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Enhanced magnetism of SiC with He defects
Physics Letters A 376 (2012) 3363–3367
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Enhanced magnetism of SiC with He defects Wei Cheng a,b , Guo-Qiang Liu b , Feng-Shou Zhang a,c,d,∗ , Hong-Yu Zhou a a
Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, PR China Ningbo Institute of Materials Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, PR China c Beijing Radiation Center, Beijing 100875, PR China d Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou, Lanzhou 730000, PR China b
a r t i c l e
i n f o
Article history: Received 23 May 2012 Received in revised form 29 August 2012 Accepted 30 August 2012 Available online 1 September 2012 Communicated by R. Wu Keywords: SiC Magnetism Defects Ab initio calculations
a b s t r a c t Helium defects in silicon carbide are studied using first principle calculations. The magnetization of various defects, such as vacancies, helium interstitials, and interstitial and vacancy complexes, are investigated. There is no magnetic element in silicon carbide. However, when a silicon atom is substituted by a helium atom, a ferromagnetic ground state is found. The total magnetic moment is found to be 4.0μ B . When the silicon atom is substituted by a helium atom, the four nearest carbon atoms are separated, and the p orbitals of the four carbon atoms are localized. This results in narrow bands within the conduction and valence bands. The magnitude of the magnetic moment is related to the C–He distance. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Magnetism is a well-known quantum phenomenon [1]. Magnetism in materials is normally caused by the presence of magnetic elements, such as iron, cobalt and nickel, which contain d and f electrons. However, unexpected d0 ferromagnetism in undoped HfO2 is found in 2005 [2]. A simply explanation is proposed which attribute the d0 magnetism to the first-low elements, where the O- and N-2p is demonstrated to be almost as localized as Mn3d orbitals [3]. Similar to O and N, C has the 2p orbital which is much localized than the Si-3p orbitals, hence, the C-2p states can also induce magnetism [4–8]. The occurrence of magnetism in systems that do not contain magnetic elements has attracted considerable attention in recent years [4–8]. Recently, it was found that magnetism can be induced from vacancies [9,10]. This finding will aid in the development of new types of magnetic semiconductors. Both the electron motions and spin motions can be controlled in magnetic semiconductors, which make them candidates for quantum computation units. To utilize the magnetic signal in semiconductors, the magnetic origin of various defects needs to be understood. SiC is important for fusion applications [11–13]. The induced radioactivity of silicon carbide (SiC) is very low [14]. Recently, the
*
Corresponding author at: Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, PR China. Tel.: +86 10 6220 5602; fax: +86 10 6223 1765. E-mail address: [email protected] (F.-S. Zhang). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.08.053
helium defects could make 6H-SiC magnetized [9]. Magnetism was also found to occur in both 4H-SiC and 6H-SiC from di-vacancies [9,15]. Many magnetic defects have been found in carbon systems, including at the surface of graphene [5–7] and in the di-vacancy containing SiC. In addition, magnetic signals have been found to slightly increase when there are helium impurities in graphite [16,17]. Knowledge of the types of defects in SiC that are magnetic defects could provide information on the defects in SiC. Since counting the number of defects is not possible because the defects transform in different ways at low and high temperatures [18]. In this Letter, the magnetism of defects induced by He are studied in detail using the MedeA1 and Vienna ab initio simulation package (VASP) [19–22]. One of the magnetic defects in our study is one He interstitial located in a Si vacancy. The magnetism arises from p orbitals of the four nearest carbon atoms surrounding the helium atom. The distances between the carbon atoms increases after the nearest carbon atoms are pushed apart by the impurity atoms, which results in an increase in the magnetic moment. The increase in the magnetic moment is related to the localization of electrons in the carbon atoms. 2. Computational approach There are many polytypes of SiC, including 3C-SiC [23,24], 4HSiC [25,26], and 6H-SiC [27,28]. The “C” indicates that the crystal structure is cubic. One of the common forms of SiC is 6H-SiC,
1
MedeA is a trade mark of Materials Design, Inc.
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W. Cheng et al. / Physics Letters A 376 (2012) 3363–3367
Table 1 The four He–C distances surrounding the defect. The first row shows the distances in perfect SiC. For the third case of vacancy, the four distances are measured from the previous He position (length in nm).
