A simple approach to polytypes of SiC and its application to nanowires

Thin Solid Films 508 (2006) 243 – 246 www.elsevier.com/locate/tsf

A simple approach to polytypes of SiC and its application to nanowires Tomonori Ito *, Kosuke Sano, Toru Akiyama, Kohji Nakamura Department of Physics Engineering, Mie University, Tsu 514-8507, Japan Available online 8 November 2005

Abstract SiC polytypes in bulk form and nanowire are systematically investigated using our empirical potential that is based on a simple approach, and which incorporates electrostatic energies due to bond charges and ionic charges. Using the empirical potential, the system energies of 3C (zinc blende), 6H, 4H and 2H (wurtzite) structured SiC in bulk form are calculated and compared with ab initio calculations and experimental results. Our calculated results reveal that 3C – SiC is the most stable while 2H – SiC is unstable among these structures at 0 K. This is consistent with experimental results. The appearance of polytypes in bulk form is qualitatively discussed by considering ionicity of semiconductors based on our simple approach. Furthermore, we clarify the versatility of our simple approach to nanostructures considering SiC nanowire. Hexagonal SiC nanowire stabilizes a 2H structure in the diameter range of D < 20 (nm), whereas 3C – SiC is stabilized only at a large diameter range beyond 20 (nm). This is also consistent with experimental findings for InAs and InP nanowires. SiC polytypes in nanowire are discussed in terms of the ratio of the number of surface dangling bonds to the total number of interatomic bonds. D 2005 Elsevier B.V. All rights reserved. Keywords: Silicon carbide; Structural properties; Nanostructures; Computer simulations

1. Introduction Many high-performance silicon carbide (SiC)-based devices are now commercially available, including Schottky rectifiers and metal semiconductor field-effect transistors (MESFETs). This is because intensive development efforts have resulted in a dramatic improvement in the quality of SiC materials. From the crystallographic viewpoints, SiC is well known as having dozens of polytypes that consist of different ordered stacking sequences of SiC atomic double layers [1]. Generally it was thought that these polytypes were non-equilibrium structures arising from special growth mechanisms. For example, polytype-controlled epitaxial growth of 4H – and 6H – SiC has been achieved by utilizing step-flow growth on off-oriented SiC(0001) substrates, mainly by chemical vapor deposition [2]. On the other hand, theoretical investigations of polytypes for semiconductors have been performed by ab initio total energy calculations. Cheng et al. [3] and Shaw and Heine [4] theoretically investigated polytypes of SiC using ab initio pseudopotential calculations in conjunction with the energy of

* Corresponding author. Tel./fax: +81 59 231 9724. E-mail address: [email protected] (T. Ito). 0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.07.349

the sequence of layers. Their results imply that 6H – SiC is the most stable among 3C (zinc blende), 6H, 4H and 2H (wurtzite) structures. Engel and Needs performed similar calculations for polytypes of ZnS, where 3C –ZnS is the stable low temperature modification in contrast to SiC [5]. Although satisfying results have been obtained in some aspects for polytypes, systematic understanding for the origin of polytypes has not been achieved. This is because the ab initio calculations tend to obscure the physical interpretation. In our previous studies, a simple systematization of structural trends in bulk form has been accomplished by a simple description of energy difference between 3C and 2H structures [6]. The simple description gave physical interpretation for relative stability between 3C and 2H structures, which is clearly determined by competition between bond charge interaction stabilizing 3C structure and ionic interaction favorable for 2H structure. Based on this idea, newly developed empirical interatomic potential has been successfully applied to investigate the structural stability of compound semiconductors in bulk form and thin films [7,8]. In this study, SiC polytypes in bulk form are systematically interpreted by our simple approach. Furthermore, its applicability to nanostructures is exemplified by the system energy calculations for SiC nanowire to clarify the origin of

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polytypes for semiconductor nanowires, which often exhibit polytypes such as 2H –InAs [9] and include many rotational twin blocks in InP [10]. 2. Computational methods The relative energy differences among 3C, 6H, 4H and 2H structured SiC in bulk form are obtained by first-principles calculations. In the present work, we use the generalized gradient approximation (GGA) [11] and the norm-conserving pseudopotentials [12]. The valence electrons are expanded by the plane-wave basis set with a cutoff energy of 49 Ry which gives enough convergence of total energies. We employ six-double-layer (12 atom) unit cells for 3C, 6H and 2H structured SiC, while the primitive four-double-layer (8 atom) unit cell is used for 4H – SiC. To maximize the accuracy in the Brillouin zone integration, we use 29-k and 58-k points in the irreducible zone for six- and four-double layers, respectively: The k-point grid perpendicular to the layers (in the zdirection) in the four-double layer cell is equivalent to those in the six-double layer unit cell. The total energies are obtained with the lattice constants and structural parameters that are optimized so as not to introduce artificial strain in the unit cells. In order to investigate SiC polytypes in bulk form and nanowire, we employ our simple system energy formula, which is given by the following Eq. [6]: E ¼ E0 þ Ees ;

