- Email: [email protected]
A simple approach to the polytypism in SiC
Journal of Crystal Growth 362 (2013) 207–210
Contents lists available at SciVerse ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
A simple approach to the polytypism in SiC Tomonori Ito n, Toru Akiyama, Kohji Nakamura Department of Physics Engineering, Mie University, Tsu 514-8507, Japan
a r t i c l e i n f o
abstract
Available online 1 August 2012
A simple approach to the polytypism in SiC is proposed on the basis of phenomenological energy formula incorporating electrostatic interactions between interatomic bond charges and between ionic charges. The versatility of our approach is confirmed by calculating the energy differences among 3C (zinc blende), 6H, 4H and 2H (wurtzite) structures with/without vacancy. When the ionicity fi ¼ 0.232 for SiC is employed, the calculated respective energy differences DE for 6H, 4H, and 2H are very small such as 0.18 meV/atom, 0.27 meV/atom, and 5.7 meV/atom where the energy origin is the energy for the most stable 3C. Furthermore, it is found that vacancy formation stabilizes 4H– or 6H–SiC. This is because the vacancy formation reduces the interatomic bond charges to destabilize 3C–SiC. These results are consistent with our previous ab initio calculations and experimental findings where 3C appears at low temperatures while 6H and 4H for SiC are favored at high temperatures enhancing vacancy formation. Our simple approach is also applied to the polytypism in SiC with impurities. & 2012 Elsevier B.V. All rights reserved.
Keywords: A1. Computer simulations A1. Crystal structure A1. Defects B2. Semiconducting silicon compounds
1. Introduction Silicon carbide (SiC) has been considered as one of the alternative power semiconductor materials due to its excellent properties such as wide band gap, high break down field, and high thermal conductivity. In particular, SiC with 3C (zinc blende structure) is suitable for high-frequency power devices from its high electron mobility and high electron-saturation-velocity. From the crystallographic viewpoints, however, SiC is well known as having dozens of polytypes including 4H and 6H in addition to 2H (wurtzite) and 3C that consist of different ordered stacking sequences of SiC atomic double layers [1]. In order to investigate the polytypism in SiC, there have been some theoretical studies using the ab initio total energy calculations. Cheng et al. [2] and Shaw and Heine [3] theoretically investigated polytypes of SiC in conjunction with the energy of the sequence of layers. Their results imply that 6H–SiC is the most stable among 3C, 6H, 4H and 2H structures. Similar results were obtained by Raffy et al. [4]. On the other hand, another ab initio pseudopotential calculations suggested that 4H and 3C are respectively the most stable structures [5,6]. More recently, Kobayashi and Komatsu [7] performed ab initio molecular dynamics calculations to obtain the total energies E of SiC polytypes with the lowest value in 4H–SiC. Although satisfying results have been obtained in some aspects for polytypes, systematic understanding for the origin of
n
Corresponding author. Tel./fax: þ 81 59 231 9724. E-mail address: [email protected] (T. Ito).
0022-0248/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jcrysgro.2012.07.031
polytypes has not been achieved. This is because of the difficulty for identifying the most stable structure of SiC due to very small energy differences 1 meV/atom among 3C, 6H, 4H, and 2H. From the experimental viewpoints, it was found that the stability of SiC polytypes is primarily dependent on temperature. The cubic form of SiC (3C–SiC) is believed to be more stable than the hexagonal structure (6H–SiC) below 2373 K [8], although some studies dispute this finding [9,10]. In contrast, 2H–SiC, which has the simplest stacking sequence, is rarely observed at higher temperatures. Krishna et al. [11] reported that single crystal 2H–SiC can be easily transformed to 3C–SiC on annealing in argon at temperatures above 1673 K. The temperature dependence of SiC polytypes inspired us to investigate the contribution of the vacancy formation in SiC to its polytypism. Our recent ab initio calculations revealed that the most stable structure is 3C without vacancy while 6H with Si-vacancy and the 4H with C-vacancy are also favorable [12,13]. This qualitatively agrees well with experimental results where 3C appears at low temperatures while 6H and 4H are favored at high temperatures enhancing vacancy formation. However, the ab initio calculations tend to obscure the physical interpretation of polytypism in SiC because of their complicated formulation. In order to make up the deficiency, we employ the simple energy formula successfully applied to the relative stability between 3C and 2H [14] to investigate SiC polytypes in this study. Using the simple approach incorporating electrostatic interactions between interatomic bond charges and between ionic charges, the energy differences among polytypes with/without vacancy and impurities are investigated to clarify the origin of polytypism in SiC.