Fig. 1. Sketch of ferromagnetic (a) and anti-ferromagnetic (b) defects in the unit cell of 6H-SiC, in which one Si (black ball) atom is replaced with one He (pink ball) atom, and all C (gold ball) atoms remain the same in defects. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
which has the hexagonal structure of a unit cell, as shown in Fig. 1 [29]. There are 6 bilayers in z direction and 6 Si and 6 C atoms in one unit cell. The supercell used in our calculations for perfect SiC contains 96 Si and 96 C atoms; a 4 × 4 × 1 section of the unit cell is shown in Fig. 1. There are two types of bonds, with bond lengths of 0.1886 nm along the bi-layer and 0.1899 nm along the stacking direction [30]. All of the defects are created near the center of the 192 atom supercell. The calculations are performed using the first principle density functional theory with the Perdew–Burke–Ernzerhof function (GGA-PBE) form of the generalized gradient approximation [19–22]. The default cutoff energy is chosen to be 478.9 eV. The k-points are chosen to be Monkhorst–Pack 3 × 3 × 3 [31]. When calculating the partial density of states, the k-points are chosen to be Monkhorst–Pack 9 × 9 × 9. The formation energy of a defect in charge q is defined as [32]:
E f X q = E tot X q − E tot (perfect)
−
n i μi + q ( E F + E v + V )
i
where E tot ( X q ) and E tot (perfect) are the total energy of a defect in charge q and perfect crystal from supercell calculation respectively, ni is the number of atom of type i that have been added or removed from the supercell, μi are the corresponding chemical potential, E F is the Fermi level measured from valence band maximum ( E v ), and V is a correction term. 3. Results and discussion The magnetic moment of perfect 6H-SiC without defects is calculated first. In this system, all of the electrons in each Si–C bond are paired. 6H-SiC is a non-magnetic, wide-bandgap semiconductor. There are no narrow impurity bands near the Fermi level from the calculation. The calculated magnetic moment of perfect 6H-SiC is exactly zero, as expected. The magnetic moments of 6H-SiC containing a single C vacancy (Vc ) is calculated and is found to be zero. With the exception of the four Si atoms surrounding the vacancy, the bonds of all of the other atoms are kept the same. There are relaxations surrounding the vacancy following the energy optimization. This relaxation does not affect the final magnetism. The
Structure
d1
d2
d3
d4
SiC [30] HeSi95 C96 Si95 C96
0.1886 0.2194 0.2065
0.1886 0.2194 0.2065
0.1886 0.2194 0.2065
0.1899 0.2236 0.2124
results suggest that the four Si atoms with four dangling bonds surrounding the vacancy do not contribute to magnetism, regardless of whether it is relaxed. Magnetic silicon, without magnetic elements, is very useful for spintronic applications; however, it is difficult to examine experimentally. The present calculated magnetism of the four separated Si atoms is consistent with that observed experimentally [33]. Fe-based magnetic silicon is recently reported [33], where the magnetism arises from the Fe atom instead of the VSi defect. It is interesting to determine whether there are other magnetic mechanisms without the presence of magnetic elements. The magnetic moment of VC VSi in 6H-SiC is then calculated. The magnetic moment is found to be 2μ B . The agreement between the calculated and reported magnetic moments confirms that the calculated magnetic moment is correct. This is one example of magnetism occurring without the presence of magnetic elements. The above calculation indicates that the three Si atoms are unlikely to contribute to the magnetism. The three C atoms surrounding the Si vacancy are likely to contribute to the magnetism in both 4H- and 6H-SiC [9,15]. This fact indicates that the VC VSi is a universal magnetic defect in various SiC systems and that the local structure surrounding the vacancy accounts for the magnetism. It is localized. Otherwise, the second and third neighboring atoms will affect the magnetism. In our calculations, most of the magnetic moments arise from the nearest three C atoms surrounding the Si vacancy. This observation is consistent with the conclusions of a study examining 4H-SiC in Ref. [15]. Silicon and carbon are of the same group, IV, in the periodic table. Therefore, their chemical properties are similar. The VC is not the magnetic defect. The VC VSi is the magnetic defect. From our localized opinion, the VSi is likely to be magnetic. This observation is demonstrated by our further calculations. Therefore, we try to understand why the VC VSi differs from VC , and how the presence of a non-magnetic helium atom can induce magnetism. Calculations were performed with several He induced defects, and the structure where the He atom stays close to the center of a SiC cage is found to have the lowest possible energy [29]. The magnetic moments of a single He interstitial in the center of a SiC (IHe ) is then calculated. No magnetic moment is found for the HeSi96 C96 structure. This observations suggests that the He atom does not contribute to magnetism and is in agreement with experimental results [16,17]. The reason is that the He atom induces magnetic defects. The injection of He can create not only a stable defect but also a pseudo-stable defect. In this Letter, we found a pseudo-stable magnetic defect. Part of the HeSi95 C96 with He and the nearest C atoms are shown in Fig. 1. The defect can be denoted as VSi IHe (HeSi ). The four He–C distances are three 0.2194 nm and one 0.2236 nm. The He–C distances are listed in Table 1. Compared with the experimental Si–C bond lengths, which consist of three 0.1886 nm and one 0.1899 nm [30], the He atom expels the C atoms and enlarges the size of the Si vacancy. The partial atomic charges do not differ significantly as compared with the bulk material. The spin resolved total density of states and partial density of states for the four nearest carbon atoms in HeSi95 C96 is shown in Fig. 2. As compared with the perfect 6H-SiC, there is a downward narrow band with peaks above the Fermi level. The narrow band width is approximately 0.69 eV. The peaks arise from the four car-
W. Cheng et al. / Physics Letters A 376 (2012) 3363–3367
Fig. 3. Band structure of VSi and HeSi , F, , B, G, are (0
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1 2
0), (0 0 0), ( 12 0 0),
Fig. 2. Spin-resolved partial DOS (a) for the four C atoms, which are closest to the He atom, and the total DOS (b) for HeSi95 C96 .