ð1Þ

1 ~ Vij ; 2 i;j

ð2Þ

E0 ¼

 Ees ¼ K

 3 Z2 Z2 ð1  fi Þ b  fi i ; 2 rbb rii

ð3Þ

where E 0 is the cohesive energy estimated by Khor – Das Sarma type empirical interatomic potential Vij within the

3C

second nearest neighbors [13 – 17]. E es is the electrostatic interaction between bond charges Z b (= 2) and that between ionic charges Z i (= 4 for group IV, 3 for III – V and 2 for II – VI semiconductors) beyond the second nearest neighbors depending on ionicity f i , which corresponds to the firstneighbor interlayer interaction J 1 [6 –8]. The coefficient K is ˚ ) which was determined by reproducing the 8.7 (meV&A energy difference of 25.3 (meV/atom) between 3C and 2H for C with f i = 0 obtained by ab initio calculations [6]. We use the values of f i = 0.240 and 0.273 for SiC fitted to the energy differences between 3C and 2H structures obtained by ab initio calculations as listed in Table 1. Using Eq. (1), the system energy is calculated for 3C – , 6H – , 4H – and 2H – SiC using six-double layer unit cells in bulk form and nanowire. In the calculation for nanowire, we consider (111)oriented nanowire, since nanowire growth for compound semiconductors usually occurs along the (111) direction [18]. The structure of nanowire is constructed according to Wulff’s theorem, where the surface energy was estimated by counting surface density of dangling bonds on the assumption that all the dangling bonds contribute equally to the surface energy at any surface orientation. As a result, we employ the structure consisting of hexagonal rings, which minimize surface density of dangling bonds, with different ordered stacking sequences of SiC atomic double layers with respect to 3C, 6H, 4H and 2H structures. Fig. 1 shows the schematic of hexagonal nanowires of 3C – and 2H – SiC considered in this study. Atomic displacements of Si and C in the unit cell are varied to minimize the system energy. The system energy is calculated as a function of nanowire diameter D. 3. Results and discussion Table 1 lists the calculated relative energy differences DE among 3C, 6H, 4H and 2H structured SiC in bulk form, where the system energy for 3C structure is employed as an

D (nm)

2H

Fig. 1. Schematic of hexagonal nanowire with 3C and 2H structures considered in this study. Diameter of nanowire D (nm) is defined as that of a circle circumscribing the hexagonal wire.

T. Ito et al. / Thin Solid Films 508 (2006) 243 – 246 Table 1 Calculated energy differences DE among 3C, 6H, 4H and 2H structured SiC in bulk form (in meV/atom) obtained by our simple approach with f i = 0.273 and ab initio calculations Material

Method

3C

6H

4H

2H

SiC

Simple approach Ab initio Ab initio—Cheng Ab initio—Engel

0 0 0 0

0.92 0.16 0.60 0.56

1.47 0.09 0.39 0.86

2.95 2.95 4.35 1.87

ZnS

Ab initio denotes the calculated results obtained by our pseudopotential calculations based on GGA and Ab initio – Cheng is the results previously reported by pseudopotential calculations based on local density approximation (LDA). The results for ZnS obtained by Engel and Needs are also listed as Ab initio – Engel in this table.

energy origin. The results previously reported for ZnS [5] with similar polytypes are also shown in this table for comparison. The results obtained by our simple approach with f i = 0.273 reveal that 3C –SiC is the most stable while 2H – SiC is unstable among these structures at 0 K. This qualitatively agrees well with experimental results, where 3C – SiC is stable at low temperatures whereas 6H – and 4H – SiC often appear at high temperatures [19]. Furthermore, the order in energies such as E es (3C) < E es (6H) < E es(4H) < Ees(2H) is also consistent with that in previously reported Ewald energies according to hexagonality [20]. In contrast with this, ab initio calculations show that the most stable structure is 6H –SiC, while the results for ZnS show that 3C– ZnS is the most stable similar to our simple approach. Although further careful ab initio calculations are necessary because of very small energy difference ¨0.1 (meV/atom) among 3C, 6H and 4H structures, it is clear that 3C – SiC is more stable than 2H – SiC. Polytypes of various semiconductors can be systematically interpreted by extending our simple approach in conjunction with interaction energy J n between nth neighbor layers. 1