208
T. Ito et al. / Journal of Crystal Growth 362 (2013) 207–210
2. Computational methods In order to investigate SiC polytypes, we employ our simple system energy formula, which is given by the following equation [14]: E ¼ E0 þ DEes
ð1Þ
1X V 2 i,j ij
ð2Þ
E0 ¼
where E0 is the cohesive energy estimated by interatomic potential Vij within the second nearest neighbors. Since Vij gives the same energy contribution to the polytypes of SiC, the relative stability among 3C, 6H, 4H, and 2H is determined by DEes beyond the second nearest neighbors. DEes is the electrostatic interaction between bond charges Zb(¼2) and that between ionic charges Zi (¼4 for group IV, 3 for III–V and 2 for II–VI semiconductors) beyond the second nearest neighbors depending on ionicity fi, which corresponds to the first-neighbor interlayer interaction J1 [2,3,14]. The energy formula for DEes has systematically given good estimates of the energy differences between 3C and 2H for various compound semiconductors [14]. Fig. 1 depicts the polytypes such as 3C, 6H, 4H, and 2H in SiC considered in this study, where letters A, B, and C denote the stacking sequence of layers. The schematic of electrostatic interactions considered in this study are also shown in this figure with the number of hexagonal unit NHexa and that of inter-unit interaction between the hexagonal units NInter in each structure. Here the inter-unit interaction specifying the periodicity of the hexagonal unit is incorporated only for 2H while those for 6H and 4H are negligible because of their large inter-unit distances. The shaded squares denote the region to incorporate electrostatic interactions in each structure. Considering the formulation of DEes, the energy differences DE6H–3C between 6H and 3C, DE4H–3C between 4H and 3C, and DE2H–3C between 2H and 3C are easily given by following equation in six-double layer unit cell considering NHexa and NInter. ( ) Z 2b Z 2i DE6H3C ¼ 2K ð1f i Þ f i ð3Þ r bb r ii (
Z2 Z2 DE4H3C ¼ 3K ð1f i Þ b f i i r bb r ii
)
( ) 3 Z 2b Z 2i ð1f i Þ DE2H3C ¼ 4K f i 2 r bb r ii
ð4Þ
Here rii ( ¼5c/8) is the distance between ionic charges located at the third nearest neighbor lattice sites and rbb ( ¼c/2) the distance between bond charges located at the center of interatomic bonds. The coefficient K is 8.7 meV A˚ that was determined by reproducing the energy difference of 25.3 meV/atom between 3C and 2H for C with fi ¼0 obtained by ab initio calculations [14,15]. We use the values of fi ¼0.232 for SiC fitted to the energy differences between 3C and 2H structures obtained by our ab initio calculations [12,13]. This value is consistent with values of 0.168 estimated by Pauling’s electronegativity [14,16], 0.177 after Phillips [17], 0.240 and 0.273 fitted to another ab initio calculations [2,6]. In the calculation for SiC with vacancy and impurities, we employ the value of bond charges Zb and ionic charges Zi such as Zb ¼1 and Zi ¼0 for vacancy and Zb ¼2.25 and Zi ¼5 for substitutional N impurity. The energy differences of SiC polytypes are calculated as a function of ionicity fi.
3. Results and discussion Fig. 2 shows the calculated relative energy differences DE among 3C, 6H, 4H and 2H structured C, Si, and SiC, where the energy for 3C structure is employed as an energy origin. The results obtained by our ab initio calculations (dotted line) [12,13] are also shown in this figure for comparison. This figure implies that 3C–SiC is the most favorable for all the semiconductors. However, the DE among four structures for SiC are very small without the explicit dependence on the hexagonality whereas the DE for C and Si linearly depend on the hexagonality in order of DE3C–3C ¼0 o DE6H–3C o DE4H–3C o DE2H–3C. This is consistent with experimental findings where SiC exhibits polytypes with 6H, 4H, and 3C depending on temperature while 3C only appears in C and Si among these polytypes. Furthermore, it should be noted that the results of DE6H–3C ¼0.18 meV/atom and DE4H–3C ¼0.27 meV/ atom estimated by simple energy formula are favorably compared with those of DE6H–3C ¼0.23 meV/atom and DE4H–3C ¼ 0.02 meV/ atom obtained by our previous ab initio calculations. The small energy differences are due to the fact that covalent contribution described by the first term in Eqs. (3) and (4) is competitive with ionic contribution corresponding to the second term in Eqs. (3) and (4) in SiC with fi ¼ 0.232. Therefore, these calculated results suggest that this simple approach is feasible for investigating the polytypism in SiC.