(0 0 12 ) in reciprocal space respectively. The black thick line indicate spin down, the red thin line indicate spin up. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
bon atoms that are closest to the He atom, instead of arising from the He atom itself. The peaks are from the p orbitals. The narrow band between the conduction band and valence band is a signature of magnetism. The total magnetic moments of HeSi are found to be 4.0μ B . There are four impurity bands between the valence band and the conduction band. The four bands are from the four dangling bonds of the nearest C atoms surrounding the He atom. In our calculations, the spin-polarized energy is less than spinunpolarized energy by 0.76 eV. This difference suggests that the spin-polarized states are stable at room temperature. The atomic partial magnetic moments are analyzed. The four closed C atoms have a magnetic moment of 0.52μ B , in which 0.50μ B is from the p orbitals. Contributions from other atoms are relatively small. The small contribution is owing to the bonding of other atoms remains almost the same as in the perfect SiC, where the two electrons within each bond are paired with opposite spin orientations. The bond lengths are very close to those where there is no defect, except near the four C atoms that surround the He atom. What is the origin of the huge magnetic moment in the above defect complex? We answer this question by performing the following calculations. First, we remove the He atom from the above structure and leave all of the other atom positions in Si95 C96 unchanged. The defect is denoted by VSi (n). The symbol (n) denotes that the atom locations are frozen and not optimized. The calculated magnetic moment is still 4.0μ B . This calculation suggests that the He atom has almost no contribution to the magnetic moments. This result is consistent with the partial DOS shown in Fig. 2. The He atom has s orbitals that lays at very low energy regions. The atom locations determine the charge distribution around the defect. With and without He atom, the four dangling bonds of the four C atoms remain the same, and as a result, the magnetic moment is the same. Second, we optimize the above structure without He atom; after relaxation, the defect is denoted as VSi . As a result, the magnetic moment drops to 2.0μ B . The four distances of the four C atoms and the initial He atom are listed in Table 1 in order to show the differences in the positions of the four neighboring C atoms surrounding the vacancy. The magnetic moment is exactly the same as that in VC VSi [9]. VSi is shown as a promising quantum system in experiments [34]. The calculation shows that the VSi
is a still a magnetic defect. The distances of the C atoms and the He atom affect the total magnetic moment. As the distances increase, the interactions decrease and the atomic p orbitals tend to be more localized. Therefore, the magnetic moment will increase. Third, we further replace all of the Si atoms with C atoms from the non-optimized Si95 C96 (n); this structure is denoted as C191 (n). The calculated magnetic moment is still 4.0μ B . The atomic partial magnetic moments still arise from the four closest C atoms. This observation shows that the magnetic moment surrounding the vacancy is independent of the second neighboring Si or C atoms. The magnetic moment depends only on the nearest four C atom locations. The result also suggests the impurity band from the four C atoms is localized around the vacancy and that the second neighboring atoms will not influence the charge distribution around the vacancy. Finally, we replace all of the C atoms with Si atoms; this structure is denoted as Si191 (n). The calculated magnetic moment is now 0μ B . Therefore, it is confirmed that the Si atoms contribute nothing around the vacancy. This result also suggests that the Si vacancy is unlikely to be a magnetic defect, regardless of whether it is relaxed and whether the Si atoms are close or far away. The lack of a magnetic defect explains why it is difficult to find magnetic Si in experiments [33]. The above four calculations show that only the nearest four C atoms in SiC contribute the most to the magnetic moments in the HeSi95 C96 . The interactions of the p electrons in the C atoms are stronger than those in the Si atoms. As a result, the dangling C bonds are more sensitive to the same changes in bond length. A magnetic moment will arise if dangling C bonds are formed. Why does a single Si vacancy produce a magnetic moment of only 2μ B whereas the magnetic moment is enhanced to 4μ B with He? The stabilization of the FM phase observed for the two defects can be understood using the phenomenological band-coupling model [3]. The band-coupling tends to make all the spins parallel for the four C atoms. The bonding of C atoms tends to make one pair of spins anti-parallel. The competition is determined by the distances of the C atoms. The four nearest carbon atoms form tetrahedral with edge of 0.34 nm and 0.36 nm and for VSi and HeSi respectively. The dispersions of VSi and HeSi are shown in Fig. 3. The electronic structure differences of the two defects can
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W. Cheng et al. / Physics Letters A 376 (2012) 3363–3367
Table 2 The energy differences in the unit cell of HeSi96 C95 , and (HeSi96 C95 )2 , and the different charge states: neutral (0), one positive charge (+), and one negative charge (−). The energy differences in parenthesis are from VC VSi in Ref. [9]. E AFM − E FM (meV)
Total magnetic moment of FM structure (μ B )
HeSi95 C96 (0) (+1) (−1)
553 523 307
4 .0 3 .0 3 .0
(HeSi96 C95 )2 (0, 0) (+1, +1) (−1, −1)
−75.3 (−78.8) −39.4 (16.8) 81.0 (68.2)
8 .0 6 .0 6 .0
System (charge states)
Fig. 4. Defect formation energy of HeSi as a function of Fermi level E F for different charge states.
