk Si bondlength (0.4– 1.4% compressed). Identical calculations were carried out in 3C-SiC giving almost identical results. Therefore, the stacking seems to have no major influence and we are confident that our results are representative for 4H and 6H as well. From linear elasticity theory, it is known that the elastic energy contained in a cylinder of radius R around the dislocation depends logarithmically
[0001]
partial glide motion are obtained by modelling the formation and migration of kinks along the dislocation line. The main emphasis lies on the 30 Shockley partial, as the 90 partial has been investigated earlier [10]. However, to facilitate comparison, selected results for the 90 partials are reproduced.
161
: faulted region
Fig. 1. The relaxed core structure of the 30 silicon terminated partial. The structure is shown both looking along the dislocation line ½1 1 2% 0 and in the basal plane (0001). The region of the intrinsic stacking fault is shaded.
ARTICLE IN PRESS A.T. Blumenau et al. / Physica B 340–342 (2003) 160–164
on the radius [15]: EðRÞ ¼
kðbÞjb~j2 R ln þ Ec ; Rc 4p
RXRc :
ð1Þ
The energy factor kðbÞ and hence the gradient of ( can be obtained from the energy vs. lnðR=AÞ elastic constants, the line direction and the Burgers vector b~ [15]. Anisotropic linear elasticity theory using the DFTB elastic constants yields kðbÞ ¼ 195 and 249 GPa for the 30 and 90 partials in 3C. For 2H, we find 191 and 246 GPa; respectively, indicating the errors within our calculations. The core energy Ec can be obtained in comparison with the atomistic calculations: The DFTB method allows an easy definition of the formation energy projected onto atom sites. This is used to define a radial formation energy Ef ðRÞ which can be compared directly with EðRÞ in Eq. (1). Table 1 gives the resulting energy factors and core energies in 3C-SiC and 2H-SiC, the latter given as formation energies with respect to stoichiometric bulk SiC. To reduce the computational effort, periodic cluster models smaller than those used to obtain the core energies, containing about 120 atoms, are used to calculate the electronic band structures with the DFT-pseudopotential code AIMPRO. The structures are relaxed, using a MonkhorstPack 2 1 1 set of k~-points [16]. The projected electronic band structure of the relaxed structure is then calculated at 21 different k~-points along the dislocation axis. Fig. 2 shows the projected band structures of the 30 partials in 2H-SiC. Both for
the C and the Si terminated partial, the localised gap states below the conduction band minimum (CBM) are related to the stacking fault. For 2H, we find these states as deep as B0:5–0:6 eV below the CBM. Similar bands have been found in 4H and 6H (B0:2 eV below the CBM) [17,18] and for intrinsic faults in 4H (B0:3 eV below the CBM) [19]. The striking difference between the C and the Si terminated partial is the presence of a localised defect band (reaching as far as 0:4 eV above the valence band maximum) related to the Si–Si reconstruction bonds. No such band is found at the C partial. Hence, results are very similar to those for the 90 partials [10].
(eV)
162
4
2
0
-2 30° Si
30° C
Fig. 2. The projected bandstructures of the silicon and the carbon terminated 30 partial (aligned at the valence band maximum).
Table 1 30 Si
kðbÞ ( Ec ð6 AÞ
(GPa) ( ðeV=AÞ
90 Si (SP)
3C
2H
3C
2H
3C
2H
203 0.48
194 0.47
249 0.68
251 0.69
250 0.59
242 0.60
30 C
kðbÞ ( Ec ð6 AÞ
(GPa) ( ðeV=AÞ
90 Si (DP)
90 C (SP)
90 C (DP)
3C
2H
3C
2H
3C
2H
194 0.86
197 0.85
242 1.27
237 1.27
244 1.17
244 1.17
( of the investigated Shockley partials in stoichiometric 3C Energy factors kðbÞ; and core energies Ec (corresponding to a radius of 6 A) and 2H SiC. kðbÞ is obtained as fit to Er ðRÞ vs. InðR=Rc Þ plot following Eq. (1).