ð5Þ
Fig. 1. Schematic of the polytypes such as 3C, 6H, 4H, and 2H in SiC considered in this study. Letters A, B, and C denote the stacking sequence of layers in each structure. Electrostatic interactions between bond charges and between ionic charges incorporated in our simple approach are also shown by shaded area and schematics. NHexa and NInter denote the numbers of hexagonal unit and interaction between the hexagonal units in each structure, respectively.
Fig. 2. Calculated energy differences for 3C, 6H, 4H and 2H structured SiC (open square), Si (open triangle), and C (open circle) as a function of hexagonality. Solid and dotted lines denote the energy differences obtained by our simple approach and ab initio calculations, respectively.
T. Ito et al. / Journal of Crystal Growth 362 (2013) 207–210
Our previous ab initio calculations revealed that vacancy formation strongly contributes to the polytypism in SiC where the vacancy formation relatively destabilizes 3C to 6H and 4H [12,13]. Based on the results and bond charge calculations, we conjectured that the decrease in bond charge around vacancy is closely related to the stabilization of 6H and 4H. In order to clarify this, the simple approach is applied to SiC with vacancy. Fig. 3 shows the calculated energy differences DE (solid line) among 3C, 6H, 4H and 2H structured SiC with vacancy. These results imply that 6H– and 4H–SiC with larger energy profits than those in SiC without vacancy are more favorable than 3C–SiC with vacancy. This is qualitatively consistent with experimental results where 3C appears at low temperatures while 6H and 4H are favored at high temperatures enhancing vacancy formation. However, the calculated results are not quantitatively compared with averaged value for SiC with Si- and C-vacancy obtained by our ab initio calculations (dotted line) [12,13] as shown in this figure. The discrepancy can be interpreted by considering ionicity around vacancy. In this calculation, we employ ionicity value of fi ¼0.232 for perfect Si–C bond that seems to be smaller than that around vacancy. Assuming fi ¼0.7 for the interatomic bond around vacancy, for example, the energetic trend denoted by dashed line in Fig. 3 is favorably compared with that in ab initio calculations. Consequently, vacancy formation inducing large deficit of bond charges favors 6H– and 4H–SiC. In order to check the versatility of our simple approach, we investigate the polytypism in SiC with impurities including nitrogen. This is because several publications reported observation of stacking faults transforming to 3C–SiC in heavily nitrogen-doped 4H–SiC polytype [18–22]. Fig. 4 shows the calculated energy differences for SiC with group V impurity as a function of ionicity of interatomic bond around the impurity. The calculated results suggest that 6H– or 4H–SiC transforms to 3C–SiC when ionicity for interatomic bonds around impurities is less than 0.33. According to Pauling’s formula to determine the ionicity for A–B interatomic bond using the electronegativity w such as fi ¼1 exp{-1/ 4(wA wB)2} [16], the ionicity of Si–N is estimated to be 0.30. Therefore, N-doped SiC may change its structure from 4H to 3C. This is consistent with our recent ab initio calculations where 3C– SiC is stabilized by N substituting for C with formation of Si–N [23]. In our calculations, however, vacancy and N concentrations are very large such as 10% inconsistent with experimental
209
Fig. 4. Calculated energy differences for 3C, 6H, 4H and 2H structured SiC as a function of ionicity for interatomic bond between group IV constituent and group V impurity atoms. Solid, dashed and dotted lines denote the energy differences for 6H–, 4H–, and 2H–SiC, respectively. Critical ionicity for phase transformation between 3C–SiC and 6H/4H–SiC is denoted by arrow (fi ¼ 0.33).
conditions. Thus larger number of atom unit cells should be employed in the calculations for quantitative discussions to clarify the critical defect concentrations for polytypism in SiC. Moreover, our simple approach incorporates only electrostatic contribution without strain contribution. Therefore, further study using our simple energy formula with the aid of the empirical interatomic potentials incorporating strain contribution is necessary for quantitatively investigating the polytypism in SiC with vacancy formation and impurity doping in a large unit cell.