245

According to Cheng et al. [3], the system energies E for 3C, 6H, 4H and 2H structures are given by: E3C ¼ E0  J1  J2  J3 ;

ð4Þ

1 1 E6H ¼ E0  J1 þ J2 þ J3 ; 3 3

ð5Þ

E4H ¼ E0 þ J2 ;

ð6Þ

E2H ¼ E0 þ J1  J2 þ J3 :

ð7Þ

Based on our simple approach as shown in Eq. (1), we assume that J n (n = 2, 3) is simply described as functions of ionic charges Z i and ionicity f i as follows:   Zb2 Zi2 ð8Þ Jn ¼ K ð1  fi Þ nth  fi nth ; rbb rii where r bbnth and r iinth are the distance between bond charges and ionic charges in the nth neighbor layers, respectively. Using Eqs. (4) – (8), phase boundaries between 3C and 6H, 3C and 4H, and 3C and 2H as functions of Z i and f i can be easily obtained as shown in Fig. 2. This figure also includes the data of f i fitted to the energy difference between 3C and 2H structures [21] for various semiconductors including SiC ( f i = 0.240 and 0.273), ZnS ( f i = 0.541) and CdSe ( f i = 0.593) with polytypes. It should be noted that the data of SiC, ZnS and CdSe are closely located near the phase boundaries between 3C and 6H for SiC and ZnS and between 3C and 4H or 2H for CdSe. These results are qualitatively consistent with experimental findings, where 3C structure is stable at low temperatures while high temperature phases are 6H or 4H for SiC [19], more complicated modification for ZnS [19] and 2H for CdSe [22]. Consequently, polytypes can be interpreted by ionicity and only certain compounds near phase boundaries can form polytypes. These successful results for polytypes inspire us to make our simple approach apply to polytypes in nanostructures such as SiC nanowire. Fig. 3 shows the prototypical results of the

3C-2H

-5.8

3C- 4H

0.8 ZnO

-5.9

3C-6H

2H-SiC CdSe ZnS

GaN

-6 3C-SiC

ZnSe

0.4

Energy (eV)

Ionicity fi

0.6

2H SiC 0.2

Polytypes

GaP

3C

GaAs Si

0 1

2

3

-6.1

-6.2

-6.3 4

Ionic charge Zi

-6.4 0

Fig. 2. Calculated phase boundaries for 3C – 2H (thick solid line), 3C – 4H (thin solid line), and 3C – 6H (dotted line). Closed square, open triangle and closed diamond denote semiconductors with 3C, polytypes and 2H, respectively.

4

8

12

16

20

Diameter (nm) Fig. 3. Calculated System energy E (eV/atom) of 3C – and 2H – SiC nanowire as a function of nanowire diameter D (nm).

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T. Ito et al. / Thin Solid Films 508 (2006) 243 – 246

system energy E as a function of nanowire diameter D for 3C – and 2H –SiC nanowires with f i = 0.273. The calculated results suggest that 2H –SiC is more stable in the diameter range of D < 20 (nm) than 3C –SiC, whereas 3C – SiC is stabilized only at a large diameter range beyond 20 (nm). Similar results were obtained in the case of f i = 0.240, where the phase transition diameter is estimated to be 18 (nm). The stability of 2H – SiC at small D contradicts the fact in bulk form, where 3C –SiC is more stable than 2H – SiC. Instability of 3C –SiC at small diameter is due to the larger number of surface dangling bonds in its structure than those in 2H –SiC. Some of surface atoms in 3C – SiC nanowire have two dangling bonds, whereas surface of 2H – SiC nanowire consists of atoms with one dangling bond. Consequently, the larger the number of dangling bonds in the structure, the less stable the SiC nanowire at small D. This is consistent with experimental findings, where semiconductor nanowires exhibit different polytypes such as 2H – InAs [9] and include many rotational twin blocks in InP [10]. On the other hand, 4H – and 6H – SiC do not appear in the nanowire, since the number of dangling bonds in the 4H – and 6H – stackings is similar to that of 3C –stacking in the nanowire structure. Consequently, SiC nanowire has the form of 2H – SiC at small diameter less than ¨20 (nm) and 3C –SiC beyond ¨20 (nm). These results suggest that polytypes of SiC can be controlled by varying nanowire diameter D. Although further calculations for nanowire shape incorporating the electronic contribution in surface dangling bonds are necessary, we believe that our simple approach gives correct qualitative trends for polytypes of semiconductor nanowires. 4. Conclusion In this paper, we theoretically investigate SiC polytypes using our empirical potential that is based on a simple approach, and which incorporates electrostatic energies due to bond charges and ionic charges. The system energy calculations in bulk form reveal that 3C – SiC is the most stable while 2H –SiC is unstable among 3C, 6H, 4H and 2H structures at 0 K. This qualitatively agrees with experimental results. Polytypes of semiconductors in bulk form can be systematically understood by considering phase boundaries between 3C and other structures based on our simple approach. Polytypes appear in semiconductors with ionicity near the phase boundaries such as SiC, ZnS and CdSe. Furthermore, we clarify that SiC nanowire stabilizes 2H structure at small diameter D¨20 (nm), whereas 3C – SiC is stabilized only at larger diameter. This is because the number