be understood by multi-center bonding which are described in Ref. [35]. Without He atom support, the four C atoms are close to each other, and the initial four dangling bonds around the vacancy tend to form 1/6-order 4-center bonds (effectively one bond) around the six edges of the tetrahedral for VSi . As a result, there are two parallel spins from two dangling bonds and two anti-parallel spins from a bond above the Fermi energy level as shown in Fig. 3(a). With He support, bonding is suppressed as the C atoms are far away, four parallel spin orbitals appear above the Fermi energy level as shown in Fig. 3(b). The magnetic moment is doubled for HeSi as a result of the above bonding differences. In a real semiconductor, the defects could either be positively charged or negatively charged. We then calculate the stability of the HeSi95 C96 structure under different charge states and different spin orientations. The defect formation energies are calculated to be 3.92, 3.68, 3.49, 3.37, 3.36, 4.42, 5.91, 7.70, and 9.67 eV, for charge states from positive four to negative four with Fermi energy at valence band maximum. The defect formation energy as a function of the Fermi energy level is shown in Fig. 4. The defects with more than one negative charges are relatively difficult to form for p-type SiC. The defect with four electrons is the most stable form in n-type SiC. The neutral defect as well as defect with one hole are the most stable in p-type SiC. We keep the minimized HeSi95 C96 structure fixed and then put the magnetic moments of the four C atoms in parallel and anti-parallel configurations, as shown in Fig. 1. The AFM–FM energy differences are listed in Table 2. As shown in Table 2, the energy minimized structure surrounding the defects is ferromagnetic. After the defects are charged with one electron and one hole, it is found that the ferromagnetic structure is still stable. However the three ferromagnetic configurations are different from each other. The typical one for the neutral defect is shown in Fig. 1(a). In order to study the magnetic coupling between these local moments, we double the size of the supercells by placing two 192atom or 384-atom cells side by side. The two HeSi are separated by 1.24 nm. In the FM structure, the spins of the two HeSi in both cells are the same as shown in Fig. 1(a), and in the AFM structure, the spins of one HeSi are the same as shown in Fig. 1(a), and the spins of the other HeSi are totally inverted. The FM and AFM energy differences for HeSi in 384-atom cell and the corresponding values for VSi VC are listed in Table 2. The magnetic coupling tends to form AFM structure for defects in neutral and positive
charge states, and tends to form FM structure for defects in negative charge states. That is slightly different from VSi VC case. In experiment, the charge states of HeSi might be altered by ion implantation techniques, and magnetism is likely to form. The above discussions show that the magnetism highly depends on the carbon distances. If all of the carbons are bonding, the magnetism disappears. If the bonds are broken, then the carbon– carbon distance increases and magnetism appears. If the carbon atoms are further separated, the magnetism increases even more. The carbon atom separation and the magnetism relations can also explain experiments examining graphite [16,17]. The bond lengths of the C atoms within a graphite sheet are shorter than the bond lengths in SiC. It is difficult to insert He atoms into the sheet. He is likely to stay within two graphite sheets. The change in bond lengths as a result of He insertion is less, and the magnetic effects are weaker in He irradiated graphite. 4. Conclusions In summary, the enhanced magnetism of a defect complex, HeSi , with a He interstitial and a Si vacancy is found. The magnetic moment of HeSi is two times greater than that of VC VSi in 6H-SiC. He is an atom that enlarges the vacancy size. C is the element that provides the magnetic moment. Charge can affect the stability of the magnetic states. The carbon atom distances are related to the final magnetic moment. Acknowledgements The authors acknowledge Professor D.S. Wang at the Institute of Physics for helpful discussion and his help in making the computation arrangement through his grant (NSFC-10634070). Part of the computation described in this research is carried out on the ScGrid of Supercomputing Center, Computer Network Information Center of Chinese Academy of Sciences. The work is supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the National Natural Science Foundation of China under Grants No. 11025524, and No. 11161130520, and the National Basic Research Program of China under Grant No. 2010CB832903. References [1] C. Kittel, Introduction to Solid State Physics, 7th ed., Willey, New York, 1996. [2] J.M.D. Coey, M. Venkatesan, P. Stamenov, C.B. Fitzgerald, L.S. Dorneles, Phys. Rev. B 72 (2005) 024450. [3] H.W. Peng, H.J. Xiang, S.H. Wei, S.-S. Li, J.B. Xia, J.B. Li, Phys. Rev. Lett. 102 (2009) 017201. [4] T.L. Makarova, F. Palacio (Eds.), Carbon Based Magnetism: An Overview of the Magnetism of Metal Free Carbon-based Compounds and Materials, Elsevier, Amsterdam, 2006. [5] J.H. Chen, L. Li, W.G. Cullen, E.D. Williams, M.S. Fuhrer, Nature Phys. 7 (2011) 535.