4. Conclusion In this paper, we have theoretically investigated the polytypes in SiC using a simple energy formula with/without vacancy and impurity in 3C, 6H, 4H and 2H structures. The calculated results imply that 3C–SiC is the most favorable. However, the energy differences DE among four structures for SiC are very small without explicit dependence on the hexagonality in contrast with C and Si without polytypes. The small energy differences are due to the fact that covalent contribution is competitive with ionic contribution for SiC with fi ¼ 0.232. The DE with vacancy explicitly reveals that the vacancy formation changes the stable structure from 3C to 6H and 4H in SiC. This is because vacancy formation inducing large deficit of bond charges destabilizes 3C to favor 6H and 4H. These results qualitatively agree well with our previously reported ab initio calculations and experimental results where 3C appears at low temperatures while 6H and 4H are favored at high temperatures enhancing vacancy formation. Furthermore, the versatility of our approach for impurity effect was exemplified by investigating the polytypism in N-doped SiC. Although using the simple energy formula with the aid of empirical interatomic potential should be necessary in the calculations for quantitative discussions, these results suggest that our simple approach is feasible for predicting the polytypes in SiC incorporating vacancy formation and impurity doping using a large number of atom unit cells.
Acknowledgments Fig. 3. Calculated energy differences for 3C, 6H, 4H and 2H structured SiC as a function of hexagonality. Solid and dashed lines denote the energy differences obtained by our simple approach using ionicity value of fi ¼ 0.232 and fi ¼ 0.7 for interatomic bonds around vacancy, respectively. The results obtained by ab initio calculations (dotted line) are also shown in this figure for comparison.
This work was partly supported by the Grant-in-Aid for Scientific Research Grant no. 21560032 of the Ministry of Education, Science, and Culture of Japan.
210
T. Ito et al. / Journal of Crystal Growth 362 (2013) 207–210
References [1] A.R. Verma, P. Krishna, Polymorphism and Polytypism in Crystals, John Wiley and Sons, Inc, New York, 1966. [2] C. Cheng, R.J. Needs, V. Heine, Journal of Physics C 21 (1988) 1049. [3] J.J.A. Shaw, V. Heine, Journal of Physics Condensed Matter 2 (1990) 4351. ¨ [4] C. Raffy, J. Furthmuller, F. Bechstedt, Physical Review B 66 (2002) 075201. ¨ [5] P. Kackell, B. Wenzien, F. Bechstedt, Physical Review B 50 (1994) 17037. [6] T. Ito, K. Sano, T. Akiyama, K. Nakamura, Thin Solid Films 508 (2006) 243. [7] K. Kobayashi, S. Komatsu, Journal of the Physical Society of Japan 77 (2008) 084703. [8] W.F. Knippenberg, Philips Research Reports 18 (1963) 161. [9] W. Weltner Jr., Journal of Chemical Physics 51 (1969) 2469. [10] L.B. Griffiths, Journal of Physics and Chemistry of Solids 27 (1966) 257. [11] P. Krishna, R. Marshall, C.E. Ryan, Journal of Crystal Growth 8 (1971) 129. [12] T. Ito, T. Kondo, T. Akiyama, K. Nakamura, Physica Status Solidi C 8 (2011) 583.
[13] T. Ito, T. Kondo, T. Akiyama, K. Nakamura, Journal of Crystal Growth 318 (2011) 141. [14] T. Ito, Japanese Journal of Applied Physics 37 (1998) L1217. [15] C.-Y. Yeh, Z.W. Lu, S. Froyen, A. Zunger, Physical Review B 46 (1992) 10086. [16] L. Pauling, The Nature of the Chemical Bond, Cornell University Press, New York, 1960. [17] J.C. Phillips, Reviews of Modern Physics 42 (1970) 317. [18] R.S. Okojie, M. Xhang, P. Pirouz, S. Tumlha, G. Jessem, L.J. Brillson, Applied Physics Letters 79 (2001) 3056. [19] B.J. Skromme, K. Palle, C.D. Poweleit, L.R. Bryant, W.M. Vetter, M. Dudley, K. Moore, T. Gehoski, Materials Science Forum 383–393 (2002) 455. [20] J.Q. Liu, H.J. Chung, T. Kur, Q. Li, M. Skowronski, Applied Physics Letters 80 (2002) 2111. [21] L.J. Brillson, S. Tumakha, G.H. Jessen, R.S. Okojie, M. Zhang, P. Pirouz, Applied Physics Letters 81 (2002) 2785. [22] S. Juillaguet, C. Balloud, V. Souliere, C. Sartel, J. Camassel, Y. Monteil, Physica Status Solidi A 202 (2005) 593. [23] T. Ito, T. Akiyama, K. Nakamura, Physica Status Solidi C, in preparation.