of surface dangling bonds in 2H –SiC is the smallest among 3C, 6H, 4H and 2H nanowire structures. This is consistent with experimental findings for semiconductor nanowires. Although the contribution of temperature and strain inevitable for fabricating crystals, which is left out of our analyses, should be taken into account particularly for quantitative discussion, our simple approach based on empirical interatomic potential is feasible not only for investigating qualitative trends for polytypes of compound semiconductors but also for incorporating these contributions in subsequent improvement performing molecular dynamics simulation with large unit cell including substrate layers. Acknowledgments This work was partly supported by the Grant-in-Aid for Scientific Research Grant No. 16560020 of the Ministry of Education, Science, and Culture of Japan. Computations were performed at RCCS (National Institute of Natural Science) and ISSP (University of Tokyo). References [1] A.R. Verma, P. Krishna, Polymorphism and Polytypism in Crystals, John Wiley & Sons, Inc., New York, 1966. [2] H. Matsunami, T. Kimoto, Mater. Sci. Eng., R Rep. 20 (1997) 125. [3] C. Cheng, R.J. Needs, V. Heine, J. Phys., C 21 (1988) 1049. [4] J.J.A. Shaw, V. Heine, J. Phys., Condens. Matter 2 (1990) 4351. [5] G.E. Engel, R.J. Needs, J. Phys., Condens. Matter 2 (1990) 367. [6] T. Ito, Jpn. J. Appl. Phys. 37 (1998) L1217. [7] T. Ito, Y. Kangawa, J. Cryst. Growth 235 (2002) 149. [8] T. Suda, Y. Kangawa, K. Nakamura, T. Ito, J. Cryst. Growth 258 (2003) 277. [9] M. Yazawa, M. Koguchi, K. Hiruma, Appl. Phys. Lett. 58 (1991) 1080. [10] S. Bhunia, T. Kawamura, Y. Watanabe, S. Fujikawa, K. Tokushima, Appl. Phys. Lett. 83 (2003) 3371. [11] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [12] N. Troullier, J.L. Martins, Phys. Rev., B 43 (1991) 1993. [13] K.E. Khor, S. Das Sarma, Phys. Rev., B 38 (1988) 3318. [14] T. Ito, K.E. Khor, S. Das Sarma, Phys. Rev., B 40 (1989) 9715. [15] T. Ito, J. Appl. Phys. 77 (1995) 4845. [16] T. Ito, Y. Kangawa, J. Cryst. Growth 237 – 239 (2002) 116. [17] T. Ito, K. Nakamura, Y. Kangawa, K. Shiraishi, A. Taguchi, H. Kageshima, Appl. Surf. Sci. 216 (2003) 458. [18] X. Duan, C.M. Lieber, Adv. Mater. 12 (2000) 298. [19] D. Pandey, P. Krishna, in: Current Topics in Materials Science, vol. 9, North Holland, 1982, p. 415. [20] D. Lenstra, A.G. Roosenbrand, P.J.H. Dentener, W. van Haeringen, Physica 138B (1986) 83. [21] C.-Y. Yeh, Z.W. Lu, S. Froyen, A. Zunger, Phys. Rev., B 46 (1992) 10086. [22] M.P. Klakov, I.V. Balyakina, J. Cryst. Growth 113 (1991) 653.