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Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Enhanced magnetism of SiC with He defects Wei Cheng a,b , Guo-Qiang Liu b , Feng-Shou Zhang a,c,d,∗ , Hong-Yu Zhou a a
Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, PR China Ningbo Institute of Materials Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, PR China c Beijing Radiation Center, Beijing 100875, PR China d Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou, Lanzhou 730000, PR China b
a r t i c l e
i n f o
Article history: Received 23 May 2012 Received in revised form 29 August 2012 Accepted 30 August 2012 Available online 1 September 2012 Communicated by R. Wu Keywords: SiC Magnetism Defects Ab initio calculations
a b s t r a c t Helium defects in silicon carbide are studied using first principle calculations. The magnetization of various defects, such as vacancies, helium interstitials, and interstitial and vacancy complexes, are investigated. There is no magnetic element in silicon carbide. However, when a silicon atom is substituted by a helium atom, a ferromagnetic ground state is found. The total magnetic moment is found to be 4.0μ B . When the silicon atom is substituted by a helium atom, the four nearest carbon atoms are separated, and the p orbitals of the four carbon atoms are localized. This results in narrow bands within the conduction and valence bands. The magnitude of the magnetic moment is related to the C–He distance. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Magnetism is a well-known quantum phenomenon [1]. Magnetism in materials is normally caused by the presence of magnetic elements, such as iron, cobalt and nickel, which contain d and f electrons. However, unexpected d0 ferromagnetism in undoped HfO2 is found in 2005 [2]. A simply explanation is proposed which attribute the d0 magnetism to the first-low elements, where the O- and N-2p is demonstrated to be almost as localized as Mn3d orbitals [3]. Similar to O and N, C has the 2p orbital which is much localized than the Si-3p orbitals, hence, the C-2p states can also induce magnetism [4–8]. The occurrence of magnetism in systems that do not contain magnetic elements has attracted considerable attention in recent years [4–8]. Recently, it was found that magnetism can be induced from vacancies [9,10]. This finding will aid in the development of new types of magnetic semiconductors. Both the electron motions and spin motions can be controlled in magnetic semiconductors, which make them candidates for quantum computation units. To utilize the magnetic signal in semiconductors, the magnetic origin of various defects needs to be understood. SiC is important for fusion applications [11–13]. The induced radioactivity of silicon carbide (SiC) is very low [14]. Recently, the
*
Corresponding author at: Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, PR China. Tel.: +86 10 6220 5602; fax: +86 10 6223 1765. E-mail address: [email protected] (F.-S. Zhang). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.08.053
helium defects could make 6H-SiC magnetized [9]. Magnetism was also found to occur in both 4H-SiC and 6H-SiC from di-vacancies [9,15]. Many magnetic defects have been found in carbon systems, including at the surface of graphene [5–7] and in the di-vacancy containing SiC. In addition, magnetic signals have been found to slightly increase when there are helium impurities in graphite [16,17]. Knowledge of the types of defects in SiC that are magnetic defects could provide information on the defects in SiC. Since counting the number of defects is not possible because the defects transform in different ways at low and high temperatures [18]. In this Letter, the magnetism of defects induced by He are studied in detail using the MedeA1 and Vienna ab initio simulation package (VASP) [19–22]. One of the magnetic defects in our study is one He interstitial located in a Si vacancy. The magnetism arises from p orbitals of the four nearest carbon atoms surrounding the helium atom. The distances between the carbon atoms increases after the nearest carbon atoms are pushed apart by the impurity atoms, which results in an increase in the magnetic moment. The increase in the magnetic moment is related to the localization of electrons in the carbon atoms. 2. Computational approach There are many polytypes of SiC, including 3C-SiC [23,24], 4HSiC [25,26], and 6H-SiC [27,28]. The “C” indicates that the crystal structure is cubic. One of the common forms of SiC is 6H-SiC,
1
MedeA is a trade mark of Materials Design, Inc.
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W. Cheng et al. / Physics Letters A 376 (2012) 3363–3367
Table 1 The four He–C distances surrounding the defect. The first row shows the distances in perfect SiC. For the third case of vacancy, the four distances are measured from the previous He position (length in nm).