Contents lists available at SciVerse ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
A simple approach to the polytypism in SiC Tomonori Ito n, Toru Akiyama, Kohji Nakamura Department of Physics Engineering, Mie University, Tsu 514-8507, Japan
a r t i c l e i n f o
abstract
Available online 1 August 2012
A simple approach to the polytypism in SiC is proposed on the basis of phenomenological energy formula incorporating electrostatic interactions between interatomic bond charges and between ionic charges. The versatility of our approach is confirmed by calculating the energy differences among 3C (zinc blende), 6H, 4H and 2H (wurtzite) structures with/without vacancy. When the ionicity fi ¼ 0.232 for SiC is employed, the calculated respective energy differences DE for 6H, 4H, and 2H are very small such as 0.18 meV/atom, 0.27 meV/atom, and 5.7 meV/atom where the energy origin is the energy for the most stable 3C. Furthermore, it is found that vacancy formation stabilizes 4H– or 6H–SiC. This is because the vacancy formation reduces the interatomic bond charges to destabilize 3C–SiC. These results are consistent with our previous ab initio calculations and experimental findings where 3C appears at low temperatures while 6H and 4H for SiC are favored at high temperatures enhancing vacancy formation. Our simple approach is also applied to the polytypism in SiC with impurities. & 2012 Elsevier B.V. All rights reserved.
Keywords: A1. Computer simulations A1. Crystal structure A1. Defects B2. Semiconducting silicon compounds
1. Introduction Silicon carbide (SiC) has been considered as one of the alternative power semiconductor materials due to its excellent properties such as wide band gap, high break down field, and high thermal conductivity. In particular, SiC with 3C (zinc blende structure) is suitable for high-frequency power devices from its high electron mobility and high electron-saturation-velocity. From the crystallographic viewpoints, however, SiC is well known as having dozens of polytypes including 4H and 6H in addition to 2H (wurtzite) and 3C that consist of different ordered stacking sequences of SiC atomic double layers [1]. In order to investigate the polytypism in SiC, there have been some theoretical studies using the ab initio total energy calculations. Cheng et al. [2] and Shaw and Heine [3] theoretically investigated polytypes of SiC in conjunction with the energy of the sequence of layers. Their results imply that 6H–SiC is the most stable among 3C, 6H, 4H and 2H structures. Similar results were obtained by Raffy et al. [4]. On the other hand, another ab initio pseudopotential calculations suggested that 4H and 3C are respectively the most stable structures [5,6]. More recently, Kobayashi and Komatsu [7] performed ab initio molecular dynamics calculations to obtain the total energies E of SiC polytypes with the lowest value in 4H–SiC. Although satisfying results have been obtained in some aspects for polytypes, systematic understanding for the origin of
n
Corresponding author. Tel./fax: þ 81 59 231 9724. E-mail address: [email protected] (T. Ito).
0022-0248/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jcrysgro.2012.07.031
polytypes has not been achieved. This is because of the difficulty for identifying the most stable structure of SiC due to very small energy differences 1 meV/atom among 3C, 6H, 4H, and 2H. From the experimental viewpoints, it was found that the stability of SiC polytypes is primarily dependent on temperature. The cubic form of SiC (3C–SiC) is believed to be more stable than the hexagonal structure (6H–SiC) below 2373 K [8], although some studies dispute this finding [9,10]. In contrast, 2H–SiC, which has the simplest stacking sequence, is rarely observed at higher temperatures. Krishna et al. [11] reported that single crystal 2H–SiC can be easily transformed to 3C–SiC on annealing in argon at temperatures above 1673 K. The temperature dependence of SiC polytypes inspired us to investigate the contribution of the vacancy formation in SiC to its polytypism. Our recent ab initio calculations revealed that the most stable structure is 3C without vacancy while 6H with Si-vacancy and the 4H with C-vacancy are also favorable [12,13]. This qualitatively agrees well with experimental results where 3C appears at low temperatures while 6H and 4H are favored at high temperatures enhancing vacancy formation. However, the ab initio calculations tend to obscure the physical interpretation of polytypism in SiC because of their complicated formulation. In order to make up the deficiency, we employ the simple energy formula successfully applied to the relative stability between 3C and 2H [14] to investigate SiC polytypes in this study. Using the simple approach incorporating electrostatic interactions between interatomic bond charges and between ionic charges, the energy differences among polytypes with/without vacancy and impurities are investigated to clarify the origin of polytypism in SiC.