ARTICLE IN PRESS A.T. Blumenau et al. / Physica B 340–342 (2003) 160–164
and angles due to the different character of the Si–Si and C–C bonds. The kink formation energies are obtained by introducing kink pairs on a dislocation embedded in a cluster as described in Ref. [10]. The elastic kink–kink interaction energy within the cluster is subtracted following Ref. [15]. For the smallest possible double kink, the interaction energy is found as 0:49 eV: Since the structures of left and right kinks are very different, one cannot expect similar formation energies. Hence, the single-kink energies are not individually found, but only the sum of two kink formation energies. We calculated the sum of formation energies for all possible four combinations of left with right kinks. LK1–RK1 occurs as the lowest energy pair. The energies of other combinations can be found by adding the appropriate D terms given in Table 2. To find the intermediate saddlepoint structure between two different kinks—and hence the migration barrier between them—the elementary kink migration step was parameterised by the coordinates of each of two core (primary) atoms at the kink projected onto the connecting line between their initial and final positions. Varying these two parameters independently, yields a twodimensional energy surface. Within the vicinity of the saddle point, the parameter mesh was chosen 1 as 100 : For details, see Ref. [10] where the method is explained. Table 2 displays the resulting barriers Wm found for right and left kinks. In the case when obstacles (point defects and impurities) are not present, and for short dislocation segments, the glide velocity is controlled by the sum of the double-kink formation energy 2Ef and the kink migration barrier Wm ; whereas for
4. Thermally activated glide Dislocation glide arises from an external stress acting on the dislocation. When the stress is insufficient to overcome the Peierls barrier, kinks must be generated at the dislocation by a thermal process and motion occurs by their migration along the dislocation line. Due to the doubleperiodicity present in the reconstructed 30 dislocation shown in Fig. 1, two varieties of left (LK1 and LK2) and right kinks (RK1 and RK2) occur. Fig. 3 shows the four different kink structures at the silicon-terminated 30 partial. All atoms in these structures are four-fold coordinated suggesting relatively low kink formation energies. Qualitatively, the structures at the carbon partial are very similar, with only differences in bond lengths
LK1
LK2
RK1
163
RK2
Fig. 3. The two different left kinks (LK) and the two right kinks (RK) of the 30 silicon terminated partial. All structures are shown in the basal plane and the region of the intrinsic stacking fault is shaded.
Table 2
Si C
2Ef ð1Þ
DðLKÞ
Wm ðLKÞ
DðRKÞ
Wm ðRKÞ
Q% 30
Q90
1.62 2.21
2.87 2.11
3.79 3.00
0.28 0.33
2.87 1.78
4.95 (4.14) 4.60 (3.50)
4.09 (3.58) 2.74 (2.29)
Kink formation energies Ef and migration barriers Wm for the 30 Shockley partials. 2Ef ð1Þ ¼ Ef ðLK1Þ þ Ef ðRK1Þ denotes the sum of the formation energies of the low-energy left and the low-energy right kink. DðLKÞ gives the energy difference between the low-energy (LK1) and the high-energy left kinks (LK2) and DðRKÞ denotes the same quantity for right kinks. To facilitate comparison with the 90 partials, an average activation energy Q% 30 ¼ ð2Ef ðLK1Þ þ Wm ðLKÞ þ 2Ef ðRK1Þ þ Wm ðRKÞÞ=2 is given. The glide activation energy assumes short dislocation segments and the values in brackets give the corresponding values for long segments. For comparison the corresponding values for the 90 partials are given in the last column [10].
ARTICLE IN PRESS A.T. Blumenau et al. / Physica B 340–342 (2003) 160–164
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long segments the glide activation energy is given as Q ¼ Ef þ Wm [15]. Having two different barriers for left and right kink migration, respectively, results in a modified expression for the partial velocity [20]
experiment [4]. Such a mechanism would further explain the experimental observation that only one type of partial—the Si partial—is highly mobile under forward bias [21].
vdisl p eðEf ðLK1ÞþEf ðRK1ÞÞ=kT
½eWm ðLKÞ=kT þ eWm ðRKÞ=kT :
ð2Þ
The kinks with the lower migration barrier dominate the glide process. However, to allow an easy comparison, as a crude approximation an average activation barrier Q% 30 ¼ ð2Ef ðLK1Þ þ Wm ðLKÞ þ 2Ef ðRK1Þ þ Wm ðRKÞÞ=2 can be defined for short segments. Values for short and long segments are given in Table 2. At both the 30 and the 90 partials, the carbon cores are found with lower activation energies than the silicon cores. This might appear to be counter intuitive as the bonds in diamond are undoubtedly stronger than those in silicon. However, one has to keep in mind that the C–C reconstruction bonds at the partial are stretched considerably when compared with diamond, and it is this effect which lowers the energy required to break the reconstructed bond necessary for kink migration.
5. Conclusions For all partials, we found stretched C–C reconstruction bonds or, respectively, bulk like Si–Si reconstruction bonds, depending on the type of partial (C or Si terminated). This results in a higher thermal mobility of the C-terminated partials. Electrically, however, the C-terminated partials appear to be inert—only the Si partials induce filled bands low in the gap. This points towards an REDG mechanism based on a nonradiative recombination between the stacking fault-related states and those induced by the Si partial. Assuming the latter states in 4H to be around 0:3 eV below the CBM together with a gap of 3:3 eV; and the states of the Si partials to be 0:4 eV above the VBM, gives a rough estimate of 2:6 eV for the non-radiative recombination. Considering the band gap error in DFT, this is in reasonable agreement with the 2:2 eV suggested by
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