Fig. 1. Sketch of ferromagnetic (a) and anti-ferromagnetic (b) defects in the unit cell of 6H-SiC, in which one Si (black ball) atom is replaced with one He (pink ball) atom, and all C (gold ball) atoms remain the same in defects. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
which has the hexagonal structure of a unit cell, as shown in Fig. 1 [29]. There are 6 bilayers in z direction and 6 Si and 6 C atoms in one unit cell. The supercell used in our calculations for perfect SiC contains 96 Si and 96 C atoms; a 4 × 4 × 1 section of the unit cell is shown in Fig. 1. There are two types of bonds, with bond lengths of 0.1886 nm along the bi-layer and 0.1899 nm along the stacking direction [30]. All of the defects are created near the center of the 192 atom supercell. The calculations are performed using the first principle density functional theory with the Perdew–Burke–Ernzerhof function (GGA-PBE) form of the generalized gradient approximation [19–22]. The default cutoff energy is chosen to be 478.9 eV. The k-points are chosen to be Monkhorst–Pack 3 × 3 × 3 [31]. When calculating the partial density of states, the k-points are chosen to be Monkhorst–Pack 9 × 9 × 9. The formation energy of a defect in charge q is defined as [32]:
E f X q = E tot X q − E tot (perfect)
−
n i μi + q ( E F + E v + V )
i
where E tot ( X q ) and E tot (perfect) are the total energy of a defect in charge q and perfect crystal from supercell calculation respectively, ni is the number of atom of type i that have been added or removed from the supercell, μi are the corresponding chemical potential, E F is the Fermi level measured from valence band maximum ( E v ), and V is a correction term. 3. Results and discussion The magnetic moment of perfect 6H-SiC without defects is calculated first. In this system, all of the electrons in each Si–C bond are paired. 6H-SiC is a non-magnetic, wide-bandgap semiconductor. There are no narrow impurity bands near the Fermi level from the calculation. The calculated magnetic moment of perfect 6H-SiC is exactly zero, as expected. The magnetic moments of 6H-SiC containing a single C vacancy (Vc ) is calculated and is found to be zero. With the exception of the four Si atoms surrounding the vacancy, the bonds of all of the other atoms are kept the same. There are relaxations surrounding the vacancy following the energy optimization. This relaxation does not affect the final magnetism. The
Structure
d1
d2
d3
d4
SiC [30] HeSi95 C96 Si95 C96
0.1886 0.2194 0.2065
0.1886 0.2194 0.2065
0.1886 0.2194 0.2065
0.1899 0.2236 0.2124
results suggest that the four Si atoms with four dangling bonds surrounding the vacancy do not contribute to magnetism, regardless of whether it is relaxed. Magnetic silicon, without magnetic elements, is very useful for spintronic applications; however, it is difficult to examine experimentally. The present calculated magnetism of the four separated Si atoms is consistent with that observed experimentally [33]. Fe-based magnetic silicon is recently reported [33], where the magnetism arises from the Fe atom instead of the VSi defect. It is interesting to determine whether there are other magnetic mechanisms without the presence of magnetic elements. The magnetic moment of VC VSi in 6H-SiC is then calculated. The magnetic moment is found to be 2μ B . The agreement between the calculated and reported magnetic moments confirms that the calculated magnetic moment is correct. This is one example of magnetism occurring without the presence of magnetic elements. The above calculation indicates that the three Si atoms are unlikely to contribute to the magnetism. The three C atoms surrounding the Si vacancy are likely to contribute to the magnetism in both 4H- and 6H-SiC [9,15]. This fact indicates that the VC VSi is a universal magnetic defect in various SiC systems and that the local structure surrounding the vacancy accounts for the magnetism. It is localized. Otherwise, the second and third neighboring atoms will affect the magnetism. In our calculations, most of the magnetic moments arise from the nearest three C atoms surrounding the Si vacancy. This observation is consistent with the conclusions of a study examining 4H-SiC in Ref. [15]. Silicon and carbon are of the same group, IV, in the periodic table. Therefore, their chemical properties are similar. The VC is not the magnetic defect. The VC VSi is the magnetic defect. From our localized opinion, the VSi is likely to be magnetic. This observation is demonstrated by our further calculations. Therefore, we try to understand why the VC VSi differs from VC , and how the presence of a non-magnetic helium atom can induce magnetism. Calculations were performed with several He induced defects, and the structure where the He atom stays close to the center of a SiC cage is found to have the lowest possible energy [29]. The magnetic moments of a single He interstitial in the center of a SiC (IHe ) is then calculated. No magnetic moment is found for the HeSi96 C96 structure. This observations suggests that the He atom does not contribute to magnetism and is in agreement with experimental results [16,17]. The reason is that the He atom induces magnetic defects. The injection of He can create not only a stable defect but also a pseudo-stable defect. In this Letter, we found a pseudo-stable magnetic defect. Part of the HeSi95 C96 with He and the nearest C atoms are shown in Fig. 1. The defect can be denoted as VSi IHe (HeSi ). The four He–C distances are three 0.2194 nm and one 0.2236 nm. The He–C distances are listed in Table 1. Compared with the experimental Si–C bond lengths, which consist of three 0.1886 nm and one 0.1899 nm [30], the He atom expels the C atoms and enlarges the size of the Si vacancy. The partial atomic charges do not differ significantly as compared with the bulk material. The spin resolved total density of states and partial density of states for the four nearest carbon atoms in HeSi95 C96 is shown in Fig. 2. As compared with the perfect 6H-SiC, there is a downward narrow band with peaks above the Fermi level. The narrow band width is approximately 0.69 eV. The peaks arise from the four car-
W. Cheng et al. / Physics Letters A 376 (2012) 3363–3367
Fig. 3. Band structure of VSi and HeSi , F, , B, G, are (0
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1 2
0), (0 0 0), ( 12 0 0),
Fig. 2. Spin-resolved partial DOS (a) for the four C atoms, which are closest to the He atom, and the total DOS (b) for HeSi95 C96 .