208
T. Ito et al. / Journal of Crystal Growth 362 (2013) 207–210
2. Computational methods In order to investigate SiC polytypes, we employ our simple system energy formula, which is given by the following equation [14]: E ¼ E0 þ DEes
ð1Þ
1X V 2 i,j ij
ð2Þ
E0 ¼
where E0 is the cohesive energy estimated by interatomic potential Vij within the second nearest neighbors. Since Vij gives the same energy contribution to the polytypes of SiC, the relative stability among 3C, 6H, 4H, and 2H is determined by DEes beyond the second nearest neighbors. DEes is the electrostatic interaction between bond charges Zb(¼2) and that between ionic charges Zi (¼4 for group IV, 3 for III–V and 2 for II–VI semiconductors) beyond the second nearest neighbors depending on ionicity fi, which corresponds to the first-neighbor interlayer interaction J1 [2,3,14]. The energy formula for DEes has systematically given good estimates of the energy differences between 3C and 2H for various compound semiconductors [14]. Fig. 1 depicts the polytypes such as 3C, 6H, 4H, and 2H in SiC considered in this study, where letters A, B, and C denote the stacking sequence of layers. The schematic of electrostatic interactions considered in this study are also shown in this figure with the number of hexagonal unit NHexa and that of inter-unit interaction between the hexagonal units NInter in each structure. Here the inter-unit interaction specifying the periodicity of the hexagonal unit is incorporated only for 2H while those for 6H and 4H are negligible because of their large inter-unit distances. The shaded squares denote the region to incorporate electrostatic interactions in each structure. Considering the formulation of DEes, the energy differences DE6H–3C between 6H and 3C, DE4H–3C between 4H and 3C, and DE2H–3C between 2H and 3C are easily given by following equation in six-double layer unit cell considering NHexa and NInter. ( ) Z 2b Z 2i DE6H3C ¼ 2K ð1f i Þ f i ð3Þ r bb r ii (
Z2 Z2 DE4H3C ¼ 3K ð1f i Þ b f i i r bb r ii
)
( ) 3 Z 2b Z 2i ð1f i Þ DE2H3C ¼ 4K f i 2 r bb r ii
ð4Þ
Here rii ( ¼5c/8) is the distance between ionic charges located at the third nearest neighbor lattice sites and rbb ( ¼c/2) the distance between bond charges located at the center of interatomic bonds. The coefficient K is 8.7 meV A˚ that was determined by reproducing the energy difference of 25.3 meV/atom between 3C and 2H for C with fi ¼0 obtained by ab initio calculations [14,15]. We use the values of fi ¼0.232 for SiC fitted to the energy differences between 3C and 2H structures obtained by our ab initio calculations [12,13]. This value is consistent with values of 0.168 estimated by Pauling’s electronegativity [14,16], 0.177 after Phillips [17], 0.240 and 0.273 fitted to another ab initio calculations [2,6]. In the calculation for SiC with vacancy and impurities, we employ the value of bond charges Zb and ionic charges Zi such as Zb ¼1 and Zi ¼0 for vacancy and Zb ¼2.25 and Zi ¼5 for substitutional N impurity. The energy differences of SiC polytypes are calculated as a function of ionicity fi.
3. Results and discussion Fig. 2 shows the calculated relative energy differences DE among 3C, 6H, 4H and 2H structured C, Si, and SiC, where the energy for 3C structure is employed as an energy origin. The results obtained by our ab initio calculations (dotted line) [12,13] are also shown in this figure for comparison. This figure implies that 3C–SiC is the most favorable for all the semiconductors. However, the DE among four structures for SiC are very small without the explicit dependence on the hexagonality whereas the DE for C and Si linearly depend on the hexagonality in order of DE3C–3C ¼0 o DE6H–3C o DE4H–3C o DE2H–3C. This is consistent with experimental findings where SiC exhibits polytypes with 6H, 4H, and 3C depending on temperature while 3C only appears in C and Si among these polytypes. Furthermore, it should be noted that the results of DE6H–3C ¼0.18 meV/atom and DE4H–3C ¼0.27 meV/ atom estimated by simple energy formula are favorably compared with those of DE6H–3C ¼0.23 meV/atom and DE4H–3C ¼ 0.02 meV/ atom obtained by our previous ab initio calculations. The small energy differences are due to the fact that covalent contribution described by the first term in Eqs. (3) and (4) is competitive with ionic contribution corresponding to the second term in Eqs. (3) and (4) in SiC with fi ¼ 0.232. Therefore, these calculated results suggest that this simple approach is feasible for investigating the polytypism in SiC.
ð5Þ
Fig. 1. Schematic of the polytypes such as 3C, 6H, 4H, and 2H in SiC considered in this study. Letters A, B, and C denote the stacking sequence of layers in each structure. Electrostatic interactions between bond charges and between ionic charges incorporated in our simple approach are also shown by shaded area and schematics. NHexa and NInter denote the numbers of hexagonal unit and interaction between the hexagonal units in each structure, respectively.
Fig. 2. Calculated energy differences for 3C, 6H, 4H and 2H structured SiC (open square), Si (open triangle), and C (open circle) as a function of hexagonality. Solid and dotted lines denote the energy differences obtained by our simple approach and ab initio calculations, respectively.
T. Ito et al. / Journal of Crystal Growth 362 (2013) 207–210
Our previous ab initio calculations revealed that vacancy formation strongly contributes to the polytypism in SiC where the vacancy formation relatively destabilizes 3C to 6H and 4H [12,13]. Based on the results and bond charge calculations, we conjectured that the decrease in bond charge around vacancy is closely related to the stabilization of 6H and 4H. In order to clarify this, the simple approach is applied to SiC with vacancy. Fig. 3 shows the calculated energy differences DE (solid line) among 3C, 6H, 4H and 2H structured SiC with vacancy. These results imply that 6H– and 4H–SiC with larger energy profits than those in SiC without vacancy are more favorable than 3C–SiC with vacancy. This is qualitatively consistent with experimental results where 3C appears at low temperatures while 6H and 4H are favored at high temperatures enhancing vacancy formation. However, the calculated results are not quantitatively compared with averaged value for SiC with Si- and C-vacancy obtained by our ab initio calculations (dotted line) [12,13] as shown in this figure. The discrepancy can be interpreted by considering ionicity around vacancy. In this calculation, we employ ionicity value of fi ¼0.232 for perfect Si–C bond that seems to be smaller than that around vacancy. Assuming fi ¼0.7 for the interatomic bond around vacancy, for example, the energetic trend denoted by dashed line in Fig. 3 is favorably compared with that in ab initio calculations. Consequently, vacancy formation inducing large deficit of bond charges favors 6H– and 4H–SiC. In order to check the versatility of our simple approach, we investigate the polytypism in SiC with impurities including nitrogen. This is because several publications reported observation of stacking faults transforming to 3C–SiC in heavily nitrogen-doped 4H–SiC polytype [18–22]. Fig. 4 shows the calculated energy differences for SiC with group V impurity as a function of ionicity of interatomic bond around the impurity. The calculated results suggest that 6H– or 4H–SiC transforms to 3C–SiC when ionicity for interatomic bonds around impurities is less than 0.33. According to Pauling’s formula to determine the ionicity for A–B interatomic bond using the electronegativity w such as fi ¼1 exp{-1/ 4(wA wB)2} [16], the ionicity of Si–N is estimated to be 0.30. Therefore, N-doped SiC may change its structure from 4H to 3C. This is consistent with our recent ab initio calculations where 3C– SiC is stabilized by N substituting for C with formation of Si–N [23]. In our calculations, however, vacancy and N concentrations are very large such as 10% inconsistent with experimental
209
Fig. 4. Calculated energy differences for 3C, 6H, 4H and 2H structured SiC as a function of ionicity for interatomic bond between group IV constituent and group V impurity atoms. Solid, dashed and dotted lines denote the energy differences for 6H–, 4H–, and 2H–SiC, respectively. Critical ionicity for phase transformation between 3C–SiC and 6H/4H–SiC is denoted by arrow (fi ¼ 0.33).
conditions. Thus larger number of atom unit cells should be employed in the calculations for quantitative discussions to clarify the critical defect concentrations for polytypism in SiC. Moreover, our simple approach incorporates only electrostatic contribution without strain contribution. Therefore, further study using our simple energy formula with the aid of the empirical interatomic potentials incorporating strain contribution is necessary for quantitatively investigating the polytypism in SiC with vacancy formation and impurity doping in a large unit cell.