(0 0 12 ) in reciprocal space respectively. The black thick line indicate spin down, the red thin line indicate spin up. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
bon atoms that are closest to the He atom, instead of arising from the He atom itself. The peaks are from the p orbitals. The narrow band between the conduction band and valence band is a signature of magnetism. The total magnetic moments of HeSi are found to be 4.0μ B . There are four impurity bands between the valence band and the conduction band. The four bands are from the four dangling bonds of the nearest C atoms surrounding the He atom. In our calculations, the spin-polarized energy is less than spinunpolarized energy by 0.76 eV. This difference suggests that the spin-polarized states are stable at room temperature. The atomic partial magnetic moments are analyzed. The four closed C atoms have a magnetic moment of 0.52μ B , in which 0.50μ B is from the p orbitals. Contributions from other atoms are relatively small. The small contribution is owing to the bonding of other atoms remains almost the same as in the perfect SiC, where the two electrons within each bond are paired with opposite spin orientations. The bond lengths are very close to those where there is no defect, except near the four C atoms that surround the He atom. What is the origin of the huge magnetic moment in the above defect complex? We answer this question by performing the following calculations. First, we remove the He atom from the above structure and leave all of the other atom positions in Si95 C96 unchanged. The defect is denoted by VSi (n). The symbol (n) denotes that the atom locations are frozen and not optimized. The calculated magnetic moment is still 4.0μ B . This calculation suggests that the He atom has almost no contribution to the magnetic moments. This result is consistent with the partial DOS shown in Fig. 2. The He atom has s orbitals that lays at very low energy regions. The atom locations determine the charge distribution around the defect. With and without He atom, the four dangling bonds of the four C atoms remain the same, and as a result, the magnetic moment is the same. Second, we optimize the above structure without He atom; after relaxation, the defect is denoted as VSi . As a result, the magnetic moment drops to 2.0μ B . The four distances of the four C atoms and the initial He atom are listed in Table 1 in order to show the differences in the positions of the four neighboring C atoms surrounding the vacancy. The magnetic moment is exactly the same as that in VC VSi [9]. VSi is shown as a promising quantum system in experiments [34]. The calculation shows that the VSi
is a still a magnetic defect. The distances of the C atoms and the He atom affect the total magnetic moment. As the distances increase, the interactions decrease and the atomic p orbitals tend to be more localized. Therefore, the magnetic moment will increase. Third, we further replace all of the Si atoms with C atoms from the non-optimized Si95 C96 (n); this structure is denoted as C191 (n). The calculated magnetic moment is still 4.0μ B . The atomic partial magnetic moments still arise from the four closest C atoms. This observation shows that the magnetic moment surrounding the vacancy is independent of the second neighboring Si or C atoms. The magnetic moment depends only on the nearest four C atom locations. The result also suggests the impurity band from the four C atoms is localized around the vacancy and that the second neighboring atoms will not influence the charge distribution around the vacancy. Finally, we replace all of the C atoms with Si atoms; this structure is denoted as Si191 (n). The calculated magnetic moment is now 0μ B . Therefore, it is confirmed that the Si atoms contribute nothing around the vacancy. This result also suggests that the Si vacancy is unlikely to be a magnetic defect, regardless of whether it is relaxed and whether the Si atoms are close or far away. The lack of a magnetic defect explains why it is difficult to find magnetic Si in experiments [33]. The above four calculations show that only the nearest four C atoms in SiC contribute the most to the magnetic moments in the HeSi95 C96 . The interactions of the p electrons in the C atoms are stronger than those in the Si atoms. As a result, the dangling C bonds are more sensitive to the same changes in bond length. A magnetic moment will arise if dangling C bonds are formed. Why does a single Si vacancy produce a magnetic moment of only 2μ B whereas the magnetic moment is enhanced to 4μ B with He? The stabilization of the FM phase observed for the two defects can be understood using the phenomenological band-coupling model [3]. The band-coupling tends to make all the spins parallel for the four C atoms. The bonding of C atoms tends to make one pair of spins anti-parallel. The competition is determined by the distances of the C atoms. The four nearest carbon atoms form tetrahedral with edge of 0.34 nm and 0.36 nm and for VSi and HeSi respectively. The dispersions of VSi and HeSi are shown in Fig. 3. The electronic structure differences of the two defects can
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W. Cheng et al. / Physics Letters A 376 (2012) 3363–3367
Table 2 The energy differences in the unit cell of HeSi96 C95 , and (HeSi96 C95 )2 , and the different charge states: neutral (0), one positive charge (+), and one negative charge (−). The energy differences in parenthesis are from VC VSi in Ref. [9]. E AFM − E FM (meV)
Total magnetic moment of FM structure (μ B )
HeSi95 C96 (0) (+1) (−1)
553 523 307
4 .0 3 .0 3 .0
(HeSi96 C95 )2 (0, 0) (+1, +1) (−1, −1)
−75.3 (−78.8) −39.4 (16.8) 81.0 (68.2)
8 .0 6 .0 6 .0
System (charge states)
Fig. 4. Defect formation energy of HeSi as a function of Fermi level E F for different charge states.