4. Conclusion In this paper, we have theoretically investigated the polytypes in SiC using a simple energy formula with/without vacancy and impurity in 3C, 6H, 4H and 2H structures. The calculated results imply that 3C–SiC is the most favorable. However, the energy differences DE among four structures for SiC are very small without explicit dependence on the hexagonality in contrast with C and Si without polytypes. The small energy differences are due to the fact that covalent contribution is competitive with ionic contribution for SiC with fi ¼ 0.232. The DE with vacancy explicitly reveals that the vacancy formation changes the stable structure from 3C to 6H and 4H in SiC. This is because vacancy formation inducing large deficit of bond charges destabilizes 3C to favor 6H and 4H. These results qualitatively agree well with our previously reported ab initio calculations and experimental results where 3C appears at low temperatures while 6H and 4H are favored at high temperatures enhancing vacancy formation. Furthermore, the versatility of our approach for impurity effect was exemplified by investigating the polytypism in N-doped SiC. Although using the simple energy formula with the aid of empirical interatomic potential should be necessary in the calculations for quantitative discussions, these results suggest that our simple approach is feasible for predicting the polytypes in SiC incorporating vacancy formation and impurity doping using a large number of atom unit cells.
Acknowledgments Fig. 3. Calculated energy differences for 3C, 6H, 4H and 2H structured SiC as a function of hexagonality. Solid and dashed lines denote the energy differences obtained by our simple approach using ionicity value of fi ¼ 0.232 and fi ¼ 0.7 for interatomic bonds around vacancy, respectively. The results obtained by ab initio calculations (dotted line) are also shown in this figure for comparison.
This work was partly supported by the Grant-in-Aid for Scientific Research Grant no. 21560032 of the Ministry of Education, Science, and Culture of Japan.
210
T. Ito et al. / Journal of Crystal Growth 362 (2013) 207–210
References [1] A.R. Verma, P. Krishna, Polymorphism and Polytypism in Crystals, John Wiley and Sons, Inc, New York, 1966. [2] C. Cheng, R.J. Needs, V. Heine, Journal of Physics C 21 (1988) 1049. [3] J.J.A. Shaw, V. Heine, Journal of Physics Condensed Matter 2 (1990) 4351. ¨ [4] C. Raffy, J. Furthmuller, F. Bechstedt, Physical Review B 66 (2002) 075201. ¨ [5] P. Kackell, B. Wenzien, F. Bechstedt, Physical Review B 50 (1994) 17037. [6] T. Ito, K. Sano, T. Akiyama, K. Nakamura, Thin Solid Films 508 (2006) 243. [7] K. Kobayashi, S. Komatsu, Journal of the Physical Society of Japan 77 (2008) 084703. [8] W.F. Knippenberg, Philips Research Reports 18 (1963) 161. [9] W. Weltner Jr., Journal of Chemical Physics 51 (1969) 2469. [10] L.B. Griffiths, Journal of Physics and Chemistry of Solids 27 (1966) 257. [11] P. Krishna, R. Marshall, C.E. Ryan, Journal of Crystal Growth 8 (1971) 129. [12] T. Ito, T. Kondo, T. Akiyama, K. Nakamura, Physica Status Solidi C 8 (2011) 583.
[13] T. Ito, T. Kondo, T. Akiyama, K. Nakamura, Journal of Crystal Growth 318 (2011) 141. [14] T. Ito, Japanese Journal of Applied Physics 37 (1998) L1217. [15] C.-Y. Yeh, Z.W. Lu, S. Froyen, A. Zunger, Physical Review B 46 (1992) 10086. [16] L. Pauling, The Nature of the Chemical Bond, Cornell University Press, New York, 1960. [17] J.C. Phillips, Reviews of Modern Physics 42 (1970) 317. [18] R.S. Okojie, M. Xhang, P. Pirouz, S. Tumlha, G. Jessem, L.J. Brillson, Applied Physics Letters 79 (2001) 3056. [19] B.J. Skromme, K. Palle, C.D. Poweleit, L.R. Bryant, W.M. Vetter, M. Dudley, K. Moore, T. Gehoski, Materials Science Forum 383–393 (2002) 455. [20] J.Q. Liu, H.J. Chung, T. Kur, Q. Li, M. Skowronski, Applied Physics Letters 80 (2002) 2111. [21] L.J. Brillson, S. Tumakha, G.H. Jessen, R.S. Okojie, M. Zhang, P. Pirouz, Applied Physics Letters 81 (2002) 2785. [22] S. Juillaguet, C. Balloud, V. Souliere, C. Sartel, J. Camassel, Y. Monteil, Physica Status Solidi A 202 (2005) 593. [23] T. Ito, T. Akiyama, K. Nakamura, Physica Status Solidi C, in preparation.