be understood by multi-center bonding which are described in Ref. [35]. Without He atom support, the four C atoms are close to each other, and the initial four dangling bonds around the vacancy tend to form 1/6-order 4-center bonds (effectively one bond) around the six edges of the tetrahedral for VSi . As a result, there are two parallel spins from two dangling bonds and two anti-parallel spins from a bond above the Fermi energy level as shown in Fig. 3(a). With He support, bonding is suppressed as the C atoms are far away, four parallel spin orbitals appear above the Fermi energy level as shown in Fig. 3(b). The magnetic moment is doubled for HeSi as a result of the above bonding differences. In a real semiconductor, the defects could either be positively charged or negatively charged. We then calculate the stability of the HeSi95 C96 structure under different charge states and different spin orientations. The defect formation energies are calculated to be 3.92, 3.68, 3.49, 3.37, 3.36, 4.42, 5.91, 7.70, and 9.67 eV, for charge states from positive four to negative four with Fermi energy at valence band maximum. The defect formation energy as a function of the Fermi energy level is shown in Fig. 4. The defects with more than one negative charges are relatively difficult to form for p-type SiC. The defect with four electrons is the most stable form in n-type SiC. The neutral defect as well as defect with one hole are the most stable in p-type SiC. We keep the minimized HeSi95 C96 structure fixed and then put the magnetic moments of the four C atoms in parallel and anti-parallel configurations, as shown in Fig. 1. The AFM–FM energy differences are listed in Table 2. As shown in Table 2, the energy minimized structure surrounding the defects is ferromagnetic. After the defects are charged with one electron and one hole, it is found that the ferromagnetic structure is still stable. However the three ferromagnetic configurations are different from each other. The typical one for the neutral defect is shown in Fig. 1(a). In order to study the magnetic coupling between these local moments, we double the size of the supercells by placing two 192atom or 384-atom cells side by side. The two HeSi are separated by 1.24 nm. In the FM structure, the spins of the two HeSi in both cells are the same as shown in Fig. 1(a), and in the AFM structure, the spins of one HeSi are the same as shown in Fig. 1(a), and the spins of the other HeSi are totally inverted. The FM and AFM energy differences for HeSi in 384-atom cell and the corresponding values for VSi VC are listed in Table 2. The magnetic coupling tends to form AFM structure for defects in neutral and positive
charge states, and tends to form FM structure for defects in negative charge states. That is slightly different from VSi VC case. In experiment, the charge states of HeSi might be altered by ion implantation techniques, and magnetism is likely to form. The above discussions show that the magnetism highly depends on the carbon distances. If all of the carbons are bonding, the magnetism disappears. If the bonds are broken, then the carbon– carbon distance increases and magnetism appears. If the carbon atoms are further separated, the magnetism increases even more. The carbon atom separation and the magnetism relations can also explain experiments examining graphite [16,17]. The bond lengths of the C atoms within a graphite sheet are shorter than the bond lengths in SiC. It is difficult to insert He atoms into the sheet. He is likely to stay within two graphite sheets. The change in bond lengths as a result of He insertion is less, and the magnetic effects are weaker in He irradiated graphite. 4. Conclusions In summary, the enhanced magnetism of a defect complex, HeSi , with a He interstitial and a Si vacancy is found. The magnetic moment of HeSi is two times greater than that of VC VSi in 6H-SiC. He is an atom that enlarges the vacancy size. C is the element that provides the magnetic moment. Charge can affect the stability of the magnetic states. The carbon atom distances are related to the final magnetic moment. Acknowledgements The authors acknowledge Professor D.S. Wang at the Institute of Physics for helpful discussion and his help in making the computation arrangement through his grant (NSFC-10634070). Part of the computation described in this research is carried out on the ScGrid of Supercomputing Center, Computer Network Information Center of Chinese Academy of Sciences. The work is supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the National Natural Science Foundation of China under Grants No. 11025524, and No. 11161130520, and the National Basic Research Program of China under Grant No. 2010CB832903. References [1] C. Kittel, Introduction to Solid State Physics, 7th ed., Willey, New York, 1996. [2] J.M.D. Coey, M. Venkatesan, P. Stamenov, C.B. Fitzgerald, L.S. Dorneles, Phys. Rev. B 72 (2005) 024450. [3] H.W. Peng, H.J. Xiang, S.H. Wei, S.-S. Li, J.B. Xia, J.B. Li, Phys. Rev. Lett. 102 (2009) 017201. [4] T.L. Makarova, F. Palacio (Eds.), Carbon Based Magnetism: An Overview of the Magnetism of Metal Free Carbon-based Compounds and Materials, Elsevier, Amsterdam, 2006. [5] J.H. Chen, L. Li, W.G. Cullen, E.D. Williams, M.S. Fuhrer, Nature Phys. 7 (2011) 535.
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